def _solveTangentProblem(self, dstate, m_hat, x):
"""
The predictor step for the parameter continuation algorithm.
"""
state_fun = vector2Function(x[STATE], self.Vh[STATE])
statel_fun = vector2Function(x[STATE], self.Vh[STATE])
param_fun = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
param_hat_fun = vector2Function(m_hat, self.Vh[PARAMETER])
x_test = [
dl.TestFunction(self.Vh[STATE]),
dl.TestFunction(self.Vh[PARAMETER])
]
x_trial = [
dl.TrialFunction(self.Vh[STATE]),
dl.TrialFunction(self.Vh[PARAMETER])
]
res = self.model.residual(state_fun, x_test[STATE], dl.Constant(0.),
param_fun, None, statel_fun, True)
Aform = dl.derivative(res, state_fun, x_trial[STATE]) + dl.derivative(
res, statel_fun, x_trial[STATE])
b_form = dl.derivative(res, param_fun, param_hat_fun)
A, b = dl.assemble_system(Aform,
b_form,
self.bcs0,
form_compiler_parameters=self.fcp)
solver = dl.PETScLUSolver()
solver.set_operator(A)
solver.solve(dstate, -b)
def __init__(self, model, particle, variation, kernel, options, comm):
self.model = model
self.particle = particle # number of particles
self.variation = variation
self.kernel = kernel
self.options = options
self.comm = comm
self.rank = comm.Get_rank()
self.nproc = comm.Get_size()
self.save_kernel = options["save_kernel"]
self.type_Hessian = options["type_Hessian"]
self.ncalls = 0
self.gls = GlobalLocalSwitch(self.model, self.particle, self.options,
self.comm)
if not self.options["is_projection"]:
if self.particle.number_particles_all == 1:
z_trial = dl.TrialFunction(self.gls.Vh_Local)
z_test = dl.TestFunction(self.gls.Vh_Local)
self.Hessian = dl.assemble(z_trial * z_test * dl.dx)
else:
z_vec = [None] * particle.number_particles_all
c_vec = [None] * particle.number_particles_all
for n in range(particle.number_particles_all):
z_trial = dl.TrialFunction(self.gls.Vh_Global.sub(0))
z_test = dl.TestFunction(self.gls.Vh_Global.sub(0))
z_vec[n] = z_trial
c_vec[n] = z_test
z_vec = dl.as_vector(z_vec)
c_vec = dl.as_vector(c_vec)
self.Hessian = dl.assemble(dl.dot(c_vec, z_vec) * dl.dx)
def __init__(self,
parameters,
mesh_name,
facet_name,
bc_dict={
"obstacle": 2,
"channel_walls": 1,
"inlet": 3,
"outlet": 4
}):
"""
Create the required function spaces, functions and boundary conditions
for a channel flow problem
"""
self.bc_dict = bc_dict
self.mesh = df.Mesh()
with df.XDMFFile(mesh_name) as infile:
infile.read(self.mesh)
mvc = df.MeshValueCollection("size_t", self.mesh,
self.mesh.topology().dim() - 1)
with df.XDMFFile(facet_name) as infile:
infile.read(mvc, "name_to_read")
self.mf = mf = df.cpp.mesh.MeshFunctionSizet(self.mesh, mvc)
self.V = V = df.VectorFunctionSpace(self.mesh, 'P',
parameters["degree velocity"])
self.Q = Q = df.FunctionSpace(self.mesh, 'P',
parameters["degree pressure"])
self.rho = df.Constant(parameters["density [kg/m3]"])
self.mu = df.Constant(parameters["viscosity [Pa*s]"])
self.dt = df.Constant(parameters["dt [s]"])
self.g = df.Constant((0, 0))
self.vu, self.vp = df.TestFunction(V), df.TestFunction(Q)
self.u_, self.p_ = df.Function(V), df.Function(Q)
self.u_1, self.p_1 = df.Function(V), df.Function(Q)
self.u_k, self.p_k = df.Function(V), df.Function(Q)
self.u, self.p = df.TrialFunction(V), df.TrialFunction(Q) # unknown!
