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Piecewise Constant Level Set Algorithm for an Inverse Elliptic Problem in Nonlinear Electromagnetism

Published online by Cambridge University Press:  28 May 2015

Xiangyin Kong ,
Zhengfang Zhang  and
Zhengda Huang
Xiangyin Kong
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Zhengfang Zhang
Affiliation:
Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China
Zhengda Huang*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
*
*Corresponding author. Email: xiangyinkong@163.com (X. Kong), zhengfangzhang@hdu.edu.cn (Z. Zhang), zhengfangzhang@hdu.edu.cn (Z. Huang)
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Abstract

An inverse problem of identifying inhomogeneity or crack in the workpiece made of nonlinear magnetic material is investigated. To recover the shape from the local measurements, a piecewise constant level set algorithm is proposed. By means of the Lagrangian multiplier method, we derive the first variation w.r.t the piecewise constant level set function and obtain the descent direction by the adjoint variable method. Numerical results show the robustness and effectiveness of our algorithm applied to reconstruct some complex shapes.

Keywords

65K1035Q6074M10Nonlinear electromagnetismadjoint variable methodshape reconstructionpiecewise constant level set algorithm
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 3 , June 2015 , pp. 387 - 406
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Giguère, S., Lepine, B. AND Dubois, J.Pulsed eddy current technology: characterizing material loss with gap and lift-off variations, Research in Nondestructive Evaluation, 13 (2001), pp. 119–129.Google Scholar
[2]Osher, S. AND Sethian, J.A.Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12–49.Google Scholar
[3]Chen, S., Merriman, B., Kang, M., Caflisch, R.-E., Ratsch, C., Cheng, L.-T, Gyure, M., Fedkiw, R.-P., Anderson, C. AND Osher, S., A level set method for thin film epitaxial growth, J. Comput. Phys., 167 (2001), pp. 475–500.Google Scholar
[4]Burger, M. AND Osher, S., A survey on level set methods for inverse problems and optimal design, Euro. J. Appl. Math., 16 (2005), pp. 263–301.Google Scholar
[5]Vese, L.A. AND Chan, T.F., A multiphase level set framework for image segmentation using the mumford and shah model, int. J. Comput. Vis., 50 (2002), pp. 271–293.Google Scholar
[6]Wang, M.Y., Wang, X. AND Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 227–246.Google Scholar
[7]Tanushev, N.M. AND Vese, L.A.A Piecewise-constant binary model for electrical impedance tomography, inverse Probl. imag., 2 (2007), pp. 423–435.Google Scholar
[8]Sussman, M., Smereka, P. AND Osher, S.A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), pp. 146–159.CrossRefGoogle Scholar
[9]Cimrák, I. AND Keer, R.V., Level set method for the inverse elliptic problem in nonlinear electro-magnetism, J. Comput. Phys., 229 (2010), pp. 9269–9283.CrossRefGoogle Scholar
[10]Cimrák, I.Inverse thermal imaging inmaterials with nonlinear conductivity bymaterial and shape derivative method, Math. Methods Appl. Sci., 34 (2011), pp. 2303–2317.Google Scholar
[11]Cimrák, I.Material and shape derivative method for quasi-linear elliptic systems with applications in inverse electromagnetic interface problems, SIAM J. Numer. Anal., 50 (2012), pp. 1086–1110.Google Scholar
[12]Allaire, G., Jouve, F. AND Toader, A.-M., A level-set method for shape optimization, C.R. Acad. Sci. Paris, Ser. I, 334 (2002), pp. 1125–1130.Google Scholar
[13]Allaire, G., Jouve, F. AND Toader, A.-M.Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), pp. 363–393.Google Scholar
[14]He, L., Kao, C.-Y. AND Osher, S., Incorporating topological derivatives into shape derivatives based level set methods, J. Comput. Phys., 225 (2007), pp. 891–909.Google Scholar
[15]Allaire, G., Jouve, F. AND Toader, A.-M.Structural optimization using topological and shape sensitivity via a level set method, Contr. Cybernet., 34 (2005), pp. 59–80.Google Scholar
[16]Amstutz, S. AND Andrae, H.A new algorithm for topology optimization using a level-set method, J. Comput. Phys., 216 (2006), pp. 573–588.Google Scholar
[17]Burger, M., Hackl, B. AND Ring, W.Incorporating topological derivatives into level set methods, J. Comput. Phys., 194 (2004), pp. 344–362.Google Scholar
[18]Wang, X., Mei, Y. AND Wang, M.Y.Incorporating topological derivatives into level set methods for structural topology optimization, in: T.L. et al. (Eds.), Optimal Shape Design and Modeling, Polish Academy of Sciences, Warsaw, 2004, pp. 145–157.Google Scholar
[19]Tai, X.C. AND Chan, T.F., A survey on multiple level set methods with applications for identifying piecewise constant functions, int. J. Numer. Anal. Model., 1 (2004), pp. 25–47Google Scholar
[20]Lie, J., Lysaker, M. AND Tai, X.C.A variant of the level set method and applications to image segmentation, Math. Comput. 75 (2006), pp. 1155–1174.Google Scholar
[21]Lie, J., Lysaker, M. AND Tai, X.C.A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. image Process., 15 (2006), pp. 1171–1181.Google Scholar
[22]Lie, J., Lysaker, M. AND Tai, X.C., Piecewise constant level set methods and image segmentation, in Scale Space and PDE Methods in Computer Vision, Lectures notes in Computer Sciences, pp. 573–584, Springer.Google Scholar
[23]Lie, J., Lysaker, M. AND Tai, X.C.A piecewise constant level set framework, int. J. Numer. Anal. Model., 2 (2005), pp. 422–438.Google Scholar
[24]Tai, X.C. AND Li, H.W.A piecewise constant level set method for elliptic inverse problems, Appl. Numer. Math., 57 (2007), pp. 686–696.CrossRefGoogle Scholar
[25]Tai, X.C. AND Li, H.W.Piecewise constant level set method for interface problems, free boundary problems, int. Ser. Numer. Math., 154 (2007), pp. 307–316.Google Scholar
[26]Wei, P. AND Wang, M.Y. , Piecewise constant level set method for structural topology optimization, int. J. Numer. Methods Eng., 78 (2009), pp. 379–402.Google Scholar
[27]Zhu, S.F, Wu, Q.B. AND Liu, C.X., Variational piecewise constant level set methods for shape optimization of a two-density drum, J. Comput. Phys., 229 (2010), pp. 5062–5089.Google Scholar
[28]Zhang, Z.F. AND Cheng, X.L.A boundary piecewise constant level set method for boundary control of eigenvalue optimization problems, J. Comput. Phys., 230 (2011), pp. 458–473.Google Scholar
[29]Muller, J.L. AND Sitanen, S.Direct reconstruction of conductivities from boundary measurements, SIAM J. Sci. Comput., 24 (2003), pp. 1232–1266.Google Scholar
[30]Gilbarg, D. AND Trudinger, N.S.Elliptic Partial Differential Equations of Second Order, Die Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer, Berlin, 1977.Google Scholar
[31]Ladyzhenskaya, O.A. AND Ural’tseva, N.N.Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, Vol. 46, Academic press, New York, 1968.Google Scholar
[32]Srinath, S. AND Mittal, D.N.An adjoint method for shape optimization in unsteady viscous flows, J. Comput. Phys., 229 (2010), pp. 1994–2008.Google Scholar
[33]Park, H. AND Shin, H.Shape identification for natural convection problems using the adjoint variable method, J. Comput. Phys., 186 (2003), pp. 198–211.Google Scholar
[34]Durand, S., Cimrâk, I. AND Sergeant, P.Adjoint variable method for time-harmonic Maxwell equations, COMPEL: int. J. Comput. Math. Electrical Electric Eng., 28 (2009), pp. 1202–1215.Google Scholar
[35]Sergeant, P., Cimrâk, I., Melicher, V., Dupré, L. AND Van Keer, R.Adjoint variable method for the study of combined active and passive magnetic shielding, Math. Probl. Eng., (2008).Google Scholar
[36]Cimrâk, I. AND Melicher, V.Determination of precession and dissipation parameters in the micromagnetism, J. Comput. Appl. Math., 234 (2010), pp. 2239–2249.Google Scholar
[37]Cimrâk, I. AND Melicher, V.Sensitivity analysis framework for micromagnetism with application to optimal shape design of magnetic random access memories, inverse Problems, 23 (2007), pp. 563–588.Google Scholar