"""
Filters for operations on regular grids.
The grids are defined as (x,y,z,...) where x is fastest and z is slowest.
This convention is consistent with the layout in grid vti files.
When converting to/from a plain list (e.g. storage in ASCII table),
the following operations are required for tensorial data:
- D3 = D1.reshape(cells+(-1,),order='F').reshape(cells+(3,3))
- D1 = D3.reshape(cells+(-1,)).reshape(-1,9,order='F')
"""
from typing import Tuple as _Tuple
from scipy import spatial as _spatial
import numpy as _np
from ._typehints import FloatSequence as _FloatSequence, IntSequence as _IntSequence
def _ks(size: _FloatSequence,
cells: _IntSequence,
first_order: bool = False) -> _np.ndarray:
"""
Get wave numbers operator.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
cells : sequence of int, len (3)
Number of cells.
first_order : bool, optional
Correction for first order derivatives, defaults to False.
"""
k_sk = _np.where(_np.arange(cells[0])>cells[0]//2,
_np.arange(cells[0])-cells[0],_np.arange(cells[0]))/size[0]
if cells[0]%2 == 0 and first_order: k_sk[cells[0]//2] = 0 # Nyquist freq=0 for even cells (Johnson, MIT, 2011)
k_sj = _np.where(_np.arange(cells[1])>cells[1]//2,
_np.arange(cells[1])-cells[1],_np.arange(cells[1]))/size[1]
if cells[1]%2 == 0 and first_order: k_sj[cells[1]//2] = 0 # Nyquist freq=0 for even cells (Johnson, MIT, 2011)
k_si = _np.arange(cells[2]//2+1)/size[2]
return _np.stack(_np.meshgrid(k_sk,k_sj,k_si,indexing = 'ij'), axis=-1)
[docs]def curl(size: _FloatSequence,
f: _np.ndarray) -> _np.ndarray:
u"""
Calculate curl of a vector or tensor field in Fourier space.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
f : numpy.ndarray, shape (:,:,:,3) or (:,:,:,3,3)
Periodic field of which the curl is calculated.
Returns
-------
∇ × f : numpy.ndarray, shape (:,:,:,3) or (:,:,:,3,3)
Curl of f.
"""
n = _np.prod(f.shape[3:])
k_s = _ks(size,f.shape[:3],True)
e = _np.zeros((3, 3, 3))
e[0, 1, 2] = e[1, 2, 0] = e[2, 0, 1] = +1.0 # Levi-Civita symbol
e[0, 2, 1] = e[2, 1, 0] = e[1, 0, 2] = -1.0
f_fourier = _np.fft.rfftn(f,axes=(0,1,2))
curl_ = (_np.einsum('slm,ijkl,ijkm ->ijks', e,k_s,f_fourier)*2.0j*_np.pi if n == 3 else # vector, 3 -> 3
_np.einsum('slm,ijkl,ijknm->ijksn',e,k_s,f_fourier)*2.0j*_np.pi) # tensor, 3x3 -> 3x3
return _np.fft.irfftn(curl_,axes=(0,1,2),s=f.shape[:3])
[docs]def divergence(size: _FloatSequence,
f: _np.ndarray) -> _np.ndarray:
u"""
Calculate divergence of a vector or tensor field in Fourier space.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
f : numpy.ndarray, shape (:,:,:,3) or (:,:,:,3,3)
Periodic field of which the divergence is calculated.
Returns
-------
∇ · f : numpy.ndarray, shape (:,:,:,1) or (:,:,:,3)
Divergence of f.
"""
n = _np.prod(f.shape[3:])
k_s = _ks(size,f.shape[:3],True)
f_fourier = _np.fft.rfftn(f,axes=(0,1,2))
div_ = (_np.einsum('ijkl,ijkl ->ijk', k_s,f_fourier)*2.0j*_np.pi if n == 3 else # vector, 3 -> 1
_np.einsum('ijkm,ijklm->ijkl',k_s,f_fourier)*2.0j*_np.pi) # tensor, 3x3 -> 3
return _np.fft.irfftn(div_,axes=(0,1,2),s=f.shape[:3])
[docs]def gradient(size: _FloatSequence,
f: _np.ndarray) -> _np.ndarray:
u"""
Calculate gradient of a scalar or vector field in Fourier space.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
f : numpy.ndarray, shape (:,:,:,1) or (:,:,:,3)
Periodic field of which the gradient is calculated.
