Mesh Operators

The tokamesh.operators module provides functions for constructing linear operators (i.e. matrices) which act on a vector of field values at each mesh vertex to compute a new vector.

tokamesh.operators.edge_difference_matrix(R: ~numpy.ndarray, z: ~numpy.ndarray, triangles: ~numpy.ndarray, normalised=False, sparse_array_type=<class 'scipy.sparse._csr.csr_array'>) → sparray

Generates a sparse matrix which, when operating on a vector of field values at each vertex, produces a vector of the differences in those field values along each edge in the mesh.

Parameters

  • R – The major radius of each mesh vertex as a 1D numpy array.

  • z – The z-height of each mesh vertex as a 1D numpy array.

  • triangles – A 2D numpy array of integers specifying the indices of the vertices which form each of the triangles in the mesh. The array must have shape (N,3) where N is the total number of triangles.

  • normalised (bool) – If set as True, the difference in the value across each edge is normalised by the length of the edge, yielding an estimate of the gradient along that edge.

  • sparse_array_type – A sparse array type from scipy.sparse which will be used as the returned type of the matrix operator.

Returns

The edge-difference matrix operator as a sparse array.

tokamesh.operators.umbrella_matrix(R: ~numpy.ndarray, z: ~numpy.ndarray, triangles: ~numpy.ndarray, ignore_boundary=True, inverse_distance_weighting=True, normalised=False, sparse_array_type=<class 'scipy.sparse._csr.csr_array'>) → sparray

Returns a sparse ‘umbrella’ matrix operator, which finds the difference between the value of every internal vertex and the average value of the other vertices to which it is connected.

Parameters

  • R – The major radius of each mesh vertex as a 1D numpy array.

  • z – The z-height of each mesh vertex as a 1D numpy array.

  • triangles – A 2D numpy array of integers specifying the indices of the vertices which form each of the triangles in the mesh. The array must have shape (N,3) where N is the total number of triangles.

  • ignore_boundary (bool) – If set as True, the operator will ignore vertices on any boundary of the mesh, instead returning a value of zero for those vertices.

  • inverse_distance_weighting (bool) – If set as True, the sum over the neighbouring vertices is weighted by their inverse-distance to the central vertex. This tends to produce better results for irregular meshes with significant variation of edge-lengths.

  • normalised (bool) – If set as True the value produced by the operator for each vertex is normalised by the square of the mean distance between the neighbouring vertices and the central vertex.

  • sparse_array_type – A sparse array type from scipy.sparse which will be used as the returned type of the matrix operator.

Returns

The umbrella matrix operator as a sparse array.

tokamesh.operators.parallel_derivative(R: ~numpy.ndarray, z: ~numpy.ndarray, index_grid: ~numpy.ndarray, order=2, sparse_array_type=<class 'scipy.sparse._csr.csr_array'>) → sparray

Constructs a sparse matrix operator which estimates the derivative (with respect to poloidal distance) of a given set of field values at each vertex in direction parallel to the magnetic field.

Parameters

  • R – The major radius of each mesh vertex as a 1D numpy array.

  • z – The z-height of each mesh vertex as a 1D numpy array.

  • index_grid – A 2D numpy array specifying the indices of triangular mesh vertices corresponding to each cell of the rectangular field-aligned grid.

  • order – The order of the derivative to be estimated. Must be either 1 or 2.

  • sparse_array_type – A sparse array type from scipy.sparse which will be used as the returned type of the matrix operator.

Returns

The parallel derivative matrix operator as a sparse array.

tokamesh.operators.perpendicular_derivative(R: ~numpy.ndarray, z: ~numpy.ndarray, index_grid: ~numpy.ndarray, order=2, sparse_array_type=<class 'scipy.sparse._csr.csr_array'>) → sparray

Constructs a sparse matrix operator which estimates the derivative (with respect to poloidal distance) of a given set of field values at each vertex in direction perpendicular to the magnetic field.

Parameters

  • R – The major radius of each mesh vertex as a 1D numpy array.

  • z – The z-height of each mesh vertex as a 1D numpy array.

  • index_grid – A 2D numpy array specifying the indices of triangular mesh vertices corresponding to each cell of the rectangular field-aligned grid.

  • order – The order of the derivative to be estimated. Must be either 1 or 2.

  • sparse_array_type – A sparse array type from scipy.sparse which will be used as the returned type of the matrix operator.

Returns

The perpendicular derivative matrix operator as a sparse array.