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.

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.

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.

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

2015 ◽  
Vol 17 (3) ◽  
pp. 867-886 ◽  
Author(s):  
C. S. Chen ◽  
Xinrong Jiang ◽  
Wen Chen ◽  
Guangming Yao
Keyword(s):  
Helmholtz Equation ◽  
Large Scale ◽  
Solution Process ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Dense Matrix ◽  
Modified Helmholtz Equation ◽  
Large Scale Problems ◽  
Particular Solution

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.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

2011 ◽  
Vol 3 (5) ◽  
pp. 572-585 ◽  
Author(s):  
A. Tadeu ◽  
C. S. Chen ◽  
J. António ◽  
Nuno Simões
Keyword(s):  
Fourier Transform ◽  
Helmholtz Equation ◽  
Initial Conditions ◽  
Inverse Fourier Transform ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Static Response ◽  
Particular Solution ◽  
The Fourier Transform ◽  
Transfer Problems

AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341007 ◽  
Author(s):  
CHIA-CHENG TSAI ◽  
PO-HO LIN
Keyword(s):  
Exponential Convergence ◽  
Fundamental Solutions ◽  
Solution Procedure ◽  
Method Of Fundamental Solutions ◽  
System Matrix ◽  
Node Number ◽  
Discontinuous Boundary ◽  
Multiple Precision ◽  
Ill Conditioning ◽  
Discontinuous Boundary Data

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.

ON THE CONVERGENCE OF THE MFS–MPS SCHEME FOR 1D POISSON'S EQUATION

2013 ◽  
Vol 10 (02) ◽  
pp. 1341006 ◽  
Author(s):  
C. S. CHEN ◽  
C.-S. HUANG ◽  
K. H. LIN
Keyword(s):  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Basis Functions ◽  
Poisson’S Equation ◽  
Method Of Fundamental Solutions ◽  
Poisson's Equation ◽  
Numerical Process ◽  
Particular Solution ◽  
Inhomogeneous Problems ◽  
Lagrange Interpolating Polynomial

The method of fundamental solutions (MFS) has been an effective meshless method for solving homogeneous partial differential equations. Coupled with radial basis functions (RBFs), the MFS has been extended to solve the inhomogeneous problems through the evaluation of the approximate particular solution and homogeneous solution. In this paper, we prove the the approximate solution of the above numerical process for solving 1D Poisson's equation converges in the sense of Lagrange interpolating polynomial using the result of Driscoll and Fornberg [2002].

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