Hybrid Octree/Quadtree AMR for Anisotropic Domains Tobin Isaac Institute for Computational Engineering & Sciences The University of Texas at Austin SIAM PP 2014 Portland, Oregon February 19, 2014 T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 1 / 23
Thin-domain Problems 1 2 Hybrid Octree/Quadtree AMR Example 3 My collaborators on this work are Carsten Burstedde and Omar Ghattas. These slides can be found at www.ices.utexas.edu/~tisaac/slides. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 2 / 23
Three-dimensional models in thin domains are common Ocean [Kanarska et al., 2007] Subsurface [Maxwell et al., 2008] Ice [Isaac et al.] Models with 3D implicit subsystems (e.g. pressure Poisson solves) are becoming more common. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 3 / 23
Many of these problems benefit from local mesh refinement They benefit from local horizontal mesh refinement [Durand et al., 2009] Ice-shelf/ice-sheet interface location, with AMR (black) and fixed resolutions (gray). T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 4 / 23
Many of these problems benefit from local mesh refinement They benefit from local vertical mesh refinement Boundary singularities, e.g. Dirichlet-Neumann transitions. Spatially varying boundary layer requirements, e.g. in ocean modeling, can be treated with AMR instead of/in addition to hybrid vertical coordinates [Pain et al., 2005]. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 5 / 23
Isotropic mesh refinement has drawbacks The aspect ratio of the discretization is fixed It cannot adaptively match the length scales of the solution: affects accuracy; affects numerics, e.g. smoother convergence rate. Forcing horizontal refinement when only vertical refinement is desired (and vice versa) increases the mesh size by much more than necessary: e.g. by a factor of 4 for octrees (8 new octants vs. 2 new layers). Limits available numerical methods, e.g. geometric multigrid with semi-coarsening. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 6 / 23
Isotropic mesh refinement has drawbacks There is no natural support for column integrity Many numerical schemes require degrees of freedom (dofs) to be organized into columns: for solver convergence, e.g. line-smoothers; for high-throughput computing. Distributing a mesh based on graph-partitioning or a space-filling curve can split a column of dofs between subproblems: leads to less-compact subdomains, worse convergence of domain decomposition methods [Farhat et al., 1994]. Ω 1 vs. Ω 0 Ω 1 Ω 0 T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 7 / 23
Fully anisotropic AMR has drawbacks Unstructured anisotropic AMR [Loseille et al., 2010] Necessary for some problems (e.g. supersonic flows), but metric-tensor based adaptivity is complex and is less scalable than structured or tree-based AMR. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 8 / 23
Fully anisotropic AMR has drawbacks Hierarchical anisotropic AMR [Williamschen and Groth, 2013] More complex than isotropic counterpart, cannot adapt to arbitrary target metric tensors without mesh movement. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 9 / 23
The hybrid AMR scheme partition 0 partition 1 A p4est forest-of-quadtrees to manage columns, with each column stored as a flat, linear binary tree of layers, which guarantees column integrity. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 10 / 23
Some good qualities of isotropic AMR are inherited Well-shaped partitions quickly generated by a space filling curve, using weighted partitioning algorithm already present in p4est [Burstedde et al., 2011]. In-place, single-sweep vertical refinement and vertical coarsening algorithms. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 11 / 23
Some good qualities of isotropic AMR are lost Partitioning has coarser granularity Load-balancing can be poor, but only if there are columns with N column � N total / P layers. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 12 / 23
Some good qualities of isotropic AMR are lost Horizontal adaptivity is less flexible Individual layers cannot be horizontally refined: the mesh size grows quickly with horizontal refinement. needs refinement refinement unnecessary Coarsened columns are combined into least-common-ancestor leaves. level 1 level 2 level 3 T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 13 / 23
Mesh quality conditions Neighboring refinement level is the only free quality condition in isotropic hierarchical AMR, hence the emphasis on 2:1 condition algorithms [Isaac et al., 2012]. In hybrid AMR, the layer aspect ratio is also free. Maximum and minimum ratios can be enforced at the same time as a 2:1 condition: a single mesh conditioning algorithm. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 14 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem fine are wrong C 0 finite elements with limited hanging-node stencils The 2:1 condition is satisfied in terms of adjacency, but there is a hanging node ( � ) that depends on another hanging node ( � ). T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 15 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem fine are wrong C 0 finite elements with limited hanging-node stencils The 2:1 condition is satisfied in terms of adjacency, but there is a hanging node ( � ) that depends on another hanging node ( � ). T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 15 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem fine are wrong C 0 finite elements with limited hanging-node stencils The 2:1 condition is satisfied in terms of adjacency, but there is a hanging node ( � ) that depends on another hanging node ( � ). T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 15 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem wrong are fine Cross-refined neighbors There is no parent/master face, but C 0 hanging nodes can be treated as if there were, and face integrals can be calculated as if there were four smaller faces, all using 2:1 interpolation rules. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 16 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem wrong are fine Cross-refined neighbors There is no parent/master face, but C 0 hanging nodes can be treated as if there were, and face integrals can be calculated as if there were four smaller faces, all using 2:1 interpolation rules. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 16 / 23
What are 2:1 conditions for hybrid AMR? Some situations that seem wrong are fine Cross-refined neighbors There is no parent/master face, but C 0 hanging nodes can be treated as if there were, and face integrals can be calculated as if there were four smaller faces, all using 2:1 interpolation rules. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 16 / 23
The full mesh conditioning algorithm Mesh conditioning (‘‘balance’’) 1 Enforce maximum flatness W / H by refining columns. 2 Enforce 2:1 between columns. This does not increase flatness. 3 Enforce minimum flatness by refining layers. 4 Enforce 2:1 between layers. This does not decrease flatness, and does not increase maximum flatness. The quadtree/octree ‘‘insulation layer’’ property does not apply. Iterative or all-to-all communication is required in parallel. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 17 / 23
The full mesh conditioning algorithm Mesh conditioning (‘‘balance’’) 1 Enforce maximum flatness W / H by refining columns. 2 Enforce 2:1 between columns. This does not increase flatness. 3 Enforce minimum flatness by refining layers. 4 Enforce 2:1 between layers. This does not decrease flatness, and does not increase maximum flatness. The quadtree/octree ‘‘insulation layer’’ property does not apply. Iterative or all-to-all communication is required in parallel. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 17 / 23
The full mesh conditioning algorithm Mesh conditioning (‘‘balance’’) 1 Enforce maximum flatness W / H by refining columns. 2 Enforce 2:1 between columns. This does not increase flatness. 3 Enforce minimum flatness by refining layers. 4 Enforce 2:1 between layers. This does not decrease flatness, and does not increase maximum flatness. The quadtree/octree ‘‘insulation layer’’ property does not apply. Iterative or all-to-all communication is required in parallel. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 17 / 23
The full mesh conditioning algorithm Mesh conditioning (‘‘balance’’) 1 Enforce maximum flatness W / H by refining columns. 2 Enforce 2:1 between columns. This does not increase flatness. 3 Enforce minimum flatness by refining layers. 4 Enforce 2:1 between layers. This does not decrease flatness, and does not increase maximum flatness. The quadtree/octree ‘‘insulation layer’’ property does not apply. Iterative or all-to-all communication is required in parallel. T. Isaac (ICES, UT Austin) p6est SIAM PP 2014 17 / 23
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