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

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 accuracy of the MFS for… 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

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 solution can be obtained and… 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. In NMFS no artificial boundary… 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 the shear modulus is presented… More >

Y.C. Liu, C.M. Fan, H.F. Chan, S.S. Hsiao

The International Conference on Computational & Experimental Engineering and Sciences, Vol.19, No.1, pp. 31-32, 2011, DOI:10.3970/icces.2011.019.031

Abstract The harbor oscillation problem, which is governed by inhomogeneous Helmholtz equation, is analyzed by the combination of the method of fundamental solutions (MFS) and method of particular solutions (MPS). The governed inhomogeneous Helmholtz equation is derived from the mild-slope equation and potential theory. The numerical solutions of the velocity potential of the harbor oscillation problem are decomposed as the homogeneous solution and the particular solution. While the particular solution is obtained by the MPS, the MFS is adopted to analyze the homogeneous solution. The particular solution is expressed as the linear combination of radial basis function, as the homogeneous solution… More >

Jeng-Tzong Chen^1,2, S. C. Shieh¹, Y. T. Lee¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.11, No.4, pp. 95-96, 2009, DOI:10.3970/icces.2009.011.095

Abstract Green's functions of Laplace problems with circular boundaries are solved by using three approaches, analytical, semi-analytical and numerical solutions. For the analytical solution, we derive the Green's function by using the bipolar coordinates. Three kinds of semi-analytical solutions using (a). image method, (b). the null field BIE using the Green's third identity, and (c). the null field BIE in conjunction with superposition technique using the addition theorem are considered. A numerical method using the image concept is also employed to study the optimal location of MFS. It is interesting to find that the two frozen images appear on the two… More >

S.P. Hu¹, D.L. Young^1,2, C.M. Fan¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.6, No.1, pp. 25-50, 2008, DOI:10.3970/icces.2008.006.025

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… More >

J. António¹ , A. Tadeu¹, L. Godinho

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.3, pp. 157-162, 2007, DOI:10.3970/icces.2007.003.157

Abstract A frequency dependent formulation based on the Method of Fundamental Solutions (MFS) is used to simulate the sound wave propagation in a 3D acoustic space. This solution is approximated by a linear combination of fundamental solutions generated by virtual sources placed outside the domain in order to avoid singularities. The coating materials can be assumed to be absorbent. This is achieved in the model prescribing the impedance that is defined as a function of the absorption coefficient. The model is first verified against analytical solutions, provided by the image source technique for a parallelepiped room bounded by rigid walls. The… More >

C.W. Chen¹, C.M. Fan¹, D.L. Young^1,2, K. Murugesan¹, C.C Tsai³

CMES-Computer Modeling in Engineering & Sciences, Vol.8, No.1, pp. 29-44, 2005, DOI:10.3970/cmes.2005.008.029

Abstract We use a meshless numerical method to analyze the eigenanalysis of thin circular membranes with degenerate boundary conditions, composed by different orientations and structures of stringers. The membrane eigenproblem is studied by solving the two-dimensional Helmholtz equation utilizing the method of fundamental solutions and domain decomposition technique as well. The method of singular value decomposition is adopted to obtain eigenvalues and eigenvectors of the resulting system of global linear equation. The proposed novel numerical scheme was first validated by three circular membranes which are structured with a single edge stringer, two opposite edge stringers and an internal stringer. Present results… More >

D.L. Young^1,2, J.W. Ruan²

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 223-232, 2005, DOI:10.3970/cmes.2005.007.223

Abstract The applications of the method of fundamental solutions (MFS) for modeling the scattering of time-harmonic electromagnetic fields, which are governed by vector Helmholtz equations with coupled boundary conditions, are described. Various perfectly electric conductors are considered as the scatterers to investigate the accuracy of the numerical performance of the proposed procedure by comparing with the available analytical solutions. It is also the intention of this study to reveal the characteristics of the algorithms by comparisons with other numerical methods. The model is first validated to the exact solutions of the electromagnetic wave propagation problems for the scatterers of a circular… More >