API Reference

AbstractSPHKernel

Supertype for all SPH kernels.

Cubic(T::DataType=Float64, dim::Integer=3)

Set up a Cubic kernel for a given DataType T and dimensin dim.

Cubic(dim::Integer)

Define Cubic kernel with dimension dim for the native DataType of the OS.

DoubleCosine(T::DataType=Float64, dim::Integer=3)

Set up a DoubleCosine kernel for a given DataType T.

struct DoubleCosine{T} <: AbstractSPHKernel
    dim::Int64
    norm::T
end

DoubleCosine(dim::Integer)

Define DoubleCosine kernel with dimension dim for the native DataType of the OS.

Quintic(T::DataType=Float64, dim::Integer=3)

Set up a Quintic kernel for a given DataType T.

Quintic(dim::Integer)

Define Quintic kernel with dimension dim for the native DataType of the OS.

Tophat(dim::Integer)

Define Tophat kernel with dimension dim for the native DataType of the OS.

WendlandC2(T::DataType=Float64, dim::Integer=3)

Set up a WendlandC2 kernel for a given DataType T.

WendlandC2(dim::Integer)

Define WendlandC2 kernel with dimension dim for the native DataType of the OS.

WendlandC4(T::DataType=Float64, dim::Integer=3)

Set up a WendlandC4 kernel for a given DataType T.

WendlandC4(dim::Integer)

Define WendlandC4 kernel with dimension dim for the native DataType of the OS.

WendlandC6(T::DataType=Float64, dim::Integer=3)

Set up a WendlandC6 kernel for a given DataType T.

WendlandC6(dim::Integer)

Define WendlandC6 kernel with dimension dim for the native DataType of the OS.

WendlandC8(T::DataType=Float64, dim::Integer=3)

Set up a WendlandC8 kernel for a given DataType T.

WendlandC8(dim::Integer)

Define WendlandC8 kernel with dimension dim for the native DataType of the OS.

bias_correction( kernel::Cubic{T}, 
                 density::Real, m::Real, h_inv::Real, 
                 n_neighbours::Integer )  where T

Does not do anything for the BSplines. Implemented for stability.

bias_correction( kernel::DoubleCosine{T}, 
                 density::Real, m::Real, h_inv::Real, 
                 n_neighbours::Integer )  where T

Does not do anything for the DoubleCosine. Implemented for stability.

bias_correction( kernel::Quintic{T}, 
                 density::Real, m::Real, h_inv::Real, 
                 n_neighbours::Integer ) where T

Does not do anything for the BSplines. Implemented for stability.

bias_correction(kernel::Tophat{T},
                density::Real, m::Real, h_inv::Real,
                n_neighbours::Integer) where {T}

Corrects the density estimate for the kernel bias. Not implemented for Tophat.

bias_correction( kernel::Union{WendlandC2_1D{T}, WendlandC2{T}}, 
                 density::Real, m::Real, h_inv::Real,
                 n_neighbours::Integer ) where T

Corrects the density estimate for the kernel bias. See Dehnen&Aly 2012, eq. 18+19.

bias_correction( kernel::Union{WendlandC4_1D{T}, WendlandC4{T}}, 
                 density::Real, m::Real, h_inv::Real, 
                 n_neighbours::Int64 ) where T

Corrects the density estimate for the kernel bias. See Dehnen&Aly 2012, eq. 18+19.

bias_correction( kernel::Union{WendlandC6_1D{T}, WendlandC6{T}}, 
                 density::Real, m::Real, h_inv::Real, 
                 n_neighbours::Integer ) where T

Corrects the density estimate for the kernel bias. See Dehnen&Aly 2012, eq. 18+19.

bias_correction(kernel::WendlandC8, density::Real, m::Real, h_inv::Real)

Corrects the density estimate for the kernel bias. See Dehnen&Aly 2012, eq. 18+19.

d𝒩(kernel::AbstractSPHKernel, h_inv::Real)

Calculate the normalisation factor for the kernel derivative.

d𝒲(kernel::AbstractSPHKernel, u::Real, h_inv::Real)

Evaluate derivative at position $u = \frac{x}{h}$.

d𝒲(kernel::AbstractSPHKernel, u::Real)

Evaluate derivative at position $u = \frac{x}{h}$, without normalisation.

