On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion
Abstract
We show the convergence of a commonly used mesh-free discretization of a dynamic non-linear bond-based peridynamic model with irreversible damage implemented via a bond-breakage criterion for a fixed horizon δ>0. The model chosen here is a mathematically well-posed regularization of a discontinuous peridynamic operator often found in the literature, which assumes that fracture results from strain in tension only. We explicitly include the one-point quadrature used in this scheme in the analysis. The convergence is with respect to the natural norm of the solution space given by C1([0,T],L∞(Ω)d), and therefore L∞(Ω)d in the spatial coordinates. This result heavily constrains the spatial regularity of the continuous solution which we prove afterwards. More specifically, for locally L-Lipschitz continuous input data, local Lipschitz bounds are derived for the displacement, regardless of damage. This is consistent with interpreting the peridynamic model used here as an approximation to a truly discontinuous fracture model with a length-scale parameter given by the horizon δ>0. We note that we do not deal with the limit of the solutions as δsearrow 0.
Related
- Numerical Investigation of Soft-to-Hard Convergence: Discretization, Admissibility, and Spectral Regularity
- Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization
- Mathematical Classification of Switching Types and Hierarchical Structures: Three Switching Types, Six Hierarchical Structures, Coefficient of Variation Lower Bound, and Discretization Plateau Effect in the k-Regime Linear Switching Model
- A hyperbolic nonlinear Schrödinger system in multiple spatial dimensions: structure, convergence, and numerical discretization
- A Criterion for Convergence of Diffusion Processes on Graphs
BibTeX
@misc{pflug2026convergence,
title = {{On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion}},
author = {Lukas Pflug and Michael Stingl and Max Zetzmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13355},
eprint = {2610.13355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13355},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion
AU - Lukas Pflug
AU - Michael Stingl
AU - Max Zetzmann
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13355
UR - https://arxiv.org/abs/2610.13355
AB - We show the convergence of a commonly used mesh-free discretization of a dynamic non-linear bond-based peridynamic model with irreversible damage implemented via a bond-breakage criterion for a fixed horizon δ>0. The model chosen here is a mathematically well-posed regularization of a discontinuous peridynamic operator often found in the literature, which assumes that fracture results from strain in tension only. We explicitly include the one-point quadrature used in this scheme in the analysis. The convergence is with respect to the natural norm of the solution space given by C^1([0,T],L^∞(Ω)^d), and therefore L^∞(Ω)^d in the spatial coordinates. This result heavily constrains the spatial regularity of the continuous solution which we prove afterwards. More specifically, for locally L-Lipschitz continuous input data, local Lipschitz bounds are derived for the displacement, regardless of damage. This is consistent with interpreting the peridynamic model used here as an approximation to a truly discontinuous fracture model with a length-scale parameter given by the horizon δ>0. We note that we do not deal with the limit of the solutions as δsearrow 0.
ER -