Hyperelasticity

Keywords: hyperelasticity, finite strain, large deformations, Newton's method, conjugate gradient, automatic differentiation

Tip

This example is also available as a Jupyter notebook: hyperelasticity.ipynb

Introduction

In this example we will solve a problem in a finite strain setting using an hyperelastic material model. In order to compute the stress we will use automatic differentiation, to solve the non-linear system we use Newton's method, and for solving the Newton increment we use conjugate gradient.

The weak format is expressed in terms of the first Piola-Kirchoff stress $\mathbf{P}$ as follows: Find $u \in \mathbb{U}$ such that

$\int_\Omega [\delta \mathbf{u} \otimes \nabla] : \mathbf{P}(\mathbf{u})\ \mathrm{d}\Omega = \int_\Omega \delta \mathbf{u} \cdot \mathbf{b}\ \mathrm{d}\Omega + \int_{\Gamma^\mathrm{N}} \mathbf{u} \cdot \mathbf{t}\ \mathrm{d}\Gamma \quad \forall \delta \mathbf{u} \in \mathbb{U}^0,$

where $\mathbf{u}$ is the unknown displacement field, $\mathbf{b}$ is the body force, $\mathbf{t}$ is the traction on the Neumann part of the boundary, and where $\mathbb{U}$ and $\mathbb{U}^0$ are suitable trial and test sets.

using JuAFEM, Tensors, TimerOutputs, ProgressMeter
import KrylovMethods, IterativeSolvers

Hyperelastic material model

The stress can be derived from an energy potential, defined in terms of the right Cauchy-Green tensor $\mathbf{C}$. We shall use a neo-Hookean model, where the potential can be written as

$\Psi(\mathbf{C}) = \frac{\mu}{2} (I_C - 3) - \mu \ln(J) + \frac{\lambda}{2} \ln(J)^2,$

where $I_C = \mathrm{tr}(C)$, $J = \sqrt{\det(C)}$ and $\mu$ and $\lambda$ material parameters. From the potential we obtain the second Piola-Kirchoff stress $\mathbf{S}$ as

$\mathbf{S} = 2 \frac{\partial \Psi}{\partial \mathbf{C}},$

and the tangent of $\mathbf{S}$ as

$\frac{\partial \mathbf{S}}{\partial \mathbf{C}} = 4 \frac{\partial \Psi}{\partial \mathbf{C}}.$

We can implement the material model as follows, where we utilize automatic differentiation for the stress and the tangent, and thus only define the potential:

struct NeoHooke
μ::Float64
λ::Float64
end

function Ψ(C, mp::NeoHooke)
μ = mp.μ
λ = mp.λ
Ic = tr(C)
J = sqrt(det(C))
return μ / 2 * (Ic - 3) - μ * log(J) + λ / 2 * log(J)^2
end

function constitutive_driver(C, mp::NeoHooke)
# Compute all derivatives in one function call
∂²Ψ∂C², ∂Ψ∂C = Tensors.hessian(y -> Ψ(y, mp), C, :all)
S = 2.0 * ∂Ψ∂C
∂S∂C = 4.0 * ∂²Ψ∂C²
return S, ∂S∂C
end;

Finally, for the finite element problem we need $\mathbf{P}$ and $\frac{\partial \mathbf{P}}{\partial \mathbf{F}}$, which can be obtained by the following transformations

\begin{align*} \mathbf{P} &= \mathbf{F} \cdot \mathbf{S},\\ \frac{\partial \mathbf{P}}{\partial \mathbf{F}} &= [\mathbf{F} \bar{\otimes} \mathbf{I}] : \frac{\partial \mathbf{S}}{\partial \mathbf{C}} : [\mathbf{F}^\mathrm{T} \bar{\otimes} \mathbf{I}] + \mathbf{I} \bar{\otimes} \mathbf{S}. \end{align*}

Finite element assembly

The element routine for assembling the residual and tangent stiffness is implemented as usual, with loops over quadrature points and shape functions:

function assemble_element!(ke, ge, cell, cv, fv, mp, ue)
# Reinitialize cell values, and reset output arrays
reinit!(cv, cell)
fill!(ke, 0.0)
fill!(ge, 0.0)

b = Vec{3}((0.0, -0.5, 0.0)) # Body force
t = Vec{3}((0.1, 0.0, 0.0)) # Traction
ndofs = getnbasefunctions(cv)

dΩ = getdetJdV(cv, qp)
# Compute deformation gradient F and right Cauchy-Green tensor C
F = one(∇u) + ∇u
C = tdot(F)
# Compute stress and tangent
S, ∂S∂C = constitutive_driver(C, mp)
P = F ⋅ S
I = one(S)
∂P∂F = otimesu(F, I) ⊡ ∂S∂C ⊡ otimesu(F', I) + otimesu(I, S)

