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Development of a Numerical Optimisation Method for Blowing Glass Parison Shapes. Hans Groot. Overview. Introduction Glass Blow Simulation Model Optimisation Method Results Conclusions. Simulation Model. Optimisation. Results. Conclusions. Introduction. Glass Manufacturing. - PowerPoint PPT Presentation
Development of a Numerical Optimisation Method for Blowing Glass Parison ShapesHans Groot
OverviewIntroductionGlass Blow Simulation ModelOptimisation MethodResultsConclusions
Glass Manufacturing
Glass Forming
Surface Treatment
Glass Melting
Glass Conditioning
Automatic Inspection
Glass Forming
presspress-blow blow-blow
Blow-Blow Processringpreformmould
Blow ModelFlow of glass and airStokes flow problemViscous forces dominateTemperature dependent glass viscosityEnergy exchange in glass and airConvection diffusion problemNo viscous dissipation Evolution of glass-air interfacesConvection problem for level sets
motivation:fixed finite element meshtopological changes are naturally dealt withinterfaces implicitly definedlevel sets maintained as signed distancesLevel Set Method
Computer Simulation ModelFinite element discretisationOne fixed mesh for entire flow domain2D axi-symmetricAt equipment boundaries:no-slip of glassair is allowed to flow out
Bottle Blowing SimulationTemperatureGlass-air interfaces
Glass Distribution for JarPreform 2: breaks!Preform 1: thickenings!
Given container g find preform
Optimisation:Find preform that minimises difference in glass distribution between model container and container obtained by blow process
Inverse Problem
Residual:
Minimise objective function:
Least Squares Minimisation Problemtrue interfaceapproximate interface
Describe interfaces by parametric curvese.g. splines, Bezier curvesDefine parameters:
Compute signed distanceMinimiseOptimisation Strategy
iterative method to minimise objective function
J: Jacobian matrixl: Levenberg-Marquardt parameterH: Hessian of penalty functions:zi = wi /ci , wi : weight, ci >0: geometric constraintg: gradient of penalty functionsDp: parameter incrementr: residualModified Levenberg-Marquardt Method
Finite difference approximation:requires p function evaluations,p: number of parametersSecant method:updates Jacobian in incremental directionno function evaluationsmay fail to find descent directionfinite difference approximationComputation of Jacobian
Hybrid Broyden Method[Martinez, Ochi]
ExampleConclusions:similar number of iterationssimilar objective function valueFinite Differences takes approx. 3 times longer than Hybrid Broyden
Optimal preformPreform Optimisation for JarModel jarInitial guess
Preform Optimisation for JarModel jarApproximate jarRadius: 1.0Mean distance: 0.019Max. distance: 0.104
ConclusionsGlass Blow Simulation Modelfinite element methodlevel set techniques for interface tracking2D axi-symmetric problemsOptimisation method for preform in glass blowingpreform described by parametric curvescontrol points optimised by nonlinear least squaresApplication to blowing of jarmean distance < 2% of radius jar
Short Term PlansExtend simulation modelimprove switch free-stress to no-slip boundary conditionsone level set problem vs. two level set problemsWell-posedness of inverse problemSensitivity analysis of inverse problem
Thank you for your attention
Blowing Model: 2nd blow
Blowing Model: 1st BlowAxi-symmetric blowing of parison 1165 oC glass500 oC mould
ringpreformmould
Re-initialisation by FMM Algorithm
Preform Optimisation for JarModel jarInitial guessOptimal preform
Who are you?When did you do this work?With who did you do this work?Where did you do this work?
[Keep in mind that not all know what is the process of glass blowing.]
I will give a brief introduction about glass manufacturing processes and in particular the glass blowing process in the manufacturing circle. I will talk about our glass blowing simulation models that is based on level-set methods.I will show you our results and conclusions so far.I will start off by showing you the different glass manufacturing processes in the order of their application.
Firstly, raw materials and recycled glass are molten in a huge furnace at an elevated temperature between 1200 and 1600 degrees Celcius.
Once the glass substance leaves the furnace it is cut into uniform gobs, which are then sent to a forming machine, where they are forced into the desired shape, e.g. a bottle or a jar. After the forming process the glass containers are rapidly cooled down on a cooling plate.
In the glass conditioning process the glass containers are again heated to 600 degrees Celcius and then gradually cooled in order to reduce the stresses.
Subsequently a surface treatment is applied to reduce the surface flaws.
Finally the glass containers go through an automatic inspection. Glass containers that are rejected straightly go back to the furnace.
In this presentation we are mainly concerned with the glass forming process.In this movie you can see a forming machine. Here a so-called glass parison or preform is brought into a mould and subsequently blown into a container shape.
