Approximate cloaking for the heat equation via transformation optics

Hoai-Minh Nguyen; Tu Nguyen

doi:10.3934/mine.2019.4.775

Mathematics in Engineering, 2019, 1(4): 775-788. doi: 10.3934/mine.2019.4.775. Research article Special Issues

Export file:

Format

RIS(for EndNote,Reference Manager,ProCite)
BibTex
Text

Content

Citation Only
Citation and Abstract

Approximate cloaking for the heat equation via transformation optics

Hoai-Minh Nguyen, Tu Nguyen

1 Department of Mathematics, EPFL SB CAMA, Station 8, CH-1015 Lausanne, Switzerland
2 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

† This contribution is part of the Special Issue: Inverse problems in imaging and engineering science
Guest Editors: Lauri Oksanen; Mikko Salo
Link: https://www.aimspress.com/newsinfo/1270.html

Received: , Accepted: , Published:

Special Issues: Inverse problems in imaging and engineering science

Abstract Full Text(HTML) Figure/Table Related pages

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order ε in three dimensions and |lnε|⁻¹ in two dimensions, where ε is the regularization parameter. To this end, we first transform the problem in time domain into a family of problems in frequency domain by taking the Fourier transform with respect to time, and then derive appropriate estimates in the frequency domain.

Figure/Table

Supplementary

Article Metrics

Keywords: heat equation; approximate cloaking; frequency analysis

Citation: Hoai-Minh Nguyen, Tu Nguyen. Approximate cloaking for the heat equation via transformation optics. Mathematics in Engineering, 2019, 1(4): 775-788. doi: 10.3934/mine.2019.4.775

References:

1. Allaire G (2007) Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation. Oxford: Oxford University Press.
2. Alu A, Engheta N (2005) Achieving transparency with plasmonic and metamaterial coatings. Phys Rev E 72: 016623.
3. Ammari H, Iakovleva E, Kang H, et al. (2005) Direct algorithms for thermal imaging of small inclusions. Multiscale Model Simul 4: 1116–1136.
4. Ammari H, Kang H, Lee H, et al. (2013) Enhancement of near cloaking for the full maxwell equations. SIAM J Appl Math 73: 2055–2076.
5. Amstutz S, Takahashi T, Vexler B (2008) Topological sensitivity analysis for time-dependent problems. ESAIM Control Optim Calc Var 14: 427–455.
6. Craster RV, Guenneau S, Hutridurga H, et al. (2018) Cloaking via mapping for the heat equation. Multiscale Model Simul 16: 1146–1174.
7. Girault V, Raviart PA (1986) Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Berlin: Springer-Verlag.
8. Guenneau S, Amra C, Veynante D (2012) Transformation thermodynamics: Cloaking and concentrating heat flux. Opt Express 20: 8207–8218.
9. Greenleaf A, Kurylev Y, Lassas M, et al. (2007) Full-wave invisibility of active devices at all frequencies. Comm Math Phys 275: 749–789.
10. Greenleaf A, Lassas M, Uhlmann G (2003) On nonuniqueness for Calderon's inverse problem. Math Res Lett 10: 685–693.
11. Kohn RV, Shen H, Vogelius MS, et al. (2008) Cloaking via change of variables in electric impedance tomography. Inverse Problem 24: 015–016.
12. Kohn RV, Onofrei D, Vogelius MS, et al. (2010) Cloaking via change of variables for the Helmholtz equation. Comm Pure Appl Math 63: 973–1016.
13. Lai Y, Chen H, Zhang Z, et al. (2009) Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. Phys Rev Lett 102: 093901.
14. Lassas M, Zhou T (2016) The blow-up of electromagnetic fields in 3-dimensional invisibility cloaking for Maxwell's equations. SIAM J Appl Math 76: 457–478.
15. Lebedev NN (1965) Special Functions and Their Applications. Englewood Cliffs: Prentice-Hall.
16. Leonhardt U (2006) Optical conformal mapping. Science 312: 1777–1780.
17. Milton GW, Nicorovici NA (2006) On the cloaking effects associated with anomalous localized resonance. Proc R Soc Lond Ser A 462: 3027–3059.
18. Nédélec JC (2000) Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer.
19. Nguyen H-M (2010) Cloaking via change of variables for the Helmholtz equation in the whole space. Comm Pure Appl Math 63: 1505–1524.
20. Nguyen H-M (2012) Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking. Comm Pure Appl Math 65: 155–186.
21. Nguyen H-M (2015) Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime. J Eur Math Soc 17: 1327–1365.
22. Nguyen H-M, (2016) Cloaking using complementary media in the quasistatic regime. Ann Inst H Poincaré Anal Non Linéaire 33: 1509–1518.
23. Nguyen H-M (2017) Cloaking an arbitrary object via anomalous localized resonance: The cloak is independent of the object. SIAM J Math Anal 49: 3208–3232.
24. Nguyen H-M, Tran L (2019) Approximate cloaking for electromagnetic waves via transformation optics: Cloaking vs infinite energy. Math Models Methods Appl Sci 29: 1511–1552.
25. Nguyen H-M, Vinoles V (2018) Electromagnetic wave propagation in media consisting of dispersive metamaterials. C R Math 356: 757–775.
26. Nguyen H-M, Vogelius MS (2009) A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity. Ann Inst H Poincaré Anal Non Linéaire 26: 2283–2315.
27. Nguyen H-M, Vogelius MS (2012) Full range scattering estimates and their application to cloaking. Arch Rational Mech Anal 203: 769–807.
28. Nguyen H-M, Vogelius MS (2012) Approximate cloaking for the wave equation via change of variables. SIAM J Math Anal 44: 1894–1924.
29. Nguyen H-M, Vogelius MS (2016) Approximate cloaking for the full wave equation via change of variables: The Drude-Lorentz model. J Math Pures Appl 106: 797–836.
30. Pendry JB, Schurig D, Smith DR (2006) Controlling electromagnetic fields. Science 321: 1780–1782.
31. Vasquez FG, Milton GW, Onofrei D (2009) Active exterior cloaking for the 2D Laplace and Helmholtz equations. Phys Rev Lett 103: 073901.
32. Weder R (2008) A rigorous analysis of high-order electromagnetic invisibility cloaks. J Phys A: Math Theor 41: 065207.
33. Weder R (2008) The boundary conditions for point transformed electromagnetic invisibility cloaks. J Phys A: Math Theor 41: 415401.

show all references

This article has been cited by:

1. Hoai-Minh Nguyen, Loc X. Tran, Approximate Cloaking for Time-dependent Maxwell Equations via Transformation Optics, SIAM Journal on Mathematical Analysis, 2019, 51, 5, 4142, 10.1137/18M1232395

Reader Comments

your name: * your email: *

Download full text in PDF

Associated material

Metrics