1. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 1 R.A. Van den Braembussche von Karman Institute for Fluid Dynamics Tuning of Optimization Strategies

2. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 2 Improving Convergence of GA Performance Database Geometry GA NS Navier- Stokes Metafunction NS, HT, FEA Predict Learn Requirements Start FEA Stress analysis HT Heat transfer Parallel computing

3. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 3 Improving Convergence of GA Performance Database Geometry GA NS Navier- Stokes Metafunction NS, HT, FEA Predict Learn Requirements Start FEA Stress analysis HT Heat transfer

4. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 4 1. Population size N 2. Substring length l 3. Crossover Probability Pc 4. Mutation Probability Pm 5. Number of children ch Optimal parameter setting ( to accelerate evolution ) Genetic Algorithm Optimal parameter setting

5. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 5 Genetic Algorithm Optimal parameter setting OF defined by test function Tests on 7 and 27 parameter function GA = non-deterministic Conclusions based on: 5 optimization Result of given effort 5000 OF evaluations Six hump camel back test function

6. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 6 Genetic Algorithm (1) Population size # evaluations = 5000 = N * t t number of generations N population size Small populations Premature convergence Local optimum Low number of feasible geometries Large populations Low number of generations No evolution 10 < N < 20 Population size

7. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 7 Genetic Algorithm (2) Substring length l = # of bits / variable 2 l values / variable ε = desired resolution Global minimum Best possible solution average OF x min x max 00 0110 01

8. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 8 N variables l bits / variable L < 3 too low resolution L > 10 too large design space (slower convergence) Substring length (# of bits) Genetic Algorithm (2) Substring length

9. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 9 Uniform crossover Swap with probability p c Genetic Algorithm Cross over 0000011111 00011 # of function evaluations Single crossover One random swap / individual 0000011111 01101

10. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 10 p m probability a bit is swapped Mutation probability Total string length ( l.N ) Optimal p m p m =1/( l.n ) p m =2/( l.n ) Genetic Algorithm Mutation 0000011111 00011 01011 mutation Optimal p m ______ p m = 1/(l.N) _ _ _ _ p m = 2/(l.N)

11. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 11 Genetic Algorithm New generation (n,ch)n best of ch offspring's replace the old population (best individuals can be lost) (n+ch)n best of (ch offspring's + n old population) replace the old population (elitism) (n/i+ch)n/i best contribute to new generation diversity

12. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 12 Genetic Algorithm Optimal number of children

13. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 13 Optimization Convergence Performance Database Geometry GA NS Navier- Stokes Metafunction NS, HT, FEA Predict Learn Requirements Start FEA Stress analysis HT Heat transfer

14. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 14 Metafunction Type (ANN, RBF) Structure (# hidden l) RBF 5 hidden neurons De Jong 2D test function ANN 2 hidden layers 10 hidden neurons Database # of samples Distribution of samples

15. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 15 Database x x x x Systematic scanning 2 values /variable n variables full factorial 2 n evaluations 7 variables 128 NS evaluations 27 variables 10 7 NS evaluations

16. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 16 Database merit function merit function objective function m(x) = f(x) - m d m (x)

17. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 17 Database Alt: Latin Hypercube – Random selection 3 variables 2 values (+ -) / variable 2 3 = 8 combinations 1, 2, 3 and 4 : main effect 5, 6, 7 and 8 : interaction D esign O f E xperiment DOE

18. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 18 Database Random DOE 64321688a168b8c8d 6 parameters Full factorial = 2 6 = 64 Error in 64 points

19. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 19 Statistical analysis 2 k factorial =64 2 k-2 factorial =16 k=6 Database

20. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 20 Statistical analysis 2 k-3 factorial = 8 k=6 2 level variables (25% and 75% of non dimensional range) 1 central variable (all variables at 50% of range) 12 to 15 variables 16 runs 16 to 31 variables 32 runs Database

21. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 21 Metafunction ANN Learning : define W (weight) and b (bias) Navier Stokes results Geometry & bound. cond.

22. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 22 Metafunction Kriging Linear least square approximation Predicts value and uncertainty Accurate evaluations in regions of high uncertainty Very time consuming

23. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 23 Metafunction RBF Learning : define W (weight) and b (bias) Gausian activation function

24. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 24 Multi-objective optimization Convergence

25. von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 2009 25 Genetic Algorithm Gray coding 1001 2011 3010 4110 5111 6101 7100 Gray coding ValueCode 1001 2010 3011 4100 5101 6110 7111 Binary coding ValueCode No real advantage observed