A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Mobile disks in hyperbolic space and minimization of conformal capacity
Tekijät: Hakula Harri, Nasser Mohamed M. S., Vuorinen Matti
Kustantaja: KENT STATE UNIVERSITY
Kustannuspaikka: KENT
Julkaisuvuosi: 2024
Journal: Electronic Transactions on Numerical Analysis
Tietokannassa oleva lehden nimi: ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
Lehden akronyymi: ELECTRON T NUMER ANA
Vuosikerta: 60
Aloitussivu: 1
Lopetussivu: 19
Sivujen määrä: 19
ISSN: 1068-9613
DOI: https://doi.org/10.1553/etna_vol60s1
Verkko-osoite: https://doi.org/10.1553/etna_vol60s1
Rinnakkaistallenteen osoite: https://arxiv.org/abs/2303.00145
Preprintin osoite: https://arxiv.org/abs/2303.00145v1
Tiivistelmä
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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