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LS-OPT User's Manual - Version 2

1. a deformed 50ms b undeformed Figure 17 6 Small car impacting a pole 17 2 2 Design criteria and design variables The objective is to minimize the Head Injury Criterion HIC over a 15ms interval of a selected point subject to an intrusion constraint of 550mm of the pole into the vehicle at 50ms The HIC is based on linear head acceleration and is widely used in occupant safety regulations in the automotive industry as a brain injury criterion In summary the criteria of interest are the following e Head injury criterion HIC of a selected point 15ms e Peak acceleration of a chosen point filtered at 60Hz SAE LS OPT Version 2 187 CHAPTER 17 EXAMPLE PROBLEMS e Component Mass of the structural components bumper front hood and underside e Intrusion computed using the relative motion of two points Units are in mm and sec The design variables are the shell thickness of the car front t_ hood and the shell thickness of the bumper t_bumper see Figure 17 6 17 2 3 Design formulation The design formulation is as follows Minimize HIC 15ms 17 4 subject to Intrusion 50ms lt 550mm The intrusion is measured as the difference between the displacement of nodes 167 and 432 Remark e The mass is computed but not constrained This is useful for monitoring the mass changes 17 2 4 Modeling The simulation is performed using LS D
2. Constraint Value Constraint E crepe Bere re er ner Maximum Violation Stress 0 Stress Smallest Margin Stress 0 2073 Stress 17 1 4 Reducing the region of interest for further refinement Upper Viol 1 033e 06 1 033e 06 It seems that further accuracy can only be obtained by reducing the size of the subregion In the following analysis the current optimum 1 766 0 4086 was used as a starting point while the region of interest was cut in half The order of the approximation is quadratic The modified statements are 2BAR3 Two Bar Truss Reducing the region of interest Created on Thu Jul 11 07 46 24 2002 5 DESIGN VARIABLES Range Area 2 Range Base 0 8 The approximations have been significantly improved Approximating Response Weight using 10 points ITERATION 1 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Mean response value u 2 RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted c R 2 prediction Determinant of X x DO OG OG OG O Approximating Response Stress using 0282 0209 1 03 0385 1 90 0157 0 77 0697 3 44 0009 9995 9995 9944 0071 10 points ITERATION 1 Mean response value 1 RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted R 2 prediction Determinant of X X ooooo0o00o00o
3. Isf Load Sharing Facility loadleveler IBM LoadLeveler pbs PBS nge NQE H 8 Point selection Experimental design Experiment Description Identifier Default approximation Linear Koshal lin koshal linear Quadratic Koshal quad _ koshal quadratic Central Composite composite quadratic Latin Hypercube latin hypercube linear Monte Carlo batch version only monte_carlo linear Plan plan linear User defined own linear D optimal dopt linear Space filling space filling 2 Factorial Designs 2 2toK Linear 3 3toK quadratic 11 11toK quadratic Solver order linear elliptic interaction quadratic FF kriging Solver experimental design design Solver basis experiment design Solver number of basis experiments number Solver number experiment number Solver update doe Solver experiment duplicate name Type of approximating function Experimental design to use Basis experiment for D optimal design points selection scheme Number of experimental points Number of experimental points Updating of experimental points Duplicate previously defined experiment LS OPT Version 2 345 APPENDIX H QUICK REFERENCE MANUAL H 9 Design problem formulation History name string Defines history function History name expression Defines history function Historysize number Defines maximum number of data points in history function Response name string Defines response function Response name expression Defines res
4. 236 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS cons lower bound con upper bound con cons train train vel train train anun o vel lower bound constrain upper bound constrain cons train vel50_4 lower bound constrain upper bound constrain cons train disp10_4 lower bound constrain upper bound constrain cons train disp15_4 lower bound constrain upper bound constrain cons train disp20_4 lower bound constrain upper bound constrain cons traint disp25_4 lower bound constrain upper bound constrain cons train disp30 _4 lower bound constrain upper bound constrain cons train disp35_4 lower bound constrain upper bound constrain cons train disp40 4 lower bound constrain upper bound constrain cons train disp45 4 lower bound constrain upper bound constrain cons train disp50_4 lower bound constrain upper bound constrain cons train accl0_5 lower bound constrain upper bound constrain cons train accl5 5 lower bound constrain upper bound constrain cons train acc20 5 lower bound constrain upper bound constrain cons train acce25 5 lower bound constrain upper bound constrain cons train acc30_5 lower bound constrain upper bound constrain cons train acc35_5 lower bound constrain uppe
5. Baseline ss Optimum Experiment m 20 Time 30 40 50 240 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 0 5 0 5 Baseline eeen Baseline weee Optimum Optimum 04 Experiment Wm 4 0 4 Experiment m 5 03 L 5 03 L i Te 3 02 Q Q lt lt 0 1 F 0 0 10 20 30 40 50 0 10 20 30 40 50 Time Time a Chest form velocity 4m s b Chest form velocity 5m s Figure 17 28 Comparison of experimental and computational displacement velocity and acceleration results 10 variables Formulation 2 LSR Airbag material identification The optimization history of the objective residual and of one of the variables are shown in Figure 17 29 and Figure 17 30 respectively In Figure 17 29 it can be seen that the case with less design variables converges more rapidly while the activation of first the panning and secondly the zooming heuristic in LS OPT is clearly evident in the optimization history of the variable Leakage_5 Figure 17 30 5 5 5 variables predicted 5 variables computed 5 10 variables predicted 4 5 10 variables computed S 4 pa 35 gt 3 5 2 2 1 5 afede fe Epe Enh 0 2 4 6 8 10 12 14 16 18 20 Iteration Number Figure 17 29 Optimization history Residual LSR Formulation LS OPT Version 2 241 CHAPTER 17 EXAMPLE PROBLEMS 4e 07 3
6. 150 LS OPT Version 2 14 Applications of Optimization This chapter provides a brief description of some of the applications of optimization that can be performed using LS OPT It should be read in conjunction with Chapter the Examples chapter where the applications are illustrated with practical examples 14 1 Multidisciplinary Design Optimization MDO The MDO capability in LS OPT implies that the user has the option of assigning different variables experimental designs and job specification information to different solvers or disciplines The directory structure change that has been incorporated in this version separates the number of experiments that needs to be run for each solver by creating separate Experiments AnalysisResults DesignFunctions and ExtendedResults files in each solver directory Command file syntax mdo mdotype The only mdotype available is mdf or multidisciplinary feasible 14 1 1 Command file All variable definitions are defined first as when solving non MDO problems regardless of whether they belong to all disciplines or solvers This means that the variable starting value bounds minimum and maximum and range sub region size are defined together If a variable is not shared by all disciplines however i e it belongs to some but not all of the disciplines solvers then it is flagged using the syntax local variable name At this stage no mention is made in the command file to which solver s t
7. 0 1 2 3 4 5 Iteration Number Figure 17 47 Comparison of optimization history of maximum knee force for full and reduced variable sets 274 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 9 Optimization with analytical design sensitivities This example demonstrates how analytical gradients Section provided by a solver can be used for optimization using the SLP algorithm and the domain reduction scheme 61 Section 2 12 The solver a Perl program is shown below followed by the command file for optimization In this example the input variables are read from the file XPoint placed in the run directory by LS OPT The input variables can also be read by defining this file as an input file and using the lt lt variable_name gt gt format to label the variable locations for substitution Note that each response requires a unique Gradient file Solver program Open output files for response results open FOUT gt fsol open G10OUT gt glsol open G20UT gt g2sol Output files for gradients open DF gt Grad open DG1 gt Gradg1 open DG2 gt Gradg2 Open the input file XPoint automatically placed by LS OPT in the run directory open X lt XPoint Compute results and write to the files i e conduct the simulation while lt X gt x1 x2 split print FOUT x1 x1 4 x2 0 5 x2 0 5 n Derivative of
8. 2BAR1 Two Bar Truss A first approximation linear Created on Wed Jul 10 17 41 03 2002 DESIGN VARIABLES variables 2 Variable Area 2 Lower bound variable Area 0 2 Upper bound variable Area 4 Range Area 4 Variable Base 0 8 Lower bound variable Base 0 1 Upper bound variable Base 1 6 Range Base 1 6 solvers 1 responses 2 5 NO HISTORIES ARE DEFINED DEFINITION OF SOLVER RUNS solver own RUNS solver command 2bar_com RESPONSES FOR SOLVER RUNS response Weight 1 0 cat wt response Weight linear response Stress 1 0 cat str response Stress linear NO HISTORIES DEFINED FOR SOLVER RUNS OBJECTIVE FUNCTIONS objectives 1 objective Weight 1 CONSTRAINT DEFINITIONS constraints 1 constraint Stress upper bound constraint Stress 1 EXPERIMENTAL DESIGN Order linear Experimental design dopt 174 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Basis experiment 3toK Number experiment 5 JOB INFO 5 concurrent jobs 4 iterate param design 0 01 iterate param objective 0 01 iterate 1 STOP The input is echoed in the file lsopt_input The output is given in lsopt_output and in the View panel of LS OPTui A summary of the response surface statistics from the output file is given Approximating Response Weight using 5 points ITERATION 1 Mean response value 2 9413 RMS error 0 7569 25 73 Maximum
9. Optimization History Optimization History For Variable Base For Variable Base Beco saben Liss ee eh ee EESE le hl 4 4 4 t call fer ge MS co hen oy he he ee Hz lee 1 2 1 8 aseg eigenen 10 11 9 2 3 4 5 6 7 8 Number of Iterations d Optimization history of Base Quadratic Number of Iterations c Optimization history of Base Linear ght Wei Optimization History For Response Optimization History For Response Weight 9 1O N ie 1y oy esuodsey 10 11 9 2 3 4 5 6 7 8 Number of Iterations f Optimization history of Weight Quadratic Number of Iterations e Optimization history of Weight Linear 185 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Optimization History For Response Stress I I I I 1 I I I i I 1 L 1 I I I I I 1 1 1 1 I I ee et ee et ee Response Stress 0 8 0 6 0 4 0 2 1 1 1 L I I 1 1 t i I I 1 1 ee te Optimization History For Response Stress a a a a a ee Gas 2 3 1 4 5 6 Number of Iterations h Optimization history of Stress Quadratic Figure 17 5 Optimization history of design variables and responses Linear and Quadratic Note that the more accurate but more expensive quadratic approximation converges in about 3 design iterations 30 simulations w
10. RESPONSES FOR SOLVER 5MPS TIMESTEP 40 TIMESTEP 45 TIMESTEP 50 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP 10322 TIMESTEP response acc10 5 1000 0 DynaASCII nodout X_ACC 10322 TIMESTEP 10 response acc15_5 1000 0 DynaASCII nodout X_ACC 10322 TIMESTEP 15 LOR 15 20 254 30 4 35 40 45 50 232 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS response acc20 _5 1000 0 response acc25_5 1000 0 response acc30_5 1000 0 response acc35_5 1000 0 response acc40_5 1000 0 response acc45_5 1000 0 response acc50_5 1000 0 DynaASCII nodou DynaASCII nodou DynaASCII nodou DynaASCII nodou DynaASCII nodou DynaASCII nodou DynaASCII nodou X ACC 10322 TIMESTEP 20 X ACC 10322 TIMESTEP 25 X ACC 10322 TIMESTEP 30 X ACC 10322 TIMESTEP 35 X ACC 10322 TIMESTEP 40 X ACC 10322 TIMESTEP 45 X ACC 10322 TIMESTEP 50 xxx bd Bd ME Bd XM tte a ee response vell0_5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 10 response vel15 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 15 response vel20 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 20 response vel25 5 1 0 DynaASCII nodout X VEL 10322 TIMESTE
11. e Noise variables Variables that are difficult or impossible to control at the design and production level but can be controlled at the analysis level for example loads and material variation A noise variable will have the nominal value as specified by the distribution that is follow the distribution exactly A variable is declared probabilistic by e Creating it as a noise variable e Assigning a distribution to a control variable e Creating it as linked to an existing probabilistic variable Three associations between probabilistic variables are possible e Their nominal values are the same but their distributions differ e Their nominal values and distributions are the same e Their nominal values differ but they refer to the same distribution Command file syntax noise variable variableName distribution distributionName variable variableName distribution distributionName variable variableName link variable variableName Item Description variableName Variable identifier distributionName Distribution identifier Example S Create a noise variable Noise Variable windLoadScatter distribution windLoadData Assigning a distribution to an existing control variable Variable Var D 1 Distribution dist 1 Creating a variable by linking it to another Variable Var D 2 Link variable Var D 1 15 4 1 Setting the Nominal Value of a Probabilistic Vari
12. A user defined program may be used to extract a history file from the database The program must produce an output file with the reserved name LsoptHistory This file contains two columns of data separated by whitespace a space or tab or the following characters comma semi colon or equal sign Lines that do not have the recognizable format will be ignored so that files with headers or footers do not need to be specially modified Example 1 history displacement _1 DynaASCII nodout r disp 12789 TIMESTEP 0 0 SAE 60 history displacement_2 DynaASCII nodout r disp 26993 TIMESTEP 0 0 SAE 60 history deformation expression displacement 2 displacement_1 response final deform expression deformation 200 Example 2 constant v0 15 65 history bumper velocity DynaASCII nodout X_VEL 73579 TIMESTEP 0 0 SAE 30 history Apillar_velocity_1 DynaASCII nodout X VEL 41195 TIMESTEP 0 0 SAE 30 history Apillar_velocity 2 DynaASCII nodout X VEL 17251 TIMESTEP 0 0 SAE 30 history global velocity DynaASCII glstat X VEL 0 TIMESTEP LS OPT Version 2 107 CHAPTER 10 HISTORY AND RESPONSE RESULTS history Apillar velocity average expression Apillar_velocity_1 Apillar_ velocity 2 2 response time to bumper zero expression Lookup bumper_velocity t 0 response vel A bumper zero expression Apillar velocity average ti
13. LS OPT Version 2 199 CHAPTER 17 EXAMPLE PROBLEMS solvers 2 variables 4 Variable t_hood_m 1 Lower bound variable t_hood_m 1 Upper bound variable t_hood_m 6 range t_hood m 2 Local t_hood_m Variable t_ bumper m 3 Lower bound variable t_bumper m 1 Upper bound variable t bumper m 6 range t_bumper m 2 Local t_bumper_m Variable t_hood_s 0 Lower bound variable t_hood_s 05 Upper bound variable t_hood_s 05 range t_hood_s 0 1 iterate param rangelimit t_hood_s 0 1 Local t_hood_s Variable t_ bumper _s 0 Lower bound variable t_bumper_s 05 Upper bound variable t_bumper_s 05 range t_bumper_s 0 1 iterate param rangelimit t_bumper_s 0 1 Local t_bumper s dependents 2 DEPENDENTS Dependent t hood t hood m t_hood s Dependent t_bumper t_bumper_m t_bumper_s responses 10 NO HISTORIES ARE DEFINED DEFINITION OF SOLVER MEAN solver dyna MEAN solver command lsdyna solver input file car5 k solver append file rigid2 RESPONSES FOR SOLVER 1 response Intru_2 m 1 0 DynaASCII Nodout X DISP 432 Timestep response Intru_1_m 1 0 DynaASCII Nodout X_DISP 167 Timestep response Intrusion m expression Intru 1 m Intru_2_m response HIC_m 1 0 DynaASCII Nodout HIC15 9810 LOCAL VARIABLES solver variable t_hood_m solver variable t_bumper_m EXPERIMENTAL DESIGN Solver Order linear Solver Experimental desig
14. upper bound constraint HIC_scaled 1 constraint Freq scaled lower bound constraint Freq_ scaled 1 constraint Intrusion_scaled upper bound constraint Intrusion_scaled 1 JOB INFO iterate param design 0 001 iterate param objective 0 001 iterate param stoppingtype or iterate 40 STOP The results are presented in a to f The random search method converges after approximately 105 crash 105 NVH simulations 7 iterations while the response surface approach requires about 60 crash 30 NVH simulations 6 iterations These numbers might vary because of the random nature of the methods involved Note the effects of mode tracking in Figure 17 34 e Optimization History For Response Mass Optimization History For Response HIC 18 16 2 5 14 12 10 Response Mass Response HIC x109 0 5 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Number of Iterations Number of Iterations a Objective Mass b HIC 252 LS OPT Version 2 EXAMPLE PROBLEMS Optimization History For Composite Intrusion CHAPTER 17 Optimization History For Response Frequency vt ise N eee dio eee ee 1 1 I I 7 Er ou nb y suods y 10 12 14 16 Number of Iterations Optimization History For Error Term Max Constr Violation d Intrusion 10 12 14 16 8 Number of Iterations Optimization History For Response Mode c Torsional mode frequency vonelol AIS
15. Bakker T M D Design Optimization with Kriging Models WBBM Report Series 47 Ph D thesis Delft University Press 2000 Barthelemy J F M Function Approximation In Structural Optimization Status and Promise Ed Kamat M P 1993 Basu A Frazer L N Rapid determination of the critical temperature in simulated annealing inversion Science 249 pp 1409 1412 1990 Bishop C M Neural Networks for Pattern Recognition Oxford University Press 1995 Bounds D G New optimization methods from physics and biology Nature 329 pp 215 218 1987 Box G E P Draper N R A basis for the selection of a response surface design Journal of the American Statistical Association 54 pp 622 654 1959 Box G E P Draper N R Empirical Model Building and Response Surfaces Wiley New York 1987 Burgee S Giunta A A Narducci R Watson L T Grossman B and Haftka R T A coarse grained parallel variable complexity multidisciplinary optimization paradigm The International Journal of Supercomputer Applications and High Performance Computing 10 4 pp 269 299 1996 Cohn D Neural network exploration using optimal experiment design Neural Networks 9 6 pp 1071 1083 1996 Craig K J Stander N Dooge D Varadappa S MDO of automotive vehicle for crashworthiness and NVH using response surface methods Paper AIAA2002_5607 9 AIAA ISSMO Symposium on Multidisciplinary Analysis and Optimization 4 6 Sept 2002 Atlanta
16. CHAPTER 10 HISTORY AND RESPONSE RESULTS Remarks 1 Histories are used by response definitions see Section to define a response surface They are therefore intermediate entities and cannot be used directly to define a response surface Only response can define a response surface 2 For LS DYNA history definition and syntax please refer to Section 10 5 In LS OPTui histories are defined in the Histories panel Figure 10 1 File Tasks Help Info Solvers Variables Point Selection Histories Responses Objective Constraints Run View USER DEFINED Interface Identifier FILE 4 Force2 Case2 BinoutHists EXPRESSION ABSTAT Force Component BNDOUT DEFORC ELOUT v Y master force v slave force GCEOUT GLSTAT xy Z master force Z slave force xy X master force wv Xslave force JNTFORC xy Resultant master force Resultant slave force NCFORC NODOUT NODFOR RBDOUT None Pr Filtering RWFORC SECFORC SPCFORC SWFORC ad a Solver Casel W Use Binary Output Database Binout History Name Forcet Figure 10 1 Histories panel in LS OPTui 10 2 Defining a response scalar The extraction of responses consists of a definition for each response and a single extraction command or mathematical expression A response is often the result of a mathematical operation of a response history LS OPT Version 2 109 CHAPTER 10 HISTORY AND RESPONSE RESULTS but can be extracted
17. LS OPT Version 2 267 CHAPTER 17 EXAMPLE PROBLEMS a WAT KA L Ay Non visible optimizable structural part Simplified knee forms Figure 17 45 Typical instrument panel prepared for a Bendix component test Right EA Width Right Bracket Gauge Left Bracket Gauge Yoke Cross section Radius N Left EA Width Left EA Depth To p Oblong Hole Radius Left EA Inner Flange Width Left EA Left EA Depth Knee Bolster Gauge Depth Front Bottom Figure 17 46 Typical major components of a knee bolster system and definition of design variables 268 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS The simulation is carried out for a 40 ms duration by which time the knees have been brought to rest It may be mentioned here that the Bendix component test is used mainly for knee bolster system development for certification purposes a different physical test representative of the full vehicle is performed Since the simulation used herein is at a subsystem level the results reported here may be used mainly for illustration purposes 17 8 2 Definition of optimization problem The optimization problem is defined as follows Minimize max Knee_F L Knee F R Subject to Left Knee intrusion lt 115mm Right Knee intrusion lt 115mm Yoke displacement lt 85mm Minimization over both knee forces is achieved by constraining them to impossibly low values The optimization algorithm will there
18. SPCFORC DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z_FORCE z_force Z force R_FORCE Resultant force X_RES x_resultant Total X force Y_RES y_resultant Total Y force Z_RES z_resultant Total Z force X MOMENT x_moment X moment Y_MOMENT y_moment Y moment Z MOMENT z_ moment Z moment R_MOMENT Resultant moment 316 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Spotweld and Rivet Forces DynaASCH Binout Description Keyword Component AXIAL axial Axial force SHEAR shear Shear force failure flag Failure flag LS OPT Version 2 317 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS 318 LS OPT Version 2 Appendix D Database files D 1 Design flow Source Database file Process Output Database file Level of directory for output database Command file com Point selection Experiments Solver Experiments Simulation runs Solver output files Run Solver output files Result extraction AnalysisResults Solver StatResults Work AnalysisResults Approximation DesignFunctions Solver Net DesignFunctions Optimize OptimumResults Work OptimizationHistory Work D 2 Database file formats The Experiments file This file appears in the solver directory and is used to save the experimental point coordinates for the analysis runs The file consists of lines having the following format repeated for each exper
19. a rear Ve ee RESPONSE Computed Predicted Computed Predicted ee ence nee ee ee ae eee cel Internal Energy 7914 8778 7914 8778 Rigid_Wall_Force 4 789e 04 7e 04 4 789e 04 7e 04 208 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 3 3 Refining the design model using a second iteration During the previous optimization step the Radius variable was reduced from 75 to 50 on the boundary of the region of interest It was also apparent that the approximations were fairly inaccurate Therefore in the new iteration the region of interest is reduced from 50 2 to 35 1 5 while retaining a quadratic approximation order The starting point is taken as the current optimum 50 2 978 The modified commands in the input file are as follows 5 DESIGN VARIABLES variables 2 Variable Radius 50 Lower bound variable Radius 20 Upper bound variable Radius 100 Range Radius 35 Variable Wall Thickness 2 9783 Lower bound variable Wall Thickness 2 Upper bound variable Wall_Thickness 6 Range Wall_ Thickness 1 5 As shown below the accuracy of fit improves but the average rigid wall force is still inaccurate Approximating Response Internal _Energy using 10 points ITERATION 1 Mean response value 8640 2050 RMS error 526 9459 6 10 Maximum Residual 890 0759 10 30 Average Error u 388 4472 4 50 Square Root PRESS Residual 1339 4046 15 50 Variance 555344 0180 R 2 0 9
20. response time _to_zero velocity expression Lookup global _velocity t 0 5 Find the average A pillar velocity where global velocity is zero 5 response velocity final Apillar velocity average time_to_zero velocity response PULSE 2 expression Integral Apillar_velocity_ average t time_to_engine_zero 330 LS OPT Version 2 APPENDIX E MATHEMATICAL EXPRESSIONS time_to_ zero velocity time_to_zero velocity time_to_ engine zero LS OPT Version 2 331 APPENDIX E MATHEMATICAL EXPRESSIONS 332 LS OPT Version 2 Appendix F Simulated Annealing The Simulated Annealing SA algorithm for global optimization can be viewed as an extension to local stochastic optimization techniques The basic idea is very simple SA takes a biased random walk through the space and aims to find a global optimum from among multiple local solutions In trying to minimize a function instead of always going downhill SA algorithm goes downhill most of the time It means that the SA process sometimes goes uphill This allows simulated annealing to move consistently towards lower function values yet still jump out of local minima and globally explore different states of the optimized system The SA algorithm was first formulated for various combinatorial problems bs The approach was later extended to continuous optimization problems In the simulated annealing algorithm was adopted to search for opt
21. 2 1 10 Optimization History For Objective Internal_Energy Optimization History For Variable Radius Number of Iterations a Radius sn pey a qeuer Wall Force For Constraint Rigid ioe x N oO foe co x 14 12 10 8 6 4 p01 AByauz eulalu aanoelqo 10 12 14 16 213 Number of Iterations d Rigid Wall Force 10 12 14 16 Energy Number of Iterations c Internal LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Optimization History Optimization History For Response Internal_Energy For Response Rigid_Wall_Force 20 45 18 40 T S 16 2 35 x 2S ZT 14 S 30 D 12 w w 25 g 10 z D ke A 5 20 5 5 15 gt 4 2 10 2 5 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Number of Iterations Number of Iterations e RMS error of Internal_Energy f RMS error ofRigid_Wall_ Force Figure 17 19 Optimization history of automated design filtered force The optimization process steadily reduces the infeasibility but the force constraint is still slightly violated when convergence is reached The internal energy is significantly lower than previously DESIGN POINT Variable Name Lower Bound Value Upper Bound Radius 20 20 51 100 Wall Thickness 2 4 342 6 Scaled Unscaled E ARER a eg RESPONSE Computed Predicted Computed Predicted ee ee eae eee ee ee Internal Energy 8344 8645 8344 8645 Rigid_Wall_Force 8 112e 04 8e 04
22. 4 5 1 Neural Nets and Kriging point selection scheme Section 6 are still considered to be advanced tools for optimal design However some experimentation with this methodology has been done to suggest the following procedure The use of neural networks Section 2 10 1 or Kriging Section combined with the space filling 1 Conduct a variable screening procedure to determine important variables Linear polynomials can be used for this purpose 2 Augment the existing design points with a space filling method using approximately the same number of points as would be required for a quadratic approximation The greater the number of points the better the approximation To fully exploit the updating feature and the potential for tradeoff studies it is recommended that the entire design space be used to construct the initial space filling design 60 LS OPT Version 2 CHAPTER 4 DESIGN OPTIMIZATION PROCESS 3 Execute the simulation runs and find the predicted optimum point Verify the design accuracy by simulating the predicted design 4 Combine with a second iteration if the first appears to be inaccurate More iterations can be done depending on the convergence requirement The NN Kriging based optimization is automated using the same domain reduction scheme see Section as the successive polynomial response surface method However in the case of neural networks or Kriging approximations it makes sense to update the experimental d
23. 7 ZS 7 ZS LIT II EC LL ma aa a If zu PEST Lz 7 IT 7 Deploying Impact 33ms Rebound 50ms Figure 17 26 Snapshots of mass impacting a deploying airbag 17 5 2 Least squares residual LSR formulation For this formulation a targeted composite function see Equation 10 1 in Section in LS OPT is used to construct the residual LS OPT Version 2 229 CHAPTER 17 EXAMPLE PROBLEMS ESR T jal where F are the experimental targets and I are scaling factors required for the normalization or weighting of each respective response In addition the simulated acceleration and displacement data f x are scaled to match the experimental units while the displacement data are offset to ensure that the quantity IF T is of the same order for all 54 responses 17 5 3 Maximum violation formulation In this formulation the deviations from the respective target values are incorporated as constraint violations so that the optimization problem for parameter identification becomes Minimize e subject to FE F T J se j 1 54 e gt 0 This formulation is automatically activated in LS OPT without specifying the objective function as the maximum constraint violation This is due to the fact that an auxiliary problem is solved internally whenever an infeasible design is found ignoring the objective function until a feasible design is found When used in param
24. 84 9810 2 184 response Stage2Pulse expression Integral_184 _334 9810 334 184 response Stage3Pulse expression Integral 334 max 9810 Disp 334 5 HISTORIES AND RESPONSES DEFINED BY EXPRESSIONS composite Disp_scaled type targeted composite Disp_ scaled response Disp 0 scale weight 1 composite StagelPulse scaled StagelPulse 14 3 composite Stage2Pulse scaled Stage2Pulse 17 5 composite Stage3Pulse scaled Stage3Pulse 20 7 SOLVER SPECIFIC JOB INFO FOR SOLVER CRASH solver concurrent jobs 4 DEFINITION OF SOLVER NVH solver dyna NVH VARIABLES FOR SOLVER NVH EXPERIMENTAL DESIGN OF SOLVER NVH Solver Order linear Solver Experimental design dopt Solver Basis experiment 3toK Solver Number experiment 8 FOR SOLVER CRASH 551 81 4 7 6 SOLVER AND PREPROCESSOR COMMANDS OF SOLVER NVH solver command lsdyna double solver input file dyna_biw input RESPONSES FOR SOLVER NVH response Frequency 1 0 DynaFreq 1 FREQ response Frequency linear NO HISTORIES DEFINED FOR SOLVER NVH HISTORIES AND RESPONSES DEFINED BY EXPRESSIONS FOR SOLVER NVH LS OPT Version 2 259 CHAPTER 17 EXAMPLE PROBLEMS composite Frequency scaled type targeted composite Frequency_scaled response Frequency 0 scale 38 77 weight 1 SOLVER SPECIFIC JOB INFO FOR SOLVER NVH Solv
25. CRASH Frequency NVYH 1 0 Mode NVH 1 0 Dyna Generalized_Mass NVH 1 Disp_scaled N A Co Frequeney_scaled N A StagelPulse_scaled N A Stage2Pulse_scaled N A Stage3Pulse scaled N A ml Solver CRASH Use Binary Output Database Binout Seale Factor 1 Offset o Response Name Disp Figure 10 2 Reponses panel in LS OPTui 10 4 Composite Functions Composite functions can be used to combine response surfaces and variables as well as other composites The objectives and constraints can then be constructed using the composite functions There are three types 1 Expression composite A general expression can be specified for a composite The composite can therefore consist of constants variables dependent variables responses and other composites 2 Standard composite a Targeted composite This is a special composite in which a target is specified for each response or variable The composite is formulated as the distance to the target using a Euclidean norm formulation The components can be weighted and normalized LS OPT Version 2 111 CHAPTER 10 HISTORY AND RESPONSE RESULTS 10 1 where o and y are scale factors and W and are weight factors These are typically used to formulate a multi objective optimization problem in which F is the distance to the target values of
26. DynaASCII rbdout X_ACC 21 AVE 5 0 Response x_acc DynaASCII rbdout X_ACC 21 AVE 5 0 80 0 Response xX acc DynaASCII RBDOUT X ACC 21 AVE 5 0 80 0 SAK Response xX acc DynaASCII rbdout X ACC 21 AVE 5 0 80 0 SAE 40 0 Response Z acc DynaASCII NODOUT Z ACC 27 TIMESTEP Response Z acc DynaASCII NODOUT Z ACC 27 TIMESTEP 10 0 Response Z acc DynaASCII NODOUT Z ACC 27 TIMESTEP 10 0 SAK Response Z acc DynaASCII NODOUT beam Z ACC 7 TIMESTEP 10 0 SAE 50 History 72 ace DynaASCII NODOUT beam Z ACC 7 TIMESTEP History Z acc DynaASCII NODOUT beam Z_ ACC 7 TIMESTEP 0 0 SAE 50 Response HIC DynaASCII nodout HIC36 9 81 2 40096 Response HIC DynaASCII nodout HIC36 00981 1 40096 Response Sigma_yy DynaASCII elout YY STRESS 989 2 MAX Response Eps xx DynaASCII elout XX STRAIN 2989 Upper MAX Response Eps xx DynaASCII elout XX_STRAIN 2989 LOWER Min Response Energy I DynaASCII glstat I_ENER 0 TIMESTEP 10 7 Extracting Response Quantities From the LS DYNA d3plot file A generic interface exists for the extraction of binary data from the LS DYNA d3plot files All results produced are representative of the end of the simulation All the quantities can be specified on a part basis as 120 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS defined in the input deck for LS DYNA The user must ensure that the d3plot files are produced by the LS DYNA
27. Shell thickness FLD General FLD Principal stress Std deviation Modal data Binout Binout Translate 346 LS OPT Version 2 APPENDIX H QUICK REFERENCE MANUAL H 11 Solution tasks Iterate n Analyze Monte Carlo Analyze Metamodel Monte Carlo H 12 Intrinsic functions for mathematical expressions Iterate over n successive approximations 141 Monte Carlo evaluation Monte Carlo evaluation with metamodel int a integer nint a nearest integer abs a absolute value mod a b remainder of a b sign a b transfer of sign from b to a max a b maximum of a and b min a b minimum of a and b sqrt a square root exp a e pow a b a log a natural logarithm log10 a base 10 logarithm sin a sine cos a cosine tan a tangent asin a arc sine acos a arc cosine atan a arc tangent atan2 a b arc tangent of a b sinh a hyperbolic sine cosh a hyperbolic cosine tanh a hyperbolic tangent asinh a arc hyperbolic sine acosh a arc hyperbolic cosine atanh a arc hyperbolic tangent sec a secant csc a cosecant ctn a Cotangent 1 LS OPT Version 2 347 APPENDIX H QUICK REFERENCE MANUAL H 13 Special functions for mathematical expressions Expression Symbols b Integral expression t lower t upper variable f as Derivative
28. history n files are kept only in the run directories and is not available elsewhere 2 In most cases after a failed run the optimization run can be restarted as if starting from the beginning There are a few notable exceptions 71 CHAPTER 6 PROGRAM EXECUTION a A single iteration has been carried out but the design formulation is incorrect and must be changed b Incorrect data was extracted e g for the wrong node or in the wrong direction c The user wants to change the response surface type but keep the original experimental design In the above cases all the history n and response n files must be deleted After restarting the data will then be newly extracted and the subsequent phases will be executed A restart will only be able to retain the data of the first iteration if more than one iteration was completed The directories of the other higher iterations must be deleted in their entirety Unless the database was deleted by e g using the clean file see Section 6 10 no simulations will be unnecessarily repeated and the simulation run should continue normally A restart can be made from any particular iteration by selecting the Specify Starting Iteration button on the Run panel and entering the iteration number The subdirectories representing this iteration and all higher numbered iterations will be deleted after selecting the Run button and confirming the selection Version 2 1 The database files Experim
29. stop Suspend all jobs cont Continue all jobs c Continue the program without taking any action Program will resume in 15 seconds if you do not enter a choice switch If v is selected more detailed information of the jobs is provided namely event time time step internal energy ratio of total to internal energy kinetic energy and total velocity 6 5 Result extraction Each simulation run is immediately followed by a result extraction to create the history n and response n files for that particular design point For distributed simulation runs this extraction process is executed on the remote machine The history n and response n files are subsequently transferred to the local run directory 6 6 Restarting Restarting is conducted by giving the command lsopt command_file_name or by selecting the Run button in the Run panel of LS OPTui Completed simulation runs will be ignored while half completed runs will be restarted automatically However the user must ensure that an appropriate restart file is dumped by the solver by specifying its name and dump frequency The following procedure must be followed when restarting a design run 1 Asa general rule the run directory structure should not be erased The reason is that on restart LS OPT will determine the status of progress made during a previous run from status and output files in the directories Important data such as response values response n files response histories
30. wn S i ake aie Tiel S Ta 1 5 68 1 ac 3 co I g g 70 2 2 5 s T 2 O i 6 72 1 O I i 0 74 0 1 2 3 4 5 6 7 8 74 72 70 68 66 64 62 60 Predicted Response Value x102 Predicted Response Value x101 a HIC response b Intru _2 response Figure 17 7 Computed vs predicted responses Linear approximation The summary data for the first iteration is LS OPT Version 2 191 CHAPTER 17 EXAMPLE PROBLEMS Baseline ITERATION NUMBER Baseline Variable Name Lower Bound Value Upper Bound ee ote aes en ae Bee t_hood 1 1 5 t_bumper 1 3 5 en a SE u 2 RESPONSE FUNCTIONS Scaled Unscaled ee a oeeeeicas Reese en a RESPONSE Computed Predicted Computed Predicted Segue ean as a ae ee as ee Gimuee seen Acc_max 8 345e 04 1 162e 05 8 345e 04 1 162e 05 Mass 0 4103 0 4103 0 4103 0 4103 Intru_2 736 7 738 736 7 738 Intru 1 161 160 7 161 160 7 HIC 68 26 74 68 68 26 74 68 en en En and 1 optimum DESIGN POINT Variable Name Lower Bound Value Upper Bound rn ee aaa ea t_hood 1 1 549 5 t_bumper 1 5 5 na a a e RESPONSE FUNCTIONS Scaled Unscaled Erna achaeae arena aca RESPONSE Computed Predicted Computed Predicted Sec RE NE den a en A E ee Acc_max 1 248e 05 1 781e 05 1 248e 05 1 781e 05 Mass 0 6571 0 657 0 6571 0 657 Intru_2 713 7 711 4 713 7 711 4 Intru_1 164 6 161 4 164 6 161 4 HIC 126 7 39 47 126 7 39 47 Ss pat re gic e
31. z_coordinate Z coordinate Rotational components RX_DISP rx_acceleration XX rotation RY_DISP rx_displacement YY rotation RZ_DISP rx_velocity ZZ rotation RX_VEL ry_acceleration XX rotational velocity RY_VEL ry_displacement YY rotational velocity RZ VEL ry_velocity ZZ rotational velocity RX_ACC rz acceleration XX rotational acceleration RY_ACC rz displacement YY rotational acceleration RZ ACC rz_velocity ZZ rotational acceleration Injury coefficients CSI CSI Chest Severity Index HIC15 HIC15 Head Injury Coefficient 15 ms HIC36 HIC36 Head Injury Coefficient 36 ms LS OPT Version 2 311 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Nodal Forces NODFOR DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z FORCE z_force Z force R_ FORCE Resultant force X TOTAL x_total X total force Y_TOTAL y_total Y total force Z TOTAL z_total Z total force R_ TOTAL Total resultant force x_local y_local z_local energy Energy etotal Total Energy 312 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Rigid Body Data RBDOUT DynaASCH Binout Description Keyword Component X_DISP global_dx X displacement Y_DISP global_dy Y displacement Z_DISP global dz Z displacement R_DISP Resultant displacement X_VEL global_vx X velocity Y_VEL global_vy Y velocity Z VEL global vz Z velocity
32. 1 4 1 1 i J 1 2 3 4 5 6 7 8 74 72 70 68 66 64 62 60 Predicted Response Value x102 Predicted Response Value x10 a HIC response b Intru _2 response LS OPT Version 2 193 CHAPTER 17 EXAMPLE PROBLEMS Tradeoff Plot Constraint Intrusion vs Objective HIC I I I 1 4 I 1 1 1 1 1 t A F aA 1 1 1 1 Bagger eae eee Objective HIC x102 gt 46 48 50 52 54 56 58 60 Constraint Intrusion x101 c Trade off of HIC versus Intrusion Figure 17 8 Computed vs predicted responses and trade off Quadratic approximation The summary data for the first iteration is Baseline ITERATION NUMBER Baseline Variable Name Lower Bound Value Upper Bound t_hood 1 1 5 t_bumper 1 3 5 Scaled Unscaled peenes iee EET E RESPONSE Computed Predicted Computed Predicted r ve eau eines Ve ee are ee ewer Acc_max 8 345e 04 1 385e 05 8 345e 04 1 385e 05 Mass 0 4103 0 4103 0 4103 0 4103 Intru_2 736 7 736 736 7 736 Intru 1 161 160 3 161 160 3 HIC 68 26 10 72 68 26 10 72 194 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS and 1 optimum DESIGN POINT Variable Name Lower Bound Value Upper Bound t_hood 1 1 653 5 t_bumper 1 3 704 5 Scaled Unscaled EEEE EE RESPONSE Computed Predicted Computed Predicted saibe e ep ore A pe pese cenie er a e Acc _ max 1 576e 05 1 985e 05 1 576e 05 1 985e 05 Mass 0 6017
33. 4r 1 001 The value of vr should be large enough so that thermal equilibrium is achieved before reducing the temperature A rule of thumb is to take vr proportional to the size of neighborhood of the current solution Often the cooling schedule F 6 also provides a condition for terminating SA iterations T lt T min F 7 Some of the convergence results for SA rely on the fact that the support of the next candidate distribution is the whole feasible region though in some cases the probability of sampling states far from the current one decreases to 0 as the iteration counter increases For these convergence results it is often only required that the temperature decreases to 0 no matter at which rate For some other convergence results the support of the next candidate distribution is only a neighborhood of the current state and to make the algorithm able to climb the barriers separating the different local minima it is required that the temperature decreases to 0 slowly enough It is clear that the selection of the initial temperature Tmax has a profound influence on the rate of convergence of the SA algorithm At temperatures much higher than the effective temperature the algorithm behaves very much like a random search while at temperatures much lower than the effective temperature it behaves like an inefficient implementation of a deterministic algorithm for local optimization Intuitively the cooling schedule F 6 should be
34. Energy 4963 0 move Upper bound constraint Energy 5790 0 stay Lower bound constraint Force 1 2 Upper bound constraint Force 1 2 The example above shows the lines required to determine a set of points that will be bounded by an upper bound on the Energy Example 2 Variable Radius _1 20 0 Varia Compo ble Radius_2 20 0 site TotalR Radius _1 Radius_2 move Constraint TotalR Upper bound constraint TotalR 50 This specification of the move command ensures that the points are selected such that the sum of the two variables does not exceed 50 Remarks 1 For constraints that are dependent on simulation results a reasonable design space can only be created if response functions have already been created by a previous iteration The mechanism is as follows Automated design After each iteration the program converts the database file DesignFunctions to file DesignFunctions PRE in the solver directory DesignFunctions PRE then defines a reasonable design space and is read at the beginning of the next design iteration Manual semi automated Procedure If a reasonable design space is to be used the user must ensure that a file DesignFunctions PRE solver_name is resident in the working directory 102 LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION before starting an iteration This file can be copied from the DesignFunctions file resulting from a previous
35. LFOPC should be sufficient for most optimization applications The following sections describe how to modify the default settings These can only be modified using the command language 16 1 Selecting an optimization algorithm There are two optimization algorithms available namely the Successive Response Surface Method SRSM and the Sequential Random Search SRS method The syntax is as follows Command file syntax Optimization method srsm randomsearch SRSM is the default 16 2 Setting the subdomain parameters To automate the successive subdomain scheme for both SRSM and Sequential Random Search the size of the region of interest as defined by the range of each variable is adapted based on the accuracy of the previous optimum and for SRSM also on the occurrence of oscillation see theory in Section 2 12 The following parameters can be adjusted refer also to Section 2 12 A suitable default has been provided for each parameter but the user should not find it necessary to change any of these parameters Available in Version 2 1 LS OPT Version 2 165 CHAPTER 16 OPTIMIZATION ALGORITHM SELECTION AND SETTINGS Table 16 1 Subdomain parameters and default values Item Parameter Default SRSM SRS objective Tolerance on objective function 0 01 0 01 accuracy amp design Tolerance on design accuracy amp 0 01 0 01 stoppingtype and objective and design and and or objective or design p
36. Shell Elements ELOUT Description Binout subdirectory shell stress components DynaASCH Binout Keyword Component XX_STRESS sig xx YY_STRESS sig_yy ZZ_STRESS SIg_ZZ XY_STRESS sig xy YZ_STRESS sig_yz ZX_STRESS sig Zx P_STRAIN plastic_strain PRESSURE E_STRESS MAX_SHEAR MAX P STRESS MIN P STRESS XX stress YY stress ZZ stress XY stress YZ stress ZX stress Plastic strain Pressure Effective stress Maximum shear stress Maximum principal stress Minimum principal stress Binout subdirectory shell strain components XX STRAIN YY_ STRAIN ZZ_STRAIN XY_STRAIN YZ_ STRAIN ZX_STRAIN E STRAIN MAX S STRAIN MAX P STRAIN MIN P STRAIN upper_eps_xx lower_eps_ xx upper_eps_yy lower_eps_yy upper_eps_zz lower_eps zz upper_eps_xy lower_eps_ xy upper_eps_ yz lower_eps_yz upper_eps_zx lower_eps_zx XX strain YY strain ZZ strain XY strain YZ strain ZX strain Effective strain Maximum shear strain Maximum principal strain Minimum principal strain 306 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS DynaASCH Binout Description Keyword Component Binout subdirectory thickshell TXX_STRESS sig XX XX stress TYY STRESS sig_yy YY stress TZZ_STRESS sig zz ZZ stress TXY _ STRESS sig xy XY stress TYZ STRESS sig_yz YZ stress TZX_STRESS Sig_Zx ZX stress yield Yield TP_STRAIN Plastic strain TPRESSURE Pressure TE_ST
37. This results in an irregular shape of the design space Therefore once the first approximation has been established all the designs will be contained in the new region of interest This region of interest is thus defined by approximate bounds 20 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY One way of establishing a reasonable set of designs is to move the points of the basis experimental design to the boundaries of the reasonable design space in straight lines connecting to the central design x so that x x a x Xx 2 25 where is determined by conducting a line search along x x This step may cause near duplicity of design points that can be addressed by removing points from the set that are closer than a fixed fraction typically 5 of the design space size The D optimality criterion is then used to attempt to find a well conditioned design from the basis set of experiments in the reasonable design space Using the above approach a poor distribution of the basis points may result in a poorly designed subset 2 8 Model adequacy checking As indicated in the previous sections response surfaces are useful for interactive trade off studies For the trade off surface to be useful its capability of predicting accurate response must be known Error analysis is therefore an important aspect of using response surfaces in design Inaccuracy is not always caused by random error noise only but modeling error sometimes
38. Variables Point Selection Histories Responses Objective Constraints Run View LS OPT User Interface Version 2 2 by u Livermore Software Technology Corporation C Copyright 2000 2003 All Rights Reserved Problem Description Small Car Problem Five variables Partial variables Crash NVH MDF Author AB Current Working Directory home nielen temp optQA MDO small_mdo_polynomial Current Command File com Last Modified Mon Sep 15 15 32 17 2003 Figure 5 1 Main panel in LS OPTui 5 2 Problem description and author name In LS OPTui the Info main panel has fields for the entering of the problem description and author information Command file syntax problem description author author_name A description of the problem can be given in double quotes This description is echoed in the 1sopt _ input and lsopt_output files and in the plot file titles 64 LS OPT Version 2 CHAPTER 5 GRAPHICAL USER INTERFACE AND COMMAND LANGUAGE Example Frontal Impact author Jim Brown The number of variables and constraints are echoed from the graphical user input These can be modified by the user in the command file Command file syntax solvers number of solvers lt 1 gt constants number_of_constants lt 0 gt variables number of variables dependents number of dependent variables lt 0 gt histories number of response histories lt 0 gt responses number of responses composite
39. design and response variables A suitable application is parameter identification In this application the target values F are the experimental results that have to be reproduced by a numerical model as accurately as possible The scale factors oj and y are used to normalize the responses The second component which uses the variables can be used to regularize the parameter identification problem Only independent variables can be included See Figure 10 3 for an example of a targeted composite response definition b Weighted composite Weighted response functions and independent variables are summed in this standard composite Each function component or variable is scaled and weighted FW 7 10 2 i 0 i l Ki These are typically used to construct objectives or constraints in which the responses and variables appear in linear combination The expression composite is a simple alternative to the weighted composite Remarks l 2 An expression composite can be a function of any other composite An objective definition involving more than one response or variable requires the use of a composite function In addition to specifying more than one function per objective multiple objectives can be defined see Section 11 2 112 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS File Tasks Help Info Solvers Variables Point Selection Histories Responses Objective Constraints Run Vi
40. f x1 x2 Hacken walls print DF 2 Sx1 df dx1 print DF 8 x2 0 5 n df f dx2 print G1OUT x1 x2 n Derivative of g1 x1 x2 Ue eee ease acne print DG1 1 print DG1 1 n print G20OUT 2 x1 x2 n Derivative of g2 x1 x2 Us Sas ceeds eesceeeues sees print DG2 2 print DG2 1 n LS OPT Version 2 275 CHAPTER 17 EXAMPLE PROBLEMS Signal normal termination print Norma _1 n Command file Example 2b QP problem analytical sensitivity analysis solvers 1 responses 3 DESIGN VARIABLES variables 2 Variable x1 1 Lower bound variable x1 3 Upper bound variable x1 3 Range x1 1 Variable x2 1 Lower bound variable x2 0 Upper bound variable x2 2 Range x2 1 SSSSSSSSSSSSssssssssssssssssssss SOLVER 1 SSSSSSSSSSSSssssssssssssssssssss solver own 1 solver command home LSOPT _EXE perl ex2 solver experimental design analytical_DSA RESPONSES FOR SOLVER 1 The Gradf Gradgl and Gradg2 files are individually copied to Gradient response f 1 0 cp Gradf Gradient cat fsol response gl 1 0 cp Gradgl Gradient cat glsol response g2 1 0 cp Gradg2 Gradient cat g2sol OBJECTIVE FUNCTIONS objectives 1 maximize objective f 1 CONSTRAINT DEFINITIONS constraints 2 constraint gl upper bound constraint gl 1 constraint g2 upper bound constraint g2 2 JOB INFO iterate par
41. gt 8 35 D Cc c fo fo B 4 2 6 oO o ox 3 3 3 a a 4 5 Qo 2 ro 2 4 0 Predicted Response Value Prediction accuracy of Weight Iteration 1 Quadratic Response Surface Accuracy For Response Function Stress Predicted Response Value Prediction accuracy of Stress Iteration 1 Quadratic Figure 17 3 Prediction accuracy of Weight and Stress Iteration 1 Quadratic An improved design is predicted with the constraint value stress changing from a computed 0 734 to 1 0 the approximate constraint becomes active Due to inaccuracy the actual constraint value of the optimum is a feasible 0 793 The weight changes from 2 561 to 1 925 1 907 computed DESIGN POINT Variable Name Lower Bound Value Upper Bound Area 1 766 4 Base 0 4068 1 6 Scaled Unscaled a SES Bee ENO RESPONSE Computed Predicted Computed Predicted Deri we cea ope as gorges eps eee el Weight 1 907 1 925 1 907 1 925 Stress 0 7927 1 0 7927 1 Computed Value Predicted Value LS OPT Version 2 179 CHAPTER 17 EXAMPLE PROBLEMS OBJECTIVE FUNCTIONS OBJECTIVE NAME Computed Predicted WT oo eo ae a eee rs eee Gea een Bere CONSTRAINT FUNCTIONS CONSTRAINT NAME oa Computed Predicted Lower pe ee ee ae ee a ele Lower Upper Lower Aamaaaaaeamananaan nennen ener 1444220444 Stress in a E Berger ee ee gen MAXIMUM VIOLATION Computed Predicted Quantity
42. points per variable the number of points 3 3 5 10 2 The generation of design points for all approximation types but especially for neural networks and Kriging 3 The augmentation of an existing experimental design This means that points can be added for each iteration while maintaining uniformity and equidistance with respect to pre existing points LS OPT contains 6 algorithms to generate space filling designs see Table 2 2 Only Algorithm 5 has been made available in the graphical interface LS OPTuwi Algorithm 0 Algorithm 1 Algorithm 2 min p 0 19 CL 0 046 min p 0 28 CL 0 012 min p 0 39 CL 0 030 Algorithm 3 Algorithm 4 Algorithm 5 min p 0 45 CL 0 011 min p 0 61 CL 0 030 min p 0 71 CL 0 108 Figure 2 1 Six space filling designs 5 points in a 2 dimensional box region 18 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY Table 2 2 Description of space filling algorithms Algorithm Description Number 0 Random 1 Central point Latin Hypercube Sampling LHS design with random pairing 2 Generalized LHS design with random pairing 3 Given an LHS design permutes the values in each column of the LHS matrix so as to optimize the maximin distance criterion taking into account a set of existing fixed design points This is done using simulated annealing Fixed points influence the maximin distance criterion but are not allowed to be changed by Simulated Annealing moves 4
43. sscanf argv i 1 Slf amp x i if flag 1 printf Error in calculation of constrainti n exit 1 x2 1 x 1 x 1 val 0 124 sqrt x2 8 x 0 1 x 0 x 1 printf Slf n val fprintf stderr Norma 1 n exit 0 The UNIX script program 2bar_com runs the C programs gw and gss using the design variable file XPoint which is resident in each run directory as input For practical purposes 2bar_com gw and gs have been placed in a directory above the working directory or three directories above the run directory Hence the references 2bar_com gw etc in the LS OPT input file Note the output of the string N o r m a 1 so that the completion status may be recognized 2bar_com gw cat XPoint swt gss cat XPoint gt str The UNIX extraction scripts get_wt andget_str are defined as user interfaces get_wt cat wt get_str cat str In Sections 17 1 2 to 17 1 4 a typical semi automated optimization procedure is illustrated Section 17 1 5 shows how a trade off study can be conducted while the last subsection 17 1 6 shows how an automated procedure can be specified for this example problem LS OPT Version 2 173 CHAPTER 17 EXAMPLE PROBLEMS 17 1 2 A first approximation using linear response surfaces The first iteration is chosen to be linear The input file for LS OPT given below The initial design is located at x 2 0 0 8
44. 0 9301 R 2 adjusted 0 9301 R 2 prediction 0 3303 Determinant of X X 0 0131 Because the size of the region of interest remained the same the curve fitting results show only a slight change because of the new location in this case an improvement However as the optimization results below show the design is much improved i e the objective value has increased whereas the approximate constraint is active Unfortunately due to the poor fit of the Rigid_Wall_Force the simulation result exceeds the force constraint by about 10kN 14 Further reduction of the region of interest is required to reduce the error or filtering of the force can be considered to reduce the noise on this response LS OPT Version 2 211 CHAPTER 17 EXAMPLE PROBLEMS DESIGN POINT Lower Bound Value Upper Bound Wall_Thickness 2 4 478 6 RESPONSE FUNCTIONS Scaled Unscaled ee ee ee ee RESPONSE Computed Predicted Computed Predicted eee T ee A en Internal_Energy 1 129e 04 1 075e 04 1 129e 04 1 075e 04 Rigid_Wall_Force 8 007e 04 7e 04 8 007e 04 7e 04 The table below gives a summary of the three iterations of the step by step procedure Table 17 2 Comparison of results Cylinder impact Variable Initial Iteration 1 Iteration 2 Iteration 3 Radius 75 50 42 43 36 51 Wall thickness 3 2 978 3 728 4 478 Energy Computed 12960 7914 9777 11290 Force Computed 174900 47
45. 1 response Acc_max 1 0 DynaASCII Nodout X_ACC 432 Max SAE 60 response Acc max linear response Mass 1 0 DynaMass 2 3 4 5 MASS response Mass linear response Intru_2 1 0 DynaASCII Nodout X DISP 432 Timestep response Intru_2 linear response Intru_1 1 0 DynaASCII Nodout X DISP 167 Timestep response Intru_1 linear response HIC 1 0 DynaASCII Nodout HIC15 9810 1 432 response HIC linear NO HISTORIES DEFINED FOR SOLVER 1 HISTORIES AND RESPONSES DEFINED BY EXPRESSIONS composites 1 composite Intrusion type weighted 190 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS composite Intrusion response Intru_2 1 scale 1 composite Intrusion response Intru_1 1 scale 1 5 OBJECTIVE FUNCTIONS objectives 1 objective HIC 1 CONSTRAINT DEFINITIONS constraints 1 constraint Intrusion upper bound constraint Intrusion 550 JOB INFO iterate param design 0 01 iterate param objective 0 01 iterate 1 STOP The computed vs predicted HIC and Intru_2 responses are given in Figure 17 7 The corresponding R value for HIC is 0 9248 while the RMS error is 27 19 For Intru 2 the R value is 0 9896 while the RMS error is 0 80 Response Surface Accuracy Response Surface Accuracy For Response Function HIC For Response Function Intru _2 8 60 i 7 r _ 62 E E 6 F7 x 64 8 ER EN lt v 1 4 amp gt 1 gt 66 o 1 1 fab 7 4
46. 1 5 Conducting a trade off study The present region of interest 2 0 8 is chosen in order to conduct a study in which the weight is traded off against the stress constraint The trade off is performed by selecting the Trade off option in the View panel of LS OPTui The upper bound of the stress constraint is varied from 0 2 to 2 0 with 20 increments Select Constraint as the Trade off option and enter the bounds and number of increments Generate the trade off This initiates the solution of a series of optimization problems using the response surface generated in Section with the constraint in each constant coefficient of the constraint response surface polynomial being varied between the limits selected The resulting curve is also referred to as a Pareto optimality curve When plotting select the Constraint Stress and not the Response Stress as the latter represents only the left hand side of the constraint equation 17 2 The resulting trade off diagram Figure 17 4 shows the compromise in weight when the stress constraint is tightened 182 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Objective Weight Tradeoff Plot Constraint Stress vs Objective Weight B X 1 I Li 1 1 1 I 0 5 1 1 5 2 2 5 3 Constraint Stress Figure 17 4 Trade off of stress and weight 17 1 6 Automating the design process This section illustrates the automation of the design pro
47. 1 workpiece block 152 2 724 2 15 0 100 0 100 z4 thick thl mate 1 endpart merge write end The error parameters for the fitted functions are given in the following output from lsopt_output file Approximating Response Thinning using 16 points ITERATION 1 222 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Mean response value RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted R 2 prediction Determinant of X x Approximating Response using 16 27 8994 6657 2932 5860 0126 0130 9913 9826 9207 5965 HoooHNohH Oo N N W points NIN BND ITERATION 1 Mean response value RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted R 2 prediction Determinant of X x 0 0698 0121 0247 0103 0332 0003 29774 9542 8272 5965 Hooooo0o0o0o0 N N W EEE EN Bly 35 14 47 oe w w U1 w U N lo eA oA oA o The thinning has a reasonably accurate response surface but the FLD approximation requires further refinement The initial design has the following response surface results which fail the criteria for maximum thinning but not for FLD DESIGN POINT Radius_1 Radius_2 Radius_3 Lower Bound 0 09123 1 5 Value 0 1006 1 5 Upper Bound LS OPT Version 2 223 CHAPTER 17 EXAM
