Development and Application of Agglomerated Multigrid Methods for Complex Geometries

Abstract

We report progress in the development of agglomerated multigrid techniques for fully unstructured grids in three dimensions, building upon two previous studies focused on efficiently solving a model diffusion equation. We demonstrate a robust fully-coarsened agglomerated multigrid technique for 3D complex geometries, incorporating the following key developments: consistent and stable coarse-grid discretizations, a hierarchical agglomeration scheme, and line-agglomeration/relaxation using prismatic-cell discretizations in the highly-stretched grid regions. A significant speed-up in computer time is demonstrated for a model diffusion problem, the Euler equations, and the Reynolds-averaged Navier-Stokes equations for 3D realistic complex geometries.

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].…”

“…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]. Further modifications have been included though; the agglomeration process isn't limited across the sub-domains' internal boundaries, allowing virtual ghost supernodes to be generated, while an additional limitation prohibits the fusion process of the control cells at prismatic layers in order the topology of the initial grid to be preserved more accurately [1].…”

“…Further modifications have been included though; the agglomeration process isn't limited across the sub-domains' internal boundaries, allowing virtual ghost supernodes to be generated, while an additional limitation prohibits the fusion process of the control cells at prismatic layers in order the topology of the initial grid to be preserved more accurately [1]. Similarly to isotropic agglomeration, the directional one begins with the construction of the initial seed list, containing though the boundary nodes of only the prismatic elements [1,5]. Since they are examined and merged in supernodes, a new list is created, filled with the nodes of the next prismatic layer; the latter are fused according to the agglomeration path of their corresponding boundary nodes, constructing the so-called implicit lines [1,5].…”

“…of incompressible fluid flow problems in CFD (Computational Fluid Dynamics). Since then it has been applied though to various types of computational simulations, e.g., compressible fluid flow [3][4][5], radiative heat transfer [6][7][8], etc. Its main concept depends on the construction of coarser resolutions, in order the lowfrequency errors at the finest discretization to become high-frequency ones at the coarser ones, and as such to be damped in a more efficient way [1,3,6].…”

“…As far as the different V-cycle types are concerned the V(1,0), V(1,1), V(2,0) and V(2,1) are assessed; the first number in parentheses denotes the number of relaxations performed prior to restriction, while the second one denotes the corresponding number of relaxations after the prolongation [6]. For simulations involving hybrid grids, in order increased accuracy to be achieved in regions with prismatic elements [18], a fullcoarsening directional agglomeration strategy is additionally developed [1,4,5]. It succeeds greater reduction of DoFs and consequently greater acceleration, comparing to this obtained with isotropic agglomeration methodology.…”