Homogenization of a Continuously Microperiodic Multidimensional Medium

Marcos Pinheiro Lima, Leslie Darien Pérez Fernández, Julián Bravo Castillero

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.


Full Text:

PDF

References


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

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.



TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

Indexed in:

                        

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia