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On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion

Lukas Pflug, Michael Stingl, Max Zetzmann

arXiv:2610.13355v1math.NA

TL;DR

Rigorous convergence results for mesh-free discretizations of non-linear peridynamic models with irreversible damage have been limited, particularly for schemes using one-point quadrature. This paper analyzes a regularized bond-breakage model and proves convergence at fixed horizon, while showing that damage does not destroy data-inherited local Lipschitz regularity. The horizon limit remains outside the analysis.

  • Problem

    Rigorous convergence analysis for mesh-free discretizations of non-linear models with irreversible damage remains limited, and prior results generally do not address the one-point quadrature used in practice.

    Source: §1 · p. 2

  • Method

    The paper analyzes the mesh-free discretization, including one-point quadrature, of a well-posed Lipschitz-regularized bond-breakage model with irreversible damage.

    Source: §1 · pp. 2–3

  • Results

    Convergence is proved in C1([0, T], L∞(Ω)d), with rate O(κ + 1/N) for suitable input data, and displacement continuity and local Lipschitz bounds persist regardless of damage at fixed δ > 0.

    Source: §1 · p. 3

  • Takeaways & Limitations

    At fixed horizon, damage in regions with regular input data manifests through steep slopes rather than jump discontinuities.

    Source: §1 · p. 3

    The analysis does not establish convergence as δ ↘0, and the local Lipschitz bounds diverge as the horizon shrinks.

    Source: §5 · p. 32

Abstract

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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 $C^1([0,T],L^\infty(Ω)^d)$, and therefore $L^\infty(Ω)^d$ 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 $δ\searrow 0$.

1 Introduction

The paper addresses gaps in rigorous convergence analysis for mesh-free discretizations of non-linear peridynamic models with irreversible damage, including the one-point quadrature used in practice. It proves convergence at fixed horizon and establishes continuity and local Lipschitz regularity of the continuous displacement under suitable data conditions.

1 Introduction

  • The studied model is a regularized, well-posed version of the bond-breakage model of, whose original operator is discontinuous.The regularization relaxes the damage factor and force scalar to be Lipschitz continuous.Source: §1 · p. 2
  • Rigorous convergence analysis for a non-linear model with irreversible damage, including one-point quadrature, addresses gaps in prior mesh-free results.The analysis concerns the model of and covers two and three space dimensions.Source: §1 · p. 2
  • Convergence holds in C1([0, T], L∞(Ω)d) for the semi-discrete and fully discrete schemes; suitable input data yield a rate of O(κ + 1/N).The fully discrete scheme uses central difference time discretization; the rate requires suitable, e.g. Lipschitz-continuous, input data.Source: §1 · p. 3
  • At fixed δ > 0, displacement inherits continuity and local Lipschitz continuity from the input data, regardless of damage.The local Lipschitz bounds depend on δ and diverge as δ ↘0; the paper does not establish convergence as δ ↘0.Source: §1 · p. 3
  • The work is limited to the discretization limit κ ↘0 at fixed horizon δ > 0, not the horizon limit or joint limit.For suitable input data, the convergence rate is O(κ + 1/N), with N the number of time steps.Source: §1 · p. 3

2 The Continuous Bond-Breakage Model

The continuous model uses a Lipschitz-relaxed bond-breakage law studied for well-posedness in. Its relaxation supports analysis but also modifies bond behavior under extreme compression.

2 The Continuous Bond-Breakage Model

  • The model is a Lipschitz relaxation of the bond-breakage criterion in, whose original operator is discontinuous.The relaxation makes the force scalar and damage factor Lipschitz continuous.Source: §2 · p. 4
  • Bonds are characterized by their stretch and deformed direction; the model treats bonds below a critical stretch as sub-critical.The force operator does not depend on the direction value for degenerate bonds.Source: §2 · pp. 4–5
  • The relaxation permits continuous weakening before breaking, but its compression-side cutoff weakens interactions beyond a threshold.The authors describe that compression behavior as physically undesirable; the cutoff also avoids degeneracy of the deformed bond direction in the analysis.Source: §2 · p. 6
  • For admissible data and a weighting function, the continuous equations have a unique solution in C2([0, T], L∞(Ω)d).The well-posedness result is taken from [12].Source: §2 · p. 7

3 The Semi-Discrete System

This section defines a mesh-free, one-point-quadrature semi-discretization, establishes its well-posedness, and develops bounds for comparing discrete and continuous solutions. It proves convergence in the natural $C^1([0,T],L^infty(\Omega)^d)$ norm when the initial data and forcing are approximated with $O(\kappa)$ errors, under restrictive regularity requirements.

