def __init__(self,
mesh: fe.Mesh,
density: fe.Expression,
constitutive_model: ConstitutiveModelBase,
bf: fe.Expression = fe.Expression('0', degree=0)):
self._mesh = mesh
self._density = density
self._constitutive_model = constitutive_model
self.bf = bf
element_v = fe.VectorElement("P", mesh.ufl_cell(), 1)
element_s = fe.FiniteElement("P", mesh.ufl_cell(), 1)
mixed_element = fe.MixedElement([element_v, element_v, element_s])
W = fe.FunctionSpace(mesh, mixed_element)
# Unknowns, values at previous step and test functions
w = fe.Function(W)
self.u, self.v, self.p = fe.split(w)
w0 = fe.Function(W)
self.u0, self.v0, self.p0 = fe.split(w0)
self.a0 = fe.Function(fe.FunctionSpace(mesh, element_v))
self.ut, self.vt, self.pt = fe.TestFunctions(W)
self.F = kin.def_grad(self.u)
self.F0 = kin.def_grad(self.u0)
def setup_element(self):
""" Implement the mixed element per @cite{danaila2014newton}. """
pressure_element = fenics.FiniteElement(
"P", self.mesh.ufl_cell(), self.temperature_element_degree)
velocity_element = fenics.VectorElement(
"P", self.mesh.ufl_cell(), self.temperature_element_degree + 1)
temperature_element = fenics.FiniteElement(
"P", self.mesh.ufl_cell(), self.temperature_element_degree)
self.element = fenics.MixedElement(
[pressure_element, velocity_element, temperature_element])
def MeshDefinition(Dimensions, NumberElements, Type='Lagrange', PolynomDegree=1):
# Mesh
Mesh = fe.BoxMesh(fe.Point(-Dimensions[0]/2, -Dimensions[1]/2, -Dimensions[2]/2), fe.Point(Dimensions[0]/2, Dimensions[1]/2, Dimensions[2]/2), NumberElements, NumberElements, NumberElements)
# Functions spaces
V_ele = fe.VectorElement(Type, Mesh.ufl_cell(), PolynomDegree)
V = fe.VectorFunctionSpace(Mesh, Type, PolynomDegree)
# Finite element functions
du = fe.TrialFunction(V)
v = fe.TestFunction(V)
u = fe.Function(V)
return [Mesh, V, u, du, v]
def make_mixed_fe(cell):
""" Define the mixed finite element.
MixedFunctionSpace used to be available but is now deprecated.
To create the mixed space, I'm using the approach from https://fenicsproject.org/qa/11983/mixedfunctionspace-in-2016-2-0
"""
velocity_degree = pressure_degree + 1
velocity_element = fenics.VectorElement("P", cell, velocity_degree)
pressure_element = fenics.FiniteElement("P", cell, pressure_degree)
temperature_element = fenics.FiniteElement("P", cell, temperature_degree)
mixed_element = fenics.MixedElement(
[velocity_element, pressure_element, temperature_element])
return mixed_element
def __init__(self,
mesh: fe.Mesh,
constitutive_model: ConstitutiveModelBase,
u_order=1,
p_order=0):
# TODO a lot here...
element_v = fe.VectorElement("P", mesh.ufl_cell(), u_order)
element_s = fe.FiniteElement("DG", mesh.ufl_cell(), p_order)
# mixed_element = fe.MixedElement([element_v, element_v, element_s])
mixed_element = fe.MixedElement([element_v, element_s])
self.W = fe.FunctionSpace(mesh, mixed_element)
self.V, self.Q = self.W.split()
self.w = fe.Function(self.W)
self.u, self.p = fe.split(self.w)
w0 = fe.Function(self.W)
self.u0, self.p0 = fe.split(w0)
self.ut, self.pt = fe.TestFunctions(self.W)
self.F = kin.def_grad(self.u)
self.F0 = kin.def_grad(self.u0)
S_iso = constitutive_model.iso_stress(self.u)
mod_p = constitutive_model.p(self.u)
J = fe.det(self.F)
F_inv = fe.inv(self.F)
if mod_p is None:
mod_p = J**2 - 1.
