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.
