1. 1 Numerical Shape Optimisation in Blow Moulding Hans Groot

2. 2Overview 1.Blow molding 2.Forward Problem 3.Inverse Problem 4.Optimisation Method 5.Conclusions & Future Work

3. Inverse Problem Optimization Method Conclusions Blow Molding Inverse Problem 3 Blow Molding/Forming  glass bottles/jars  polymer containers mould pre-form container

4. 4 Glass Bottle Forming Machine Inverse Problem Optimization Method Conclusions Blow Molding Inverse Problem

5. 5Problem Forward problem Inverse problem pre-formcontainer Blow Molding Optimization Method Conclusions Forward Problem Inverse Problem

6. 6 Forward Problem R1R1 R2R2 RiRi RmRm Surfaces R 1 and R 2 given Surface R m fixed (mould wall) Surface R i unknown Forward problem Blow MoldingConclusions Forward Problem Optimization MethodInverse Problem

7. 7 Constitutive Equations 1)Mechanics  Stokes flow problem 2)Thermodynamics  Convection diffusion problem 3)Evolution of surfaces  Convection problem Blow MoldingConclusions Forward Problem Optimization MethodInverse Problem

8. 8 Level Set Method glass air θ > 0 θ < 0 θ = 0  motivation: fixed finite element mesh topological changes are naturally dealt with surfaces implicitly defined level sets maintained as signed distances Blow MoldingConclusions Forward Problem Optimization MethodInverse Problem

9. 9 Computer Simulation Model  Finite element method  One fixed mesh for entire flow domain  2D axi-symmetric  At equipment boundaries:  no-slip of material  air is allowed to “ flow out ” Blow MoldingConclusions Forward Problem Optimization MethodInverse Problem

10. 10 Glass Blowing Blow MoldingConclusions Forward Problem Optimization MethodInverse Problem

11. 11 R1R1 Inverse Problem Inverse problem R2R2 RiRi RmRm Surfaces R i and R m given Surface R 1 fixed Surface R 2 unknown Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method

12. 12 Blowing corner No surface tension:  infinite time Surface tension  not possible: equilibrium Blowing round cavity Possible if: radius of mould cavity < radius of curvature Mould Requirements Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method

13. 13 Surface in equilibrium: Glass Surface in Equilibrium Contact angle φ between surface and mould Unknowns: curve z = f(r) contact radius r c contact depth z c Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

14. 14 Surface in equilibrium: Glass Surface in Equilibrium 2D curve z(r) : Boundary conditions: Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

15. 15 Surface in equilibrium: Glass Surface in Equilibrium Second order ODE: Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

16. 16 Surface in equilibrium: Glass Surface in Equilibrium Multiplication by z’ : Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

17. 17 Surface in equilibrium: Glass Surface in Equilibrium Integration: Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

18. 18 Glass Surface in Equilibrium First order ODE: Boundary conditions: Constants: Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method zczc rcrc -H φ L r z φ

19. 19 corner Example corner in (1,-10)

20. 20 Time Scale Equilibrium (no gravity): Time scale: Typical values: Typical blow process takes ~1s zczc rcrc -H L r z Blow Molding Forward Problem Conclusions Inverse Problem Optimization Method

21. 21Optimization Find pre-form for approximate container with minimal distance from container design mould wall container design approximate container Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

22. 22Optimization mould wall container design approximate container Minimize objective function RiRi d Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

23. 23 Computation of Objective Function  Objective function:  Line integral:  Composite Gaussian quadrature: m +1 control points ( ) → m intervals n weights w i per interval ( ˣ ) Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

24. 24 Parameterization of Pre-Form P1P1 P5P5 P4P4 P3P3 P2P2 P0P0 O R,φ 1.Describe unknown surface by parametric curve e.g. spline, Bezier curve 2.Define parameters as spherical radii of control points: 3.Optimization problem:  Find p as to minimize Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

25. 25  iterative method to minimize objective function  J : Jacobian matrix  : Levenberg-Marquardt parameter  H : Hessian of penalty functions:   i  w i /c i, w i : weight, c i >0: geometric constraint  g : gradient of penalty functions   p : parameter increment  d : distance between containers Modified Levenberg-Marquardt Method Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

26. 26 Tolerance Tolerance should not be smaller than total error of optimisation method

27. 27 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Truncation error: ε T  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I

28. 28 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Model simplifications  Discretisation of forward problem  Truncation error: ε T  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I

29. 29 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Discretisation of forward problem  Truncation error: ε T  Numerical differentiation of residual  Numerical integration (objective function)  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I

30. 30 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Discretisation of forward problem  Truncation error: ε T  Numerical differentiation of residual  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I

31. 31 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Discretisation of forward problem  Truncation error: ε T  Numerical differentiation of residual  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I

