Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball

2007 ◽  
Vol 29 (4) ◽  
pp. 393-413 ◽  
Author(s):  
Xin Li
Keyword(s):  
Unit Ball ◽  
Rate Of Convergence ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Helmholtz Equations ◽  
Modified Helmholtz Equations

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 On Inverse Problems for Characteristic Sources in Helmholtz Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Carlos J. S. Alves ◽  
Roberto Mamud ◽  
Nuno F. M. Martins ◽  
Nilson C. Roberty
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
External Boundary ◽  
Equivalent Formulation ◽  
Helmholtz Equations ◽  
Functional Formulation ◽  
Boundary Measurements

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.

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.

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