Theodorsen’s and Garrick’s Method for Conformal Mapping of the Unit Circle into an Ellipse

1985 ◽  
pp. 274-276
Author(s):  
A. M. Ostrowski
Keyword(s):  
Conformal Mapping ◽  
Unit Circle

A Point Dislocation Interacting with an Elliptical Hole Located at a Bi-Material Interface

2012 ◽  
Vol 151 ◽  
pp. 75-79 ◽  
Author(s):  
Xian Feng Wang ◽  
Feng Xing ◽  
Norio Hasebe ◽  
P.B.N. Prasad
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Half Plane ◽  
Elliptical Hole ◽  
Mapping Function ◽  
Complex Stress ◽  
Complex Variables ◽  
Material Interface ◽  
Rational Mapping ◽  
Point Dislocation

The problem of a point dislocation interacting with an elliptical hole at the interface of two bonded half-planes is studied. Complex stress potentials are obtained by applying the methods of complex variables and conformal mapping. A rational mapping function that maps a half plane with a semi-elliptical notch onto a unit circle is used for mapping the bonded half-planes. The solution derived can serve as Green’s function to study internal cracks interacting with an elliptical interfacial cavity.

scholarly journals The problem of extreme decomposition of a complex plane with fixed poles on a circle

2018 ◽  
Vol 32 ◽  
pp. 10-17
Author(s):  
Liudmyla Vyhivska
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Complex Plane ◽  
Complex Analysis ◽  
Function Theory ◽  
Geometric Function Theory ◽  
Holomorphic Dynamics ◽  
Geometric Function ◽  
Inner Radius ◽  
Overlapping Domains

The problem of extreme decomposition of a complex plane with fixed poles on a circle. Investigation on geometric function theory has been conducted by several researchers, however, few studies have reported on the problem considering extremal configurations the product of inner radii of non-overlapping domains with respect to fixed poles. The paper describes the problem of finding the maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to points on a unit circle multiply by a certain positive degree \(\gamma\) of the inner radius of the domain with respect to the zero. The problem was studied using the method of separating transformation. Proving the theorem shows that the maximum is obtained if \(\gamma\in(1,n^2]\) and for all \(n\geqslant 2\). Its results and the method for the obtaining of these results can be used in the theory of potential, approximations, holomorphic dynamics, estimation of the distortion problems in conformal mapping, and complex analysis.

Some Simple Transformation Functions for Square and Triangular Holes with Rounded Corners

1960 ◽  
Vol 11 (2) ◽  
pp. 195-199 ◽  
Author(s):  
W. H. Wittrick
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Equilateral Triangle ◽  
Simple Transformation ◽  
Transformation Functions

SummaryTransformation functions, in the form of three-term polynomials, are derived for the conformal mapping of the region outside a square or an equilateral triangle, having rounded corners with varying and controllable degrees of curvature, on to the region outside the unit circle.

Semi-analytical study of the Voinovs problem

2018 ◽  
Vol 30 (2) ◽  
pp. 298-337 ◽  
Author(s):  
E. A. KARABUT ◽  
A. G. PETROV ◽  
E. N. ZHURAVLEVA
Keyword(s):  
Free Boundary ◽  
Power Series ◽  
Conformal Mapping ◽  
Unit Circle ◽  
Ideal Fluid ◽  
Laurent Series ◽  
Analytical Study ◽  
Padé Approximants ◽  
Boundary Shape ◽  
Change Of Variables

A problem from the class of unsteady plane flows of an ideal fluid with a free boundary is considered. A conformal mapping of the exterior of a unit circle onto the region occupied by the fluid is sought. The solution is constructed in the form of power series in time or Laurent series which are analytically continued with the use of Padé approximants and change of variables of a certain special type. The free boundary shape and the pressure and velocity distributions are found. Singularities of the solution are studied.

scholarly journals Relative Distance and Quasi-Conformal Mappings

1960 ◽  
Vol 16 ◽  
pp. 111-117
Author(s):  
D. A. Storvick
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Distance Function ◽  
Connected Domain ◽  
General Class ◽  
Conformal Mappings ◽  
Relative Distance ◽  
Simply Connected Domain ◽  
Simply Connected

1. Introduction. M. A. Lavrentiev made use of a relative distance function to establish some important results concerning the correspondence between the frontiers under a conformal mapping of a simply connected domain onto the unit circle. The purpose of this note is to show that some of these results are valid for the boundary correspondences induced by the more general class of quasi-conformal mappings.

A novel analytical calculation of magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping

2020 ◽  
pp. 1-15
Author(s):  
Jianqi Li ◽  
Yu Zhou ◽  
Jianying Li
Keyword(s):  
Magnetic Field ◽  
Conformal Mapping ◽  
Permanent Magnet ◽  
Flux Density ◽  
Analytical Method ◽  
Electromotive Force ◽  
Analytical Calculation ◽  
Air Gap ◽  
Permanent Magnet Machines ◽  
Peak Value

This paper presented a novel analytical method for calculating magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping. Firstly, flux density without slots and complex relative air-gap permeance of slotted air gap are derived from conformal transformation separately. Secondly, they are combined in order to obtain normalized flux density taking account into the slots effect. The finite element (FE) results confirmed the validity of the analytical method for predicting magnetic field and back electromotive force (BEMF) in the slotted air gap of spoke-type permanent-magnet machines. In comparison with FE result, the analytical solution yields higher peak value of cogging torque.

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