def calculate_darcy_flow(self):
# equation 5a
du = self.mech_velocity
p = self.pressure
rho = Constant(self.parameters['rho'])
phi = [Constant(phi) for phi in self.parameters['phi']]
F = df.variable(kinematics.DeformationGradient(self.displacement))
J = kinematics.Jacobian(F)
dx = self.geometry.dx
N = self.parameters['N']
# Calculate endo to epi permeability gradient
w = [TrialFunction(self.vector_space) for i in range(N)]
v = [TestFunction(self.vector_space) for i in range(N)]
a = [(1 / phi[i]) * df.inner(F * J * df.inv(F) * w[i], v[i]) * dx
for i in range(N)]
# a = [phi[i]*df.inner((w[i]-du), v[i])*dx for i in range(N)]
# porous dynamics
if self.parameters['mechanics']:
A = [-J * self.K[i] * df.inv(F.T) for i in range(N)]
else:
A = [self.K[i] for i in range(N)]
L = [-df.dot(A[i] * df.grad(p[i]), v[i]) * dx for i in range(N)]
[
df.solve(a[i] == L[i],
self.darcy_flow[i], [],
solver_parameters={
"linear_solver": "cg",
"preconditioner": "sor"
}) for i in range(N)
]
def Coupled_setup(d_, v_, p_, psi_v, psi_p, psi_d, dS2, n, mu_f, body_force,
dx_s, dx_f, mu_s, rho_s, lamda_s, k, mesh_file, rho_f,
**semimp_namespace):
F_coupled_linear = rho_s / k**2 * inner(
d_["n"] - 2 * d_["n-1"] + d_["n-2"], psi_d) * dx_s
F_coupled_linear -= rho_s * inner(body_force, psi_d) * dx_s
F_coupled_linear = rho_f / k * J_(d_["tilde"]) * inner(
v_["n"] - v_["tilde"], psi_v) * dx_f
F_coupled_linear += J_(d_["tilde"]) * inner(
inv(F_(d_["tilde"])).T * grad(p_["n"]), psi_v) * dx_f
F_coupled_linear += inner(
div(J_(d_["tilde"]) * inv(F_(d_["tilde"])) * v_["n"]), psi_p) * dx_f
# Non-linear part
F_coupled_nonlinear = Constant(0.5) * inner(Piola1(d_["n"], lamda_s, mu_s), grad(psi_d))*dx_s \
+ Constant(0.5) * inner(Piola1(d_["n-2"], lamda_s, mu_s), grad(psi_d))*dx_s
# Impose BC weakly
F_coupled_nonlinear -= inner(J_(d_["tilde"]("+"))*sigma_f(p_["n"]("+"), v_["tilde"]("+"),
d_["tilde"]("+"), mu_f) \
* inv(F_(d_["tilde"]("+"))).T*n("+"), psi_d("+"))*dS2(5)
# TODO: This might be better strongly...
F_coupled_linear += J_(d_["tilde"]("+")) * inner(
dot(v_["n"]("+") - -(d_["n"]("-") - d_["n-1"]("-")) / k, n("+")),
psi_p("+")) * dS2(5)
return dict(F_coupled_linear=F_coupled_linear,
F_coupled_nonlinear=F_coupled_nonlinear)
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, dS, mu_f, rho_f, k, v_deg,
**semimp_namespace):
F_fluid_linear = (rho_f / k) * inner(
J_(d_["n"]) * (v_["n"] - v_["n-1"]), psi) * dx_f
F_fluid_linear -= inner(div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]),
gamma) * dx_f
F_fluid_linear += inner(
J_(d_["n"]) * sigma_f_new(v_["n"], p_["n"], d_["n"], mu_f) *
inv(F_(d_["n"])).T, grad(psi)) * dx_f
F_fluid_linear -= inner(dot(J_(d_["n"]("-"))*\
sigma_f_new(v_["n"]("-"), p_["n"]("-"), d_["n"]("-"), mu_f)*inv(F_(d_["n"]("-"))).T, n("+")) , psi("-") )*dS(5)
F_fluid_nonlinear = rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) *
(v_["n"] - ((d_["n"] - d_["n-1"]) / k)), psi) * dx_f
if v_deg == 1:
F_fluid -= beta * h * h * inner(
J_(d_["n"]) * inv(F_(d_["n"]).T) * grad(p), grad(gamma)) * dx_f
print "v_deg", v_deg
