Skip to content

Utility Functions

device_inductance.grid

Grid input definitions

Extent module-attribute 

Extent = tuple[float, float, float, float]

[m] rmin, rmax, zmin, zmax rectangular bounds

GridSpec module-attribute 

GridSpec = tuple[float, int, float, int]

[m] rmin, nr, zmin, nz exact specification of regular grid, as an alternative to approximate minimum extent

Resolution module-attribute 

Resolution = tuple[float, float]

[m] dr, dz spatial resolution of computational grid

device_inductance.utils

calc_flux_density_from_flux 

calc_flux_density_from_flux(
    psi: NDArray, rmesh: NDArray, zmesh: NDArray
) -> tuple[NDArray, NDArray]

Back-calculate B-field from poloidal flux per Wesson eqn 3.2.2 by 4th-order finite difference, modified to use total poloidal flux instead of flux per radian.

This avoids an expensive sum over filamentized contributions at the expense of some numerical error.

Errors

* If the input grids are not regular
* If any input grid dimensions have size less than 5

References

* [1] J. Wesson, Tokamaks. Oxford, New York: Clarendon Press, 1987.

Parameters:

Name Type Description Default
psi NDArray

[Wb] poloidal flux

 required
rmesh NDArray

[m] 2D r-coordinates

 required
zmesh NDArray

[m] 2D z-coordinates

 required

Returns:

Type Description
tuple[NDArray, NDArray]

(br, bz) [T] 2D arrays of poloidal flux density
Source code in src/device_inductance/utils.py 
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
def calc_flux_density_from_flux(psi: NDArray, rmesh: NDArray, zmesh: NDArray) -> tuple[NDArray, NDArray]:
    """
    Back-calculate B-field from poloidal flux per Wesson eqn 3.2.2 by 4th-order finite difference,
    modified to use total poloidal flux instead of flux per radian.

    This avoids an expensive sum over filamentized contributions at the expense of some numerical error.

    # Errors

        * If the input grids are not regular
        * If any input grid dimensions have size less than 5

    # References

        * [1] J. Wesson, Tokamaks. Oxford, New York: Clarendon Press, 1987.

    Args:
        psi: [Wb] poloidal flux
        rmesh: [m] 2D r-coordinates
        zmesh: [m] 2D z-coordinates

    Returns:
        (br, bz) [T] 2D arrays of poloidal flux density
    """

    dpsidr, dpsidz = gradient_order4(psi, rmesh, zmesh)

    r_inv = rmesh**-1

    br = -r_inv * dpsidz / (2.0 * np.pi)  # [T]
    bz = r_inv * dpsidr / (2.0 * np.pi)  # [T]

    return (br, bz)

flux_solver 

flux_solver(
    grids: tuple[NDArray, NDArray],
) -> Callable[[NDArray], NDArray]

Linear solver for extracting a flux field from a toroidal current density distribution using a 4th-order finite difference approximation of the Grad-Shafranov PDE. For jtor toroidal current density shaped like (nr, nz), call like psi = flux_solver(rhs) to get psi in [Wb] or [V-s], where rhs = -2.0 * np.pi * mu_0 * rmesh * jtor with the boundary values set to the circular-filament solved flux.

Parameters:

Name Type Description Default
grids tuple[NDArray, NDArray]

[m] regular 1D r,z grids

 required

Returns:

Name Type Description
solver Callable[[NDArray], NDArray]

factorized solver for Grad-Shafranov differential operator
Source code in src/device_inductance/utils.py 
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
def flux_solver(grids: tuple[NDArray, NDArray]) -> Callable[[NDArray], NDArray]:
    """
    Linear solver for extracting a flux field from a toroidal current density distribution
    using a 4th-order finite difference approximation of the Grad-Shafranov PDE.
    For `jtor` toroidal current density shaped like (nr, nz), call like `psi = flux_solver(rhs)`
    to get `psi` in [Wb] or [V-s], where `rhs = -2.0 * np.pi * mu_0 * rmesh * jtor` with the boundary
    values set to the circular-filament solved flux.

    Args:
        grids: [m] regular 1D r,z grids

    Returns:
        solver: factorized solver for Grad-Shafranov differential operator
    """
    # Build Grad-Shafranov Delta* linear operator for finite difference
    # as a sparse matrix
    _ = _check_regular(grids)
    rgrid, zgrid = grids
    nr = rgrid.size
    nz = zgrid.size
    vals, rows, cols = gs_operator_order4(*grids)
    operator = csc_matrix((vals, (rows, cols)), shape=(nr * nz, nr * nz))
    # Store LU factorization of operator matrix to allow fast, reusable
    # solves using different right-hand-side (different current density)
    return factorized(operator)

gradient_order4 

gradient_order4(
    z: NDArray, xmesh: NDArray, ymesh: NDArray
) -> tuple[NDArray, NDArray]

Calculate gradient by 4th-order finite difference.

numpy.gradient exists and is fast and convenient, but only uses a second-order difference, which produces unacceptable error in B-fields (well over 1% for typical geometries).

Errors
* If the input grids are not regular
* If any input grid dimensions have size less than 5
References
* [1] “Finite difference coefficient,” Wikipedia. Aug. 22, 2023.
      Accessed: Mar. 29, 2024. [Online].
      Available: https://en.wikipedia.org/w/index.php?title=Finite_difference_coefficient

Parameters:

Name Type Description Default
z NDArray

[] 2D array of values on which to calculate the gradient

 required
xmesh NDArray

[m] 2D array of coordinates of first dimension

 required
ymesh NDArray

[m] 2D array of coordinates of second dimension

 required

Returns:

Type Description
tuple[NDArray, NDArray]

(dzdx, dzdy) [/m] 2D arrays of gradient components
Source code in src/device_inductance/utils.py 
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
def gradient_order4(z: NDArray, xmesh: NDArray, ymesh: NDArray) -> tuple[NDArray, NDArray]:
    """
    Calculate gradient by 4th-order finite difference.

    `numpy.gradient` exists and is fast and convenient, but only uses a second-order difference,
    which produces unacceptable error in B-fields (well over 1% for typical geometries).

    ## Errors

        * If the input grids are not regular
        * If any input grid dimensions have size less than 5

    ## References

        * [1] “Finite difference coefficient,” Wikipedia. Aug. 22, 2023.
              Accessed: Mar. 29, 2024. [Online].
              Available: https://en.wikipedia.org/w/index.php?title=Finite_difference_coefficient

    Args:
        z: [<xunits>] 2D array of values on which to calculate the gradient
        xmesh: [m] 2D array of coordinates of first dimension
        ymesh: [m] 2D array of coordinates of second dimension

    Returns:
        (dzdx, dzdy) [<xunits>/m] 2D arrays of gradient components
    """
    nx, ny = z.shape
    dx = xmesh[1][0] - xmesh[0][0]
    dy = ymesh[0][1] - ymesh[0][0]

    # Check regular grid assumption
    assert np.all(np.abs(np.diff(xmesh[:, 0]) - dx) / dx < 1e-6), (
        "This method is only implemented for a regular grid"
    )
    assert np.all(np.abs(np.diff(ymesh[0, :]) - dy) / dy < 1e-6), (
        "This method is only implemented for a regular grid"
    )

    dzdx = np.zeros_like(z)
    for offs, w in _DDX_CENTRAL_ORDER4:
        start = int(2 + offs)
        end = int(nx - 2 + offs)
        dzdx[2:-2, :] += w * z[start:end, :] / dx  # Central difference on interior points
    for offs, w in _DDX_FWD_ORDER4:
        offs = int(offs)
        dzdx[0:2, :] += w * z[offs : offs + 2, :] / dx  # One-sided difference on left side
    for offs, w in _DDX_BWD_ORDER4:
        start = int(-2 + offs)
        end = int(nx + offs)
        dzdx[-2:, :] += w * z[start:end, :] / dx  # right side

    dzdy = np.zeros_like(z)
    for offs, w in _DDX_CENTRAL_ORDER4:
        start = int(2 + offs)
        end = int(ny - 2 + offs)
        dzdy[:, 2:-2] += w * z[:, start:end] / dy  # Interior points
    for offs, w in _DDX_FWD_ORDER4:
        offs = int(offs)
        dzdy[:, 0:2] += w * z[:, offs : offs + 2] / dy  # One-sided difference on bottom
    for offs, w in _DDX_BWD_ORDER4:
        start = int(-2 + offs)
        end = int(ny + offs)
        dzdy[:, -2:] += w * z[:, start:end] / dy  # top

    return dzdx, dzdy

solve_flux_axisymmetric 

solve_flux_axisymmetric(
    grids: tuple[NDArray, NDArray],
    meshes: tuple[NDArray, NDArray],
    current_density: NDArray,
    solver: Callable[[NDArray], NDArray] | None = None,
) -> NDArray

Calculate the flux field associated with a given toroidal current density distribution, by solving the Grad-Shafranov PDE.

