API

Incident

Some parameters related to the incident field.

Parameters

• k: the wave number
• θ: the incident field
• α: α = k * sin(θ)
• β: β = k * cos(θ)

Incident(k::Float64, θ::Float64)

Define the incident field with the wavenumber k and the incident field θ.

PML

Information on PML layer and coordinate transformation.

allocate_matrices(dof::DofHandler, cst::ConstraintHandler)

Allocate the complex-valued stiffness matrices.

allocate_stiff_matrix(dofhandler::DofHandler, csthandler::ConstraintHandler, dofs)

Create a sparse pattern for the stiffness matrix. We need to add extra entries due to the DtN map by hand.

apply_all_bds!(K::Union{SparseMatrixCSC, Symmetric}, ch::ConstraintHandler)
apply_all_bds!(KK::Union{SparseMatrixCSC, Symmetric}, f::AbstractVector, ch::ConstraintHandler, applyzero::Bool = false)

Impose the periodic boundary condition and the Dirichlet boundary condition on the matrices.

assemble_A(cv::CellValues, dh::DofHandler, A::SparseMatrixCSC, inc::Incident)

assemble_A0(cv::CellValues, dh::DofHandler, A₀::SparseMatrixCSC, medium::Function, k)

Assemble the zero order term $\mathbf{A}_{0}$ in the Nonlinear eigenvalue problem.

$A_{0}(u, v) = \int \nabla u \cdot \nabla \bar{v} - k^{2}n(x_{1}, x_{2}) u \bar{v} d x.$

Argument

• cv: CellValues
• dh: DofHandler
• A₀: an empty sparse pattern preallocated for A₀
• medium: refractive index function which describes the properties of the medium
• k: the wavenumber

See also assemble_A1, assemble_A2.

assemble_A1(cv::CellValues, dh::DofHandler, A₁::SparseMatrixCSC)

Assemble the first order term $\mathbf{A}_{1}$ in the Nonlinear eigenvalue problem.

$A_{1}(u, v) = -\int 2i \partial_{1} u \bar{v} d x.$

See also assemble_A0, assemble_A2.

assemble_A2(cv::CellValues, dh::DofHandler, A₂::SparseMatrixCSC)

Assemble the second order term in the Nonlinear eigenvalue problem.

$A_{2}(u, v) = \int u \bar{v} d x.$

See also assemble_A0, assemble_A1.

assemble_load(fv::FacetValues, dh::DofHandler, facetset, f, inc::Incident, height)

Assemble the load vector due to the incident field.

assemble_pml_A0(cv::CellValues, dh::DofHandler, A₀::SparseMatrixCSC, medium::Function, p::PML, k)

Assemble the zero order term in the quadratic eigenvalue problem constructed by PML-FEM.

\[A_{0}(u, v) = \int s(x_{2}) \frac{\partial u}{\partial x_{1}} \frac{\partial \bar{v}}{\partial x_{1}} + \frac{1}{s(x_{2})} \frac{\partial u}{\partial x_{2}} \frac{\partial \bar{v}}{\partial x_{2}} - k^{2} n(x_{1}, x_{2}) s(x_{2}) u \bar{v} dx.\]

Arguments

• cv: CellValues
• dh: DofHandler
• A₀: Sparse matix preallocated for A₀
• medium: the refractive index
• p: the information about PML, see PML
• k: the wavenumber

See also assemble_pml_A1, assemble_pml_A2.

assemble_pml_A1(cv::CellValues, dh::DofHandler, A₁::SparseMatrixCSC, p::PML)

Assemble the first order term in the quadratic eigenvalue problem constructed by PML-FEM.

\[A_{1}(u, v) = -\int 2i s(x_{2}) \frac{\partial u}{\partial x_{1}} \bar{v} dx.\]

Arguments

• cv: CellValues
• dh: DofHandler
• A₁: Sparse matix preallocated for A₁
• p: the information about PML, see PML

See also assemble_pml_A0, assemble_pml_A2.

assemble_pml_A2(cv::CellValues, dh::DofHandler, A₂::SparseMatrixCSC, p::PML)

Assemble the second order term in the quadratic eigenvalue problem from the PML-FEM method.

\[A_{2}(u, v) = \int s(x_{2}) u \bar{v} dx.\]

Arguments

• cv: CellValues
• dh: DofHandler
• A₂: Sparse matix preallocated for A₂
• p: the information about PML, see PML

See also assemble_pml_A0, assemble_pml_A1.

assemble_tbc(fv::FacetValues, dh::DofHandler, inc::Incident, facetset, F, N, dofsDtN; period = 2π)

Assemble the TBC matrix.

beta_n(k, α, n)

Compute $\beta_{n}$ for complex $\alpha$. We need this function when we construct nonlinear eigenvalue problems. Please note that we take the negative imaginary axis as the branch cut for square root. So we use csqrt_negimag instead of the default square root in Julia. See also csqrt_negimag.

