Stokes flow

Keywords: periodic boundary conditions, multiple fields, mean value constraint

Tip

This example is also available as a Jupyter notebook: stokes-flow.ipynb.

Figure 1: Left: Computational domain $\Omega$ with boundaries $\Gamma_1$, $\Gamma_3$ (periodic boundary conditions) and $\Gamma_2$, $\Gamma_4$ (homogeneous Dirichlet boundary conditions). Right: Magnitude of the resulting velocity field.

Introduction and problem formulation

This example is a translation of the step-45 example from deal.ii which solves Stokes flow on a quarter circle. In particular it shows how to use periodic boundary conditions, how to solve a problem with multiple unknown fields, and how to enforce a specific mean value of the solution. For the mesh generation we use Gmsh.jl and then use FerriteGmsh.jl to import the mesh into Ferrite's format.

The strong form of Stokes flow with velocity $\boldsymbol{u}$ and pressure $p$ can be written as follows:

\[\begin{align*} -\Delta \boldsymbol{u} + \boldsymbol{\nabla} p &= \bigl(\exp(-100||\boldsymbol{x} - (0.75, 0.1)||^2), 0\bigr) =: \boldsymbol{b} \quad \forall \boldsymbol{x} \in \Omega,\\ -\boldsymbol{\nabla} \cdot \boldsymbol{u} &= 0 \quad \forall \boldsymbol{x} \in \Omega, \end{align*}\]

where the domain is defined as $\Omega = \{\boldsymbol{x} \in (0, 1)^2: \ ||\boldsymbol{x}|| \in (0.5, 1)\}$, see Figure 1. For the velocity we use periodic boundary conditions on the inlet $\Gamma_1$ and outlet $\Gamma_3$:

\[\begin{align*} u_x(0,\nu) &= -u_y(\nu, 0) \quad & \nu\ \in\ [0.5, 1],\\ u_y(0,\nu) &= u_x(\nu, 0) \quad & \nu\ \in\ [0.5, 1], \end{align*}\]

and homogeneous Dirichlet boundary conditions for $\Gamma_2$ and $\Gamma_4$:

\[\boldsymbol{u} = \boldsymbol{0} \quad \forall \boldsymbol{x}\ \in\ \Gamma_2 \cup \Gamma_4 := \{ \boldsymbol{x}:\ ||\boldsymbol{x}|| \in \{0.5, 1\}\}.\]

The corresponding weak form reads as follows: Find $(\boldsymbol{u}, p) \in \mathbb{U} \times \mathrm{L}_2$ s.t.

\[\begin{align*} \int_\Omega \Bigl[[\delta\boldsymbol{u}\otimes\boldsymbol{\nabla}]:[\boldsymbol{u}\otimes\boldsymbol{\nabla}] - (\boldsymbol{\nabla}\cdot\delta\boldsymbol{u})\ p\ \Bigr] \mathrm{d}\Omega &= \int_\Omega \delta\boldsymbol{u} \cdot \boldsymbol{b}\ \mathrm{d}\Omega \quad \forall \delta \boldsymbol{u} \in \mathbb{U},\\ \int_\Omega - (\boldsymbol{\nabla}\cdot\boldsymbol{u})\ \delta p\ \mathrm{d}\Omega &= 0 \quad \forall \delta p \in \mathrm{L}_2, \end{align*}\]

where $\mathbb{U}$ is a suitable function space, that, in particular, enforces the Dirichlet boundary conditions, and the periodicity constraints. This formulation is a saddle point problem, and, just like the example with Incompressible Elasticity, we need our formulation to fulfill the LBB condition. We ensure this by using a quadratic approximation for the velocity field, and a linear approximation for the pressure.

With this formulation and boundary conditions for $\boldsymbol{u}$ the pressure will only be determined up to a constant. We will therefore add an additional constraint which fixes this constant (see deal.ii step-11 for some more discussion around this). In particular, we will enforce the mean value of the pressure on the boundary to be 0, i.e. $\int_{\Gamma} p\ \mathrm{d}\Gamma = 0$. One option is to enforce this using a Lagrange multipler. This would give a contribution $\lambda \int_{\Gamma} \delta p\ \mathrm{d}\Gamma$ to the second equation in the weak form above, and a third equation $\delta\lambda \int_{\Gamma} p\ \mathrm{d}\Gamma = 0$ so that we can solve for $\lambda$. However, since we in this case are not interested in computing $\lambda$, and since the constraint is linear, we can directly embed this constraint using an AffineConstraint in Ferrite.

