Worst scenario method in homogenization. Linear case

Luděk Nechvátal

Applications of Mathematics (2006)

  • Volume: 51, Issue: 3, page 263-294
  • ISSN: 0862-7940

Abstract

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The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain.

How to cite

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Nechvátal, Luděk. "Worst scenario method in homogenization. Linear case." Applications of Mathematics 51.3 (2006): 263-294. <http://eudml.org/doc/33254>.

@article{Nechvátal2006,
abstract = {The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain.},
author = {Nechvátal, Luděk},
journal = {Applications of Mathematics},
keywords = {homogenization; two-scale convergence; worst-scenario; reliable solution; homogenization; two-scale convergence; worst-scenario; reliable solution},
language = {eng},
number = {3},
pages = {263-294},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Worst scenario method in homogenization. Linear case},
url = {http://eudml.org/doc/33254},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Nechvátal, Luděk
TI - Worst scenario method in homogenization. Linear case
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 263
EP - 294
AB - The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain.
LA - eng
KW - homogenization; two-scale convergence; worst-scenario; reliable solution; homogenization; two-scale convergence; worst-scenario; reliable solution
UR - http://eudml.org/doc/33254
ER -

References

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  1. 10.1137/0523084, SIAM J.  Math. Anal. 23 (1992), 1482–1518. (1992) Zbl0770.35005MR1185639DOI10.1137/0523084
  2. Homogenization and its application. Mathematical and computational problems, Numerical Solution of Partial Differential Equations III (SYNSPADE  1975, College Park), Academic Press, New York, 1976, pp. 89–116. (1976) MR0502025
  3. Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, Vol. 5, North Holland, Amsterdam, 1978. (1978) MR0503330
  4. 10.1007/BF02925757, Rend. Sem. Mat. Fis. Milano 47 (1977), 269–328. (1977) Zbl0402.35005MR0526888DOI10.1007/BF02925757
  5. Homogenization of Multiple Integrals. Oxford Lecture Series in Mathematics and its Application, Vol. 12, Oxford University Press, Oxford, 1998. (1998) MR1684713
  6. 10.1016/S0362-546X(99)00274-6, Nonlinear Anal., Theory Methods Appl. 44 (2001), 375–388. (2001) Zbl1002.35041MR1817101DOI10.1016/S0362-546X(99)00274-6
  7. Γ -convergenza e G -convergenza, Boll. Unione Mat. Ital. ser. V 14A (1977), 213–220. (1977) Zbl0389.49008MR0458348
  8. Homogenization of linear elasticity equations, Apl. Mat. 27 (1982), 96–117. (1982) MR0651048
  9. 10.1006/jmaa.1997.5518, J.  Math. Anal. Appl. 212 (1997), 452–466. (1997) MR1464890DOI10.1006/jmaa.1997.5518
  10. 10.1016/S0362-546X(96)00236-2, Nonlinear Anal., Theory Methods Appl. 30 (1997), 3879–3890. (1997) MR1602891DOI10.1016/S0362-546X(96)00236-2
  11. 10.1023/A:1023049608047, Appl. Math. 42 (1997), 321–343. (1997) Zbl0898.35008MR1467553DOI10.1023/A:1023049608047
  12. Multiobjective optimum design of structures, In: Structural Optimization, 4th International Conference on Computer Aided Optimum Design of Structures, Computational Mechanics Publication, , 1995, pp. 35–42. (1995) 
  13. Two-scale convergence, Int.  J.  Pure Appl. Math. 2 (2002), 35–86. (2002) MR1912819
  14. NAG Foundation Toolbox User’s Guide, The MathWorks, Natick, 1996. (1996) 
  15. 10.1137/0520043, SIAM J.  Math. Anal. 20 (1989), 608–623. (1989) Zbl0688.35007MR0990867DOI10.1137/0520043
  16. G -convergence and Homogenization of Nonlinear Partial Differential Operators, Mathematics and its Applications Vol.  422, Kluwer Academic Publishers, Dordrecht, 1997. (1997) Zbl0883.35001MR1482803
  17. Partial Differential Toolbox User’s guide, The MathWorks, Natick, 1996. (1996) 
  18. 10.1137/S0895479891219216, SIAM J. Matrix Anal. Appl. 15 (1994), 175–184. (1994) Zbl0796.65065MR1257627DOI10.1137/S0895479891219216
  19. Non-homogeneous media and vibration theory, Lecture Notes in Physics  127, Springer-Verlag, Berlin-Heidelberg-New York, 1980. (1980) Zbl0432.70002MR0578345
  20. G -convergence of parabolic operators, Uspekhi Mat. Nauk 36 (1981), 11–58. (1981) MR0608940

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