Which electric fields are realizable in conducting materials?

Marc Briane; Graeme W. Milton; Andrejs Treibergs

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 307-323
  • ISSN: 0764-583X

Abstract

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In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

How to cite

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Briane, Marc, Milton, Graeme W., and Treibergs, Andrejs. "Which electric fields are realizable in conducting materials?." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 307-323. <http://eudml.org/doc/273130>.

@article{Briane2014,
abstract = {In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.},
author = {Briane, Marc, Milton, Graeme W., Treibergs, Andrejs},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {isotropic conductivity; electric field; gradient system; laminate; realizable electric field; isotropic realizability; realizable electric matrix field; rank- laminate; dynamical systems approach},
language = {eng},
number = {2},
pages = {307-323},
publisher = {EDP-Sciences},
title = {Which electric fields are realizable in conducting materials?},
url = {http://eudml.org/doc/273130},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Briane, Marc
AU - Milton, Graeme W.
AU - Treibergs, Andrejs
TI - Which electric fields are realizable in conducting materials?
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 307
EP - 323
AB - In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.
LA - eng
KW - isotropic conductivity; electric field; gradient system; laminate; realizable electric field; isotropic realizability; realizable electric matrix field; rank- laminate; dynamical systems approach
UR - http://eudml.org/doc/273130
ER -

References

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  1. [1] G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal.158 (2001) 155–171. Zbl0977.31006MR1838656
  2. [2] G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Appl. Math. Sci. Springer-Verlag, New-York (2002) 456. Zbl0990.35001MR1859696
  3. [3] A. Ancona, Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J.165 (2002) 123–158. Zbl1028.31003MR1892102
  4. [4] V.I. Arnold, Ordinary differential equations, translated from the third Russian edition by R. Cooke, Springer Textbook. Springer-Verlag, Berlin (1992) 334. Zbl0744.34001MR1162307
  5. [5] N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Mathematical Problems in the Mechanics of Composite Materials, translated from the Russian by D. Leĭtes, vol. 36 of Math. Appl. (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1989) 366. Zbl0692.73012MR1112788
  6. [6] P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J.50 (2001) 747–757. Zbl1330.35121MR1871388
  7. [7] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in vol. 5 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978) 700. Zbl0404.35001MR503330
  8. [8] M. Briane, Correctors for the homogenization of a laminate. Adv. Math. Sci. Appl.4 (1994) 357–379. Zbl0829.35009MR1294225
  9. [9] M. Briane, G.W. Milton and V. Nesi, Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal.173 (2004) 133–150. Zbl1118.78009MR2073507
  10. [10] M. Briane, and V. Nesi, Is it wise to keep laminating? ESAIM: COCV 10 (2004) 452–477. Zbl1072.74057MR2111075
  11. [11] A. Cherkaev and Y. Zhang, Optimal anisotropic three-phase conducting composites: Plane problem. Int. J. Solids Struct.48 (2011) 2800–2813. 
  12. [12] B. Dacorogna, Direct Methods in the Calculus of Variations, in vol. 78 of Appl. Math. Sci. Springer-Verlag, Berlin-Heidelberg (1989) 308. Zbl0703.49001MR990890
  13. [13] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, translated from the Russian by G.A. Yosifian. Springer-Verlag, Berlin (1994) 570. Zbl0838.35001MR1329546
  14. [14] G.W. Milton, Modelling the properties of composites by laminates, Homogenization and Effective Moduli of Materials and Media, in vol. 1 of IMA Math. Appl. Springer-Verlag, New York (1986) 150–174. Zbl0631.73011MR859415
  15. [15] G.W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002) 719. Zbl0993.74002MR1899805
  16. [16] F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, in vol. 31 of Progr. Nonlinear Differ. Equ. Appl., edited by L. Cherkaev and R.V. Kohn. Birkhaüser, Boston (1997) 21–43. Zbl0920.35019MR1493039
  17. [17] V. Nesi, Bounds on the effective conductivity of two-dimensional composites made of n ≥ 3 isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1219–1239. Zbl0852.35016MR1363001
  18. [18] U. Raitums, On the local representation of G-closure. Arch. Rational Mech. Anal.158 (2001) 213–234. Zbl1123.35320MR1842345

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