Dr.-Ing. Johannes Will CAD-FEM GmbH/DYNARDO GmbH dynamic software & engineering GmbH

Transcription

1 Evolutionary and Genetic Algorithms in OptiSLang Dr.-Ing. Johannes Will CAD-FEM GmbH/DYNARDO GmbH dynamic software & engineering GmbH

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3 Genetic Algorithms (GA) versus Evolutionary Algorithms (EA) Schwefel, & other Rechenberg, & other design improvement due to combination of genes design improvement due to heredity of genes weak selection (clean gene pool) different crossover methods for reproduction/recombination (main evolution process) weak mutation (fresh up gene pool) strong selection copy for reproduction of individuals strong mutation (main evolution process due to mutation) due to collective memory higher/faster convergence to global optimum

4 Reselection/Replace procedures of GA selection of n best design according to high/low rank for parent generation select elits with roulette whell logic according to elit selection pressure

5 SLang Crossover procedures of GA multipoint segmented uniform OptiSLang arithmetic

6 Mutation procedures of GA -a new value for the mutant gene is stochastically selected assuming: - Gauss distribution σ = standard deviation strongly influence the mutation step - standard deviation σ - old gene value = mean value = µ σ = 0.1

7 OptiSLang Subflow Library Optimization stochastic search for optima/design improvements using GA general settings number of unmodified best designs number designs to be replaced uniform Elit weak Elitism strong Elitism

8 OptiSLang Subflow Library Optimization stochastic search for optima/design improvements using GA Genetic Advance Parameter Setting interpolation function between start/end deviation gene mutation percentage Gauss distribution smaller variation for converging GA recommended

9 using combined GA RSM Methodology A) use of higher order approximation functions (weighted interpolation) for RSM - optimization on possibly non-smooth, non-convex RS with GA - adaptive DOE sampling with deterministic /central composite or stochastic DOE schemes (Monte Carlo, Latin hypercube) - possibly better local approximation with weighted interpolation, but still evaluation of the full parameter range local and global refinement of RS B) use of weighted radii interpolation function for re-evaluation of GA - reevaluation GA with modified objectives - use of all available support points for RS with weighted interpolation recycling of solver runs for re-evaluations

10 Approximation with Response Surface Methods (RSM) - example of different approximation functions with 4 support points linear interpolation weak approximation quadratic interpolation good smoothness, good convexity, but possibly weak approximation near and far of the support points weighted interpolation good smoothness, less convexity but possibly better approximation near and far support points

11 Approximation with Response Surface Methods (RSM) -Problem: Parameter Identification of frame structure - two Parameter (Young Modulus of vertical/horizontal bars) - five load cases Solution: 2 / 1.8 MPa calculation of exact RS with Monte Carlo Simulation

12 Approximation with Response Surface Methods (RSM) DOE Scheme Number Support points Results on RS linear /2.63 quadratic /2.01 m= /2.09 central composite /2.06 m=4 adaptive, m= / /1.84 solution 2/1.8 linear quadratic m=2

13 Approximation with Response Surface Methods (RSM) DOE Scheme Number Support points Results on RS linear /2.63 quadratic /2.01 m= /2.09 central composite /2.06 m=3 adaptiv, m= /2.04 (9) /1.84 solution 2/1.8 central composite, m=3, adaptiv, m=2

14 Approximation with Response Surface Methods (RSM) using weighted interpolation functions DOE Scheme Number Support points Interpolation function Results on RS m=3 adaptiv, m=2 16 quadratic 2.23/2.04 (9) + 5 quadratic 1.99/1.84 m=3 adaptiv, m=3 16 weighted interpolation coeff.=2 32 weighted interpolation coeff.=2 2.37/ /1.85 solution 2/1.8, m=3, adaptiv, m=3

