Matteo Strano Università di Cassino, Dip. Ingegneria Industriale Cassino (FR), Italy
description
Transcript of Matteo Strano Università di Cassino, Dip. Ingegneria Industriale Cassino (FR), Italy
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
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xN
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Output Output response response variablesvariables
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dN
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Control Control variablesvariables
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uN
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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
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xN
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-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
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dN
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Numisheet ’05 benchmark #2Numisheet ’05 benchmark #2
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IDEF model: uncertaintyIDEF model: uncertainty
FEMsimulation
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xN
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uu 0 ,x N x random input vectorrandom input vector
meanmeancovariancecovariance
? ?d random outputrandom output
unknownunknowndistribution &distribution &
momentsmoments
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Incorporate uncertainty into Incorporate uncertainty into xx
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dN
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The FEM simulation setupThe FEM simulation 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)
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FEMsimulation
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xN
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High dimensional High dimensional problemproblem
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The random input vector The random input vector xx
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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estimatingestimating
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The random input vector The random input vector xx
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
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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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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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FEMsimulation
dd
max d++ ==
max d
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Calculating Calculating : positioning the sheet: positioning the sheet
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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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Calculating Calculating : positioning the sheet: positioning the sheet
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after springbackafter springback
Method Method AA 2 reference points 2 reference points
+ symmetry plane+ symmetry plane
B: sampled B: sampled geometrygeometry
BB
AA
A: A: reference reference geometrygeometry
distance distance dd--
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Calculating Calculating : positioning the sheet: positioning the sheet
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after springbackafter springback
Method Method AA 2 reference points 2 reference points
+ symmetry plane+ symmetry plane
B: sampled B: sampled geometrygeometry
BB
AA
A: A: reference reference geometrygeometry
distance distance dd++
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Calculating Calculating : positioning the sheet: positioning the sheet
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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after springbackafter springback
ZZ
XX
YY
Symmetry Symmetry planeplane
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Calculating Calculating : positioning the sheet: positioning the sheet
Method Method CC
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1 point fixed in 1 point fixed in spacespace
after springbackafter springback
YY
Positioning planePositioning plane
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RO
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STNESS
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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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STNESS
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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
;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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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......
Evaluating the geometrical robustnessEvaluating the geometrical robustness
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STNESS
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STNESS
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[mm
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NNmcmc
UCL
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[mm
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NNmcmc
UCL
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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
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STNESS
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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
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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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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
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STNESS
xx11
xx22
33
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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 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)
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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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Approximate upper bound methodApproximate upper bound method
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
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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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BUSTN
ESSComparing the different methodsComparing the different methods
CO
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ON
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Available resultsAvailable results Montecarlo provides allMontecarlo provides all
d d UCL
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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
d d UCLMO
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ON
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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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# 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
d d UCLMO
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Available resultsAvailable results UB provides only UB provides only
UCL
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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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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
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LUSIO
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LUSIO
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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
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LUSIO
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CO
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LUSIO
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