SPH Functions

We provide some basic functions to compute the gradient, divergence and curl of the kernel with a particle quantity if required.

Please note that these functions are not terribly flexible and/or performance optimized, as we believe that this should be done in a seperate SPHLoop.jl package. These functions are more meant as a starting point for a more efficient implementation or code that is not used in performance-relevant applications.

Notation

The derivative of a quantity in gradient, divergence and curl can be reduced to the derivative of the kernel combined with the remaining mathematical operation (see e.g. Price (2012)).

To provide some helper functions we split this into two functions:

One to calculate only the kernel derivative and use as a gradient for divergence and curl with the particle quantity.

The other to directly calculate the contribution of particle j to the gradient, divergence and curl of the SPH quantity for particle i.

Quantities

You can compute the contribution of particle j to the SPH quantity A for particle i via (see e.g. Price 2012):

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

To do this you can loop over the function

SPHKernels.kernel_quantity — Function

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.

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

or its more compact form

SPHKernels.𝒜 — Function

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.

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

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.

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

Gradient

The gradient of a quantity in SPH can be reduced to the gradient of the kernel (see e.g. Price 2012):

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

We provide two functionalities

Kernel

You can compute the gradient of the kernel at position x_j $∇W(x_{ij}, h_i) = \frac{dW}{dx}\vert_{x_j} \frac{Δx_{ij}}{||x_{ij}||} \frac{1}{h_i}$

by using

SPHKernels.kernel_gradient — Function

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.

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

or its more compact form

SPHKernels.∇𝒲 — Function

∇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.

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

∇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.

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

where xᵢ and xⱼ are the positions of particles i and j in 1D-3D space.

Quantity

To compute the gradient of a SPH quantity for particle i $∇\vec{A}_i(x) ≈ - \sum_j m_j \frac{\vec{A}_j}{\rho_j} ∇W(\vec{x}_i - \vec{x}_j, h_i)$

you can loop over

SPHKernels.quantity_gradient — Function

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.

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

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.

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

or its compact form

SPHKernels.∇𝒜 — Function

∇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.

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

∇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.

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

Divergence

The divergence of a quantity in SPH can be calculated as (see e.g. Price 2012):

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

We provide two functionalities

Kernel

You can compute the divergence of the kernel at position x_j by using

SPHKernels.kernel_div — Function

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.

or its more compact form

SPHKernels.∇̇dot𝒲 — Function

∇̇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.

where xᵢ and xⱼ are the positions of particles i and j in 1D-3D space.

Quantity

To compute the gradient of a SPH quantity for particle i you can loop over

SPHKernels.quantity_divergence — Function

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.

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

or its compact form

SPHKernels.∇dot𝒜 — Function

∇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.

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

Curl

The curl of a quantity in SPH can be calculated as (see e.g. Price 2012):

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

We provide two functionalities

Kernel

You can compute the divergence of the kernel at position x_j by using

SPHKernels.kernel_curl — Function

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.

or its more compact form

SPHKernels.∇x𝒲 — Function

∇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.

where xᵢ and xⱼ are the positions of particles i and j in 1D-3D space.

Quantity

To compute the gradient of a SPH quantity for particle i you can loop over

SPHKernels.quantity_curl — Function

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.

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

or its compact form

SPHKernels.∇x𝒜 — Function

∇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.

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