a Corresponding author: guenhael.lequilliec@univ-tours.fr Shape Optimization in Metal Forming Applications Based on the Shape- Manifold Approach Guenhael Le Quilliec 1,a , Balaji Raghavan 2 and Piotr Breitkopf 3 1 Laboratoire de Mécanique et Rhéologie - EA 2640, Polytech Tours, 7 avenue Marcel Dassault, 37200 Tours, France 2 Laboratoire de Génie Civil et Génie Mécanique - EA 3913, INSA de Rennes, 35708 Rennes, France 3 Sorbonne Universités, Université de Technologie de Compiègne, Laboratoire Roberval, UMR 7337, 60200 Compiègne, France Abstract. The concept of shape-manifold is applied here to the optimization of a deep drawn blank considering parameterized tool geometries. By adopting this concept, a realistic post-springback shape of the blank closest to a given ideal target shape is obtained. The global approach is fully automated, from the generation of the tool geometries, to the identification of the optimized design parameters. This approach was successfully applied to both simple 2D and complex 3D industrial test cases. 1 Introduction Due to elastic springback, the shape of a formed blank differs from the shape of the tools: i.e. the die and the punch. Various optimization approaches (often based on Finite Element simulation loops) are used to find the best shape of the tools for obtaining a desired shape, referred to as the target shape in the remainder of this paper. Geometries are usually described with a set of primitives (i.e. lengths, radii and angles). Although it can be very easy to describe simple geometries with primitives, it often becomes very difficult in the case of more complex shapes, as those obtained in deep drawing, after springback. Another problem with complex shapes is the number of parameters involved. When these parameters are chosen at random, they may not fully describe the geometry. Some parameters can also be highly correlated together. Finally, the number of parameters is usually greater than the intrinsic dimensionality of the assessed shape. It is common knowledge that the fewer the number of parameters is, the more efficient the optimization approach will be. In the present work, we propose to use a Level Set approach to fully describe springback shapes. The dimensionality reduction is achieved by Proper Orthogonal Decomposition. After detecting the intrinsic dimensionality, interpolation between shapes is performed by Diffuse Approximation [1]. The obtained response surface is called a Shape-Manifold [2] which represents any admissible shape of the shape-space. Finally, a Walking Algorithm [3] is used to efficiently approach the target shape by constructing the shape manifold only locally, step by step, until convergence. 2 Shape-manifold approach 2.1 Level set approach The level set approach is used to fully describe springback shapes (Fig. 1), regardless of their complexity. This Eulerian approach allows us to represent not only the target shape (usually designed with a CAD software) but also the springback shapes (either meshes coming from Finite Element simulations or even actual 3D scanned parts). However, Level Sets imply a very high dimensionality, due to the resolution of the grid, which still needs to be decreased. Figure 1. Example of a 2D level set function whose zero level set represents half of the profile of a U-shaped sheet after springback.