A technique for FEM optimization A technique for FEM optimization under uncertainty of time-dependent under uncertainty of time-dependent

process variables in sheet metal process variables in sheet metal formingforming

ESAFORM 2006ESAFORM 2006

M. StranoM. Strano

Università di CassinoUniversità di CassinoDip. Ingegneria IndustrialeDip. Ingegneria Industriale

http://webuser.unicas.it/tslhttp://webuser.unicas.it/tsl

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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Executive summaryExecutive summary• A method is proposed for the FEM optimization of the time-A method is proposed for the FEM optimization of the time-

dependent design variables of sheet metal forming processesdependent design variables of sheet metal forming processes– useful when the non-controllable process parameters can be useful when the non-controllable process parameters can be

modelled as random variablesmodelled as random variables– suited for problems with large computational times, small process suited for problems with large computational times, small process

window and non-uniform variance of resultswindow and non-uniform variance of results• The problem is formulated as the minimization of a cost The problem is formulated as the minimization of a cost

function, subject to a reliability constraintfunction, subject to a reliability constraint– The cost function is optimized through a “metamodel”, built by The cost function is optimized through a “metamodel”, built by

“Kriging” interpolation“Kriging” interpolation– The reliability is assessed by a binary logistic regression analysis of The reliability is assessed by a binary logistic regression analysis of

the resultsthe results• The method is applied to the Numisheet ’93 benchmark The method is applied to the Numisheet ’93 benchmark

problem (u-channel forming and springback)problem (u-channel forming and springback)– modified by handling the blankholder force as a time-dependent modified by handling the blankholder force as a time-dependent

variablevariable

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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Overview of the problemOverview of the problem

Optimization under uncertaintyOptimization under uncertainty– The solution must The solution must

1.1. Optimize the expected value of an Optimize the expected value of an objective functionobjective function

2.2. Be robust & reliableBe robust & reliable

OptimizationOptimizationUncertaintyUncertainty

Two alternativesTwo alternatives– Coupled approachCoupled approach

– Build an objective function that incorporates both mean value Build an objective function that incorporates both mean value and standard deviation of the performanceand standard deviation of the performance

– Decoupled approachDecoupled approach– Find a potentially optimal solutionFind a potentially optimal solution– Assess its reliability (use reliability as a constraint)Assess its reliability (use reliability as a constraint)

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Overview of the problemOverview of the problem

General literature on methods for General literature on methods for stochastic optimization with reliability stochastic optimization with reliability assessment is very extensiveassessment is very extensive

– Still, not much is available in the field of Still, not much is available in the field of numerical methods for sheet metal formingnumerical methods for sheet metal forminge.g. [Sahai, Schramm, Buranathiti, Chen, Cao, Xia e.g. [Sahai, Schramm, Buranathiti, Chen, Cao, Xia

/ Numiform ’04, Columbus]/ Numiform ’04, Columbus]– decoupled methodology for optimization of a decoupled methodology for optimization of a

metamodel function and for reliability assessmentmetamodel function and for reliability assessment

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Decoupled approach: Decoupled approach: optimizationoptimizationThe main issue with optimization is the The main issue with optimization is the very high computational costvery high computational cost

– computational burden is even computational burden is even worse if uncertainty is consideredworse if uncertainty is considered

A A metamodelmetamodel is required is required

FEMFEM MetamodelMetamodel

f(f(xx)) f’(f’(xx)=0)=0OptimizerOptimizer

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Decoupled approach: Decoupled approach: optimizationoptimization

MetamodelingMetamodeling– Generally, RSM (Response Surface Method) is Generally, RSM (Response Surface Method) is

used for metamodelingused for metamodeling• RSM is a RSM is a regressionregression (approximation) method, (approximation) method,

which may result quite inaccurate when the which may result quite inaccurate when the response is deterministicresponse is deterministic

• Not perfectly suited to (deterministic) computer Not perfectly suited to (deterministic) computer simulations simulations [Bonte, Van den Boogaard, Huétink / Esaform ’05 Cluj-[Bonte, Van den Boogaard, Huétink / Esaform ’05 Cluj-

Napoca]Napoca]

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Decoupled approach: Decoupled approach: optimizationoptimization

MetamodelingMetamodeling– InterpolationInterpolation can be better suited to can be better suited to

computer simulations resultscomputer simulations results• DACE (Design and Analysis of Computer DACE (Design and Analysis of Computer

Experiments) and Kriging are recently the method Experiments) and Kriging are recently the method of choiceof choice

interpolation vs. regressioninterpolation vs. regression

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Decoupled approach: Decoupled approach: reliability reliability assessmentassessmentSeveral methods are generally used for Several methods are generally used for reliability assessmentreliability assessment– Most available approaches are based on Most available approaches are based on locallocal

approximationapproximation ( (regressionregression) of results, in ) of results, in order to estimate the standard deviation order to estimate the standard deviation

– For all available methods (also in the coupled For all available methods (also in the coupled approach), the approach), the standard deviationstandard deviation of of response is requiredresponse is required• Usually this is Usually this is

– either inaccurate, when not enough data is usedeither inaccurate, when not enough data is used– or computationally expensive, otherwiseor computationally expensive, otherwise

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Decoupled approach: Decoupled approach: reliability reliability assessmentassessmentA proposal for A proposal for globalglobal reliability reliability assessment when the process window is assessment when the process window is not very large (failure probability is not not very large (failure probability is not very small)very small)– Logistic regression analysisLogistic regression analysis

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Description of the proposed methodDescription of the proposed methodFEMFEM

,pN

output variablesoutput variables

,y y x Non-controllable randomNon-controllable randomprocess parametersprocess parameters

Deterministic control variablesDeterministic control variablesx

Cost function (affected byCost function (affected byuncertainty)uncertainty)