Returns
-------
∇ f : numpy.ndarray, shape (:,:,:,3) or (:,:,:,3,3)
Divergence of f.
"""
n = _np.prod(f.shape[3:])
k_s = _ks(size,f.shape[:3],True)
f_fourier = _np.fft.rfftn(f,axes=(0,1,2))
grad_ = (_np.einsum('ijkl,ijkm->ijkm', f_fourier,k_s)*2.0j*_np.pi if n == 1 else # scalar, 1 -> 3
_np.einsum('ijkl,ijkm->ijklm',f_fourier,k_s)*2.0j*_np.pi) # vector, 3 -> 3x3
return _np.fft.irfftn(grad_,axes=(0,1,2),s=f.shape[:3])
[docs]def coordinates0_point(cells: _IntSequence,
size: _FloatSequence,
origin: _FloatSequence = _np.zeros(3)) -> _np.ndarray:
"""
Cell center positions (undeformed).
Parameters
----------
cells : sequence of int, len (3)
Number of cells.
size : sequence of float, len (3)
Physical size of the periodic field.
origin : sequence of float, len(3), optional
Physical origin of the periodic field. Defaults to [0.0,0.0,0.0].
Returns
-------
x_p_0 : numpy.ndarray, shape (:,:,:,3)
Undeformed cell center coordinates.
"""
size_ = _np.array(size,float)
start = origin + size_/_np.array(cells,_np.int64)*.5
end = origin + size_ - size_/_np.array(cells,_np.int64)*.5
return _np.stack(_np.meshgrid(_np.linspace(start[0],end[0],cells[0]),
_np.linspace(start[1],end[1],cells[1]),
_np.linspace(start[2],end[2],cells[2]),indexing = 'ij'),
axis = -1)
[docs]def displacement_fluct_point(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Cell center displacement field from fluctuation part of the deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_p_fluct : numpy.ndarray, shape (:,:,:,3)
Fluctuating part of the cell center displacements.
"""
integrator = 0.5j*_np.array(size,float)/_np.pi
k_s = _ks(size,F.shape[:3],False)
k_s_squared = _np.einsum('...l,...l',k_s,k_s)
k_s_squared[0,0,0] = 1.0
displacement = -_np.einsum('ijkml,ijkl,l->ijkm',
_np.fft.rfftn(F,axes=(0,1,2)),
k_s,
integrator,
) / k_s_squared[...,_np.newaxis]
return _np.fft.irfftn(displacement,axes=(0,1,2),s=F.shape[:3])
[docs]def displacement_avg_point(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Cell center displacement field from average part of the deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_p_avg : numpy.ndarray, shape (:,:,:,3)
Average part of the cell center displacements.
"""
F_avg = _np.average(F,axis=(0,1,2))
return _np.einsum('ml,ijkl->ijkm',F_avg - _np.eye(3),coordinates0_point(F.shape[:3],size))
[docs]def displacement_point(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Cell center displacement field from deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_p : numpy.ndarray, shape (:,:,:,3)
Cell center displacements.
"""
return displacement_avg_point(size,F) + displacement_fluct_point(size,F)
[docs]def coordinates_point(size: _FloatSequence,
F: _np.ndarray,
origin: _FloatSequence = _np.zeros(3)) -> _np.ndarray:
"""
Cell center positions.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
origin : sequence of float, len(3), optional
Physical origin of the periodic field. Defaults to [0.0,0.0,0.0].
Returns
-------
x_p : numpy.ndarray, shape (:,:,:,3)
Cell center coordinates.
"""
return coordinates0_point(F.shape[:3],size,origin) + displacement_point(size,F)
[docs]def cellsSizeOrigin_coordinates0_point(coordinates0: _np.ndarray,
ordered: bool = True) -> _Tuple[_np.ndarray,_np.ndarray,_np.ndarray]:
"""
Return grid 'DNA', i.e. cells, size, and origin from 1D array of point positions.
Parameters
----------
coordinates0 : numpy.ndarray, shape (:,3)
Undeformed cell center coordinates.
ordered : bool, optional
Expect coordinates0 data to be ordered (x fast, z slow).
Defaults to True.
Returns
-------
cells, size, origin : Three numpy.ndarray, each of shape (3)
Information to reconstruct grid.
