2D Stokes Using Firedrake¶

MIT License

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$$ \let\b=\boldsymbol \def\ip#1#2{\left\langle #1, #2\right\rangle} \begin{eqnarray*} \Delta \b{u}+ \nabla p & = & -\b{f} \quad (x \in \Omega),\\ \nabla \cdot \b{u} & = & 0 \quad (x \in \Omega),\\ \b{u} & = & \b{u}_0 \quad (x \in \partial \Omega) . \end{eqnarray*} $$

import numpy as np
import numpy.linalg as la
import firedrake.mesh as fd_mesh
import matplotlib.pyplot as plt

from firedrake import *

import meshpy.triangle as triangle

def round_trip_connect(start, end):
    return [(i, i+1) for i in range(start, end)] + [(end, start)]

reentrant_corner = 1

def make_mesh():
    if not reentrant_corner:
        # tube
        points = [
                  (-1, -1), 
                  (1,-1),
                  (1, 1),
                  (-1, 1)]
    else:
        # reentrant corner
        points = [
                  (-1, 0), 
                  (0,0), 
                  (0, -1), 
                  (1,-1),
                  (1, 1),
                  (-1, 1)]
        
    facets = round_trip_connect(0, len(points)-1)
    # 1 for "prescribed 0 velocity"
    # 2 for "prescribed velocity"
    facet_markers = [1] * len(facets)
    facet_markers[-1] = 2
    facet_markers[-3] = 3

    def needs_refinement(vertices, area):
        bary = np.sum(np.array(vertices), axis=0)/3
        if reentrant_corner:
            max_area = 0.0001 + la.norm(bary, np.inf)*0.01
        else:
            max_area = 0.01
        return bool(area > max_area)

    info = triangle.MeshInfo()
    info.set_points(points)
    info.set_facets(facets, facet_markers=facet_markers)

    built_mesh = triangle.build(info, refinement_func=needs_refinement)
    plex = fd_mesh._from_cell_list(
        2, np.array(built_mesh.elements), np.array(built_mesh.points), COMM_WORLD)

    import firedrake.cython.dmplex as dmplex

    v_start, v_end = plex.getDepthStratum(0)   # vertices
    for facet, fmarker in zip(built_mesh.facets, built_mesh.facet_markers):
        vertices = [fvert + v_start for fvert in facet]
        join = plex.getJoin(vertices)
        plex.setLabelValue(dmplex.FACE_SETS_LABEL, join[0], fmarker)

    return Mesh(plex)
    
mesh = make_mesh()

triplot(mesh)
plt.gca().set_aspect("equal")
plt.legend()

<matplotlib.legend.Legend at 0x7ff21c3e8810>

Choose some function spaces:

if 0:
    # "P1-P1"
    V = VectorFunctionSpace(mesh, "CG", 1)
    W = FunctionSpace(mesh, "CG", 1)
elif 1:
    # MINI
    P1 = FiniteElement("CG", cell=mesh.ufl_cell(), degree=1)
    B = FiniteElement("B", cell=mesh.ufl_cell(), degree=3)
    mini = P1 + B
    V = VectorFunctionSpace(mesh, mini)
    W = FunctionSpace(mesh, 'CG', 1)
else:
    # Taylor-Hood
    V = VectorFunctionSpace(mesh, "CG", 2)
    W = FunctionSpace(mesh, "CG", 1)
    
Z = V * W

Set up the weak form: $$ \begin{align*} a (\b{u}, \b{v}) = \int_{\Omega} J_{\b{u}} : J_{\b{v}}, \\ b (\b{v}, q) = \int_{\Omega} \nabla \cdot \b{v}q, \end{align*} $$ where $A : B = \operatorname{tr} (AB^T)$. Find $(\b{u}, p) \in X \times M$ so that $$ \begin{eqnarray*} a (\b{u}, \b{v}) + b (\b{v}, p) & = & \ip{\b{f}}{\b{v}}_{L^2} \quad (\b{v} \in X),\\ b (\b{u}, q) & = & 0 \quad (q \in M), \end{eqnarray*} $$

u, p = TrialFunctions(Z)
v, q = TestFunctions(Z)

a = (inner(grad(u), grad(v)) - p * div(v) + div(u) * q)*dx

L = inner(Constant((0, 0)), v) * dx

Pick boundary conditions:

bcs = [
    DirichletBC(Z.sub(0), Constant((1, 0)), (2,)),
    DirichletBC(Z.sub(0), Constant((0.5 if reentrant_corner else 1, 0)), (3,)),
    DirichletBC(Z.sub(0), Constant((0, 0)), (1,))
]

Let the linear solver know about the nullspace:

nullspace = MixedVectorSpaceBasis(
    Z, [Z.sub(0), VectorSpaceBasis(constant=True)])

Solve:

upsol = Function(Z)
usol, psol = upsol.split()

solve(a == L, upsol, bcs=bcs,
      nullspace=nullspace,
      solver_parameters={'pc_type': 'fieldsplit',
                         'ksp_rtol': 1e-15,
                         'pc_fieldsplit_type': 'schur',
                         'fieldsplit_schur_fact_type': 'diag',
                         'fieldsplit_0_pc_type': 'redundant',
                         'fieldsplit_0_redundant_pc_type': 'lu',
                         'fieldsplit_1_pc_type': 'none',
                         'ksp_monitor_true_residual': None,
                         'mat_type': 'aij'})

Plot the velocity:

plt.figure(figsize=(8, 8))
ax = plt.gca()
ax.set_aspect("equal")
triplot(mesh, axes=ax, interior_kw=dict(alpha=0.05))
quiver(usol, axes=ax)

<matplotlib.quiver.Quiver at 0x7ff1ff1d3790>

Plot the pressure and the divergence of $u$:

div_usol = project(div(usol), W)

plt.figure(figsize=(12,6))
plt.subplot(121)
ax = plt.gca()
l = tricontourf(psol, axes=ax)
triplot(mesh, axes=ax, interior_kw=dict(alpha=0.05))
plt.colorbar(l)
plt.title("Pressure")

plt.subplot(122)
ax = plt.gca()
l = tricontourf(div_usol, axes=ax)
triplot(mesh, axes=ax, interior_kw=dict(alpha=0.05))
plt.colorbar(l)
plt.title(r"$\nabla\cdot u$")

Text(0.5, 1.0, '$\\nabla\\cdot u$')