DAMASK_EICMD/python/damask/_grid.py

import os
import copy
import warnings
import multiprocessing as mp
from functools import partial

import numpy as np
import pandas as pd
import h5py
from scipy import ndimage, spatial

from . import VTK
from . import util
from . import grid_filters
from . import Rotation


class Grid:
    """
    Geometry definition for grid solvers.

    Create and manipulate geometry definitions for storage as VTK
    image data files ('.vti' extension). A grid contains the
    material ID (referring to the entry in 'material.yaml') and
    the physical size.
    """

    def __init__(self,material,size,origin=[0.0,0.0,0.0],comments=[]):
        """
        New geometry definition for grid solvers.

        Parameters
        ----------
        material : numpy.ndarray of shape (:,:,:)
            Material indices. The shape of the material array defines
            the number of cells.
        size : list or numpy.ndarray of shape (3)
            Physical size of grid in meter.
        origin : list or numpy.ndarray of shape (3), optional
            Coordinates of grid origin in meter.
        comments : list of str, optional
            Comments, e.g. history of operations.

        """
        self.material = material
        self.size     = size
        self.origin   = origin
        self.comments = comments


    def __repr__(self):
        """Basic information on grid definition."""
        mat_min = np.nanmin(self.material)
        mat_max = np.nanmax(self.material)
        mat_N   = self.N_materials
        return util.srepr([
               f'cells  a b c: {util.srepr(self.cells, " x ")}',
               f'size   x y z: {util.srepr(self.size,  " x ")}',
               f'origin x y z: {util.srepr(self.origin,"   ")}',
               f'# materials: {mat_N}' + ('' if mat_min == 0 and mat_max+1 == mat_N else
                                          f' (min: {mat_min}, max: {mat_max})')
              ])


    def __copy__(self):
        """Create deep copy."""
        return copy.deepcopy(self)

    copy = __copy__


    def __eq__(self,other):
        """
        Test equality of other.

        Parameters
        ----------
        other : damask.Grid
            Grid to compare self against.

        """
        return (np.allclose(other.size,self.size)
            and np.allclose(other.origin,self.origin)
            and np.all(other.cells == self.cells)
            and np.all(other.material == self.material))


    @property
    def material(self):
        """Material indices."""
        return self._material

    @material.setter
    def material(self,material):
        if len(material.shape) != 3:
            raise ValueError(f'invalid material shape {material.shape}')
        elif material.dtype not in np.sctypes['float'] + np.sctypes['int']:
            raise TypeError(f'invalid material data type {material.dtype}')
        else:
            self._material = np.copy(material)

            if self.material.dtype in np.sctypes['float'] and \
               np.all(self.material == self.material.astype(int).astype(float)):
                self._material = self.material.astype(int)


    @property
    def size(self):
        """Physical size of grid in meter."""
        return self._size

    @size.setter
    def size(self,size):
        if len(size) != 3 or any(np.array(size) < 0):
            raise ValueError(f'invalid size {size}')
        else:
            self._size = np.array(size)

    @property
    def origin(self):
        """Coordinates of grid origin in meter."""
        return self._origin

    @origin.setter
    def origin(self,origin):
        if len(origin) != 3:
            raise ValueError(f'invalid origin {origin}')
        else:
            self._origin = np.array(origin)

    @property
    def comments(self):
        """Comments, e.g. history of operations."""
        return self._comments

    @comments.setter
    def comments(self,comments):
        self._comments = [str(c) for c in comments] if isinstance(comments,list) else [str(comments)]


    @property
    def cells(self):
        """Number of cells in x,y,z direction."""
        return np.asarray(self.material.shape)


    @property
    def N_materials(self):
        """Number of (unique) material indices within grid."""
        return np.unique(self.material).size


    @staticmethod
    def load(fname):
        """
        Load from VTK image data file.

        Parameters
        ----------
        fname : str or or pathlib.Path
            Grid file to read. Valid extension is .vti, which will be appended
            if not given.