kernel_curl(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2) where {T1,T2}

Compute the kernel curl ∇x𝒲 between particle i and neighbour j for some SPH quantity A.

kernel_deriv(kernel::AbstractSPHKernel, u::Real, h_inv::Real) where T

Evaluate the derivative of the kernel at position $u = \frac{x}{h}$.

kernel_deriv(kernel::Cubic{T}, u::Real) where T

Evaluate the derivative of the Cubic spline at position $u = \frac{x}{h}$, without normalisation.

kernel_deriv(kernel::DoubleCosine, u::Real)

Evaluate the derivative of the DoubleCosine spline at position $u = \frac{x}{h}$, without normalisation.

kernel_deriv(kernel::Quintic{T}, u::Real) where T

Evaluate the derivative of the Quintic spline at position $u = \frac{x}{h}$, without normalisation.

kernel_deriv_1D(kernel::WendlandC2{T}, u::Real) where T

Evaluate the derivative of the WendlandC2 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv(kernel::WendlandC4_1D{T}, u::Real) where T

Evaluate the derivative of the WendlandC4 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv(kernel::WendlandC6_1D{T}, u::Real) where T

Evaluate the derivative of the WendlandC6 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv(kernel::WendlandC8_1D{T}, u::Real, h_inv::Real) where T

Evaluate the derivative of the WendlandC8 spline at position $u = \frac{x}{h}$.

kernel_deriv_1D(kernel::Tophat{T}, u::Real) where T

Evaluate the derivative of the Tophat spline at position $u = \frac{x}{h}$, without normalisation.

kernel_deriv(kernel::WendlandC2{T}, u::Real) where T

Evaluate the derivative of the WendlandC2 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv(kernel::WendlandC4{T}, u::Real) where T

Evaluate the derivative of the WendlandC4 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv(kernel::WendlandC6, u::Real)

Evaluate the derivative of the WendlandC6 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_deriv_2D(kernel::WendlandC8{T}, u::Real) where T

Evaluate the derivative of the WendlandC8 spline at position $u = \frac{x}{h}$, without normalisation.

kernel_deriv_norm(kernel::AbstractSPHKernel, h_inv::Real)

Calculate the normalisation factor for the kernel derivative.

kernel_div( k::AbstractSPHKernel, h_inv::T1, 
            xᵢ::T2, xⱼ::T2, Aⱼ::T2) where {T1,T2}

Compute the kernel divergence ∇⋅𝒲 between particle i and neighbour j for some SPH quantity A.

kernel_gradient( k::AbstractSPHKernel, r::T1, h_inv::T1, Δx::T2) where {T1,T2}

Computes the gradient of the kernel k at the distance r along the distance vector Δx of the neighbour j.

$∇W(x_{ij}, h_i) = \frac{dW}{dx}\vert_{x_j} \frac{Δx_{ij}}{||x_{ij}||} \frac{1}{h_i}$

kernel_gradient( k::AbstractSPHKernel, h_inv::Real, xᵢ::T, xⱼ::T ) where T

Computes the gradient of the kernel k at the position of the neighbour xⱼ.

$∇W(x_{ij}, h_i) = \frac{dW}{dx}\vert_{x_j} \frac{Δx_{ij}}{||x_{ij}||} \frac{1}{h_i}$

kernel_norm(kernel::AbstractSPHKernel, h_inv::Real) where {T}

Calculate the normalisation factor for the kernel.

kernel_norm(kernel::WendlandKernel, h_inv::Real) where {T}

Calculate the normalisation factor for the WendlandC2 kernel.

kernel_norm(kernel::Tophat, h_inv::Real) where {T}

Calculate the normalisation factor for the Tophat kernel.

kernel_norm(kernel::WendlandC2_1D{T}, h_inv::Real) where {T}

Calculate the normalisation factor for the WendlandC2 kernel.

kernel_quantity(k::AbstractSPHKernel, r::T1, h_inv::T1, 
                Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the SPH quantity A for particle i. Based on Euclidean distance r between the particles. Useful if many quantities need to be computed for the same particle pair.

$\vec{A}_i(x) ≈ \sum_j m_j \frac{\vec{A}_j}{\rho_j} W(\vec{x}_i - \vec{x}_j, h_i)$

kernel_quantity(k::AbstractSPHKernel, h_inv::T1, 
                xᵢ::T2, xⱼ::T2, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the SPH quantity A for particle i. Based on positions xᵢ and xⱼ.

