Optimal local truncation error method for 2‐D elastodynamics problems on irregular domains and unfitted Cartesian meshes

Abstract

The optimal local truncation error method (OLTEM) has been developed for 2‐D time‐dependent elasticity problems on irregular domains and trivial unfitted Cartesian meshes. Compact nine‐point uniform and nonuniform stencils (similar to those for linear finite elements on uniform meshes) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for nine‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The cases of the nondiagonal and diagonal mass matrices are considered for OLTEM. The results of numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM with the nondiagonal mass matrix is much more accurate than linear finite elements and than quadratic and cubic finite elements (up to the engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. The proposed numerical technique can be efficiently used for many engineering applications including geomechanics.