def test_only_first_order(V):
u = Coefficient(V)
v = TestFunction(V)
c = Coefficient(V)
F = (inner(c, c) * inner(Dt(Dt(u)), v) + inner(grad(u), grad(v)) +
inner(c, v) + inner(Dt(u), v)) * dx
with pytest.raises(ValueError):
check_integrals(F.integrals(), expect_time_derivative=True)
def wave(n, deg, butcher_tableau, splitting=AI):
N = 2**n
msh = UnitIntervalMesh(N)
params = {
"snes_type": "ksponly",
"ksp_type": "preonly",
"mat_type": "aij",
"pc_type": "lu"
}
V = FunctionSpace(msh, "CG", deg)
W = FunctionSpace(msh, "DG", deg - 1)
Z = V * W
x, = SpatialCoordinate(msh)
t = Constant(0.0)
dt = Constant(2.0 / N)
up = project(as_vector([0, sin(pi * x)]), Z)
u, p = split(up)
v, w = TestFunctions(Z)
F = (inner(Dt(u), v) * dx + inner(u.dx(0), w) * dx + inner(Dt(p), w) * dx -
inner(p, v.dx(0)) * dx)
E = 0.5 * (inner(u, u) * dx + inner(p, p) * dx)
stepper = TimeStepper(F,
butcher_tableau,
t,
dt,
up,
solver_parameters=params,
splitting=splitting)
energies = []
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
energies.append(assemble(E))
return np.array(energies)
def heat_subdomainbc(N, deg, butcher_tableau, splitting=AI):
dt = Constant(1.0 / N)
t = Constant(0.0)
msh = UnitSquareMesh(N, N)
V = FunctionSpace(msh, "CG", 2)
x, y = SpatialCoordinate(msh)
uexact = t*(x+y)
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))
u = interpolate(uexact, V)
v = TestFunction(V)
n = FacetNormal(msh)
F = inner(Dt(u), v)*dx + inner(grad(u), grad(v))*dx - inner(rhs, v)*dx - inner(dot(grad(uexact), n), v)*ds
bc = DirichletBC(V, uexact, [1, 2])
luparams = {"mat_type": "aij",
"snes_type": "ksponly",
"ksp_type": "preonly",
"pc_type": "lu"}
stepper = TimeStepper(F, butcher_tableau, t, dt, u, bcs=bc,
solver_parameters=luparams)
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
return norm(u-uexact)
def StokesTest(N, butcher_tableau, stage_type="deriv", splitting=AI):
mesh = UnitSquareMesh(N, N)
Ve = VectorElement("CG", mesh.ufl_cell(), 2)
Pe = FiniteElement("CG", mesh.ufl_cell(), 1)
Ze = MixedElement([Ve, Pe])
Z = FunctionSpace(mesh, Ze)
t = Constant(0.0)
dt = Constant(1.0/N)
(x, y) = SpatialCoordinate(mesh)
uexact = as_vector([x*t + y**2, -y*t+t*(x**2)])
pexact = Constant(0, domain=mesh)
u_rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact)) + grad(pexact)
p_rhs = -div(uexact)
z = Function(Z)
test_z = TestFunction(Z)
(u, p) = split(z)
(v, q) = split(test_z)
F = (inner(Dt(u), v)*dx
+ inner(grad(u), grad(v))*dx
- inner(p, div(v))*dx
- inner(q, div(u))*dx
- inner(u_rhs, v)*dx
- inner(p_rhs, q)*dx)
bcs = [DirichletBC(Z.sub(0), uexact, "on_boundary")]
nsp = [(1, VectorSpaceBasis(constant=True))]
u, p = z.split()
u.interpolate(uexact)
lu = {"mat_type": "aij",
"snes_type": "newtonls",
"snes_linesearch_type": "l2",
"snes_linesearch_monitor": None,
"snes_monitor": None,
"snes_rtol": 1e-8,
"snes_atol": 1e-8,
"snes_force_iteration": 1,
