Direct design to stress mapping for cellular structures

This paper aims to instantly predict within any accuracy the stress distribution of cellular structures under parametric design, including the shapes or distributions of the cell geometries, or the magnitudes of external loadings. A classical model reduction technique has to balance the simulation accuracy and interaction speed, and has difficulty achieving this goal. We achieve this by computing offline a design-to-stress mapping that ultimately expresses the stress distribution as an explicit function in terms of its design parameters. The mapping is determined as a solution to an extended finite element analysis problem in a high-dimension space, including both the spatial coordinates and the design parameters. The well-known curse of dimensionality intrinsic to the high-dimension problem is (partly) resolved through a spatial separation using two main techniques. First, the target mapping takes a reduced form as a sum of the products of separated one-variable functions, extending the proper generalized decomposition technique. Second, the simulation problem in a varied computation domain is reformulated as that in a fixed-domain, taking an integration function as the sum of the products of separated one-variable functions, in combination with high-order singular value decomposition. Extensive 2D and 3D examples are shown to demonstrate the approach's performance.