Source code for damask.grid_filters

"""
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' if n == 3 else 'slm,ijkl,ijknm->ijksn',e,k_s,f_fourier)*2.0j*_np.pi) # vector 3->3, 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)) divergence_ = (_np.einsum('ijkl,ijkl ->ijk' if n == 3 else 'ijkm,ijklm->ijkl', k_s,f_fourier)*2.0j*_np.pi) # vector 3->1, tensor 3x3->3 return _np.fft.irfftn(divergence_,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) Gradient 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)) gradient_ = (_np.einsum('ijkl,ijkm->ijkm' if n == 1 else 'ijkl,ijkm->ijklm',f_fourier,k_s)*2.0j*_np.pi) # scalar 1->3, vector 3->3x3 return _np.fft.irfftn(gradient_,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. """ 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, _np.array([0.5j/_np.pi]*3), ) / 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_n : 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 unravel_index(idx: _np.ndarray) -> _np.ndarray: """ Convert flat indices to coordinate indices. Parameters ---------- idx : numpy.ndarray, shape (:,:,:) Grid of flat indices. Returns ------- unravelled : numpy.ndarray, shape (:,:,:,3) Grid of coordinate indices. Examples -------- Unravel a linearly increasing sequence of material indices on a 3 × 2 × 1 grid. >>> import numpy as np >>> import damask >>> seq = np.arange(6).reshape((3,2,1),order='F') >>> (coord_idx := damask.grid_filters.unravel_index(seq)) array([[[[0, 0, 0]], [[0, 1, 0]]], [[[1, 0, 0]], [[1, 1, 0]]], [[[2, 0, 0]], [[2, 1, 0]]]]) >>> coord_idx[1,1,0] array([1, 1, 0]) """ cells = idx.shape idx_ = _np.expand_dims(idx,3) return _np.block([ idx_ %cells[0], (idx_//cells[0]) %cells[1], ((idx_//cells[0])//cells[1])%cells[2]])
[docs]def ravel_index(idx: _np.ndarray) -> _np.ndarray: """ Convert coordinate indices to flat indices. Parameters ---------- idx : numpy.ndarray, shape (:,:,:,3) Grid of coordinate indices. Returns ------- ravelled : numpy.ndarray, shape (:,:,:) Grid of flat indices. Examples -------- Ravel a reversed sequence of coordinate indices on a 2 × 2 × 1 grid. >>> import numpy as np >>> import damask >>> (rev := np.array([[1,1,0],[0,1,0],[1,0,0],[0,0,0]]).reshape((2,2,1,3))) array([[[[1, 1, 0]], [[0, 1, 0]]], [[[1, 0, 0]], [[0, 0, 0]]]]) >>> (flat_idx := damask.grid_filters.ravel_index(rev)) array([[[3], [2]], [[1], [0]]]) """ cells = idx.shape[:3] return idx[:,:,:,0] \ + idx[:,:,:,1]*cells[0] \ + idx[:,:,:,2]*cells[0]*cells[1]
[docs]def regrid(size: _FloatSequence, F: _np.ndarray, cells: _IntSequence) -> _np.ndarray: """ Map a deformed grid A back to a rectilinear grid B. The size of grid B is chosen as the average deformed size of grid A. Parameters ---------- size : sequence of float, len (3) Physical size of grid A. F : numpy.ndarray, shape (:,:,:,3,3) Deformation gradient field on grid A. cells : sequence of int, len (3) Cell count along x,y,z of grid B. Returns ------- idx : numpy.ndarray of int, shape (cells) Flat index of closest point on deformed grid A for each point on grid B. """ box = _np.dot(_np.average(F,axis=(0,1,2)),size) c = coordinates_point(size,F)%box tree = _spatial.cKDTree(c.reshape((-1,3),order='F'),boxsize=box) return tree.query(coordinates0_point(cells,box))[1]