numerical shape optimisation in blow moulding
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Numerical Shape Optimisation Numerical Shape Optimisation in Blow Mouldingin Blow Moulding
Hans GrootHans Groot
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OverviewOverview
1.1. Blow moldingBlow molding
2.2. Inverse ProblemInverse Problem
3.3. Optimization MethodOptimization Method
4.4. Application to Glass BlowingApplication to Glass Blowing
5.5. Conclusions & future workConclusions & future work
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Inverse Problem Glass Blowing ConclusionsBlow Molding Optimization Method
Blow MoldingBlow Molding
glass bottles/jars
plastic/rubber containers
mould
pre-form
container
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Example: JarExample: Jar
Inverse Problem Glass Blowing ConclusionsBlow Molding Optimization Method
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ProblemProblem
Forward problem
Inverse problem
pre-form container
Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method
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Forward ProblemForward Problem
1
2
i
m
•Surfaces 1 and 2 given•Surface m fixed (mould wall)•Surface i unknown
Forward problem
Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method
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1
Inverse ProblemInverse Problem
•Surfaces i and m given
•Either 1 or 2 unknown
Inverse problem
2
i
m
Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method
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Construction of Pre-Form by PressingConstruction of Pre-Form by Pressing
1
2
Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method
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OptimizationOptimization
Find pre-form for approximate container with minimal distance from model container
mould wallmodel container
approximate container
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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OptimizationOptimizationmould wallmodel containerapproximate container
Minimize objective function2
2
2
i
dd d
idOptimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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Computation of Objective FunctionComputation of Objective Function Objective Function:
Composite Gaussian quadrature:
• m+1 control points (•) → m intervals•n weights wi per interval (ˣ)
2
i
dd
2 2
i
d ( )m n
i nj ij i
d w d
x
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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Parameterization of Pre-FormParameterization of Pre-Form
P1
P5P4
P3
P2
P0
OR,φ1. 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
1 2 5P P P( , ,..., )R R Rp
)(p
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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iterative method to minimize objective function
J: Jacobian matrix
: Levenberg-Marquardt parameter
H: Hessian of penalty functions:
iwi /ci , wi : weight, ci >0: geometric
constraint
g: gradient of penalty functions
p: parameter increment
d: distance between containers
Modified Levenberg-Marquardt Method
T T
i i i i i i i i J J I H p J d g
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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Function Evaluations per Iteration
Distance function d:o one function evaluation
Jacobian matrix:
1. Finite difference approximation:
o p function evaluations (p: number of parameters)
2. Broyden’s method:o no function evaluations, but less accurate
function evaluation = solve forward problem
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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Neglect mass flow in azimuthal direction (uf≈0)
Given R1(f), determine R2(f)
Volume conservation:
•R(f) radius of interface
Approximation for InitialApproximation for Initial GuessGuess
streamlines
3 3 3 32 1 m i( ) ( ) ( ) ( )R R R R
f r
Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions
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InitialInitial GuessGuess
approximate inverse problem
initial guess of pre-form
model container
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Glass BlowingGlass Blowing
Blow Molding Inverse Problem ConclusionsGlass BlowingOptimization Method
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Forward ProblemForward 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 ConclusionsGlass BlowingOptimization Method
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Level Set MethodLevel Set Method
glass
airair
θ > 0
θ < 0θ < 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 ConclusionsGlass BlowingOptimization Method
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Computer Simulation ModelComputer 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 ConclusionsGlass BlowingOptimization Method
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Comparison Approximation with Comparison Approximation with Simulation ModelSimulation Model
forward problem
pre-form container
simulation
approximation (uf≈0)
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Optimization of Pre-Form
inverse problem
initial guess
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Optimization of Pre-Form
initial guess
inverse problem
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Optimization of Pre-Form
optimal pre-form
inverse problem
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Signed Distance between Approximate and Model Container
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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
moldBlow Molding Inverse Problem Glass Blowing ConclusionsOptimization Method
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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 ConclusionsOptimization Method
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Parison Optimization for Ellipse
model container optimal containerinitial guess
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Blow MoldingBlow Molding
mould
ring
parison
container
e.g. glass bottles/jars
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ApproximationApproximation
Initial guess
pre-form model container
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Incompressible medium:
•R(f) radius of interface G
Simple example → axial symmetry:
•If R1 is known, R2 is uniquely determined and vice versa
1 12 2
3 3 3 32 1 m i
0 0
( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R
Initial GuessInitial Guess
3 3 3 32 1 m iR R R R
R(f)
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Inverse ProblemInverse Problem
1 given (e.g. plunger)
m, i given
•determine 2 2
1
•Optimization:•Find pre-form for container with minimal difference in glass distribution with respect to desired container
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Inverse Problem
1
2i
m
i and m given
1 and 2 unknown
Inverse problem
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1 12 2
3 3 3 32 1 m i
0 0
( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R
Volume Conservation (incompressibility)
R1
R2Ri
Rm
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1 12 2
3 3 3 32 1 m i
0 0
( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R
Volume Conservation (incompressibility)
•Rm fixed
•Ri variable
with R1 and R2
•R1, R2??
Ri
Rm
R1
R2
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Blow Moulding
preform container
Forward problem
Inverse problem
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Hybrid Broyden Method
Optimisation ResultsIntroduction Simulation Model Conclusions
iii
ii
ii
iii
iiii
iii
ii
ii
ii
ii
iiii
rrr
JJ
JJ
pJr
rpJr
pJr
rr
pp
pp
pp
ppJr
1
1
111
with
otherwise ,
:method bad sBroyden'
if ,
:method good sBroyden'
[Martinez, Ochi]
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Example (p = 13)
Optimisation ResultsIntroduction Simulation Model Conclusions
Method # function evaluations
# iterations
Hybrid Broyden 32 8 1.75
Finite Differences 98 9 1.36
Conclusions:
• similar number of iterations
• similar objective function value
• Finite Differences takes approx. 3 times longer
than Hybrid Broyden
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Optimal preform
Preform Optimisation for Jar
Model jar Initial guess
ResultsLevel Set MethodIntroduction Simulation Model Conclusions
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Preform Optimisation for Jar
Model jar
ResultsLevel Set MethodIntroduction Simulation Model Conclusions
Approximate jar
Radius: 1.0
Mean distance: 0.019Max. distance: 0.104
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Conclusions
ConclusionsOptimisationIntroduction 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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Thank you for your attention
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ComparisonComparisonInverse problem Forward problem
two unknown intefaces one unknown interface
Inverse problem
Forward problem1
2
i
m
Inverse problem under-determined or forward problem over-determined?
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Inverse Problem
Optimisation ResultsIntroduction Simulation Model Conclusions
preform container
Unknown surfaces
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Forward Problem
Optimisation ResultsIntroduction Simulation Model Conclusions
preform container
Rm knownRi unknown
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Incompressible medium:
•R(f) radius of interface G
Simple example → axial symmetry:
•If R1 is known, R2 is uniquely determined and vice versa
1 12 2
3 3 3 32 1 m i
0 0
( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R
Volume ConservationVolume Conservation
3 3 3 32 1 m iR R R R
R(f)
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Incompressible medium:
•R(f) radius of interface G
Simple example → axial symmetry:
•If R1 is known, R2 is uniquely determined and vice versa
1 12 2
3 3 3 32 1 m i
0 0
( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R
Volume ConservationVolume Conservation
3 3 3 32 1 m iR R R R
R(f)
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