On Semi-Analytical Integration Specified for Mass Matrix of Finite Elements

TL;DR

This paper introduces a semi-analytical integration method for finite element mass matrices that improves accuracy and efficiency over traditional quadrature, especially on coarse meshes, for eight-node brick elements.

Contribution

It presents a novel semi-analytical approach for mass matrix evaluation, providing explicit formulas that are easy to implement and more accurate than standard quadrature methods.

Findings

Achieves comparable computational efficiency to traditional schemes.

Provides explicit one- and four-point formulas for brick elements.

Demonstrates superior accuracy on coarse meshes.

Abstract

Spatial numerical integration is essential for finite element analysis. Currently, numerical integration schemes, mostly based on Gauss quadrature, are widely used. Herein, we present an alternative semi-analytical approach for mass matrix evaluation, resulting in an accurate, efficient and easy-to-implement integration rule. To this end, integrands of mass matrix entries get separated to two multiplicative parts. The first depends on natural coordinates while the second depends also on element parameters (mesh). Second part is approximated using polynomials and function evaluations at sampling (integration) points, allowing later analytical integration to precompute the weight matrices. Resulting formulas possess typical form such that computational efficiency equivalence to traditional schemes is demonstrated, namely our n-point formula is computationally equivalent to the conventional use of n-point quadrature. Specifically, one and four-point semi-analytical formulas are explicitly derived for an eight node brick element. Our one-point rule describes exactly constant metric and density elements; four-point scheme correctly admits the linear metric density multiplication. Dramatic superiority in terms of accuracy is established based on coarse mesh study. Wherefore test sets are generated with the help of ABAQUS, and then analyzed via variety of formulas. Authors of the report intend to achieve the next goals: * Introduction of an alternative way to compute the mass matrix of solid finite elements, including inhomogeneous continua, consistent as well as inconsistent mass matrices. * New easy-to-implement, efficient, formulas specified for widely-used eight-node brick element. * Demonstration of accuracy advantage based on coarse mesh generated with commercial software.