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

Original Research


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.


Kriging Metamodel Binary logistic regression Reliability 



values of the blankholder force vs. time curve


operator of expectation


hardening coefficient


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


hardening exponent


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


probability density function


probability of failure


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


maximum thinning of the initial sheet thickness t 0


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


objective function


Young’s modulus

\(\underline z \)

vector of output variables of interest


maximum tolerable probability of failure

\(\bar \varepsilon \)

effective strain

\(\underline \varphi \)

mean square error of the Kriging predictor


weights of the objective function


expected value

\(\bar \sigma \)

flow stress


standard deviation


covariance matrix


feasibility window

\(\underline \xi \)

vector of random process variables


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Copyright information

© Springer/ESAFORM 2008

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria IndustrialeUniversità di CassinoCassino (FR)Italy

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