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Journal articles on the topic 'Method of fundamental solution'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Hrubá, M. "dCAPS method: advantages, troubles and solution." Plant, Soil and Environment 53, No. 9 (January 7, 2008): 417–20. http://dx.doi.org/10.17221/2293-pse.

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In our work, we focus on the evolutionary studies of sex chromosomes. As model organisms we use several species of the plant genus <i>Silene</i>. An important part of our research is represented by genetic mapping based on the assays of DNA length or sequence polymorphisms. Apart from the other methods we also use the dCAPS method, which is very useful for detection of the sequence polymorphisms (SNPs). This method is unique as it is able to detect SNPs that are not situated in any restriction site; a fundamental principle of this method is usage of primer designed with one or two mismatches that bring into the target sequence the mutation in vicinity of SNP. Using this method, we found out some improvements that can make analyses more cost-effective.

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2

Kitagawa, Takashi. "Asymptotic stability of the fundamental solution method." Journal of Computational and Applied Mathematics 38, no. 1-3 (December 1991): 263–69. http://dx.doi.org/10.1016/0377-0427(91)90175-j.

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3

GÁSPÁR, CSABA. "REGULARIZATION TECHNIQUES FOR THE METHOD OF FUNDAMENTAL SOLUTIONS." International Journal of Computational Methods 10, no. 02 (March 2013): 1341004. http://dx.doi.org/10.1142/s0219876213410041.

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A special regularization method based on higher-order partial differential equations is presented. Instead of using the fundamental solution of the original partial differential operator with source points located outside of the domain, the original second-order partial differential equation is approximated by a higher-order one, the fundamental solution of which is continuous at the origin. This allows the use of the method of fundamental solutions (MFS) for the approximate problem. Due to the continuity of the modified operator, the source points and the boundary collocation points are allowed to coincide, which makes the solution process simpler. This regularization technique is generalized to various problems and combined with the extremely efficient quadtree-based multigrid methods. Approximation theorems and numerical experiences are also presented.

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4

Wei, H., J. L. Zheng, J. Sladek, V. Sladek, and P. H. Wen. "Method of fundamental solution using Erdogan's solution: Static and dynamic." Engineering Analysis with Boundary Elements 148 (March 2023): 176–89. http://dx.doi.org/10.1016/j.enganabound.2022.12.035.

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5

Nath, D., and M. S. Kalra. "Solution of Grad–Shafranov equation by the method of fundamental solutions." Journal of Plasma Physics 80, no. 3 (February 19, 2014): 477–94. http://dx.doi.org/10.1017/s0022377814000026.

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In this paper we have used the Method of Fundamental Solutions (MFS) to solve the Grad–Shafranov (GS) equation for the axisymmetric equilibria of tokamak plasmas with monomial sources. These monomials are the individual terms appearing on the right-hand side of the GS equation if one expands the nonlinear terms into polynomials. Unlike the Boundary Element Method (BEM), the MFS does not involve any singular integrals and is a meshless boundary-alone method. Its basic idea is to create a fictitious boundary around the actual physical boundary of the computational domain. This automatically removes the involvement of singular integrals. The results obtained by the MFS match well with the earlier results obtained using the BEM. The method is also applied to Solov'ev profiles and it is found that the results are in good agreement with analytical results.

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6

Hu, S. P., C. M. Fan, C. W. Chen, and D. L. Young. "Method of Fundamental Solutions for Stokes' First and Second Problems." Journal of Mechanics 21, no. 1 (March 2005): 25–31. http://dx.doi.org/10.1017/s1727719100000514.

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AbstractThis paper describes the applications of the method of fundamental solutions (MFS) as a mesh-free numerical method for the Stokes' first and second problems which prevail in the semi-infinite domain with constant and oscillatory velocity at the boundary in the fluid-mechanics benchmark problems. The time-dependent fundamental solutions for the semi-infinite problems are used directly to obtain the solution as a linear combination of the unsteady fundamental solution of the diffusion operator. The proposed numerical scheme is free from the conventional Laplace transform or the finite difference scheme to deal with the time derivative term of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. It is not necessary to locate and specify the condition at the infinite domain such as other numerical methods. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to the domain-oriented numerical schemes. Test results obtained for the Stokes' first and second problems show good comparisons with the analytical solutions. Thus the present numerical scheme has provided a promising mesh-free numerical tool to solve the unsteady semi-infinite problems with the space-time unification for the time-dependent fundamental solution.

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7

Liu, Chein-Shan, Zhuojia Fu, and Chung-Lun Kuo. "Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation." Journal of Mathematics Research 9, no. 6 (November 8, 2017): 112. http://dx.doi.org/10.5539/jmr.v9n6p112.

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We propose a simple extension of the two-dimensional method of fundamental solutions (MFS) to a two-dimensional like MFS for the numerical solution of the three-dimensional Laplace equation in an arbitrary interior domain. In the directional MFS (DMFS) the directors are planar orientations, which can take the geometric anisotropy of the problem domain into account, and more importantly the order of the logarithmic singularity with $\ln R$ of the new fundamental solution is reduced than that of the conventional three-dimensional fundamental solution with singularity $1/r$. Some numerical examples are used to validate the performance of the DMFS.

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8

Shidfar, A., and Z. Darooghehgimofrad. "Numerical solution of two backward parabolic problems using method of fundamental solutions." Inverse Problems in Science and Engineering 25, no. 2 (January 30, 2016): 155–68. http://dx.doi.org/10.1080/17415977.2016.1138947.

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9

Lin, Ji, C. S. Chen, and Chein-Shan Liu. "Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions." Communications in Computational Physics 20, no. 2 (July 21, 2016): 512–33. http://dx.doi.org/10.4208/cicp.060915.301215a.

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AbstractThis paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

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10

Chen, C. S., Xinrong Jiang, Wen Chen, and Guangming Yao. "Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions." Communications in Computational Physics 17, no. 3 (March 2015): 867–86. http://dx.doi.org/10.4208/cicp.181113.241014a.

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AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

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11

Mužík, Juraj, and Roman Bulko. "Free surface groundwater flow solution using boundary collocation methods." MATEC Web of Conferences 196 (2018): 03026. http://dx.doi.org/10.1051/matecconf/201819603026.

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In this paper, two meshless numerical algorithms are developed for the solution of two-dimensional steady-state diffusion equation that describes the stationary groundwater flow. The proposed numerical methods, which are truly meshless, quadrature-free and boundary only, are based on the method of fundamental solutions and singular boundary method respectively. The diffusion equation is transformed into a Poisson-type equation with a known fundamental solution. Numerical examples with moving boundary are presented and compared to the solutions obtained by the finite element method.

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12

Zuosheng, Yang. "The fundamental solution method for incompressible Navier-Stokes equations." International Journal for Numerical Methods in Fluids 28, no. 3 (September 15, 1998): 565–68. http://dx.doi.org/10.1002/(sici)1097-0363(19980915)28:3<565::aid-fld749>3.0.co;2-j.

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13

Tracinà, Rita. "Fundamental solution in classical elasticity via Lie group method." Applied Mathematics and Computation 218, no. 9 (January 2012): 5132–39. http://dx.doi.org/10.1016/j.amc.2011.10.079.

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14

Hon, Y. C., and T. Wei. "A fundamental solution method for inverse heat conduction problem." Engineering Analysis with Boundary Elements 28, no. 5 (May 2004): 489–95. http://dx.doi.org/10.1016/s0955-7997(03)00102-4.

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15

KARAGEORGHIS, A., and G. FAIRWEATHER. "The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems." IMA Journal of Numerical Analysis 9, no. 2 (1989): 231–42. http://dx.doi.org/10.1093/imanum/9.2.231.

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16

Karageorghis, Andreas, and Graeme Fairweather. "The method of fundamental solutions for the numerical solution of the biharmonic equation." Journal of Computational Physics 69, no. 2 (April 1987): 434–59. http://dx.doi.org/10.1016/0021-9991(87)90176-8.

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17

Cao, Changyong, and Qing-Hua Qin. "Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications." Advances in Mathematical Physics 2015 (2015): 1–38. http://dx.doi.org/10.1155/2015/916029.

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An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.

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18

Tkachev, Alexander, and Dmitry Chernoivan. "Combined Adaptive Meshfree Method for Modeling Potential Physical Fields." Известия высших учебных заведений. Электромеханика 64, no. 1 (2021): 5–12. http://dx.doi.org/10.17213/0136-3360-2021-1-5-12.

