Optimal Design by Adaptive Mesh Refinement on Shape Optimization of Flow Fields Considering Proper Orthogonal Decomposition

This paper presents optimal design using Adaptive Mesh Refinement (AMR) with shape optimization method. The method suppresses time periodic flows driven only by the non-stationary boundary condition at a sufficiently low Reynolds number using Snapshot Proper Orthogonal Decomposition (Snapshot POD). For shape optimization, the eigenvalue in Snapshot POD is defined as a cost function. The main problems are non-stationary Navier– Stokes problems and eigenvalue problems of POD. An objective functional is described using Lagrange multipliers and finite element method. Two-dimensional cavity flow with a disk-shaped isolated body is adopted. The non-stationary boundary condition is defined on the top boundary and non-slip boundary condition respectively for the side and bottom boundaries and for the disk boundary. For numerical demonstration, the disk boundary is used as the design boundary. Using H gradient method for domain deformation, all triangles over a mesh are deformed as the cost function decreases. To avoid decreasing the numerical accuracy based on squeezing triangles, AMR is applied throughout the shape optimization process to maintain numerical accuracy equal to that of a mesh in the initial domain. The combination of eigenvalues that can best suppress the time periodic flow is investigated.