Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients
Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients
复制标题
通过非局部梯度对非局部模型的正交配置近似进行鲁棒后验应力分析
DOI：
10.1016/j.cma.2016.07.023
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发表时间：
2016
影响因子：
7.2
通讯作者：
Jiang Yang
中科院分区：
文献类型：
--
作者：
Q. Du;Yunzhe Tao;Xiaochuan Tian;Jiang Yang
作为偏微分方程 (PDE) 的替代方案，以积分形式给出的非局部连续体模型避免了传统空间导数的显式使用，并允许解表现出所需的奇异行为。开发鲁棒的数值方案不仅对于非局部模型的数值解而且对于评估适当定义的解的导数也具有实际意义。后者促进了偏微分方程数值解的梯度恢复非局部模拟的发展。对于结构力学模型，这会导致后验非局部应力分析。我们说明，当在非局部模型中找到平滑解时，可以使用偏微分方程等传统技术来计算非局部解的局部梯度。然而，更一般地说，我们提出了一个基于非局部梯度算子及其渐近兼容离散化的非局部解的应力分析框架。我们证明，非局部梯度恢复在局部极限内是一致的，并且比使用非局部解的局部梯度更有优势。针对非局部连续体模型，识别了一些特殊非局部梯度算子的超收敛性质。此外，还提出了在数值离散化中保留这些特征的方法。提供计算观察和理论见解来证实我们的发现。
As alternatives to partial differential equations (PDEs), nonlocal continuum models given in integral forms avoid the explicit use of conventional spatial derivatives and allow solutions to exhibit desired singular behavior. It is of practical interest to develop robust numerical schemes not only for the numerical solution of nonlocal models but also for the evaluation of suitably defined derivatives of solutions. The latter motivates the development of a nonlocal analog of gradient recovery for numerical solution of PDEs. For structure mechanical models, this leads to a posteriori nonlocal stress analysis. We illustrate that when smooth solutions are found in nonlocal models, one may compute local gradients of nonlocal solutions using conventional techniques like that for PDEs. More generically however, we present a framework for stress analysis of nonlocal solutions based on nonlocal gradient operators and their asymptotically compatible discretization. We demonstrate that the nonlocal gradient recovery is consistent in the local limit and is more advantageous than using local gradients of nonlocal solutions. Superconvergence properties of some special nonlocal gradient operators are identified for nonlocal continuum models. Moreover, methods are presented to preserve such features in the numerical discretization. Both computational observations and theoretical insights are provided to substantiate our findings.