LS-OPT Current development: A perspective on multi-level optimization, MOO and classification methods

Transcription

1 LS-OPT Current development: A perspective on multi-level optimization, MOO and classification methods Nielen Stander, Anirban Basudhar, Imtiaz Gandikota LSTC, Livermore, CA LS-DYNA Developers Forum, Gothenburg, Sweden October 10,

2 Overview Background New developments Multi-level Optimization Multi-objective Optimization Classification Methods 2

3 Integrated Computational Materials Engineering $4m project funded by DOE ( ) with participation of US Automotive (Chrysler, Ford, GM), Steel industries and 6 Universities (material modeling) Goal: Reduce the lead time in applying lightweight 3GAHSS to integrated vehicle manufacturing Material model development, validation and LS-DYNA integration Forming analysis and validation Crash and NVH analysis of vehicle assembly Cost analysis Optimization: Goal: 35% (body) 25% (chassis) weight reduction 3

4 Integrated Computational Materials Engineering LSTC Collaborate with universities/government labs (e.g. PNNL) to integrate 3GAHSS material models Support forming analysis and vehicle crash analysis (consultant) Provide a common ICME framework for integrated design: material identification (and design?) Stamping MDO (crash, NVH) LS-OPT Development work: Interfaces to additional Third Party solvers (Matlab, Excel, Abaqus, LS- OPT) Additional features for multistage (flowchart) interface, e.g. variable transfer between stages Framework for multi-level optimization Parameterization of LS-OPT attributes (analogous to *PARAMETER) GUI enhancements, e.g. multilevel job monitoring 4

5 Multi-level Optimization Variables LS-OPT Stage type Optimized Variables/Responses OUTER Subdivision of problem into levels Nesting the optimization problem Variables and responses are transferred between levels Inner level optimization is done for each outer level sample INNER LS-DYNA Stage type LS-DYNA Stage type 5

6 Multi-level Optimization: Why? Organization. Easier to organize the problem as a collection of subsystems Efficiency. Solution algorithm takes advantage of the subproblem type Can match optimization methods with variable types, e.g. materials (categorical), sizing/shape (continuous). Robustness and accuracy. Smaller sub-problems are typically solved in a relatively low-dimensional space Critical framework for rational decomposition methods: Analytical Target Cascading Computing power. ~ 20,000 node clusters in operation Applications: System Optimization (component sublevels), Product families, Tolerance optimization 6

7 Multi-level Optimization: Why? Organization. Easier to organize the problem as a collection of subsystems Efficiency. Solution algorithm takes advantage of the subproblem type Can match optimization methods with variable types, e.g. materials (categorical), sizing/shape (continuous). Robustness and accuracy. Smaller sub-problems are typically solved in a relatively low-dimensional space Critical framework for rational decomposition methods: Analytical Target Cascading Computing power. ~ 20,000 node clusters in operation Applications: System Optimization (component sublevels), Product families, Tolerance optimization 7

8 Analytical Target Cascading Allows the individual design tasks to be conducted separately Consistency : Linked variables (which appear in multiple tasks) are solved iteratively Targets are identified first. These are propagated or cascaded to the rest of the system. Incorporated in penalty or Lagrangian formulations responses are determined which are fed back to the system task for the next iteration Formulated as a multilevel optimization problem Suitable for collaborative optimization 15

9 ATC: Simple analytical example Min f = (x 1 10) 2 +5(x 2 12) 2 +x x x x 62 + x 74 4x 6 x 7 10x 6 8x 7 g 1 = g 1 (x 1, x 2, x 3, x 4, x 5 ) g 2 = g 2 (x 1, x 2, x 3, x 4, x 5 ) g 3 = g 3 (x 1, x 2, x 3, x 4, x 5 ) g 4 = g 4 (x 1, x 2, x 3, x 4, x 5 ) All-in-one 10 x i 10, i = 1,, 7 P11: f = (x 1 10) 2 + 5(x 2 12) 2 +x 3 4 Decomposed x 1,x 2, x 3 P22 : f = 3 x x 6 5 g 1, g 2 x 4, x 5 x 1,x 2, x 3 P23 : f = 7x 62 + x 74 4x 6 x 7 10x 6 8x 7 g 3, g 4 x 6, x 7 17