3 The Semi-Discrete System

  • The semi-discrete model uses mesh-free points and one-point quadrature to represent the continuous model’s terms.The domain is partitioned into equal-sized cubes or squares, retaining cells with positive-volume intersection with Ω.Source: §3 · p. 7
  • The L∞(Ω)^d solution norm makes the convergence assumptions restrictive, requiring sufficiently regular material parameters and L∞-norm convergence of numerical input data.For discontinuous input data, discontinuities must lie within the closures of discrete-element boundaries along the chosen mesh sequence; offsets can mitigate this for simple sets.Source: §3 · p. 8
  • Lipschitz-continuous input data yield linearly convergent discrete input data when sampled at grid points or projected, while partial-area quadrature weights can be incorporated under stated assumptions.The weights are symmetric and account for whether cells intersect or lie fully within the horizon; the text says the estimates can be adapted to include them.Source: §3 · pp. 9, 12
  • Under Assumptions 3.3, the semi-discrete problem has a unique solution.The proof uses a Picard iteration after rewriting the system as an integral equation.Source: §3 · pp. 11–12

3.1 Convergence of the Semi-Discrete Systems

  • The subsection aims to prove convergence of solutions of (6) to the continuous solution (5), and Proposition 3.13 states a discretization-error bound for all t in [0, T], with β independent of κ as κ ↘ 0.Source: §3.1 · pp. 12, 16
  • Theorem 3.15 concerns convergence of semi-discrete solutions to the continuous solution as κ_n ↘ 0 under Assumptions 3.3; the supplied fragments do not state the convergence norm.The theorem’s listed conditions include O(κ) errors in the initial displacement, initial velocity, and body-force approximation.Source: §3.1 · p. 18
  • The convergence result requires O(κ) approximation errors for the initial displacement, initial velocity, and time-dependent body force.These are the consistency conditions stated for the semi-discrete data in the convergence theorem.Source: §3.1 · p. 18
  • The proof transports the reduced semi-discrete equations to L∞(Ω)^d and uses a Grönwall estimate to establish convergence.The transport uses Kκ, which is bounded for each κ.Source: §3.1 · pp. 18–19

4 Convergence of the Fully Discrete System

The fully discrete central-difference scheme converges as spatial and temporal discretization are refined. Under stronger regularity conditions, the paper gives a rate and extends convergence to the integrity function.

4 Convergence of the Fully Discrete System

  • The time discretization uses a central-difference scheme, and the discrete operator reduces the history maximum to a finite maximum evaluable in code.The time step is h = T/N.Source: §4 · p. 20
  • As κ→0 and N→∞, the fully discrete solutions converge to the continuous solution under Assumptions 3.3.The theorem compares the solution at the discrete time nodes with the fully discrete approximation.Source: §4 · p. 22
  • With time-Lipschitz b and input condition (30), the fully discrete error is O(1/N + κ).The estimate combines an O(κ) spatial contribution with a time-discretization contribution.Source: §4 · pp. 22, 24
  • The integrity function also converges under Assumptions 3.3 and 3.9 when Ω satisfies an interior cone condition.The proof uses the resulting uniform lower bound on vol(Ω∩Bδ(x)).Source: §4 · pp. 25–26
  • An Lp(Ω)^d reformulation remains unresolved because the spatial Lp norm cannot be interchanged with the temporal essential supremum in the damage term.This obstruction prevents the presented Lipschitz-continuity proof for the peridynamic operator from carrying over.Source: §4 · pp. 26–27

5 Properties of the Continuous Solution

The section establishes spatial continuity and local Lipschitz regularity of the continuous displacement under corresponding regularity of the input data. These results explain the spatial regularity required by the convergence analysis while keeping the claims at fixed horizon δ > 0.

