Improving mass matrix and inverse mass matrix computations of hexahedral elements

We employ a semi-analytical approach to derive new practical schemes for mass matrix computation of 8-node and 20-node hexahedral elements. The new schemes offer accuracy equivalent to that of the conventional numerical integration (quadrature rule) with a significantly smaller number of integration points. Specifically, for the 8-node hexahedral element, we propose a 4-point rule to replace the currently used 8-point quadrature. Also, for the 20-node hexahedral element, we propose a 4-point scheme to replace the 14-point quadrature adopted by ANSYS and a 10-point scheme to replace the 27-point quadrature adopted by ABAQUS. In addition, we develop a novel approach for direct computation of the inverse mass matrix of 8-node hexahedral elements. This new approach requires a computational effort equivalent to standard numerical integration and eliminates the high computational cost associated with matrix inversion.