A technique for FEM optimization under uncertainty of time-dependent process variables in sheet metal forming

1. ESAFORM 2006 A technique for FEM optimization under uncertainty of time-dependent process variables in sheet metal forming M. Strano Università di Cassino Dip. Ingegneria Industriale http://webuser.unicas.it/tsl

2. Executive summary • A method is proposed for the FEM optimization of the time-dependent design variables of sheet metal forming processes • useful when the non-controllable process parameters can be modelled as random variables • suited for problems with large computational times, small process window and non-uniform variance of results • The problem is formulated as the minimization of a cost function, subject to a reliability constraint • The cost function is optimized through a “metamodel”, built by “Kriging” interpolation • The reliability is assessed by a binary logistic regression analysis of the results • The method is applied to the Numisheet ’93 benchmark problem (u-channel forming and springback) • modified by handling the blankholder force as a time-dependent variable

3. Overview of the problem Optimization under uncertainty • The solution must • Optimize the expected value of an objective function • Be robust & reliable Optimization Uncertainty • Two alternatives • Coupled approach • Build an objective function that incorporates both mean value and standard deviation of the performance • Decoupled approach • Find a potentially optimal solution • Assess its reliability (use reliability as a constraint)

4. Overview of the problem General literature on methods for stochastic optimization with reliability assessment is very extensive • Still, not much is available in the field of numerical methods for sheet metal forming e.g. [Sahai, Schramm, Buranathiti, Chen, Cao, Xia / Numiform ’04, Columbus] • decoupled methodology for optimization of a metamodel function and for reliability assessment

5. A metamodel is required Optimizer FEM Metamodel f’(x)=0 Decoupled approach: optimization The main issue with optimization is the very high computational cost • computational burden is even worse if uncertainty is considered

6. Decoupled approach: optimization Metamodeling • Generally, RSM (Response Surface Method) is used for metamodeling • RSM is a regression (approximation) method, which may result quite inaccurate when the response is deterministic • Not perfectly suited to (deterministic) computer simulations [Bonte, Van den Boogaard, Huétink / Esaform ’05 Cluj-Napoca]

7. Decoupled approach: optimization Metamodeling • Interpolation can be better suited to computer simulations results • DACE (Design and Analysis of Computer Experiments) and Kriging are recently the method of choice interpolation vs. regression

8. Decoupled approach: reliability assessment Several methods are generally used for reliability assessment • Most available approaches are based on localapproximation (regression) of results, in order to estimate the standard deviation • For all available methods (also in the coupled approach), the standard deviation of response is required • Usually this is • either inaccurate, when not enough data is used • or computationally expensive, otherwise

9. Decoupled approach: reliability assessment A proposal for global reliability assessment when the process window is not very large (failure probability is not very small) • Logistic regression analysis

10. Description of the proposed method Deterministic control variables FEM output variables Non-controllable randomprocess parameters Cost function (affected byuncertainty) Reliability constraint Technologicalconstraints Minimization of the expected value

11. Description of the proposed method 4-step procedure Latin hypercube or similar space-filling for x

12. Step 2: reliability assessment The failure probability Pf(x)=1-Ps(x) is calculated thanks to: binary logistic regression “Logit” link function Polynomial model • A binary value is assigned to each simulation run • 0 if the final part is sound or the process feasible • 1 if the final part is failed or the process unfeasible • It does not require local calculation of standard deviation

13. Step 3: metamodeling A metamodel of the cost function C(y), i.e. a predictor (x,x), is built using Kriging interpolation The mean square error of the predictor is unlike regression models (such as RSM), the mean square error f is zero at the design points f is larger where the design space has not been filled

14. Step 4: optimization The expected value of the cost function is minimized under the technological constraints and the reliability constraints However, in order to ensure accuracy, the mean square error fcould be taken into account by modifying the objective function

15. Application to u-channel forming • Numisheet ’93 benchmark (u-channel forming) • a time-dependent blankholder force BHF has been optimized, in order to minimise the amount of springback, given a constraint on the maximum tolerable percentage thinning of the sheet

16. Application to u-channel forming • Components of the random vector x • Technological constraints 9 kN  xj 40 kN; j=1, ... 5

17. Application to u-channel forming • Step 1: design and execution of simulations • 97 simulations have been planned • Step 2: reliability assessment • A part is considered to be failed if the limit of 15% for maximum thinning is exceeded Ps(x)=P[y115%]1- (a=0.01) • Step 3: metamodeling • The output y2 (a measure of springback) is used as the main parameter in the cost function. • A metamodel of the cost is built using Kriging interpolation

18. Application to u-channel forming 97 runs 500 runs

19. Application to u-channel forming • Theresults With nominal values of random parameters

20. Conclusions • An innovative method has been proposed for optimization of sheet metal forming processes • suited for multiple design variables and affected by randomness of a few non-controllable parameters • The proposed procedure is divided in 4 steps: • design of simulations • reliability assessment using logistic regression • metamodeling with Kriging interpolation • constrained optimization • It has been successfully applied to the Numisheet ’93 u-channel forming benchmark, which has been re-formulated as an optimization problem with uncertainty