Topology Optimization of Transport Problems Modeled by the Boltzmann Equation

Design optimization provides a mechanism to create novel and non-intuitive optimal designs in a formal and mathematical process. The current paradigm for design optimization is to relax the discrete description of a material layout and vary continuously the density of a fictitious porous material. This dissertation builds on a new design optimization paradigm using the level set method (LSM) and the extended finite element method (XFEM). The LSM and the XFEM allow crisp descriptions of the material layout and address numerical artifacts typically seen in density methods. The LSM and XFEM approach is applied to structural problems and fluid-thermal transport problems. Fluid flow is predicted by the Boltzmann equation, which has a simpler numerical formulation than the Navier-Stokes equations and is valid in a larger flow regime. The most popular approach is the lattice Boltzmann method (LBM). This dissertation initially explores the current LBM and density approach to optimization. The theoretical basis to allow LSM and XFEM topology optimization with an alternative to the LBM is presented including a streamline upwind Petrov-Galerkin finite element formulation for the Boltzmann equation and consistent material interpolation in the XFEM. Finally, the finite element hydrodynamic Boltzmann approach is explored for topology optimization of transport problems. The finite element Boltzmann approach is found to address several numerical issues with the LBM at the cost of solving a sparse linear system of equations. The LSM and XFEM approach does not suffer from the numerical artifacts seen in the LBM and density approach. However, a robust regularization strategy has not been developed