jInv.Mesh
Discretization of differential operators is an essential ingredient of PDE parameter estimation problems. Depending on the problem and computational resources, different mesh types are needed in practical applications. The Mesh submodule of jInv provides different mesh geometry under the abstract type AbstractMesh. Currently, the Mesh submodule methods and provides operators on regular and tensor meshes but is easily extensible.
Regular Meshes
List of types and methods
#
jInv.Mesh.RegularMesh — Type.
type jInv.Mesh.RegularMesh <: AbstractTensorMesh
Regular mesh in 1D, 2D, and 3D
Fields:
domain::Vector{Float64} - physical domain [min(x1) max(x1) min(x2) max(x2)]
h::Vector{Float64} - cell size
x0::Vector{Float64} - origin
dim::Int - dimension of mesh
n::Vector{Int64} - number of cells in each dimension
nc::Int - total number of cells
nf::Vector{Int64} - number of faces in each dimension
ne::Vector{Int64} - number of edges in each dimension
Persistent Operators:
Operators should not be accessed directly. They will be built, if needed,
when accessing them using specified method. clear!(M) will release all
memory.
Div::SparseMatrixCSC - divergence (faces -> cell-centers)
Access via: getDivergenceMatrix(M)
Grad::SparseMatrixCSC - gradient (nodal -> edges)
Access via: getNodalGradientMatrix(M)
Curl::SparseMatrixCSC - curl (edges -> faces)
Access via: getCurlMatrix(M)
Af::SparseMatrixCSC - face average (faces -> cell-centers)
Access via: getFaceAverageMatrix(M)
Ae::SparseMatrixCSC - edge average (edges -> cell-centers)
Access via: getEdgeAverageMatrix(M)
An::SparseMatrixCSC - nodal average (nodes -> cell-centers)
Access via: getNodalAverageMatrix(M)
V::SparseMatrixCSC - cell volumes (diagonal matrix)
Access via: getVolume(M)
F::SparseMatrixCSC - face area (diagonal matrix)
Access via: getFaceArea(M)
L::SparseMatrixCSC - edge length (diagonal matrix)
Access via: getLength(M)
Vi::SparseMatrixCSC - inverse cell volumes (diagonal matrix)
Access via: getVolumeInv(M)
Fi::SparseMatrixCSC - inverse face area (diagonal matrix)
Access via: getFaceAreaInv(M)
Li::SparseMatrixCSC - inverse edge length (diagonal matrix)
Access via: getLengthAreaInv(M)
nLap::SparseMatrixCSC - nodal Laplacian
Access via: getNodalLaplacian(M)
Examples:
M2D = getRegularMesh([1.2 2.4 2.2 5.0],[3,4])
M3D = getRegularMesh([1.2 2.4 2.2 5.0 0 1],[3,4,7])
#
jInv.Mesh.TensorMesh3D — Type.
type jInv.Mesh.TensorMesh3D <: AbstractTensorMesh
Fields:
h1::Vector{Float64} - cell size in x1 direction
h2::Vector{Float64} - cell size in x2 direction
h3::Vector{Float64} - cell size in x3 direction
x0::Vector{Float64} - origin
dim::Int - dimension (dim=3)
n::Vector{Int64} - number of cells in each direction
nc::Int - nc total number of cells (nc=prod(n))
nf::Vector{Int64} - number of faces
ne::Vector{Int64} - number of edges
Persistent Operators:
Operators should not be accessed directly. They will be built, if needed,
when accessing them using specified method. clear!(M) will release all
memory.
