Concept

Asymptotic homogenization

Summary
In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as : \nabla\cdot\left(A\left(\frac{\vec x}{\epsilon}\right)\nabla u_{\epsilon}\right) = f where \epsilon is a very small parameter and
A\left(\vec y\right) is a 1-periodic coefficient: A\left(\vec y+\vec e_i\right)=A\left(\vec y\right), 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 mat
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