Anisotropy in electrostatics - Solutions for inclusions with canonical shapes

Tommi Rimpiläinen

Anisotropy in electrostatics - Solutions for inclusions with canonical shapes

Julkaisun otsikon käännös: Anisotropia sähköstatiikassa

Tutkimustuotos: Doctoral Thesis › Collection of Articles

Abstrakti

Muodoltaan kanonisilla inkluusioilla on sähköstatiikassa keskeinen merkitys. Tämän kaltaisten inkluusioitten analysoimisessa sovelletaan matemaattisia menetelmiä, jotka ovat luonteeltaan perusteellisia ja täsmällisiä. Siinä, missä mielivaltaisen muotoisia inkluusioita pystyy tutkimaan vain likimääräisin menetelmin, joitakin kanonisia muotoja pystyy tutkimaan menetelmin, jotka antavat täsmällisen suljetun muodon kaavan. Kanonisista muodoista tekee erityisiä se, että näillä muodoilla on poikkeuksellisen yksinkertainen esitystapa tärkeässä joukossa koordinaatistoja. Matematiikassa tunnetaan yksitoista ortogonaalista koordinaatistoa, jonka koordinaattikäyrät määrittelee toisen asteen yhtälö. Kaikilla näillä yhdellätoista koordinaatistolla on se ominaisuus, että niitten puitteissa Laplace-yhtälö separoituu. Tämä väitöskirja tarkastelee kanonisista muodoista erityisesti palloja, sferoideja ja ellipsejä. Omissa koordinaatistoissaan nämä muodot vastaavat pintoja, joilla jokin koordinaatin arvo on vakio. Koska Laplace yhtälö separoituu valittujen muotojen kannalta olennaisissa koordinaatistoissa, suuri osa sähköstaattisen ongelman ratkaisemisesta saadaan aikaan analyyttisin menetelmin, ilman likiarvoistamista. Väitöskirjassa esitetään sähköstaattisten ongelmien ratkaisuja, joista jotkin ovat puolianalyyttisia ja loput kauttaaltaan matemaattisen täsmällisiä. Vaikka kirjallisuus tunteekin valmiita ratkaisuja kaikille kolmelle väitöskirjan kannalta olennaiselle kanoniselle muodolle, olemassa oleva tutkimus jättää monta kysymystä avoimeksi. Väitöskirja poikkeaa aikaisemmasta tutkimuksesta siten, että se käsittelee materiaaliparametreiltaan aikaisempaa yleistetympiä inkluusioita. Väitöskirjan tutkimus pohjaa aikaisempaan radiaalisesti anisotrooppisia palloja koskevaan tutkimukseen, mutta luopuu siitä olettamuksesta, että pallon tangentiaalisten permittivisyyskomponenttien on oltava samat. Tämän yleistyksen aikaansaamaa uutta inkluusiota kutsutaan väitöskirjassa systrooppiseksi palloksi.

Julkaisun otsikon käännös	Anisotropia sähköstatiikassa
Alkuperäiskieli	Englanti
Pätevyys	Tohtorintutkinto
Myöntävä instituutio	Aalto-yliopisto
Valvoja/neuvonantaja	Sihvola, Ari, Vastuuprofessori Wallén, Henrik, Ohjaaja
Kustantaja	Aalto University
Painoksen ISBN	978-952-60-7184-8
Sähköinen ISBN	978-952-60-7183-1
Tila	Julkaistu - 2016
OKM-julkaisutyyppi	G5 Artikkeliväitöskirja

Tutkimusalat

anisotropia
Laplace-yhtälö
potentiaali

Pääsy asiakirjaan

http://urn.fi/URN:ISBN:978-952-60-7183-1Lisenssi: Määrittelemätön

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@phdthesis{b22e884fee254a07af05e18fdd0cfd82,

title = "Anisotropy in electrostatics - Solutions for inclusions with canonical shapes",

abstract = "Inclusions that take a canonical shape have a special place in electrostatics. Their analysis involves methods with great mathematical subtlety and rigor. Where an arbitrarily shaped inclusion can only be solved with a method that involves approximation, some of the canonical shapes permit an approach that leads to an exact solution in a closed form.What makes canonical shapes special, is that they have a particularly simple presentation in an important class of coordinate systems. There are 11 orthogonal coordinate systems where the coordinate curves are defined by equations of second degree. In these 11 systems, the Laplace equation is separable. The particular canonical shapes that the thesis studies are spheres, spheroids, and ellipses. In their corresponding coordinate systems, they correspond to surfaces with a fixed coordinate. Because these three systems allow the Laplace equation to separate, much work can be done analytically before involving approximation or computation. Some of the presented solution in the thesis are exact and others are semi-analytical. Although solutions for inclusions with the three canonical shapes exist in the literature, the thesis goes beyond the existing work by generalizing the materials of the inclusions. In all of the enclosed articles, the material of the inclusion is assumed to be anisotropic and inhomogeneous. More specifically, it is assumed that the axes of anisotropy correspond to the coordinate unit vectors of the relevant coordinate system. The term radial anisotropy has been used in the literature to refer to anisotropy in the spherical coordinates when the tangential components are equal but possibly differ from the normal component. The thesis employs a similar concept in spheroidal and elliptic systems. In the spherical coordinate system, the thesis relaxes the assumption that the two tangential components must be equal. The resulting inclusions is referred to as the systropic sphere.",

