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 引用本文: 傅卓佳, 李明娟, 习强, 徐文志, 刘庆国. 物理信息依赖核函数配点法的研究进展. 力学学报, 2022, 54(12): 3352-3365.
Fu Zhuojia, Li Mingjuan, Xi Qiang, Xu Wenzhi, Liu Qingguo. Research advances on the collocation methods based on the physical-informed kernel functions. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3352-3365.
 Citation: Fu Zhuojia, Li Mingjuan, Xi Qiang, Xu Wenzhi, Liu Qingguo. Research advances on the collocation methods based on the physical-informed kernel functions. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3352-3365.

## RESEARCH ADVANCES ON THE COLLOCATION METHODS BASED ON THE PHYSICAL-INFORMED KERNEL FUNCTIONS

• 摘要: 在过去几十年里, 尽管有限元等传统计算方法已被成功用于众多科学与工程领域, 但是其在数值模拟无限域波传播、大尺寸比结构、工程反演和移动边界问题时仍面临计算量大、计算效率低、网格生成困难等计算难题. 本文介绍一类基于物理信息依赖核函数的无网格配点法及其在上述难点问题中的应用. 物理信息依赖核函数配点法的关键在于构建能反映问题微分控制方程物理信息的基函数. 基于这些物理信息依赖核函数, 该方法无需/仅需少量配点对所求微分控制方程进行离散, 即可有效提高计算效率. 本文首先介绍满足常见齐次微分方程的基本解、调和函数、径向Trefftz函数以及T完备函数等典型物理信息依赖核函数. 接着依次介绍非齐次、非均质、非稳态以及隐式微分方程构造物理信息依赖核函数的方法. 随后, 根据所求问题特点, 选用全域配点或局部配点技术, 建立相应的物理信息依赖核函数配点法. 最后, 通过几个典型算例验证所提物理信息依赖核函数配点法的有效性.

Abstract: In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods is to construct the related basis functions, which includes the physical information of the considered differential governing equation. Based on these physical-informed kernel functions, these methods do not need/only need a few collocation nodes to discretize the considered differential governing equations, which may effectively improve the computational efficiency. In this paper, several typical physical-informed kernel functions that satisfy common-used homogeneous differential equations, such as the fundamental solutions, the harmonic functions, the radial Trefftz functions and the T-complete functions and so on, are firstly introduced. After that, the ways to construct the physical-informed kernel functions for nonhomogeneous differential equations, inhomogeneous differential equations, unsteady-state differential equations and implicit differential equations are introduced in turn. Then according to the characteristics of the considered problems, the global collocation scheme or the localized collocation scheme is selected to establish the corresponding physical-informed kernel function collocation method. Finally, four typical examples are given to verify the effectiveness of the physical-informed kernel function collocation methods proposed in this paper.

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