Optimal local truncation error method for solution of elasticity problems for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Abstract

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes was recently developed for PDEs with homogeneous materials on regular and irregular domains as well as for the scalar time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. Here, OLTEM is extended to a system of time-independent elastic PDEs for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes. We show the development of OLTEM for the 2D elasticity equations using compact 9-point stencils that are similar to those for linear quadrilateral finite elements. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. There are no unknowns on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal second order of accuracy for OLTEM with 9-point stencils on unfitted Cartesian meshes. Numerical experiments for elastic heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom: a) OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; b) up to engineering accuracy of OLTEM with unfitted Cartesian meshes is even more computationally efficient than quadratic and cubic finite elements with much wider stencils and conformed meshes. The proposed technique yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems.