On Detecting the Shape of an Unknown Object in an Electric Field

Jukka-Pekka Humaloja; Timo Hämäläinen; Seppo Pohjolainen

doi:10.1007/978-3-319-23413-7

On Detecting the Shape of an Unknown Object in an Electric Field

Jukka-Pekka Humaloja, Timo Hämäläinen, Seppo Pohjolainen

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

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Abstract

The problem discussed in this paper is detecting the shape of an unknown object in a 2-dimensional static electric field. For simplicity, the problem is defined in a partially rectangular domain, where on a part of the boundary the potential and/or its normal derivative are known. On the other part of the boundary the boundary curve is unknown, and this curve is to be determined. The unknown part of the boundary curve describes the shape of the unknown object.
The problem is defined in the complex plane by an analytic function w=f(z) = u(x,y)+iv(x,y) with the potential u as its real part. Then the inverse function is given as f^{-1}(w) = x(u,v)+iy(u,v), where the functions x and y are harmonic in a rectangle with an unknown boundary condition on one boundary. The alternating-field technique is used to solve the unknown boundary condition.

Original language	English
Title of host publication	Progress in Industrial Mathematics at ECMI 2014
Editors	G. Russo, V. Capasso, G. Nicosia, V. Romano
Publisher	Springer International Publishing
ISBN (Electronic)	978-3-319-23413-7
ISBN (Print)	978-3-319-23412-0
DOIs	https://doi.org/10.1007/978-3-319-23413-7
Publication status	Published - 2016
Publication type	A4 Article in conference proceedings
Event	EUROPEAN CONFERENCE ON MATHEMATICS FOR INDUSTRY - Duration: 1 Jan 2000 → …

Publication series

Name	Mathematics in Industry
Publisher	Springer-Verlag
Volume	22
ISSN (Electronic)	1612-3956

Conference

Conference	EUROPEAN CONFERENCE ON MATHEMATICS FOR INDUSTRY
Abbreviated title	ECMI
Period	1/01/00 → …

Keywords

free boundary problem
industrial mathematics

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On Detecting the Shape of an Unknown 2016
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abstract = "The problem discussed in this paper is detecting the shape of an unknown object in a 2-dimensional static electric field. For simplicity, the problem is defined in a partially rectangular domain, where on a part of the boundary the potential and/or its normal derivative are known. On the other part of the boundary the boundary curve is unknown, and this curve is to be determined. The unknown part of the boundary curve describes the shape of the unknown object.The problem is defined in the complex plane by an analytic function w=f(z) = u(x,y)+iv(x,y) with the potential u as its real part. Then the inverse function is given as f^{-1}(w) = x(u,v)+iy(u,v), where the functions x and y are harmonic in a rectangle with an unknown boundary condition on one boundary. The alternating-field technique is used to solve the unknown boundary condition.",

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Humaloja, J-P, Hämäläinen, T & Pohjolainen, S 2016, On Detecting the Shape of an Unknown Object in an Electric Field. in G Russo, V Capasso, G Nicosia & V Romano (eds), Progress in Industrial Mathematics at ECMI 2014. Mathematics in Industry, vol. 22, Springer International Publishing, EUROPEAN CONFERENCE ON MATHEMATICS FOR INDUSTRY, 1/01/00. https://doi.org/10.1007/978-3-319-23413-7

On Detecting the Shape of an Unknown Object in an Electric Field. / Humaloja, Jukka-Pekka; Hämäläinen, Timo ; Pohjolainen, Seppo.
Progress in Industrial Mathematics at ECMI 2014. ed. / G. Russo; V. Capasso; G. Nicosia; V. Romano. Springer International Publishing, 2016. (Mathematics in Industry; Vol. 22).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

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N2 - The problem discussed in this paper is detecting the shape of an unknown object in a 2-dimensional static electric field. For simplicity, the problem is defined in a partially rectangular domain, where on a part of the boundary the potential and/or its normal derivative are known. On the other part of the boundary the boundary curve is unknown, and this curve is to be determined. The unknown part of the boundary curve describes the shape of the unknown object.The problem is defined in the complex plane by an analytic function w=f(z) = u(x,y)+iv(x,y) with the potential u as its real part. Then the inverse function is given as f^{-1}(w) = x(u,v)+iy(u,v), where the functions x and y are harmonic in a rectangle with an unknown boundary condition on one boundary. The alternating-field technique is used to solve the unknown boundary condition.

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On Detecting the Shape of an Unknown Object in an Electric Field

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