After FE discretization we obtain a linear system of the form $\underline{\underline{K}}\ \underline{a} = \underline{f}$, where

\[\underline{\underline{K}} = \begin{bmatrix} \underline{\underline{K}}_{uu} & \underline{\underline{K}}_{pu}^\textrm{T} \\ \underline{\underline{K}}_{pu} & \underline{\underline{0}} \end{bmatrix}, \quad \underline{a} = \begin{bmatrix} \underline{a}_{u} \\ \underline{a}_{p} \end{bmatrix}, \quad \underline{f} = \begin{bmatrix} \underline{f}_{u} \\ \underline{0} \end{bmatrix},\]

and where

\[\begin{align*} (\underline{\underline{K}}_{uu})_{ij} &= \int_\Omega [\boldsymbol{\phi}^u_i\otimes\boldsymbol{\nabla}]:[\boldsymbol{\phi}^u_j\otimes\boldsymbol{\nabla}] \mathrm{d}\Omega, \\ (\underline{\underline{K}}_{pu})_{ij} &= \int_\Omega - (\boldsymbol{\nabla}\cdot\boldsymbol{\phi}^u_j)\ \phi^p_i\ \mathrm{d}\Omega, \\ (\underline{f}_{u})_{i} &= \int_\Omega \boldsymbol{\phi}^u_i \cdot \boldsymbol{b}\ \mathrm{d}\Omega. \end{align*}\]

The affine constraint to enforce zero mean pressure on the boundary is obtained from $\underline{\underline{C}}_p\ \underline{a}_p = \underline{0}$, where

\[(\underline{\underline{C}}_p)_{1j} = \int_{\Gamma} \phi^p_j\ \mathrm{d}\Gamma.\]

Note

The constraint matrix $\underline{\underline{C}}_p$ is the same matrix we would have obtained when assembling the system with the Lagrange multiplier. In that case the full system would be

\[\underline{\underline{K}} = \begin{bmatrix} \underline{\underline{K}}_{uu} & \underline{\underline{K}}_{pu}^\textrm{T} & \underline{\underline{0}}\\ \underline{\underline{K}}_{pu} & \underline{\underline{0}} & \underline{\underline{C}}_p^\mathrm{T} \\ \underline{\underline{0}} & \underline{\underline{C}}_p & 0 \\ \end{bmatrix}, \quad \underline{a} = \begin{bmatrix} \underline{a}_{u} \\ \underline{a}_{p} \\ \underline{a}_{\lambda} \end{bmatrix}, \quad \underline{f} = \begin{bmatrix} \underline{f}_{u} \\ \underline{0} \\ \underline{0} \end{bmatrix}.\]

Commented program

What follows is a program spliced with comments. The full program, without comments, can be found in the next section.

using Ferrite, FerriteGmsh, Gmsh, Tensors, LinearAlgebra, SparseArrays

Geometry and mesh generation with Gmsh.jl

In the setup_grid function below we use the Gmsh.jl package for setting up the geometry and performing the meshing. We will not discuss this part in much detail but refer to the Gmsh API documentation instead. The most important thing to note is the mesh periodicity constraint that is applied between the "inlet" and "outlet" parts using gmsh.model.set_periodic. This is necessary to later on apply a periodicity constraint for the approximated velocity field.

function setup_grid(h=0.05)
    # Initialize gmsh
    Gmsh.initialize()
    gmsh.option.set_number("General.Verbosity", 2)

    # Add the points
    o = gmsh.model.geo.add_point(0.0, 0.0, 0.0, h)
    p1 = gmsh.model.geo.add_point(0.5, 0.0, 0.0, h)
    p2 = gmsh.model.geo.add_point(1.0, 0.0, 0.0, h)
    p3 = gmsh.model.geo.add_point(0.0, 1.0, 0.0, h)
    p4 = gmsh.model.geo.add_point(0.0, 0.5, 0.0, h)

    # Add the lines
    l1 = gmsh.model.geo.add_line(p1, p2)
    l2 = gmsh.model.geo.add_circle_arc(p2, o, p3)
    l3 = gmsh.model.geo.add_line(p3, p4)
    l4 = gmsh.model.geo.add_circle_arc(p4, o, p1)