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A procedure for numerical solution of the Dirichlet problem for the Laplace equation, which is often re-duced to modeling potential physical fields in homogeneous media, is described. The approximate solution is proposed to be found using a combined meshfree Monte Carlo method and fundamental solutions, which is implemented in two stages. At the first stage, the element of the best approximation in the linear shell of the fundamental solutions of the Laplace equation is determined. At the second stage, the solution is refined using the potential values found by the Monte Carlo method at individual points in the computational domain. Algorithms are given for finding the defining parameters of both methods used to reduce the error. The procedure for evaluating the accuracy of the found approximate solution of the problem is described. An example of calculating the potential distribution in the angular zone under specified boundary conditions using the combined meshfree method is given. The accuracy of the approximate solution is estimated by comparing it with the exact solution. It is shown that the use of the meshfree method leads to a decrease in the error without a significant increase in the computational resources required for its implementation

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19

Protektor, Denys, and Iryna Hariachevska. "SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS." Grail of Science, no. 12-13 (May 28, 2022): 356–59. http://dx.doi.org/10.36074/grail-of-science.29.04.2022.059.

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The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

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20

Niyazbek, Muheyat, M. O. Nogaybaeva, Kuenssaule Talp, and A. A. Kudaikulov. "Solution of Thermoelectricity Problems Energy Method." E3S Web of Conferences 38 (2018): 04002. http://dx.doi.org/10.1051/e3sconf/20183804002.

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On the basis of the fundamental laws of conservation of energy in conjunction with local quadratic spline functions was developed a universal computing algorithm, a method and associated software, which allows to investigate the Thermophysical insulated rod, with limited length, influenced by local heat flow, heat transfer and temperature

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21

KAJISHIMA, Takeo, Susumu MURATA, and Yutaka MIYAKE. "Boundary element method on the basis of Oseen's fundamental solution." Transactions of the Japan Society of Mechanical Engineers Series B 51, no. 467 (1985): 2345–50. http://dx.doi.org/10.1299/kikaib.51.2345.

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22

KAJISHIMA, Takeo, Susumu MURATA, and Yutaka MIYAKE. "Boundary Element Method on the Basis of Oseen's Fundamental Solution." Bulletin of JSME 29, no. 249 (1986): 810–15. http://dx.doi.org/10.1299/jsme1958.29.810.

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23

Gu, Zhi-jie, and Yong-ji Tan. "Fundamental solution method for inverse source problem of plate equation." Applied Mathematics and Mechanics 33, no. 12 (November 5, 2012): 1513–32. http://dx.doi.org/10.1007/s10483-012-1641-6.

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24

TSAI, CHIA-CHENG, and PO-HO LIN. "ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS." International Journal of Computational Methods 10, no. 02 (March 2013): 1341007. http://dx.doi.org/10.1142/s0219876213410077.

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It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.

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25

Poullikkas, A., A. Karageorghis, and G. Georgiou. "The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions." Engineering Analysis with Boundary Elements 25, no. 3 (March 2001): 221–27. http://dx.doi.org/10.1016/s0955-7997(01)00007-8.

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26

Karageorghis, Andreas. "The method of fundamental solutions for the solution of steady-state free boundary problems." Journal of Computational Physics 93, no. 2 (April 1991): 486. http://dx.doi.org/10.1016/0021-9991(91)90201-u.

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27

Karageorghis, Andreas. "The method of fundamental solutions for the solution of steady-state free boundary problems." Journal of Computational Physics 98, no. 1 (January 1992): 119–28. http://dx.doi.org/10.1016/0021-9991(92)90178-2.

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28

Fursov, V., A. Gavrilov, Ye Goshin, and K. Pugachev. "Conforming identification of the fundamental matrix in image matching problem." Computer Optics 41, no. 4 (2017): 559–63. http://dx.doi.org/10.18287/2412-6179-2017-41-559-563.

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The article considers the conforming identification of the fundamental matrix in the image matching problem. The method consists in the division of the initial overdetermined system into lesser dimensional subsystems. On these subsystems, a set of solutions is obtained, from which a subset of the most conforming solutions is defined. Then, on this subset the resulting solution is deduced. Since these subsystems are formed by all possible combinations of rows in the initial system, this method demonstrates high accuracy and stability, although it is computationally complex. A comparison with the methods of least squares, least absolute deviations, and the RANSAC method is drawn.

