Quadrature

abstract type QuadraturePoints{T <: AbstractFloat} end

Abstract base type for all sets of quadrature points

struct QuadPts2D{T} <: QuadraturePoints{T}
    num   ::Int         # number of points
    x     ::Vector{T}   # x values
    y     ::Vector{T}   # y values
    weight::Vector{T}   # weights
end

Quadrature points and weights for triangles

struct QuadPts3D{T} <: QuadraturePoints{T}
    num   ::Int         # number of points
    x     ::Vector{T}   # x values
    y     ::Vector{T}   # y values
    z     ::Vector{T}   # z values
    weight::Vector{T}   # weights
end

Quadrature points and weights for tetrahedra

abstract type ElementQuad{T <: AbstractFloat} end

Abstract base type for quadrature points on specific elements

struct TriangleQuad{T} <: ElementQuad{T}
    elem   ::Triangle{T}   # given element
    qpts   ::Matrix{T}     # quadrature points on the element
    weights::Vector{T}     # weights for the quadrature points
end

Quadrature points on a specific surface triangle, including weights.

Special constructors

julia TriangleQuad(elem::Triangle{T})` Computes quadrature points and weights for the given triangle.

quadraturepoints(::Type{Tetrahedron{Float64}})
quadraturepoints(::Type{Tetrahedron{Float32}})
quadraturepoints(::Type{Triangle{Float64}})
quadraturepoints(::Type{Triangle{Float32}})

Generator function for quadrature points:

• Triangles: 7 points per element [Rad48]
• Tetrahedra: 5 points per element [Kea86]

Return type

QuadPts2D or QuadPts3D

laplacecoll(
    ptype::Type{<: PotentialType},
    ξ    ::AbstractVector{T},
    elem ::AbstractVector{Triangle{T}};
    # kwargs
    dat  ::AbstractVector{T} = Vector{T}(undef, 12)
)

Analytical solution for the single or double layer Laplace potential for a given triangle and observation point ξ [Rja90].

Arguments

• dat Writable vector for internal use (pre-allocate and reuse if possible)

Return type

Void

laplacecoll!(
    ptype   ::Type{<: PotentialType},
    dest    ::AbstractVector{T},
    elements::AbstractVector{Triangle{T}},
    Ξ       ::AbstractVector{Vector{T}},
    fvals   ::AbstractVector{T}
)

laplacecoll!(
    ptype   ::Type{<: PotentialType},
    dest    ::AbstractMatrix{T},
    elements::AbstractVector{Triangle{T}},
    Ξ       ::AbstractVector{Vector{T}}
)

Analytical solution for the single or double layer Laplace potential for a given list of triangles and observation points Ξ [Rja90].

The first version of this function uses a vector as destination dest, where each element represents the dot product of the corresponding coefficient matrix row and the fvals vector.

Return type

Void

regularyukawacoll(
          ::Type{<: PotentialType},
    ξ     ::AbstractVector{T},
    elem  ::Union{Triangle{T}, TriangleQuad{T}},
    yukawa::T
)

Computes the regular part of the single or double layer Yukawa potential (that is, Yukawa minus Laplace) using a seven-point Radon cubature [Rad48] for a given triangle and observation point ξ.

Arguments

• yukawa Exponent of the Yukawa operator's fundamental solution

Return type

Void

regularyukawacoll!(
            ::Type{<: PotentialType},
    dest    ::AbstractVector{T},
    elements::AbstractVector{Triangle{T}},
    Ξ       ::AbstractVector{Vector{T}},
    yukawa  ::T,
    fvals   ::AbstractVector{T}
)

regularyukawacoll!(
            ::Type{<: PotentialType},
    dest    ::AbstractMatrix{T},
    elements::AbstractVector{Triangle{T}},
    Ξ       ::AbstractVector{Vector{T}},
    yukawa  ::T
)

Computes the regular part of the single or double layer Yukawa potential (that is, Yukawa minus Laplace) using a seven-point Radon cubature [Rad48] for a given list of triangles and observation points Ξ.

The first version of this function uses a vector as destination dest, where each element represents the dot product of the corresponding coefficient matrix row and the fvals vector.

Arguments

• yukawa Exponent of the Yukawa operator's fundamental solution

Return type

Void