Previous |  Up |  Next

Article

Title: Worst scenario method in homogenization. Linear case (English)
Author: Nechvátal, Luděk
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940
Volume: 51
Issue: 3
Year: 2006
Pages: 263-294
Summary lang: English
.
Category: math
.
Summary: 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. (English)
Keyword: homogenization
Keyword: two-scale convergence
Keyword: worst-scenario
Keyword: reliable solution
MSC: 35B27
MSC: 35B40
MSC: 35J25
MSC: 35R05
MSC: 49J20
idZBL: Zbl 1164.35317
idMR: MR2228666
DOI: 10.1007/s10492-006-0015-9
.
Date available: 2009-09-22T18:25:57Z
Last updated: 2015-05-17
Stable URL: http://hdl.handle.net/10338.dmlcz/134640
.
Reference: [1] G. Allaire: Homogenization and two-scale convergence.SIAM J.  Math. Anal. 23 (1992), 1482–1518. Zbl 0770.35005, MR 1185639, 10.1137/0523084
Reference: [2] I.  Babuška: 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. MR 0502025
Reference: [3] A.  Bensoussan, J.-L.  Lions, G.  Papanicolau: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, Vol. 5.North Holland, Amsterdam, 1978. MR 0503330
Reference: [4] M.  Biroli: $G$-convergence for elliptic equations, variational inequalities and quasi-variational inequalities.Rend. Sem. Mat. Fis. Milano 47 (1977), 269–328. Zbl 0402.35005, MR 0526888, 10.1007/BF02925757
Reference: [5] A.  Braides, A.  Defranceschi: Homogenization of Multiple Integrals. Oxford Lecture Series in Mathematics and its Application, Vol. 12.Oxford University Press, Oxford, 1998. MR 1684713
Reference: [6] J.  Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients: Sensitivity analysis and numerical examples.Nonlinear Anal., Theory Methods Appl. 44 (2001), 375–388. Zbl 1002.35041, MR 1817101, 10.1016/S0362-546X(99)00274-6
Reference: [7] E.  De Giorgi: $\Gamma $-convergenza e $G$-convergenza.Boll. Unione Mat. Ital. ser. V 14A (1977), 213–220. Zbl 0389.49008, MR 0458348
Reference: [8] J.  Franců : Homogenization of linear elasticity equations.Apl. Mat. 27 (1982), 96–117. MR 0651048
Reference: [9] I.  Hlaváček: Reliable solution of a quasilinear nonpotential elliptic problem of a nonmonotone type with respect to uncertainty in coefficients.J.  Math. Anal. Appl. 212 (1997), 452–466. MR 1464890, 10.1006/jmaa.1997.5518
Reference: [10] I.  Hlaváček: Reliable solutions of elliptic boundary value problems with respect to uncertain data. Proceedings of the second  WCNA.Nonlinear Anal., Theory Methods Appl. 30 (1997), 3879–3890. MR 1602891, 10.1016/S0362-546X(96)00236-2
Reference: [11] A.  Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results.Appl. Math. 42 (1997), 321–343. Zbl 0898.35008, MR 1467553, 10.1023/A:1023049608047
Reference: [12] W.  Li, Y.  Han: 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.
Reference: [13] D.  Lukkassen, G.  Nguetseng, P.  Wall: Two-scale convergence.Int.  J.  Pure Appl. Math. 2 (2002), 35–86. MR 1912819
Reference: [14] : NAG Foundation Toolbox User’s Guide.The MathWorks, Natick, 1996.
Reference: [15] G.  Nguetseng: A general convergence result for a functional related to the theory of homogenization.SIAM J.  Math. Anal. 20 (1989), 608–623. Zbl 0688.35007, MR 0990867, 10.1137/0520043
Reference: [16] A. A. Pankov: $G$-convergence and Homogenization of Nonlinear Partial Differential Operators.Mathematics and its Applications Vol.  422, Kluwer Academic Publishers, Dordrecht, 1997. Zbl 0883.35001, MR 1482803
Reference: [17] : Partial Differential Toolbox User’s guide.The MathWorks, Natick, 1996.
Reference: [18] J.  Rohn: Positive definiteness and stability of interval matrices.SIAM J. Matrix Anal. Appl. 15 (1994), 175–184. Zbl 0796.65065, MR 1257627, 10.1137/S0895479891219216
Reference: [19] E.  Sanchez-Palencia: Non-homogeneous media and vibration theory.Lecture Notes in Physics  127, Springer-Verlag, Berlin-Heidelberg-New York, 1980. Zbl 0432.70002, MR 0578345
Reference: [20] V. V.  Zhikov, S. M.  Kozlov, and O. A.  Oleĭnik: $G$-convergence of parabolic operators.Uspekhi Mat. Nauk 36 (1981), 11–58. MR 0608940
.

Files

Files Size Format View
AplMat_51-2006-3_4.pdf 437.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo