M. R. Hematiyan^1,*, B. Jamshidi¹, M. Mohammadi²

CMES-Computer Modeling in Engineering & Sciences, Vol.130, No.3, pp. 1349-1369, 2022, DOI:10.32604/cmes.2022.018235 - 30 December 2021

Abstract The method of fundamental solutions (MFS) is a boundary-type and truly meshfree method, which is recognized as an efficient numerical tool for solving boundary value problems. The geometrical shape, boundary conditions, and applied loads can be easily modeled in the MFS. This capability makes the MFS particularly suitable for shape optimization, moving load, and inverse problems. However, it is observed that the standard MFS lead to inaccurate solutions for some elastostatic problems with stress concentration and/or highly anisotropic materials. In this work, by a numerical study, the important parameters, which have significant influence on the… More >

Kue-Hong Chen^{1, *}, Cheng-Tsung Chen^{2, 3}

CMC-Computers, Materials & Continua, Vol.64, No.1, pp. 145-160, 2020, DOI:10.32604/cmc.2020.08864 - 20 May 2020

Abstract 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 More >

M. R. Hematiyan^1,*, M. Arezou¹, N. Koochak Dezfouli¹, M. Khoshroo¹

CMES-Computer Modeling in Engineering & Sciences, Vol.121, No.2, pp. 661-686, 2019, DOI:10.32604/cmes.2019.08275

Abstract In this paper, some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made. First, the effects of the distance between pseudo and main boundaries on the solution are investigated and by a numerical study a lower bound for the distance of each source point to the main boundary is suggested. In some cases, the resulting system of equations becomes ill-conditioned for which, the truncated singular value decomposition with a criterion based on the accuracy of the imposition of boundary conditions is used. Moreover, a procedure for normalizing 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 More >

E. Sincich¹, B. Šarler^1,2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.5, pp. 393-415, 2014, DOI:10.3970/cmes.2014.099.393

Abstract In this paper, a solution of a two-dimensional (2D) Stokes flow problem, subject to Dirichlet and fluid traction boundary conditions, is developed based on the Non-singular Method of Fundamental Solutions (NMFS). The Stokes equation is decomposed into three coupled Laplace equations for modified components of velocity, and pressure. The solution is based on the collocation of boundary conditions at the physical boundary by the fundamental solution of Laplace equation. The singularities are removed by smoothing over on disks around them. The derivatives on the boundary in the singular points are calculated through simple reference solutions. 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 More >

P.G. Santos¹, J. Carbajo², L. Godinho³, J. Ramis²

CMC-Computers, Materials & Continua, Vol.43, No.2, pp. 109-136, 2014, DOI:10.3970/cmc.2014.043.109

Abstract The study of periodic structures, namely sonic crystals, for sound attenuation purposes has been a topic of intense research in the last years. Some efficient methods are available in literature to solve the problem of sound propagation in the presence of this kind of structures such as those based in the Multiple Scattering Theory (MST) or the Finite Element Method (FEM). In this paper a solution based on the Method of Fundamental Solutions (MFS) which presents advantages, namely in computational discretization and calculation costs, is presented. The proposed formulation considers the presence of elastic ring 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 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… More >