Accurate computation of partial volumes in 3D peridynamics

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.