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29

Tkachev, Alexander, Dmitry Chernoivan, and Nikolay Savelov. "A Combined Mesh-Free Method for Solving the Mixed Boundary Value Problems in Modeling the Potential Physical Fields." Известия высших учебных заведений. Электромеханика 65, no. 4 (2022): 3–14. http://dx.doi.org/10.17213/0136-3360-2022-4-3-14.

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The article describes a combined mesh-free method for solving mixed boundary value problems for the Laplace equa-tion arising from the analysis of potential physical fields in homogeneous media. The solution using mesh-free meth-ods of fundamental solutions and the Monte Carlo method is found. The modification of these methods is carried out taking into account the features that arise when setting mixed boundary conditions at the computational domain boundary. The conjugate fundamental solutions of the Laplace equation and the procedure of random walk by spheres are used, taking into account the features of constructing the trajectory of motion, close to the boundary part on which the normal derivative is set. The article proposes a procedure for the combined implementation of both two mesh-free methods of fundamental solutions and the Monte Carlo method which provides higher calculations accura-cy. The combined method was verified using benchmarks which are the results obtained using the reference solution squared and the solution obtained by the finite element method using the FEMM 4.2 software package when analyzing the electromagnet gap field in the vicinity of the.

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30

Wolf, John P., and Chongmin Song. "The scaled boundary finite-element method – a fundamental solution-less boundary-element method." Computer Methods in Applied Mechanics and Engineering 190, no. 42 (August 2001): 5551–68. http://dx.doi.org/10.1016/s0045-7825(01)00183-9.

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31

FUJITANI, YOSHINOBU. "FINITE ELEMENT ANALYSIS OF THREE-DIMENSIONAL ELASTIC FUNDAMENTAL SOLUTION : AXI-SYMMETRICAL SOLUTION : Studies on analysis of elastic fundamental solutions by finite element method, Part 2." Journal of Structural and Construction Engineering (Transactions of AIJ) 407 (1990): 71–78. http://dx.doi.org/10.3130/aijsx.407.0_71.

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32

Mužík, Juraj, and Roman Bulko. "Lid-driven cavity flow using dual reciprocity." MATEC Web of Conferences 313 (2020): 00043. http://dx.doi.org/10.1051/matecconf/202031300043.

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The paper presents the use of the multi-domain dual reciprocity method of fundamental solutions (MD-MFSDR) for the analysis of the laminar viscous flow problem described by Navier-Stokes equations. A homogeneous part of the solution is solved using the method of fundamental solutions with the 2D Stokes fundamental solution Stokeslet. The dual reciprocity approach has been chosen because it is ideal for the treatment of the non-homogeneous and nonlinear terms of Navier-Stokes equations. The presented DR-MFS approach to the solution of the 2D flow problem is demonstrated on a standard benchmark problem - lid-driven cavity.

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33

Liu, Dongdong, Xing Wei, Chengbin Li, Chunguang Han, Xiaxi Cheng, and Linlin Sun. "Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method." Mathematics 10, no. 22 (November 17, 2022): 4323. http://dx.doi.org/10.3390/math10224323.

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In this paper, the singular boundary method (SBM) in conjunction with the exponential window method (EWM) is firstly extended to simulate the transient dynamic response of two-dimensional saturated soil. The frequency-domain (Fourier space) governing equations of Biot theory is solved by the SBM with a linear combination of the fundamental solutions. In order to avoid the perplexing fictitious boundary in the method of fundamental solution (MFS), the SBM places the source point on the physical boundary and eliminates the source singularity of the fundamental solution via the origin intensity factors (OIFs). The EWM is carried out for the inverse Fourier transform, which transforms the frequency-domain solutions into the time-domain solutions. The accuracy and feasibility of the SBM-EWM are verified by three numerical examples. The numerical comparison between the MFS and SBM indicates that the SBM takes a quarter of the time taken by the MFS.

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ZHANG, ZE-WEI, HUI WANG, and QING-HUA QIN. "METHOD OF FUNDAMENTAL SOLUTIONS FOR NONLINEAR SKIN BIOHEAT MODEL." Journal of Mechanics in Medicine and Biology 14, no. 04 (July 3, 2014): 1450060. http://dx.doi.org/10.1142/s0219519414500602.

