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.

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

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

kernel_curl( k::AbstractSPHKernel,       h_inv::Real, 
             xᵢ::Vector{<:Real}, xⱼ::Vector{<:Real},
             Aᵢ::Vector{<:Real}, Aⱼ::Vector{<:Real} )

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

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

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

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

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

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

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

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

Evaluate the derivative of the WendlandC2 spline at position $u = \frac{x}{h}$.

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

Evaluate the derivative of the WendlandC4 spline at position $u = \frac{x}{h}$.

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

Evaluate the derivative of the WendlandC6 spline at position $u = \frac{x}{h}$.

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, h_inv::Real) where T

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

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

Evaluate the derivative of the WendlandC2 spline at position $u = \frac{x}{h}$.

kernel_deriv_2D(kernel::WendlandC4{T}, u::Real, h_inv::Real) where T

Evaluate the derivative of the WendlandC4 spline at position $u = \frac{x}{h}$.

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

Evaluate the derivative of the WendlandC6 spline at position $u = \frac{x}{h}$.

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

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

kernel_div( k::AbstractSPHKernel,       h_inv::Real, 
            xᵢ::Vector{<:Real}, xⱼ::Vector{<:Real},
            Aᵢ::Vector{<:Real}, Aⱼ::Vector{<:Real} )

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

kernel_gradient( 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 xⱼ.

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

kernel_gradient( k::AbstractSPHKernel, r::Real, h_inv::Real, 
                 Δx::Vector{<:Real})

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::Union{Real, Vector{<:Real}}, 
                 xⱼ::Union{Real, Vector{<:Real}} )

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

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

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

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

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, h_inv::Real) where T

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

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

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

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

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

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

Evaluate WendlandC2 spline at position $u = \frac{x}{h}$.

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

Evaluate WendlandC4 spline at position $u = \frac{x}{h}$.

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

Evaluate WendlandC6 spline at position $u = \frac{x}{h}$.

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

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

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

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

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

Evaluate WendlandC2 spline at position $u = \frac{x}{h}$.

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

Evaluate WendlandC4 spline at position $u = \frac{x}{h}$.

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

Evaluate WendlandC6 spline at position $u = \frac{x}{h}$.

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

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

quantity_curl( k::AbstractSPHKernel, h_inv::Real, 
      xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
      Aⱼ::Vector{<:Real},
      mⱼ::Real,             ρⱼ::Real )

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

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

quantity_divergence( k::AbstractSPHKernel, h_inv::Real, 
                      xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
                      Aᵢ::Vector{<:Real},   Aⱼ::Vector{<:Real},
                      mⱼ::Real,             ρⱼ::Real )

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

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

quantity_gradient( k::AbstractSPHKernel, 
                   r::Real,  h_inv::Real, 
                   Δx::Union{Real, Vector{<:Real}},
                   Aⱼ::Union{Real, Vector{<:Real}},
                   mⱼ::Real, ρⱼ::Real )

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.

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

quantity_gradient( k::AbstractSPHKernel, h_inv::Real, 
                   xᵢ::Union{Real, Vector{<:Real}},   
                   xⱼ::Union{Real, Vector{<:Real}},
                   Aⱼ::Union{Real, Vector{<:Real}},
                   mⱼ::Real,             ρⱼ::Real )

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

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

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::Real, 
       xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
       Aᵢ::Vector{<:Real},   Aⱼ::Vector{<:Real},
       mⱼ::Real,             ρⱼ::Real )

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

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

∇x𝒜( k::AbstractSPHKernel, h_inv::Real, 
      xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
      Aᵢ::Vector{<:Real},   Aⱼ::Vector{<:Real},
      mⱼ::Real,             ρⱼ::Real )

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

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

∇x𝒲( k::AbstractSPHKernel, h_inv::Real, 
      xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
      Aⱼ::Vector{<:Real})

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

∇̇dot𝒲( k::AbstractSPHKernel, h_inv::Real, 
        xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
        Aᵢ::Vector{<:Real},   Aⱼ::Vector{<:Real})

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

∇𝒜( k::AbstractSPHKernel, 
     r::Real,  h_inv::Real, 
     Δx::Union{Real, Vector{<:Real}},
     Aⱼ::Union{Real, Vector{<:Real}},
     mⱼ::Real, ρⱼ::Real )

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.

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

∇𝒜( k::AbstractSPHKernel, 
     r::Real,  h_inv::Real, 
     Δx::Union{Real, Vector{<:Real}},
     Aⱼ::Union{Real, Vector{<:Real}},
     mⱼ::Real, ρⱼ::Real )

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.

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

∇𝒜( k::AbstractSPHKernel, h_inv::Real, 
    xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
    Aⱼ::Vector{<:Real},   
    mⱼ::Real=1,           ρⱼ::Real=1 )

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

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

∇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::Real, xᵢ::Union{Real, Vector{<:Real}}, xⱼ::Union{Real, Vector{<:Real}} )

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::Real,  h_inv::Real, 
    Aⱼ::Union{Real, Vector{<:Real}},
    mⱼ::Real, ρⱼ::Real )

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::Real, 
    xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
    Aⱼ::Vector{<:Real},
    mⱼ::Real,             ρⱼ::Real )

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

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

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

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

𝒜( k::AbstractSPHKernel, h_inv::Real, 
    xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
    Aᵢ::Vector{<:Real},   Aⱼ::Vector{<:Real},
    mⱼ::Real,             ρⱼ::Real )

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

See e.g. Price 2012: $\vec{A}_i(x) ≈ \sum_j m_j \frac{\vec{A}_j}{\rho_j} W(\vec{x}_i - \vec{x}_j, h_i)$

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.

kernel_quantity( k::AbstractSPHKernel, h_inv::Real, 
                 xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
                 Aⱼ::Vector{<:Real},
                 mⱼ::Real,             ρⱼ::Real )

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::Real, 
                 xᵢ::Vector{<:Real},   xⱼ::Vector{<:Real},
                 Aⱼ::Vector{<:Real},
                 mⱼ::Real,             ρⱼ::Real )

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

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