def test_formulation_2_traps_1_material():
'''
Test function formulation() with 2 intrinsic traps
and 1 material
'''
# Set parameters
dt = 1
traps = [{
"energy": 1,
"density": 2,
"materials": [1]
},
{
"energy": 1,
"density": 2,
"materials": [1]
}]
materials = [{
"alpha": 1,
"beta": 2,
"density": 3,
"borders": [0, 1],
"E_diff": 4,
"D_0": 5,
"id": 1
}]
extrinsic_traps = []
# Prepare
mesh = fenics.UnitIntervalMesh(10)
V = fenics.VectorFunctionSpace(mesh, 'P', 1, len(traps)+1)
u = fenics.Function(V)
u_n = fenics.Function(V)
v = fenics.TestFunction(V)
solutions = list(fenics.split(u))
previous_solutions = list(fenics.split(u_n))
testfunctions = list(fenics.split(v))
mf = fenics.MeshFunction('size_t', mesh, 1, 1)
dx = fenics.dx(subdomain_data=mf)
temp = fenics.Expression("300", degree=0)
flux_ = fenics.Expression("1", degree=0)
F, expressions = FESTIM.formulation(
traps, extrinsic_traps, solutions, testfunctions,
previous_solutions, dt, dx, materials, temp, flux_)
# Transient sol
expected_form = ((solutions[0] - previous_solutions[0]) / dt) * \
testfunctions[0]*dx
# Diffusion sol
expected_form += 5 * fenics.exp(-4/8.6e-5/temp) * \
fenics.dot(
fenics.grad(solutions[0]), fenics.grad(testfunctions[0]))*dx(1)
# Source sol
expected_form += -flux_*testfunctions[0]*dx
# Transient trap 1
expected_form += ((solutions[1] - previous_solutions[1]) / dt) * \
testfunctions[1]*dx
# Trapping trap 1
expected_form += - 5 * fenics.exp(-4/8.6e-5/temp)/1/1/2 * \
solutions[0] * (2 - solutions[1]) * \
testfunctions[1]*dx(1)
# Detrapping trap 1
expected_form += 1e13*fenics.exp(-1/8.6e-5/temp)*solutions[1] * \
testfunctions[1]*dx(1)
# Source detrapping sol
expected_form += ((solutions[1] - previous_solutions[1]) / dt) * \
testfunctions[0]*dx
# Transient trap 2
expected_form += ((solutions[2] - previous_solutions[2]) / dt) * \
testfunctions[2]*dx
# Trapping trap 2
expected_form += - 5 * fenics.exp(-4/8.6e-5/temp)/1/1/2 * \
solutions[0] * (2 - solutions[2]) * \
testfunctions[2]*dx(1)
# Detrapping trap 2
expected_form += 1e13*fenics.exp(-1/8.6e-5/temp)*solutions[2] * \
testfunctions[2]*dx(1)
# Source detrapping 2 sol
expected_form += ((solutions[2] - previous_solutions[2]) / dt) * \
testfunctions[0]*dx
assert expected_form.equals(F) is True
def create_step_functionals(self):
u, v = self._sp.u, self._sp.v
u0, v0 = self._sp.u0, self._sp.v0
a0 = self._sp.a0
bf = self._sp.bf
ut = self._sp.ut
F, F0 = self._sp.F, self._sp.F0
density = self._sp.density
constitutive_model = self._sp.constitutive_model
self.d_LHS = fe.inner(u, ut) * density * fe.dx \
- pow(self._dt, 2) * self._beta * fe.inner(F * constitutive_model.stress(F), fe.grad(ut)) * fe.dx
self.d_RHS = density * (fe.inner(u0, ut) * fe.dx
+ self._dt * fe.inner(v0, ut) * fe.dx
+ pow(self._dt, 2) * (0.5 - self._beta) * fe.inner(a0, ut) * fe.dx) \
- pow(self._dt, 2) * self._beta * (fe.inner(bf, ut) * fe.dx)
# Define boundary condition
u_D = fs.Expression(u_code, degree=2)
def boundary(x, on_boundary):
return on_boundary
bc = fs.DirichletBC(V, u_D, boundary)
# Define variational problem
u = fs.Function(V)
v = fs.TestFunction(V)
f = fs.Expression(f_code, degree=2)
F = q(u) * fs.dot(fs.grad(u), fs.grad(v)) * fs.dx - f * v * fs.dx
a, L = fs.lhs(F), fs.rhs(F)
# Create VTK file for saving solution
vtkfile = fs.File('solution.pvd')
#Compute Solution
fs.solve(F == 0, u, bc)
#plot solution
fs.plot(u)
