A novel energy-fitted hexagonal quadrature scheme enables low-cost and high-fidelity peridynamic computations

Abstract

In this contribution, we propose a novel hexagonal quadrature scheme for one-neighbor interactions in continuum-kinematics-inspired peridynamics equivalent to bond-based peridynamics. The hexagonal quadrature scheme is fitted to correctly integrate the stored energy density within the nonlocal finite-sized neighborhood of a continuum point subject to affine expansion. Our proposed hexagonal quadrature scheme is grid-independent by relying on appropriate interpolation of pertinent quantities from collocation to quadrature points. In this contribution, we discuss linear and quadratic interpolations and compare our novel hexagonal quadrature scheme to common grid-dependent quadrature schemes. For this, we consider both, tetragonal and hexagonal discretizations of the domain. The accuracy of the presented quadrature schemes is first evaluated and compared by computing the stored energy density of various prescribed affine deformations within the nonlocal neighborhood. Furthermore, we perform three different boundary value problems, where we measure the effective Poisson’s ratio resulting from each quadrature scheme and evaluate the deformation of a unit square under extension and beam bending. Key findings of our studies are: The Poisson’s test is a good indicator for the convergence behavior of quadrature schemes with respect to the grid density. The accuracy of quadrature schemes depends, as expected, on their ability to appropriately capture the deformation within the nonlocal neighborhood. Our novel hexagonal quadrature scheme, rendering the correct effective Poisson’s ratio of $1/3$ for small deformations, together with quadratic interpolation consequently yields the most accurate results for the studies presented in this contribution, thereby effectively reducing the computational cost.