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- Asymptotic_homogenization abstract "In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such aswhere 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 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 formwhere is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed asfrom 1-periodic functions satisfying:This process of replacing an equation with a highly oscillatory coefficient 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 , providedin 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. 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 above.".
- Asymptotic_homogenization wikiPageID "23852047".
- Asymptotic_homogenization wikiPageRevisionID "594361010".
- Asymptotic_homogenization hasPhotoCollection Asymptotic_homogenization.
- Asymptotic_homogenization subject Category:Asymptotic_analysis.
- Asymptotic_homogenization subject Category:Partial_differential_equations.
- Asymptotic_homogenization type Abstraction100002137.
- Asymptotic_homogenization type Communication100033020.
- Asymptotic_homogenization type DifferentialEquation106670521.
- Asymptotic_homogenization type Equation106669864.
- Asymptotic_homogenization type MathematicalStatement106732169.
- Asymptotic_homogenization type Message106598915.
- Asymptotic_homogenization type PartialDifferentialEquation106670866.
- Asymptotic_homogenization type PartialDifferentialEquations.
- Asymptotic_homogenization type Statement106722453.
- Asymptotic_homogenization comment "In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such aswhere 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.".
- Asymptotic_homogenization label "Asymptotic homogenization".
- Asymptotic_homogenization sameAs m.06_xq24.
- Asymptotic_homogenization sameAs Q4812185.
- Asymptotic_homogenization sameAs Q4812185.
- Asymptotic_homogenization sameAs Asymptotic_homogenization.
- Asymptotic_homogenization wasDerivedFrom Asymptotic_homogenization?oldid=594361010.
- Asymptotic_homogenization isPrimaryTopicOf Asymptotic_homogenization.