Application of Fast Multipole VBEM to Two-Dimensional Potential Problems

2009 ◽  
Author(s):  
Zhijia Zhang ◽  
Qiang Xu
Keyword(s):  
Two Dimensional ◽  
Fast Multipole ◽  
Potential Problems

scholarly journals The influence of vortices upon the resitance experienced by solids moving through a liquid

1928 ◽  
Vol 119 (781) ◽  
pp. 146-156 ◽  
Keyword(s):  
Perfect Fluid ◽  
Relative Motion ◽  
Constant Velocity ◽  
Fluid Motion ◽  
Resultant Force ◽  
The Other ◽  
Two Dimensional ◽  
Perfect Fluids ◽  
Potential Problems ◽  
The Moment

(1) It is not so long ago that it was generally believed that the "classical" hydrodynamics, as dealing with perfect fluids, was, by reason of the very limitations implied in the term "perfect," incapable of explaining many of the observed facts of fluid motion. The paradox of d'Alembert, that a solid moving through a liquid with constant velocity experienced no resultant force, was in direct contradiction with the observed facts, and, among other things, made the lift on an aeroplane wing as difficult to explain as the drag. The work of Lanchester and Prandtl, however, showed that lift could be explained if there was "circulation" round the aerofoil. Of course, in a truly perfect fluid, this circulation could not be produced—it does need viscosity to originate it—but once produced, the lift follows from the theory appropriate to perfect fluids. It has thus been found possible to explain and calculate lift by means of the classical theory, viscosity only playing a significant part in the close neighbourhood ("grenzchicht") of the solid. It is proposed to show, in the present paper, how the presence of vortices in the fluid may cause a force to act on the solid, with a component in the line of motion, and so, at least partially, explain drag. It has long been realised that a body moving through a fluid sets up a train of eddies. The formation of these needs a supply of energy, ultimately dissipated by viscosity, which qualitatively explains the resistance experienced by the solid. It will be shown that the effect of these eddies is not confined to the moment of their birth, but that, so long as they exist, the resultant of the pressure on the solid does not vanish. This idea is not absolutely new; it appears in a recent paper by W. Müller. Müller uses some results due to M. Lagally, who calculates the resultant force on an immersed solid for a general fluid motion. The result, as far as it concerns vortices, contains their velocities relative to the solid. Despite this, the term — ½ ρq 2 only was used in the pressure equation, although the other term, ρ ∂Φ / ∂t , must exist on account of the motion. (There is, by Lagally's formulæ, no force without relative motion.) The analysis in the present paper was undertaken partly to supply this omission and partly to check the result of some work upon two-dimensional potential problems in general that it is hoped to publish shortly.

scholarly journals Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jufeng Wang ◽  
Fengxin Sun ◽  
Ying Xu
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Potential Problems ◽  
Element Free Method ◽  
Error Estimations ◽  
Element Free

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

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