Topology optimization on two-dimensional manifolds
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Topology optimization is one of the most-used method to inversely determine the geometrical configurations of structures. In the current density method-based topology optimization, the design variable used to represent the structure was usually defined on three-dimensional domains or reduced two-dimensional planes, with the deficiency on the general two-dimensional manifolds. Therefore, this article focuses on developing a density method-based topology optimization approach implemented on two-dimensional manifolds. In this approach, material interpolation is implemented on a material parameter in the partial differential equation used to describe a physical field, when this physical field is defined on a two-dimensional manifold; the density variable is used to formulate a mixed boundary condition of a physical field and implement the penalization between two different types of boundary conditions, when this physical field is defined on a three-dimensional domain and its boundary conditions are defined on a two-dimensional manifold, an interface imbedded in the three-dimensional space. Because of the homeomorphic property of two-dimensional manifolds, several typical two-dimensional manifolds, e.g., sphere, torus, Möbius strip and Klein bottle, are included in the numerical tests. This approach has been tested and demonstrated by the problems in the areas of soft matter, heat transfer and electromagnetics. Because the derived structural pattern on a two-dimensional manifold is a fiber bundle with a corresponding base manifold and topology optimization is a typical inverse design method, this developed topology optimization approach is an inverse design method of fiber bundles.
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