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  Glass Melting  Glass Conditioning  Automatic Inspection Glass Manufacturing  Glass Forming  Surface Treatment Introduction ResultsSimulation ModelOptimisationConclusions

4 Glass Forming press press-blow blow-blow Introduction ResultsSimulation ModelOptimisationConclusions

5 Blow-Blow Process glass mould ring preform mould ring Introduction ResultsSimulation ModelOptimisationConclusions

6 Blow Model 1)Flow of glass and air  Stokes flow problem Viscous forces dominate Temperature dependent glass viscosity 2)Energy exchange in glass and air  Convection diffusion problem No viscous dissipation 3)Evolution of glass-air interfaces  Convection problem for level sets Simulation Model ResultsIntroductionOptimisationConclusions

7 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 Simulation Model ResultsIntroductionOptimisationConclusions

8 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 ” Simulation Model ResultsIntroductionOptimisationConclusions

9 Bottle Blowing Simulation TemperatureGlass-air interfaces Simulation Model ResultsIntroductionOptimisationConclusions

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

11 Given container  find preform  Optimisation: Find preform that minimises difference in glass distribution between model container and container obtained by blow process Inverse Problem Optimisation ResultsIntroduction Simulation Model Conclusions

12 Least Squares Minimisation Problem  Residual:  Minimise objective function: Optimisation ResultsIntroduction Simulation Model Conclusions d true interface approximate interface

13 Optimisation Strategy 1.Describe interfaces by parametric curves e.g. splines, Bezier curves 2.Define parameters: 3.Compute signed distance 4.Minimise Optimisation ResultsIntroduction Simulation Model Conclusions P2P2 P1P1 Q3Q3 Q2Q2 Q1Q1 Q0Q0 Q4Q4 Q5Q5 P0P0 z r

14  iterative method to minimise 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  r : residual Modified Levenberg-Marquardt Method Optimisation ResultsIntroduction Simulation Model Conclusions

15 Computation of Jacobian 1.Finite difference approximation:  requires p function evaluations, p : number of parameters 2.Secant method:  updates Jacobian in incremental direction  no function evaluations  may fail to find descent direction  finite difference approximation Optimisation ResultsIntroduction Simulation Model Conclusions

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

17 Example 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

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

19 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

20 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

21 Short Term Plans Conclusions Optimisation Introduction Simulation Model Results  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

22 Thank you for your attention

23 Blowing Model: 2 nd blow

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

25 ring preform mould

26 Re-initialisation by FMM Algorithm CB A ΔxΔx Level Set Method ResultsIntroduction Simulation Model Conclusions

27 Preform Optimisation for Jar Level Set Method ResultsIntroduction Simulation Model Conclusions Model jar Initial guessOptimal preform