195 research outputs found
Two-dimensional shapes and lemniscates
A shape in the plane is an equivalence class of sufficiently smooth Jordan
curves, where two curves are equivalent if one can be obtained from the other
by a translation and a scaling. The fingerprint of a shape is an equivalence of
orientation preserving diffeomorphisms of the unit circle, where two
diffeomorphisms are equivalent if they differ by right composition with an
automorphism of the unit disk. The fingerprint is obtained by composing Riemann
maps onto the interior and exterior of a representative of a shape in a
suitable way. In this paper, we show that there is a one-to-one correspondence
between shapes defined by polynomial lemniscates of degree n and nth roots of
Blaschke products of degree n. The facts that lemniscates approximate all
Jordan curves in the Hausdorff metric and roots of Blaschke products
approximate all orientation preserving diffeomorphisms of the circle in the
C^1-norm suggest that lemniscates and roots of Blaschke products are natural
objects to study in the theory of shapes and their fingerprints
Matrix equation solving of PDEs in polygonal domains using conformal mappings
We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz-Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation
Novel applications of complex analysis to effective parameter quantification in transport theory
This thesis proposes the application of complex analysis to the calculation of effective
parameters of transport problems in multiply connected domains. This can be done by
using special functions called Schottky-Klein prime functions. The effective parameters
focused on in this thesis are electrical resistivity, electrical capacity, and slip lengths of
channels. The prime function is a powerful mathematical function invented by Crowdy for
solving problems in multiply connected domains including transport problems governed
by Laplace’s equation and Poisson’s equation in domains with multiple boundaries. The
functional properties of the prime function make it possible to analyse effective parameters
in multiply connected domains.
First, a new method for solving a new class of boundary value problems in multiply
connected domains is explained. An explicit solution can be derived by multiplying of the
boundary data with a radial slit map written in terms of the prime functions.
We then focus on two electrical transport problems called “the van der Pauw method”
and “electrical capacity”. For the van der Pauw method, the prime function allows us to
derive new formulas for calculating the resistivity of holey samples. A new method for
the electrical capacity of multiply connected domains is formulated by coupling the prime
function with asymptotic matching.
We next construct explicit solutions for flows through superhydrophobic surfaces in
periodic channels and calculate the slip length of these channels. We end the thesis by
mentioning that the new methodology gives accurate estimates for so-called “accessory
parameter problems” associated with conformal maps of multiply connected domains.Open Acces
Simulating the Hele-Shaw flow in the presence of various obstacles and moving particles
Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is
performed. The method of solution utilises the uniform asymptotic formulas for
Green's and Neumann functions recently obtained by V. Maz'ya and A. Movchan.
Theoretical results of the paper are illustrated by the numerical simulations.Comment: 29 pages, 14 figure
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
Here shape space is either the manifold of simple closed smooth
unparameterized curves in or is the orbifold of immersions from
to modulo the group of diffeomorphisms of . We
investige several Riemannian metrics on shape space: -metrics weighted by
expressions in length and curvature. These include a scale invariant metric and
a Wasserstein type metric which is sandwiched between two length-weighted
metrics. Sobolev metrics of order on curves are described. Here the
horizontal projection of a tangent field is given by a pseudo-differential
operator. Finally the metric induced from the Sobolev metric on the group of
diffeomorphisms on is treated. Although the quotient metrics are
all given by pseudo-differential operators, their inverses are given by
convolution with smooth kernels. We are able to prove local existence and
uniqueness of solution to the geodesic equation for both kinds of Sobolev
metrics.
We are interested in all conserved quantities, so the paper starts with the
Hamiltonian setting and computes conserved momenta and geodesics in general on
the space of immersions. For each metric we compute the geodesic equation on
shape space. In the end we sketch in some examples the differences between
these metrics.Comment: 46 pages, some misprints correcte
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