# Loop over test functions
for i in 1:ndofs
δui = shape_value(cv, qp, i)
# Add contribution to the residual from this test function
ge[i] += ( ∇δui ⊡ P - δui ⋅ b ) * dΩ

∇δui∂P∂F = ∇δui ⊡ ∂P∂F # Hoisted computation
for j in 1:ndofs
# Add contribution to the tangent
ke[i, j] += ( ∇δui∂P∂F ⊡ ∇δuj ) * dΩ
end
end
end

# Surface integral for the traction
for face in 1:nfaces(cell)
if onboundary(cell, face)
reinit!(fv, cell, face)
dΓ = getdetJdV(fv, q_point)
for i in 1:ndofs
δui = shape_value(fv, q_point, i)
ge[i] -= (δui ⋅ t) * dΓ
end
end
end
end
end;

Assembling global residual and tangent

function assemble_global!(K, f, dh, cv, fv, mp, u)
n = ndofs_per_cell(dh)
ke = zeros(n, n)
ge = zeros(n)

# start_assemble resets K and f
assembler = start_assemble(K, f)

# Loop over all cells in the grid
@timeit "assemble" for cell in CellIterator(dh)
global_dofs = celldofs(cell)
ue = u[global_dofs] # element dofs
@timeit "element assemble" assemble_element!(ke, ge, cell, cv, fv, mp, ue)
assemble!(assembler, global_dofs, ge, ke)
end
end;

Define a main function, with a loop for Newton iterations

function solve()
reset_timer!()

# Generate a grid
N = 10
L = 1.0
left = zero(Vec{3})
right = L * ones(Vec{3})
grid = generate_grid(Tetrahedron, (N, N, N), left, right)

# Material parameters
E = 10.0
ν = 0.3
μ = E / (2(1 + ν))
λ = (E * ν) / ((1 + ν) * (1 - 2ν))
mp = NeoHooke(μ, λ)

# Finite element base
ip = Lagrange{3, RefTetrahedron, 1}()
cv = CellVectorValues(qr, ip)
fv = FaceVectorValues(qr_face, ip)

# DofHandler
dh = DofHandler(grid)
push!(dh, :u, 3) # Add a displacement field
close!(dh)

function rotation(X, t, θ = deg2rad(60.0))
x, y, z = X
return t * Vec{3}(
(0.0,
L/2 - y + (y-L/2)*cos(θ) - (z-L/2)*sin(θ),
L/2 - z + (y-L/2)*sin(θ) + (z-L/2)*cos(θ)
))
end

dbcs = ConstraintHandler(dh)
# Add a homogenous boundary condition on the "clamped" edge
dbc = Dirichlet(:u, getfaceset(grid, "right"), (x,t) -> [0.0, 0.0, 0.0], [1, 2, 3])
dbc = Dirichlet(:u, getfaceset(grid, "left"), (x,t) -> rotation(x, t), [1, 2, 3])
close!(dbcs)
t = 0.5
JuAFEM.update!(dbcs, t)

# Pre-allocation of vectors for the solution and Newton increments
_ndofs = ndofs(dh)
un = zeros(_ndofs) # previous solution vector
u  = zeros(_ndofs)
Δu = zeros(_ndofs)
ΔΔu = zeros(_ndofs)
apply!(un, dbcs)

# Create sparse matrix and residual vector
K = create_sparsity_pattern(dh)
g = zeros(_ndofs)

# Perform Newton iterations
newton_itr = -1
NEWTON_TOL = 1e-8
prog = ProgressMeter.ProgressThresh(NEWTON_TOL, "Solving:")

while true; newton_itr += 1
u .= un .+ Δu # Current guess
assemble_global!(K, g, dh, cv, fv, mp, u)
normg = norm(g[JuAFEM.free_dofs(dbcs)])
apply_zero!(K, g, dbcs)
ProgressMeter.update!(prog, normg; showvalues = [(:iter, newton_itr)])

if normg < NEWTON_TOL
break
elseif newton_itr > 30
error("Reached maximum Newton iterations, aborting")
end

