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Concept
Asymptotic homogenization
Summary
In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as
where is a very small parameter and
is a 1-periodic coefficient:
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 length scale which is far bigger than the characteristic length scale of the microstructure. In this situation, one can often replace the equation above with an equation of the form
where is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as
from 1-periodic functions satisfying:
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
for small enough , provided
in some appropriate norm as .
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" in homogenization and micromechanics.
Official source
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