        Returns
        -------
        loaded : damask.Grid
            Grid-based geometry from file.

        """
        v = VTK.load(fname if str(fname).endswith(('.vti','.vtr')) else str(fname)+'.vti')          # compatibility hack
        comments = v.get_comments()
        cells = np.array(v.vtk_data.GetDimensions())-1
        bbox  = np.array(v.vtk_data.GetBounds()).reshape(3,2).T

        return Grid(material = v.get('material').reshape(cells,order='F'),
                    size = bbox[1] - bbox[0],
                    origin = bbox[0],
                    comments=comments)


    @staticmethod
    def load_ASCII(fname):
        """
        Load from geom file.

        Storing geometry files in ASCII format is deprecated.
        This function will be removed in a future version of DAMASK.

        Parameters
        ----------
        fname : str, pathlib.Path, or file handle
            Geometry file to read.

        Returns
        -------
        loaded : damask.Grid
            Grid-based geometry from file.

        """
        warnings.warn('Support for ASCII-based geom format will be removed in DAMASK 3.1.0', DeprecationWarning,2)
        try:
            f = open(fname)
        except TypeError:
            f = fname

        f.seek(0)
        try:
            header_length,keyword = f.readline().split()[:2]
            header_length = int(header_length)
        except ValueError:
            header_length,keyword = (-1, 'invalid')
        if not keyword.startswith('head') or header_length < 3:
            raise TypeError('header length information missing or invalid')

        comments = []
        content = f.readlines()
        for i,line in enumerate(content[:header_length]):
            items = line.split('#')[0].lower().strip().split()
            key = items[0] if items else ''
            if   key == 'grid':
                cells  = np.array([  int(dict(zip(items[1::2],items[2::2]))[i]) for i in ['a','b','c']])
            elif key == 'size':
                size   = np.array([float(dict(zip(items[1::2],items[2::2]))[i]) for i in ['x','y','z']])
            elif key == 'origin':
                origin = np.array([float(dict(zip(items[1::2],items[2::2]))[i]) for i in ['x','y','z']])
            else:
                comments.append(line.strip())

        material = np.empty(cells.prod())                                                           # initialize as flat array
        i = 0
        for line in content[header_length:]:
            items = line.split('#')[0].split()
            if len(items) == 3:
                if   items[1].lower() == 'of':
                    items = np.ones(int(items[0]))*float(items[2])
                elif items[1].lower() == 'to':
                    items = np.linspace(int(items[0]),int(items[2]),
                                        abs(int(items[2])-int(items[0]))+1,dtype=float)
                else:                        items = list(map(float,items))
            else:                            items = list(map(float,items))
            material[i:i+len(items)] = items
            i += len(items)

        if i != cells.prod():
            raise TypeError(f'invalid file: expected {cells.prod()} entries, found {i}')

        if not np.any(np.mod(material,1) != 0.0):                                                   # no float present
            material = material.astype('int') - (1 if material.min() > 0 else 0)

        return Grid(material.reshape(cells,order='F'),size,origin,comments)


    @staticmethod
    def load_Neper(fname):
        """
        Load from Neper VTK file.

        Parameters
        ----------
        fname : str, pathlib.Path, or file handle
            Geometry file to read.

        Returns
        -------
        loaded : damask.Grid
            Grid-based geometry from file.

        """
        v = VTK.load(fname,'vtkImageData')
        cells = np.array(v.vtk_data.GetDimensions())-1
        bbox  = np.array(v.vtk_data.GetBounds()).reshape(3,2).T

        return Grid(v.get('MaterialId').reshape(cells,order='F') - 1, bbox[1] - bbox[0], bbox[0],
                    util.execution_stamp('Grid','load_Neper'))


    @staticmethod
    def load_DREAM3D(fname,
                     feature_IDs=None,cell_data=None,
                     phases='Phases',Euler_angles='EulerAngles',
                     base_group=None):
        """
        Load DREAM.3D (HDF5) file.

