Alleviating Adjoint Solver Robustness Issues Via Mimetic Cfd Discretization Schemes

Abstract

Adjoint methods allow to compute the gradient of a cost function at an expense that is independent of the number of design variables, thus representing an appealing tool to those who deal with large-scale gradient-based optimization processes. However, researchers in the field are currently facing difficulties related to poor convergence and high storage memory requirements. Both continuous and discrete adjoints require a fully converged solution to the primal problem. It is argued that, specifically in Computational Fluid Dynamics, the presence of numerical adjustments (e.g. non-orthogonal correctors) and segregated solution algorithms yields a solution that, while acceptable as a mere aerodynamics study, is not accurate enough to produce a robust adjoint system. In the past decade some new PDE discretization schemes have emerged aiming to remove some constraints imposed by classical finite volumes. The main goal is to allow for more freedom in both the physical model (e.g. strong anisotropy in a material property) and its corresponding numerical model (e.g. the possibility to use polyhedral meshes with strongly non-orthogonal and/or non-convex cells), while improving solution accuracy and convergence properties at the same time. One of such new schemes is known as Mimetic Finite Differences; the present work introduces a specific implementation of it and extends the scheme to cater for convection-diffusion reaction problems. Preliminary numerical results are also included.