48. 2 67 subject to fF T J lt e j 1 p e20 This formulation is automatically activated in LS OPT when specifying both the lower and upper bounds of f T equal to F T There is therefore no need to define an objective function This is due to the fact that an auxiliary problem is solved internally whenever an infeasible design is found ignoring the objective function until a feasible design is found When used in parameter identification the constraint set is in general never completely satisfied due to the typically over determined systems used 2 15 4 Worst case design Worst case design involves minimizing an objective with respect to certain variables while maximizing the objective with respect to other variables The solution lies in the so called saddle point of the objective function and represents a worst case design This definition of a worst case design is different to what is sometimes referred to as min max design where one multi objective component is minimized while another is maximized both with respect to the same variables There is an abundance of examples of worst case scenarios in mechanical design One class of problems involves minimization design variables and maximization case or condition variables One example in automotive design is the minimization of head injury with respect to the design variables of the interior trim while maximizing over a range of head orientation angles Therefore the wors
49. 3 dependent Shear modulus Youngs modulus 2 1 Poisson_ratio 92 LS OPT Version 2 CHAPTER 8 DESIGN VARIABLES CONSTANTS AND DEPENDENTS File Tasks Help Info Solvers Variables Point Selection Histories Responses Objective Constraints Run View u ee Minimum Range Variable CE rail_ back ci mm zu sein Direction Minimize Variable t hood le 2 A Is Solvers Constant t bumper E v All Variable it roof le 2 A e List Dependent Definition BBrUMLONZT Select All Ce ca Add a Variable Delete Variable arr Figure 8 2 Variables panel in LS OPTui with Constants and Dependents 8 8 Worst case design Worst case or saddle point design is where the objective function is minimized or maximized with respect to some variables while it is maximized or minimized with respect to the remaining variables in the variable set The maximization variables are set using the Maximize option in the Saddle Direction field of the Variables panel The default selection is Minimize Command file syntax Variable variable name max Example variable head_orientation max LS OPT Version 2 93 CHAPTER 8 DESIGN VARIABLES CONSTANTS AND DEPENDENTS 94 LS OPT Version 2 9 Metamodels and Point Selection This chapter describes the specification of the metamodel types and point selection schemes design of experiments or DOE Th
50. First linear iteration First quadratic iteration Automated run Trade off using neural network approximation Reliability based design optimization Impact of a cylinder 2 variables Problem statement A first approximation Refining the design model using a second iteration Third iteration Response filtering using the peak force as a constraint Sheet metal forming 3 variables Problem statement First Iteration Automated design Material identification airbag 10 variables Problem statement Least squares residual LSR formulation Maximum violation formulation Implementation xii LS OPT Version 2 CONTENTS Results Small car crash and NVH MDO 5 variables Parameterization and Variable screening MDO with D optimal experimental design and SRSM Sequential random search Large car crash and NVH MDO 7 variables Modeling Formulation of optimization problem Implementation in LS OPT Simulation results Optimization history results Comparison of optimum designs Convergence and computational cost Knee impact with variable screening 11 variables Problem statement Definition of optimization problem Implementation Variable screening ptimization with reduced variables Optimization with analytical design sensitivities Probabilistic Example Overview Problem Description Monte Carlo evaluation Monte Carlo using Metamodel LS OPT Version 2 xiii CONTENTS Bibliograph
51. Given an LHS design moves the points within each LHS subinterval preserving the starting LHS structure optimizing the maximin distance criterion and taking into consideration a set of fixed points 5 given an arbitrary design and a set of fixed points randomly moves the points so as to optimize the maximin distance criterion using simulated annealing see Appendix F Discussion of algorithms The Maximin distance space filling algorithms 3 4 and 5 minimize the energy function defined as the negative minimal distance between any two design points Theoretically any function that is a metric can be used to measure distances between points although in practice the Euclidean metric is usually employed The three algorithms 3 4 and 5 differ in their selection of random Simulated Annealing moves from one state to a neighboring state For algorithm 3 the next design is always a central point LHS design Eq 2 21 The algorithm swaps two elements of J Sy and Sy where i and k are random integers from 1 to N and jis a random integer from 1 to n Each step of algorithm 4 makes small random changes to a LHS design point preserving the initial LHS structure Algorithm 5 transforms the design completely randomly one point at a time In more technical terms algorithms 4 and 5 generate a candidate state S by modifying a randomly chosen element Sy of the current design S according to S S 2 23 where is
52. H H H H E E H H EE A HEER HHHH ERR HHRH HHHH Variable T1 Distribution Information Number of points 1000000 Mean Value 1 Standard Deviation 0 04997 Coef of Variation 0 04997 Maximum Value 1 227 Minimum Value 0 7505 Variable YS Distribution Information Number of points 1000000 Mean Value 1 Standard Deviation 0 09994 Coef of Variation 0 09994 Maximum Value 1 472 Minimum Value 0 5187 PEHE HEHE HEHE HEHE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE FE HE HE HE HE HE HE HE FE HE FE HE HE HEH HH HH HHH STATISTICS OF RESPONSES PEHE HEHE HEHE HEHE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE FE HE FE HE FE HEHE HEH HH HH HHH Response NodDisp Distribution Information Number of points 1000000 Mean Value 141 4 Standard Deviation 14 95 Coef of Variation 0 1058 Maximum Value 68 5 Minimum Value 206 3 284 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Response DispT Distribution Information Number of points 1000000 Mean Value 8 7 68 Standard Deviation 0 546 Coef of Variation 0 0711 Maximum Value 9 267 Minimum Value 2 565 HHRHHHPTEE HHP EEE HRP EEE HRP HERE EEE HERE HHHH STATISTICS OF COMPOSITES E HEH E E H H HH E A H H EEEH H REE EE HRP HERE HERE EEE RE E A R EA H EHHH RRHH RRHH RRHH HERE EEE STATISTICS OF CONSTRAINTS ERARE HHRHH HAAHR RREH RHEE RH HEEREH RHEE ERE Con
53. I I I Computed Response Value x1 02 gt Computed Response Value x1 0 1 i I I 1 I I 4 1 I I 1 T I 1 1 4 I ar otoia toea ea ala free rFr 1 2 3 4 5 6 7 8 72 70 68 66 64 62 60 Predicted Response Value x102 Predicted Response Value x101 a HIC response b Intru _ 2 constraint Figure 17 11 Response surface accuracy using neural network approximation A trade off study considers a variation in the Intrusion constraint originally fixed at 550mm between 450 and 600mm the same as in The experimental design used for the responses in Figure 17 11 LS OPT Version 2 197 EXAMPLE PROBLEMS CHAPTER 17 is shown in Figure 17 12 The effect of the Space Filling algorithm in maximizing the minimum distance between the experimental design points can clearly be seen from the evenly distributed design The resulting hood and t bumper can be seen in It can be seen that a tightening of the Intrusion constraint increases the HIC value through an 13 Pareto optimality curves for HIC and the two design variables t Figure 17 increase of the hood thickness in the optimal design Tradeoff Plot Variable t_hood vs Variable t_bumper Jedwng ajqeueA Variable t_hood Figure 17 12 Experimental design points used for trade off Tradeoff Plot Constraint Intrusion vs Variable t_bum Tradeoff Plot Constraint Intrusion vs Objective HIC per ie lt mM 2
54. MAX AVE Minimum maximum or average computed over all the elements of the selected parts LS OPT Version 2 121 CHAPTER 10 HISTORY AND RESPONSE RESULTS Example History VonMisesStress DynaD3plotHistory 38 NODE MAX History Thickness DynaD3plotHistory 371 1 2 4 ELEMENT MAX Response VonMisesStressMax Max VonMisesStress t Response ThicknessFinal Final Thickness t Response TimeAtMinStress LookupMin VonMisesStress t 10 9 Extracting Metal Forming Response Quantities LS DYNA Responses directly related to sheet metal forming can be extracted namely the final sheet thickness or thickness reduction Forming Limit criterion and principal stress All the quantities can be specified on a part basis as defined in the input deck for LS DYNA Mesh adaptivity can be incorporated into the simulation run The user must ensure that the d3plot files are produced by the LS DYNA simulation 10 9 1 Thickness and thickness reduction Either thickness or thickness reduction can be specified as follows Command file syntax DynaThick THICKNESS REDUCTION pl p2 pm MIN MAX AVE Table 10 8 DynaThick item description Item Description THICKNESS Final thickness of part REDUCTION A percentage thickness reduction of the part pl pn The parts as defined in LS DYNA If they are omitted all the parts are used MIN MAX AVE Minimum maximum or average computed
55. METHODOLOGY for n 20 50 and 100 In Figure 2 6 the successive linear response surface method is compared with the random search method for 20 50 and 100 variable optimization problems In this example SRSM uses the default number of simulations per iteration namely 32 77 and 152 respectively D optimal point selection is used The random search uses 20 LHS simulations per iteration As expected the cost increases with n for both SRSM and SRS Note the logarithmic trends of the convergence for both methods Each interval on the vertical axis represents an order or magnitude in accuracy The user should be aware that the search method picks the best observed design while the metamodeling methods especially NN and RSM are designed to find the best average design More intelligent search methods will be available in future versions 0 01 0 001 0 0001 oO gt c e 5 8 c gt i oO 2 oO Q O 0 00001 0 000001 500 1000 1500 Number of simulations Figure 2 6 Minimization of a quadratic polynomial Efficiency comparison of linear response surface m and random search methods for 20 50 and 100 variables Each point is an iteration 38 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 14 Summary of the optimization process The following tasks may be identified in the process of an optimization cycle using response surfaces Table 2 3 Summary of optimiza
56. OPTui constraints are defined in the Constraints panel Figure 11 2 Figure 11 2 Constraints panel in LS OPTui 136 LS OPT Version 2 CHAPTER 11 OBJECTIVES AND CONSTRAINTS 11 4 Bounds on the constraint functions Upper and lower bounds may be placed on the constraint functions Command file syntax Lower bound constraint constraint name value lt 10 gt 30 Upper bound constraint constraint name value lt 10 gt Example Lower bound constraint Stress 1 e 6 Upper bound constraint Stress 20 2 Remark 1 A flag can be set to identify specific constraint bounds to define a reasonable design space For this purpose the move environment must be specified See Section 9 5 11 5 Minimizing the maximum response or violation Refer to Section for the theory regarding strict and slack constraints To specify hard strict or soft slack constraints the following syntax is used Command file syntax strict strictness factor lt 1 gt slack Each command functions as an environment Therefore all lower bound constraint or upper bound constraint commands which appear aftera strict slack command will be classified as strict orslack In the following example the first two constraints are slack while the last three are strict The purpose of the formulation is to compromise only on the knee forces if a feasible design cannot be found Example This formulation minimizes the average knee force
57. Result Files and Components Airbag Statistics ABSTAT Keyword Description VOLUME Volume PRESSURE Pressure I ENER Internal energy IN_FLOW_RATE Input mass flow rate OUT_FLOW_RATE Output mass flow rate MASS Mass TEMP Temperature DENSITY Density AREA Area Boundary Nodal Forces BNDOUT Keyword Description X FORCE X force Y_ FORCE Y force Z FORCE Z force Discrete Element Forces DEFORC Keyword Description X_FORCE X force Y_FORCE Y force Z FORCE Z force R_FORCE Resultant force 291 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Element Output Brick and Beam Elements ELOUT Keyword Element Description BXX_STRESS Brick XX stress BYY_STRESS YY stress BZZ_STRESS ZZ stress BXY_STRESS XY stress BYZ STRESS YZ stress BZX STRESS ZX stress YIELD Yield function BE STRESS Effective stress BPRESSURE Pressure BMAX SHEAR Maximum shear stress BMAX P STRESS BMIN P STRESS Maximum principal stress Minimum principal stress AXIAL S SHEAR T SHEAR S MOMENT T MOMENT TORSION SIG 11 SIG_12 SIG 31 PLASTIC Beam Axial force resultant s Shear resultant t Shear resultant s Moment resultant t Moment resultant Torsional resultant O11 02 031 Plastic strain 292 LS OPT Version 2 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Element Output Shell Elements ELOUT Keyword Element Description XX_STRESS S
58. S3 Oi O2 O3 MEAN 0i 02 03 3 pl pn Part numbers of the model Omission implies the entire model MIN MAX AVE Minimum maximum or average computed over all the elements of the selected parts Example Response Stress 1 DynaPStress MEAN 14 17 MAX 10 10 Extracting data from the LS DYNA Binout file From Version 970 of LS DYNA the ASCII output can be written to a binary file the Binout file The Binout commands differ from the DynaASCII commands in that a keyword must be specified to delimit an item in a command For example res type nodout instead of just nodout The position of the items in a command is therefore not important for the Binout extraction commands The LS PREPOST Binout capability can be used for the graphical exploration and troubleshooting of the data 126 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS The response options are an extension of the history options a history will be extracted as part of the response extraction 10 10 1 Binout Histories Results can be extracted for the whole model or a finite element entity such as a node or element For shell and beam elements the through thickness position can be specified as well Command file syntax BinoutHistory res type res type sub sub cmp component invariant invariant id id pos position side side Item Description Default Remarks res_type Re
59. X_ACC response acc15_4 1000 0 DynaASCII nodout X_ACC response acc20 4 1000 0 DynaASCII nodout X_ACC response acc25 4 1000 0 DynaASCII nodout X_ACC response acc30 4 1000 0 DynaASCII nodout X ACC response acc35 4 1000 0 DynaASCII nodout X_ACC response acc40 4 1000 0 DynaASCII nodout X_ACC response acc45 4 1000 0 DynaASCII nodout X_ACC response acc50_4 1000 0 DynaASCII nodout X_ACC response vell0_4 1 0 DynaASCII nodout X VEL 1 response vell5_4 1 0 DynaASCII nodout X VEL 1 response vel20 4 1 0 DynaASCII nodout X VEL 1 response vel25 4 1 0 DynaASCII nodout X VEL 1 response vel30 4 1 0 DynaASCII nodout X VEL 1 response vel35 4 1 0 DynaASCII nodout X VEL 1 response vel40 4 1 0 DynaASCII nodout X VEL 1 response vel45 4 1 0 DynaASCII nodout X VEL 1 response vel50 4 1 0 DynaASCII nodout X VEL 1 response displ0 4 1 124 833 DynaASCII nodout response disp15 4 1 124 833 DynaASCII nodout response disp20 4 1 124 833 DynaASCII nodout response disp25_4 1 124 833 DynaASCII nodout response disp30 4 1 124 833 DynaASCII nodout response disp35 4 1 124 833 DynaASCII nodout response disp40 4 1 124 833 DynaASCII nodout response disp45 4 1 124 833 DynaASCII nodout response disp50 4 1 124 833 DynaASCII nodout NO HISTORIES DEFINED FOR SOLVER 4MPS DEFINITION OF SOLVER 5MPS solver dyna 5MPS solver command lsdyna solver input file sim5mpros inp
60. a random number sampled from a normal distribution with zero mean and standard deviation 6 Omin Cmax In algorithm 4 it is required that both S and S in Eq 2 23 belong to the same Latin hypercube subinterval Notice that maximin distance energy function does not need to be completely recalculated for every iterative step of simulated annealing The perturbation in the design applies only to some of the rows and columns of S After each step we can recompute only those nearest neighbor distances that are affected by the stepping procedures described above This reduces the calculation and increased the speed of the algorithm To perform an annealing run for the algorithms 3 4 and 5 the values for Tmax and Tmin can be adapted to the scale of the objective function according to LS OPT Version 2 19 CHAPTER 2 OPTIMIZATION METHODOLOGY Ts Tyo XAE 2 24 Tain Trin XAE where AE gt 0 is the average value of E E observed in a short preliminary run of simulated annealing and Tmax and Tmin are positive parameters The basic parameters that control the simulated annealing in algorithms 3 4 and 5 can be summarized as follows 1 Energy function negative minimal distance between any two points in the design 2 Stepping scheme depends on whether the LHS property is preserved or not 3 Scalar parameters 1 Parameters for the cooling schedule scaling factor for the initial maximal temperature Tmax in 2 24
61. and blank The design problem is formulated to minimize the maximum tool radius while also specifying an FLD constraint and a maximum thickness reduction of 20 thinning constraint Since the user wants to enforce the FLD and thinning constraints strictly these constraints are defined as strict To minimize the maximum radius a small upper bound for the radii has been specified arbitrarily chosen as a number close to the lower bound of the design space namely 1 1 The optimization solver will then minimize the maximum difference between the radii and their respective bounds The radius constraints must not be enforced strictly This translates to the following mathematical formulation Minimize e with 1 5 lt 7 lt 4 5 1 5 lt r lt 4 5 1 5 lt r lt 4 5 subject to g x lt 0 0 At x lt 20 N r l l lt e rn 1 l lt e e gt 0 The design variables r r2 and r3 are the radii of the work piece as indicated in Figure 17 21 At is the thickness reduction which is positive when the thickness is reduced The FLD constraint is feasible when smaller than zero LS OPT Version 2 217 CHAPTER 17 EXAMPLE PROBLEMS 17 4 2 First Iteration The initial run is a quadratic analysis designed as an initial investigation of the following issues e The dependency of the through thickness strain constraint on the radii e The dependency of the FLD constraint on the radii e The location of the optimal design point The subregio
62. but S constrains the forces to 6500 If a feasible design is not available the maximum violation S will be minimized 5 Objective composite Knee Forces type weighted composite Knee Forces response Left Knee Force 0 5 composite Knee Forces response Right Knee Force 0 5 LS OPT Version 2 137 CHAPTER 11 OBJECTIVES AND CONSTRAINTS objective Knee Forces S Constraints SLACK Constraint Left Knee Force Upper bound constraint Left Knee Force 6500 Constraint Right Knee Force Upper bound constraint Right Knee Force 6500 STRICT Constraint Left Knee Displacement Lower bound constraint Left Knee Displacement 8233 Constraint Right Knee Displacement Lower bound constraint Right Knee Displacement 81 33 5 Constraint Kinetic Energy Upper bound constraint Kinetic Energy 154000 The composite function is explained in Section Note that the same response functions appear both in the objective and the constraint definitions This is to ensure that the violations to the knee forces are minimized but if they are both feasible their average will be minimized as defined by the composite The constraint bounds of all the soft constraints can also be set to a number that is impossible to comply with e g zero This will force the optimization procedure to always ignore the objective and it will minimize the maximum respo
63. but they are not random Stochastic methods have also been touted as design improvement methods In a typical approach the user iteratively selects the best design results of successive stochastic simulations to improve the design These design methods being dependent on chance are generally not as efficient as response surface methods However an iterative design improvement method based on stochastic simulation is available in LS OPT Stochastic methods have an important purpose when conducted directly or on the surrogate approximated design response in reliability based design optimization and robustness improvement This methodology is currently under development and will be available in future versions of LS OPT 2 2 Theory of Optimization Optimization can be defined as a procedure for achieving the best outcome of a given operation while satisfying certain restrictions 119 This objective has always been central to the design process but is now assuming greater significance than ever because of the maturity of mathematical and computational tools available for design Mathematical and engineering optimization literature usually presents the above phrase in a standard form as min f x 2 1 subject to g x lt 0 PJM and REN amp kei where f g and h are functions of independent variables x x2 x3 Xn The function f referred to as the cost or objective function identifies the quantity to be minimized or maximiz
64. called bias error especially in a large subregion or where there is strong non linearity present could play a very significant role There are several error measures available to determine the accuracy of a response surface 2 8 1 Residual sum of squares For the predicted response y and the actual response y this error is expressed as 2 E v 3 2 26 Mb Il l If applied only to the regression points this error measure is not very meaningful unless the design space is oversampled E g 0 if the number of points P equals the number of basis functions Z in the approximation 2 8 2 RMS error The residual sum of squares is sometimes used in its square root form and called the RMS error 2 27 LS OPT Version 2 21 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 8 3 Maximum residual This is the maximum residual considered over all the design points and is given by yi Vi 2 28 E max max 2 8 4 Prediction error The same as the RMS error but using only responses at preselected prediction points independent of the regression points This error measure is an objective measure of the prediction accuracy of the response surface since it is independent of the number of construction points It is important to know that the choice of a larger number of construction points will for smooth problems diminish the prediction error The prediction points can be determined by adding
65. command or response response name scale factor offset command line history history name command line 10 6 Extracting response quantities from ASCII output LS DYNA 10 6 1 Mass Command file syntax DynaMass pl p2 p3 pn mass attribute Table 10 1 Mass item description Item Description pl pn Part numbers of the model Omission implies the entire model Mass_attribute Type of mass quantity see table below 116 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS Table 10 2 Mass attribute description Attribute Description MASS Mass I11 Principal inertias I22 133 IXX Components of inertia tensor IXY XZ YX IYY IYZ IZX IZY IZZ X_COORD x coordinate of mass center Y_COORD y coordinate of mass center Z_COORD z coordinate of mass center Example Specify the mass of material number 13 14 and 16 as the response Component mass response Component mass DynaMass 3 13 14 16 Mass Specify the total principal inertial moment about the x axis response Inertia DynaMass Ixx Remarks 1 A Perl utility is used to extract the desired responses The output file d3hsp must be produced by LS DYNA 3 Values are summed if more than one part is specified so only the mass value will be correct However for the full model part specification omitted the correct values are given for all the quantities 10 6 2 Frequency of
66. conditions are not used explicitly in LS OPT and are not tested for at optima They are more of theoretical interest in this manual although the user should be aware that some optimization algorithms are based on these conditions 2 3 Gradient Computation and the Solution of Optimization Problems Solving the optimization problem requires an optimization algorithm The list of optimization methods is long and the various algorithms are not discussed in any detail here For this purpose the reader is referred to the texts on optimization e g or 119 It should however be mentioned that the Sequential Quadratic Programming method is probably the most popular algorithm for constrained optimization and is considered to be a state of the art approach for structural optimization falles In LS OPT the subproblem is optimized by an accurate and robust gradient based algorithm the dynamic leap frog method 60 Both these algorithms and most others have in common that they are based on first order formulations i e they require the first derivatives of the component functions df dx and dg dx 8 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY in order to construct the local approximations These gradients can be computed either analytically or numerically In order for gradient based algorithms such as SQP to converge the functions must be continuous with continuous first derivatives Analytical differentiation requires the formul
67. effective plastic strain 45 50 lower surface x y Z xy yz zx strain 51 56 upper surface x y Z xy yz zx strain 57 62 middle surface x y Z xy yz zx strain 63 internal energy density 64 xy displacement 65 yz displacement 66 zx displacement 67 shell thickness 68 shell thickness reduction 69 80 lower upper middle principal effective strains 81 n n th history variable 507 FLD criterion brick 508 509 in plane Ist 2nd principal strains 543 544 lower upper surface FLD criterion 546 547 lower upper surface in plane 1st principal strain 549 550 lower upper surface in plane 2nd principal strain 302 LS OPT Version 2 Appendix C Airbag Statistics LS DYNA Binout Result File and Components ABSTAT DynaASCH Binout Description Keyword Component VOLUME volume Volume PRESSURE pressure Pressure I ENER internal energy Internal energy IN FLOW _RATE dm_dt in Input mass flow rate OUT _FLOW RATE dm dt out Output mass flow rate MASS total_mass Mass TEMP gas_temp Temperature DENSITY density Density AREA surface_area Area reaction Reaction 303 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Boundary Nodal Forces BNDOUT DynaASCH Binout Description Keyword Component Binout subdirectory discrete nodes X_FORCE x_force X force Y_FORCE y_force Y force Z_FORCE z_force Z force x_ total Total X force y_total Total Y force z_total
68. enough to meet all the crash constraints The reduction in the allowable frequency band made the NVH performance more interesting in Phase 1 than Phase 2 It can be seen in Figure 17 43 that the lower bound becomes active during the optimization process but that the optimizer then pulls the torsional mode frequency within the prescribed range The final design iteration considered iteration 9 was repeated see point 10 in Figure 17 41 through with the variables rounded to the nearest 0 1mm due to the 0 1mm manufacturing tolerance typically used in the stamping of automotive parts It is shown that the design is lighter by 4 75 from the baseline but at the cost of a 2 4 violation in the Stage 2 pulse The other constraints are satisfied A summary result for the heaviest and lightest starting designs 2 and 3 is given in Figure 17 44 for the objective function In both cases an ANOVA was performed after one iteration of full sharing only in order to reduce the number of discipline specific variables using variable screening The optimization was then restarted using the variable sets as defined above As expected both designs converge to an intermediate mass in an attempt to satisfy all the constraints The heaviest design history exhibits the largest mass change because of the significant increase in the thickness of the components over the baseline design The initial allowable range or move limit on the design variables was doubled in the heav
69. file descriptions Appendix D a mathematical expression library Appendix E advanced theory Appendix F a Glossary Appendix G and a Quick Reference Manual Sections containing advanced topics are indicated with an asterisk How to read this manual Most users will start learning LS OPT by consulting the User s Manual section beginning with Chapter a The design optimization process The Theoretical Manual Chapter 2 serves mainly as an in depth reference section for the underlying methods The Examples section is included to demonstrate the features and capabilities and can be read together with Chapters 3 to 14 to help the user to set up a problem formulation The items in the Appendices are included for reference to detail while the Quick Reference Manual provides an overview of all the features and command file syntax Features that are only available in Version 2 1 are indicated The most important of these are 1 extraction from the binout database ii stochastic search iii probabilistic modeling and iv Kriging INTRODUCTION 2 LS OPT Version 2 THEORETICAL MANUAL LS OPT Version 2 2 Optimization Methodology 2 1 Introduction In the conventional design approach a design is improved by evaluating its response and making design changes based on experience or intuition This approach does not always lead to the desired result that of a best design since design objectives are sometimes in
70. files reside and where output is produced See also Run directory 342 LS OPT Version 2 Appendix H LS OPT Commands Quick Reference Manual Note All commands are case insensitive The commands which are definitions are given in boldface Page reference numbers of the syntax definition are given in the last column Command phrases in are optional string Extraction command solver preprocessor command or file name in double quotes name Name in single quotes expression Mathematical expression in curly brackets H 1 Problem description Constants number The number of constants in the problem Variables number The number of variables in the problem Dependents number The number of dependent variables Histories number The number of histories Responses number The number of responses Composites number The number of composite functions Objectives number The number of objectives Constraints number The number of constraints Solvers number The number of solvers Distribution number The number of probabilistic distributions H 2 Parameter definition Constant name value constant P1 343 APPENDIX H QUICK REFERENCE MANUAL H 3 Probabilistic distributions Distribution name type values type values NORMAL mu sigma UNIFORM lower upper USER DEFINED PDF filename USER DEFINED CDF filename LOGNORMAL mu sigma WEIBULL scale shape H 4 Design space and region of interest Variab
71. for the large scale multidimensional problems an algorithm which always or often obtains a solution near the global optimum is valuable since various local deterministic optimization methods allow quick refinement of a nearly correct solution In summary simulated annealing is a powerful method for global optimization in challenging real world problems Certainly some trial and error experimentation is required for an effective implementation of the algorithm The energy cost function should employ some heuristic related to the problem at hand clearly reflecting how good or bad is a given solution Random perturbations of the system state and corresponding cost change calculations should be simple enough so that SA algorithm can perform its iterations very fast The scalar parameters of the simulated annealing algorithm Tmax uT Vr in particular have to be chosen carefully If the parameters are chosen such that the optimization evolves too fast the solution converges directly to some possibly good solution depending on the initial state of the problem 336 LS OPT Version 2 Appendix G Glossary ANOVA Analysis of variance Used to perform variable screening by identifying insignificant variables Variable regression coefficients are ranked based on their significance as obtained through a partial F test See also variable screening Bias error The total error the difference between the exact and computed response
72. functions return the value of at the minimum or maximum respectively The implied variable represented in the first column of any history file is Therefore all history files produced by the DynaASCII extraction command contain functions of t The fourth argument of the Integral function defaults to t The variable t must increase monotonically The derivative assumes a piecewise linear function defined by the points in the history n file T constant in the Derivative function defaults to the end time If a time is specified smaller than the smallest time value of the computed history the first value is returned same as Initial Ifa time is specified larger than the largest time value of the computed history the last value is returned same as Final For derivatives the first or last slopes are returned respectively E 5 Constants associated with histories The following commands can be given to override defaults for history operations Constant Explanation Default variable fdstepsize Finite difference step size for 0 0001 Upper bound Lower bound numerical derivatives with respect to variables historysize Number of time points for new 10000 history Command file syntax variable fdstepsize value historysize integer value e The variable fdstepsize is used to find the gradients of expression composite functions These are used in the optimization process e The historysize is used when n
73. if a polynomial response surface method is selected The ANOVA information can be used to screen variables remove insignificant variables at the start of or during the optimization process The ANOVA method a more sophisticated version of what is sometimes termed Sensitivities or DOE determines the significance of main and interaction effects through a partial F test equivalent to Student s t test B3 This screening is especially useful to reduce the number of design variables for different disciplines see Sections b 152 theory and 17 7 example The ANOVA results are viewed in bar chart format by clicking on the ANOVA button The ANOVA panel is shown in Figure 13 4 148 LS OPT Version 2 CHAPTER 13 VIEWING RESULTS E Type of Plot x Response Surface Accuracy wv Optimization History v Tradeoff ANOVA ANOVA Plot for R Knee Force Lower half of 90 confidence interval in red Solver Yolk_Radius Bolster_gauge R_Bracket_Radius R_Flange Width R_Bracket_Gauge L_Flange_ Width I_Flange_ Width B_Flange Depth F_ Flange Depth T_Flange Depth L_Bracket_Gauge 0 25 0 3 0 35 Terms in expansion of R_ Knee Force Figure 13 4 ANOVA plot in View panel in LS OPTui 13 5 Plot generation Plots can be generated in LS OPTui by selecting File gt Export The current supported format is postscript both color and monochrome either to a device or file LS OPT Version 2 149 CHAPTER 13 VIEWING RESULTS
74. indicator and the absolute move distance d 36 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY Accuracy The accuracy is estimated using the proximity of the predicted optimum of the current iteration to the starting previous design The smaller the distance between the starting and optimum designs the more rapidly the region of interest will diminish in size If the solution is on the bound of the region of interest the optimal point is estimated to be beyond the region Therefore a new subregion which is centered on the current point does not change its size This is called panning Figure 2 4 a If the optimum point coincides with the previous one the subregion is stationary but reduces its size zooming Figure 2 44b Both panning and zooming may occur if there is partial movement Figure 2 4 c The range r for the new subregion in the k 1 th iteration is then determined by ei G 1 nan k 0 niter 2 59 where A represents the contraction rate for each design variable To determine A d is incorporated by scaling according to a zoom parameter n that represents pure zooming and the contraction parameter y to yield the contraction rate A n d y m 2 60 for each variable see Figure 2 5 When used in conjunction with neural networks or Kriging the same heuristics are applied as described above However the nets are constructed using all the available points including those belonging to p
75. input dir sed s g we awk print 2 gt precnt expr local_ent 1 local pre echo input dir sed s g awk print Sprecnt local dir echo input dir sed s g awk print local_cnt cd input_dir export WORKDIR net o300 Sinput_dir e cho DN Bene Fe EE cr ah a re rn EE a m Ban kr AA ee ah An eer ye an En NT ak an tn re ran Se n echo This job will run under directory WORKDIR e cho W na te Se ee Ss eg es a ee ee Ge ee cdg ee OS es ee eis Sc Gy er rt os Sd as a SS a W cat gt jobname local_dir job lt lt EOF BSUB J local_pre local_dir bin sh cd SWORKDIR export LSTC_FILE usr company license LSTC_FILE echo i Sinputfile memory 5000000 gt commandline if test precision single then vclass bin wrappers wrapper hp 11 vclass tools mppscript jobname inputfile 2 else vclass bin wrappers wrapper hp 11 vclass bin 1s970 double i inputfile memory 2000000 2 gt jobname err fi exit EOF LSF queuing command bsub m vclass lstc com lt jobname local_dir job amp 6 14 Database conversion A database which was produced by versions prior to 2a can be converted for use with LS OPTui Version 2a The command is lsopt lt command_file name gt convert ver20 This command produces a result database in a subdirectory called newdatabase The Version 2a LS OPTui can be executed from inside the newdat abase dir
76. is placed and the number of levels is the same as the number of runs The levels are assigned to runs either randomly or so as to optimize some criterion e g so that the minimal distance between any two design points is maximized maximin distance criterion Restricting the design in this way tends to produce better Latin Hypercubes However the computational cost of obtaining these designs is high In multidimensional problems the search for an optimal Latin hypercube design using traditional deterministic methods e g the optimization algorithm described in H5 may be computationally prohibitive This situation motivates the search for alternatives Probabilistic search techniques simulated annealing and genetic algorithms are attractive heuristics for approximating the solution to a wide range of optimization problems In particular these techniques are frequently used to solve combinatorial optimization problems such as the traveling salesman problem Morris and Mitchell adopted the simulated annealing algorithm to search for optimal Latin hypercube designs LS OPT Version 2 17 CHAPTER 2 OPTIMIZATION METHODOLOGY In LS OPT space filling designs can be useful for constructing experimental designs for the following purposes 1 The generation of basis points for the D optimality criterion This avoids the necessity to create a very large number of basis points using e g the full factorial design for large n E g for n 20 and 3
77. is straightforward to show that the derivative of the network Eq 2 46 with respect to any of its inputs is given by LS OPT Version 2 29 CHAPTER 2 OPTIMIZATION METHODOLOGY a 4 7 I SW Waf Mm EW k 1 K 2 47 Ox h 1 h 1 Neural networks have been mathematically shown to be universal approximators of continuous functions and their derivatives on compact sets 25 In other words when a network Eq 2 46 converges towards the underlying function all the derivatives of the network converge towards the derivatives of this function Standard non linear optimization techniques including a variety of gradient algorithms the steepest descent RPROP Levenberg Marquardt etc are applied to adjust FF network s weights and biases For _neural networks the gradients are easily obtained using a chain rule technique called backpropagation 51 The second order Levenberg Marquardt algorithm appears to be the fastest method for training moderate sized FF neural networks up to several hundred adjustable weights 8 However when training larger networks the first order RPROP algorithm becomes preferable for computational reasons 47 Regularization For FF networks regularization may be done by controlling the number of network weights model selection by imposing penalties on the weights ridge regression or by various combinations of these strategies Model selection requires choosing the number of hidd
78. length of the cylinder respectively The problem is simulated using LS DYNA The following TrueGrid input file including the lt lt name gt gt statements is used to create the FE input deck with the FE model as shown in Figure 17 16 Note that the design variables have been scaled c cyl2 crush cylinder constant volume lsdyna3d keyword lsdyopts secforc 00002 rwforc 00002 lsdyopts endtim 02 d3plot dtcycl 0001 lsdyopts thkchg 2 lsdyopts elout 0 001 lsdyopts glstat 0 001 lsdymats 1 3 rho 2880 shell elfor bt tsti 4 e 71 38e9 pr 33 sigy 102 0e6 etan 0 2855e9 lsdymats 2 20 rho 14 3e6 e 7 138e10 pr 33 cmo con 4 7 shell elfor bt tsti 4 para r lt lt Radius gt gt 1000 0 1 3 0e 1 lt lt Radius gt gt lt lt Wall_Thickness gt gt h lt lt Wall_Thickness gt gt 1000 0 204 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 12 75 0 lt lt Radius gt gt 0 02 h2 002 vo 10 n 33 pi 3 14159 plane 1 0 O 002 0 0 1 001 ston pen 2 stick sid 1 lsdsi 13 slvmat 1 scoef 4 dcoef 4 sfsps 1 5 C RKKEKKKKKKKKKKK part 1 mat L RRR KKKKKEKKEK shell cylinder 1 1 60 1 50 51 Sr 0 360 0 1 12 1 dom 1 1 1 1 2 3 X X 01 Sh sin Spi z 57 3 Spir Sr Sr Sh Sh 12 1 Sn Sn 25 thick h thi 2 3 h2 bi 9 30 3z dx 1 dy 1 rx I try i rz L3 c interrupt swi 1 velocity 0 0 v0 mate 1 mti 2 3 2 c element spring block epb 111123 endpart merge stp 000001 write e
79. maximum values The starting design can be run by selecting 0 as the number of iterations in the Run panel Check that the design variables dependents and or constants are substituted into the input file as intended Modify the input to define the experimental design for a full analysis For a time dependent analysis or non linear analysis reduce the termination time or load significantly to test the logistics and features of the problem and solution procedure Execute LS OPT with the full problem specified and monitor the process Also refer to Sectionf 2 4 4 Pitfalls in design optimization A number of pitfalls or potential difficulties with optimization are highlighted here The perils of using numerical sensitivity analysis have already been discussed and will not be repeated in detail Global optimality The Karush Kuhn Tucker conditions Eqs 2 3 govern the local optimality of a point However there may be more than one optimum in the design space This is typical of most designs and even the simplest design problem such as the well known 10 bar truss with 10 design variables may have more than one optimum The objective is of course to find the global optimum Many gradient based as well as discrete optimal design methods have been devised to address global optimality rigorously but as there is no mathematical criterion available for global optimality nothing short of an exhaustive search method can determine wheth
80. of iteration 3 The creation of subdirectories is automated and the user only needs to deal with the working directory In the case of simulation runs being conducted on remote nodes a replica of the run directory is automatically created on the remote machine The response nand history n files will automatically be transferred back to the local run directory at the end of the simulation run These are the only files required by LS OPT for further processing 6 4 Job Monitoring The job status is automatically reported at a regular interval The user can also specify the interval The interface LS OPTui reports the progress of the jobs in the Run panel see Section 12 6 The text screen output while running both the batch and the graphical version also reports the status as follows JobID Status PID Remaining 1 Norma 1 termination 2 Running 8427 00 01 38 91 complete 3 Running 8428 00 01 16 93 complete 4 Running 8429 00 00 21 97 complete 5 Running 8430 00 01 13 93 complete 70 LS OPT Version 2 CHAPTER 6 PROGRAM EXECUTION 6 Running 8452 00 21 59 0 complete 7 Waiting 8 Waiting In the batch version the user may also type control C to get the following response Jobs started Got control C Trying to pause scheduler Enter the type of sense switch swl Terminate all running jobs sw2 Get a current job status report for all jobs t Set the report interval v Toggle the reporting status level to verbose
81. of objectives Two methods for achieving this are given 40 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY Euclidean Distance Function Designs often contain objectives that are in conflict so that they cannot be achieved simultaneously If the one objective is improved the other deteriorates and vice versa The preference function p f x combines the various objectives f The Euclidean distance function allows the designer to find the design with the smallest distance to a specified set of target responses or design variables p SA 2 62 i The symbols F represent the target values of the responses A value I is used to normalize each response i Weights W are associated with each quantity and can be chosen by the designer to convey the relative importance of each normalized response Maximum distance Another approach to target responses is by using the maximum distance to a target value p max LOE 2 63 i This form belongs to the same category of preference functions as the Euclidean distance function and is referred to as the Tchebysheff distance function A general distance function for target values F is defined as f x F r L i l yr p F 2 64 with r 2 for the Euclidean metric and r gt for the min max formulation Tchebysheff metric The approach for dealing with the Tchebysheff formulation differs somewhat from the explicit formul
82. on the upper bound suggesting an optimal value larger than 3 728 DESIGN POINT Variable Name Lower Bound Value Upper Bound Radius 20 42 43 100 Wall_Thickness 2 3 728 6 Scaled Unscaled ee ee Seed oc aa RESPONSE Computed Predicted Computed Predicted ee E ee ee eee Internal Energy 9777 9575 9777 9575 Rigid_Wall_Force 6 417e 04 7e 04 6 417e 04 7e 04 210 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 3 4 Third iteration Because of the large change in the wall_Thickness on to the upper bound of the region of interest a third iteration is conducted keeping the region of interest the same The starting point is the previous optimum Variable Radius 42 43 Variable Wall_Thickness 3 728 The approximation improves as shown below Approximating Response Internal_Energy using 10 points ITERATION 1 Mean response value 9801 0070 RMS error z 439 8326 4 49 Maximum Residual 834 5960 8 52 Average Error 372 3133 3 80 Square Root PRESS Residual 1451 3233 14 81 Variance 386905 5050 R 2 0 9618 R 2 adjusted 0 9618 R 2 prediction 0 5842 Determinant of X X 0 0131 Approximating Response Rigid Wall Force using 10 points ITERATION 1 Mean response value 81576 0534 RMS error 12169 4703 14 92 Maximum Residual 26348 0687 32 30 Average Error 10539 2275 12 92 Square Root PRESS Residual 37676 3033 46 19 Variance 296192016 4365 R 2
83. opposite corners of the design space hypercube i e the lightest design and heaviest design possible with the design variables used 260 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS accel_vs_displacement baseline accel_vs_displacement Iteration 6 StagePulses baseline StagePulses Iteration 6 Deceleration g 0 100 200 300 400 500 600 Displacement mm Figure 17 39 Deceleration Filtered SAE 60Hz versus displacement of baseline and Iteration 6 design Partially shared variables Starting design 1 Table 17 4 Bounds on design variables and starting designs for optimization Rail_ Rail_ Cradle Aprons Shotgun Shotgun Cradle cross inner outer rail mm inner outer member mm mm mm mm mm mm Lower bound 1 1 1 1 1 1 1 Upper bound 3 3 2 5 2 5 3 3 2 5 Starting design 1 2 1 5 1 93 1 3 1 3 1 3 1 93 Baseline Starting design 2 Minit 1 1 1 1 1 1 1 weight Starting Re 3 3 Da 25 3 3 2 5 Maximum weight LS OPT Version 2 261 CHAPTER 17 EXAMPLE PROBLEMS Table 17 5 Comparison of objective and constraints for all optimization cases Case It Mass Maximum Stage 1 Stage 2 Stage 3 Frequency No kg displacement pulse pulse pulse Hz mm e tel Ie Constraint 37 77 39 77 or 551 8 14 34 17 57 20 76 38 27 39 27 Fully Starting shared design 1 9 42 9 552 0 14
84. optimization of the approximate problem kriging A Metamodeling technique using Bayesian regression see e g 5 19 Latin Hypercube Sampling The use of a constrained random experimental design as a point selection scheme for response approximation Least Squares Approximation The determination of the coefficients in a mathematical expression so that it approximates certain experimental results by the minimization of the sum of the squares of the approximation errors Used to determine response surfaces as well as calibrating analysis models Local Approximation See Gradient vector Local variable A variable of which the scope is limited to a particular discipline or disciplines Used in the MDO context Material identification See parameter identification MDO Multidisciplinary design optimization Metamodeling The construction of surrogate design models such as polynomial response surfaces Artificial Neural Networks or Kriging surfaces from simulations at a set of design points Min Max optimization problem An optimization problem in which the maximum value considering several responses or functions is minimized Model calibration The optimal adjustment of parameters in a numerical model to simulate the physical model as closely as possible Modeling error See bias error LS OPT Version 2 339 APPENDIX G GLOSSARY Multidisciplinary design optimization MDO The inclusion of multiple disciplines in the des
85. penalty term Ew Using the modified performance function Eq 2 48 will cause the network to have smaller weights and this will force the network response to be smoother and less likely to overfit This eliminates the guesswork required in determining the 30 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY optimum network size Unfortunately finding the optimal value for and is not a trivial task If we make a B too small we may get over fitting If is too large the network will not adequately fit the training data A rule of thumb is that a little regularization usually helps 52 It is important that weight decay regularization does not require that a validation subset be separated out of the training data It uses all of the data This advantage is especially noticeable in small sample size situations Another nice property of weight decay regularization is that it can lend numerical robustness to the Levenberg Marquardt algorithm The L M approximation to the Hessian of Eq 2 48 is moved further away from singularity due to a positive addend to its diagonal A H al 2 49 where P gt H VE g g x i l SE EEE f OW OW In B9 and Ro the Bayesian evidence framework or type II maximum likelihood approach to regularization is discussed The Bayesian re estimation algorithm is formulated as follows At first we choose the initial values for and 2 Then a neural network is trained using a stand
86. points at least of the order of the minimum required for a quadratic approximation To converge to an optimum use the iterative scheme with domain reduction as with any other approximations but choose to update the experimental design and response surfaces after each iteration this is the default method for neural nets and Kriging The response surface will be built using the total number of points At the end of the iterative optimization procedure the trade off can be based on neural networks or Kriging surfaces using results from all the points calculated See also Sections f 5 1and Random search methods These methods can be used in very much the same way as first order approximations but are bound to be more expensive Section L 3 2 15 Applications of optimization 2 15 1 Multicriteria Design Optimization A typical design formulation is somewhat distinct from the standard formulation for mathematical optimization Eq 2 3 Most design problems present multiple objectives design targets and design constraints whereas the standard mathematical programming problem is defined in terms of a single objective and multiple constraints The standard formulation of Eq 2 3 has been modified to represent the more general approach as applied in LS OPT Minimize the function PAF 2 61 subject to the inequality constraint functions L lt g x lt U j 12 m The preference function p can be formulated to incorporate target values
87. procedure and design tools well in advance The following points are considered important 1 The user should be familiar with and have confidence in the accuracy of the model e g finite element model used for the design Without a reliable model the design would make little or no sense Select suitable criteria to formulate the design The responses represented in the criteria must be produced by the analyses and be accessible to LS OPT Request the necessary output from the analysis program and set appropriate time intervals for time dependent output Avoid unnecessary output as a high rate of output will rapidly deplete the available storage space Run at least one simulation using LS OPT To save time the termination time of the simulation can be reduced substantially This exercise will test the response extraction commands and various other features Just as in the case of traditional simulation it is advisable to dump restart files for long simulations LS OPT will automatically restart a design simulation if a restart file is available For this purpose the runrsf file is required when using LS DYNA as solver Determine suitable design parameters In the beginning it is important to select many rather than few design variables If more than one discipline is involved in the design some interdisciplinary discussion is required with regard to the choice of design variables Determine suitable starting values for the design
88. results Pareto optimal A multi objective design is Pareto optimal if none of the objectives can be improved without at least one objective being affected adversely Also referred to as functionally efficient Preference function A function of objectives used to combine several objectives into a single one suitable for the standard MP formulation Preprocessor A graphical tool used to prepare the input for a solver Random error The total error the difference between the exact and computed response is composed of a random and a bias component The random component is as the name implies a random deviation from the nominal value of the exact response often assumed to be normally distributed around the nominal value See also bias error Reasonable design space A subregion of the design space within the region of interest It is bounded by lower and upper bounds of the response values 340 LS OPT Version 2 APPENDIX G GLOSSARY Region of interest A sub region of the design space Usually defined by a mid point design and a range of each design variable Usually dynamic Reliability based design optimization RBDO The performing of design optimization while considering reliability based failure criteria in the constraints of the design optimization formulation This implies the inclusion of random variables in the generation of responses and then extracting the standard deviation of the responses about their mean valu
89. rows to X X X Be 2 29 max X7X max X X A A 2 30 and solving for Xp 2 8 5 PRESS residuals The prediction sum of squares residual PRESS uses each possible subset of P 1 responses as a regression data set and the remaining response in turn is used to form a prediction set 43 PRESS can be computed from a single regression analysis of all P points axe lt 2 PRESS X 211 2 31 i l i h where A are the diagonal terms of H X X X X 2 32 H is the hat matrix the matrix that maps the observed responses to the fitted responses i e j Hy 2 33 SPRESS 2 34 For a saturated design H equals the unit matrix J so that the PRESS indicator becomes undefined 22 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 8 6 The coefficient of multiple determination R The coefficient of determination R is defined as R5 2 35 o 7 where P is the number of design points and y y and y represent the mean of the responses the predicted response and the actual response respectively This indicator which varies between 0 and 1 represents the ability of the response surface to identify the variability of the design response A low value of R usually means that the region of interest is either too large or too small and that the gradients are not trustworthy The value of 1 0 for R indicates a perfect fit However the value will not warn against an overfitted mod
90. scaling factor for the minimal temperature Tmin in 2 24 damping factor for temperature ur in Eq F 5 Appendix F number of iterations at each temperature vr App 2 Parameters that control the standard deviation of in 2 23 upper bound Omax lower bound Omin 3 Termination criteria maximal number of energy function evaluations Ni 2 6 7 Random number generator A portable random number generator is used for the construction of D optimal designs using the genetic algorithm and to generate Monte Carlo designs such as the Latin Hypercube The algorithm employs a scheme by Bays and Durham as described in Knuth 29 and resembles the Maclaren Marsaglia method It greatly increases the period of the basic linear congruential generator In certain applications the Mersenne Twister is used The Mersenne Twister MT19937 is a pseudorandom number generator developed by Matsumoto and Nishimura and has the merit that it has a far longer period and far higher order of equidistribution than any other implemented generators It has been proved that the period is 2 1 and a 623 dimensional equidistribution property is assured LS OPT will probably standardize on this generator in future versions 2 7 Reasonable experimental designs A reasonable design space refers to a region of interest which in addition to having specified bounds on the variables is also bounded by specified values of the responses
91. selects designs that are optimally distributed throughout the design space to construct approximate surfaces or design formulae Thus the local effect caused by noise is alleviated and the method attempts to find a representation of the design response within a bounded design space or smaller region of interest This extraction of global information allows the designer to explore the design space using alternative design formulations For instance in vehicle design the designer may decide to investigate the effect of varying a mass constraint while monitoring the crashworthiness responses of a vehicle The designer might also decide to constrain the crashworthiness response while minimizing or maximizing any other criteria such as mass ride comfort criteria etc These criteria can be weighted differently according to importance and therefore the design space needs to be explored more widely Part of the challenge of developing a design program is that designers are not always able to clearly define their design problem In some cases design criteria may be regulated by safety or other considerations and therefore a response has to be constrained to a specific value These can be easily defined as mathematical constraint equations In other cases fixed criteria are not available but the designer knows whether the CHAPTER 2 OPTIMIZATION METHODOLOGY responses must be minimized or maximized In vehicle design for instance crashworth
92. simulation The results are nodal data If element data is required the DynaD3plotHistory command must be used with the option for elements selected Command file syntax Dyna cn pl p2 pn MIN MAX AVE Table 10 6 Dyna item description Item Description cn Component number of binary data See Appendix B pl pn Part numbers of the model Omission implies the entire model MIN MAX AVE Minimum maximum or average computed over all the elements of the selected parts Example Response x_vel Dyna 21 MAX Response x vel Dyna 21 1 2 4 MAX The former command requests the maximum value of component 21 over all parts while the latter command requests the same but only over parts 1 2 and 4 10 8 Extracting Histories From The LS DYNA d3plot file A generic interface exists for the extraction of histories from the LS DYNA d3plot files All the quantities can be specified on a part basis as defined in the input deck for LS DYNA A history point is created at each state available in the d3plot file The user must therefore ensure that a sufficient number of states is produced by the LS DYNA simulation Command file syntax DynaD3plotHistory cn pl p2 pn ELEMENT NODE MIN MAX AVE Table 10 7 Dyna item description Item Description cn Component number of binary data See pl pn Part numbers of the model Omission implies the entire model ELEMENT NODE Element or nodal values MIN
