def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
VFS = VectorFunctionSpace(mesh, "DG", 1)
FS = FunctionSpace(mesh, "DG", 1)
ic = project(Expression((mms.u(),mms.v()), degree=5), VFS)
u = project(ic, VFS)
f = project(Expression((mms.f_u(),mms.f_v()), degree=5), VFS)
nu = project(Expression(mms.nu(), degree=5), FS)
uFile = File("u.pvd")
nuFile = File("nu.pvd")
uFile << u
nuFile << nu
i, j, k, l = indices(4)
p, q, r, s = indices(4)
v = TestFunction(VFS)
# PARTIAL STRESS FORM
# A = as_tensor(
# [ [ [ [ 2*nu, 0.0 ], [0.0 , nu ] ],
# [ [ 0.0 , nu ], [0.0 , 0.0 ] ] ],
# [ [ [ 0.0 , 0.0 ], [nu , 0.0 ] ],
# [ [ nu , 0.0 ], [0.0 , 2*nu] ] ] ]
# )
# TENSOR FORM
A = as_tensor(
[ [ [ [ 1.0, 0.0 ], [0.0, 0.0] ],
[ [ 0.0, 1.0 ], [0.0, 0.0] ] ],
[ [ [ 0.0, 0.0 ], [1.0, 0.0] ],
[ [ 0.0, 0.0 ], [0.0, 1.0] ] ] ]
)
un = jump(u)[i]*n("+")[j]
un_ij = as_tensor(un, (i,j))
vn = jump(v)[i]*n("+")[j]
vn_ij = as_tensor(vn, (i,j))
un_b = (u-ic)[i]*n[j]
un_bij = as_tensor(un_b, (i,j))
vn_b = v[i]*n[j]
vn_bij = as_tensor(vn_b, (i,j))
F_z = 0
F_u = 0
for i_ in range(2):
F = (-
(
grad(u)[i_,j]*grad(v)[i_,j]*dx
)+
(
avg(grad(v)[i_,j])*un_ij[i_,j]*dS +
avg(grad(u)[i_,j])*vn_ij[i_,j]*dS +
grad(v)[i_,j]*un_bij[i_,j]*ds +
grad(u)[i_,j]*vn_bij[i_,j]*ds
)+
(
un_ij[i_,j]*vn_ij[i_,j]*dS +
un_bij[i_,j]*vn_bij[i_,j]*ds
)-
(
v[i_]*u[i_]*dx
)-
(
f[i_]*v[i_]*dx
)
)
print u.vector().array()
solve(F == 0, u)
uFile << u
Eu = errornorm(u, ic, norm_type="L2", degree_rise=3)
print Eu
return Eu
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
VFS = VectorFunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, VFS, FS, VFS])
w = Function(W)
w_ic = project((Expression(
(
mms.u(),
'0.0','0.0',
mms.v(),
'0.0','0.0'
)
, W.ufl_element())), W)
f = project((Expression(
(
mms.f_u(),
'0.0','0.0',
mms.f_v(),
'0.0','0.0',
)
, W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0; a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i])*dx
L += inner(test[i], w_ic[i])*dx
solve(a == L, w)
u_0, z_0, u_1, z_1 = w.split()
u0File = File("u0.pvd")
z0File = File("z0.pvd")
u1File = File("u1.pvd")
z1File = File("z1.pvd")
u0File << u_0
z0File << z_0
u1File << u_1
z1File << z_1
(v0, q0, v1, q1) = TestFunctions(W)
(u0, z0, u1, z1) = (w[0], as_vector((w[1], w[2])), w[3], as_vector((w[4], w[5])))
def F_z(z, u, q):
return (inner(z, q)*dx -
inner(q, grad(u))*dx +
inner(jump(u)*n("+"), avg(q))*dS
)
def F_u(z, u, u_b, f_, v):
return (- inner(z, grad(v))*dx +
inner(jump(v)*n("+"), avg(z))*dS +
(u-u_b)*v*ds -
v*u*dx -
f_*v*dx
)
