1 Numerical Shape Optimisation in Blow Moulding Hans Groot

2Overview 1.Blow molding 2.Inverse Problem 3.Optimization Method 4.Application to Glass Blowing 5.Conclusions & future work

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

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

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

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

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

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

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

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

11 Computation of Objective Function  Objective Function:  Composite Gaussian quadrature: m +1 control points ( ) → m intervals n weights w i per interval ( ˣ ) Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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

13  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 Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

14 Function Evaluations per Iteration  Distance function d : oone function evaluation  Jacobian matrix: 1.Finite difference approximation: op function evaluations ( p : number of parameters) 2.Broyden’s method: ono function evaluations, but less accurate function evaluation = solve forward problem Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

15  Neglect mass flow in azimuthal direction ( u f ≈ 0)  Given R 1 (f), determine R 2 (f)  Volume conservation: R( f ) radius of interface Approximation for Initial Guess streamlines f r Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

16 Initial Guess approximate inverse problem initial guess of pre-form model container

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

18 Forward Problem 1)Flow of glass and air  Stokes flow problem 2)Energy exchange in glass and air  Convection diffusion problem 3)Evolution of glass-air interfaces  Convection problem Blow Molding Inverse Problem Conclusions Glass Blowing Optimization Method

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

20 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 ” Blow Molding Inverse Problem Conclusions Glass Blowing Optimization Method

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

22 Optimization of Pre-Form  inverse problem  initial guess

23 Optimization of Pre-Form  initial guess  inverse problem

24 Optimization of Pre-Form  optimal pre-form  inverse problem

25 Signed Distance between Approximate and Model Container

26 Summary  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 Glass Blowing Conclusions Optimization Method

27 Short Term Plans  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 Glass Blowing Conclusions Optimization Method

28 Parison Optimization for Ellipse model containeroptimal containerinitial guess

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

30Approximation Initial guess pre-form model container

31  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)

32 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

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

34 Volume Conservation (incompressibility) R1R1 R2R2 RiRi RmRm

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

36 Blow Moulding preform container Forward problem Inverse problem

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

38 Example (p = 13) Optimisation ResultsIntroduction Simulation Model Conclusions Method# function evaluations# iterations  Hybrid Broyden Finite Differences  Conclusions: similar number of iterations similar objective function value Finite Differences takes approx. 3 times longer than Hybrid Broyden

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

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

41 Conclusions Optimisation Introduction Simulation Model Results  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

42 Thank you for your attention

43Comparison Inverse problemForward problem two unknown intefacesone unknown interface Inverse problem Forward problem 11 22 ii mm Inverse problem under-determined or forward problem over-determined?

44 Inverse Problem Optimisation ResultsIntroduction Simulation Model Conclusions preformcontainer Unknown surfaces

45 Forward Problem Optimisation ResultsIntroduction Simulation Model Conclusions preformcontainer R m known R i un known

46  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 Volume Conservation R(f)R(f)

47  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 Volume Conservation R(f)R(f)