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, mu_f, rho_f, k, dt, v_deg,
theta, **semimp_namespace):
J_theta = theta * J_(d_["n"]) + (1 - theta) * J_(d_["n-1"])
d_cn = Constant(theta) * d_["n"] + Constant(1 - theta) * d_["n-1"]
v_cn = Constant(theta) * v_["n"] + Constant(1 - theta) * v_["n-1"]
p_cn = Constant(theta) * p_["n"] + Constant(1 - theta) * p_["n-1"]
F_fluid_linear = rho_f / k * inner(J_theta *
(v_["n"] - v_["n-1"]), psi) * dx_f
F_fluid_nonlinear = rho_f*inner(J_(d_cn)*grad(v_cn)*inv(F_(d_cn))* \
(0.5*(3*v_["n-1"] - v_["n-2"]) - \
0.5/k*((3*d_["n-1"] - d_["n-2"]) - (3*d_["n-2"] - d_["n-3"]))), psi)*dx_f
#(d_["n"]-d_["n-1"])/k), psi)*dx_f
F_fluid_nonlinear += inner(
J_(d_cn) * sigma_f_p(p_cn, d_cn) * inv(F_(d_cn)).T, grad(psi)) * dx_f
F_fluid_nonlinear += inner(
J_(d_cn) * sigma_f_u(v_cn, d_cn, mu_f) * inv(F_(d_cn)).T,
grad(psi)) * dx_f
#OrgF_fluid_nonlinear +=inner(div(J_(d_["n"])*inv(F_(d_["n"]))*v_["n"]), gamma)*dx_f
F_fluid_nonlinear += inner(div(J_(d_cn) * inv(F_(d_cn)) * v_cn),
gamma) * dx_f
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def SecondPiolaStress(self, F, p=None, deviatoric=False):
import dolfin
from pulse import kinematics
material = self.material
I = kinematics.SecondOrderIdentity(F)
f0 = material.f0
f0f0 = dolfin.outer(f0, f0)
I1 = dolfin.variable(material.active.I1(F))
I4f = dolfin.variable(material.active.I4(F))
Fe = material.active.Fe(F)
Fa = material.active.Fa
Ce = Fe.T * Fe
# fe = Fe*f0
# fefe = dolfin.outer(fe, fe)
# Elastic volume ratio
J = dolfin.variable(dolfin.det(Fe))
# Active volume ration
Ja = dolfin.det(Fa)
dim = self.geometry.dim()
Ce_bar = pow(J, -2.0 / float(dim)) * Ce
w1 = material.W_1(I1, diff=1, dim=dim)
w4f = material.W_4(I4f, diff=1)
# Total Stress
S_bar = Ja * (2 * w1 * I + 2 * w4f * f0f0) * dolfin.inv(Fa).T
if material.is_isochoric:
# Deviatoric
Dev_S_bar = S_bar - (1.0 / 3.0) * dolfin.inner(
S_bar, Ce_bar) * dolfin.inv(Ce_bar)
S_mat = J**(-2.0 / 3.0) * Dev_S_bar
else:
S_mat = S_bar
# Volumetric
if p is None or deviatoric:
S_vol = dolfin.zero((dim, dim))
else:
psi_vol = material.compressibility(p, J)
S_vol = J * dolfin.diff(psi_vol, J) * dolfin.inv(Ce)
# Active stress
wactive = material.active.Wactive(F, diff=1)
eta = material.active.eta
S_active = wactive * (f0f0 + eta * (I - f0f0))
S = S_mat + S_vol + S_active
return S
def Fluid_correction_variation(v, p, v_, d_, vp_, dw_, psi, eta, dx_f, \
mu_f, rho_f, k, dt, n, dS, **semimp_namespace):
# Pressure update
F_correction = rho_f/k*J_(d_["tilde"])*inner(v - v_["tilde"], psi)*dx_f
F_correction += J_(d_["tilde"])*inner(inv(F_(d_["tilde"])).T*grad(p), psi)*dx_f
#F_correction -= p*J_(d_["tilde"])*inner(inv(F_(d_["tilde"])).T, grad(psi))*dx_f
F_correction += inner(div(J_(d_["n"])*inv(F_(d_["n"]))*v), eta)*dx_f
return dict(F_correction=F_correction)
def SecondPiolaStress(self, F, p=None, deviatoric=False, *args, **kwargs):
# First Piola Kirchoff
if deviatoric:
p = None
P = self.FirstPiolaStress(F, p)
S = dolfin.inv(F) * P * dolfin.inv(F).T
return S
def permeability_tensor(self, K):
FS = self.geometry.f0.function_space()
