An asymptotically compatible bond-based peridynamics with Gaussian kernel

In this paper, we introduce a novel bond-based peridynamic model that utilizes a Gaussian kernel function. Previous peridynamic models, when directly discretized, have exhibited a lack of asymptotically compatibility with their corresponding local elastic solutions. Additionally, these models have faced challenges in accurately computing the volume of intersecting regions between the horizon of the material point and its neighboring cells. These difficulties in numerical simulations within peridynamics have spurred numerous efforts to develop corrective numerical methods. While such corrective methods have addressed certain issues, they remain complex to formulate and computationally intensive. Instead of addressing these challenges through modified numerical discretization, this paper presents a novel approach: bond-based peridynamics with a Gaussian kernel . This model replaces the bounded kernel function commonly used in existing peridynamic models with an unbounded Gaussian kernel function. Through direct meshfree discretization, we demonstrate that the solution of the proposed model , without the need for compatibility correction or volume correction, aligns with the corresponding local elastic solution. Furthermore, we derive the relevant material parameters based on energy equivalence to an elastic continuum model. Building upon this foundation, we propose a damage model and a linearized version of the proposed peridynamic model. We validate our proposed model through simulations of 2D smooth problems, demonstrating its convergence and accuracy. Finally, we assess the applicability of our approach to realistic scenarios by predicting displacements caused by a 2D plate with pre-existing cracks and replicating high-velocity impact results from the Kalthoff-Winkler experiments.

💡 Research Summary

This paper presents a significant advancement in peridynamics (PD) by introducing a novel bond-based peridynamic model that employs an unbounded Gaussian kernel function, termed BBPD-GK. The work directly addresses two long-standing numerical challenges in standard PD models: the lack of asymptotic compatibility between the discretized PD solution and the corresponding local elastic continuum solution, and the difficulty in accurately computing the partial volume of material points intersecting the horizon boundary, which necessitates complex volume correction algorithms. Previous efforts to solve these issues have focused on developing corrective numerical discretizations, which are often intricate and computationally expensive.

The core innovation of BBPD-GK lies in its fundamental redefinition of the interaction kernel. Instead of the bounded, constant or conical kernels defined within a finite horizon radius δ, the model utilizes a Gaussian function, G(ξ) = exp(-|ξ|²/σ²)/(2πσ²), as its kernel. Although this kernel has infinite support, its value decays rapidly with distance. The parameter σ controls the spread of the nonlocal interaction. The authors formulate the nonlinear BBPD-GK governing equations for both 2D and 3D. Through energy equivalence with classical linear elasticity, they derive the explicit expressions for the model’s micromodulus constant (β) in terms of Young’s modulus E and the variance σ². A damage model is also formulated by defining a critical bond stretch (s_c), which is linked to the material’s fracture energy G0.

A key theoretical contribution is the analysis of the linearized version of BBPD-GK. By assuming small deformations, the force function is linearized, and the resulting linear nonlocal operator is analyzed. Utilizing the properties of Gaussian moments (where odd moments vanish and fourth-order moments are proportional to σ⁴), the authors rigorously demonstrate that in the limit of discretization refinement, the linear BBPD-GK equation converges asymptotically to the classical Navier equations of linear elasticity. This proves the model’s inherent asymptotic compatibility without requiring any auxiliary correction schemes.

From a computational perspective, the Gaussian kernel’s fast decay allows for a practical solution: truncation. While the theoretical interaction region is infinite, a finite cut-off distance can be chosen beyond which interactions are negligible. This replaces the infinite domain with a manageable computational region. Crucially, because the interaction weight decays smoothly to near-zero at the truncation boundary, the problematic “partial volume” issue inherent to sharp-bounded horizons is effectively eliminated, removing the need for specialized volume correction algorithms.

The paper validates the proposed model through comprehensive numerical examples. First, smooth 2D problems (like a static plate under tension) are used to demonstrate the model’s convergence and accuracy, showing that the solution converges to the classical elastic solution as the discretization is refined. Subsequently, to assess practical applicability, the model is tested on more challenging scenarios. It successfully predicts displacement fields in a 2D plate with pre-existing cracks. Furthermore, it replicates the results of the well-known Kalthoff-Winkler experiment, a benchmark for dynamic fracture involving high-velocity impact, demonstrating its capability to handle complex fracture mechanics problems.

In summary, this research introduces a fundamentally redesigned bond-based peridynamic model that achieves asymptotic compatibility and circumvents volume correction challenges through the use of a Gaussian kernel. The model is theoretically sound, computationally efficient due to truncation, and validated on both smooth and discontinuous deformation problems, offering a robust and simpler alternative to traditional PD models for fracture and failure simulation.