self.U_m = U_m = parameters["velocity [m/s]"]
x = [0, .41 / 2] # center of the channel
Ucenter = 4. * U_m * x[1] * (.41 - x[1]) / (.41 * .41)
U0_str = "4.*U_m*x[1]*(.41-x[1])/(.41*.41)"
self.U_mean = np.mean(2 / 3 * Ucenter)
U0 = df.Expression((U0_str, "0"), U_m=U_m, degree=2)
bc0 = df.DirichletBC(V, df.Constant((0, 0)), mf, bc_dict["obstacle"])
bc1 = df.DirichletBC(V, df.Constant((0, 0)), mf,
bc_dict["channel_walls"])
bc2 = df.DirichletBC(V, U0, mf, bc_dict["inlet"])
bc3 = df.DirichletBC(Q, df.Constant(0), mf, bc_dict["outlet"])
self.bcu = [bc0, bc1, bc2]
self.bcp = [bc3]
self.ds_ = df.Measure("ds", domain=self.mesh, subdomain_data=mf)
return
def setUp(self):
mesh = dl.UnitSquareMesh(10, 10)
self.mpi_rank = dl.MPI.rank(mesh.mpi_comm())
self.mpi_size = dl.MPI.size(mesh.mpi_comm())
Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
uh, vh = dl.TrialFunction(Vh1), dl.TestFunction(Vh1)
mh, test_mh = dl.TrialFunction(Vh1), dl.TestFunction(Vh1)
## Set up B
ndim = 2
ntargets = 200
np.random.seed(seed=1)
targets = np.random.uniform(0.1, 0.9, [ntargets, ndim])
B = assemblePointwiseObservation(Vh1, targets)
## Set up Asolver
alpha = dl.Constant(1.0)
varfA = ufl.inner(ufl.grad(uh), ufl.grad(vh))*ufl.dx +\
alpha*ufl.inner(uh,vh)*ufl.dx
A = dl.assemble(varfA)
Asolver = PETScKrylovSolver(A.mpi_comm(), "cg", amg_method())
Asolver.set_operator(A)
Asolver.parameters["maximum_iterations"] = 100
Asolver.parameters["relative_tolerance"] = 1e-12
## Set up C
varfC = ufl.inner(mh, vh) * ufl.dx
C = dl.assemble(varfC)
self.Hop = Hop(B, Asolver, C)
## Set up RHS Matrix M.
varfM = ufl.inner(mh, test_mh) * ufl.dx
self.M = dl.assemble(varfM)
self.Minv = PETScKrylovSolver(self.M.mpi_comm(), "cg", amg_method())
self.Minv.set_operator(self.M)
self.Minv.parameters["maximum_iterations"] = 100
self.Minv.parameters["relative_tolerance"] = 1e-12
myRandom = Random(self.mpi_rank, self.mpi_size)
x_vec = dl.Vector(mesh.mpi_comm())
self.Hop.init_vector(x_vec, 1)
k_evec = 10
p_evec = 50
self.Omega = MultiVector(x_vec, k_evec + p_evec)
self.k_evec = k_evec
myRandom.normal(1., self.Omega)
def setup_fnc_spaces(self):
"""
Call me again if the mesh is modified, like after adaptive meshing.
"""
# Linear Lagrange triangles for the velocity.
self.v_fnc_space = dfn.FunctionSpace(self.mesh, "CG", 1)
self.v = dfn.TrialFunction(self.v_fnc_space)
self.vt = dfn.TestFunction(self.v_fnc_space)
# We use piecewise constant discontinuous elements so that
# no matrix inversion is required in order to update the stress
self.S_fnc_space = dfn.VectorFunctionSpace(self.mesh, "DG", 0)
self.S = dfn.TrialFunction(self.S_fnc_space)
self.St = dfn.TestFunction(self.S_fnc_space)
def __init__(self, adjoint = False):
self.dt = dol.Constant(0)
self.uold = dol.interpolate(self.u0_expr, self.V)
self.u0 = dol.interpolate(self.u0_expr, self.V)
self.unew = dol.Function(self.V)
self.ulow = dol.Function(self.V)
self.v = dol.TestFunction(self.V)
self.utrial = dol.TrialFunction(self.V)
if adjoint:
self.zold = dol.interpolate(self.z0_expr, self.V)
self.z0 = dol.interpolate(self.z0_expr, self.V)
self.znew = dol.Function(self.V)
self.ztrial = dol.TrialFunction(self.V)
def testblockdiagonal():
mesh = dl.UnitSquareMesh(40, 40)
V1 = dl.FunctionSpace(mesh, "Lagrange", 2)
test1, trial1 = dl.TestFunction(V1), dl.TrialFunction(V1)
V2 = dl.FunctionSpace(mesh, "Lagrange", 2)
test2, trial2 = dl.TestFunction(V2), dl.TrialFunction(V2)
V1V2 = createMixedFS(V1, V2)
test12, trial12 = dl.TestFunction(V1V2), dl.TrialFunction(V1V2)
bd = BlockDiagonal(V1, V2, mesh.mpi_comm())
mpirank = dl.MPI.rank(mesh.mpi_comm())
if mpirank == 0: print 'mass+mass'
M1 = dl.assemble(dl.inner(test1, trial1) * dl.dx)
M2 = dl.assemble(dl.inner(test1, trial2) * dl.dx)
M12bd = bd.assemble(M1, M2)
M12 = dl.assemble(dl.inner(test12, trial12) * dl.dx)
diff = M12bd - M12
nn = diff.norm('frobenius')
if mpirank == 0:
print '\nnn={}'.format(nn)
if mpirank == 0: print 'mass+2ndD'
D2 = dl.assemble(
dl.inner(dl.nabla_grad(test1), dl.nabla_grad(trial2)) * dl.dx)
M1D2bd = bd.assemble(M1, D2)
tt1, tt2 = test12
tl1, tl2 = trial12
M1D2 = dl.assemble(
dl.inner(tt1, tl1) * dl.dx +
dl.inner(dl.nabla_grad(tt2), dl.nabla_grad(tl2)) * dl.dx)
diff = M1D2bd - M1D2
nn = diff.norm('frobenius')
if mpirank == 0:
print 'nn={}'.format(nn)
if mpirank == 0: print 'wM+wM'
u11 = dl.interpolate(dl.Expression("x[0]*x[1]", degree=10), V1)
u22 = dl.interpolate(dl.Expression("11+x[0]+x[1]", degree=10), V2)
M1 = dl.assemble(dl.inner(u11 * test1, trial1) * dl.dx)
M2 = dl.assemble(dl.inner(u22 * test1, trial2) * dl.dx)
M12bd = bd.assemble(M1, M2)
ua, ub = dl.interpolate(dl.Expression(("x[0]*x[1]", "11+x[0]+x[1]"),\
degree=10), V1V2)
M12 = dl.assemble(
dl.inner(ua * tt1, tl1) * dl.dx + dl.inner(ub * tt2, tl2) * dl.dx)
diff = M12bd - M12
nn = diff.norm('frobenius')
if mpirank == 0:
print 'nn={}'.format(nn)
def get_initial_conditions(self,rho_solute=const.rhom):
"""
Set the initial condition at the end of melting (melting can be solved analytically
"""
# --- Initial states --- #
# ice temperature
self.u0_i = dolfin.Function(self.ice_V)
T,lam,self.R_melt,self.t_melt = analyticalMelt(np.exp(self.ice_coords[:,0])*self.R_melt,self.T_inf,self.Q_initialize,R_target=self.R_melt)
self.u0_i.vector()[:] = T/abs(self.T_inf)
# solution temperature
self.u0_s = dolfin.Function(self.sol_V)
T,lam,self.R_melt,self.t_melt = analyticalMelt(np.exp(self.sol_coords[:,0])*self.R_melt,self.T_inf,self.Q_initialize,R_target=self.R_melt)
self.u0_s.vector()[:] = T/abs(self.T_inf)
# solution concentration
self.u0_c = dolfin.interpolate(dolfin.Constant(self.C_init),self.sol_V)
# --- Time Array --- #
# Now that we have the melt-out time, we can define the time array
self.ts = np.arange(self.t_melt,self.t_final+self.dt,self.dt)/self.t0
self.dt /= self.t0
# Define the ethanol source
self.source_timing = self.ts[np.argmin(abs(self.ts-self.source_timing/self.t0))]
if 'gaussian_source' in self.flags:
self.source_duration /= self.t0
self.source = self.source_mass_final/(self.source_duration*np.sqrt(np.pi))*np.exp(-((self.ts-self.source_timing)/self.source_duration)**2.)
else:
self.source = np.zeros_like(self.ts)
self.source[np.argmin(abs(self.ts-self.source_timing))] = self.source_mass_final/self.dt
# --- Define the test and trial functions --- #
self.u_i = dolfin.TrialFunction(self.ice_V)
self.v_i = dolfin.TestFunction(self.ice_V)
self.T_i = dolfin.Function(self.ice_V)
self.u_s = dolfin.TrialFunction(self.sol_V)
self.v_s = dolfin.TestFunction(self.sol_V)
self.T_s = dolfin.Function(self.sol_V)
self.C = dolfin.Function(self.sol_V)
self.Tf = Tf_depression(self.C,linear=True)/abs(self.T_inf)
self.Tf_wall = dolfin.project(self.Tf,self.sol_V).vector()[self.sol_idx_wall]
# Get the updated solution properties
self.rhos = dolfin.project(dolfin.Expression('C + rhow*(1.-C/rho)',degree=1,C=self.C,rhow=const.rhow,rho=rho_solute),self.sol_V)
self.cs = dolfin.project(dolfin.Expression('ce*(C/rho) + cw*(1.-C/rho)',degree=1,C=self.C,cw=const.cw,ce=const.ce,rho=rho_solute),self.sol_V)
self.ks = dolfin.project(dolfin.Expression('ke*(C/rho) + kw*(1.-C/rho)',degree=1,C=self.C,kw=const.kw,ke=const.ke,rho=rho_solute),self.sol_V)
self.rhos_wall = self.rhos.vector()[self.sol_idx_wall]
self.cs_wall = self.cs.vector()[self.sol_idx_wall]
self.ks_wall = self.ks.vector()[self.sol_idx_wall]
self.flags.append('get_ic')
def setUp(self):
mesh = dl.UnitSquareMesh(10, 10)
self.mpi_rank = dl.MPI.rank(mesh.mpi_comm())
self.mpi_size = dl.MPI.size(mesh.mpi_comm())
Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
uh, vh = dl.TrialFunction(Vh1), dl.TestFunction(Vh1)
mh = dl.TrialFunction(Vh1)
# Define B
ndim = 2
ntargets = 10
np.random.seed(seed=1)
targets = np.random.uniform(0.1, 0.9, [ntargets, ndim])