In essence there are three different ways to form a glass container. Firstly by directly pressing the glass into its desired shape; this is particularly used for relatively flat products. Secondly, by a two stage press-blow process. Here the glass is first pressed into a preform and then the preform is blown into the final shape. And thirdly by a blow-blow process, which consists of two blowing stages. In this presentation we are primary interested in the final blow stage of either the press-blow or the blow-blow process.By way of illustration, let me explain to you how the blow-blow process works. In the first blow stage a glass gob enters a mould and is then blown from above in order to form the neck of the bottle. Subsequently, pressurised air is blown in from below in order to form the preform. Next this preform is transported by the ring to another mould for the second blow stage, in which the preform is blown into its final shape, for example a bottle or a jar. In order to simulate this process we have developed a computer simulation model.In this simulation model the following characteristic physical problems are solved. Firstly, the flow of glass and air during the blow process is modelled. Since the viscous forces dominate this can be done by solving a Stokes flow problem. Here the glass viscosity is highly temperature dependent. Secondly, the energy exchange in glass, air and equipment is modelled. This is done by solving a convection-diffusion problem involving the heat equation. Here the influence of viscous dissipation can be neglected.Thirdly, the evolution of the glass-air interfaces is modelled by solving a so-called level set problem, which I will explain next. It should be mentioned that these three problems are coupled to each other and cannot be solved independently.Solutions of the level set problem are so-called level set functions. They are smaller than zero in air, greater than zero in glass and equal to zero on the glass-air interfaces. In this way level set functions can be used to track the location of the interfaces and to determine the local physical properties.
Our motivation to use level set methods is the following: one fixed finite element mesh can be used for our numerical computations, topological changes are naturally dealt with, the interfaces are implicitly defined in the problem formulation and by reinitialisation techniques the level set function is maintained as a signed distance function to the glass air interfaces. This signed distance property will be relevant later on for our optimisation strategy.The characteristic problems are solved by means of finite element methods. For the finite element discretisation one fixed mesh is used for the entire flow domain. Both structured and unstructured meshes can be used; for the simulations in this presentation we have used unstructured meshes. Up to now the simulation model is restricted to 2D axi-symmetrical problems. Finally, in our simulations we assume at the equipment boundaries no-slip conditions for glass and free-stress conditions for air.Here you can see simulations of the blowing of a beer bottle. On the left you can see the flow of glass and air; glass is red, air is blue. First we let the preform sag half a second, before we start blowing air in the mould. On the right you can see the temperature distribution during the blow process.
However, a uniform glass distribution over the mould wall is not a matter of course. That the parison shape is crucial for the right glass distribution of the container can be seen in the following jar blowing simulations. Suppose we try to blow a jar with a nice uniform glass distribution. First we blow a thick preform. We observe that the container has some undesired thickenings. Next we blow a thin preform. Now the glass breaks and we do not have a full jar at all. It is possible to obtain a nice glass distribution in this way by trial-and-error, but this is not very efficient. So we finally arrive at the following inverse problem.Given the level set function of desired glass container, what is the level set function of the optimal preform. In other words, we would like to find the preform, for which the difference in glass distribution between the desired container, the so-called model container, and the container obtained by the blow process is as small as possible. In order to solve this inverse problem numerically, we will formulate the problem as a nonlinear least squares problem.First we consider those nodes of the finite element mesh within a narrow band around the glass air interfaces of the model container, say within a distance delta.
Then we take the difference between the level set values corresponding to the approximate container, which is obtained by blowing the parison shape, and the level set values corresponding to the model container in these nodes. This difference is the least squares residual. During our simula`tions a re-initialisation technique is applied each time step to maintain the level set function as a signed distance function to the nearest glass-air interface. Our objective is to minimise the objective function, which is one half times the squared Euclidean norm of the residuals.The following strategy is used to minimise the objecive function. First the glass-air interfacesof the preform are described by parametric curves, for example splines or Bezier curves.
By doing this the number of parameters to be optimised can be restricted to the number of control points used to define the curves. Consider the approximation of the preform by parametric curves in the right-hand figure. In this example cubic splines are used with six control points to describe the outer interface and three control points to describe the inner interface. The top most control points are fixed to the ring. The left most control points are on the symmetry axis, so they are fixed in radial direction. Furthermore, the height of the preform is fixed, which means that the bottom control point on the symmetry axis is also fixed in axial direction. The remaining coordinates of the control points are subject to optimisation and therefore appear in the parameter vector.
Next the level set function of the preform is computed as the signed distance to the parametric curves.
