1 numerical shape optimisation in blow moulding hans groot

1

Numerical Shape Optimisation Numerical Shape Optimisation in Blow Mouldingin Blow Moulding

Hans GrootHans Groot

2

OverviewOverview

1.1. Blow moldingBlow molding

2.2. Forward ProblemForward Problem

3.3. Inverse ProblemInverse Problem

4.4. Optimisation MethodOptimisation Method

5.5. Conclusions & Future WorkConclusions & Future Work

Inverse Problem Optimization Method ConclusionsBlow Molding Inverse Problem

3

Blow Molding/FormingBlow Molding/Forming

glass bottles/jars

polymer containers

mould

pre-form

container

4

Glass Bottle Forming Glass Bottle Forming MachineMachine

Inverse Problem Optimization Method ConclusionsBlow Molding Inverse Problem

5

ProblemProblem

Forward problem

Inverse problem

pre-form container

Blow Molding Optimization Method ConclusionsForward Problem Inverse Problem

6

Forward ProblemForward Problem

R1

R2

Ri

Rm

•Surfaces R1 and R2 given•Surface Rm fixed (mould wall)•Surface Ri unknown

Forward problem

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

7

Constitutive EquationsConstitutive Equations

1)Mechanics Stokes flow problem

2)ThermodynamicsConvection diffusion

problem

3)Evolution of surfacesConvection problem

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

8

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• surfaces implicitly defined• level sets maintained as signed

distances

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

9

Computer Simulation ModelComputer Simulation Model Finite element method One fixed mesh for

entire flow domain 2D axi-symmetric At equipment

boundaries: no-slip of material air is allowed to “flow

out”

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

10

Glass BlowingGlass Blowing

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

11

R1

Inverse ProblemInverse Problem

Inverse problem

R2

Ri

Rm

•Surfaces Ri and Rm given•Surface R1 fixed•Surface R2 unknown

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

12

Blowing corner•No surface tension:infinite time

•Surface tensionnot possible: equilibrium

Blowing round cavity•Possible if:

radius of mould cavity < radius of curvature

Mould RequirementsMould Requirements

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

13

•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

•Contact angle φ between surface and mould

•Unknowns:•curve z = f(r)•contact radius rc

•contact depth zc

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

14

•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

3/22

''( )( )

1 '( )

z rr

z r

•2D curve z(r) :

•Boundary conditions:

c cd d

cd d

( ) , ( )( ) tan , ( ) cotz z

r r

z r H z L zr L

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

15

•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

0,p

g g

3/22

''

1 ( ')

zz

z

•Second order ODE:

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

16

•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

0,p

g g

3/22

''( )

1 ( ')

''

z zzz

z

•Multiplication by z’:

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

17

•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

0,p

g g

1/22 22 1 ( ') ( )z z E

•Integration:

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

18

Glass Surface in EquilibriumGlass Surface in Equilibrium•First order ODE:

•Boundary conditions:

c cd d

cd d

( ) , ( )( ) tan , ( ) cotz z

r r

z r H z L zr L

2

22

4' 1

( )z

z E

•Constants:

1c

2 2

22

2

21c 4

4c

2 cos ( )

2 sin ( )

1 dz

z EH

E H

z H

r L z

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

zc

rc

-HφLr

z

φ

19

corner

Example

corner in (1,-10)

0

0.2

20

9

gp

g

20

Time ScaleTime Scale•Equilibrium (no gravity):

•Time scale:L

tV

pL LV

•Typical values:5

2 1

10 Pa s1 Pa m

10 m

310 st

•Typical blow process takes ~1s

zc

rc

-HLr

z

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

210 s

10s3s1s

21

OptimizationOptimization

Find pre-form for approximate container with minimal distance from container design

mould wallcontainer design

approximate container

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

22

OptimizationOptimizationmould wallcontainer designapproximate container

Minimize objective function2

2

2

i

dd d

RidOptimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

23

Computation of Objective FunctionComputation of Objective Function Objective function:

Line integral:

Composite Gaussian quadrature:

• m+1 control points (•) → m intervals•

n weights wi per interval (ˣ)

2

i

dd

2'( ) ( ( ))m n

i nj i nj ij i

w s d s x x

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

1 2

0'( ) ( ( )) ds d s s x x

24

Parameterization of Pre-FormParameterization of Pre-Form

P1

P5P4

P3

P2

P0

OR,φ1. Describe unknown surface

by parametric curve• e.g. spline, Bezier curve

2. Define parameters as spherical radii of control points:

3. Optimization problem: Find p as to minimize

1 2 5P P P( , ,..., )R R Rp

)(pOptimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

25

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

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

26

( ) ( ) p 1

Tolerance

Tolerance should not be smaller than

total error of optimisation method

27

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Truncation error: εT

Rounding error: εR

Measurement error: εM

Interpolation error: εI

28

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Model simplifications

Discretisation of forward problem

Truncation error: εT

Rounding error: εR

Measurement error: εM

Interpolation error: εI

L( ) ( ) r p r p ε

29

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Discretisation of forward problem

