Robustness Evaluation and Tolerance Prediction for a Stamping Process with Springback Calculation by the FEM Matteo Strano Università di Cassino, Dip

Robustness Evaluation and Robustness Evaluation and Tolerance Prediction for a Stamping Tolerance Prediction for a Stamping

Process with Springback Calculation by Process with Springback Calculation by the FEMthe FEM

Matteo StranoMatteo StranoUniversità di Cassino, Dip. Ingegneria Industriale Università di Cassino, Dip. Ingegneria Industriale Cassino (FR), ItalyCassino (FR), [email protected]@unicas.ithttp://webuser.unicas.it/tslhttp://webuser.unicas.it/tsl

M. Strano Robustness Evaluation and Tolerance Prediction…

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Outline of the presentationOutline of the presentation

Objectives of the researchObjectives of the research An IDEF model of the FEM simulationsAn IDEF model of the FEM simulations

The model setupThe model setup The FEM simulation setupThe FEM simulation setup Variables of the IDEF modelVariables of the IDEF model

The random input vector The random input vector xx The output geometrical responseThe output geometrical response

Evaluating the geometrical robustnessEvaluating the geometrical robustness Sensitivity analysisSensitivity analysis Montecarlo simulationMontecarlo simulation Response Surface MethodologyResponse Surface Methodology A new Upper Bound methodA new Upper Bound method

Comparing the different methodsComparing the different methods ConclusionsConclusions


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IDEF Model of the FEM SimulationIDEF Model of the FEM Simulation

FEMsimulation

Input process Input process parametersparameters

1

2

...

xN

x

x

x

xx

Output Output response response variablesvariables

1

2

...

dN

d

d

d

dd

Control Control variablesvariables

1

2

...

uN

u

u

u

uu

OB

JEC

TIVE

OB

JEC

TIVE

SS

Numisheet ’05 benchmark #2Numisheet ’05 benchmark #2


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deterministic deterministic modelmodel

IDEF model: the benchmark ’05 - #2IDEF model: the benchmark ’05 - #2

FEMsimulation

1

2

...

xN

x

x

x

1

2

...

uN

u

u

u

-Sheet thickness-Sheet thickness-Young modulus-Young modulus-Anisotropy r-values-Anisotropy r-values-Friction f-values-Friction f-valuesetc.etc.

-Binder force-Binder force-Binder travel-Binder travel

-Deviation from -Deviation from reference geometryreference geometryxx

dd

uu

OB

JEC

TIVE

OB

JEC

TIVE

SS

1

2

...

dN

d

d

d

Numisheet ’05 benchmark #2Numisheet ’05 benchmark #2


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IDEF model: uncertaintyIDEF model: uncertainty

FEMsimulation

1

2

...

xN

x

x

x

1

2

...

uN

u

u

u

xx

dd

uu 0 ,x N x random input vectorrandom input vector

meanmeancovariancecovariance

? ?d random outputrandom output

unknownunknowndistribution &distribution &

momentsmoments

OB

JEC

TIVE

OB

JEC

TIVE

SS

Incorporate uncertainty into Incorporate uncertainty into xx

1

2

...

dN

d

d

d


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The FEM simulation setupThe FEM simulation setup

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SETUP

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SETUP

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SolverSolver Pam-Stamp 2g, with springbackPam-Stamp 2g, with springback

DrawbeadsDrawbeads PhysicalPhysical

MeshMesh Quadrangular Belytschko-Tsay,Quadrangular Belytschko-Tsay,

5 integration points, initial size 10 mm5 integration points, initial size 10 mm

MaterialMaterial isotropic Hill ’48 hardening, orthotropic material isotropic Hill ’48 hardening, orthotropic material

with given with given rr00, , rr4545 and and rr9090, ,

Flow stress lawFlow stress law nK 0


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The random input vector The random input vector xx

