Polytope-based topology optimization using a mimetic-inspired method

摘要: Topology optimization refers to the optimum distribution of materials, so as to achieve certain prescribed design objectives while simultaneously satisfying constraints. Engineering applications often require unstructured meshes to capture the domain and boundary conditions accurately and to ensure reliable solutions. Hence, unstructured polyhedral elements are becoming increasingly popular. Since the pioneering work of Wachspress, many interpolants for polytopes have come forth; such as, mean value coordinates, natural neighbor-based coordinates, metric coordinate method and maximum entropy shape functions. The extension of the shape functions to three-dimensions, however, has been relatively slow partly due to the fact that these interpolants are subject to restrictions on the topology of admissible elements (eg, convexity, maximum valence count) and can be sensitive to geometric degeneracies. More importantly, calculating these functions and their gradients are in general computationally expensive. Numerical evaluation of weak form integrals with sufficient accuracy poses yet another challenge due to the non-polynomial nature of these functions as well as the arbitrary domain of integration. Virtual Element Method (VEM), which has evolved from Mimetic Finite Difference methods, addresses both the issues of accuracy and efficiency. In this work, a VEM framework for three-dimensional elasticity is presented. Even though VEM is a conforming Galerkin formulation, it differs from tradition finite element methods in the fact that it does not require explicit computation of approximation spaces. In VEM, the deformation states of an …