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13
2007a): A modified Trefftz method for twodimensional Laplace equation considering the domain’s characteristic length
 CMES: Computer Modeling in Engineering & Sciences
"... Abstract: A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for twodimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by th ..."
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Cited by 20 (9 self)
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Abstract: A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for twodimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equation, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accurate as one desired. Then, we use the collocation method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to explore why the conventional Trefftz method is failure and the modified one still survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numerical results with absolute errors about in the orders from 10−10 to 10−16. The new method is powerful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary.
C.S.(2007): A highly accurate solver for the mixedboundary potential problem and singular problem in arbitrary plane domain. CMES:Computer Modeling
 in Engineering & Sciences
"... Abstract: A highly accurate new solver is developed to deal with interior and exterior mixedboundary value problems for twodimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circular artificial boundary which is uniquely determined by the phy ..."
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Cited by 16 (9 self)
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Abstract: A highly accurate new solver is developed to deal with interior and exterior mixedboundary value problems for twodimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circular artificial boundary which is uniquely determined by the physical problem domain, and derive a Dirichlet to Robin mapping on that artificial circle, which is an exact boundary condition described by the first kind Fredholm integral equation. As a consequence, we obtain a modified Trefftz method equipped with a characteristic length factor, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the Fourier coefficients. We find that the new method is powerful even for the problem with complex boundary shape and with adding random noise on the boundary data. It is also applicable to the singular problem of Motz type, resulting to a highly accurate result never seen before.
Improving the illconditioning of the method of fundamental solutions for 2D Laplace equation
 CMES: Computer Modeling in Engineering & Sciences
, 2008
"... Abstract: The method of fundamental solutions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are determined from the bo ..."
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Cited by 5 (2 self)
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Abstract: The method of fundamental solutions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are determined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its illconditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Trefftz method and MFS for circular domain, while the second proposed a modified Trefftz method (MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the wellposedness of MTM, we can alleviate the illconditioning of MFS through a new linear equations system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Numerical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10−10. The present method is fully not similar to the preconditioning technique as used to solve the illconditioned linear equations system.
A Fictitious Time Integration Method for TwoDimensional Quasilinear Elliptic Boundary Value Problems
"... Abstract: Dirichlet boundary value problem of quasilinear elliptic equation is numerically solved by using a new concept of fictitious time integration method (FTIM). We introduce a fictitious time coordinate t by transforming the dependent variable u(x,y) into a new one by (1+ t)u(x,y) =: v(x,y, t) ..."
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Cited by 5 (2 self)
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Abstract: Dirichlet boundary value problem of quasilinear elliptic equation is numerically solved by using a new concept of fictitious time integration method (FTIM). We introduce a fictitious time coordinate t by transforming the dependent variable u(x,y) into a new one by (1+ t)u(x,y) =: v(x,y, t), such that the original equation is naturally and mathematically equivalently written as a quasilinear parabolic equation, including a viscous damping coefficient to enhance stability in the numerical integration of spatially semidiscretized equation as an ordinary differential equations set on grid points. Six examples of Laplace, Poisson, reaction diffusion, Helmholtz, the minimal surface, as well as the explosion equations are tested. It is interesting that the FTIM can easily treat the nonlinear boundary value problems without any iteration and has high efficiency and high accuracy. Due to the dissipation nature of the resulting parabolic equation, the FTIM is insensitive to the guess of initial conditions and approaches the true solution very fast.
Solving the Laplace equation by meshless collocation using harmonic kernels, Adv
 in Comp. Math
"... We present a meshless technique which can be seen as an alternative to the Method of Fundamental Solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic ..."
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Cited by 4 (2 self)
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We present a meshless technique which can be seen as an alternative to the Method of Fundamental Solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artifical boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples. 1
2008): FDMFS for diffusion equation with unsteady forcing function
 CMES: Computer Modeling in Engineering & Sciences
"... Abstract: In this paper, a novel numerical scheme called (FDMFS), which combines the finite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most meshless methods are confined to t ..."
