def S(self, u_, p_, ivar=None, tang=False):
C_ = variable(self.kin.C(u_))
stress = constantvalue.zero((3, 3))
# volumetric (kinematic) growth
if self.mat_growth:
theta_ = ivar["theta"]
# material has to be evaluated with C_e only, however total S has
# to be computed by differentiating w.r.t. C (S = 2*dPsi/dC)
self.mat = materiallaw(self.C_e(C_, theta_), self.I)
else:
self.mat = materiallaw(C_, self.I)
m = 0
for matlaw in self.matmodels:
stress += self.add_stress_mat(matlaw, self.matparams[m], ivar, C_)
m += 1
# add remodeled material
if self.mat_growth and self.mat_remodel:
self.stress_base = stress
self.stress_remod = constantvalue.zero((3, 3))
m = 0
for matlaw in self.matmodels_remod:
self.stress_remod += self.add_stress_mat(
matlaw, self.matparams_remod[m], ivar, C_)
m += 1
# update the stress expression: S = (1-phi(theta)) * S_base + phi(theta) * S_remod
stress = (
1. - self.phi_remod(theta_)
) * self.stress_base + self.phi_remod(theta_) * self.stress_remod
# if we have p (hydr. pressure) as variable in a 2-field functional
if self.incompr_2field:
if self.mat_growth:
# TeX: S_{\mathrm{vol}} = -2 \frac{\partial[p(J^{\mathrm{e}}-1)]}{\partial \boldsymbol{C}}
stress += -2. * diff(
p_ * (sqrt(det(self.C_e(C_, theta_))) - 1.), C_)
else:
# TeX: S_{\mathrm{vol}} = -2 \frac{\partial[p(J-1)]}{\partial \boldsymbol{C}} = -Jp\boldsymbol{C}^{-1}
stress += -2. * diff(p_ * (sqrt(det(C_)) - 1.), C_)
if tang:
return 2. * diff(stress, C_)
else:
return stress
def stress(self, strain, alpha):
# Differentiate the elastic energy w.r.t. the strain tensor
eps_ = ufl.variable(strain)
# Derivative of energy w.r.t. the strain tensor to obtain the stress
# tensor
sigma = ufl.diff(self.elastic_energy_density_strain(eps_, alpha), eps_)
return sigma
def hyperelasticity(domain, q, p, nf=0):
# Based on https://github.com/firedrakeproject/firedrake-bench/blob/experiments/forms/firedrake_forms.py
V = ufl.FunctionSpace(domain, ufl.VectorElement('P', domain.ufl_cell(), q))
P = ufl.FunctionSpace(domain, ufl.VectorElement('P', domain.ufl_cell(), p))
v = ufl.TestFunction(V)
du = ufl.TrialFunction(V) # Incremental displacement
u = ufl.Coefficient(V) # Displacement from previous iteration
B = ufl.Coefficient(V) # Body force per unit mass
# Kinematics
I = ufl.Identity(domain.ufl_cell().topological_dimension())
F = I + ufl.grad(u) # Deformation gradient
C = F.T*F # Right Cauchy-Green tensor
E = (C - I)/2 # Euler-Lagrange strain tensor
E = ufl.variable(E)
# Material constants
mu = ufl.Constant(domain) # Lame's constants
lmbda = ufl.Constant(domain)
# Strain energy function (material model)
psi = lmbda/2*(ufl.tr(E)**2) + mu*ufl.tr(E*E)
S = ufl.diff(psi, E) # Second Piola-Kirchhoff stress tensor
PK = F*S # First Piola-Kirchoff stress tensor
# Variational problem
it = ufl.inner(PK, ufl.grad(v)) - ufl.inner(B, v)
f = [ufl.Coefficient(P) for _ in range(nf)]
return ufl.derivative(reduce(ufl.inner,
list(map(ufl.div, f)) + [it])*ufl.dx, u, du)
def s(e):
"""
Stress as function of strain from strain energy function
"""
e = ufl.variable(e)
W = lamé_μ * e * e + lamé_λ / 2 * e**2 # Saint-Venant Kirchhoff
s = ufl.diff(W, e)
return s
def S(E):
"""
Stress as function of strain from strain energy function
"""
E = ufl.variable(E)
W = lamé_μ * ufl.inner(E, E) + lamé_λ / 2 * ufl.tr(E)**2 # Saint-Venant Kirchhoff
S = ufl.diff(W, E)
return S
def eigenstate_legacy(A):
"""Eigenvalues and eigenprojectors of the 3x3 (real-valued) tensor A.