93. specified for an individual reponse Command file syntax response response name linear interaction elliptic quadratic FF kriging The default is the metamodel specified in Section FF refers to the feed forward neural network approximation method see Sections 2 10 1l and HS Example response Displacement kriging 110 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS In LS OPTui responses are defined in the Responses panel Figure 10 2 File Tasks Help Into Solvers Variables Point Selection Histories Responses Objective Constraints Run View USER DEFINED Node Number Maximum Time Value tl Filtering EXPRESSION 26730 ABSTAT BNDOUT Response Type Select a Component DEFORC Displacement 4 X Component ELOUT FLD v Velocity x Y Component GCEOUT f GLSTAT v Acceleration v Z Component JNTFORC v Rotational Displacement v Resultant MASS MATSUM wv Rotational Velocity NCFORC 3 x Rotational Acceleration NODFOR v Injury Coefficient PSTRESS RBDOUT Time Evaluation Option Sampling Interval RCFORC RWFORC SECFORC SPCFORC SWFORC THICK FREQUENCY Composite Composite Expression Vehicle Mass crash CRAS time_to_184 CRASH wee time_to_334 CRASH time_to_ max CRASH Integral_0 184 CRASH Integral_184 334 CRASH Integral_334 max CRASH StagelPulse CRASH Stage2Pulse CRASH Stage3Pulse
94. substring e g elout_beam 2 Parameters g and u apply only to the injury components HIC15 HIC36 CSI Not applicable to histories LS OPT Version 2 119 CHAPTER 10 HISTORY AND RESPONSE RESULTS 3 Integration point for the elout results Relates only to the shell element stress thick shell element and beam element types For shell element strain the positions UPPER or LOWER must be specified See ELOUT Appendix A 4 a The time dependent response is integrated to find the average AVE response ty J f Odt t t b The upper time limit is not applicable to the history command while the time will be ignored in e g History Z acc DynaASCII NODOUT Z ACC 27 TIMESTEP 0 0 SAE 60 5 The SAE BUTT Butterworth or point wise averaging AVER filters can be applied If the filter attribute is not specified no filtering will be done If the attribute is specified without a value a 60 cycles time unit or 5 point averaging filter is applied Note The user should be careful when specifying the filtering frequency for instance if a filter of e g 60Hz is desired but the time units are milliseconds a value of 60 1000 0 06 must be specified Examples Response x acc DynaASCII rbdout X ACC 21 MAX Response xX acc DynaASCII rbdout X ACC 21 MAX SAE 40 0 Response x acc DynaASCII rbdout X ACC 21 AVE Response x acc DynaASCII rbdout X ACC 21 AVE SAE 40 0 Response x_acc
95. terminate with an appropriate error message 6 12 Parallel processing Runs can be executed simultaneously The user has to specify how many processors are available Command file syntax concurrent jobs number_of jobs If a parallel solver is used the number of concurrent jobs used for the solution will be number_of jobs times the number of cpu s specified for the solver Example concurrent jobs 16 6 13 Using an external queuing or job scheduling system The LS OPT Queuing Interface interfaces with load sharing facilities e g LSF or LoadLeveler to enable running simulation jobs across a network LS OPT will automatically copy the simulation input files to each remote node extract the results on the remote directory and transfer the extracted results to the local directory This feature allows the progress of each simulation run to be monitored via LS OPTui The README queue file should be consulted for more up to date information about the queuing interface Registered Trademark of Platform Computing Inc Registered Trademark of International Business Machines Corporation 75 CHAPTER 6 PROGRAM EXECUTION Command file syntax Solver queuer queuer name Table 6 4 Queuing options queuer_ name Description lsf LSF loadleveler LoadLeveler pbs PBS nqe NQE To run LS OPT with a queuing load sharing facility the following binary files are provided in the bin directory which un t
96. the construction of the metamodel 4 The probabilistic analysis to be executed must be specified for example a reliability analysis LS OPT Version 2 153 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION 15 3 Probabilistic Distributions The probabilistic component of a design variable is described using a probabilistic distribution The distributions are created without referring to a variable Many design variables can refer to a single distribution 15 3 1 Normal Distribution The normal distribution is symmetric and centered about the mean u with a standard deviation of 0 2 ai p o 0 16 fz 0 14 N 0 12 p ol j 008 006 004 002 j R 10 4 6 4 0 2 4 6 3 L x Figure 15 1 Normal Distribution Command file syntax distribution name NORMAL mu sigma Item Description name Distribution name mu Mean value sigma Standard deviation Example distribution normalDist NORMAL 12 2 1 1 154 LS OPT Version 2 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION 15 3 2 Uniform distribution The uniform distribution has a constant value over a given range 1Xb a f x x Figure 15 2 Uniform Distribution Command file syntax distribution name UNIFORM lower upper Item Description name Distribution name lower Lower bound upper Upper bound Example distribution rangeX UNIFORM 1 2 3 4 15 3
97. the space filling scheme is the default for the Neural Net and Kriging methods Figure 9 2 File Tasks Help Into Solvers Variables Point Selection Histories Responses Objective Constraints Run View CRASH A Metamodel Point Selection Basis Type Polynomial v Linear Koshal Full Factorial wv Sensitivity y Quadratic Koshal v Latin Hypercube xy Neural Net xy Composite v Space Filling Order x Full Factorial Points Per Variable For Basis 4 Linear lt D Optimal v2 v6 vi Linear with Interaction v Latin Hypercube 3 V7 vil vy Quadratic vy Space Filling v4 v8 v Elliptic wv Duplicate v5 v Number of Experiments blank for default 5 Current Default 5 Figure 9 1 Metamodel and Point Selection panel in LS OPTui 98 LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION 9 2 2 D Optimal point selection The D optimal design criterion can be used to select the best optimal set of points for a response surface from a given set of points The basis set can be determined using any of the other point selection schemes and is referred to here as the basis experiment The order of the functions used has an influence on the distribution of the optimal experimental design The following must be defined to select D optimal points Order The order of the functions that will be used Linear linear with interaction elliptic or quadratic Number experiments The number of experimental points that
98. the trade off between mass or energy efficiency and safety Adding to the complexity is the fact that mechanical design is really an interdisciplinary process involving a variety of modeling and analysis tools To facilitate this process and allow the designer to focus on creativity and refinement it is important to provide suitable interfacing utilities to integrate these design tools Designs are bound to become more complex due to the legislation of safety and energy efficiency as well as commercial competition It is therefore likely that in future an increasing number of disciplines will have be integrated into a particular design This approach of multidisciplinary design requires the designer to run more than one case often using more than one type of solver For example the design of a vehicle may require the consideration of crashworthiness ride comfort noise level as well as durability Moreover the crashworthiness analysis may require more than one analysis case e g frontal and side impact It is therefore likely that as computers become more powerful the integration of design tools will become more commonplace requiring a multidisciplinary design interface Modern architectures often feature multiple processors and all indications are that the demand for distributed computing will strengthen into the future This is causing a revolution in computing as single analyses that took a number of days in the recent past can now be done w
99. the variables are shared E g t_bumper in Figure 8 1 is only associated with the solver CRASH File Tasks Help Info Solvers Variables Point Selection Histories Responses Objective Constraints Run View Design Variables Minimum Range Name Starting Range Minimum Type Maximum fi gt Af 2 Variable t rail_back 2 2 q g Saddle Direction E 5 m 5 Minimize Variable t hood 2 2 Aa 6 Solvers Variable t root 2 2 4 6 List Variable rail_front 5 2 a e setan Add a Variable Delete Variable aer Figure 8 1 Variables panel in LS OPTui 89 CHAPTER 8 DESIGN VARIABLES CONSTANTS AND DEPENDENTS 8 1 Selection of design variables The variable command is the identification command for each variable Command file syntax variable variable name value Example DEFINE THE VARIABLE Area Variable Area 0 8 The value assigned is the initial value of the variable 8 2 Definition of upper and lower bounds of the design space Command file syntax Lower bound variable variable name value lt 10 Upper bound variable variable name value lt 10 gt 3 0 Example Lower bound variable Area 0 1 Upper bound variable Area 2 0 Both the lower and upper bounds must be specified as they are used for scaling 8 3 Size and location of region of interest range Command file syntax range variable name subregion size E
100. to a given data set and the central problem is that of not enough data The minimal number of data points required for network training is related to the unknown complexity of the underlying function and the dimensionality of the design space In reality the more design variables the more training samples are required In the statistical and neural network literature this problem is known as the curse of dimensionality Most forms of neural networks in particular feed forward networks actually suffer less from the curse of dimensionality than some other methods as they can concentrate on a lower dimensional section of the high dimensional space For example by setting the outgoing weights from a particular input to zero a network can entirely ignore that input Figure 52 Nevertheless the curse of dimensionality is still a problem and the performance of a network can certainly be improved by eliminating unnecessary input variables network output network input X X X weights and weights and biases of bias of hidden layer output layer Figure 2 2 Schematic of a neural network with 2 inputs and a hidden layer of 4 neurons with activation function f It is clear that if the number of network free parameters is sufficiently large and the training optimization algorithm is run long enough it is possible to drive the training MSE error as close as one likes to zero However it is also clear that driving MSE all the wa
101. tpl This file is specified as the prepro input file 2 Control nodes file This is a nodal template file used by Templex to produce the nodal output file using the current values of the variables This file is specified using the prepro controlnodes command The default name is nodes tpl 3 A coefficient file that contains original coordinates and motion vectors specified in two columns must be available The command used is prepro coefficient file and the default file name is nodes shp 4 Templex produces a nodal output file that is specified under the solver append file command The default name is nodes include Example 5 DEFINITION OF SOLVER 1 5 solver dyna 1 solver command lsdyna Registered Trademark of Altair Engineering Inc LS OPT Version 2 85 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR solver append file nodes include solver input file dyna k prepro templex prepro command origin 2 user mytemplex templex prepro input file a tpl prepro coefficient file a dynakey node tpl prepro controlnodes file a shp In the example several files can be defaulted Table 7 1 Templex solver and prepro files and defaults Command Description Default prepro input file Templex input file input tpl prepro coefficient file Coefficient file nodes shp prepro controlnodes file Control Nodes file nodes tpl solver append file Append file same as templex output file node
102. vel50_5 disp10_5 disp10_5 disp15_5 disp15_5 disp20_5 disp20 _5 disp25_5 disp25_5 disp30 5 disp30 5 disp35_ 5 disp35_5 disp40 5 disp40 5 disp45 5 disp45 5 517805 517805 123231 123231 L 705 L 705 24697 WwW N 0 0 HH HH HH oO oO 51584 51584 24697 59243 59243 03072 03072 393792 393792 33087 33087 76819 76819 91663 91663 917493 917493 694946 694946 497515 497515 342808 342808 253116 253116 2386 2386 284147 284147 363297 363297 238 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS constraint disp50_5 lower bound constraint disp50_5 0 456215 upper bound constraint disp50_5 0 456215 move constraint C1 strict lower bound constraint C1 0 constraint C2 lower bound constraint C2 0 constraint C3 lower bound constraint C3 0 constraint C4 lower bound constraint C4 0 EXPERIMENTAL DESIGN Order linear Experimental design dopt Basis experiment 3toK Number experiment 10 JOB INFO concurrent jobs 10 iterate param design 0 001 iterate param objective 0 001 iterate 20 STOP The residual is retained in the formulation as the objective although it is not used in the optimization process This is so that the residual can be monitore
103. x acceleration id 432 select MAX filter SAE filter freq 10 Remarks 1 The maximum minimum average or value at a specific time must be selected If selection is TIME then the end_time history value will be used If end_time is not specified the last value end of analysis will be used Binout Injury Criteria Injury criteria such as HIC can be specified as the result component The acceleration components will be extracted the magnitude computed and the injury criteria computed from the acceleration magnitude history Command file syntax BinoutResponse history_options res type res type gravity gravity units units Item Description Default history_options All available history options including filtering and slicing res_type HIC15 HIC36 or CSI g gravity Gravitational acceleration 9 81 units S seconds MS milliseconds S Example response HIC ms 1 0 BinoutResponse res type Nodout cmp HIC15 gravity 9810 units MS id 432 10 11 Translating ASCII output commands to Binout commands Translation of DynaASCII commands to Binout commands can be done in the GUI or specified in the command file The translated commands will be available in the GUI and the Isopt_input file Not all components are available for both the DynaASCII and the Binout extraction routines In particular invariants such as the maximum principle stress may not be available in Binout Some of these invarian
104. xiv LS OPT Version 2 Preface to Version 1 LS OPT originated in 1995 from research done within the Department of Mechanical Engineering University of Pretoria South Africa The original development was done in collaboration with colleagues in the Department of Aerospace Engineering Mechanics and Engineering Science at the University of Florida in Gainesville Much of the later development at LSTC was influenced by industrial partners particularly in the automotive industry Thanks are due to these partners for their cooperation and also for providing access to high end computing hardware At LSTC the author wishes to give special thanks to colleague and co developer Dr Trent Eggleston Thanks are due to Mr Mike Burger for setting up the examples Nielen Stander Livermore CA August 1999 Preface to Version 2 Version 2 of LS OPT evolved from Version 1 and differs in many significant regards These can be summarized as follows The addition of a mathematical library of expressions for composite functions The addition of variable screening through the analysis of variance The expansion of the multidisciplinary design optimization capability of LS OPT The expansion of the set of point selection schemes available to the user The interface to the LS DYNA binary database Additional features to facilitate the distribution of simulation runs on a network The addition of Neural Nets and Kriging as metamodeling techniques Probabil
105. 0 2 45 RMS ger MSE ger MSE nMSE gcy La 02 RM NRMS Gey ee where vis the effective number of model parameters In theory GCV estimates should be related to v As a very rough approximation to v we can assume that all of the network free parameters are well determined so that v M where M is the total number of network weights and biases This is what we would expect to be the case for large P so that P gt gt M Note that GCV is undefined when vis equal to the number of training points P Feed forward neural networks Feed forward FF neural networks have a distinct layered topology Each unit performs a biased weighted sum of their inputs and passes this value through a transfer activation function to produce the output The outputs of each layer of neurons are the inputs to the next layer In a feed forward network the activation function of intermediate hidden layers is generally a sigmoidal function Figure 2 3 network input and output layers being linear Consider a FF network with K inputs one hidden layer with H sigmoid units and a linear output unit For a given input vector x x x and network weights W W W Wi Wio Wi Wue the output of the network is H K W x W W SW S Wr pa DMX 2 46 h 1 k l where 1 f x l e The computational graph of Eq 2 46 is shown schematically in The extension to the case of more than one hidden layers can be obtained accordingly It
106. 0 6018 0 6017 0 6018 Intru_2 712 7 711 9 712 7 711 9 Intru_1 163 3 161 9 163 3 161 9 HIC 171 4 108 2 171 4 108 2 A En a a a ls 17 2 7 Automated run An automated optimization is performed with a linear approximation The LS OPT input file is modified as follows Order linear Experimental design dopt Basis experiment 3toK Number experiment 5 iterate 8 It can be seen in that the objective function HIC and intrusion constraint are approximately optimized at the 5 iteration It takes about 8 iterations for the approximated solid line and computed square symbols HIC to correspond The approximation improves through the contraction of the subregion As the variable t_hood never moves to the edge of the subregion during the optimization process the heuristic in LS OPT enforces pure zooming see Figure 17 10 For t_bumper panning occurs as well due to the fact that the linear approximation predicts a variable on the edge of the subregion LS OPT Version 2 195 EXAMPLE PROBLEMS CHAPTER 17 timization History omposite Intrusion r For Optimization History For Objective HIC N O foe o 01 OIH ennoelgo Number of Iterations Number of Iterations b Optimization history of Intrusion 9 Optimization history of HIC and Intrusion Figure 17 a Optimization history of HIC per Optimization History For Variable t_bum Optimization History For Variable t
107. 00 and 400 to component numbers 1 through 16 component numbers for infinitesimal strains Green St Venant strains Almansi strains and strain rates are obtained respectively Element Type Number Component 1 6 X Y Z XY YZ ZX 7 effective plastic strain 8 pressure or average strain Solids Membranes 9 von Mises stress Shells amp Brick shells 10 first principal deviator maximum 11 second principal deviator 12 third principal deviator minimum 13 maximum shear stress 14 Ist principal maximum stress 15 2nd principal stress 16 3rd principal min 17 x displacement 18 y displacement 19 z displacement 20 maximum displacement 21 x velocity 22 y velocity 23 z velocity 24 maximum velocity 25 temperature LS TOPAZ 26 M bending resultant 27 M bending resultant 28 Mz bending resultant 29 Qx shear resultant Membranes amp Shells 30 O shear resultant 31 N normal resultant 32 N normal resultant 301 APPENDIX B LS DYNA BINARY RESULT COMPONENTS Element type Number Component 33 Ny normal resultant 34 Surface stress N t 6M t 35 Surface stress Nyx t 6Mo t 36 Surface stress N t 6M t 37 Surface stress N t 6Myy t 38 Surface stress Ny t 6M t 39 Surface stress Ny t 6May t 40 effective upper surface stress 41 effective lower surface stress 42 maximum effective surface stress 43 lower surface effective plastic strain 44 upper surface
108. 01x OIH ayoafqO eee E sc SE ES 1 L 1 1 T 1 1 L 1 ee eee beeen a 50 52 54 56 58 60 48 46 48 50 52 54 56 58 60 46 Constraint Intrusion x101 Constraint Intrusion x101 LS OPT Version 2 198 CHAPTER 17 EXAMPLE PROBLEMS a Objective HIC versus Intrusion constraint b t_bumper versus Intrusion constraint Tradeoff Plot Constraint Intrusion vs Variable t_hood gt a are Variable t hood 46 48 50 52 54 56 58 60 Constraint Intrusion x101 c t_hood versus Intrusion constraint Figure 17 13 Trade off results Small car 2 variables 17 2 9 Reliability based design optimization The limited reliability based design optimization in LS OPT is illustrated in this example The optimization problem is modified as follows Minimize HIC 17 5 subject to Intrusion lt 550mm 60 intrusion where Opce and Omi are the standard deviations of the HIC and Intrusion responses respectively The design space for the two variables are increased from 1 5 to 1 6 The formulation in Eq 17 5 implies that the car is made safer by 6 standard deviations of the intrusion The standard deviation of both the HIC and Intrusion responses is calculated using the procedure outlined in Section 2 15 5 The resulting command input file is as follows Small Car Problem EX4a Reliability based design Created on Tue Mar 5 14 03 45 2002 DESIGN VARIABLES
109. 02 0 7 045 0 0 lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt 0 PART material type 3 Kinematic Isotropic Elastic Plastic 4 4 4 0 4 0 DEFINITION OF MATERIAL 5 MAT PLASTIC KINEMATIC 5 7 800E 08 2 000E 05 0 300 400 0 0 0 0 0 HOURGLASS 5 0 0 0 0 0 SECTION SHELL 5 2 0 0 0 0 0 lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt 0 PART material type 3 Kinematic Isotropic Elastic Plastic py Dp oy Ot by O LS OPT Version 2 189 CHAPTER 17 EXAMPLE PROBLEMS 17 2 5 First linear iteration A design space of 1 5 is used for both design variables with no range specified This means that the range defaults to the whole design space The LS OPT input file is as follows Small Car Problem EX4a Created on Mon Aug 26 19 11 06 2002 solvers 1 responses 5 NO HISTORIES ARE DEFINED DESIGN VARIABLES variables 2 Variable t_hood 1 Lower bound variable t_hood 1 Upper bound variable t_hood 5 Variable t_bumper 3 Lower bound variable t_bumper 1 Upper bound variable t_bumper 5 DEFINITION OF SOLVER 1 solver dyna 1 solver command lsdyna solver input file car5 k solver append file rigid2 solver order linear solver experiment design dopt solver number experiments 5 solver basis experiment 3toK solver concurrent jobs 1 RESPONSES FOR SOLVER
110. 0o0 The results after one iteration are as follows DESIGN POINT 2293 0966 1831 0826 3159 0186 9830 9830 8182 0071 Variable Name Lower Bound Value Upper Bound ee ae Bee wa ee Area 0 2 1 444 4 Base 0 1 0 5408 1 6 ee a ee esse jene RESPONSE FUNCTIONS Scaled Unscaled ee E pree e ee RESPONSE Computed Predicted Computed Predicted ee ehe en ae ee Weight 1 642 1 627 1 642 1 627 Stress 0 9614 1 0 9614 1 Computed Value Predicted Value LS OPT Version 2 181 CHAPTER 17 EXAMPLE PROBLEMS OBJECTIVE FUNCTIONS Computed Predicted WT I I I 1 I I I I I 1 l l l I I I I I j I l I I I Weight 1 642 1 627 1 Bat ten es er Sheth as Sas Sah PN a ee Oona Sart ty ee hall ch beet the att a en Men re CONSTRAINT FUNCTIONS CONSTRAINT NAME oa Computed Predicted Lower Upper Viol pe ee ee nd er aman rg ro Ko a H gt H l O w oO He 5 O CONSTRAINT NAME Sess ats ae Lower Upper Lower Upper An improved design is predicted with the constraint value stress changing from an approximate 0 8033 0 7928 computed to 1 0 the approximate constraint becomes active Due to inaccuracy the actual constraint value of the optimum is a feasible 0 961 This value is now much closer to the value of the simulation result The weight changes from 1 909 1 907 computed to 1 627 1 642 computed 17
111. 1 3973 The initial design below shows that the constraint is severely exceeded DESIGN POINT Variable Name Lower Bound Value Upper Bound Radius 20 75 100 Wall_Thickness 2 3 6 Scaled Unscaled pee eri oa en Ee es u RESPONSE Computed Predicted Computed Predicted Bar a FERNE Er EUREN duet ened ee en See ee Internal _Energy 1 296e 04 1 142e 04 1 296e 04 1 142e 04 Rigid_Wall_Force 1 749e 05 1 407e 05 1 749e 05 1 407e 05 LS OPT Version 2 207 CHAPTER 17 EXAMPLE PROBLEMS Response Surface Accuracy For Response Function Internal_Energy oa m wo _ oO Computed Response Value x1 03 5 6 7 8 9 10 11 12 18 Predicted Response Value x1 03 Response Surface Accuracy For Response Function Rigid_Wall_Force 18 16 14 x g 12 oO gt g 10 Cc oO 2 8 oc 86 Q S 4 O 2 0 20 Predicted Response Value x104 Figure 17 17 Prediction accuracy of Internal Energy and Rigid Wall Force One Quadratic iteration Despite the relatively poor approximation a prediction of the optimum is made based on the approximation response surface The results are shown below The fact that the optimal Radius is on the lower bound of the subregion specified Range 50 suggests an optimal value below 50 DESIGN POINT Variable Name Lower Bound Value Upper Bound ete ac a ee a eas ee Radius 20 50 100 Wall Thickness 2 978 6 Se ar esse Pre ee RESPONSE FUNCTIONS Scaled Unscaled
112. 2 11 Because of the possible ill conditioning of R a small constant number is adaptively added to its diagonal during optimization The net effect is that the approximating functions no longer interpolate the observed response values exactly However these observations are still closely approximated LS OPT Version 2 33 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 10 3 Concluding remarks which metamodel There is little doubt that the polynomial based response surfaces are the most robust A negative aspect is the fact that the user has to choose the order of the polynomial and a greater possibility exists for bias error of a nonlinear response Therefore linear approximations may only be useful within a certain subregion and quadratic polynomials may be required for greater global accuracy However the linear SRSM method has proved to be excellent for optimization and can be used with confidence 616253 Neural Networks function well as global approximations and no serious deficiencies have been observed when used as prescribed in Section 4 5 NN s have been used successfully for optimization 63 and can be updated during the process Although the literature seems to indicate that Kriging is one of the more accurate methods 52 there is evidence of Kriging having fitting problems with certain types of experimental designs 4 Kriging is very sensitive to noise since it interpolates the data Be The authors of this manual have also
113. 2 23 2003 Also www stc com Stander N Snyman J A Coster J E On the robustness and efficiency of the SAM algorithm for structural optimization International Journal for Numerical Methods in Engineering 38 pp 119 135 1995 Sunar M Belegundu A D Trust region methods for structural optimization using exact second order sensitivity International Journal for Numerical Methods in Engineering 32 pp 275 293 1991 Thanedar P B Arora J S Tseng C H Lim O K Park G J Performance of some SQP algorithms on structural design problems International Journal for Numerical Methods in Engineering 23 pp 2187 2203 1986 Toropov V V Simulation approach to structural optimization Structural Optimization 1 pp 37 46 1989 Tu J and Choi K K Design potential concept for reliability based design optimization Technical report R99 07 Center for computer aided design and department of mechanical engineering College of engineering University of Iowa December 1999 Van Campen D H Nagtegaal R Schoofs A J G Approximation methods in structural optimization using experimental designs for multiple responses In Eschenauer H Koski J Osyczka A Eds Multicriteria Design Optimization Procedures and Applications Springer Verlag Berlin Heidelberg New York pp 205 228 1990 Vanderplaats G N Numerical Optimization Techniques for Engineering Design with Applications McGraw Hill New York 1984 Wahba G Sp
114. 20 Upper bound variable T Flange Depth 50 Range T Flange Depth 10 Variable F Flange Depth 27 5 Lower bound variable F Flange Depth 20 Upper bound variable F Flange Depth 50 Range F Flange Depth 10 Variable B Flange Depth 22 3 Lower bound variable B Flange Depth 15 Upper bound variable B Flange Depth 50 Range B Flange Depth 10 Variable I_ Flange Width 7 Lower bound variable I Flange Width 5 Upper bound variable I Flange Width 25 Range I_ Flange Width 5 Variable L Flange Width 32 Lower bound variable L Flange Width 20 Upper bound variable L Flange Width 50 Range L Flange Width 10 Variable R Bracket _Gauge 1 1 Lower bound variable R Bracket _Gauge Upper bound variable R Bracket _Gauge Range R Bracket _Gauge 2 Variable R_ Flange Width 32 Lower bound variable R Flange Width 20 Upper bound variable R Flange Width 50 Range R_ Flange Width 10 Variable R Bracket Radius 15 Lower bound variable R_Bracket Radius 10 Upper bound variable R Bracket Radius 25 Range R Bracket Radius 5 Variable Bolster gauge 3 5 Lower bound variable Bolster gauge 1 Upper bound variable Bolster gauge 6 Range Bolster gauge 3 Variable Yolk Radius 4 Lower bound variable Yolk Radius 2 Upper bound variable Yolk Radius 8 Range Yolk Radius 2 solvers 1 responses 7 DEFINITION OF SOLVER 1 solver dyna 1 solver command lsdyna solver input file trugrdo solver insert file for
115. 21 R_ Flange Width 0 1345 0 2268 0 04219 0 2497 0 01929 96 R_Bracket_Radius 0 08237 0 01147 0 1762 0 03475 0 1995 84 Bolster gauge 0 4067 0 3275 0 4859 0 3078 0 5055 100 Yolk_ Radius 0 1723 0 08353 0 2611 0 0615 0 2831 99 Hager a nn a a cubsel er euse LS OPT Version 2 273 CHAPTER 17 EXAMPLE PROBLEMS Ranking of terms based on coefficient bounds Coeff Absolute Value 90 10 Scale Gh Se ee re L_Bracket_Gauge 0 4929 10 0 Bolster gauge 0 3275 6 6 L Flange_Width 0 1358 2 8 Yolk_Radius 0 08353 1 7 R_Flange width 0 04219 0 9 I Flange Width 0 03974 0 8 F Flange Depth 0 00908 0 2 T Flange Depth Insignificant 0 0 R_Bracket_Radius Insignificant 0 0 B Flange Depth Insignificant 0 0 R_Bracket_Gauge Insignificant 0 0 This result is based on one iteration only Reducing the number of variables from 11 to 7 reduces the number of LS DYNA simulations from 19 to 13 when using the default D optimal design settings 17 8 5 Optimization with reduced variables The optimization history of the knee forces using the original 11 variables and the reduced set 7 variables is shown in Figure 17 47 10000 11 Variables 9000 Predicted 8000 11 Variables 5 7000 Computed im g 6000 2 7 Variables 5000 Predicted 3 E 4000 x EB 7 Variables s 3000 Computed 2000 1000
116. 3 4 5 Range Radius_3 1 The number of D optimal experiments is reduced because of the linear approximation used Order linear Experimental design dopt Basis experiment 3toK Number experiment 7 The optimization is run for 10 iterations iterate 10 The optimization history is shown in Figure 17 23 for the design variables and responses LS OPT Version 2 225 EXAMPLE PROBLEMS CHAPTER 17 Optimization History For Variable Radius_2 Optimization History For Variable Radius_1 sn pey eIgeueA 7 4 5 6 3 3 Number of Iterations Number of Iterations b Optimization history of variable Radius 2 a Optimization history of variable Radius _ 1 Optimization History For Response Thinning Optimization History For Variable Radius_3 snipey ajqeve 3 Number of Iterations Number of Iterations d Optimization history of response Thinning c Optimization history of variable Radius 3 LS OPT Version 2 226 CHAPTER 17 EXAMPLE PROBLEMS Optimization History For Response FLD 0 12 l 1 0 1 De I 1 I I 0 08 eur l 1 I 1 O 0 06 T r LL Il 1 Ged l I B 0 04 ots O I I F ae 0 02 to r I 1 l 1 9 Aa I 1 0 02 er I I 0 04 12 3 4 5 6 7 8 9 10 Number of Iterations e Optimization history of response FLD Figure 17 23 Optimization history of design variables and responses automated design The details o
117. 3 User defined distribution A user defined distribution is specified by referring to the file containing the distribution data The probability density is to be assumed piecewise uniform and the cumulative distribution to be piecewise linear Either the PDF or the CDF data can be given LS OPT Version 2 155 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION e PDF distribution The value of the distribution and the probability at this value must be provided for a given number of points along the distribution The probability density is assumed to be piecewise uniform at this value to halfway to the next value both the first and last probability must be zero e CDF distribution The value of the distribution and the cumulative probability at this value must be provided for a given number of points along the distribution It is assumed to vary piecewise linearly The first and last value in the file must be 0 0 and 1 0 respectively PDF File 1 00 0 0 02 0 1 03 0 2 00 0 P X CDF File 0 50 0 05 04 1510 m Figure 15 3 User defined distribution Lines in the data file starting with the character will be ignored Command file syntax distribution name USER DEFINED PDF fileName distribution name USER DEFINED CDF fileName Item Description name Distribution name filename Name of file containing the distribution data Example distribution bendDi
118. 632 R 2 adjusted 0 9632 R 2 prediction 0 7622 Determinant of X X 0 0556 Approximating Response Rigid_Wall_Force using 10 points ITERATION 1 Mean response value 82483 2224 RMS error 19905 3990 24 13 Maximum Residual 35713 1794 43 30 Average Error 17060 6074 20 68 Square Root PRESS Residual 54209 4513 65 72 Variance 792449819 5138 LS OPT Version 2 209 CHAPTER 17 EXAMPLE PROBLEMS R 2 0 8949 R 2 adjusted 0 8949 R 2 prediction 0 2204 Determinant of X X 0 0556 The goodness of fit diagrams are shown in Figure 17 18 Response Surface Accuracy Response Surface Accuracy For Response Function Internal_Energy For Response Function Rigid_Wall_Force 14 225 13 200 a 12 a 175 zZ gt ee g 2 150 10 S Q Q 2 2 125 9 fa Q Qa 2 2 r 8 100 ke a I g D 1 a 7 2 75 Bn SRS E E 1 1 S 6 S g 50 H H F t I I 5 1 I I i i 25 EEE ET E 5 6 7 8 9 10 11 12 13 14 2 4 6 8 10 12 14 16 18 20 22 Predicted Response Value x109 Predicted Response Value x104 Figure 17 18 Prediction accuracy of Internal Energy and Rigid Wall Force One Quadratic iteration Nevertheless an optimization is conducted of the approximate subproblem yielding a much improved feasible result The objective function increases to 9575 9777 computed whereas the constraint is active at 70 000 The computed constraint is lower at 64 170 However the wall_Thickness is now
119. 74 17 46 20 73 38 48 base Starting design 1 9 42 4 551 6 14 62 17 53 20 77 38 26 D base Starting gt design 2 8 43 2 552 5 14 66 17 56 20 69 38 15 E min Starting design 3 6 43 8 553 7 14 46 17 48 20 61 39 07 max Beginning with starting design 1 the optimization history of the objective and constraints are shown for the full and partially shared variable cases in Figure 17 40 through Figure 17 43 Most of the reduction in mass occurs in the first iteration Figure 17 40 although this results in a significant violation of the maximum displacement and second stage pulse constraints especially in the fully shared variable case The second iteration corrects this and from here the optimizer tries to reconcile four constraints that are marginally active Most of the intermediate constraint violations see e g can be ascribed to the difference between the value predicted by the response surface and the value computed by the simulation The torsional frequency remains within the bounds set during the optimization for the full shared case Fully shared Partially shared Mass kg Iteration Figure 17 40 Optimization history of component mass Objective Starting design 1 262 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 556 E 554 E 552 S 7 550 2 ke E 548 Ma
120. 77 7 8 13 128 36 55 2187 143 8 9 14 256 45 68 6561 273 9 10 16 512 55 83 19683 531 10 11 17 1024 66 100 59049 1045 2 6 5 Latin Hypercube Sampling LHS The Latin Hypercube design is a constrained random experimental design in which for n points the range of each design variable is subdivided into n non overlapping intervals on the basis of equal probability One value from each interval is then selected at random with respect to the probability density in the interval The n values of the first value are then paired randomly with the n values of variable 2 These n pairs are then combined randomly with the n values of variable 3 to form n triplets and so on until k tuplets are formed LS OPT Version 2 15 CHAPTER 2 OPTIMIZATION METHODOLOGY Latin Hypercube designs are independent of the mathematical model of the approximation and allow estimation of the main effects of all factors in the design in an unbiased manner On each level of every design variable only one point is placed There are the same number of levels as points and the levels are assigned randomly to points This method ensures that every variable is represented no matter if the response is dominated by only a few ones Another advantage is that the number of points to be analyzed can be directly defined Let P denote the number of points and n the number of design variables each of which is uniformly distributed between 0 and 1
121. 8 112e 04 8e 04 Figure 17 20 below confirms that the final design is only slightly infeasible when the maximum filtered force exceeds the specified limit for a short duration at around 9ms 214 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Normal Force E 3 0 0 005 0 01 0 015 Time sec Figure 17 20 Cylinder Constrained rigid wall force F t lt 80000 SAE 300Hz filtered LS OPT Version 2 215 CHAPTER 17 EXAMPLE PROBLEMS 17 4 Sheet metal forming 3 variables A sheet metal forming example in which the design involves thinning and FLD criteria is demonstrated in this chapter The example has the following features The maximum of all the design variables is minimized Adaptive meshing is used in the finite element analysis The binary LS DYNA database is used The example employs the sheet metal forming interface utilities Composite functions are used An appended file containing extra input is used The example utilizes the independent parametric preprocessor Truegrid 17 4 1 Problem statement The design parameterization for the sheet metal forming example is shown in Figure 17 21 Fi punch S t al F F2 die S Figure 17 21 Parameterization of cross section 1 Registered Trademark of XYZ Scientific Applications Inc 216 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS The FE model is shown in Figure 17 22 Figure 17 22 Quarter segment of FE model tools
122. 890 64170 80070 It is apparent that the result of the second iteration is a dramatic improvement on the starting design and a good approximation to the converged optimum design 17 3 5 Response filtering using the peak force as a constraint Because of the poor accuracy of the response surface fit for the rigid wall force above it was decided to modify the force constraint so that the peak filtered force is used instead Therefore the previous response definition for Rigid _Wall_ Force is replaced with a command that extracts the maximum rigid wall force from a response from which frequencies exceeding 300Hz are excluded The upper bound of the force constraint is changed to 80000 response Rigid Wall Force DynaASCII RWForc Normal 1 Max SAE 300 20 iterations are specified with a 1 tolerance for convergence As expected the response histories Figure 17 19 show that the baseline design is severely infeasible the first peak force is about 1 75 x 10 vs the constraint value of 0 08 x 10 A steady reduction in the error of the response surfaces is observed up to about iteration 5 The optimization terminates after 16 iterations having reached the 1 threshold for both objective and design variable changes 212 LS OPT Version 2 14 16 2 1 10 EXAMPLE PROBLEMS Number of Iterations Optimization History b Wall Thickness Optimization History For Variable Wall_Thickness CHAPTER 17 2 5 14 16
123. D RESPONSES DEFINED BY EXPRESSIONS composites 4 composite Radl type weighted composite Radl variable Radius_1 1 scale 1 composite Rad2 type weighted composite Rad2 variable Radius_ 2 1 scale 1 composite Rad3 type weighted composite Rad3 variable Radius_3 1 scale 1 composite Thinning scaled Thinning 100 NO OBJECTIVES DEFINED objectives 0 CONSTRAINT DEFINITIONS constraints 5 constraint FLD strict upper bound constraint FLD 0 0 constraint Radl1 slack upper bound constraint Radl 1 1 constraint Rad2 upper bound constraint Rad2 1 1 constraint Rad3 upper bound constraint Rad3 1 1 constraint Thinning scaled strict upper bound constraint Thinning scaled 0 2 EXPERIMENTAL DESIGN Order quadratic Experimental design dopt Basis experiment 3toK Number experiment 16 JOB INFO concurrent jobs 8 iterate param design 0 01 iterate param objective 0 01 iterate 1 STOP LS OPT Version 2 219 CHAPTER 17 EXAMPLE PROBLEMS The file ShellSetList contains commands for LS DYNA in addition to the preprocessor output It is slotted into the input file Adaptive meshing is chosen as an analysis feature for the simulation The FLD curve data is also specified in this file The extra commands are DATABASE BINARY RUNRSF 70 DATABASE EXTENT BINARY 0 O O 1 0 O O 1 0 0 O O O O SLIDING INTERFACE DEFINITIONS TrueGrid
124. E FUNCTIONS objectives 1 maximize objective Internal Energy 1 CONSTRAINT DEFINITIONS constraints 1 constraint Rigid Wall Force upper bound constraint Rigid Wall Force 70000 EXPERIMENTAL DESIGN Order quadratic Experimental design dopt Basis experiment 5toK Number experiment 10 JOB INFO concurrent jobs 5 iterate param design 0 01 iterate param objective 0 01 iterate 1 STOP The curve fitting results below show that the internal energy is approximated reasonably well whereas the average force is poorly approximated The accuracy plots confirm this result Figure 17 17 206 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Approximating Response Internal_Energy using 10 points ITERATION 1 Mean response value 10686 0081 RMS error 790 3291 7 40 Maximum Residual 1538 9208 14 40 Average Error 654 4415 6 12 Square Root PRESS Residual 2213 7994 20 72 Variance 1249240 2552 R 2 0 9166 R 2 adjusted 0 9166 R 2 prediction 0 3453 Determinant of X X 1 3973 Approximating Response Rigid Wall Force using 10 points ITERATION 1 Mean response value 121662 9474 RMS error 24730 1732 20 33 Maximum Residual 48569 4162 39 92 Average Error 21111 3307 17 35 Square Root PRESS Residual 75619 5531 62 15 Variance 1223162932 2092 R 2 z 0 8138 R 2 adjusted 0 8138 R 2 prediction 0 7406 Determinant of X X
125. Energy 1 5 EXPERIMENTAL DESIGN Order linear Experimental design dopt Basis experiment 3toK Number experiment 19 5 JOB INFO 5 concurrent jobs 5 iterate param design 0 01 iterate param objective 0 01 iterate 5 STOP LS OPT Version 2 271 CHAPTER 17 EXAMPLE PROBLEMS Results of initial optimization shape and size Table 17 8 Knee ne isaac results 11 variables 4 variables Left Bracket Gauge mm Right Bracket Gauge mm Knee Bolster Gauge mm Yoke Cross Section Radius mm Oblong Hole Radius mm Right EA Width mm Left EA Depth Top mm Left EA Depth Front mm Left EA Depth Bottom mm Left EA Inner Flange Width mm Left EA Width mm Maximum Left Knee Force N Maximum Right Knee Force N Maximum Left Knee Disp mm Maximum Right Knee Disp mm Yoke displacement mm 17 8 4 Variable screening Using the ANOVA technique Section P 9 the number of design variables are reduced from 11 to 7 An extract from the lsopt_ anova file is given below where the ranked factors more detail below are rounded to the nearest 10 From this output it was decided to eliminate the variables T Flange Depth F_Flange Depth B Flange Depth and I Flange Width from the optimization process 272 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Summary significance L_Bracket_Gauge T_Flange_Depth F_Flange_Depth B Flange Depth I_F
126. GA Daberkow D D and Mavris D N An investigation of metamodeling techniques for complex systems design Symposium on Multidisciplinary Analysis and Design Atlanta October 2002 Eschenauer H Koski J Osyczka A Multicriteria Design Optimization Procedures and Applications Springer Verlag Berlin 1990 Fedorova N N Terekhoff S A Space Filling Designs Internal Report April 2002 Foresee F D Hagan M T Gauss Newton approximation to Bayesian regularization Proceedings of the 1997 International Joint Conference on Neural Networks pp 1930 1935 1997 Forsberg J Simulation Based Crashworthiness Design Accuracy Aspects of Structural optimization using Response Surfaces Thesis No 954 Division of Solid Mechanics Department of Mechanical Engineering Link ping University Sweden 2002 Giger M Redhe M and Nilsson L Division of Mechanics Department of Mechanical Engineering Link ping University Sweden Personal Communication January 2003 287 BIBLIOGRAPHY 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Giger M An Investigation of Structural Optimization in Crashworthiness Design Using a Stochastic Approach A Comparison of Stochastic Optimization and Response Surface Methodology Thesis Division of Mechanics Department of Mechanical Engineering Link ping University Sweden 2003 Haftka R T G rdal A El
127. GENMASS attribute see able 10 4 Rigid body inertia and coupling will be incorporated in a later version Example Obtain the frequency of the current mode corresponding to the baseline mode shape number 15 as the response Frequency response Frequency DynaFreq 15 FREQ Obtain the number sequence of the current mode corresponding to the baseline mode shape number 15 as the response Number of mode response Modal number DynaFregq 15 NUMBER Remarks 1 The user must identify which baseline mode is of interest by viewing the baseline d3eigv file in LSPOST LS OPT must then be run for the baseline iterate 0 to generate the required modal output files These include massvectout eigvectout and eigvalout where correspond to the modal sequence number e g 03 for mode 3 The DynaFreq command must be omitted in the baseline run since an error will occur in the absence of the baseline data files referred to in point 2 below 2 The three files massvectout eigvectout and eigvalout must be renamed massvectBaseline eigvalBaseline and eigvectBaseline where refers to the number of the mode of interest selected above i e 05 for mode 5 These files must be placed in the working directory before the optimization starts 118 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS 3 The optimization run can now be started with the activated DynaFreg command 4 A
128. ION METHODOLOGY The points are used to fit a second order function The value of amp 4 9 2 6 4 D optimal design This method uses a subset of all the possible design points as a basis to solve max X EX The subset is usually selected from an factorial design where Z is chosen a priori as the number of grid points in any particular dimension Design regions of irregular shape and any number of experimental points can be considered BB The experiments are usually selected within a sub region in the design space thought to contain the optimum A genetic algorithm is used to solve the resulting discrete maximization problem See References 68 The numbers of required experimental designs for linear as well as quadratic approximations are summarized in the table below The value for the D optimality criterion is chosen to be 1 5 times the Koshal design value plus one This seems to be a good compromise between prediction accuracy and computational cost a8 The factorial design referred to below is based on a regular grid of 2 points linear or 3 points quadratic Table 2 1 Number of experimental points required for experimental designs Number of Linear approximation Quadratic approximation Central Variables n Koshal D optimal Factorial Koshal D optimal Factorial Composite 1 2 4 2 3 5 3 3 2 3 5 4 6 10 9 9 3 4 7 8 10 16 27 15 4 5 8 16 15 23 81 25 5 6 10 32 21 32 243 43 6 7 11 64 28 43 729
129. ISTORIES FOR SOLVER SOLVER 1 WU Ur Ur histories 1 history NHist BinoutHistory res_type nodout cmp z displacement id 486 RESPONSES FOR SOLVER SOLVER _1 responses 2 response NodDisp 1 0 BinoutResponse res_ type nodout cmp z displacement id 486 select MIN response DispT LookupMin NHist t constraints 1 constraint NodDisp lower bound constraint NodDisp 150 JOB INFO analyze monte carlo STOP The LS OPT output EE E E E H E H E HE E H E H E E E E EHARA EPH HH HH HH HH HH HH HH RoE HEHE Direct Monte Carlo simulation considering 2 stochastic variables HH HHHHHRRHHEHHHHHRR HEHE HHHERH HRP EEE HH HH EEE HH HH EEE EERE EE HEE FEE EHEHE HE FE E HEHE HE HE HE E HE HE HE HE E HE H HE H HE H H HH HH HH HHHH RR HH HHHFHIHIHFEH STATISTICS OF VARIABLES FEFE HE HE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE H H HH EH HH HHHH HHHH Variable T1 Distribution Information Number of points 5 60 Mean Value 1 Standard Deviation 0 0 Coef of Variation 0 0 Maximum Value 1 Minimum Value 0 8 280 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Variable YS Distribution Information Number of points 60 Mean Value 1 Standard Deviation 0 09895 Coef of Variation 0 09895 Maximum Value 1 239 Minimum Value 0 7606 HHRHHHHTEE HHP EEE HRP EEE HRP EH RRP EEE HERE EEE RE STATISTICS OF RE
130. If print steps 1 then the printing is done on step 0 and exit only The values of the design variables are suppressed on intermediate steps if print lt 0 LS OPT Version 2 167 CHAPTER 16 OPTIMIZATION ALGORITHM SELECTION AND SETTINGS Command file syntax lfop param parameter identifier value Example lfop param eg 1 0e 6 In the case of an infeasible optimization problem the solver will find the most feasible design within the given region of interest bounded by the simple upper and lower bounds A global solution is attempted by multiple starts from a set of random points 168 LS OPT Version 2 EXAMPLES 169 170 LS OPT Version 2 17 Example Problems 17 1 Two bar truss 2 variables This example has the following features A user defined solver is used Extraction is performed using user defined scripts First and second order response surface approximations are compared The effect of subregion size is investigated A trade off study is performed The design optimization process is automated 17 1 1 Description of problem This example problem as shown in Figure 17 1 has one geometric and one element sizing variable Figure 17 1 The two bar truss example 171 CHAPTER 17 EXAMPLE PROBLEMS The problem is statically determinate The forces on the members depend only on the geometric variable Only one load case is considered F F F 24 8KN 198 4kN There are two des
131. JN with o the standard deviation of f x and N the number of sampling points The error is therefore unrelated to the number of design variables 0 The error of estimating p the probability of an event is a random value with the following variance amp p p x N Which can be manipulated to provide a minimum sampling A suggestion for the minimum sampling size provided by Tu and Choi is 10 N Fla lt 0 The above indicates that for a 10 estimated probability of failure about 100 structural evaluations are required with some confidence on the first digit of failure prediction To verify an event having a 1 probability about a 1000 structural analyses are required which usually would be too expensive A general procedure of obtaining the minimum number of sampling points for a given accuracy is illustrated using an example at the end of this section For more information a statistics text for example reference Ho should be consulted A collection of statistical tables and formulae such as the CRC reference will also be useful The variance of the probability estimation must be taken into consideration when comparing two different designs The error of estimating the difference of the mean values is a random variable with a variance of o Pee N 0 N with the subscripts 1 and 2 referring to the different design evaluations The error of estimating the difference of sample proportions is a random variable with a v
132. LS OPT User s Manual A DESIGN OPTIMIZATION AND PROBABILISTIC ANALYSIS TOOL FOR THE ENGINEERING ANALYST NIELEN STANDER Ph D TRENT EGGLESTON Ph D KEN CRAIG Ph D WILLEM ROUX Ph D October 2003 Version 2 Copyright 1999 2003 LIVERMORE SOFTWARE TECHNOLOGY CORPORATION All Rights Reserved Mailing address Livermore Software Technology Corporation 2876 Waverley Way Livermore California 94551 Support Address Livermore Software Technology Corporation 7374 Las Positas Road Livermore California 94551 FAX 925 449 2507 TEL 925 449 2500 EMAIL Copyright 1999 2003 by Livermore Software Technology Corporation All Rights Reserved Contents iii CONTENTS iv LS OPT Version 2 CONTENTS LS OPT Version 2 y CONTENTS vi LS OPT Version 2 CONTENTS LS OPT Version 2 Vil CONTENTS viii LS OPT Version 2 CONTENTS LS OPT Version 2 1X CONTENTS X LS OPT Version 2 CONTENTS 17 Example Problems 171 Two bar truss 2 variables Description of problem A first approximation using linear response surfaces Updating the approximation to second order Reducing sion of interest for further refinement Conducting a trade off study Automating the design process LS OPT Version 2 x1 CONTENTS Small car crash 2 variables Introduction Design criteria and design variables Design formulation Modeling
133. Latin hypercube sampling LHS provides a P by n matrix S Sy that randomly samples the entire design space broken down into P equal probability regions Si m 6 yp 2 20 where 7 7 are uniform random permutations of the integers 1 through P and independent random numbers uniformly distributed between 0 and 1 A common simplified version of LHS has centered points of P equal probability sub intervals S n 0 5 P 2 21 LHS can be thought of as a stratified Monte Carlo sampling Latin hypercube samples look like random scatter in any bivariate plot though they are quite regular in each univariate plot Often in order to generate an especially good space filling design the Latin hypercube point selection S described above is taken as a starting experimental design and then the values in each column of matrix is permuted so as to optimize some criterion Several such criteria are described in the literature Maximin One approach is to maximize the minimal distance between any two points i e between_any two rows of S This optimization could be performed using for example Simulated Annealing see Appendix Fh The maximin strategy would ensure that no two points are too close to each other For small P maximin distance designs will generally lie on the exterior of the design space and fill in the interior as P becomes larger See Section for more detail Centered L2 discrepancy Another strategy is to mi
134. MS 17 5 Material identification airbag 10 variables Example by courtesy of DaimlerChrysler A methodology for deriving material parameters from experimental results known as material parameter identification is applied here using optimization The example has the following features e Composite functions are used e The problem is unconstrained e Two formulations are used A least squares residual and a maximum violation approach 17 5 1 Problem statement The problem is illustrated in Figure 17 26 Shown is an impacting mass chest form and a deploying airbag The experimental results contain the acceleration of the mass for two impacting velocities namely 4 and 5m s The velocity and displacement data are derived from the acceleration through time integration Altogether 54 responses 9 per time history curve acceleration velocity and displacement for both impacting velocities are used in the regression This represents a monitoring increment of 5ms The design variables x are the ordinates on the leakage coefficient pressure curve that is used as a material load curve in LS DYNA when simulating the impact depicted in Results are shown for both 5 and 10 design variables The load curve used is implemented as a piece wise linear table lookup of the leakage curve data 2 ZZ EL 22 ee ae se RE RRF I UI Z ZL LLL 22 Ly 2255 gt AON FE EEE 2 7 Se AS 7
135. Note how the two disciplines crash and NVH are treated separately Variables are flagged as local with the Local variable_name statement and then linked to a solver using the Solver variable variable_name command Full Vehicle MDO Crash and NVH 5 DEFINITION OF MULTIDISCIPLINARY QUANTITIES solvers 2 variables 7 responses 12 histories 2 composites 5 SHARED DESIGN VARIABLES Variable cradle rails 1 93 Lower bound variable cradle rails 1 Upper bound variable cradle rails 3 Range cradle rails 0 4 LS OPT Version 2 257 CHAPTER 17 EXAMPLE PROBLEMS Variable cradle csmbr 1 93 Lower bound variable cradle_csmbr 1 Upper bound variable cradle_csmbr 3 Range cradle_csmbr 0 4 Variable shotgun_inner 1 3 Lower bound variable shotgun_inner 1 Upper bound variable shotgun inner 2 5 Range shotgun_inner 0 3 Variable shotgun_outer 1 3 Lower bound variable shotgun_outer 1 Upper bound variable shotgun_outer 2 5 Range shotgun_outer 0 3 Variable rail_inner 2 Lower bound variable rail_inner 1 Upper bound variable rail_inner 3 Range rail_inner 0 4 Local rail_inner Variable rail_outer 1 5 Lower bound variable rail_outer 1 Upper bound variable rail_ outer 3 Range rail_outer 0 4 Local rail_outer Variable aprons 1 3 Lower bound variable aprons 1 Upper bound variable aprons 2 5 Range aprons 0 3 Local aprons 5 DEFINITION OF SOLVER CRASH solver d
136. P 25 response vel30_5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 30 response vel35_ 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 35 response vel40 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 40 response vel45 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 45 response vel50 5 1 0 DynaASCII nodout X VEL 10322 TIMESTEP 50 response disp10_5 1 140 841 DynaASCII nodout X_DISP 10322 TIMESTEP 10 response disp15_5 1 140 841 DynaASCII nodout X DISP 10322 TIMESTEP 15 response disp20 5 1 140 841 DynaASCII nodout X_DISP 10322 TIMESTEP 20 response disp25_5 1 140 841 DynaASCII nodout X_DISP 10322 TIMESTEP 25 response disp30 _ 5 1 140 841 DynaASCII nodout X DISP 10322 TIMESTEP 30 response disp35_5 1 140 841 DynaASCII nodout X DISP 10322 TIMESTEP 35 response disp40 5 1 140 841 DynaASCII nodout X_DISP 10322 TIMESTEP 40 response disp45 5 1 140 841 DynaASCII nodout X DISP 10322 TIMESTEP 45 response disp50 5 1 140 841 DynaASCII nodout X DISP 10322 TIMESTEP 50 NO HISTORIES DEFINED FOR SOLVER 5MPS HISTORIES AND RESPONSES DEFINED BY EXPRESSIONS composites 10 composite Residual type targeted composite Residual response accl10 4 44 9417 scale 100 composite Residual response acc15_4 75 4247 scale 100 composite Residual response acc20 4 118 58 scale 100 composite Residual response acc25 4 176 239 scale 100 composite Residual response acc30_ 4 221 678 scale 100 c
137. PLE PROBLEMS Rad2 Rad3 Thinning_scaled Rad Rad2 Rad3 Thinning_scaled 1 5 1 5 1 5 1 5 0 2957 0 3078 0 09123 0 4 0 4 0 4 0 09567 le 30 1 1 YES le 30 1 1 YES le 30 0 2 YES Upper As shown below after 1 iteration a feasible design is generated The simulation response of the optimum is closely approximated by the response surface DESIGN POINT Radius 1 Radius 2 Radius 3 Radl Rad2 Rad3 Thinning scaled Radl Rad2 Rad3 Thinning scaled Lower Bound Value 1 3 006 1 3 006 1 3 006 0 04308 0 03841 3 006 3 006 3 006 3 006 3 006 3 006 0 2172 0 2 Upper Bound 4 5 4 5 4 5 Lower Upper Viol le 30 O no 1e 30 1 1 YES 1e 30 1 1 YES 1e 30 1 1 YES le 30 0 2 no 224 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 4 3 Automated design The optimization process can also be automated so that no user intervention is required The starting design lower and upper bounds and region of interest is modified from the 1 iteration study above The input file is modified as follows The variable definitions are as follows Variable Radius_1 1 5 Lower bound variable Radius_1 Upper bound variable Radius_1 4 5 Range Radius_1 1 Variable Radius_2 1 5 Lower bound variable Radius_2 Upper bound variable Radius_2 4 5 Range Radius_2 1 Variable Radius_3 1 5 Lower bound variable Radius_3 1 Upper bound variable Radius_
138. R 17 EXAMPLE PROBLEMS Figure 17 38 Body in white model of vehicle in torsional vibration mode 38 7Hz 17 7 2 Formulation of optimization problem To illustrate the effect of coupling between the disciplines both full and partial sharing of the design variables are considered In addition different starting designs are considered in a limited investigation of the global optimality of the design The optimization problem for the different starting designs considered is defined as follows Minimize Mass subject to Maximum intrusion Xerash gt 551 8mm Fully shared variables Maximum intrusion Xerash 551 8mm Partially shared variables Stage 1 pulse amp crasn gt 14 348 Stage 2 pulse amp crash gt 17 578 Stage 3 pulse amp crash gt 20 708 37 77Hz lt Torsional mode frequency xnvn lt 39 77Hz Fully shared variables 38 27Hz lt Torsional mode frequency xnvu lt 39 27Hz Partially shared variables Fully shared variables Xcrash Xnvu rail_inner rail outer cradle rails aprons shotgun inner shotgun_outer T cradle_crossmember Partially shared variables Starting design I Baseline Xcrash rail_inner rail_outer cradle_rails aprons shotgun_inner shotgun outer xnvu cradle_rails shotgun_inner shotgun_outer cradle_crossmember 256 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Partially shared variables Starting design 2 Minimum weight Xerash rail_inner r
139. R kinetic_energy Kinetic energy I ENER internal energy Internal energy X MOMENTUM Y MOMENTUM Z MOMENTUM X momentum y_ momentum Z momentum X momentum Y momentum Z momentum MOMENTUM Momentum XRB_VEL x_rbvelocity X rigid body velocity YRB_VEL y_rbvelocity Y rigid body velocity ZRB_VEL z_rbvelocity Z rigid body velocity RB VEL Rigid body velocity TK_ENER Total kinetic energy TI ENER Total internal energy hourglass energy Contact Node Forces NCFORC DynaASCII Binout Description Keyword Component Binout subdirectory master_00001 and slave_00001 X_FORCE x_force X force Y_FORCE y_force Y force Z_FORCE z_force Z force R_FORCE Resultant Force PRESSURE pressure Pressure x X coordinate y Y coordinate Z Z coordinate 310 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Nodal Point Response NODOUT DynaASCH Binout Description Keyword Component X_DISP x_displacement X displacement Y_DISP y_displacement Y displacement Z_DISP z_displacement Z displacement R_DISP Resultant displacement X_VEL x_velocity X velocity Y_VEL y_velocity Y velocity Z VEL z_velocity Z velocity R VEL Resultant velocity X _ ACC x_acceleration X acceleration Y_ACC y_acceleration Y acceleration Z ACC z_acceleration Z acceleration R ACC Resultant acceleration x_coordinate X coordinate y_coordinate Y coordinate
140. RESS Effective stress TMAX SHEAR Maximum shear stress TMAX P STRESS Maximum principal stress TMIN P STRESS Minimum principal stress upper_eps xx XX strain lower_eps_ xx upper_eps_yy YY strain lower_eps_yy upper_eps_ zz ZZ strain lower_eps zz upper_eps_ xy XY strain lower_eps xy upper_eps_yz YZ strain lower_eps_ yz upper_eps_ zx ZX strain lower_eps_zx LS OPT Version 2 307 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Contact Entities Resultants GCEOUT DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z FORCE z_force Z force R FORCE force magnitude Force magnitude X MOMENT x_ moment X moment Y_MOMENT y_moment Y moment Z MOMENT z_ moment Z moment R_MOMENT moment magnitude Moment magnitude Global Statistics GLSTAT DynaASCH Binout Description Keyword Component K_ENER kinetic_energy Kinetic energy I ENER internal energy Internal energy T ENER total energy Total energy RATIO energy_ratio Ratio SW_ENER stonewall energy Stonewall energy D_ ENER spring and_damper_energy Spring amp Damper energy HG_ENER hourglass _energy Hourglass energy SI ENER sliding interface_energy Sliding interface energy EW_ENER external_ work External work X_VEL global x velocity Global x velocity Y_VEL global y_velocity Global y velocity Z VEL global z velocity Global z velocity T_VEL Velocity system
141. R_VEL Resultant velocity X ACC global ax X acceleration Y_ACC global _ay Y acceleration Z ACC global az Z acceleration R_ACC Resultant acceleration global_x X coordinate global_y Y coordinate global z Z coordinate local_dx Local X displacement local_dy Local Y displacement local _dz Local Z displacement local_vx Local X velocity local_vy Local Y velocity local_vz Local Z velocity local_ax Local X acceleration local_ay Local Y acceleration local_az Local Z acceleration LS OPT Version 2 313 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Rigid Body Data RBDOUT Rotational components DynaASCH Binout Description Keyword Component RX_DISP global_rax X rotation RY_DISP global_ray Y rotation RZ_DISP global raz Z rotation RX_VEL global_rdx X velocity RY_VEL global_rdy Y velocity RZ_VEL global_rdz Z velocity RX_ACC global_rvx X acceleration RY_ACC global rvy Y acceleration RZ_ACC global_rvz Z acceleration local_rdx Local X rotation local_rdy Local Y rotation local_rdz Local Z rotation local_rvx Local X velocity local_rvy Local Y velocity local_rvz Local Z velocity local_rax Local X acceleration local_ray Local Y acceleration local_raz Local Z acceleration Direction cosines dircos 11 11 direction cosine dircos 12 12 direction cosine dircos_ 13 13 direction cosine dircos 21 21 direction cosine dircos 22 22 direction cosine d
142. Residual 0 8978 30 52 Average Error 0 7131 24 24 Square Root PRESS Residual u 2 5054 85 18 Variance 0 9549 R 2 0 9217 R 2 adjusted _ 0 9217 R 2 prediction 0 1426 Determinant of X X S 3 5615 Approximating Response Stress using 5 points ITERATION 1 Mean response value 4 6210 RMS error 2 0701 44 80 Maximum Residual 4 1095 88 93 Average Error 1 6438 35 57 Square Root PRESS Residual u 3 9077 84 56 Variance 7 1420 R 2 0 8243 R 2 adjusted 0 8243 R 2 prediction 0 3738 Determinant of X X 3 5615 The accuracy of the response surfaces can also be illustrated by plotting the predicted results vs the computed results Figure 17 2 LS OPT Version 2 175 CHAPTER 17 EXAMPLE PROBLEMS Response Surface Accuracy For Response Function Weight 8 T 1 I 7 l o 6 Ss o 9 nn Cc 2 4 N e cc o 3 2 2 e 2 o Oo 4 0 0 1 2 3 4 5 6 Predicted Response Value Prediction accuracy of Weight Iteration 1 Linear Computed Response Value Response Surface Accuracy For Response Function Stress Predicted Response Value Prediction accuracy of Stress Iteration 1 Linear Figure 17 2 Prediction accuracy of Weight and Stress Iteration 1 Linear The R values are large However the prediction accuracy especially for weight seems to be poor so that a higher order of approximation will be required Nevertheless an i
143. SPONSES HHRHHHTTEE HHP HERAA AHHH HRP HRP EE HERE ER Response NodDisp Distribution Information Number of points 60 Mean Value 141 8 Standard Deviation 15 21 Coef of Variation 0 1073 Maximum Value 102 3 Minimum Value 168 9 Response DispT Distribution Information Number of points 60 Mean Value 2 7 726 Standard Deviation 0 6055 Coef of Variation 0 07837 Maximum Value 8 4 Minimum Value 5 5 FEFE HEHE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE FE HE HE HE HE HE E H H H H H H H EH H HHHH HH HHHH H HHHH STATISTICS OF COMPOSITES EEHEEHE HE E E HEHE HE HE E HE H HE H HE H H EE H A HRP HERE HH ERE EE ER HHRFRRRRHHHRR AHHAR RRHH EEE HRP RHEE REEERE HHHH HHHH STATISTICS OF CONSTRAINTS HHRHHHHTEE H ARARE EE HH REE EEE HRA EE HERE EEE HERE EEE RE Constraint NodDisp Distribution Information Number of points 60 Mean Value 141 8 Standard Deviation 15 21 Coef of Variation 0 1073 Maximum Value 102 3 LS OPT Version 2 281 CHAPTER 17 EXAMPLE PROBLEMS Minimum Value 168 9 Lower Bound BOUNA EEEE aks u El tate Ae 150 Evaluations exceeding this bound 20 Probability of exceeding bound 0 3333 Confidence Interval on Probability Standard Deviation of Prediction Error 0 06086 Lower Bound Probability Higher Bound 0 2116 0 3333 0 455 Confidence Interval of 95 assuming Normal Distribution Confidence Interval of 75 using Tchebysheff s T
144. Second order response surface approximations are compared using different subregions The design optimization process is automated Noisy response variables are improved using filtering The example in this chapter is modeled on one by Yamazaki 76 17 3 1 Problem statement The problem consists of a tube impacting a rigid wall as shown in The energy absorbed is maximized subject to a constraint on the rigid wall impact force The cylinder has a constant mass of 0 54 kg with the design variables being the mean radius and thickness The length of the cylinder is thus dependent on the design variables because of the mass constraint A concentrated mass of 500 times the cylinder weight is attached to the end of the cylinder not impacting the rigid wall The deformed shape at 20ms is shown in Figure 17 16 for a typical design a Figure 17 15 Impacting cylinder LS OPT Version 2 203 CHAPTER 17 EXAMPLE PROBLEMS KIN WAAN TER RK S SEES s NG N ON VE Se Figure 17 16 Deformed finite element model time 20ms The optimization problem is stated as Maximize E ma X1 X gt 1 0 02 subject to Fret 215 average lt 70000 2790X x where the design variables x and x are the radius and the thickness of the cylinder respectively Ente 02 iS the objective function and constraint functions F x normal average and U x are the average normal force on the rigid wall and the
145. Sliding Interface 1 CONTACT FORMING ONE WAY SURFACE TO SURFACE workpiece vs punch 0 1000000 0 000 0 000 1 2 3 3 1 0 0 CONTACT FORMING ONE WAY SURFACE TO SURFACE workpiece vs die 1 3 3 3 1 I 0 1000000 0 000 0 000 0 0 5 CONTACT FORMING ONE WAY SURFACE TO SURFACE workpiece vs blankholder 1 4 3 3 1 T 0 1000000 0 000 0 000 0 0 CONTROL ADAPTIVE ADPFREQ ADPTOL ADPOPT MAXLVL TBIRTH TDEATH LCADP IOFLAG 0 100E 03 5 000 2 3 0 000E 00 1 0000000 0 al ADPSIZE ADPASS IREFLG ADPENE 0 0000000 1 0 3 0000 LOAD RIGID BODY rbID dir lcID scale 2 3 2 1 0000000 LOAD RIGID BODY rbID dir lcID scale 4 3 3 1 0000000 DEFINE CURVE FLD curve 90 1 2 083 0 25 1 75 END The input file file m3 tg opt used to generate the FE mesh in Truegrid is c generate LS DYNA input deck for sheet metal example lsdyna keyword lsdyopts endtim 0009 nodout 1 e 6 d3plot dtcycl 0001 lsdyopts istupd 1 c lsdymats 1 37 shell elfor bt rho 7 8e 9 e 2 e5 pr 28 220 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS c sigy 200 etan 572 er 1 4 lsdymats 2 20 shell elfor bt rho 7 8e 9 e 2 e5 pr 28 shth 1 cmo con 4 7 lsdymats 3 20 shell elfor bt rho 7 8e 9 e 2 e5 pr 28 shth 1 cmo con 7 7 lsdymats 4 20 shell elfor bt rho 7 8e 9 e 2 e5 pr 28 shth 1 cmo con 4 7 plane 2 0002100 01 symm plane 3 000010 0 01 symm c sid 1 1sdsi alO slvmat 1l mstmat 2 scoef 1 c sid 2 lsdsi alO slvmat 1 m
146. Space filling designs Space filling 5 recommended space filling Algorithm 5 Section 2 6 6 Space filling 0 monte carlo Space filling 1 lhd centralpoint Space filling 2 lhd generalized Space filling 3 maximin permute Space filling 4 maximin subinterval User defined designs User defined own Plan plan Command file syntax Solver Solver Solver Solver Solver Example 1 Solver order quadratic Solver experimental design dopt Solver basis experiment 5toK Example 2 Solver order linear Solver Solver Solver Solver experimental design dopt basis experiment latin hypercube number experiments 40 number basis experiments 1000 order linear interaction elliptic quadratic FF kriging experimental design point selection scheme basis experiment basis experiment number experiment number experimental_points number basis experiments number basis experimental_points LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION In Example 1 the default number of experiments will be selected depending on the number of design variables In Example 2 40 points are selected from a total number of 1000 In LS OPTui the point selection scheme is selected using the Point Selection panel Figure 9 1 The default options are preset e g the D optimal point selection scheme basis type Full Factorial 3 points per variable is the default for linear polynomials Figure 9 1 and
147. T FILES AND COMPONENTS Reaction Forces RCFORC Keyword Description X_FORCE X force Y_FORCE Y force Z FORCE Z force R FORCE R force XS FORCE X slave force YS FORCE Y slave force ZS FORCE Z slave force RS FORCE R slave force RigidWall Forces RWFORC Keyword Description NORMAL normal X FORCE X force Y_ FORCE Y force Z FORCE Z force Section Forces SECFORC Keyword Description X FORCE X force Y_FORCE Y force Z FORCE Z force X MOMENT X moment Y_MOMENT Y moment Z MOMENT Z moment X CENTER X center Y_CENTER Y center Z CENTER Z center R FORCE R force R MOMENT R moment 298 LS OPT Version 2 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Single Point Constraint Reaction Forces SPCFORC Keyword Description X_FORCE X force Y_FORCE Y force Z FORCE Z force R FORCE R force X RES Total X force Y_RES Total Y force Z RES Total Z force X MOMENT X moment Y_MOMENT Y moment Z MOMENT Z moment R_ MOMENT R moment Spotweld and Rivet Forces SWFORC Keyword Description AXIAL Axial force SHEAR Shear force LS OPT Version 2 299 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS 300 LS OPT Version 2 Appendix B LS DYNA Binary Result Components The table contains component numbers for element variables These can be specified in the Dyna interface command to extract response variables By adding 100 200 3
148. Total Z force energy Energy etotal Total Energy Discrete Element Forces DEFORC DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z_FORCE z_force Z force R_FORCE resultant_force Resultant force displacement Change in length 304 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Element Output Brick and Beam Elements ELOUT DynaASCH Binout Description Keyword Component Binout subdirectory solid BXX_STRESS sig XxX XX stress BYY_ STRESS sig xy YY stress BZZ_STRESS sig_yy ZZ stress BXY_STRESS sig_yz XY stress BYZ STRESS Sig_Zx YZ stress BZX_STRESS SIg_ZZ ZX stress YIELD yield Yield function BE_STRESS effsg Effective stress BPRESSURE Pressure BMAX SHEAR Maximum shear stress BMAX P STRESS Maximum principal stress BMIN_P_ STRESS Minimum principal stress eps_xx XX strain eps xy YY strain eps_yy ZZ strain eps_yz XY strain eps _Zx YZ strain eps Zz ZX strain Binout subdirectory beam AXIAL axial Axial force resultant S_ SHEAR shear_s s Shear resultant T SHEAR shear_t t Shear resultant S MOMENT moment_s s Moment resultant T MOMENT moment t t Moment resultant TORSION torsion Torsional resultant SIG 11 O11 SIG_12 Gin SIG 31 Cs PLASTIC z Plastic strain LS OPT Version 2 305 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Element Output
149. UOD Xey wie 10413 N oO po ssuodss 10 12 14 16 Number of Iterations 253 f Maximum constraint violation 10 12 14 16 Number of Iterations e Mode sequence Figure 17 34 Optimization histories Small car MDO Latin Hypercube Sampling SRS LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 7 Large car crash and NVH MDO 7 variables Example by courtesy of DaimlerChrysler This example has the following features e LS DYNA is used for both explicit full frontal crash and implicit NVH simulations e Miultidisciplinary design optimization MDO is illustrated with a realistic full vehicle example e Extraction is performed using standard LS DYNA interfaces This example illustrates a realistic application of Multidisciplinary Design Optimization MDO and concerns the coupling of the crash performance of a large vehicle with one of its Noise Vibration and Harshness NVH criteria namely the torsional mode frequency 114 The MDO formulation used is depicted in Figure 17 35 Design variables Multidisciplinary Analyses Crashworthiness analysis System level Optimizer Goal Minimize Mass s t Crashworthiness and NVH constraints NVH analysis State variables Figure 17 35 Multidisciplinary feasible MDF MDO architecture 17 7 1 Modeling The crashworthiness simulation considers a model containing approximately 30 000 elements of a National Highway Transportation
150. YNA An extract from the parameterized input deck is shown below Note how the design variables are labeled for substitution through the characters lt lt gt gt The cylinder for impact is modeled as a rigid wall DEFINITION OF MATERIAL 1 MAT PLASTIC KINEMATIC 1 1 000E 07 2 000E 05 0 300 400 0 0 Obs 70200 HOURGLASS 1 05 0537 050 0 SECTION SHELL 1 2 0 0 0 0 0 2 00 2 00 2 00 2 00 0 PART material type 3 Kinematic Isotropic Elastic Plastic 1 1 1 0 1 0 5 DEFINITION OF MATERIAL 2 5 MAT PLASTIC KINEMATIC 2 7 800E 08 2 000E 05 0 300 400 0 0 188 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 0 0 0 HOURGLASS 2 07 0 5 0 70 4 0 SECTION SHELL 24 ea 0 rye 0 ara ear Ora lt lt t_bumper gt gt lt lt t_bumper gt gt lt lt t_bumper gt gt lt lt t_bumper gt gt 0 PART material type 3 Kinematic Isotropic Elastic Plastic 2 2 0 2 0 DEFINITION OF MATERIAL 3 WU UN MAT PLASTIC KINEMATIC 3 7 800E 08 2 000E 05 0 300 400 0 0 0 0 0 HOURGLASS 3 0 0 0 0 0 SECTION SHELL 320 25 9 ig Oe Os 0 lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt lt lt t_hood gt gt 0 PART material type 3 Kinematic Isotropic Elastic Plastic 34374390434 0 DEFINITION OF MATERIAL 4 MAT PLASTIC KINEMATIC 4 7 800E 08 2 000E 05 0 300 400 0 0 0 0 0 HOURGLASS 4 0 0 0 0 0 SECTION SHELL 4 2
151. _hood Jeduing ajqeue 1 1 1 I T t 1 1 1 1 Te Te ron Te x m N pooy eigeue Number of Iterations Number of Iterations b Optimization history of t_bumper a Optimization history of t_hood Figure 17 10 Optimization history of design variables LS OPT Version 2 196 CHAPTER 17 EXAMPLE PROBLEMS 17 2 8 Trade off using neural network approximation In order to build a more accurate response surface for trade off studies the Neural Net method is chosen under the ExpDesign panel This results in a feed forward FF neural network Section D being solved for the points selected The recommended point selection scheme Space Filling is used One iteration is performed to analyze only one experimental design with 25 points The modifications to the command input file are as follows 5 DEFINITION OF SOLVER 1 solver dyna 1 solver command lsdyna solver input file car5 k solver append file rigid2 solver order FF solver update doe solver experiment design space filling solver number experiments 25 iterate 1 The response surface accuracy is illustrated in Figure 17 11 for the HIC and Intru_2 responses The HIC has more scatter than Intru_2 for the 25 design points used Response Surface Accuracy Response Surface Accuracy For Response Function HIC For Response Function Intru_2 8 I I L 60 I i i i i I I 1 I 2 62 4 I I I I L I e A P 64 3 I L I P Boa 66 4
152. a design variable An example of an input line in a LS DYNA structured input file is shfact z integr printout quadrule 0 5 07 10 2 0 thickni thickn2 thickn3 thickn4 ref surf lt lt Thick_1 gt gt lt lt Thick_1 gt gt lt lt Thick_1 gt gt lt lt Thick_1 gt gt 0 0 If a solver but no preprocessor has been specified only the relevant solver utility routines will be executed The field width of the substituted variable has been set to 10 with three digits after the decimal point The C language notation is 10 3e Care must be taken not to exceed the maximum field width tolerated by the simulation package Consult the relevant User s manual for rules regarding input format 7 2 Interfacing to a Solver In LS OPTui solvers are specified in the Solver panel Figure 7 1 Both the preprocessor and solver input and append files are specified in this panel Multiple solvers as used in multi case or multi disciplinary applications are defined by selecting Add solver The Replace button must be used after the modification of current data The solver name is used as the name for the subdirectory Execution command The command to execute the solver must be specified The command depends on the solver type and could be a script but typically excludes the solver input file name argument as this is specified using a separate command Input template files An input template file in which the design variables have been re
153. able If no nominal value is specified for a control variable then the nominal value of the distribution is used If the nominal value of a control variable is specified then this value is used the associated distribution will be used to describe the variation around this nominal value For example a variable with a nominal value of 160 LS OPT Version 2 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION 7 is assigned a normal distribution with u 0 and o 2 the results values of the variable will be normally distributed around a nominal value of 7 with a standard deviation of 2 This behavior is only applicable to control variables noise variables will always follow the specified distribution exactly 15 4 2 Bounds on Probabilistic Variable Values Assigning a distribution to a control value may result in designs exceeding the bounds on the control variables The default is not to enforce the bounds The user can control this behavior A noise variable is bounded by the distribution specified and does not have upper and lower bounds similar to control variables However bounds are required for the construction of the approximating functions and are chosen as described in the next subsection Command file syntax set variable distribution bound state Item Description state Whether the bounds must be enforced for the probabilistic component of the variable Example ignore bounds on control va
154. acceleration Rotational components RX DISP XX rotation RY_DISP YY rotation RZ_DISP ZZ rotation RX_VEL XX rotational velocity RY_VEL YY rotational velocity RZ_VEL ZZ rotational velocity RX_ACC XX rotational acceleration RY_ACC YY rotational acceleration RZ ACC ZZ rotational acceleration Injury coefficients CSI Chest Severity Index HIC15 Head Injury Coefficient 15 ms HIC36 Head Injury Coefficient 36 ms 296 LS OPT Version 2 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Nodal Forces NODFOR Keyword Description X_FORCE X force Y_FORCE Y force Z FORCE Z force R_FORCE Resultant force X TOTAL X total force Y_TOTAL Y total force Z TOTAL Z total force R TOTAL Total resultant force Rigid Body Data RBDOUT Keyword Description X DISP X displacement Y_DISP Y displacement Z_DISP Z displacement R_DISP R displacement X VEL X velocity Y_VEL Y velocity Z VEL Z velocity R_VEL Resultant velocity X ACC X acceleration Y_ACC Y acceleration Z ACC Z acceleration R_ACC R acceleration Rotational components RX_ DISP X rotation RY_DISP Y rotation RZ_DISP Z rotation RX_VEL X velocity RY_VEL Y velocity RZ_VEL Z velocity RX_ACC X acceleration RY_ACC Y acceleration RZ_ACC Z acceleration Injury coefficients CSI Chest Severity Index HIC15 Head Injury Coefficient 15 ms HIC36 Head Injury Coefficient 36 ms LS OPT Version 2 297 APPENDIX A LS DYNA ASCII RESUL
155. action of the design space Numerical sensitivities are computed by perturbing n points relative to the current design point xo where the j th perturbed point is xj z X elx x 6 0 ifi j and 1 0 if i j The perturbation constant is relative to the design space size The same value applies to all the variables and is specified as Command file syntax Solver perturb perturbation value 104 LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION Example Solver experimental design numerical_DSA Solver perturb 0 01 Figure 9 3 Selecting Sensitivities in the Point Selection panel LS OPT Version 2 105 CHAPTER 9 METAMODELS AND POINT SELECTION 106 LS OPT Version 2 10 History and Response Results This chapter describes the specification of the history or response results to be extracted from the solver database The chapter focuses on the standard response interfaces for LS DYNA 10 1 Defining a response history vector A response history can be defined by using the history command with an extraction or a mathematical expression The extraction of the result can be done using a standard LS DYNA interface see Section 10 5 or with a user defined program Command file syntax history history name string history history name expression math expression The string is an interface definition in double quotes while the math_expression is a mathematical expression in curly brackets
156. adial basis functions and Kriging 77 Space filling points can be also submitted as the basis set for constructing an optimal D Optimal etc design for a particular model e g polynomial Some space filling designs are random Latin Hypercube Sampling LHS Orthogonal Arrays and Orthogonal Latin Hypercubes The key to space filling experimental designs is in generating good random points and achieving reasonably uniform coverage of sampled volume for a given user specified number of points In practice however we can only generate finite pseudorandom sequences which particularly in higher dimensions can lead to a clustering of points which limits their uniformity To find a good space filling design is a nonlinear programming hard problem which from a theoretical point of view is difficult to solve exactly This problem however has a representation which might be within the reach of currently available tools To reduce the search time and still generate good designs the popular approach is to restrict the search within a subset of the general space filling designs This subset typically has some good built in properties with respect to the uniformity of a design The constrained randomization method termed Latin Hypercube Sampling LHS and proposed in B7 has become a popular strategy to generate points on the box hypercube design region The method implies that on each level of every design variable only one point
157. age 2 4e 07 Range Leakage_2 le 07 Variable Leakage_3 6e 08 Lower bound variable Leakage_3 2e 08 Upper bound variable Leakage_3 4e 07 Range Leakage_3 1e 07 Variable Leakage 4 6e 08 Lower bound variable Leakage 4 Upper bound variable Leakage 4 Range Leakage 4 le 07 Variable Leakage_5 6e 08 Lower bound variable Leakage_5 2e 08 Upper bound variable Leakage_5 4e 07 Range Leakage_5 le 07 Variable Leakage_6 6e 08 Lower bound variable Leakage_6 2e 08 Upper bound variable Leakage_6 4e 07 Range Leakage_6 le 07 Variable Leakage_7 6e 08 Lower bound variable Leakage_7 2e 08 Upper bound variable Leakage_7 4e 07 Range Leakage_7 le 07 Variable Leakage_8 6e 08 Lower bound variable Leakage_8 2e 08 Upper bound variable Leakage_8 4e 07 Range Leakage_8 le 07 Variable Leakage_9 6e 08 Lower bound variable Leakage_9 2e 08 Upper bound variable Leakage_ 9 4e 07 Range Leakage 9 le 07 Variable Leakage_10 6e 08 Lower bound variable Leakage_10 2e 08 2e 08 4e 07 LS OPT Version 2 231 CHAPTER 17 EXAMPLE PROBLEMS Upper bound variable Leakage_10 4e 07 Range Leakage_10 le 07 solvers 2 responses 54 NO HISTORIES ARE DEFINED DEFINITION OF SOLVER 4MPS solver dyna 4MPS solver command lsdyna solver input file sim4mpros inp RESPONSES FOR SOLVER 4MPS response accl0 4 1000 0 DynaASCII nodout
158. ail_outer cradle_rails aprons xnvu cradle _rails shotgun_inner shotgun_outer cradle_crossmember Partially shared variables Starting design 3 Maximum weight Xerash rail_inner rail_outer cradle_rails aprons shotgun_inner shotgun_outer cradle crossmember xnvu cradle_rails shotgun_inner shotgun_outer cradle_crossmember The different variables set above were obtained by using ANOVA variable screening Section 13 4 The Mass objective in each case incorporates all the components defined in The allowable torsional mode frequency band is reduced to 1Hz for the partially shared cases to provide an optimum design that is more similar to the baseline The three stage pulses are calculated from the SAE filtered 60Hz acceleration and displacement of a left rear sill node in the following fashion Stage i pulse k j adx k 0 5 for i 1 1 0 otherwise with the limits di d2 0 184 184 334 334 Max displacement for i 1 2 3 respectively all displacement units in mm and the minus sign to convert acceleration to deceleration The Stage 1 pulse is represented by a triangle with the peak value being the value used The constraints are scaled using the target values to balance the violations of the different constraints This scaling is only important in cases where multiple constraints are violated as in the current problem 17 7 3 Implementation in LS OPT The LS OPT input file is given below
159. aints Composite values Number of composites Responses computed Number of responses Max constraint violation l Composites computed Number of composites Constraints computed Number of constraints Objectives computed Number of objectives Multi objective computed 1 Max constraint violation computed 1 Constants Number of constants Dependents Number of dependents Values of 2 0 10 are assigned to responses of error terminations LS OPT Version 2 321 APPENDIX D DATABASE FILES The ExtendedResults file This file contains all points represented in the AnalysisResults file and appears in the solver directory All values are based on the simulation results A line has the following format Values of 2 0 10 are assigned to responses of simulations with error terminations g p Entities Count Objective weights Number of objectives Objective values Number of objectives Variables Number of solver variables Responses Number of solver responses Multi objective 1 Constraint values Number of constraints Composite values Number of composites Max constraint violation 1 Constants N umber of constants Dependents Number of dependents The OptimumResults file This file contains just the optimum design point data metamodel values i e interpolated and appears in the solver directory All values are Entities Cou
160. al 1 96 standard deviations wide is required The resulting standard deviation is 0 0051 and the minimum number of sampling points is accordingly _ Pa _ O20 _ 3457 o 0 051 3 3 2 Estimation assuming a Normally Distributed Response For these computations we assume a linear expansion of the limit state function LSF The reliability index is then computed as _ EIG X S DIG X with E and D the expected value and standard deviation operators respectively A normally distributed response can be assumed for the estimation of the probability of failure that is the central limit theorem is assumed to be valid The probability of failure is then computed as P f with x the cumulative distribution function of the normal distribution Caution is advised in the following cases e Nonlinear responses Say we have a normally distributed stress responses this implies that fatigue failure is not normally distributed e The variables are not normally distributed for example one is uniformly distributed In which case 50 LS OPT Version 2 CHAPTER 3 PROBABILISTIC FUNDAMENTALS o A small number of variables may not sum up to a normally distributed response even for a linear response o The response may be strongly dependent on the behavior of a single variable The distribution associated with this variable may then dominate the variation of the response The assumption is less valid at the tail regions of the dist
161. ally applied to generate a set of design points Slack constraint A constraint with a slack variable The violation of this constraint can be minimized Slack variable The variable which is minimized to find a feasible solution to an optimization problem e g ein min e subject to g x lt e e20 See Strictness Simulation The analysis of a physical process or entity in order to compute useful responses See Function evaluation LS OPT Version 2 341 APPENDIX G GLOSSARY Solver A computational tool used to analyze a structure or fluid using a mathematical model See Discipline Solver directory A subdirectory of the work directory that bears the name of a solver and where database files resulting from extraction and the optimization process are stored Space Filling Experimental Design A class of experimental designs that employ an algorithm to maximize the minimum distance between any two points Space Mapping A technique which uses a fine design model to improve a coarse surrogate model The hope is that if the misalignment between the coarse and fine models is not too large only a few fine model simulations will be required to significantly improve the coarse model The coarse model can be a response surface Stochastic Involving or containing random variables Involving probability or chance Stopping Criterion A mathematical criterion for terminating an iterative procedure Strictness A number between 0 and 1
162. am design 0 01 iterate param objective 0 01 iterate param stoppingtype and iterate 5 STOP 276 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Typical Gradient file e g for f 1 8000000000 3 20000000000 The optimization results are shown in the plots below An iteration represents a single simulation The dots represent the computed results while the solid line represents a linear approximation constructed from the gradient information of the previous point Optimization History Optimization History For Objective f For Variable x2 Optimization History For Variable x1 2 5 3 2 i 2 t Po ee Rene ead zes aa eee te 1 1 5 a a ac an 1 5 a 3 S a 0 i I 1 i 7 3 BS 0 41 77 1 7 D S 1 A 05 8 i 1 I gt i s O I 1 I 1 1 1 Ferne et Inn i i 0 i i i 0 5 ak Set ae A PAE RE E PEREA ESE 0 5 Li Li I I i 1 i i i i 1 f f 9 T 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Number of Iterations Number of Iterations Number of Iterations Optimization History Optimization History For Response g1 For Response g2 2 0 1 5 0 5 S v 2 q 2 4 Q o a 2 Q X 0 5 1 5 2 1 2 3 4 5 1 2 3 4 5 Number of Iterations Number of Iterations LS OPT Version 2 277 CHAPTER 17 EXAMPLE PROBLEMS 17 10 Probabilistic Example 17 10 1 Overview This example has the following features e Probabilistic analysis e Monte Carlo analy
163. ance 10 28877 por 11 _ 28898 12 _ 28900 13 Kinetic Energy Total Energy Fractional Energy Global X Velocity Global Y Velocity Global Z Velocity Total CPU Time Time to Completion Internal Energy x10 N ol Zea Simulation Time x10 2 Figure 12 1 Run panel in LS OPTuwi 12 3 Restarting When a solution is interrupted through the Stop button or if a previous optimization run is to be repeated from a certain starting iteration this can be specified in the appropriate field in the Run panel Figure 12 1 12 4 Job concurrency When LS OPT is run on a multi processor machine the user can select how many simulations jobs can run concurrently on different processors see Figure 12 1 Only the solver process and response extraction are parallellized The preprocessor processes run serially The number of Concurrent Jobs is ignored for jobs that are run by a queuing system 142 LS OPT Version 2 CHAPTER 12 RUNNING THE OPTIMIZATION PROBLEM 12 5 Job distribution When a queuing system is available its operation can be specified in the Run panel Figure 12 1 12 6 Job and analysis monitoring The Run panel allows a graphical indication of the job progress with the green horizontal bars linked to estimated completion time This progress is only available for LS DYNA jobs The job monitoring is also visible when running remotely through a supported job distribution queuing system Whe
164. and Safety Association NHTSA vehicle undergoing a full frontal impact A modal analysis is performed on a so called body in white model containing approximately 18 000 elements The crash model for the full vehicle is shown in Figure 17 36 for the undeformed and deformed time 78ms states and with only the structural components affected by the design variables both in the undeformed and deformed time 72ms states in The NVH model is depicted in 17 38 jin the first torsion vibrational mode Only body parts that are crucial to the vibrational mode shapes are retained in this model The design variables are all thicknesses or gages of structural components in the 254 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS engine compartment of the vehicle Figure 17 37 parameterized directly in the LS DYNA input file Twelve parts are affected comprising aprons rails shotguns cradle rails and the cradle cross member Figure 17 37 LS DYNA v 960 is used for both the crash and NVH simulations in explicit and implicit modes respectively b Figure 17 36 Crash model of vehicle showing road and wall a Undeformed b Deformed 78ms Shotgun outer Left and right ang InnET cradle rails Inner and outer rail Front cradle upper and lower cross members a b Figure 17 37 Structural components affected by design variables a Undeformed and b deformed time 72ms LS OPT Version 2 255 CHAPTE
165. aram rangelimit Radius 10 16 3 Setting parameters in the LFOPC optimization algorithm The values of the responses are scaled with the values at the initial design The default parameters in LFOPC should therefore be adequate Should the user have more stringent requirements the following parameters may be set for LFOPC These are only available in the command input file Table 16 2 LFOPC parameters and default values Item Parameter Default value Remark mu Initial penalty value u 1 0E 2 mumax Maximum penalty value U max 1 0E 4 1 xtol Convergence tolerance amp on the step movement 1 0E 8 2 eg Convergence tolerance gon the norm of the gradient 1 0E 5 2 delt Maximum step size 6 See remark 3 steps Maximum number of steps per phase 1000 1 print Printing interval 10 4 Remarks 1 For higher accuracy at the expense of economy the value of U max can be increased Since the optimization is done on approximate functions economy is usually not important The value of steps must then be increased as well 2 The optimization is terminated when either of the convergence criteria becomes active that is when A x lt e or Molse 3 It is recommended that the maximum step size 6 be of the same order of magnitude as the diameter of the region of interest To enable a small step size for the successive approximation scheme the value of delt has been defaulted to 0 054 9 range 4
166. ard non linear optimization algorithm to minimize the error function Eq 2 48 After training i e in the minimum of Eq 2 48 the values for and J are re estimated and training restarts with the new performance function Regularization hyperparameters are computed in a sequence of 3 steps yt where Am m 1 M are positive eigenvalues of matrix H in Eq 2 49 v is the estimate of the effective number of parameters of a neural network v ad 2 51 TB 2 51 P v B 2E It should be noted that the algorithm Eqs 2 50 and 2 51 relies on numerous simplifications and assumptions which hold only approximately in typical real world problems 113 In the Bayesian formalism a trained network is described in terms of the posterior probability distribution of weight values The method typically assumes a simple Gaussian prior distribution of weights governed by an inverse variance hyperparameter 1 De If we present a new input vector to such a network then the distribution of weights gives rise to a distribution of network outputs There will be also an addend to the output distribution arising from the assumed g 1 Gaussian noise on the output variables noise y y x N 0 0 se 2 52 LS OPT Version 2 31 CHAPTER 2 OPTIMIZATION METHODOLOGY With these assumptions the negative log likelihood of network weights W given P training points x 1 x P is proportional to MSE Eq 2 46
167. ariables In the quest for accuracy increased hardware capacity has been consumed by greater modeling detail and therefore optimization methods have remained largely on the periphery of the area of mechanical design In lieu of formal methods designers have traditionally resorted to experience and intuition to improve designs This is seldom effective and also manually intensive Moreover design objectives are often in conflict making conventional methods difficult to apply and therefore more analysts are formalizing their design approach by using optimization 10 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 5 1 Approximating the response Response Surface Methodology or RSM requires the analysis of a predetermined set of designs A design surface is fitted to the response values using regression analysis Least squares approximations are commonly used for this purpose The response surfaces are then used to construct an approximate design subproblem which can be optimized The response surface method relies on the fact that the set of designs on which it is based is well chosen Randomly chosen designs may cause an inaccurate surface to be constructed or even prevent the ability to construct a surface at all Because simulations are often time consuming and may take days to run the overall efficiency of the design process relies heavily on the appropriate selection of a design set on which to base the approximations Fo
168. ariance of C p p f pl p f N N The Monte Carlo method can therefore become prohibitively expensive for computing events with small probabilities more so if you need to compare different designs The procedure can be sped up using Latin Hypercube sampling which is available in LS OPT These sampling techniques are described elsewhere in the LS OPT manual The experimental design will first be LS OPT Version 2 49 CHAPTER 3 PROBABILISTIC FUNDAMENTALS computed in a normalized uniformly distributed design space and then transformed to the distributions specified for the design variables Example The reliability of a structure is being evaluated The probability of failure is estimated to be 0 1 and must be computed to an accuracy of 0 01 with a 95 confidence The minimum number of function evaluations must be computed For an accuracy of 0 01 we use a confidence interval having a probability of containing the correct value of 0 95 The accuracy of 0 01 is taken as 4 5 standard deviations large using the Tchebysheff s theorem see appendix A which gives a standard deviation of 0 0022 The minimum number of sampling points is therefore N 24 De 18595 o 0 0022 Tchebysheff s theorem is quite conservative If we consider the response to be normally distributed then for an accuracy of 0 01 and a corresponding confidence interval having a probability of containing the correct value of 0 95 the a confidence interv
169. ars from the tar file during installation of LS OPT bin wrappers wrapper bin wrappers perl bin wrappers taurus bin runqueuer bin DynaMass The represents platform details e g wrapper hp or wrapper redhat72 The following instructions should be followed For all remote machines running LS DYNA 1 Create a directory on the remote machine for keeping all the executables including Isdyna a Copy the appropriate executable wrapper program located in the bin wrappers directory to this directory E g if you are running LS DYNA on HP place wrapper _hp on this machine Rename it to wrapper b Do the same for the appropriate perl_ program and rename it to perl c Do the same for the appropriate taurus_ program and rename it to taurus d Copy the DynaMass script to the same directory Local installation 2 Select the queuer option in LS OPTuwi or add a statement in the LS OPT command file to identify the queuing system e g Solver queuer 1sf or Solver queuer loadleveler Portable Batch System Registered Trademark of Veridian Systems Network Queuing Environment Registered Trademark of Cray Inc 76 LS OPT Version 2 CHAPTER 6 PROGRAM EXECUTION for each solver Example solver command rundyna hp DynaOpt inp single 970 solver input file care crash k solver queuer 1sf Change the script you use to run the solver via the queuing facility by prepending wrapper to the solver execu
170. ation In case 3a different approaches can be taken Firstly the user should try to identify the source of the noise E g when considering acceleration related responses was filtering performed Are sufficient significant digits available for the response in the extraction database Is mesh adaptivity used correctly Secondly if the noise cannot be attributed to a specific numerical source the process being modeled may be chaotic or random leading to a noisy response In this case the user could implement reliability based design optimization techniques as described in Section 2 15 5 Thirdly other less noisy but still relevant design responses could be considered as alternative objective or constraint functions in the formulation of the optimization problem In case 3b the subregion can be made smaller In most cases the source of discrepancy cannot be identified so in either case a further iteration would be required to determine whether the design can be improved Optimize the approximate subproblem The solution will be either in the interior or on the boundary of the subregion If the approximate solution is in the interior the solution may be good enough especially if it is close to the starting point It is recommended to analyze the optimum design to verify its accuracy If the accuracy of any of the functions in the current subproblem is poor another iteration is required with a reduced subregion size If the solut
171. ation The alternative formulation becomes Minimize e 2 65 subject to F F ee its lt T l y e i 1 p j l m i i i e 0 LS OPT Version 2 41 CHAPTER 2 OPTIMIZATION METHODOLOGY In the above equation T is a normalization factor e represents the constraint violation or target discrepancy and represents the strictness factor If 0 the constraint is slack or soft and will allow violation If amp 1 the constraint is strict or hard and will not allow violation of the constraint The effect of distinguishing between strict and soft constraints on the above problem is that the maximum violation of the soft constraints is minimized Because the user is seldom aware of the feasibility status of the design problem at the start of the investigation the solver will automatically solve the above problem first to find a feasible region If the solution to e is zero or within a small tolerance the problem has a feasible region and the solver will immediately continue to minimize the design objective using the feasible point as a starting point A few points are notable e The variable bounds of both the region of interest and the design space are always hard This is enforced to prevent extrapolation of the response surface and the occurrence of impossible designs e Soft constraints will always be strictly satisfied if a feasible design is possible e Ifa feasible design is not possible the most feasibl
172. ation a noise variable may be less at a high process temperature a control variable can be used to selected control variables for a more robust manufacturing process 47 CHAPTER 3 PROBABILISTIC FUNDAMENTALS 3 2 1 Variable linking A single design parameter can apply to several statistically independent components in a system for example one joint design may be applicable to several joints in the structure The components will then all follow the same distribution but the actual value of each component will differ Each duplicate component is in effect an additional variable and will result in additional computational cost contribute to the curse of dimensionality for techniques requiring an experimental design to build an approximation or requiring the derivative information such as FORM Direct Monte Carlo simulation on the other hand does not suffer from the curse of dimensionality but is expensive when evaluating events with a small probability Design variables can be linked to have the same expected nominal value but allowed to vary independently according to the statistical distribution during a probabilistic analysis One can therefore have one design variable associated with many probabilistic variables Three probabilistic associations between variables are possible e Their nominal values and distributions are the same e Their nominal values differ but they refer to the same distribution e Their nominal values are th
173. ation and implementation of derivatives with respect to the design variables in the simulation code Because of the complexity of this task analytical gradients also known as design sensitivities are mostly not readily available Numerical differentiation is typically based on forward difference methods that require the evaluation of n perturbed designs in addition to the current design This is simple to implement but is expensive and hazardous because of the presence of round off error As a result it is difficult to choose the size of the intervals of the design variables without risking spurious derivatives the interval is too small or inaccuracy the interval is too large Some discussion on the topic is presented in Reference 119 As a result gradient based methods are typically only used where the simulations provide smooth responses such as linear structural analysis and certain types of nonlinear analysis In non linear dynamic analysis such as the analysis of impact or metal forming the derivatives of the response functions are mostly severely discontinuous This is mainly due to the presence of friction and contact The response and therefore the sensitivities may also be highly nonlinear due to the chaotic nature of impact phenomena and therefore the gradients may not reveal much ofthe overall behavior Furthermore the accuracy of numerical sensitivity analysis may also be adversely affected by round off error Analytical sensiti
174. be done using the metamodels response surfaces neural networks or Kriging as prescribed by the user The number of function evaluations using the metamodels can be set by the user The default value is 10 The designs to be analyzed are chosen randomly respecting the distributions of the design variables The following data will be collected e Statistics such as the mean and standard deviation for all responses constraints and variables e The reliability information for each constraint o The number of times a specific constraint was violated during the simulation o The probability of violating the bounds and the confidence region of the probability Command file syntax analyze metamodel monte carlo Example analyze metamodel monte carlo 15 5 3 Accuracy of Metamodel Based Monte Carlo The number of function evaluations to be analyzed can be set by the user The default value is 10 Command file syntax set reliability resolution m Item Description m Number of sample values Example set reliability resolution 1000 LS OPT Version 2 163 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION 164 LS OPT Version 2 16 Optimization Algorithm Selection and Settings This chapter describes the parameter settings for the domain reduction and LFOPC methods that are used in LS OPT The default parameters for both the domain reduction scheme and the core optimization algorithm
175. ce Delete Clear Figure 7 1 Solver panel in LS OPTui Command file syntax solver software package identifier solver_name solver input file solver input file name solver command solver program name solver append file solver append file name interval Time_interval_between_progress_ reports lt 15 gt not available in LS OPTui The following software package identifiers are available own user defined solver dyna LS DYNA Versions prior to 960 dyna960 LS DYNA Version 960 970 LS OPT Version 2 81 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR 7 2 1 Interfacing with LS DYNA The first command demarcates the beginning of the solver environment Example Define the solver software to be used solver dyna960 SIDE IMPACT the data deck to be read by the solver solver input file ingrido the command to execute the solver solver command alpha6 2 usr 1ls dyna bin 1s970 single Extra commands to the solver solver append file ShellSetList More than one analysis case may be run using the same solver If a new solver is specified the data items not specified will assume previous data as default All commands assume the current solver Remarks e The name of the solver will be used as the name of the sub directory to the working directory e The command solver package identifier name initializes a new solver environment All subsequent commands up to the next solve
176. ced after every iteration of anormal optimization procedure 2 9 1 The confidence interval of the regression coefficients The 100 1 0 confidence interval for the regression coefficients b j 0 1 L is determined by the inequality Ab Ab b 7 lt p lt b 5 2 38 where Ab O 2t y P L 0 C 2 39 and 6 is an unbiased estimator of the variance o given by P A Pee DR P L P L 2 40 C y is the diagonal element of X TX corresponding to b and tan r is Student s 1 Distribution 100 1 amp therefore represents the level of confidence that b will be in the computed interval 2 9 2 The significance of a regression coefficient 5 The contribution of a single regressor variable to the model can also be investigated This is done by means of the partial F test where F is calculated to be 2 2 f er complete r Fe ER P gt L 2 41 where r 1 and the reduced model is the one in which the regressor variable in question has been removed Each of the terms represents the sum of squared residuals for the reduced and complete models respectively It turns out that the computation can be done without analyzing a reduced model by computing 2 b i Ci Eis P L 2 42 F can be compared with the F statistic Fa so that if F gt Fo1p1 A is non zero with 100 9 confidence The confidence level that is not zero can also be determined by computing the fo
177. cess for both a linear and a quadratic response surface approximation order 10 iterations are performed for the linear approximation with only 5 iterations performed for the more expensive quadratic approximation The modified statements in the input file are as follows Variable Area 2 Range Area 4 Variable Base 0 8 Range Base 1 6 EXPERIMENTAL DESIGN Order linear Number experiment 5 JOB INFO iterate 10 for the linear approximation and EXPERIMENTAL DESIGN Order quadratic LS OPT Version 2 183 CHAPTER 17 EXAMPLE PROBLEMS Number experiment 10 5 JOB INFO iterate 5 The final results of the two types of approximations are as follows Table 17 1 Summary of final results 2 bar truss Linear Quadratic Number of iterations 10 5 Number of simulations 51 51 Area 1 414 1 408 Base 0 3737 0 3845 Weight 1 51 1 509 Stress 0 9993 1 000 The optimization histories have been plotted to illustrate convergence in Figure 17 5 Optimization History For Variable Area Variable Area N 1 2 3 4 5 6 7 8 Number of Iterations 9 10 11 a Optimization history of Area Linear Variable Area I I I 4 I 1 1 4 I Li woe Optimization History For Variable Area Number of Iterations b Optimization history of Area Quadratic 184 LS OPT Version 2 EXAMPLE PROBLEMS CHAPTER 17
178. ch execution lsopt info Create a log file for licensing lsopt env Check the LSOPT environment setting viewer command file name Execute the graphical postprocessor The LSOPT environment is automatically set to the location of the 1sopt executable 6 3 Directory structure When conducting an analysis in which response evaluations are done for each of the design points a sub directory will automatically be created for each analysis 69 CHAPTER 6 PROGRAM EXECUTION Command file Output files Work directory Plot files Database files Solver 1 Solver 2 Simulation files m keme ee files 14 l4 211 3 1 4 1 5 c 2 al nal 2 Plot files e g FLD Run directories Figure 6 1 Directory structure in LS OPT These sub directories are named solver_name mmm nnnn where mmm represents the iteration number and nnnn is a number starting from 1 solver_ name represents the solver interface specified with the command e g solver dyna side impact In this case dyna is a reserved package name and side impact isa solver name chosen by the user The work directory needs to contain at least the command file and the template input files Various other files may be required such as a command file for a preprocessor An example of a sub directory name defined by LS OPT is side_impact 3 11 where 3 11 represents the design point number
179. cl0 4 0 9417 constraint accl15 4 lower bound constraint acc15_4 0 754247 upper bound constraint acc15_4 0 754247 constraint acc20 4 lower bound constraint acc20 4 1 1858 upper bound constraint acc20 4 1 1858 constraint acc25_4 lower bound constraint acc25 4 1 76239 upper bound constraint acc25_4 1 76239 constraint acc30_4 lower bound constraint acc30 4 2 21678 upper bound constraint acc30 4 2 21678 constraint acc35_4 lower bound constraint acc35 4 2 27923 upper bound constraint acc35 4 2 27923 constraint acc40_4 lower bound constraint acc40 4 1 82374 upper bound constraint acc40 4 1 82374 constraint acc45 4 lower bound constraint acc45 4 1 218 upper bound constraint acc45 4 1 218 constraint acc50_4 lower bound constraint acc50 4 0 727288 upper bound constraint acc50_4 0 727288 constraint vell0_4 lower bound constraint vell0_4 3 492 upper bound constraint vell0_4 3 492 constraint vell5_4 lower bound constraint vel15 4 3 19588 upper bound constraint vel15 4 3 19588 constraint vel20 4 lower bound constraint vel20 4 2 71324 upper bound constraint vel20 4 2 71324 constraint vel25 4 lower bound constraint vel25 4 1 9779 upper bound constraint vel25_4 1 9779 constraint vel30_4 lower bound constraint vel30_4 0 973047 upper bound constraint vel30 4 0 973047 constraint vel35 4 lower bound constraint vel35 4 0 169702 upper bound constraint vel35 4 0 169702
180. conflict and it is not always clear how to change the design to achieve the best compromise of these objectives A more systematic approach can be obtained by using an inverse process of first specifying the criteria and then computing the best design The procedure by which design criteria are incorporated as objectives and constraints into an optimization problem that is then solved is referred to as optimal design The state of computational methods and computer hardware has only recently advanced to the level where complex nonlinear problems can be analyzed routinely Many examples can be found in the simulation of impact problems and manufacturing processes The responses resulting from these time dependent processes are as a result of behavioral instability often highly sensitive to design changes Program logic as for instance encountered in parallel programming or adaptivity may cause spurious sensitivity Roundoff error may further aggravate these effects which if not properly addressed in an optimization method could obstruct the improvement of the design by way of corrupting the function gradients Among several methodologies available to address optimization in this design environment response surface methodology RSM a statistical method for constructing smooth approximations to functions in a multi dimensional space has achieved prominence in recent years Rather than relying on local information such as a gradient only RSM
181. d and used for comparison with the LSR formulation results 17 5 5 Results The result of the optimization is shown in Figure 17 27 Shown are the leakage curves for a 5 and 10 variable model It can be seen that the introduction of more points on the leakage curve allows more resolution in the low pressure range of the material model LS OPT Version 2 239 CHAPTER 17 EXAMPLE PROBLEMS 5e 07 4 5e 07 4e 07 3 5e 07 3e 07 2 5e 07 2e 07 1 5e 07 le 07 5e 08 F Leakage coefficient 225 250 Pressure kPa 5 variables Maximum violation sees u 10 variables LSR Starting values all cases Figure 17 27 Leakage curve Airbag material identification As an example the matching between the experimental and simulated acceleration velocity and displacement is depicted for both 4 and 5m s chest form velocities in Figure 17 28 for the 10 variable LSR Formulation case It can be seen that the displacement and velocity curves are closely matched by the optimum curve parameters and that the discrepancy as exhibited by the residual is mainly due to the acceleration curve not being matched exactly 0 r r 10 Optimum Experiment m r Baseline 177 v Q k amp By a Time ar Baseline eeren Optimum Experiment m at gt ms gt Time 50 Displacement Velocity Baseline Pereeeeeetetts Optimum Experiment m Time 50
182. d in the LS OPT command file 7 3 4 User defined preprocessor In its simplest form the prepro own preprocessor can be used in combination with the design point file XPoint to read the design variables from the run directory Only the prepro command statement will therefore be used and no input file prepro input file will be specified The user defined prepro command will be executed with the standard preprocessor input file UserPreproOpt inp appended to the command The UserPreproOpt inp file is generated after performing the substitutions intheprepro input file specified by the user Example prepro own prepro command gambit r1 3 id casefile in prepro input file setup jou The executed command is gambit r1 3 id casefile in setup jou Alternatively a script can be executed with the prepro command to perform any number of command line commands that result in the generation of a file called UserOpt inp for use by an own solver or DynaOpt inp for use by LS DYNA LS OPT Version 2 87 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR 88 LS OPT Version 2 8 Design Variables Constants and Dependents This chapter describes the definition of the input variables constants and dependents design space and the initial subregion All the items in this chapter are specified in the Variables panel in LS OPTui Figure 8 1 Shown is a multidisciplinary design optimization MDO case where not all
183. d7 k prepro truegrid 0 3 prepro command cp curves cp node cp bar tg prepro input file s7 tg DESIGN FUNCTIONS FOR SOLVER 1 elem sog cp elem response L Knee Force 0 000153846 0 DynaASCII rcforc R_FORCE 1 MAX SAE 60 0 270 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS response R Knee Force 0 000153846 0 DynaASCII rcforc R_FORCE 2 MAX SAE 60 0 response L Knee Disp 0 00869565 0 DynaASCII Nodout R_DISP 24897 MAX response R Knee Disp 0 00869565 0 DynaASCII Nodout R_DISP 25337 MAX response Yoke Disp 0 0117647 0 DynaASCII Nodout R_DISP 28816 MAX response Kinetic_Energy 6 49351le 06 0 DynaASCII glstat K ENER 0 TIMESTEP response Mass 638 162 0 DynaMass 7 8 48 62 MASS DUMMY OBJECTIVE FUNCTION objectives 1 objective Mass response Mass 1 CONSTRAINT DEFINITIONS constraints 6 constraint L Knee Force response L Knee Force upper bound constraint L Knee Force 0 5 constraint R Knee Force response R Knee Force upper bound constraint R Knee Force 0 5 constraint L Knee Disp response L Knee Disp strict upper bound constraint L Knee Disp 1 constraint R Knee Disp response R Knee Disp upper bound constraint R Knee Disp 1 constraint Yoke Disp response Yoke Disp upper bound constraint Yoke Disp 1 constraint Kinetic Energy upper bound constraint response Kinetic_Energy Kinetic
