A Product to Sum Approach for Matrix Filling in a Hierarchical Finite-Element Method

TL;DR

This paper introduces a product-to-sum method that accelerates matrix filling in hierarchical finite-element methods by reducing multidimensional integrations to summations, especially effective with rapidly varying coupling factors.

Contribution

It proposes a novel product-to-sum approach that significantly decreases the computational effort in matrix assembly for inhomogeneous FEM elements.

Findings

Reduces multidimensional integrations to summations

Improves efficiency for inhomogeneous elements with rapid coupling variations

Enables faster higher-order FEM computations

Abstract

This paper presents a product to sum approach for a fast and efficient matrix filling in a hierarchical finite-element method (FEM). Due to the existence of a coupling factor arising from the material and Jacobian inhomogeneities in curved inhomogeneous elements, the calculation of the FEM matrix elements should be carried out through full multidimensional integrations. This reduces the efficiency of the higher order FEM solvers especially when the coupling factor varies rapidly inside the elements. In the product to sum approach, every product of the basis and weighting polynomials is replaced with a sum of appropriate polynomials. This reduces the number of required multidimensional integrals significantly and converts the integration into a summation. Therefore, the method will be efficient if the number of summation terms in the product to sum conversion is as small as possible.