Mean-field homogenization replaces a heterogeneous microstructure — a matrix carrying inclusions, pores, or cracks — by an equivalent homogeneous medium with the same overall response. MeanFieldHom computes that response from phase properties, geometries, volume fractions, and a scheme encoding an assumption about how the phases interact.
Simplest first within each group. Pages under generated/ are produced from the runnable demos in scripts/ by Literate.jl.
The porous material gets two pages: the simplest non-trivial microstructure, and the one where the choice of scheme matters most.
| Page | What it shows |
|---|
| Hill polarization tensors in practice | hill_tensor on four geometries; residues vs cubature on an anisotropic matrix; the Eshelby tensor against its closed form; P → a dilute estimate |
| Cracks and crack density | volume fraction → crack density; the COD tensor |
| Layered spheres | Hervé–Zaoui n-layer localization and layer averages |
| Layered spheroids: geometry and effective conductivity | the confocal n-layer spheroid, the equivalent particle, harmonic-series accuracy |
| Imperfect interfaces: what they do to the local fields | pointwise temperature and flux, streamlines, conductance sweep, 3-D view |
| Highly conducting interfaces | equivalent conductivity of an HC-coated particle vs aspect ratio |
| Symmetrization | exact rotation-group average vs best-fit projection |
| The custom-inclusion contract | plugging an arbitrary morphology into every scheme — the three entry gates, the density seam, free orientation averaging |
| An inclusion whose response is a neural network | both phases — how a surrogate is trained (schematics of the network and of the fitting loop, the recorded learning curve) and how a trained one is used: what stays exact whatever the fit, accuracy against the closed form, every scheme, and the derivative with respect to the morphology |
| Replacing a finite-element solve by a surrogate | the case the machinery exists for: the eccentric-core sphere, whose localization tensors have no closed form. Gate B with the 6-component transversely isotropic pair, the contrast ratios that replace gate A's homogeneity, the accuracy against the finite elements, the speed-up, and a derivative with respect to the eccentricity |
Composite inclusions carry no Hill tensor at all: they enter the schemes through their volume-averaged concentration tensors instead.
| Page | What it shows |
|---|
| Validating a finite-element crack | what the corrected boundary condition buys, and convergence to the closed-form COD |
| Transport properties | 2nd-order homogenization: diffusivity of a porous medium, anisotropy from oriented pores |
| Symbolic spheres | the same tensor algebra on SymPy / Symbolics expressions: Eshelby/Hill tensors and the closed-form estimates |