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 (I A) has been developed to estimate the large elasto-plastic strains in thin metallic panels obtained by deep drawing. This paper deals with the main re cent developments introduced by the authors on the IA to improve its effici ency in the analysis and optimum design of blank contours of complicated in dustrial 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 possibl e to estimate the strains and stresses in the final workpiece. Important as sumptions are adopted regarding the constitutive equations (the deformation theory of plasticity) and the action of the tools (the punch, die and blan k holders). The IA implies only two degrees of freedom per node even if ben ding effects are considered. In this paper, we present several recent devel opments: (1) The bending effects are taken into account using a simple tria ngular shell element without increasing the number of dof per node. (2) Som e analytical formulas are introduced to consider the restraining forces due to the drawbeads. (3) Some improvements of resolution algorithms such as t he introduction of a relaxation coefficient, a damping factor and a good in itial solution are realized. (4) Shape optimization of blank contours is pe rformed using a numerical procedure based on the coupling of the IA and a s equential quadratic programming method (SQP). In this work, all sensitiviti es are computed analytically using the adjoint variable method. The numeric al results of the IA on two benchmark tests are compared with experimental and other numerical results. The optimization procedure is applied to the b lank optimum design of the Renault/Twingo dashpot cup where the objective f unction is defined to minimize the maximum of the thickness variations. (C) 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.