| 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 (e.g., 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 element are kinematically decomposed into rigid body, linear and higher order modes. The discrete bilinear form is constructed to capture the linear deformations exactly which ensures that the displacement patch test is passed and optimum convergence is achieved. The present work focuses on first-order VEM with degrees of freedom associated with the vertices of the elements. Construction of the stiffness matrix reduces to the evaluation of surface integrals, in contrast to the volume integrals encountered in the conventional finite element method (FEM), thus reducing the overall computational cost.;By means of the aforementioned approach, a framework for three-dimensional topology optimization is developed for polyhedral meshes. In the literature, topology optimization problems are typically solved with either tetrahedral or brick meshes. Numerical anomalies, such as checkerboard patterns and one-node connections, are present in such formulations. Constraints in the geometrical features of spatial discretization can also result in mesh dependent sub-optimal designs. In the current work, polyhedral meshes are proposed as a means to address the geometric features of the domain discretization. Polyhedral meshes not only provide greater flexibility in discretizing complicated domains but also alleviate the aforementioned numerical anomalies. For topology optimization problems, many approaches are available; which can mainly be classified as density-based methods and differential equation-driven methods (further subclassified as level-set and phase-field methods). Before choosing density-based methods for polyhedral topology optimization, a couple of differential equation-driven methods; which are representative of the literature, are exhaustively analyzed in two-dimensions. Finally, we also investigate aesthetics in topology optimization designs. In this work, two-dimensional topology optimization on tessellations is investigated as a means to coalesce art and engineering. M.C. Escher's tessellations using recognizable figures are mainly utilized. The aforementioned Mimetic Finite Difference-inspired approach (VEM) facilitates accurate numerical analysis on any non self-intersecting closed polygons such as tessellations. |