# Compute increment using cg! from IterativeSolvers.jl
@timeit "linear solve (KrylovMethods.cg)" ΔΔu′, flag, relres, iter, resvec = KrylovMethods.cg(K, g; maxIter = 1000)
@assert flag == 0
@timeit "linear solve (IterativeSolvers.cg!)" IterativeSolvers.cg!(ΔΔu, K, g; maxiter=1000)

apply_zero!(ΔΔu, dbcs)
Δu .-= ΔΔu
end

# Save the solution
@timeit "export" begin
vtk_grid("hyperelasticity", dh) do vtkfile
vtk_point_data(vtkfile, dh, u)
end
end

print_timer(title = "Analysis with $(getncells(grid)) elements", linechars = :ascii) return u end solve (generic function with 1 method) Run the simulation u = solve(); Solving: (thresh = 1e-08, value = 0.76197)[K iter: 0[K [A [K[A Solving: (thresh = 1e-08, value = 0.194526)[K iter: 1[K [A [K[A Solving: (thresh = 1e-08, value = 0.013799)[K iter: 2[K [A [K[A Solving: (thresh = 1e-08, value = 0.000329859)[K iter: 3[K [A [K[A Solving: (thresh = 1e-08, value = 1.92988e-07)[K iter: 4[K [A [K[A Solving: Time: 0:00:34 (6 iterations)[K iter: 5[K ------------------------------------------------------------------------------ Analysis with 6000 elements Time Allocations ---------------------- ----------------------- Tot / % measured: 35.3s / 99.1% 11.3GiB / 99.4% Section ncalls time %tot avg alloc %tot avg ------------------------------------------------------------------------------ assemble 6 34.7s 99.3% 5.78s 11.2GiB 100% 1.87GiB element assemble 36.0k 34.6s 98.9% 960μs 11.2GiB 100% 326KiB linear solve (Iter... 5 161ms 0.46% 32.1ms 474KiB 0.00% 94.8KiB linear solve (Kryl... 5 47.3ms 0.14% 9.45ms 669KiB 0.01% 134KiB export 1 42.4ms 0.12% 42.4ms 4.27MiB 0.04% 4.27MiB ------------------------------------------------------------------------------ Plain Program Below follows a version of the program without any comments. The file is also available here: hyperelasticity.jl using JuAFEM, Tensors, TimerOutputs, ProgressMeter import KrylovMethods, IterativeSolvers struct NeoHooke μ::Float64 λ::Float64 end function Ψ(C, mp::NeoHooke) μ = mp.μ λ = mp.λ Ic = tr(C) J = sqrt(det(C)) return μ / 2 * (Ic - 3) - μ * log(J) + λ / 2 * log(J)^2 end function constitutive_driver(C, mp::NeoHooke) # Compute all derivatives in one function call ∂²Ψ∂C², ∂Ψ∂C = Tensors.hessian(y -> Ψ(y, mp), C, :all) S = 2.0 * ∂Ψ∂C ∂S∂C = 4.0 * ∂²Ψ∂C² return S, ∂S∂C end; function assemble_element!(ke, ge, cell, cv, fv, mp, ue) # Reinitialize cell values, and reset output arrays reinit!(cv, cell) fill!(ke, 0.0) fill!(ge, 0.0) b = Vec{3}((0.0, -0.5, 0.0)) # Body force t = Vec{3}((0.1, 0.0, 0.0)) # Traction ndofs = getnbasefunctions(cv) for qp in 1:getnquadpoints(cv) dΩ = getdetJdV(cv, qp) # Compute deformation gradient F and right Cauchy-Green tensor C ∇u = function_gradient(cv, qp, ue) F = one(∇u) + ∇u C = tdot(F) # Compute stress and tangent S, ∂S∂C = constitutive_driver(C, mp) P = F ⋅ S I = one(S) ∂P∂F = otimesu(F, I) ⊡ ∂S∂C ⊡ otimesu(F', I) + otimesu(I, S) # Loop over test functions for i in 1:ndofs # Test function + gradient δui = shape_value(cv, qp, i) ∇δui = shape_gradient(cv, qp, i) # Add contribution to the residual from this test function ge[i] += ( ∇δui ⊡ P - δui ⋅ b ) * dΩ ∇δui∂P∂F = ∇δui ⊡ ∂P∂F # Hoisted computation for j in 1:ndofs ∇δuj = shape_gradient(cv, qp, j) # Add contribution to the tangent ke[i, j] += ( ∇δui∂P∂F ⊡ ∇δuj ) * dΩ end end end # Surface integral for the traction for face in 1:nfaces(cell) if onboundary(cell, face) reinit!