C y

min ,pR

C y c y x pdf d

Minimization Minimization

of the of the expected expected

valuevalueTechnologicalTechnological

constraintsconstraintsReliability constraintReliability constraint , 1s sP x P y x

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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Description of the proposed methodDescription of the proposed method4-step 4-step procedureprocedurex

y

1 sP

c

Latin hypercube or Latin hypercube or similar space-filling similar space-filling for for xx

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Step 2: Step 2: reliability assessmentreliability assessment• The failure probability The failure probability PPff((xx)=1-P)=1-Pss((xx)) is is

calculated thanks to:calculated thanks to:– binary logistic regressionbinary logistic regression ( )

ln1 ( )

s

s

P xL x

P x

Polynomial Polynomial modelmodel

““Logit” link Logit” link functionfunction

• A binary value is assigned to each simulation runA binary value is assigned to each simulation run– 00 if the final part is sound or the process feasible if the final part is sound or the process feasible– 11 if the final part is failed or the process unfeasible if the final part is failed or the process unfeasible

• It does not require local calculation of standard It does not require local calculation of standard deviationdeviation

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Step 3: Step 3: metamodelingmetamodeling• A metamodel of the cost function A metamodel of the cost function C(C(yy)), i.e. a , i.e. a

predictor predictor ((xx,, )), is built using Kriging , is built using Kriging interpolationinterpolation

• The mean square error of the predictor isThe mean square error of the predictor is

– unlike regression models (such as RSM), the mean unlike regression models (such as RSM), the mean square error square error is zero at the design points is zero at the design points

– is larger where the design space has not been is larger where the design space has not been filledfilled

c

2ˆ, ,x C y c x

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Step 4: Step 4: optimizationoptimization• The expected value of the cost function is minimized The expected value of the cost function is minimized

under the technological constraints and the reliability under the technological constraints and the reliability constraintsconstraints– However, in order to ensure accuracy, the mean square error However, in order to ensure accuracy, the mean square error

could be taken into account by modifying the objective could be taken into account by modifying the objective functionfunction

c

mod ˆmin E , ,pR

C c x x pdf d

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Application to u-channel formingApplication to u-channel forming• Numisheet ’93 benchmark (u-channel forming)Numisheet ’93 benchmark (u-channel forming)

– a time-dependent blankholder force a time-dependent blankholder force BHF BHF has been optimized, has been optimized, in order to minimise the amount of springback, given a in order to minimise the amount of springback, given a constraint on the maximum tolerable percentage thinning of constraint on the maximum tolerable percentage thinning of the sheetthe sheet

BHF vs. time

0

5

10

15

20

25

30

0 2 4 6 8 10time (sec)

BH

F (k

N)

curve Acurve B

BHF 1BHF 2

BHF 3BHF 4 BHF 5

x

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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Application to u-channel formingApplication to u-channel forming

• Components of the random vector Components of the random vector

• Technological constraintsTechnological constraints9 kN 9 kN x xjj 40 kN; j=1, ... 5 40 kN; j=1, ... 5

harden. exp. young’s modul. initial thick. 1=n 2=E (GPa102) 3=t0 (mm)

1 1 2 2 3 3 0.347 0.01 0.690 0.02 0.810 0.01

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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Application to u-channel formingApplication to u-channel forming• Step 1: Step 1: design and execution of simulationsdesign and execution of simulations

– 9797 simulations have been planned simulations have been planned• Step 2: Step 2: reliability assessmentreliability assessment

– A part is considered to be failed if the limit of 15% A part is considered to be failed if the limit of 15% for maximum thinning is exceededfor maximum thinning is exceededPPss(x)=P[y(x)=P[y1115%]15%]1-1-

• Step 3: Step 3: metamodelingmetamodeling– The output The output yy22 (a measure of springback) is used as (a measure of springback) is used as

the main parameter in the cost function. the main parameter in the cost function. – A metamodel of the cost is built using Kriging A metamodel of the cost is built using Kriging

interpolationinterpolation 1 2 3 4 5 1 2 3ˆ , , , , , , ,c x x x x x

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Application to u-channel formingApplication to u-channel forming97 runs 500 runs

“true” metamodel “reduced” metamodel E[C] E[Cmod] Ps E[C] E[Cmod] Ps xA 0.252 0.314 0.990 0.304 0.413 0.926 xB 0.246 0.306 0.981 0.270 0.369 0.990

BHF vs. time

0

5

10

15

20

25

30

0 2 4 6 8 10time (sec)

BH

F (k

N)

curve Acurve B

BHF 1BHF 2

BHF 3BHF 4 BHF 5

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Application to u-channel formingApplication to u-channel forming• TheThe

resultsresults With nominal With nominal

values of random values of random parametersparameters

FEM optimization under uncertainty – FEM optimization under uncertainty – M. StranoM. StranoESAFORM 2006ESAFORM 2006

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ConclusionsConclusions• An innovative method has been proposed for An innovative method has been proposed for

optimization of sheet metal forming processesoptimization of sheet metal forming processes– suited for multiple design variables and affected by suited for multiple design variables and affected by

randomness of a few non-controllable parametersrandomness of a few non-controllable parameters• The proposed procedure is divided in 4 steps:The proposed procedure is divided in 4 steps:

1.1. design of simulationsdesign of simulations2.2. reliability assessment using logistic regressionreliability assessment using logistic regression3.3. metamodeling with Kriging interpolationmetamodeling with Kriging interpolation4.4. constrained optimizationconstrained optimization

• It has been successfully applied to the It has been successfully applied to the Numisheet ’93 u-channel forming benchmark, Numisheet ’93 u-channel forming benchmark, which has been re-formulated as an which has been re-formulated as an optimization problem with uncertaintyoptimization problem with uncertainty