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In this paper, the method of fundamental solution (MFS) coupling with the dual reciprocity method (DRM) is developed to solve nonlinear steady state bioheat transfer problems. A two-dimensional nonlinear skin model with temperature-dependent blood perfusion rate is studied. Firstly, the original bioheat transfer governing equation with nonlinear term induced by temperature-dependent blood perfusion rate is linearized with the Taylor's expansion technique. Then, the linearized governing equation with specified boundary conditions is solved using a meshless approach, in which the DRM and the MFS are employed to obtain particular and homogeneous solutions, respectively. Several numerical examples involving linear, quadratic and exponential relations between temperature and blood perfusion rate are tested to verify the efficiency and accuracy of the proposed meshless model in solving nonlinear steady state bioheat transfer problems, and also the sensitivity of coefficients in the expression of temperature-dependent blood perfusion rate is analyzed for investigating the influence of blood perfusion rate to temperature distribution in skin tissues.

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35

Chen, Wen, and Yan Gu. "An Improved Formulation of Singular Boundary Method." Advances in Applied Mathematics and Mechanics 4, no. 5 (October 2012): 543–58. http://dx.doi.org/10.4208/aamm.11-m11118.

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AbstractThis study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.

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Fan, Zhuowan, Yancheng Liu, Anyu Hong, Fugang Xu, and Fuzhang Wang. "The Localized Method of Fundamental Solution for Two Dimensional Signorini Problems." Computer Modeling in Engineering & Sciences 132, no. 1 (2022): 341–55. http://dx.doi.org/10.32604/cmes.2022.019715.

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37

Othman, M. I. A., and I. A. Abbas. "Fundamental solution of generalized thermo-viscoelasticity using the finite element method." Computational Mathematics and Modeling 23, no. 2 (April 2012): 158–67. http://dx.doi.org/10.1007/s10598-012-9127-0.

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38

Shanazari, Kamal, and Mahmood Fallahi. "A quasi-linear technique applied to the method of fundamental solution." Engineering Analysis with Boundary Elements 34, no. 4 (April 2010): 388–92. http://dx.doi.org/10.1016/j.enganabound.2009.11.002.

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39

Zhou, Junchen, Keyong Wang, Peichao Li, and Xiaodan Miao. "Hybrid fundamental solution based finite element method for axisymmetric potential problems." Engineering Analysis with Boundary Elements 91 (June 2018): 82–91. http://dx.doi.org/10.1016/j.enganabound.2018.03.009.

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40

Barrios, Carlos P., Facundo Almeraya-Calderon, Rosa Elba Nuñez-Jaquez, Citalli Gaona, Alberto Martínez-Villafañe, and Jose Chacon Nava. "Numerical Simulation of Corrosion Cells by the Method of Fundamental Solution." ECS Transactions 3, no. 24 (December 21, 2019): 19–23. http://dx.doi.org/10.1149/1.2753232.

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41

Yu-ying, Huang, and Yin Lei-fang. "A direct method for deriving fundamental solution of half-plane problem." Applied Mathematics and Mechanics 8, no. 12 (December 1987): 1181–90. http://dx.doi.org/10.1007/bf02450912.

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42

Šarler, Božidar. "Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions." Engineering Analysis with Boundary Elements 33, no. 12 (December 2009): 1374–82. http://dx.doi.org/10.1016/j.enganabound.2009.06.008.

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43

Wang, Kai, Shiting Wen, Rizwan Zahoor, Ming Li, and Božidar Šarler. "Method of regularized sources for axisymmetric Stokes flow problems." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 3/4 (May 3, 2016): 1226–39. http://dx.doi.org/10.1108/hff-09-2015-0397.

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Purpose – The purpose of this paper is to find solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient non-singular method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space. Design/methodology/approach – Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions. Findings – An axisymmetric driven cavity numerical example and the flow in a hollow tube and flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary. Originality/value – Analytical expressions for the axisymmetric blobs are derived. The method of regularized sources is for the first time applied to axisymmetric Stokes flow problems.

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Fu, Zhuo-Jia, Wen Chen, Ji Lin, and Alexander H. D. Cheng. "Singular Boundary Method for Various Exterior Wave Applications." International Journal of Computational Methods 12, no. 02 (March 2015): 1550011. http://dx.doi.org/10.1142/s0219876215500115.

