Output-based mesh adaptation using geometric multi-grid for error estimation

Abstract

The adjoint method in computational fluid dynamics (CFD) made shape optimisation affordable. However, the typical cost of the process is still at least an order of magnitude higher than obtaining a ow solution only. In this work, the author presents methods that help to further reduce the computational effort in optimisation. The fi rst method involves reducing the run-time of the flow solver; the second involves developing a low-cost error estimate that could be used to create a computationally less expensive grid without affecting the accuracy of an objective function. Implicit solvers are well-established in CFD, but their performance is often limited by the instabilities that arise in the initial convergence stage of the code. To address this issue, a methodology to stabilise an implicit solver using adaptive CFL number adjustment technique is implemented in the in-house code STAMPS. The CFL number is altered at each solver iteration based on the outcome of a line-search algorithm - the Armijo rule. It is shown that the building blocks of a line-search algorithm can be accurately and easily evaluated using automatic differentiation of the Tapenade source code transformation tool without a need to approximate derivatives of discrete system of ow equations. The line-search algorithm is also used to control re-evaluation of Jacobian/preconditioner between solver iterations, by detecting when the linear convergence regime was reached, and the spectra of system matrix eigenvalues are contractive. This work shows that the proposed combination of automatic CFL adjustment and system matrix re-evaluation control result in improvements in solver stability and reductions of the overall run-time of the code. A method of manufactured solution is used by the author for veri fication of the discretisation accuracy of the STAMPS solver, as well as for the development of local error estimation. The truncation error, which is defi ned as a difference between the continuous PDEs and its discrete approximation, can be evaluated exactly using a known manufactured solution and used for verifi cation of error estimation methodology. In this work, a novel low-cost method is presented that estimates the truncation error using building blocks of the geometric multi-grid solver. The methodology requires little implementation effort and uses the same set of multi-grid meshes as the solver. It is shown that a reasonable indication of high-error regions can be achieved, even though the coarse and fine meshes are topologically inconsistent. Although the truncation error can be directly used to obtain an adaptation sensor it is benefi cial to apply adjoint-weighting beforehand. The adjoint-weighting of the local truncation error gives an outputbased sensor that determines the effect of the local error on the objective function of interest. The output-based sensor can be effectively used for the goal-driven mesh adaptation/coarsening process. This work presents example applications of mesh refi nement driven by output-based sensor and mesh regeneration technique.