# Compute maximum error at vertices. This computation illustrates
# an alternative to using compute_vertex_values as in poisson.py.
u_e = fs.interpolate(u_D, V)
error_max = np.abs(u_e.vector() - u.vector()).max()
VW = fn.TestFunctions(
V) # test function potential concentration lagrange multi
v, w, mu = VW[0], VW[1:M + 1], VW[M + 1:]
#lets try rot
r = fn.Expression('x[0]', degree=0)
# changing concentrations charges
Rho = 0
for i in range(M):
if i % 2:
Rho += -c[i]
else:
Rho += c[i]
PoissonLeft = (fn.dot(fn.grad(u),
fn.grad(v))) * fn.dx # weak solution Poisson left
PoissonRight = -(Rho) * v * fn.dx # weak solution Poisson right
NernstPlanck = 0
for i in range(M):
if i % 2:
NernstPlanck += fn.dot(
(-fn.grad(c[i]) + c[i] * fn.grad(u)), fn.grad(
w[i])) * fn.dx # weak solution Nernst-Planck
else:
NernstPlanck += fn.dot(
(-fn.grad(c[i]) - c[i] * fn.grad(u)), fn.grad(
w[i])) * fn.dx # weak solution Nernst-Planck
constraint = 0
for i in range(M):
def eps(u):
return fn.sym(fn.grad(u))
def explicit(d_, w_, k):
E = grad(d_["n-1"]) + grad(d_["n-1"]).T \
+ grad(d_["n-1"]).T*grad(d_["n-1"])
return 0.5 * E
def cn_before_ab_higher_order(d_, w_, k):
E = 0.5*(grad(d_["n"]).T + grad(d_["n-1"]).T + grad(d_["n"]) + grad(d_["n-1"])) \
+ 0.5*((grad(d_["n-1"] + k*(23./12*w_["n-1"] - 4./3*w_["n-2"] + 5/12.*w_["n-3"])).T \
* grad(d_["n"])) + (grad(d_["n-1"]).T*grad(d_["n-1"])))
return 0.5 * E
def k(u, v):
return inner(sigma(u), sym(grad(v))) * dx
"(x[0] >= -0.2) && (x[0] <= 0.2) && near(x[1], 0.75) && on_boundary")
GammaU = fn.CompiledSubDomain(
"(near(x[0], -0.5) || near(x[0], 0.5) ) && on_boundary")
GammaU.mark(bdry, 31)
FooT.mark(bdry, 32)
ds = fn.Measure("ds", subdomain_data=bdry)
bcU = fn.DirichletBC(Hh.sub(0), u_g, bdry, 31)
bcP = fn.DirichletBC(Hh.sub(2), p_g, bdry, 32)
bcs = [bcU, bcP]
# weak form
PLeft = 2*mu*fn.inner(strain(u),strain(v)) * fn.dx \
- fn.div(v) * phi * fn.dx \
+ (c0/alpha + 1.0/lmbda)* p * q * fn.dx \
+ kappa/(alpha*nu) * fn.dot(fn.grad(p),fn.grad(q)) * fn.dx \
- 1.0/lmbda * phi * q * fn.dx \
- fn.div(u) * psi * fn.dx \
+ 1.0/lmbda * psi * p * fn.dx \
- 1.0/lmbda * phi * psi * fn.dx
PRight = fn.dot(f, v) * fn.dx + fn.dot(h_g, v) * ds(32)
Sol = fn.Function(Hh)
# solve linear system
fn.solve(PLeft == PRight, Sol, bcs)
u, phi, p = Sol.split()
u.rename("u", "u")
fileU << u
# -- 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])
A_fenics = fenics.PETScMatrix(A_petsc)
return A_fenics
if perform_tests:
import numpy as np
from time import time
# Make Laplacian matrix in fenics
n = 10
mesh = fenics.UnitCubeMesh(n, n, n)
V = fenics.FunctionSpace(mesh, 'CG', 1)
u = fenics.TrialFunction(V)
v = fenics.TestFunction(V)