else:
mod_p -= self.p
S = S_iso + J * self.p * F_inv * F_inv.T
self.d_LHS = fe.inner(fe.grad(self.ut), self.F * S) * fe.dx \
+ fe.inner(mod_p, self.pt) * fe.dx
# + fe.inner(mod_p, fe.tr(fe.nabla_grad(self.ut)*fe.inv(self.F))) * fe.dx
self.d_RHS = (fe.inner(fe.Constant((0., 0., 0.)), self.ut) * fe.dx)
from fenics import div, grad, curl, inner, dot, inv, tr
import numpy as np
#import matplotlib.pyplot as plt
# some fenics settings
fe.parameters['form_compiler']['cpp_optimize'] = True
fe.parameters['form_compiler']['optimize'] = True
#fe.parameters["form_compiler"]["quadrature_degree"] = 5
# Create mesh and define function space
n = 20
mesh = fe.UnitCubeMesh(n, n, n)
# Init function spaces
element_3 = fe.VectorElement("P", mesh.ufl_cell(), 1)
element = fe.FiniteElement("P", mesh.ufl_cell(), 2)
# Mixed function space
TH = element_3 * element
V = fe.FunctionSpace(mesh, TH)
# Define Boundaries
left = fe.CompiledSubDomain("near(x[0], side) && on_boundary", side=0.0)
right = fe.CompiledSubDomain("near(x[0], side) && on_boundary", side=1.0)
# Define Dirichlet boundary (x = 0 or x = 1)
u_left = fe.Expression(("0.0", "0.0", "0.0"), element=element_3)
u_right = fe.Expression(("0.0", "0.0", "0.0"), element=element_3)
p_left = fe.Constant(0.)
EPSILON = 1.0e-14
DEG = 2
mesh = fe.Mesh('step.xml')
# Control pannel
MODEL = False # flag to use SA model
b = fe.Expression(('0', '0'), degree=DEG) # forcing
nu = fe.Constant(2e-6)
rho = fe.Constant(1)
RE = 0.01
lmx = 1 # mixing length
dt = 0.1
# Re = 10 / 1e-4 = 1e5
V = fe.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
NU = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
if MODEL: M = fe.MixedElement([V, P, NU])
else: M = fe.MixedElement([V, P])
W = fe.FunctionSpace(mesh, M)
W0 = fe.Function(W)
We = fe.Function(W)
u0, p0 = fe.split(We)
#u0 = fe.Function((W0[0], W0[1]), 'Velocity000023.vtu')
#p0 = fe.Function(W0[2])
v, q = fe.TestFunctions(W)
#u, p = fe.split(W0)
u, p = (fe.as_vector((W0[0], W0[1])), W0[2])
"""Inverse permeability as a function of rho"""
return alphabar + (alphaunderbar - alphabar) * rho * (1 + q) / (rho + q)
N = 20
delta = 1.5 # The aspect ratio of the domain, 1 high and \delta wide
V = (
fenics_adjoint.Constant(1.0 / 3) * delta
) # want the fluid to occupy 1/3 of the domain
mesh = fenics_adjoint.Mesh(
fenics.RectangleMesh(fenics.Point(0.0, 0.0), fenics.Point(delta, 1.0), N, N)
)
A = fenics.FunctionSpace(mesh, "CG", 1) # control function space
U_h = fenics.VectorElement("CG", mesh.ufl_cell(), 2)
P_h = fenics.FiniteElement("CG", mesh.ufl_cell(), 1)
W = fenics.FunctionSpace(mesh, U_h * P_h) # mixed Taylor-Hood function space
# Define the boundary condition on velocity
(x, y) = ufl.SpatialCoordinate(mesh)
l = 1.0 / 6.0 # noqa: E741
gbar = 1.0