keywords = "anisotropy, Laplace equation, potential, anisotropia, Laplace-yht{\"a}l{\"o}, potentiaali, anisotropy, Laplace equation, potential",

author = "Tommi Rimpil{\"a}inen",

year = "2016",

language = "English",

isbn = "978-952-60-7184-8",

series = "Aalto University publication series DOCTORAL DISSERTATIONS",

publisher = "Aalto University",

number = "260",

address = "Finland",

school = "Aalto University",

}

TY - BOOK

T1 - Anisotropy in electrostatics - Solutions for inclusions with canonical shapes

AU - Rimpiläinen, Tommi

PY - 2016

Y1 - 2016

N2 - Inclusions that take a canonical shape have a special place in electrostatics. Their analysis involves methods with great mathematical subtlety and rigor. Where an arbitrarily shaped inclusion can only be solved with a method that involves approximation, some of the canonical shapes permit an approach that leads to an exact solution in a closed form.What makes canonical shapes special, is that they have a particularly simple presentation in an important class of coordinate systems. There are 11 orthogonal coordinate systems where the coordinate curves are defined by equations of second degree. In these 11 systems, the Laplace equation is separable. The particular canonical shapes that the thesis studies are spheres, spheroids, and ellipses. In their corresponding coordinate systems, they correspond to surfaces with a fixed coordinate. Because these three systems allow the Laplace equation to separate, much work can be done analytically before involving approximation or computation. Some of the presented solution in the thesis are exact and others are semi-analytical. Although solutions for inclusions with the three canonical shapes exist in the literature, the thesis goes beyond the existing work by generalizing the materials of the inclusions. In all of the enclosed articles, the material of the inclusion is assumed to be anisotropic and inhomogeneous. More specifically, it is assumed that the axes of anisotropy correspond to the coordinate unit vectors of the relevant coordinate system. The term radial anisotropy has been used in the literature to refer to anisotropy in the spherical coordinates when the tangential components are equal but possibly differ from the normal component. The thesis employs a similar concept in spheroidal and elliptic systems. In the spherical coordinate system, the thesis relaxes the assumption that the two tangential components must be equal. The resulting inclusions is referred to as the systropic sphere.

AB - Inclusions that take a canonical shape have a special place in electrostatics. Their analysis involves methods with great mathematical subtlety and rigor. Where an arbitrarily shaped inclusion can only be solved with a method that involves approximation, some of the canonical shapes permit an approach that leads to an exact solution in a closed form.What makes canonical shapes special, is that they have a particularly simple presentation in an important class of coordinate systems. There are 11 orthogonal coordinate systems where the coordinate curves are defined by equations of second degree. In these 11 systems, the Laplace equation is separable. The particular canonical shapes that the thesis studies are spheres, spheroids, and ellipses. In their corresponding coordinate systems, they correspond to surfaces with a fixed coordinate. Because these three systems allow the Laplace equation to separate, much work can be done analytically before involving approximation or computation. Some of the presented solution in the thesis are exact and others are semi-analytical. Although solutions for inclusions with the three canonical shapes exist in the literature, the thesis goes beyond the existing work by generalizing the materials of the inclusions. In all of the enclosed articles, the material of the inclusion is assumed to be anisotropic and inhomogeneous. More specifically, it is assumed that the axes of anisotropy correspond to the coordinate unit vectors of the relevant coordinate system. The term radial anisotropy has been used in the literature to refer to anisotropy in the spherical coordinates when the tangential components are equal but possibly differ from the normal component. The thesis employs a similar concept in spheroidal and elliptic systems. In the spherical coordinate system, the thesis relaxes the assumption that the two tangential components must be equal. The resulting inclusions is referred to as the systropic sphere.

KW - anisotropy

KW - Laplace equation

KW - potential

KW - anisotropia

KW - Laplace-yhtälö

KW - potentiaali

KW - anisotropy

KW - Laplace equation

KW - potential

M3 - Doctoral Thesis

SN - 978-952-60-7184-8

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

Anisotropy in electrostatics - Solutions for inclusions with canonical shapes

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Tutkimusalat

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