Fz = F_z(z = z0, u = u0, q = q0) + \
F_z(z = z1, u = u1, q = q1)
Fu = F_u(z = z0, u = u0, u_b = Expression(mms.u()), f_ = f[0], v = v0) + \
F_u(z = z1, u = u1, u_b = Expression(mms.v()), f_ = f[3], v = v1)
F = Fz + Fu
solve(F == 0, w)
u_0, z_0, u_1, z_1 = w.split()
u0File << u_0
z0File << z_0
u1File << u_1
z1File << z_1
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, FS, FS, FS, FS, FS])
w = Function(W)
w_ic = project((Expression(
(mms.u(), mms.v(), '0.0', '0.0', '0.0', '0.0'), W.ufl_element())), W)
f = project((Expression((
mms.f_u(),
mms.f_v(),
'0.0',
'0.0',
'0.0',
'0.0',
), W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0
a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i]) * dx
L += inner(test[i], w_ic[i]) * dx
solve(a == L, w)
nu = project(Expression(mms.nu()), FS)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File = File("u0.pvd")
u1File = File("u1.pvd")
nuFile = File("nu.pvd")
z00File = File("z00.pvd")
z01File = File("z01.pvd")
z10File = File("z10.pvd")
z11File = File("z11.pvd")
u0File << u_0
u1File << u_1
nuFile << nu
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
i, j, k, l = indices(4)
p, q, r, s = indices(4)
(v, q) = (as_vector(
(test[0], test[1])), as_tensor(
((test[2], test[3]), (test[4], test[5]))))
(u, z) = (as_vector((w[0], w[1])), as_tensor(((w[2], w[3]), (w[4], w[5]))))
# PARTIAL STRESS FORM
A = as_tensor([[[[2 * nu, 0.0], [0.0, nu]], [[0.0, nu], [0.0, 0.0]]],
[[[0.0, 0.0], [nu, 0.0]], [[nu, 0.0], [0.0, 2 * nu]]]])
# TENSOR FORM
# A = as_tensor(
# [ [ [ [ 1.0, 0.0 ], [0.0, 0.0] ],
# [ [ 0.0, 1.0 ], [0.0, 0.0] ] ],
# [ [ [ 0.0, 0.0 ], [1.0, 0.0] ],
# [ [ 0.0, 0.0 ], [0.0, 1.0] ] ] ]
# )
u = as_vector((w_ic[0], w_ic[1]))
un = jump(u)[i] * n("+")[j]
un_ij = as_tensor(un, (i, j))
vn = jump(v)[i] * n("+")[j]
vn_ij = as_tensor(vn, (i, j))
u_b = as_vector((w_ic[0], w_ic[1]))
un_b = (u - u_b)[i] * n[j]
un_bij = as_tensor(un_b, (i, j))
vn_b = v[i] * n[j]
vn_bij = as_tensor(vn_b, (i, j))
F_z = 0
F_u = 0
for i_ in range(2):
F_z += A[i_,j,r,s]*z[i_,j]*q[r,s]*dx - \
q[i_,j]*grad(u)[i_,j]*dx + \
un_ij[i_,j]*avg(q[i_,j])*dS + \
un_bij[i_,j]*q[i_,j]*ds
F_u = v[i_] * u[i_] * dx - v[i_] * u[i_] * dx
# F_u += - z[i_,j]*grad(v)[i_,j]*dx + \
# vn_ij[i_,j]*avg(z[i_,j])*dS + \
# vn_bij[i_,j]*z[i_,j]*ds - \
# v[i_]*u[i_]*dx - \
# f[i_]*v[i_]*dx
F = F_z + F_u
solve(F == 0, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, FS, FS, FS, FS, FS])
w = Function(W)
w_ic = project((Expression(
(
mms.u(),
mms.v(),
'0.0',
'0.0',
'0.0',
'0.0'
)
, W.ufl_element())), W)
f = project((Expression(
(
mms.f_u(),
mms.f_v(),
'0.0',
'0.0',
'0.0',
'0.0',
)
, W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0; a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i])*dx
L += inner(test[i], w_ic[i])*dx
solve(a == L, w)
nu = project(Expression(mms.nu()), FS)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File = File("u0.pvd")
u1File = File("u1.pvd")
nuFile = File("nu.pvd")
z00File = File("z00.pvd")
z01File = File("z01.pvd")
z10File = File("z10.pvd")
z11File = File("z11.pvd")
u0File << u_0
u1File << u_1
nuFile << nu
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
i, j, k, l = indices(4)
p, q, r, s = indices(4)
(v, q) = (as_vector((test[0], test[1])),
as_tensor(((test[2], test[3]),(test[4], test[5]))))
(u, z) = (as_vector((w[0], w[1])),
as_tensor(((w[2], w[3]),(w[4], w[5]))))
# PARTIAL STRESS FORM
A = as_tensor(
[ [ [ [ 2*nu, 0.0 ], [0.0 , nu ] ],
[ [ 0.0 , nu ], [0.0 , 0.0 ] ] ],
[ [ [ 0.0 , 0.0 ], [nu , 0.0 ] ],
[ [ nu , 0.0 ], [0.0 , 2*nu] ] ] ]
)
# TENSOR FORM
# A = as_tensor(
# [ [ [ [ 1.0, 0.0 ], [0.0, 0.0] ],
# [ [ 0.0, 1.0 ], [0.0, 0.0] ] ],
# [ [ [ 0.0, 0.0 ], [1.0, 0.0] ],
# [ [ 0.0, 0.0 ], [0.0, 1.0] ] ] ]
# )
u = as_vector((w_ic[0],w_ic[1]))
un = jump(u)[i]*n("+")[j]
un_ij = as_tensor(un, (i,j))
vn = jump(v)[i]*n("+")[j]
vn_ij = as_tensor(vn, (i,j))
u_b = as_vector((w_ic[0],w_ic[1]))
un_b = (u-u_b)[i]*n[j]
un_bij = as_tensor(un_b, (i,j))
vn_b = v[i]*n[j]
vn_bij = as_tensor(vn_b, (i,j))
F_z = 0
F_u = 0
for i_ in range(2):
F_z += A[i_,j,r,s]*z[i_,j]*q[r,s]*dx - \
q[i_,j]*grad(u)[i_,j]*dx + \
un_ij[i_,j]*avg(q[i_,j])*dS + \
un_bij[i_,j]*q[i_,j]*ds
F_u = v[i_]*u[i_]*dx - v[i_]*u[i_]*dx
# F_u += - z[i_,j]*grad(v)[i_,j]*dx + \
# vn_ij[i_,j]*avg(z[i_,j])*dS + \
# vn_bij[i_,j]*z[i_,j]*ds - \
# v[i_]*u[i_]*dx - \
# f[i_]*v[i_]*dx
F = F_z + F_u
solve(F == 0, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
VFS = VectorFunctionSpace(mesh, "DG", 1)
FS = FunctionSpace(mesh, "DG", 1)
ic = project(Expression((mms.u(), mms.v()), degree=5), VFS)
u = project(ic, VFS)
f = project(Expression((mms.f_u(), mms.f_v()), degree=5), VFS)
nu = project(Expression(mms.nu(), degree=5), FS)
uFile = File("u.pvd")
nuFile = File("nu.pvd")
uFile << u
nuFile << nu
i, j, k, l = indices(4)
p, q, r, s = indices(4)
v = TestFunction(VFS)
# PARTIAL STRESS FORM
# A = as_tensor(
# [ [ [ [ 2*nu, 0.0 ], [0.0 , nu ] ],
# [ [ 0.0 , nu ], [0.0 , 0.0 ] ] ],
# [ [ [ 0.0 , 0.0 ], [nu , 0.0 ] ],
# [ [ nu , 0.0 ], [0.0 , 2*nu] ] ] ]
# )
# TENSOR FORM
A = as_tensor([[[[1.0, 0.0], [0.0, 0.0]], [[0.0, 1.0], [0.0, 0.0]]],