cond1 = ufl.And(ufl.gt(y, (1.0 / 4 - l / 2)), ufl.lt(y, (1.0 / 4 + l / 2)))
val1 = gbar * (1 - (2 * (y - 0.25) / l) ** 2)
cond2 = ufl.And(ufl.gt(y, (3.0 / 4 - l / 2)), ufl.lt(y, (3.0 / 4 + l / 2)))
val2 = gbar * (1 - (2 * (y - 0.75) / l) ** 2)
inflow_outflow = ufl.conditional(cond1, val1, ufl.conditional(cond2, val2, 0))
inflow_outflow_bc = fenics_adjoint.project(inflow_outflow, W.sub(0).sub(0).collapse())
solve_templates = (fenics_adjoint.Function(A),)
import fenics as fn
fn.parameters["form_compiler"]["representation"] = "uflacs"
fn.parameters["form_compiler"]["cpp_optimize"] = True
# Create mesh and define function space
size = width, height = 1.0, 0.75
mesh = fn.RectangleMesh(fn.Point(-width / 2, 0.0), fn.Point(width / 2, height),
52, 39) # 52 * 0.75 = 39, elements are square
nn = fn.FacetNormal(mesh)
# Defintion of function spaces
Vh = fn.VectorElement("CG", mesh.ufl_cell(), 2)
Zh = fn.FiniteElement("CG", mesh.ufl_cell(), 1)
Qh = fn.FiniteElement("CG", mesh.ufl_cell(), 2)
# spaces for displacement and total pressure should be compatible
# whereas the space for fluid pressure can be "anything". In particular the one for total pressure
Hh = fn.FunctionSpace(mesh, fn.MixedElement([Vh, Zh, Qh]))
(u, phi, p) = fn.TrialFunctions(Hh)
(v, psi, q) = fn.TestFunctions(Hh)
fileU = fn.File(mesh.mpi_comm(), "Output/2_5_1_Footing_wall_removed/u.pvd")
filePHI = fn.File(mesh.mpi_comm(), "Output/2_5_1_Footing_wall_removed/phi.pvd")
fileP = fn.File(mesh.mpi_comm(), "Output/2_5_1_Footing_wall_removed/p.pvd")
# ******** Model constants ********** #
E = 3.0e4
nu = 0.4995
def navierStokes(projectId, mesh, faceSets, boundarySets, config):
log("Navier Stokes Analysis has started")
# this is the default directory, when user request for download all files in this directory is being compressed and sent to the user
resultDir = "./Results/"
if len(config["steps"]) > 1:
return "more than 1 step is not supported yet"
# config is a dictionary containing all the user inputs for solver configurations
t_init = 0.0
t_final = float(config['steps'][0]["finalTime"])
t_num = int(config['steps'][0]["iterationNo"])
dt = ((t_final - t_init) / t_num)
t = t_init
#
# Viscosity coefficient.
#
nu = float(config['materials'][0]["viscosity"])
rho = float(config['materials'][0]["density"])
#
# Declare Finite Element Spaces
# do not use triangle directly
P2 = fn.VectorElement("P", mesh.ufl_cell(), 2)
P1 = fn.FiniteElement("P", mesh.ufl_cell(), 1)
TH = fn.MixedElement([P2, P1])
V = fn.VectorFunctionSpace(mesh, "P", 2)
Q = fn.FunctionSpace(mesh, "P", 1)
W = fn.FunctionSpace(mesh, TH)
#
# Declare Finite Element Functions
#
(u, p) = fn.TrialFunctions(W)
(v, q) = fn.TestFunctions(W)
w = fn.Function(W)
u0 = fn.Function(V)
p0 = fn.Function(Q)