        Data in DREAM.3D files can be stored per cell ('CellData') and/or
        per grain ('Grain Data'). Per default, cell-wise data is assumed.

        damask.ConfigMaterial.load_DREAM3D gives the corresponding material definition.

        Parameters
        ----------
        fname : str
            Filename of the DREAM.3D (HDF5) file.
        feature_IDs : str
            Name of the dataset containing the mapping between cells and
            grain-wise data. Defaults to 'None', in which case cell-wise
            data is used.
        cell_data : str
            Name of the group (folder) containing cell-wise data. Defaults to
            None in wich case it is automatically detected.
        phases : str
            Name of the dataset containing the phase ID. It is not used for
            grain-wise data, i.e. when feature_IDs is not None.
            Defaults to 'Phases'.
        Euler_angles : str
            Name of the dataset containing the crystallographic orientation as
            Euler angles in radians It is not used for grain-wise data, i.e.
            when feature_IDs is not None. Defaults to 'EulerAngles'.
        base_group : str
            Path to the group (folder) that contains geometry (_SIMPL_GEOMETRY),
            and grain- or cell-wise data. Defaults to None, in which case
            it is set as the path that contains _SIMPL_GEOMETRY/SPACING.

        Returns
        -------
        loaded : damask.Grid
            Grid-based geometry from file.

        """
        b = util.DREAM3D_base_group(fname)      if base_group is None else base_group
        c = util.DREAM3D_cell_data_group(fname) if cell_data  is None else cell_data
        f = h5py.File(fname, 'r')

        cells  = f['/'.join([b,'_SIMPL_GEOMETRY','DIMENSIONS'])][()]
        size   = f['/'.join([b,'_SIMPL_GEOMETRY','SPACING'])] * cells
        origin = f['/'.join([b,'_SIMPL_GEOMETRY','ORIGIN'])][()]

        if feature_IDs is None:
            phase = f['/'.join([b,c,phases])][()].reshape(-1,1)
            O = Rotation.from_Euler_angles(f['/'.join([b,c,Euler_angles])]).as_quaternion().reshape(-1,4) # noqa
            unique,unique_inverse = np.unique(np.hstack([O,phase]),return_inverse=True,axis=0)
            ma = np.arange(cells.prod()) if len(unique) == cells.prod() else \
                 np.arange(unique.size)[np.argsort(pd.unique(unique_inverse))][unique_inverse]
        else:
            ma = f['/'.join([b,c,feature_IDs])][()].flatten()

        return Grid(ma.reshape(cells,order='F'),size,origin,util.execution_stamp('Grid','load_DREAM3D'))


    @staticmethod
    def from_table(table,coordinates,labels):
        """
        Create grid from ASCII table.

        Parameters
        ----------
        table : damask.Table
            Table that contains material information.
        coordinates : str
            Label of the vector column containing the spatial coordinates.
            Need to be ordered (1./x fast, 3./z slow).
        labels : str or list of str
            Label(s) of the columns containing the material definition.
            Each unique combination of values results in one material ID.

        Returns
        -------
        new : damask.Grid
            Grid-based geometry from values in table.

        """
        cells,size,origin = grid_filters.cellsSizeOrigin_coordinates0_point(table.get(coordinates))

        labels_ = [labels] if isinstance(labels,str) else labels
        unique,unique_inverse = np.unique(np.hstack([table.get(l) for l in labels_]),return_inverse=True,axis=0)

        ma = np.arange(cells.prod()) if len(unique) == cells.prod() else \
             np.arange(unique.size)[np.argsort(pd.unique(unique_inverse))][unique_inverse]

        return Grid(ma.reshape(cells,order='F'),size,origin,util.execution_stamp('Grid','from_table'))


    @staticmethod
    def _find_closest_seed(seeds, weights, point):
        return np.argmin(np.sum((np.broadcast_to(point,(len(seeds),3))-seeds)**2,axis=1) - weights)

    @staticmethod
    def from_Laguerre_tessellation(cells,size,seeds,weights,material=None,periodic=True):
        """
        Create grid from Laguerre tessellation.