[[[0.0, 0.0], [1.0, 0.0]], [[0.0, 0.0], [0.0, 1.0]]]])
un = jump(u)[i] * n("+")[j]
un_ij = as_tensor(un, (i, j))
vn = jump(v)[i] * n("+")[j]
vn_ij = as_tensor(vn, (i, j))
un_b = (u - ic)[i] * n[j]
un_bij = as_tensor(un_b, (i, j))
vn_b = v[i] * n[j]
vn_bij = as_tensor(vn_b, (i, j))
F_z = 0
F_u = 0
for i_ in range(2):
F = (-(grad(u)[i_, j] * grad(v)[i_, j] * dx) +
(avg(grad(v)[i_, j]) * un_ij[i_, j] * dS + avg(grad(u)[i_, j]) *
vn_ij[i_, j] * dS + grad(v)[i_, j] * un_bij[i_, j] * ds +
grad(u)[i_, j] * vn_bij[i_, j] * ds) +
(un_ij[i_, j] * vn_ij[i_, j] * dS +
un_bij[i_, j] * vn_bij[i_, j] * ds) - (v[i_] * u[i_] * dx) -
(f[i_] * v[i_] * dx))
print u.vector().array()
solve(F == 0, u)
uFile << u
Eu = errornorm(u, ic, norm_type="L2", degree_rise=3)
print Eu
return Eu
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, FS, FS, FS, FS, FS])
w = Function(W)
w_ic = project((Expression(
(mms.u(), mms.v(), '0.0', '0.0', '0.0', '0.0'), W.ufl_element())), W)
f = project((Expression((
mms.f_u(),
mms.f_v(),
'0.0',
'0.0',
'0.0',
'0.0',
), W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0
a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i]) * dx
L += inner(test[i], w_ic[i]) * dx
solve(a == L, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File = File("u0.pvd")
u1File = File("u1.pvd")
z00File = File("z00.pvd")
z01File = File("z01.pvd")
z10File = File("z10.pvd")
z11File = File("z11.pvd")
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
i, j, k, l = indices(4)
p, q, r, s = indices(4)
(v, q) = (as_vector(
(test[0], test[1])), as_tensor(
((test[2], test[3]), (test[4], test[5]))))
(u, z) = (as_vector((w[0], w[1])), as_tensor(((w[2], w[3]), (w[4], w[5]))))
# int0ij = z[:,j]*q[:,j]
# int0 = as_tensor(int0ij, (i,j))
# int1 = q[i,j]*grad(u)[i,j]
# int1i = as_tensor(int1, (i,j))
un = jump(u)[i] * n("+")[j]
un_ij = as_tensor(un, (i, j))
vn = jump(v)[i] * n("+")[j]
vn_ij = as_tensor(vn, (i, j))
# int_2 = int2_0ij[i,j]*avg(q)[i,j]
# int2i = as_vector(int2, (i))
F_z = 0
F_u = 0
for i_ in range(2):
F_z += z[i_,j]*q[i_,j]*dx - \
q[i_,j]*grad(u)[i_,j]*dx + \
un_ij[i_,j]*avg(q)[i_,j]*dS
F_u += - z[i_,j]*grad(v)[i_,j]*dx + \
vn_ij[i_,j]*avg(z)[i_,j]*dS + \
(u[i_]-w_ic[i_])*v[i_]*ds - \
v[i_]*u[i_]*dx - \
f[i_]*v[i_]*dx
