S.Yu. Reutskiy¹

CMC-Computers, Materials & Continua, Vol.2, No.3, pp. 177-188, 2005, DOI:10.3970/cmc.2005.002.177

Abstract In this paper a new meshless method for eigenproblems with Laplace and biharmonic operators in simply and multiply connected domains is presented. The solution of an eigenvalue problem is reduced to a sequence of inhomogeneous problems with the differential operator studied. These problems are solved using the method of fundamental solutions. The method presented shows a high precision in simply and multiply connected domains. The results of the numerical experiments justifying the method are presented. More >

W. Chen^1,2, C. J. Liu¹, Y. Gu^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.4, pp. 251-270, 2015, DOI:10.3970/cmes.2015.105.251

Abstract The singular boundary method (SBM) is a recently-developed meshless boundary collocation method. This method overcomes the well-known fictitious boundary issue associated with the method of fundamental solutions (MFS) while remaining the merits of the later of being truly meshless, integral-free, and easy-to-program. Similar to the MFS, this method, however, produces dense and unsymmetrical coefficient matrix, which although much smaller in size compared with domain discretization methods, requires O(N²) operations in the iterative solution of the resulting algebraic system of equations. To remedy this bottleneck problem for its application to large-scale problems, this paper makes the first attempt to develop a… More >

Csaba Gáspár¹

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.6, pp. 365-386, 2014, DOI:10.3970/cmes.2014.101.365

Abstract The Method of Fundamental Solutions (MFS) is investigated for 3D potential problem in the case when the source points are located along the boundary of the domain of the original problem and coincide with the collocation points. This generates singularities at the boundary collocation points, which are eliminated in different ways. The (weak) singularities due to the singularity of the fundamental solution at the origin are eliminated by using approximate but continuous fundamental solution instead of the original one (regularization). The (stronger) singularities due to the singularity of the normal derivatives of the fundamental solution are eliminated by solving special… More >

Chia-Cheng Tsai^1,2, D. L. Young³

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.2, pp. 175-205, 2013, DOI:10.3970/cmes.2013.094.175

Abstract It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In this study, the exponential convergence of the MFS is demonstrated by obtaining the eigensolutions of the Helmholtz equation. In the solution procedure, the sought solution is approximated by a superposition of the Helmholtz fundamental solutions and a system matrix is resulted after imposing the boundary condition. A golden section determinant search method is applied to the matrix for finding exponentially convergent eigenfrequencies. In addition, the least-squares method of fundamental solutions is applied for solving the corresponding eigenfunctions. In the solution procedure,… More >

C. Gáspár¹

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.1, pp. 103-121, 2013, DOI:10.3970/cmes.2013.092.103

Abstract Some regularized versions of the Method of Fundamental Solutions are investigated. The problem of singularity of the applied method is circumvented in various ways using truncated or modified fundamental solutions, or higher order fundamental solutions which are continuous at the origin. For pure Dirichlet problems, these techniques seem to be applicable without special additional tools. In the presence of Neumann boundary condition, however, they need some desingularization techniques to eliminate the appearing strong singularity. Using fundamental solutions concentrated to lines instead of points, the desingularization can be omitted. The method is illustrated via numerical examples. More >

Q. G. Liu¹, B. Šarler^1,2,3,4

CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.4, pp. 235-266, 2013, DOI:10.3970/cmes.2013.091.235

Abstract The purpose of the present paper is development of a Non-singular Method of Fundamental Solutions (NMFS) for two-dimensional isotropic linear elasticity problems. The NMFS is based on the classical Method of Fundamental Solutions (MFS) with regularization of the singularities. This is achieved by replacement of the concentrated point sources by distributed sources over circular discs around the singularity, as originally suggested by [Liu (2010)] for potential problems. The Kelvin’s fundamental solution is employed in collocation of the governing plane strain force balance equations. In case of the displacement boundary conditions, the values of distributed sources are calculated directly and analytically.… More >

L. Godinho¹, D. Soares Jr.²

CMES-Computer Modeling in Engineering & Sciences, Vol.87, No.4, pp. 327-354, 2012, DOI:10.3970/cmes.2012.087.327

Abstract In this work, a coupling strategy between the Method of Fundamental Solutions (MFS) and the Kansa's Method (KM) for the analysis of fluid-solid interaction problems in the frequency domain is proposed. In this approach, the MFS is used to model the acoustic fluid medium, while KM accounts for the elastodynamic solid medium. The coupling between the two methods is performed iteratively, with independent discretizations being used for the two methods, without requiring matching between the boundary nodes along the solid-fluid interface. Two application examples, with single and multiple solid sub-domains, are presented, illustrating the behavior of the proposed approach. More >

Y. Hu¹, T. Wei^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.87, No.4, pp. 307-326, 2012, DOI:10.3970/cmes.2012.087.307

Abstract This paper employ the method of fundamental solutions for determining an unknown heat source term in a heat equation from overspecified boundary measurement data. By a function transformation, the inverse source problem is changed into an inverse initial data problem which is solved by a method of fundamental solutions. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by five numerical examples in one-dimensional and two-dimensional cases. More >

Z. C. Li², T. T. Lu³, H. T. Huang⁴, A. H.-D. Cheng⁵

CMES-Computer Modeling in Engineering & Sciences, Vol.52, No.1, pp. 39-82, 2009, DOI:10.3970/cmes.2009.052.039

Abstract For Laplace's equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coefficients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Workshops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there… More >

LiviuMarin ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.37, No.3, pp. 203-242, 2008, DOI:10.3970/cmes.2008.037.203

Abstract In this paper, a meshless method for the stable solution of direct and inverse problems associated with the two-dimensional Laplace equation in the presence of boundary singularities and noisy boundary data is proposed. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS), whilst the existence of the boundary singularity is taken into account by subtracting from the original MFS solution the corresponding singular solutions, as given by the asymptotic expansion of the solution near the singular point. However, even in the case when the boundary singularity is accounted for, the numerical solutions obtained by… More >