×

Level set method for the inverse elliptic problem in nonlinear electromagnetism. (English) Zbl 1207.78045

The paper deals with the inverse problem for a quasilinear PDE in a 2-dimensional domain, where a spatial inhomogeneity is to be identified from certain boundary values of the solution of the PDE. A typical example related to this problem is the identification of a crack inside a magnetic material (hard magnetic steel), where measurements of the magnetic induction are given on the outer boundary of the workpiece. The reconstruction is realized by the minimization of a cost function using the steepest descent method, where the gradient directions are evaluated using the sensitivity equation and the adjoint variable method. The efficiency of the proposed method is illustrated by numerical simulations in the cases of a single as well as multiple inhomogeneities.

MSC:

78M50 Optimization problems in optics and electromagnetic theory
35R30 Inverse problems for PDEs
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Giguère, S.; Lepine, B.; Dubois, J., Pulsed eddy current technology: characterizing material loss with gap and lift-off variations, Research in Nondestructive Evaluation, 13, 3, 119-129 (2001)
[2] Brühl, M.; Hank, M., Recent progress in electrical impedance tomography, Inverse Problems, 19, 65-90 (2003) · Zbl 1048.92022
[3] Chen, W.; Cheng, J.; Lin, J.; Wang, L., A level set method to reconstruct the discontinuity of the conductivity in EIT, Science in China Series A: Mathematics, 52, 1, 29-44 (2009) · Zbl 1181.35274
[4] Chan, T. F.; Tai, X.-C., Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, Journal of Computational Physics, 193, 1, 40-66 (2004) · Zbl 1036.65086
[5] Isakov, V., Inverse obstacle problems, Inverse Problems, 25, 12, 123002 (2009), 18 pp · Zbl 1181.35334
[6] Muller, J. L.; Sitanen, S., Direct reconstruction of conductivities from boundary measurements, SIAM Journal of Scientific Computing, 24, 1232-1266 (2003) · Zbl 1031.78008
[7] Osher, S.; Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces. Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153 (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1026.76001
[8] Chung, E. T.; Chan, T. F.; Tai, X.-C., Electrical impedance tomography using level set representation and total variational regularization, Journal of Computational Physics, 205, 1, 357-372 (2005) · Zbl 1072.65143
[9] Ito, K.; Kunisch, K.; Li, Z., Level-set function approach to an inverse interface problem, Inverse Problems, 17, 5, 1225-1242 (2001) · Zbl 0986.35130
[10] Luo, Z.; Tong, L.; Luo, J.; Wei, P.; Wang, M. Y., Design of piezoelectric actuators using a multiphase level set method of piecewise constants, Journal of Computational Physics, 228, 7, 2643-2659 (2009), http://dx.doi.org/10.1016/j.jcp.2008.12.019 · Zbl 1160.78322
[11] Luo, Z.; Tong, L.; Ma, H., Shape and topology optimization for electrothermomechanical microactuators using level set methods, Journal of Computational Physics, 228, 9, 3173-3181 (2009) · Zbl 1163.65045
[12] Zhu, S.; Wu, Q.; Liu, C., Variational piecewise constant level set methods for shape optimization of a two-density drum, Journal of Computational Physics, 229, 13, 5062-5089 (2010) · Zbl 1194.65087
[13] van den Doel, K.; Ascher, U. M., On level set regularization for highly ill-posed distributed parameter estimation problems, Journal of Computational Physics, 216, 2, 707-723 (2006) · Zbl 1097.65112
[14] Santosa, F., A level-set approach for inverse problems involving obstacles, ESAIM Contrôle Optimisation and Calculus of Variations, 1, 17-33 (1995/96), (electronic) · Zbl 0870.49016
[15] Park, W.; Lesselier, D., Reconstruction of thin electromagnetic inclusions by a level-set method, Inverse Problems, 25, 8, 085010 (2009), 24 pp · Zbl 1173.35737
[16] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order. Elliptic Partial Differential Equations of Second Order, Die Grundlehren der Mathematischen Wissenschaften, vol. 224 (1977), Springer: Springer Berlin · Zbl 0361.35003
[17] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, vol. 46 (1968), Academic press: Academic press New York, NY · Zbl 0164.13002
[18] S. Durand, I. Cimrak, P. Sergeant, A. Abdallh, Analysis of a non-destructive evaluation technique for defect characterization in magnetic materials using local magnetic measurements, Mathematical Problems in Engineering, in press.; S. Durand, I. Cimrak, P. Sergeant, A. Abdallh, Analysis of a non-destructive evaluation technique for defect characterization in magnetic materials using local magnetic measurements, Mathematical Problems in Engineering, in press.
[19] Srinath, S.; Mittal, D. N., An adjoint method for shape optimization in unsteady viscous flows, Journal of Computational Physics, 229, 6, 1994-2008 (2010) · Zbl 1303.76091
[20] Park, H.; Shin, H., Shape identification for natural convection problems using the adjoint variable method, Journal of Computational Physics, 186, 1, 198-211 (2003) · Zbl 1017.65076
[21] Durand, S.; Cimrák, I.; Sergeant, P., Adjoint variable method for time-harmonic Maxwell’s equations, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 28, 5, 1202-1215 (2009) · Zbl 1177.78064
[22] Sergeant, P.; Cimrák, I.; Melicher, V.; Dupré, L.; Van Keer, R., Adjoint variable method for the study of combined active and passive magnetic shielding, Mathematical Problems in Engineering, 2008 (2008), 15 pp · Zbl 1152.78305
[23] Cimrák, I.; Melicher, V., Determination of precession and dissipation parameters in the micromagnetism, Journal of Computational and Applied Mathematics, 234, 7, 2239-2249 (2010) · Zbl 1206.78007
[24] Cimrák, I.; Melicher, V., Sensitivity analysis framework for micromagnetism with application to optimal shape design of magnetic random access memories, Inverse Problems, 23, 2, 563-588 (2007) · Zbl 1115.35146
[25] DeCezaro, A.; Leitao, A.; Tai, X.-C., On multiple level-set regularization methods for inverse problems, Inverse Problems, 25, 035004 (2009) · Zbl 1169.65045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.