# A technique for FEM optimization under reliability constraint of process variables in sheet metal forming

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- Received:
- Accepted:

DOI: 10.1007/s12289-008-0001-8

- Cite this article as:
- Strano, M. Int J Mater Form (2008) 1: 13. doi:10.1007/s12289-008-0001-8

- 4 Citations
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## Abstract

A method is proposed for the optimization, by finite element analysis, of design variables of sheet metal forming processes. The method is useful when the non-controllable process parameters (e.g. the coefficient of friction or the material properties) can be modelled as random variables, introducing a degree of uncertainty into any process solution. The method is suited for problems with large FEM computational times and small process window. The problem is formulated as the minimization of a cost function, subject to a reliability constraint. The cost function is indirectly optimized through a “metamodel”, built by “Kriging” interpolation. The reliability, i.e. the failure probability, is assessed by a binary logistic regression analysis of the simulation results. The method is applied to the u-channel forming and springback problem presented in Numisheet 1993, modified by handling the blankholder force as a time-dependent variable.

### Keywords

KrigingMetamodelBinary logistic regressionReliability### Abbreviation

- BHF
_{i} values of the blankholder force vs. time curve

*E*operator of expectation

*K*hardening coefficient

*m*number of components of deterministic vector \(\underline x \)

*n*hardening exponent

*p*number of components of vector \(\underline \xi \)

probability density function

*P*_{F}probability of failure

*q*number of components of vector \(\underline z \)

*t*_{%}maximum thinning of the initial sheet thickness

*t*_{0}*w*_{%}percentage increment of the initial channel width

*w*_{0}after springback- \(\underline w \)
vector of Kriging input variables

- \(\underline x \)
vector of deterministic control variables

*y*objective function

*Y*Young’s modulus

- \(\underline z \)
vector of output variables of interest

*α*maximum tolerable probability of failure

- \(\bar \varepsilon \)
effective strain

- \(\underline \varphi \)
mean square error of the Kriging predictor

*λ*weights of the objective function

*μ*expected value

- \(\bar \sigma \)
flow stress

*σ*standard deviation

*Σ*covariance matrix

*Ω*_{s}feasibility window

- \(\underline \xi \)
vector of random process variables