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

The precise triangulation of the domain assumes a critical role in calculating numerical approximations of differential operators utilizing a collocation method. A well-executed triangulation contributes significantly to the reduction of discretization errors. Conventional collocation techniques typically represent the smooth curved domain by triangulating a mesh, wherein boundary points are approximated using polygons. However, this methodology frequently introduces geometrical errors that adversely impact the accuracy of the numerical approximation. To mitigate such geometrical inaccuracies, isoparametric, subparametric, and iso-geometric methods have been proposed, facilitating the approximation of curved surfaces or line segments. This paper proposes an efficient finite element method tailored to approximate the elliptic boundary value problem (BVP) solution that governs the response of an elastic solid containing a V-notch and inclusions. The algebraically nonlinear constitutive equation and the balance of linear momentum are reduced to a second-order quasi-linear elliptic partial differential equation. Our approach encompasses the representation of complex curved boundaries through a smooth, distinctive point transformation. The principal objective is to utilize higher-order shape functions to accurately compute the entries within the finite element matrices and vectors and obtain a precise approximate solution to the BVP. A Picard-type linearization addresses the nonlinearities inherent in the governing differential equation. Numerical results derived from the test cases demonstrate a significant enhancement in accuracy.