184. damping energy System damping energy energy ratio wo_eroded Energy ratio w o eroded eroded_internal_energy Eroded internal energy eroded _ kinetic energy Eroded kinetic energy 308 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Joint Element Forces JNTFORC DynaASCH Keyword Binout Component Description Binout subdirectory joints X_FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT R_ FORCE R_ MOMENT x_ force y_force z_force x_moment y_moment z moment resultant_force resultant moment X force Y force Z force X moment Y moment Z moment R force R moment Binout subdirectory typed d phi _dt d psi dt d theta dt joint_energy phi_degrees phi_moment_damping phi moment stiffness phi moment total psi_degrees psi moment damping psi moment stiffness psi moment total theta_degrees theta moment damping theta moment _ stiffness theta moment total d phi dt d psi dt degrees d theta dt degrees joint energy phi degrees phi moment damping phi moment stiffness phi moment total psi degrees psi moment damping psi moment stiffness psi moment total theta degrees theta moment damping theta moment stiffness theta moment total LS OPT Version 2 309 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Material Summary MATSUM DynaASCH Binout Description Keyword Component K_ENE
185. dard interfaces that are used to extract any particular data item from the database specialized responses for metal forming are also available The computation and extraction of these secondary responses are discussed in Section The user must ensure that the LS DYNA program will provide the output files required by LS OPT LS OPT Version 2 115 CHAPTER 10 HISTORY AND RESPONSE RESULTS As multiple result output sets are generated during a parallel run the user must be careful not to generate unnecessary output The following rules should be considered To save space only those output files that are absolutely necessary should be requested A significant amount of disk space can be saved by judiciously specifying the time interval between outputs DT E g in many cases only the output at the final event time may be required In this case the value of DT can be set slightly smaller than the termination time The result extraction is done immediately after completion of each simulation run Database files can be deleted immediately after extraction ifrequested by the user clean file see also Section 6 10 If the simulation runs are executed on remote nodes the responses of each simulation are extracted on the remote node and transferred to the local run directory For more specialized responses the Perl programs provided can be used as templates for the development of own routines All the utilities can be specified through the
186. dditional files are generated by LS DYNA and placed in the run directories to perform the scalar product and extract the modal frequency and number 5 mode_original cannot exceed 999 10 6 3 Response history This is a generic interface for the extraction of simulation response histories from ASCII data files Any variable available in any ASCII data file can be extracted as either a response or a history Command file syntax DynaASCII res type cmp g u id pos time att t1 t2 filter att n Table 10 5 DynaASCII database description and defaults Item Description Default Remarks res_type Name of ASCII result file 1 cmp Component of result 1 g Gravitational acceleration 2 u Time units 1 seconds 2 milliseconds 2 id ID number of entity pos elout Through thickness shell position at which 3 stress strain is evaluated element output time_att Time attribute MAX MIN AVE TIMESTEP tl Lower time limit AVE MIN MAX time 0 AVE MIN MAX 4 TIMESTEP tmax TIMESTEP t2 Upper time limit AVE MIN MAX bna 4 filter_att Filtering attribute SAE BUTT AVER SAE 5 n Frequency SAE BUTT Number of points 60 cycles time unit AVER SAE BUTT 5 points AVER Remarks The history and response syntax is the same except that for a history the time attribute is always TIMESTEP 1 See Appendix A The res_type is allowed to have the valid keyword as a
187. directly using the standard LS DYNA interface see Section or a user defined interface Each extracted response is identified by a name and the command line for the program that extracts the results The command line must be enclosed in double quotes If scaling and or offsetting of the response is required the final response is computed as the extracted response X scale factor offset This operation can also be achieved with a simple mathematical expression A mathematical expression for a response is defined in curly brackets after the response name Command file syntax response response name scale factor offset string response response name expression math expression Example response Displacement x 25 4 0 0 DynaASCII nodout r disp 63 TIMESTEP 0 1 response Force SHOME ownbin calculate force response Displacement_y calc constraint2 response Disp expression Displacement_x Displacement_y Remarks 1 The first command will use a standard interface for the specified solver package The standard interfaces for LS DYNA are described in Section 2 The middle two commands are used for a user supplied interface program see Section 10 13 The interface name must either be in the path or the full path name must be specified Aliases are not allowed 3 For the last command the second argument expression is a reserved name 10 3 Specifying the metamodel type The metamodel type can be
188. e composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite 100 100 100 100 100 00 100 100 100 IH pa po Residual response disp40 4 38 6093 scale 100 Residual response disp45 4 46 6906 scale 100 Residual response disp50 4 57 8468 scale 100 Residual response acc10 5 72 3756 scale Residual response acc15 5 97 7004 scale Residual response acc20 5 168 931 scale Residual response acc25 5 264 071 scale Residual response acc30 5 311 405 scale Residual response acc35 5 244 28 scale 1 Residual response acc40 5 130 942 scale Residual response acc45_5 51 7805 scale Residual response acc50_ 5 12 3231 scale Residual response vell0_5 4 705 scale 1 Residual response vell5_5 4 24697 scale Residual response vel20 5 3 59243 scale Residual response vel25_5 2 51584 scale Residual response vel30_5 1 03072 scale 1 Residual response vel35_5 0 393792 scale 1 Residual response vel40_5 1 33087 scale Residual response vel45 5 1 76819 scale Residual response vel50_5 1 91663 scale Residual response disp10 5 91 7493 scale R
189. e 01 1 6000000000000001e 00 2 79 Weight get_wt 0 0 1 0000000000000000e 30 1 0000000000000000e 30 1 7313148666666667e 01 9 0171633333333390e 01 8 4697225964912348e 01 1 5711848567878486e 16 5 8787263157894765e 01 4 9821866666666648e 01 AA lee E i 0 79 Stress get _ str 0 0 1 0000000000000000e 30 1 0000000000000000e 30 1 2313574304761465e 01 7 3525990078485721e 00 4 3560129742690190e 00 1 1417587257617741e 00 1 8766964912280690e 01 2 1619635555555621e 00 1 1 1 1 1 320 LS OPT Version 2 APPENDIX D DATABASE FILES 0 The flags for active coefficients exclude the constant ap The OptimizationHistory file This file is used to save the optimization history results and appears in the work directory Each line contains the values at the optimum point of an iteration Entities Count Objective values Number of objectives Variables Number of variables Variable lower bounds Number of variables Variable upper bounds Number of variables RMS errors Number of responses Average errors Number of responses Maximum errors Number of responses R errors Number of responses Adjusted R errors Number of responses PRESS errors Number of responses Prediction R Number of responses Maximum prediction error Number of responses Responses Number of responses Multi objective 1 Constraint values Number of constr
190. e 07 2e 07 Variable Leakage_5 0 2 4 6 8 10 12 14 16 18 20 Iteration Number Figure 17 30 Optimization history Leakage_5 Formulation 1 Maximum violation 5 variables Airbag material identification 242 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 6 Small car crash and NVH MDO 5 variables This example has the following features LS DYNA is used for both explicit crash and implicit NVH simulations Variable screening is performed Multidisciplinary design optimization MDO is illustrated with a simple example Mode tracking is used The standard LS DYNA interface is used to extract the results 17 6 1 Parameterization and Variable screening To illustrate a relatively simple example of multidisciplinary design optimization MDO the small car of Section is extended to have five variables In addition one Noise Vibration and Harshness NVH parameter is considered as a constraint i e the first torsional vibrational mode frequency Figure 17 31 shows the modified small car Rails are added and the combined bumper hood section is separated into a grill hood and bumper The mass of the affected components in the initial design is 1 328 units while the torsional mode frequency is 2 281Hz This corresponds to mode number 15 The Head Injury Criterion HIC based on a 15ms interval is initially 17500 The initial intrusion of the bumper is 531 mm Hood thickness Grill thickness Roof
191. e computed If feasibility must be compromised there is no feasible design the solver will automatically use the slackness of the soft constraints to try and achieve feasibility of the hard constraints However there is always a possibility that hard constraints must still be violated even when allowing soft constraints In this case the variable bounds may be violated which is highly undesirable as the solution will lie beyond the region of interest and perhaps beyond the design space This could cause extrapolation of the response surface or worse a future attempt to analyze a design which is not analyzable e g a sizing variable might have become zero or negative Soft and strict constraints can also be specified for search methods If there are feasible designs with respect to hard constraints but none with respect to all the constraints including soft constraints the most feasible design will be selected If there are no feasible designs with respect to hard constraints the problem is hard infeasible and the optimization terminates with an error message LS OPT Version 2 139 CHAPTER 11 OBJECTIVES AND CONSTRAINTS 140 LS OPT Version 2 12 Running the Optimization Problem This chapter explains simulation job related information and how to start an optimization run from the graphical user interface The optimization process is triggered by the iterate command in the input file or by the Run command in the Run
192. e design formulae rather than a specific design If this can be achieved and the proper design parameters have been used the design remains flexible and changes can still be made at a late stage before verification of the final design This also allows multidisciplinary design to proceed with a smaller risk of having to repeat simulations As designers are moving towards computational prototyping and as parallel computers or network computing are becoming more commonplace the paradigm of design exploration is becoming more important Response surface methods can thus be used for global exploration in a parallel computational setting For instance interactive trade off studies can be conducted e Global optimization Response surfaces have a tendency to capture globally optimal regions because of their smoothness and global approximation properties Local minima caused by noisy response are thus avoided 2 5 4 Other types of response surfaces Neural network and Kriging approximations can also be used as response surfaces and are discussed in Sections 2 10 1 and 2 10 2 12 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 6 Experimental design Experimental design is the selection procedure for finding the points in the design space that must be analyzed Many different types are available 43 The factorial Koshal composite D optimal and Latin Hypercube designs are detailed here 2 6 1 Factorial design This is an Z
193. e design will be computed e If feasibility must be compromised there is no feasible design the solver will automatically use the slackness of the soft constraints to try and achieve feasibility of the hard constraints However even when allowing soft constraints there is always a possibility that some hard constraints must still be violated In this case the variable bounds could be violated which is highly undesirable as the solution will lie beyond the region of interest and perhaps beyond the design space If the design is reasonable the optimizer remains robust and finds such a compromise solution without terminating or resorting to any specialized procedure Soft and strict constraints can also be specified for search methods If there are feasible designs with respect to hard constraints but none with respect to all the constraints including soft constraints the most feasible design will be selected If there are no feasible designs with respect to hard constraints the problem is hard infeasible and the optimization terminates with an error message In the following cases the use of the Min Max formulation can be considered 1 Minimize the maximum of several responses e g minimize the maximum knee force in a vehicle occupant simulation problem This is specified by setting both the knee force constraints to have zero upper bounds The violation then becomes the actual knee force 2 Minimize the maximum design variable e
194. e formula pani e 2 10 Au T Xi is used to transform each variable x to a normalized variable When using LS OPT to minimize maximum violations the responses must be normalized by the user This method is chosen to give the user the freedom in selecting the importance of different responses when e g performing parameter identification Section 2 15 3 will present this application in more detail 2 5 Response Surface Methodology An authoritative text on Response Surface Methodology defines the method as a collection of statistical and mathematical techniques for developing improving and optimizing processes Although an established statistical method for several decades 110 it has only recently been actively applied to mechanical design 167 Due to the importance of weight as a criterion and the multidisciplinary nature of aerospace design the application of optimization and RSM to design had its early beginnings in the aerospace industry A large body of pioneering work on RSM was conducted in this and other mechanical design areas during the eighties and nineties 68 RSM can be categorized as a Metamodeling technique see Section for other Metamodeling techniques namely Neural Networks and Kriging available in LS OPT Although inherently simple the application of response surface methods to mechanical design has been inhibited by the high cost of simulation and the large number of analyses required for many design v
195. e if an iterative scheme is used An example is given in Section 17 2 9 to illustrate the method Care must be taken in interpreting the resulting reliability of the responses Accuracy can be especially poor at the tail ends of the response distribution For a very conservative estimate of the failure Tchebysheff s Theorem can be considered 2 PY weko i or equivalently P y ul lt ko gt 1 7 with Y the stochastic response with mean 4 and variance 0 and k a positive constant The Theorem says that the probability that the outcome lies outside k standard deviations of the mean is less than Va Conversely it also says that the probability that the outcome lies within k standard deviations of the mean is at least 1 Va The relationship is valid for any probability distribution Note that both tails of the distribution are considered in the above The choice of a uniform distribution for the design variable also contributes to the error in the estimated reliability The analyst can adjust the distribution parameters lower and upper bounds for a more conservative estimate of failure 46 LS OPT Version 2 3 Probabilistic Fundamentals 3 1 Introduction No system will be manufactured and operated exactly as designed Adverse combinations of design and loading variation may lead to undesirable behavior or failure therefore if significant variation exists a probabilistic evaluation may be desirable Sources of varia
196. e parameters of a monotonic load curve which in some of the parameter sets proposed by the experimental design may be non monotonic This may cause unexpected behavior and possible failure of the simulation process This is almost always an indication that the design formulation is non robust In most cases poor design formulations can be eliminated by providing suitable constraints to the problem and using these to limit future experimental designs to a reasonable design space see Section 2 7 e Impossible designs The set of impossible designs represents a hole in the design space A simple example is a two bar truss structure with each of the truss members being assigned a length parameter An impossible design occurs when the design variables are such that the sum of the lengths becomes smaller than the base measurement and the truss becomes unassemblable It can also occur if the design space is violated resulting in unreasonable variables such as non positive sizes of members or angles outside the range of operability In complex structures it may be difficult to formulate explicit bounds of impossible regions or holes The difference between a non robust design and an impossible one is that the non robust design may show unexpected behavior causing the run to be aborted while the impossible design cannot be synthesized at all Impossible designs are common in mechanism design 4 5 Advanced methods for design optimization
197. e refers to a region of interest that in addition to having specified bounds on the variables is also bounded by arbitrary constraints This may result in an irregular shape of the design space Therefore once the first approximation has been established all the designs will be contained in the new region of interest This region of interest is thus defined by approximate constraint bounds and by variable bounds The purpose of an irregular design space is to avoid designs which may be impossible to analyze The move stay commands can be used to define an environment in which the constraint bound commands Section can be used to double as bounds for the reasonable design space If a reasonable experimental design is required from the start a DesignFunctions PRE solver name file can be provided by the user This is however not necessary if explicit constraints i e constraints that do not require simulations are specified for the reasonable design space An explicit constraint may be a simple relationship between the design variables LS OPT Version 2 101 CHAPTER 9 METAMODELS AND POINT SELECTION The move start option moves the designs to the starting point instead of the center point see Section 2 7 This option removes the requirement for having the center point inside the reasonable design space Command file syntax move stay move start Example 1 SET THE BOUNDS ON THE REASONABLE DESIGN SPACE Lower bound constraint
198. e same but their distributions differ 3 3 Probabilistic Methods The reliability the probability of exceeding a constraint value can be computed using probabilistic methods The current version of LS OPT provides only Monte Carlo evaluation of using approximations Methods considering the Most Probable Point MPP of failure will be included in future versions of LS OPT The accuracy can be limited by the accuracy of the data used in the computations as well as the accuracy of the simulation The choice of methods depends on the desired accuracy and use of the reliability information 3 3 1 Monte Carlo Analysis In a Monte Carlo analysis we approximate the nominal value of a response using the mean of a number of computer experiments The values of the random variables are selected considering their probability density function Under the law of large numbers the solution will eventually converge Applications of a Monte Carlo investigation are e Compute the distribution of the responses in particular the mean and standard deviation e Compute reliability e Investigate design space search for outliers The approximation to the nominal value is ASO LL SX 48 LS OPT Version 2 CHAPTER 3 PROBABILISTIC FUNDAMENTALS If the X are independent then the laws of large numbers allow us any degree of accuracy by increasing N The error of estimating the nominal value is a random variable with standard deviation oO
199. e terms point selection and experimental design are used interchangeably 9 1 Metamodel definition The user can select from three metamodel types in LS OPT The standard and default selection is the polynomial response surface method RSM where response surfaces are fitted to results at data points using polynomials For global approximations neural network or Kriging approximations are available Sensitivity data analytical or numerical can also be used for optimization This method is more suitable for linear analysis solvers Command file syntax Solver order linear interaction elliptic quadratic FF kriging The linear interaction linear with interaction effects elliptic and quadratic options are for polynomials FF represents the Feedforward Neural network 9 1 1 Response Surface Methodology When polynomial response surfaces are constructed the user can select from different approximation orders The available options are linear linear with interaction elliptic and quadratic Increasing the order of the polynomial results in more terms in the polynomial and therefore more coefficients In LSOPT i the approximation order is set in the Order field See The polynomial terms can be used during the variable screening process see Section to determine the significance of certain variables main effects and the cross influence interaction effects between variables when determining responses These results can be viewed graphica
200. e the output Activation function of intermediate hidden layers is generally a sigmoidal function Figure 2 3 while network input and output layers are usually linear transparent In theory such networks can model functions of almost arbitrary complexity see 73 All of the parameters in a feed forward network are usually determined at the same time as part of a single non linear optimization strategy based on the standard gradient algorithms the steepest descent RPROP Levenberg Marquardt etc The gradient information is typically obtained using a technique called backpropagation which is known to be computationally effective 51 For feed forward networks regularization may be done by controlling the number of network weights model selection by imposing penalties on the weights ridge regression or by various combinations of these strategies Model adequacy checking Nature is rarely if ever perfectly predictable Real data never exactly fit the model that is being used One must take into consideration that the prediction errors not only come from the variance error due to the intrinsic noise and unreliability in the measurement of the dependent variables but also from the systematic bias error due to model miss specification According to George E P Box s famous maxim all models are wrong some are useful To be genuinely useful a fitting procedure should provide the means to assess whether or not the m
201. ecting All Previous or All The All selection shows the final verification point in green see Figure 13 1 LS OPT Version 2 145 CHAPTER 13 VIEWING RESULTS R_Knee_ Force 1 i Results of All Iterations File For Response Function Response Surface Accuracy Type of Plot Sees ses sees ieee as gt Dee L en _ a 14 __ 135 12 nfeA asuodsay paynduroy R Knee Force xy Optimization History xy Tradeoff v ANOVA B 5 E Response Value to Plot Iterations Sorte sera bettas It eM IC ed pitted Nia felt TIC Ian A tr IRIRE 1 UB Sie 105 11 115 12 125 13 135 14 145 15 155 16 165 17 1 0 9 0 95 Predicted Response Value Figure 13 1 Computed vs Predicted plot in View panel in LS OPTui history ion imizat 13 2 Opt on the Optimization History button Figure 13 2 For the variables the upper and lower bounds subregion are also displayed For all the dependents responses objectives constraints and maximum violation a black point of each iteration For the error parameters only one solid red line of the optimization history is plotted RMS Maximum and R error indicators are available By clicking on any of the red squares the data of the selected design point is listed For LS DYNA results solid line indicates the predicted values while the red squares represent the computed values at the starti
202. ectives The curve can be plotted with any of the variables responses composites constraints or objectives on either of the two axes Care should be taken when selecting e g a certain constraint for plotting as it may also be either a response or composite and that this value maybe different from the constraint value depending on whether the constraint is active during the trade off process The example in the picture below has Constraint Intrusion selected for the X Axis Entity and not Composite Intrusion LS OPT Version 2 147 CHAPTER 13 VIEWING RESULTS An example of trade off is given in Section and K viewer File Tradeoff Plot Constraint Intrusion vs Objective HIC Type of Plot v Response Surface Accuracy w Optimization History Tradeoff X Axis Entity Response Intru_1 Composite Intrusion f r 1 aiL Objective HIC Y Axis Entity Composite Intrusion Constraint Intrusion a A Multi Objective Constraints w Objectives Tradeoff Constraint Intrusion 1 Bound to Vary Upper Min 450 Max 600 Reset Reset All No of Pts 2 Generate z Ran 9 ae a D D 2 eo 53 54 Constraint Intrusion eto Figure 13 3 Trade off plot in View panel in LS OPTui 13 4 Variable screening The Analysis of Variance ANOVA refer to Section of the approximation to the experimental design is automatically performed
203. ectory 78 LS OPT Version 2 7 Interfacing to a solver or preprocessor This chapter describes how to interface LS OPT with a simulation package and or a parametric preprocessor Standard interfaces as well as interfaces for user defined executables are discussed 7 1 Identifying design variables in a solver and preprocessor In addition to the existing LS OPT features routines facilitating the use and execution of the preprocessor and Finite Element solver have been incorporated into the program The design variables must be identified using the keywords lt lt expression gt gt in the input file where expression is an expression which incorporates constants design variables or dependents Inserting the relevant design variable or expression into the preprocessor command file requires that a preprocessor command such as create fillet radius 5 0 line 77 line 89 be replaced with create fillet radius lt lt Radius gt gt line 77 line 89 where the design variable named Radius is the radius of the fillet Similarly if the design variables are to be specified using a Finite Element input deck then data lines such as SECTION SHELL 1 10 3 000 0 002 0 002 0 002 0 002 can be replaced with SECTION SHELL 1 10 3 000 lt lt Thickness 3 gt gt lt lt Thickness 3 gt gt lt lt Thickness 3 gt gt lt lt Thickness 3 gt gt 79 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR to make the shell thickness
204. ed The functions g and h are constraint functions which represent the design restrictions The variables collectively described by the vector x are often referred to as design variables or design parameters The two sets of functions g and h define the constraints of the problem The equality constraints do not appear in any further formulations presented here because algorithmically each equality constraint can be represented by two inequality constraints in which the upper and lower bounds are set to the same number e g h x 0 0 lt h x lt 0 2 2 Equations 2 1 then become min f x 2 3 subject to g x lt 0 j 1 2 m LS OPT Version 2 7 CHAPTER 2 OPTIMIZATION METHODOLOGY The necessary conditions for the solution x to Eq 2 3 are the Karush Kuhn Tucker optimality conditions Vf x A V g x 0 2 4 A glx 0 a x lt 0 A 0 These conditions are derived by differentiating the Lagrangian function of the constrained minimization problem L x f x A g x 2 5 and applying the conditions V fax 20 optimality 2 6 and V gox lt 0 feasibility 2 7 to a perturbation ox A are the Lagrange multipliers which may be nonzero only if the corresponding constraint is active i e g x 0 For x to be a local constrained minimum the Hessian of the Lagrangian function V f x A v z x on the subspace tangent to the active constraint g must be positive definite at x These
205. el with poor prediction capabilities 2 8 7 R for Prediction For the purpose of prediction accuracy the R indicator has been devised hol prediction PRE R 1 gt 2 36 S Iy where P 2 u 5 i l Syp y ar 2 37 R oiia represents the ability of the model to detect the variability in predicting new responses ksl 2 8 8 Iterative design and prediction accuracy In an iterative scheme with a shrinking region the R value tends to be small at the beginning then approaches unity as the region of interest shrinks thereby improving the modeling ability It may then reduce again as the noise starts to dominate in a small region causing the variability to become indistinguishable In the same progression the prediction error will diminish as the modeling error fades but will stabilize at above zero as the modeling error is replaced by the random error noise 2 9 ANOVA Since the number of regression coefficients determines the number of simulation runs it is important to remove those coefficients or variables which have small contributions to the design model This can be done by doing a preliminary study involving a design of experiments and regression analysis The statistical results are used in an analysis of variance ANOVA to rank the variables for screening purposes The LS OPT Version 2 23 CHAPTER 2 OPTIMIZATION METHODOLOGY procedure requires a single iteration using polynomial regression but results are produ
206. ements of Structural Optimization Kluwer 1992 Hajela P Berke L Neurobiological computational models in structural analysis and design Proceedings of the 31st AIAA ASME ASCE AHS ASC Structures Structural Dynamics and Materials Conference Long Beach CA April 1990 Hock W Schittkowski K Test examples for nonlinear programming codes Springer Verlag Berlin Germany 1981 Hornik K Stinchcombe M White H Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks Neural Networks 3 pp 535 549 1990 Jin R Chen W and Simpson T W Comparative studies of metamodeling techniques under multiple modeling criteria AIAA Paper AIAA 2000 4801 Kaufman M D Balabanov V Burgee S L Giunta A A Grossman B Haftka R T Mason W H Watson L T Variable complexity response surface approximations for wing structural weight in HSCT design Computational Mechanics 18 pp 112 126 1996 Kirkpatrick S Gelatt C D Vecchi M P Science 220 pp 671 680 1983 Knuth D E Seminumerical Algorithms 2nd ed Vol 2 of The Art of Computer Programming Reading MA Addison Wesley 3 2 3 3 1981 Kok S Stander N Optimization of a Sheet Metal Forming Process using Successive Multipoint Approximations Structural Optimization 18 4 pp 277 295 1999 Kok S Stander N Roux W J Thermal optimization in transient thermoelasticity using response surface approximation
207. en units and sometimes the number of network hidden layers Most straightforward is to search for an optimal network architecture that minimizes MSEgcy MSErpg or MSEcyx Often it is feasible to loop over 1 2 hidden units and finally select the network with the smallest GCV error In any event in order for the GCV measure to be applicable the number of training points P should not be too small compared to the required network size M Over fitting To prevent over fitting it is always desirable to find neural solutions with the smallest number of parameters In practice however networks with a very parsimonious number of weights are often hard to train The addition of extra parameters i e degrees of freedom can aid convergence and decrease the chance of becoming stuck in local minima or on plateaus B2 Weight decay regularization involves modifying the performance function F which is normally chosen to be the mean sum of squares of the network errors on the training set Eq 2 43 When minimizing MSE Eq 2 43 the weight estimates tend to be exaggerated We can impose a penalty for this tendency by adding a term that consists of the sum of squares of the network weights see also Eq 2 43 F BE 0Ey 2 48 where gt P in 2 Bae I G 2 M 2 aes Wn Ey 5 gt where M is the number of weights and P the number of points in the training set Notice that network biases are usually excluded from the
208. ents iter AnalysisResults iter DesignFunctions iter Net funcname iter in the case of Neural Nets or Kriging and Optimumresults iter are used for restarting see Section 6 7hnd Appendix D 1 In each phase LS OPT will therefore first try to read these files before looking e g for results of individual runs in the run directories This feature has become necessary to accommodate restarting when using random processes such as Latin Hypercube Sampling Section P 6 5 Space Filling Designs Section 2 6 6 and Neural p 10 1 Nets Section P 10 1 These experimental designs approximations will differ each time a new optimization is started and must therefore rely on the database for repeatability Note Starting with Version 2 1 the user must delete these files when attempting a clean start 6 7 Output files The following files are intermediate database files containing ASCII data Table 6 1 Intermediate ASCII database files Database file Description Directory Trial designs computed as a result of the Experiments a Solver experimental design The same trial designs and the responses Anal R lt lver sen extracted from the solver database Pee DesignFunctions Parameters of the approximate functions Solver i hist f Gueinicat souwickors Variable response and error history o Work the successive approximation process All variabl t ExtendedResults yarianles Men and ve ended Solver results at each trial desig
209. er a design is optimal or not Most global optimization methods require large numbers of function evaluations simulations In LS OPT global optimality is treated on the level of the approximate subproblem through a multi start method originating at all the experimental design points Noise Although noise may evince the same problems as global optimality the term refers more to a high frequency randomly jagged response than an undulating one This may be largely due to numerical 59 CHAPTER 4 DESIGN OPTIMIZATION PROCESS round off and or chaotic behavior Even though the application of analytical or semi analytical design sensitivities for noisy problems is currently an active research subject suitable gradient based optimization methods which can be applied to impact and metal forming problems are not likely to be forthcoming This is largely because of the continuity requirements of optimization algorithms and the increased expense of the sensitivity analysis Although fewer function evaluations are required analytical sensitivity analysis is costly to implement and probably even more costly to parallelize e Non robust designs Because RSM is a global approximation method the experimental design may contain designs in the remote corners of the region of interest which are prone to failure during simulation aside from the fact that the designer may not be remotely interested in these designs An example is the identification of th
210. er concurrent jobs 1 OBJECTIVE FUNCTIONS objectives 1 objective Vehicle Mass _crash 1 CONSTRAINT DEFINITIONS constraints 5 constraint Disp scaled lower bound constraint Disp_scaled 1 upper bound constraint Disp_scaled 1 constraint StagelPulse_scaled lower bound constraint StagelPulse _scaled 1 constraint Stage2Pulse scaled lower bound constraint Stage2Pulse_scaled 1 constraint Stage3Pulse_ scaled lower bound constraint Stage3Pulse _scaled 1 constraint Frequency scaled lower bound constraint Frequency_scaled 0 98710 upper bound constraint Frequency_scaled 1 01290 MULTIDISCIPLINARY JOB INFO iterate param design 0 01 iterate param objective 0 01 iterate 10 STOP 17 7 4 Simulation results The deceleration versus displacement curves of the baseline crash model and Iteration 6 design are shown in Figure 17 39 for the partially shared variable case The stage pulses as calculated by Equation 4 are also shown with the optimum values only differing slightly from the baseline The reduction in displacement at the end of the curve shows that there is spring back or rebound at the end of the simulation 17 7 5 Optimization history results The bounds on the design variables are given in Table 17 4 together with the different initial designs or starting locations used Starting design 1 corresponds to the baseline model as shown in through while the other two designs correspond to the
211. erefore be minimized with respect to the controlled variables and maximized with respect to the uncontrollable variables This requires a special flag in the optimization algorithm and the formulation of Equation 2 1 becomes min max f y z ye RN zE RI 2 68 y z subject to g y z lt 0 5 FSL at The algorithm remains a minimization algorithm but with modified gradients mod __ V Vy Vr Vz For a maximization problem the min and max are switched 3 The dependent set the subset of y and z that are dependent on each other x y z must be defined as input for each simulation e g if the manufacturing tolerance on a thickness is specified as the uncontrollable component it is defined as a variation added to a mean value i e t tmean deviation where t is the dependent variable 2 15 5 Reliability based design optimization LS OPT allows a limited reliability based design capability by computing the standard deviation of any response The procedure uses the multidisciplinary optimization capability to separate the mean and random responses components The procedure is as follows 1 Choose the design variables and divide them into pairs where each pair is represented by a mean component and a random component Uncontrollable random variables must be included in the list 2 Assign the mean variables to a solver MEAN and the random variables to a solver RANDOM 3 Change those variables defined in the MEAN solver and tha
212. ersatile it has the ability to take on various shapes The probability density function is skewed to the right especially for low values of the shape parameter 14 Shape 0 25 Scale 1 0 12 l fix iza arn 0s I By KA Shape 2 0 06 F Ss hape 1 0 A ry h ma 04 5 Figure 15 5 Weibull distribution 158 LS OPT Version 2 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION Command file syntax distribution name WEIBULLOG scale shape Item Description name Distribution name scale Scale parameter shape Shape parameter Example distribution wDist WEIBULL 2 3 3 1 15 4 Probabilistic Variables A probabilistic variable has a mean or nominal value a variation around this nominal value according to a statistical distribution an optional upper bound and an optional lower bound Lower Bound Nominal Value Upper Bound Probability Variable Value Figure 15 6 Probabilistic Variable A distinction is made between control and noise variables e Control variables Variables that can be controlled in the design analysis and production level for example a shell thickness It can therefore be assigned a nominal value and will have a variation around this nominal value The nominal value can be adjusted during the design phase in order to have a more suitable design LS OPT Version 2 159 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION
213. erse33 expression Lookup hisl t his2 t 2 75 response MaxI expression max Inversell Inverse2l response MinT expression min Inversell Inverse2l response hist expression his3 Inverse31 response hist66 expression his3 66 1 0 1 response nhist66 expression nint hist66 response ihist66 expression int hist66 response Integll expression Integral hisl t response Integl14 expression Integral his1 t 11 ul t response Integl5 expression Integral his1 t 11 UPPER t response Integ22 expression Integral his2 t 11 ul t response Integ32 expression Integral his3 t 11 ul t response Integ33 expression Integral hisl t his2 t 2 11 ul t response Integ34 expression Integral his3 t response Integ35 expression Integral his3 t 11 response Integ36 expression Integral his3 t 11 ul 5 Cross functional integrals 5 response Integ2 expression Integral hisl t 11 ul on response Integ3a expression Integral his1 t 0 30 his2 t response Integ3b expression Integral hist 30 100 hie2 t 4 response Integ4 expression Integl Integ2 response Integ5 expression Integral sin t hisl t his2 t 11 ul t response Integ7 expression Integral sin t hisl t his2 t response Velocityl expressio
214. es However linear experimental designs can be easily augmented to incorporate higher order terms If a large parallel computer or nodes on a network are available for distributed computing it may not be worth the trouble of first constructing a linear approximation then continuing on to a higher order one Before using neural nets or Kriging surfaces as approximations please consult Section After suitable preparation the optimization process may now be commenced At this point the user has to decide whether to use an automated iterative procedure or whether to firstly perform variable screening through ANOVA based on one or a few iterations Variable screening is important for reducing the number of design variables and therefore the overall computational time An automated iterative procedure can be conducted with any choice of approximating function It automatically adjusts the size of the subregion and automatically terminates whenever the stopping criterion is satisfied If a single optimal point is desired this is probably the procedure to use If there is a large number of design variables a linear approximation can be chosen However a step by step semi automated procedure can be just as useful since it allows the designer to proceed more resourcefully Computer time can be wasted with iterative methods especially if handled carelessly It mostly pays to pause after every iteration especially the first to allow verification of t
215. es solver experiments and solver related job information defined within this environment are associated with the particular solver e strict slack soft Pertains to the strictness of constraints See Sections e move stay Pertains to whether constraints should be used to define a reasonable design space or not for the experimental design See Section 5 3 6 Expressions Each entity can be defined as a standard formula a mathematical expression or can be computed with a user supplied program that reads the values of known entities The bullets below indicate which options apply to the various entities Variables are initialized as specified numbers Table 5 1 Expression options of optimization entities Entity Standard Expression User defined Variable Dependent History Response e e Composite A list of mathematical and special function expressions that may be used is given in Appendix F Mathematical Expressions 68 LS OPT Version 2 6 Program Execution This chapter describes the directory structure output and status files and logistical handling of a simulation based optimization run 6 1 Work directory Create a work directory to keep the main command file input files and other command files as well as the LS OPT program output 6 2 Execution commands lsoptui command file name Execute the graphical user interface lsopt command file name LS OPT bat
216. es due to the random variance and including the standard deviation in the constraint s calculation Residual The difference between the computed response using simulation and the predicted response using a response surface Response quantity See response Response Surface A mathematical expression which relates the response variables to the design parameters Typically computed using statistical methods Response A numerical indicator of the performance of the design A function of the design variables approximated using response surface methodology which can be considered for optimization Symbolized by f Collected over all design iterations for plotting See also history RSM Response Surface Methodology Run directory The directory in which the simulations are done Two levels below the Work directory The run directory contains status files the design coordinate file XPoint and all the simulation output Saturated design An experimental design in which the number of points equals the number of unknown coefficients of the approximation For a saturated design no test can be made for the lack of fit Scale factor A factor which is specified as a divisor of a response in order to normalize the response Sensitivity See Design sensitivity Sequential Random Search An iterative method in which the best design is selected from all the simulation results of each iteration A Monte Carlo based point selection scheme is typic
217. esidual response disp15 5 69 4946 scale Residual response disp20 5 49 7515 scale Residual response disp25_5 34 2808 scale Residual response disp30 5 25 3116 scale Residual response disp35 5 23 8644 scale Residual response disp40_ 5 28 4147 scale Residual response disp45 5 36 3297 scale Residual response disp50_5 45 6215 scale C1 type weighted c1 variable Leakage_1 1 scale 1le 07 c1 variable Leakage_2 1 scale le 07 C2 type weighted C2 variable Leakage_2 1 scale 1le 07 C2 variable Leakage_ 3 1 scale le 07 C3 type weighted C3 variable Leakage_3 1 scale le 07 C3 variable Leakage 4 1 scale le 07 C4 type weighted C4 variable Leakage 4 1 scale 1e 07 C4 variable Leakage_5 1 scale 1le 07 C5 type weighted C5 variable Leakage_5 1 scale 1le 07 C5 variable Leakage 6 1 scale le 07 C6 type weighted C6 variable Leakage 6 1 scale 1le 07 c6 variable Leakage_7 1 scale le 07 C7 type weighted C7 variable Leakage_7 1 scale 1le 07 C7 variable Leakage_ 8 1 scale le 07 C8 type weighted C8 variable Leakage_ 8 1 scale 1e 07 C8 variable Leakage_9 1 scale le 07 C9 type weighted c9 variable Leakage_9 1 scale 1le 07 C9 variable Leakage_10 1 scale 1le 07 234 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS OBJECTIVE FUNCTIONS objectives 1 Residual 1 objective CONSTRAINT DEFINITIONS c
218. esign so updated the default should be selected An example is given in Section 17 2 8 The advantage of using neural networks or Kriging surfaces is the avoidance of having to choose a polynomial order the adaptability of the response surface and the global nature of the final surface that can subsequently be used for trade off studies or reliability investigations 61 CHAPTER 4 DESIGN OPTIMIZATION PROCESS 62 LS OPT Version 2 5 Graphical User Interface and Command Language This chapter introduces the graphical user interface the command language and describes syntax rules for names of variables strings and expressions 5 1 LS OPT user interface LS OPTui LS OPT can be operated in one of two modes The first is through a graphical user interface LS OPTui and the second through the command line using the Design Command Language DCL The user interface is launched with the command lsoptui command file The layout of the menu structure Figure 5 1 mimics the optimization setup process starting from the problem description through the selection of design variables and experimental design the definition and responses and finally the formulation of the optimization problem objectives and constraints The run information number of processors monitoring and termination criteria is also controlled via LS OPTuwi 63 CHAPTER 5 GRAPHICAL USER INTERFACE AND COMMAND LANGUAGE File Tasks Help m Solvers
219. esponse name scale factor offset command line Examples 1 The user has an own executable program ExtractForce which is kept in the directory SHOME own bin The executable extracts a value from a result output file The relevant response definition command must therefore be as follows response Force SHOME own bin ExtractForce 2 If Perl is to be used to execute the user script DynaFLD2 the command may be response Acc SLSOPT perl LSOPT DynaFLD2 0 5 0 25 1 833 Remark 1 An alias must not be used for an interface program LS OPT Version 2 131 CHAPTER 10 HISTORY AND RESPONSE RESULTS 132 LS OPT Version 2 11 Objectives and Constraints This chapter describes the specification of objectives and constraints for the design formulation 11 1 Formulation Multi criteria optimal design problems can be formulated These typically consist of the following e Multiple objectives multi objective formulation e Multiple constraints Mathematically the problem is defined as follows Minimize F subject to L lt g SU L Sg SU L lt Ba U where F represents the multi objective function x x x represent the various objective functions and g gj X x represent the constraint functions The symbols x represent the n design variables In order to generate a trade off design curve involving objective functions more than one objective mus
220. eter identification all the constraints are in general never completely satisfied due to typically over determined systems that are used Both formulations have one additional set of constraints in this example namely to ensure monotonicity of the load curve to be developed i e Ku gt X A l 2 p l where p 54 is the number of experimental collocation points The monotonicity constraints are strictly enforced using the strict option in LS OPT i e they do not contain the slack variable e This means that the constraints at the collocation points are compromised at the cost of satisfying the monotonicity constraints 230 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 5 4 Implementation The LS OPT input files for this problem are shown below LSR formulation 10 design variables Note the use of a Targeted Composite for the objective Residual The initial leakage curve is horizontal The two LS DYNA solvers 4MPS and 5MPS with input decks sim4mpros inp and sim5mpros inp refer to the two cases i e 4 and 5 m s approach velocity respectively Parameter Estimation Airbag 10 variables Created on Thu Aug 30 17 16 28 2001 DESIGN VARIABLES variables 10 Variable Leakage_1 6e 08 Lower bound variable Leakage_1 2e 08 Upper bound variable Leakage_1 4e 07 Range Leakage _1 le 07 Variable Leakage 2 6e 08 Lower bound variable Leakage 2 2e 08 Upper bound variable Leak
221. ew USER DEFINED EXPRESSION Composite Components Responses Variables ABSTAT Response Weight Scale Target BNDOUT Al DEFORC F1 R fa R 7500 ELOUT z FLD zn i ee 19554 F3 R fa 22000 F4 R fa 26832 Expression Fi Expression Fi Expression Fi Expression Fi GCEOUT GLSTAT JNTFORC MASS MATSUM NCFORC NODOUT NODFOR 3 p PSTRESS Composite Function Type Targeted RBDOUT RCFORC RWFORC SECFORC SPCFORC SWFORC THICK FREQUENCY Z Composite Expression 4 eT a Response Name Residual Adal Replace Delete Figure 10 3 Definition of targeted composite response in LS OPTui 10 4 1 Defining the composite function This command identifies the composite function The type of composite is specified as weighted targeted or expression The expression composite type does not have to be declared and can simply be stated as an expression Command file syntax composite composite name type targeted weighted Example composite Damage type targeted composite Acceleration type weighted The expression composite is defined as follows LS OPT Version 2 113 CHAPTER 10 HISTORY AND RESPONSE RESULTS Command file syntax composite composite name math expression The math_expression is a mathematical expression given in curly brackets see Appendix E The number of composite functions to be employed must be specified in the pr
222. ew histories are generated 326 LS OPT Version 2 APPENDIX E MATHEMATICAL EXPRESSIONS E 6 Generic expressions Expressions can be specified for any floating point number In some cases previously defined parameters can be used as follows The parameter type represents the highest entity in the hierarchy Thus constants are included in the variable parameters In LS OPT expressions can be entered for variables constants dependents histories responses constraints and objectives Example constant Targetl 12756 333 1000 constant Target2 966002 1000 variable Emod 1le7 composite Residual type targeted composite Residual response F1 hea scale De composite Residual response F2 Target2 scale Target2 objective Residual 5 variable fdstepsize 1 500 time fdstepsize 1 300 history size 10000 LS OPT Version 2 327 APPENDIX E MATHEMATICAL EXPRESSIONS E 7 Examples illustrating syntax of expressions Example 1 The following example shows a simple evaluation of variables and functions The histories are specified in plot files his1 and his2 A third function his3 is constructed from the files by averaging File hisl 0 0 0 100 1000 200 500 300 500 File his2 0 0 0 100 2000 200 2000 300 2000 Input file Mathematical Expressions S CONSTANTS constants 3 constant lowerlimit 0 constant upperlimit 200 co
223. experienced fitting problems with non smooth surfaces Z x observed to peak at data points in some cases apparently due to large values of that may be due to local optima of the maximum likelihood function The model construction can be very time consuming also experienced with LS OPT Furthermore the slight global altering of the Kriging surface due to local updating has also been observed 63 Reference compares the use of the three metamodeling techniques for crashworthiness optimization This paper which incorporates three case studies in crashworthiness optimization concludes that while RSM NN and Kriging were similar in performance RSM and NN are the most robust for this application 2 11 Core optimization algorithm LFOPC The optimization algorithm used to solve the approximate subproblem is the LFOPC algorithm of Snyman 60 It is a gradient method that generates a dynamic trajectory path from any given starting point towards a local optimum This method differs conceptually from other gradient methods such as SQP in that no explicit line searches are performed The original leap frog method for unconstrained minimization problems seeks the minimum of a function of n variables by considering the associated dynamic problem of a particle of unit mass in an n dimensional conservative force field in which the potential energy of the particle at point x at time t is taken to be the function f x to be minimized The soluti
224. expression T constant Af At _ df dt r Min expression t_lower t_upper Smin min f Max expression t_lower t_ upper Fe max f Initial expression First function value on record Final expression Last function value on record Lookup expression value Inverse function f F LookupMin expression t lower t upper Inverse function f fmin LookupMax expression t lower t upper Inverse function t f fmax H 14 Selecting an optimization method Optimization method srsm Successive Response Surface Method SRSM 165 Optimization method randomsearch Sequential Random Search SRS 165 H 15 Setting parameters for optimization algorithm iterate param identifier value Define parameters in LFOPC 166 iterate param rangelimit variable value Define minimum range of variable in SRSM 167 348 LS OPT Version 2
225. f the 10 iteration have been extracted DESIGN POINT Variable Name Lower Bound Value Upper Bound Radius 1 1 2 653 4 5 Radius 2 1 2 286 4 6 5 Radius 3 1 2 004 gt Scaled Unscaled 2 RESPONSE Computed Predicted Computed Predicted 7 11 121m 422224 14440202042 iia 444044000 Thinning 19 92 19 6 19 92 19 6 FLD 0 000843 0 002907 0 000843 0 002907 4221124114 442 4 4444 45 ones ences i ee A comparison between the starting and the final values is tabulated below LS OPT Version 2 227 CHAPTER 17 EXAMPLE PROBLEMS Table 17 3 Comparison of results Sheet metal forming Variable Start Computed Optimal Predicted Optimal Computed Thinning 29 57 19 92 19 6 FLD 0 09123 0 000843 0 002907 Radius _1 1 5 2 653 Radius 2 1 5 2 286 Radius 3 1 5 2 004 The FLD diagrams Figure 17 24 for the baseline design and the optimum illustrate the improvement of the FLD feasibility FLD diagram 1 0 8 0 6 0 4 Major Strain 0 2 0 0 2 1 0 5 0 Minor Strain Baseline FLD diagram Major Strain FLD diagram 0 5 0 0 5 1 Minor Strain FLD diagram of 10 iteration Figure 17 24 FLD diagrams of baseline and 10 iteration A typical deformed state is depicted in Figure 17 25 below Figure 17 25 Deformed state 228 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLE
226. files must remain intact to ensure that a restart can be executed if necessary A brief explanation is given below 73 CHAPTER 6 PROGRAM EXECUTION Table 6 3 Status files generated by LS OPT prepro The preprocessing has been done replace The variables have been replaced in the input files started The run has been started finished The run has been completed The completion status is given in the file response n Response number n has been extracted history n History number n has been extracted EXIT_STATUS Error message after termination The user interface LS OPTui uses the message in the EXIT_STATUS file as a pop up message The 1fop log file contains a log of the core optimization solver solution The simulation run extraction log is saved in a file called Lognnnnnn in the local run directory where nnnnnn represents the process ID number of the run An example of a logfile name is 109234771 Please refer to Section 6 for restarting an optimization run 6 10 Managing disk space during run time During a successive approximation procedure superfluous data can be erased after each run while keeping allthe necessary data and status files see above and example below For this purpose the user can provide a file named clean containing the required erase statements such as rm rf d3 rm rf elout rm rf nodout rm rf rcforc The clean file will be executed immediately after eac
227. fore always try to minimize the maximum knee force The knee forces have been filtered SAE 60 Hz to improve the approximation accuracy 17 8 3 Implementation Truegrid is used to parameterize the geometry The section of the Truegrid input file s7 tg where the design variables are substituted is shown below para wl lt lt L Flange Width gt gt w2 lt lt R_Flange Width gt gt thickl lt lt L_ Bracket _Gauge gt gt thick2 lt lt R_Bracket_Gauge gt gt thick3 lt lt Bolster_gauge gt gt fl lt lt T_Flange Depth gt gt Left EA flange width Right EA flange width Left bracket gauge Right bracket gauge Knee bolster gauge Left EA Depth Top 2 lt lt F Flange Depth gt gt Left EA Depth Front f3 lt lt B Flange Depth gt gt Left EA Depth Bottom 4 lt lt I_Flange Width gt gt Left EA Inner Flange Width Yolk bar radius Oblong hole radius rl lt lt Yolk_Radius gt gt r2 lt lt R_Bracket_Radius gt gt aanaaaanaaaaaa The LS OPT input file is shown below for the 11 variable shape optimization case Knee Impact Simulation Shape Optimization Created on Wed Oct 4 13 31 36 2000 5 DESIGN VARIABLE DEFINITIONS variables 11 Variable L Bracket _Gauge 1 1 Lower bound variable L Bracket _Gauge 0 7 Upper bound variable L Bracket Gauge 3 Range L Bracket _Gauge 2 LS OPT Version 2 269 CHAPTER 17 EXAMPLE PROBLEMS Variable T Flange Depth 28 3 Lower bound variable T Flange Depth
228. g minimize the maximum of several radii in a sheet metal forming problem The radii are all incorporated into composite functions which in turn are incorporated into constraints which have zero upper bounds 3 Find the most feasible design For cases in which a feasible design region does not exist the user may be content with allowing the violation of some of the constraints but is still interested in minimizing this violation 42 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 15 2 Multidisciplinary Design Optimization There is increasing interest in the coupling of other disciplines into the optimization process especially for complex engineering systems like aircraft and automobiles 35 The aerospace industry was the first to embrace multidisciplinary design optimization MDO 56 because of the complex integration of aerodynamics structures control and propulsion during the development of air and spacecraft The automobile industry has followed suit B8 In 58 the roof crush performance of a vehicle is coupled to its Noise Vibration and Harshness NVH characteristics modal frequency static bending and torsion displacements in a mass minimization study Different methods have been proposed when dealing with MDO The conventional or standard approach is to evaluate all disciplines simultaneously in one integrated objective and constraint set by applying an optimizer to the multidisciplinary analysis MDA sim