Div::SparseMatrixCSC - divergence (faces -> cell-centers)
Access via: getDivergenceMatrix(M)
Grad::SparseMatrixCSC - gradient (nodal -> edges)
Access via: getNodalGradientMatrix(M)
Curl::SparseMatrixCSC - curl (edges -> faces)
Access via: getCurlMatrix(M)
Af::SparseMatrixCSC - face average (faces -> cell-centers)
Access via: getFaceAverageMatrix(M)
Ae::SparseMatrixCSC - edge average (edges -> cell-centers)
Access via: getEdgeAverageMatrix(M)
An::SparseMatrixCSC - nodal average (nodes -> cell-centers)
Access via: getNodalAverageMatrix(M)
V::SparseMatrixCSC - cell volumes (diagonal matrix)
Access via: getVolume(M)
F::SparseMatrixCSC - face area (diagonal matrix)
Access via: getFaceArea(M)
L::SparseMatrixCSC - edge length (diagonal matrix)
Access via: getLength(M)
Vi::SparseMatrixCSC - inverse cell volumes (diagonal matrix)
Access via: getVolumeInv(M)
Fi::SparseMatrixCSC - inverse face area (diagonal matrix)
Access via: getFaceAreaInv(M)
Li::SparseMatrixCSC - inverse edge length (diagonal matrix)
Access via: getLengthAreaInv(M)
nLap::SparseMatrixCSC - nodal Laplacian
Access via: getNodalLaplacian(M)
Example:
h1 = rand(4); h2 = rand(6); h3 = rand(5);
M = getTensorMesh2D(h1,h2,h3)
#
jInv.Mesh.getBoundaryNodes — Method.
function jInv.Mesh.getBoundaryNodes(Mesh)
Returns an array tuple (boundary node indices, inner node indices)
Input:
Mesh::Abstract Mesh
Output:
Tuple{Array{Int64,1},Array{Int64,1}}
#
jInv.Mesh.getEdgeAverageMatrix — Method.
function jInv.Mesh.getEdgeAverageMatrix
Returns Edge-to-CellCenter average matrix from Mesh.Ae.
Matrix is constructed if Mesh.Ae is empty.
For 3D Mesh: Ae = [A1 A2 A3]
Input:
Mesh::Abstract Mesh
#
jInv.Mesh.getEdgeIntegralOfPolygonalChain — Method.
function jInv.Mesh.getEdgeIntegralOfPolygonalChain
s = getEdgeIntegralPolygonalChain(mesh,polygon)
s = getEdgeIntegralPolygonalChain(mesh,polygon,normalize)
Compute the integral of a piecewise linear edge grid basis projected onto
the edges of a polygonal chain. This function can be used to evaluate the
source term of a line current carried by the polygonal chain in electromagnetic
modelling.
The piecewise linear edge grid basis consists of the functions
phix_i (i = 1 ... numXEdges)
phiy_i (i = 1 ... numYEdges)
phiz_i (i = 1 ... numZEdges)
where phix_i, phiy_i, phiz_i = 1 at the i-th x/y/z-directed edge
and phix_i, phiy_i, phiz_i = 0 at all other edges.
INPUT
mesh ...... Tensor mesh
polygon ... vertices of polygonal chain as numVertices x 3 array
normalize . divide line integral by length of integration path (boolean, optional)
OUTPUT
s ......... source vector
For a closed current loop, specify polygon such that
polygon[1,:] == polygon[end,:]
#
jInv.Mesh.getEdgeMassMatrix — Method.
function jInv.Mesh.getEdgeMassMatrix(mesh,sigma)
Returns mass matrix on cell edges, weighted by vector sigma.
Matrix is always constructed. Uses pre-constructed edge averaging
and cell volume matrices if available.
Input:
Mesh::Abstract Mesh
sigma::Vector
Output:
SparseMatrixCSC{Float64,Int64}
#
jInv.Mesh.getFaceAverageMatrix — Method.
function jInv.Mesh.getFaceAverageMatrix
Returns Face-to-CellCenter average matrix from Mesh.Af.
Matrix is constructed if Mesh.Af is empty.
For 2D Mesh: Af = [A1 A2]
For 3D Mesh: Af = [A1 A2 A3]
Input:
Mesh::Abstract Mesh
#
jInv.Mesh.getFaceMassMatrix — Method.
function jInv.Mesh.getFaceMassMatrix(M,sigma)
Returns face mass matrix, weighted by vector sigma.
Matrix is always constructed. Uses pre-constructed face averaging
and cell volume matrices if available.