        Parameters
        ----------
        cells : int numpy.ndarray of shape (3)
            Number of cells in x,y,z direction.
        size : list or numpy.ndarray of shape (3)
            Physical size of the grid in meter.
        seeds : numpy.ndarray of shape (:,3)
            Position of the seed points in meter. All points need to lay within the box.
        weights : numpy.ndarray of shape (seeds.shape[0])
            Weights of the seeds. Setting all weights to 1.0 gives a standard Voronoi tessellation.
        material : numpy.ndarray of shape (seeds.shape[0]), optional
            Material ID of the seeds.
            Defaults to None, in which case materials are consecutively numbered.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        new : damask.Grid
            Grid-based geometry from tessellation.

        """
        if periodic:
            weights_p = np.tile(weights,27)                                                         # Laguerre weights (1,2,3,1,2,3,...,1,2,3)
            seeds_p = np.vstack((seeds  -np.array([size[0],0.,0.]),seeds,  seeds  +np.array([size[0],0.,0.])))
            seeds_p = np.vstack((seeds_p-np.array([0.,size[1],0.]),seeds_p,seeds_p+np.array([0.,size[1],0.])))
            seeds_p = np.vstack((seeds_p-np.array([0.,0.,size[2]]),seeds_p,seeds_p+np.array([0.,0.,size[2]])))
        else:
            weights_p = weights
            seeds_p   = seeds

        coords = grid_filters.coordinates0_point(cells,size).reshape(-1,3)

        pool = mp.Pool(int(os.environ.get('OMP_NUM_THREADS',4)))
        result = pool.map_async(partial(Grid._find_closest_seed,seeds_p,weights_p), coords)
        pool.close()
        pool.join()
        material_ = np.array(result.get()).reshape(cells)

        if periodic: material_ %= len(weights)

        return Grid(material = material_ if material is None else material[material_],
                    size     = size,
                    comments = util.execution_stamp('Grid','from_Laguerre_tessellation'),
                   )


    @staticmethod
    def from_Voronoi_tessellation(cells,size,seeds,material=None,periodic=True):
        """
        Create grid from Voronoi tessellation.

        Parameters
        ----------
        cells : int numpy.ndarray of shape (3)
            Number of cells in x,y,z direction.
        size : list or numpy.ndarray of shape (3)
            Physical size of the grid in meter.
        seeds : numpy.ndarray of shape (:,3)
            Position of the seed points in meter. All points need to lay within the box.
        material : numpy.ndarray of shape (seeds.shape[0]), optional
            Material ID of the seeds.
            Defaults to None, in which case materials are consecutively numbered.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        new : damask.Grid
            Grid-based geometry from tessellation.

        """
        coords = grid_filters.coordinates0_point(cells,size).reshape(-1,3)
        tree = spatial.cKDTree(seeds,boxsize=size) if periodic else \
               spatial.cKDTree(seeds)
        try:
            material_ = tree.query(coords, workers = int(os.environ.get('OMP_NUM_THREADS',4)))[1]
        except TypeError:
            material_ = tree.query(coords, n_jobs = int(os.environ.get('OMP_NUM_THREADS',4)))[1]    # scipy <1.6

        return Grid(material = (material_ if material is None else material[material_]).reshape(cells),
                    size     = size,
                    comments = util.execution_stamp('Grid','from_Voronoi_tessellation'),
                   )


