Analysis of Convergence for the IPA-AC Method

TL;DR

This paper provides a theoretical convergence analysis of the IPA-AC meshfree discretization method for peridynamic models, revealing its second-order accuracy in mesh size and sensitivity to the nonlocal horizon, with implications for its use in local limit approximations.

Contribution

It establishes a unified convergence framework for the IPA-AC method, deriving explicit error estimates and clarifying its performance across different scaling regimes.

Findings

Achieves second-order convergence $ ext{O}(h^2)$ for fixed horizon $oldsymbol{ ext{δ}}$

Discretization error scales as $ ext{O}(oldsymbol{ ext{δ}}^{-2})$ for fixed mesh

Does not satisfy the Asymptotic Compatibility (AC) condition

Abstract

The Improved Partial Area-Analytical Calculation (IPA-AC) method represents a leading meshfree discretization strategy for peridynamic models, distinguished by its rigorous geometric treatment of boundary intersections via dual corrections of integration weights and quadrature points. Despite its empirical success in suppressing boundary-induced geometric errors, a systematic theoretical characterization of its convergence behaviors under distinct scaling limits has remained elusive. This work establishes a unified convergence framework for the IPA-AC method applied to both scalar and tensor kernels. By leveraging the Lax Equivalence Theorem, we explicitly derive error estimates that reveal the method's performance across three critical limiting regimes. The theoretical analysis, substantiated by numerical validation, demonstrates that: (1) for a fixed horizon $\delta$, the method achieves robust second-order convergence $\mathcal{O}(h ^{2})$ with respect to the mesh size $h$; (2) for a fixed mesh, the discretization error scales as $\mathcal{O}(\delta^{-2})$, indicating a sensitivity to the nonlocal length scale; and (3) the method does not satisfy the Asymptotic Compatibility (AC) condition. These findings clarify that while the IPA-AC method offers superior accuracy for simulating fixed nonlocal models, it requires a sufficiently large horizon-to-mesh ratio to mitigate intrinsic discretization errors when approximating the local limit.