32. 32 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Discretisation of forward problem  Truncation error: ε T  Numerical differentiation of residual  Rounding error: ε R  Measurement error: ε M  Interpolation error: ε I  Interpolation of known surfaces (through data)  Interpolation of unknown surface (parametrisation)

33. 33 Numerical Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L  Discretisation of forward problem  Truncation error: ε T  Numerical differentiation of residual  Measurement error: ε M  Interpolation error: ε I  Interpolation of unknown surface (parametrisation)  Total Error: ε = ε L + ε T + ε M + ε I

34. 34 Restrictions on Errors Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Mesh size : h Linear elements: ε L =O(h 2 ) h 2  ε M  Distance between control points of unknown surface: ξ Cubic spline interpolation : ε I =O( ξ 4 ) ξ ~ C l /(m-1) o C : constant o l : curve length o m : number of control points ξ 4  h 2  m  C l h -1/2 +1 ξ

35. 35 Truncation Error Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Forward Difference Approximation:  Error:  Funtion evaluations: p = number of parameters  Central Difference Approximation:  Error:  Funtion evaluations: 2 p = number of parameters  Broyden Update:  Error: ?  No function evaluations

36. 36 Forward Difference Approximation Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Error bounded by  Minimum:  Conclusion:

37. 37 Broyden Update Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Jacobian Lipschitz continuous:  Error bound:  Conclusion:

38. 38 Error Analysis Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Types:  Model error: ε L = O(h 2 )  Truncation error: ε T = O(h)  Measurement error: ε M = O(h 2 ) (by choice of h )  Interpolation error: ε I = O(h 2 )  Total Error: ε ~ε T = O(h)  Tolerance:

39. 39  Assumptions:  Negligible mass flow in azimuthal direction ( u φ ≈ 0)  Constant viscosity  Given R 1 ( φ ), determine R 2 ( φ )  Volume conservation: R( φ ) radius of interface Approximation for Initial Guess streamlines φ r Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions

40. 40 Initial Guess for Bottle Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions R φ φ R streamlines r φ r

41. 41 Initial Guess approximate inverse problem initial guess of pre-formmodel container

42. 42 forward problem pre-formcontainer simulation approximation ( u φ ≈0) Comparison Approximation with Simulation

43. 43 Optimization of Pre-Form (no sagging) inverse probleminitial guess

44. 44 inverse probleminitial guess Optimization of Pre-Form (no sagging)

45. 45 inverse problemoptimal preform Optimization of Pre-Form (no sagging)

46. 46 Error Signed Distance between Approximate and Model Container topbottom Absolute Tollerance: h ≈ 0.059

47. 47 Summary  Inverse problem: find preform corresponding to container  Shape optimization method for pre- form in blow molding Pre-from surface described by parametric curve Approximation for initial guess Error in approximation of Jacobian is dominant  Application to glass blowing Blow Molding Forward Problem Optimization Method Conclusions Inverse Problem

48. 48 Future Work  Initial guess with sagging  Sensitivity analysis (w.r.t. perturbations in thickness)  Comparison finite difference with derivative-free optimisation  Adaptive optimisation strategy T-splines Adaptive mesh  Volume constraint  Application to polymers Blow Molding Forward Problem Optimization Method Conclusions Inverse Problem

49. 49 Blow Molding Forward Problem Optimization Method Conclusions Inverse Problem Thank you for your attention Questions?

50. 50  Incompressible medium: R( f ) radius of interface G  Simple example → axial symmetry: If R 1 is known, R 2 is uniquely determined and vice versa Initial Guess R(f)R(f)

51. 51 Inverse Problem  1 given (e.g. plunger)  m,  i given determine  2 22 11 Optimization: Find pre-form for container with minimal difference in glass distribution with respect to desired container

52. 52 Inverse Problem 11 22 ii mm  i and  m given  1 and  2 unknown Inverse problem

53. 53 Volume Conservation (incompressibility) R1R1 R2R2 RiRi RmRm

54. 54 Volume Conservation (incompressibility) R m fixed R i variable with R 1 and R 2 R 1, R 2 ?? RiRi RmRm R1R1 R2R2

55. 55 Blow Moulding preform container Forward problem Inverse problem

56. 56 Hybrid Broyden Method Optimisation ResultsIntroduction Simulation Model Conclusions [Martinez, Ochi]

57. 57 Error Analysis Optimization Method Inverse ProblemBlow Molding Inverse Problem Conclusions  Mesh size : h Linear elements: ε L =O(h 2 ) h 2   ε M  Distance between control points of unknown surface: ξ Cubic spline interpolation : ε I =O( ξ 4 ) ξ ~ C l /(m-1) o C : constant o l : curve length o m : number of control points ξ 4   h 2  m  C l h -1/2 +1