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, F::SparseMatrixCSC, facetset, N, dofsDtN)

Assemble the TBC matrix.

beta_n(inc::Incident, n)

Compute the βₙ.

compute_coef!(fv::FacetValues, dh::DofHandler, θ::SparseVector, facetset, n)

Compute Θⁿ on the facetset. Actually the computation of the TBC matrix reduces to the computation of the vector Θⁿ.

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.

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.

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.