    # Create the closed curve loop and the surface
    loop = gmsh.model.geo.add_curve_loop([l1, l2, l3, l4])
    surf = gmsh.model.geo.add_plane_surface([loop])

    # Synchronize the model
    gmsh.model.geo.synchronize()

    # Create the physical domains
    gmsh.model.add_physical_group(1, [l1], -1, "Γ1")
    gmsh.model.add_physical_group(1, [l2], -1, "Γ2")
    gmsh.model.add_physical_group(1, [l3], -1, "Γ3")
    gmsh.model.add_physical_group(1, [l4], -1, "Γ4")
    gmsh.model.add_physical_group(2, [surf])

    # Add the periodicity constraint using 4x4 affine transformation matrix,
    # see https://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations
    transformation_matrix = zeros(4, 4)
    transformation_matrix[1, 2] = 1  # -sin(-pi/2)
    transformation_matrix[2, 1] = -1 #  cos(-pi/2)
    transformation_matrix[3, 3] = 1
    transformation_matrix[4, 4] = 1
    transformation_matrix = vec(transformation_matrix')
    gmsh.model.mesh.set_periodic(1, [l1], [l3], transformation_matrix)

    # Generate a 2D mesh
    gmsh.model.mesh.generate(2)

    # Save the mesh, and read back in as a Ferrite Grid
    grid = mktempdir() do dir
        path = joinpath(dir, "mesh.msh")
        gmsh.write(path)
        togrid(path)
    end

    # Finalize the Gmsh library
    Gmsh.finalize()

    return grid
end

Degrees of freedom

As mentioned in the introduction we will use a quadratic approximation for the velocity field and a linear approximation for the pressure to ensure that we fulfill the LBB condition. We create the corresponding FE values with interpolations ipu for the velocity and ipp for the pressure. Note that we use linear geometric interpolation (ipg) for both the velocity and pressure, this is because our grid contains linear triangles. We also construct face-values for the pressure since we need to integrate along the boundary when assembling the constraint matrix $\underline{\underline{C}}$.

function setup_fevalues(ipu, ipp, ipg)
    qr = QuadratureRule{2,RefTetrahedron}(2)
    cvu = CellVectorValues(qr, ipu, ipg)
    cvp = CellScalarValues(qr, ipp, ipg)
    qr_face = QuadratureRule{1,RefTetrahedron}(2)
    fvp = FaceScalarValues(qr_face, ipp, ipg)
    return cvu, cvp, fvp
end

The setup_dofs function creates the DofHandler, and adds the two fields: a vector field :u with interpolation ipu, and a scalar field :p with interpolation ipp.

function setup_dofs(grid, ipu, ipp)
    dh = DofHandler(grid)
    add!(dh, :u, 2, ipu)
    add!(dh, :p, 1, ipp)
    close!(dh)
    return dh
end

Boundary conditions and constraints

Now it is time to setup the ConstraintHandler and add our boundary conditions and the mean value constraint. This is perhaps the most interesting section in this example, and deserves some attention.

Let's first discuss the assembly of the constraint matrix $\underline{\underline{C}}$ and how to create an AffineConstraint from it. This is done in the setup_mean_constraint function below. Assembling this is not so different from standard assembly in Ferrite: we loop over all the faces, loop over the quadrature points, and loop over the shape functions. Note that since there is only one constraint the matrix will only have one row. After assembling C we construct an AffineConstraint from it. We select the constrained dof to be the one with the highest weight (just to avoid selecting one with 0 or a very small weight), then move the remaining to the right hand side. As an example, consider the case where the constraint equation $\underline{\underline{C}}_p\ \underline{a}_p$ is

\[w_{10} p_{10} + w_{23} p_{23} + w_{154} p_{154} = 0\]

i.e. dofs 10, 23, and 154, are the ones located on the boundary (all other dofs naturally gives 0 contribution). If $w_{23}$ is the largest weight, then we select $p_{23}$ to be the constrained one, and thus reorder the constraint to the form

\[p_{23} = -\frac{w_{10}}{w_{23}} p_{10} -\frac{w_{154}}{w_{23}} p_{154} + 0,\]

which is the form the AffineConstraint constructor expects.