Motivation

  • The continuity result explains the spatial regularity required by convergence in the C0([0, T], L∞(Ω)^d) norm.The section motivates its pointwise analysis by noting this implication of the piecewise constant discretization's convergence.Source: §5 · p. 27

Proposition 5.4

  • Continuous input data yield a displacement continuous in space for every time, with the result stated for a pointwise representation of the L∞ solution.This regularity result applies under Assumptions 5.1.Source: §5 · p. 31
  • Locally Lipschitz input data yield a displacement that is locally Lipschitz in space, uniformly over time; globally Lipschitz data yield a globally Lipschitz displacement.The local Lipschitz constant is independent of t ∈ [0, T].Source: §5 · p. 31

Remark 5.5

  • The convergence result requires input-data discontinuities to lie inside the closures of discrete-element boundaries along the sequence of discretizations.This is enforced by the stated strict convergence requirements.Source: §5 · pp. 31–32

Remark 5.5–5.6

  • The regularity conclusions are qualitative at fixed δ: the bounds diverge as δ ↘ 0, and the analysis does not establish convergence to a sharp-discontinuity limit.The authors also note that the divergence alone does not imply that solutions become steeper as δ shrinks.Source: §5 · p. 32

6 A Numerical Study of the Convergence

The numerical study illustrates the continuity and convergence results with a two-dimensional example. As resolution and horizon vary, it examines damage patterns and convergence behavior in L∞ and L2 norms.

6 A Numerical Study of the Convergence

  • The numerical study is presented to illustrate the theoretical results, not as a rigorous numerical analysis.The example model is used for both the continuity visualization and the convergence study.Source: §6 · p. 32

6.1 Visualization of the Continuity

  • At fixed horizon, the example shows a steep displacement near discontinuous loading but a smoothed displacement across the central damaged region.Elements fill the gap at the center line, consistent with the continuity result; the theoretical bounds are too large to quantify the observed slope.Source: §6.1 · p. 33, Figure 1 · p. 34

6.2 About the Convergence

  • The observed EOC behavior suggests that shrinking δ requires higher horizon resolution to reach the asymptotic rate, while L2 EOCs are better than L∞ EOCs as errors localize.The authors describe the results as suggesting a pre-asymptotic regime, not contradicting the convergence theorem.Source: §6.2 · p. 35, Table 1 · p. 35
  • For δ = 1/20, the higher-resolution plot at κ^-1 = 640 is compared with the lower-resolution plot at κ^-1 = 80; the damage region remains fairly stable as error localizes.The caption relates this localization to the gap between the p = 2 and p = ∞ rates and notes that the L∞ norm does not capture the stable damage region in the same way as L2.Source: Figure 2 · p. 36
  • The study concludes that resolving displacement near damage in the L∞ norm requires very large resolutions in this example, making adaptive schemes relevant given the performance cost of large m.This is the authors' stated practical implication of the numerical results.Source: §6.2 · p. 36

7 Homogeneous Dirichlet Volume Constraint

The section imposes homogeneous Dirichlet volume constraints by setting displacement to zero on a disjoint region and incorporating its force on the domain. The corresponding force and discretizations retain the convergence results established for the unconstrained schemes.

7 Homogeneous Dirichlet Volume Constraint

  • Setting displacement to zero on the adjacent, disjoint constraint region is simple but can create artificial boundary effects at its interface with the domain.The section identifies alternative approaches but does not analyze them here.Source: §7 · p. 37
  • The analysis rewrites external body forces as a Dirichlet force representing the force exerted on Ω by the constrained portion of Ω∪ΩI.Bonds crossing the interface between ΩI and Ω may break under the definition of this force.Source: §7 · p. 37
  • The discrete construction duplicates indices for cells intersecting both Ω and ΩI, allowing the two portions to carry different displacement values.The copies distinguish the domain displacement from the zero displacement assigned on ΩI.Source: §7 · p. 37
  • The Dirichlet-force properties establish well-posedness of the constrained equations, and the section obtains convergence of the semi-discrete solutions to the continuous solution.The text notes that well-posedness of the continuous problem follows already from, while the bound is needed later.Source: §7 · pp. 38–39
  • The fully discrete scheme with boundary conditions also converges as the time-step parameter tends to its limit and the spatial parameter κ tends to zero.The corollary states this result under Assumptions 3.3 and for an open, bounded ΩI disjoint from Ω.Source: §7 · p. 39

A.1 Proofs for Section 3

The appendix supplies bounds used in the Section 3 analysis for discrete integrals and weighting functions with partial area weights.

A.1 Proofs for Section 3

  • Lemma A.1 gives uniform bounds on discrete integrals with partial area weights.Its stated scope is for indices i∈Iκ.Source: §A.1 · p. 45
  • The proof arguments use the non-increasing, finite weighting function and the triangle inequality to establish convergence.The appendix also reduces the weighting-error argument to bounding an error term.Source: §A.1 · p. 45
  • Lemma A.2 bounds the discrete weighting-function error by Dκ, with Dκ=O(κ) as κ decreases.The bound is stated uniformly in x and i.Source: §A.1 · p. 45
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