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 引用本文: 李志远, 黄丹, Timon Rabczuk. 近场动力学算子方法. 力学学报, 2023, 55(7): 1593-1603.
Li Zhiyuan, Huang Dan, Timon Rabczuk. Peridynamic operator method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(7): 1593-1603.
 Citation: Li Zhiyuan, Huang Dan, Timon Rabczuk. Peridynamic operator method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(7): 1593-1603.

## PERIDYNAMIC OPERATOR METHOD

• 摘要: 提出一种基于非局部思想求解物理学问题的近场动力学算子方法(peridynamic operator method, PDOM). 运用PDOM可将任意阶局部微分及其乘积转化为相应的非局部积分形式, 且无需额外地特殊处理间断点与奇异点等问题. 近年来研究较多的两种非局部算子: 近场动力学微分算子(peridynamic differential operator, PDDO)和非局部算子方法(nonlocal operator method, NOM), 均可视为PDOM的一种特例. 以弹性力学问题为例, 采用变分原理和拉格朗日方程, 建立了适用于分析静/动态弹性力学问题的PDOM模型. 理论分析表明, 当分别限定相互作用域为与位置无关或位置相关的圆形域时, 该PDOM弹性模型即可退化为近年来文献中常见的近场动力学(peridynamics, PD)模型或对偶域近场动力学(dual-horizon peridynamics, DH-PD)模型. 通过3个典型实例: 杆的拉伸与波动、亥姆霍兹方程和含孔板的拉伸, 说明本方法的计算精度、收敛性与数值稳定性. PDOM方法适用于任意均匀或非均匀离散, 且能有效避免零能模式以及由其引起的数值振荡, 可望为各种物理学问题特别是不连续问题的非局部建模求解提供一种新选择.

Abstract: A new method based on non-local theories, named as peridynamic operator method (PDOM), for solving ordinary and partial differential equations in physics, is proposed in the present work. By using the peridynamic operator method, the local differentiations of any orders as well as their products can be converted into corresponding nonlocal integral forms without any extra remedies or special treatment in the presence of discontinuities or singularities. It can be proved that both the so-called peridynamic differential operator (PDDO) and the nonlocal operator method (NOM), two nonlocal operators which have gained much concern of researchers in the field of computational mechanics in recent years, can be seen as special cases of this proposed PDOM. As a typical application example, linear elastic PDOM model for static and dynamic elasticity problems is developed by employing the variational principles and Lagrange's equations. Theoretical analysis shows that when the nonlocal interaction domain is defined by position-independent or dependent circles, PDOM elasticity model can be simplified to the classical peridynamic (PD) model or the dual-horizon peridynamic (DH-PD) model in literature correspondingly. The accuracy, convergence, as well as the numerical stability of the presented method are validated by analyzing three typical examples, including tension and wave motion in a bar, Helmholtz equation, and tensile deformation of plates with hole. It is shown that the proposed method can be effectively used with both uniform and non-uniform discretization, and it can naturally avoid the zero-energy modes and numerical oscillations which will occur when the original non-ordinary state-based peridynamic simulations, the peridynamic differential operator or the nonlocal operator method is employed without extra remedies. PDOM can provide a potential alternative to develop the nonlocal models for various physical problems especially involving discontinuities.

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