a = fenics.inner(fenics.grad(u),
fenics.grad(v)) * fenics.dx + u * v * fenics.dx
f = fenics.Function(V)
f.vector()[:] = np.random.randn(V.dim())
b = f * v * fenics.dx
b_fenics = fenics.assemble(b)
b_numpy = b_fenics[:]
A_fenics = fenics.assemble(a)
# Test correctness of matrix converters
A_scipy = convert_fenics_csr_matrix_to_scipy_csr_matrix(A_fenics)
A_fenics2 = convert_scipy_csr_matrix_to_fenics_csr_matrix(A_scipy)
Ct4 = fe.Constant(2)
d1 = fe.Expression('x[0] - 0', degree=1)
#d2 = fe.Expression('1 - x[0]', degree=1)
#d3 = fe.Expression('1 - x[1]', degree=1)
d = fe.Expression('x[1] - 0', degree=1)
#d = fe.Constant(10)
#d = Min(d1, d4)
xi = nu_trial / nu
fv1 = xi**3 / (xi**3 + Cv1 * Cv1 * Cv1)
#fv1 = fe.elem_pow(xi, 3) / (fe.elem_pow(xi, 3) + Cv1 * Cv1 * Cv1)
fv2 = 1 - xi / (1 + xi * fv1)
ft2 = Ct3 * fe.exp(-Ct4 * xi * xi)
Omega = fe.Constant(0.5) * (fe.grad(u) - fe.grad(u).T)
S = fe.sqrt(EPSILON + fe.Constant(2) * fe.inner(Omega, Omega))
Stilde = S + nu_trial / (kappa * kappa * d * d) * fv2
ft2 = Ct3 * fe.exp(fe.Constant(-1) * Ct4 * xi * xi)
r = Min(nu_trial / (Stilde * kappa * kappa * d * d), 10)
g = r + Cw2 * (r * r * r * r * r * r - r)
fw = fe.elem_pow(
EPSILON +
abs(g * ((1 + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3) /
(g * g * g * g * g * g + Cw3 * Cw3 * Cw3 * Cw3 * Cw3 * Cw3))),
fe.Constant(1. / 6.))
ns_conv = fe.inner(v, fe.grad(u) * u) * fe.dx
ns_press = p * fe.div(v) * fe.dx
s = fe.grad(u) + fe.grad(u).T
if MODEL:
UC = fn.Function(V)
uc = fn.split(UC) # trial function potential concentration lagrange multi
u, c, lam = uc[0], uc[1:M+1], uc[M+1:]
VW = fn.TestFunctions(V) # test function potential concentration lagrange multi
v, w, mu = VW[0], VW[1:M+1], VW[M+1:]
# changing concentrations charges
Rho = 0
for i in range(M):
if i%2:
Rho += -c[i]
else:
Rho += c[i]
PoissonLeft = (fn.dot(alpha*fn.grad(u), fn.grad(v)))*fn.dx # weak solution Poisson left
PoissonRight = -(Rho)*beta*v*fn.dx # weak solution Poisson right
NernstPlanck = 0
for i in range(M):
if i%2:
NernstPlanck += fn.dot(-gamma*fn.grad(c[i]) + c[i]*delta*fn.grad(u),fn.grad(w[i]))*fn.dx # weak solution Nernst-Planck
else:
NernstPlanck += fn.dot(-gamma*fn.grad(c[i]) - c[i]*delta*fn.grad(u),fn.grad(w[i]))*fn.dx # weak solution Nernst-Planck
constraint = 0
for i in range(M):
constraint += lam[i] * w[i] * fn.dx + (c[i] - c_avg) * mu[i] * fn.dx #constraint a la hoermann
PNP_xy = PoissonLeft + PoissonRight + NernstPlanck + constraint # PNP system
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 main():
"""Main function. Organizes workflow."""
fname = str(INPUTS['filename'])
term = Terminal()
print(
term.yellow
+ "Working on file {}.".format(fname)
+ term.normal
)