$\vec{A}_i(x) ≈ \sum_j m_j \frac{\vec{A}_j}{\rho_j} W(\vec{x}_i - \vec{x}_j, h_i)$

kernel_value( k::AbstractSPHKernel, h_inv::Real, 
              xᵢ::Real, xⱼ::Real )

Computes the value of the kernel k at the position of the neighbour xⱼ.

$W(x_i - x_j, h_i)$

kernel_value(kernel::AbstractSPHKernel, u::Real, h_inv::Real) where T

Evaluate the kernel at position $u = \frac{x}{h}$.

kernel_value( k::AbstractSPHKernel, h_inv::T1, 
                   xᵢ::T2, xⱼ::T2 ) where {T1,T2}

Computes the value of the kernel k at the position of the neighbour xⱼ.

$W(\vec{x}_i - \vec{x}_j, h_i)$

kernel_value_1D(kernel::Cubic{T}, u::Real) where T

Evaluate cubic spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::DoubleCosine, u::Real)

Evaluate DoubleCosine spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::Quintic{T}, u::Real) where T

Evaluate quintic spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::WendlandC2_1D{T}, u::Real) where T

Evaluate WendlandC2 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_value(kernel::WendlandC4{T}, u::Real) where T

Evaluate WendlandC4 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_value(kernel::WendlandC6_1D{T}, u::Real, h_inv::Real) where T

Evaluate WendlandC6 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_value(kernel::WendlandC8_1D{T}, u::Real) where T

Evaluate WendlandC8 spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::Tophat{T}, u::Real) where T

Evaluate Tophat spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::WendlandC2{T}, u::Real, h_inv::Real) where T

Evaluate WendlandC2 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_value(kernel::WendlandC4{T}, u::Real) where T

Evaluate WendlandC4 spline at position $u = \frac{x}{h}$ without normalisation.

kernel_value(kernel::WendlandC6{T}, u::Real) where T

Evaluate WendlandC6 spline at position $u = \frac{x}{h}$, without normalisation.

kernel_value(kernel::WendlandC8{T}, u::Real) where T

Evaluate WendlandC8 spline at position $u = \frac{x}{h}$, without normalisation.

quantity_curl(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the curl of the SPH quantity A for particle i.

$∇×\vec{A}_i(x) ≈ - \sum_j \frac{m_j}{\rho_j} \vec{A}_j \times ∇W(\vec{x}_i - \vec{x}_j, h_i)$

quantity_divergence(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the divergence of the SPH quantity A for particle i.

$∇\cdot\vec{A}_i(x) ≈ \sum_j \frac{m_j}{\rho_j} \vec{A}_j \cdot ∇W(\vec{x}_i - \vec{x}_j, h_i)$

quantity_gradient(k::AbstractSPHKernel,
                  r::T1, h_inv::T1,
                  Δx::T2, Aⱼ::T2,
                  mⱼ::T1, ρⱼ::T1) where {T1<:Real,T2}

Compute the contribution of particle j to the gradient of the SPH quantity A for particle i. Based on Euclidean distance r and distance vector Δx between the particles. Useful if many quantities need to be computed for the same particle pair.

$∇\vec{A}_i(x) ≈ \sum_j \frac{m_j}{\rho_j} \vec{A}_j \: ∇W(||\vec{x}_i - \vec{x}_j||, h_i)$

quantity_gradient(k::AbstractSPHKernel, h_inv::T1, 
                  xᵢ::T2, xⱼ::T2, Aⱼ::T2,
                  mⱼ::T1, ρⱼ::T1 ) where {T1<:Real, T2}

Compute the contribution of particle j to the gradient of the SPH quantity A for particle i. Based on positions xᵢ and xⱼ.

$∇\vec{A}_i(x) ≈ \sum_j \frac{m_j}{\rho_j} \vec{A}_j \: ∇W(||\vec{x}_i - \vec{x}_j||, h_i)$

δρ₁(kernel::AbstractSPHKernel, density::Real, m::Real, h_inv::Real)

Corrects the density estimate for the kernel bias. See Dehnen&Aly 2012, eq. 18+19.