TS = TensorFunctionSpace(self.geometry.mesh, 'P', 1)
d = self.geometry.dim()
fibers = Function(FS)
fibers.vector()[:] = self.geometry.f0.vector().get_local()
fibers.vector()[:] /= df.norm(self.geometry.f0)
if self.geometry.s0 is not None:
# normalize vectors
sheet = Function(FS)
sheet.vector()[:] = self.geometry.s0.vector().get_local()
sheet.vector()[:] /= df.norm(self.geometry.s0)
if d == 3:
csheet = Function(FS)
csheet.vector()[:] = self.geometry.n0.vector().get_local()
csheet.vector()[:] /= df.norm(self.geometry.n0)
else:
return Constant(1)
from ufl import diag
factor = 10
if d == 3:
ftensor = df.as_matrix(((fibers[0], sheet[0], csheet[0]),
(fibers[1], sheet[1], csheet[1]),
(fibers[2], sheet[2], csheet[2])))
ktensor = diag(df.as_vector([K, K / factor, K / factor]))
else:
ftensor = df.as_matrix(
((fibers[0], sheet[0]), (fibers[1], sheet[1])))
ktensor = diag(df.as_vector([K, K / factor]))
permeability = df.project(
df.dot(df.dot(ftensor, ktensor), df.inv(ftensor)), TS)
return permeability
def Fluid_correction_variation(v, p, v_, d_, vp_, dw_, psi, eta, dx_f, \
mu_f, rho_f, k, dt, n, dS2, **semimp_namespace):
# Pressure update
F_correction = rho_f/k*J_(d_["tilde"])*inner(v - v_["tilde"], psi)*dx_f
# This gives no pressure gradient thorugh the pipe
#F_correction -= J_(d_["tilde"])*inner(p*inv(F_(d_["tilde"])).T, grad(psi))*dx_f
F_correction += J_(d_["tilde"])*inner(inv(F_(d_["tilde"])).T*grad(p), psi)*dx_f
F_correction += inner(div(J_(d_["tilde"])*inv(F_(d_["tilde"]))*v), eta)*dx_f
#F_correction += J_(d_["tilde"]("+"))*inner(dot(v("+") - \
# (d_["n"]("-") - d_["n-1"]("-"))/k, n("+")), eta("+"))*dS2(5)
return dict(F_correction=F_correction)
def Fe(self, F):
if self.model == ActiveModels.active_stress:
return F
Fa = self.Fa
Fe = F * dolfin.inv(Fa)
return Fe
def __call__(self, u=None):
if u is None:
u = dolfin.Function(
self._displacement_space,
name="Zero displacement from strain observation",
)
u_int = map_displacement(
u, self._displacement_space, self._interpolation_space, self._approx
)
# We need to correct for th reference deformation
F = pulse.kinematics.DeformationGradient(u_int) * dolfin.inv(self._F_ref)
# Compute the strains
if self._tensor == "gradu":
tensor = pulse.kinematics.EngineeringStrain(F, isochoric=self._isochoric)
elif self._tensor == "E":
tensor = pulse.kinematics.GreenLagrangeStrain(F, isochoric=self._isochoric)
elif self._tensor == "almansi":
tensor = pulse.kinematics.EulerAlmansiStrain(F, isochoric=self._isochoric)
form = dolfin.inner(tensor * self.field, self.field)
strain = dolfin_adjoint.Function(self._V, name="Simulated Strain")
dolfin_adjoint.solve(
dolfin.inner(self._trial, self._test) / self._vol * self._dmu
== dolfin.inner(self._test, form) * self._dmu,
strain,
solver_parameters={"linear_solver": "gmres"},
)
return strain
def __call__(self, u=None):
"""
Arguments
---------
u : :py:class:`dolfin.Function`
The displacement
"""
if u is None:
volume_form = (-1.0 / 3.0) * dolfin.dot(self._X, self._N)
else:
u_int = map_displacement(
u, self._displacement_space, self._interpolation_space, self._approx
