Optimal shape of a variable condenser

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova-Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor-Saffman bubble. In the limit of high capacitance, the Polubarinova-Kochina contour tends to the Saffman-Taylor finger, which in its own turn coincides with the Morse-Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum).

Original languageEnglish
Pages (from-to)485-494
Number of pages10
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume457
Issue number2006
DOIs
Publication statusPublished - Feb 8 2001

Fingerprint

Optimal Shape
Concrete dams
condensers
Boundary value problems
range (extremes)
Capacitance
Extremum
Hydraulics
Polynomials
Cauchy Integral
Dirichlet Boundary Value Problem
Curve
Isoperimetric
dams
Chebyshev Polynomials
curves
Fourier coefficients
boundary value problems
Quadratic form
hydraulics

Keywords

  • Dirichlet's boundary-value problem
  • Global extremum
  • Laplace's equation
  • Optimization
  • Shaping

ASJC Scopus subject areas

  • General

Cite this

Optimal shape of a variable condenser. / Kacimov, Anvar.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 457, No. 2006, 08.02.2001, p. 485-494.

Research output: Contribution to journalArticle

@article{0844dc2964b449f4922d330167595472,
title = "Optimal shape of a variable condenser",
abstract = "The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova-Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor-Saffman bubble. In the limit of high capacitance, the Polubarinova-Kochina contour tends to the Saffman-Taylor finger, which in its own turn coincides with the Morse-Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum).",
keywords = "Dirichlet's boundary-value problem, Global extremum, Laplace's equation, Optimization, Shaping",
author = "Anvar Kacimov",
year = "2001",
month = "2",
day = "8",
doi = "10.1098/rspa.2000.0677",
language = "English",
volume = "457",
pages = "485--494",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "0080-4630",
publisher = "Royal Society of London",
number = "2006",

}

TY - JOUR

T1 - Optimal shape of a variable condenser

AU - Kacimov, Anvar

PY - 2001/2/8

Y1 - 2001/2/8

N2 - The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova-Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor-Saffman bubble. In the limit of high capacitance, the Polubarinova-Kochina contour tends to the Saffman-Taylor finger, which in its own turn coincides with the Morse-Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum).

AB - The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova-Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor-Saffman bubble. In the limit of high capacitance, the Polubarinova-Kochina contour tends to the Saffman-Taylor finger, which in its own turn coincides with the Morse-Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum).

KW - Dirichlet's boundary-value problem

KW - Global extremum

KW - Laplace's equation

KW - Optimization

KW - Shaping

UR - http://www.scopus.com/inward/record.url?scp=57249114434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57249114434&partnerID=8YFLogxK

U2 - 10.1098/rspa.2000.0677

DO - 10.1098/rspa.2000.0677

M3 - Article

VL - 457

SP - 485

EP - 494

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2006

ER -