B = assemblePointwiseObservation(Vh1, targets)
# Define Asolver
alpha = dl.Constant(0.1)
varfA = dl.inner(dl.nabla_grad(uh), dl.nabla_grad(vh))*dl.dx +\
alpha*dl.inner(uh,vh)*dl.dx
A = dl.assemble(varfA)
Asolver = dl.PETScKrylovSolver(A.mpi_comm(), "cg", amg_method())
Asolver.set_operator(A)
Asolver.parameters["maximum_iterations"] = 100
Asolver.parameters["relative_tolerance"] = 1e-12
# Define M
varfC = dl.inner(mh, vh) * dl.dx
C = dl.assemble(varfC)
self.J = J_op(B, Asolver, C)
self.k_evec = 10
p_evec = 50
myRandom = Random(self.mpi_rank, self.mpi_size)
x_vec = dl.Vector(C.mpi_comm())
C.init_vector(x_vec, 1)
self.Omega = MultiVector(x_vec, self.k_evec + p_evec)
myRandom.normal(1., self.Omega)
y_vec = dl.Vector(C.mpi_comm())
B.init_vector(y_vec, 0)
self.Omega_adj = MultiVector(y_vec, self.k_evec + p_evec)
myRandom.normal(1., self.Omega_adj)
def delta_dg(mesh, expr):
CG = df.FunctionSpace(mesh, "CG", 1)
DG = df.FunctionSpace(mesh, "DG", 0)
CG3 = df.VectorFunctionSpace(mesh, "CG", 1)
DG3 = df.VectorFunctionSpace(mesh, "DG", 0)
m = df.interpolate(expr, DG3)
n = df.FacetNormal(mesh)
u_dg = df.TrialFunction(DG)
v_dg = df.TestFunction(DG)
u3 = df.TrialFunction(CG3)
v3 = df.TestFunction(CG3)
a = u_dg * df.inner(v3, n) * df.ds - u_dg * df.div(v3) * df.dx
mm = m.vector().array()
mm.shape = (3, -1)
K = df.assemble(a).array()
L3 = df.assemble(df.dot(v3, df.Constant([1, 1, 1])) * df.dx).array()
f = np.dot(K, mm[0]) / L3
fun1 = df.Function(CG3)
fun1.vector().set_local(f)
df.plot(fun1)
a = df.div(u3) * v_dg * df.dx
A = df.assemble(a).array()
L = df.assemble(v_dg * df.dx).array()
h = np.dot(A, f) / L
fun = df.Function(DG)
fun.vector().set_local(h)
df.plot(fun)
res = []
for x in xs:
res.append(fun(x, 5, 0.5))
"""
fun2 =df.interpolate(fun, df.VectorFunctionSpace(mesh, "CG", 1))
file = df.File('field2.pvd')
file << fun2
"""
return res
def _get_rtmass(self,output_petsc=True):
"""
Get the square root of assembled mass matrix M using lumping.
--credit to: Umberto Villa
"""
test = df.TestFunction(self.V)
V_deg = self.V.ufl_element().degree()
try:
V_q_fe = df.FiniteElement('Quadrature',self.V.mesh().ufl_cell(),2*V_deg,quad_scheme='default')
V_q = df.FunctionSpace(self.V.mesh(),V_q_fe)
except:
print('Use FiniteElement in specifying FunctionSpace after version 1.6.0!')
V_q = df.FunctionSpace(self.V.mesh(), 'Quadrature', 2*self.V._FunctionSpace___degree)
trial_q = df.TrialFunction(V_q)
test_q = df.TestFunction(V_q)
M_q = df.PETScMatrix()
df.assemble(trial_q*test_q*df.dx,tensor=M_q,form_compiler_parameters={'quadrature_degree': 2*V_deg})
ones = df.interpolate(df.Constant(1.), V_q).vector()
dM_q = M_q*ones
M_q.zero()
dM_q_2fill = ones.array() / np.sqrt(dM_q.array() )
dM_q.set_local( dM_q_2fill )
M_q.set_diagonal(dM_q)
mixedM = df.PETScMatrix()
df.assemble(trial_q*test*df.dx,tensor=mixedM,form_compiler_parameters={'quadrature_degree': 2*V_deg})
if output_petsc and df.has_petsc4py():
rtM = df.PETScMatrix(df.as_backend_type(mixedM).mat().matMult(df.as_backend_type(M_q).mat()))
else:
rtM = sps.csr_matrix(mixedM.array()*dM_q_2fill)
csr_trim0(rtM,1e-12)
return rtM
def __init__(self, simulation, u_conv):
"""
Given a velocity in DG, e.g DG2, produce a velocity in DGT0,
i.e. a constant on each facet
"""
V = u_conv[0].function_space()
V_dgt0 = dolfin.FunctionSpace(V.mesh(), 'DGT', 0)
u = dolfin.TrialFunction(V_dgt0)
v = dolfin.TestFunction(V_dgt0)
ndim = simulation.ndim
w = u_conv
w_new = dolfin.as_vector(
[dolfin.Function(V_dgt0) for _ in range(ndim)])
dot, avg, dS, ds = dolfin.dot, dolfin.avg, dolfin.dS, dolfin.ds
a = dot(avg(u), avg(v)) * dS + dot(u, v) * ds
L = []
for d in range(ndim):
L.append(avg(w[d]) * avg(v) * dS + w[d] * v * ds)
self.lhs = [dolfin.Form(Li) for Li in L]
self.A = dolfin.assemble(a)
self.solver = dolfin.PETScKrylovSolver('cg')
self.velocity = simulation.data['u_conv_dgt0'] = w_new
def test_basic_interior_facet_assembly():
ghost_mode = dolfin.cpp.mesh.GhostMode.none
if (dolfin.MPI.size(dolfin.MPI.comm_world) > 1):
ghost_mode = dolfin.cpp.mesh.GhostMode.shared_facet
mesh = dolfin.RectangleMesh(
dolfin.MPI.comm_world,
[numpy.array([0.0, 0.0, 0.0]),
numpy.array([1.0, 1.0, 0.0])], [5, 5],
cell_type=dolfin.cpp.mesh.CellType.Type.triangle,
ghost_mode=ghost_mode)
V = dolfin.function.FunctionSpace(mesh, ("DG", 1))
u, v = dolfin.TrialFunction(V), dolfin.TestFunction(V)
a = ufl.inner(ufl.avg(u), ufl.avg(v)) * ufl.dS
A = dolfin.fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
L = ufl.conj(ufl.avg(v)) * ufl.dS
b = dolfin.fem.assemble_vector(L)
b.assemble()
assert isinstance(b, PETSc.Vec)
def linear_solver(self, phi_k):
# cast params as constant functions so that, if they are set to 0, FEniCS still understand
# what is being integrated
mu, M = Constant(self.physics.mu), Constant(self.physics.M)
lam = Constant(self.physics.lam)
mn, mf = Constant(self.mn), Constant(self.mf)
D = self.physics.D
# boundary condition
Dirichlet_bc = self.get_Dirichlet_bc()
# trial and test function
phi = d.TrialFunction(self.fem.S)
v = d.TestFunction(self.fem.S)
# r^(D-1)
rD = Expression('pow(x[0],D-1)', D=D, degree=self.fem.func_degree)
# bilinear form a and linear form L
a = - inner( grad(phi), grad(v) ) * rD * dx + ( (mu/mn)**2 \
- 3.*lam*(mf/mn)**2*phi_k**2 ) * phi * v * rD * dx
L = ((mn**(D - 2.) / (mf * M)) * self.source.rho - 2. * lam *
(mf / mn)**2 * phi_k**3) * v * rD * dx
# define a vector with the solution
sol = d.Function(self.fem.S)
# solve linearised system
pde = d.LinearVariationalProblem(a, L, sol, Dirichlet_bc)
solver = d.LinearVariationalSolver(pde)
solver.solve()
return sol
def assembleA(self, x, assemble_adjoint=False, assemble_rhs=False):
"""
Assemble the matrices and rhs for the forward/adjoint problems
"""
trial = dl.TrialFunction(self.Vh[STATE])
test = dl.TestFunction(self.Vh[STATE])
c = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
Avarf = dl.inner(
dl.exp(c) * dl.nabla_grad(trial), dl.nabla_grad(test)) * dl.dx
if not assemble_adjoint:
bform = dl.inner(self.f, test) * dl.dx
Matrix, rhs = dl.assemble_system(Avarf, bform, self.bc)
else:
# Assemble the adjoint of A (i.e. the transpose of A)
s = vector2Function(x[STATE], self.Vh[STATE])
bform = dl.inner(dl.Constant(0.), test) * dl.dx
Matrix, _ = dl.assemble_system(dl.adjoint(Avarf), bform, self.bc0)
Bu = -(self.B * x[STATE])
Bu += self.u_o
rhs = dl.Vector()
self.B.init_vector(rhs, 1)
self.B.transpmult(Bu, rhs)
rhs *= 1.0 / self.noise_variance
if assemble_rhs:
return Matrix, rhs
else:
return Matrix
def run_model(kappa,forcing,function_space,boundary_conditions=None):
"""
Solve complex valued Helmholtz equation by solving coupled system, one
for the real part of the solution one for the imaginary part.
"""
mesh = function_space.mesh()
kappa_sq = kappa**2
if boundary_conditions==None:
bndry_obj = dl.CompiledSubDomain("on_boundary")
boundary_conditions = [['dirichlet',bndry_obj,[0,0]]]
num_bndrys = len(boundary_conditions)
boundaries = mark_boundaries(mesh,boundary_conditions)
dirichlet_bcs = collect_dirichlet_boundaries(
function_space,boundary_conditions,boundaries)
# To express integrals over the boundary parts using ds(i), we must first
# redefine the measure ds in terms of our boundary markers:
ds = dl.Measure('ds', domain=mesh, subdomain_data=boundaries)
#dx = dl.Measure('dx', domain=mesh)
dx = dl.dx
(pr, pi) = dl.TrialFunction(function_space)
(vr, vi) = dl.TestFunction(function_space)
# real part
bilinear_form = kappa_sq*(pr*vr - pi*vi)*dx
bilinear_form += (-dl.inner(dl.nabla_grad(pr),dl.nabla_grad(vr))+
dl.inner(dl.nabla_grad(pi),dl.nabla_grad(vi)))*dx
# imaginary part
bilinear_form += kappa_sq*(pr*vi + pi*vr)*dx
bilinear_form += -(dl.inner(dl.nabla_grad(pr),dl.nabla_grad(vi))+
dl.inner(dl.nabla_grad(pi),dl.nabla_grad(vr)))*dx
for ii in range(num_bndrys):
if (boundary_conditions[ii][0]=='robin'):
alpha_real, alpha_imag = boundary_conditions[ii][3]
bilinear_form -= alpha_real*(pr*vr-pi*vi)*ds(ii)
bilinear_form -= alpha_imag*(pr*vi+pi*vr)*ds(ii)
forcing_real,forcing_imag=forcing
rhs = (forcing_real*vr+forcing_real*vi+forcing_imag*vr-forcing_imag*vi)*dx
for ii in range(num_bndrys):
if ((boundary_conditions[ii][0]=='robin') or
(boundary_conditions[ii][0]=='neumann')):
beta_real, beta_imag=boundary_conditions[ii][2]
# real part of robin boundary conditions
rhs+=(beta_real*vr - beta_imag*vi)*ds(ii)
# imag part of robin boundary conditions
rhs+=(beta_real*vi + beta_imag*vr)*ds(ii)
# compute solution
p = dl.Function(function_space)
#solve(a == L, p)
dl.solve(bilinear_form == rhs, p, bcs=dirichlet_bcs)
return p
def solve_initial_pressure(w_NSp, p, q, u, v, bcs_NSp, M_, g_, phi_, rho_,
rho_e_, V_, drho, sigma_bar, eps, grav, dveps,
enable_PF, enable_EC):
V = u.function_space()
grad_p = df.TrialFunction(V)
grad_p_out = df.Function(V)
F_grad_p = (df.dot(grad_p, v) * df.dx - rho_ * df.dot(grav, v) * df.dx)
if enable_PF:
F_grad_p += -drho * M_ * df.inner(df.grad(u), df.outer(df.grad(g_),
v)) * df.dx
F_grad_p += -sigma_bar * eps * df.inner(
df.outer(df.grad(phi_), df.grad(phi_)), df.grad(v)) * df.dx
if enable_EC and rho_e_ != 0:
F_grad_p += rho_e_ * df.dot(df.grad(V_), v) * df.dx
if enable_PF and enable_EC:
F_grad_p += dveps * df.dot(df.grad(phi_), v) * df.dot(
df.grad(V_), df.grad(V_)) * df.dx
info_red("Solving initial grad_p...")
df.solve(df.lhs(F_grad_p) == df.rhs(F_grad_p), grad_p_out)
F_p = (df.dot(df.grad(q), df.grad(p)) * df.dx -
df.dot(df.grad(q), grad_p_out) * df.dx)
info_red("Solving initial p...")
df.solve(df.lhs(F_p) == df.rhs(F_p), w_NSp, bcs_NSp)
def get_initial_conditions(self,melt_velocity=-1./3600.):
"""
Set the initial condition to no melted ice and all at the bulk ice temperature
Parameters
----------
melt_velocity: float
velocity of the downgoing drill (m/s)
"""
# --- Initial states --- #
# ice temperature
self.u0_i = dolfin.interpolate(dolfin.Constant(self.Tstar),self.ice_V)
# the upward velocity is equal to negative the melt rate (the mesh is Lagrangian following the drill)
self.velocity = dolfin.Expression('melt',melt=melt_velocity*self.t0,degree=1)
# --- Time Array --- #
# Now that we have the melt-out time, we can define the time array
self.ts = np.arange(0.,self.t_final+self.dt,self.dt)/self.t0
self.dt /= self.t0
# --- Define the test and trial functions --- #
self.u_i = dolfin.TrialFunction(self.ice_V)
self.v_i = dolfin.TestFunction(self.ice_V)
self.T_i = dolfin.Function(self.ice_V)
self.flags.append('get_ic')
def solveFwd(self, state, x):
""" Solve the possibly nonlinear forward problem:
Given :math:`m`, find :math:`u` such that
.. math:: \\delta_p F(u, m, p;\\hat{p}) = 0,\\quad \\forall \\hat{p}."""
self.n_calls["forward"] += 1
if self.solver is None:
self.solver = self._createLUSolver()
if self.is_fwd_linear:
u = dl.TrialFunction(self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
A_form = ufl.lhs(res_form)
b_form = ufl.rhs(res_form)
A, b = dl.assemble_system(A_form, b_form, bcs=self.bc)
self.solver.set_operator(A)
self.solver.solve(state, b)
else:
u = vector2Function(x[STATE], self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
dl.solve(res_form == 0, u, self.bc)
state.zero()
state.axpy(1., u.vector())
def make_mass_matrix_unlumped(function_space_V):
V = function_space_V
u_trial = dl.TrialFunction(V)
v_test = dl.TestFunction(V)
mass_form = u_trial * v_test * dl.dx
M = dl.assemble(mass_form)
return M
def solve(self, vkscale=1., lscale=1., sscale=1.):
"""This approach only works for small grids since the memory requirements
rapidly become prohibitive. (nl, nk, nv) = (8, 45, 40) is just barely
doable on my laptop. If the memory is available, computation time is
not significant.
