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Analysis of Convergence for the IPA-AC Method

Xiuzhu Yang, Xiaobo Yin

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 $δ$, 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 $\mathcal{O}(δ^{-2})$, 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.

Analysis of Convergence for the IPA-AC Method

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 , the method achieves robust second-order convergence with respect to the mesh size ; (2) for a fixed mesh, the discretization error scales as , 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.
Paper Structure (25 sections, 7 theorems, 55 equations, 7 figures, 6 tables)

This paper contains 25 sections, 7 theorems, 55 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Predynamic Model
  3. Kernel function
  4. Scalar kernel
  5. Tensor kernel
  6. Numerical Implementation via IPA-AC Algorithm
  7. Spatial discretization and nodal representation
  8. The IPA-AC algorithm
  9. Convergence Analysis
  10. Error decomposition Framework
  11. Convergence analysis of scalar kernel
  12. Stability analysis of scalar kernel
  13. Consistency analysis of scalar kernel
  14. Convergence analysis of scalar kernel
  15. Convergence analysis of tensor kernel
  16. ...and 10 more sections

Key Result

Lemma 4.1

(Invertibility) Provided that the horizon $\delta$ is sufficiently resolved by the mesh to ensure graph connectivity, the discrete stiffness matrix $\mathbf{K}$ is a non-singular M-matrix. Consequently, the inverse operator $(\mathcal{L}_{\delta}^{h})^{-1}$ exists and is non-negative.

Figures (7)

  • Figure 1: The red border delineates the physical body $\mathcal{B}$ from the surrounding fictitious layer. Note that the neighborhoods of interior nodes reside entirely within $\mathcal{B}$, whereas those of nodes near the boundary partially intersect the fictitious layer.
  • Figure 2: Classification of material points within the interaction neighborhood of point $\mathbf{x}_{i}$. The interior of the circle denotes the interaction domain $\mathcal{H}$.
  • Figure 3: IPA-AC algorithm: Shifting quadrature points of partial cells to the centroids of the intersected regions.
  • Figure 4: Geometric symmetry of the neighborhood of node $\mathbf{x}_{i}$. Each point $\mathbf{x}_{i+m}$ admits a symmetric counterpart $\mathbf{x}_{i-m}$ with an identical cell area. The two vectors formed by each symmetric pair and the source point $\mathbf{x}_{i}$ have equal magnitudes and opposite directions.
  • Figure 5: The $h$-convergence of the discretization error for a fixed horizon $\delta$. The solid lines with symbols represent numerical results for Cases 1–3, while the dashed reference line indicates the theoretical $\mathcal{O}(h^{2})$ convergence rate.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • Lemma 4.4
  • Lemma 4.5
  • Theorem 4.2