The vector The vector xx has 11 components has 11 components NNxx=11=11

xx11==KK, , xx22==nn, , xx33==00 ( (flow stressflow stress parameters) parameters)

xx44==tt (initial sheet (initial sheet thicknessthickness))

xx55==rr00, x, x66==rr4545, x, x77==rr9090 ( (anisotropyanisotropy parameters) parameters)

xx88==EE ((youngyoung modulus) modulus)

xx99==ffbb, , xx1010==ffdd, x, x1111==ffpp ((friction coefficientsfriction coefficients

between the blank and the binder, the between the blank and the binder, the upper die and the lower punch)upper die and the lower punch)

MO

DEL

SETUP

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SETUP

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FEMsimulation

1

2

...

xN

x

x

x

High dimensional High dimensional problemproblem


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Mean vectorMean vector Given nominal valuesGiven nominal values

Covariance matrixCovariance matrix ANOVA + correlation analysis for ANOVA + correlation analysis for xx11 to to xx88

No data available for No data available for xx99 to to xx11 11 (friction (friction

coefficients)coefficients)Assumptions on mean and standard deviationAssumptions on mean and standard deviation

0 ,x N x 0x

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SETUP

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SETUP

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estimatingestimating


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x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 units MPa - - mm - - - MPa - - -

x0 1032 0.1595 0.000344 1.622 0.7377 0.9230 0.8970 203125 0.12 0.12 0.12

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x1 3.832E+02 0 0 0 0 0 0 0 0 0 0 x2 0 2.20E-05 6.30E-07 -4.71E-06 0 0 0 0 0 0 0 x3 0 6.30E-07 2.665E-08 -1.80E-07 0 0 0 0 0 0 0 x4 0 -4.71E-06 -1.80E-07 1.90E-06 0 0 0 0 0 0 0 x5 0 0 0 0 2.233E-05 0 0 0 0 0 0 x6 0 0 0 0 0 1.2E-05 0 0 0 0 0 x7 0 0 0 0 0 0 3.1E-05 0 0 0 0 x8 0 0 0 0 0 0 0 1.563E+04 0 0 0 x9 0 0 0 0 0 0 0 0 1.296E-03 0 0 x10 0 0 0 0 0 0 0 0 0 1.296E-03 0 x11 0 0 0 0 0 0 0 0 0 0 1.296E-03

MO

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SETUP

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SETUP

OBJE

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VES Mean vectorMean vector

Covariance matrixCovariance matrix

0x

0 ,x N x


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The output responseThe output response

Geometrical deviationGeometrical deviation

For every simulation run, the position For every simulation run, the position of the formed sheet afterof the formed sheet afterspringback must bespringback must befixedfixed

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SETUP

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SETUP

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The The reference geometryreference geometry is obtained by running is obtained by running

a simulation with nominal values of a simulation with nominal values of xx00

FEMsimulation

dd


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The output responseThe output response

Geometrical deviationGeometrical deviation

max maxd d

For every simulation run, the position For every simulation run, the position of the formed sheet afterof the formed sheet afterspringback must bespringback must befixedfixed

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SETUP

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SETUP

OBJE

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FEMsimulation

dd

max d++ ==

max d


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Calculating Calculating : positioning the sheet: positioning the sheet

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SETUP

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SETUP

OBJE

CTI

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totally totally fixedfixed

after springbackafter springback

X and Y X and Y fixedfixed

Method Method AA 2 reference points 2 reference points

+ symmetry plane+ symmetry plane


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SETUP

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SETUP

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B: sampled B: sampled geometrygeometry

BB

AA

A: A: reference reference geometrygeometry

distance distance dd--


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SETUP

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SETUP

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B: sampled B: sampled geometrygeometry

BB

AA

A: A: reference reference geometrygeometry

distance distance dd++


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Method Method BB rotating and translating rotating and translating

each shape until the each shape until the

error error is minimized is minimized

exactexact estimation of estimation of but computationally but computationally expensiveexpensive

