Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach

A simplified efficient finite element method called the inverse approach (IA) has been developed to estimate the large elasto-plastic strains in thin metallic panels obtained by deep drawing. This paper deals with the main recent developments introduced by the authors on the IA to improve its efficiency in the analysis and optimum design of blank contours of complicated industrial parts. The IA mainly exploits the knowledge of the 3D shape of the final workpiece. An iterative scheme is used to find the original position of each material point in the initial flat blank after which it is possible to estimate the strains and stresses in the final workpiece. Important assumptions are adopted regarding the constitutive equations (the deformation theory of plasticity) and the action of the tools (the punch, die and blank holders). The IA implies only two degrees of freedom per node even if bending effects are considered. In this paper, we present several recent developments: (1) The bending effects are taken into account using a simple triangular shell element without increasing the number of dof per node. (2) Some analytical formulas are introduced to consider the restraining forces due to the drawbeads. (3) Some improvements of resolution algorithms such as the introduction of a relaxation coefficient, a damping factor and a good initial solution are realized. (4) Shape optimization of blank contours is performed using a numerical procedure based on the coupling of the IA and a sequential quadratic programming method (SQP). In this work, all sensitivities are computed analytically using the adjoint variable method. The numerical results of the IA on two benchmark tests are compared with experimental and other numerical results. The optimization procedure is applied to the blank optimum design of the Renault/Twingo dashpot cup where the objective function is defined to minimize the maximum of the thickness variations.