An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics
An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics
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DOI：
10.1016/j.cma.2018.08.016
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发表时间：
2018-01
影响因子：
7.2
通讯作者：
Nathaniel Trask;Huaiqian You;Yue Yu;M. Parks
中科院分区：
文献类型：
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作者：
Nathaniel Trask;Huaiqian You;Yue Yu;M. Parks
提出了一种求解径向核紧支撑非局部积分微分方程的无网格求积法则。我们应用这一规则来开发一个无网格离散化的周向固体力学模型，不需要背景网格。现有的离散化的周波模型已被证明缺乏渐近相容性，相应的线弹性局部解。通过将求积法则转化为等式约束最小二乘问题，引入多项式再生约束，得到了渐近相容收敛性。我们的方法自然地处理无牵引力条件，表面效应和静态和动态问题的损伤建模。我们证明了高阶收敛到本地理论比较制造的解决方案和裂纹奇异性的情况下，解析解是可用的。最后，我们验证了该方法的适用性，以现实的问题，再现高速冲击的结果从Kalthoff-Winkler实验。
We present a meshfree quadrature rule for compactly supported nonlocal integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a meshfree discretization of a peridynamic solid mechanics model that requires no background mesh. Existing discretizations of peridynamic models have been shown to exhibit a lack of asymptotic compatibility to the corresponding linearly elastic local solution. By posing the quadrature rule as an equality constrained least squares problem, we obtain asymptotically compatible convergence by introducing polynomial reproduction constraints. Our approach naturally handles traction-free conditions, surface effects, and damage modeling for both static and dynamic problems. We demonstrate high-order convergence to the local theory by comparing to manufactured solutions and to cases with crack singularities for which an analytic solution is available. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff–Winkler experiments.