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