Optimization under uncertainty of sheet-metal-forming processes by the finite element method

In this paper, a novel method is proposed for the optimization, by finite element method (FEM) analysis, of the design variables of sheet-metal-forming processes. The method is useful when one or more non-controllable process parameters can be modelled as a random variable that introduces a degree of uncertainty into a process solution or, in other words, when the effect of any combination of design variables on the process response is disturbed by the non-controllable effect of a random vector of parameters. The method is particularly suited to problems with large FEM computational times, small process window, and large non-uniform variance of results. The process window is the region in the space of the design variables where the failure probability (i.e. the probability that wrinkling or excessive thinning occur) is lower than a given a value. 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 described and applied to a real production problem, a flexforming operation with large probability of failure by wrinkling, where the fluid pressure curve must be optimized. This particular industrial example has been chosen because several solutions found by process engineers did not prove to be robust in respect of a random variation in the coefficient of friction. A simple and conventional design approach would lead to an unreliable solution.