Development of a Numerical Optimisation Method for Blowing Glass Parison Shapes

1. Development of a Numerical Optimisation Method for Blowing Glass Parison Shapes Hans Groot

2. Overview • Introduction • Glass Blow Simulation Model • Optimisation Method • Results • Conclusions

3. Simulation Model Optimisation Results Conclusions Introduction Glass Manufacturing • Glass Forming • Surface Treatment • Glass Melting • Glass Conditioning • Automatic Inspection

4. Simulation Model Optimisation Results Conclusions Introduction Glass Forming • press • press-blow • blow-blow

5. glass mould ring Simulation Model Optimisation Results Conclusions Introduction Blow-Blow Process ring preform mould

6. Introduction Optimisation Results Conclusions Simulation Model Blow Model • Flow of glass and air • Stokes flow problem • Viscous forces dominate • Temperature dependent glass viscosity • Energy exchange in glass and air • Convection diffusion problem • No viscous dissipation • Evolution of glass-air interfaces • Convection problem for level sets

7. θ = 0 air air θ < 0 θ < 0 θ > 0 glass Introduction Optimisation Results Conclusions Simulation Model Level Set Method • motivation: • fixed finite element mesh • topological changes are naturally dealt with • interfaces implicitly defined • level sets maintained as signed distances

8. Introduction Optimisation Results Conclusions Simulation Model Computer Simulation Model • Finite element discretisation • One fixed mesh for entire flow domain • 2D axi-symmetric • At equipment boundaries: • no-slip of glass • air is allowed to “flow out”

9. Introduction Optimisation Results Conclusions Simulation Model Bottle Blowing Simulation Glass-air interfaces Temperature

10. Glass Distribution for Jar Preform 1: thickenings! Preform 2: breaks!

11. Introduction Simulation Model Results Conclusions Optimisation Inverse Problem Given container g find preform • Optimisation: • Find preform that minimises difference in glass distribution between model container and container obtained by blow process ?

12. d Introduction Simulation Model Results Conclusions Optimisation Least Squares Minimisation Problem true interface approximate interface • Residual: • Minimise objective function:

13. P0 Q0 Q1 Q2 Q3 P1 P2 Q4 Q5 Introduction Simulation Model Results Conclusions Optimisation z r Optimisation Strategy • Describe interfaces by parametric curves • e.g. splines, Bezier curves • Define parameters: • Compute signed distance • Minimise

14. Introduction Simulation Model Results Conclusions Optimisation Modified Levenberg-Marquardt Method • iterative method to minimise 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 • r: residual

15. Introduction Simulation Model Results Conclusions Optimisation Computation of Jacobian • Finite difference approximation: • requirespfunction evaluations, p:number ofparameters • Secant method: • updates Jacobian inincremental direction • no function evaluations • may fail to find descent direction • finite difference approximation

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

17. Introduction Simulation Model Results Conclusions Optimisation Example • Conclusions: • similar number of iterations • similar objective function value • Finite Differences takes approx. 3 times longer than Hybrid Broyden

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

19. 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

20. 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

21. 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 Introduction Simulation Model Optimisation Results Conclusions Short Term Plans

22. Thank you for your attention

23. Blowing Model: 2nd blow

24. m 2 s 1 i Blowing Model: 1st Blow Axi-symmetric blowing of parison 1165 oC glass 500 oC mould

25. ring preform mould

26. A Δx B C Introduction Simulation Model Results Conclusions Level Set Method Re-initialisation by FMM Algorithm

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