Hierarchical Modeling of Heterogeneous Plates
DOI:
https://doi.org/10.5540/tema.2007.08.02.0219Abstract
We describe here the modeling of highly heterogeneous plates, whenthree different lenght scales are present: the area of the middle surface, the platethickness, and the heterogeneity scale. We derive a two-dimensional PDE model forsuch problem, which turns out to have rough coefficients. We employ asymptotictechniques to estimate the modeling error with respect to the thickness. To tamethe numerical troubles of the resulting model we use finite elements methods ofmultiscale type.References
[1] S.M. Alessandrini, D.N. Arnold, R.S. Falk, A.L. Madureira, Derivation and justification of plate models by variotional methods, in “Plates and shells” (Qu´ebec, QC, 1996), pp.1-20, CRM Proc. Lecture Notes, vol. 21, Amer. Math. Soc., Providence, RI, 1999.
D.N. Arnold, A.L. Madureira, Asymptotic estimates of hierarchical modeling, Math. Models Methods Appl. Sci., 13 (2003), 1325-1350.
D.N. Arnold, A.L. Madureira, S. Zhang, On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models, J. Elasticity, 67 (2002), 171-185.
F. Auricchio, C. Lovadina, A.L. Madureira, An asymptotically optimal model for isotropic heterogeneous linearly elastic plates, Math. Model. Numer. Anal., 38 (2004), 877-897.
F. Brezzi, A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci., 4 (1994), 571-587.
F. Brezzi, L.P. Franca, T.J.R. Hughes, A. Russo, b =
R g, Comput. Methods Appl. Mech. Engrg., 145 (1997), 329-339.
A.C. Carius, “Modelagem Hier´arquica para a Equa¸c˜ao do Calor em uma Placa Heterogˆenea”, Disserta¸c˜ao de Mestrado, LNCC, Petr´opolis, RJ, 2006.
P.G. Ciarlet, “Mathematical Elasticity, vol I”, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1998.
D. Cioranescu, J.S.J. Paulin, “Homogenization of Reticulated Structures”, Applied Mathematical Sciences, Springer-Verlag, New York, 1999.
L.P. Franca, An overview of the residual-free bubbles method, in “Numerical Methods in Mechanics” (Concepci´on, 1995), pp.83-92, Pitman Res. Notes Math., Longman, Harlow, 1997.
T.Y. Hou, Numerical approximations to multiscale solutions in partial differential equations, in “Frontiers in Numerical Analysis” (Durham, 2002), pp. 241-301, Universitext, Springer, Berlin, 2003.
T.Y. Hou, X.Wu, Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), 913-943.
T.Y. Hou, X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169-189.
A.L. Madureira, Hierarchical modeling based on mixed principles: asymptotic error estimates, Math. Models Methods Appl. Sci., 15 (2005), 985-1008.
G. Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul., 1 (2003), 485-503.
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