Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations

The peridynamics theory is a reformulation of the classical theory of continuum mechanics, based on long-range interactions, suitable for the description of material failure and damage. Integration plays a central role in this theory. We study one-point quadrature algorithms for the discretization of two-dimensional integrals in peridynamics. These algorithms are closely related to meshfree methods; they assume an underlying reference lattice, where each lattice point within a body is assigned a cell. A main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell. We provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas, leading to improved integration accuracy. To address convergence issues, geometric centers of intersecting regions are taken as quadrature points, replacing commonly used cell centers. We present analytical derivations for the exact calculation of those geometric centers, based on a decomposition of intersecting regions into subdomains of simple geometry. By using exact values for the areas and geometric centers of intersecting regions, we achieve an asymptotically monotonic convergence in numerical integration; in contrast, other algorithms from the literature exhibit a highly-oscillatory behavior. Numerical results compare the proposed algorithms with others from the literature, for different quantities of interest, and demonstrate their improved accuracy and convergence. Error estimates for the quadrature algorithms are derived, and extended and hybrid algorithms are discussed. Additional numerical studies, using influence functions with a finite support and a controlled regularity, suggest an alternative means to improve the convergence of discretization algorithms.