# def F_z(z, u, q):
# return (inner(z, q)*dx -
# inner(q, grad(u))*dx +
# inner(jump(u)*n("+"), avg(q))*dS
# )
# def F_u(z, u, u_b, f_, v):
# i,j = indices(2)
# if z[2] == 0:
# A0 = as_matrix([[2.0,0.0],[0.0,1.0]])
# A1 = as_matrix([[1.0,0.0],[0.0,0.0]])
# if z[2] == 1:
# A0 = as_matrix([[0.0,0.0],[0.0,1.0]])
# A1 = as_matrix([[1.0,0.0],[0.0,2.0]])
# Az0 = A0*z[0]
# # Az0 = as_vector(Az0i, i)
# Az1 = A1*z[1]
# # Az1 = as_vector(Az1i, i)
# return (- (inner(Az0,grad(v))*dx# +
# # inner(Az1,grad(v))*dx
# ) +
# (inner(jump(v)*n("+"), avg(Az0))*dS# +
# # inner(jump(v)*n("+"), avg(Az1))*dS
# ) +
# (u-u_b)*v*ds -
# v*u*dx -
# f_*v*dx
# )
# Fz = F_z(z = z0, u = u0, q = q0) + \
# F_z(z = z1, u = u1, q = q1)
# Fu = F_u(z = [z0,z1,0], u = u0, u_b = Expression(mms.u()), f_ = f[0], v = v0) + \
# F_u(z = [z0,z1,1], u = u1, u_b = Expression(mms.v()), f_ = f[3], v = v1)
F = F_z + F_u
solve(F == 0, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, FS, FS, FS, FS, FS])
w = Function(W)
w_ic = project((Expression((mms.u(), mms.v(), "0.0", "0.0", "0.0", "0.0"), W.ufl_element())), W)
f = project((Expression((mms.f_u(), mms.f_v(), "0.0", "0.0", "0.0", "0.0"), W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0
a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i]) * dx
L += inner(test[i], w_ic[i]) * dx
solve(a == L, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File = File("u0.pvd")
u1File = File("u1.pvd")
z00File = File("z00.pvd")
z01File = File("z01.pvd")
z10File = File("z10.pvd")
z11File = File("z11.pvd")
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
i, j, k, l = indices(4)
p, q, r, s = indices(4)
(v, q) = (as_vector((test[0], test[1])), as_tensor(((test[2], test[3]), (test[4], test[5]))))
(u, z) = (as_vector((w[0], w[1])), as_tensor(((w[2], w[3]), (w[4], w[5]))))
# int0ij = z[:,j]*q[:,j]
# int0 = as_tensor(int0ij, (i,j))
# int1 = q[i,j]*grad(u)[i,j]
# int1i = as_tensor(int1, (i,j))
un = jump(u)[i] * n("+")[j]
un_ij = as_tensor(un, (i, j))
vn = jump(v)[i] * n("+")[j]
vn_ij = as_tensor(vn, (i, j))
# int_2 = int2_0ij[i,j]*avg(q)[i,j]
# int2i = as_vector(int2, (i))
F_z = 0
F_u = 0
for i_ in range(2):
F_z += z[i_, j] * q[i_, j] * dx - q[i_, j] * grad(u)[i_, j] * dx + un_ij[i_, j] * avg(q)[i_, j] * dS
F_u += (
-z[i_, j] * grad(v)[i_, j] * dx
+ vn_ij[i_, j] * avg(z)[i_, j] * dS
+ (u[i_] - w_ic[i_]) * v[i_] * ds
- v[i_] * u[i_] * dx
- f[i_] * v[i_] * dx
)
