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Grid.md

FaADE.Grid — Module
Grid

Submodule containing the grid data structure and functions.

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One thing to note if editing code is that the Grid.jl module has its own internal copy of first derivative operators for constructing curvilinear coordinates.

1D Problems

FaADE.Grid.Grid1D — Type
Grid1D{TT<:Real,MET<:MetricType,GT<:Vector{TT},DT<:Union{Real,Vector{TT}}} <: LocalGridType{TT,1,MET}

Grid data structure for 1 dimensional problems.

Constructors:

Grid1D(𝒟::Vector{TT},n::Integer) where TT
Grid1D(𝒟::Vector{TT}) where TT

Grid points can be accessed as if the grid object is an array;

G = Grid2D([0.0,0.5],[0.0,1.0],5,5)
G[1]

Inputs:

  • Vector of domain boundaries [x_0,x_N]
  • Number of grid points.

Returns:

  • Struct for 1D grid object containing vector of grid points, $\Delta x$ and $n$.
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2D

FaADE.Grid.Grid2D — Type
Grid2D{TT,MET<:MetricType,GT<:AbstractArray{TT}} <: LocalGridType{TT,2,MET}

Grid data structure for 2 dimensional problems.

Grid points can be accessed as if the grid object is an array;

G = Grid2D([0.0,0.5],[0.0,1.0],5,5)
G[1,1]

Inputs:

  • Domain boundaries in $x$
  • Domain boundaries in $y$
  • Number of nodes in $x$
  • Number of nodes in $y$

Returns:

  • Struct for 2D grid object containing grid points in $x$ and $y$, $\Delta x$ and $\Delta y$, and $n_x$ and $n_y$.
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Multiblock grids

Solvers for these methods are not implemented yet (SATs required)

The following is used for constructing multiblock problems.

FaADE.Grid.GridMultiBlock — Type
GridMultiBlock{TT,DIM,MET,TG,TJ,IT} <: GridType{TT,DIM,MET}

Grid data for multiblock problems

Grabbing a particular subgrid can be done by G.Grids[i] which indexes in the order the grids are given. Indexing can be performed by G[i] for 1D or G[i,j] for 2D multiblock problems. GridMultiBlock.Joint contains the information on how to connect grids. If periodic boundary conditions are being used, do not specify the joint across that boundary.

Example grid creation,

    D1  = Grid2D([0.0,0.5],[0.0,1.0],5,5)
    D2  = Grid2D([0.5,1.0],[0.0,1.0],5,5)
    D3  = Grid2D([1.0,1.5],[0.0,1.0],5,5)

    glayout = ([(2,Right)],
                [(1,Left),(3,Right)],
                [(2,Left)])

    G = GridMultiBlock((D1,D2,D3),glayout)
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Curvilinear grid generation

The meshgrid function can be used to generate domains in curvilinear coordinates.

FaADE.Grid.meshgrid — Function

Generates the 2D grid of nx and ny points in a domain given a set of functions which bound a domain.

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meshgrid(cbottom::Function,cleft::Function,cright::Function,ctop::Function,nx::Int,ny::Int)

Construct a grid of nx by ny points in a domain bounded by the functions cbottom, cleft, cright, and ctop.

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meshgrid(𝒟x::Vector{TT},𝒟y::Vector{TT}) where TT

Generate matrix of coordinates from vectors of coordinates this is useful for packed grids.

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meshgrid(cinner::Function,couter::Function,nx,ny)

Meshgrid for annular domains where the inner and outer boundaries are parameterised boundaries

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meshgrid(inner::Torus,outer::Torus,ζ,nr,nθ)

Take two tori and generate a meshgrid between them at a given toroidal angle ζ with nr radial points and nθ poloidal points.

See also: Torus

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Toroidal surfaces

For generating meshgrid functions in an annulus one can use the following commands to generate two torii.

FaADE.Grid.Torus — Type
Torus{TT}

Represents a torus in boundary using Fourier series in R and Z coordinates as,

$R(\theta,\zeta) = \sum_{i,j} R_{i,j} cos(m_j\theta - n_i\zeta), \qquad Z(\theta,\zeta) = \sum_{i,j} Z_{i,j} sin(m_j\theta - n_i\zeta)$

where Rᵢⱼ and Zᵢⱼ are the Fourier coefficients and mⱼ and nᵢ are the Fourier modes.

Example:

Rin = [3e-1]; Zin=[-3e-1]
Rout = [6e-1]; Zout=[-6e-1]

inner = Torus(Rout,Zout,[1],[0])
outer = Torus(Rout,Zout,[1],[0])

meshgrid(inner,outer,0.0,11,21)

Coordinates at a given $\theta\in[0,R]$ and $\zeta\in[0,2\pi)$ can be computed using the call syntax

inner(θ,ζ)
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