Optimal local truncation error method to solution of 2‐D time‐independent elasticity problems with optimal accuracy on irregular domains and unfitted Cartesian meshes

Abstract

Recently we have developed the optimal local truncation error method (OLTEM) for scalar PDEs on irregular domains and unfitted Cartesian meshes. Here, OLTEM is extended to a much more general case of a system of PDEs for the 2-D time-independent elasticity equations on irregular domains. Compact 9-point uniform and nonuniform stencils (with the computational costs of linear finite elements) 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 9-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 numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM is much more accurate than linear finite elements and than quadratic finite elements (up to engineering accuracy of 0.1%-1%) at the same numbers of degrees of freedom. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular domains, the proposed technique with no remeshing for the shape change of irregular domains will be effective for many engineering applications.

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

“…However, the use of regular domains is a significant limitation for these stencils. Recently, we have developed OLTEM with nine‐point stencils for a system of PDEs for the 2‐D time‐independent elasticity on irregular domains; see our paper 44 . Here, we extend OLTEM to a system of PDEs for the 2‐D time‐dependent elasticity on irregular domains that have numerous engineering applications.…”

“…Recently, we have developed OLTEM with nine-point stencils for a system of PDEs for the 2-D time-independent elasticity on irregular domains; see our paper. 44 Here, we extend OLTEM to a system of PDEs for the 2-D time-dependent elasticity on irregular domains that have numerous engineering applications. In contrast to OLTEM for the time-independent PDEs on irregular domains considered in our papers, [35][36][37][38][39][40]44 the derivation of OLTEM for elastodynamics is totally different because it includes the manipulations with the time derivatives; see below Section 2.…”

“…44 Here, we extend OLTEM to a system of PDEs for the 2-D time-dependent elasticity on irregular domains that have numerous engineering applications. In contrast to OLTEM for the time-independent PDEs on irregular domains considered in our papers, [35][36][37][38][39][40]44 the derivation of OLTEM for elastodynamics is totally different because it includes the manipulations with the time derivatives; see below Section 2. In contrast to OLTEM for the time-dependent scalar wave and heat equations on irregular domains considered in our papers, [35][36][37]39,40 OLTEM for elastodynamics includes twice the number of unknown stencil coefficients for the same nine-point stencils, the optimal order of accuracy for OLTEM with the 9-point stencils is different for the scalar wave (heat) equation and for the elastodynamics equations, the different orders of terms in a Taylor series expansion of the local truncation error are used for the derivation of OLTEM for the scalar wave (heat) equation and for the elastodynamics equations.…”