Provides the spectral decomposition A = sum_{a=0}^{2} λ_a * E_a
with eigenvalues λ_a and their associated eigenprojectors E_a = n_a^R x n_a^L
ordered by magnitude.
The eigenprojectors of eigenvalues with multiplicity n are returned as 1/n-fold projector.
Note: Tensor A must not have complex eigenvalues!
"""
if ufl.shape(A) != (3, 3):
raise RuntimeError(
f"Tensor A of shape {ufl.shape(A)} != (3, 3) is not supported!")
#
eps = 1.0e-10
#
A = ufl.variable(A)
#
# --- determine eigenvalues λ0, λ1, λ2
#
# additively decompose: A = tr(A) / 3 * I + dev(A) = q * I + B
q = ufl.tr(A) / 3
B = A - q * ufl.Identity(3)
# observe: det(λI - A) = 0 with shift λ = q + ω --> det(ωI - B) = 0 = ω**3 - j * ω - b
j = ufl.tr(
B * B
) / 2 # == -I2(B) for trace-free B, j < 0 indicates A has complex eigenvalues
b = ufl.tr(B * B * B) / 3 # == I3(B) for trace-free B
# solve: 0 = ω**3 - j * ω - b by substitution ω = p * cos(phi)
# 0 = p**3 * cos**3(phi) - j * p * cos(phi) - b | * 4 / p**3
# 0 = 4 * cos**3(phi) - 3 * cos(phi) - 4 * b / p**3 | --> p := sqrt(j * 4 / 3)
# 0 = cos(3 * phi) - 4 * b / p**3
# 0 = cos(3 * phi) - r with -1 <= r <= +1
# phi_k = [acos(r) + (k + 1) * 2 * pi] / 3 for k = 0, 1, 2
p = 2 / ufl.sqrt(3) * ufl.sqrt(j + eps**2) # eps: MMM
r = 4 * b / p**3
r = ufl.Max(ufl.Min(r, +1 - eps), -1 + eps) # eps: LMM, MMH
phi = ufl.acos(r) / 3
# sorted eigenvalues: λ0 <= λ1 <= λ2
λ0 = q + p * ufl.cos(phi + 2 / 3 * ufl.pi) # low
λ1 = q + p * ufl.cos(phi + 4 / 3 * ufl.pi) # middle
λ2 = q + p * ufl.cos(phi) # high
#
# --- determine eigenprojectors E0, E1, E2
#
E0 = ufl.diff(λ0, A).T
E1 = ufl.diff(λ1, A).T
E2 = ufl.diff(λ2, A).T
#
return [λ0, λ1, λ2], [E0, E1, E2]
def eigenstate(A):
"""Eigenvalues and eigenprojectors of the 3x3 (real-valued) tensor A.
Provides the spectral decomposition A = sum_{a=0}^{2} λ_a * E_a
with (ordered) eigenvalues λ_a and their associated eigenprojectors E_a = n_a^R x n_a^L.
Note: Tensor A must not have complex eigenvalues!
"""
if ufl.shape(A) != (3, 3):
raise RuntimeError(
f"Tensor A of shape {ufl.shape(A)} != (3, 3) is not supported!")
