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Published byAlaina James Modified over 9 years ago
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1 Numerical Shape Optimisation in Blow Moulding Hans Groot
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2Overview 1.Blow molding 2.Inverse Problem 3.Optimization Method 4.Application to Glass Blowing 5.Conclusions & future work
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3 Inverse Problem Glass Blowing Conclusions Blow Molding Optimization Method Blow Molding glass bottles/jars plastic/rubber containers mould pre-form container
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4 Example: Jar Inverse Problem Glass Blowing Conclusions Blow Molding Optimization Method
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5Problem Forward problem Inverse problem pre-formcontainer Blow Molding Glass Blowing Conclusions Inverse Problem Optimization Method
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6 Forward Problem 11 22 ii mm Surfaces 1 and 2 given Surface m fixed (mould wall) Surface i unknown Forward problem Blow Molding Glass Blowing Conclusions Inverse Problem Optimization Method
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7 11 Inverse Problem Surfaces i and m given Either 1 or 2 unknown Inverse problem 22 ii mm Blow Molding Glass Blowing Conclusions Inverse Problem Optimization Method
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8 Construction of Pre-Form by Pressing 11 22 Blow Molding Glass Blowing Conclusions Inverse Problem Optimization Method
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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
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10Optimization mould wall model container approximate container Minimize objective function ii d Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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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
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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
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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
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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
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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
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16 Initial Guess approximate inverse problem initial guess of pre-form model container
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17 Glass Blowing Blow Molding Inverse Problem Conclusions Glass Blowing Optimization Method
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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
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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
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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
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21 Comparison Approximation with Simulation Model forward problem pre-form container simulation approximation ( u f ≈0)
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22 Optimization of Pre-Form inverse problem initial guess
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23 Optimization of Pre-Form initial guess inverse problem
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24 Optimization of Pre-Form optimal pre-form inverse problem
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25 Signed Distance between Approximate and Model Container
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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
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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
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28 Parison Optimization for Ellipse model containeroptimal containerinitial guess
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29 Blow Molding mould ring parison container e.g. glass bottles/jars
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30Approximation Initial guess pre-form model container
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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)
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32 Inverse Problem 1 given (e.g. plunger) m, i given determine 2 22 11 Optimization: Find pre-form for container with minimal difference in glass distribution with respect to desired container
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33 Inverse Problem 11 22 ii mm i and m given 1 and 2 unknown Inverse problem
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34 Volume Conservation (incompressibility) R1R1 R2R2 RiRi RmRm
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35 Volume Conservation (incompressibility) R m fixed R i variable with R 1 and R 2 R 1, R 2 ?? RiRi RmRm R1R1 R2R2
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36 Blow Moulding preform container Forward problem Inverse problem
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37 Hybrid Broyden Method Optimisation ResultsIntroduction Simulation Model Conclusions [Martinez, Ochi]
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38 Example (p = 13) Optimisation ResultsIntroduction Simulation Model Conclusions Method# function evaluations# iterations Hybrid Broyden3281.75 Finite Differences9891.36 Conclusions: similar number of iterations similar objective function value Finite Differences takes approx. 3 times longer than Hybrid Broyden
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39 Optimal preform Preform Optimisation for Jar Model jar Initial guess Results Level Set Method Introduction Simulation Model Conclusions
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40 Preform Optimisation for Jar Model jar Results Level Set Method Introduction Simulation Model Conclusions Approximate jar Radius: 1.0 Mean distance: 0.019 Max. distance: 0.104
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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
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42 Thank you for your attention
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43Comparison Inverse problemForward problem two unknown intefacesone unknown interface Inverse problem Forward problem 11 22 ii mm Inverse problem under-determined or forward problem over-determined?
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44 Inverse Problem Optimisation ResultsIntroduction Simulation Model Conclusions preformcontainer Unknown surfaces
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45 Forward Problem Optimisation ResultsIntroduction Simulation Model Conclusions preformcontainer R m known R i un known
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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)
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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)
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