scholarly journals The Schwarz-Christoffel Conformal Mapping for “Polygons” with Infinitely Many Sides

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
Gonzalo Riera ◽  
Hernán Carrasco ◽  
Rubén Preiss
Keyword(s):  
Conformal Mapping ◽  
Mapping Function ◽  
Mathematical Software ◽  
Explicit Formulas ◽  
Line Segments ◽  
Hyperelliptic Surfaces ◽  
Open Questions ◽  
Upper Half Plane ◽  
Koch Snowflake ◽  
Boundary Curves

The classical Schwarz-Christoffel formula gives conformal mappings of the upper half-plane onto domains whose boundaries consist of a finite number of line segments. In this paper, we explore extensions to boundary curves which in one sense or another are made up of infinitely many line segments, with specific attention to the “infinite staircase” and to the Koch snowflake, for both of which we develop explicit formulas for the mapping function and explain how one can use standard mathematical software to generate corresponding graphics. We also discuss a number of open questions suggested by these considerations, some of which are related to differentials on hyperelliptic surfaces of infinite genus.

scholarly journals Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130848 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Slit Domain ◽  
Combined Use ◽  
Polygonal Region ◽  
Upper Half Plane ◽  
Indefinite Integral ◽  
Detailed Derivation

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

New Characterizations of Polyhedral Cones

1957 ◽  
Vol 9 ◽  
pp. 1-4 ◽  
Author(s):  
H. Mirkil
Keyword(s):  
Half Plane ◽  
Single Point ◽  
Circular Cone ◽  
Half Line ◽  
Polyhedral Cones ◽  
Horizontal Projection ◽  
Line Segments ◽  
Upper Half Plane

A pyramid clearly has all its projections closed, even when the line segments from vertex to base are extended to infinite half-lines. Not so a circular cone. For if the cone is on its side and supported by the (x, y) plane in such a way that its infinite half-line of support coincides with the positive x axis, then its horizontal projection on the (y, z) plane is the open upper half-plane y > 0, together with the single point (0, 0).

An Application of Conformal Mapping to Fracture Analysis of Composite Plate with Curved Crack

2011 ◽  
Vol 121-126 ◽  
pp. 1759-1763
Author(s):  
Jun Lin Li ◽  
Hai Xia Cheng
Keyword(s):  
Stress Intensity Factor ◽  
Conformal Mapping ◽  
Composite Plate ◽  
Mapping Function ◽  
Fracture Analysis ◽  
Curved Crack ◽  
Special Stress ◽  
Parabolic Crack ◽  
Stress Boundary Conditions ◽  
Stress Boundary

Fracture problem about anisotropic composite plate of curved crack is researched with the help of appropriate conformal mapping. The crack curve is transformed into a unit circle by the quoted conformal mapping function. Based on the stress boundary conditions, the special stress function is found and the expressions for the stress field and the stress intensity factor near the parabolic crack tip are obtained.

Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusion

2021 ◽  
pp. 108128652110134
Author(s):  
Ping Yang ◽  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Conformal Mapping ◽  
Connected Domain ◽  
Mapping Function ◽  
Elliptical Inhomogeneity ◽  
Surrounding Matrix ◽  
Eshelby Inclusion ◽  
Doubly Connected Domain ◽  
The Matrix ◽  
Uniformity Property ◽  
Complex Cases

We establish the uniformity of stresses inside both a non-parabolic open inhomogeneity and a non-elliptical closed inhomogeneity interacting with a nearby circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Our procedure involves the introduction of a conformal mapping function for the doubly connected domain occupied by the matrix and the circular Eshelby inclusion. Two conditions are established in order to achieve the uniformity property inside each of the two inhomogeneities. Our results indicate that: (a) the internal uniform stresses are independent of the specific shapes of the two inhomogeneities and the existence of the nearby circular Eshelby inclusion; (b) the open and closed shapes of the respective inhomogeneities are significantly affected by the presence of the circular Eshelby inclusion. We also consider the two more complex cases involving: (a) an arbitrary number of circular Eshelby inclusions undergoing uniform eigenstrains; (b) a circular Eshelby inclusion undergoing linear eigenstrains. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed theory.

scholarly journals Approximate Conformal Mappings and Elasticity Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Pyotr N. Ivanshin ◽  
Elena A. Shirokova
Keyword(s):  
Conformal Mapping ◽  
Elasticity Theory ◽  
Unit Disk ◽  
Connected Domain ◽  
Smooth Boundary ◽  
Mapping Function ◽  
Conformal Mappings ◽  
New Method ◽  
Taylor Polynomial ◽  
Elasticity Problems

Here, we present the new method of approximate conformal mapping of the unit disk to a one-connected domain with smooth boundary without auxiliary constructions and iterations. The mapping function is a Taylor polynomial. The method is applicable to elasticity problems solution.

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