AI 中文总结
该研究针对空间变 horizon 的键基近场动力学，从 Lagrange-d'Alembert 原理出发推导双 horizon 公式，构建异步变分积分器，消除非均匀 horizon 的虚假反射，减少内力评估量，适用于局部加密的动态断裂模拟。
AI 中文摘要
键基近场动力学提供了一种用于建模断裂的非局部框架，无需位移场的空间导数。然而，当空间变 horizon 与非均匀离散化结合使用时，经典的单 horizon 键基近场动力学公式会导致物质点之间的非对称相互作用。这些非对称相互作用违反了平衡定律，并可能引入非物理伪影，如幽灵力和虚假波反射。在这项工作中，我们开发了具有空间变 horizon 的键基近场动力学的变分公式。从 Lagrange-d'Alembert 原理出发，我们推导了运动控制方程，并表明双 horizon 近场动力学公式是从内能的变分中自然产生的。基于该变分结构，我们构建了异步变分积分器，允许在域的不同区域使用不同的时间步长。这对于具有局部加密的动态断裂模拟特别有用，其中仅在高分辨率区域或预期裂纹扩展区域附近需要小时间步长。涉及波传播、受拉预裂纹板和 Kalthoff-Winkler 冲击实验的数值例子表明，所提出的框架消除了由非均匀 horizon 引起的虚假反射，保留了物理一致的断裂模式，并取得了与均匀加密模拟相当的结果。同时，与标准速度-Verlet 方法相比，异步变分积分器减少了内力评估的数量。因此，所提出的方法为具有空间变 horizon 的键基近场动力学模拟提供了一致的变分基础和高效的时间积分策略。
英文摘要
Bond-based peridynamics provides a non-local framework for modelling fracture without requiring spatial derivatives of the displacement field. However, when spatially varying horizons are used together with non-uniform discretisations, the classical single-horizon bond-based peridynamics formulation leads to asymmetric interactions between material points. These asymmetric interactions violate balance laws and can introduce non-physical artefacts such as ghost forces and spurious wave reflections. In this work, we develop a variational formulation for bond-based peridynamics with spatially varying horizons. Starting from the Lagrange-d'Alembert principle, we derive the governing equations of motion and show that the dual-horizon peridynamics formulation emerges naturally from the variation of the internal energy. Building on this variational structure, we construct asynchronous variational integrators that allow different time step sizes in different regions of the domain. This is particularly useful for dynamic fracture simulations with local refinement, where small time steps are required only near regions of high resolution or expected crack growth. Numerical examples involving wave propagation, a pre-cracked plate under tension, and the Kalthoff-Winkler impact experiment demonstrate that the proposed framework removes spurious reflections caused by non-uniform horizons, preserves physically consistent fracture patterns, and achieves results comparable to uniformly refined simulations. At the same time, the asynchronous variational integrator reduces the number of internal force evaluations compared to the standard velocity-Verlet method. The proposed approach therefore provides a consistent variational foundation and an efficient time-integration strategy for bond-based peridynamic simulations with spatially varying horizons.
Comments33 pages, 17 figures, 2 tables; author name corrected, additional sections added at the end
