A two-point Green tensor between offset frames

The Green operator linking two points that belong to different spherical frames, offset along . This is the kernel used when scattering by a cluster of inclusions is treated inclusion by inclusion, each with its own local frame.

It is also a compact demonstration that tensors carrying different bases can be combined: the position vectors live in two distinct spherical charts and are subtracted after conversion to a common frame.

Theory: Bases and variance, Curvilinear differential calculus.

using TensND
using LinearAlgebra
using SymPy

𝕀, 𝕁, 𝕂 = iso_projectors(Val(3), Val(Sym))
𝟏 = tens_Id2(Val(3), Val(Sym))

E = symbols("E", positive = true)
ν = symbols("ν", real = true)
k = E / (3(1 - 2ν))
μ = E / (2(1 + ν))

Two spherical frames

The "field" frame and the "source" frame , the latter translated by along .

θ, ϕ, r = symbols("θ", real = true), symbols("ϕ", real = true), symbols("r", positive = true)
S = coorsys_spherical((θ, ϕ, r))
𝐱 = getOM(S)

θ′, ϕ′, r′ = symbols("θ′", real = true), symbols("ϕ′", real = true), symbols("r′", positive = true)
S′ = coorsys_spherical((θ′, ϕ′, r′))

Cartesian = coorsys_cartesian(symbols("x y z", real = true))
𝐞₁, 𝐞₂, 𝐞₃ = unitvec(Cartesian)
R = symbols("R", positive = true)

𝐱′ = getOM(S′) + R * 𝐞₃

The separation vector

Both position vectors are expressed in their own chart; subtracting them requires a common frame, which TensND supplies automatically.

Δ𝐱 = 𝐱 - 𝐱′
ρ = tsimplify(norm(Δ𝐱))

The distance between the two points, as a function of both sets of spherical coordinates and the offset:

ρ

The unit vector along the separation, brought back to the canonical basis:

𝐍 = Δ𝐱 / ρ
𝐧 = tsimplify(change_tens_canon(𝐍))

The Green operator

The same closed form as in Elastic Green's functions, with the separation direction in place of the radial vector and in place of :

ℾ = 1 / (16PI * μ * (1 - ν) * ρ^3) * (
    -3𝕁 + 2(1 - 2ν) * 𝕀 + 3(𝟏 ⊗ 𝐧 ⊗ 𝐧 + 𝐧 ⊗ 𝐧 ⊗ 𝟏)
        + 12ν * 𝐧 ⊗ˢ 𝟏 ⊗ˢ 𝐧 - 15𝐧 ⊗ 𝐧 ⊗ 𝐧 ⊗ 𝐧
)

Checks

is a unit vector, whatever the two sets of coordinates:

tsimplify(𝐧 ⋅ 𝐧)

On the axis    the two frames align and the separation reduces to   :

tsimplify(subs(ρ, θ => Sym(0), θ′ => Sym(0)))

and the separation direction becomes :

tsimplify(subs(get_array(𝐧)[3], θ => Sym(0), θ′ => Sym(0)))

Coincident frames

Setting   and   reduces the two-point kernel to the ordinary radial one — the consistency check that the construction must pass:

ρ_same = tsimplify(subs(ρ, R => Sym(0), θ′ => θ, ϕ′ => ϕ))
tsimplify(ρ_same - abs(r - r′))

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