def demo16(self, permeability, obs_case=1):
"""This demo program solves the mixed formulation of Poisson's
equation:
sigma + grad(u) = 0 in Omega
div(sigma) = f in Omega
du/dn = g on Gamma_N
u = u_D on Gamma_D
The corresponding weak (variational problem)
<sigma, tau> + <grad(u), tau> = 0
for all tau
- <sigma, grad(v)> = <f, v> + <g, v>
for all v
is solved using DRT (Discontinuous Raviart-Thomas) elements
of degree k for (sigma, tau) and CG (Lagrange) elements
of degree k + 1 for (u, v) for k >= 1.
"""
mesh = UnitSquareMesh(15, 15)
ak_values = permeability
flux_order = 1
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((16, 16))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [1, 3, 5, 7, 9, 11, 13, 15]
position = [1, 3, 5, 7, 9, 11, 13, 15]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, ak_values, pos
def demo32(self, permeability, obs_case=1):
mesh = UnitSquareMesh(31, 31)
ak_values = permeability
flux_order = 3
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((32, 32))
vx = x2[:1024].reshape((32, 32))
vy = x2[1024:].reshape((32, 32))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [2, 6, 10, 14, 18, 22, 26, 30]
position = [2, 6, 10, 14, 18, 22, 26, 30]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, vx, vy, ak_values, pos
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})
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
lmbda = E * nu / ((1. + nu) * (1. - 2. * nu))
mu = E / (2. * (1. + nu))
c0 = 1.0e-3
kappa = 1.0e-6
alpha = 0.1
fileE = fn.File("Output/1_4_Karma/E.pvd")
filen = fn.File("Output/1_4_Karma/n.pvd")
t = 0.0; dt = 0.3; Tfinal = 600.0; frequency = 100;
# mesh
L = 6.72; nps = 64;
mesh = fn.RectangleMesh(fn.Point(0, 0), fn.Point(L, L),
nps, nps, "crossed")
# element and function spaces
Mhe = fn.FiniteElement("CG", mesh.ufl_cell(), 1)
Mh = fn.FunctionSpace(mesh, Mhe)
Nh = fn.FunctionSpace(mesh, fn.MixedElement([Mhe,Mhe]))
# trial and test functions
v, n = fn.TrialFunctions(Nh)
w, m = fn.TestFunctions(Nh)
# solution
Ksol = fn.Function(Nh)
# model constants
diffScale = fn.Constant(1e-3)
D0 = 1.1 * diffScale
tauv = fn.Constant(2.5)
taun = fn.Constant(250.0)
Re = fn.Constant(1.0)
M = fn.Constant(5.0)
beta = fn.Constant(0.008)
vstar = fn.Constant(1.5415)
vh = fn.Constant(3.0)
Qh = fn.FiniteElement("CG", mesh.ufl_cell(), 2)
Mh = fn.FiniteElement("CG", mesh.ufl_cell(), 1)
Wh = fn.TensorElement("CG", mesh.ufl_cell(), 1)
# function spaces
Vhf = fn.FunctionSpace(mesh, Vh)
Zhf = fn.FunctionSpace(mesh, Zh)
Qhf = fn.FunctionSpace(mesh, Qh)
Mhf = fn.FunctionSpace(mesh, Mh)
Whf = fn.FunctionSpace(mesh, Wh)
Hh = fn.FunctionSpace(mesh, fn.MixedElement([Vh,Zh,Qh]))
Nh = fn.FunctionSpace(mesh, fn.MixedElement([Mh, Mh]))
# functions and test functions
u, phi, p = fn.TrialFunctions(Hh)
v, psi, q = fn.TestFunctions(Hh)
E, n = fn.TrialFunctions(Nh)
F, m = fn.TestFunctions(Nh)
# pvd files
pvdU = fn.File(mesh.mpi_comm(), "Output/3_Combined_models/u.pvd")
