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localized method of fundamental solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

Advances in Applied Mathematics and Mechanics 2023年第1期15卷 182-201页

作者： Zengtao Chen Fajie Wang College of Mechanical and Electrical Engineering National Engineering Research Center for Intelligent Electrical Vehicle Power SystemQingdao UniversityQingdaoShandong 266071China Institute of Mechanics for Multifunctional Materials and Structures Qingdao UniversityQingdaoShandong 266071China

This paper documents the first attempt to apply a localized method of fundamental solutions(LMFS)to the acoustic analysis of car cavity containing soundabsorbing *** LMFS is a recently developed meshless approach with the merits of being mathematically simple,numerically accurate,and requiring less computer time and *** with the traditional method of fundamental solutions(MFS)with a full interpolation matrix,the LMFS can obtain a sparse banded linear algebraic system,and can circumvent the perplexing issue of fictitious boundary encountered in the MFS for complex solution *** the LMFS,only circular or spherical fictitious boundary is *** on these advantages,the method can be regarded as a competitive alternative to the standard method,especially for high-dimensional and large-scale *** benchmark numerical examples are provided to verify the effectiveness and performance of the present method for the solution of car cavity acoustic problems with impedance conditions.

关键词： Acoustic analysis localized method of fundamental solutions car cavity soundabsorbing material

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localized method of fundamental solutions for two- and three-dimensional transient convection-diffusion-reaction equations

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2021年 124卷 237-244页

作者： Liu, Shuainan Li, Po-Wei Fan, Chia-Ming Gu, Yan Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan

This paper investigates the use of the localized method of fundamental solutions (LMFS) for the numerical solution of general transient convection-diffusion-reaction equation in both two-(2D) and three-dimensional (3D) materials. The method is developed as a generalization of the author's earlier work on Laplace's equation to transient convection-diffusion-reaction equation. The popular Crank-Nicolson (CN) time-stepping technology is adopted to perform the temporal simulations. The LMFS approach is then introduced for solving the resulting inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. As compared with the classical MFS and boundary element method (BEM), the present CN-CCS-LMFS approach produces sparse and banded stiffness matrix which makes the method possible to perform large-scale dynamic simulations. Several benchmark numerical examples are presented to demonstrate the efficiency and feasibility of the present method.

关键词： localized method of fundamental solutions Meshless boundary collocation method Transient convection-diffusion-reaction Particular solutions fundamental solutions

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localized method of fundamental solutions for two-dimensional anisotropic elasticity problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2021年 125卷 59-65页

作者： Liu, Q. G. Fan, C. M. Sarler, B. Inst Met & Technol Lab Simulat Mat & Proc Ljubljana Slovenia Univ Ljubljana Fac Mech Engn Dept Fluid Dynam & Thermodynam Ljubljana Slovenia Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung Taiwan Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung Taiwan

The purpose of the presented paper is to develop further the localized method of fundamental solutions (LMFS) for solving two-dimensional anisotropic elasticity problems. The computational domain is divided into overlapping subdomains. In the LMFS, the classical method of fundamental solutions (MFS) is employed in each of these local subdomains to get an expression of the solution for the main node of this subdomain. The expression is structured by a linear combination of the solutions of the other nodes in this subdomain. Displacement or traction boundary conditions are satisfied at the boundary nodes. The solution is calculated from an equation set formed by the boundary conditions for the boundary nodes and expressions in the subdomain for the interior nodes. The presented three numerical examples demonstrate that the novel method is suitable for solving large-scale problems, and especially, the problems with complicated domains.