This calculation is most commonly used for the plasma, but is in fact more general, and applies to anything with an equivalent toroidal current density and axisymmetry.

Parameters:

Name Type Description Default
grids tuple[NDArray, NDArray]

[m] 1D r,z regular coordinate grids

 required
meshes tuple[NDArray, NDArray]

[m] 2D meshgrids made from grids like np.meshgrid(*grids, indexing="ij")

 required
current_density NDArray

[A/m^2], shape (nr, nz), toroidal current density on finite-difference mesh

 required
solver Callable[[NDArray], NDArray] | None

Optionally, provide a pre-initialized linear solver. See cfsem.utils.flux_solver.

 None

Returns:

Type Description
NDArray

poloidal flux field, [Wb] with shape (nr, nz)
Source code in src/device_inductance/utils.py 
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
def solve_flux_axisymmetric(
    grids: tuple[NDArray, NDArray],
    meshes: tuple[NDArray, NDArray],
    current_density: NDArray,
    solver: Callable[[NDArray], NDArray] | None = None,
) -> NDArray:
    """
    Calculate the flux field associated with a given toroidal current density distribution,
    by solving the Grad-Shafranov PDE.

    This calculation is most commonly used for the plasma, but is in fact more general,
    and applies to anything with an equivalent toroidal current density and axisymmetry.

    Args:
        grids: [m] 1D r,z regular coordinate grids
        meshes: [m] 2D meshgrids made from grids like np.meshgrid(*grids, indexing="ij")
        current_density: [A/m^2], shape (nr, nz), toroidal current density on finite-difference mesh
        solver: Optionally, provide a pre-initialized linear solver. See `cfsem.utils.flux_solver`.

    Returns:
        poloidal flux field, [Wb] with shape (nr, nz)
    """
    # Build the differential operator, if needed
    solver = solver or flux_solver(grids)

    # Unpack and filter down to just useful inputs
    dr, dz = _check_regular(grids)  # [m] grid spacing
    area = dr * dz  # [m^2]
    rmesh, zmesh = meshes  # [m]
    nonzero_inds = np.where(current_density != 0.0)
    current_density_nonzero = np.ascontiguousarray(current_density[nonzero_inds])  # [A/m^2]
    rmesh_nonzero = np.ascontiguousarray(rmesh[nonzero_inds])  # [m]
    zmesh_nonzero = np.ascontiguousarray(zmesh[nonzero_inds])  # [m]
    # Solve `Delta* @ psi = -mu_0 * 2pi * rmesh * jtor`
    #   Set up right-hand-side of Grad-Shafranov
    rhs = -(2.0 * np.pi * mu_0) * rmesh * current_density  # [Wb/m^2]
    #   Set flux boundary condition
    #   For most relevant grid sizes (up to 500 X 500), doing the O(N^3)
    #   circular-filament flux calc is faster than the linear solve
    #   and therefore faster than doing an extra fixed-boundary linear solve
    #   in order to use Von Hagenow's asymptotically-O(N^2logN) method.
    ifil = (area * current_density_nonzero).flatten()  # [A] plasma filament current
    rfil = rmesh_nonzero.flatten()
    zfil = zmesh_nonzero.flatten()
    for s in [[0, ...], [-1, ...], [..., 0], [..., -1]]:  # All boundary slices
        rhs[s[0], s[1]] = flux_circular_filament(ifil, rfil, zfil, rmesh[s[0], s[1]], zmesh[s[0], s[1]])
    #   Do the actual linear solve
    psi = solver(rhs.flatten()).reshape(rmesh.shape)  # [Wb]

    return psi