∇dot𝒜(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the divergence of the SPH quantity A for particle i. Compact notation of quantity_divergence.

$∇\cdot\vec{A}_i(x) ≈ \sum_j \frac{m_j}{\rho_j} \vec{A}_j \cdot ∇W(\vec{x}_i - \vec{x}_j, h_i)$

∇dot𝒲( k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2) where {T1,T2}

Compute the kernel divergence ∇⋅𝒲 between particle i and neighbour j for some SPH quantity A. Compact notation of kernel_div.

∇x𝒜(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the curl of the SPH quantity A for particle i.

$∇×\vec{A}_i(x) ≈ - \sum_j \frac{m_j}{\rho_j} \vec{A}_j \times ∇W(\vec{x}_i - \vec{x}_j, h_i)$

∇x𝒲(k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2, Aⱼ::T2) where {T1,T2}

Compute the kernel curl ∇x𝒲 between particle i and neighbour j for some SPH quantity A.

∇𝒜( k::AbstractSPHKernel, h_inv, xᵢ, xⱼ, Aⱼ, mⱼ, ρⱼ)

Compute the contribution of particle j to the gradient of the SPH quantity A for particle i. Compact notation of quantity_gradient.

$∇\vec{A}_i(x) ≈ \sum_j \frac{m_j}{\rho_j} \vec{A}_j \: ∇W(||\vec{x}_i - \vec{x}_j||, h_i)$

∇𝒲( k::AbstractSPHKernel, h_inv::Real, xᵢ::Union{Real, Vector{<:Real}}, xⱼ::Union{Real, Vector{<:Real}} )

Computes the gradient of the kernel k at the position of the neighbour j. Based on Euclidean distance r and distance vector Δx between the particles. Useful if many quantities need to be computed for the same particle pair. Compact notation of kernel_gradient.

$∇W(x_{ij}, h_i) = \frac{dW}{dx}\vert_{x_j} \frac{Δx_{ij}}{||x_{ij}||} \frac{1}{h_i}$

∇𝒲( k::AbstractSPHKernel, h_inv::T1, xᵢ::T2, xⱼ::T2 ) where {T1<:Real,T2}

Computes the gradient of the kernel k at the position of the neighbour xⱼ. Compact notation of kernel_gradient.

$∇W(x_{ij}, h_i) = \frac{dW}{dx}\vert_{x_j} \frac{Δx_{ij}}{||x_{ij}||} \frac{1}{h_i}$

𝒜(k::AbstractSPHKernel, r::T1,  h_inv::T1, Aⱼ::T2, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the SPH quantity A for particle i. Based on Euclidean distance r between the particles. Useful if many quantities need to be computed for the same particle pair.

$\vec{A}_i(x) ≈ \sum_j m_j \frac{\vec{A}_j}{\rho_j} W(r, h_i)$

𝒜(k::AbstractSPHKernel, h_inv::T1, 
  xᵢ::Union{T1, T2}, xⱼ::Union{T1, T2},
  Aⱼ::Union{T1, T2}, mⱼ::T1, ρⱼ::T1 ) where {T1,T2}

Compute the contribution of particle j to the SPH quantity A for particle i. Based on positions xᵢ and xⱼ.

$\vec{A}_i(x) ≈ \sum_j m_j \frac{\vec{A}_j}{\rho_j} W(\vec{x}_i - \vec{x}_j, h_i)$

𝒩(kernel::AbstractSPHKernel, h_inv::Real)

Calculate the normalisation factor for the kernel.

𝒲( k::AbstractSPHKernel, h_inv, xᵢ, xⱼ)

Computes the value of the kernel k at the position of the neighbour xⱼ.

$W(\vec{x}_i - \vec{x}_j, h_i)$

𝒲( kernel::AbstractSPHKernel, u::Real, h_inv::Real)

Evaluate kernel at position $u = \frac{x}{h}$.

𝒲( kernel::AbstractSPHKernel, u::Real)

Evaluate kernel at position $u = \frac{x}{h}$, without normalisation.

get_r(xᵢ::Real, xⱼ::Real)

Eukledian distance between xᵢ and xⱼ.

get_r(xᵢ::Vector{<:Real}, xⱼ::Vector{<:Real})

Eukledian distance between xᵢ and xⱼ.

WendlandKernel

Supertype for Wendland kernels.