Note

If all nodes along the boundary are equidistant all the weights would be the same. In this case we can construct the constraint without having to do any integration by simply finding all degrees of freedom that are located along the boundary (and using 1 as the weight). This is what is done in the deal.ii step-11 example.

function setup_mean_constraint(dh, fvp)
    assembler = start_assemble()
    # All external boundaries
    set = union(
        getfaceset(dh.grid, "Γ1"),
        getfaceset(dh.grid, "Γ2"),
        getfaceset(dh.grid, "Γ3"),
        getfaceset(dh.grid, "Γ4"),
    )
    # Allocate buffers
    range_p = dof_range(dh, :p)
    element_dofs = zeros(Int, ndofs_per_cell(dh))
    element_dofs_p = view(element_dofs, range_p)
    element_coords = zeros(Vec{2}, 3)
    Ce = zeros(1, length(range_p)) # Local constraint matrix (only 1 row)
    # Loop over all the boundaries
    for (ci, fi) in set
        Ce .= 0
        getcoordinates!(element_coords, dh.grid, ci)
        reinit!(fvp, element_coords, fi)
        celldofs!(element_dofs, dh, ci)
        for qp in 1:getnquadpoints(fvp)
            dΓ = getdetJdV(fvp, qp)
            for i in 1:getnbasefunctions(fvp)
                Ce[1, i] += shape_value(fvp, qp, i) * dΓ
            end
        end
        # Assemble to row 1
        assemble!(assembler, [1], element_dofs_p, Ce)
    end
    C = end_assemble(assembler)
    # Create an AffineConstraint from the C-matrix
    _, J, V = findnz(C)
    _, constrained_dof = findmax(abs2, V)
    V ./= V[constrained_dof]
    mean_value_constraint = AffineConstraint(
        constrained_dof,
        Pair{Int,Float64}[J[i] => -V[i] for i in 1:length(J) if J[i] != constrained_dof],
        0.0,
    )
    return mean_value_constraint
end

We now setup all the boundary conditions in the setup_constraints function below. Since the periodicity constraint for this example is between two boundaries which are not parallel to each other we need to i) compute the mapping between each mirror face and the corresponding image face (on the element level) and ii) describe the dof relation between dofs on these two faces. In Ferrite this is done by defining a transformation of entities on the image boundary such that they line up with the matching entities on the mirror boundary. In this example we consider the inlet $\Gamma_1$ to be the image, and the outlet $\Gamma_3$ to be the mirror. The necessary transformation to apply then becomes a rotation of $\pi/2$ radians around the out-of-plane axis. We set up the rotation matrix R, and then compute the mapping between mirror and image faces using collect_periodic_faces where the rotation is applied to the coordinates. In the next step we construct the constraint using the PeriodicDirichlet constructor. We pass the constructor the computed mapping, and also the rotation matrix. This matrix is used to rotate the dofs on the mirror surface such that we properly constrain $\boldsymbol{u}_x$-dofs on the mirror to $-\boldsymbol{u}_y$-dofs on the image, and $\boldsymbol{u}_y$-dofs on the mirror to $\boldsymbol{u}_x$-dofs on the image.

For the remaining part of the boundary we add a homogeneous Dirichlet boundary condition on both components of the velocity field. This is done using the Dirichlet constructor, which we have discussed in other tutorials.

function setup_constraints(dh, fvp)
    ch = ConstraintHandler(dh)
    # Periodic BC
    R = rotation_tensor(π / 2)
    periodic_faces = collect_periodic_faces(dh.grid, "Γ3", "Γ1", x -> R ⋅ x)
    periodic = PeriodicDirichlet(:u, periodic_faces, R, [1, 2])
    add!(ch, periodic)
    # Dirichlet BC
    Γ24 = union(getfaceset(dh.grid, "Γ2"), getfaceset(dh.grid, "Γ4"))
    dbc = Dirichlet(:u, Γ24, (x, t) -> [0, 0], [1, 2])
    add!(ch, dbc)
    # Compute mean value constraint and add it
    mean_value_constraint = setup_mean_constraint(dh, fvp)
    add!(ch, mean_value_constraint)
    # Finalize
    close!(ch)
    update!(ch, 0)
    return ch
end

Global and local assembly

Assembly of the global system is also something that we have seen in many previous tutorials. One interesting thing to note here is that, since we have two unknown fields, we use the dof_range function to make sure we assemble the element contributions to the correct block of the local stiffness matrix ke.

