Recent Advances in Agglomerated Multigrid

Abstract

We report recent advancements of the agglomerated multigrid methodology for complex flow simulations on fully unstructured grids. An agglomerated multigrid solver is applied to a wide range of test problems from simple two-dimensional geometries to realistic threedimensional configurations. The solver is evaluated against a single-grid solver and, in some cases, against a structured-grid multigrid solver. Grid and solver issues are identified and overcome, leading to significant improvements over single-grid solvers. I. Nomenclature R Global residual vector II. Introduction Multigrid techniques1, 2 are used to accelerate convergence of current Reynolds-Averaged Navier-Stokes (RANS) solvers for both steady and unsteady flow solutions, particularly for structured-grid applications. Lallemand et al. 3 and Smith 4 introduced Agglomerated Multigrid (AgMG) methods for unstructured-grid applications, and Mavriplis et al. [5][6][7][8] developed AgMG methods for large-scale unstructured-grid applications. The AgMG methods have been implemented in several practical unstructured grid codes and applied to configurations used for the Drag and High Lift Prediction Workshops.9, 10 The AgMG methods accelerated convergence in many instances but problems were encountered, such as grid-dependent convergence to smaller residual levels, discrepancies between single-grid and multigrid solutions, and convergence problems for cases at the edges of the flight envelope. In fact, analysis and systematic computations with these techniques showed convergence degradation on highly-refined grids even for simple model problems. To overcome the difficulty, we critically studied AgMG techniques 11, 12 for isotropic and highly-stretched grids, and developed quantitative analysis methods and computational techniques to achieve fast grid-independent convergence for a model equation representing laminar diffusion in the incompressible limit. It was found that consistent coarse-grid discretizations are essential for * Senior Research Scientist (hiro@nianet.org), National Institute of Aerospace, 100 Exploration Way, Hampton grid-independent multigrid convergence on isotropic grids, 11 and that prismatic cells, line-agglomeration, and line-implicit subiterations are essential for grid-independent multigrid convergence on highly-stretched grids. 12Building upon these fundamental studies, we extended and demonstrated these techniques for inviscid, viscous, and turbulent flow equations over complex geometries using a serial code 13 and a parallel code. 14The developed AgMG method has been shown to provide a significant speedup over a state-of-the-art singlegrid (SG) solver for the complex flows considered in the previous papers. However, several difficulties were encountered when the AgMG method was applied to some other problems. For example, multigrid convergence degraded in the presence of nearly-degenerate stencils on surface grids, grids with a polar singularity, disjoint grids in grid partitioning, etc. In this paper, we probe these and other issue...

Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030

“…Incipient or large scale unsteadiness may require the use of time-accurate simulation or continuation to obtain the desired solution when multiple solutions or hysteresis are observed. 15 For example, Nishikawa et al 84 demonstrate this behavior for the Drag Prediction Workshop configuration in Fig. 7.…”

Comparison of Different Spatial/Angular Agglomeration Multigrid Schemes for Radiative Heat Transfer Computations

“…As mentioned in Introduction, the isotropic agglomeration strategy appears to be less effective in test cases involving hybrid grids, with highly stretched elements (prisms or hexahedra) at boundary surfaces [1,4,5,[14][15][16][17]; the latter are employed in order increased accuracy to be obtained at those areas [18]. Thus, a full-coarsening directional agglomeration procedure was also incorporated to the proposed algorithm, which is based on the methodology of Nishikawa and Diskin [5].…”

“…The first number in parentheses denotes the number of relaxations performed prior to restriction, while the second denotes the corresponding number after the prolongation [1,3]. Depending mainly on the implemented iterative procedure, as well as on the encountered problem's nature, various V-cycle types have been reported in the open literature as the most appropriate for them [1,3,5,[14][15][16][17]. For example, if an explicit iterative scheme is applied for an inviscid compressible fluid flow simulation a V(1,0)-cycle is usually preferred, while in case of a turbulent flow problem along with an implicit method, a V(2,1) appears to be the commonly preferred choice [1,3].…”

“…As a result, the grid anisotropy, caused by the utilization of prismatic or hexahedral elements, is moderated [15]. Another popular approach is the fullcoarsening directional agglomeration [1,4,16,17], according to which the boundary control cells are fused initially, while a line-agglomeration step is then performed, for merging the prismatic or hexahedral control volumes along implicit lines, directly above the corresponding already agglomerated boundary ones. As a result, a deeper reduction of DoFs (Degrees of Freedom) is succeeded and consequently further acceleration is gained, without though the produced discretizations to differentiate significantly from the initial topology.…”

CFL3D, FUN3D, and NSU3D Contributions to the Fifth Drag Prediction Workshop

“…It is computed with a first-order Frechet derivative to enhance the standard defect correction approach. 49 The solution of this exact linearization is also wrapped with GCR for stabilization and convergence acceleration. The CFL specification for the nonlinear psudo-time term and the preconditioner can be adjusted with an adaptive approach.…”