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Ph.D. Theses

Adaptive Local Refinement with Octree Load-Balancing for the Parallel Solution of Three-Dimensional

By Raymond M. Loy
Advisor: Joseph E. Flaherty
May 4, 1998

An adaptive technique for a partial differential system automatically adjusts a computational mesh or varies the order of a numerical procedure to obtain a solution satisfying prescribed accuracy criteria in an optimal fashion. Conservation laws are solved by a discontinuous Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. The Courant stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalances, introduced at adaptive enrichment steps, are corrected by using traversals of an octree representing a spatial decomposition of the domain. The octree structure facilitates a rapid load-balancing procedure by performing tree traversals that (i) appraise subtree costs and (ii) partition spatial regions accordingly. This partitioning strategy is applicable to three-dimensional octree-structured meshes as well as three-dimensional meshes generated by other means. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octree and inertial partitioning procedures are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.

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