)
# Compute volume
F = pulse.kinematics.DeformationGradient(u_int)
J = pulse.kinematics.Jacobian(F)
volume_form = (-1.0 / 3.0) * dolfin.dot(
self._X + u_int, J * dolfin.inv(F).T * self._N
)
volume = dolfin_adjoint.Function(self._V, name="Simulated volume")
# Make a project for dolfin-adjoint recording
dolfin_adjoint.solve(
dolfin.inner(self._trial, self._test) / self._endoarea * self._dmu
== dolfin.inner(volume_form, self._test) * self._dmu,
volume,
)
return volume
def Structure_setup(d_, w_, v_, p_, phi, gamma, dS, n, mu_f, \
vp_, dx_s, mu_s, rho_s, lamda_s, k, mesh_file, theta, **semimp_namespace):
delta = 1E10
theta = 1.0
F_solid_linear = rho_s/k*inner(w_["n"] - w_["n-1"], phi)*dx_s \
+ delta*(1./k)*inner(d_["n"] - d_["n-1"], gamma)*dx_s \
- delta*inner(Constant(theta)*w_["n"] \
+ Constant(1 - theta)*w_["n-1"], gamma)*dx_s
F_solid_nonlinear = inner(Piola1(Constant(theta)*d_["n"] \
+ Constant(1 - theta)*d_["n-1"], lamda_s, mu_s), grad(phi))*dx_s
#F_solid_nonlinear -= inner(J_(d_["n"]("+")) * \
#3sigma_f(p_["n"]("+"),v_["tilde"]("+"), d_["n"]("+"), mu_f) \
#*inv(F_(d_["n"]("+"))).T*n("+"), phi("+"))*dS(5)
#org
F_solid_nonlinear -= inner(J_(d_["tilde"]("+")) * \
sigma_f(p_["n"]("+"),v_["tilde"]("+"), d_["tilde"]("+"), mu_f) \
*inv(F_(d_["tilde"]("+"))).T*n("+"), phi("-"))*dS(5)
return dict(F_solid_linear=F_solid_linear,
F_solid_nonlinear=F_solid_nonlinear)
def Structure_setup(d_, w_, v_, p_, phi, gamma, dS2, n, mu_f, body_force, \
vp_, dx_s, mu_s, rho_s, lamda_s, k, mesh_file, theta, **semimp_namespace):
delta = 1E10
#theta = 1.0
theta = 0.5
F_solid_linear = rho_s/k*inner(w_["n"] - w_["n-1"], phi)*dx_s \
+ delta*(1./k)*inner(d_["n"] - d_["n-1"], gamma)*dx_s \
- delta*inner(Constant(theta)*w_["n"] \
+ Constant(1 - theta)*w_["n-1"], gamma)*dx_s
F_solid_linear -= rho_s * inner(body_force, phi) * dx_s
# Non-linear part
F_solid_nonlinear = Constant(theta)*inner(Piola1(d_["n"], lamda_s, mu_s), grad(phi))*dx_s \
+ Constant(1 - theta)*inner(Piola1(d_["n-1"], lamda_s, mu_s), grad(phi))*dx_s
u = vp_["tilde"].sub(0)
p = vp_["n"].sub(1)
F_solid_nonlinear -= inner(J_(d_["tilde"]("+"))*sigma_f(p("+"), u("+"), d_["tilde"]("+"), mu_f) \
*inv(F_(d_["tilde"]("+"))).T*n("+"), phi("+"))*dS2(5)
return dict(F_solid_linear=F_solid_linear,
F_solid_nonlinear=F_solid_nonlinear)
def get_volume(geometry, u=None, chamber="lv"):
if "ENDO" in geometry.markers:
lv_endo_marker = geometry.markers["ENDO"]
rv_endo_marker = None
else:
lv_endo_marker = geometry.markers["ENDO_LV"]
rv_endo_marker = geometry.markers["ENDO_RV"]
marker = lv_endo_marker if chamber == "lv" else rv_endo_marker
if marker is None:
return None
if hasattr(marker, "__len__"):
marker = marker[0]
ds = df.Measure(
"exterior_facet", subdomain_data=geometry.ffun, domain=geometry.mesh
)(marker)
X = df.SpatialCoordinate(geometry.mesh)
N = df.FacetNormal(geometry.mesh)
if u is None:
vol = df.assemble((-1.0 / 3.0) * df.dot(X, N) * ds)
else:
F = df.grad(u) + df.Identity(3)
J = df.det(F)
vol = df.assemble((-1.0 / 3.0) * df.dot(X + u, J * df.inv(F).T * N) * ds)
return vol
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, mu_f, rho_f, k, dt, v_deg,
theta, **semimp_namespace):
#J_theta = theta*J_(d_["n"]) + (1 - theta)*J_(d_["n-1"])
#F_fluid_linear = rho_f/k*inner(J_theta*(v_["n"] - v_["n-1"]), psi)*dx_f
#F_fluid_nonlinear = rho_f/k*inner(J_theta*(v_["n"] - v_["n-1"]), psi)*dx_f
J_theta = theta * J_(d_["n"]) + (1 - theta) * J_(d_["n-1"])
F_fluid_nonlinear = rho_f / k * inner(
J_theta * (v_["n"]) - theta * J_(d_["n"]) * v_["n-1"], psi) * dx_f
F_fluid_linear = rho_f / k * inner(
Constant(1 - theta) * J_(d_["n-1"]) * -(v_["n-1"]), psi) * dx_f
F_fluid_nonlinear = Constant(theta) * rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) * v_["n"], psi) * dx_f
F_fluid_nonlinear += inner(
J_(d_["n"]) * sigma_f_p(p_["n"], d_["n"]) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_nonlinear += Constant(theta) * inner(
J_(d_["n"]) * sigma_f_u(v_["n"], d_["n"], mu_f) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_linear += Constant(1 - theta) * inner(
J_(d_["n-1"]) * sigma_f_u(v_["n-1"], d_["n-1"], mu_f) *
inv(F_(d_["n-1"])).T, grad(psi)) * dx_f
F_fluid_nonlinear += inner(div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]),
gamma) * dx_f
F_fluid_linear += Constant(1 - theta) * rho_f * inner(
J_(d_["n-1"]) * grad(v_["n-1"]) * inv(F_(d_["n-1"])) * v_["n-1"],
psi) * dx_f
F_fluid_nonlinear -= rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) *
((d_["n"] - d_["n-1"]) / k), psi) * dx_f
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def Fluid_correction_variation(v_f, p_f, v_, p_, d_, dvp_, psi, gamma, dx_f, \
mu_f, rho_f, k, dt, n, dS, **semimp_namespace):
F_correction = rho_f/k*J_(d_["tilde"])*inner(v_f - v_["tilde"], psi)*dx_f
F_correction -= p_f*J_(d_["tilde"])*inner(inv(F_(d_["tilde"])).T, grad(psi))*dx_f
F_correction += J_(d_["tilde"])*inner(grad(v_f), inv(F_(d_["tilde"])).T)*gamma*dx_f
#Use newly computed "n" from step 3.2m first time "tilde"
F_correction += J_(d_["tilde"]("-"))*dot(v_f("-") - \
(d_["n"]("-")-d_["n-1"]("-"))/k, n("-"))*gamma("-")*dS(5)
#F_correction += J_(d_["tilde"]("+"))*dot(v_f("+") - \
#(d_["n"]("-")-d_["n-1"]("-"))/k, n("+"))*gamma("+")*dS(5)
#F_correction += dot(v_f("+") - \
#(d_["n"]("-")-d_["n-1"]("-"))/k, n("+"))*gamma("+")*dS(5)
return dict(F_correction=F_correction)
def Fluid_tentative_variation(v_tent, v_, d_, beta, dx_f, mu_f, rho_f, k, dS2,
**semimp_namespace):
F_tentative = rho_f / k * J_(d_["tilde"]) * inner(v_tent - v_["n-1"],
beta) * dx_f
F_tentative += rho_f*inner(J_(d_["tilde"])*grad(v_tent)*inv(F_(d_["tilde"])) \
* (v_["tilde-1"] - (d_["tilde"] - d_["n-1"]) / k), beta)*dx_f
F_tentative += 2*mu_f*J_(d_["tilde"])*inner(eps(d_["tilde"], v_tent), \
eps(d_["tilde"], beta))*dx_f
F_tentative -= inner(Constant((0, 0)), beta) * dx_f
return dict(F_tentative=F_tentative)
def fluid_setup(v_, p_, d_, psi, gamma, dx_f, mu_f, rho_f, k, theta,
**namespace):
"""
ALE formulation (theta-scheme) of the incompressible Navier-Stokes flow problem:
du/dt + u * grad(u - w) = grad(p) + nu * div(grad(u))
div(u) = 0
"""
theta0 = Constant(theta)
theta1 = Constant(1 - theta)