#
# Macros needed for weak formulation.
#
def contract(u, v):
return fn.inner(fn.nabla_grad(u), fn.nabla_grad(v))
def b(u, v, w):
return 0.5 * (fn.inner(fn.dot(u, fn.nabla_grad(v)), w) -
fn.inner(fn.dot(u, fn.nabla_grad(w)), v))
# Define boundaries
bcs = []
for BC in config['BCs']:
if BC["boundaryType"] == "wall":
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(0),
fn.Constant((0.0, 0.0, 0.0)),
boundarySets,
int(edge),
method='topological'))
if BC["boundaryType"] == "inlet":
vel = json.loads(BC['value'])
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(0),
fn.Expression(
(str(vel[0]), str(vel[1]), str(vel[2])),
degree=2),
boundarySets,
int(edge),
method='topological'))
if BC["boundaryType"] == "outlet":
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(1),
fn.Constant(float(BC['value'])),
boundarySets,
int(edge),
method='topological'))
f = fn.Constant((0.0, 0.0, 0.0))
# weak form NSE
NSE = (1.0/dt)*fn.inner(u, v)*fn.dx + b(u0, u, v)*fn.dx + nu * \
contract(u, v)*fn.dx - fn.div(v)*p*fn.dx + q*fn.div(u)*fn.dx
LNSE = fn.inner(f, v) * fn.dx + (1. / dt) * fn.inner(u0, v) * fn.dx
velocity_file = fn.XDMFFile(resultDir + "/vel.xdmf")
pressure_file = fn.XDMFFile(resultDir + "/pressure.xdmf")
velocity_file.parameters["flush_output"] = True
velocity_file.parameters["functions_share_mesh"] = True
pressure_file.parameters["flush_output"] = True
pressure_file.parameters["functions_share_mesh"] = True
#
# code for projecting a boundary condition into a file for visualization
#
# for bc in bcs:
# bc.apply(w.vector())
# fn.File("para_plotting/bc.pvd") << w.sub(0)
for jj in range(0, t_num):
t = t + dt
# print('t = ' + str(t))
A, b = fn.assemble_system(NSE, LNSE, bcs)
fn.solve(A, w.vector(), b)
# fn.solve(NSE==LNSE,w,bcs)
fn.assign(u0, w.sub(0))
fn.assign(p0, w.sub(1))
# Save Solutions to Paraview File
if (jj % 20 == 0):
velocity_file.write(u0, t)
pressure_file.write(p0, t)
sendFile(projectId, resultDir + "vel.xdmf")
sendFile(projectId, resultDir + "vel.h5")
sendFile(projectId, resultDir + "pressure.xdmf")
sendFile(projectId, resultDir + "pressure.h5")
statusUpdate(projectId, "STARTED", {"progress": jj / t_num * 100})
import fenics
import matplotlib
N = 4
mesh = fenics.UnitSquareMesh(N, N)
P2 = fenics.VectorElement('P', mesh.ufl_cell(), 2)
P1 = fenics.FiniteElement('P', mesh.ufl_cell(), 1)
P2P1 = fenics.MixedElement([P2, P1])
W = fenics.FunctionSpace(mesh, P2P1)
psi_u, psi_p = fenics.TestFunctions(W)
w = fenics.Function(W)
u, p = fenics.split(w)
dynamic_viscosity = 0.01
mu = fenics.Constant(dynamic_viscosity)
inner, dot, grad, div, sym = fenics.inner, fenics.dot, fenics.grad, fenics.div, fenics.sym
momentum = dot(psi_u, dot(grad(u), u)) - div(psi_u) * p + 2. * mu * inner(
sym(grad(psi_u)), sym(grad(u)))
mass = -psi_p * div(u)
def monolithic_solve(self):
self.U = fe.VectorElement('CG', self.mesh.ufl_cell(), 1)
self.W = fe.FiniteElement("CG", self.mesh.ufl_cell(), 1)
self.M = fe.FunctionSpace(self.mesh, self.U * self.W)
self.WW = fe.FunctionSpace(self.mesh, 'DG', 0)
m_test = fe.TestFunctions(self.M)
m_delta = fe.TrialFunctions(self.M)
m_new = fe.Function(self.M)
self.eta, self.zeta = m_test
self.x_new, self.d_new = fe.split(m_new)
self.H_old = fe.Function(self.WW)
vtkfile_u = fe.File('data/pvd/{}/u.pvd'.format(self.case_name))