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SETUP

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SETUP

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ZZ

XX

YY

Symmetry Symmetry planeplane


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Method Method CC

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SETUP

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SETUP

OBJE

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1 point fixed in 1 point fixed in spacespace


YY

Positioning planePositioning plane


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SETUP

OBJE

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RO

BU

STNESS

RO

BU

STNESS

GoalGoal estimating the variation of theestimating the variation of the

geometrical deviationgeometrical deviation

average valuesaverage values

Evaluating the geometrical robustnessEvaluating the geometrical robustness

max maxd d

1122334455

;d ;d


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SETUP

OBJE

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RO

BU

STNESS

RO

BU

STNESS

GoalGoal estimating the variation of theestimating the variation of the

geometrical deviationgeometrical deviation

average valuesaverage values


max maxd d

;d ;d

Upper Confidence Upper Confidence Limit of Limit of atat 99.7%99.7%

3

width of width of 66 tolerance tolerance intervalinterval

of the final of the final shapeshape

UCL==


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SETUP

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Alternative methodsAlternative methods Inexpensive and roughInexpensive and rough

Sensitivity analysis Sensitivity analysis – changing 1 parameters each simulationchanging 1 parameters each simulation

…… Approximate upper bound methodApproximate upper bound method …… Expensive and preciseExpensive and precise

Montecarlo simulationMontecarlo simulationResponse Surface MethodologyResponse Surface Methodology......


RO

BU

STNESS

RO

BU

STNESS


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0

1

2

3

4

5

6 11 16 21 26 31 36

[mm

]

NNmcmc

UCL

d

d

0

1

2

3

4

5

6 11 16 21 26 31 36

[mm

]

NNmcmc

UCL

d

d

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SETUP

OBJE

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Montecarlo simulationMontecarlo simulation

Sampling Sampling NNmcmc combinations from the combinations from the

multinormalmultinormal

All statistics canAll statistics canbe calculatedbe calculated

Average valuesAverage valuesand confidenceand confidencelimitslimitsstabilize stabilize as as NNmc mc

increasesincreases

0 ,x N x

RO

BU

STNESS

RO

BU

STNESS


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MO

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SETUP

OBJE

CTI

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Response Surface MethodologyResponse Surface Methodology

Full second order polynomial regression Full second order polynomial regression model for model for as a function of as a function of xx reduced dimensionality for reduced dimensionality for xx using normal using normal

anisotropyanisotropy

A new vector can be formedA new vector can be formed

The “metamodel” can be used for The “metamodel” can be used for calculating all statistics, includingcalculating all statistics, including

RO

BU

STNESS

RO

BU

STNESS

4

2 67512

xxxxr

1 2 3 4 8 9 10 11 12; ; ; ; ; ; ; ;z x x x x x x x x x

UCL


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MO

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SETUP

OBJE

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Approximate upper bound methodApproximate upper bound method

Hp: components of Hp: components of xx standardized and standardized and

independently distributedindependently distributed probability density function is a spheroidprobability density function is a spheroid take the spheroid with radius 3 (6take the spheroid with radius 3 (6 interval) interval) sample a (small) number of points on this spheroidsample a (small) number of points on this spheroid

extreme conditions are selectedextreme conditions are selected geometrical deviation of final shape geometrical deviation of final shape

will be larger than for any other pointwill be larger than for any other point

falling within the falling within the 66 sphere sphere

RO

BU

STNESS

RO

BU

STNESS

xx11

xx22

33


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SETUP

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Hp: components of Hp: components of xx standardized and standardized and independently distributedindependently distributed probability density function is a spheroidprobability density function is a spheroid

take the sphere with radius 3 (6take the sphere with radius 3 (6 interval) interval) sample a (small) number of points on this spheresample a (small) number of points on this sphere calculate average values of this calculate average values of this

boundary sample boundary sample (not the population)(not the population)

