Finite element modeling of V-notched thermoelastic strain-limiting solids containing inclusions

TL;DR

This paper develops a specialized finite element method for accurately modeling thermoelastic solids with V-notches and inclusions, effectively handling complex boundaries and nonlinearities to improve solution precision.

Contribution

It introduces a tailored FEM approach that efficiently solves complex thermoelastic models with nonlinearities and curved boundaries, enhancing accuracy in numerical approximations.

Findings

Significant improvement in solution accuracy demonstrated.

Effective handling of nonlinearities via Picard linearization.

Accurate modeling of complex geometries with curved boundaries.

Abstract

A precise domain triangulation is recognized as indispensable for the accurate numerical approximation of differential operators within collocation methods, leading to a substantial reduction in discretization errors. An efficient finite element method (FEM) is presented in this paper, meticulously developed to solve a complex mathematical model. This model governs the behavior of thermoelastic solids containing both a V-notch and inclusions. The system of partial differential equations underlying this model consists of two primary components: a linear elliptic equation, which is used to describe the temperature distribution, and a quasilinear equation, which governs the mechanical behavior of the body. Through the application of this specifically tailored FEM, accurate and efficient solutions are able to be obtained for these intricate thermoelastic problems. The algebraically nonlinear constitutive equation, alongside the balance of linear momentum, is effectively reduced to a second-order quasi-linear elliptic partial differential equation. Complex curved boundaries are represented through the application of a smooth, distinctive point transformation. Furthermore, higher-order shape functions are employed to ensure the accurate computation of entries within the FEM matrices and vectors, from which a highly precise approximate solution to the BVP is subsequently obtained. The inherent nonlinearities in the governing differential equation are addressed through the implementation of a Picard-type linearization scheme. Numerical results, derived from a series of test cases, have consistently demonstrated a significant enhancement in accuracy, a crucial achievement for the nuanced analysis of thermoelastic solids.