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The method of fundamental solutions for Steady-State Heat Conduction in Nonlinear Materials

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Communications in Computational Physics 2008年第9期4卷 911-928页

作者： A.Karageorghis D.Lesnic Department of Mathematics and Statistics University of Cyprus1678 NicosiaCyprus. Department of Applied Mathematics University of LeedsLeeds LS29JTUK.

The steady-state heat conduction in heat conductors with temperature dependent thermal conductivity and mixed boundary conditions involving radiation is investigated using the method of fundamental solutions.Various computational issues related to the method are addressed and numerical results are presented and discussed for problems in two and three dimensions.

关键词： Nonlinear heat conduction radiation method of fundamental solutions

A Practical Algorithm for Determining the Optimal Pseudo-Boundary in the method of fundamental solutions

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Advances in Applied Mathematics and Mechanics 2009年第4期1卷 510-528页

作者： A.Karageorghis Department of Mathematics and Statistics University of CyprusP.O.Box 205371678 NicosiaCyprus

One of the main difficulties in the application of the method of fundamental solutions(MFS)is the determination of the position of the pseudo-boundary on which are placed the singularities in terms of which the approximation is expressed.In this work,we propose a simple practical algorithm for determining an estimate of the pseudo-boundary which yields the most accurate MFS approximation when the method is applied to certain boundary value problems.Several numerical examples are provided.

关键词： method of fundamental solutions elliptic boundary value problems function minimization

Galerkin Formulations of the method of fundamental solutions

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Advances in Applied Mathematics and Mechanics 2013年第4期5卷 423-441页

作者： J.R.Berger Andreas Karageorghis Department of Mechanical Engineering Colorado School of MinesGoldenCO 80401USA Department of Mathematics and Statistics University of CyprusP.O.Box 205371678 NicosiaCyprus

In this paper,we introduce two Galerkin formulations of themethod of fundamental solutions(MFS).In contrast to the collocation formulation of the MFS,the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration.On the other hand,these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases.Several numerical examples are considered by these methods and their various features compared.

关键词： Galerkin formulation Laplace equation method of fundamental solutions

The method of fundamental solutions for Solving Convection-Diffusion Equations with Variable Coefficients

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Advances in Applied Mathematics and Mechanics 2009年第2期1卷 215-230页

作者： C.M.Fan C.S.Chen J.Monroe Department of Mathematics University of Southern MississippiHattiesburgMS 39406USA Department of Mathematics Spring Hill CollegeMobileAL 36608USA

A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization and the onestage MFS for spatial discretization.In contrast to the traditional two-stage process,the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations.The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easy to find than the traditional approach.The numerical results show that the one-stage approach is robust and stable.

关键词： Meshless method method of fundamental solutions particular solution singular value decomposition time-dependent partial differential equations

The method of fundamental solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

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Advances in Applied Mathematics and Mechanics 2013年第4期5卷 477-493页

作者： Wen Chen Ji Lin C.S.Chen College of Engineering Mechanics Hohai UniversityNanjing 210098JiangsuChina Department of Mathematics University of Southern MississippiHattiesburgMS 39406USA

In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fromtheMFS approximation has a block circulant structure,it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space.Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

关键词： method of fundamental solutions exterior Helmholtz problem circulant matrix fast Fourier transform axisymmetric domain

The method of fundamental solutions for the Cauchy problem of Helmholtz-type equations with variable coefficients

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The method of fundamental solutions for the Cauchy problem o...

A Novel Technique for Estimating the Numerical Error in Solving the Helmholtz Equation

第十一届南方计算力学学术会议（SCCM-11）

作者： Fuzhang Wang Congcong Li School of Mathematical and Sciences Huaibei Normal University

In this paper, the application of the method of fundamental solutions to the Cauchy problem for Helmholtz-type equations with variable coefficients is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the regularization methods and regularization parameters. Numerical results are presented for Cauchy problems insmooth and piecewise smooth geometries, as well as simply and doubly connected domains. The accuracy, convergenceand stability of the proposed numerical method with respect to various approximating functions, various MFSdiscretizations and various levels of noise added in the boundary data, respectively, are also analyzed.

关键词： method of fundamental solutions Cauchy problem Regularization method

来源： cnki会议评论

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Computers, Materials & Continua 2020年第7期64卷 145-160页

作者： Kue-Hong Chen Cheng-Tsung Chen Department of Civil Engineering National Ilan UniversityIlan26047Taiwan Department of Civil Engineering Yancheng Institute of TechnologyYancheng224051China Center for Innovative Research on Aging Society National Chung Cheng UniversityChia-yi621301Taiwan

In this study,we applied a defined auxiliary problem in a novel error estimation technique to estimate the numerical error in the method of fundamental solutions(MFS)for solving the Helmholtz equation.The defined auxiliary problem is substituted for the real problem,and its analytical solution is generated using the complementary solution set of the governing equation.By solving the auxiliary problem and comparing the solution with the quasianalytical solution,an error curve of the MFS versus the source location parameters can be obtained.Thus,the optimal location parameter can be identified.The convergent numerical solution can be obtained and applied to the case of an unavailable analytical solution condition in the real problem.Consequently,we developed a systematic error estimation scheme to identify an optimal parameter.Through numerical experiments,the optimal location parameter of the source points and the optimal number of source points in the MFS were studied and obtained using the error estimation technique.

关键词： Error estimation auxiliary problem optimal parameter method of fundamental solutions complementary solution quasianalytical solution

The Pressure-Streamfunction MFS Formulation for the Detection of an Obstacle Immersed in a Two-Dimensional Stokes Flow

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Advances in Applied Mathematics and Mechanics 2010年第2期2卷 183-199页

作者： A.Karageorghis D.Lesnic Department of Mathematics and Statistics University of CyprusP.O.Box 205371678 NicosiaCyprus Department of Applied Mathematics University of LeedsLeeds LS29JTUK

In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a Stokes slow viscous stationary flow of an incompressible fluid,from a single set of Cauchy(fluid velocity and stress force)boundary measurements.The numerical reconstruction is based on the method of fundamental solutions(MFS)for the pressure and streamfunction in two dimensions combined with regularization.Numerical results are presented and discussed.

关键词： Stokes flow method of fundamental solutions regularization streamfunction stress force

A Regularized Singular Boundary method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1459-1477页

作者： Aixia Zhang Yan Gu Qingsong Hua Wen Chen Chuanzeng Zhang School of Mathematics and Statistics Qingdao UniversityQingdao 266071ShandongChina School of Mechanical Engineering Qingdao UniversityQingdao 266071ShandongChina Department of Engineering Mechanics College of Mechanics and MaterialsHohai UniversityNanjing 210098JiangsuChina Department of Civil Engineering University of SiegenPaul-Bonatz-Str.9-11D-57076 SiegenGermany

The application of the singular boundary method(SBM),a relatively new meshless boundary collocation method,to the inverse Cauchy problem in threedimensional(3D)linear elasticity is investigated.The SBM involves a coupling between the non-singular boundary element method(BEM)and the method of fundamental solutions(MFS).The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS.Due to the boundary-only discretizations and its semi-analytical nature,the method can be viewed as an ideal candidate for the solution of inverse problems.The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique,while the optimal regularization parameter is determined by the L-curve criterion.Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.

关键词： Meshless method singular boundary method method of fundamental solutions elastostatics inverse problem