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Merged
merged 4 commits into from
Sep 19, 2025
Merged

Analytical derivatives for eight-chain model #9

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f727688
added derivatives
miguelmaso Sep 19, 2025
8481e33
added test
miguelmaso Sep 19, 2025
fbf536f
removed redefinition
miguelmaso Sep 19, 2025
04591ad
added function for benchmarking
miguelmaso Sep 19, 2025
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53 changes: 47 additions & 6 deletions src/PhysicalModels/PhysicalModels.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -909,16 +909,57 @@ struct EightChain{A} <: Elasto
end

function (obj::EightChain)(Λ::Float64=1.0)
_, H, J = get_Kinematics(obj.Kinematic; Λ=Λ)
μ, N = obj.μ, obj.N

Ψ(F) = begin
C = F' * F
C_iso = det(C)^(-2/3) * C
β = sqrt(tr(C_iso) / 3 / obj.N)
C_iso = J(F)^(-2/3) * C
β = sqrt(tr(C_iso) / 3 / N)
L = β * (3.0 - β^2) / (1.0 - β^2)
μ * N * (β * L + log(L / sinh(L)))
end

∂Ψ∂F(F) = begin
C = F' * F
C_iso = J(F)^(-2 / 3) * C
β = sqrt(tr(C_iso) / 3 / N)
L = β * (3.0 - β^2) / (1.0 - β^2)
obj.μ * obj.N * (β*L + log(L / sinh(L)))
∂β∂I1_ = 0.5 / sqrt(tr(C_iso) * 3 * N)
∂L∂I1_ = ((3 * (1 - β^2)^2 + 2 * β * (3 * β - β^3)) / (1 - β^2)^2) * ∂β∂I1_
n = (∂L∂I1_ * sinh(L) - L * cosh(L) * ∂L∂I1_)
d = (L * sinh(L))
∂Ψ∂I1_ = μ * N * (∂β∂I1_ * L + β * ∂L∂I1_ + n / d)
∂I1_∂F = 2 * J(F)^(-2 / 3) * F
∂I1_∂J = -(2 / 3) * J(F)^(-5 / 3) * tr(C)
∂Ψ∂I1_ * (∂I1_∂F + ∂I1_∂J * H(F))
end
∂Ψ∂F(F) = ForwardDiff.gradient(Ψ, get_array(F))
∂Ψ∂FF(F) = ForwardDiff.jacobian(∂Ψ∂F, get_array(F))
return (Ψ, TensorValue ∘ ∂Ψ∂F, TensorValue ∘ ∂Ψ∂FF)

∂Ψ∂FF(F) = begin
H_ = H(F)
C = F' * F
C_iso = det(F)^(-2 / 3) * C
β = sqrt(tr(C_iso) / 3 / N)
L = β * (3.0 - β^2) / (1.0 - β^2)
∂β∂I1_ = 0.5 / sqrt(tr(C_iso) * 3 * N)
∂L∂I1_ = ((3 * (1 - β^2)^2 + 2 * β * (3 * β - β^3)) / (1 - β^2)^2) * ∂β∂I1_
∂β∂I1I1_ = -(3 * N) / (4 * (3 * N * tr(C_iso))^(3 / 2))
∂L∂I1I1_ = ((4 * β * (β^2 + 3)) / (1 - β^2)^3) * ∂β∂I1_^2 + ((3 * (1 - β^2)^2 + 2 * β * (3 * β - β^3)) / (1 - β^2)^2) * ∂β∂I1I1_
∂I1_∂F = 2 * det(F)^(-2 / 3) * F
∂I1_∂J = -(2 / 3) * det(F)^(-5 / 3) * tr(C)
∂I1_∂F∂F = 2 * det(F)^(-2 / 3) * I9
∂I1_∂J∂J = (10 / 9) * det(F)^(-8 / 3) * tr(C)
∂I1_∂F∂J = -(4 / 3) * det(F)^(-5 / 3) * F
n = (∂L∂I1_ * sinh(L) - L * cosh(L) * ∂L∂I1_)
d = (L * sinh(L))
∂n∂I1_ = ∂L∂I1I1_ * sinh(L) + ∂L∂I1_ * ∂L∂I1_* cosh(L) - ∂L∂I1_^2 * cosh(L) - L * sinh(L) * ∂L∂I1_^2 - L * cosh(L) * ∂L∂I1I1_
∂d∂I1_ = ∂L∂I1_ * sinh(L) + L * ∂L∂I1_* cosh(L)
∂Ψ∂I1_ = μ * N * (∂β∂I1_ * L + β * ∂L∂I1_ + n / d)
∂Ψ∂I1I1_ = μ * N * (∂β∂I1I1_ * L + 2 * ∂β∂I1_ * ∂L∂I1_ + β * ∂L∂I1I1_ + (∂n∂I1_ * d - n * ∂d∂I1_) / d^2)
∂Ψ∂I1I1_ * ((∂I1_∂F + ∂I1_∂J * H_) ⊗ (∂I1_∂F + ∂I1_∂J * H_)) + ∂Ψ∂I1_ * (∂I1_∂F∂F + ∂I1_∂F∂J ⊗ H_ + H_ ⊗ ∂I1_∂F∂J + ∂I1_∂J∂J * (H_ ⊗ H_) + I9 × (∂I1_∂J * F))
end