"""
D = d.Function(self.tensor_space) # diffusion tensor
Q = d.Function(self.scalar_space) # source term
u = d.TrialFunction(self.scalar_space) # u is my 'f'
v = d.TestFunction(self.scalar_space)
soln = d.Function(self.scalar_space)
a = d.dot(d.dot(D, d.grad(u)), d.grad(v)) * d.dx
L = Q * v * d.dx
equation = (a == L)
soln = d.Function(self.scalar_space)
bc = d.DirichletBC(self.scalar_space, d.Constant(0), direct_boundary)
dbuf = np.zeros(D.vector().size()).reshape((-1, 3, 3))
dbuf[:,0,0] = self.D_VV * vkscale
dbuf[:,1,0] = self.D_VK * vkscale
dbuf[:,0,1] = self.D_VK * vkscale
dbuf[:,1,1] = self.D_KK * vkscale
dbuf[:,2,2] = self.D_LL * lscale
D.vector()[:] = dbuf.reshape((-1,))
Q.vector()[:] = self.source_term * sscale
d.solve(equation, soln, bc)
return self.to_cube(soln.vector().array())
def _init_forms(self):
logger.debug("Initialize forms mechanics problem")
# Displacement and hydrostatic_pressure
u, p = dolfin.split(self.state)
v, q = dolfin.split(self.state_test)
# Some mechanical quantities
F = dolfin.variable(kinematics.DeformationGradient(u))
J = kinematics.Jacobian(F)
dx = self.geometry.dx
internal_energy = (self.material.strain_energy(F, ) +
self.material.compressibility(p, J))
self._virtual_work = dolfin.derivative(
internal_energy * dx,
self.state,
self.state_test,
)
external_work = self._external_work(u, v)
if external_work is not None:
self._virtual_work += external_work
self._set_dirichlet_bc()
self._jacobian = dolfin.derivative(
self._virtual_work,
self.state,
dolfin.TrialFunction(self.state_space),
)
self._init_solver()
def test_custom_mesh_loop_cffi_rank2(set_vals):
"""Test numba assembler for bilinear form"""
mesh = dolfin.generation.UnitSquareMesh(dolfin.MPI.comm_world, 64, 64)
V = dolfin.FunctionSpace(mesh, ("Lagrange", 1))
# Test against generated code and general assembler
u, v = dolfin.TrialFunction(V), dolfin.TestFunction(V)
a = inner(u, v) * dx
A0 = dolfin.fem.assemble_matrix(a)
A0.assemble()
A0.zeroEntries()
start = time.time()
dolfin.fem.assemble_matrix(A0, a)
end = time.time()
print("Time (C++, pass 2):", end - start)
A0.assemble()
# Unpack mesh and dofmap data
c = mesh.topology.connectivity(2, 0).connections()
pos = mesh.topology.connectivity(2, 0).pos()
geom = mesh.geometry.points
dofs = V.dofmap().dof_array
A1 = A0.copy()
for i in range(2):
A1.zeroEntries()
start = time.time()
assemble_matrix_cffi(A1.handle, (c, pos), geom, dofs, set_vals, PETSc.InsertMode.ADD_VALUES)
end = time.time()
print("Time (Numba, pass {}): {}".format(i, end - start))
A1.assemble()
assert (A1 - A0).norm() == pytest.approx(0.0)
def __init__(self, Vh, dX, bc, form=None):
"""
Constructor:
- Vh: the finite element space for the state variable.
- dX: the integrator on subdomain X where observation are presents.
E.g. dX = dl.dx means observation on all \Omega and dX = dl.ds means observations on all \partial \Omega.
- bc: If the forward problem imposes Dirichlet boundary conditions u = u_D on \Gamma_D;
bc is a dl.DirichletBC object that prescribes homogeneuos Dirichlet conditions u = 0 on \Gamma_D.
- form: if form = None we compute the L^2(X) misfit:
\int_X (u - ud)^2 dX,
otherwise the integrand specified in form will be used.
"""
if form is None:
u, v = dl.TrialFunction(Vh), dl.TestFunction(Vh)
self.W = dl.assemble(dl.inner(u, v) * dX)
else:
self.W = dl.assemble(form)
if bc is not None:
Wt = Transpose(self.W)
self.bc.zero(Wt)
self.W = Transpose(Wt)
self.bc.zero(self.W)
self.d = dl.Vector()
self.W.init_vector(self.d, 1)
self.noise_variance = 0
def __init__(self,V,sigma=1.25,s=0.0625,mean=None,rel_tol=1e-10,max_iter=100,**kwargs):
self.V=V
self.dim=V.dim()
self.dof_coords=get_dof_coords(V)
self.sigma=sigma
self.s=s
self.mean=mean
self.mpi_comm=kwargs['mpi_comm'] if 'mpi_comm' in kwargs else df.mpi_comm_world()
# mass matrix and its inverse
M_form=df.inner(df.TrialFunction(V),df.TestFunction(V))*df.dx
self.M=df.PETScMatrix()
df.assemble(M_form,tensor=self.M)
self.Msolver = df.PETScKrylovSolver("cg", "jacobi")
self.Msolver.set_operator(self.M)
self.Msolver.parameters["maximum_iterations"] = max_iter
self.Msolver.parameters["relative_tolerance"] = rel_tol
self.Msolver.parameters["error_on_nonconvergence"] = True
self.Msolver.parameters["nonzero_initial_guess"] = False
# square root of mass matrix
self.rtM=self._get_rtmass()
