An Output-Based Dynamic Order Refinement Strategy for Unsteady Aerodynamics

Abstract

An output-based dynamic order refinement strategy is presented for unsteady simulations using the discontinuous Galerkin finite element method in space and time. A discrete unsteady adjoint solution provides scalar output error estimates and drives adaptive refinement of the space-time mesh. Space-time anisotropy is measured using projection of the adjoint onto semi-refined spaces and is used to allocate degrees of freedom to additional time slabs or increased spatial order of individual space-time elements. The spatial refinement is dynamic in that the solution approximation order can change between time slabs. Results for the compressible Euler equations demonstrate benefits of the dynamic order refinement strategy in terms of total degrees of freedom: at strict error tolerances, the savings approaches two orders of magnitude compared to uniform refinement, and a factor of two to three compared to output-based refinement with a static spatial mesh.

Output Error Estimation for Chaotic Flows

“…Over the last few years, several groups have made progress on various fronts in tackling unsteady problems, including: temporal-only error estimation and adaptation; 9, 10 spatial-only error estimation and adaptation; 11,15,16 combined temporal and spatial mesh refinement with a static geometry and mesh; 13, 17 combined temporal and dynamic spatial refinement on static geometries; 14,18,19 combined temporal and dynamic-order spatial refinement on deformable domains. [20][21][22] In our previous work we have employed space-time discontinuous Galerkin (DG) and hybridized discontinuous Galerkin (HDG) 23-28 finite element discretizations using time slabs and an approximate space-time solver.…”

An Output-Based Adaptive Hybridized Discontinuous Galerkin Method on Deforming Domains

“…[20][21][22] Unsteady problems pose additional challenges and computational costs, namely in the unsteady adjoint solution, yet output-based adaptive methods have also been explored, with various modes of adaptation, including static-mesh, dynamic-mesh, space-only, and combined space-time. [23][24][25][26][27][28][29][30][31] In this work, we extend unsteady output-based adaptation techniques to high-order compressible Navier-Stokes simulations on deforming domains, discretized with HDG. We build on our previous work with DG, 31 and we note requisite discretization modifications for HDG in an ALE formulation.…”

Output-based space–time mesh optimization for unsteady flows using continuous-in-time adjoints

“…For nonlinear problems, the requisite linearizations also require the state at each time, and this is typically saved to disk at a cost of storage and computational time. In spite of these costs, output-based adaptive methods have been extended to unsteady problems, with various adaptation mechanics, including static-mesh, dynamic-mesh, space-only, and combined space-time refinement [9,10,11,12,13,14,15,16,17,18].…”

“…In this case, the adjoint solution, which is at the core of output-based methods, can be solved using a discrete approach in which the adjoint equations are obtained systematically from the primal discrete system by transposing the operator. This is the approach taken in many previous works [9,10,16,13,17,18]. While the same approach can be applied to non-variational discretizations, it has pitfalls as the relationship between the resulting discrete adjoint coefficients and the underlying continuous adjoint solution may not be clear, making difficult the computation of the adjoint-weighted residual error estimate and the application of approximate adjoint solvers that rely on smoothness of the adjoint solution.…”