232 lines
6.6 KiB
Python
232 lines
6.6 KiB
Python
"""
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Matrix operators for finite difference simulations
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Basic discrete calculus etc.
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"""
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from typing import List, Callable, Tuple, Dict
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import numpy
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import scipy.sparse as sparse
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from .types import fdfield_t, vfdfield_t
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def rotation(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
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"""
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Utility operator for performing a circular shift along a specified axis by a
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specified number of elements.
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Args:
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axis: Axis to shift along. x=0, y=1, z=2
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shape: Shape of the grid being shifted
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shift_distance: Number of cells to shift by. May be negative. Default 1.
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Returns:
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Sparse matrix for performing the circular shift.
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"""
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if len(shape) not in (2, 3):
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raise Exception('Invalid shape: {}'.format(shape))
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if axis not in range(len(shape)):
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raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
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shifts = [abs(shift_distance) if a == axis else 0 for a in range(3)]
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shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
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ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
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n = numpy.prod(shape)
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i_ind = numpy.arange(n)
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j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
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vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
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d = sparse.csr_matrix(vij, shape=(n, n))
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if shift_distance < 0:
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d = d.T
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return d
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def shift_with_mirror(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
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"""
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Utility operator for performing an n-element shift along a specified axis, with mirror
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boundary conditions applied to the cells beyond the receding edge.
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Args:
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axis: Axis to shift along. x=0, y=1, z=2
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shape: Shape of the grid being shifted
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shift_distance: Number of cells to shift by. May be negative. Default 1.
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Returns:
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Sparse matrix for performing the shift-with-mirror.
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"""
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if len(shape) not in (2, 3):
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raise Exception('Invalid shape: {}'.format(shape))
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if axis not in range(len(shape)):
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raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
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if shift_distance >= shape[axis]:
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raise Exception('Shift ({}) is too large for axis {} of size {}'.format(
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shift_distance, axis, shape[axis]))
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def mirrored_range(n, s):
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v = numpy.arange(n) + s
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v = numpy.where(v >= n, 2 * n - v - 1, v)
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v = numpy.where(v < 0, - 1 - v, v)
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return v
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shifts = [shift_distance if a == axis else 0 for a in range(3)]
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shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
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ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
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n = numpy.prod(shape)
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i_ind = numpy.arange(n)
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j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
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vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
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d = sparse.csr_matrix(vij, shape=(n, n))
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return d
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def deriv_forward(dx_e: List[numpy.ndarray]) -> List[sparse.spmatrix]:
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"""
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Utility operators for taking discretized derivatives (forward variant).
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Args:
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dx_e: Lists of cell sizes for all axes
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`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
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Returns:
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List of operators for taking forward derivatives along each axis.
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"""
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shape = [s.size for s in dx_e]
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n = numpy.prod(shape)
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dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
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def deriv(axis):
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return rotation(axis, shape, 1) - sparse.eye(n)
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Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
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for a, dx in enumerate(dx_e_expanded)]
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return Ds
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def deriv_back(dx_h: List[numpy.ndarray]) -> List[sparse.spmatrix]:
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"""
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Utility operators for taking discretized derivatives (backward variant).
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Args:
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dx_h: Lists of cell sizes for all axes
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`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
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Returns:
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List of operators for taking forward derivatives along each axis.
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"""
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shape = [s.size for s in dx_h]
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n = numpy.prod(shape)
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dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
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def deriv(axis):
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return rotation(axis, shape, -1) - sparse.eye(n)
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Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
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for a, dx in enumerate(dx_h_expanded)]
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return Ds
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def cross(B: List[sparse.spmatrix]) -> sparse.spmatrix:
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"""
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Cross product operator
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Args:
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B: List `[Bx, By, Bz]` of sparse matrices corresponding to the x, y, z
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portions of the operator on the left side of the cross product.
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Returns:
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Sparse matrix corresponding to (B x), where x is the cross product.
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"""
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n = B[0].shape[0]
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zero = sparse.csr_matrix((n, n))
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return sparse.bmat([[zero, -B[2], B[1]],
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[B[2], zero, -B[0]],
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[-B[1], B[0], zero]])
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def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
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"""
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Vector cross product operator
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Args:
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b: Vector on the left side of the cross product.
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Returns:
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Sparse matrix corresponding to (b x), where x is the cross product.
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"""
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B = [sparse.diags(c) for c in numpy.split(b, 3)]
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return cross(B)
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def avg_forward(axis: int, shape: List[int]) -> sparse.spmatrix:
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"""
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Forward average operator `(x4 = (x4 + x5) / 2)`
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Args:
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axis: Axis to average along (x=0, y=1, z=2)
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shape: Shape of the grid to average
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Returns:
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Sparse matrix for forward average operation.
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"""
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if len(shape) not in (2, 3):
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raise Exception('Invalid shape: {}'.format(shape))
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n = numpy.prod(shape)
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return 0.5 * (sparse.eye(n) + rotation(axis, shape))
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def avg_back(axis: int, shape: List[int]) -> sparse.spmatrix:
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"""
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Backward average operator `(x4 = (x4 + x3) / 2)`
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Args:
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axis: Axis to average along (x=0, y=1, z=2)
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shape: Shape of the grid to average
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Returns:
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Sparse matrix for backward average operation.
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"""
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return avg_forward(axis, shape).T
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def curl_forward(dx_e: List[numpy.ndarray]) -> sparse.spmatrix:
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"""
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Curl operator for use with the E field.
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Args:
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dx_e: Lists of cell sizes for all axes
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`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
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Returns:
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Sparse matrix for taking the discretized curl of the E-field
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"""
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return cross(deriv_forward(dx_e))
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def curl_back(dx_h: List[numpy.ndarray]) -> sparse.spmatrix:
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"""
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Curl operator for use with the H field.
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Args:
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dx_h: Lists of cell sizes for all axes
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`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
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Returns:
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Sparse matrix for taking the discretized curl of the H-field
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"""
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return cross(deriv_back(dx_h))
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