# kernel matrix and its square root
self.K=self._get_ker()
self.rtK = _get_sqrtm(self.K,'K')
# set solvers
for op in ['K']:
operator=getattr(self, op)
solver=self._set_solver(operator,op)
setattr(self, op+'solver', solver)
if mean is None:
self.mean=df.Vector()
self.init_vector(self.mean,0)
def custom_newton_solver(self):
V = self.my_function_space_cls.V
F = self.my_var_prob_cls.F
#self.u = my_function_space_cls.u
du = df.TrialFunction(V) #TrialFunctions --> wrong, without 's'
Jac = df.derivative(F, self.u, du)
absolute_tol = 1E-5
relative_tol = 9E-2
u_inc = df.Function(V)
nIter = 0
relative_residual = 1
CONVERGED = True
MAX_ITER = 5000
while relative_residual > relative_tol and nIter < MAX_ITER and CONVERGED:
nIter += 1
A, b = df.assemble_system(Jac, -(F), [])
df.solve(A, u_inc.vector(), b)
#df.solve(F == 0, u_inc.vector(), [], J=Jac, solver_parameters={'newton_solver' : {'linear_solver' : 'mumps'}})
relative_residual = u_inc.vector().norm('l2')
#a = df.assemble(F)
residual = b.norm('l2')
lmbda = 0.8
self.u.vector()[:] += lmbda * u_inc.vector()
if residual > 1000:
CONVERGED = False
print("Did not converge after {} Iterations".format(nIter))
elif residual < absolute_tol:
CONVERGED = False
print("converged after {} Iterations".format(nIter))
print('{0:2d} {1:3.2E} {2:5e}'.format(nIter, relative_residual,
residual))
def test_basic_assembly():
mesh = dolfin.generation.UnitSquareMesh(dolfin.MPI.comm_world, 12, 12)
V = dolfin.FunctionSpace(mesh, ("Lagrange", 1))
u, v = dolfin.TrialFunction(V), dolfin.TestFunction(V)
a = 1.0 * inner(u, v) * dx
L = inner(1.0, v) * dx
# Initial assembly
A = dolfin.fem.assemble(a)
b = dolfin.fem.assemble(L)
assert isinstance(A, dolfin.cpp.la.PETScMatrix)
assert isinstance(b, dolfin.cpp.la.PETScVector)
# Second assembly
A = dolfin.fem.assemble(A, a)
b = dolfin.fem.assemble(b, L)
assert isinstance(A, dolfin.cpp.la.PETScMatrix)
assert isinstance(b, dolfin.cpp.la.PETScVector)
# Function as coefficient
f = dolfin.Function(V)
a = f * inner(u, v) * dx
A = dolfin.fem.assemble(a)
assert isinstance(A, dolfin.cpp.la.PETScMatrix)
def get_distance_function(config, domains):
V = dolfin.FunctionSpace(config.domain.mesh, "CG", 1)
v = dolfin.TestFunction(V)
d = dolfin.TrialFunction(V)
sol = dolfin.Function(V)
s = dolfin.interpolate(Constant(1.0), V)
domains_func = dolfin.Function(dolfin.FunctionSpace(config.domain.mesh, "DG", 0))
domains_func.vector().set_local(domains.array().astype(numpy.float))
def boundary(x):
eps_x = config.params["turbine_x"]
eps_y = config.params["turbine_y"]
min_val = 1
for e_x, e_y in [(-eps_x, 0), (eps_x, 0), (0, -eps_y), (0, eps_y)]:
try:
min_val = min(min_val, domains_func((x[0] + e_x, x[1] + e_y)))
except RuntimeError:
pass
return min_val == 1.0
bc = dolfin.DirichletBC(V, 0.0, boundary)
# Solve the diffusion problem with a constant source term
log(INFO, "Solving diffusion problem to identify feasible area ...")
a = dolfin.inner(dolfin.grad(d), dolfin.grad(v)) * dolfin.dx
L = dolfin.inner(s, v) * dolfin.dx
dolfin.solve(a == L, sol, bc)
return sol
def __init__(me, function_space_V, initial_points=None):
me.V = function_space_V
me.N = me.V.dim()
u_trial = dl.TrialFunction(me.V)
v_test = dl.TestFunction(me.V)
mass_form = u_trial * v_test * dl.dx
me.M = dl.assemble(mass_form)
me.constant_one_function = dl.interpolate(dl.Constant(1.0), me.V)
me.NPPSS = NeumannPoissonSolver(me.V)
me.solve_neumann_poisson = me.NPPSS.solve
me.solve_neumann_point_source = me.NPPSS.solve_point_source
me.points = list()
me.impulse_responses = list()
me.solve_S = lambda x: np.nan
me.eta = np.zeros(me.num_pts)
me.mu = np.nan
me.smooth_basis = list()
me.weighting_functions = list()
if initial_points is not None:
me.add_points(initial_points)
def __init__(self, V, kappa):
'''
Parameters
----------
V : dolfin.FunctionSpace
Function space.
kappa : float
Filter diffusivity constant.
'''
if not isinstance(V, dolfin.FunctionSpace):
raise TypeError(
'Parameter `V` must be of type `dolfin.FunctionSpace`.')
v = dolfin.TestFunction(V)
f = dolfin.TrialFunction(V)
x = V.mesh().coordinates()
l = (x.max(0) - x.min(0)).min()
k = Constant(float(kappa) * l**2)
a_M = f * v * dx
a_D = k * dot(grad(f), grad(v)) * dx
A = assemble(a_M + a_D)
self._M = assemble(a_M)
self.solver = dolfin.LUSolver(A, "mumps")
self.solver.parameters["symmetric"] = True