International Journal of Material Forming

, Volume 1, Issue 1, pp 13–20

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

Original Research

## 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

Kriging Metamodel Binary logistic regression Reliability

### Abbreviation

BHFi

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 $$\underline \xi$$

pdf

probability density function

PF

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

Ωs

feasibility window

$$\underline \xi$$

vector of random process variables

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