#
eps = 3.0e-16 # slightly above 2**-(53 - 1), see https://en.wikipedia.org/wiki/IEEE_754
#
A = ufl.variable(A)
#
# --- determine eigenvalues λ0, λ1, λ2
#
I1, I2, I3 = invariants_principal(A)
dq = 2 * I1**3 - 9 * I1 * I2 + 27 * I3
#
Δx = [
A[0, 1] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 1],
A[0, 1]**2 * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 1] +
A[0, 1] * A[0, 2] * A[2, 2] - A[0, 2]**2 * A[2, 1],
A[0, 0] * A[0, 1] * A[2, 1] - A[0, 1]**2 * A[2, 0] -
A[0, 1] * A[2, 1] * A[2, 2] + A[0, 2] * A[2, 1]**2,
A[0, 0] * A[0, 2] * A[1, 2] + A[0, 1] * A[1, 2]**2 -
A[0, 2]**2 * A[1, 0] - A[0, 2] * A[1, 1] * A[1, 2],
A[0, 0] * A[0, 1] * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 0] -
A[0, 1] * A[1, 2] * A[2, 2] +
A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501
A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1] * A[0, 2] * A[2, 0] +
A[0, 1] * A[1, 2] * A[2, 1] -
A[0, 2] * A[1, 1] * A[2, 1], # noqa: E501
A[0, 1] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0] * A[1, 1] +
A[0, 2] * A[1, 0] * A[2, 2] -
A[0, 2] * A[1, 2] * A[2, 0], # noqa: E501
A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] -
A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] +
A[0, 1] * A[1, 0] * A[1, 2] + A[0, 2] * A[1, 0] * A[2, 2] +
A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501
A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] -
A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] +
A[0, 2] * A[1, 0] * A[1, 1] + A[0, 2] * A[1, 2] * A[2, 0] +
A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] -
A[0, 1]**2 * A[1, 0] + A[0, 1] * A[0, 2] * A[2, 0] -
A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[2, 2]**2 +
A[0, 2] * A[1, 1] * A[2, 1] -
A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] +
A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1]**2 * A[1, 0] -
A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[1, 2] * A[2, 1] +
A[0, 1] * A[2, 2]**2 - A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 1] +
A[0, 0] * A[0, 2] * A[2, 2] - A[0, 1] * A[1, 1] * A[1, 2] -
A[0, 2]**2 * A[2, 0] + A[0, 2] * A[1, 1]**2 -
A[0, 2] * A[1, 1] * A[2, 2] +
A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501
A[0, 0] * A[0, 2] * A[1, 1] - A[0, 0] * A[0, 2] * A[2, 2] -
A[0, 1] * A[0, 2] * A[1, 0] + A[0, 1] * A[1, 1] * A[1, 2] -
A[0, 1] * A[1, 2] * A[2, 2] + A[0, 2]**2 * A[2, 0] -
A[0, 2] * A[1, 1]**2 + A[0, 2] * A[1, 1] * A[2, 2], # noqa: E501
A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] -
A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] -
A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 +
A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] +
A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] -
A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2]
] # noqa: E501
Δy = [
A[0, 2] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 2] * A[2, 0],
A[1, 0]**2 * A[2, 1] - A[1, 0] * A[1, 1] * A[2, 0] +
A[1, 0] * A[2, 0] * A[2, 2] - A[1, 2] * A[2, 0]**2,
A[0, 0] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0]**2 -
A[1, 0] * A[1, 2] * A[2, 2] + A[1, 2]**2 * A[2, 0],
A[0, 0] * A[2, 0] * A[2, 1] - A[0, 1] * A[2, 0]**2 +
A[1, 0] * A[2, 1]**2 - A[1, 1] * A[2, 0] * A[2, 1],
A[0, 0] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 0] * A[2, 0] -
A[1, 0] * A[2, 1] * A[2, 2] +
A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 0] +
A[1, 0] * A[1, 2] * A[2, 1] -
A[1, 1] * A[1, 2] * A[2, 0], # noqa: E501
A[0, 1] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 1] * A[2, 0] +
A[0, 1] * A[2, 0] * A[2, 2] -
A[0, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] -
A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] +
A[0, 1] * A[1, 0] * A[2, 1] + A[0, 1] * A[2, 0] * A[2, 2] +
A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501
A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] -
A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] +
A[0, 1] * A[1, 1] * A[2, 0] + A[0, 2] * A[2, 0] * A[2, 1] +
A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501
A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] -
A[0, 1] * A[1, 0]**2 + A[0, 2] * A[1, 0] * A[2, 0] -
A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[2, 2]**2 +
A[1, 1] * A[1, 2] * A[2, 0] -
A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] +
A[0, 0] * A[1, 2] * A[2, 0] - A[0, 1] * A[1, 0]**2 -
A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[1, 2] * A[2, 1] +
A[1, 0] * A[2, 2]**2 - A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0] * A[1, 0] * A[2, 1] - A[0, 0] * A[1, 1] * A[2, 0] +
A[0, 0] * A[2, 0] * A[2, 2] - A[0, 2] * A[2, 0]**2 -
A[1, 0] * A[1, 1] * A[2, 1] + A[1, 1]**2 * A[2, 0] -
A[1, 1] * A[2, 0] * A[2, 2] +
A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0] * A[1, 1] * A[2, 0] - A[0, 0] * A[2, 0] * A[2, 2] -
