Matteo Strano Università di Cassino, Dip. Ingegneria Industriale Cassino (FR), Italy

1. 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 m.strano@unicas.it http://webuser.unicas.it/tsl

2. Outline of the presentation • Objectives of the research • An IDEF model of the FEM simulations • The model setup • The FEM simulation setup • Variables of the IDEF model • The random input vector x • The output geometrical response • Evaluating the geometrical robustness • Sensitivity analysis • Montecarlo simulation • Response Surface Methodology • A new Upper Bound method • Comparing the different methods • Conclusions

3. Control variables u Output response variables x Input process parameters d IDEF Model of the FEM Simulation Numisheet ’05 benchmark #2 OBJECTIVES FEM simulation

4. u x d IDEF model: the benchmark ’05 - #2 Numisheet ’05 benchmark #2 OBJECTIVES -Binder force -Binder travel deterministic model -Deviation from reference geometry -Sheet thickness -Young modulus -Anisotropy r-values -Friction f-values etc. FEM simulation

5. random output random input vector u unknowndistribution & moments covariance x mean d IDEF model: uncertainty Incorporate uncertainty into x OBJECTIVES FEM simulation

6. The FEM simulation setup • Solver • Pam-Stamp 2g, with springback • Drawbeads • Physical • Mesh • Quadrangular Belytschko-Tsay,5 integration points, initial size 10 mm • Material • isotropic Hill ’48 hardening, orthotropic material with given r0, r45 and r90, • Flow stress law OBJECTIVES MODEL SETUP

7. FEM simulation The random input vector x • The vector x has 11 components Nx=11 • x1=K, x2=n, x3=e0 (flow stress parameters) • x4=t(initial sheet thickness) • x5=r0, x6=r45, x7=r90 (anisotropy parameters) • x8=E(young modulus) • x9=fb, x10=fd, x11=fp(friction coefficients between the blank and the binder, the upper die and the lower punch) OBJECTIVES MODEL SETUP High dimensional problem

8. estimating The random input vector x • Mean vector • Given nominal values • Covariance matrix • ANOVA + correlation analysis for x1 to x8 • No data available for x9 to x11 (friction coefficients) • Assumptions on mean and standard deviation OBJECTIVES MODEL SETUP

9. The random input vector x • Mean vector • Covariance matrix OBJECTIVES MODEL SETUP

10. d FEM simulation The reference geometry is obtained by running a simulation with nominal values of x0 The output response • Geometrical deviation OBJECTIVES MODEL SETUP For every simulation run, the position of the formed sheet afterspringback must befixed

11. d FEM simulation = The output response • Geometrical deviation OBJECTIVES MODEL SETUP For every simulation run, the position of the formed sheet afterspringback must befixed +

12. totally fixed X and Y fixed Calculating d: positioning the sheet • Method A • 2 reference points + symmetry plane OBJECTIVES MODEL SETUP after springback

13. A: reference geometry B: sampled geometry B A Calculating d: positioning the sheet • Method A • 2 reference points + symmetry plane OBJECTIVES MODEL SETUP distance d- after springback

14. A: reference geometry B: sampled geometry B A Calculating d: positioning the sheet • Method A • 2 reference points + symmetry plane OBJECTIVES MODEL SETUP distance d+ after springback

15. Calculating d: positioning the sheet Symmetry plane • Method B • rotating and translating each shape until the error d is minimized • exact estimation of d but computationally expensive OBJECTIVES MODEL SETUP Z Y X after springback

16. 1 point fixed in space Calculating d: positioning the sheet • Method C OBJECTIVES MODEL SETUP Positioning plane Y after springback

17. d5 d4 d2 d3 d1 Evaluating the geometrical robustness • Goal • estimating the variation of thegeometrical deviation • average values OBJECTIVES MODEL SETUP ROBUSTNESS

18. Upper Confidence Limit of dat99.7% = Evaluating the geometrical robustness • Goal • estimating the variation of thegeometrical deviation • average values OBJECTIVES MODEL SETUP ROBUSTNESS width of 6s tolerance intervalof the final shape 

19. Evaluating the geometrical robustness • Alternative methods • Inexpensive and rough • Sensitivity analysis • changing 1 parameters each simulation • … • Approximate upper bound method • … • Expensive and precise • Montecarlo simulation • Response Surface Methodology • ... OBJECTIVES MODEL SETUP ROBUSTNESS

20. Montecarlo simulation • Sampling Nmc combinations from the multinormal • All statistics canbe calculated • Average valuesand confidencelimitsstabilize as Nmc increases OBJECTIVES MODEL SETUP ROBUSTNESS

21. Response Surface Methodology • Full second order polynomial regression model for d as a function of x • reduced dimensionality for xusing normal anisotropy • A new vector can be formed • The “metamodel” can be used for calculating all statistics, including OBJECTIVES MODEL SETUP ROBUSTNESS

22. x2 3 x1 Approximate upper bound method Hp: components of x standardized and independently distributed • probability density function is a spheroid • take the spheroid with radius 3 (6s interval) • sample a (small) number of points on this spheroid • extreme conditions are selected • geometrical deviation of final shape will be larger than for any other pointfalling within the 6s sphere OBJECTIVES MODEL SETUP ROBUSTNESS

23. x2 3 x1 Approximate upper bound method Hp: components of x standardized and independently distributed • probability density function is a spheroid • take the sphere with radius 3 (6s interval) • sample a (small) number of points on this sphere • calculate average values of this boundary sample (not the population) OBJECTIVES MODEL SETUP ROBUSTNESS can be taken as an upper bound estimate of

24. x2 x2 3 x1 x1 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 6s boundary of the ellipsoid OBJECTIVES MODEL SETUP ROBUSTNESS

25. Comparing the different methods • Available results • Montecarlo provides all OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON

26. Comparing the different methods • Available results • RSM may provide and only if a regression model is built OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON

27. Comparing the different methods • Available results • UB provides only OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON

28. Comparing the different methods • Accuracy and cost • UB with 20 runs is close to RSM and MC OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON

29. Conclusions • A method has been proposed for evaluating robustness of sheet metal forming operations • Estimating the width dUCL of the tolerance band for the final part shape, requires: • Preliminary estimation of the covariance matrix S of the random input vector x • A method for calculating the geometrical deviation d of each simulation from the reference geometry • A statistical method for calculating dUCL, the 6s interval of d OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON CONCLUSIONS

30. Conclusions • method for calculating d • Method A (benchmark) • Method B (exact, minimization of d) • Method C (proposed) • Less expensive but not exact (overestimates d) • method for calculating dUCL, the 6s interval of d • Montecarlo • RSM • proposed UB approach • Less expensive, provides close upper bound OBJECTIVES MODEL SETUP ROBUSTNESS COMPARISON CONCLUSIONS