"ksp_type": "preonly",
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps"}
stepper = TimeStepper(F, butcher_tableau, t, dt, z,
stage_type=stage_type,
bcs=bcs, solver_parameters=lu, nullspace=nsp)
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
(u, p) = z.split()
return errornorm(uexact, u) + errornorm(pexact, p)
def test_can_split_mixed(W):
u = Coefficient(W)
v = TestFunction(W)
c = Coefficient(W)
F = (inner(c, c) * inner(Dt(u), v) + inner(grad(u), grad(v)) +
inner(c, v) + inner(Dt(u), v)) * dx
split = extract_terms(F)
expect_t = (inner(c, c) * inner(Dt(u), v) + inner(Dt(u), v)) * dx
expect_no_t = inner(grad(u), grad(v)) * dx + inner(c, v) * dx
assert sig(expect_t) == sig(split.time)
assert sig(expect_no_t) == sig(split.remainder)
def Fubc(V, uexact, rhs):
u = interpolate(uexact, V)
v = TestFunction(V)
F = inner(Dt(u), v) * dx + inner(grad(u), grad(v)) * dx - inner(rhs,
v) * dx
bc = DirichletBC(V, uexact, "on_boundary")
return (F, u, bc)
def NSETest(butch, stage_type, splitting):
N = 16
M = UnitSquareMesh(N, N)
V = VectorFunctionSpace(M, "CG", 2)
W = FunctionSpace(M, "CG", 1)
Z = V * W
up = Function(Z)
u, p = split(up)
v, q = TestFunctions(Z)
Re = Constant(100.0)
F = (inner(Dt(u), v) * dx
+ 1.0 / Re * inner(grad(u), grad(v)) * dx
+ inner(dot(grad(u), u), v) * dx
- p * div(v) * dx
+ div(u) * q * dx
)
bcs = [DirichletBC(Z.sub(0), Constant((1, 0)), (4,)),
DirichletBC(Z.sub(0), Constant((0, 0)), (1, 2, 3))]
nullspace = [(1, VectorSpaceBasis(constant=True))]
solver_parameters = {"mat_type": "aij",
"snes_type": "newtonls",
"snes_linesearch_type": "bt",
"snes_linesearch_monitor": None,
"snes_monitor": None,
"snes_rtol": 1e-10,
"snes_atol": 1e-10,
"snes_force_iteration": 1,
"ksp_type": "preonly",
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps"}
t = Constant(0.0)
dt = Constant(0.3/N)
stepper = TimeStepper(F, butch,
t, dt, up,
bcs=bcs,
stage_type="value",
solver_parameters=solver_parameters,
nullspace=nullspace)
tfinal = 1.0
while (float(t) < tfinal):
if (float(t) + float(dt) > tfinal):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
u, p = up.split()
return norm(u)
def test_can_split_mixed_split(W):
u = Coefficient(W)
from ufl import split as splt
u0, u1 = splt(u)
v = TestFunction(W)
v0, v1 = splt(v)
c = Coefficient(W)
F = (inner(c, c) * inner(Dt(u0), v0) + inner(grad(u), grad(v)) +
inner(c, v) + inner(Dt(u), v)) * dx
split = extract_terms(F)
expect_t = (inner(c, c) * inner(Dt(u0), v0) + inner(Dt(u), v)) * dx
expect_no_t = inner(grad(u), grad(v)) * dx + inner(c, v) * dx
assert sig(expect_t) == sig(split.time)
assert sig(expect_no_t) == sig(split.remainder)
def test_1d_heat_dirichletbc(butcher_tableau):
# Boundary values
u_0 = Constant(2.0)
u_1 = Constant(3.0)
N = 100
x0 = 0.0
x1 = 10.0
msh = IntervalMesh(N, x1)
V = FunctionSpace(msh, "CG", 1)
dt = Constant(10.0 / N)
t = Constant(0.0)
(x,) = SpatialCoordinate(msh)
