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.

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.

scholarly journals Influence of fish backbone model geometrical features on the numerical target strength of swimbladdered fish

2020 ◽  
Author(s):  
I Pérez-Arjona ◽  
L Godinho ◽  
V Espinosa
Keyword(s):  
Atlantic Salmon ◽  
Salmo Salar ◽  
Sparus Aurata ◽  
Fundamental Solutions ◽  
Sea Bream ◽  
Method Of Fundamental Solutions ◽  
Target Strength ◽  
Simplified Models ◽  
Geometrical Features ◽  
Proper Estimation

Abstract The method of fundamental solutions has been applied to evaluate the influence of fish models geometrical features on the target strength (TS) directivity and TS frequency response of swimbladdered fish. Simplified models were considered for two fish species: gilt-head sea bream (Sparus aurata, Linnaeus 1758) and Atlantic salmon (Salmo salar, Linnaeus 1758), and different geometrical details of their morphology were studied, such as backbone presence, and its curvature or the inclusion of vertebrae modulation. Swimbladder shape and tilt, together with the inclusion of backbone (and its realistic curvature) for dorsal measurements were the most important features for proper estimation of mean TS. The estimation of mean TS is considered including the effect of fish tilt, the echosounder frequency, and the fish-to-transducer distance.

Sign in / Sign up

Export Citation Format

Share Document