A[0, 1] * A[1, 0] * A[2, 0] + A[0, 2] * A[2, 0]**2 +
A[1, 0] * A[1, 1] * A[2, 1] - A[1, 0] * A[2, 1] * A[2, 2] -
A[1, 1]**2 * A[2, 0] + A[1, 1] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] -
A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] -
A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 +
A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] +
A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] -
A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2]
] # noqa: E501
Δd = [9, 6, 6, 6, 8, 8, 8, 2, 2, 2, 2, 2, 2, 1]
Δ = 0
for i in range(len(Δd)):
Δ += Δx[i] * Δd[i] * Δy[i]
Δxp = [
A[1, 0], A[2, 0], A[2, 1], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2],
-A[1, 1] + A[2, 2]
]
Δyp = [
A[0, 1], A[0, 2], A[1, 2], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2],
-A[1, 1] + A[2, 2]
]
Δdp = [6, 6, 6, 1, 1, 1]
dp = 0
for i in range(len(Δdp)):
dp += 1 / 2 * Δxp[i] * Δdp[i] * Δyp[i]
# Avoid dp = 0 and disc = 0, both are known with absolute error of ~eps**2
# Required to avoid sqrt(0) derivatives and negative square roots
dp += eps**2
Δ += eps**2
phi3 = ufl.atan_2(ufl.sqrt(27) * ufl.sqrt(Δ), dq)
# sorted eigenvalues: λ0 <= λ1 <= λ2
λ = [(I1 + 2 * ufl.sqrt(dp) * ufl.cos((phi3 + 2 * ufl.pi * k) / 3)) / 3
for k in range(1, 4)]
#
# --- determine eigenprojectors E0, E1, E2
#
E = [ufl.diff(λk, A).T for λk in λ]
return λ, E
def stress(self, eps, alpha):
eps_ = variable(eps)
sigma = diff(self.elastic_energy_density(eps_, alpha), eps_)
return sigma
# unless otherwise specified. # # Next, we will be needing functions for the boundary source ``B``, the # traction ``T`` and the displacement solution itself ``u``:: # Functions u = Coefficient(element) # Displacement from previous iteration # B = Coefficient(element) # Body force per unit volume # T = Coefficient(element) # Traction force on the boundary # Now, we can define the kinematic quantities involved in the model:: # Kinematics d = len(u) I = Identity(d) # Identity tensor F = variable(I + grad(u)) # Deformation gradient C = F.T*F # Right Cauchy-Green tensor # Invariants of deformation tensors Ic = tr(C) J = det(F) # Before defining the energy density and thus the total potential # energy, it only remains to specify constants for the elasticity # parameters:: # Elasticity parameters E = 10.0 nu = 0.3 mu = E/(2*(1 + nu)) lmbda = E*nu/((1 + nu)*(1 - 2*nu))
# -
# The first line creates an object of type `InitialConditions`. The
# following two lines make `u` and `u0` interpolants of `u_init`
# (since `u` and `u0` are finite element functions, they may not be
# able to represent a given function exactly, but the function can be
# approximated by interpolating it in a finite element space).
#
# ```{index} automatic differentiation
# ```
#
# The chemical potential $df/dc$ is computed using UFL automatic
# differentiation:
# Compute the chemical potential df/dc
c = ufl.variable(c)
f = 100 * c**2 * (1 - c)**2
dfdc = ufl.diff(f, c)
# The first line declares that `c` is a variable that some function can
# be differentiated with respect to. The next line is the function
# $f$ defined in the problem statement, and the third line performs
# the differentiation of `f` with respect to the variable `c`.
#
# It is convenient to introduce an expression for $\mu_{n+\theta}$:
# mu_(n+theta)
mu_mid = (1.0 - theta) * mu0 + theta * mu
# which is then used in the definition of the variational forms:
def dJedC(self, u_, theta_):
C_ = variable(self.kin.C(u_))
Je = sqrt(det(self.C_e(C_, theta_)))
return diff(Je,C_)
# Configuration gradient
I = ufl.Identity(u.geometric_dimension()) # noqa: E741
F = I + ufl.grad(u) # deformation gradient as function of displacement
# Strain measures
E = 1 / 2 * (F.T * F - I) # E = E(F), total Green-Lagrange strain
E_el = E - P # E_el = E(F) - P, elastic strain
# Stress
S = 2 * mu * E_el + la * ufl.tr(
E_el) * I # S = S(E_el), PK2, St.Venant-Kirchhoff
# Wrap variable around expression (for diff)
S, B, h = ufl.variable(S), ufl.variable(B), ufl.variable(h)
# Yield function
f = ufl.sqrt(3) * rJ2(ufl.dev(S) - ufl.dev(B)) - (Sy + h)
# Plastic potential
g = f
# Total differential of yield function
df = + ufl.inner(ufl.diff(f, S), S - S0) \
+ ufl.inner(ufl.diff(f, h), h - h0) \
+ ufl.inner(ufl.diff(f, B), B - B0)
# Derivative of plastic potential wrt stress
dgdS = ufl.diff(g, S)
def assemble_test(cell_batch_size: int):
mesh = dolfin.UnitCubeMesh(MPI.comm_world, 40, 40, 40)
def isochoric(F):
C = F.T*F
I_1 = tr(C)
I4_f = dot(e_f, C*e_f)
I4_s = dot(e_s, C*e_s)
I8_fs = dot(e_f, C*e_s)
def cutoff(x):
return 1.0/(1.0 + ufl.exp(-(x - 1.0)*30.0))
def scaled_exp(a0, a1, argument):
return a0/(2.0*a1)*(ufl.exp(b*argument) - 1)
E_1 = scaled_exp(a, b, I_1 - 3.)