Finally the objective function is minimised as a function of the parameters.A modified Levenberg-Marquardt algorithm is used to solve the nonlinear least squares problem. The Levenberg-Marquardt method is a numerical optimisation method and it is modified in the sense that it accounts for geometric constraints on the preform. Each iteration the parameters are updated as to decrease the objective function until convergence. The Levenberg-Marquardt method requires the evaluation of a Jacobian matrix of the residual with respect to the parameters.
In essence there are two different methods to estimate the Jacobian matrix.
Firstly, the Jacobian matrix can be computed by finite differences. A finite difference scheme approximates the derivatives of the residuals with respect to all p parameters. As a result p times a forward problem has to be solved, which takes a considerable amount of computational time if p is large.
Secondly, the Jacobian matrix can be updated each iteration by a secant method, for example Broydens method. In this case the Jacobian matrix is only computed in the last search direction for the parameter vector. Broydens method does not require any function evaluations, however, after a few iterations it may fail to find the descent direction. A descent direction is a search direction for the parameter vector for which the objective function is decreased. If the search direction is not a descent direction, the Levnberg-Marquardt method ceases to converge. In this case there is an easy fix: use a finite difference approximation. In our case combining Broydens method and finite differences gives us a considerable saving on computational effort. Firstly, the Jacobian matrix can be computed by finite differences. A finite difference scheme approximates the derivatives of the residuals with respect to all p parameters. As a result p times a forward problem has to be solved, which takes a considerable amount of computational time if p is large.
Secondly, the Jacobian matrix can be updated each iteration by a secant method, for example Broydens method. In this case the Jacobian matrix is only computed in the last search direction for the parameter vector. Broydens method does not require any function evaluations, however, after a few iterations it may fail to find the descent direction. A descent direction is a search direction for the parameter vector for which the objective function is decreased. If the search direction is not a descent direction, the Levnberg-Marquardt method ceases to converge. In this case there is an easy fix: use a finite difference approximation. In our case combining Broydens method and finite differences gives us a considerable saving on computational effort. Firstly, the Jacobian matrix can be computed by finite differences. A finite difference scheme approximates the derivatives of the residuals with respect to all p parameters. As a result p times a forward problem has to be solved, which takes a considerable amount of computational time if p is large.
Secondly, the Jacobian matrix can be updated each iteration by a secant method, for example Broydens method. In this case the Jacobian matrix is only computed in the last search direction for the parameter vector. Broydens method does not require any function evaluations, however, after a few iterations it may fail to find the descent direction. A descent direction is a search direction for the parameter vector for which the objective function is decreased. If the search direction is not a descent direction, the Levnberg-Marquardt method ceases to converge. In this case there is an easy fix: use a finite difference approximation. In our case combining Broydens method and finite differences gives us a considerable saving on computational effort. Finally, I will show you a result of the optimisation method for jar blowing. Suppose one would like to blow the following model jar.
We will start with a initial guess of the preform for which the glass preferably does not break during the simulation.
Let the interfaces be described by cubic splines with five control points for the outer interface and four control points for the inner interface.
Starting from the initial guess the position of the variable control points are updated each iteration until convergence.
Finally we arrive at a preform for which a jar can be blown with a much better glass distribution.
Indeed for the final preform we obtain a reduction of the residual by 40 percent with respect to the initial guess.Finally, I will show you a result of the optimisation method for jar blowing. Suppose one would like to blow the following model jar.
We will start with a initial guess of the preform for which the glass preferably does not break during the simulation.
Let the interfaces be described by cubic splines with five control points for the outer interface and four control points for the inner interface.
Starting from the initial guess the position of the variable control points are updated each iteration until convergence.
Finally we arrive at a preform for which a jar can be blown with a much better glass distribution.
Indeed for the final preform we obtain a reduction of the residual by 40 percent with respect to the initial guess.In conclusion: A glass blow simulation model has been presented that is based on finite element methods and uses level set techniques to track the glass-air interfaces. Up to now only 2D axi-symmetric problems can be modelled, but in the future we hope to extend the model to 3D. Furthermore, a method to optimise the preform in blowing glass containers has been developed. This optimisation method describes the glass-air interfaces by parametric curves and finds the optimal positions of the control points by solving a nonlinear least squares problem. In conclusion: A glass blow simulation model has been presented that is based on finite element methods and uses level set techniques to track the glass-air interfaces. Up to now only 2D axi-symmetric problems can be modelled, but in the future we hope to extend the model to 3D. Furthermore, a method to optimise the preform in blowing glass containers has been developed. This optimisation method describes the glass-air interfaces by parametric curves and finds the optimal positions of the control points by solving a nonlinear least squares problem.