pvdPHI = fn.File(mesh.mpi_comm(), "Output/3_Combined_models/phi.pvd")
pvdP = fn.File(mesh.mpi_comm(), "Output/3_Combined_models/p.pvd")
pvdE = fn.File(mesh.mpi_comm(), "Output/3_Combined_models/E.pvd")
pvdN = fn.File(mesh.mpi_comm(), "Output/3_Combined_models/n.pvd")
# constants for poroelasticity model (taken from footing-scaled-BCs.py)
EE = fn.Constant(3.0e4)
nu = fn.Constant(0.4995)
fn.parameters["form_compiler"]["cpp_optimize"] = True
# mesh setup
nps = 64 # number of points
mesh = fn.UnitSquareMesh(nps, nps)
fileu = fn.File(mesh.mpi_comm(), "Output/1_3_Schnackenberg/u.pvd")
filev = fn.File(mesh.mpi_comm(), "Output/1_3_Schnackenberg/v.pvd")
# function spaces
P1 = fn.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
Mh = fn.FunctionSpace(mesh, "Lagrange", 1)
Nh = fn.FunctionSpace(mesh, fn.MixedElement([P1, P1]))
Sol = fn.Function(Nh)
# trial and test functions
u, v = fn.TrialFunctions(Nh)
uT, vT = fn.TestFunctions(Nh)
# model constants
a = fn.Constant(0.1305)
b = fn.Constant(0.7695)
c1 = fn.Constant(0.05)
c2 = fn.Constant(1.0)
d = fn.Constant(170.0)
# initial value
uinit = fn.Expression('a + b + 0.001 * exp(-100.0 * (pow(x[0] - 1.0/3, 2)' +
' + pow(x[1] - 0.5, 2)))',
a=a,
b=b,
degree=3)
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(
'================================================================================='
)
# -- Spatial domain
ne = 2**7
L = 15
mesh = fe.IntervalMesh(int(ne),0,L)
degQ = 1
degA = 1
QE = fe.FiniteElement("Lagrange", cell=mesh.ufl_cell(), degree=degQ)
AE = fe.FiniteElement("Lagrange", cell=mesh.ufl_cell(), degree=degA)
ME = fe.MixedElement([AE,QE])
V = fe.FunctionSpace(mesh,ME)
V_A = V.sub(0)
V_Q = V.sub(1)
(v1,v2) = fe.TestFunctions(V)
dv1 = fe.grad(v1)[0]
dv2 = fe.grad(v2)[0]
(u1,u2) = fe.TrialFunctions(V)
U0 = fe.Function(V)
U0.assign( fe.Expression( ( 'A0', 'Q0' ) , A0=A0, Q0=Q0, degree=1 ) )
Un = fe.Function(V)
Un.assign( fe.Expression( ( 'A0', 'Q0' ) , A0=A0, Q0=Q0, degree=1 ) )
(u01,u02) = fe.split(U0)
(un1,un2) = fe.split(Un)
du01 = fe.grad(u01)[0] ; du02 = fe.grad(u02)[0]
dun1 = fe.grad(un1)[0] ; dun2 = fe.grad(un2)[0]
B0 = getB(u01,u02)
H0 = getH(u01,u02)
HdUdz0 = matMult(H0,[du01, du02])
Bn = getB(un1,un2)
Hn = getH(un1,un2)
def half_exp_dyn(w0, dt=1.e-5, t_end=1.e-4, show_plots=False):
u0 = w0.u
p0 = w0.p
v0 = w0.v
bcs_u, bcs_p, bcs_v = load_2d_muscle_bc(V_u, V_pv.sub(0), V_pv.sub(1),
boundaries)
F = deformation_grad(u0)
I_1, I_2, J = invariants(F)
F_iso = isochronic_deformation_grad(F, J)
#I_1_iso, I_2_iso = invariants(F_iso)[0:2]
W = material_mooney_rivlin(I_1, I_2, c_10, c_01)
g = incompr_constr(J)
# Lagrange function (without constraint)
L = -W
P = first_piola_stress(L, F)
G = incompr_stress(g, F)
# a_dyn_u = inner(u1 - u0, eta) * dx - dt * inner(v1, eta) * dx
u1 = fe.TrialFunction(V_u)
eta = fe.TestFunction(V_u)
#u11 = fe.Function(V_u)
#F1 = deformation_grad(u11)
#g1 = incompr_constr(fe.det(F1))