# Note that we here split the equation into a linear and nonlinear part for faster
# computation of the Jacobian matrix.
# Temporal derivative
F_fluid_nonlinear = rho_f / k * inner(
J_(d_["n"]) * theta0 * (v_["n"] - v_["n-1"]), psi) * dx_f
F_fluid_linear = rho_f / k * inner(
J_(d_["n-1"]) * theta1 * (v_["n"] - v_["n-1"]), psi) * dx_f
# Convection
F_fluid_nonlinear += theta0 * rho_f * inner(
grad(v_["n"]) * inv(F_(d_["n"])) * J_(d_["n"]) * v_["n"], psi) * dx_f
F_fluid_linear += theta1 * rho_f * inner(
grad(v_["n-1"]) * inv(F_(d_["n-1"])) * J_(d_["n-1"]) * v_["n-1"],
psi) * dx_f
# Stress from pressure
F_fluid_nonlinear += inner(
J_(d_["n"]) * sigma_f_p(p_["n"], d_["n"]) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
# Stress from velocity
F_fluid_nonlinear += theta0 * inner(
J_(d_["n"]) * sigma_f_u(v_["n"], d_["n"], mu_f) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_linear += theta1 * inner(
J_(d_["n-1"]) * sigma_f_u(v_["n-1"], d_["n-1"], mu_f) *
inv(F_(d_["n-1"])).T, grad(psi)) * dx_f
# Divergence free term
F_fluid_nonlinear += inner(div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]),
gamma) * dx_f
# ALE term
F_fluid_nonlinear -= rho_f / k * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) *
(d_["n"] - d_["n-1"]), psi) * dx_f
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def fluid_setup(d, v, p, v_, p_, d_, n, psi, gamma, dx_f, ds, mu_f, rho_f, k,
dt, v_deg, theta, **semimp_namespace):
#First Order
"""
dvdt = 1./k*(d - v_["n-1"])
dudt = 1./(2*k)*(d_["n-1"] - d_["n-3"])
J_tilde = J_(d_["n-1"])
F_tilde = F_(d_["n-1"])
d_tilde = d_["n-1"]
"""
#Second Order
dvdt = 1. / (2 * k) * (3. * v - 4. * v_["n-1"] + v_["n-2"])
dudt = 1. / (2 * k) * (3. * d_["n-1"] - 4. * d_["n-2"] + d_["n-3"])
J_tilde = 2. * J_(d_["n-1"]) - J_(d_["n-2"])
F_tilde = 2. * F_(d_["n-1"]) - F_(d_["n-2"])
d_tilde = 2. * d_["n-1"] - d_["n-2"]
F_fluid = rho_f * inner(J_tilde * dvdt, psi) * dx_f
F_fluid += rho_f * inner(
J_tilde * grad(v) * inv(F_tilde) * (v_["n-1"] - dudt), psi) * dx_f
F_fluid += J_tilde * inner(2 * mu_f * D_U(d_tilde, v), D_U(d_tilde,
psi)) * dx_f
F_fluid -= inner(J_tilde * p * inv(F_tilde).T, grad(psi)) * dx_f
#F_fluid += inner(div(J_tilde*inv(F_tilde)*v), gamma)*dx_f #Check 2.13 paper
F_fluid += inner(J_tilde * grad(v),
inv(F_tilde).T * gamma) * dx_f #Check 2.13 paper
#Not a must
#djdt = 1./(2*k)*(3.*J_(d_["n-1"]) - 4.*J_(d_["n-2"]) + J_(d_["n-3"]) )
#F_fluid += rho_f/2.*inner(djdt*v, psi)*dx_f
#F_fluid += rho_f/2.*inner(div(J_(d_["n-1"])*inv(F_tilde)\
# *(v_["n-1"] - dudt))*v, psi)*dx_f
return dict(F_fluid=F_fluid)
def get_cavity_volume_form(mesh, u=None, xshift=0.0):
from . import kinematics
shift = Constant((xshift, 0.0, 0.0))
X = dolfin.SpatialCoordinate(mesh) - shift
N = dolfin.FacetNormal(mesh)
if u is None:
vol_form = (-1.0 / 3.0) * dolfin.dot(X, N)
else:
F = kinematics.DeformationGradient(u)
J = kinematics.Jacobian(F)
vol_form = (-1.0 / 3.0) * dolfin.dot(X + u, J * dolfin.inv(F).T * N)
return vol_form
def Fluid_tentative_variation(v_tent, v_, p_, d_, dw_, vp_, v, \
beta, dx_f, mu_f, rho_f, k, dt, dS2, **semimp_namespace):
#Reuse of TrialFunction w, TestFunction psi
#used in extrapolation assuming same degree
F_tentative = rho_f/k*J_(d_["tilde"])*inner(v_tent - v_["n-1"], beta)*dx_f
