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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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 mutation Optimal p m ______ p m = 1/(l.N) _ _ _ _ p m = 2/(l.N)

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

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

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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

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

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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, 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

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

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

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

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

von Karman Institute for Fluid Dynamics RTO, AVT 167, October, Genetic Algorithm Gray coding Gray coding ValueCode Binary coding ValueCode No real advantage observed