Hi,
After following this FEniCSx tutorial (2D pressure driven flow between 2 walls), I tried to replace the top wall (see geometry below) by a symmetry condition but could not achieve to do so. I followed this post to define the symmetry, but this gives me a strange velocity and pressure field I can not explain. Flow is left to right (imposed pressure on the inlet and outlet), bottom wall is a no-slip condition and top wall is a symmetry condition.
Here is the code I am using for this.
from mpi4py import MPI
from petsc4py import PETSc
import numpy as np
from dolfinx.fem import Constant,Function, FunctionSpace, assemble_scalar, dirichletbc, form, locate_dofs_geometrical
from dolfinx.fem.petsc import assemble_matrix, assemble_vector, apply_lifting, create_vector, set_bc
from dolfinx.io import XDMFFile
from dolfinx.mesh import create_unit_square
from dolfinx.plot import create_vtk_mesh
from ufl import (FacetNormal, FiniteElement, Identity,TestFunction, TrialFunction, VectorElement,
div, dot, ds, dx, inner, lhs, nabla_grad, rhs, sym)
#### Geometry and parameters ####
mesh = create_unit_square(MPI.COMM_WORLD, 30, 30)
t = 0
T = 10
num_steps = 500
dt = T/num_steps
#### Elements and FunctionSpaces ####
v_cg2 = VectorElement("CG", mesh.ufl_cell(), 2)
s_cg1 = FiniteElement("CG", mesh.ufl_cell(), 1)
V = FunctionSpace(mesh, v_cg2)
Q = FunctionSpace(mesh, s_cg1)
u = TrialFunction(V)
v = TestFunction(V)
p = TrialFunction(Q)
q = TestFunction(Q)
def top(x):
return np.isclose(x[1],1)
def bot(x):
return np.isclose(x[1],0)
def inflow(x):
return np.isclose(x[0], 0)
def outflow(x):
return np.isclose(x[0], 1)
#Symmetry on top plane
top_dofs = locate_dofs_geometrical(V, top)
u_sym = np.array((0), dtype=PETSc.ScalarType)
bc_top = dirichletbc(u_sym, top_dofs, V.sub(1))
##Wall on top plane
#top_dofs = locate_dofs_geometrical(V, top)
#u_noslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)
#bc_top = dirichletbc(u_noslip, top_dofs, V)
#Wall on bot plane
bot_dofs = locate_dofs_geometrical(V, bot)
u_noslip = np.array((0,) * mesh.geometry.dim, dtype=PETSc.ScalarType)
bc_bot = dirichletbc(u_noslip, bot_dofs, V)
#Pressure Gradient between inlet and outlet
inflow_dofs = locate_dofs_geometrical(Q, inflow)
bc_inflow = dirichletbc(PETSc.ScalarType(8), inflow_dofs, Q)
outflow_dofs = locate_dofs_geometrical(Q, outflow)
bc_outflow = dirichletbc(PETSc.ScalarType(0), outflow_dofs, Q)
#Collect boundary conditions on u and p
bcu = [bc_bot,bc_top]
bcp = [bc_outflow,bc_inflow]
#### Variationnal forms ####
u_n = Function(V)
u_n.name = "u_n"
U = 0.5 * (u_n + u)
n = FacetNormal(mesh)
f = Constant(mesh, PETSc.ScalarType((0,0)))
k = Constant(mesh, PETSc.ScalarType(dt))
mu = Constant(mesh, PETSc.ScalarType(1))
rho = Constant(mesh, PETSc.ScalarType(1))
# Define strain-rate tensor
def epsilon(u):
return sym(nabla_grad(u))
# Define stress tensor
def sigma(u, p):
return 2*mu*epsilon(u) - p*Identity(u.geometric_dimension())
# Define the variational problem for the first step
p_n = Function(Q)
p_n.name = "p_n"
F1 = rho*dot((u - u_n) / k, v)*dx
