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Adjoint variable method for time-harmonic Maxwell equations

Stephane Durand UGent, Ivan Cimrak UGent and Peter Sergeant UGent (2009) COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING. 28(5). p.1202-1215
abstract
Purpose - The purpose of this paper is to study the optimization problem of low-frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods to calculate the gradient. Design/methodology/approach - The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well-posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented. Findings - Using Friedrichs' inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two. Research limitations/implications - The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy. Originality/value - Theoretical results for eddy Currents model are often based on a non-vanishing conductivity. The theoretical value of this paper is the presence of non-conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (proceedingsPaper)
publication status
published
subject
keyword
Eddy currents, Optimization techniques, Numerical analysis, OPTIMAL SHAPE DESIGN
journal title
COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING
Compel-Int. J. Comp. Math. Electr. Electron. Eng.
volume
28
issue
5
pages
1202 - 1215
conference name
10th International workshop on Optimization and Inverse Problems in Electromagnetism
conference location
Ilmenau, Germany
conference start
2009-09-14
conference end
2009-09-17
Web of Science type
Proceedings Paper
Web of Science id
000270210500010
JCR category
ENGINEERING, ELECTRICAL & ELECTRONIC
JCR impact factor
0.46 (2009)
JCR rank
182/244 (2009)
JCR quartile
3 (2009)
ISSN
0332-1649
DOI
10.1108/03321640910969458
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
782166
handle
http://hdl.handle.net/1854/LU-782166
date created
2009-11-17 09:19:03
date last changed
2014-02-27 15:23:03
@article{782166,
  abstract     = {Purpose - The purpose of this paper is to study the optimization problem of low-frequency magnetic shielding using the adjoint variable method (AVM). This method is compared with conventional methods to calculate the gradient.
Design/methodology/approach - The equation for the vector potential (eddy currents model) in appropriate Sobolev spaces is studied to obtain well-posedness. The optimization problem is formulated in terms of a cost functional which depends on the vector potential and its rotation. Convergence of a steepest descent algorithm to a stationary point of this functional is proved. Finally, some numerical results for an axisymmetric induction heater are presented.
Findings - Using Friedrichs' inequality, the existence and uniqueness of the vector potential, its gradient and the corresponding adjoint variable can be proved. From the numerical results, it is concluded that the AVM is advantageous if the number of parameters to optimize is larger than two.
Research limitations/implications - The AVM is only faster than conventional methods if the gradients can be calculated with sufficient accuracy.
Originality/value - Theoretical results for eddy Currents model are often based on a non-vanishing conductivity. The theoretical value of this paper is the presence of non-conducting materials in the domain. From a practical viewpoint, it has been demonstrated that the AVM can yield a significant reduction of computational time for advanced optimization problems.},
  author       = {Durand, Stephane and Cimrak, Ivan and Sergeant, Peter},
  issn         = {0332-1649},
  journal      = {COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING},
  keyword      = {Eddy currents,Optimization techniques,Numerical analysis,OPTIMAL SHAPE DESIGN},
  language     = {eng},
  location     = {Ilmenau, Germany},
  number       = {5},
  pages        = {1202--1215},
  title        = {Adjoint variable method for time-harmonic Maxwell equations},
  url          = {http://dx.doi.org/10.1108/03321640910969458},
  volume       = {28},
  year         = {2009},
}

Chicago
Durand, Stephane, Ivan Cimrak, and Peter Sergeant. 2009. “Adjoint Variable Method for Time-harmonic Maxwell Equations.” Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering 28 (5): 1202–1215.
APA
Durand, S., Cimrak, I., & Sergeant, P. (2009). Adjoint variable method for time-harmonic Maxwell equations. COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING, 28(5), 1202–1215. Presented at the 10th International workshop on Optimization and Inverse Problems in Electromagnetism.
Vancouver
1.
Durand S, Cimrak I, Sergeant P. Adjoint variable method for time-harmonic Maxwell equations. COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING. 2009;28(5):1202–15.
MLA
Durand, Stephane, Ivan Cimrak, and Peter Sergeant. “Adjoint Variable Method for Time-harmonic Maxwell Equations.” COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING 28.5 (2009): 1202–1215. Print.