Input:
M::Abstract Mesh
sigma::Vector
Output:
SparseMatrixCSC{Float64,Int64}
#
jInv.Mesh.getInterpolationMatrix — Method.
function jInv.Mesh.getInterpolationMatrix
computes bi/trilinear interpolation matrix P for cell-centered data from Mesh1 to Mesh2. If I1 is a cell-centerd discretization of some function on Mesh1 then, its interpolant on Mesh2 is given by
I2 = P*I1
Required Input:
M1::AbstractTensorMesh
M2::AbstractTensorMesh
Example:
In mesh-decoupling, we use different meshes for the inverse solution
and the different forward problems.
Mesh2Mesh = getInterpolationMatrix(Minv,Mfor)
#
jInv.Mesh.getNodalAverageMatrix — Method.
function jInv.Mesh.getNodalAverageMatrix
Returns Nodal-to-CellCenter average matrix from Mesh.An.
Matrix is constructed if Mesh.An is empty.
Input:
Mesh::Abstract Mesh
#
jInv.Mesh.getNodalMassMatrix — Method.
function jInv.Mesh.getNodalMassMatrix(M,sigma)
Returns nodal mass matrix, weighted by vector sigma.
Matrix is always constructed. Uses pre-constructed nodal averaging
and cell volume matrices if available.
Input:
M::Abstract Mesh
sigma::Vector
Output:
SparseMatrixCSC{Float64,Int64}
#
jInv.Mesh.getRegularMesh — Method.
function jInv.Mesh.getRegularMesh
Constructs regular mesh
Input:
domain - physical domain rectangular
n - number of cells in each dimension
Examples:
M2D = getRegularMesh([1.2 2.4 2.2 5.0],[3,4])
M3D = getRegularMesh([1.2 2.4 2.2 5.0 0 1],[3,4,7])
#
jInv.Mesh.getStraightLineCurrentIntegral — Method.
function jInv.Mesh.getStraightLineCurrentIntegral
[sx,sy,sz] = getStraightLineCurrentIntegral(hx,hy,hz,ax,ay,az,bx,by,bz)
Compute integral int(W . J dx^3) in brick of size hx x hy x hz
where W denotes the 12 local bilinear edge basis functions
and where J prescribes a unit line current
between points (ax,ay,az) and (bx,by,bz).
#
jInv.Mesh.getTensorMesh3D — Function.
function jInv.Mesh.getTensorMesh3D
constructs TensorMesh3D
Required Input:
h1::Array - cell-sizes in x1 direction
h2::Array - cell-sizes in x2 direction
h3::Array - cell-sizes in x3 direction
Optional Input
x0::Array - origin (default = zeros(3))
#
jInv.Mesh.getdEdgeMassMatrix — Method.
function jInv.Mesh.getdEdgeMassMatrix(M,v)
Returns directional derivative of edge mass matrix w.r.t. sigma, i.e.,
d_sigma (M(sigma)*v)
Matrix is always constructed. Uses pre-constructed edge averaging
and cell volume matrices if available.
Input:
Mesh::Abstract - Mesh
v::Vector - edge vector defining derivative
Output:
SparseMatrixCSC{Float64,Int64}
#
jInv.Mesh.getdFaceMassMatrix — Method.
function jInv.Mesh.getdFaceMassMatrix(M,v)
Returns directional derivative of face mass matrix w.r.t sigma, i.e.
d_sigma (M(sigma)*v)
Matrix is always constructed. Uses pre-constructed face averaging
and cell volume matrices if available.
Input:
M::Abstract - Mesh
v::Vector - face vector defining derivative
Output:
SparseMatrixCSC{Float64,Int64}
#
jInv.Mesh.getdNodalMassMatrix — Method.
function jInv.Mesh.getdNodalMassMatrix(M,v)
Returns directional derivative of nodal mass matrix w.r.t. sigma, i.e.,
d_sigma (M(sigma)*v)
Matrix is always constructed. Uses pre-constructed nodal averaging
and cell volume matrices if available.
Input:
M::Abstract - Mesh
v::Vector - nodal vector defining derivative
Output:
SparseMatrixCSC{Float64,Int64}