beta_n(inc::Incident, n; period = 2π)

Compute the $βₙ$.

compute_coef!(Θ::SparseVector, fv::FacetValues, dh::DofHandler, facetset, n; period = 2π)

Compute $\Theta^{n}$ on the facetset. Actually the computation of the TBC matrix reduces to the computation of the vector $\Theta^{n}$.

compute_rayleigh_n(fv::FacetValues, dh::DofHandler, facetset, u, n)

Compute the $n$-th order Rayleigh coefficient for the solution $u$. Be careful that we need to compute the Rayleigh coefficients for scattered solutions.

\[u_{n} = \int_{\text{boundary}} u e^{-inx_{1}} ds.\]

compute_scaling(A₀::M, A₁::M, A₂::M; scaling=:nothing::Symbol) where {T,M<:AbstractMatrix{T}}

TBW

coord_transform(x₂, p::PML)

Coordinate transformation of PML method.

csqrt_negimag(z::Complex{T}) where T<:AbstractFloat

Square root for complex numbers which takes the negative imaginary axis as the branch cut. Note that sqrt for complex number in julia takes the negative real axis as the default branch cut.

dofs_on_dtn(dh::DofHandler, field::Symbol, facetset)

Extract the global indices of Dofs associated to the artificial boundary.

get_width(p::PML)

Get the width of the PML layer.

integral_nuv̄(cv::CellValues, dh::DofHandler, n::Function, u, v)

Compute the integral

\[\int n(x_{1}, x_{2}) u v dx, \]

where $n(x_{1}, x_{2})$ is a function defined in our computational domain (normally, the refraction index). See also integral_uv̄ and integral_∂₁uv̄.

integral_uv(cv::CellValues, dh::DofHandler, u, v)

Compute the integral

\[\int u v dx, \]

where $u$ and $v$ are vectors indexed by degrees of freedom. Normally, they are results obtained by FEM.

integral_uv̄(cv::CellValues, dh::DofHandler, u, v)

Compute the integral

\[\int u \bar{v} dx, \]

where $\bar{v}$ is the conjugate of the function $v$. See also integral_nuv̄ and integral_∂₁uv̄.

integral_∂₁uv(cv::CellValues, dh::DofHandler, u, v)

Compute the integral

\[\int \frac{\partial u}{\partial x_{1}} v dx,\]

where $u$ and $v$ are vectors indexed by degrees of freedom. Normally, they are results obtained by FEM.

integral_∂₁uv̄(cv::CellValues, dh::DofHandler, u, v)

Compute the integral

\[\int \frac{\partial u}{\partial x_{1}} \bar{v} dx, \]

where $\bar{v}$ is the conjugate of the function $v$. See also integral_uv̄ and integral_nuv̄.

periodic_cell(;lc=0.5, period=2π, height=2.0)

Generate a mesh for the periodic cell.

Arguments

• lc: the mesh size near points
• period: the period of the periodic cell
• height: the height of the periodic cell

Points

 4 ------------ 3
 |              |
 |              |
 1 ------------ 2

Lines

 . ---- l3 ---- .
 l4             l2
 . ---- l1 ---- .

setup_bcs(dh::DofHandler; period=2π)

Set the periodic boundary condition ("left" and "right") and Dirichelt boundary condition ("bottom").

Arguments

• dh: DofHandler
• period: the period of the periodic cell, see periodic_cell

setup_bdcs(dof::DofHandler, d)

Set the periodic boundary condition ("left" and "right") and Dirichelt boundary condition ("bottom" and "top").

setup_dofs(grid::Grid, ip)

Setup degree of freedoms.

setup_fevs(ip)

Define quadrature rules on Reference triangle and setup FE values.

setup_grid(;d=2π, ĥ=1.5, δ=2.0, lc=0.5, lp=0.5, vtk=false)

Generate a mesh by julia interface for Gmsh and read the .msh file by FerriteGmsh.

Arguments

• d: the period of the periodic layer
• ĥ: the start of the PML layer
• δ: the height of the PML layer
• lc: mesh size near the periodic layer
• lp: mesh size in the PML layer
• vtk: write .vtk file, default is false

Points

 5 ------------- 4
 |               |
 6               3
 |               |
 1 ------------- 2

Lines

 . ----- l4 ----- . 
 l5               l3
 .                .
 l6               l2 
 . ----- l1 ----- .

Misc

• We can generate a *.vtk file by replace the do end block with gmsh.write(*.vtk)

setup_vals(ip)

Set up CellValues and FacetValues by using interpolation ip. Here we need to define FacetValues because the load vector and the TBC matrix contains integral on the boundary.

sub_preserve_structure(A::SparseMatrixCSC, B::SparseMatrixCSC)

Subtract two sparse matrix without changing the sparse pattern. A and B should have the same sparse pattern.