    _minimal_surface = \
        {'Schwarz P':        lambda x,y,z: np.cos(x) + np.cos(y) + np.cos(z),
         'Double Primitive': lambda x,y,z: ( 0.5 * (np.cos(x)*np.cos(y) + np.cos(y)*np.cos(z) + np.cos(z)*np.cos(x))
                                           + 0.2 * (np.cos(2*x) + np.cos(2*y) + np.cos(2*z)) ),
         'Schwarz D':        lambda x,y,z: ( np.sin(x)*np.sin(y)*np.sin(z)
                                           + np.sin(x)*np.cos(y)*np.cos(z)
                                           + np.cos(x)*np.cos(y)*np.sin(z)
                                           + np.cos(x)*np.sin(y)*np.cos(z) ),
         'Complementary D':  lambda x,y,z: ( np.cos(3*x+y)*np.cos(z) - np.sin(3*x-y)*np.sin(z) + np.cos(x+3*y)*np.cos(z)
                                           + np.sin(x-3*y)*np.sin(z) + np.cos(x-y)*np.cos(3*z) - np.sin(x+y)*np.sin(3*z) ),
         'Double Diamond':   lambda x,y,z: 0.5 * (np.sin(x)*np.sin(y)
                                                + np.sin(y)*np.sin(z)
                                                + np.sin(z)*np.sin(x)
                                                + np.cos(x) * np.cos(y) * np.cos(z) ),
         'Dprime':           lambda x,y,z: 0.5 * ( np.cos(x)*np.cos(y)*np.cos(z)
                                                 + np.cos(x)*np.sin(y)*np.sin(z)
                                                 + np.sin(x)*np.cos(y)*np.sin(z)
                                                 + np.sin(x)*np.sin(y)*np.cos(z)
                                                 - np.sin(2*x)*np.sin(2*y)
                                                 - np.sin(2*y)*np.sin(2*z)
                                                 - np.sin(2*z)*np.sin(2*x) ) - 0.2,
         'Gyroid':           lambda x,y,z: np.cos(x)*np.sin(y) + np.cos(y)*np.sin(z) + np.cos(z)*np.sin(x),
         'Gprime':           lambda x,y,z : ( np.sin(2*x)*np.cos(y)*np.sin(z)
                                            + np.sin(2*y)*np.cos(z)*np.sin(x)
                                            + np.sin(2*z)*np.cos(x)*np.sin(y) ) + 0.32,
         'Karcher K':        lambda x,y,z: ( 0.3 * ( np.cos(x)           + np.cos(y)           + np.cos(z)
                                                   + np.cos(x)*np.cos(y) + np.cos(y)*np.cos(z) + np.cos(z)*np.cos(x) )
                                           - 0.4 * ( np.cos(2*x)         + np.cos(2*y)         + np.cos(2*z) ) ) + 0.2,
         'Lidinoid':         lambda x,y,z: 0.5 * ( np.sin(2*x)*np.cos(y)*np.sin(z)
                                                 + np.sin(2*y)*np.cos(z)*np.sin(x)
                                                 + np.sin(2*z)*np.cos(x)*np.sin(y)
                                                 - np.cos(2*x)*np.cos(2*y)
                                                 - np.cos(2*y)*np.cos(2*z)
                                                 - np.cos(2*z)*np.cos(2*x) ) + 0.15,
         'Neovius':          lambda x,y,z: ( 3 * (np.cos(x)+np.cos(y)+np.cos(z))
                                           + 4 * np.cos(x)*np.cos(y)*np.cos(z) ),
         'Fisher-Koch S':    lambda x,y,z: ( np.cos(2*x)*np.sin(  y)*np.cos(  z)
                                           + np.cos(  x)*np.cos(2*y)*np.sin(  z)
                                           + np.sin(  x)*np.cos(  y)*np.cos(2*z) ),
      }


    @staticmethod
    def from_minimal_surface(cells,size,surface,threshold=0.0,periods=1,materials=(0,1)):
        """
        Create grid from definition of triply periodic minimal surface.

        Parameters
        ----------
        cells : int numpy.ndarray of shape (3)
            Number of cells in x,y,z direction.
        size : list or numpy.ndarray of shape (3)
            Physical size of the grid in meter.
        surface : str
            Type of the minimal surface. See notes for details.
        threshold : float, optional.
            Threshold of the minimal surface. Defaults to 0.0.
        periods : integer, optional.
            Number of periods per unit cell. Defaults to 1.
        materials : (int, int), optional
            Material IDs. Defaults to (0,1).

