Conductivity homogenization

Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

Table of contents
- Polarization field of a conductivity inclusion Conductivity homogenization
  - Exciting field of a conductivity inclusion Polarization field of a conductivity inclusion
  - Spherical conductivity inclusion Polarization field of a conductivity inclusion
    - Dilute spherical-inclusion conductivity Spherical conductivity inclusion
      - Maxwell approximation for conductivity Dilute spherical-inclusion conductivity
      - Dipole interaction correction to effective conductivity Dilute spherical-inclusion conductivity
- Periodic conductivity cell problem Conductivity homogenization
  - Two-scale expansion for periodic conductivity Periodic conductivity cell problem
    - Boundary corrector in periodic homogenization Two-scale expansion for periodic conductivity

Polarization field of a conductivity inclusion ( $P = (a - a_{0} I) \nabla u$ )

 0  0

Relative to a uniform reference conductivity

a_{0}

, this polarization is the excess positive conductivity flux. The equation becomes

a_{0} Δ u + \nabla \cdot P = 0

. For a prescribed mean gradient

E

, averaging gives

a^{*} E = a_{0} E + ⟨ P ⟩

. This sign convention differs from the physical heat-flux sign.

Exciting field of a conductivity inclusion ( $E_{exc} = E + e_{regular} (y)$ )

 0  0

The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre

y

. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.

Spherical conductivity inclusion ( $β = 3 a_{0} (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

 0  0

A sphere of conductivity

a_{1}

in a host

a_{0}

under a uniform incident gradient

E

has interior gradient

3 a_{0} E / (a_{1} + 2 a_{0})

. Outside a sphere of radius

R

, its potential correction is

- R^{3} (a_{1} - a_{0}) (E \cdot x) / [(a_{1} + 2 a_{0}) ∣ x ∣^{3}]

. Its polarization is

βE

inside and zero outside. Temperature and normal-flux continuity determine these coefficients.

Dilute spherical-inclusion conductivity ( $a^{*} = a_{0} I + pβ I + o (p)$ )

 0  0

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

Maxwell approximation for conductivity ( $a^{*} = a_{0} (1 + 2 p k) / (1 - p k), k = (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

 0  0

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is

a_{0} + pβ + p^{2} β^{2} / (3 a_{0}) + O (p^{3})

. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.

Dipole interaction correction to effective conductivity ( $a^{*} \approx a_{0} I + pβ I - β^{2} \sum_{i} Λ_{i}$ )

 0  0

If the first exciting-field correction at inclusion

i

- β Λ_{i} E / p_{i}

, its revised polarization is

β χ_{i} (E - β Λ_{i} E / p_{i})

. Averaging weights it by

p_{i}

and gives the displayed second-reflection approximation. Further dipole reflections and finite-size multipoles are neglected.

Periodic conductivity cell problem ( $\nabla_{y} \cdot [a (y) (e_{j} + \nabla_{y} w_{j})] = 0$ )

 0  0

The periodic zero-mean corrector

w_{j}

adjusts a unit imposed gradient so microscopic flux is divergence-free. For bounded uniformly elliptic coefficients, the weak problem has a unique solution modulo constants. The averaged flux is

a^{*} e_{j} = ⟨ a (e_{j} + \nabla w_{j})⟩

. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.

Two-scale expansion for periodic conductivity ( $\nabla = \nabla_{x} + ϵ^{- 1} \nabla_{y}$ )

 0  0

A periodic expansion with

y = x / ϵ

separates the macroscopic variable from the cell variable. Its leading cell equation forces

u_{0}

to be independent of

y

, the next equation gives

u_{1} = w_{j} (y) \partial_{x_{j}} u_{0}

, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.

Boundary corrector in periodic homogenization ( $b^{ϵ} ∣_{\partial Ω} = - ϵ w_{j} (x / ϵ) \partial_{j} u_{0}$ )

 0  0

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.

Conductivity homogenization

Polarization field of a conductivity inclusion ( $P = (a - a_{0} I) \nabla u$ )

Exciting field of a conductivity inclusion ( $E_{exc} = E + e_{regular} (y)$ )

Spherical conductivity inclusion ( $β = 3 a_{0} (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

Dilute spherical-inclusion conductivity ( $a^{*} = a_{0} I + pβ I + o (p)$ )

Maxwell approximation for conductivity ( $a^{*} = a_{0} (1 + 2 p k) / (1 - p k), k = (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

Dipole interaction correction to effective conductivity ( $a^{*} \approx a_{0} I + pβ I - β^{2} \sum_{i} Λ_{i}$ )

Periodic conductivity cell problem ( $\nabla_{y} \cdot [a (y) (e_{j} + \nabla_{y} w_{j})] = 0$ )

Two-scale expansion for periodic conductivity ( $\nabla = \nabla_{x} + ϵ^{- 1} \nabla_{y}$ )

Boundary corrector in periodic homogenization ( $b^{ϵ} ∣_{\partial Ω} = - ϵ w_{j} (x / ϵ) \partial_{j} u_{0}$ )

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