scholarly journals Structure‐preserving numerical scheme for the one‐phase Hele‐Shaw problems by the method of fundamental solutions

2019 ◽  
Vol 1 (6) ◽  
Author(s):  
Koya Sakakibara ◽  
Shigetoshi Yazaki
Keyword(s):  
Numerical Scheme ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Structure Preserving ◽  
The One

Method of Fundamental Solutions for Stokes Problems by the Pressure-Stream Function Formulation

2008 ◽  
Vol 24 (2) ◽  
pp. 137-144 ◽  
Author(s):  
D. L. Young ◽  
J. T. Wu ◽  
C. L. Chiu
Keyword(s):  
Stream Function ◽  
Numerical Scheme ◽  
Numerical Procedure ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Circular Cavity ◽  
Stokes Flows ◽  
Function Formulation ◽  
Stream Function Formulation ◽  
Harmonic Equation

ABSTRACTThe main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.

scholarly journals Method of Fundamental Solutions for Stokes' First and Second Problems

2005 ◽  
Vol 21 (1) ◽  
pp. 25-31 ◽  
Author(s):  
S. P. Hu ◽  
C. M. Fan ◽  
C. W. Chen ◽  
D. L. Young
Keyword(s):  
Fundamental Solution ◽  
Numerical Scheme ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Memory Storage ◽  
Time Dependent ◽  
Method Of Fundamental Solutions ◽  
Benchmark Problems ◽  
Infinite Domain ◽  
Mesh Free

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.

A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP

2009 ◽  
Vol 06 (02) ◽  
pp. 181-197 ◽  
Author(s):  
Y. C. HON ◽  
M. LI
Keyword(s):  
Heat Conduction Problem ◽  
Linear Equations ◽  
Fundamental Solutions ◽  
Computational Method ◽  
Method Of Fundamental Solutions ◽  
Discrepancy Principle ◽  
Regularization Technique ◽  
One Dimensional ◽  
Curve Method ◽  
The One

Based on the discrepancy principle, we develop in this paper a new method of choosing the location of source points to solve the backward heat conduction problem (BHCP) by using the method of fundamental solutions (MFS). The standard Tikhonov regularization technique with the L curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Numerical verifications of the proposed computational method are presented for both the one-dimensional and the two-dimensional BHCP.

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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