(fv, cell, face) for q_point in 1:getnquadpoints(fv) dΓ = getdetJdV(fv, q_point) for i in 1:ndofs δui = shape_value(fv, q_point, i) ge[i] -= (δui ⋅ t) * dΓ end end end end end; function assemble_global!(K, f, dh, cv, fv, mp, u) n = ndofs_per_cell(dh) ke = zeros(n, n) ge = zeros(n) # start_assemble resets K and f assembler = start_assemble(K, f) # Loop over all cells in the grid @timeit "assemble" for cell in CellIterator(dh) global_dofs = celldofs(cell) ue = u[global_dofs] # element dofs @timeit "element assemble" assemble_element!(ke, ge, cell, cv, fv, mp, ue) assemble!(assembler, global_dofs, ge, ke) end end; function solve() reset_timer!() # Generate a grid N = 10 L = 1.0 left = zero(Vec{3}) right = L * ones(Vec{3}) grid = generate_grid(Tetrahedron, (N, N, N), left, right) # Material parameters E = 10.0 ν = 0.3 μ = E / (2(1 + ν)) λ = (E * ν) / ((1 + ν) * (1 - 2ν)) mp = NeoHooke(μ, λ) # Finite element base ip = Lagrange{3, RefTetrahedron, 1}() qr = QuadratureRule{3, RefTetrahedron}(1) qr_face = QuadratureRule{2, RefTetrahedron}(1) cv = CellVectorValues(qr, ip) fv = FaceVectorValues(qr_face, ip) # DofHandler dh = DofHandler(grid) push!(dh, :u, 3) # Add a displacement field close!(dh) function rotation(X, t, θ = deg2rad(60.0)) x, y, z = X return t * Vec{3}( (0.0, L/2 - y + (y-L/2)*cos(θ) - (z-L/2)*sin(θ), L/2 - z + (y-L/2)*sin(θ) + (z-L/2)*cos(θ) )) end dbcs = ConstraintHandler(dh) # Add a homogenous boundary condition on the "clamped" edge dbc = Dirichlet(:u, getfaceset(grid, "right"), (x,t) -> [0.0, 0.0, 0.0], [1, 2, 3]) add!(dbcs, dbc) dbc = Dirichlet(:u, getfaceset(grid, "left"), (x,t) -> rotation(x, t), [1, 2, 3]) add!(dbcs, dbc) close!(dbcs) t = 0.5 JuAFEM.update!(dbcs, t) # Pre-allocation of vectors for the solution and Newton increments _ndofs = ndofs(dh) un = zeros(_ndofs) # previous solution vector u = zeros(_ndofs) Δu = zeros(_ndofs) ΔΔu = zeros(_ndofs) apply!(un, dbcs) # Create sparse matrix and residual vector K = create_sparsity_pattern(dh) g = zeros(_ndofs) # Perform Newton iterations newton_itr = -1 NEWTON_TOL = 1e-8 prog = ProgressMeter.ProgressThresh(NEWTON_TOL, "Solving:") while true; newton_itr += 1 u .= un .+ Δu # Current guess assemble_global!(K, g, dh, cv, fv, mp, u) normg = norm(g[JuAFEM.free_dofs(dbcs)]) apply_zero!(K, g, dbcs) ProgressMeter.update!(prog, normg; showvalues = [(:iter, newton_itr)]) if normg < NEWTON_TOL break elseif newton_itr > 30 error("Reached maximum Newton iterations, aborting") end # Compute increment using cg! from IterativeSolvers.jl @timeit "linear solve (KrylovMethods.cg)" ΔΔu′, flag, relres, iter, resvec = KrylovMethods.cg(K, g; maxIter = 1000) @assert flag == 0 @timeit "linear solve (IterativeSolvers.cg!)" IterativeSolvers.cg!(ΔΔu, K, g; maxiter=1000) apply_zero!(ΔΔu, dbcs) Δu .-= ΔΔu end # Save the solution @timeit "export" begin vtk_grid("hyperelasticity", dh) do vtkfile vtk_point_data(vtkfile, dh, u) end end print_timer(title = "Analysis with$(getncells(grid)) elements", linechars = :ascii)
return u
end

u = solve();

# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl