vedo/examples/other/dolfin/stokes1.py

"""This demo solves the Stokes equations, using quadratic elements for
the velocity and first degree elements for the pressure (Taylor-Hood elements)"""
# Credits:
# https://github.com/pf4d/fenics_scripts/blob/master/cbc_block/stokes.py
from dolfin import *
import numpy as np
from vedo.dolfin import plot
from vedo import Latex, dataurl, download

# Load mesh and subdomains
fpath = download(dataurl + "dolfin_fine.xml")
mesh = Mesh(fpath)

fpath = download(dataurl + "dolfin_fine_subdomains.xml.gz")
sub_domains = MeshFunction("size_t", mesh, fpath)

# Define function spaces
P2 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = P2 * P1
W = FunctionSpace(mesh, TH)

# No-slip boundary condition for velocity
noslip = Constant((0, 0))
bc0 = DirichletBC(W.sub(0), noslip, sub_domains, 0)

# Inflow boundary condition for velocity
inflow = Expression(("-sin(x[1]*pi)", "0.0"), degree=2)
bc1 = DirichletBC(W.sub(0), inflow, sub_domains, 1)
bcs = [bc0, bc1]

# Define variational problem
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
f = Constant((0, 0))
a = (inner(grad(u), grad(v)) - div(v) * p + q * div(u)) * dx
L = inner(f, v) * dx
w = Function(W)

solve(a == L, w, bcs)

# Split the mixed solution using a shallow copy
(u, p) = w.split()

##################################################################### vedo
f = r"-\nabla \cdot(\nabla u+p I)=f ~\mathrm{in}~\Omega"
formula = Latex(f, pos=(0.55, 0.45, -0.05), s=0.1)

plot(
    u,
    N=2,
    mode="mesh and arrows",
    scale=0.03,
    wireframe=True,
    scalarbar=False,
    style=1,
).close()

plot(p, text="pressure", cmap="rainbow").close()