10 ATC formulation with penalty P11: min f 11 = (x 1 10) 2 + 5(x 2 12) 2 +x 34 + π(c 22, c 23 ) with t 22 = x 1, x 2, x 3, t 23 = x 1, x 2, x 3 P22: min f 22 = 3 x x 56 + π(c 22 ) s.t. g 2, g 2 with r 22 = x 1, x 2, x 3, x 22 = x 4, x 5 P23: min f 23 = 7x 62 + x 74 4x 6 x 7 10x 6 8x 7 + π(c 23 ) s.t. g 3, g 4 with r 23 = x 1, x 2, x 3, x 23 = x 6, x 7 Consistency constraints: c ij = t ij r ij Relaxation function: π DorMohammadi, S., Rais-Rohani, M. Analytical Target Cascading using the Exponential Method of Multipliers, 12 th AIAA ATIO Conference and 14 th AIAA/ISSMO Conference, September,

11 Crashworthiness Design Problem Full Frontal Impact (FFI) Side Impact (SI) Minimize both mass and overall occupant injury Combination of injury criteria of FFI and SI used as objective function Multilevel optimization approach to satisfy FFI and SI crash requirements (constraints) Surrogate models of crash responses in FFI and SI used as design constraints 19

12 Analytical Target Cascading: Decomposition Element 11 Min f 1 x 11 = FFI IC + SI IC Element 22 Min f 2 x 22 = M FFI g i x 22 = R i R b i 0, i = 1,, 7 g 8 x 22 = R 8 b R 8 0 Element 23 Min f 3 x 23 = M SI g j x 23 = R j R b j 0, j = 9, 12 g 13 x 23 = R 13 b R 13 0 FFI Full Frontal Impact SI Side Impact M - Mass R Injury Criteria, Energy Gandikota, I., Rais-Rohani, M., DorMohammadi, S., Kiani, M. Multilevel vehicledummy design optimization for mass and injury criteria minimization, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 20

13 Crashworthiness Design Problem g 1 x g 7 x = FFI occupant responses, g 8 x = internal energy of FFI g 9 x g 12 x = SI occupant responses, g 13 x = SI internal energy FFI and SI response targets cascaded down to lower levels WIC and TTI injury criteria combined to form composite objective function Equal weight given to WIC and TTI, coefficients changed accordingly 21

14 Crash problem: ATC formulation r L22 t L22 Min f 11 =.3G G G G G G G 11 +π(c 22, c 23, c L22, c L23 ) s.t. 0 G i 1, i = 1,2,6,7,9,10,11 with t 22 = G i ; i = 1,2,6,7 ; t 23 = G j ; j = 9,10,11 t 22 = [G 1, G 2, G 6, G 7] t L22 = t L23 = x k ; k = 1,4,5,12 x 11 = [t 22, t 23, t L22, t L23 ] t 23 = [G 9, G 10, G 11] r 22 = [g 1, g 2, g 6, g 7 ] r 23 = [g 9, g 10, g 11 ] r L23 t L23 Min f 22 = f(x 1 11 ) + π(c 22, c L22 ) s.t. g 1,2,6,7 (r L22, x 22 ) 1 0 g 3,4,5 (r L22, x 22 ) 1 0 g 8 (r L22, x 22 ) 1 0 x L x x U With r 22 = g ; i = 1,2,6,7 i r L22 = x j ; j = 1,4,5,12 x 22 = x 2, x 3, x 6 11, x x 22 = [r 22, r L22, x 22 ] Min f 23 = f(x 1,4,5 x ) + π(c 23, c L23 ) s.t. g 8,9,10 (r L23, x 22 ) 1 0 g 12 (r L23, x 23 ) 1 0 g 13 (r L23, x 23 ) 1 0 x L x x U With r 23 = g ; i = 9,10,11 i r L23 = x j ; j = 1,4,5,12 x 23 = x x 23 = [r 23, r L23, x 23 ] 22

15 Crashworthiness Optimization example Gandikota, I., Rais-Rohani, M., DorMohammadi, S., Kiani, M. Multilevel vehicle-dummy design optimization for mass and injury criteria minimization, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering DorMohammadi, S., Rais-Rohani, M. Exponential penalty function formulation for multilevel optimization using the analytical target cascading framework. Structural and Multidisciplinary Optimization, DOI /s x, January

16 Multi-Objective Optimization Multi-Objective optimization is expensive NSGA-II algorithm in LS-OPT uses thousands of simulations for accurate results To improve efficiency, LS-OPT uses metamodels Uniform global sampling (Space Filling) Iterative solver: Global accuracy is improved at the expense of local accuracy Slows down convergence For single objectives, LS-OPT has adaptive metamodel-based methods (e.g. SRSM) Samples optimal region Benchmarked and widely tested in production environments A good starting point for extension to MOO. Requirement MOO method must be unified with SRSM: it becomes the same method for a single objective 26

17 Design Variable 2 Domain reduction (single objective) Region of Interest (for sampling) 2 3 start optimum RSM (polynomials) Use only points from current iteration NN, RBF, SVR Use all available points Design Space Design Variable 1 27

18 Domain Reduction Contraction rate as a function of accuracy and oscillation ( cˆ,abs( d)) pan =1 ĉ 0 1 osc =0.6 Panning -1 Oscillation 1 d 28

19 Pareto Domain Reduction Iteration i + + Subregion + Iteration i+k POF kernel (i-1) Exploratory design

20 Typical PDR Sampling (LS-OPT GUI) 30

21 Benchmarks: ZDT1 & ZDT2 1.2 Theory 1 PDR (computed) variables Metamodel (25 iter. * 110 simulations/iter.) Pareto simulations = 2850 simulations Direct (NSGA-II) ~20,000 simulations User selected Minimize X f ( X ) x ; 1 ZDT1: g( X ) 1 9( N 1) h( X ) 1 ( f1 g); x 1 ZDT2 : g( X ) 1 9( N 1) h( X ) 1 ( f 1 1 f 2 2 g) ; x ( X ) g( X ) h N 1 i2 x ; [0,1]. i2 x ; [0,1] g( X ), f i i N i i 1 ( X ), F2 1.2 Theory 1 PDR (computed) F1 31

22 Example: Direct MOO Crash Min. (Mass, Intrusion) Subject to: Intrusion 551mm Stage 1 pulse > 14.5g Stage 2 pulse > 17.6g Stage 3 pulse > 20.7g 41.38Hz freq 42.38Hz Vibration 7 Crash variables 7 Vibration variables 32

23 Strategies compared 1. Direct Optimization using NSGA-II algorithm. Tolerance on dominated hypervolume = Good reference solution. 2. Sequential Uniform Global optimization stopping at 30 iterations. Default metamodel strategy in LS-OPT. 3. Pareto Domain Reduction stopping at 30 iterations 33

24 Direct GA Convergence (benchmark) 34

25 Run Statistics Generations Simulations Per iteration Total number of simulations Run configur ation Direct NSGA-II , Uniform Global Sampling PDR (Adaptive Sampling) * * 13 4 *User-selected population of trade-off curve Solver: LS-DYNA MPP (explicit and implicit) 768 core cluster (LSTC) Queued through SunGridEngine (Oracle) 36

26 Convergence history (Direct vs. PDR) NSGA-II PDR 37

27 Uniform Global Sampling vs. Direct GA Direct GA Uniform Global Sampling (computed) Intrusion Mass 38

28 PDR vs. Direct GA Direct GA PDR (computed) Intrusion Mass 39

29 Feasibility (max. constraint violation) 40

30 Remarks and Conclusions (MOO) The PDR method has the potential for significantly reducing simulation cost for MOO SRSM and PDR are unified. If the Pareto Front has one point, the method automatically becomes SRSM Population choice not simple, may lead to suboptimal solutions 41