def increase_order(V: function.FunctionSpace) -> function.FunctionSpace:
"""For a given function space, return the same space, but with
polynomial degree increase by 1.
"""
e = ufl.algorithms.elementtransformations.increase_order(V.ufl_element())
return function.FunctionSpace(V.mesh, e)
def test_ghost_mesh_assembly(mode, dx, ds):
mesh = UnitSquareMesh(MPI.comm_world, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
dx = dx(mesh)
ds = ds(mesh)
f = Function(V)
with f.vector.localForm() as f_local:
f_local.set(10.0)
a = inner(f * u, v) * dx + inner(u, v) * ds
L = inner(f, v) * dx + inner(2.0, v) * ds
# Initial assembly
A = fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
b = fem.assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert isinstance(b, PETSc.Vec)
# Check that the norms are the same for all three modes
normA = A.norm()
assert normA == pytest.approx(0.6713621455570528, rel=1.e-6, abs=1.e-12)
normb = b.norm()
assert normb == pytest.approx(1.582294032953906, rel=1.e-6, abs=1.e-12)
def change_regularity(V: function.FunctionSpace,
family: str) -> function.FunctionSpace:
"""For a given function space, return the corresponding space with
the finite elements specified by 'family'. Possible families are
the families supported by the form compiler
"""
e = ufl.algorithms.elementtransformations.change_regularity(
V.ufl_element(), family)
return function.FunctionSpace(V.mesh, e)
def test_ghost_mesh_dS_assembly(mode, dS):
mesh = UnitSquareMesh(MPI.comm_world, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
dS = dS(mesh)
a = inner(avg(u), avg(v)) * dS
# Initial assembly
A = fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
# Check that the norms are the same for all three modes
normA = A.norm()
print(normA)
assert normA == pytest.approx(2.1834054713561906, rel=1.e-6, abs=1.e-12)
def ode_1st_nonlinear_odeint(a=1.0, b=1.0, c=1.0, nT=10, dt=0.1, **kwargs):
"""
Create 1st order ODE problem and solve with `ODEInt` time integrator.
First order nonlinear non-autonomous ODE:
t * dot u - a * cos(c*t) * u^2 - 2 * u - a * b^2 * t^4 * cos(c*t) = 0 with initial condition u(t=1) = 0
"""
mesh = UnitCubeMesh(MPI.COMM_WORLD, 1, 1, 1)
U = FunctionSpace(mesh, ("DG", 0))
u = Function(U, name="u")
ut = Function(U, name="ut")
u.vector.set(0.0) # initial condition
ut.vector.set(
a * b**2 *
numpy.cos(c)) # exact initial rate of this ODE for generalised alpha
u.vector.ghostUpdate()
ut.vector.ghostUpdate()
δu = ufl.TestFunction(U)
dx = ufl.Measure("dx", domain=mesh)
# Global time
t = dolfinx.Constant(mesh, 1.0)
# Time step size
dt = dolfinx.Constant(mesh, dt)
# Time integrator
odeint = dolfiny.odeint.ODEInt(t=t, dt=dt, x=u, xt=ut, **kwargs)
# Weak form (as one-form)
f = δu * (t * ut - a * ufl.cos(c * t) * u**2 - 2 * u -
a * b**2 * t**4 * ufl.cos(c * t)) * dx
# Overall form (as one-form)
F = odeint.discretise_in_time(f)
# Overall form (as list of forms)
F = dolfiny.function.extract_blocks(F, [δu])
# # Options for PETSc backend
from petsc4py import PETSc
opts = PETSc.Options()
opts["snes_type"] = "newtonls"
opts["snes_linesearch_type"] = "basic"
opts["snes_atol"] = 1.0e-10
opts["snes_rtol"] = 1.0e-12
# Silence SNES monitoring during test
dolfiny.snesblockproblem.SNESBlockProblem.print_norms = lambda self, it: 1
# Create nonlinear problem
problem = dolfiny.snesblockproblem.SNESBlockProblem(F, [u])
# Book-keeping of results
u_avg = numpy.zeros(nT + 1)
u_avg[0] = u.vector.sum() / u.vector.getSize()
dolfiny.utils.pprint(f"+++ Processing time steps = {nT}")
# Process time steps
for time_step in range(1, nT + 1):
# Stage next time step
odeint.stage()
# Solve nonlinear problem
u, = problem.solve()
# Assert convergence of nonlinear solver
assert problem.snes.getConvergedReason(
) > 0, "Nonlinear solver did not converge!"
