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Robustness Evaluation and Tolerance Prediction for a Stamping Process with Springback Calculation by the FEM Matteo Strano Università di Cassino, Dip. Ingegneria Industriale Cassino (FR), Italy

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’052/30 Outline of the presentation Objectives of the research Objectives of the research An IDEF model of the FEM simulations The model setup The model setup The FEM simulation setup Variables of the IDEF model The random input vector x The random input vector x The output geometrical response The output geometrical response Evaluating the geometrical robustness Evaluating the geometrical robustness Sensitivity analysis Montecarlo simulation Response Surface Methodology A new Upper Bound method Comparing the different methods Comparing the different methods Conclusions Conclusions

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’053/30 IDEF Model of the FEM Simulation FEM simulation Input process parameters x Output response variables d Control variables u OBJECTIVES Numisheet ’05 benchmark #2

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’054/30 deterministic model IDEF model: the benchmark ’05 - #2 FEM simulation -Sheet thickness -Young modulus -Anisotropy r-values -Friction f-values etc. -Binder force -Binder travel -Deviation from reference geometry x duOBJECTIVES Numisheet ’05 benchmark #2

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’055/30 IDEF model: uncertainty FEM simulation x du random input vector mean covariance random output unknown distribution & moments OBJECTIVES Incorporate uncertainty into x

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’056/30 The FEM simulation setup MODEL SETUP OBJECTIVES Solver Solver Pam-Stamp 2g, with springback Drawbeads Drawbeads Physical Mesh Mesh Quadrangular Belytschko-Tsay, 5 integration points, initial size 10 mm Material Material isotropic Hill ’48 hardening, orthotropic material with given r 0, r 45 and r 90, Flow stress law

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’057/30 The random input vector x The vector x has 11 components N x =11 The vector x has 11 components N x =11 x 1 =K, x 2 =n, x 3 = 0 (flow stress parameters) x 4 =t (initial sheet thickness) x 5 =r 0, x 6 =r 45, x 7 =r 90 (anisotropy parameters) x 8 =E (young modulus) x 9 =f b, x 10 =f d, x 11 =f p (friction coefficients between the blank and the binder, the upper die and the lower punch) MODEL SETUP OBJECTIVES FEM simulation High dimensional problem

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’058/30 The random input vector x Mean vector Mean vector Given nominal values Covariance matrix Covariance matrix ANOVA + correlation analysis for x 1 to x 8 No data available for x 9 to x 11 (friction coefficients) Assumptions on mean and standard deviation Assumptions on mean and standard deviation MODEL SETUP OBJECTIVES estimating

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’059/30 The random input vector x MODEL SETUP OBJECTIVES Mean vector Mean vector Covariance matrix Covariance matrix

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0510/30 The output response Geometrical deviation Geometrical deviation For every simulation run, the position of the formed sheet after springback must be fixed MODEL SETUP OBJECTIVES The reference geometry is obtained by running a simulation with nominal values of x 0 FEM simulationd

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0511/30 The output response Geometrical deviation Geometrical deviation For every simulation run, the position of the formed sheet after springback must be fixed MODEL SETUP OBJECTIVES FEM simulationd+=

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0512/30 Calculating : positioning the sheet MODEL SETUP OBJECTIVES totally fixed after springback X and Y fixed Method A Method A 2 reference points + symmetry plane

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0513/30 Calculating : positioning the sheet MODEL SETUP OBJECTIVES after springback Method A Method A 2 reference points + symmetry plane B: sampled geometry B A A: reference geometry distance d -

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0514/30 Calculating : positioning the sheet MODEL SETUP OBJECTIVES after springback Method A Method A 2 reference points + symmetry plane B: sampled geometry B A A: reference geometry distance d +

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0515/30 Calculating : positioning the sheet Method B Method B rotating and translating each shape until the error is minimized exact estimation of but computationally expensive MODEL SETUP OBJECTIVES after springback Z X Y Symmetry plane

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0516/30 Calculating : positioning the sheet Method C Method C MODEL SETUP OBJECTIVES 1 point fixed in space after springback Y Positioning plane

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0517/30 MODEL SETUP OBJECTIVES ROBUSTNESS Goal eeeestimating the variation of the geometrical deviation aaaaverage values Evaluating the geometrical robustness 1111 2222 3333 4444 5555