function assemble_system!(K, f, dh, cvu, cvp)
    assembler = start_assemble(K, f)
    ke = zeros(ndofs_per_cell(dh), ndofs_per_cell(dh))
    fe = zeros(ndofs_per_cell(dh))
    range_u = dof_range(dh, :u)
    ndofs_u = length(range_u)
    range_p = dof_range(dh, :p)
    ndofs_p = length(range_p)
    ϕᵤ = Vector{Vec{2,Float64}}(undef, ndofs_u)
    ∇ϕᵤ = Vector{Tensor{2,2,Float64,4}}(undef, ndofs_u)
    divϕᵤ = Vector{Float64}(undef, ndofs_u)
    ϕₚ = Vector{Float64}(undef, ndofs_p)
    for cell in CellIterator(dh)
        reinit!(cvu, cell)
        reinit!(cvp, cell)
        ke .= 0
        fe .= 0
        for qp in 1:getnquadpoints(cvu)
            dΩ = getdetJdV(cvu, qp)
            for i in 1:ndofs_u
                ϕᵤ[i] = shape_value(cvu, qp, i)
                ∇ϕᵤ[i] = shape_gradient(cvu, qp, i)
                divϕᵤ[i] = shape_divergence(cvu, qp, i)
            end
            for i in 1:ndofs_p
                ϕₚ[i] = shape_value(cvp, qp, i)
            end
            # u-u
            for (i, I) in pairs(range_u), (j, J) in pairs(range_u)
                ke[I, J] += ( ∇ϕᵤ[i] ⊡ ∇ϕᵤ[j] ) * dΩ
            end
            # u-p
            for (i, I) in pairs(range_u), (j, J) in pairs(range_p)
                ke[I, J] += ( -divϕᵤ[i] * ϕₚ[j] ) * dΩ
            end
            # p-u
            for (i, I) in pairs(range_p), (j, J) in pairs(range_u)
                ke[I, J] += ( -divϕᵤ[j] * ϕₚ[i] ) * dΩ
            end
            # rhs
            for (i, I) in pairs(range_u)
                x = spatial_coordinate(cvu, qp, getcoordinates(cell))
                b = exp(-100 * norm(x - Vec{2}((0.75, 0.1)))^2)
                bv = Vec{2}((b, 0.0))
                fe[I] += (ϕᵤ[i] ⋅ bv) * dΩ
            end
        end
        assemble!(assembler, celldofs(cell), ke, fe)
    end
    return K, f
end

Running the simulation

We now have all the puzzle pieces, and just need to define the main function, which puts them all together.

function main()
    # Grid
    h = 0.05 # approximate element size
    grid = setup_grid(h)
    # Interpolations
    ipu = Lagrange{2,RefTetrahedron,2}() # quadratic
    ipp = Lagrange{2,RefTetrahedron,1}() # linear
    # Dofs
    dh = setup_dofs(grid, ipu, ipp)
    # FE values
    ipg = Lagrange{2,RefTetrahedron,1}() # linear geometric interpolation
    cvu, cvp, fvp = setup_fevalues(ipu, ipp, ipg)
    # Boundary conditions
    ch = setup_constraints(dh, fvp)
    # Global tangent matrix and rhs
    coupling = [true true; true false] # no coupling between pressure test/trial functions
    K = create_sparsity_pattern(dh, ch; coupling=coupling)
    f = zeros(ndofs(dh))
    # Assemble system
    assemble_system!(K, f, dh, cvu, cvp)
    # Apply boundary conditions and solve
    apply!(K, f, ch)
    u = K \ f
    apply!(u, ch)
    # Export the solution
    vtk_grid("stokes-flow", grid) do vtk
        vtk_point_data(vtk, dh, u)
    end
    return
end

Run it!

main()

The resulting magnitude of the velocity field is visualized in Figure 1.

Plain program

Here follows a version of the program without any comments. The file is also available here: stokes-flow.jl.

using Ferrite, FerriteGmsh, Gmsh, Tensors, LinearAlgebra, SparseArrays

function setup_grid(h=0.05)
    # Initialize gmsh
    Gmsh.initialize()
    gmsh.option.set_number("General.Verbosity", 2)

    # Add the points
    o = gmsh.model.geo.add_point(0.0, 0.0, 0.0, h)
    p1 = gmsh.model.geo.add_point(0.5, 0.0, 0.0, h)
    p2 = gmsh.model.geo.add_point(1.0, 0.0, 0.0, h)
    p3 = gmsh.model.geo.add_point(0.0, 1.0, 0.0, h)
    p4 = gmsh.model.geo.add_point(0.0, 0.5, 0.0, h)

    # Add the lines
    l1 = gmsh.model.geo.add_line(p1, p2)
    l2 = gmsh.model.geo.add_circle_arc(p2, o, p3)
    l3 = gmsh.model.geo.add_line(p3, p4)
    l4 = gmsh.model.geo.add_circle_arc(p4, o, p1)

    # Create the closed curve loop and the surface
    loop = gmsh.model.geo.add_curve_loop([l1, l2, l3, l4])
    surf = gmsh.model.geo.add_plane_surface([loop])

    # Synchronize the model
    gmsh.model.geo.synchronize()

    # Create the physical domains
    gmsh.model.add_physical_group(1, [l1], -1, "Γ1")
    gmsh.model.add_physical_group(1, [l2], -1, "Γ2")
    gmsh.model.add_physical_group(1, [l3], -1, "Γ3")
    gmsh.model.add_physical_group(1, [l4], -1, "Γ4")
    gmsh.model.add_physical_group(2, [surf])

    # Add the periodicity constraint using 4x4 affine transformation matrix,
    # see https://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations
    transformation_matrix = zeros(4, 4)
    transformation_matrix[1, 2] = 1  # -sin(-pi/2)
    transformation_matrix[2, 1] = -1 #  cos(-pi/2)
    transformation_matrix[3, 3] = 1
    transformation_matrix[4, 4] = 1
    transformation_matrix = vec(transformation_matrix')
    gmsh.model.mesh.set_periodic(1, [l1], [l3], transformation_matrix)

    # Generate a 2D mesh
    gmsh.model.mesh.generate(2)

    # Save the mesh, and read back in as a Ferrite Grid
    grid = mktempdir() do dir
        path = joinpath(dir, "mesh.msh")
        gmsh.write(path)
        togrid(path)
    end

    # Finalize the Gmsh library
    Gmsh.finalize()

    return grid
end

function setup_fevalues(ipu, ipp, ipg)
    qr = QuadratureRule{2,RefTetrahedron}(2)
    cvu = CellVectorValues(qr, ipu, ipg)
    cvp = CellScalarValues(qr, ipp, ipg)
    qr_face = QuadratureRule{1,RefTetrahedron}(2)
    fvp = FaceScalarValues(qr_face, ipp, ipg)
    return cvu, cvp, fvp
end

function setup_dofs(grid, ipu, ipp)
    dh = DofHandler(grid)
    add!(dh, :u, 2, ipu)
    add!(dh, :p, 1, ipp)
    close!(dh)
    return dh
end

function setup_mean_constraint(dh, fvp)
    assembler = start_assemble()
    # All external boundaries
    set = union(
        getfaceset(dh.grid, "Γ1"),
        getfaceset(dh.grid, "Γ2"),
        getfaceset(dh.grid, "Γ3"),
        getfaceset(dh.grid, "Γ4"),
    )
    # Allocate buffers
    range_p = dof_range(dh, :p)
    element_dofs = zeros(Int, ndofs_per_cell(dh))
    element_dofs_p = view(element_dofs, range_p)
    element_coords = zeros(Vec{2}, 3)
    Ce = zeros(1, length(range_p)) # Local constraint matrix (only 1 row)
    # Loop over all the boundaries
    for (ci, fi) in set
        Ce .= 0
        getcoordinates!(element_coords, dh.grid, ci)
        reinit!(fvp, element_coords, fi)
        celldofs!(element_dofs, dh, ci)
        for qp in 1:getnquadpoints(fvp)
            dΓ = getdetJdV(fvp, qp)
            for i in 1:getnbasefunctions(fvp)
                Ce[1, i] += shape_value(fvp, qp, i) * dΓ
            end
        end
        # Assemble to row 1
        assemble!(assembler, [1], element_dofs_p, Ce)
    end
    C = end_assemble(assembler)
    # Create an AffineConstraint from the C-matrix
    _, J, V = findnz(C)
    _, constrained_dof = findmax(abs2, V)
    V ./= V[constrained_dof]
    mean_value_constraint = AffineConstraint(
        constrained_dof,
        Pair{Int,Float64}[J[i] => -V[i] for i in 1:length(J) if J[i] != constrained_dof],
        0.0,
    )
    return mean_value_constraint
end

