930 research outputs found
Numerical computation of the conformal map onto lemniscatic domains
We present a numerical method for the computation of the conformal map from
unbounded multiply-connected domains onto lemniscatic domains. For -times
connected domains the method requires solving boundary integral
equations with the Neumann kernel. This can be done in
operations, where is the number of nodes in the discretization of each
boundary component of the multiply connected domain. As demonstrated by
numerical examples, the method works for domains with close-to-touching
boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains
of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a
set without symmetr
Surface-tension-driven Stokes flow: a numerical method based on conformal geometry
AbstractA novel numerical scheme is presented for solving the problem of two dimensional Stokes flows with free boundaries whose evolution is driven by surface tension. The formulation is based on a complex variable formulation of Stokes flow and use of conformal mapping to track the free boundaries. The method is motivated by applications to modelling the fabrication process for microstructured optical fibres (MOFs), also known as “holey fibres”, and is therefore tailored for the computation of multiple interacting free boundaries. We give evidence of the efficacy of the method and discuss its performance
Magnetic states in multiply-connected flat nano-elements
Flat magnetic nano-elements are an essential component of current and future
spintronic devices. By shaping an element it is possible to select and
stabilize chosen metastable magnetic states, control its magnetization
dynamics. Here, using a recent significant development in mathematics of
conformal mapping, complex variable based approach to the description of
magnetic states in planar nano-elements is extended to the case when elements
are multiply-connected (that is, contain holes or magnetic anti-dots). We show
that presence of holes implies a certain restriction on the set of magnetic
states of nano-element.Comment: 5 pages, 7 figure
Stress fields around two pores in an elastic body: exact quadrature domain solutions
Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores
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An orthonormalization method for the approximate conformal mapping of multiply-connected domains
We consider the use of an orthonormalization method for constructing approximations to one of the standard conformal maps for multiply-connected domains. The method has been used successfully in [12], but only for the mapping of doubly-connected domains. Our purpose here is to consider its application to the mapping of domains whose connectivity is greater than two
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