J-Derivatives (JD) integrator for the 10-node tetrahedral element

Abstract

In the field of the Finite Element Method (FEM) for computational solid mechanics, there is a strong need to develop efficient integrators for element-level quantities, such as mass and stiffness matrices, internal and external forces, and so on. Currently, the Standard (ST) approach to numerical integration is employed. Thus, generally speaking, the more accurate (the higher order) quadrature (cubature) is chosen, the more integration points it includes, and the more computationally expensive it becomes. Therefore, the practical rule of thumb is to select the least costly scheme that is sufficiently accurate for global convergence.

This study is dedicated to the derivation of a highly efficient, yet sufficiently accurate, integrator for the element-level consistent mass matrix evaluation of the widely used 10-node tetrahedral element. We start by adopting the recently proposed JDerivatives (JD) approach and, for the first time, specifying it for the quadratic tetrahedral element. We explicitly provide all the necessary details for the in-code implementation of the resulting formula. We account for both constant and varying initial density.

The accuracy of the suggested integrator is examined using five coarse-mesh geometries. Findings reveal that the suggested JD integrator outperforms the 11-point ST cubature by a wide margin and is practically equivalent to the 15-point ST scheme. Those results support the sufficient accuracy of our proposal. The relative ‘polynomial’ complexity is examined using a Computer Algebra System (CAS). Accordingly, the complexity of the complete, sufficiently accurate JD formula is 25% higher than that of the only 1-point ST integration. Also, we compare the CPU time of the in-code implementation. CPU time of the complete JD integration is 29% higher than that of only 1-point ST integration. All those outcomes convincingly support the proposed scheme as a practically desirable formula.