Email this article Homogenization of a Continuously Microperiodic Multidimensional Medium
Abstract
The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.
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S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties. New York: Springer-Verlag, 2002.
M. H. Sadd, Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier Academic Press, 2005.
J. C. Maxwell, Treatise on Electricity and Magnetism. Oxford: Clarendon Press, 1873.
L. Rayleigh, ''On the inuence of obstacles arranged in a rectangular order upon the properties of medium,'' Philosophical Magazine, vol. 34, pp. 481-502, 1892.
A. Einstein, ''Eine neue bestimmung der moleküldimensionen,'' Annalen der Physik, vol. 19, pp. 289-306, 1906.
N. S. Bakhvalov and G. P. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Dordrecht: Kluwer Academic Publishers, 1989.
Y. Benveniste, ''Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases,'' Physical Review B, vol. 51, pp. 16424-16427, 1995.
J. Bravo-Castillero, L. M. Sixto-Camacho, R. Brenner, R. Guinovart-Díaz, L. D. Pérez-Fernández, and F. J. Sabina, ''Temperature-related effective properties and exact relations for thermo-magneto-electro-elastic fibrous composites,'' Computers and Mathematics with Applications, vol. 69, pp. 980-996, 2015.
M. P. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods and Applications. New York: Springer-Verlag, 2003.
S. Torquato, ''Optimal design of heterogeneous materials,'' Annual Review of Materials Research, vol. 40, pp. 101-129, 2010.
W. J. Parnell and Q. Grimal, ''The inuence of mesoscale porosity on cortical bone anisotropy,'' Journal of the Royal Society Interface, vol. 6, pp. 97-109, 2009.
L. D. Pérez-Fernández and A. T. Beck, ''Failure detection in umbilical cables via electroactive elements - a mathematical homogenization approach,'' International Journal of Modeling and Simulation for the Petroleum Industry, vol. 8, pp. 34-39, 2014.
Y. Capdeville, L. Guillot, and J. J. Marigo, ''1-d non-periodic homogenization for the seismic wave equation,'' Geophysical Journal International, vol. 181, pp. 897-910, 2010.
G. Allaire and G. Bal, ''Homogenization of the criticality spectral equation in neutron transport,'' Mathematical Modelling and Numerical Analysis, vol. 33, no. 4, pp. 721-746, 1999.
C.-O. Ng, ''Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions,'' Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 462, no. 2066, pp. 481-515, 2006.
L. D. Kudriavtsev, Curso de Análisis Matemático: Tomo I. Moscou: Mir, 1983.
L. C. Evans, Partial Differential Equations. New York: American Mathematical Society, 2010.
D. Cioranescu and P. Donato, An Introduction to Homogenization. New York: Oxford University Press, 1999.
H. J. Choe, K.-B. Kong, and C.-O. Lee, ''Convergence in Lp space for the homogenization problems of elliptic and parabolic equations in the plane,'' Journal of Mathematical Analysis and Applications, vol. 287, pp. 321-336, 2003.
A. N. Kolmogorov and S. V. Fomin, Elementos de la Teoria de Funciones y del Análisis Funcional. Moscou: Mir, 1975.
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin: Walter de Gruyeter, 2007.
R. L. Burden and J. D. Faires, Numerical Analysis. Canada: Cengage Learning, 2010.
H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods. New York: Dover, 1995.
DOI: https://doi.org/10.5540/tema.2018.019.01.15
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Trends in Computational and Applied Mathematics
A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
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