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Localized collocation schemes and their applications

局部配点法及其应用

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Abstract

This paper presents a summary of various localized collocation schemes and their engineering applications. The basic concepts of localized collocation methods (LCMs) are first introduced, such as approximation theory, semianalytical collocation methods and localization strategies. Based on these basic concepts, five different formulations of localized collocation methods are introduced, including the localized radial basis function collocation method (LRBFCM) and the generalized finite difference method (GFDM), the localized method of fundamental solutions (LMFS), the localized radial Trefftz collocation method (LRTCM), and the localized collocation Trefftz method (LCTM). Then, several additional schemes, such as the generalized reciprocity method, Laplace and Fourier transformations, and Krylov deferred correction, are introduced to enable the application of the LCM to large-scale engineering and scientific computing for solving inhomogeneous, nonisotropic and time-dependent partial differential equations. Several typical benchmark examples are presented to show the recent developments and applications on the LCM solution of some selected boundary value problems, such as numerical wave flume, potential-based inverse electrocardiography, wave propagation analysis and 2D phononic crystals, elasticity and in-plane crack problems, heat conduction problems in heterogeneous material and nonlinear time-dependent Burgers’ equations. Finally, some conclusions and outlooks of the LCMs are summarized.

摘要

本文总结了局部配点法的各种离散格式及工程应用. 首先介绍了局部配点法(LCMs)的基本概念, 如逼**理论、半解析配点 法和局部化策略. 基于这些基本概念, 介绍了局部径向基函数配点法(LRBFCM)和广义有限差分法(GFDM)、局部基本解法 (LMFS)、局部径向Trefftz配点法(LRTCM), 局部Trefftz配点法(LCTM). 随后引入了一些诸如如广义互易法(GRM)、拉普拉斯/傅 里叶变换、Krylov延迟校**法等技术, 使得上述局部配点法能用于基于非均质、各向异性和瞬态偏微分方程的大规模工程和科学 计算. 通过几个典型的基准算例, 展示了局部配点法求解边值问题的最新研究进展和应用, 如数值波浪水槽、基于电位势的反向肌 电图生成、波传播分析和声子晶体波传播特性、弹性力学和面内裂纹, 各向异性材料热传导和非线性Burgers方程等. 最后给出局 部配点法的一些研究结论及未来展望.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12122205 and 11772119), and the Six Talent Peaks Project in Jiangsu Province of China (Grant No. 2019-KTHY-009).

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Authors and Affiliations

  1. Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nan**g, 210098, China

    Zhuojia Fu  (傅卓佳) & Zhuochao Tang  (汤卓超)

  2. Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nan**g, 211100, China

    Zhuojia Fu  (傅卓佳), Zhuochao Tang  (汤卓超) & Qiang **  (****)

  3. Department of Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, 1000, Slovenia

    Qingguo Liu  (刘庆国)

  4. Laboratory for Simulation of Materials and Processes, Institute of Metals and Technology, Ljubljana, 1000, Slovenia

    Qingguo Liu  (刘庆国)

  5. School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071, China

    Yan Gu  (谷岩)

  6. College of Mechanical and Electrical Engineering, National Engineering Research Center for Intelligent Electrical Vehicle Power System, Qingdao University, Qingdao, 266071, China

    Fajie Wang  (王发杰)

Authors
  1. Zhuojia Fu  (傅卓佳)
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  2. Zhuochao Tang  (汤卓超)
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  3. Qiang **  (****)
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  4. Qingguo Liu  (刘庆国)
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  5. Yan Gu  (谷岩)
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  6. Fajie Wang  (王发杰)
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Corresponding authors

Correspondence to Zhuojia Fu  (傅卓佳) or Zhuochao Tang  (汤卓超).

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Author contributions

Zhuochao Tang wrote the first draft of this review article and Zhuojia Fu revised the manuscript and approved the final version. The programmes of localized collocation methods including LRBFCM, GFDM, LMFS, LRTCM and LCTM were realized respectively by Zhuochao Tang, Zhuojia Fu, Qiang **, Qingguo Liu, Yan Gu and Fajie Wang. The visualizations are made by Zhuochao Tang. The research of LCMs for various applications in this paper is under the management and supervision of Zhuojia Fu.

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Fu, Z., Tang, Z., **, Q. et al. Localized collocation schemes and their applications. Acta Mech. Sin. 38, 422167 (2022). https://doi.org/10.1007/s10409-022-22167-x

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