为非局部问题设计的渐近兼容无网格求积规则及其在Peridynamics中的应用
An asymptotically compatible meshfree quadrature rule for non-local problems with applications to peridynamics
AI总结：
本文提出一种无网格求积规则，用于求解具有径向核的非局部积分微分方程，通过可重复约束实现渐近兼容性，适用于Peridynamics固体力学模型，处理自由面效应和损伤建模。
摘要：
我们提出了一种用于紧支撑非局部积分微分方程（IDEs）的无网格求积规则，具有径向核。我们将此规则应用于开发一种不需要背景网格的Peridynamics固体力学模型的强形式无网格离散化方法。现有Peridynamics模型的离散化方法已被证明缺乏对相应线性弹性局部解的渐近兼容性。通过将求积规则视为等式约束最小二乘问题，我们通过可重复约束获得渐近兼容的收敛性。我们的方法自然处理自由面条件、表面效应和损伤建模，适用于静态和动态问题。我们通过与制造解和具有裂纹奇异性且有解析解的案例进行比较，展示了高阶收敛性到局部理论。最后，我们通过重现Kalthoff-Winkler实验中的高速冲击结果来验证该方法在现实问题中的适用性。
英文摘要：
We present a meshfree quadrature rule for compactly supported non-local integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a strong-form 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 via reproducability 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.
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