vtkfile_d = fe.File('data/pvd/{}/d.pvd'.format(self.case_name))
self.build_weak_form_monolithic()
dG = fe.derivative(self.G, m_new)
self.set_bcs_monolithic()
p = fe.NonlinearVariationalProblem(self.G, m_new, self.BC, dG)
solver = fe.NonlinearVariationalSolver(p)
for i, (disp, rp) in enumerate(
zip(self.displacements, self.relaxation_parameters)):
print('\n')
print(
'================================================================================='
)
print('>> Step {}, disp boundary condition = {} [mm]'.format(
i, disp))
print(
'================================================================================='
)
self.H_old.assign(
fe.project(
history(self.H_old, self.psi(strain(fe.grad(self.x_new))),
self.psi_cr), self.WW))
self.presLoad.t = disp
newton_prm = solver.parameters['newton_solver']
newton_prm['maximum_iterations'] = 100
newton_prm['absolute_tolerance'] = 1e-4
newton_prm['relaxation_parameter'] = rp
solver.solve()
self.x_plot, self.d_plot = m_new.split()
self.x_plot.rename("u", "u")
self.d_plot.rename("d", "d")
vtkfile_u << self.x_plot
vtkfile_d << self.d_plot
force_upper = float(fe.assemble(self.sigma[1, 1] * self.ds(1)))
print("Force upper {}".format(force_upper))
self.delta_u_recorded.append(disp)
self.sigma_recorded.append(force_upper)
print(
'================================================================================='
)
import time
from load_meshes import load_2d_muscle_geo, load_2d_muscle_bc
from plotting_tools import *
mesh, dx, ds, boundaries = load_2d_muscle_geo()
cell = mesh.ufl_cell()
dim = 2
# Notation
# u: displacement field
# p: lagrange multiplier, pressure
# v: velocity field
element_u = fe.VectorElement('CG', cell, 2)
element_p = fe.FiniteElement('CG', cell, 1)
element_v = fe.VectorElement('CG', cell, 2)
# define mixed elements
mix_up = fe.MixedElement([element_u, element_p])
mix_uv = fe.MixedElement([element_u, element_v])
mix_pv = fe.MixedElement([element_p, element_v])
mix_upv = fe.MixedElement([element_u, element_p, element_v])
# define function spaces
V_u = fe.FunctionSpace(mesh, element_u)
V_p = fe.FunctionSpace(mesh, element_p)
V_v = fe.FunctionSpace(mesh, element_v)
V_up = fe.FunctionSpace(mesh, mix_up)
V_uv = fe.FunctionSpace(mesh, mix_uv)
def xest_implement_2d_myosin(self):
#Parameters
total_time = 1.0
number_of_time_steps = 100
delta_t = total_time / number_of_time_steps
nx = ny = 100
domain_size = 1.0
lambda_ = 5.0
mu = 2.0
gamma = 1.0
eta_b = 0.0
eta_s = 1.0
k_b = 1.0
k_u = 1.0
# zeta_1 = -0.5
zeta_1 = 0.0
zeta_2 = 1.0
mu_a = 1.0
K_0 = 1.0
K_1 = 0.0
K_2 = 0.0
K_3 = 0.0
D = 0.25
alpha = 3
c = 0.1
# Sub domain for Periodic boundary condition
class PeriodicBoundary(fenics.SubDomain):
# Left boundary is "target domain" G
def inside(self, x, on_boundary):
# return True if on left or bottom boundary AND NOT on one of the two corners (0, 1) and (1, 0)
return bool(
(fenics.near(x[0], 0) or fenics.near(x[1], 0)) and
(not ((fenics.near(x[0], 0) and fenics.near(x[1], 1)) or
(fenics.near(x[0], 1) and fenics.near(x[1], 0))))
and on_boundary)
def map(self, x, y):
if fenics.near(x[0], 1) and fenics.near(x[1], 1):
y[0] = x[0] - 1.
y[1] = x[1] - 1.
elif fenics.near(x[0], 1):
y[0] = x[0] - 1.
y[1] = x[1]
else: # near(x[1], 1)
y[0] = x[0]
y[1] = x[1] - 1.