"""
coords = [_np.unique(coordinates0[:,i]) for i in range(3)]
mincorner = _np.array(list(map(min,coords)))
maxcorner = _np.array(list(map(max,coords)))
cells = _np.array(list(map(len,coords)),_np.int64)
size = cells/_np.maximum(cells-1,1) * (maxcorner-mincorner)
delta = size/cells
origin = mincorner - delta*.5
# 1D/2D: size/origin combination undefined, set origin to 0.0
size [_np.where(cells==1)] = origin[_np.where(cells==1)]*2.
origin[_np.where(cells==1)] = 0.0
if cells.prod() != len(coordinates0):
raise ValueError(f'data count {len(coordinates0)} does not match cells {cells}')
start = origin + delta*.5
end = origin - delta*.5 + size
atol = _np.max(size)*5e-2
if not (_np.allclose(coords[0],_np.linspace(start[0],end[0],cells[0]),atol=atol) and \
_np.allclose(coords[1],_np.linspace(start[1],end[1],cells[1]),atol=atol) and \
_np.allclose(coords[2],_np.linspace(start[2],end[2],cells[2]),atol=atol)):
raise ValueError('non-uniform cell spacing')
if ordered and not _np.allclose(coordinates0.reshape(tuple(cells)+(3,),order='F'),
coordinates0_point(list(cells),size,origin),atol=atol):
raise ValueError('input data is not ordered (x fast, z slow)')
return (cells,size,origin)
[docs]def coordinates0_node(cells: _IntSequence,
size: _FloatSequence,
origin: _FloatSequence = _np.zeros(3)) -> _np.ndarray:
"""
Nodal positions (undeformed).
Parameters
----------
cells : sequence of int, len (3)
Number of cells.
size : sequence of float, len (3)
Physical size of the periodic field.
origin : sequence of float, len(3), optional
Physical origin of the periodic field. Defaults to [0.0,0.0,0.0].
Returns
-------
x_n_0 : numpy.ndarray, shape (:,:,:,3)
Undeformed nodal coordinates.
"""
return _np.stack(_np.meshgrid(_np.linspace(origin[0],size[0]+origin[0],cells[0]+1),
_np.linspace(origin[1],size[1]+origin[1],cells[1]+1),
_np.linspace(origin[2],size[2]+origin[2],cells[2]+1),indexing = 'ij'),
axis = -1)
[docs]def displacement_fluct_node(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Nodal displacement field from fluctuation part of the deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_n_fluct : numpy.ndarray, shape (:,:,:,3)
Fluctuating part of the nodal displacements.
"""
return point_to_node(displacement_fluct_point(size,F))
[docs]def displacement_avg_node(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Nodal displacement field from average part of the deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_n_avg : numpy.ndarray, shape (:,:,:,3)
Average part of the nodal displacements.
"""
F_avg = _np.average(F,axis=(0,1,2))
return _np.einsum('ml,ijkl->ijkm',F_avg - _np.eye(3),coordinates0_node(F.shape[:3],size))
[docs]def displacement_node(size: _FloatSequence,
F: _np.ndarray) -> _np.ndarray:
"""
Nodal displacement field from deformation gradient field.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
Returns
-------
u_p : numpy.ndarray, shape (:,:,:,3)
Nodal displacements.
"""
return displacement_avg_node(size,F) + displacement_fluct_node(size,F)
[docs]def coordinates_node(size: _FloatSequence,
F: _np.ndarray,
origin: _FloatSequence = _np.zeros(3)) -> _np.ndarray:
"""
Nodal positions.
Parameters
----------
size : sequence of float, len (3)
Physical size of the periodic field.
F : numpy.ndarray, shape (:,:,:,3,3)
Deformation gradient field.
origin : sequence of float, len(3), optional
Physical origin of the periodic field. Defaults to [0.0,0.0,0.0].
Returns
-------
x_n : numpy.ndarray, shape (:,:,:,3)
Nodal coordinates.
"""
return coordinates0_node(F.shape[:3],size,origin) + displacement_node(size,F)
[docs]def cellsSizeOrigin_coordinates0_node(coordinates0: _np.ndarray,
ordered: bool = True) -> _Tuple[_np.ndarray,_np.ndarray,_np.ndarray]:
"""
Return grid 'DNA', i.e. cells, size, and origin from 1D array of nodal positions.
Parameters
----------
coordinates0 : numpy.ndarray, shape (:,3)
Undeformed nodal coordinates.
ordered : bool, optional
Expect coordinates0 data to be ordered (x fast, z slow).
Defaults to True.
