Level set topology optimization of scalar transport problems

This paper studies level set topology optimization of scalar transport problems, modeled by an advection-diffusion equation. Examples of such problems include the transport of energy or mass in a fluid. The geometry is defined via a level set method (LSM). The flow field is predicted by a hydrodynamic Boltzmann transport model and the scalar transport by a standard advection-diffusion model. Both models are discretized by the extended Finite Element Method (XFEM). The hydrodynamic Boltzmann equation is well suited for the XFEM as it allows for convenient enforcement of boundary conditions along immersed boundaries. In contrast, Navier Stokes models require more complex approaches to impose Dirichlet boundary conditions, such as stabilized Lagrange multiplier and Nitsche methods. The combination of the LSM and the XFEM is an alternative to density-based topology optimization methods which have been applied previously to scalar transport problems. Density methods often suffer from a fuzzy description of boundaries, spurious diffusion through "void" regions, and the presence of fictitious material in the optimized design. This paper illustrates that the LSM/XFEM approach addresses these three concerns. The proposed approach is studied with two dimensional problems at steady state conditions. Both "fluid-void" and "fluid-solid" optimization problems are considered. For the "fluid-void" case, optimization results are obtained without spurious diffusion through "void" regions. For the "fluid-solid" case, the analysis recovers strong gradients of the flow and scalar fields at the fluid-solid interface, using moderately refined meshes.