In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients,[1][2][3] such as where
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.
Under this assumption, materials such as fluids, solids, etc.
Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are subjected to loads or forcings which vary on a length scale which is far bigger than the characteristic length scale of the microstructure.
is a constant tensor coefficient and is known as the effective property associated with the material in question.
satisfying: This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as homogenization.
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
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] 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
The homogenized equation is obtained and the effective coefficients are determined by solving the so-called "cell problems" for the function