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An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model

Ram Manohar, S. M. Mallikarjunaiah

Abstract

We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and $L^2$-norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack.

An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model

Abstract

We propose an -adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and -norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack.

Paper Structure

This paper contains 11 sections, 118 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Formulation of the model problem.
  3. Background and Motivation.
  4. Our Contribution.
  5. Precursory
  6. Discretization
  7. Existence of a unique discrete solution $u_h^{dg}$ in $\mathcal{S}^{\mathtt{n}}_h(\mathcal{T}_h)$
  8. Error Analysis
  9. Numerical Assessments
  10. Conclusion
  11. Acknowledgement.

Figures (7)

  • Figure 1: Red color dotes denote the hanging nodes in both families of subdivisions $\mathcal{T}_h$.
  • Figure 2: A domain and the boundary indicators.
  • Figure 3: Uniform mesh with mesh size $128\times 128$ and the corresponding adaptive meshes.
  • Figure 5: Plots of $L^2$-norm and energy ($\mathcal{E}$)-norm convergence in different polynomial spaces P1dc, P2dc, and P3dc, respectively
  • Figure 8: A square domain containing a single edge crack.
  • ...and 2 more figures

Theorems & Definitions (6)

  • proof
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