Title: Finite Element Basis Compression for Extreme-Scale Simulations and R-Adaptive Mesh Optimization to Enable Compression

Abstract: To push finite element method (FEM) applications to extreme scales, modern approaches such as sum factorization and hierarchical bases combat scaling with respect to dimension and polynomial degree. This enables more efficient simulation of higher polynomial degree and higher dimension simulations, but does not improve the scaling with respect to the number of mesh cells. To this end, we develop an approach for exploiting redundancy in mesh cell shapes by compressing basis evaluations through a library-based compression scheme based on shape, which may be performed as a mesh preprocessing step. This approach identifies unique cell shapes through the reference to physical mapping Jacobian and performs a single set of basis evaluations for every unique shape, resulting in an outer-rank tensor decomposition on the FEM basis data tensor. As this depends on the reference to physical mapping Jacobian, it also respects the discretization regardless of finite element space. We show this approach results in both reduced memory and increased speed for extreme-scale FEM simulations, including the ability to simulate 1 billion cells in under 100GB of application memory in the open-source Trilinos-based code MrHyDE. Additionally, we present an r-adaptive mesh optimization approach to enhance the structure in a given mesh to enable such compression. This approach utilizes data-driven shape clustering combined with an augmented Lagrangian sequential quadratic programming method to move mesh nodes and increase the compression that may be obtained for a given mesh. We show examples for several meshes and discuss future directions of interest.

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