Asymptotic homogenization

In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients,[1] [2] [3] such as

${\displaystyle \nabla \cdot \left(A\left({\frac {\vec {x}}{\epsilon }}\right)\nabla u_{\epsilon }\right)=f}$

where ${\displaystyle \epsilon }$ is a very small parameter and ${\displaystyle A\left({\vec {y}}\right)}$ is a 1-periodic coefficient: ${\displaystyle A\left({\vec {y}}+{\vec {e}}_{i}\right)=A\left({\vec {y}}\right)}$, ${\displaystyle i=1,\dots ,n}$.

It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics. Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc.

Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are subjected to loads or forcings which vary on a lengthscale which is far bigger than the characteristic lengthscale of the microstructure. In this situation, one can often replace the equation above with an equation of the form

${\displaystyle \nabla \cdot \left(A^{*}\nabla u\right)=f}$

where ${\displaystyle A^{*}}$ is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as

${\displaystyle A_{ij}^{*}=\int _{(0,1)^{n}}A({\vec {y}})\left(\nabla w_{j}({\vec {y}})+{\vec {e}}_{j}\right)\cdot {\vec {e}}_{i}\,dy_{1}\dots dy_{n},\qquad i,j=1,\dots ,n}$

from 1-periodic functions ${\displaystyle w_{j}}$ satisfying:

${\displaystyle \nabla _{y}\cdot \left(A({\vec {y}})\nabla w_{j}\right)=-\nabla _{y}\cdot \left(A({\vec {y}}){\vec {e}}_{j}\right).}$

This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as homogenization. This subject is inextricably linked with the subject of micromechanics for this very reason.

In homogenization one equation is replaced by another if ${\displaystyle u_{\epsilon }\approx u}$ for small enough ${\displaystyle \epsilon }$, provided ${\displaystyle u_{\epsilon }\to u}$ in some appropriate norm as ${\displaystyle \epsilon \to 0}$.

As a result of the above, homogenization can therefore be viewed as an extension of the continuum concept to materials which possess microstructure. The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that material), is known as the "Representative Volume Element"[4] in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. Therefore averaging over this element gives an effective property such as ${\displaystyle A^{*}}$ above.

Classical results of homogenization theory[1][2][3] were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical properties are the same at every point in space.[5][6] In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this end the methods of the homogenization theory have been extended to partial differential equations, which coefficients are neither periodic nor statistically homogeneous (so-called arbitrarily rough coefficients).[7][8]

The method of asymptotic homogenization

Mathematical homogenization theory dates back to the French, Russian and Italian schools.[1][2][3][9] The method of asymptotic homogenization proceeds by introducing the fast variable ${\displaystyle {\vec {y}}={\vec {x}}/\epsilon }$ and posing a formal expansion in ${\displaystyle \epsilon }$:

${\displaystyle u_{\epsilon }({\vec {x}})=u({\vec {x}},{\vec {y}})=u_{0}({\vec {x}},{\vec {y}})+\epsilon u_{1}({\vec {x}},{\vec {y}})+\epsilon ^{2}u_{2}({\vec {x}},{\vec {y}})+O(\epsilon ^{3})\,}$

which generates a hierarchy of problems. The homogenized equation is obtained and the effective coefficients are determined by solving the so-called "cell problems" for the function ${\displaystyle u_{1}({\vec {x}},{\vec {x}}/\epsilon )}$.

Notes

1. Sanchez-Palencia, E: "Non-homogeneous media and vibration theory, Springer Verlag, 1980
2. Bakhvalov, N. and Panasenko, G: "Homogenization: Averaging Processes in Periodic Media", Kluwer, Dordrecht, 1989
3. Bensoussan, A., Lions, J.L. and Papanicolaou, G., "Asymptotic Analysis for Periodic Structures", North-Holland, Amsterdam, 1978
4. Ostoja-Starzewski, M., Microstructural randomness and scaling in materials, Chapman and Hall/CRC Press, 2007
5. S.M. Kozlov. Homogenization of Random Operators. Matem. Sbornik, 1979, 109(151), 188-202. (English transl.: Math. USSR, Sb. 37:2, 1980, pp. 167-180)
6. G. C. Papanicolaou, S.R. Varadhan. Boundary Value Problems with Rapidly Oscillating Coefficients. In: Seria Colloq. Math. Society Janos Bolyai, vol. 27, p. 835-873, Amsterdam, 1981
7. L. Berlyand and H. Owhadi, Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast, Archive for Rational Mechanics and Analysis, November 2010, Volume 198, Issue 2, pp 677-721
8. A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Mathematics of Computation, 2014, Volume 83, Number 290, pp 2583-2603
9. Dal Maso, G., "An introduction to ${\displaystyle \Gamma }$-convergence", Birkhauser, 1992

References

• Kozlov, S.M.; Oleinik, O.A.; Zhikov, V.V. (1994), Homogenization of differential operators and integral functionals, Berlin-Heidelberg-New York City: Springer-Verlag, ISBN 3-540-54809-2, Zbl 0838.35001
• Oleinik, O.A.; Shamaev, A.S.; Yosifian, G.A. (1991), Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, 26, Amsterdam - London - New York City - Tokyo: North-Holland, ISBN 0-444-88441-6, Zbl 0768.73003
• Hornung, Ulrich (Ed.). (1997), Homogenization and Porous Media, Interdisciplinary Applied Mathematics, 6, Springer-Verlag, doi:10.1007/978-1-4612-1920-0, ISBN 978-1-4612-7339-4
• Bakhvalov, N. S.; Panasenko, G. P. (1984), Averaging of Processes in Periodic Media (English translation: Kluwer,1989), Moscow: Nauka, Zbl 0607.73009
• Braides, A.; Defranceschi, A. (1998), Homogenization of Multiple Integrals, Oxford Lecture Series in Mathematics and Its Applications, Oxford: Clarendon Press, ISBN 978-0-198-50246-3