Modification of unfolding approach to two-scale convergence

Jan Franců

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 4, page 403-412
  • ISSN: 0862-7959

Abstract

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Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator which satisfies the property and thus simplifies the theory. The properties of two-scale convergence are surveyed.

How to cite

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Franců, Jan. "Modification of unfolding approach to two-scale convergence." Mathematica Bohemica 135.4 (2010): 403-412. <http://eudml.org/doc/196694>.

@article{Franců2010,
abstract = {Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator which satisfies the property and thus simplifies the theory. The properties of two-scale convergence are surveyed.},
author = {Franců, Jan},
journal = {Mathematica Bohemica},
keywords = {homogenization; two-scale convergence; periodic unfolding; homogenization; two-scale convergence; periodic unfolding},
language = {eng},
number = {4},
pages = {403-412},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Modification of unfolding approach to two-scale convergence},
url = {http://eudml.org/doc/196694},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Franců, Jan
TI - Modification of unfolding approach to two-scale convergence
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 4
SP - 403
EP - 412
AB - Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ``dual'' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator which satisfies the property and thus simplifies the theory. The properties of two-scale convergence are surveyed.
LA - eng
KW - homogenization; two-scale convergence; periodic unfolding; homogenization; two-scale convergence; periodic unfolding
UR - http://eudml.org/doc/196694
ER -

References

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  5. Cioranescu, D., Damlamian, A., Griso, G., 10.1016/S1631-073X(02)02429-9, C. R. Math. Acad. Sci. Paris 335 (2002), 99-104. (2002) Zbl1001.49016MR1921004DOI10.1016/S1631-073X(02)02429-9
  6. Cioranescu, D., Damlamian, A., Griso, G., 10.1137/080713148, SIAM J. Math. Anal. 40 (2008), 1585-1620. (2008) Zbl1167.49013MR2466168DOI10.1137/080713148
  7. Damlamian, A., An elementary introduction to periodic unfolding, Proceedings of the Narvic Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo (2006), 119-136. (2006) Zbl1204.35038MR2233174
  8. Franců, J., On two-scale convergence, Proceeding of the 6th Mathematical Workshop, Faculty of Civil Engineering, Brno University of Technology, Brno, October 18, 2007, CD, 7 pages. 
  9. Holmbom, A., Silfver, J., Svanstedt, N., Wellander, N., 10.1007/s10492-006-0014-x, Appl. Math. 51 (2006), 247-262. (2006) Zbl1164.40304MR2228665DOI10.1007/s10492-006-0014-x
  10. Lukkassen, D., Nguetseng, G., Wall, P., Two-scale convergence, Int. J. Pure Appl. Math. 2 (2002), 35-86. (2002) Zbl1061.35015MR1912819
  11. Nechvátal, L., 10.1023/B:APOM.0000027218.04167.9b, Appl. Math. (Praha) 49 (2004), 97-110. (2004) MR2043076DOI10.1023/B:APOM.0000027218.04167.9b
  12. Nguetseng, G., 10.1137/0520043, SIAM J. Math. Anal. 20 (1989), 608-623. (1989) Zbl0688.35007MR0990867DOI10.1137/0520043

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