def _abs(self):
return Abs(self)
def transient_adv_2d_R22_FV(mesh_2d, s, init_cond, dt_num, steps):
# Input
# mu: diffusivity, constant
# order: element order. 1 = linear elements. 2 = quadratic elements.
# nx: number of elements
# Output
# u: the solution, dolfin function
P1 = FiniteElement("DG", mesh_2d.ufl_cell(), 0)
TH = MixedElement([P1, P1])
V = FunctionSpace(mesh_2d, TH)
V0 = V.sub(0).collapse()
u = TrialFunction(V)
w = TestFunction(V)
#u0 = Function(V0)
u0 = project(init_cond, V0)
DG_space = FunctionSpace(mesh_2d, 'DG', 0)
Vec = VectorFunctionSpace(mesh_2d, 'CR', 1)
adv = project(Expression( ('sin(pi*x[0])*cos(pi*x[1])', '-cos(pi*x[0])*sin(pi*x[1])'), degree=1 ), Vec )
n = FacetNormal(mesh_2d)
x_ = interpolate(Expression("x[0]", degree=1), DG_space)
y_ = interpolate(Expression("x[1]",degree=1), DG_space)
Delta_h = sqrt(jump(x_)**2 + jump(y_)**2)
u_list = [u0.copy()]
def boundary_L(x, on_boundary):
return on_boundary and near(x[0], 0, DOLFIN_EPS)
def boundary_R(x, on_boundary):
return on_boundary and near(x[0], 1, DOLFIN_EPS)
bc = []
dt = Constant(dt_num)
one = Constant(1.0)
adv_np = ( dot ( adv, n ) + Abs ( dot ( adv, n ) ) ) / 2.0
adv_nm = ( dot ( adv, n ) - Abs ( dot ( adv, n ) ) ) / 2.0
#adv_n = dot ( adv, n )
def L(w, u):
#return Constant(0.5)*dot(jump(w), adv_np('+')*u('+') + adv_nm('+')*u('-') )\
# + Constant(0.5)*jump(w)*avg(dot( adv, n ))*jump(u)/Delta_h
#return dot(w('-'), (dot(adv_n, u('-') - u('+') )))
return dot(jump(w), adv_np('+')*u('+') - adv_np('-')*u('-') )
delta_u = as_vector([(u[0] - u0), (u[1] - u[0])])
W = as_matrix([[7.0/24, -1.0/24], [13.0/24, 5.0/24]])
W_inv = inv(W)
WU = W*delta_u
#ww = as_vector( [0.5*(w[0]*s-L(w[0], u[1])), 0.5*(w[1]*s-L(w[1], u0))] )
LL = as_vector([L(w[0], u[0] - u0), L(w[1], u[1] - u[0])])
ww = as_vector([0.5, 0.5])
F = dot( delta_u/dt, as_vector([w[0], w[1]]) )*dx\
+ (L(w[0], WU[0]) + L(w[1], WU[1]))*dS \
+ 0.5*(L(w[0], u0) + L(w[1], u0))*dS \
- (0.5*(w[0]*s) + 0.5*(w[1]*s))*dx
a, L = lhs(F), rhs(F)
u = Function(V)
problem = LinearVariationalProblem(a, L, u, bcs=bc)
solver = LinearVariationalSolver(problem)
prm = solver.parameters
prm['krylov_solver']['absolute_tolerance'] = 1e-12
prm['krylov_solver']['relative_tolerance'] = 1e-10
prm['krylov_solver']['maximum_iterations'] = 1000
#if iterative_solver:
prm['linear_solver'] = 'gmres'
prm['preconditioner'] = 'ilu'
for i in range(steps):
solver.solve()
u1, u2 = u.split(True)
u0.assign(u2)
u_list.append(u0.copy())
return u_list
def steady_adv_diff_1d_FV(mu, nx, s, left_bc, right_bc):
# Input
# mu: diffusivity, constant
# nx: number of elements
# Output
# u: the solution, dolfin function
# mesh of 10 uniform linear elements
mesh_1d = IntervalMesh(nx, 0, 1)
h = 1.0 / nx
adv = as_vector([1.0])
eta = Constant(1e5)
one = Constant(1.0)
Pe = 1.0 * h / (2 * mu)
print('Peclet number = ', Pe)
V = FunctionSpace(mesh_1d, 'DG', 0)
u = TrialFunction(V)
w = TestFunction(V)
n = FacetNormal(mesh_1d)
x_ = interpolate(Expression("x[0]", degree=1), V)
Delta_h = sqrt(jump(x_)**2)
boundary_markers = MeshFunction('size_t', mesh_1d, dim=0)