关键词： Anisotropic elasticity problems Displacement and traction boundary conditions fundamental solution localized method of fundamental solutions

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Three-dimensional numerical wave tank containing submerged breakwaters based on the localized method of fundamental solutions

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MATHEMATICS AND COMPUTERS IN SIMULATION 2025年 229卷 273-287页

作者： Li, Lanlan Fu, Zhuojia Qin, Ming Liu, Shuainan Zeng, Weihong Liu, Xiaoting Hohai Univ Key Lab Minist Educ Coastal Disaster & Protect Nanjing 210098 Peoples R China China Three Gorges Corp Inst Sci & Technol Res Beijing 101199 Peoples R China Bauhaus Univ Weimar Inst Struct Mech Marienstr 15 Weimar Germany

This paper presents a meshless computational framework to simulate nonlinear water wave propagation behaviors of three-dimensional (3D) numerical wave tank containing submerged trapezoidal breakwaters. In the present computational framework, the localized method of fundamental solutions (LMFS) is used to spatial discretization, which is a localized meshless collocation method based on fundamental solutions and moving least square (MLS) technique, and fourth-order predictor-corrector scheme is used to temporal discretization. 3D numerical wave tanks containing single submerged trapezoidal breakwater and double submerged trapezoidal breakwaters are established by using LMFS. The LMFS results are compared with analytical results and experimental data through benchmarks. The effects of variations in incident wave parameters and shape parameters of the submerged trapezoidal breakwater on wave propagation are further analyzed. Additionally, the influence of double submerged trapezoidal breakwaters on the nonlinear water wave propagation behavior is investigated in comparison with the single submerged trapezoidal breakwater. Moreover, the effect of breakwater orientation on wave propagation is presented.

关键词： Nonlinear water waves localized method of fundamental solutions Numerical wave tank Submerged trapezoidal breakwater Fourth-order predictor-corrector scheme

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The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2022年 139卷 208-220页

作者： Zhao, Shengdong Gu, Yan Fan, Chia-Ming Wang, Xiao Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China Qingdao Univ Inst Mech Multifunct Mat & Struct Qingdao 266071 Peoples R China Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan

In this paper, a new framework for the numerical solutions of general nonlinear problems is presented. By employing the analog equation method, the actual problem governed by a nonlinear differential operator is converted into an equivalent problem described by a simple linear equation with unknown fictitious body forces. The solution of the substitute problem is then obtained by using the localized method of fundamental solutions, where the fictitious nonhomogeneous term is approximated using the dual reciprocity method using the radial basis functions. The main difference between the classical and the present localized method of fundamental solutions is that the latter produces sparse and banded stiffness matrix which makes the method very suitable for large-scale nonlinear simulations, since sparse matrices are much cheaper to inverse at each iterative step of the Newton's method. The present method is simple in derivation, efficient in calculation, and may be viewed as a completive alternative for nonlinear analysis, especially for large-scale problems with complex-shape geometries. Preliminary numerical experiments involving second-order nonlinear boundary value problems in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method.

关键词： Nonlinear problems localized method of fundamental solutions Meshless collocation method Analog equation method Large-scale problem Radial basis function

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The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

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MATHEMATICS AND COMPUTERS IN SIMULATION 2022年第0期200卷 504-524页

作者： Zhang, Junli Yang, Chenchen Zheng, Hui Fan, Chia-Ming Fu, Ming-Fu Nanchang Univ Sch Civil Engn & Architecture Nanchang 330031 Peoples R China Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan

In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multidimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme. (c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： localized method of fundamental solutions Meshless method Recursive composite multiple reciprocity method Multi-dimensional boundary value problems Inhomogeneous partial differential equations

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Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the localized method of fundamental solutions

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Advances in Applied Mathematics and Mechanics 2023年第4期15卷 880-900页

作者： Yan Gu Ji Lin Chia-Ming Fan Key Laboratory of Road Construction Technology&Equipment Chang’an UniversityXi’anShaanxi 710064China School of Mathematics and Statistics Qingdao UniversityQingdaoShandong 266071China Department of Engineering Mechanics College of Mechanics and MaterialsHohai UniversityNanjingJiangsuChina Department of Harbor and River Engineering&Computation and Simulation Center Taiwan Ocean UniversityKeelung 20224TaiwanChina