F_tentative += rho_f*inner(J_(d_["tilde"])*grad(v_tent)*inv(F_(d_["tilde"])) \
* (v_["tilde-1"] - 1./k*(d_["tilde"] - d_["n-1"])), beta)*dx_f
F_tentative += J_(d_["tilde"])*inner(2*mu_f*eps(d_["tilde"], v_tent), eps(d_["tilde"], beta))*dx_f
F_tentative -= inner(Constant((0, 0)), beta)*dx_f
return dict(F_tentative=F_tentative)
def FirstPiolaStress(self, F, p=None, *args, **kwargs):
F = dolfin.variable(F)
# First Piola Kirchoff
psi_iso = self.strain_energy(F)
P = dolfin.diff(psi_iso, F)
if p is not None:
J = dolfin.variable(kinematics.Jacobian(F))
psi_vol = self.compressibility(p, J)
# PiolaTransform
P_vol = J * dolfin.diff(psi_vol, J) * dolfin.inv(F).T
P += P_vol
return P
def extrapolate_setup(F_fluid_linear, extype, mesh_file, d_, phi, gamma, dx_f,
**semimp_namespace):
def F_(U):
return Identity(len(U)) + grad(U)
def J_(U):
return det(F_(U))
def eps(U):
return 0.5 * (grad(U) * inv(F_(U)) + inv(F_(U)).T * grad(U).T)
def STVK(U, alfa_mu, alfa_lam):
return alfa_lam * tr(eps(U)) * Identity(
len(U)) + 2.0 * alfa_mu * eps(U)
#return F_(U)*(alfa_lam*tr(eps(U))*Identity(len(U)) + 2.0*alfa_mu*eps(U))
alfa = 1.0 # holder value if linear is chosen
if extype == "det":
#alfa = inv(J_(d_["n"]))
alfa = 1. / (J_(d_["n"]))
if extype == "smallconst":
alfa = 0.01 * (mesh_file.hmin())**2
if extype == "const":
alfa = 1.0
F_extrapolate = alfa * inner(grad(d_["n"]), grad(phi)) * dx_f
if extype == "linear":
hmin = mesh_file.hmin()
#E_y = 1./(J_(d_["n"]))
#nu = -0.2 #(-1, 0.5)
E_y = 1. / CellVolume(mesh_file)
nu = 0.25
alfa_lam = nu * E_y / ((1. + nu) * (1. - 2. * nu))
alfa_mu = E_y / (2. * (1. + nu))
#alfa_lam = hmin*hmin ; alfa_mu = hmin*hmin
F_extrapolate = inner(
J_(d_["n"]) * STVK(d_["n"], alfa_mu, alfa_lam) *
inv(F_(d_["n"])).T, grad(phi)) * dx_f
#F_extrapolate = inner(STVK(d_["n"],alfa_mu,alfa_lam) , grad(phi))*dx_f
F_fluid_linear += F_extrapolate
return dict(F_fluid_linear=F_fluid_linear)
def extrapolate_setup(F_fluid_linear, mesh, d_, phi, gamma, dx_f, **namespace):
"""
Elastic lifting operator solving the equation of linear elasticity.
div(sigma(d)) = 0 in the fluid domain
d = 0 on the fluid boundaries other than FSI interface
d = solid_d on the FSI interface
"""
E_y = 1.0 / CellVolume(mesh)
nu = 0.25
alpha_lam = nu * E_y / ((1.0 + nu) * (1.0 - 2.0 * nu))
alpha_mu = E_y / (2.0 * (1.0 + nu))
F_extrapolate = inner(
J_(d_["n"]) * S_linear(d_["n"], alpha_mu, alpha_lam) *
inv(F_(d_["n"])).T, grad(phi)) * dx_f
F_fluid_linear += F_extrapolate
return dict(F_fluid_linear=F_fluid_linear)
def initialize_with_field(self, u):
super().initialize_with_field(u)
d = self.deformation_measures.d
F = self.deformation_measures.F
J = self.deformation_measures.J
I1 = self.deformation_measures.I1
# I2 = self.deformation_measures.I2
# I3 = self.deformation_measures.I3
for m in self.material_parameters:
E = m.get('E', None)
nu = m.get('nu', None)
mu = m.get('mu', None)
lm = m.get('lm', None)
if mu is None:
if E is None or nu is None:
raise RuntimeError(
'Material model requires parameter "mu"; '
'otherwise, require parameters "E" and "nu".')
mu = E / (2 * (1 + nu))
if lm is None:
if E is None or nu is None:
raise RuntimeError(
'Material model requires parameter "lm"; '
'otherwise, require parameters "E" and "nu".')
