Topology optimization on two-dimensional manifolds

Abstract

This paper presents topology optimization on general two-dimensional manifolds for phenomena described by second-order partial differential equations, where the material interpolation is implemented by using the material distribution method. When a physical field is defined on a two-dimensional manifold, the material interpolation is implemented on a material parameter in the partial differential equation used to describe the distribution of the physical field. When the physical field is defined on a three-dimensional domain with its boundary conditions defined on a two-dimensional manifold corresponding a surface or an interface of this three-dimensional domain, the material density is used to formulate a mixed boundary condition of the partial differential equation for the physical field and implement the penalization between two different boundary types. Based on the homeomorphic property of two-dimensional manifolds, typical two-dimensional manifolds, e.g., sphere, torus, Möbius strip and Klein bottle, are included in the numerical tests, which are used to demonstrate this topology optimization approach for the design problems of fluidic mechanics, heat transfer and electromagnetics.