F1 += rho*dot(dot(u_n, nabla_grad(u_n)), v)*dx
F1 += inner(sigma(U, p_n), epsilon(v))*dx
F1 += dot(p_n*n, v)*ds - dot(mu*nabla_grad(U)*n, v)*ds
F1 -= dot(f, v)*dx
a1 = form(lhs(F1))
L1 = form(rhs(F1))
A1 = assemble_matrix(a1, bcs=bcu)
A1.assemble()
b1 = create_vector(L1)
# Define variational problem for step 2
u_ = Function(V)
a2 = form(dot(nabla_grad(p), nabla_grad(q))*dx)
L2 = form(dot(nabla_grad(p_n), nabla_grad(q))*dx - (1/k)*div(u_)*q*dx)
A2 = assemble_matrix(a2, bcs=bcp)
A2.assemble()
b2 = create_vector(L2)
# Define variational problem for step 3
p_ = Function(Q)
a3 = form(dot(u, v)*dx)
L3 = form(dot(u_, v)*dx - k*dot(nabla_grad(p_ - p_n), v)*dx)
A3 = assemble_matrix(a3)
A3.assemble()
b3 = create_vector(L3)
# Solver for step 1
solver1 = PETSc.KSP().create(mesh.comm)
solver1.setOperators(A1)
solver1.setType(PETSc.KSP.Type.BCGS)
pc1 = solver1.getPC()
pc1.setType(PETSc.PC.Type.HYPRE)
pc1.setHYPREType("boomeramg")
# Solver for step 2
solver2 = PETSc.KSP().create(mesh.comm)
solver2.setOperators(A2)
solver2.setType(PETSc.KSP.Type.BCGS)
pc2 = solver2.getPC()
pc2.setType(PETSc.PC.Type.HYPRE)
pc2.setHYPREType("boomeramg")
# Solver for step 3
solver3 = PETSc.KSP().create(mesh.comm)
solver3.setOperators(A3)
solver3.setType(PETSc.KSP.Type.CG)
pc3 = solver3.getPC()
pc3.setType(PETSc.PC.Type.SOR)
xdmf = XDMFFile(mesh.comm, "poiseuille.xdmf", "w")
xdmf.write_mesh(mesh)
xdmf.write_function(u_n, t)
xdmf.write_function(p_n, t)
for i in range(num_steps):
t += dt
print("Iteration "+str(i)+"/"+str(num_steps))
# Step 1: Tentative veolcity step
with b1.localForm() as loc_1:
loc_1.set(0)
assemble_vector(b1, L1)
apply_lifting(b1, [a1], [bcu])
b1.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
set_bc(b1, bcu)
solver1.solve(b1, u_.vector)
u_.x.scatter_forward()
# Step 2: Pressure corrrection step
with b2.localForm() as loc_2:
loc_2.set(0)
assemble_vector(b2, L2)
apply_lifting(b2, [a2], [bcp])
b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, bcp)
solver2.solve(b2, p_.vector)
p_.x.scatter_forward()
# Step 3: Velocity correction step
with b3.localForm() as loc_3:
loc_3.set(0)
assemble_vector(b3, L3)
b3.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
solver3.solve(b3, u_.vector)
u_.x.scatter_forward()
# Update variable with solution form this time step
u_n.x.array[:] = u_.x.array[:]
p_n.x.array[:] = p_.x.array[:]
# Write solutions to file
xdmf.write_function(u_n, t)
xdmf.write_function(p_n, t)
# Close xmdf file
xdmf.close()
The restult is fine when using a wall on the top boundary (commented section), but the issue appears when using the symmetry condition.
Finer mesh and smaller time steps do not seem to help and I am running out of ideas, so I’d appreciate any help as I am not really experienced using FEniCSx and Finite Elements in general.
Is my symmetry condition well defined ? Is it causing an issue with the weak formulation or some DOFs because the symmetry only constraints 1 component of the velocity ?
Is this solving algorithm well suited for what I am trying to do ?
I am using dolfinx 0.5.2 installed with conda if that can help.
Thanks