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  • Open Access

    PROCEEDINGS

    Solving the Time-Dependent Diffusion Problems by the Method of Fundamental Solutions and the Particle Swarm Optimization

    Tan Phat Lam1,2, Chia-Ming Fan1,*, Chiung-Lin Chu1, Fu-Li Chang1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.32, No.1, pp. 1-1, 2024, DOI:10.32604/icces.2024.012160

    Abstract In this study, the combination of the Method of Fundamental Solutions (MFS) and the Particle Swarm Optimization (PSO) is proposed to accurately and stably analyze the multi-dimensional diffusion equations. The MFS, truly free from mesh generation and numerical quadrature, is one of the most promising meshless methods. In the implementation of the MFS, only field points and sources, which are located out of the computational domain, are required. The numerical solutions of the MFS is expressed as a linear combination of diffusion fundamental solutions with different strengths. The unknown coefficients in the solution expressions can… More >

  • Open Access

    PROCEEDINGS

    Research Advances on the Collocation Methods Based on the PhysicalInformed Kernel Functions

    Zhuojia Fu1,*, Qiang Xi2, Wenzhi Xu1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.27, No.1, pp. 1-1, 2023, DOI:10.32604/icces.2023.09393

    Abstract In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function (PIKF) to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods (PIKFCMs) is to… More >

  • Open Access

    ARTICLE

    The Method of Fundamental Solutions for Two-Dimensional Elastostatic Problems with Stress Concentration and Highly Anisotropic Materials

    M. R. Hematiyan1,*, 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 >

  • Open Access

    ARTICLE

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

    Kue-Hong Chen1, *, Cheng-Tsung Chen2, 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 >

  • Open Access

    ARTICLE

    Some Remarks on the Method of Fundamental Solutions for Two Dimensional Elasticity

    M. R. Hematiyan1,*, 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 >

  • Open Access

    ARTICLE

    A Fast Multipole Accelerated Singular Boundary Method for Potential Problems

    W. Chen1,2, C. J. Liu1, Y. Gu2,3

    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 >

  • Open Access

    ARTICLE

    Non-Singular Method of Fundamental Solutions based on Laplace decomposition for 2D Stokes flow problems

    E. Sincich1, B. Šarler1,2,3

    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 >

  • Open Access

    ARTICLE

    Analysis of 3D Anisotropic Solids Using Fundamental Solutions Based on Fourier Series and the Adaptive Cross Approximation Method

    R. Q. Rodríguez1,2, C. L. Tan2, P. Sollero1, E. L. Albuquerque3

    CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.5, pp. 359-372, 2014, DOI:10.3970/cmes.2014.102.359

    Abstract The efficient evaluation of the fundamental solution for 3D general anisotropic elasticity is a subject of great interest in the BEM community due to its mathematical complexity. Recently, Tan, Shiah, andWang (2013) have represented the algebraically explicit form of it developed by Ting and Lee (Ting and Lee, 1997; Lee, 2003) by a computational efficient double Fourier series. The Fourier coefficients are numerically evaluated only once for a specific material and are independent of the number of field points in the BEM analysis. This work deals with the application of hierarchical matrices and low rank More >

  • Open Access

    ARTICLE

    Inverse Green Element Solutions of Heat Conduction Using the Time-Dependent and Logarithmic Fundamental Solutions

    Akpofure E. Taigbenu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.4, pp. 271-289, 2014, DOI:10.3970/cmes.2014.102.271

    Abstract The solutions to inverse heat conduction problems (IHCPs) are provided in this paper by the Green element method (GEM), incorporating the logarithmic fundamental solution of the Laplace operator (Formulation 1) and the timedependent fundamental solution of the diffusion differential operator (Formulation 2). The IHCPs addressed relate to transient problems of the recovery of the temperature, heat flux and heat source in 2-D homogeneous domains. For each formulation, the global coefficient matrix is over-determined and ill-conditioned, requiring a solution strategy that involves the least square method with matrix decomposition by the singular value decomposition (SVD) method, More >

  • Open Access

    ARTICLE

    A Regularized Method of Fundamental Solutions for 3D and Axisymmetric Potential Problems

    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 >

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