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This paper presents an improved singular boundary method (ISBM) to various exterior wave applications. The SBM is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors to eliminate the singularity of the fundamental solutions. In this study, we first derive the source intensity factors of the exterior Helmholtz equation by means of the source intensity factors of the exterior Laplace equation due to the same order of the singularities on their fundamental solutions. The source intensity factors of the exterior Laplace equation can be determined using the reference technique [Chen, W. and Gu, Y. [2011] "Recent advances on singular boundary method," Joint international workshop on Trefftz method VI and method of fundamental solution II, Taiwan]. Numerical illustrations demonstrate the efficiency and accuracy of the proposed scheme on four benchmark exterior wave examples.

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Tsai, C. C. "The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings." Journal of Mechanics 24, no. 2 (June 2008): 163–71. http://dx.doi.org/10.1017/s1727719100002197.

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ABSTRACTThis paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

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Šarler, B. "Solution of a two-dimensional bubble shape in potential flow by the method of fundamental solutions." Engineering Analysis with Boundary Elements 30, no. 3 (March 2006): 227–35. http://dx.doi.org/10.1016/j.enganabound.2005.09.007.

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Mužík, Juraj, and Roman Bulko. "Regularized singular boundary method for groundwater flow in a cofferdam." MATEC Web of Conferences 196 (2018): 03025. http://dx.doi.org/10.1051/matecconf/201819603025.

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The boundary integral methods form one group of methods for solving differential equations. The boundary element method (BEM) is the basic method of this group. However, it requires the boundary mesh of elements and the evaluation of improper singular integrals, that arise due to fundamental solution singularity. Therefore, boundary meshless methods have recently have come into focus to remove these shortcomings. One of the most promising boundary collocation numerical schemes is the singular boundary method (SBM). To tackle the singularity of the fundamental solution, this method adopts the concept of original intensity factors (OIFs). The application of the proposed SBM scheme to groundwater flow in the cofferdam structure is presented and compared to the finite element method (FEM) solution.

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Zhang, Junli, Hui Zheng, Chia-Ming Fan, and Ming-Fu Fu. "Localized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problems." Mathematics 10, no. 9 (April 27, 2022): 1464. http://dx.doi.org/10.3390/math10091464.

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Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is ill posed, a small disturbance will lead to great errors in the numerical simulations. More accurate numerical methods are needed in the inverse Cauchy problem. In this work, the LMFS is firstly proposed to analyze the inhomogeneous inverse Cauchy problem. The recursive composite multiple reciprocity method (RC-MRM) is adopted to change original inhomogeneous problem into a higher-order homogeneous problem. Then, the high-order homogeneous problem can be solved directly by the LMFS. Several numerical experiments are carried out to demonstrate the efficiency of the LMFS for the inhomogeneous inverse Cauchy problems.

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FUJITANI, YOSHINOBU. "FINITE ELEMENT ANALYSIS OF TWO-DIMENSIONAL ELASTIC FUNDAMENTAL SOLUTION : Studies on analysis of elastic fundamental solutions by finite element method, Part 1." Journal of Structural and Construction Engineering (Transactions of AIJ) 393 (1988): 54–61. http://dx.doi.org/10.3130/aijsx.393.0_54.

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Klekiel, Tomasz. "Application of the Fundamental Solution Method to Object Recognition in the Pictures." Image Processing & Communications 22, no. 3 (September 1, 2017): 13–22. http://dx.doi.org/10.1515/ipc-2017-0014.

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Abstract Recognition of objects in pictures and movies requires the use of techniques, such as filtering, segmentation and classification. Image filtering is required to remove all artifacts that hinder the unequivocal identification and sharpen interesting objects. Segmentation refers to finding areas of images respected to individual objects. For the selected areas corresponding to objects in the selected picture, the classification of objects finally gives information about the type of object which orientation is made. This paper presents a method for the classification of objects from drawings as a bitmap using the method of fundamental solutions (MFS). The MFS was tested on the selected bitmap depicting simple geometric shapes. The correlations between errors occurring on the boundary for particular shapes are used for the selection of geometric shape figures. Due to this correlation, it is possible to recognize the shape of the image appearing on the drawing by an analysis consisting of the comparison of recognized points describing the shape of contour to a database containing solutions of boundary value problems for the selected shape. In one way, the comparison of the pattern can determine which shape from database it is most similar to in terms of contour. This article appear that this approach is very simple and clearly. In result, this method can be used to recognition of the objects in the systems of real-time processing.

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