Multigrid and Mesh Refinement Methods

PhD thesisGradient-based aerodynamic shape optimisation using adjoint methods have received major interest in the engineering community, as cost of computing CFD output gradients becomes almost independent of the number of design variables. However, slow convergence of well-established RANS implicit solvers on fixed high resolution grids is a major drawback. Cost reductions can be made by using multigrid, which employs a sequence of coarser grids where slowly convergent low frequency errors are quickly reduced. Further savings can be made by adaptively refining grid regions to achieve superior accuracy and convergence of flow and adjoint solver compared to fixed high resolution grids. Challenges of combining adaptive mesh refinement and multigrid for production-level RANS problems are primarily threefold: (i) The sequential anisotropic coarsening and refinement algorithms advised for better capture of boundary layer flows are expensive and can quickly become a bottleneck with increasing mesh sizes. (ii) Available Goal-oriented adaptation sensors, which predict the refinement field, are either expensive or fail to address viscous effects and or grid anisotropy. Finally, (iii) mesh adaptation introduces refined and unrefined regions in the grid. When highly stretched grids typical of turbulent flows are involved, triangulation between different grid resolutions produces stretched cells with poor orthogonality, which stalls convergence on adapted grids. These challenges are addressed in this work to enable the application of a combined Adaptive Mesh Refinement and Multigrid (AMR-MG) strategy for production-scale problems. To address (i), the author revisits multigrid strategy to better harness the strength of implicit solvers. As presented in this work, only one coarse level for an implicit solver is often sufficient for production-scale grids contrary to explicit solvers, which require several grid levels for multigrid to be effective. Furthermore, robust multigrid performance using isotropic coarse grids with standard 1:2 and aggressive 1:4 coarsening ratios for turbulent turbomachinery flows is reported here. Hence, a novel combined AMR-MG strategy employing one level of isotropically coarsened and adapted grid is proposed. This is demonstrated by isotropically coarsening a hexahedral initial grid, typical in turbomachinery, using a topology preserving framework to obtain a multigrid sequence. Then the adapted grid is generated by isotropic element subdivision. The existing coarse grid is also efficiently linked with the adapted grid to accelerate convergence. Secondly (ii), an affordable goal oriented edge-based sensor is proposed. Sensor field is obtained by efficiently employing reverse mode differentiation to estimate the impact of edge contraction fields on the goal function. The sensor is unique as its edge-based formulation addresses grid anisotropy, while capturing the viscous effects in turbulent flow problems. Comparable performances have been demonstrated when used to drive adaptation of transonic and supersonic inviscid NACA0012 simulations. Reasonable refinement field for turbulent turbomachinery flows has also been demonstrated. Finally, (iii) is addressed by improving the orthogonality of finite volume dual mesh by employing a 3D Containment-Dual algorithm. The approach is unique as it enables standard solvers, which require triangulation between refinement levels, to generate improved dual volume configurations similar to that of a non-triangulated refined grid