The Method of Fundamental Solutions applied to 3D structures with body forces using particular solutions

2005 ◽  
Vol 36 (4) ◽  
pp. 245-254 ◽  
Author(s):  
George S. A. Fam ◽  
Youssef F. Rashed
Keyword(s):  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
3D Structures ◽  
Particular Solutions ◽  
Body Forces

Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

2016 ◽  
Vol 20 (2) ◽  
pp. 512-533 ◽  
Author(s):  
Ji Lin ◽  
C. S. Chen ◽  
Chein-Shan Liu
Keyword(s):  
Three Dimensional ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Singular Problem ◽  
Helmholtz Equations ◽  
Particular Solutions ◽  
Radial Basis ◽  
Particular Solution ◽  
Polyharmonic Spline

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.

scholarly journals THE METHOD OF APPROXIMATE PARTICULAR SOLUTIONS FOR SOLVING ELLIPTIC PROBLEMS WITH VARIABLE COEFFICIENTS

2011 ◽  
Vol 08 (03) ◽  
pp. 545-559 ◽  
Author(s):  
C. S. CHEN ◽  
C. M. FAN ◽  
P. H. WEN
Keyword(s):  
Solution Process ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Left Hand ◽  
Particular Solutions ◽  
Particular Solution ◽  
The Right ◽  
Close Form

A new version of the method of approximate particular solutions (MAPSs) using radial basis functions (RBFs) has been proposed for solving a general class of elliptic partial differential equations. In the solution process, the Laplacian is kept on the left-hand side as a main differential operator. The other terms are moved to the right-hand side and treated as part of the forcing term. In this way, the close-form particular solution is easy to obtain using various RBFs. The numerical scheme of the new MAPSs is simple to implement and yet very accurate. Three numerical examples are given and the results are compared to Kansa's method and the method of fundamental solutions.

scholarly journals The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

2008 ◽  
Vol 24 (2) ◽  
pp. 163-171 ◽  
Author(s):  
C. C. Tsai
Keyword(s):  
Lateral Displacement ◽  
Fundamental Solutions ◽  
Solution Procedure ◽  
Real Coefficient ◽  
Method Of Fundamental Solutions ◽  
Thin Plates ◽  
Complex Coefficient ◽  
Particular Solutions ◽  
Analytical Formulas ◽  
Polyharmonic Splines

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.

Particular Solution of Polyharmonic Spline Associated with Reissner Plate Problems

2011 ◽  
Vol 27 (4) ◽  
pp. 493-501 ◽  
Author(s):  
C. C. Tsai ◽  
M. E. Quadir ◽  
H. H. Hwung ◽  
T. W. Hsu
Keyword(s):  
Homogeneous Problem ◽  
Coupled System ◽  
Fundamental Solutions ◽  
Solution Procedure ◽  
Method Of Fundamental Solutions ◽  
Reissner Plate ◽  
Particular Solutions ◽  
Arbitrary Loading ◽  
Particular Solution ◽  
Polyharmonic Spline

ABSTRACTIn this paper, analytical particular solutions of the augmented polyharmonic spline (APS) associated with Reissner plate model are explicitly derived in order to apply the dual reciprocity method. In the derivations of the particular solutions, a coupled system of three second-ordered partial differential equations (PDEs), which governs problems of Reissner plates, is initially transformed into a single six-ordered PDE by the Hörmander operator decomposition technique. Then the particular solutions of the coupled system can be found by using the particular solution of the six-ordered PDE derived in the first author's previous study. These formulas are further implemented for solving problems of Reissner plates under arbitrary loadings. In the solution procedure, an arbitrary loading measured at some scattered points is first interpolated by the APS and a corresponding particular solution can then be approximated by using the prescribed formulas. After that the complementary homogeneous problem is formally solved by the method of fundamental solutions (MFS). Numerical experiments are carried out to validate these particular solutions.

DERIVATION OF PARTICULAR SOLUTIONS USING CHEBYSHEV POLYNOMIAL BASED FUNCTIONS

2007 ◽  
Vol 04 (01) ◽  
pp. 15-32 ◽  
Author(s):  
C. S. CHEN ◽  
SUNGWOOK LEE ◽  
C.-S. HUANG
Keyword(s):  
Differential Equations ◽  
Power Series Expansion ◽  
Numerical Procedure ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Partial Differential ◽  
Irregular Boundary ◽  
Particular Solutions ◽  
Particular Solution ◽  
The Given

In this paper, we propose a simple and direct numerical procedure to obtain particular solutions for various types of differential equations. This procedure employs the power series expansion of a differential operator. Chebyshev polynomials are selected as basis functions for the approximation of the inhomogeneous terms of the given partial differential equation. This numerical scheme provides a highly efficient and accurate approximation for the evaluation of a particular solution for a variety of classes of partial differential equations. To demonstrate the effectiveness of the proposed scheme, we couple the method of fundamental solutions to solve a modified Helmholtz equation with irregular boundary configuration. The solutions were observed to have high accuracy.

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