229. gin one order of magnitude higher than the effective temperature and end one order of magnitude lower 7 It is difficult to give the initial temperature directly because this value depends on the neighborhood structure the scale of the objective function the initial solution etc In a suitable initial temperature is one that results in an average uphill move acceptance probability of about 0 8 This Tmax can be estimated by LS OPT Version 2 335 APPENDIX F SIMULATED ANNEALING conducting an initial search in which all uphill moves are accepted and calculating the average objective increase observed In some other papers it is suggested that parameter Tmax is set to a value which is larger than the expected value of E E that is encountered from move to move In f it is suggested to spend most of the computational time in short sample runs with different Tmax in order to detect the effective temperature In practice the optimal control of 7 may require physical insight and trial and error experiments According to 94 choosing an annealing schedule for practical purposes is still something of a black art Simulated annealing has proved surprisingly effective for a wide variety of hard optimization problems in science and engineering Many of the applications in our list of references attest to the power of the method This is not to imply that a serious implementation of simulated annealing to a difficult real world problem wil
230. given modal shape number Command file syntax DynaFreq mode_original modal_attribute Table 10 3 Frequency item description Item Description mode_original The number sequence of the baseline modal shape to be tracked modal attribute Type of modal quantity See table below LS OPT Version 2 117 CHAPTER 10 HISTORY AND RESPONSE RESULTS Table 10 4 Frequency attribute description Attribute Description Frequency of current mode corresponding in modal shape to FREO baseline mode specified eee Number of current mode corresponding in modal shape to baseline mode specified GENMASS max u 2d i mo Theory Mode tracking is required during optimization using modal analyses as mode switching a change in the sequence of modes can occur as the optimizer modifies the design variables In order to extract the frequency of a specified mode LS OPT performs a scalar product between the baseline modal shape mass orthogonalized eigenvector and each mode shape of the current design The maximum scalar product indicates the mode most similar in shape to the original mode selected To adjust for the mass orthogonalization the maximum scalar product is found in the following manner max mi a vie 10 3 where M is the mass matrix excluding all rigid bodies is the mass orthogonalized eigenvector and the subscript 0 denotes the baseline mode This product can be extracted with the
231. gn The selection of designs to enable the construction of a design response surface Sometimes referred to as the Point Selection Scheme Feasible Design A design which complies with the constraint bounds Function A mathematical expression for a response variable in terms of design variables Often used interchangeably with response Symbolized by f Functionally efficient See Pareto optimal Function evaluation Using a solver to analyze a single design and produce a result See Simulation Global variable A variable of which the scope spans across all the design disciplines or solvers Used in the MDO context 338 LS OPT Version 2 APPENDIX G GLOSSARY Global approximation A design function which is representative of the entire design space Global Optimization The mathematical procedure for finding the global optimum in the design space E g Genetic Algorithm Particle Swarm etc Gradient vector A vector consisting of the derivatives of a function fin terms of a number of variables x to xn df dx See Design Sensitivity History Response history containing two columns of usually time data generated by a simulation Importance See Weight Infeasible Design A design which does not comply with the constraint functions An entire design space or region of interest can sometimes be infeasible Iteration A cycle involving an experimental design function evaluations of the designs approximation and
232. gn when the design variables are substituted See response surface Design space A region in the n dimensional space of the design variables x through x to which the design is limited The design space is specified by upper and lower bounds on the design variables Response variables can also be used to bound the design space Design surface The response variable as a function of the design variables used to construct the formulation of a design problem See also response surface design rule Design sensitivity The gradient vector of the response The derivatives of the response function in terms of the design variables df dx Design variable An independent design parameter which is allowed to vary in order to change the design Symbolized by x or x vector containing several design variables Discipline An area of analysis requiring a specific set of simulation tools usually because of the unique nature of the physics involved e g structural dynamics or fluid dynamics In the context of MDO often used interchangeably with solver DOE Design of Experiments See experimental design D optimal The state of an experimental design in which the determinant of the moment matrix x X of the least squares formulation is maximized DSA Design sensitivity analysis Elliptic approximation An approximation in which only the diagonal Hessian terms are used Experiment Evaluation of a single design Experimental Desi
233. gorithm Appropriate choices are essential to guarantee the efficiency of the algorithm Many different definitions of the above entities have been given in the existing literature about SA These will be discussed in the next few paragraphs trying to emphasize some key ideas that have driven the choices of the researches in this field In the existing literature about SA algorithms very few acceptance functions have been employed In most cases the acceptance function is the so called Metropolis function A x x T min CE 2h F 4 Another possibility is the so called Barker criterion 1 EC zo T A x x T F 5 The theoretical motivation for such a restricted choice of acceptance functions can be found in 55 It is shown that under appropriate assumptions many acceptance functions which share some properties are equivalent to F 4 or F 5 after a monotonic transformation of the temperature T Due to the difficult nature of the problems solved by SA algorithms it is hard if not impossible to define a general stopping rule which guarantees to stop when the global optimum has been detected or when there is a sufficiently high probability of having detected it Thus the stopping rules proposed in the literature about SA all have a heuristic nature and are in fact more problem dependent than SA algorithm dependent These 334 LS OPT Version 2 APPENDIX F SIMULATED ANNEALING heuristics are
234. grid of designs and forms the basis of many other designs Z is the number of grid points in one dimension It can be used as a basis set of experiments from which to choose a D optimal design In LSOPT the 3 and 5 designs are used by default as the basis experimental designs for first and second order D optimal designs respectively Factorial designs may be expensive to use directly especially for a large number of design variables 2 6 2 Koshal design This family of designs are saturated for modeling of any response surface of order d First order model For n 3 the coordinates are oO m Oo or Oo Co Oo Oo As aresult four coefficients can be estimated in the linear model 1 2 18 Second order model For n 3 the coordinates are LS OPT Version 2 13 CHAPTER 2 OPTIMIZATION METHODOLOGY 4 os X So O O O m m O O O O O m O O O O O o o Lo As a result ten coefficients can be estimated in the quadratic model OH bio ee 2 19 2 6 3 Central Composite design This design uses the 2 factorial design the center point and the face center points and therefore consists of P 2 2n 1 experimental design points For n 3 the coordinates are A A H 0 0 0 a 0 0 0 a 0 0 0 a Q 0 0 0 a 0 0 0 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 l 1 1 1 1 1 1 1 14 LS OPT Version 2 CHAPTER 2 OPTIMIZAT
235. h simulation and will clean all the run directories except the baseline first or 1 1 and the optimum last runs Care should be taken not to delete the lowest level directories or the log files prepro started replace finished response n or history n which must remain in the lowest level directories These directories and log files indicate different levels of completion status which are essential for effective restarting Each file response response_ number contains the extracted value for the response response_number E g the file response 2 contains the extracted value of response 2 The data is thus preserved even if all solver data files are deleted The response_number starts from 0 Complete histories are similarly kept in history history_number The minimal list to ensure proper restarting is prepro XPoint 74 LS OPT Version 2 CHAPTER 6 PROGRAM EXECUTION replace started finished response O response l history 0 history 1 Remarks 1 The clean file must be created in the work directory 2 Ifthe clean file is absent all data will be kept for all the iterations 3 For remote simulations the clean file will be executed on the remote machine 6 11 Error termination of a solver run The job scheduler will mark an error terminated job to avoid termination of LS OPT Results of abnormally terminated jobs are ignored If there are not enough results to construct the approximate design surfaces LS OPT will
236. hanges from 15 to 18 LS OPT Version 2 247 EXAMPLE PROBLEMS CHAPTER 17 Optimization History For Response HIC Optimization History For Response Mass 175 150 Ke oO LO oO N oO N LO OLX DIH esuodsay Sse esuodsay 10 Number of Iterations Number of Iterations a Objective Mass b HIC Optimization History For Composite Intrusion Optimization History For Response Frequency aS a Se ker Ww st mM K a Aouanba l asuodsay N l l 4 l rm 12 3 45 6 7 8 9 10 10 Number of Iterations Number of Iterations d Intrusion c Torsional mode frequency LS OPT Version 2 248 CHAPTER 17 EXAMPLE PROBLEMS Optimization History Optimization History For Response Mode For Error Term Max Constr Violation 20 20 E 2 5 O 15 5 15 0b ra 8 2 S 3 10 Q 10 S wo 4 E fe 5 5 5 5 In 2 4 6 8 10 2 4 6 8 10 Number of Iterations Number of Iterations e Mode sequence f Maximum constraint violation Figure 17 33 Optimization histories Small car MDO D optimal SRSM LS OPT Version 2 249 CHAPTER 17 EXAMPLE PROBLEMS 17 6 3 Sequential random search The small car crash design problem was also optimized using the Sequential Random Search procedure ME described in Section Because of the multidisciplinary nature of the problem all the variables were selected to be fully shared The design points for
237. he particular variable belongs This reference is made under the solver context where the syntax Solver variable variable name is used see next paragraph and example below To limit the scope of a variable an experimental design or job information to a particular solver the prefix solver should be applied to the commands below The solver definition must precede any commands having the solver prefix Omission of the prefix implies that the specification is multidisciplinary i e it is shared between all the specified solvers LS OPT Version 2 151 CHAPTER 14 APPLICATIONS OF OPTIMIZATION Variable Concurrent jobs Order Experiment design Basis experiment Number Basis experiment Number experiment Queuer Update doe Experiment duplicate See the examples in Sections 17 6Jand for the command file format 14 2 Worst case design The default setting in LS OPT is that all design variables are treated as minimization variables This means that the objective function is minimized or maximized with respect to all the variables Maximization variables are selected in the Variables panel see by toggling the required variables from Minimize to Maximize 14 3 Reliability based design optimization The methodology is described in Section An example is given in Section 17 2 9 152 LS OPT Version 2 15 Probabilistic Modeling and Monte Carlo Simulation 15 1 Introduction Probabilistic evaluations
238. he data and design formulation and inspection of the results including ANOVA data In many cases it takes only 2 to 3 iterations to achieve a reasonably optimal design An improvement of the design can usually be achieved within one iteration A suggested step by step semi automated procedure is outlined as follows 4 2 2 A step by step design optimization procedure 1 Evaluate as many points as required to construct a linear approximation Assess the accuracy of the linear approximation using any of the error parameters Inspect the main effects by looking at the 57 CHAPTER 4 DESIGN OPTIMIZATION PROCESS ANOVA results This will highlight insignificant variables that may be removed from the problem An ANOVA is simply a single iteration run typically using a linear response surface to investigate main and or interaction effects The ANOVA results can be viewed in the post processor If the linear approximation is not accurate enough add enough points to enable the construction of a quadratic approximation Assess the accuracy of the quadratic approximation Intermediate steps can be added to assess the accuracy of the interaction and or elliptic approximations Ifthe second order approximation is not accurate enough the problem may be twofold a There is significant noise in the design response b There is a modeling error i e the function is too nonlinear and the subregion is too large to enable an accurate quadratic approxim
239. hell element XX stress YY_STRESS stress YY stress ZZ_STRESS ZZ stress XY_STRESS XY stress YZ STRESS YZ stress ZX_STRESS ZX stress P STRAIN Plastic strain PRESSURE Pressure E_STRESS Effective stress MAX SHEAR Maximum shear stress MAX P STRESS MIN P STRESS Maximum principal stress Minimum principal stress XX_STRAIN Shell element XX strain YY_STRAIN strain YY strain ZZ STRAIN ZZ strain XY_STRAIN XY strain YZ STRAIN YZ strain ZX_STRAIN ZX strain E_STRAIN Effective strain MAX S_ STRAIN Maximum shear strain MAX P_ STRAIN Maximum principal strain MIN P STRAIN Minimum principal strain TXX STRESS Thick shell XX stress TYY _ STRESS stress YY stress TZZ STRESS ZZ stress TXY_STRESS XY stress TYZ_ STRESS YZ stress TZX_STRESS ZX stress TP_STRAIN Plastic strain TPRESSURE Pressure TE_STRESS Effective stress TMAX_SHEAR Maximum shear stress TMAX P STRESS TMIN P STRESS Maximum principal stress Minimum principal stress LS OPT Version 2 293 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Contact Entities Resultants GCEOUT Keyword Description X_FORCE X force Y_FORCE Y force Z_FORCE Z force R_FORCE Force magnitude X_MOMENT X moment Y_MOMENT Y moment Z MOMENT Z moment R_MOMENT Moment magnitude Global Statisties GLSTAT Keyword Description K_ENER Kinetic energy I ENER Internal energy T ENER Total energy RATIO Ratio SW_ENER Stonewall energy D_ ENER S
240. heorem Reliability Assuming Normal Distribution Lower Bound BOUNG ti sh nein ee ce 150 Probability of exceeding Bound 0 2956 Reliability Index Beta 0 5372 ANALYSIS COMPLETED 17 10 4 Monte Carlo using Metamodel The bounds on the design variables are set to be two standard distributions away from the mean the default for noise variables Noise variables are not used because of the need to have more control over the variable bounds specifically we want to change the standard deviation of some variables without affecting the variable bounds the metamodel is computed scaled with respect to the upper and lower bounds on the variables The command file for using a metamodel is Tube Crush Metamodel Monte Carlo Created on Tue Apr 1 11 26 07 2003 solvers 1 5 distribution 2 distribution t NORMAL 1 0 0 05 distribution y NORMAL 1 0 0 10 5 DESIGN VARIABLES variables 2 variable T1 1 0 upper bound variable T1 1 1 lower bound variable T1 0 9 variable T1 distribution t variable YS 1 0 upper bound variable YS 1 2 282 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS lower bound variable YS 0 8 variable YS distribution y 5 DEFINITION OF SOLVER SOLVER_1 solver dyna960 SOLVER_1 solver command 1s970 single solver input file tube k solver experiment design 3toK solver order quadratic HISTORIES FOR SOLVER SOLVER_1 UW Ur Ur histories 1 history NH
241. hile it takes about 7 iterations 35 simulations for the objective 4 7 3 I I I I 1 1 p a a tt i E I I I I 1 Q 1 l i 1 1 1 2 2 RA O A a 1 I I I I L L I I I I I I roto r i 1 I I I I 1 1 1 1 u a I I I I I I I L 1 2 3 4 5 6 7 8 9 10 11 Number of Iterations g Optimization history of Stress Linear Remarks 1 of the linear case to converge 2 In general the lower the order of the approximation the more iterations are required to refine the optimum 186 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 2 Small car crash 2 variables This example has the following features An LS DYNA explicit crash simulation is performed Extraction is performed using standard LS DYNA interfaces First and second order response surface approximations are compared The design optimization process is automated A trade off study is performed using both a quadratic and neural network approximation A limited reliability based design optimization study is performed 17 2 1 Introduction This example considers the crashworthiness of a simplified small car model A simplified vehicle moving at a constant velocity of 15 64m s 35mph impacts a rigid pole See The thickness of the front nose above the bumper is specified as part of the hood LS DYNA is used to perform a simulation of the crash for a simulation duration of 50ms
242. his file is normally created in the run directory by LS OPT after substitution of the variables or creation by a preprocessor The original template file can have a different name and is specified as the input file in the solver input file command 2 lsdynampp is the name of the MPP executable 3 The file dumpbdb for creating the ASCII database must be executable 4 The script must be specified in one of the following formats a path relative to the run directory two levels above the run directory see example above b absolute path e g origin users john crash runmpp c in a directory which is in the path In this case the command is solver command runmpp 5 LS DYNA MPP will only give continual progress feedback through LS OPT for version 960 and later 7 2 3 Interfacing with a user defined solver An own solver can be specified using the solver own solvername command or selecting User defined in LS OPTui The solver command can either execute a command or a script The substituted input file UserOpt inp will automatically be appended to the command or script Variable substitution will be performed in the solver input file which will be renamed UserOpt inp and the solver append file If the own solver does not generate a Normal termination command to standard output the solver command must execute a script that has as its last statement the command echo Norma lL Example solver own Analyzer so
243. i e the maximum likelihood estimate for W is that which minimizes Eq 2 46 or equivalently Ep In order for Bayes estimates of and J to do a good job of minimizing the generalization in practice it is usually necessary that the priors on which they are based are realistic The Bayesian formalism also allows us to calculate error bars on the network outputs instead of just providing a single best guess output y Given an unbiased model minimization of the performance function Eq 2 46 amounts to minimizing the variance of the model The estimate for output variance o je of the network at a particular point x is given by Oi g x A g x 2 53 Equation 2 53 is based on a second order Taylor series expansion of Eq 2 48 around its minimum and assumes that dy dW is locally linear 2 10 2 Kriging Kriging is named after D G Krige B3 who applied empirical methods for determining true ore grade distributions from distributions based on sampled ore grades In recent years the Kriging method has found wider application as a spatial prediction method in engineering design Detailed mathematical formulations of Kriging are given by Simpson and Bakker 5 The basic postulate of this formulation is V X f x Z x where y is the unknown function of interest f x is a known polynomial and Z x the stochastic component with mean zero and covariance Cov Z x Z 2 o 7RYRXx With L the number of sampli
244. iables are substituted in the RANDOM_ solver input deck car5s k only 3 The aim of the RANDOM_ solver is only to provide the standard deviations of the responses 4 The reliability of the design is improved by increasing the Intrusion response before it is compared to the 550mm upper constraint limit The result of the optimization process is given in Figure 17 14 Shown are both the Intrusion and HIC responses The reliability limit on the Intrusion response is shown as a dashed line This corresponds to the left hand side of the constraint in Eq 17 5 rewritten as Intrusion 60 lt 550mm Intrusion A 6 sigma range is given in the plot for the HIC response It can be seen that the mean HIC value is reduced in the presence of the reliability constraint 300 630 a HIC_mean HIC Mean 6sigma 620 HIC Mean 6sigma 4 Intrusion_mean 610 Intrusion Mean 6sigma 600 590 580 Intrusion 570 560 550 540 530 Iteration Number Figure 17 14 Reliability based design results Small car 2 variables 202 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 3 Impact of a cylinder 2 variables This example has the following features An LS DYNA explicit impact simulation is performed An independent parametric preprocessor is used to incorporate shape optimization Extraction is performed using standard ASCII LS DYNA interfaces
245. ictive qualities or near singularity during the regression procedure It is therefore better to use the D optimal experimental design for RSM Example Solver order linear Solver experimental design latin_hypercube Solver number experiment 20 LS OPT Version 2 99 CHAPTER 9 METAMODELS AND POINT SELECTION The Latin Hypercube point selection scheme is also well suited to sequential random search methods see Section 2 13 9 2 4 Space filling Only algorithm 5 see Section is available in LS OPTui This algorithm maximizes the minimum distance between experimental design points The only item of information that the user must provide for this point selection scheme is the number of experimental design points Space filling is useful when applied in conjunction with the Neural Net neural network and Kriging methods see Section File Tasks Help Into Solvers Variables I Point Selection Histories Responses Objective Constraints Run View CRASH Metamodel Point Selection NVH x Polynomial x Full Factorial wv Sensitivity v Latin Hypercube Neural Net Space Filling Neural Net Option v Duplicate M Updated Number of Experiments 20 Figure 9 2 Selecting the Neural network approximation method in the Point Selection panel 100 LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION 9 3 User defined experiments To retain existing expensive simulation data in the optimization process i
246. iest design case to investigate the effect of the initial subregion size on the convergence rate It can be seen that this resulted in the objective being minimized in relatively few iterations 264 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 8 Max Design Min Design Mass kg Iteration Figure 17 44 Optimization history of component mass Objective Starting designs 2 and 3 Table 17 6 Comparison of design variables for all optimization cases Case Rail_ Rail_ Cradle Aprons Shotgun Shotgun Cradle inner outer rail mm inner outer cross member mm mm mm mm mm mm Fully Starting design 1 2 322 1 286 1 842 1 158 1 196 1 614 1 486 shared base Starting ie design 1 1 948 1 475 1 275 1 992 1 346 1 383 1 2 base J Starting gt design 2 2 04 1 884 1 507 1 441 1 11 1 372 1 161 E lightest Starting design 3 1 95 1 765 1 469 1 303 2 123 1 391 1 208 heaviest 17 7 6 Comparison of optimum designs The optimum designs obtained in each case above are compared in Table 17 5 for the objective function and constraints and in Table 17 6 for the design variables Note how the partially shared variable Starting design 1 case has the lowest mass while performing the best as far as the constraints are concerned The extreme starting designs gave interesting results After rapidly improv
247. ign optimization process In general only some design variables need to be shared between the disciplines to provide limited coupling in the optimization of a multidisciplinary target or objective Multi objective An objective function which is constituted of more than one objective Symbolized by F Multi criteria Refers to optimization problems in which several criteria are considered MP Mathematical Programming Mathematical optimization Neural network approximation The use of trained feed forward neural networks to perform non linear regression thereby constructing a non linear response surface Numerical sensitivity A derivative ofa function computed by using finite differences Noise See random error Objective A function of the design variables that the designer wishes to minimize or maximize If there exists more than one objective the objectives have to be combined mathematically into a single objective Symbolized by Optimal design The methodology of using mathematical optimization tools to improve a design iteratively with the objective of finding the best design in terms of predetermined criteria Point selection scheme Same as experimental design Parameter identification A procedure in which a numerical model is calibrated by optimizing selected parameters in order to minimize the response error with respect to certain targeted responses The targeted responses are usually derived from experimental
248. ign variables x the cross sectional area of the bars and x half of the distance m between the supported nodes The lower bounds on the variables are 0 2cm and 0 1m respectively The upper bounds on the variables are 4 0cm and 1 6m respectively The objective function is the weight of the structure f x C x Jlt 23 17 1 The stresses in the members are constrained to be less than 100 MPa o x C V 1 x EASE 17 2 X 1 AX ee si 17 3 X AX where C 1 0 and C 0 124 Only the first stress constraint is considered since it will always have the larger value The C language is used for the simulation program The following two programs simulate the weight response and stress response respectively W C include lt stdlib h gt include lt stdio h gt include lt math h gt define NUMVAR 2 main int argc char argv int i flag double x NUMVAR val for i 0 i lt NUMVAR i flag sscanf argv i 1 Slf amp x i if flag 1 printf Error in calculation of Objective Function n exit 1 val x 0 sqrt 1 x 1 x 1 printf Slf n val fprintf stderr Norma 1 n 172 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS exit 0 S C include lt stdlib h gt include lt stdio h gt include lt math h gt define NUMVAR 2 main int argc char argv int i flag double x NUMVAR val double x2 for i 0 i lt NUMVAR i flag
249. ilar to that followed in single discipline optimization The standard method has been called multidisciplinary feasible MDF as it maintains feasibility with respect to the MDA but as it does not imply feasibility with respect to the disciplinary constraints is has also been called fully integrated optimization FIO A number of MDO formulations are aimed at decomposing the MDF problem The choice of MDO formulation depends on the degree of coupling between the different disciplines and the ratio of shared to total design variables 78 It was decided to implement the MDF formulation in this version of LS OPT as it ensures correct coupling between disciplines albeit at the cost of seamless integration being required between different disciplines that may contain diverse simulation software and different design teams In LS OPT the user has the capability of assigning different variables experimental designs and job specification information to the different solvers or disciplines The file locations in Version 2 have been altered to accommodate separate Experiments AnalysisResults and DesignFunctions files in each solver s directory An example of job specific information is the ability to control the number of processors assigned to each discipline separately This feature allows allocation of memory and processor resources for a more efficient solution process Refer to the user s manual Section 14 1 for the details of implementing an MDO p
250. imal Latin hypercube designs The term simulated annealing derives from the rough analogy of the way that the liquids freeze and crystallize or metals cool and anneal starting at a high temperature 28 When the liquid is hot the molecules move freely and very many changes of energy can occur When the liquid is cooled this thermal mobility is partially lost If the rate of cooling is sufficiently slow the atoms are often able to line themselves up and form a pure crystal which is the state of minimum most stable energy for this physical system If a liquid metal is cooled quickly or quenched it usually does not reach this state but rather ends up in a polycrystalline or amorphous state having somewhat higher energy So the essence of the whole process is slow cooling Nature s minimization algorithm is based on the fact that a system in thermal equilibrium at temperature 7 has its energy E probabilistically distributed among all different energy states as determined by the Boltzmann distribution Probability E exp E KBT F 1 Hence even at low temperature there is a chance albeit very small of a system being in a high energy state This slight probability of choosing a state that gives higher energy is what allows the physical system to get out of local i e amorphous minima in favor of finding a better more stable orientation The quantity Kg Boltzmann s constant is a constant of nature that relates temperature
251. imental point x 1 x 2 x n where x 1 to x n are the values of the n solver design variables at the experimental point The AnalysisResults file This file is used to save the responses at the experimental points and appears in the solver directory Every line describes an experimental point and gives the response values at the experimental point The file consists of lines having the following format repeated for each experimental point x 1 x 2 x n RespVal 1 RespVal 2 RespVal m where x 1 to x n are the values of the n solver design variables at the experimental point RespVal 1 to RespVal m are the values of the m solver responses Values of 2 0 10 are assigned to responses of simulations with error terminations The AnalysisResults file is synchronous with the Experiments file 319 APPENDIX D DATABASE FILES The DesignFunctions file The DesignFunctions file which appears in the solver directory is used to save a description of the polynomial design functions Entities Remark Number of variables n n rows Lower bound Upper bound Number of responses m Lines below repeated m times Response type Number of constants in a row Response name Number of constants 1 in a row Response command Solver ID Unused Unused Unused Polynomial constants Flags for active constants Example 2 2 0000000000000001e 01 4 0000000000000000e 00 1 0000000000000001
252. iness can be constrained because of regulation while other parameters such as mass cost and ride comfort can be treated as objectives to be weighted according to importance In these cases the designer may have target values in mind for the various response and or design parameters so that the objective formulation has to be formulated to approximate the target values as closely as possible Because the relative importance of various criteria can be subjective the ability to visualize the trade off properties of one response vs another becomes important Trade off curves are visual tools used to depict compromise properties where several important response parameters are involved in the same design They play an extremely important role in modern design where design adjustments must be made accurately and rapidly Design trade off curves are constructed using the principle of Pareto optimality This implies that only those designs of which the improvement of one response will necessarily result in the deterioration of any other response are represented In this sense no further improvement of a Pareto optimal design can be made it is the best compromise The designer still has a choice of designs but the factor remaining is the subjective choice of which feature or criterion is more important than another Although this choice must ultimately be made by the designer these curves can be helpful in making such a decision An example in vehicle design is
253. ing a quadratic approximation The following statements differ from the input file above 2BAR2 Two Bar Truss Updating the approximation to 2nd order response Weight quadratic response Stress quadratic EXPERIMENTAL DESIGN Order quadratic Experimental design dopt Basis experiment 5toK Number experiment 10 LS OPT Version 2 177 CHAPTER 17 EXAMPLE PROBLEMS The approximation results have improved considerably but the stress approximation is still poor Approximating Response Weight Mean response value RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted R 2 prediction Determinant of X X Approximating Response Stress Mean response value RMS error Maximum Residual Average Error Square Root PRESS Residual Variance R 2 R 2 adjusted R 2 prediction Determinant of X X The fit is illustrated below in Figure 17 3 oooo0o000o0 He gt D OON N O N He gt 8402 0942 1755 0737 2815 0177 9983 9983 9851 6629 4592 0291 0762 8385 4797 1182 9378 9378 6387 6629 N O W nt ee Fee SS using 10 points using 10 points 29 60 24 7L 32 18 x59 IL oe oe a an oe ITERATION 1 ITERATION 1 178 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Response Surface Accuracy For Response Function Weight 8 12 7 10 36 E Gy oO gt
254. ing from the initial violations they both converged to local minima The maximum design Starting design 3 started the furthest away from the optimum design but converged rapidly due to the increased initial move limit This highlights the need for a global optimization algorithm even for these costly simulation based MDO problems All the designs LS OPT Version 2 265 CHAPTER 17 EXAMPLE PROBLEMS converge to different design vectors with different combinations of the component thicknesses resulting in similar performance 17 7 7 Convergence and computational cost Comparing the fully and partially shared variable cases for starting design 1 it can be seen that the optimization process converged in 9 iterations in the first case while in the latter a good compromised design was found in only 6 iterations Coupled to the reduction in design variables especially for the NVH simulations the reduction in the number of simulations as shown in rable 17 7 is the result To explain the number of simulations clarification of the experimental design used is in order A 50 over sampled D optimal experimental design is used whereby the number of experimental points for a linear approximation is determined from the formula 1 5 n 1 1 where n refers to the number of design variables Consequently for the full sharing 7 variables imply 13 experimental design points while for the partial sharing 6 variables for crash imply 11 design poin
255. investigate the effects of variations of the system parameters on the system responses The variation of the system parameters is described using design variables with probabilistic distributions describing their variation around the mean design variable value The concept of noise variables which vary only according to a statistical distribution and of which the value is not under the control of the analyst is introduced in addition to the control variables Accordingly the system responses will vary according to some statistical distribution From this distribution information such as the nominal value of the response reliability and extreme values is inferred More background on the probabilistic methods is given in the theoretical manual Section B The use of the probabilistic analysis techniques is clarified using examples 15 2 Probabilistic problem modeling Introducing the probabilistic effects into analysis requires the specification of 1 Statistical distributions 2 Assigning the statistical distributions to design variables 3 Specification of the experimental design For a Monte Carlo analysis a suitable strategy for selecting the experimental points must be specified for example a Latin Hypercube experimental design can be used to minimize the number of runs required to approximate the mean and standard deviation However if the Monte Carlo analysis is done using a metamodel then the experimental design pertains to
256. ion is on the boundary of the subregion the desired solution is probably beyond the region Therefore if the user wants to explore the design space more fully a new approximation has to be built The accuracy of the current response surfaces can be used as an indication of whether to reduce the size of the new region The whole procedure can then be repeated for the new subregion and is repeated automatically when selecting a larger number of iterations initially 58 LS OPT Version 2 CHAPTER 4 DESIGN OPTIMIZATION PROCESS 4 3 Recommended test procedure A full optimization run can be very costly It is therefore recommended to proceed with care Check that the LS OPT optimization run is set up correctly before commencing to the full run By far the most of the time should be spent in checking that the optimization runs will yield useful results A common problem is to not check the robustness of the design so that some of the solver runs are aborted due to unreasonable parameters which may cause distortion of the mesh interference of parts or undefinable geometry The following general procedure is therefore recommended l ay Test the robustness of the analysis model by running a few perhaps two or three designs in the extreme corners of the chosen design space Run these designs to their full term in the case of time dependent analysis Two important designs are those with all the design variables set at their minimum and
257. ion of subregion in SRSM a pure panning b pure zooming and c a combination of panning and zooming The starting point x will form the center point of the first region of interest The lower and upper bounds x x of the initial subregion are calculated using the specified initial range value r so that L L x x 05r and x x 0 57 i 1 n 2 54 L LS OPT Version 2 35 CHAPTER 2 OPTIMIZATION METHODOLOGY where n is the number of design variables The modification of the ranges on the variables for the next iteration depends on the oscillatory nature of the solution and the accuracy of the current optimum Oscillation A contraction parameter yis firstly determined based on whether the current and previous designs x and x are on the opposite or the same side of the region of interest Thus an oscillation indicator c may be determined in iteration k as a d Pd 2 55 where dP 2A4x r Ax x x de Ets 2 56 The oscillation indicator purposely omitting indices i and k is normalized as c where c lc sign c 2 57 The contraction parameter yis then calculated as Aal I led y 5 2 58 is typically 0 5 0 7 representing shrinkage to dampen oscillation osc See Figure 2 5 The parameter v whereas y pan represents the pure panning case and therefore unity is typically chosen A Figure 2 5 The sub region contraction rate A as a function of the oscillation
258. ional command may be given Command file syntax weight weight value lt 1 gt Example composite damage type targeted composite damage response intrusion 3 20 weight 1 5 composite damage response intrusion 4 35 is used to specify F age Lf 20 f 35 The weight applies to the last specified composite and response 10 5 Extracting History and Response Quantities LS DYNA In LS OPT the general functionality for reading histories and responses from the simulation output is achieved through the history and response definitions see Section and Section 10 2 respectively For ASCII files the syntax for the extraction commands for responses and histories is identical except for the time attributes The history function is included so that operations such as subtracting two histories can first be performed after which a scalar such as maximum over time can be extracted from the resulting history There are two types of interfaces namely 1 Standard LS DYNA result interfaces This interface provides access to the ASCII and binary databases d3plot or LSDA of LS DYNA The interface is an integral part of LS OPT except for the extraction of mass properties that relies on a Per program DynaMass The perl compiler is included in the same directory as 1sopt during installation 2 User specified interface programs These can reside anywhere The user specifies the full path Aside of the stan
259. ions A popular choice is the quadratic approximation Del Ben 2 17 but any suitable function can be chosen LS OPT allows linear elliptical linear and diagonal terms interaction linear and off diagonal terms and quadratic functions 2 5 2 Factors governing the accuracy of the response surface Several factors determine the accuracy of a response surface 43 1 The size of the subregion For problems with smooth responses the smaller the size of the subregion the greater the accuracy For the general problem there is a minimum size at which there is no further gain in accuracy Beyond this size the variability in the response may become indistinguishable due to the presence of noise 2 The choice of the approximating function Higher order functions are generally more accurate than lower order functions Theoretically over fitting the use of functions of too high complexity may occur and result in suboptimal accuracy but there is no evidence that this is significant for polynomials up to second order 43 3 The number and distribution of the design points For smooth problems the prediction accuracy of the response surface improves as the number of points is increased However this is only true up to roughly 50 oversampling 43 very roughly 2 5 3 Advantages of the method e Design exploration As design is a process often requiring feedback and design modifications designers are mostly interested in suitabl
260. ircos 23 23 direction cosine dircos 31 31 direction cosine dircos 32 32 direction cosine dircos_ 33 33 direction cosine Injury coefficients CSI CSI Chest Severity Index HIC15 HIC15 Head Injury Coefficient 15 ms HIC36 HIC36 Head Injury Coefficient 36 ms 314 LS OPT Version 2 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Reaction Forces RCFORC DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z FORCE z_force Z force R_FORCE Resultant force XS FORCE X slave force YS FORCE Y slave force ZS FORCE Z slave force RS FORCE R slave force mass Mass RigidWall Forces RWFORC DynaASCH Binout Description Keyword Component Binout subdirectory forces NORMAL normal_force normal X_FORCE x_force X force Y_FORCE y_force Y force Z_FORCE z_force Z force LS OPT Version 2 315 APPENDIX C LS DYNA BINOUT RESULT FILE AND COMPONENTS Section Forces SECFORC DynaASCH Binout Description Keyword Component X_FORCE x_force X force Y_FORCE y_force Y force Z FORCE z_force Z force X MOMENT x_moment X moment Y_MOMENT y_moment Y moment Z MOMENT z_ moment Z moment X_CENTER x_centroid X center Y_CENTER y_centroid Y center Z_CENTER z_centroid Z center R_FORCE total_force Resultant force R_MOMENT total moment Resultant moment area Area Single Point Constraint Reaction Forces
261. is composed of a random and a bias component The bias component is a systematic deviation between the chosen model approximation type and the exact response of the structure FEA analysis is usually considered to be the exact response Also known as the modeling error See also random error Binout The name of the binary output file generated by LS DYNA Version 970 onwards Composite function A function constructed by combining responses and design variables into a single value Symbolized by F Concurrent simulation The running of simulation tasks in parallel without message passing between the tasks Confidence interval The interval in which a parameter may occur with a specified level of confidence Computed using Student s t test Typically applied to accompany the significance of a variable in the form of an error bar Constraint An absolute limit on a response variable specified in terms of an upper or lower limit Constrained optimization The mathematical optimization of a function subject to specified limits on other functions Conventional Design The procedure of using experience and or intuition and or ad hoc rules to improve a design Dependent A function which is dependent on variables Dependent variable Design of Experiments See experimental design Design parameter See design variable 337 APPENDIX G GLOSSARY Design formula A simple mathematical expression which gives the response of a desi
262. ist BinoutHistory res_type nodout cmp z displacement id 486 RESPONSES FOR SOLVER SOLVER_1 responses 2 response NodDisp 1 0 BinoutResponse res_ type nodout cmp z displacement id 486 select MIN response DispT LookupMin NHist t constraints 1 constraint NodDisp lower bound constraint NodDisp 150 0 JOB INFO analyze metamodel monte carlo STOP The accuracy of the response surface is of interest Approximating Response NodDisp ITERATION 1 Polynomial approximation using 9 points Global error parameters of response surface Mean response value 142 0087 RMS error S 2 0840 1 47 Maximum Residual 3 3633 2 37 Average Error 1 6430 1 16 Square Root PRESS Residual 6 2856 4 43 Variance u 13 0296 R 2 0 9928 R 2 adjusted 0 9856 R 2 prediction 0 9346 LS OPT Version 2 283 CHAPTER 17 EXAMPLE PROBLEMS The probabilistic evaluation results PEHE EHE HEHE HEHE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE HE FE HE FE HE FE HE FE HE FE HE HH HE H HE H HH HE H HH HH HHHH HHHH HHHH Monte Carlo simulation considering 2 stochastic variables Computed using 1000000 simulations EE E E E E E EE H E HE EE HE E HE A HE A E HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH FEFE EE HE HE FE FE EHE HE HE HE FE HEE HE HE HE FE HE HE HE HE H H H H H H H HERE HERE HHHH STATISTICS OF VARIABLES EEEE HE HE FE FE HEHE HE HE HE E HE HE HE HE
263. istic modeling and Monte Carlo simulation A sequential search method OO ON es to As in the past these developments have been influenced by industrial partners particularly in the automotive industry Several developments were also contributed by Nely Fedorova and Serge Terekhoff of SFTI Invaluable research contributions have been made by Professor Larsgunnar Nilsson and his group in the Mechanical Engineering Department at Link ping University Sweden and by Professor Ken Craig s group in the Department of Mechanical Engineering at the University of Pretoria South Africa The authors also wish to give special thanks to Mike Burger at LSTC for setting up further examples for Version 2 Nielen Stander Ken Craig Trent Eggleston and Willem Roux Livermore CA January 2003 LS OPT Version 2 XV PREFACE xvi LS OPT Version 2 1 Introduction This LS OPT manual consists of three parts In the first part the Theoretical Manual Chapter 2 the theoretical background is given for the various features in LS OPT The next part is the User s Manual Chapters Rlthrough 16 which guides the user in the use of LS OPTui the graphical user interface These chapters also describe the command language syntax The final part of the manual is the Examples section Chapter 17 where eight examples illustrate the application of LS OPT to a variety of practical applications Appendices contain interface features and Appendix C database
264. iteration 2 A reasonable design space can only be created using the D optimal experimental design 3 The reasonable design space will only be created if the center point or the starting point in the case of move start of the region of interest is feasible Feasibility is determined within a tolerance of 0 001 fnax fmin Where fmax and fmin are the maximum and minimum values of the interpolated response over all the points 4 The move feature should be used with extreme caution since a very tightly constrained experimental design may generate a poorly conditioned response surface 9 6 Updating an experimental design Updating the experimental design involves augmenting an existing design with new points Updating only makes sense if the response surface can be successfully adapted to the augmented points such as for neural nets or Kriging surfaces in combination with a space filling scheme Command file syntax solver update doe The new points have the following properties e They are located within the current region of interest e The minimum distance between the new points and between the new and existing points is maximized space filling only 9 7 Duplicating an experimental design When executing a search method see e g Section for a multi case or multidisciplinary optimization problem the design points of the various disciplines must be duplicated so that all the responses can be evaluated for any particular design
265. ithin a few hours Optimization and RSM in particular lend themselves very well to being applied in distributed computing environments because of the low level of message passing Response surface methodology is efficiently handled since each design can be analyzed independently during a particular iteration Needless to say sequential methods have a smaller advantage in distributed computing environments than global search methods such as RSM The present version of LS OPT also features Monte Carlo based point selection schemes and optimization methods The respective relevance of stochastic and response surface based methods may be of interest In a pure response surface based method the effect of the variables is distinguished from chance events while Monte Carlo simulation is used to investigate the effect of these chance events The two methods should be used in a complimentary fashion rather than substituting the one for the other In the case of events in which chance plays a significant role responses of design interest are often of a global nature being averaged or integrated over time These responses are mainly deterministic in character The full vehicle crash example 6 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY in this manual can attest to the deterministic qualities of intrusion and acceleration pulses These types of responses may be highly nonlinear and have random components due to uncontrollable noise variables
266. l be easy In the real life conditions the energy trajectory i e the sequence of energies following each move accepted and the energy landscape itself can be terrifically complex Note that state space which consists of wide areas with no energy change and a few deep narrow valleys or even worse golf holes is not suited for simulated annealing because in a long narrow valley almost all random steps are uphill Choosing a proper stepping scheme is crucial for SA in these situations However experience has shown that simulated annealing algorithms get more likely trapped in the largest basin which is also often the basin of attraction of the global minimum or of the deep local minimum Anyway the possibility which can always be employed with simulated annealing is to adopt a multistart strategy i e to perform many different runs of the SA algorithm with different starting points Another potential drawback of using SA for hard optimization problems is that finding a good solution can often take an unacceptably long time While SA algorithms may detect quickly the region of the global optimum they often require many iterations to improve its approximation For small and moderate optimization problems one may be able to construct effective procedures that provide similar results much more quickly especially in cases when most of the computing time is spent on calculations of values of the objective function But it should be noted that
267. lange Width L Flange Width R_Bracket_ Gauge R_Flange Width R_Bracket_ Radius Bolster gauge Yolk Radius More detailed results based on the 90 and 95 confidence intervals show e g that the left knee force is mostly influenced by the left bracket gauge bolster gauge and left flange width as would be expected If the spread in the data as denoted by the upper and lower limits of the respective confidence intervals causes the sensitivity coefficient value to change sign as e g in the case of T_ Flange Depth in the table below then that variable s contribution to the respective response is deemed insignificant Approximating Response L Knee Force using 19 points Individual regression coefficients significance and confidence Coeff Confidence Int 90 Confidence Int 95 Confidence Coeff not Value Lower Upper Lower Upper zero EE ea ee pc ee ur che ee u een re L_Bracket_Gauge 0 5736 0 4929 0 6543 0 4729 0 6743 100 T Flange Depth 0 09883 0 002844 0 2005 0 02807 0 2257 87 F Flange Depth 0 07274 0 00908 0 1364 0 006714 0 1522 92 B Flange Depth 0 05128 0 1224 0 01982 0 14 0 03746 76 I_Flange Width 0 1106 0 03974 0 1815 0 02216 0 1991 97 L Flange Width 0 1988 0 1358 0 2618 0 1202 0 2775 100 R_Bracket_Gauge 0 01627 0 123 0 09045 0 1495 0 1169
268. le name value Range name value Lower bound variable name value Upper bound variable name value Dependent name expression Variable name max Constant name value Local name Starting value for design variable Range of variable to define region of interest Lower bound of Variable Upper bound of Variable Dependent variable Saddle direction flag Value of constant Variable is not global H 5 Miultidisciplinary environment Solver package name name Solver input file name Solver command string Solver append file string Prepro name Prepro command string Prepro input file name Prepro controlnodes file name Prepro coefficient file name Queuer queuer type Interval value Concurrent jobs number Solver variable software package identifier solver input file name solver command line name of file to be appended to input software package identifier pre processor command file pre processor input file control nodes file for Templex shape vector file for Templex queuer for workload scheduling time interval for progress reports number of concurrent jobs Flag for solver variable m 54 bs N S 5 5 S Ne OO COT SOT OO TOO SAA SIE 344 LS OPT Version 2 APPENDIX H QUICK REFERENCE MANUAL H 6 Package identifiers ingrid LS INGRID truegrid TrueGrid templex Templex dyna LS DYNA versions prior to 960 dyna960 LS DYNA Version 960 970 own user defined H 7 Queuer identifiers
269. line Models for Observational Data Volume 59 of Regional Conference Series in Applied Mathematics SIAM Press Philadelphia 1990 Wall L Christiansen T Schwartz R Programming Perl O Reilly amp Associates Inc Cambridge 1991 White H Hornik K Stinchcombe M Universal approximation of an unknown mapping and its derivatives Artificial Neural Networks Approximations and Learning Theory H White ed Oxford UK Blackwell 1992 Wilson B Cappelleri D J Frecker M I and Simpson T W Efficient Pareto Frontier Exploration using surrogate approximations Optimization and Engineering 2 1 pp 31 50 2001 Xu Q S Liang Y Z Fang K T The effects of different experimental designs on parameter estimation in the kinetics of a reversible chemical reaction Chemometrics and Intelligent Laboratory Systems 52 pp 155 166 2000 Yamazaki K Han J Ishikawa H Kuroiwa Y Maximation of crushing energy absorption of cylindrical shells simulation and experiment Proceedings of the OPTI 97 Conference Rome Italy September 1997 Ye K Li W Sudjianto A Algorithmic construction of optimal symmetric Latin Hypercube designs Journal of Statistical Planning and Inferences 90 pp 145 159 2000 Zang T A Green L L Multidisciplinary Design Optimization techniques Implications and opportunities for fluid dynamics research AZAA Paper 99 3798 1999 290 LS OPT Version 2 Appendix A LS DYNA ASCII
270. lly Section 13 4 The recommended point selection scheme for polynomial response surfaces is the D optimal scheme Section 95 CHAPTER 9 METAMODELS AND POINT SELECTION 9 1 2 Neural Networks and Kriging To apply neural network or Kriging approximations select the appropriate option in the Metamodel field in LS OPTui See The recommended Point Selection Scheme for neural networks and Kriging is the space filling method The user can select either a sub region local approach or update the set of points for each iteration to form a global approximation An updated network is fitted to all the points See Section P for more detail on updating Please refer to Section 4 5 for recommendations on how to use neural network and Kriging approximations 9 2 Point Selection Schemes 9 2 1 Overview able 9 1 shows the available point selection schemes experimental design methods Table 9 1 Point selection schemes Experiment Description Identifier Remark Linear Koshal lin_koshal For polynomials Quadratic Koshal quad_koshal Central Composite composite D optimal designs D optimal dopt Polynomials Factorial Designs 2 2toK 3 3toK 11 11toK 96 LS OPT Version 2 CHAPTER 9 METAMODELS AND POINT SELECTION Random designs Latin Hypercube Monte Carlo latin_hypercube monte carlo For probabilistic analysis or random search
271. lver command run_this script solver input file setup jou LS OPT Version 2 83 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR 7 3 Preprocessors The preprocessor must be identified as well as the command used for the execution The command file executed by the preprocessor to generate the input deck must also be specified The preprocessor specification is valid for the current solver environment Command file syntax prepro software package identifier prepro command prepro program name prepro input file pre file name The interfacing of a preprocessor involves the specification of the design variables input files and the preprocessor run command Interfacing with LS INGRID TrueGrid and AutoDV is detailed in this section The identification of the design variables in the input file is detailed in Section 1 7 3 1 LS INGRID The identifier in the prepro section for the use of LS INGRID is ingrid The file ingridopt inp is created from the LS INGRID input template file Example the preprocessor software to be used prepro ingrid the command to execute the preprocessor prepro command ingrid the input file to be used by the preprocessor prepro input file p9i This will allow the execution of LS INGRID using the command ingrid i ingridopt inp d TTY The file ingridopt inp is created by replacing the lt lt name gt gt keywords in the p9i file with the relevant values of the desig
272. me_to_ bumper zero response PULSE 1 expression Integral Apillar_ velocity _average t 0 time_to_ bumper zero time_to_bumper zero response time _to_ zero velocity expression Lookup global_velocity t 0 response velocity final expression Apillar_ velocity average time _to zero velocity response PULSE 2 expression Integral Apillar_ velocity _average t time_to_bumper_ zero time_to_zero_velocity time_to_ zero velocity time to bumper zero Example 3 constant Event_time 200 Results from a physical experiment history experiment_vel cp users john experiments chest_results LsoptHistory LS DYNA results history velocity DynaASCII nodout X_VEL 12667 TIMESTEP response RMS error expression Integral experiment_vel velocity 2 0 Event_ time Example 4 In this example a user defined program the post processor LS PREPOST is used to produce a history file from the LS DYNA database The LS PREPOST command file get_ force open d3plot d3plot ascii rcforc open rcforc 0 ascii rcforc plot 4 Ma 1 xyplot 1 savefile xypair LsoptHistory 1 deletewin 1 quit produces the LsoptHistory file history Force lspost c get_force response Forcel expression Force 002 response Force2 expression Force 004 response Force3 expression Force 006 response Force4 expression Force 008 108 LS OPT Version 2
273. mes Entities such as variables responses etc are identified by their names The following entities must be given unique names solver constant variable dependent history response composite objective constraint A name is specified in single quotes e g solver dyna DYNA_side impact constant Young modulus 50000 0 variable Delta 1 5 dependent new modulus Young modulus Delta History y vel DynaASCII nodout Y VEL 187705 TIMESTEP 0 SAE 30 Response x acc DynaASCII rbdout X ACC 21 AVE composite deformation type targeted composite sqdef sqrt deformation objective deformation composite deformation 1 0 constraint Mass response Mass In addition to numbers 0 9 upper or lower case letters a name can contain any of the following characters The leading character must be alphabetical Spaces are not allowed Note Because mathematical expressions can be constructed using various entities in the same formula duplication of names is not allowed 66 LS OPT Version 2 CHAPTER 5 GRAPHICAL USER INTERFACE AND COMMAND LANGUAGE 5 3 2 Command lines Preprocessor commands solver commands or response extraction commands are enclosed in double quotes e g SPECIFICATION OF PREPROCESSOR AND SOLVER preprocessor command usr ls dyna ingrid solver command alpha6_2 usr ls dyna bin ls dyna_9402_dec_40 IDENTIFICATION OF THE RESPONSE re
