A1 Refereed original research article in a scientific journal
Mobile disks in hyperbolic space and minimization of conformal capacity
Authors: Hakula Harri, Nasser Mohamed M. S., Vuorinen Matti
Publisher: KENT STATE UNIVERSITY
Publishing place: KENT
Publication year: 2024
Journal: Electronic Transactions on Numerical Analysis
Journal name in source: ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
Journal acronym: ELECTRON T NUMER ANA
Volume: 60
First page : 1
Last page: 19
Number of pages: 19
ISSN: 1068-9613
DOI: https://doi.org/10.1553/etna_vol60s1
Web address : https://doi.org/10.1553/etna_vol60s1
Self-archived copy’s web address: https://arxiv.org/abs/2303.00145
Preprint address: https://arxiv.org/abs/2303.00145v1
Abstract
Our focus is to study constellations of disjoint disks in the hyperbolic space, i.e., the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set E which is the union of m > 2 disks with hyperbolic radii r(j )> 0, j = 1, ... , m. The centers of the disks are not fixed, and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when m = 3 or m = 4. In the absence of analytic methods, our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the hp-FEM method, respectively. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas.
Our focus is to study constellations of disjoint disks in the hyperbolic space, i.e., the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set E which is the union of m > 2 disks with hyperbolic radii r(j )> 0, j = 1, ... , m. The centers of the disks are not fixed, and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when m = 3 or m = 4. In the absence of analytic methods, our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the hp-FEM method, respectively. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas.
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