Optimal local truncation error method for 3-D elasticity interface problems

The paper deals with a new effective numerical technique on unfitted Cartesian meshes for simulations of heterogeneous elastic materials. We develop the optimal local truncation error method (OLTEM) with 27-point stencils (similar to those for linear finite elements) for the 3-D time-independent elasticity equations with irregular interfaces. Only displacement unknowns at each internal Cartesian grid point are used. The interface conditions are added to the expression for the local truncation error and do not change the width of the stencils. The unknown stencil coefficients are calculated by the minimization of the local truncation error of the stencil equations and yield the optimal second order of accuracy for OLTEM with the 27-point stencils on unfitted Cartesian meshes. A new post-processing procedure for accurate stress calculations has been developed. Similar to basic computations it uses OLTEM with the 27-point stencils and the elasticity equations. The post-processing procedure can be easily extended to unstructured meshes and can be independently used with existing numerical techniques (e.g., with finite elements). Numerical experiments show that at an accuracy of 0.1% for stresses, OLTEM with the new post-processing procedure significantly (by 105−109 times) reduces the number of degrees of freedom compared to linear finite elements. OLTEM with the 27-point stencils yields even more accurate results than high-order finite elements with wider stencils.