15 Approximation with Response Surface Methods (RSM) using weighted interpolation functions DOE Scheme Number Support points Interpolation function Results on RS m=3 adaptiv, m=2 16 quadratic 2.23/2.04 (9) + 5 quadratic 1.99/1.84 m=3 adaptiv, m=3 16 weighted radii interpolation coeff.=2 32 weighted radii interpolation coeff.=4 2.37/ /1.64 solution 2/1.8, adaptiv, m=3

16 using Genetic Algorithms (GA) & Evolutionary Algorithms (EA) for fast design improvements - stochastic search strategy - flexible implementation of genetic and evolutionary algorithms and programming or combinations GA/EA [bit & float (Riedel)] - constrains treated as penalties (cliffs) recommended area of application: 0/1-problems, continuous, discrete & binary variables, up to some 1000 variables + always valid solution, arbitrary restart/replay, arbitrary constrains, no gradients necessary - slow convergence,

17 Genetic Optimization of spotwelds GENETIC ALGORITHM (EA) II Optimization of spotweld number and position due to stiffness & fatigue 134 binary Variables, two load cases, stress constrains - strong Elitism to reach global optima Design Evaluations: 1300 Design Improvement: 76 % - Fatigue related stress evaluation in all spotwelds σ R( φ ) = σ FX cosφ + σ FY sinφ σ FZ σ MX sinφ + σ MY cosφ 140N / mm 2 Solver: ANSYS (using automatic spotweld meshing procedure in ANSYS)

18 Genetic Optimization of spotwelds GENETIC ALGORITHM (EA) II Optimization of spotweld number and position due to stiffness & fatigue 134 binary Variables, torsion loading, stress constrains - weak Elitism to reach fast Design Improvement Design Evaluations: 200 Design Improvement: 47 % - Fatigue related stress evaluation in all spotwelds σ R( φ ) = σ FX cosφ + σ FY sinφ σ FZ σ MX sinφ + σ MY cosφ 140N / mm 2 Solver: ANSYS (using automatic spotweld meshing procedure in ANSYS)

19 Genetic Optimization of Boxbeam structure GENETIC ALGORITHM (EA) III weighted multi criteria optimization (total weight, spotweld number and position, energy dissipation due to stiffness (dynamic/static) & fatigue & crash Constrains: 1 & 2 Eigenfrequency, Spotweld Stress, deformation due to torsion and bending, compression length, force magnitude integral function 23 binary Variables (spotwelds), 1 discrete variable (wall thickness) - Modal analysis, two static & one crash load case, - GA with strong Elitism Solver: ANSYS /LS-DYNA

20 after 2 optimization cycle with 100 Design evaluations New Design: 13 % weight reduction, 37 % Spot weld reduction

21 How fast is the design improvement? (100) = 40 spot welds mm sheet thickness Start (40 SWP, 1.5 mm) Referenz 2. Optimization (107) with improved robustness constraints Generation (10 Runs) 90,7 86,7 2 Generation (20 Runs) 3 Generation (30 Runs) 85,4 85,4 85,4 85,4 82,4 81,8 81,4 79,6 4 Generation (40 Runs) Referenz1. Optimization (94) 5 Generation (50 Runs) 1 Optimierung 2 Optimierung The main design improvements resulted from the first 3 Generation

22 Benchmark German Automotive Industry for Non-linear Optimization and Robustness Evaluation of crash worthiness Design Evaluations: 174 Design Improvement: 58 % BEST_HISTORY Object BEST_HISTORY 8.5E E1 Design Evaluations: 173 Design Improvement: 27 % BEST_HISTORY SLang the Structural Language, Version Created on 16 Sep SLang the Structural Language, Version Created on 16 Sep 2001.

23 Genetic Optimization of very large Ship vessel EVOLUTIONARY ALGORITHM (EA) I Optimization of total weight of two load cases with constrains (stresses) discrete Variables self regulating Evolutionary strategy - Population of 4, uniform crossover for reproduction - active search for dominant genes with different mutation rates Design Evaluations: 3000 Design Improvement: > 10 % Solver: ANSYS on SGI 24 proc.