Tutorials
Mean-field homogenization replaces a heterogeneous microstructure — a matrix carrying inclusions, pores, or cracks — by an equivalent homogeneous medium with the same overall response. MeanFieldHomogenization computes that response from phase properties, geometries, volume fractions, and a scheme encoding an assumption about how the phases interact.
The groups follow the chapters of Theory, in the same order, so that a tutorial sits where its theory is: the problems of a single heterogeneity first, then the schemes that turn them into effective properties, then the extensions to conductivity and viscoelasticity. Two groups have no theory chapter of their own, differentiation and symbolic computation. Simplest first within each group. Pages under generated/ are produced from the runnable demos in scripts/ by Literate.jl.
First steps
| Page | What it shows |
|---|---|
| A first homogenization | build an RVE; dilute vs Mori–Tanaka |
| Bounds and classical schemes | Voigt/Reuss bounds, self-consistent, where each scheme sits |
The Eshelby problem
One inclusion in an infinite matrix: its Hill tensor, how an orientation average acts on it, and inclusions that go beyond the ellipsoid. Theory: The Eshelby inclusion problem and Hill polarization tensors.
| 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 |
| Hill tensors by reduction to a sphere | every Hill tensor is that of a sphere in a transformed medium: the same tensor computed both ways, on a spheroid in a transversely isotropic matrix, a triaxial ellipsoid in a triclinic one and an elliptically orthotropic matrix |
| 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 concentration 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 |
Cracks and degenerate shapes
The flat limit of the ellipsoid, where the volume fraction gives way to a crack density. Theory: Crack opening displacement and compliance.
| Page | What it shows |
|---|---|
| Cracks and crack density | volume fraction → crack density; the COD tensor |
| Two routes to a crack: the COD tensor and the flattened ellipsoid | the identity |
| Crack distributions: isotropic or parallel | the same density, two orientation rules; the two self-consistent forms and their two percolation thresholds (9/16 exactly for the compliance form, ≈ 1.158 for the stiffness one); the local-versus-global frame of a TensRotated result |
| Validating a finite-element crack | the same crack solved by finite elements: what the corrected boundary condition buys, and convergence to the closed-form COD |
The generalized Eshelby problem — morphological patterns
Composite inclusions carry no Hill tensor at all: they enter the schemes through their volume-averaged concentration tensors instead. Theory: the layered sphere and the layered spheroid.
| Page | What it shows |
|---|---|
| Layered spheres | Hervé–Zaoui n-layer concentration and layer averages |
| n-layer sphere: pointwise fields | the strain, stress and displacement at any point inside and outside an n-layer sphere, perfect or imperfect interfaces, consistent with the 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 |
| A layered spheroid, meshed | the same body by Fourier axisymmetric finite elements: calibrated against the two closed forms, taken to nests neither covers, then replaced by a surrogate trained on the cell so the morphology becomes differentiable |
| Nanocomposites: the equivalent particle | a Gurtin-Murdoch interface condensed into a particle stiffness, its three limiting shapes reproduced exactly, and the size effect it produces through an ordinary Mori-Tanaka estimate — no new scheme needed |
Homogenization schemes
The schemes that turn those solutions into effective properties: the mean-field estimates, the exact laminate, and the two N-body schemes, which resolve the interactions inclusion by inclusion and so need a ParticleAssembly, a cell that carries positions, rather than an RVE. The cluster model on cubic arrays and the equivalent inclusion method against a published table are under Applications: each exists to reproduce one paper's numbers rather than to teach the library. Theory: Homogenization schemes; for the N-body schemes, interaction tensors, the cluster model and the equivalent inclusion method, with the API on the particle-assembly manual page.
| Page | What it shows |
|---|---|
| Porous materials: the self-consistent trap and every scheme | why soft pores break the naive self-consistent iteration, then every scheme on the canonical porosity sweep, spheres and oblate pores |
| The differential scheme and path dependence | incremental homogenization; why mixing order matters, and four trajectories to the same fractions watched τ by τ |
| Periodic multilayer: the exact laminate solution | a cell with no matrix and no Eshelby problem: the exact effective tensor, per-layer concentration, the bounds it saturates |
| Imperfect interfaces in a laminate, and the size effect | the four interface models on a planar interface, and the interface density that makes the period matter |
| Multiscale chaining: explicit and declarative, side by side | the two ways of chaining scales, the same answer, and the ForwardDiff pass that goes through both |
| Chaining scales through an N-body scheme | the declarative multiscale seam with an assembly on either side, and why chaining two N-body estimates needs the anisotropic Green operator: a cluster estimate on a cubic array is cubic, not isotropic. Three scales end to end, plus a sensitivity across them |
| Thermal expansion and Biot tensor | Levin's theorem on the concentration tensors of a scheme: the thermal expansion of a two-phase composite under five schemes, anisotropic with aligned platelets, the Biot tensor of a solid with pores and pressurized cracks, and both in closed form |
Extension to conductivity
| Page | What it shows |
|---|---|
| Transport properties | the same algebra at tensor order 2: diffusivity of a porous medium, anisotropy from oriented pores |
Extension to viscoelasticity
| Page | What it shows |
|---|---|
| Viscoelastic composites | complex moduli in the frequency domain; a first taste of aging creep |
| The rheological model catalog | the classical chains, the fractional family and the bituminous models side by side; master curves, Cole-Cole and Black diagrams; the exact 2S2P1D pair in both domains; and one object driving both homogenization routes |
| Generalized Kelvin ⇄ generalized Maxwell | the exact conversion between the two representations: the interlacing that makes it unconditionally robust, the round trip at twenty branches, and two independent closed forms — the Zener relations and the Burgers relaxation function — as oracles |
| Choosing a numerical Laplace inversion | four algorithms measured against four exact pairs; why branch cuts are not the problem and oscillation is; the Gaver-Stehfest optimum and why more terms is worse; ForwardDiff straight through |
| Frequency or time? | the complex-modulus and time-domain ALV routes, cross-checked on the same non-aging composite |
| Aging viscoelastic schemes side by side | Dilute / Mori-Tanaka / Maxwell / PCW on one creep test; the aspect ratio; where the distribution shape decides the answer |
| Aging creep: loading age against inclusion shape | aging and morphology on the same output: three loading ages × three aspect ratios, and why the shape effect is an offset independent of the age |
| Derivatives through the aging-viscoelastic pipeline | ForwardDiff through the Volterra assembly: the set_param lens for RVE parameters, closure capture for moduli and relaxation times |
| A creeping laminate | the exact multilayer in aging viscoelasticity: the same algebra with Volterra products in place of tensor products |
Differentiation and solvers
| Page | What it shows |
|---|---|
| Derivatives and sensitivities | differentiate any result with ForwardDiff, no finite differences |
| From derivatives to a strength criterion | those derivatives as a macroscopic strength criterion |
| Nonlinear solvers for the self-consistent fixed point | NonlinearSolve.jl instead of Picard, and sensitivities that agree either way |
| Nonlinear homogenization by the secant method | elastic–perfectly plastic porous solid, closed by second moments |
Symbolic computation
| Page | What it shows |
|---|---|
| Symbolic spheres | the same tensor algebra on SymPy / Symbolics expressions: Eshelby/Hill tensors and the closed-form estimates |
| Symbolic laminates: arithmetic and harmonic averages | the one microstructure whose effective behavior is a closed form, derived symbolically |
| Symbolic viscoelasticity | models with symbolic parameters, and symbolic Laplace-Carson inversion: the Burgers relaxation function derived rather than transcribed, and precisely where the symbolic route stops |