        Returns
        -------
        new : damask.Grid
            Grid-based geometry from definition of minimal surface.

        Notes
        -----
        The following triply-periodic minimal surfaces are implemented:
          - Schwarz P
          - Double Primitive
          - Schwarz D
          - Complementary D
          - Double Diamond
          - Dprime
          - Gyroid
          - Gprime
          - Karcher K
          - Lidinoid
          - Neovius
          - Fisher-Koch S

        References
        ----------
        S.B.G. Blanquer et al., Biofabrication 9(2):025001, 2017
        https://doi.org/10.1088/1758-5090/aa6553

        M. Wohlgemuth et al., Macromolecules 34(17):6083-6089, 2001
        https://doi.org/10.1021/ma0019499

        M.-T. Hsieh and L. Valdevit, Software Impacts 6:100026, 2020
        https://doi.org/10.1016/j.simpa.2020.100026

        Examples
        --------
        Minimal surface of 'Gyroid' type.

        >>> import numpy as np
        >>> import damask
        >>> damask.Grid.from_minimal_surface(np.array([64]*3,int),np.ones(3),
        ...                                  'Gyroid')
        cells  a b c: 64 x 64 x 64
        size   x y z: 1.0 x 1.0 x 1.0
        origin x y z: 0.0   0.0   0.0
        # materials: 2

        Minimal surface of 'Neovius' type. non-default material IDs.

        >>> import numpy as np
        >>> import damask
        >>> damask.Grid.from_minimal_surface(np.array([80]*3,int),np.ones(3),
        ...                                  'Neovius',materials=(1,5))
        cells  a b c: 80 x 80 x 80
        size   x y z: 1.0 x 1.0 x 1.0
        origin x y z: 0.0   0.0   0.0
        # materials: 2 (min: 1, max: 5)

        """
        x,y,z = np.meshgrid(periods*2.0*np.pi*(np.arange(cells[0])+0.5)/cells[0],
                            periods*2.0*np.pi*(np.arange(cells[1])+0.5)/cells[1],
                            periods*2.0*np.pi*(np.arange(cells[2])+0.5)/cells[2],
                            indexing='ij',sparse=True)
        return Grid(material = np.where(threshold < Grid._minimal_surface[surface](x,y,z),materials[1],materials[0]),
                    size     = size,
                    comments = util.execution_stamp('Grid','from_minimal_surface'),
                   )


    def save(self,fname,compress=True):
        """
        Save as VTK image data file.

        Parameters
        ----------
        fname : str or pathlib.Path
            Filename to write. Valid extension is .vti, it will be appended if not given.
        compress : bool, optional
            Compress with zlib algorithm. Defaults to True.

        """
        v = VTK.from_image_data(self.cells,self.size,self.origin)
        v.add(self.material.flatten(order='F'),'material')
        v.add_comments(self.comments)

        v.save(fname if str(fname).endswith('.vti') else str(fname)+'.vti',parallel=False,compress=compress)


    def save_ASCII(self,fname):
        """
        Save as geom file.

        Storing geometry files in ASCII format is deprecated.
        This function will be removed in a future version of DAMASK.

        Parameters
        ----------
        fname : str or file handle
            Geometry file to write with extension '.geom'.
        compress : bool, optional
            Compress geometry with 'x of y' and 'a to b'.

        """
        warnings.warn('Support for ASCII-based geom format will be removed in DAMASK 3.1.0', DeprecationWarning,2)
        header =  [f'{len(self.comments)+4} header'] + self.comments \
                + ['grid   a {} b {} c {}'.format(*self.cells),
                   'size   x {} y {} z {}'.format(*self.size),
                   'origin x {} y {} z {}'.format(*self.origin),
                   'homogenization 1',
                  ]

        format_string = '%g' if self.material.dtype in np.sctypes['float'] else \
                        '%{}i'.format(1+int(np.floor(np.log10(np.nanmax(self.material)))))
        np.savetxt(fname,
                   self.material.reshape([self.cells[0],np.prod(self.cells[1:])],order='F').T,
                   header='\n'.join(header), fmt=format_string, comments='')


    def show(self):
        """Show on screen."""
        VTK.from_rectilinear_grid(self.cells,self.size,self.origin).show()


    def add_primitive(self,dimension,center,exponent,
                      fill=None,R=Rotation(),inverse=False,periodic=True):
        """
        Insert a primitive geometric object at a given position.