274. mission implies the entire model intercept The FLD curve value at amp 0 negative_slope The absolute value of the slope of the FLD curve value at amp lt 0 positive_slope The absolute value of the slope of the FLD curve value at amp gt 0 Example Specify the FLD Constraint to be used Response FLD DynaFLD 1 2 3 0 25 1 833 0 5 124 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS General FLD Constraint A more general FLD criterion is available if the forming limit is represented by a general curve Any of the upper lower or middle shell surfaces can be considered Remarks 1 A piece wise linear curve is defined by specifying a list of interconnected points The abscissae amp of consecutive points must increase or an error termination will occur Duplicated points are therefore not allowed 2 The curve is extrapolated infinitely in both the negative and positive directions of amp The first and last segments are used for this purpose 3 The computation of the constraint value is the same as shown in Figure 10 4 Command file syntax DynaFLDg LOWER CENTER UPPER pl p2 pn load curve id The following must be defined for the model and FLD curve Table 10 10 DynaFLDg item description Item Description LOWER Lower surface of the sheet UPPER Upper surface of the sheet CENTER Middle surface of the sheet pl pn Part numbers of the model Omission implies the entire m
275. mproved design is predicted with the constraint value stress changing from an approximate 4 884 severely violated to 1 0 the constraint is active Due to inaccuracy the actual constraint value of the optimum is 0 634 The weight changes from 2 776 to 4 137 3 557 computed to accommodate the violated stress DESIGN POINT Weight Lower Bound Value Upper Bound 539 4 1 1 6 Unscaled ee Predicted ee Be 4 137 3 557 4 137 1 0 6338 1 176 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS OBJECTIVE Il W ul u N Computed Value Predicted Value 4 137 OBJECTIVE FUNCTIONS OBJECTIVE NAME Computed Predicted WT a ee Weight 3 557 4 137 1 ee oe gas aa esse re e CONSTRAINT FUNCTIONS CONSTRAINT NAME Computed Predicted Lower Upper Viol eg Beer ee Stress 0 6338 1 1e 30 1 no un oor a ee ee CONSTRAINT VIOLATIONS Computed Violation Predicted Violation CONSTRAINT NAME Lower Upper Lower Upper A es pare ere ee a eee Stress nn Seren MAXIMUM VIOLATION Computed Predicted Quantity Constraint Value Constraint Value re sense ee Zee ee Maximum Violation Stress 0 Stress 6 995e 08 Smallest Margin Stress 0 3662 Stress 6 995e 08 17 1 3 Updating the approximation to second order To improve the accuracy a second run is conducted us
276. must be selected Basis experiment The set of points from which the D optimal design points must be chosen e g 3tok Number basis experiments The number of basis experimental points only random latin hypercube and space filling The default basis experiment for the D optimal design is 5 for quadratic and elliptic approximations and 3 for linear The default number of points selected is int 1 5 n 1 1 for linear int 1 5 2 1 1 for elliptic int 0 75 n n 2 1 for interaction and int 0 75 n 1 n 2 1 for quadratic As a result about 50 more points than the minimum required are generated If the user wants to override this number of experiments the command solver number experiments is required after solver order function order 9 2 3 Latin Hypercube Sampling The Latin Hypercube point selection scheme is typically used for probabilistic analysis The Latin Hypercube design is also useful to construct a basis experimental design for the D optimal design for a large number of variables where the cost of using a full factorial design is excessive E g for 15 design variables the number of basis points for a 3 design is more than 14 million The Monte Carlo Latin Hypercube and Space Filling point selection schemes require a user specified number of experiments Even if the Latin Hypercube design has enough points to fit a response surface there is a likelihood of obtaining poor pred
277. n Derivative Displacement t 0 08 response Velocity2 expression Derivative Displacement t COMPOSITE FUNCTIONS composites 1 composite 5 Integ6 Integ3a 4 Maximum11 Integ2 2 5 LS OPT Version 2 329 APPENDIX E MATHEMATICAL EXPRESSIONS OBJECTIVE FUNCTIONS objectives 1 objective Integ6 S CONSTRAINT FUNCTIONS constraints 1 constraint Integl iterate 0 STOP Example 2 constant vO 15 65 history engine velocity DynaASCII nodout X_VEL 73579 TIMESTEP 0 0 SAE 30 history Apillar_velocity_1 DynaASCII nodout X VEL 41195 TIMESTEP 0 0 SAE 30 history Apillar_velocity 2 DynaASCII nodout X VEL 17251 TIMESTEP 0 0 SAE 30 history global velocity DynaASCII glstat X VEL 0 TIMESTEP 0 0 history Apillar velocity average Apillar velocity 1 Apillar_ velocity 2 2 Find the time when the engine velocity 0 response time to engine zero expression Lookup engine_velocity t 0 Find the average velocity at time of engine velocity 0 response vel A engine zero expression Apillar velocity average time_to engine zero Integrate the average A pillar velocity up to zero engine velocity Divide by the time to get the average Sane PULSE _1 expression Integral Apillar_velocity_ average t ee ER Find the time at which the global velocity is zero
278. n expression t_lower t_ upper Inverse function t f fmin LookupMax expression t_lower t_ upper Inverse function t f fmax The arguments used in the expressions have the following explanations Argument Explanation Symbol Type t lower lower limit of integration or range a generic t_upper upper limit of integration or range b generic variable integration variable g t generic expression history defined as an expression string f t generic value value for which lookup is required F generic T constant specific time T generic Generic implies that the quantity can be an expression a previously defined entity or a constant number An entity which may be specified in an expression can be any previously defined LS OPT entity Thus constant variable dependent history response and composite are acceptable An expression is given in double quotes e g Displacement t E 4 Reserved variable names Name Explanation t Time LowerLimit 0 0 UpperLimit Maximum event time over all histories of all solvers Omitting the lower and upper bounds implies operation over the entire available history LS OPT Version 2 325 APPENDIX E MATHEMATICAL EXPRESSIONS The Lookup function allows finding the value of t for a specified value of f t F If such a value cannot be found the largest value of in the history is returned The LookupMin and LookupMax
279. n considered for this study is 2 0 large in r1 72 and r3 and is centered about 1 5 1 5 15 The FLD constraint formulation tested in this phase is based on the maximum perpendicular distance of a point violating the FLD constraint to the FLD curve see Section 10 9 2 The LS OPT command file used to run the problem is Sheet Minimization of Maximum Tool Radius Author Aaron Spelling Created on Wed May 29 19 23 20 2002 DESIGN VARIABLES variables 3 Variable Radius_1 1 5 Lower bound variable Radius_1 1 Upper bound variable Radius_1 4 5 Range Radius_1 4 Variable Radius_2 1 5 Lower bound variable Radius_2 1 Upper bound variable Radius_2 4 5 Range Radius_2 4 Variable Radius_3 1 5 Lower bound variable Radius_3 Upper bound variable Radius_3 4 5 Range Radius_3 4 solvers 1 responses 2 NO HISTORIES ARE DEFINED DEFINITION OF SOLVER DYNA1 solver dyna DYNAI solver command lsdyna solver input file trugrdo solver append file ShellSetList prepro truegrid prepro command net src ultra4 4 common hp tg2 1 tg prepro input file m3 tg opt UW UW RESPONSES FOR SOLVER DYNA1 218 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS response Thinning 1 0 DynaThick REDUCTION MAX response Thinning linear response FLD 1 0 DynaFLDg CENTER 1 2 3 90 response FLD linear NO HISTORIES DEFINED FOR SOLVER DYNA1 HISTORIES AN
280. n dopt 1 432 200 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS Solver Basis experiment 5toK Solver Number experiment 5 5 JOB INFO concurrent jobs 1 DEFINITION OF SOLVER RANDOM_ solver dyna RANDOM _ solver command lsdyna solver input file car5s k solver append file rigid2 LOCAL VARIABLES solver variable t_hood_s solver variable t_bumper_s RESPONSES FOR SOLVER 1 response Intru_2_s 1 0 DynaASCII Nodout X DISP 432 Timestep response Intru_1_s 1 0 DynaASCII Nodout X DISP 167 Timestep response Intrusion s expression Intru 1 s Intru 2_s response HIC_s 1 0 DynaASCII Nodout HIC15 9810 1 432 response Intrusion _dev DynaStat STDDEV Intrusion s response HIC dev DynaStat STDDEV HIC_s EXPERIMENTAL DESIGN Solver Experimental design latin_hypercube Solver Number experiment 20 JOB INFO concurrent jobs 1 COMPOSITES composites 1 composite Intrusion Intru 1 m Intru_2 m 6 Intrusion_ dev OBJECTIVE FUNCTIONS objectives 1 objective HIC m 1 CONSTRAINT DEFINITIONS constraints 1 constraint Intrusion upper bound constraint Intrusion 550 OPTIMIZATION METHOD S iterate param design 0 01 iterate param objective 0 01 LS OPT Version 2 201 CHAPTER 17 EXAMPLE PROBLEMS iterate 15 STOP Remarks 1 Note that the stochastic random solver name must be RANDOM 2 The dependent var
281. n point 72 LS OPT Version 2 CHAPTER 6 PROGRAM EXECUTION Statistical properties of each response in Work AnalysisResults Parameters of the neural net Kriging StatResults Nena surface of function with name funcname Poet A more detailed description of the database is available in Appendix D Thefollowing files are output files Table 6 2 Output files lsopt_input Input in a formatted style Work lsopt output Results and some logging information Work Table of the objective and constraint history design values for each iteration e g for Work plotting Table of the design variables responses history variables and composites for each iteration e g Work for plotting 6 8 Using a table of existing results to conduct an analysis A table of existing results can be used to conduct an optimization or probabilistic analysis The required format for the file is as described for the AnalysisResults file in A single line is required for each point to be analyzed The file must be named AnalysisResults PRE solver name and placed in the work directory 6 9 Log files and status files The LS OPT logfile is named lsopt_ logfile Status files prepro replace started finished history n response n and EXIT STATUS are placed in the run directories to indicate the status of the solution progress The directories can be cleaned to free disk space but selected status
282. n using LS DYNA the user can also view the progress time history of the analysis by selecting one of the available quantities Time Step Kinetic Energy Internal Energy etc LS OPT Version 2 143 CHAPTER 12 RUNNING THE OPTIMIZATION PROBLEM 144 LS OPT Version 2 13 Viewing Results This chapter describes the viewing of metamodeling accuracy optimization history trade off and ANOVA results The View panel in LS OPTui is used to view the results of the optimization process The results include the metamodelling accuracy data optimization history of the variables dependents responses constraints and objective s Trade off data can be generated using the existing response surfaces and ANOVA results can be viewed There are three options for viewing accuracy and tradeoff anthill plots namely viewing data for the current iteration for all previous iterations simultaneously all iterations see e g Figure 13 1 The last option will also show the last verification point optimal design in green 13 1 Metamodel accuracy The accuracy of the metamodel fit is illustrated in a Computed vs Predicted plot Figure 13 1 By clicking on any of the red squares the data of the selected design point is listed For LS DYNA results LS PREPOST can then be launched to investigate the simulation results The results of each iteration are displayed separately using the slider bar The iterations can be viewed simultaneously by sel
283. n variables 7 3 2 TrueGrid The identifier in the prepro section for the use of TrueGrid is truegrid This will allow the execution of TrueGrid using the command prepro program name i TruOpt inp The file TruOpt inp is created by replacing the lt lt name gt gt keywords in the TrueGrid input template file with the relevant values of the design variables Registered Trademark of XYZ Scientific Applications Inc Registered Trademark of Altair Engineering Inc 84 LS OPT Version 2 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR Example the preprocessor software to be used prepro truegrid the command to execute the preprocessor prepro command tgx the input file to be used by the preprocessor prepro input file cyl These lines will execute TrueGrid using the command tgx i cy1 having replaced all the keyword names lt lt name gt gt in cyl with the relevant values of the design variables The TrueGrid input file requires the line write end at the very end 7 3 3 AutoDV The geometric preprocessor AutoDV can be interfaced with LS OPT which allows shape variables to be specified The identifier in the prepro section for the use of AutoDV is templex the name of an auxiliary product Templex The use of AutoDV requires several input files to be available 1 Input deck At the top the variables are defined as DVAR1 DVAR2 etc along with their current values The default name is input
284. nd 17 3 2 A first approximation In the first iteration a quadratic approximation is chosen from the beginning The ASCII database is suitable for this analysis as the energy and impact force can be extracted from the glstat and rwforc databases respectively Five processors are available The region of interest is arbitrarily chosen to be about half the size of the design space The following LS OPT command input deck was used to find the approximate optimum solution Cylinder Impact Problem Created on Thu Jul 11 11 37 33 2002 DESIGN VARIABLES variables 2 Variable Radius 75 Lower bound variable Radius 20 Upper bound variable Radius 100 Range Radius 50 Variable Wall Thickness 3 Lower bound variable Wall_Thickness 2 Upper bound variable Wall_Thickness 6 LS OPT Version 2 205 CHAPTER 17 EXAMPLE PROBLEMS Range Wall_Thickness 2 solvers 1 responses 2 NO HISTORIES ARE DEFINED DEFINITION OF SOLVER RUN1 solver dyna960 RUNI solver command lsdyna solver input file trugrdo prepro truegrid prepro command net src ultra4 4 common hp tg2 1 tg prepro input file cyl2 RESPONSES FOR SOLVER RUN1 response Internal Energy 1 0 DynaASCII Glstat I_ Ener 0 Timestep response Internal Energy quadratic response Rigid Wall Force 1 0 DynaASCII rwforc normal 1 ave response Rigid Wall Force quadratic NO HISTORIES DEFINED FOR SOLVER RUN1 OBJECTIV
285. nd b min a b minimum of a and b sqrt a square root exp a e pow a b a log a natural logarithm log10 a base 10 logarithm sin a sine cos a cosine tan a tangent asin a arc sine acos a arc cosine atan a arc tangent atan2 a b arc tangent of a b sinh a hyperbolic sine cosh a hyperbolic cosine tanh a hyperbolic tangent asinh a arc hyperbolic sine acosh a arc hyperbolic cosine atanh a arc hyperbolic tangent sec a secant csc a cosecant ctn a cotangent 324 LS OPT Version 2 APPENDIX E MATHEMATICAL EXPRESSIONS E 3 Special functions Special response functions can be specified to apply to response histories These include integration minima and maxima and finding the time at a specific value of the function General expressions in double quotes can be used for limits and for the integration variable Histories must be defined as strings in double quotes and functions of time using the symbol t e g Velocity t Expression Symbols Integral expression t lower t upper variable b R ee Ode Derivative expression T constant Af At _ Of dtl _ Min expression t_lower t_ upper fain min f t t Max expression t_lower t upper foa maxl f 2 t Initial expression First function value on record Final expression Last function value on record Lookup expression value Inverse function t f F LookupMi
286. ng LS PREPOST can then be launched to investigate the simulation results The optimization history of a variable dependent response constraint objective multi objective or the approximation error parameters of pure responses not composites or expressions can be plotted by clicking LS OPT Version 2 146 CHAPTER 13 VIEWING RESULTS K viewer File Type of Plot wv Response Surface Accuracy Optimization History v Tradeoff Entity to Monitor Response Acc_max Response Mass Response Intru_2 N Response Intru_1 Response Frequency Value to Plot Value oO x o ai a i Qa a a DE wv RMS Error wv MAX Error wv R2 Error En RE Fee oe EEE ot SESS Se E N GEIS EI pe eee pacar Pad Number of Iterations Figure 13 2 Optimization History plot in View panel in LS OPTui 13 3 Trade off and anthill plots The results of all the simulated points appear as dots on the trade off plots This feature allows the two dimensional plotting of any variable response against any other variable response Trade off studies can also be conducted based on the results of an optimization run This is because the response surfaces for each response are at that stage available at each iteration for rapid evaluation Trade off is performed in LS OPTwi using the View panel and selecting Trade off Figure 13 3 Trade off curves can be developed using either constraints or obj
287. ng points R is the L x L correlation matrix with Rxx the correlation function between data points x and x R is symmetric positive definite with unit diagonal Two commonly applied correlation functions used are Tal Exponential R Je and k 1 0 4 Gaussian R e k l 32 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY where n is the number of variables and dy xp xy the distance between the k components of points x and x There are n unknown values to be determined The default function in LS OPT is Gaussian Once the correlation function has been selected the predicted esitimate of the response y x is given by p B rOR YEL where r x is the correlation vector length L between a prediction point x and the L sampling points y represents the responses at the L points and f is an Z vector of ones in the case that f x is taken as a constant The vector r and scalar 2 are given by r x Ro ROAD ROT A B f TR Bw TR ly The estimate of variance from the underlying global model is pee Y A The maximum likelihood estimates for k 1 n can be found by solving the following constrained maximization problem itae In R Max subject to O gt 0 A where both and R are functions of This is the same as minimizing a2 1 o IR st gt 0 This optimization problem is solved using the LFOPC algorithm Section
288. nimize the centered L2 discrepancy measure The discrepancy is a quantitative measure of non uniformity of the design points on an experimental domain Intuitively for a uniformly distributed set in the n dimensional cube 7 0 1 we would expect the same number of points to be in all subsets of Z having the same volume Discrepancy is defined by considering the number of points in the subsets of Z Centered L2 CL2 takes into account not only the uniformity of the design points over the n dimensional box region J but also the uniformity of all the projections of points over lower dimensional subspaces 16 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY E12 En i 3 5 i 1 P j l S 0 5 apa 2 22 1 Is 0 5 S 0 5 S S ed fi F z P k 1 Pi 1 P j1 2 6 6 Space filling designs In the modeling of an unknown nonlinear relationship when there is no persuasive parametric regression model available and the constraints are uncertain one might believe that a good experimental design is a set of points that are uniformly scattered on the experimental domain design space Space filling designs impose no strong assumptions on the approximation model and allow a large number of levels for each variable with a moderate number of experimental points These designs are especially useful in conjunction with nonparametric models such as neural networks feed forward networks r
289. nse In the following example the objective is to minimize the maximum of Left Knee Force or Right Knee Force The displacement and energy constraints are strict Example This formulation minimizes the maximum knee force Because the knee forces are always positive the objective will be ignored and the knee force minimized Objective composite Knee Forces type weighted composite Knee Forces response Left Knee Force 0 5 composite Knee Forces response Right Knee Force 0 5 objective Knee Forces S Constraints OEE EEE see SLACK Constraint Left_Knee_Force Upper bound constraint Left_Knee_Force 0 138 LS OPT Version 2 CHAPTER 11 OBJECTIVES AND CONSTRAINTS Constraint Right Knee Force Upper bound constraint Right Knee Force 0 STRICT Constraint Left_Knee Displacement Lower bound constraint Left Knee Displacement 8133 Constraint Right Knee Displacement Lower bound constraint Right Knee Displacement 81 33 5 Constraint Kinetic Energy Upper bound constraint Kinetic Energy 154000 Remarks 1 The objective function is ignored if the problem is infeasible 2 The variable bounds of both the region of interest and the design space are always hard Soft constraints will be strictly satisfied if a feasible design is possible If a feasible design is not possible the most feasible design will b
290. nstant angle 30 5 DESIGN VARIABLE DEFINITIONS variables 2 Variable x1 45 Lower bound variable x1 10 Upper bound variable x1 50 Variable x2 45 Lower bound variable x2 10 Upper bound variable x2 50 DEPENDENT VARIABLES dependents 2 dependent 11 lowerlimit 1000 dependent ul upperlimit 1000 328 LS OPT Version 2 APPENDIX E MATHEMATICAL EXPRESSIONS 5 HISTORIES history 3 history hisl cat hisl gt LsoptHistory history his2 cat his2 gt LsoptHistory history his3 hisl t his2 t 2 S RESPONSES responses 42 response LOWER expression LowerLimit response UPPER expression UpperLimit response UL expression ul response First expression Initial his1 t response Last expression Final his1 t response Last3 expression Final hisl t his2 t 2 response Max1 expression Max his1 t response Max2 expression Max hisl1 t 11 1 0 response Maximuml1 expression Max hisl1 t 11 ul response Maximum32 expression Max his3 t 11 ul response Minimum32 expression Min his3 t 11 u response Inversell expression Lookup his1 t 75 response Inverse21 expression Lookup his2 t 5 response Inverse3l expression Lookup his3 t ne response Inv
291. nt Objective weights Number of objectives Objective values Number of objectives Variables Number of variables Responses Number of responses Multi objective Constraint values Number of constraints Composite values Number of composites Max constraint violation Constants Number of constants Dependents Number of dependents 322 LS OPT Version 2 Appendix E Mathematical Expressions Mathematical expressions are available for the following entities dependent history response composite multiobjective E l Syntax rules 1 Mathematical expressions are placed in curly brackets in the command file or in double angular brackets e g lt lt Thickness 25 4 gt gt in the input template files 2 Expressions consist of parameters and constants A parameter can be any previously defined entity 3 Expressions can be wrapped to appear on multiple lines 4 Mathematical expressions can be used for any floating point number e g upper bound of constraint convergence tolerance objective weight etc 5 An expression is limited to 1024 characters 323 APPENDIX E MATHEMATICAL EXPRESSIONS E 2 Intrinsic functions int a integer nint a nearest integer abs a absolute value mod a b remainder of a b sign a b transfer of sign from b to a max a b maximum of a a
292. nterfaces The minimum of two maxima is obtained in the objective multi criteria or multi objective problem 17 8 1 Problem statement Figure 17 45 shows the finite element model of a typical automotive instrument panel IP R For model simplification and reduced per iteration computational times only the driver s side of the IP is used in the analysis and consists of around 25 000 shell elements Symmetry boundary conditions are assumed at the centerline and to simulate a bench component Bendix test body attachments are assumed fixed in all 6 directions Also shown in Figure 17 45 are simplified knee forms which move in a direction as determined from prior physical tests As shown in the figure this system is composed of a knee bolster steel plastic or both that also serves as a steering column cover with a styled surface and two energy absorption EA brackets usually steel attached to the cross vehicle IP structure The brackets absorb a significant portion of the lower torso energy of the occupant by deforming appropriately Sometimes a steering column isolator also known as a yoke may be used as part of the knee bolster system to delay the wrap around of the knees around the steering column The last three components are non visible and hence their shape can be optimized The 11 design variables are shown in The three gauges and the yoke cross sectional radius are also considered in a separate sizing 4 variable optimization
293. nvergence in Phase 1 The penalty parameters have default values as listed in the User s manual Section 16 3 In addition the step size of the algorithm and termination criteria of the subproblem solver are listed The values of the responses are scaled with the values at the initial design The variables are scaled internally by scaling the design space to the 0 1 interval The default parameters in LFOPC as listed in Section 16 3 should therefore be adequate The termination criteria are also listed in Section In the case of an infeasible optimization problem the solver will find the most feasible design within the given region of interest bounded by the simple upper and lower bounds A global solution is attempted by multiple starts from the experimental design points 2 12 Successive response surface method SRSM The purpose of the SRSM method is to allow convergence of the solution to a prescribed tolerance The SRSM method uses a region of interest a subspace of the design space to determine an approximate optimum A range is chosen for each variable to determine its initial size A new region of interest centers on each successive optimum Progress is made by moving the center of the region of interest as well as reducing its size Figure 2 4 shows the possible adaptation of the subregion pan pan amp zoom X 0 subregion range r range r subregion a x2 b x2 c x2 Figure 2 4 Adaptat
294. oblem description 10 4 2 Assigning design variable or response components to the composite Command file syntax composite name response response name value lt l gt scale scale factor lt l gt composite name variable variable name value scale scale factor lt 1 gt The value is the target value for type targeted and the weight value for the type weighted The scale_factor is a divisor Example composite damage type targeted composite damage response intrusion 3 20 scale 30 composite damage response intrusion 4 35 scale 25 2 2 for the composite function Fuge A i A The equivalent code using the expression composite is composite damage sqrt intrusion 3 20 30 2 intrusion 4 35 25 2 Example S X10 gt x9 composite C9 type weighted composite C9 variable x 9 el composite C9 variable x_10 als constraint C9 Lower bound constraint C9 0 for the composite function which defines the inequality x19 gt xo The equivalent code using the expression composite is Sooo SOLO s X9 OSs en a composite C9 x 10 x 9 114 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS constraint C9 Lower bound constraint C9 0 Needless to say this is the preferable way to describe this composite If weights are required for the targeted function an addit
295. obs 1 RESPONSES FOR SOLVER CRASH response Acc max 1 0 DynaASCII Nodout X_ACC 432 Max SAE 60 response Mass 1 0 DynaMass 2 3 4 5 6 7 MASS response Intru_2 1 0 DynaASCII Nodout X DISP 432 Timestep response Intru_1 1 0 DynaASCII Nodout X DISP 184 Timestep response HIC 1 0 DynaASCII Nodout HIC15 9810 1 432 SSSSSSSSSSSSSSSSssSSsssssssssssss SOLVER NVH SSSSSSSSSSSSSSSSssSSsssssssssssss DEFINITION OF SOLVER NVH solver dyna960 NVH solver command 1s970 double solver input file car6 NVH k solver concurrent jobs 1 solver experiment duplicate CRASH RESPONSES FOR SOLVER NVH response response response Frequency Mode 1 0 Generalized_Mass 1 0 DynaFreq 15 FREQ DynaFreq 15 NUMBER 1 0 DynaFreq 15 GENMASS composites 4 COMPOSITE RESPONSES 5 composite Intrusion type weighted composite Intrusion response Intru_2 1 scale 1 composite Intrusion response Intru_1 1 scale 1 composite HIC_scaled type targeted composite HIC_scaled response HIC 0 scale 900 weight 1 composite Freq_scaled type targeted composite Freq_scaled response Frequency 0 scale 3 weight 1 5 COMPOSITE EXPRESSIONS composite Intrusion_scaled Intrusion 500 OBJECTIVE FUNCTIONS objectives 1 objective Mass CONSTRAINT DEFINITIONS constraints 3 constraint HIC_scaled 1 LS OPT Version 2 251 CHAPTER 17 EXAMPLE PROBLEMS
296. odel load_curve_id Identification number of a load curve in the LS DYNA input file The DEFINE_CURVE keyword must be used Refer to the LS DYNA User s Manual for an explanation of this keyword Example Specify the general FLD Constraint to be used Response FLDL DynaFLDg LOWER 1 2 3 23 Response FLDU DynaFLDg UPPER 1 2 3 23 Response FLDC DynaFLDg CENTER 23 For all three specifications load curve 23 is used In the first two specifications only parts 1 2 and 3 are considered LS OPT Version 2 125 CHAPTER 10 HISTORY AND RESPONSE RESULTS Remarks 1 The interface program produces an output file FLD_curve which contains the amp and amp values in the first and second columns respectively Since the program first looks for this file it can be specified in lieu of the keyword specification The user should take care to remove an old version of the FLD curve if the curve specification is changed in the keyword input file If a structured input file is used for LS DYNA input data FLD_curve must be created by the user 2 The scale factor and offset values feature of the DEFINE CURVE keyword are not utilized 10 9 3 Principal Stress Any of the principal stresses or the mean can be computed The values are nodal stresses Command file syntax DynaPStress S1 S2 S3 MEAN pl p2 pn MIN MAX AVE Table 10 11 DynaPStress item description Item Description S1 S2
297. odel is appropriate and to test the goodness of fit against some statistical standard There are several error measures available to determine the accuracy of the model Among them are LS OPT Version 2 27 CHAPTER 2 OPTIMIZATION METHODOLOGY gt MSE f y IP 2 43 RMS MSE nMSE us oO nRMS a Oo P prane Xi ViVi R P TE 3 S Zo y i l i l where P denotes the number of data points y is the observed response value target value is the model s prediction of response 3 is the mean average value of p y is the mean average value of y and G is P aA TE De 6 2 44 PSL P L given by Mean squared error MSE for short and root mean squared error RMS summarize the overall model error Unique or rare large error values can affect these indicators Sometimes MSE and RMS measures are normalized with sample variance of the target value see formulae for nMSE and nRMS to allow for comparisons between different datasets and underlying functions R and R are relative measures The coefficient of multiple determination R R square is explained variance relative to total variance in the target value This indicator is widely used in linear regression analysis R represents the amount of response variability explained by the model R is the correlation coefficient between the network response and the target It is a measure of how well the variation in the
298. ombining the results of successive iterations 2 10 1 Neural network approximations Neural methods are natural extensions and generalizations of regression methods Neural networks have been known since the 1940 s but it took the dramatic improvements in computers to make them practical B Neural networks just like regression techniques model relationships between a set of input variables and an outcome They can be thought of as computing devices consisting of numerical units neurons whose inputs and outputs are linked according to specific topologies A neural model is defined by its free parameters the inter neuron connection strengths weights and biases These parameters are typically Available in Version 2 1 LS OPT Version 2 25 CHAPTER 2 OPTIMIZATION METHODOLOGY learned from the training data by some appropriate optimization algorithm The training set consists of pairs of input design vectors and associated outputs responses The training algorithm tries to steer network parameters towards minimizing some distance measure typically the mean squared error MSE of the model computed on the training data Several factors determine the predictive accuracy of a neural network approximation and if not properly addressed may adversely affect the solution For a neural network as well as for any other data derived model the most critical factor is the quality of training data In practical cases we are limited
299. omials although perhaps less accurate are highly suitable for variable screening Section P 9 At the core these techniques differ in the regression methods that they employ to construct the surrogate models The polynomial response surface method uses linear regression while neural networks use nonlinear regression methods requiring optimization algorithms Kriging is considered to be a Gaussian Process 15 which uses Bayesian regression also requiring optimization Not mentioned so far are other types of metamodeling techniques such as Space Mapping ke This technique make use of coarse approximate models which are iteratively refined using fine model simulations to improve the coarse model approximation locally In Space Mapping any of the first three approximation types can be used as coarse models When using polynomials the user is faced with the choice of deciding which monomial terms to include In addition polynomials by way of their nature as Taylor series approximations are not natural for the creation of updateable surfaces This means that if an existing set of point data is augmented by a number of new points which have been selected in a local subregion e g in the vicinity of a predicted optimum better information could be gained from a more flexible type of approximation that will keep global validity while allowing refinement in a subregion of the parameter space Such an approximation provides a more natural approach for c
300. omposite Residual response acc35 4 227 923 scale 100 composite Residual response acc40 4 182 374 scale 100 composite Residual response acc45 4 121 8 scale 100 composite Residual response acc50_4 72 7288 scale 100 composite Residual response vell0_4 3 49244 scale 1 composite Residual response vel15 4 3 19588 scale 1 composite Residual response vel20_4 2 71324 scale 1 composite Residual response vel25 4 1 9779 scale 1 composite Residual response vel30 4 0 973047 scale 1 composite Residual response vel35 4 0 169702 scale 1 composite Residual response vel40 4 1 21011 scale 1 composite Residual response vel45 4 1 97152 scale 1 composite Residual response vel50 4 2 44973 scale 1 88 0516 scale 100 71 2695 scale 100 56 4066 scale 100 44 5582 scale 100 37 0841 scale 100 35 0629 scale 100 composite Residual response disp10 4 composite Residual response disp15 4 composite Residual response disp20_ composite Residual response disp25 4 composite Residual response disp30 4 composite Residual response disp35 4 LS OPT Version 2 233 CHAPTER 17 EXAMPLE PROBLEMS composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composite composit
301. on can be made possible Standard interfaces are provided for a number of well known geometrical preprocessors e The LS DYNA interface is comprehensive Response variables of the ASCII and binary databases including the LS DYNA Ver 970 Binout database can be extracted The time dependent responses are available at any time Average maximum and minimum responses over time can be extracted e Time dependent LS DYNA response can be post processed by filtering or integration e Metal forming criteria are provided e Job execution and result extraction can be conducted on remote nodes in parallel Finished jobs are immediately replaced by waiting jobs until completion File transfer to and from the remote nodes is automatic and occurs at standard file transfer ftp speed e Since designs are typically target oriented the post processing utilities were designed to include a comprehensive array of multi criteria optimization features Design trade off curves can be constructed Min Max problems in which the objective is to minimize the maximum value of several variables can be addressed e The design space can be bounded by the design variables as well as the response variables Thus only reasonable designs need to be analyzed E g massive designs need not be incorporated e Miultidisciplinary optimization MDO can be conducted using more than one solver and more than one analysis case for each solver Variables can be defined to be excl
302. on to the unconstrained problem may be approximated by applying the unconstrained minimization algorithm to a penalty function formulation of the original algorithm The LFOPC algorithm uses a penalty function formulation to incorporate constraints into the optimization problem This implies that when constraints are violated active the violation is magnified and added to an augmented objective function which is solved by the gradient based dynamic leap frog method LFOP The algorithm uses three phases Phase 0 Phase 1 and Phase 2 In Phase 0 the active constraints are introduced as mild penalties through the pre multiplication of a moderate penalty parameter value This allows for the solution of the penalty function formulation where the violation of the active constraints are premultiplied by the penalty value and added to the objective function in the minimization process After the 34 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY solution of Phase 0 through the leap frog dynamic trajectory method some violations of the constraints are inevitable because of the moderate penalty In the subsequent Phase 1 the penalty parameter is increased to more strictly penalize violations of the remaining active constraints Finally and only if the number of active constraints exceed the number of design variables a compromised solution is found to the optimization problem in Phase 2 Otherwise the solution terminates having reached co
303. ons upper bound cons constraint vel30_ lower bound cons upper bound cons constraint vel35_ lower bound cons upper bound cons constraint vel40_ lower bound cons upper bound cons constraint vel45 lower bound cons upper bound cons constraint vel50_ lower bound cons upper bound cons constraint displ lower bound cons upper bound cons constraint displ lower bound cons upper bound cons constraint disp2 lower bound cons upper bound cons constraint disp2 lower bound cons upper bound cons constraint disp3 lower bound cons upper bound cons constraint disp3 lower bound cons upper bound cons constraint disp4 lower bound cons upper bound cons constraint disp4 lower bound cons upper bound cons 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 5 train train 0 5 train train 5 5 train train 0 5 train train 5 5 train train 0 5 train train 5 5 train train 0 5 train train 5 5 train train Erg Ta fs u et qT CT Ct Gh Ta tr ET m a qc ct ara Cr er Fe EE ra ER EN cT ct ct ct ie Et acc45 5 acc45 5 acc50_5 acc50_5 vel10_5 vel10_5 vel1l5_5 vell5_5 vel20_5 vel20_5 vel25_5 yvel25 5 vel30_5 vel30_5 vel35_5 yvel35_ 5 vel40 5 vel40 5 vel45 5 vel45 5 vel50_5
304. onstraints 9 move constraint C1 lower bound cons constraint C2 lower bound cons constraint C3 lower bound cons constraint C4 lower bound cons constraint C5 lower bound cons constraint C6 lower bound cons constraint C7 lower bound cons constrain lower bound cons constrain lower bound cons t C8 E eo traint traint traint traint traint traint traint traint traint EXPERIMENTAL DESIGN Order linear Experimental design dopt Basis experiment 3toK Number experiment 17 JOB INFO concurrent jobs 9 C1 1021 Cc3 C4 CORI C 6 IOTI cg c9 iterate param design 0 01 iterate param objective 0 01 iterate 20 STOP LS OPT Version 2 235 CHAPTER 17 EXAMPLE PROBLEMS Maximum violation formulation 5 design variables The LS OPT input file is the same for this formulation as before for all sections but the constraints As the formulation is driven by the violation of the constraints they are defined as equality constraints identical upper and lower bounds In addition the monotonicity constraints are defined as hard by using the strict option CONSTRAINT DEFINITIONS constraints 58 constraint acc10_4 lower bound constraint acc10_4 0 9417 upper bound constraint ac
305. output is explained by the targets If this number is equal to 1 then there is a perfect correlation between targets and outputs Outliers can greatly affect the magnitudes of correlation coefficients Of course the larger the sample size the smaller is the impact of one or two outliers Training accuracy measures MSE RMS R R etc are computed along all the data points used for training As mentioned above the performance of a good model on the training set does not necessarily mean good prediction of new unseen data The objective measures of the prediction accuracy of the model are test errors computed along independent testing points i e not training points This is certainly true provided that we have an infinite number of testing points In practice however test indicators are usable only if treated with appropriate caution Actual problems are often characterized by the limited availability 28 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY of data and when the training datasets are quite small and the test sets are even smaller only quite large differences in performance can be reliably discerned by comparing training and test indicators The generalized cross validation GCV and Akaike s final prediction error FPE provide computationally feasible means of estimating the appropriateness of the model GCV and FPE estimates combine the training MSE with a measure of the model complexity MSE MSE I
306. over all the elements of the selected parts Example Response Thickness 1 DynaThick THICK 1 2 MAXIMUM Response Thickness 1 DynaThick REDU 1 2 MINIMUM 10 9 2 FLD Constraint The FLD constraint is shown in Two cases are distinguished for the FLD constraint e The values of some strain points are located above the FLD curve In this case the constraint is computed as 122 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS g dmax with dmax the maximum smallest distance of any strain point above the FLD curve to the FLD curve e All the values of the strain points are located below the FLD curve In this case the constraint is computed as 27 Amin with din the minimum smallest distance of any strain value to the FLD curve Figure 10 4 d Constraint Active g dmax amp a FLD Constraint active LS OPT Version 2 123 CHAPTER 10 HISTORY AND RESPONSE RESULTS E Constraint Inactive 2S Amin d amp b FLD Constraint inactive Figure 10 4 FLD curve constraint definition Bilinear FLD Constraint The values of both the principle upper and lower surface in plane strains are used for the FLD constraint Command file syntax DynaFLD pi p2 pn intercept negative slope positive slope The following must be defined for the model and FLD curve Table 10 9 DynaFLD item description Item Description pl pn Part numbers of the model O
307. panel in LS OPTui Figure 12 1 The optimization history is written to the OptimizationHistory file and can be viewed using the View panel 12 1 Number of iterations The number of iterations are specified in the appropriate field in the Run panel If previous results exist LS OPT will recognize this through the presence of results files in the Run directories and not rerun these simulations If the termination criteria described below are reached first LS OPT will terminate and not perform the maximum number of iterations Command file syntax iterate maximum_number of iterations 12 2 Termination criteria The user can specify tolerances on both the design change Ax and the objective function change Af and whether termination is reached if either or both these criteria are met The default selection is and but the user can modify this by selecting or Refer to Section for the modification of the stopping type in the Command File LS OPT Version 2 141 CHAPTER 12 RUNNING THE OPTIMIZATION PROBLEM File Tasks Help Into Solvers Variables Point Selection Histories Responses Objective Constraints Ron View JobID PID Progress Concurrent Jobs BEGIN OPTIMIZATION Tolerance Required gt Queuer for Termination Number of Iterations z i wv Design OR Objective 5 a Design AND Objective _ Specify Starting Iteration zz Design Change Tolerance 8 28858 bon Em 28860 Objective Function Toler
308. parameters The starting values are an estimate of the optimum design These values can be acquired from a present design if it exists The starting design will form the center point of the first region of interest 56 LS OPT Version 2 CHAPTER 4 DESIGN OPTIMIZATION PROCESS 8 Choose a design space This is represented by absolute bounds on the variables that you have chosen The responses may also be bounded if previous information of the functional responses is available Even a simple approximation of the design response can be useful to determine approximate function bounds for conducting an analysis 9 Choose a suitable starting design range for the design variables The range should be neither too small nor too large A small design region is conservative but may require many iterations to converge or may not allow convergence of the design at all It may be too small to capture the variability of the response because of the dominance of noise It may also be too large such that a large modeling error is introduced This is usually less serious as the region of interest is gradually reduced during the optimization process 10 Choose a suitable order for the design approximations when using polynomial response surfaces the default A good starting approximation is linear because it requires the least number of analyses to construct However it is also the least accurate The choice therefore also depends on the available resourc
309. placed by the keywords lt lt name gt gt can be specified LS OPT converts the template to an input deck for the preprocessor or solver by replacing each entire string lt lt name gt gt with a number During run time LS OPT appends a standard input deck name to the end of the execution command In the case of the standard solvers the appropriate syntax is used e g i DynaOpt inp for LS DYNA For a user defined solver the name UserOpt inp is appended The specification of an input file is not required for a user defined solver Appended file Additional solver data can be appended to the input deck using the solver_append_file_name file This file can contain variables to be substituted A report interval can be specified in the command line version of LS OPT only A progress report is given for the runs at regular intervals This report identifies jobs waiting jobs running jobs completed and jobs aborted as well as other statistics such as time remaining and relative progress toward completion 80 LS OPT Version 2 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR File Tasks Help into Solvers Variables Point Selection Histories Responses Objective Constraints Run View CRASH Pre Processor Package Name None NVH Command 1 browse Input File Ls E browse Solver Package Name LS DYNA 960970 auran 1s970 single browse eu icar6_crash k browse Appended File browse RA Solver Name CRASH Aaa Repla
310. point The command must appear in the environment of the solver requiring the duplicate points An experimental design can therefore be duplicated as follows Command file syntax solver experiment duplicate string where string is the name of the master solver in single quotes e g Solver experiment duplicate CRASH CRASH is the master experimental design that must be copied exactly See also the example in Section 17 6 3 LS OPT Version 2 103 CHAPTER 9 METAMODELS AND POINT SELECTION 9 8 Using design sensitivities for optimization Both analytical and numerical sensitivities can be used for optimization The syntax for the solver experimental design command is as follows Experiment Description Identifier Numerical Sensitivity numerical DSA Analytical Sensitivity analytical DSA 9 8 1 Analytical sensitivities If analytical sensitivities are available they must be provided for each response in its own file named Gradient The values one value for each variable in Gradient should be placed on a single line separated by spaces In LS OPTwi the Metamodel Point Selection panel must be set to Sensitivity Type gt Analytical See Example Solver experimental design analytical DSA A complete example is given in Section 9 8 2 Numerical sensitivities To use numerical sensitivities select Numerical Sensitivities in the Metamodel field in LS OPTui and assign the perturbation as a fr
311. ponse function Response linear elliptic quadratic FF kriging Type of approximation Composite name type weighted targeted Type of composite function Composite name expression Defines composite function Composite name response name value scale factor Component definition Composite name variable name value scale factor Component definition Weight value Weight only targeted Objective definition Constraint definition Lower bound on constraint Upper bound on constraint Strictness environment Reasonable space Objective name weight Constraint name Lower bound constraint name value Upper bound constraint name value Strict 0 to 1 slack Move stay move start Maximize Maximize objective value target value for type targeted weight for type weighted H 10 LS DYNA result interfaces DynaMass pl p2 p3 pn mass attribute DynaASCII rslt cmp g u id pos t_at t1 t2 f at n Dyna cn pl p2 pn MIN MAX AVE DynaD3plotHistory cn pl p2 pn ELEMENT NODE MIN MAX AVE DynaThick THICKNESS REDUCTION pl p2 pm MIN MAX AVE DynaFLD pl p2 pn intercept neg slope pos slope DynaFLDg LOWER CENTER UPPER pl p2 pn load curve id DynaPStress S1 S2 S3 MEAN pl p2 pn MIN MAX AVE DynaStat STDDEV response name DynaFreq mode_original FREQ NUMBER GENMASS BinoutHistory history_options BinoutResponse response options Set Binout Mass ASCII results d3plot 121 d3plot 121
312. pring amp Damper energy HG_ENER hourglass energy SI ENER sliding interface energy EW_ENER external work X VEL global x velocity Y_VEL global y velocity Z VEL global z velocity T_VEL velocity 294 LS OPT Version 2 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Joint Element Forces JNTFORC Keyword Description X_FORCE X force Y_FORCE Y force Z_FORCE Z force X_MOMENT X moment Y_MOMENT Y moment Z MOMENT Z moment R_FORCE R force R MOMENT R moment Material Summary MATSUM Keyword Description K ENER Kinetic energy I ENER Internal energy X MOMENTUM Y MOMENTUM Z MOMENTUM MOMENTUM XRB VEL YRB VEL ZRB VEL RB _VEL TK_ENER TI ENER X momentum Y momentum Z momentum Momentum X rigid body velocity Y rigid body velocity Z rigid body velocity Rigid body velocity Total kinetic energy Total internal energy LS OPT Version 2 295 APPENDIX A LS DYNA ASCII RESULT FILES AND COMPONENTS Contact Node Forces NCFORC Keyword Description X_FORCE X force Y_FORCE Y force Z FORCE Z force R_ FORCE R force PRESSURE Pressure Nodal Point Response NODOUT Keyword Description X_DISP X displacement Y_DISP Y displacement Z_DISP Z displacement R_DISP Resultant displacement X VEL X velocity Y_VEL Y velocity Z VEL Z velocity R_VEL Resultant velocity X ACC X acceleration Y_ACC Y acceleration Z ACC Z acceleration R_ACC R
313. r F F 1 p The importance of is therefore estimated by both the magnitude of b as well as the level of confidence in a non zero 3 24 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY The significance of regressor variables may be represented by a bar chart of the magnitudes of the coefficients b with an error bar of length 2Ab for each coefficient representing the confidence interval for a given level of confidence a The relative bar lengths allow the analyst to estimate the importance of the variables and terms to be included in the model while the error bars represent the contribution to noise or poorness of fit by the variable All terms have been normalized to the size of the design space so that the choice of units becomes irrelevant and a reasonable comparison can be made for variables of different kinds e g sizing and shape variables or different material constants 2 10 Metamodeling techniques Metamodeling techniques allow the construction of surrogate design models for the purpose of design exploration such as variable screening optimization and reliability LS OPT provides the capability of using three types of metamodeling techniques namely polynomial response surfaces already discussed see Section 2 5p Neural Networks NN s Section and Kriging Section B 10 2 All three these approaches can be useful to provide a predictive capability for optimization or reliability In addition linear polyn
314. r at each point e Collect the statistics of the responses The user must specify the experimental design strategy sampling strategy to be used in the Monte Carlo evaluation The Monte Carlo Latin Hypercube and space filling experimental designs are available The experimental design will first be computed in a normalized uniformly distributed design space and then transformed to the distributions specified for the design variables Only variables with a statistical distribution will be perturbed all other variables will be considered at their nominal value The following will be computed for all responses e Statistics such as the mean and standard deviation for all responses and constraints e Reliability information regarding all constraints o The number of times a specific constraint was violated during the simulation o The probability of violating the bounds and the confidence region of the probability o A reliability analysis for each constraint assuming a normal distribution of the response The exact value at each point will be used Composite functions referring to responses in more than one discipline will not be computed because the experimental designs will differ across disciplines Command file syntax analyze Monte Carlo 162 LS OPT Version 2 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION Example analyze Monte Carlo 15 5 2 Monte Carlo analysis using a Metamodel The Monte Carlo analysis will
315. r bound constrain cons train acc40 5 lower bound constrain upper bound constrain ara ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ie SE ct ct ct ct ct ct Cr et ct ct ct ct rT a cr 63 vel40 4 1 21011 vel40 4 1 21011 vel45 4 1 97152 vel45 4 1 97152 vel50_ 4 2 44973 vel50_4 2 44973 disp10_4 0 880516 displ0_4 0 880516 disp15_4 0 712695 disp15 4 0 712695 disp20_4 0 564066 disp20_4 0 564066 disp25_4 0 445582 disp25_4 0 445582 disp30 4 0 370841 disp30 4 0 370841 disp35 4 0 350629 disp35 4 0 350629 disp40 4 0 386093 disp40 4 0 386093 disp45_4 0 466906 disp45_4 0 466906 disp50_4 0 578468 disp50_4 0 578468 acc10_5 0 723756 acc10_5 0 723756 acc15_5 0 977004 acc15_5 0 977004 acc20 5 1 68931 acc20 5 1 68931 acc25_ 5 2 64071 acc25 5 2 64071 acc30 5 3 11405 acc30 5 3 11405 acc35_5 2 4428 acc35_5 2 4428 acc40 5 1 30942 acc40 5 1 30942 LS OPT Version 2 231 CHAPTER 17 EXAMPLE PROBLEMS constraint acc45 lower bound cons upper bound cons constraint acc50_ lower bound cons upper bound cons constraint vellO_ lower bound cons upper bound cons constraint vell5 lower bound cons upper bound cons constraint vel20_ lower bound cons upper bound cons constraint vel25 lower bound c
316. r name command will apply to that particular solver This is particularly important when specifying response name commandline commands as each response is assigned to a specific solver and is recovered from the directory bearing the name of the solver See Section 10 e Do not specify the command nohup before the solver command and do not specify the UNIX background mode symbol amp These are automatically taken into account e The solver command name must not be an alias The full path name or the full path name of a script which contains the full solver path name must be specified The LS DYNA restart command will use the same command line arguments as the starting command line replacingthei input file withr runrsf 7 2 2 Interfacing with LS DYNA MPP The LS DYNA MPP Message Passing Parallel version can be run using the LS DYNA option in the Solver window of LS OPTui same as the dyna option for the solver in the command file However the run commands must be specified in a script e g the UNIX script runmpp mpirun np 2 lsdynampp i DynaOpt inp cat dbout gt dbout dumpbdb dbout 82 LS OPT Version 2 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR The solver specification in the command file is as follows solver dyna crash solver command runmpp solver input file car5 k solver append file rigid2 Remarks l DynaOpt inp is the reserved name for the LS DYNA MPP input file name T
317. r the purpose of determining the individual designs the theory of experimental design Design of Experiments or DOE is required Several experimental design criteria are available but one of the most popular for an arbitrarily shaped design space is the D optimality criterion This criterion has the flexibility of allowing any number of designs to be placed appropriately in a design space with an irregular boundary The understanding of the D optimality criterion requires the formulation ofthe least squares problem Consider a single response variable y dependent upon a number of variables x The exact functional relationship between these quantities is y n x 2 11 The exact functional relationship is now approximated e g polynomial approximation as x f x 2 12 The approximating function fis assumed to be a summation of basis functions f x 2 ap x 2 13 where L is the number of basis functions used to approximate the model T i f SEEN The constants a a a a have to be determined in order to minimize the sum of the square error P Div F amp I Fi Lae 2 14 P is the number of experimental points and y is the exact functional response at the experimental points x The solution to the unknown coefficients is a X X X y 2 15 where X is the matrix X X 1 2 16 LS OPT Version 2 11 CHAPTER 2 OPTIMIZATION METHODOLOGY The next critical step is to choose appropriate basis funct
318. ras ant eae wc a a a ava Gua ecuces users 192 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 2 6 First quadratic iteration The LS OPT input file is modified as follows the response approximations are allquadratic not shown Order quadratic Experimental design dopt Basis experiment 5toK Number experiment 10 For very expensive simulations if previous extracted simulation is available as e g from the previous linear iteration in Section then these points can be used to reduce the computational cost of this quadratic approximation To do this the previous AnalysisResults 1 file is copied to the current work directory and renamed AnalysisResults PRE 1 As is shown in the results below the computed vs predicted HIC and Intru_2 responses are is now improved from the linear approximation The accuracy of the HIC and Intru_2 responses are given in The corresponding R value for HIC is 0 9767 while the RMS error is 10 28 For Intru_2 the R value is 0 9913 while the RMS error is 0 61 When conducting trade off studies a higher order approximation like the current one will be preferable See trade off of HIC versus intrusion in a range 450mm to 600mm in Response Surface Accuracy Response Surface Accuracy For Response Function HIC For Response Function Intru_2 N r Computed Response Value x1 02 gt Computed Response Value x10 D 1 1 I 1 r 1 I 1 r 1 1 1 1 I 1 L I 1 I 1
319. revious iterations Therefore the response surfaces are progressively updated in the region of the optimal point Refer to Section for the setting of parameters in the iterative Successive Response Surface Method The above methodology can also be applied to sequential random searches 2 13 Sequential random search SRS LS OPT allows a sequential search method by which the best design is selected from each iteration without computing response surfaces A sorting procedure is used to select the design with the lowest for minimization or highest for maximization objective from all the feasible designs If no feasible design exists the least infeasible design is chosen An experimental design such as Latin Hypercube Sampling LHS allows a sequential random search procedure LS OPT automatically moves the region of interest by centering it on the most recent best design The scheme also involves automatic subdomain reduction in which the subdomain is reduced by the zoom parameter 77 see Section 2 12 if the best design is the same as the baseline design Ropi Otherwise 7 pan is used All the variable ranges are reduced by the same amount The following example illustrates the convergence performance of the methodology The example is an unconstrained minimization problem with starting point 1 1 1 1 solution 0 0 0 0 and an initial range of 0 5 1 5 and the objective to minimize LS OPT Version 2 37 CHAPTER 2 OPTIMIZATION
