An efficient finite element method for computing the response of a strain-limiting elastic solid containing a v-notch and inclusions

Abstract

Accurate triangulation of the domain plays a pivotal role in computing the numerical approximation of the differential operators. A good triangulation is the one which aids in reducing discretization errors. In a standard collocation technique, the smooth curved domain is typically triangulated with a mesh by taking points on the boundary to approximate them by polygons. However, such an approach often leads to geometrical errors which directly affect the accuracy of the numerical approximation. To restrict such geometrical errors, \textit{isoparametric}, \textit{subparametric}, and \textit{iso-geometric} methods were introduced which allow the approximation of the curved surfaces (or curved line segments). In this paper, we present an efficient finite element method to approximate the solution to the elliptic boundary value problem (BVP), which governs the response of an elastic solid containing a v-notch and inclusions. The algebraically nonlinear constitutive equation along with the balance of linear momentum reduces to second-order quasi-linear elliptic partial differential equation. Our approach allows us to represent the complex curved boundaries by smooth \textit{one-of-its-kind} point transformation. The main idea is to obtain higher-order shape functions which enable us to accurately compute the entries in the finite element matrices and vectors. A Picard-type linearization is utilized to handle the nonlinearities in the governing differential equation. The numerical results for the test cases show considerable improvement in the accuracy.