Accurate numerical integration in 3D meshless peridynamic models

In mathematical terms, physical problems are often described by equations involving the computation of integrals, either because they appear in the weak form of differential equations or for the nature itself of the strong formulation. The recent theory of Peridynamics expresses the internal forces of mechanical problems as the integral of the nonlocal interactions of the material points. The numerical evaluation of those integrals strongly affects the accuracy of the solution of discretized peridynamic problems. The most common discretization of a peridynamic body is a meshless regular grid of points such that a cube is the volume associated to every point of the grid. Since the neighborhood of every (source) node is a sphere containing a large number of (family) nodes the intersection between the neighborhood and some of the cubic volumes associated to the family nodes is partial, and it is a complex function of the ratio between the horizon of the neighborhood (δ) and the grid spacing (h). Such a problem has affected the numerical solution of peridynamic problems since the first time Peridynamics was proposed. The paper presents the following two main results: - the exact solution to the geometrical problem of computing the partial volumes generated by all possible combinations of δ/h;- the application of the above exact evaluation to the numerical integration of peridynamic problems and the evaluation of the impact of the exact integration on the solution of structural problems. Several examples will illustrate various aspects of the newly proposed numerical integration techniques.