Analysis and shape optimization in incompressible flows with unstructured grids

Abstract

The aim of this study is to develop and validate numerical methods that perform shape optimization in incompressible flows using unstructured meshes. The three-dimensional Euler equations for compressible flow are modified using the idea of artificial compressibility and discretized on unstructured tetrahedral grids to provide estimates of pressure distributions for aerodynamic configurations. Convergence acceleration techniques like multigrid and residual averaging are used along with parallel computing platforms to enable these simulations to be performed in a few minutes. This computational frame-work is used to analyze sail geometries. The adjoint equations corresponding to the “incompressible” field equations are derived along with the functional form of gradients. The evaluation of the gradients is reduced to an integral around the boundary to circumvent hurdles posed by adjoint-based gradient evaluations on unstructured meshes. The reduced gradient evaluations provide major computational savings for unstructured grids and its accuracy and use for canonical and industrial problems is a major contribution of this study. The design process is driven by a steepest-descent algorithm with a fixed step-size. The feasibility of the design process is demonstrated for three inverse design problems, two canonical problems and one industrial problem.