320. riables set variable distribution bound 0 Respect bounds on control variables set variable distribution bound 1 15 4 3 Noise Variable Subregion Size Bounds are required for noise variables to construct the metamodels The bounds are taken to a number of standard deviations away from the mean the default being two standard deviations of the distribution The number of standard deviations can however be set by the user In general a noise variable is bounded by the distribution specified and does not have upper and lower bounds similar to control variables Command file syntax set noise variable range standardDeviations Item Description standardDeviations The subregion size in standard deviations for the noise variable LS OPT Version 2 161 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION Example Set noise var bounds to 1 5 standard deviations for defining subregion for creating approximation set noise variable range 1 5 15 5 Probabilistic Simulation The following simulation methods are provided e Monte Carlo e Monte Carlo using metamodels The upper and lower bounds on constraints will be used as failure values for the reliability computations 15 5 1 Monte Carlo Analyses The Monte Carlo evaluation will e Select the random sample points according to a user specified strategy and the statistical distributions assigned to the variables e Evaluate the structural behavio
321. ribution Usually the tail regions are of specific interest Accuracy especially at the tail regions requires the following conditions e The responses must be linear functions of the design variables Or sufficiently linear where sufficiently linear requires o The constraint associated with the response is active it is being evaluated close to the most probable point o The linearized response is sufficiently accurate over a range encompassed by the variables e Normally distributed design variables 3 3 3 Monte Carlo Analysis Using Metamodels Performing the Monte Carlo analysis using approximations to the functions instead of FE function evaluations allows a significant reduction in the cost of the procedure A very large number of function evaluations millions are possible considering that function evaluations using the metamodels are very cheap Accordingly given an exact approximation to the responses the exact probability of an event can be computed The choice of the point about which the approximation is constructed has an influence on accuracy Accuracy may suffer if the metamodel is not accurate close to the failure initiation hyperplane G x 0 A metamodel accurate at the failure initiation hyperplane more specifically the Most Probable Point of failure is desirable in the cases of nonlinear responses The results should however be exact for linear responses or quadratic responses approximated using a quadratic response s
322. roblem There are two crashworthiness modal analysis case studies in the examples chapter Sections and 17 7 2 15 3 Parameter Identification In parameter identification a computational model is calibrated to experimental results The experimental results are set up as target values for a distance function see Section B 15 1 Constraints such as to enforce monotonicity can also be applied The application of optimization to parameter identification is demonstrated in Section 17 5 Two formulations for parameter identification can be implemented using LS OPT The first is the standard least squares residual or LSR formulation while the second uses the fact that LS OPT solves an auxiliary problem to eliminate infeasibility to minimize the maximum violation of constraints These two formulations are outlined below LS OPT Version 2 43 CHAPTER 2 OPTIMIZATION METHODOLOGY Least squares residual LSR formulation For this formulation a targeted composite function see Equation 10 1 in Section in LS OPT is used to construct the residual LSR ae 2 66 J where F are the experimental targets and I are scaling factors required for the normalization or weighting of each respective response Maximum violation formulation In this formulation the deviations from the respective target values are incorporated as constraint violations so that the optimization problem for parameter identification becomes Minimize e
323. s International Journal for Numerical Methods in Engineering 43 pp 1 21 1998 Kokoska S and Zwillinger D CRC Standard Probability and Statistics Tables and Formulae Student Edition Chapman amp Hall CRC New York 2000 Krige D G A statistical approach to some mine valuation and allied problems on the Witwatersrand Masters thesis University of the Witwatersrand South Africa 1951 Lawrence S C Lee Giles Ah Chung Tsoi What size neural network gives optimal generalization Convergence Properties of Backpropogation Technical Report UMIACS TR 96 22 and CS TR 3617 University of Maryland 1996 Lewis K Mistree F The other side of multidisciplinary design optimization accommodating a mutiobjective uncertain and non deterministic world Engineering Optimization 31 pp 161 189 1998 Luenberger D G Linear and Nonlinear Programming Second Edition Addison Wesley 1984 Matsumoto M and Nishimura T Mersenne Twister A 623 Dimensionally Equidistributed Uniform Pseudo Random Number Generator ACM Transactions on Modeling and Computer Simulation 8 1 pp 3 30 1998 McKay M D Conover W J Beckman R J A comparison of three methods for selecting values of input variables in the analysis of output from a computer code Technometrics pp 239 245 1979 MacKay D J C Bayesian interpolation Neural Computation 4 3 pp 415 447 1992 Mendenhall W Wackerly D D Scheaffer R L Mathematical Statistics with Application
324. s PWS Kent Boston 1990 288 LS OPT Version 2 BIBLIOGRAPHY 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Moody J E The effective number of parameters An analysis of generalization and regularization in nonlinear learning systems in J E Moody S J Hanson and R P Lippmann editors Advances in Neural Information Processing Systems 4 Morgan Kaufmann Publishers San Mateo CA 1992 Morris M Mitchell T Exploratory design for computer experiments Journal of Statistical Planning Inference 43 pp 381 402 1995 Myers R H Montgomery D C Response Surface Methodology Process and Product Optimization using Designed Experiments Wiley 1995 National Crash Analysis Center NCAC Public Finite Element Model Archive www ncac gwu edu archives model index html 2001 Park J S Optimal Latin hypercube designs for computer experiments Journal of Statistical Planning Inference 39 pp 95 111 1994 Redhe M and Nilsson L Using space mapping and surrogate models to optimize vehicle crashworthiness design Paper 2002 5607 9 AIAA ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta September 4 6 2003 Riedmiller M Braun H A direct adaptive method for faster backpropagation learning The RPROP algorithm In H Ruspini editor Proceedings of the IEEE International Conference on Neural Networks ICNN page
325. s 586 591 San Francisco 1993 Roux W J Structural Optimization using Response Surface Approximations PhD thesis University of Pretoria April 1997 Roux W J du Preez R J Stander N The design optimization of a semi solid tire using response surface approximations Engineering Computations 16 pp 165 184 1999 Roux W J Stander N Haftka R T Response surface approximations for structural optimization International Journal for Numerical Methods in Engineering 42 pp 517 534 1998 Rummelhart D E Hinton G E Williams R J Learning internal representations by error propagation In D E Rumelhart and J L McClelland editors Parallel Distributed Processing Vol I Foundations pages 318 362 MIT Press Cambridge MA 1986 Simpson T W Lin D K J and Chen W Sampling Strategies for Computer Experiments Design and Analysis International Journal for Reliability and Applications Aug 2001 Revised Manuscript Sj berg J Ljung L Overtraining regularization and searching for minimum in neural networks Preprints of the 4th IFAC Int Symp on Adaptive Systems in Control and Signal Processing p 669 July 1992 Schoofs A J G Experimental Design and Structural Optimization PhD thesis Technische Universiteit Eindhoven August 1987 Schuur P C Classification of acceptance criteria for the simulated annealing algorithm Mathematics of Operations Research 22 2 pp 266 275 1997 Simpson T W A concept explora
326. s include The prepro command will enable LS OPT to execute the following command in the default case origin 2 john mytemplex templex input tpl gt nodes include or if the input file is specified as in the example origin 2 user mytemplex templex a tpl gt nodes include Remarks l LS OPT lt lt varname gt gt type substitutions can be specified in the Templex input file and the solver input file LS OPT uses the name of the variable on the DV ARi line of the input file parameter DVAR1 Radius_1 1 0 5 3 0 parameter DVAR2 Radius_2 1 0 5 3 0 to replace the variables and bounds at the end of each line by the current values This name e g Radius_1 must therefore also be defined in the LS OPT command file see Section B 1 The DVARi designation is not changed in any way so in general there is no relationship between the number or rank of the variable specified in LS OPT and the number or rank of the variable as represented by 7 in DVARi LS OPT can also replace variables in the Templex input file using the standard LS OPT lt lt varname gt gt notation This would normally not be required as typically only the DVARi lines are modified for the Templex run Therefore typically no manual changes are required after creation of the Templex input 86 LS OPT Version 2 CHAPTER 7 INTERFACING TO A SOLVER OR PREPROCESSOR file using variable names that are consistent with the variable names define
327. s number of composites lt 0 gt objectives number of objectives lt 0 gt constraints number of constraints lt 0 gt distributions number of distributions lt 0 gt Example variable 2 constraint 1 responses 2 objectives 2 The most important data commands are the definitions These serve to define the various entities which constitute the design problem namely solvers variables responses objectives constraints and composites The definition commands are solver package_name constant name value variable name value dependent name value history name string response name string composite name type type composite name string objective name entity weight constraint name entity name Each definition identifies the entity with a name and must be placed before any other occurrence of the name 65 CHAPTER 5 GRAPHICAL USER INTERFACE AND COMMAND LANGUAGE 5 3 Command Language The command input file is a sequence of text commands describing the design optimization process It is also written automatically by LS OPTui The Design Command Language DCL is used as a medium for defining the input to the design process This language is based on approximately 70 command phrases drawing on a vocabulary of about 70 words Names can be used to describe the various design entities The command input file combines a sequence of text commands describing the design optimization process The command syntax is not case sensitive 5 3 1 Na
328. si Pan 1 0 1 0 gamma Voss 0 6 1 0 eta Zoom parameter 7 0 6 0 5 rangelimit Minimum range 0 0 0 0 repeatlimit Limit on number of times solution is 5 5 repeated SRS only Applied when the design has not changed Command file syntax iterate param parameter identifier value The iterative process is terminated if the following convergence criteria become active kl fpa ee Fe LE and or x u lal i where x refers to the vector of design variables d is the size of the design space f denotes the value of the objective function and k and k 1 refer to two successive iteration numbers The stoppingtype parameter is used to determine whether and or or will be used e g iterate param design 0 001 iterate param objective 0 001 iterate param stoppingtype or implies that the optimization will terminate when either criterion is met The range limit can be used to specify the minimum size of the region of interest This is not a stopping criterion so that the solver will still continue to iterate until any of the other stopping criteria are met An application of the range limit is to maintain a constant tolerance on the random variables See Section 17 2 9 for an example 166 LS OPT Version 2 CHAPTER 16 OPTIMIZATION ALGORITHM SELECTION AND SETTINGS Command file syntax iterate param rangelimit variable name value Example iterate param rangelimit thickness 1 0 5 iterate p
329. sion response Intru_1 1 scale 1 composite HIC_scaled type targeted composite HIC_scaled response HIC 0 scale 900 weight 1 composite Freq_scaled type targeted composite Freq_scaled response Frequency 0 scale 3 weight 1 COMPOSITE EXPRESSIONS composite Intrusion_scaled Intrusion 500 OBJECTIVE FUNCTIONS objectives 1 246 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS objective Mass 1 CONSTRAINT DEFINITIONS constraints 3 constraint HIC_scaled upper bound constraint HIC_scaled 1 constraint Freq scaled lower bound constraint Freq_scaled 1 constraint Intrusion_scaled upper bound constraint Intrusion_scaled 1 JOB INFO iterate param design 0 01 iterate param objective 0 01 iterate param stoppingtype and iterate 10 STOP The optimization results are shown in Figure 17 33 for the objective three constraints mode sequence number and maximum constraint violation The initial design is infeasible due to too high a HIC value too high an intrusion and too low a torsional frequency To meet all these constraints the mass of the affected components has to increase substantially Convergence is obtained at about the 8 iteration and when the optimization terminates a 0 0003 violation in the scaled constraints remains The mode tracking feature in LS DYNA is used to keep track of the frequency mode Figure 17 33 e Note that the mode number c
330. sis e Monte Carlo analysis using a metamodel 17 10 2 Problem Description A symmetric short crush tube impacted by a moving wall as shown in the figure is considered The design criterion is the intrusion of the wall into the space initially occupied by the tube or alternatively how much the structure is shortened by the impact with the wall Figure 17 48 Tube impact Both the shell thickness and the yield strength of the structure follow a probabilistic distribution The shell thickness is normally distributed around a value of 1 0 with a standard deviation of 5 while the yield strength is normally distributed around a value scaled to 1 0 with a standard deviation of 10 278 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS The nominal design has an intrusion of 144 4 units The probability of the intrusion being greater than 150 units is computed The best known results are obtained using a Monte Carlo analysis of 1500 runs The problem is analyzed using a Monte Carlo evaluation of 60 runs and a quadratic response surface build using a 3 experimental design The results from the different methods are close to each other as can be seen in the following table Response Monte Carlo Monte Carlo Response Surface 1500 runs 60 runs 9 runs Average Intrusion 141 3 141 8 141 4 Intrusion Standard Deviation 15 8 15 2 15 0 Probability of Intrusion gt 150 0 32 0 33 0 29 Using the response surface
331. sponse displacement DynaRelativeDisp 0 2 response Force Myforce In addition to numbers 0 9 upper or lower case letters and spaces a command line can contain any of the following characters Sa eae In the command input file a line starting with the character is ignored A command must be specified on a single line 5 3 3 File names Input file names for the solver and preprocessor must be specified in double quotes prepro input file pl11li solver input file side impact 5 3 4 Command file structure The commands are arranged in two categories e problem data e solution tasks The only command for specifying a task is the iterate command All the remaining commands are for the specification of problem data A solution task command serves to execute a solver or processor while the other commands store the design data in memory In the following chapters the command descriptions can be easily found by looking for the large typescript bounded by horizontal lines Otherwise the reader may refer to the quick reference manual that also serves as an index The default values are given in angular brackets e g lt 1 gt 67 CHAPTER 5 GRAPHICAL USER INTERFACE AND COMMAND LANGUAGE 5 3 5 Environments Environments have been defined to represent all dependent entities that follow The only environments in LS OPT are for e solver identifier_name All responses response histories solver variabl
332. sssss DEFINITION OF SOLVER CRASH solver dyna960 CRASH solver command lsdyna single solver input file car6_crash k solver order linear solver experiment design dopt solver basis experiment 3toK LS OPT Version 2 245 CHAPTER 17 EXAMPLE PROBLEMS solver concurrent jobs 1 LOCAL DESIGN VARIABLES FOR SOLVER CRASH UW Ur Ur solver variable t_rail_back solver variable t_bumper solver variable t_roof solver variable t_rail_front RESPONSES FOR SOLVER CRASH UW Ur WU response Acc max 1 0 DynaASCII Nodout X_ ACC 432 Max SAE 60 response Mass 1 0 DynaMass 2 3 4 5 6 7 MASS response Intru_2 1 0 DynaASCII Nodout X DISP 432 Timestep response Intru_1 1 0 DynaASCII Nodout X DISP 184 Timestep response HIC 1 0 DynaASCII Nodout HIC15 9810 1 432 SSSSSSSSSSSSSSSSSSssssssssssssss SOLVER NVH SSSSSSSSSSSSSSSSSSSsssssssssssss DEFINITION OF SOLVER NVH solver dyna960 NVH solver command Isdyna double solver input file car6_NVH k solver order linear solver experiment design dopt solver basis experiment 5toK solver concurrent jobs 1 RESPONSES FOR SOLVER NVH response Frequency 1 0 DynaFreq 15 FREQ response Mode 1 0 DynaFreq 15 NUMBER response Generalized Mass 1 0 DynaFreq 15 GENMASS composites 4 COMPOSITE RESPONSES 5 composite Intrusion type weighted composite Intrusion response Intru_2 1 scale 1 composite Intru
333. st USER DEFINED PDF bendingTest pdt distribution testDat USER DEFINED CDF threePointTest dat 156 LS OPT Version 2 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION The file bendingTest pdf contains Demonstration of user defined distribution with piecewise uniform PDF values x PDF s First PDF value must be 0 5 0 00000 2 5 0 11594 0 0 14493 2 25 0 11594 Last PDF value must be 0 5 0 00000 The file threePointTest dat contains Demonstration of user defined distribution with piecewise linear CDF values S x CDF s First CDF value must be 0 5 0 00000 4 5 0 02174 3555 0 09420 12 5 0 20290 1 5 0 32609 04 5 0 46377 0 5 0 60870 145 0 73913 215 0 85507 3 5 0 94928 Last CDF value must be 1 4 5 1 00000 15 3 4 Lognormal distribution If X is a lognormal random variable with parameters u and o then the random variable Y In X has a normal distribution with mean u and variance 0 06 pl c fix Figure 15 4 Lognormal distribution LS OPT Version 2 157 CHAPTER 15 PROBABILISTIC MODELING AND MONTE CARLO SIMULATION Command file syntax distribution name LOGNORMAL mu sigma Item Description name Distribution name mu Mean value in logarithmic domain sigma Standard deviation in logarithmic domain Example distribution logDist LOGNORMAL 12 3 1 1 15 3 5 Weibull distribution The Weibull distribution is quite v
334. stmat 3 scoef 1 c sid 3 lsdsi alO slvmat 1 mstmat 4 scoef 1 lcd 1 0 000000000E 00 0 275600006E 03 0 665699990E 04 0 276100006E 03 0 136500006E 03 0 276700012E 03 0 312799990E 00 0 481799988E 03 0 469900012E 00 0 517200012E 03 0 705600023E 00 0 555299988E 03 c c die cross section para c rl lt lt Radius_1 gt gt c upper radius minimum 2 r2 lt lt Radius_2 gt gt c middle radius minimum 2 r3 lt lt Radius_3 gt gt c lower radius minimum 2 load2 100000 load3 20000 thi 1 0 c thickness of blank th3 00 c thickness of die and punch th2 1 001 th1 11 20 c length of draw 5 40 c z5 11 22 c Position of workpiece z4 z54 1 001 th1 2 th3 2 c Position of blankholder z3 z4 1 001 th1 2 th3 2 nl 25 4 0 11 n2 25 8 0 11 c part 2 z6 z5 4 th2 z7 z5 11 4 th2 c c die cross section LS OPT Version 2 221 CHAPTER 17 EXAMPLE PROBLEMS c punch cross section closed configuration ld 2 lod 1 th2 th3 c punch cross section withdrawn configuration ld 3 lstl 2 0 z5 26 endpart C FKKKHKHKKHKKAKKKEKERK Dart 2 mat 2 punch cylinder 1 8 35 40 67 76 76 n1 70 n1 10 1 41 1 001 17 23 36 44 50 75 100 0 90 S27 thick th3 mate 2 endpart C RKKKKKKKKKK part 3 mat 4 kkkkkkkkk blankholder cylinder 1 10 1 41 1 80 100 0 90 z3 b 0O 0 0 0 0 0O dx 1 dy 1 rx 1 ry 1 rz 1 thick th3 mate 4 endpart C kkkkkkkkk k Dart 4 mat
335. straint NodDisp Distribution Information Number of points 1000000 Mean Value 141 4 Standard Deviation 14 95 Coef of Variation 0 1058 Maximum Value 68 5 Minimum Value 206 3 Lower Bound BOUNG Ar ee ea er 150 Evaluations exceeding this bound 285347 Probability of exceeding bound 0 2853 Confidence Interval on Probability Standard Deviation of Prediction Error 0 0004516 Lower Bound Probability Higher Bound 0 2844 0 2853 0 2863 Confidence Interval of 95 assuming Normal Distribution Confidence Interval of 75 using Tchebysheff s Theorem ANALYSIS COMPLETED LS OPT Version 2 285 CHAPTER 17 EXAMPLE PROBLEMS 286 LS OPT Version 2 Bibliography 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Akaike H Statistical predictor identification Ann Inst Statist Math 22 pp 203 217 1970 Akkerman A Thyagarajan R Stander N Burger M Kuhn R Rajic H Shape optimization for crashworthiness design using response surfaces Proceedings of the 1 International Workshop on Multidisciplinary Design Optimization Pretoria South Africa 8 10 August 2000 pp 270 279 Arora J S Introduction to Optimum Design Ist ed McGraw Hill 1989 Arora J S Sequential linearization and quadratic programming techniques In Structural Optimization Status and Promise Ed Kamat M P AIAA 1993
336. sult type name 1 sub Result subdirectory 1 cmp Component of result 2 invariant Invariant of results Only MAGNITUDE is currently available 3 id ID number of entity pos Through thickness shell position at which results are computed 1 4 side Interface side for RCFORC data MASTER or SLAVE SLAVE Example history ELOUT1 BinoutHistory res_type Elout sub shell cmp sig XxX 10 1 pos 1 history invarHis BinoutHistory res_type nodout cmp displacement invariant MAGNITUDE id 432 Remarks l The result types and subdirectories are as documented for the DATABASE OPTION LS DYNA keyword 2 The component names are as listed in Appendix Cj LS DYNA Binout Result File and Components 3 The individual components required to compute the invariant will be extracted automatically for example cmp displacement invariant MAGNITUDE will result in the automatic extraction ofthe x y and z components of the displacement 4 For the shell and thickshell strain results the upper and lower surface results are written to the database using the component names such as lower eps xxand upper eps xx LS OPT Version 2 127 CHAPTER 10 HISTORY AND RESPONSE RESULTS Averaging Filtering and Slicing Binout histories These operations will be applied in the following order averaging filtering and slicing Command file syntax BinoutHistory history options filter filter type filter freq filter freq units units a
337. t are uncontrollable to constants 4 Also change the random variables that are not desired to be random to constants LS OPT Version 2 45 CHAPTER 2 OPTIMIZATION METHODOLOGY 5 Select suitable lower and upper bounds for the MEAN variables as for a standard design formulation 6 The random variables are centered on the starting point so that the starting vector is 0 0 0 Omit the range but choose the upper and lower bounds to conform to the design tolerance e g 2 2 for a tolerance The tolerance allows a uniformly distributed scatter of the input variables 7 The most suitable experimental design for the random variables is Latin Hypercube Sampling LHS 8 Define a dependent variable as the sum of the mean and random variables There may be some constants in both categories 9 Substitute the mean variable names in the MEAN solver simulation deck and the dependent variable names in the RANDOM _ solver deck 10 Define responses for the design criteria for both the MEAN and _ RANDOM solvers Also compute the standard deviations of the RANDOM_ responses 11 Add or subtract fractions of the standard deviations to from the mean corresponding responses to compensate for the sensitivity of the design The method above has the advantage that actual simulations are used to model stochastic behavior The standard deviations are assumed to be constant over the sub region but the method should converg
338. t be specified so that the multi objective F a 11 1 A component function must be assigned to each objective function where the component function can be defined as a composite function F see Section or a response function f The number of objectives N must be specified in the problem description see Section 5 2 LS OPT Version 2 133 CHAPTER 11 OBJECTIVES AND CONSTRAINTS 11 2 Defining an objective function This command identifies each objective function The name of the objective is the same as the component which can be a response or composite Command file syntax objective name weight lt 1 gt Examples objective Intrusion 1 objective Intrusion 2Y 2 objective Acceleration 3 for Multi objective F 20 3 F 2F 3f Remarks 1 The distinction between objectives is made solely for the purpose of constructing a Pareto optimal curve involving multiple objectives However it is still better to construct a Pareto optimal curve using a varying constraint bound instead of varying weights See Sections 11 4 and 2 Objectives can be specified in terms of composite functions and or response functions 3 The weight applies to each objective as represented by ax in Equation 11 1 The default is to minimize the objective function The program can however be set to maximize the objective function In LS OPTui maximization is activated in the Objective panel Command file s
339. t case design represents the optimal trim design for the worst case head orientation Another example is the minimization of crashworthiness related criteria injury intrusion etc during a frontal impact while maximizing the same criteria for a range of off set angles in an oblique impact situation 44 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY Another class of problems involves the introduction of uncontrollable variables z i 1 n in addition to the controlled variables y op RR The controlled variables can be set by the designer and therefore optimized by the program The uncontrollable variables are determined by the random variability of manufacturing processes loadings materials etc Controlled and uncontrollable variables can be independent but can also be associated with one another i e a controlled variable can have an uncontrollable component The methodology requires three features 1 The introduction of a constant range p of the region of interest for the uncontrollable variables This constant represents the possible variation of each uncontrollable parameter or variable In LS OPT this is introduced by specifying a lower limit on the range as being equal to the initial range p The lower and upper bounds of the design space are set to p 2 for the uncontrollable variables 2 The controlled and uncontrollable variables must be separated as minimization and maximization variables The objective will th
340. t is advantageous to be able to augment an existing design with additional experimental points This can be performed by constructing a user defined experiment as follows User defined experiments can be placed in a file named Experiments PRE solver name in the work directory These will be used in the first iteration only for the solver with name solver name The user can augment this set D optimally by requesting a number of experiments greater than the number of lines in Experiments PRE solver_ name Each experiment must appear on a separate line with spaces commas or tabs between values If the user wants to specify an experimental design plan in all iterations a file Experiments PLAN must be supplied The point coordinates must be normalized to the bounds 1 1 This unit experimental design will be scaled to the region of interest in each iteration 9 4 Remarks Point selection 1 The number of points specified in the solver number experiment num command is reduced by the number already available in the Experiments PRE solver name or AnalysisResults PRE solver_name files 2 The files Experiments and AnalysisResults are synchronous i e they will always have the same experiments after extraction of results Both these files also mirror the result directories for a specific iteration 3 Design points that replicate the starting point are omitted 9 5 Specifying an irregular design space An irregular reasonable design spac
341. t time integration limits etc 2 to convert a variable to a constant This requires only changing the designation variable to constant in the command file without having to modify the input template The number of optimization variables is thus reduced without interfering with the template files Command file syntax constant constant_name value LS OPT Version 2 91 CHAPTER 8 DESIGN VARIABLES CONSTANTS AND DEPENDENTS Example constant Youngs modulus 2 07e8 constant Poisson ratio 0 3 dependent Shear modulus Youngs modulus 2 1 Poisson ratio In this case the dependent is of course not a variable but a constant as well 8 7 Dependent Variables Dependent variables see Figure 8 2 for example of definition in Variables panel are functions of the basic variables and are required to define quantities that have to be replaced in the input template files but which are dependent on the optimization variables They do therefore not contribute to the size of the optimization problem Dependent variables are specified using mathematical expressions see Appendix E Command file syntax dependent variable name expression The string must conform to the rules for expressions and be placed in curly brackets The dependent variables can be specified in an input template and will therefore be replaced by their actual values Example variable Youngs modulus 2 0e08 variable Poisson ratio 0
342. t_rail_back t_rail_back 1 2 3 4 5 6 0204 0608 1 12 1 4 c d Figure 17 32 ANOVA plots for small car response 244 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 6 2 MDO with D optimal experimental design and SRSM The LS OPT input file below illustrates how the variables were reduced for the NVH solver Note how the two solvers i e crash and NVH are specified Variables are flagged as local with the Local variable name statement and then linked to a solver using the Solver variable variable name command Small Car Problem Five variables Partial variables Crash NVH MDF Created on Sun Jan 5 16 12 00 2003 solvers 2 responses 8 NO HISTORIES ARE DEFINED DESIGN VARIABLES variables 5 Variable t_rail_back 2 Lower bound variable t_rail_back 1 Upper bound variable t_rail_back 6 Range t_rail_back 2 Local t_rail_back Variable t_hood 2 Lower bound variable t_hood 1 Upper bound variable t hood 6 Range t_hood 2 Variable t_bumper 3 Lower bound variable t_bumper 1 Upper bound variable t bumper 6 Range t_bumper 2 Local t_bumper Variable t_roof 2 Lower bound variable t_roof 1 Upper bound variable t_roof 6 Range t_roof 2 Local t_roof Variable t_rail_front 5 Lower bound variable t_rail_front 1 Upper bound variable t rail front 6 Range t_rail_front 2 Local t_rail_front SSSSSSSSSSSSSSSSSSSsssssssssssss SOLVER CRASH SSSSSSSSSSSSSSSSSSSssssssss
343. the NVH discipline were forced to be coincident with the points of the crash discipline see Section P 7 The number of simulations per iteration is 16 The input file is therefore as follows Small Car Problem Five variables 4 17 27 53 2003 Created on Tue Feb solvers 2 responses 8 NO HISTORIES ARE DEFINED DESIGN VARIABLES variables 5 Variable t_rail_back Lower bound variable Upper bound variable Range t_rail_back Variable t_hood 2 Lower bound variable Upper bound variable Range t_hood 2 Variable t_bumper 3 Lower bound variable Upper bound variable Range t_bumper 2 Variable t_roof 2 Lower bound variable Upper bound variable Range t_roof 2 Variable t_rail_front Lower bound variable Upper bound variable Range t_rail_front 2 2 t_ rail back 1 t_ rail back 6 t hood 1 t hood 6 t bumper 1 t bumper 6 t_ roof 1 t_ roof 6 5 t rail front 1 t rail front 6 SEQUENTIAL RANDOM SEARCH optimization method randomsearch SSSSSSSSSSSSSSSSSSSsssssssssssss SOLVER CRASH SSSSSSSSSSSSSSSSSSSsssssssssssss DEFINITION OF SOLVER CRASH solver dyna960 CRASH solver command 1s970 single solver input file car6_crash k Partial variables Crash NVH MDF 250 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS solver experiment design latin_hypercube solver number experiments 16 solver concurrent j
344. the derivatives of the intrusions with respect to the design variables are computed as given in the following table Variable Intrusion derivative Shell Thickness 208 Yield Strength 107 The quadratic response surface also allows the investigation of the dependence of the response variation on each design variable variation The values of the intrusion standard deviation given in the following table are computed considering the variable as the only source of variation in the structure the variation of the other design variables are set to zero Source of variation Intrusion Standard Deviation Shell Thickness 10 4 Yield Strength 10 7 The details of the analyses are given the following subsections 17 10 3 Monte Carlo evaluation The probabilistic variation is described by specifying statistical distributions and assigning the statistical distributions to noise variables Tube Crush Monte Carlo Created on Tue Apr 1 11 26 07 2003 solvers 1 5 distribution 2 distribution t NORMAL 1 0 0 05 distribution y NORMAL 1 0 0 10 5 DESIGN VARIABLES variables 2 noise variable T1 distribution t LS OPT Version 2 279 CHAPTER 17 EXAMPLE PROBLEMS noise variable YS distribution y 5 DEFINITION OF SOLVER SOLVER_1 5 solver dyna960 SOLVER_1 solver command 1s970 single solver input file tube k solver experiment design lhd centralpoint solver number experiments 60 H
345. thickness Bumper thickness Rail_front thickness Rail_back thickness Figure 17 31 Small car with crash rails definition of design variables LS OPT Version 2 243 CHAPTER 17 EXAMPLE PROBLEMS The optimization problem is defined as follows Minimize Mass Xcrash subject to HIC Xcrash lt 900 Intrusion Xcrash lt 500mm Torsional mode frequency anvn gt 3Hz where Xerash t_hood t_bumper t_ rail _back t rail _ front t_roof and XnvH t_hood These variables were selected by first conducting an LS OPT run in which all five variables were included for both crash and NVH linear response surfaces shows the ANOVA charts produced from these results using the Viewer Using these charts all five variables were selected for crash while only t_hood was kept for NVH Note that in this first iteration there is a significant error in the intrusion prediction while mass has no error because of its linearity ANOVA Plot for Mass ANOVA Plot for Intru_2 Lower half of 90 confidence interval in red Lower half of 90 confidence interval in red t_rail_front t_rail_front t_roof t_roof t_bumper t_bumper 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 5 10 15 20 25 30 35 40 45 50 a b ANOVA Plot for HIC ANOVA Plot for Frequency Lower half of 90 confidence interval in red Lower half of 90 confidence interval in red t_rail_ front t_rail_front t_roof t_roof t_bumper t_bumper t_hood t_hood
346. tion are e Variation in structural properties for example variation in yield stress e Variation in the environment for example variation in a load e Variation in the problem modeling and analysis for example buckling initiation mesh density or results output frequency From the probabilistic analysis we want to infer e Distribution of the response values e Probability of failure e Properties of the designs associated with failure o Variable screening identify important noise factors o Dispersion factors factors whose settings may increase variability of the responses e Efficient redesign strategies 3 2 Probabilistic Variables The probabilistic component of a parameter is described using a probability distribution for example a normal distribution The parameter will therefore have a mean or nominal value as specified by the distribution though in actual use the parameter will have a value randomly chosen according to the probability density function of the distribution The relationship between the control variables and the variance can be used to adjust the control process variables in order to have an optimum process The variance of the control and noise variables can be used to predict the variance of the system which may then be used for redesign Knowledge of the interaction between the control and noise variables can be valuable for example information such as that the dispersion effect of the material vari
347. tion command Use full path names for both the wrapper and executable or make sure the path on the remote machine includes the directory where the executables are kept The argument for the input deck specified in the script must always be the LS OPT reserved name for the chosen solver e g for LS DYNA use DynaOpt inp Examples a Running an SMP Shared Memory Parallel LS DYNA job vclass user bin wrappers wrapper hp vclass dyna 1s970 i DynaOpt inp memory 2m 2 gt a err b Running an MPP Message Passing Parallel LS DYNA job The following command is executed within a UNIX script which executes a second script to run the DYNA MPP job vclass user bin wrappers wrapper hp vclass dyna mppscript mppopt DynaOpt inp 2 where mppscript isas follows bin sh opt mpi bin mpirun vclass dyna mpp970 np 3 i 2 2 gt l err cat dbout gt dbout vclass dyna dumpbdb dbout The wrapper will not execute multiple commands and therefore a script is required in this case Wrapping the entire script ensures i all the result files are available for extraction as soon as the wrapper is terminated and ii the sense switches will alow job termination An example of a LSF script rundyna hp is as follows bin sh inputfile 1 precision 2 version 3 T1 CHAPTER 6 PROGRAM EXECUTION jobname echo inputfile sed s g awk print 1 gt version 970 input _dir pwd local cnt echo
348. tion method for product family design Ph D Thesis Georgia Institute of Technology 1998 Sobieszcezanski Sobieski J Haftka R T Multidisciplinary aerospace design optimization Survey of recent developments Structural Optimization 14 No 1 pp 1 23 1997 Sobieszczanski Sobieski J Kodiyalam S Yang R J Optimization of car body under constraints of noise vibration and harshness NVH and crash AZAA Paper 2000 1521 2000 Snyman J A An improved version of the original leap frog dynamic method for unconstrained minimization LFOPI b Appl Math Modelling 7 pp 216 218 1983 Snyman J A The LFOPC leap frog algorithm for constrained optimization Comp Math Applic 40 pp 1085 1096 2000 LS OPT Version 2 289 BIBLIOGRAPHY 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 Stander N Craig K J On the robustness of a simple domain reduction scheme for simulation based optimization Engineering Computations 19 4 pp 431 450 2002 Stander N Reichert R Frank T 2000 Optimization of nonlinear dynamic problems using successive linear approximations AIAA Paper 2000 4798 Stander N Roux W J Giger M Redhe M Fedorova N and Haarhoff J Crashworthiness optimization in LS OPT Case studies in metamodeling and random search techniques Proceedings of the 4 European LS DYNA Conference Ulm Germany May 2
349. tion process points Analysis programs to be scheduled Item Input Output DOE Location and size of the subregion Location of the experimental in the design space The points experimental design desired An approximation order An affordable number of points Simulation Location of the experimental Responses at the experimental points Build response surface Location of the experimental points Responses at the experimental points Function types to be fitted The approximate functions response surfaces The goodness of fit of the approximate functions at the construction points response surfaces Bounds on the responses and variables Check adequacy The approximate functions The goodness of fit of the response surfaces The location approximate functions at the of the check points The responses check points at the check points Optimization The approximate functions The approximate optimal design The approximate responses at the optimal design Pareto optimal curve data Two approaches may be taken 2 14 1 Design exploration Conduct one iteration usually by utilizing second order approximations with a large range Then assess the adequacy of the surfaces using the error parameters To improve the accuracy of the response surfaces a smaller region of interest can be chosen but the trade off properties will be artificially bounded by the range of the subregion To a
350. to energy In simulated annealing algorithm parlance the objective function of the optimization problem is often called energy The optimization algorithm proceeds in small iterative steps At each iteration SA algorithm randomly generates a candidate state and through a random mechanism controlled by a parameter called temperature in view of the analogy with the physical process decide whether to move to the candidate state or to stay in the current one at the next iteration More formally a general SA algorithm can be described as follows 333 APPENDIX F SIMULATED ANNEALING Step 0 Let x X be a given starting state of the optimized system E E x Start the sequence of observed states X x Set the starting temperature 7 to a high value 7 Tmax and initialize the counter of iterations to k 0 Step 1 Sample a point x from the candidate distribution D X and set X X U x The sequence X contains all the states observed up to iteration k Step 2 Sample a uniform random number in 0 1 and set x yf lt A x x T or F 2 x x otherwise Step 3 Apply the cooling schedule to the temperature i e set 7 CCX T Step 4 Check a stopping criterion and if it fails set k k 1 and go back to Step 1 The distribution of the next candidate state D the acceptance function A the cooling schedule C and the stopping criterion must be specified in order to define the SA al
351. ts LS OPT Version 2 129 CHAPTER 10 HISTORY AND RESPONSE RESULTS can be constructed using expressions see Appendix Ej Mathematical Expressions Error and warning messages will be generated Command file syntax set Binout Example DynaASCII commands following this command should be translated to BinoutResponse and BinoutHistory commands set Binout 10 12 DynaStat When using LS OPT for reliability based design optimization statistical quantities like the standard deviation of responses must be extracted This command is only available in the current batch version The syntax is Command file syntax DynaStat STDDEV response_name See Section 17 2 9 for an example 10 13 User Interface for Extracting Results The user may provide an own extraction routine to output a single floating point number to standard output Examples of the output statement in such a program are e The C language printf l f n output_value or fprintf stdout lf n output value e The FORTRAN language write 6 output value e The Perl script language print Soutput_ value n 130 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS The string N o r m a 1 must be written to the standard error file identifier stderr in C to signify a normal termination See Section I7 1 for an example The command to use a user defined program to extract a response is Command file syntax response r
352. ts and 4 variables for NVH imply 8 points see Chapter 8 The NVH simulations although not time consuming due to their implicit formulation involve a large use of memory due to double precision matrix operations Crashworthiness simulations on the other hand require little memory because of single precision vector operations but are time consuming due to their explicit nature It is therefore preferable to assign as many processors as possible to the crashworthiness simulations while limiting the number of simultaneous NVH simulations to the available computer memory to prevent swapping Table 17 7 Number of simulations for Fully and Partially Shared Variable Cases Starting design 1 Case Number of crash Number of NVH simulations simulations for for convergence convergence Fully shared variables 9x 13 127 9x 13 127 Partially shared variables 6x 11 66 6x 8 48 266 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS 17 8 Knee impact with variable screening 11 variables Example by courtesy of Visteon and Ford Motor Company This example has the following features e Variable screening is illustrated for a knee impact minimization study for a problem with both thickness and shape variables The use of ANOVA for variable screening is shown LS DYNA is used for the explicit impact simulation An independent parametric preprocessor is used Extraction is performed using standard LS DYNA i
353. urface Using approximations to search for improved designs can be very cost efficient Even in cases where absolute accuracy is not good the technique can still indicate whether a new design is comparatively better The number of FE evaluations required to build the approximations increases linearly with the number of variables for linear approximations the default being 1 5n points and quadratically for quadratic approximations the default being 0 75 n 2 n 1 points LS OPT Version 2 51 CHAPTER 3 PROBABILISTIC FUNDAMENTALS 52 LS OPT Version 2 USER S MANUAL 54 LS OPT Version 2 4 Design Optimization Process 4 1 LS OPT Features The program LS OPT has been designed to address the simulation based design optimization environment The following list presents some of the main features of the program e LS OPT is command file based A graphical user interface LS OPTuwi allows the entry and modification of command files The command language is flexible and easy to use Simple problems do not require complex descriptions e Design variables can be specified in either the preprocessor input files or the solver input files Solver input files are typically only useful for sizing or the variation of material constants or load curves or intensities Interfacing with a parametric preprocessor allows the designer to use the full capability thereof for optimization so that other forms of design such as shape optimizati
354. usive to disciplines but can also be shared Experimental design and job execution data can be exclusive to each discipline The software also features the restart of an aborted parallel multidisciplinary analysis schedule Error terminations are tagged for post processing purposes e Mode tracking can be activated for frequency based criteria LS DYNA interface 55 CHAPTER 4 DESIGN OPTIMIZATION PROCESS e A response surface and its optimal solution can be improved to a desired tolerance by successive iteration This process has been automated e An automated sequential random search method is available e Dependent variables responses and composite functions can be defined using C like mathematical expressions e LS OPT features a limited reliability based design capability that allows simple statistical analysis of result output The computation of an optimum and reliable design has been automated Random uncontrollable variables are allowed e Probabilistic Modeling and Monte Carlo Simulation e Version 3 of LS OPT is being designed as a full Reliability Based Design Optimization code 4 2 A modus operandi for design using response surfaces 4 2 1 Preparation for design Since the design optimization process is expensive the designer should avoid discovering major flaws in the model or process at an advanced stage of the design Therefore the procedure must be carefully planned and the designer needs to be familiar with the model
355. usually based on the idea to stop the iterative algorithm when it does not make a noticeable progress over a number of iterations The choice of the next candidate distribution and the cooling schedule for the temperature are typically the most important and strongly interrelated issues in the definition of a SA algorithm The next candidate state x is usually selected randomly among all the neighbors of the current solution x with the same probability for all neighbors However with a complicated neighbor structure a non uniformly random selection might be appropriate The choice of the size of the neighborhood typically follows the idea that when the current function value is far from the global minimum the algorithm should have more freedom i e larger step sizes are allowed The basic idea of the cooling schedule is to start the algorithm off at high temperature and then gradually to drop the temperature to zero The primary goal is to quickly reach the so called effective temperature roughly defined as the temperature at which low function values are preferred but it is still possible to explore different states of the optimized system 7 After that the simulated annealing algorithm lowers the temperature by slow stages until the system freezes and no further changes occur A straightforward and most popular strategy is to decrement T by a constant factor every vr iterations T T f F 6 where 4r is slightly greater than 1 e g
356. ve_points ave points start_time start_time end_time end_time Item Description Default history_options All available history options filter_type Type of filter to use SAE or BUTT 7 filter_freq Filter frequency 60 cycles time unit units S seconds MS milliseconds S ave_points Number of points to average start_time Start time of history interval to extract using slicing 0 end_time End time of history interval to extract using slicing tinue Example history ELOUT12 BinoutHistory res type Elout sub shell cmp sig xx id 1 pos 2 filter SAE start_time 0 02 end_time 0 04 history nodHist432acc_AVE BinoutHistory res type nodout cmp x acceleration id 432 ave points 5 10 10 2 _Binout Responses A response is extracted from a history all the history options are therefore applicable and options required for histories are required for responses as well Command file syntax BinoutResponse history options select selection Item Description Default Remarks All available history options including history_options f ae averaging filtering and slicing selection MAX MIN AVE TIME TIME 1 Example 128 LS OPT Version 2 CHAPTER 10 HISTORY AND RESPONSE RESULTS response eTime BinoutResponse res_type glstat cmp kinetic_energy select TIME end time 0 015 5 response nodeMax BinoutResponse res type nodout cmp
357. vity analysis for friction and contact problems is a subject of current research It is mainly for the above reasons that researchers have resorted to global approximation methods for smoothing the design response The art and science of developing design approximations has been a popular theme in design optimization research for decades for a review of the various approaches see e g Reference by Barthelemy Barthelemy categorizes two main global approximation methods namely response surface methodology and neural networks 23 In the present implementation the gradient vectors of general composites based on mathematical expressions of the basic response surfaces are computed using numerical differentiation A default interval of 1 1000 of the size of the design space is used in the forward difference method 2 4 Normalization of constraints and variables It is a good idea to eliminate large variations in the magnitudes of design variables and constraints by normalization In LS OPT the typical constraint is formulated as follows L Sg SU jel2 m 2 8 which when normalized becomes LS OPT Version 2 9 CHAPTER 2 OPTIMIZATION METHODOLOGY L Pa v g xo g xo g xp J 1 2 m 2 9 where xo is the starting vector The normalization is done internally The design variables have been normalized internally by scaling the design space xz xu to 0 1 where xz is the lower and xy the upper bound Th
358. void this problem points can be added to the design space and adaptable response surfaces such as neural networks or Kriging can be used to improve prediction The response surfaces are then used to explore the design space e g through trade off studies In a trade off study a constraint value may be varied for the purpose of observing how the objective function and the optimal design changes LS OPT Version 2 39 CHAPTER 2 OPTIMIZATION METHODOLOGY 2 14 2 Convergence to an optimal point First order approximations Because of the absence of curvature it is likely that perhaps 5 to 10 iterations may be required for convergence The first order approximation method turns out to be robust thanks to the successive approximation scheme which addresses possible oscillatory behavior Linear approximations may be rather inaccurate to study trade off i e in general they make poor global approximations but this is not necessarily true and must be assessed using the error parameters Second order approximations Due to the consideration of curvature a successive quadratic response surface method is likely to be more robust but can be more expensive depending on the number of design variables Other approximations Neural networks Section 2 10 1 and Kriging Section 2 10 2 provide good approximations when many design points are used A suggested approach is to start the optimization procedure in the full design space with the number of
359. which signifies the strictness with which a design constraint must be treated A zero value implies that the constraint may be violated If a feasible design is possible all constraints will be satisfied Used in the design formulation to minimize constraint violations See Slack variable Subproblem The approximate design subproblem constructed using response surfaces It is solved to find an approximate optimum Subregion See region of interest Successive Approximation Method An iterative method using the successive solution of approximate subproblems Target A desired value for a response The optimizer will not use this value as a rigid constraint Instead it will try to get as close as possible to the specified value Template An input file in which some of the data has been replaced by variable names e g lt lt Radius gt gt Trade off curve A curve constructed using Pareto optimal designs Transformed variables Variables which are transformed mapped to a different n space using a functional relationship The experimental design and optimization are performed in this space Variable screening Method to remove insignificant variables from the design optimization process based on a ranking of regression coefficients using analysis of variance ANOVA See also ANOVA Weight A measure of importance of a response function or objective Typically varies between 0 and 1 Work directory The directory is which the input
360. xample S RANGE OF Area range Area 0 4 This will allow Area to vary from 0 6 to 1 0 Remarks 1 A value of 25 50 of the design space can be chosen if the user is unsure of a suitable value 90 LS OPT Version 2 CHAPTER 8 DESIGN VARIABLES CONSTANTS AND DEPENDENTS 2 The full design space is used ifthe range is omitted 3 The region of interest is centered on a given design and is used as a sub space of the design space to define the experimental design If the region of interest protrudes beyond the design space it is moved without contraction to a location flush with the design space boundary 8 4 Local variables For multidisciplinary design optimization MDO certain variables are assigned to some but not all solvers disciplines In the command file the following syntax defines the variable as local Command file syntax local variable name See Section 17 6 for an example 8 5 Assigning variable to solver If a variable has been flagged as local it needs to be assigned to a solver The command file syntax is Command file syntax Solver variable variable name See Section 17 6 for an example 8 6 Constants Each variable above can be modified to be a constant See Figure 8 2 where this is the case for t_bumper Constants are used 1 to define constant values in the input file such as 7 e or any other constant that may relate to the optimization problem e g initial velocity even
361. ximum displacement Full E x Maximum displacement Partial S 546 Lower bound 544 0 1 2 3 4 5 6 7 8 9 10 Iteration Figure 17 41 Optimization history of maximum displacement Starting design 1 Stage1Pulse Full t Stage2Pulse Full Stage3Pulse Full Stage1Pulse Partial Stage2Pulse Partial A Stage3Pulse Partial Lower bound Stage 1 BEAS Lower bound Stage 2 Lower bound Stage 3 Acceleration g Iteration Figure 17 42 Optimization history of Stage pulses Starting design 1 LS OPT Version 2 263 CHAPTER 17 EXAMPLE PROBLEMS Frequency Full Frequency Partial Upper bound Full Lower bound Full Frequency Hz eens Upper bound Partial Kennen Lower bound Partial Iteration Figure 17 43 Optimization history of torsional mode frequency Starting design 1 l The results of the partially shared variable case for starting design 1 can be seen to be superior to the fully shared case The reason for this is that all the disciplinary responses are now sensitive to their respective variables allowing faster convergence Interestingly most of the mass reduction in this case occurs in the cradle cross member a variable that is only included in the NVH simulation The variation of the remaining variables is however
362. y to zero is not a desirable thing to do For noisy data this may indicate over fitting rather than good modeling For highly discrepant training data zero MSE makes no sense at all Regularization means that some constraints are applied to the construction of the 26 LS OPT Version 2 CHAPTER 2 OPTIMIZATION METHODOLOGY neural model with the goal of reducing the generalization error that is the ability to predict interpolate the unobserved response for new data points that are generated by a similar mechanism as the observed data A fundamental problem in modeling noisy and or incomplete data is to balance the tightness of the constraints with the goodness of fit to the observed data This tradeoff is called the bias variance tradeoff in the statistical literature 0 5 0 5 0 5 Figure 2 3 Sigmoid transfer function y 1 1 g typically used with feed forward networks A multi layer feed forward network and a radial basis function network are two of the most common neural architectures used for approximating functions Networks of both types have a distinct layered topology in the sense that their processing units neurons are divided into several groups layers the outputs of each layer of neurons being the inputs to the next layer Figure 2 2 In a feed forward network each neuron performs a biased weighted sum of their inputs and passes this value through a transfer activation function to produc
363. yna CRASH VARIABLES FOR SOLVER CRASH Solver Variable rail_inner Solver Variable rail_outer Solver Variable aprons EXPERIMENTAL DESIGN OF SOLVER CRASH Solver Order linear Solver Experimental design dopt Solver Basis experiment 3toK Solver Number experiment 13 SOLVER AND PREPROCESSOR COMMANDS OF SOLVER CRASH solver command lsdyna single solver input file dyna input HISTORIES DEFINED FOR SOLVER CRASH history XDISP DynaASCII Nodout X_ DISP 26730 TIMESTEP history XACCEL DynaASCII Nodout X_ACC 26730 TIMESTEP SAE 60 RESPONSES FOR SOLVER CRASH 258 LS OPT Version 2 CHAPTER 17 EXAMPLE PROBLEMS response Vehicle Mass_crash 2204 62 0 DynaMas response Vehicle Mass_crash linear response Disp 1 0 DynaASCII Nodout X_DISP 267 response Disp linear response time to 184 expression Lookup XDISP response time to 334 expression Lookup XDISP response time _to_max expression LookupMax XD response Integral_0 184 expression Integral XACCEL t response Integral_184 334 expression Integral XACCEL t time_to_184 time_to_334 XD response Integral_334 max expression Integral XACCEL t time_to_334 time _to_max XD response StagelPulse expression Integral_ 0 1 s 29 30 32 33 34 35 79 81 82 83 MASS 30 MAX t 184 t 334 ISP t 0 time_to_184 XDISP t ISP t ISP t
364. yntax Maximize Example Response Mass DynaMass 3 13 14 16 MASS Maximize Objective Mass Constraint Acceleration 134 LS OPT Version 2 CHAPTER 11 OBJECTIVES AND CONSTRAINTS In LS OPTui objectives are defined in the Objective panel Figure 11 1 File Tasks Help Into Solvers Variables Point Selection Histories Responses Objective Constraints Run View Maximize the Objective Function instead of minimize Response Weight Generalized_Mass Intrusion HIC scaled 1 Create the Response definitions Responses Tab 2 Select Responses to be part of an Objective 3 Enter the relative weights for the components Figure 11 1 Objective panel in LS OPTui 11 3 Defining a constraint This command identifies each constraint function The constraint has the same name as its component A component can be a response or composite Command file syntax constraint constraint_name LS OPT Version 2 135 CHAPTER 11 OBJECTIVES AND CONSTRAINTS Examples history displacement_1 DynaASCII nodout r disp 12789 TIMESTEP 0 0 SAE 60 history displacement_2 DynaASCII nodout r disp 26993 TIMESTEP 0 0 SAE 60 history Intrusion displacement 2 displacement_1 response Intrusion_80 Intrusion 80 constraint Intrusion_80 Remark 1 Constraints can be specified in terms of response functions or composite functions In LS

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