Implicit rare mesh scheme for solving 3D elasticity problems

A new implicit scheme for solving 3D dynamic elasticity problems was considered. To approximate the elasticity equations by spatial coordinates, a rare mesh FEM scheme based on a four-node finite element with a linear approximation of displacements within the element was employed. The finite elements are located in the centers of hexahedral cells, with each cell containing a single element. As a result, for meshes with the same element size, this scheme uses five times fewer finite elements and half as many nodes as traditional schemes utilizing four-node linear finite elements, which makes it highly efficient. The equations were approximated in time based on the implicit unconditionally stable Crank–Nicolson numerical scheme (trapezoidal rule). The applicability of the scheme was discussed, with a focus on the class of problems for which it outperforms the explicit scheme. An example of a test model problem solved using this scheme was provided.