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

Shylaja G., Kesavulu Naidu V., Venkatesh B., S. M. Mallikarjunaiah

TL;DR

Problem: compute the response of a strain-limiting elastic solid containing a v-notch and inclusions, where the constitutive law is algebraically nonlinear. Approach: reduce to the Mode-III anti-plane problem with a scalar PDE for the Airy function $\Phi$ given by $-\nabla \cdot(\Psi_1(\|\nabla \Phi\|) \nabla \Phi)=0$, and solve with a curved-cubic finite element method using a one-of-a-kind point transformation; apply Picard linearization for the nonlinear term. Contributions: (i) well-posed continuous and discrete formulation; (ii) curved-element FEM that eliminates geometry discretization error and supports higher-order accuracy; (iii) demonstration on square, v-notch, and v-notch-with-inclusions with rapid convergence (e.g., $<25$ Picard iterations, 21 iterations in one case); (iv) potential extension to crack propagation and multiscale problems. Significance: enables accurate, efficient simulation of nonlinear elasticity in geometrically complex domains, with applicability to material design and failure analysis.

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.

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

TL;DR

Problem: compute the response of a strain-limiting elastic solid containing a v-notch and inclusions, where the constitutive law is algebraically nonlinear. Approach: reduce to the Mode-III anti-plane problem with a scalar PDE for the Airy function given by , and solve with a curved-cubic finite element method using a one-of-a-kind point transformation; apply Picard linearization for the nonlinear term. Contributions: (i) well-posed continuous and discrete formulation; (ii) curved-element FEM that eliminates geometry discretization error and supports higher-order accuracy; (iii) demonstration on square, v-notch, and v-notch-with-inclusions with rapid convergence (e.g., Picard iterations, 21 iterations in one case); (iv) potential extension to crack propagation and multiscale problems. Significance: enables accurate, efficient simulation of nonlinear elasticity in geometrically complex domains, with applicability to material design and failure analysis.

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.
Paper Structure (12 sections, 1 theorem, 38 equations, 10 figures, 8 tables, 1 algorithm)

This paper contains 12 sections, 1 theorem, 38 equations, 10 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Implicit theory of elasticity
  3. Anti-plane Shear Problem (Mode-III)
  4. Boundary value problem and existence of solution
  5. Finite element discretization
  6. Construction of shape functions
  7. Discrete formulation and Picard's iteration
  8. Numerical Examples
  9. Square domain with a manufactured solution
  10. Domain with V-notch
  11. Domain with V-notch and inclusions
  12. Conclusion

Key Result

Theorem 1

$\Omega \subset \mathbb{R}^2$ is a simply connected domain with $C^{0,1}$ boundary consisting of two subsets $\partial \Omega_1 \cup \partial \Omega_2$ so that $\partial \Omega_1 := \left\{ \boldsymbol{x} \in \partial \Omega \colon \boldsymbol{t} (\boldsymbol{x}) \cap \Omega = \emptyset \righ

Figures (10)

  • Figure 1: Plot of the best fit for the model similar to the one in \ref{['eqn:main2']}, for a specific choice of material invariants, compared with the experimental data for gum metal and rubber-like material.
  • Figure 2: Mapping the 10-node cubic order with one side curved triangle to the standard right-angled unit triangle
  • Figure 3: A domain and the Dirichlet boundary conditions for h-convergence study.
  • Figure 4: Doamin discretization into 8, 16, and 32 elements
  • Figure 5: Plot of finite element solution with 32 elements
  • ...and 5 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1