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0518/30 MODEL SETUP OBJECTIVES ROBUSTNESS Goal Goal estimating the variation of the geometrical deviation average values Evaluating the geometrical robustness Upper Confidence Limit of at 99.7% width of 6 tolerance interval of the final shape =

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0519/30 MODEL SETUP OBJECTIVES Alternative methods Alternative methods Inexpensive and rough Sensitivity analysis Sensitivity analysis – changing 1 parameters each simulation … Approximate upper bound method … Expensive and precise Montecarlo simulation Montecarlo simulation Response Surface Methodology Response Surface Methodology Evaluating the geometrical robustness ROBUSTNESS

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0520/30 MODEL SETUP OBJECTIVES Montecarlo simulation Sampling N mc combinations from the multinormal Sampling N mc combinations from the multinormal All statistics can be calculated Average values and confidence limits stabilize as N mc increases ROBUSTNESS

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0521/30 MODEL SETUP OBJECTIVES Response Surface Methodology Full second order polynomial regression model for as a function of x Full second order polynomial regression model for as a function of x reduced dimensionality for x using normal anisotropy A new vector can be formed The “metamodel” can be used for calculating all statistics, including ROBUSTNESS

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0522/30 MODEL SETUP OBJECTIVES Approximate upper bound method Hp: components of x standardized and independently distributed probability density function is a spheroid take the spheroid with radius 3 (6 interval) sample a (small) number of points on this spheroid extreme conditions are selected extreme conditions are selected geometrical deviation of final shape will be larger than for any other point falling within the 6 sphere geometrical deviation of final shape will be larger than for any other point falling within the 6 sphere ROBUSTNESS x1x1x1x1 x2x2x2x2 3

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0523/30 MODEL SETUP OBJECTIVES Approximate upper bound method Hp: components of x standardized and independently distributed probability density function is a spheroid take the sphere with radius 3 (6 interval) sample a (small) number of points on this sphere calculate average values of this boundary sample (not the population) ROBUSTNESS x1x1x1x1 x2x2x2x23 can be taken as an upper bound estimate of

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0524/30 x1x1x1x1 x2x2x2x23 MODEL SETUP OBJECTIVES Approximate upper bound method If the components of x are correlated the density function is an ellipsoid the mahalanobis transformation can be used for sampling on the 6 boundary of the ellipsoid ROBUSTNESS x1x1x1x1 x2x2x2x2

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0525/30 # of runs method d-d+DDucl 36+1 Montecarlo RSM Upper Bound MODEL SETUP OBJECTIVES ROBUSTNESS Comparing the different methods COMPARISON Available results Available results Montecarlo provides all

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0526/30 # of runs method d-d+DDucl 36+1 Montecarlo RSM Upper Bound MODEL SETUP OBJECTIVES ROBUSTNESS Comparing the different methods COMPARISON Available results Available results RSM may provide and only if a regression model is built

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0527/30 # of runs method d-d+DDucl 36+1 Montecarlo RSM Upper Bound MODEL SETUP OBJECTIVES ROBUSTNESS Comparing the different methods COMPARISON Available results Available results UB provides only

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0528/30 MODEL SETUP OBJECTIVES ROBUSTNESS Comparing the different methods COMPARISON # of runs method Ducl 36+1 Montecarlo RSM Upper Bound4.39 Accuracy and cost Accuracy and cost UB with 20 runs is close to RSM and MC

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0529/30 MODEL SETUP OBJECTIVES ROBUSTNESS Conclusions COMPARISON A method has been proposed for evaluating robustness of sheet metal forming operations A method has been proposed for evaluating robustness of sheet metal forming operations Estimating the width UCL of the tolerance band for the final part shape, requires: Estimating the width UCL of the tolerance band for the final part shape, requires: 1.Preliminary estimation of the covariance matrix of the random input vector x 2.A method for calculating the geometrical deviation of each simulation from the reference geometry 3.A statistical method for calculating UCL, the 6 interval of CONCLUSIONS

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M. Strano Robustness Evaluation and Tolerance Prediction… NUMISHEET ’0530/30 MODEL SETUP OBJECTIVES ROBUSTNESS Conclusions COMPARISON 2. method for calculating Method A (benchmark) Method B (exact, minimization of ) Method C (proposed) Less expensive but not exact (overestimates ) Less expensive but not exact (overestimates ) 3. method for calculating UCL, the 6 interval of Montecarlo RSM proposed UB approach Less expensive, provides close upper bound Less expensive, provides close upper bound CONCLUSIONS

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