Numerical Shape Optimisation in Blow Moulding

1. Numerical Shape Optimisation in Blow Moulding Hans Groot

2. Overview • Blow molding • Inverse Problem • Optimization Method • Application to Glass Blowing • Conclusions & future work

3. Inverse Problem Glass Blowing Optimization Method Conclusions Blow Molding Blow Molding container mould pre-form • glass bottles/jars • plastic/rubber containers

4. Inverse Problem Glass Blowing Optimization Method Conclusions Blow Molding Example: Jar

5. Blow Molding Glass Blowing Optimization Method Conclusions Inverse Problem Problem Forward problem Inverse problem pre-form container

6. Blow Molding Glass Blowing Optimization Method Conclusions Inverse Problem Forward Problem • Surfaces1and2 given • Surfacem fixed (mould wall) • Surfacei unknown Forward problem m 1 i 2

7. Blow Molding Glass Blowing Optimization Method Conclusions Inverse Problem Inverse Problem • Surfacesiandm given • Either1 or 2 unknown m 1 i 2 Inverse problem

8. Blow Molding Glass Blowing Optimization Method Conclusions Inverse Problem Construction of Pre-Form by Pressing 2 1

9. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Optimization mould wall model container approximate container Find pre-form for approximate container with minimal distance from model container

10. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Optimization mould wall model container approximate container Minimize objective function d i

11. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Computation of Objective Function • Objective Function: • Composite Gaussian quadrature: • m+1 control points (•)→m intervals • n weights wiper interval (ˣ)

12. OR,φ P0 P1 P2 P3 P4 P5 Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Parameterization of Pre-Form • Describe surface by parametric curve • e.g. spline, Bezier curve • Define parameters as radii of control points: • Optimization problem: • Find p as to minimize

13. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Modified Levenberg-Marquardt Method • iterative method to minimize objective function • J: Jacobian matrix • l: Levenberg-Marquardt parameter • H: Hessian of penalty functions: • zi = wi /ci , wi : weight, ci >0: geometric constraint • g: gradient of penalty functions • Dp: parameter increment • d: distance between containers

14. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Function Evaluations per Iteration function evaluation = solve forward problem • Distance function d: • one function evaluation • Jacobian matrix: • Finite difference approximation: • pfunction evaluations (p:number of parameters) • Broyden’s method: • no function evaluations, but less accurate

15. Blow Molding Inverse Problem Glass Blowing Conclusions Optimisation Method Approximation for InitialGuess • Neglect mass flow in azimuthal direction (uf≈0) • Given R1(f), determine R2(f) • Volume conservation: • R(f) radius of interface r f streamlines

16. InitialGuess approximate inverse problem initial guess of pre-form model container

17. Blow Molding Inverse Problem Optimization Method Conclusions Glass Blowing Glass Blowing

18. Blow Molding Inverse Problem Optimization Method Conclusions Glass Blowing Forward Problem • Flow of glass and air • Stokes flow problem • Energy exchange in glass and air • Convection diffusion problem • Evolution of glass-air interfaces • Convection problem

19. θ = 0 air air θ < 0 θ < 0 θ > 0 glass Blow Molding Inverse Problem Optimization Method Conclusions Glass Blowing Level Set Method • motivation: • fixed finite element mesh • topological changes are naturally dealt with • interfaces implicitly defined • level sets maintained as signed distances

20. Blow Molding Inverse Problem Optimization Method Conclusions Glass Blowing Computer Simulation Model • Finite element method • One fixed mesh for entire flow domain • 2D axi-symmetric • At equipment boundaries: • no-slip of glass • air is allowed to “flow out”

21. Comparison Approximation with Simulation Model simulation approximation (uf≈0) forward problem pre-form container

22. ? Optimization of Pre-Form • inverse problem • initial guess

23. ? Optimization of Pre-Form • inverse problem • initial guess

24. ? Optimization of Pre-Form • inverse problem • optimal pre-form

25. Signed Distance between Approximate and Model Container

26. Shape optimization method for pre-form in blow molding describe either pre-form surface by parametric curve minimize distance from approximate container to model container find optimal radii of control points use approximation for initial guess Application to glass blowing average distance < 1% of radius mold Blow Molding Inverse Problem Optimization Method Glass Blowing Conclusions Summary

27. Extend simulation model improve switch free-stress to no-slip boundary conditions one level set problem vs. two level set problems Well-posedness of inverse problem Sensitivity analysis of inverse problem Blow Molding Inverse Problem Optimization Method Glass Blowing Conclusions Short Term Plans

28. Parison Optimization for Ellipse model container optimal container initial guess

29. container ring mould parison Blow Molding • e.g. glass bottles/jars

30. Approximation Initial guess pre-form model container

31. R(f) Initial Guess • Incompressible medium: • R(f) radius of interface G • Simple example → axial symmetry: • If R1is known, R2 is uniquely determined and vice versa

32. Inverse Problem • 1given(e.g. plunger) • m,igiven • determine 2 1 2 • Optimization: • Find pre-form for container with minimal difference in glass distribution with respect to desired container

33. m i Inverse Problem • iandm given • 1 and 2 unknown 1 2 Inverse problem

34. Volume Conservation (incompressibility) R1 Rm Ri R2

35. R1 R2 Volume Conservation (incompressibility) • Rm fixed • Ri variable • with R1 and R2 • R1,R2?? Rm Ri

36. Blow Moulding Forward problem ? Inverse problem preform container

37. Introduction Simulation Model Results Conclusions Optimisation Hybrid Broyden Method [Martinez, Ochi]

38. Introduction Simulation Model Results Conclusions Optimisation Example (p = 13) • Conclusions: • similar number of iterations • similar objective function value • Finite Differences takes approx. 3 times longer than Hybrid Broyden

39. Introduction Simulation Model Level Set Method Conclusions Results Preform Optimisation for Jar Model jar Optimal preform Initial guess

40. Introduction Simulation Model Level Set Method Conclusions Results Preform Optimisation for Jar Radius: 1.0 Mean distance: 0.019 Max. distance: 0.104 Model jar Approximate jar

41. Glass Blow Simulation Model finite element method level set techniques for interface tracking 2D axi-symmetric problems Optimisation method for preform in glass blowing preform described by parametric curves control points optimised by nonlinear least squares Application to blowing of jar mean distance < 2% of radius jar Introduction Simulation Model Optimisation Results Conclusions Conclusions

42. Thank you for your attention

43. Forward problem m 1 i 2 Inverse problem Comparison Inverse problem under-determined or forward problem over-determined?

44. Introduction Simulation Model Results Conclusions Optimisation Inverse Problem Unknown surfaces preform container

45. Introduction Simulation Model Results Conclusions Optimisation Forward Problem Rmknown Ri unknown preform container

46. R(f) Volume Conservation • Incompressible medium: • R(f) radius of interface G • Simple example → axial symmetry: • If R1is known, R2 is uniquely determined and vice versa

47. R(f) Volume Conservation • Incompressible medium: • R(f) radius of interface G • Simple example → axial symmetry: • If R1is known, R2 is uniquely determined and vice versa