A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

Abstract

The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain non-local gradients of higher-order accuracy. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence model-based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative methods accompanied by higher-order gradient corrections provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

Coupling of IGA and Peridynamics for Air-Blast Fluid-Structure Interaction Using an Immersed Approach

“…The non-BA PD results are of poor quality due to the presence of unstable modes that deteriorate the solution and eventually lead to divergence of the simulation (around t = 0.25 ms here). The results of quadratic PD model equipped with BA stabilization are in good agreement with the reference solution (better than the linear + BA version), which confirms the conclusions of [55] that the combination of BA stabilization and higher-order corrections provides the best quality results in correspondence-based PD. In the remainder of this paper, we utilize the quadratic + BA model for the PD calculations.…”

“…Unless otherwise noted, RK functions with quadratic consistency, rectangular support, and the BA stabilization technique [34,54] are employed in the PD formulation. The PD support size δ is chosen with respect to the discretization size ∆x and the order of the kernel function n is chosen such that δ/∆x = n + 1 [55]. The air properties are used for the fluid with constant viscocity µ = 1.81 × 10 −5 kg/(m s), Prandtl number 0.72, and adiabatic index γ = 1.4.…”

IGA-PD Penalty-Based Coupling for Immersed Air-Blast Fluid-Structure Interaction: A Simple and Effective Solution for Fracture and Fragmentation

“…Weak forms of the fluid and structural mechanics equations are discretized on the background and foreground domains, respectively, and are coupled by means of a volumetric penalty operator, which is the main novelty of the proposed approach. We employ IGA based on NURBS in the background domain and a correspondence-based PD solid in the foreground domain using the RK functions to define the nonlocal derivatives [57,58]. We feel that the combination of these numerical methodologies is particularly attractive for the problem class of interest due to the higher-order accuracy and smoothness of IGA and RK, the benefits of using immersed methodology in handling the fluid-structure interfaces and coupling, and the unique capabilities of PD for modeling fracture and fragmentation.…”

“…RK functions with quadratic consistency, rectangular support, and bond-associative stabilization [10,56] are employed in the PD formulation. The PD support size δ is chosen with respect to the mesh size h and the quadratic order of the method, i.e., δ = 2.5h [57]. The fluid is assumed to have properties of air at room temperature, namely, initial density ρ = 1.0 kg/m 3 viscosity µ = 1.81 × 10 −5 kg/(m s), Prandtl number 0.72, and adiabatic index γ = 1.4.…”

Coupling of non‐ordinary state‐based peridynamics and finite element method with reduced boundary effect

“…74 The accuracy and convergence of the high-order peridynamic formulation were demonstrated in both uniform and nonuniform discretizations without ghost boundary nodes. Behzadinasab et al 75 implemented the bond-associative approach to a high-order correspondence formulation and proposed a new method to enforce natural boundary conditions in correspondence peridynamic formulations.…”