#G1 = incompr_stress(g1, F1)
(p1, v1) = fe.TrialFunctions(V_pv)
(q, xi) = fe.TestFunctions(V_pv)
a_dyn_u = inner(u1 - u0, eta) * dx - dt * inner(v0, eta) * dx
a_dyn_p = fe.tr(G * grad(v1)) * q * dx
#a_dyn_v = rho*inner(v1-v0, xi)*dx + dt*(inner(P + p0*G, grad(xi))*dx - inner(B, xi)*dx)
a_dyn_v = rho * inner(v1 - v0, xi) * dx + dt * (inner(
P, grad(xi)) * dx + inner(p1 * G, grad(xi)) * dx - inner(B, xi) * dx)
a_u = fe.lhs(a_dyn_u)
l_u = fe.rhs(a_dyn_u)
a_pv = fe.lhs(a_dyn_p + a_dyn_v)
l_pv = fe.rhs(a_dyn_p + a_dyn_v)
u1 = fe.Function(V_u)
pv1 = fe.Function(V_pv)
sol = []
vol = fe.assemble(1. * dx)
A_u = fe.assemble(a_u)
A_pv = fe.assemble(a_pv)
for bc in bcs_u:
bc.apply(A_u)
t = 0
while t < t_end:
print("progress: %f" % (100. * t / t_end))
# update displacement u
L_u = fe.assemble(l_u)
fe.solve(A_u, u1.vector(), L_u)
u0.assign(u1)
L_pv = fe.assemble(l_pv)
for bc in bcs_p + bcs_v:
bc.apply(A_pv, L_pv)
fe.solve(A_pv, pv1.vector(), L_pv)
if fe.norm(pv1.vector()) > 1e8:
print('ERROR: norm explosion')
break
# update initial values for next step
w0.u = u1
w0.pv = pv1
p0.assign(w0.p)
v0.assign(w0.v)
t += dt
if show_plots:
# plot result
fe.plot(w0.sub(0), mode='displacement')
plt.show()
# save solutions
sol.append(Solution(t=t))
sol[-1].upv.assign(w0)
return sol, W
def explicit_relax_dyn(w0, kappa=1e5, dt=1.e-5, t_end=1.e-4, show_plots=False):
(u0, p0, v0) = fe.split(w0)
bcs_u, bcs_p, bcs_v = load_2d_muscle_bc(V_upv.sub(0), V_upv.sub(1),
V_upv.sub(2), boundaries)
kappa = fe.Constant(kappa)
F = deformation_grad(u0)
I_1, I_2, J = invariants(F)
F_iso = isochronic_deformation_grad(F, J)
#I_1_iso, I_2_iso = invariants(F_iso)[0:2]
W = material_mooney_rivlin(I_1, I_2, c_10, c_01) + incompr_relaxation(
p0, kappa)
g = incompr_constr(J)
# Lagrange function (without constraint)
L = -W
P = first_piola_stress(L, F)
G = incompr_stress(g, F)
(u1, p1, v1) = fe.TrialFunctions(V_upv)
(eta, q, xi) = fe.TestFunctions(V_upv)
a_dyn_u = inner(u1 - u0, eta) * dx - dt * inner(v1, eta) * dx
a_dyn_p = (p1 - p0) * q * dx - dt * kappa * div(v1) * J * q * dx
#a_dyn_v = rho*inner(v1-v0, xi)*dx + dt*(inner(P + p0*G, grad(xi))*dx - inner(B, xi)*dx)
a_dyn_v = rho * inner(v1 - v0, xi) * dx + dt * (inner(
P, grad(xi)) * dx + inner(p0 * G, grad(xi)) * dx - inner(B, xi) * dx)
a = fe.lhs(a_dyn_u + a_dyn_p + a_dyn_v)
l = fe.rhs(a_dyn_u + a_dyn_p + a_dyn_v)
w1 = fe.Function(V_upv)
w2 = fe.Function(V_upv)
sol = []
vol = fe.assemble(1. * dx)
t = 0
while t < t_end:
print("progress: %f" % (100. * t / t_end))
A = fe.assemble(a)
L = fe.assemble(l)
for bc in bcs_u + bcs_p + bcs_v:
bc.apply(A, L)
fe.solve(A, w1.vector(), L)
if fe.norm(w1.vector()) > 1e7:
print('ERROR: norm explosion')
break
# update initial values for next step
w0.assign(w1)
t += dt
if show_plots:
# plot result
fe.plot(w0.sub(0), mode='displacement')
plt.show()
# save solutions
sol.append(Solution(t=t))
sol[-1].upv.assign(w0)
return sol, W, kappa