Levelset-XFEM Topology Optimization with Applications to Convective Heat Transfer

Computational design optimization is a technique that provides designers with automated approaches to developing novel and non-intuitive optimal designs. Topology optimization is a subset of design optimization that seeks to determine the optimal geometry allowing for topologic changes during the design process. The thesis focuses on the design of devices whose performance is dominated by convective heat transfer. Convective heat transfer is a process that results from the coupling between thermal fields and fluid motion. Frequently benefitting from complicated geometries, convective design problems are an ideal case for computational design optimization. Commonly used simple engineering models of convection like Newton\u27s Law of Cooling rely on design dependent boundary conditions that may lie along immersed design edges. These boundary conditions are difficult to represent accurately with traditional density approaches for topology optimization. In this thesis Level Set Method (LSM) and the eXtended Finite Element Methods (XFEM) are developed to handle convective design problems to ensure crisp resolution of design boundaries for accurate physical modeling. The LSM is used to provide a precise definition of geometric boundaries. Here the explicit LSM is used, which updates the parameterized Level Set Field (LSF) via Nonlinear Programming methods (NLP). The XFEM is incorporated to provide for crisp resolution of the LSM geometry within the discretization of the governing equations. With accurate resolution of simplified convection boundary conditions, complicated, potentially unphysical geometries are developed. To overcome this issue this thesis develops new regularization approaches for explicit LSMs. To enforce a minimum feature size a new measure is developed that identifies violations of the minimum feature size. To demonstrate the applicability of the LSXFEM approach we study more complicated, coupled problems where the fluid motion is driven by buoyancy forces. The natural convection model is applied to both 2D and 3D steady-state design problems and 2D transient problems