Truncation error: εT

Numerical differentiation of residual

Numerical integration (objective function)

Rounding error: εR

Measurement error: εM

Interpolation error: εI

T

( ) ( )( )

r p e r pJ p e ε

30

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Discretisation of forward problem

Truncation error: εT

Numerical differentiation of residual

Rounding error: εR

Measurement error: εM

Interpolation error: εI

31

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Discretisation of forward problem

Truncation error: εT

Numerical differentiation of residual

Rounding error: εR

Measurement error: εM

Interpolation error: εI

32

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Discretisation of forward problem

Truncation error: εT

Numerical differentiation of residual

Rounding error: εR

Measurement error: εM

Interpolation error: εI

Interpolation of known surfaces (through data) Interpolation of unknown surface (parametrisation)

33

Numerical Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL

Discretisation of forward problem

Truncation error: εT

Numerical differentiation of residual

Measurement error: εM

Interpolation error: εI

Interpolation of unknown surface (parametrisation)

Total Error: ε = εL + εT + εM + εI

34

Restrictions on Errors

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Mesh size : h• Linear elements: εL =O(h2)

• h2 εM

Distance between control points of unknown surface: ξ• Cubic spline interpolation : εI =O(ξ4)

• ξ ~ C l/(m-1)o C : constanto l : curve lengtho m : number of control points

• ξ4 h2 m C l h-1/2+1

ξ

35

Truncation Error

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Forward Difference Approximation: Error:

Funtion evaluations: p = number of parameters

Central Difference Approximation: Error:

Funtion evaluations: 2p = number of parameters

Broyden Update: Error: ? No function evaluations

Tε O( )

2

Tε O( )

36

Forward Difference Approximation

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Error bounded by

Minimum:

Conclusion:

( ) ( )( ) ( )O

r p e r p

J p e

L 12

2εM

L

4ε

M

T Lε ε O( )h

37

Broyden Update

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Jacobian Lipschitz continuous:

Error bound:

Conclusion:

T

11 T

i i i i ii i

i i

r r A s sA A

s s

1( ) ( )i i i J p J p s

31 1 2

( ) ( )i i i i i A J p A J p s

1 1

( ) O( ) ( ) O( ) O( )

i ii i

i

A J pA J p

s

38

Error Analysis

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Types:

Model error: εL= O(h2)

Truncation error: εT = O(h)

Measurement error: εM = O(h2) (by choice of h)

Interpolation error: εI = O(h2)

Total Error: ε ~εT = O(h)

Tolerance:

h

39

Assumptions:Negligible mass flow in azimuthal direction (uφ ≈0)Constant viscosity

Given R1(φ), determine R2(φ)

Volume conservation:

•R(φ) radius of interface

Approximation for InitialApproximation for Initial GuessGuess

streamlines

3 3 3 32 1 m i( ) ( ) ( ) ( )R R R R

φr

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

40

InitialInitial Guess for BottleGuess for Bottle

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Rφ

φR

streamlines

r

φr

41

InitialInitial GuessGuess

approximate inverse problem

initial guess of pre-form model container

42

forward problem

pre-form container

simulation

approximation (uφ≈0)

Comparison Approximation with Comparison Approximation with SimulationSimulation

43

Optimization of Pre-Form (no sagging)

inverse problem initial guess

44

inverse problem initial guess

Optimization of Pre-Form (no sagging)

45

inverse problem optimal preform

Optimization of Pre-Form (no sagging)

46

ErrorSigned Distance between Approximate and Model Container

top bottom

Absolute Tollerance:h≈ 0.059

47

Summary Inverse problem:

• find preform corresponding to container Shape optimization method for pre-

form in blow molding• Pre-from surface described by parametric

curve• Approximation for initial guess• Error in approximation of Jacobian is

dominant Application to glass blowing

Blow Molding Forward Problem Optimization Method ConclusionsInverse Problem

48

Future Work Initial guess with sagging Sensitivity analysis (w.r.t. perturbations in

thickness)

Comparison finite difference with derivative-free optimisation

Adaptive optimisation strategy• T-splines• Adaptive mesh

Volume constraint Application to polymersBlow Molding Forward Problem Optimization Method ConclusionsInverse Problem

49

Blow Molding Forward Problem Optimization Method ConclusionsInverse Problem

Thank you for your attention

Questions?

50

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)

51

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

52

Inverse Problem

1

2i

m

i and m given

1 and 2 unknown

Inverse problem

53

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

54

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

55

Blow Moulding

preform container

Forward problem

Inverse problem

56

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]

57

Error Analysis

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

Mesh size : h• Linear elements: εL =O(h2)

• h2 εM

Distance between control points of unknown surface: ξ• Cubic spline interpolation : εI =O(ξ4)

• ξ ~ C l/(m-1)o C : constanto l : curve lengtho m : number of control points

• ξ4 h2 m C l h-1/2+1