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On The Convergence of the Discretized Linear Static State-Based Peridynamic Equations

Lukas Pflug, Michael Stingl, Max Zetzmann

Abstract

In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply one-point-quadrature to the terms in the discrete equations. The resulting system coincides with the commonly used meshfree discretization using a regular lattice, including the possibility of using partial area algorithms to improve the numerical behavior. We again prove convergence, this time for fixed choices of a weighting function commonly used in literature and stronger assumptions on the input data. We note however, that these assumptions are not significantly restrictive for practical purposes. In particular, they still allow discontinuities in the material parameters and external body forces.

On The Convergence of the Discretized Linear Static State-Based Peridynamic Equations

Abstract

In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply one-point-quadrature to the terms in the discrete equations. The resulting system coincides with the commonly used meshfree discretization using a regular lattice, including the possibility of using partial area algorithms to improve the numerical behavior. We again prove convergence, this time for fixed choices of a weighting function commonly used in literature and stronger assumptions on the input data. We note however, that these assumptions are not significantly restrictive for practical purposes. In particular, they still allow discontinuities in the material parameters and external body forces.
Paper Structure (6 sections, 152 equations, 1 figure)

This paper contains 6 sections, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Linear State-Based Model
  3. Convergence of an Analytical Model
  4. Convergence of a Numerical Model
  5. The Convergence Proof
  6. Appendix

Figures (1)

  • Figure 1: Comparison of the convergence behavior between the two choices of partial area weights as given in (\ref{['eqn:defweightsPA']}). A sequence of solutions for increasingly finer discretizations of a state-based model was calculated for both choices of the partial area weights (See (\ref{['eqn:defweightsPA']})) for two distinct problems defined by (\ref{['eqn:defprob1']}) and (\ref{['eqn:descrincl']}). The $y$-axis shows the logarithm of the $L^2(\Omega)^d$ norm of the difference between the solution $\mathbf{u}_\kappa$ and a reference solution $u^*$ calculated at $\kappa = 360^{-1}$ using the PAAC weights. The $x$-axis was chosen to be a logarithmic scale for $\kappa$, shifted such that $\kappa_0=40^{-1}$ appears at $0$. The data of the first problem is shown plotted by the circles, while the crosses correspond to the second problem. The blue markers correspond to the solutions using the $w_{ij}^{\text{PAAC}}$ weights, the red markers correspond to the solutions for the $w_{ij}^{\text{FA}}$ weights.

Theorems & Definitions (24)

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