Element-level mass matrix integration

Abstract

Many real-life engineering problems associated with solid mechanics are analyzed using the widely used Finite Element Method (FEM). An essential part of this numerical procedure is the evaluation of the element-level quantities, such as mass and stiffness matrices, internal and external forces, and other relevant functions. To this end, commercial packages adopt the Standard (ST) approach to numerical integration. Namely, a sufficiently accurate quadrature scheme (such as Gauss points) is employed. The higher the number of integration points, the higher the integration accuracy (integration order), yet the higher the complexity (i.e., the computational cost of the numerical procedure). Thus, practical guidance is to implement the least expensive quadrature (cubature) for which the global system converges. Therefore, deriving efficient and sufficiently accurate element-level integrators is an essential and practical goal.

In this contribution, we propose a novel approach to element-level mass matrix computation that does not rely on the notion of integration points. Instead, we exploit a special mathematical form of the integrand to derive a unique, highly efficient, case-specific integrator. Specifically, we rely on a systematic approximation of the Jacobian function in terms of its partial derivatives. Those partial derivative terms are efficiently expressed in terms of the systematic Jacobian matrix approximation. Finally, analytical integration is used to pre-compute the matrix coefficients (generalized weights). Our approach results in an easy-to-implement integration formula for the element mass matrix that is dramatically more efficient than the widely used ST method. Moreover, the suggested approach is roughly four times less computationally expensive than the recently proposed, state-of-the-art, Semi-Analytical (SA) approach for mass matrix evaluation. Finally, the recommended approach is general in the sense that it is suitable for all solid isoparametric elements regardless of their order (e.g., 8-node vs 20-node hexahedral, 6-node vs 15-node wedge) or dimension (e.g., quadrilateral \& triangular vs. hexahedral \& tetrahedral, etc). It allows for constant and varying initial density.

Finally, we specialize our model to a 20-node hexahedral element. We estimate the accuracy, the memory requirement, and the relative complexity of the resulting formula using the Computer Algebra System (CAS). In terms of relative 'polynomial' complexity, the resulting integrator is approximately equivalent to 1-point ST integration. In terms of accuracy, the proposed JD scheme is superior to the commercially used 14-point ST quadrature, which is sufficiently accurate for global convergence. In terms of memory storage, i.e., the number of constants which must be stored, the proposed model is several times less expensive than the 1-point ST. In conclusion, based on practical demonstration, our case-specific integration approach is a highly efficient and sufficiently accurate solution.