Adjoint-based optimization for non-linear inverse problems with high-order discretization of the compressible RANS equations

Abstract

This work presents an adjoint-based strategy to solve non-linear inverse problems discretized with high-order numerical methods. The inverse problem is defined here based on the optimization of a control parameter to minimize a cost-functional subject to the compressible RANS equations discretized with the modal discontinuous Galerkin (DG) method. The distributed control parameter is searched in the DG function space and the discrete adjoint approach, consistent with the formal problem, is used to compute the derivative of the cost function in the optimization process. The linearization of the cost-functional and of the governing equations, the expression of the gradient, as well as the numerical strategy to efficiently solve the adjoint system with flexible inner-outer GMRES solvers have been detailed. In the case of a strongly under-determined problem, regularization techniques based on the penalization of the norm of the control parameter have been introduced. The methodology is illustrated on the case of a data-assimilation (DA) problem, which aims at minimizing the discrepancy of (sparse) high-fidelity measurements with the solution of the RANS equations corrected by four different control parameters. The optimization strategy is tested progressively with measurements on the full computational domain (abundant measurements) and solid wall boundaries (sparse measurements). First, a laminar flow around a cylinder is used to validate the inverse problem resolution with a DG discretization of different approximation orders. Subsequently, results regarding a turbulent flow around a square cylinder allow to compare the optimization convergence of each corrective parameters with abundant measurements. Finally, a shock-wave/turbulent boundary-layer interaction configuration is considered. Great correction of the velocity field is obtained with one of the proposed corrective term. In the case of abundant measurements it is also possible to get accurate correction of wall variables such as the skin-friction and pressure coefficient. Regularization of the optimal space, in case of sparse measurements, is attempt through penalization techniques.