| Topology optimization is a tool for finding the best solutions to engineering design problems. Such solutions meet performance specifications while minimizing cost, weight, and/or selected responses and thus potentially offer tremendous benefits. Topology optimization has been used to determine the distribution of materials in beams and mechanisms, and to design the microstructure of periodic materials to attain, for example, extreme elastic properties.; The goal of this work is to extend topology optimization to the design of periodic materials with maximized stiffness and permeability. To achieve this, methodologies are proposed for circumventing numerical instabilities and difficulties in stiffness and fluid transport optimization. In particular, mesh dependence and checkerboard patterns in stiffness problems are overcome by imposing a minimum length scale on structural members. The proposed methodology implements nodal design variables that are projected onto element space via a regularized Heaviside function. This technique is shown to yield nearly 0--1 (void-solid) solutions that meet the length scale criterion. This methodology is also combined with established numerical homogenization techniques to design one-length-scale materials with extreme elastic properties. For the maximum fluid transport problem, the binary moving-boundary no-slip condition along the solid-fluid interface is regularized with a new Darcy-Stokes finite element. The element is scaled so that fluid flow through voids and solids is governed by Stokes flow and Darcy flow, respectively. When a low permeability material is used, the technique successfully simulates the no-slip condition and creates nearly 0--1 optimal topologies. It is also applied to the design of periodic materials, where a numerical implementation of homogenization theory is proposed and an inverse homogenization problem for designing a maximum permeability material is solved. The optimal design is found to be a solution that minimizes the fluid-structure interface.; With numerical difficulties overcome and the inverse homogenization formulation for fluids developed, the modules are combined to design a multifunctional material optimized for both effective stiffness and permeability. These properties are competing and consequently the final design is dependent on the relative importance assigned by the designer to the respective terms in the objective function. The designer selects these values according to the materials' intended use, thereby tailoring the microstructure for its specific application. |
