no. 1
On optimal cloaking-by-mapping transformations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 303-316

A central ingredient of cloaking-by-mapping is the diffeomorphism which transforms an annulus with a small hole into an annulus with a finite size hole, while being the identity on the outer boundary of the annulus. The resulting meta-material is anisotropic, which makes it difficult to manufacture. The problem of minimizing anisotropy among radial transformations has been studied in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17]. In this work, as in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17], we formulate the problem of minimizing anisotropy as an energy minimization problem. Our main goal is to provide strong evidence for the conjecture that for cloaks with circular boundaries, non-radial transformations do not lead to lower degree of anisotropy. In the final section, we consider cloaks with non-circular boundaries and show that in this case, non-radial cloaks may be advantageous, when it comes to minimizing anisotropy.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022004
Classification : 35R30, 78A46
Keywords: Cloak enhancement, pproximate cloaking, cloaking-by-mapping, optimal cloaking 
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     title = {On optimal cloaking-by-mapping transformations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {303--316},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {56},
     number = {1},
     doi = {10.1051/m2an/2022004},
     mrnumber = {4378549},
     zbl = {1494.35005},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2022004/}
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Capdeboscq, Yves; Vogelius, Michael S. On optimal cloaking-by-mapping transformations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 303-316. doi: 10.1051/m2an/2022004

[1] G. V. Alekseev, Analysis of a two-dimensional thermal cloaking problem on the basis of optimization. Comput. Math. Math. Phys. 58 (2018) 478–492. | MR | Zbl | DOI

[2] H. Ammari, J. Garnier, V. Jugnon, H. Kang, H. Lee and M. Lim, Enhancement of near-cloaking. Part III: numerical simulations, statistical stability, and related questions. In: Multi-scale and High-contrast PDE. From Modelling, to Mathematical Analysis, to Inversion. Proceedings of the Conference, University of Oxford, UK, June 28–July 1, 2011. American Mathematical Society (AMS), Providence, RI (2012) 1–24. | MR | Zbl

[3] H. Ammari, H. Kang, H. Lee and M. Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. I: the conductivity problem. Commun. Math. Phys. 317 (2013) 253–266. | MR | Zbl | DOI

[4] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón’s inverse problem. Math. Res. Lett. 10 (2003) 685–693. | MR | Zbl | DOI

[5] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems. Bull. Am. Math. Soc. New Ser. 46 (2009) 55–97. | MR | Zbl | DOI

[6] R. Griesmaier and M. S. Vogelius, Enhanced approximate cloaking by optimal change of variables. Inverse Prob. 30 (2014) 17. | MR | Zbl | DOI

[7] H. Heumann and M. S. Vogelius, Analysis of an enhanced approximate cloaking scheme for the conductivity problem. Asymp. Anal. 87 (2014) 223–246. | MR | Zbl

[8] R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography. Inverse Prob. 24 (2008) 21. | MR | Zbl

[9] G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance. Proc. Math. Phys. Eng. Sci. 462 (2006) 3027–3059. | MR | Zbl

[10] J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields. Science 312 (2006) 1780–1782. | MR | Zbl | DOI

[11] C.-W. Qiu, L. Hu, B. Zhang, B.-I. Wu, S. G. Johnson and J. D. Joannopoulos, Spherical cloaking using nonlinear transformations for improved segmentation into concentric isotropic coatings. Opt. Express 17 (2009) 13467–13478. | DOI

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