# Method of manufactured solutions copied from Heat equation demo.
S = Constant(2.0)
C = Constant(1000.0)
B = (x - Constant(x0)) * (x - Constant(x1)) / C
R = (x * x) ** 0.5
# Note end linear contribution
uexact = (
B * atan(t) * (pi / 2.0 - atan(S * (R - t)))
+ u_0
+ ((x - x0) / x1) * (u_1 - u_0)
)
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))
u = interpolate(uexact, V)
v = TestFunction(V)
F = (
inner(Dt(u), v) * dx
+ inner(grad(u), grad(v)) * dx
- inner(rhs, v) * dx
)
bc = [
DirichletBC(V, u_1, 2),
DirichletBC(V, u_0, 1),
]
luparams = {"mat_type": "aij", "ksp_type": "preonly", "pc_type": "lu"}
stepper = TimeStepper(
F, butcher_tableau, t, dt, u, bcs=bc, solver_parameters=luparams
)
t_end = 2.0
while float(t) < t_end:
if float(t) + float(dt) > t_end:
dt.assign(t_end - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
# Check solution and boundary values
assert norm(u - uexact) / norm(uexact) < 10.0 ** -5
assert isclose(u.at(x0), u_0)
assert isclose(u.at(x1), u_1)
def test_appctx():
msh = UnitIntervalMesh(2)
V = FunctionSpace(msh, "CG", 1)
u = Function(V)
v = TestFunction(V)
F = inner(Dt(u) + u, v) * dx
bt = GaussLegendre(1)
t = Constant(0.0)
dt = Constant(0.1)
stepper = TimeStepper(F, bt, t, dt, u, appctx={"hello": "world"})
assert "hello" in stepper.solver._ctx.appctx
def heat(n, deg, time_stages, stage_type="deriv", splitting=IA):
N = 2**n
msh = UnitIntervalMesh(N)
params = {
"snes_type": "ksponly",
"ksp_type": "preonly",
"mat_type": "aij",
"pc_type": "lu"
}
V = FunctionSpace(msh, "CG", deg)
x, = SpatialCoordinate(msh)
t = Constant(0.0)
dt = Constant(2.0 / N)
uexact = exp(-t) * sin(pi * x)
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))
butcher_tableau = GaussLegendre(time_stages)
u = project(uexact, V)
v = TestFunction(V)
F = (inner(Dt(u), v) * dx + inner(grad(u), grad(v)) * dx -
inner(rhs, v) * dx)
bc = DirichletBC(V, Constant(0), "on_boundary")
stepper = TimeStepper(F,
butcher_tableau,
t,
dt,
u,
bcs=bc,
solver_parameters=params,
stage_type=stage_type,
splitting=splitting)
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
return errornorm(uexact, u) / norm(uexact)
def test_expecting_time_derivative(V):
u = Coefficient(V)
v = TestFunction(V)
c = Coefficient(V)
F = (inner(grad(u), grad(v)) + inner(c, v)) * dx
with pytest.raises(ValueError):
check_integrals(F.integrals(), expect_time_derivative=True)
check_integrals(F.integrals(), expect_time_derivative=False)
F += inner(Dt(u), v) * dx
with pytest.raises(ValueError):
check_integrals(F.integrals(), expect_time_derivative=False)
def RTCFtest(N, deg, butcher_tableau, splitting=AI):
msh = UnitSquareMesh(N, N, quadrilateral=True)
Ve = FiniteElement("RTCF", msh.ufl_cell(), 2)
V = FunctionSpace(msh, Ve)
dt = Constant(0.1 / N)
t = Constant(0.0)
x, y = SpatialCoordinate(msh)
uexact = as_vector(
[t + 2 * t * x + 4 * t * y, 7 * t + 5 * t * x + 6 * t * y])
rhs = expand_derivatives(diff(uexact, t)) + grad(div(uexact))
u = project(uexact, V)
v = TestFunction(V)
F = inner(Dt(u), v) * dx + inner(div(u), div(v)) * dx - inner(rhs, v) * dx
bc = DirichletBC(V, uexact, "on_boundary")
luparams = {"mat_type": "aij", "ksp_type": "preonly", "pc_type": "lu"}
stepper = TimeStepper(F,
butcher_tableau,
t,
dt,
u,
bcs=bc,
solver_parameters=luparams,
splitting=splitting)
while (float(t) < 0.1):
if (float(t) + float(dt) > 0.1):
dt.assign(0.1 - float(t))
stepper.advance()
print(float(t))
t.assign(float(t) + float(dt))
return norm(u - uexact)
def curltest(N, deg, butcher_tableau, splitting):
msh = UnitSquareMesh(N, N)
Ve = FiniteElement("N1curl", msh.ufl_cell(), 2, variant="integral")
V = FunctionSpace(msh, Ve)
dt = Constant(0.1 / N)
t = Constant(0.0)
x, y = SpatialCoordinate(msh)
uexact = as_vector([t + 2*t*x + 4*t*y + 3*t*(y**2) + 2*t*x*y, 7*t + 5*t*x + 6*t*y - 3*t*x*y - 2*t*(x**2)])
rhs = expand_derivatives(diff(uexact, t)) + curl(curl(uexact))