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational *** the past few decades,the method of fundamental solutions(MFS)has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering *** classical MFS formulation,however,leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering *** this paper,a localized version of the MFS(LMFS)is devised for electroelastic analysis of twodimensional(2D)piezoelectric *** the LMFS,the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square(MLS)technique are *** to the classical MFS,the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale *** numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

关键词： localized method of fundamental solutions meshless methods piezoelectric structures coupled electroelastic analysis fundamental solutions

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Domain-Decomposition localized method of fundamental solutions for Large-Scale Heat Conduction in Anisotropic Layered Materials

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Advances in Applied Mathematics and Mechanics 2022年第3期14卷 759-776页

作者： Shuainan Liu Zhuojia Fu Yan Gu Center for Numerical Simulation Software in Engineering and Sciences College of Mechanics and MaterialsHohai UniversityNanjingJiangsu 211100China School of Mathematics and Statistics Qingdao UniversityQingdaoShandong 266071China

The localized method of fundamental solutions(LMFS)is a relatively new meshless boundary collocation *** the LMFS,the global MFS approxima-tion which is expensive to evaluate is replaced by local MFS formulation deﬁned in a set of overlapping *** LMFS algorithm therefore converts differential equations into sparse rather than dense matrices which are much cheaper to *** paper makes theﬁrst attempt to apply the LMFS,in conjunction with a domain-decomposition technique,for the numerical solution of steady-state heat con-duction problems in two-dimensional(2D)anisotropic layered ***,the layered material is decomposed into several subdomains along the layer-layer inter-faces,and in each of the subdomains,the solution is approximated by using the LMFS *** the subdomain interface,compatibility of temperatures and heatﬂuxes are *** numerical experiments illustrate that the proposed domain-decomposition LMFS algorithm is accurate,stable and computationally efﬁcient for the numerical solution of large-scale multi-layered materials.

关键词： Meshless method localized method of fundamental solutions heat conduction prob-lems layered materials large-scale problems

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localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2019年 101卷 188-197页

作者： Fan, C. M. Huang, Y. K. Chen, C. S. Kuo, S. R. Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung 20224 Taiwan Nanchang Univ Sch Civil Engn & Architecture Nanchang 330031 Jiangxi Peoples R China Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA

The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.

关键词： localized method of fundamental solutions Laplace equation Biharmonic equation Meshless method Complicated domain

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localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems - Theory and MATLAB code

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COMPUTERS & STRUCTURES 2019年 220卷 144-155页

作者： Gu, Yan Fan, Chia-Ming Qu, WenZhen Wang, Fajie Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung 20224 Taiwan Shandong Univ Technol Inst Appl Math Zibo 255049 Peoples R China

The method of fundamental solutions (MFS) belongs to the family of meshless boundary collocation methods and now has been successfully tried for many kinds of engineering problems. The traditional MFS based on the "global" boundary discretization, however, leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. Such an approach will be arduous, time consuming and computationally expensive for analyzing large-scale problems. In the present work, a localized version of the MFS, named as the localized MFS (LMFS), is proposed for large-scale modelling of three-dimensional (3D) anisotropic heat conduction problems. In the LMFS, the computational domain can be divided into small subdomains with a simple geometry such as circle and/or sphere. To each of the subdomains, the MFS formulation is applied and the unknown coefficients on the local simple geometric boundary are approximated by the moving least square (MLS) method. The satisfactions of governing equations at interior points and boundary conditions at boundary nodes lead to a sparse and banded system matrix. Numerical examples with up to 1,000,000 unknowns are solved successfully using the developed LMFS code. The advantages, disadvantages and potential applications of the proposed method, as compared with the traditional MFS and boundary element method (BEM), are discussed. Finally, a fast, reliable and self-contained MATLAB code is provided in the part of Supplementary Materials of the paper. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： localized method of fundamental solutions Meshless method Large-scale problem Three-dimensional anisotropic heat conduction problem Complicated domain

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