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.

中文翻译：

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一种新颖的能量拟合六边形正交方案可实现低成本和高保真度的近场动力学计算


在这项贡献中，我们提出了一种新的六边形正交方案，用于连续运动学启发的近场动力学中的单邻相互作用，相当于基于键的近场动力学。拟合六边形正交方案是为了在受仿射展开的连续体点的非局部有限大小邻域内正确积分存储的能量密度。我们提出的六边形正交方案是独立于网格的，它依赖于从搭配到正交点的相关量的适当插值。在这篇文章中，我们讨论了线性和二次插值，并将我们新颖的六边形正交方案与常见的网格相关正交方案进行了比较。为此，我们考虑了域的四边形和六边形离散化。首先，通过计算非局部邻域内各种指定仿射变形的存储能量密度，评估和比较所提出的正交方案的准确性。此外，我们执行了三个不同的边界值问题，其中我们测量了每个正交方案产生的有效泊松比，并评估了单位平方在拉伸和梁弯曲下的变形。我们研究的主要发现是：泊松检验是正交方案相对于网格密度收敛行为的良好指标。正如预期的那样，正交方案的准确性取决于它们正确捕获非局部邻域内变形的能力。 我们新颖的六边形正交方案，为小变形提供了 1/3 的正确有效泊松比，再加上二次插值，因此为本文中提出的研究产生了最准确的结果，从而有效地降低了计算成本。