Development of a perfect match system in the improvement of eigenfrequencies of free vibration

It is well known that the standard finite element method (FEM) with overly-stiff effect gives the upper bound solutions of natural frequencies in the free vibration analysis using triangular and tetrahedral elements. In this study, for the first time, this paper aims to improve the prediction of eigenfrequencies through the perfect match between the stiffness and mass matrices. With redistribution of mass in the system, we can tune the balance between stiffness and mass of a discrete model. This can be done by simply shifting the integration points away from the Gaussian locations, while ensuring the mass conservation. A number of numerical examples including 2D and 3D problems have demonstrated that the accuracy of eigenfrequencies is strongly determined by the location of integration points in the mass matrix. With appropriate selection of integration points in the mass matrix, even the exact solution of eigenfrequencies can be obtained in both FEM and smoothed finite element method (SFEM) models.