Accurate computation of partial volumes in 3D peridynamics

Abstract

The peridynamic theory is a nonlocal formulation of continuum mechanics based on integro-differential equations, devised to deal with fracture in solid bodies. In particular, the forces acting on each material point are evaluated as the integral of the nonlocal interactions with all the neighboring points within a spherical region, called “neighborhood”. Peridynamic bodies are commonly discretized by means of a meshfree method into a uniform grid of cubic cells. The numerical integration of the nonlocal interactions over the neighborhood strongly affects the accuracy and the convergence behavior of the results. However, near the boundary of the neighborhood, some cells are only partially within the sphere. Therefore, the quadrature weights related to those cells are computed as the fraction of cell volume which actually lies inside the neighborhood. This leads to the complex computation of the volume of several cube–sphere intersections for different positions of the cells. We developed an innovative algorithm able to accurately compute the quadrature weights in 3D peridynamics for any value of the grid spacing (when considering fixed the radius of the neighborhood). Several examples have been presented to show the capabilities of the proposed algorithm. With respect to the most common algorithm to date, the new algorithm provides an evident improvement in the accuracy of the results and a smoother convergence behavior as the grid spacing decreases.

An improved coupling of 3D state‐based peridynamics with high‐order 1D finite elements to reduce spurious effects at interfaces

“…The peridynamic body is discretized by the meshfree method with a uniform grid spacing h, which is arguably the most commonly used method. [27][28][29] Each node represents a cell with a volume V = h 3 . Consider a node i and its neighborhood  i , as shown in Figure 2.…”

“…The peridynamic body is discretized by the meshfree method with a uniform grid spacing $$$ h $$$, which is arguably the most commonly used method 27‐29 . Each node represents a cell with a volume $$$ V={h}^3 $$$.…”

“…Clearly, a node is considered part of the neighborhood only if the quadrature coefficient of its cell is $$$ \beta >0 $$$. The computation of the quadrature coefficients in 3D peridynamics can be carried out in several ways (see, for instance, References 29‐31). For simplicity, we adopt the method illustrated in Reference 32 for this work.…”

Surface node method for the peridynamic simulation of elastodynamic problems with Neumann boundary conditions

“…where 𝑥𝑥 𝑖𝑖 and 𝑥𝑥 𝑗𝑗 are respectively the coordinates of node 𝑖𝑖 and any node 𝑗𝑗 within the neighborhood 𝐻𝐻 𝑖𝑖 of node 𝑖𝑖, and 𝛽𝛽 𝑖𝑖𝑗𝑗 is the quadrature coefficient, namely the fraction of cell of node 𝑗𝑗 which lies within the neighborhood 𝐻𝐻 𝑖𝑖 [5]. The explicit central difference method is used for time integration [4]:…”

Peridynamic simulation of elastic wave propagation by applying the boundary conditions with the surface node method

“…In this work, the surface node method (see details in the following) is used to mitigate the PD surface effect and impose local boundary conditions in the peridynamic model. This method has been applied to quasi-static mechanical problems [9][10][11] and to a diffusion problem evolving over time [12]. Here we extend it to elastodynamic problems in 1D.…”