periodic_boundary_condition = PeriodicBoundary()
#Set up finite elements
mesh = fenics.RectangleMesh(fenics.Point(0, 0),
fenics.Point(domain_size, domain_size), nx,
ny)
vector_element = fenics.VectorElement('P', fenics.triangle, 2, dim=2)
single_element = fenics.FiniteElement('P', fenics.triangle, 2)
mixed_element = fenics.MixedElement(vector_element, single_element)
V = fenics.FunctionSpace(
mesh,
mixed_element,
constrained_domain=periodic_boundary_condition)
v, r = fenics.TestFunctions(V)
full_trial_function = fenics.Function(V)
u, rho = fenics.split(full_trial_function)
full_trial_function_n = fenics.Function(V)
u_n, rho_n = fenics.split(full_trial_function_n)
#Define non-linear weak formulation
def epsilon(u):
return 0.5 * (fenics.nabla_grad(u) + fenics.nabla_grad(u).T
) #return sym(nabla_grad(u))
def sigma_e(u):
return lambda_ * ufl.nabla_div(u) * fenics.Identity(
2) + 2 * mu * epsilon(u)
def sigma_d(u):
return eta_b * ufl.nabla_div(u) * fenics.Identity(
2) + 2 * eta_s * epsilon(u)
# def sigma_a(u,rho):
# return ( -zeta_1*rho/(1+zeta_2*rho)*mu_a*fenics.Identity(2)*(K_0+K_1*ufl.nabla_div(u)+
# K_2*ufl.nabla_div(u)*ufl.nabla_div(u)+K_3*ufl.nabla_div(u)*ufl.nabla_div(u)*ufl.nabla_div(u)))
def sigma_a(u, rho):
return -zeta_1 * rho / (
1 + zeta_2 * rho) * mu_a * fenics.Identity(2) * (K_0)
F = (gamma * fenics.dot(u, v) * fenics.dx -
gamma * fenics.dot(u_n, v) * fenics.dx +
fenics.inner(sigma_d(u), fenics.nabla_grad(v)) * fenics.dx -
fenics.inner(sigma_d(u_n), fenics.nabla_grad(v)) * fenics.dx -
delta_t *
fenics.inner(sigma_e(u) + sigma_a(u, rho), fenics.nabla_grad(v)) *
fenics.dx + rho * r * fenics.dx - rho_n * r * fenics.dx +
ufl.nabla_div(rho * u) * r * fenics.dx -
ufl.nabla_div(rho * u_n) * r * fenics.dx - D * delta_t *
fenics.dot(fenics.nabla_grad(rho), fenics.nabla_grad(r)) *
fenics.dx + delta_t *
(-k_u * rho * fenics.exp(alpha * ufl.nabla_div(u)) + k_b *
(1 - c * ufl.nabla_div(u))) * r * fenics.dx)
# F = ( gamma*fenics.dot(u,v)*fenics.dx - gamma*fenics.dot(u_n,v)*fenics.dx + fenics.inner(sigma_d(u),fenics.nabla_grad(v))*fenics.dx -
# fenics.inner(sigma_d(u_n),fenics.nabla_grad(v))*fenics.dx - delta_t*fenics.inner(sigma_e(u)+sigma_a(u,rho),fenics.nabla_grad(v))*fenics.dx
# +rho*r*fenics.dx-rho_n*r*fenics.dx + ufl.nabla_div(rho*u)*r*fenics.dx - ufl.nabla_div(rho*u_n)*r*fenics.dx -
# D*delta_t*fenics.dot(fenics.nabla_grad(rho),fenics.nabla_grad(r))*fenics.dx +delta_t*(-k_u*rho*fenics.exp(alpha*ufl.nabla_div(u))+k_b*(1-c*ufl.nabla_div(u))))
vtkfile_rho = fenics.File(
os.path.join(os.path.dirname(__file__), 'output', 'myosin_2d',
'solution_rho.pvd'))
vtkfile_u = fenics.File(
os.path.join(os.path.dirname(__file__), 'output', 'myosin_2d',
'solution_u.pvd'))
# rho_0 = fenics.Expression(((('0.0'),('0.0'),('0.0')),('sin(x[0])')), degree=1 )
# full_trial_function_n = fenics.project(rho_0, V)
time = 0.0
for time_index in range(number_of_time_steps):
# Update current time
time += delta_t
# Compute solution
fenics.solve(F == 0, full_trial_function)
# Save to file and plot solution
vis_u, vis_rho = full_trial_function.split()
vtkfile_rho << (vis_rho, time)
vtkfile_u << (vis_u, time)
full_trial_function_n.assign(full_trial_function)
