Inverse Mass Matrix For Higher-Order Finite Element Method In Linear Free-Vibration Problems

In the paper, we present a direct inverse mass matrix in the higher-order finite element method for solid mechanics. The direct inverse mass matrix is sparse, has the same structure as the consistent mass matrix and preserves the total mass. The core of derivation of the semi-discrete mixed form is based on the Hamilton's principle of least action. The cardinal issue is finding the relationship between discretized velocities and discretized linear momentum. Finally, the simple formula for the direct inverse mass matrix is presented as well as the choice of density-weighted dual shape functions for linear momentum with respect to the displacement shape function with a choice of the lumping mass method for obtaining the correct and positive definitive velocity-linear momentum operator. The application of Dirichlet boundary conditions into the direct inverse mass matrix for a floating system is achieved using the projection operator. The suggested methodology is tested on a free-vibration problem of heterogeneous bar for different orders of shape functions.