Returns
-------
cells, size, origin : Three numpy.ndarray, each of shape (3)
Information to reconstruct grid.
"""
coords = [_np.unique(coordinates0[:,i]) for i in range(3)]
mincorner = _np.array(list(map(min,coords)))
maxcorner = _np.array(list(map(max,coords)))
cells = _np.array(list(map(len,coords)),_np.int64) - 1
size = maxcorner-mincorner
origin = mincorner
if (cells+1).prod() != len(coordinates0):
raise ValueError(f'data count {len(coordinates0)} does not match cells {cells}')
atol = _np.max(size)*5e-2
if not (_np.allclose(coords[0],_np.linspace(mincorner[0],maxcorner[0],cells[0]+1),atol=atol) and \
_np.allclose(coords[1],_np.linspace(mincorner[1],maxcorner[1],cells[1]+1),atol=atol) and \
_np.allclose(coords[2],_np.linspace(mincorner[2],maxcorner[2],cells[2]+1),atol=atol)):
raise ValueError('non-uniform cell spacing')
if ordered and not _np.allclose(coordinates0.reshape(tuple(cells+1)+(3,),order='F'),
coordinates0_node(list(cells),size,origin),atol=atol):
raise ValueError('input data is not ordered (x fast, z slow)')
return (cells,size,origin)
[docs]def point_to_node(cell_data: _np.ndarray) -> _np.ndarray:
"""
Interpolate periodic point data to nodal data.
Parameters
----------
cell_data : numpy.ndarray, shape (:,:,:,...)
Data defined on the cell centers of a periodic grid.
Returns
-------
node_data : numpy.ndarray, shape (:,:,:,...)
Data defined on the nodes of a periodic grid.
"""
n = ( cell_data + _np.roll(cell_data,1,(0,1,2))
+ _np.roll(cell_data,1,(0,)) + _np.roll(cell_data,1,(1,)) + _np.roll(cell_data,1,(2,))
+ _np.roll(cell_data,1,(0,1)) + _np.roll(cell_data,1,(1,2)) + _np.roll(cell_data,1,(2,0)))*0.125
return _np.pad(n,((0,1),(0,1),(0,1))+((0,0),)*len(cell_data.shape[3:]),mode='wrap')
[docs]def node_to_point(node_data: _np.ndarray) -> _np.ndarray:
"""
Interpolate periodic nodal data to point data.
Parameters
----------
node_data : numpy.ndarray, shape (:,:,:,...)
Data defined on the nodes of a periodic grid.
Returns
-------
cell_data : numpy.ndarray, shape (:,:,:,...)
Data defined on the cell centers of a periodic grid.
"""
c = ( node_data + _np.roll(node_data,1,(0,1,2))
+ _np.roll(node_data,1,(0,)) + _np.roll(node_data,1,(1,)) + _np.roll(node_data,1,(2,))
+ _np.roll(node_data,1,(0,1)) + _np.roll(node_data,1,(1,2)) + _np.roll(node_data,1,(2,0)))*0.125
return c[1:,1:,1:]
[docs]def coordinates0_valid(coordinates0: _np.ndarray) -> bool:
"""
Check whether coordinates form a regular grid.
Parameters
----------
coordinates0 : numpy.ndarray, shape (:,3)
Array of undeformed cell coordinates.
Returns
-------
valid : bool
Whether the coordinates form a regular grid.
"""
try:
cellsSizeOrigin_coordinates0_point(coordinates0,ordered=True)
return True
except ValueError:
return False
[docs]def regrid(size: _FloatSequence,
F: _np.ndarray,
cells: _IntSequence) -> _np.ndarray:
"""
Return mapping from coordinates in deformed configuration to a regular grid.
Parameters
----------
size : sequence of float, len (3)
Physical size.
F : numpy.ndarray, shape (:,:,:,3,3), shape (:,:,:,3,3)
Deformation gradient field.
cells : sequence of int, len (3)
Cell count along x,y,z of remapping grid.
"""
c = coordinates_point(size,F)
outer = _np.dot(_np.average(F,axis=(0,1,2)),size)
for d in range(3):
c[_np.where(c[:,:,:,d]<0)] += outer[d]
c[_np.where(c[:,:,:,d]>outer[d])] -= outer[d]
tree = _spatial.cKDTree(c.reshape(-1,3),boxsize=outer)
return tree.query(coordinates0_point(cells,outer))[1].flatten()