# Update solution states for time integration
odeint.update()
# Store result
u_avg[time_step] = u.vector.sum() / u.vector.getSize()
return u_avg
def ode_1st_linear_odeint(a=1.0, b=0.5, u_0=1.0, nT=10, dt=0.1, **kwargs):
"""
Create 1st order ODE problem and solve with `ODEInt` time integrator.
First order linear ODE:
dot u + a * u - b = 0 with initial condition u(t=0) = u_0
"""
mesh = UnitCubeMesh(MPI.COMM_WORLD, 1, 1, 1)
U = FunctionSpace(mesh, ("DG", 0))
u = Function(U, name="u")
ut = Function(U, name="ut")
u.vector.set(u_0) # initial condition
ut.vector.set(
b - a * u_0) # exact initial rate of this ODE for generalised alpha
u.vector.ghostUpdate()
ut.vector.ghostUpdate()
δu = ufl.TestFunction(U)
dx = ufl.Measure("dx", domain=mesh)
# Global time
time = dolfinx.Constant(mesh, 0.0)
# Time step size
dt = dolfinx.Constant(mesh, dt)
# Time integrator
odeint = dolfiny.odeint.ODEInt(t=time, dt=dt, x=u, xt=ut, **kwargs)
# Weak form (as one-form)
f = δu * (ut + a * u - b) * dx
# Overall form (as one-form)
F = odeint.discretise_in_time(f)
# Overall form (as list of forms)
F = dolfiny.function.extract_blocks(F, [δu])
# Silence SNES monitoring during test
dolfiny.snesblockproblem.SNESBlockProblem.print_norms = lambda self, it: 1
# Create problem (although having a linear ODE we use the dolfiny.snesblockproblem API)
problem = dolfiny.snesblockproblem.SNESBlockProblem(F, [u])
# Book-keeping of results
u_avg = numpy.zeros(nT + 1)
u_avg[0] = u.vector.sum() / u.vector.getSize()
dolfiny.utils.pprint(f"+++ Processing time steps = {nT}")
# Process time steps
for time_step in range(1, nT + 1):
# Stage next time step
odeint.stage()
# Solve (linear) problem
u, = problem.solve()
# Update solution states for time integration
odeint.update()
# Store result
u_avg[time_step] = u.vector.sum() / u.vector.getSize()
return u_avg
def test_neohooke():
mesh = UnitCubeMesh(MPI.COMM_WORLD, 7, 7, 7)
V = VectorFunctionSpace(mesh, ("P", 1))
P = FunctionSpace(mesh, ("P", 1))
L = FunctionSpace(mesh, ("R", 0))
u = Function(V, name="u")
v = ufl.TestFunction(V)
p = Function(P, name="p")
q = ufl.TestFunction(P)
lmbda0 = Function(L)
d = mesh.topology.dim
Id = ufl.Identity(d)
F = Id + ufl.grad(u)
C = F.T * F
J = ufl.det(F)
E_, nu_ = 10.0, 0.3
mu, lmbda = E_ / (2 * (1 + nu_)), E_ * nu_ / ((1 + nu_) * (1 - 2 * nu_))
psi = (mu / 2) * (ufl.tr(C) -
3) - mu * ufl.ln(J) + lmbda / 2 * ufl.ln(J)**2 + (p -
1.0)**2
pi = psi * ufl.dx
F0 = ufl.derivative(pi, u, v)
F1 = ufl.derivative(pi, p, q)
# Number of eigenvalues to find
nev = 8
opts = PETSc.Options("neohooke")
opts["eps_smallest_magnitude"] = True
opts["eps_nev"] = nev
opts["eps_ncv"] = 50 * nev
opts["eps_conv_abs"] = True
# opts["eps_non_hermitian"] = True
opts["eps_tol"] = 1.0e-14
opts["eps_max_it"] = 1000
opts["eps_error_relative"] = "::ascii_info_detail"
opts["eps_monitor"] = "::ascii_info_detail"
slepcp = dolfiny.slepcblockproblem.SLEPcBlockProblem([F0, F1], [u, p],
lmbda0,
prefix="neohooke")
slepcp.solve()
# mat = dolfiny.la.petsc_to_scipy(slepcp.eps.getOperators()[0])
# eigvals, eigvecs = linalg.eigsh(mat, which="SM", k=nev)
with XDMFFile(MPI.COMM_WORLD, "eigvec.xdmf", "w") as ofile:
ofile.write_mesh(mesh)
for i in range(nev):
eigval, ur, ui = slepcp.getEigenpair(i)
# Expect first 6 eignevalues 0, i.e. rigid body modes
if i < 6:
assert np.isclose(eigval, 0.0)
for func in ur:
name = func.name
func.name = f"{name}_eigvec_{i}_real"
ofile.write_function(func)
func.name = name