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gradient.py

GradientMixin

Mixin providing methods for finite differencing signed-distance φ grid.

Methods:

fwd_back_differences staticmethod 

fwd_back_differences(
    φ: MaskedArray, Δx: float
) -> tuple[MaskedArray, MaskedArray, MaskedArray, MaskedArray]

Compute forward, backward, mean & double finite differences.

Could perhaps use np.roll to make this process more efficient?

Parameters

φ: MaskedArray Signed distance grid. Δx: float Grid spacing in both x and z directions (assumed equal).

Returns

(Δφ_x, Δφ_z, Δ2φ_x2, Δ2φ_z2,): tuple[MaskedArray, MaskedArray, MaskedArray, MaskedArray] 1st-order (center-weighted) and 2nd-order φ finite differences.

Source code in erosionfront/levelset/gradient.py 
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@staticmethod
def fwd_back_differences(
        φ: MaskedArray,
        Δx: float,
    ) -> tuple[MaskedArray, MaskedArray, MaskedArray, MaskedArray]:
    """
    Compute forward, backward, mean & double finite differences.

    Could perhaps use np.roll to make this process more efficient?

    Parameters
    ----------
    φ: MaskedArray
        Signed distance grid.
    Δx: float
        Grid spacing in both x and z directions (assumed equal).

    Returns
    -------
    (Δφ_x, Δφ_z, Δ2φ_x2, Δ2φ_z2,): tuple[MaskedArray, MaskedArray, \
        MaskedArray, MaskedArray]
        1st-order (center-weighted) and 2nd-order φ finite differences.
    """
    Δz: float = Δx
    φ_xp: MaskedArray = φ.copy()
    φ_xm: MaskedArray = φ.copy()
    φ_zp: MaskedArray = φ.copy()
    φ_zm: MaskedArray = φ.copy()

    # Generate shifted grids to make differencing cleaner
    # Remember arrays are [rows, cols] and that rows=z, cols=x
    φ_xp[:,:-1] = φ[:,1:]
    φ_xm[:,1:]  = φ[:,:-1]
    φ_zp[:-1,:] = φ[1:,:]
    φ_zm[1:,:]  = φ[:-1,:]

    # Forward and backward differences in x and z
    Δφ_xp: MaskedArray = φ_xp - φ
    Δφ_xm: MaskedArray = φ - φ_xm
    Δφ_zp: MaskedArray = φ_zp - φ
    Δφ_zm: MaskedArray = φ - φ_zm

    # Mean differences ~dφ/dx∇∇𝛻𝚺∇
    Δφ_x: MaskedArray = ((Δφ_xp + Δφ_xm)/2) / Δx
    Δφ_z: MaskedArray = ((Δφ_zp + Δφ_zm)/2) / Δz

    # Double differences ~d2φ/dx2
    Δ2φ_x2: MaskedArray = ((Δφ_xp - Δφ_xm)/2) / Δx
    Δ2φ_z2: MaskedArray = ((Δφ_zp - Δφ_zm)/2) / Δz

    return (Δφ_x, Δφ_z, Δ2φ_x2, Δ2φ_z2,)