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Cited by 4 (0 self)
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Abstract: In this paper, a novel numerical scheme called (FDMFS), which combines the finite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most meshless methods are confined to the investigations of nonhomogeneous diffusion equations with steady forcing functions due to the difficulty to find an unsteady particular solution. Therefore, we proposed a FDM with Cartesian grid to handle the unsteady nonhomogeneous term of the equations. The numerical solution in FDMFS is decomposed into a particular solution and a homogeneous solution. The particular solution is constructed using the FDM in an artificial regular domain which contains the real irregular domain without boundary conditions, and the homogeneous solution can be obtained by the timespace unification MFS in the irregular domain with boundary conditions. Besides, the Cartesian grid for particular solution is very simple to generate automatically. Our paper is the first time to propose an algorithm to solve nonhomogeneous diffusion equations with unsteady forcing functions using MFS to solve homogeneous solutions and FDM to calculate the particular solutions. Numerical experiments are presented for 2D problems in regular and irregular domains to show the high performance of this proposed scheme. Moreover, the stabilities of explicit and implicit FDM for particular solution are analyzed. Numerical studies suggest that the proposed FDMFS can speed up the simulation and save the CPU time and memory storage substantially.
The method of fundamental solutions for eigenfrequencies of plate vibrations
 CMC: Computers, Materials & Continua
, 2006
"... Abstract: This paper describes the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibrations by utilizing the direct determinant search method. The complexvalued kernels are used in the MFS in order to avoid the spurious eigenvalues. The benchmark problems of a circular p ..."
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Cited by 3 (1 self)
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Abstract: This paper describes the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibrations by utilizing the direct determinant search method. The complexvalued kernels are used in the MFS in order to avoid the spurious eigenvalues. The benchmark problems of a circular plate with clamped, simply supported and free boundary conditions are studied analytically as well as numerically using the discrete and continuous versions of the MFS schemes to demonstrate the major results of the present paper. Namely only true eigenvalues are contained and no spurious eigenvalues are included in the range of direct determinant search method. Consequently analytical derivation is carried out by using the degenerate kernels and Fourier series to obtain the exact eigenvalues which are used to validate the numerical methods. The MFS is free from meshes, singularities, and numerical integrations. As a result, the proposed numerical method can be easily used to solve plate vibrations free from spurious eigenvalues in simply connected domains.
A Matrix Decomposition MFS Algorithm for Biharmonic Problems in Annular Domains
 CMC: COMPUTERS, MATERIALS, & CONTINUA
, 2004
"... The Method of Fundamental Solutions (MFS) is a boundarytype method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in annular domains. The circulant structure of the matr ..."
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Cited by 2 (0 self)
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The Method of Fundamental Solutions (MFS) is a boundarytype method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in annular domains. The circulant structure of the matrices involved in the MFS discretization is exploited by using Fast Fourier Transforms. The algorithm is tested numerically on several examples.
THE UNDER–DETERMINED VERSION OF THE MFS: TAKING MORE SOURCES THAN COLLOCATION POINTS
"... Abstract. In this study we investigate the approximation of the solutions of certain elliptic boundary value problems by the Method of Fundamental Solutions (MFS). In particular, we study the case in which the number of singularities (sources) exceeds the number of boundary (collocation) points and ..."
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Abstract. In this study we investigate the approximation of the solutions of certain elliptic boundary value problems by the Method of Fundamental Solutions (MFS). In particular, we study the case in which the number of singularities (sources) exceeds the number of boundary (collocation) points and we propose two methods of determining which approximate solution is optimal. We also develop an efficient numerical algorithm for the Dirichlet problem for Laplace’s equation in the special case the domain of the problem is a disk. Numerical experiments are presented which reveal that in the case where the number of singularities is twice the number of boundary points, the conditioning of the MFS matrices and the concomitant accuracy of the method improve significantly. Key words. Method of fundamental solutions, Elliptic boundary value problems, Under–determined linear sys
Iterative coupling between the MFS and Kansa’s method for acoustic problems
"... In the present work, a numerical frequencydomain model based on the joint use of two distinct meshless methods (the Method of Fundamental Solutions and Kansa’s Method) is discussed. In this context, the MFS is used to model the homogeneous part of the propagation domain, while the Kansa’s Method is ..."
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In the present work, a numerical frequencydomain model based on the joint use of two distinct meshless methods (the Method of Fundamental Solutions and Kansa’s Method) is discussed. In this context, the MFS is used to model the homogeneous part of the propagation domain, while the Kansa’s Method is applied to model the presence of possible heterogeneities. For Kansa’s Method, the MQ RBF is used, and the optimal value of its free parameter is computed by minimizing the residual of the PDE throughout the subdomain. The coupling between the two parts of the propagation domain is performed iteratively, allowing totally independent spatial discretizations to be used for each of the subdomains of the model. Given this strategy, the use of matching collocation points at common surfaces is not necessary. To improve the behavior of the iterative process, an optimized algorithm, based on the use of a varying relaxation parameter, is used to speed up and/or to ensure the convergence of the iterative coupling. A set of numerical results is here presented to illustrate the