RO

BU

STNESS

RO

BU

STNESS

xx11

xx22

33;d ;d

UCL can be taken as an can be taken as an

upper bound estimate ofupper bound estimate of


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xx11

xx22

33

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SETUP

OBJE

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If the components of If the components of xx are correlated are correlated the density function is an ellipsoidthe density function is an ellipsoid the mahalanobis transformation can be used the mahalanobis transformation can be used

for sampling on the for sampling on the 66 boundary of the boundary of the ellipsoidellipsoid

RO

BU

STNESS

RO

BU

STNESS

xx11

xx22


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# of# ofrunsruns methodmethod d- d+ D Ducl

36+1 Montecarlo 1.06 0.98 2.04 4.05

67+1 RSM - - 1.32 4.14

19+1 Upper Bound - - - 4.39

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SETUP

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RO

BUSTN

ESSComparing the different methodsComparing the different methods

CO

MPA

RIS

ON

CO

MPA

RIS

ON

Available resultsAvailable results Montecarlo provides allMontecarlo provides all

d d UCL


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36+1 Montecarlo 1.06 0.98 2.04 4.05

67+1 RSM - - 1.32 4.14

19+1 Upper Bound - - - 4.39

d d UCLMO

DEL

SETUP

OBJE

CTI

VES

RO

BUSTN


CO

MPA

RIS

ON

CO

MPA

RIS

ON

Available resultsAvailable results RSM may provide and only if a RSM may provide and only if a

regression model is builtregression model is builtd d


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36+1 Montecarlo 1.06 0.98 2.04 4.05

67+1 RSM - - 1.32 4.14

19+1 Upper Bound - - - 4.39

d d UCLMO

DEL

SETUP

OBJE

CTI

VES

RO

BUSTN


CO

MPA

RIS

ON

CO

MPA

RIS

ON

Available resultsAvailable results UB provides only UB provides only

UCL


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MO

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SETUP

OBJE

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RO

BUSTN


CO

MPA

RIS

ON

CO

MPA

RIS

ON # of# of

runsruns methodmethod Ducl

36+1 Montecarlo 4.05

67+1 RSM 4.14

19+1 Upper Bound 4.39

UCL

Accuracy and costAccuracy and cost UB with 20 runs is close to RSM and MCUB with 20 runs is close to RSM and MC


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MO

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SETUP

OBJE

CTI

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RO

BUSTN

ESSConclusionsConclusions

CO

MPAR

ISO

N

A method has been proposed for evaluating A method has been proposed for evaluating robustness of sheet metal forming operationsrobustness of sheet metal forming operations

Estimating the width Estimating the width UCLUCL of the tolerance of the tolerance

band for the final part shape, requires:band for the final part shape, requires:1.1. Preliminary estimation of the covariance matrix Preliminary estimation of the covariance matrix of of

the random input vector the random input vector xx2.2. A method for calculating the geometrical deviation A method for calculating the geometrical deviation

of each simulation from the reference geometryof each simulation from the reference geometry

3.3. A statistical method for calculating A statistical method for calculating UCLUCL, the 6, the 6

interval of interval of

CO

NC

LUSIO

NS

CO

NC

LUSIO

NS


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d d UCLMO

DEL

SETUP

OBJE

CTI

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RO

BUSTN

ESSConclusionsConclusions

CO

MPAR

ISO

N

2.2. method for calculating method for calculating Method A (benchmark)Method A (benchmark) Method B (exact, minimization of Method B (exact, minimization of )) Method C (proposed)Method C (proposed)

Less expensive but not exact (overestimates Less expensive but not exact (overestimates ))

3.3. method for calculating method for calculating UCLUCL, the 6, the 6

interval of interval of MontecarloMontecarlo RSMRSM proposed UB approachproposed UB approach

Less expensive, provides close upper boundLess expensive, provides close upper bound

CO

NC

LUSIO

NS

CO

NC

LUSIO

NS

Documents

Robustness Evaluation and Tolerance Prediction for a Stamping Process with Springback Calculation by the FEM Matteo Strano Università di Cassino, Dip