E_f = cutoff(I4_f)*scaled_exp(a_f, b_f, (I4_f - 1.)**2)
E_s = cutoff(I4_s)*scaled_exp(a_s, b_s, (I4_s - 1.)**2)
E_3 = scaled_exp(a_fs, b_fs, I8_fs**2)
E = E_1 + E_f + E_s + E_3
return E
cell = mesh.ufl_cell()
lamda = dolfin.Constant(0.48, cell)
a = dolfin.Constant(1.0, cell)
b = dolfin.Constant(1.0, cell)
a_s = dolfin.Constant(1.0, cell)
b_s = dolfin.Constant(1.0, cell)
a_f = dolfin.Constant(1.0, cell)
b_f = dolfin.Constant(1.0, cell)
a_fs = dolfin.Constant(1.0, cell)
b_fs = dolfin.Constant(1.0, cell)
# For more fun, make these general vector fields rather than
# constants:
e_s = dolfin.Constant([0.0, 1.0, 0.0], cell)
e_f = dolfin.Constant([1.0, 0.0, 0.0], cell)
V = dolfin.FunctionSpace(mesh, ufl.VectorElement("CG", cell, 1))
u = dolfin.Function(V)
du = dolfin.function.argument.TrialFunction(V)
v = dolfin.function.argument.TestFunction(V)
# Misc elasticity related tensors and other quantities
F = grad(u) + ufl.Identity(3)
F = ufl.variable(F)
J = det(F)
Fbar = J**(-1.0/3.0)*F
# Define energy
E_volumetric = lamda*0.5*ln(J)**2
psi = isochoric(Fbar) + E_volumetric
# Find first Piola-Kircchoff tensor
P = ufl.diff(psi, F)
# Define the variational formulation
F = inner(P, grad(v))*dx
# Take the derivative
J = ufl.derivative(F, u, du)
a, L = J, F
if cell_batch_size > 1:
cxx_flags = "-O2 -ftree-vectorize -funroll-loops -march=native -mtune=native"
else:
cxx_flags = "-O2"
assembler = dolfin.fem.assembling.Assembler([[a]], [L], [],
form_compiler_parameters={"cell_batch_size": cell_batch_size,
"enable_cross_cell_gcc_ext": True,
"cpp_optimize_flags": cxx_flags})
t = -time.time()
A, b = assembler.assemble(
mat_type=dolfin.cpp.fem.Assembler.BlockType.monolithic)
t += time.time()
return A, b, t
def holzapfel_ogden(mesh, q, p, nf=0):
# Based on https://gist.github.com/meg-simula/3ab3fd63264c8cf1912b
#
# Original credit note:
# "Original implementation by Gabriel Balaban,
# modified by Marie E. Rognes"
from ufl import (Constant, VectorConstant, Identity, Coefficient,
TestFunction, conditional, det, diff, dot, exp,
grad, inner, tr, variable)
# Define some random parameters
a = Constant(mesh)
b = Constant(mesh)
a_s = Constant(mesh)
b_s = Constant(mesh)
a_f = Constant(mesh)
b_f = Constant(mesh)
a_fs = Constant(mesh)
b_fs = Constant(mesh)
# For more fun, make these general vector fields rather than
# constants:
e_s = VectorConstant(mesh)
e_f = VectorConstant(mesh)
# Define the isochoric energy contribution
def isochoric(C):
I_1 = tr(C)
I4_f = dot(e_f, C*e_f)
I4_s = dot(e_s, C*e_s)
I8_fs = dot(e_s, C*e_f)
def heaviside(x):
return conditional(x < 1, 0, 1)
def scaled_exp(a, b, x):
return a/(2*b)*(exp(b*x) - 1)
E_1 = scaled_exp(a, b, I_1 - 3)
E_f = heaviside(I4_f)*scaled_exp(a_f, b_f, (I4_f - 1)**2)
E_s = heaviside(I4_s)*scaled_exp(a_s, b_s, (I4_s - 1)**2)
E_3 = scaled_exp(a_fs, b_fs, I8_fs**2)
E = E_1 + E_f + E_s + E_3
return E
# Define mesh and function space
V = ufl.FunctionSpace(mesh, ufl.VectorElement("CG", mesh.ufl_cell(), q))
u = Coefficient(V)