lm = E * nu / ((1 + nu) * (1 - 2 * nu))
psi = (mu / 2) * (I1 - d - 2 * ln(J)) + (lm / 2) * ln(J)**2
pk1 = diff(psi, F)
pk2 = dot(inv(F), pk1)
self.psi.append(psi)
self.pk1.append(pk1)
self.pk2.append(pk2)
def Fluid_tentative_variation(v_, p_, d_, dvp_, w, w_f, v_tilde_n1, \
psi, beta, gamma, dx_f, mu_f, rho_f, k, dt, **semimp_namespace):
d_tent = dvp_["tilde"].sub(0, deepcopy=True)
d_n = dvp_["tilde"].sub(0, deepcopy=True)
d_n1 = dvp_["n-1"].sub(0, deepcopy=True)
v_n1 = dvp_["n-1"].sub(1, deepcopy=True)
#Reuse of TrialFunction w, TestFunction beta
#used in extrapolation assuming same degree
F_tentative = rho_f/k*J_(d_tent)*inner(w - v_n1, beta)*dx_f
F_tentative += rho_f*inner(J_(d_tent)*grad(w)*inv(F_(d_tent)) \
* (v_tilde_n1 - 1./k*(d_n - d_n1)), beta)*dx_f
F_tentative += J_(d_tent)*inner(mu_f*D_U(d_tent, w), D_U(d_tent, beta))*dx_f
F_tentative -= inner(Constant((0, 0)), beta)*dx_f
return dict(F_tentative=F_tentative)
def eps(U):
return 0.5 * (grad(U) * inv(F_(U)) + inv(F_(U)).T * grad(U).T)
def sigma_f(p, u, d, mu_f):
return -p*Identity(len(u)) +\
mu_f*(grad(u)*inv(F_(d)) + inv(F_(d)).T*grad(u).T)
def main(dt,
tEnd,
length=10.0,
height=5.0,
numElementsFilm=2.0,
reGen=True,
ratLame=5.0,
ratFilmSub=100.0):
#fileDir = ("results-dt-%.2f-tEnd-%.0f-L-%.1f-H-%.1f-ratioLame-%.0f-ratioFilm-%.0f" %
# (dt, tEnd, length, height, ratLame, ratFilmSub))
fileDir = "results-1"
# create geometry
rectDomain = Geometry(length=length, height=height, filmHeight=0.05)
rectDomain.read_mesh(numElementsFilm, reGen)
# define periodic boundary conditions
periodicBC = PeriodicBoundary(rectDomain)
# specify physical parameters
physParams = Physical_Params(lmbdaSub=ratLame, muSub=1.0, ratFilmSub=100.0)
physParams.create_Lame(rectDomain)
# create discrete function space
element = create_vector_element(rectDomain, order=1)
W = dl.FunctionSpace(rectDomain.mesh,
element,
constrained_domain=periodicBC)
# define boundaries
def left(x, on_boundary):
return dl.near(x[0], 0.) and on_boundary
def right(x, on_boundary):
return dl.near(x[0], rectDomain.length) and on_boundary
def bottom(x, on_boundary):
return dl.near(x[1], 0.0) and on_boundary
def top(x, on_boundary):
return dl.near(x[1], rectDomain.height) and on_boundary
def corner(x, on_boundary):
return dl.near(x[0], 0.0) and dl.near(x[1], 0.0)
# define fixed boundary
bcs = [
dl.DirichletBC(W.sub(0), dl.Constant(0), left),
dl.DirichletBC(W.sub(0), dl.Constant(0), right),
dl.DirichletBC(W.sub(1), dl.Constant(0), bottom),
dl.DirichletBC(W.sub(0), dl.Constant(0), corner, method="pointwise")
]
bcs = bcs[-2:]
# the variable to solve for
w = dl.Function(W, name="Variables at current step")
# test and trial function
dw = dl.TrialFunction(W)
w_ = dl.TestFunction(W)
# dealing with physics of growth
growthFactor = create_growthFactor(rectDomain, filmGrowth=1, subGrowth=0)
Fg = create_Fg(growthFactor, "uniaxial")
# kinematics
I = dl.Identity(2)
F = I + dl.grad(w)
Fe = F * dl.inv(Fg)
# write the variational form from potential energy
psi = cal_neoHookean(Fe, physParams)
Energy = psi * dl.dx
Residual = dl.derivative(Energy, w, w_)
Jacobian = dl.derivative(Residual, w, dw)
problem = BucklingProblem(w, Energy, Residual, Jacobian)
wLowerBound, wUpperBound = create_bounds(wMin=[-0.5, -0.5],
wMax=[0.5, 0.5],
functionSpace=W,
boundaryCondtions=bcs)
# Create the PETScTAOSolver
solver = dl.PETScTAOSolver()
TAOSolverParameters = {
"method": "tron",
"maximum_iterations": 1000,
"monitor_convergence": True
}
solver.parameters.update(TAOSolverParameters)
#solver.parameters["report"] = False
#solver.parameters["linear_solver"] = "umfpack"
#solver.parameters["line_search"] = "gpcg"
#solver.parameters["preconditioner"] = "ml_amg"
growthRate = 0.1
growthEnd = growthRate * tEnd
gF = 0.0
outDisp = dl.File(fileDir + "/displacement.pvd")
outDisp << (w, 0.0)
iCounter = 0
wArray = w.compute_vertex_values()
while gF <= growthEnd - tol:
gF += growthRate * dt
iCounter += 1
growthFactor.gF = gF
w.vector()[:] += 2e-04 * np.random.uniform(-1, 1,
w.vector().local_size())
nIters, converged = solver.solve(problem, w.vector(),
wLowerBound.vector(),
wUpperBound.vector())
wArray[:] = w.compute_vertex_values()
np.save(fileDir + "/w-{}.npy".format(iCounter), wArray)
print("--- growth = %.4f, niters = %d, dt = %.5f -----" %
(1 + gF, nIters, dt))
outDisp << (w, gF)