# Load mesh and physical domains from file.
mesh = fe.Mesh(fname + ".xml")
if INPUTS['saving']['mesh']:
fe.File(fname + "_mesh.pvd") << mesh
if INPUTS['plotting']['mesh']:
fe.plot(mesh, title='Mesh')
subdomains = fe.MeshFunction(
'size_t', mesh, fname + '_physical_region.xml')
if INPUTS['saving']['subdomains']:
fe.File(fname + "_subdomains.pvd") << subdomains
if INPUTS['plotting']['subdomains']:
fe.plot(subdomains, title='Subdomains')
# function space for temperature/concentration
func_space = fe.FunctionSpace(
mesh,
INPUTS['element_type'],
INPUTS['element_degree'],
constrained_domain=PeriodicDomain()
)
# discontinuous function space for visualization
dis_func_space = fe.FunctionSpace(
mesh,
'DG',
INPUTS['element_degree'],
constrained_domain=PeriodicDomain()
)
if ARGS['--verbose']:
print('Number of cells:', mesh.num_cells())
print('Number of faces:', mesh.num_faces())
print('Number of edges:', mesh.num_edges())
print('Number of vertices:', mesh.num_vertices())
print('Number of DOFs:', len(func_space.dofmap().dofs()))
# temperature/concentration field
field = fe.TrialFunction(func_space)
# test function
test_func = fe.TestFunction(func_space)
# function, which is equal to 1 everywhere
unit_function = fe.Function(func_space)
unit_function.assign(fe.Constant(1.0))
# assign material properties to each domain
if INPUTS['mode'] == 'conductivity':
mat_prop = SubdomainConstant(
subdomains,
fe.Constant(INPUTS['conductivity']['gas']),
fe.Constant(INPUTS['conductivity']['solid']),
degree=0
)
elif INPUTS['mode'] == 'diffusivity':
mat_prop = SubdomainConstant(
subdomains,
fe.Constant(INPUTS['diffusivity']['gas']),
fe.Constant(INPUTS['diffusivity']['solid']) *
fe.Constant(INPUTS['solubility'] *
gas_constant * INPUTS['temperature']),
degree=0
)
# assign 1 to gas domain, and 0 to solid domain
gas_content = SubdomainConstant(
subdomains,
fe.Constant(1.0),
fe.Constant(0.0),
degree=0
)
# define structure of foam over whole domain
structure = fe.project(unit_function * gas_content, dis_func_space)
# calculate porosity and wall thickness
porosity = fe.assemble(structure *
fe.dx) / ((XMAX - XMIN) * (YMAX - YMIN) * (ZMAX - ZMIN))
print('Porosity: {0}'.format(porosity))
dwall = wall_thickness(
porosity, INPUTS['morphology']['cell_size'],
INPUTS['morphology']['strut_content'])
print('Wall thickness: {0} m'.format(dwall))
# calculate effective conductivity/diffusivity by analytical model
if INPUTS['mode'] == 'conductivity':
eff_prop = analytical_conductivity(
INPUTS['conductivity']['gas'], INPUTS['conductivity']['solid'],
porosity, INPUTS['morphology']['strut_content'])
print('Analytical model: {0} W/(mK)'.format(eff_prop))
elif INPUTS['mode'] == 'diffusivity':
eff_prop = analytical_diffusivity(
INPUTS['diffusivity']['solid'] *
INPUTS['solubility'], INPUTS['solubility'],
porosity, INPUTS['morphology']['cell_size'], dwall,
INPUTS['temperature'], INPUTS['morphology']['enhancement_par'])
print('Analytical model: {0} m^2/s'.format(eff_prop))
# create system matrix
system_matrix = -mat_prop * \
fe.inner(fe.grad(field), fe.grad(test_func)) * fe.dx
left_side, right_side = fe.lhs(system_matrix), fe.rhs(system_matrix)
# define boundary conditions
bcs = [
fe.DirichletBC(func_space, fe.Constant(
INPUTS['boundary_conditions']['top']), top_bc),
fe.DirichletBC(func_space, fe.Constant(
INPUTS['boundary_conditions']['bottom']), bottom_bc)
]
# compute solution
field = fe.Function(func_space)
fe.solve(left_side == right_side, field, bcs)
# output temperature/concentration at the boundaries
if ARGS['--verbose']:
print('Checking periodicity:')
print('Value at XMIN:', field(XMIN, (YMIN + YMAX) / 3, (ZMIN + ZMAX) / 3))
print('Value at XMAX:', field(XMAX, (YMIN + YMAX) / 3, (ZMIN + ZMAX) / 3))
print('Value at YMIN:', field((XMIN + XMAX) / 3, YMIN, (ZMIN + ZMAX) / 3))
print('Value at YMAX:', field((XMIN + XMAX) / 3, YMAX, (ZMIN + ZMAX) / 3))
print('Value at ZMIN:', field((XMIN + XMAX) / 3, (YMIN + YMAX) / 3, ZMIN))
print('Value at ZMAX:', field((XMIN + XMAX) / 3, (YMIN + YMAX) / 3, ZMAX))
# calculate flux, and effective properties
vec_func_space = fe.VectorFunctionSpace(
mesh,
INPUTS['element_type'],
INPUTS['element_degree']
)
flux = fe.project(-mat_prop * fe.grad(field), vec_func_space)
divergence = fe.project(-fe.div(mat_prop * fe.grad(field)), func_space)
flux_x, flux_y, flux_z = flux.split()
av_flux = fe.assemble(flux_z * fe.dx) / ((XMAX - XMIN) * (YMAX - YMIN))
eff_prop = av_flux * (ZMAX - ZMIN) / (
INPUTS['boundary_conditions']['top']
- INPUTS['boundary_conditions']['bottom']
)
if INPUTS['mode'] == 'conductivity':
print('Numerical model: {0} W/(mK)'.format(eff_prop))
elif INPUTS['mode'] == 'diffusivity':
print('Numerical model: {0} m^2/s'.format(eff_prop))
# projection of concentration has to be in discontinuous function space
if INPUTS['mode'] == 'diffusivity':
sol_field = SubdomainConstant(
subdomains,
fe.Constant(1.0),
fe.Constant(INPUTS['solubility'] *
gas_constant * INPUTS['temperature']),
degree=0
)
field = fe.project(field * sol_field, dis_func_space)
# save results
with open(fname + "_eff_prop.csv", 'w') as textfile:
textfile.write('eff_prop\n')
textfile.write('{0}\n'.format(eff_prop))
fe.File(fname + "_solution.pvd") << field
fe.File(fname + "_structure.pvd") << structure
if INPUTS['saving']['flux']:
fe.File(fname + "_flux.pvd") << flux
if INPUTS['saving']['flux_divergence']:
fe.File(fname + "_flux_divergence.pvd") << divergence
if INPUTS['saving']['flux_components']:
fe.File(fname + "_flux_x.pvd") << flux_x
fe.File(fname + "_flux_y.pvd") << flux_y
fe.File(fname + "_flux_z.pvd") << flux_z
# plot results
if INPUTS['plotting']['solution']:
fe.plot(field, title="Solution")
if INPUTS['plotting']['flux']:
fe.plot(flux, title="Flux")
if INPUTS['plotting']['flux_divergence']:
fe.plot(divergence, title="Divergence")
if INPUTS['plotting']['flux_components']:
fe.plot(flux_x, title='x-component of flux (-kappa*grad(u))')
fe.plot(flux_y, title='y-component of flux (-kappa*grad(u))')
fe.plot(flux_z, title='z-component of flux (-kappa*grad(u))')
if True in INPUTS['plotting'].values():
fe.interactive()
print(
term.yellow
+ "End."
+ term.normal
)
def strain(v):
return fn.sym(fn.grad(v))
def naive_linearization(d_, w_, k):
E = grad(d_["n"]) + grad(d_["n"]).T \
+ grad(d_["n-1"]).T*grad(d_["n"])
return 0.5 * E
def eps(v):
return fnc.sym(fnc.grad(v))
def cn_before_ab(d_, w_, k):
E = 0.5*(grad(d_["n"]).T + grad(d_["n-1"]).T + grad(d_["n"]) + grad(d_["n-1"])) \
+ 0.5*((grad(d_["n-1"] + k*(3./2*w_["n-1"] - 1./2*w_["n-2"])).T * grad(d_["n"])) \
+ (grad(d_["n-1"]).T*grad(d_["n-1"])))
return 0.5 * E
def sigma(r):
return 2.0 * mu * fnc.sym(fnc.grad(r)) + lmbda * fnc.tr(
fnc.sym(fnc.grad(r))) * fnc.Identity(len(r))
# Initialize the adapter according to the specific participant
if problem is ProblemType.DIRICHLET:
precice = Adapter(adapter_config_filename="precice-adapter-config-D.json")
precice_dt = precice.initialize(coupling_boundary, read_function_space=V, write_object=f_N_function)
elif problem is ProblemType.NEUMANN:
precice = Adapter(adapter_config_filename="precice-adapter-config-N.json")
precice_dt = precice.initialize(coupling_boundary, read_function_space=W, write_object=u_D_function)
dt = Constant(0)
dt.assign(np.min([fenics_dt, precice_dt]))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression('beta - 2 - 2*alpha', degree=2, alpha=alpha, beta=beta, t=0)
F = u * v / dt * dx + dot(grad(u), grad(v)) * dx - (u_n / dt + f) * v * dx
bcs = [DirichletBC(V, u_D, remaining_boundary)]
# Set boundary conditions at coupling interface once wrt to the coupling
# expression
coupling_expression = precice.create_coupling_expression()
if problem is ProblemType.DIRICHLET:
# modify Dirichlet boundary condition on coupling interface
bcs.append(DirichletBC(V, coupling_expression, coupling_boundary))
if problem is ProblemType.NEUMANN:
# modify Neumann boundary condition on coupling interface, modify weak
# form correspondingly
F += v * coupling_expression * dolfin.ds
a, L = lhs(F), rhs(F)
def k(u, w):
return fnc.inner(sigma(u), fnc.sym(fnc.grad(w))) * fnc.dx
inclusion = Inclusion_3d()
point0 = "near(x[0], 0) && near(x[1], 0) && near(x[2], 0)"
V = FunctionSpace(mesh, "CG", order, constrained_domain=PeriodicBoundary(dim))
# setting the elements that lies in inclusion and in matrix phase
domains = MeshFunction("size_t", mesh, dim)
domains.set_all(0)
inclusion.mark(domains, 1)
dx = Measure('dx', subdomain_data=domains)
def bilf(up, vp): # bilinear form
return inner(mat * up, vp) * dx(0) + inner(inc * up, vp) * dx(1)
bc0 = DirichletBC(V, Constant(0.), point0, method='pointwise')
# SOLVER
u = TrialFunction(V)
v = TestFunction(V)
uE = Function(V)
solve(bilf(grad(u), grad(v)) == -bilf(E, grad(v)), uE, bcs=[bc0])
# POSTPROCESSING evaluation of guaranteed bound
AH11 = assemble(bilf(grad(uE) + E, grad(uE) + E))
print('homogenised component A11 = {} (FEM)'.format(AH11))
print('END')
def staggered_solve(self):
self.U = fe.VectorFunctionSpace(self.mesh, 'CG', 1)
self.W = fe.FunctionSpace(self.mesh, 'CG', 1)
self.WW = fe.FunctionSpace(self.mesh, 'DG', 0)
self.eta = fe.TestFunction(self.U)
self.zeta = fe.TestFunction(self.W)
del_x = fe.TrialFunction(self.U)
del_d = fe.TrialFunction(self.W)
self.x_new = fe.Function(self.U)
self.d_new = fe.Function(self.W)
x_old = fe.Function(self.U)
d_old = fe.Function(self.W)
self.H_old = fe.Function(self.WW)
self.build_weak_form_staggered()
J_u = fe.derivative(self.G_u, self.x_new, del_x)
J_d = fe.derivative(self.G_d, self.d_new, del_d)
self.set_bcs_staggered()
p_u = fe.NonlinearVariationalProblem(self.G_u, self.x_new, self.BC_u,
J_u)
p_d = fe.NonlinearVariationalProblem(self.G_d, self.d_new, self.BC_d,
J_d)
solver_u = fe.NonlinearVariationalSolver(p_u)
solver_d = fe.NonlinearVariationalSolver(p_d)
vtkfile_u = fe.File('data/pvd/{}/u.pvd'.format(self.case_name))
vtkfile_d = fe.File('data/pvd/{}/d.pvd'.format(self.case_name))
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_u.parameters['newton_solver']
newton_prm['maximum_iterations'] = 100
newton_prm['absolute_tolerance'] = 1e-4
newton_prm['relaxation_parameter'] = rp
iteration = 0
err = 1.
while err > self.staggered_tol:
iteration += 1
solver_d.solve()
solver_u.solve()
err_u = fe.errornorm(self.x_new,
x_old,
norm_type='l2',
mesh=None)
err_d = fe.errornorm(self.d_new,
d_old,
norm_type='l2',
mesh=None)
err = max(err_u, err_d)
x_old.assign(self.x_new)
d_old.assign(self.d_new)
print(
'---------------------------------------------------------------------------------'
)
print('>> iteration. {}, error = {:.5}'.format(iteration, err))
print(
'---------------------------------------------------------------------------------'
)
if err < self.staggered_tol or iteration >= self.staggered_maxiter:
print(
'================================================================================='
)
print('\n')
self.x_new.rename("u", "u")
self.d_new.rename("d", "d")
vtkfile_u << self.x_new
vtkfile_d << self.d_new
break
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)
# old results
uold = fn.interpolate(uinit, Mh)
vold = fn.interpolate(fn.Constant(b * pow(a + b, -2)), Mh)
# computation parameters
hsize = 1.0 / nps
t = 0.0
CFL = 5.0
dt = CFL * hsize * hsize
T = 2.0
frequencySave = 100
# weak form
Left = u/dt * uT * fn.dx + v/dt * vT * fn.dx \
+ c1 * fn.inner(fn.grad(u), fn.grad(uT)) * fn.dx \
+ c2 * fn.inner(fn.grad(v), fn.grad(vT)) * fn.dx
Right = uold * uT/dt * fn.dx + vold * vT/dt * fn.dx \
+ d * (a - uold + uold * uold * vold) * uT * fn.dx \
+ d * (b - uold * uold * vold) * vT * fn.dx
AA = fn.assemble(Left)
solver = fn.LUSolver(AA)
solver.parameters["reuse_factorization"] = True
# time loop
inc = 0
while (t <= T):
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(
'================================================================================='
)
def update_history(self):
psi_new = self.psi_plus(strain(fe.grad(self.x_new)))
return psi_new
def get_new_value(self, r: fenics.Function) -> fenics.Expression:
return 2.0*self.lame_mu*fenics.sym(fenics.grad(r)) \
+ self.lame_lambda*fenics.tr(fenics.sym(fenics.grad(r)))*fenics.Identity(len(r))
def boundary(x, on_boundary):
return on_boundary
bc = fs.DirichletBC(V, u_D, boundary)
# Define initial value
u_0 = fs.Expression('exp(-a * pow(x[0], 2) - a * pow(x[1], 2))', degree=2, a=5)
u_n = fs.interpolate(u_0, V)
# Define variational problem
u = fs.TrialFunction(V)
v = fs.TestFunction(V)
f = fs.Constant(0)
F = u * v * fs.dx + dt * fs.dot(fs.grad(u), fs.grad(v)) *fs.dx - (u_n + dt * f) * v * fs.dx
a, L = fs.lhs(F), fs.rhs(F)
# Create VTK file for saving solution
vtkfile = fs.File('heat_gaussian/solution.pvd')
# Time-stepping
u = fs.Function(V)
t = 0
for n in range(num_steps):
# Update current time
t += dt
# Compute solution
fs.solve(a == L, u, bc)
def ab_on_F(d_, w_, k):
F1 = grad(1.5 * d_["n-1"] - 0.5 * d_["n-2"]) - Identity(2)
F2 = 0.5 * grad(d_["n"] + d_["n-1"])
E = F1.T * F2 - I
return 0.5 * E
def test_formulation_1_trap_2_materials():
'''
Test function formulation() with 1 intrinsic trap
and 2 materials
'''
def create_subdomains(x1, x2):
class domain(FESTIM.SubDomain):
def inside(self, x, on_boundary):
return x[0] >= x1 and x[0] <= x2
domain = domain()
return domain
dt = 1
traps = [{
"energy": 1,
"density": 2,
"materials": [1, 2]
}]
materials = [{
"alpha": 1,
"beta": 2,
"density": 3,
"borders": [0, 0.5],
"E_diff": 4,
"D_0": 5,
"id": 1
},
{
"alpha": 2,
"beta": 3,
"density": 4,
"borders": [0.5, 1],
"E_diff": 5,
"D_0": 6,
"id": 2
}]
extrinsic_traps = []
mesh = fenics.UnitIntervalMesh(10)
mf = fenics.MeshFunction("size_t", mesh, 1, 1)
mat1 = create_subdomains(0, 0.5)
mat2 = create_subdomains(0.5, 1)
mat1.mark(mf, 1)
mat2.mark(mf, 2)
V = fenics.VectorFunctionSpace(mesh, 'P', 1, 2)
u = fenics.Function(V)
u_n = fenics.Function(V)
v = fenics.TestFunction(V)
solutions = list(fenics.split(u))
previous_solutions = list(fenics.split(u_n))
testfunctions = list(fenics.split(v))
mf = fenics.MeshFunction('size_t', mesh, 1, 1)
dx = fenics.dx(subdomain_data=mf)
temp = fenics.Expression("300", degree=0)
flux_ = fenics.Expression("1", degree=0)
F, expressions = FESTIM.formulation(
traps, extrinsic_traps, solutions, testfunctions,
previous_solutions, dt, dx, materials, temp, flux_)
# Transient sol
expected_form = ((solutions[0] - previous_solutions[0]) / dt) * \
testfunctions[0]*dx
# Diffusion sol mat 1
expected_form += 5 * fenics.exp(-4/8.6e-5/temp) * \
fenics.dot(
fenics.grad(solutions[0]), fenics.grad(testfunctions[0]))*dx(1)
# Diffusion sol mat 2
expected_form += 6 * fenics.exp(-5/8.6e-5/temp) * \
fenics.dot(
fenics.grad(solutions[0]), fenics.grad(testfunctions[0]))*dx(2)
# Source sol
expected_form += -flux_*testfunctions[0]*dx
# Transient trap 1
expected_form += ((solutions[1] - previous_solutions[1]) / dt) * \
testfunctions[1]*dx
# Trapping trap 1 mat 1
expected_form += - 5 * fenics.exp(-4/8.6e-5/temp)/1/1/2 * \
solutions[0] * (2 - solutions[1]) * \
testfunctions[1]*dx(1)
# Trapping trap 1 mat 2
expected_form += - 6 * fenics.exp(-5/8.6e-5/temp)/2/2/3 * \
solutions[0] * (2 - solutions[1]) * \
testfunctions[1]*dx(2)
# Detrapping trap 1 mat 1
expected_form += 1e13*fenics.exp(-1/8.6e-5/temp)*solutions[1] * \
testfunctions[1]*dx(1)
# Detrapping trap 1 mat 2
expected_form += 1e13*fenics.exp(-1/8.6e-5/temp)*solutions[1] * \
testfunctions[1]*dx(2)
# Source detrapping sol
expected_form += ((solutions[1] - previous_solutions[1]) / dt) * \
testfunctions[0]*dx
assert expected_form.equals(F) is True
def solve(self):
# TODO: when FEniCS ported to Python3, this should be exist_ok
try:
os.makedirs('results')
except OSError:
pass
z, w = (self.z, self.w)
u0 = d.Constant(0.0)
# Define the linear and bilinear forms
L = u0 * w * dx
# Define useful functions
cond = d.Function(self.DV)
U = d.Function(self.V)
# Initialize the max_e vector, that will store the cumulative max e values
max_e = d.Function(self.V)
max_e.vector()[:] = 0.0
max_e.rename("max_E", "Maximum energy deposition by location")
max_e_file = d.File("results/%s-max_e.pvd" % input_mesh)
max_e_per_step = d.Function(self.V)
max_e_per_step_file = d.File("results/%s-max_e_per_step.pvd" % input_mesh)
self.es = {}
self.max_es = {}
fi = d.File("results/%s-cond.pvd" % input_mesh)
potential_file = d.File("results/%s-potential.pvd" % input_mesh)
# Loop through the voltages and electrode combinations
for i, (anode, cathode, voltage) in enumerate(v.electrode_triples):
print("Electrodes %d (%lf) -> %d (0)" % (anode, voltage, cathode))
cond = d.project(self.sigma_start, V=self.DV)
# Define the Dirichlet boundary conditions on the active needles
uV = d.Constant(voltage)
term1_bc = d.DirichletBC(self.V, uV, self.patches, v.needles[anode])
term2_bc = d.DirichletBC(self.V, u0, self.patches, v.needles[cathode])
e = d.Function(self.V)
e.vector()[:] = max_e.vector()
# Re-evaluate conductivity
self.increase_conductivity(cond, e)
for j in range(v.max_restarts):
# Update the bilinear form
a = d.inner(d.nabla_grad(z), cond * d.nabla_grad(w)) * dx
# Solve again
print(" [solving...")
d.solve(a == L, U, bcs=[term1_bc, term2_bc])
print(" ....solved]")
# Extract electric field norm
for k in range(len(U.vector())):
if N.isnan(U.vector()[k]):
U.vector()[k] = 1e5
e_new = d.project(d.sqrt(d.dot(d.grad(U), d.grad(U))), self.V)
# Take the max of the new field and the established electric field
e.vector()[:] = N.array([max(*X) for X in zip(e.vector(), e_new.vector())])
# Re-evaluate conductivity
fi << cond
self.increase_conductivity(cond, e)
potential_file << U
# Save the max e function to a VTU
max_e_per_step.vector()[:] = e.vector()[:]
max_e_per_step_file << max_e_per_step
# Store this electric field norm, for this triple, for later reference
self.es[i] = e
# Store the max of this electric field norm and that for all previous triples
max_e_array = N.array([max(*X) for X in zip(max_e.vector(), e.vector())])
max_e.vector()[:] = max_e_array
# Create a new max_e function for storage, or it will be overwritten by the next iteration
max_e_new = d.Function(self.V)
max_e_new.vector()[:] = max_e_array
# Store this max e function for the cumulative coverage curve calculation later
self.max_es[i] = max_e_new
# Save the max e function to a VTU
max_e_file << max_e
self.max_e_count = i