1. Recent CASA consultancy activities By Jürgen Tas 09 - 05 - 2006

2. Consultancy Added-value for the industry: optimize process parameters improve products provide insight in production processModelling NumericalSimulations Product and Process optimization

3. Goal Presentation Three relevant practical cases Explain - problem - approach - solution / result - future work / idea's Feedback!

4. Case 1: Problem(1) For measured spectrum: Find shape parameters: Such that: -   0 (best fit) measured computed Optical diffraction Micro structure

5. Case 1: Problem(2) Forward problem (simulation) Minimize cost function Define initial shape Compute spectrumUpdate shape Goal: Software Tool (30mins → 1 sec.) Inverse problem Computed ≈ Measured No Yes START END

6. Case 1: Approach Mathematics Least Squares (Gauss-Newton) Maxwell equations Sub-sampling Multivariate data analysis Software development C++ implementation with LAPACK Caching / Library Parallel Computing

7. Case 1: Solution(1) Nonlinear Least Squares Problem: “Computed ≈ Measured” (best fit) min p ║C(p) – M║ 2 = min p ║F(p)║ 2 Gauss-Newton SymbolMath. descriptionDescription pp = [p 1,...,p n ] T n shape parameters MM = [M 1,...,M m ] T m measured points CC(p) = [C 1 (p),...,C m (p)] T m computed points, (C: R n → R m )

8. Case 1: Solution(2) Features: linearization QR (normal eqn) relaxation } while(!converged ) i = 0 do { A = (  F/  p)(p i ), b = Ap i – F(p i ) linear LS: Ap i+1 = QRp i+1 = b p i+1 = (1-λ)p i + λp i+1, 0 ≤ λ ≤ 1 i++ Disadvantages: initial guess Jacobian expensive

9. Case 1: Future Ideas Robustness: damped Gauss-Newton (LM) adaptive relaxation Speed-Up: less iterations (← good initial guess) estimate Jacobian (Quasi-Newton) Trade-off? 30mins → 15 sec. (speed-up x 120!!)

10. Case 2: Problem Feasibility Study: Bellows - hearing aid devices Maximum Stress for - given form - given material - given load Yield point x

11. Case 2: Approach COMSOL Multiphysics Benchmark!

12. Case 2: Results (1) 25μm 650μm 975 μ m 5MPa (force/area)

13. Case 2: Results (2)

14. Case 2: Results (3) spring stiffness

15. Case 2: Future Work minimize spring stiffness minimize maximum Von Mises stress? Shape optimization by: Under constraints: minimum wall thickness minimum flow rate? Formulation: inverse problem?

16. Case 3: Problem(1) Fastest path (A → B) |x´(t)| ≤ v max, |x´´(t)| ≤ a max Avoid obstacles

17. Case 3: Problem (2) Time-Optimal Control Theory x y Obstacle ● ● B A x(0), x´(0) x(T), x´(T) minimal T? |x´(t)| ≤ v max, |x´´(t)| ≤ a max

18. Case 3: Approach (2)?? Time Optimization Problem: Given x 0 єR 2, determine T>0 and a function u:[0,T]→[-a max,a max ] such that the solution of x´ = Ax + bu with initial values x(0) = x 0, x´(0) = 0, we have x(T) = 0, x´(T) = 0, | x´ (t) | ≤ v max for a minimal T. + Avoid obstacles!!

19. Case 2: Future Work Minimize cost function Parameterized Bellow Forward ProblemUpdate shape Design: optimal shape + max. stress ≤ τ max Inverse problem Optimal Shape? No Yes START END Comsol Multiphysics (White Box) Constraint: max. stress ≤ τ max