A nonlocal operator method for solving partial differential equations
A nonlocal operator method for solving partial differential equations
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DOI：
10.1016/j.cma.2019.112621
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发表时间：
2018-10
期刊：
影响因子：
--
通讯作者：
H. Ren;X. Zhuang;T. Rabczuk
中科院分区：
文献类型：
--
作者：
H. Ren;X. Zhuang;T. Rabczuk
提出了一种适用于求解力学问题偏微分方程组的非局部算子法。在求解未知场的非局域相互作用模型意义上，非局域算符可视为与微分形式“等价”的积分形式。非局部算子的变分与无网格法或有限元方法中形函数的导数起到了等价的作用，从而避开了形函数及其导数计算中的许多困难。非局部算子法可以与导致弱形式的共同过程一致地应用，即变分原理和加权残值法。在此基础上，可以方便地得到残差和切线刚度矩阵。本文还对非局域算符方法进行了改进，引入了算符能量泛函，以满足场的线性一致性。由此推广了高阶非局域算子和高阶算子能量泛函。该方法的一个突出特点是基于非局部算子的泛函可以将剩余矩阵和刚度矩阵的构造转化为使用预定义的非局部算子的一系列矩阵乘法。通过引入支集和对偶支集的概念，可以很容易地得到不同泛函的非局部强形式。最后给出了几个不同类型偏微分方程组的数值算例，验证了该方法的有效性。
A nonlocal operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems. The nonlocal operator can be regarded as the integral form “equivalent” to the differential form in the sense of a nonlocal interaction model for solving the unknown field. The variation of a nonlocal operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method, thus it circumvents many difficulties in the calculation of shape functions and their derivatives. The nonlocal operator method can consistently applied with common procedure leading to the weak forms, i.e. the variational principle and the weighted residual method. Based on these, the residual and the tangent stiffness matrix can be obtained with ease. The nonlocal operator method is enhanced here also with an operator energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order operator energy functional are hereby generalized. A highlighted of the present method is the functional derived based on the nonlocal operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual-support, the two basic elements introduced in the paper. Several numerical examples of different types of PDEs are presented in the end to show the effectiveness of the present method and also serve for validation.