Cylindrical inclusions

Infinite cylinders are handled by the dedicated Cylinder type — a subtype of AbstractEllipsoidalInclusion{3, T} that stores only the two transverse semi-axes and the local basis (the infinite axis is implicit).

Construction

using MeanFieldHom

# Elliptic cross-section — returns Cylinder{EllipticCylindrical, …}
cyl_ell = Cylinder(2.0, 1.0)

# Circular cross-section — returns Cylinder{CircularCylindrical, …}
cyl_circ = Cylinder(1.5)          # shortcut
cyl_circ2 = Cylinder(1.5, 1.5)    # equivalent, detected numerically

# Oriented cylinder — axis along the third global direction
cyl_oriented = Cylinder(2.0, 1.0; euler_angles=(π/2, 0.0, 0.0))

# Via rotation matrix
cyl_rot = Cylinder(2.0, 1.0, [0 0 1; 1 0 0; 0 1 0])

The local frame convention mirrors Prolate: the cylinder axis is the first column of the basis, and the transverse semi-axes (b, c) (with b ≥ c for real element types) are associated with the second and third columns respectively.

Redirection from Ellipsoid

Passing an infinite or zero semi-axis to the Ellipsoid constructor redirects to the appropriate dedicated type — the caller does not have to switch constructors manually:

As for ellipsoids, the detection is active only for real element types (see Ellipsoidal inclusions).

Hill tensor

using TensND
K, μ = 175.0, 80.0
C_iso = TensISO{3}(3K, 2μ)

cyl = Cylinder(2.0, 1.0)
P   = hill_tensor(cyl, C_iso)        # TensOrtho (elliptic)
P_c = hill_tensor(Cylinder(1.5), C_iso)  # TensTI{4} (circular, TI axis e₁)

All components involving the cylinder axis (P_{11kl}) are exactly zero — reflecting the fact that an infinite cylinder cannot sustain a non-uniform axial strain.

For a general anisotropic matrix the call hill_tensor(cyl, C_aniso) transparently routes to a dedicated 1D QuadGK quadrature over the transverse plane. The method=:residues option is remapped to the same routine (the residue algorithm is not applicable to a cylinder — see theory / Hill polarization tensors).

Conductivity

The 2nd-order Hill tensor is computed via the same hill_tensor(…, K₀) entry point:

K = TensISO{3}(3.5)
H = hill_tensor(cyl, K)
# H[1, 1]  ≈ 0     (axial component)
# H[2, 2]  ≈ c / (k (b + c))
# H[3, 3]  ≈ b / (k (b + c))

Anisotropic conductors use a K⁻¹/² change of variable on the transverse 2×2 sub-matrix, reusing the 2D Newton potentials.

Auxiliary tensors

tens_IA, tens_UA, tens_VA are all defined for cylinders:

IA = tens_IA(cyl)   # 2nd-order, Iₐ = 0, sum = 1
UA = tens_UA(cyl)   # 4th-order — TensTI{4} (circular) or TensOrtho (elliptic)
VA = tens_VA(cyl)

ForwardDiff and symbolic element types

Every cylinder path (isotropic, anisotropic, auxiliary tensors) is ForwardDiff- and symbolic-compatible. Derivatives with respect to b and c propagate through the analytical formulas; symbolic construction uses structural equality (isequal(b, c)) to select the circular branch. For symbolic differentiation at b = c, prefer the single-argument constructor Cylinder(b) which forces the circular trait at compile time.