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

1. Problem Statement

Peridynamics (PD) is a non-local continuum theory that replaces partial differential equations (PDEs) with integral equations, making it naturally suited for modeling discontinuities like fractures without requiring spatial derivatives. While the state-based formulation of PD allows for modeling materials with arbitrary bulk and shear moduli (overcoming the fixed Poisson's ratio limitation of bond-based models), rigorous mathematical proofs regarding the convergence of its discretized solutions to the continuous solution have been lacking, particularly under practical numerical conditions.

Specifically, existing literature had not rigorously proven the convergence of:

1. Mesh-free discretizations on regular lattices.
2. Discretizations that employ one-point quadrature (a standard simplification in numerical implementations to evaluate non-local integrals).
3. Scenarios involving discontinuous material parameters or external forces, which are common in fracture mechanics but often excluded in theoretical proofs requiring smooth inputs.

2. Methodology

The authors address the convergence problem in two distinct stages, moving from an idealized analytical discretization to a fully practical numerical model.

A. Analytical Discretization (No Quadrature)

• Model: The authors consider the linear static state-based peridynamic problem defined by a bilinear form $B(u, v)$.
• Discretization: They partition the domain $\Omega$ into small cells ($\Omega_i$) and use piecewise constant approximations (step functions) for the displacement field.
• Approach: They prove convergence using Galerkin theory. The core of the proof relies on demonstrating the density of the discrete admissible sets ($V_\kappa$) within the continuous solution space ($V$) in the $L^2(\Omega)^d$ norm.
• Key Assumption: This stage assumes the exact evaluation of all integrals involved in the bilinear form. It requires minimal regularity on input data (only $L^2$ for body forces and measurable/bounded material parameters), allowing for discontinuities.

B. Numerical Discretization (With One-Point Quadrature)

• Model: The authors introduce a practical numerical scheme where the integrals in the bilinear form are approximated using one-point quadrature (evaluating integrands at cell midpoints).
• Weighting Function: They specifically analyze the case where the weighting function is singular: $\omega(\zeta) = \frac{1}{\|\zeta\|}\chi_{[0,\delta)}(\|\zeta\|)$. This is the standard choice in many PD codes (e.g., for Prototype Microelastic Brittle materials).
• Partial Area Weights: To improve convergence behavior near boundaries, they incorporate partial area/volume weights ($w_{ij}$), which account for the fraction of a horizon intersecting a cell, rather than just using a binary characteristic function.
• Regularity Requirements: To handle the singular weighting function and quadrature error, the authors impose stronger assumptions on the input data: material parameters and body forces must be almost everywhere (a.e.) locally Lipschitz continuous. This still permits discontinuities on sets of measure zero (e.g., material interfaces).
• Proof Strategy:
  1. Operator Splitting: The discrete linear operator is split into four components ($L_1, L_2^x, L_2^{x'}, L_2^p$) to match the structure of the continuous operator.
  2. Uniform Boundedness: They prove the discrete operators are uniformly bounded with respect to the discretization parameter $\kappa$.
  3. Pointwise Convergence: They show that the discrete operators converge pointwise to the continuous operator on a dense subspace ($C^{0,1}$).
  4. Coercivity: They establish a uniform coercivity result for the discrete bilinear form, ensuring the discrete system remains well-posed and stable as $\kappa \to 0$.

3. Key Contributions

1. First Rigorous Proof for Practical PD: The paper provides the first rigorous proof of convergence for the static linear state-based peridynamic model using the standard mesh-free discretization on a regular lattice combined with one-point quadrature.
2. Handling Discontinuities: Unlike previous works that assumed smooth material parameters, this proof explicitly allows for discontinuous material parameters and external forces (provided they are a.e. locally Lipschitz), which is critical for fracture modeling.
3. Partial Area Weights: The authors integrate and prove convergence for partial area weights (PAAC), showing they are essential for consistent convergence behavior compared to simple characteristic function weights.
4. Generalization of Weighting Functions: While the main proof focuses on the singular weight $\rho(\zeta) = 1/\|\zeta\|$, the authors demonstrate that the proof framework can be adapted to other common weighting functions, including constant, conical, and polynomial weights.
5. Bond-Based Model Inclusion: The results naturally extend to bond-based models (a special case of state-based models) without additional effort.

4. Results

• Convergence Theorem: The authors prove that the sequence of discrete solutions $u_\kappa$ (embedded in $L^2$) converges strongly to the unique continuous solution $u$ as the grid size $\kappa \to 0$.
  $$\lim_{\kappa \to 0} \| u - u_\kappa \|_{L^2(\Omega)^d} = 0$$
• Numerical Validation: The paper includes numerical experiments comparing "Full Area" (FA) weights (binary) vs. "Partial Area" (PAAC) weights. The results confirm that PAAC weights eliminate oscillatory behavior and provide consistent convergence rates, validating the theoretical findings.
• Stability: The discrete system is proven to be symmetric positive definite and invertible, ensuring well-posedness for any finite discretization.

5. Significance

This work bridges a critical gap between the theoretical foundations of peridynamics and its practical numerical implementation.

• Validation of Standard Practices: It mathematically justifies the use of one-point quadrature and regular lattice discretizations, which are the industry standard in PD simulations but previously lacked rigorous convergence guarantees for state-based models.
• Robustness for Fracture: By accommodating discontinuous material parameters, the proof supports the use of PD for complex fracture problems where material properties change abruptly across crack surfaces.
• Foundation for Future Work: The rigorous framework established here (handling singular weights and partial area integration) provides a solid mathematical basis for developing more advanced numerical schemes, error estimation, and adaptive mesh refinement in peridynamics.

In summary, the paper confirms that the standard numerical methods used in state-based peridynamics are mathematically sound and converge to the correct physical solution, even in the presence of material discontinuities and singular weighting functions.