u = interpolate(uexact, V)
v = TestFunction(V)
n = FacetNormal(msh)
F = inner(Dt(u), v)*dx + inner(curl(u), curl(v))*dx - inner(curlcross(uexact, n), v)*ds - inner(rhs, v)*dx
bc = DirichletBC(V, uexact, [1, 2])
luparams = {"mat_type": "aij",
"ksp_type": "preonly",
"pc_type": "lu"}
stepper = TimeStepper(F, butcher_tableau, t, dt, u, bcs=bc,
solver_parameters=luparams,
splitting=splitting)
while (float(t) < 0.1):
if (float(t) + float(dt) > 0.1):
dt.assign(0.1 - float(t))
stepper.advance()
print(float(t))
t.assign(float(t) + float(dt))
return norm(u-uexact)
def test_Dt_linear(V, typ):
u = Coefficient(V)
v = TestFunction(V)
c = Coefficient(V)
F = (inner(grad(u), grad(v)) + inner(c, v) + inner(Dt(u), v)) * dx
if typ == "mul":
F += inner(Dt(u), c) * inner(Dt(u), v) * dx
elif typ == "div":
F += inner(Dt(u), v) / Dt(u)[0] * dx
elif typ == "sin":
F += inner(sin(Dt(u)[0]), v[0]) * dx
with pytest.raises(ValueError):
check_integrals(F.integrals(), expect_time_derivative=True)
u, p = split(up)
n = FacetNormal(msh)
dt = Constant(1.0 / 1600)
t = Constant(0.0)
nu = Constant(0.001)
rho = 1.0
r = 0.05
Umean = 1.0
L = 0.1
# define the variational form once outside the loop
gamma = Constant(1.e2)
F = (
inner(Dt(u), v) * dx + inner(dot(grad(u), u), v) * dx
#+ gamma * inner(cell_avg(div(u)), cell_avg(div(v))) * dx
+ nu * inner(grad(u), grad(v)) * dx - inner(p, div(v)) * dx +
inner(div(u), w) * dx)
# boundary conditions are specified for each subspace
UU = 1.5 * sin(pi * t / 8)
bcs = [
DirichletBC(Z.sub(0), as_vector([4 * UU * y * (y1 - y) / (y1**2), 0]),
(4, )),
DirichletBC(Z.sub(0), Constant((0, 0)), (1, 3, 5))
]
#nullspace = MixedVectorSpaceBasis(
# Z, [Z.sub(0), VectorSpaceBasis(constant=True)])
def elastodynamics(N, deg, butcher_tableau, splitting=AI):
dt = Constant(1.0 / N)
t = Constant(0.0)
msh = UnitSquareMesh(N, N)
x, y = SpatialCoordinate(msh)
V = VectorFunctionSpace(msh, "CG", 1)
VV = V * V
uu1 = Function(VV)
u1, udot1 = uu1.split()
u1.interpolate(
as_vector([sin(pi * x) * sin(pi * y),
sin(pi * x) * sin(pi * y)]))
uu2 = Function(VV)
u2, udot2 = uu2.split()
u2.interpolate(
as_vector([sin(pi * x) * sin(pi * y),
sin(pi * x) * sin(pi * y)]))
u1, udot1 = split(uu1)
u2, udot2 = split(uu2)
v, w = TestFunctions(VV)
F1 = (inner(Dt(u1) - udot1, v) * dx + inner(Dt(udot1), w) * dx +
inner(grad(u1), grad(w)) * dx)
F2 = (inner(Dt(u2) - udot2, v) * dx + inner(Dt(udot2), w) * dx +
inner(grad(u2), grad(w)) * dx)
bc1 = DirichletBC(VV.sub(0), Function(VV.sub(0)), "on_boundary")
bc2 = [
DirichletBC(
VV.sub(0).sub(i), Function(VV.sub(0).sub(i)), "on_boundary")
for i in (0, 1)
]
luparams = {
"mat_type": "aij",
"snes_type": "ksponly",
"ksp_type": "preonly",
"pc_type": "lu"
}
stepper1 = TimeStepper(F1,
butcher_tableau,
t,
dt,
uu1,
bcs=bc1,
solver_parameters=luparams,
splitting=splitting)
stepper2 = TimeStepper(F2,
butcher_tableau,
t,
dt,
uu2,
bcs=bc2,
solver_parameters=luparams,
splitting=splitting)
stepper1.advance()
stepper2.advance()
return norm(uu1 - uu2)
# This will give the exact solution at any time t. We just have
# to t.assign(time_we_want)
uexact = exp(-t) * cos(pi * x) * sin(pi * y)
# MMS works on symbolic differentiation of true solution, not weak form
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))
u = interpolate(uexact, V)
# define the variational form once outside the loop
h = CellSize(msh)
n = FacetNormal(msh)
beta = Constant(100.0)
v = TestFunction(V)
F = (inner(Dt(u), v) * dx + inner(grad(u), grad(v)) * dx - inner(rhs, v) * dx -
inner(dot(grad(u), n), v) * ds - inner(dot(grad(v), n), u - uexact) * ds +
beta / h * inner(u - uexact, v) * ds)
params = {
"mat_type": "aij",
"snes_type": "ksponly",
"ksp_type": "preonly",
"pc_type": "lu"
}
stepper = TimeStepper(F, ButcherTableau, t, dt, u, solver_parameters=params)
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))