class left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], 0.0, DOLFIN_EPS)
class right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], 1.0, DOLFIN_EPS)
b_left = left()
b_right = right()
boundary_markers.set_all(0)
b_left.mark(boundary_markers, 2)
b_right.mark(boundary_markers, 3)
bc = []
dS = Measure('dS', domain=mesh_1d, subdomain_data=boundary_markers)
ds = Measure('ds', domain=mesh_1d, subdomain_data=boundary_markers)
dx = Measure('dx', domain=mesh_1d, subdomain_data=boundary_markers)
adv_np = (dot(adv, n) + Abs(dot(adv, n))) / 2.0
adv_nm = (dot(adv, n) - Abs(dot(adv, n))) / 2.0
#adv_n = dot ( adv, n )
F = Constant(mu)*jump(w)*jump(u)/Delta_h*dS - w*s*dx\
- w*(Constant(left_bc) - u)/(x_ - 0.0)*ds(2) - w*(Constant(right_bc) - u)/(one -x_)*ds(3) \
+ jump(w)*(adv_np('+')*u('+') - adv_np('-')*u('-'))*dS(0)\
a, L = lhs(F), rhs(F)
u = Function(V)
solve(a == L, u, bcs=bc)
return u, V.dofmap().dofs(), V.tabulate_dof_coordinates().T[
0] #vertex_to_dof_map(V)
def transient_adv_2d_upwind(mesh_2d, s, init_cond, dt_num, steps, theta_num, add_diff):
# Input
# nx: number of elements
# Output
# u: the solution, dolfin function
V = FunctionSpace(mesh_2d, 'DG', 0)
Vec = VectorFunctionSpace(mesh_2d, 'CR', 1)
u = TrialFunction(V)
w = TestFunction(V)
u0 = project(init_cond, V)
adv = project(Expression( ('sin(pi*x[0])*cos(pi*x[1])', '-cos(pi*x[0])*sin(pi*x[1])'), degree=1 ), Vec )
n = FacetNormal(mesh_2d)
x_ = interpolate(Expression("x[0]", degree=1), V)
y_ = interpolate(Expression("x[1]",degree=1), V)
Delta_h = sqrt(jump(x_)**2 + jump(y_)**2)
u_list = [u0.copy()]
def boundary_L(x, on_boundary):
return on_boundary and near(x[0], 0, DOLFIN_EPS)
def boundary_R(x, on_boundary):
return on_boundary and near(x[0], 1, DOLFIN_EPS)
#bc_L = DirichletBC(V, Constant(left_bc), boundary_L)
#bc_R = DirichletBC(V, Constant(right_bc), boundary_R)
bc = []
dt = Constant(dt_num)
one = Constant(1.0)
adv_np = ( dot ( adv, n ) + Abs ( dot ( adv, n ) ) ) / 2.0
adv_nm = ( dot ( adv, n ) - Abs ( dot ( adv, n ) ) ) / 2.0
adv_n = dot ( adv, n )
def L(w, u):
return dot(jump(w), adv_np('+')*u('+') + adv_nm('+')*u('-') )
#- Constant(add_diff)*0.5*dt*jump(dot(adv, adv))*dot(jump(w, n), jump(u, n))/Delta_h
#return dot(jump(w), adv_np('+')*u('+') - adv_np('-')*u('-') )
#return Constant(0.5)*w*div(adv*u) + Constant(0.5)*w*dot(adv, grad(u))
delta_u = u - u0
W = 0.5
ww = (w*s-L(w, u0))
#F = (w*delta_u/dt + W*L(w, u - u0) - ww)*dx
#a, L = lhs(F), rhs(F)
theta = Constant(theta_num)
a = ( w*u/dt )*dx + theta*L(w, u)*dS
L = ( w*u0/dt + w*s )*dx - (one-theta)*L(w, u0)*dS
u = Function(V)
problem = LinearVariationalProblem(a, L, u, bcs=bc)
solver = LinearVariationalSolver(problem)
prm = solver.parameters
prm['krylov_solver']['absolute_tolerance'] = 1e-12
prm['krylov_solver']['relative_tolerance'] = 1e-10
prm['krylov_solver']['maximum_iterations'] = 20
#if iterative_solver:
prm['linear_solver'] = 'gmres'
prm['preconditioner'] = 'ilu'
for i in range(steps):
solver.solve()
u0.assign(u)
u_list.append(u0.copy())
return u_list