v = TestFunction(V)
# Misc elasticity related tensors and other quantities
I = Identity(mesh.ufl_cell().topological_dimension())
F = grad(u) + I
F = variable(F)
J = det(F)
Cbar = J**(-2/3)*F.T*F
# Define energy
Psi = isochoric(Cbar)
# Find first Piola-Kirchhoff tensor
P = diff(Psi, F)
# Define the variational formulation
it = inner(P, grad(v))
P = ufl.FunctionSpace(mesh, ufl.VectorElement('P', mesh.ufl_cell(), p))
f = [ufl.Coefficient(P) for _ in range(nf)]
return ufl.derivative(reduce(ufl.inner,
list(map(ufl.div, f)) + [it])*ufl.dx, u)
))
svals = np.zeros_like(stretchVals)
plt.plot(timeVals, stretchVals)
plt.savefig("stretchesVisco.png")
plt.close()
# stabilization parameters
h = FacetArea(mesh)
h_avg = avg(h)
# new variable name to take derivatives
FF = Identity(3) + grad(u)
CC = FF.T * FF
Fv = variable(FF)
S = diff(freeEnergy(Fv.T * Fv, CCv), Fv) # first PK stress
dl_interp(CC, C)
dl_interp(CC, Cn)
my_identity = grad(SpatialCoordinate(mesh))
dl_interp(my_identity, CCv)
dl_interp(my_identity, Cvn)
dl_interp(my_identity, C_quart)
dl_interp(my_identity, C_thr_quart)
dl_interp(my_identity, C_half)
a_uv = (derivative(freeEnergy(CC, CCv), u, v) * dx +
qvals / h_avg * dot(jump(u), jump(v)) * dS)
jac = derivative(a_uv, u, du)
def dJdC(self, u_):
C_ = variable(self.C(u_))
J = sqrt(det(C_))
return diff(J,C_)
# Create intial conditions and interpolate
u.interpolate(u_init)
# The first line creates an object of type ``InitialConditions``. The
# following two lines make ``u`` and ``u0`` interpolants of ``u_init``
# (since ``u`` and ``u0`` are finite element functions, they may not be
# able to represent a given function exactly, but the function can be
# approximated by interpolating it in a finite element space).
#
# .. index:: automatic differentiation
#
# The chemical potential :math:`df/dc` is computed using automated
# differentiation::
# Compute the chemical potential df/dc
c = variable(c)
f = 100 * c**2 * (1 - c)**2
dfdc = diff(f, c)
# The first line declares that ``c`` is a variable that some function can
# be differentiated with respect to. The next line is the function
# :math:`f` defined in the problem statement, and the third line performs
# the differentiation of ``f`` with respect to the variable ``c``.
#
# It is convenient to introduce an expression for :math:`\mu_{n+\theta}`::
# mu_(n+theta)
mu_mid = (1.0 - theta) * mu0 + theta * mu
# which is then used in the definition of the variational forms::
# External forces T = Coefficient(u_element) p0 = Coefficient(p_element) # Material parameters FIXME rho = Constant(cell) K = Constant(cell) c00 = Constant(cell) c11 = Constant(cell) c22 = Constant(cell) # Deformation gradient I = Identity(d) F = I + grad(u) F = variable(F) Finv = inv(F) J = det(F) # Left Cauchy-Green deformation tensor B = F * F.T I1_B = tr(B) I2_B = (I1_B**2 - tr(B * B)) / 2 I3_B = J**2 # Right Cauchy-Green deformation tensor C = F.T * F I1_C = tr(C) I2_C = (I1_C**2 - tr(C * C)) / 2 I3_C = J**2
