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

Fu Zhuojia; Li Mingjuan; Xi Qiang; Xu Wenzhi; Liu Qingguo

doi:10.6052/0459-1879-22-485

Volume 54 Issue 12

Dec. 2022

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Article Navigation > Chinese Journal of Theoretical and Applied Mechanics > 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 doi: 10.6052/0459-1879-22-485

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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 doi: 10.6052/0459-1879-22-485

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

doi: 10.6052/0459-1879-22-485

*.
Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University,Nanjing 211100, China
†.
Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana 1000, Slovenia
**.
Institute of Metals and Technology, Ljubljana 1000, Slovenia

Received Date: 2022-10-11
Accepted Date: 2022-11-16

Available Online: 2022-11-17

Publish Date: 2022-12-15

Abstract

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.
- collocation methods,
- physical-informed kernel functions,
- fundamental solutions,
- T-complete function,
- harmonic function

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References(33)

References

[1]	Zhang WW, Bernd RN. Artificial intelligence in fluid mechanics. Acta Mechanica Sinica, 2021, 37(12): 1715-1717
[2]	张雄, 刘岩. 无网格法. 北京: 清华大学出版社, 2004 Zhang Xiong, Liu Yan. Meshless Methods. Beijing: Tsinghua University Press, 2004 (in Chinese)
[3]	樊礼恒, 王东东, 刘宇翔等. 节点梯度光滑有限元配点法. 力学学报, 2021, 53(2): 467-481 (Fan Liheng, Wang Dongdong, Liu Yuxiang, et al. A finite element collocation method with smoothed nodal gradients. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 467-481 (in Chinese)
[4]	王莉华, 李溢铭, 褚福运. 基于分区径向基函数配点法的大变形分析. 力学学报, 2019, 51(3): 743-753 (Wang Lihua, Li Yiming, Chu Fuyun. Finite subdomain radial basis collocation method for large deformation analysis. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 743-753 (in Chinese)
[5]	Guo H, Zhuang X, Chen P, et al. Analysis of three-dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis. Engineering with Computers, 2022, doi: 10.1007/s00366-022-01633-6
[6]	陈增涛, 王发杰, 王超. 广义有限差分法在含阻抗边界空腔声学分析中的应用. 力学学报, 2021, 53(4): 1183-1195 (Chen Zengtao, Wang Fajie, Wang Chao. Application of generalized finite difference method in acoustic analysis of cavity with impedance boundary. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(4): 1183-1195 (in Chinese) doi: 10.6052/0459-1879-20-311
[7]	李煜冬, 傅卓佳, 汤卓超. 功能梯度碳纳米管增强复合材料板弯曲和模态的广义有限差分法. 力学学报, 2022, 54(2): 414-424 (Li Yudong, Fu Zhuojia, Tang Zhuochao. Generalized finite difference method for bending and modal analysis of functionally graded carbon nanotube-reinforced composite plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 414-424 (in Chinese)
[8]	Ren HL, Zhuang XY, Rabczuk T. A nonlocal operator method for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2020, 358: 112621 doi: 10.1016/j.cma.2019.112621
[9]	Wang LH, Qian ZH. A meshfree stabilized collocation method (SCM) based on reproducing kernel approximation. Computer Methods in Applied Mechanics and Engineering, 2020, 371: 113303 doi: 10.1016/j.cma.2020.113303
[10]	王莉华, 阮剑武. 配点型无网格法理论和研究进展. 力学季刊, 2021, 42(4): 613-632 (Wang Lihua, Ruan Jianwu. Theory and research progress of the collocation-type meshfree methods. Chinese Quarterly of Mechanics, 2021, 42(4): 613-632 (in Chinese) doi: 10.15959/j.cnki.0254-0053.2021.04.001
[11]	Hatami M. Weighted Residual Methods: Principles, Modifications and Applications. Academic Press, 2017
[12]	陈文, 傅卓佳, 魏星. 科学与工程计算中的径向基函数方法. 北京: 科学出版社, 2014 Chen Wen, Fu Zhuojia, Wei Xing. Radial Basis Function Methods in Scientific and Engineering Computing. Beijing: Science Press, 2014 (in Chinese))
[13]	Li ZC, Lu TT, Hu HY, et al. Trefftz and Collocation Methods. WIT Press, 2008
[14]	Cheng AHD, Hong Y. An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability. Engineering Analysis with Boundary Elements, 2020, 120: 118-152 doi: 10.1016/j.enganabound.2020.08.013
[15]	Young DL, Chen KH, Lee CW. Novel meshless method for solving the potential problems with arbitrary domain. Journal of Computational Physics, 2005, 209(1): 290-321 doi: 10.1016/j.jcp.2005.03.007
[16]	Šarler B. Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Engineering Analysis with Boundary Elements, 2009, 33(12): 1374-1382 doi: 10.1016/j.enganabound.2009.06.008
[17]	Chen W, Fu Z, Wei X. Potential problems by singular boundary method satisfying moment condition. Computer Modeling in Engineering and Sciences (CMES) , 2009, 54(1): 65-86
[18]	Liu YJ. A new boundary meshfree method with distributed sources. Engineering Analysis with Boundary Elements, 2010, 34(11): 914-919 doi: 10.1016/j.enganabound.2010.04.008
[19]	Takhteyev V, Brebbia CA. Analytical integrations in boundary elements. Engineering Analysis with Boundary Elements, 1990, 7(2): 95-100 doi: 10.1016/0955-7997(90)90027-7
[20]	Partridge PW, Brebbia CA. Dual Reciprocity Boundary Element Method. Springer Science & Business Media, 2012
[21]	Gao XW. A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity. International Journal for Numerical Methods in Engineering, 2006, 66(9): 1411-1431 doi: 10.1002/nme.1602
[22]	Nowak AJ, Neves AC. The Multiple Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publication, 1994
[23]	Chen W, Fu ZJ, Qin QH. Boundary particle method with high-order Trefftz functions. Computers, Materials & Continua (CMC) , 2009, 13(3): 201-217
[24]	Fu Z, Tang Z, Xi Q, et al. Localized collocation schemes and their applications. Acta Mechanica Sinica, 2022, 38(7): 422167 doi: 10.1007/s10409-022-22167-x
[25]	Benito JJ, Urena F, Gavete L. Influence of several factors in the generalized finite difference method. Applied Mathematical Modelling, 2001, 25(12): 1039-1053 doi: 10.1016/S0307-904X(01)00029-4
[26]	Guo L, Chen T, Gao XW. Transient meshless boundary element method for prediction of chloride diffusion in concrete with time dependent nonlinear coefficients. Engineering Analysis with Boundary Elements, 2012, 36(2): 104-111 doi: 10.1016/j.enganabound.2011.08.005
[27]	Fu ZJ, Yang LW, Xi Q, et al. A boundary collocation method for anomalous heat conduction analysis in functionally graded materials. Computers & Mathematics with Applications, 2021, 88: 91-109
[28]	Qiu L, Wang F, Lin J, et al. A novel combined space-time algorithm for transient heat conduction problems with heat sources in complex geometry. Computers & Structures, 2021, 247: 106495
[29]	陈文. 复杂科学与工程问题仿真的隐式微积分建模. 计算机辅助工程, 2014, 23(5): 1-6 (Chen Wen. Implicit calculus modeling for simulation of complex scientific and engineering problems. Computer Aided Engineering, 2014, 23(5): 1-6 (in Chinese) doi: 10.13340/j.cae.2014.05.001
[30]	Liang Y, Yu Y, Magin RL. Computation of the inverse Mittag–Leffler function and its application to modeling ultraslow dynamics. Fractional Calculus and Applied Analysis, 2022, 25(2): 439-452 doi: 10.1007/s13540-022-00020-8
[31]	Hansen PC. The truncated SVD as a method for regularization. BIT Numerical Mathematics, 1987, 27(4): 534-553 doi: 10.1007/BF01937276
[32]	Ohyama T, Beji S, Nadaoka K, et al. Experimental verification of numerical model for nonlinear wave evolutions. Journal of Waterway, Port, Coastal and Ocean Engineering, 1994, 120(6): 637-644 doi: 10.1061/(ASCE)0733-950X(1994)120:6(637)
[33]	Ma J, Chen W, Lin J. Crack analysis by using the enriched singular boundary method. Engineering Analysis with Boundary Elements, 2016, 72: 55-64 doi: 10.1016/j.enganabound.2016.08.004