function setup_constraints(dh, fvp)
    ch = ConstraintHandler(dh)
    # Periodic BC
    R = rotation_tensor(π / 2)
    periodic_faces = collect_periodic_faces(dh.grid, "Γ3", "Γ1", x -> R ⋅ x)
    periodic = PeriodicDirichlet(:u, periodic_faces, R, [1, 2])
    add!(ch, periodic)
    # Dirichlet BC
    Γ24 = union(getfaceset(dh.grid, "Γ2"), getfaceset(dh.grid, "Γ4"))
    dbc = Dirichlet(:u, Γ24, (x, t) -> [0, 0], [1, 2])
    add!(ch, dbc)
    # Compute mean value constraint and add it
    mean_value_constraint = setup_mean_constraint(dh, fvp)
    add!(ch, mean_value_constraint)
    # Finalize
    close!(ch)
    update!(ch, 0)
    return ch
end

function assemble_system!(K, f, dh, cvu, cvp)
    assembler = start_assemble(K, f)
    ke = zeros(ndofs_per_cell(dh), ndofs_per_cell(dh))
    fe = zeros(ndofs_per_cell(dh))
    range_u = dof_range(dh, :u)
    ndofs_u = length(range_u)
    range_p = dof_range(dh, :p)
    ndofs_p = length(range_p)
    ϕᵤ = Vector{Vec{2,Float64}}(undef, ndofs_u)
    ∇ϕᵤ = Vector{Tensor{2,2,Float64,4}}(undef, ndofs_u)
    divϕᵤ = Vector{Float64}(undef, ndofs_u)
    ϕₚ = Vector{Float64}(undef, ndofs_p)
    for cell in CellIterator(dh)
        reinit!(cvu, cell)
        reinit!(cvp, cell)
        ke .= 0
        fe .= 0
        for qp in 1:getnquadpoints(cvu)
            dΩ = getdetJdV(cvu, qp)
            for i in 1:ndofs_u
                ϕᵤ[i] = shape_value(cvu, qp, i)
                ∇ϕᵤ[i] = shape_gradient(cvu, qp, i)
                divϕᵤ[i] = shape_divergence(cvu, qp, i)
            end
            for i in 1:ndofs_p
                ϕₚ[i] = shape_value(cvp, qp, i)
            end
            # u-u
            for (i, I) in pairs(range_u), (j, J) in pairs(range_u)
                ke[I, J] += ( ∇ϕᵤ[i] ⊡ ∇ϕᵤ[j] ) * dΩ
            end
            # u-p
            for (i, I) in pairs(range_u), (j, J) in pairs(range_p)
                ke[I, J] += ( -divϕᵤ[i] * ϕₚ[j] ) * dΩ
            end
            # p-u
            for (i, I) in pairs(range_p), (j, J) in pairs(range_u)
                ke[I, J] += ( -divϕᵤ[j] * ϕₚ[i] ) * dΩ
            end
            # rhs
            for (i, I) in pairs(range_u)
                x = spatial_coordinate(cvu, qp, getcoordinates(cell))
                b = exp(-100 * norm(x - Vec{2}((0.75, 0.1)))^2)
                bv = Vec{2}((b, 0.0))
                fe[I] += (ϕᵤ[i] ⋅ bv) * dΩ
            end
        end
        assemble!(assembler, celldofs(cell), ke, fe)
    end
    return K, f
end

function main()
    # Grid
    h = 0.05 # approximate element size
    grid = setup_grid(h)
    # Interpolations
    ipu = Lagrange{2,RefTetrahedron,2}() # quadratic
    ipp = Lagrange{2,RefTetrahedron,1}() # linear
    # Dofs
    dh = setup_dofs(grid, ipu, ipp)
    # FE values
    ipg = Lagrange{2,RefTetrahedron,1}() # linear geometric interpolation
    cvu, cvp, fvp = setup_fevalues(ipu, ipp, ipg)
    # Boundary conditions
    ch = setup_constraints(dh, fvp)
    # Global tangent matrix and rhs
    coupling = [true true; true false] # no coupling between pressure test/trial functions
    K = create_sparsity_pattern(dh, ch; coupling=coupling)
    f = zeros(ndofs(dh))
    # Assemble system
    assemble_system!(K, f, dh, cvu, cvp)
    # Apply boundary conditions and solve
    apply!(K, f, ch)
    u = K \ f
    apply!(u, ch)
    # Export the solution
    vtk_grid("stokes-flow", grid) do vtk
        vtk_point_data(vtk, dh, u)
    end
    return
end

main()

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