Topology optimization of conformal structures on manifolds using extended level set methods (X-LSM) and conformal geometry theory

In this paper, we propose a new method to systematically address the issue of structural shape and topology optimization on free-form surfaces. A free-form surface, also termed manifold, is conformally mapped onto a 2D rectangle domain where the level set function is defined. With the conformal mapping, the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar. Keeping this intrinsic relation in mind, we derive the Euclidean form for the Riemannian Hamilton–Jacobi equation governing the boundary evolution on the manifold, which can be solved on a 2D plane using classical level set methods, such as the upwind finite difference or fast marching method. By reducing the dimension of the problem, the topology optimization problem on the manifold embedded in the 3D space can be recast as a 2D topology optimization problem in the Euclidean space. Compared with other approaches which need project the Euclidean differential operators to the manifold, the proposed method can not only reduce the computational cost but also preserve all the advantages of conventional level set methods. The proposed method reveals the fundamental relation between topology optimization on manifolds and Euclidean planes. It provides a unified level-set-based computational framework for the generative design of conformal structures with increasing applications in different fields of interests.