Robust die compensation in sheet metal design through the integration of dual response surface and shape function optimization

In sheet metal forming, springback represents a major drawback increasing die set-up problems, especially for ultra-high strength steels. Finite Element Analysis is a well-established method to simulate the process during design, and multicriteria optimizations, for example, via surrogate models, are investigated in order to develop integrated design. Since to take into account also springback compensation die design may involve a large number of geometric variables, this paper presents a robust design formulation, based on the adoption of the shape function optimization, to describe springback in terms of weights directly associated to global shape variations of the die shape. Doing so, multicriteria optimization, which involves also die compensation, can be set up in a more intuitive approach, as requested in the preliminary steps of die design. After the introduction of the industrial problem, the mathematical formulation of the shape function optimization is presented together with its novel extension to Robust Design, which is based on the Dual Response Surface. Through a test case derived from the head part of a B-pillar, stamped from a Dual Phase sheet 1.5 mm thick, this novel extension investigates the effect of 6% variation from nominal values of initial yield stress and thickness. Results demonstrate the feasibility of the procedure, underlying that an optimal compensation may not be optimal in terms of process robustness.