Efficient Assembly for High-Order Unstructured FEM Meshes (FPL 2015)

Abstract

The Finite Element Method (FEM) is a common numerical technique used for solving Partial Differential Equations on large and unstructured domain geometries. Numerical methods for FEM typically use algorithms and data structures which exhibit an unstructured memory access pattern. This makes acceleration of FEM on Field-Programmable Gate Arrays using an efficient, deeply pipelined architecture particularly challenging. In this work, we focus on implementing and optimising a vector assembly operation which, in the context of FEM, induces the unstructured memory access. We propose a dataflow architecture, graph-based theoretical model, and design flow for optimising the assembly operation for spectral/hp finite element method on reconfigurable accelerators. We evaluate the proposed approach on two benchmark meshes and show that the graph-theoretic method of generating a static data access schedule results in a significant improvement in resource utilisation compared to prior work. This enables supporting larger FEM meshes on FPGA than previously possible.