# def F_z(z, u, q):
# return (inner(z, q)*dx -
# inner(q, grad(u))*dx +
# inner(jump(u)*n("+"), avg(q))*dS
# )
# def F_u(z, u, u_b, f_, v):
# i,j = indices(2)
# if z[2] == 0:
# A0 = as_matrix([[2.0,0.0],[0.0,1.0]])
# A1 = as_matrix([[1.0,0.0],[0.0,0.0]])
# if z[2] == 1:
# A0 = as_matrix([[0.0,0.0],[0.0,1.0]])
# A1 = as_matrix([[1.0,0.0],[0.0,2.0]])
# Az0 = A0*z[0]
# # Az0 = as_vector(Az0i, i)
# Az1 = A1*z[1]
# # Az1 = as_vector(Az1i, i)
# return (- (inner(Az0,grad(v))*dx# +
# # inner(Az1,grad(v))*dx
# ) +
# (inner(jump(v)*n("+"), avg(Az0))*dS# +
# # inner(jump(v)*n("+"), avg(Az1))*dS
# ) +
# (u-u_b)*v*ds -
# v*u*dx -
# f_*v*dx
# )
# Fz = F_z(z = z0, u = u0, q = q0) + \
# F_z(z = z1, u = u1, q = q1)
# Fu = F_u(z = [z0,z1,0], u = u0, u_b = Expression(mms.u()), f_ = f[0], v = v0) + \
# F_u(z = [z0,z1,1], u = u1, u_b = Expression(mms.v()), f_ = f[3], v = v1)
F = F_z + F_u
solve(F == 0, w)
u_0, u_1, z_00, z_01, z_10, z_11 = w.split()
u0File << u_0
u1File << u_1
z00File << z_00
z01File << z_01
z10File << z_10
z11File << z_11
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev
def run(nx):
mesh = UnitSquareMesh(nx, nx)
n = FacetNormal(mesh)
h = CellSize(mesh)
FS = FunctionSpace(mesh, "DG", 1)
VFS = VectorFunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace([FS, VFS, FS, VFS])
w = Function(W)
w_ic = project((Expression(
(mms.u(), '0.0', '0.0', mms.v(), '0.0', '0.0'), W.ufl_element())), W)
f = project((Expression((
mms.f_u(),
'0.0',
'0.0',
mms.f_v(),
'0.0',
'0.0',
), W.ufl_element())), W)
test = TestFunction(W)
trial = TrialFunction(W)
L = 0
a = 0
for i in range(len(w_ic)):
a += inner(test[i], trial[i]) * dx
L += inner(test[i], w_ic[i]) * dx
solve(a == L, w)
u_0, z_0, u_1, z_1 = w.split()
u0File = File("u0.pvd")
z0File = File("z0.pvd")
u1File = File("u1.pvd")
z1File = File("z1.pvd")
u0File << u_0
z0File << z_0
u1File << u_1
z1File << z_1
(v0, q0, v1, q1) = TestFunctions(W)
(u0, z0, u1, z1) = (w[0], as_vector(
(w[1], w[2])), w[3], as_vector((w[4], w[5])))
def F_z(z, u, q):
return (inner(z, q) * dx - inner(q, grad(u)) * dx +
inner(jump(u) * n("+"), avg(q)) * dS)
def F_u(z, u, u_b, f_, v):
return (-inner(z, grad(v)) * dx +
inner(jump(v) * n("+"), avg(z)) * dS + (u - u_b) * v * ds -
v * u * dx - f_ * v * dx)
Fz = F_z(z = z0, u = u0, q = q0) + \
F_z(z = z1, u = u1, q = q1)
Fu = F_u(z = z0, u = u0, u_b = Expression(mms.u()), f_ = f[0], v = v0) + \
F_u(z = z1, u = u1, u_b = Expression(mms.v()), f_ = f[3], v = v1)
F = Fz + Fu
solve(F == 0, w)
u_0, z_0, u_1, z_1 = w.split()
u0File << u_0
z0File << z_0
u1File << u_1
z1File << z_1
FE = FunctionSpace(mesh, "DG", 3)
S_u = project(Expression(mms.u(), degree=5), FE)
S_v = project(Expression(mms.v(), degree=5), FE)
Eu = errornorm(u_0, S_u, norm_type="L2", degree_rise=3)
Ev = errornorm(u_1, S_v, norm_type="L2", degree_rise=3)
return Eu, Ev