return (Ψ, ∂Ψ∂F, ∂Ψ∂FF)
end
end

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37 changes: 31 additions & 6 deletions test/TestConstitutiveModels/PhysicalModels.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -4,6 +4,7 @@ using JSON
using StaticArrays
using Test
using HyperFEM.PhysicalModels
using BenchmarkTools


import Base: -
Expand Down Expand Up @@ -38,6 +39,30 @@ function test_derivatives_3D_(model::PhysicalModel; rtol=1e-14, kwargs...)
end


function benchmark_derivatives__(model::PhysicalModel, ∇u)
Ψ, ∂Ψu, ∂Ψuu = model()
∂Ψu_(F) = TensorValue(ForwardDiff.gradient(Ψ, get_array(F)))
∂Ψuu_(F) = TensorValue(ForwardDiff.hessian(Ψ, get_array(F)))

F, _, _ = get_Kinematics(model.Kinematic)
print("Analyitical ∂Ψu | ")
@btime $∂Ψu($F($∇u))
print("Numerical ∂Ψu | ")
@btime $∂Ψu_($F($∇u))
print("Analyitical ∂Ψuu | ")
@btime $∂Ψuu($F($∇u))
print("Numerical ∂Ψuu | ")
@btime $∂Ψuu_($F($∇u))
end

function benchmark_derivatives_2D_(model::PhysicalModel)
benchmark_derivatives__(model, ∇u2)
end

function benchmark_derivatives_3D_(model::PhysicalModel)
benchmark_derivatives__(model, ∇u3)
end


@testset "NonlinearMooneyRivlin_CV" begin
model = NonlinearMooneyRivlin_CV(λ=3.0, μ1=1.0, μ2=1.0, α=2.0, β=1.0, γ=6.0)
Expand Down Expand Up @@ -141,6 +166,12 @@ end
end


@testset "EightChain" begin
model = EightChain(μ=μParams[1], N=μParams[2])
test_derivatives_3D_(model, rtol=1e-13)
end


@testset "TransverseIsotropy2D" begin
∇u = TensorValue(1.0, 2.0, 3.0, 4.0) * 1e-3
∇u0 = TensorValue(1.0, 2.0, 3.0, 4.0) * 0.0
Expand Down Expand Up @@ -701,12 +732,6 @@ end
@testset "Hessian∇JRegularization" begin
∇u = TensorValue(1.0, 2.0, 3.0, 4.0) * 1e-3
∇u0 = TensorValue(1.0, 2.0, 3.0, 4.0) * 0.0

μParams = [6456.9137547089595, 896.4633794151492,
1.999999451256222,
1.9999960497608036,
11747.646562400318,
0.7841068624959612, 1.5386288924587603]
model = ARAP2D(μ=μParams[1])
modelreg = Hessian∇JRegularization(Mechano=model)

Expand Down