Topology optimization of rarefied gas flows using an adjoint discrete velocity method

In this paper, a topology optimization method for rarefied gas flows is proposed. We use the discrete velocity method (DVM) to solve the Boltzmann equation, and construct an extension of the standard DVM algorithm in order to model gas flow in a domain of mixed materials. The proposed extension consists of a correction in convection fluxes as well as an extra source term which mimics reflection boundary conditions, so that the presence of fluid or solid can be characterized by a pseudo density. To evaluate the change in objective functional due to the design variable, an adjoint problem is derived from the steady state condition of the extended DVM, which forms a linear system of equations that can be solved using iterative methods like in standard DVM. Using the design sensitivity obtained from the adjoint system, optimization of flow structures can be achieved. Validity of the proposed method is demonstrated through several numerical examples.