Helmholtz equation

In this example, we want to solve a (variant of) of the Helmholtz equation. The example is inspired by an dealii step_7 on the standard square.

\[ - \Delta u + u = f\]

With boundary conditions given by

\[u = g_1 \quad x \in \Gamma_1\]

and

\[n \cdot \nabla u = g_2 \quad x \in \Gamma_2\]

Here Γ₁ is the union of the top and the right boundary of the square, while Γ₂ is the union of the bottom and the left boundary.

We will use the following weak formulation:

\[\int_\Omega \nabla δu \cdot \nabla u \, d\Omega + \int_\Omega δu \cdot u \, d\Omega - \int_\Omega δu \cdot f \, d\Omega - \int_{\Gamma_2} δu g_2 \, d\Gamma = 0 \quad \forall δu\]

where $δu$ is a suitable test function that satisfies:

\[δu = 0 \quad x \in \Gamma_1\]

and $u$ is a suitable function that satisfies:

\[u = g_1 \quad x \in \Gamma_1\]

The example highlights the following interesting features:

• There are two kinds of boundary conditions, "Dirichlet" and "Von Neumann"
• The example contains boundary integrals
• The Dirichlet condition is imposed strongly and the Von Neumann condition is imposed weakly.

using Ferrite
using Tensors
using SparseArrays
using LinearAlgebra

const ∇ = Tensors.gradient
const Δ = Tensors.hessian;

grid = generate_grid(Quadrilateral, (150, 150))

ip = Lagrange{RefQuadrilateral, 1}()
qr = QuadratureRule{RefQuadrilateral}(2)
qr_facet = FacetQuadratureRule{RefQuadrilateral}(2)
cellvalues = CellValues(qr, ip);
facetvalues = FacetValues(qr_facet, ip);

dh = DofHandler(grid)
add!(dh, :u, ip)
close!(dh)

DofHandler{2, Grid{2, Quadrilateral, Float64}}
  Fields:
    :u, Lagrange{RefQuadrilateral, 1}()
  Dofs per cell: 4
  Total dofs: 22801

We will set things up, so that a known analytic solution is approximately reproduced. This is a good testing strategy for PDE codes and known as the method of manufactured solutions.

function u_ana(x::Vec{2, T}) where {T}
    xs = (Vec{2}((-0.5,  0.5)),
          Vec{2}((-0.5, -0.5)),
          Vec{2}(( 0.5,  -0.5)))
    σ = 1/8
    s = zero(eltype(x))
    for i in 1:3
        s += exp(- norm(x - xs[i])^2 / σ^2)
    end
    return max(1e-15 * one(T), s) # Denormals, be gone
end;

dbcs = ConstraintHandler(dh)

ConstraintHandler:
  Not closed!

The (strong) Dirichlet boundary condition can be handled automatically by the Ferrite library.

dbc = Dirichlet(:u, union(getfacetset(grid, "top"), getfacetset(grid, "right")), (x,t) -> u_ana(x))
add!(dbcs, dbc)
close!(dbcs)
update!(dbcs, 0.0)

K = allocate_matrix(dh);

function doassemble(cellvalues::CellValues, facetvalues::FacetValues,
                         K::SparseMatrixCSC, dh::DofHandler)
    b = 1.0
    f = zeros(ndofs(dh))
    assembler = start_assemble(K, f)

    n_basefuncs = getnbasefunctions(cellvalues)

    fe = zeros(n_basefuncs) # Local force vector
    Ke = zeros(n_basefuncs, n_basefuncs) # Local stiffness mastrix

    for (cellcount, cell) in enumerate(CellIterator(dh))
        fill!(Ke, 0)
        fill!(fe, 0)
        coords = getcoordinates(cell)

        reinit!(cellvalues, cell)

First we derive the non boundary part of the variation problem from the destined solution u_ana

\[\int_\Omega \nabla δu \cdot \nabla u \, d\Omega + \int_\Omega δu \cdot u \, d\Omega - \int_\Omega δu \cdot f \, d\Omega\]

        for q_point in 1:getnquadpoints(cellvalues)
            dΩ = getdetJdV(cellvalues, q_point)
            coords_qp = spatial_coordinate(cellvalues, q_point, coords)
            f_true = -LinearAlgebra.tr(hessian(u_ana, coords_qp)) + u_ana(coords_qp)
            for i in 1:n_basefuncs
                δu = shape_value(cellvalues, q_point, i)
                ∇δu = shape_gradient(cellvalues, q_point, i)
                fe[i] += (δu * f_true) * dΩ
                for j in 1:n_basefuncs
                    u = shape_value(cellvalues, q_point, j)
                    ∇u = shape_gradient(cellvalues, q_point, j)
                    Ke[i, j] += (∇δu ⋅ ∇u + δu * u) * dΩ
                end
            end
        end

Now we manually add the von Neumann boundary terms

\[\int_{\Gamma_2} δu g_2 \, d\Gamma\]

        for facet in 1:nfacets(cell)
            if (cellcount, facet) ∈ getfacetset(grid, "left") ||
               (cellcount, facet) ∈ getfacetset(grid, "bottom")
                reinit!(facetvalues, cell, facet)
                for q_point in 1:getnquadpoints(facetvalues)
                    coords_qp = spatial_coordinate(facetvalues, q_point, coords)
                    n = getnormal(facetvalues, q_point)
                    g_2 = gradient(u_ana, coords_qp) ⋅ n
                    dΓ = getdetJdV(facetvalues, q_point)
                    for i in 1:n_basefuncs
                        δu = shape_value(facetvalues, q_point, i)
                        fe[i] += (δu * g_2) * dΓ
                    end
                end
            end
        end

        assemble!(assembler, celldofs(cell), Ke, fe)
    end
    return K, f
end;

K, f = doassemble(cellvalues, facetvalues, K, dh);
apply!(K, f, dbcs)
u = Symmetric(K) \ f;

vtk = VTKGridFile("helmholtz", dh)
write_solution(vtk, dh, u)
close(vtk)
println("Helmholtz successful")

Helmholtz successful

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