        Parameters
        ----------
        dimension : int or float numpy.ndarray of shape (3)
            Dimension (diameter/side length) of the primitive. If given as
            integers, cell centers are addressed.
            If given as floats, coordinates are addressed.
        center : int or float numpy.ndarray of shape (3)
            Center of the primitive. If given as integers, cell centers are addressed.
            If given as floats, coordinates in space are addressed.
        exponent : numpy.ndarray of shape (3) or float
            Exponents for the three axes.
            0 gives octahedron (ǀxǀ^(2^0) + ǀyǀ^(2^0) + ǀzǀ^(2^0) < 1)
            1 gives sphere     (ǀxǀ^(2^1) + ǀyǀ^(2^1) + ǀzǀ^(2^1) < 1)
        fill : int, optional
            Fill value for primitive. Defaults to material.max()+1.
        R : damask.Rotation, optional
            Rotation of primitive. Defaults to no rotation.
        inverse : Boolean, optional
            Retain original materials within primitive and fill outside.
            Defaults to False.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        Examples
        --------
        Add a sphere at the center.

        >>> import numpy as np
        >>> import damask
        >>> g = damask.Grid(np.zeros([64]*3,int), np.ones(3)*1e-4)
        >>> g.add_primitive(np.ones(3)*5e-5,np.ones(3)*5e-5,1)
        cells  a b c: 64 x 64 x 64
        size   x y z: 0.0001 x 0.0001 x 0.0001
        origin x y z: 0.0   0.0   0.0
        # materials: 2

        Add a cube at the origin.

        >>> import numpy as np
        >>> import damask
        >>> g = damask.Grid(np.zeros([64]*3,int), np.ones(3)*1e-4)
        >>> g.add_primitive(np.ones(3,int)*32,np.zeros(3),np.inf)
        cells  a b c: 64 x 64 x 64
        size   x y z: 0.0001 x 0.0001 x 0.0001
        origin x y z: 0.0   0.0   0.0
        # materials: 2

        """
        # radius and center
        r = np.array(dimension)/2.0*self.size/self.cells if np.array(dimension).dtype in np.sctypes['int'] else \
            np.array(dimension)/2.0
        c = (np.array(center) + .5)*self.size/self.cells if np.array(center).dtype    in np.sctypes['int'] else \
            (np.array(center) - self.origin)

        coords = grid_filters.coordinates0_point(self.cells,self.size,
                                          -(0.5*(self.size + (self.size/self.cells
                                                              if np.array(center).dtype in np.sctypes['int'] else
                                                              0)) if periodic else c))
        coords_rot = R.broadcast_to(tuple(self.cells))@coords

        with np.errstate(all='ignore'):
            mask = np.sum(np.power(coords_rot/r,2.0**np.array(exponent)),axis=-1) > 1.0

        if periodic:                                                                                # translate back to center
            mask = np.roll(mask,((c/self.size-0.5)*self.cells).round().astype(int),(0,1,2))

        return Grid(material = np.where(np.logical_not(mask) if inverse else mask,
                                        self.material,
                                        np.nanmax(self.material)+1 if fill is None else fill),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','add_primitive')],
                   )


    def mirror(self,directions,reflect=False):
        """
        Mirror grid along given directions.

        Parameters
        ----------
        directions : iterable containing str
            Direction(s) along which the grid is mirrored.
            Valid entries are 'x', 'y', 'z'.
        reflect : bool, optional
            Reflect (include) outermost layers. Defaults to False.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        Examples
        --------
        Mirror along x- and y-direction.

        >>> import numpy as np
        >>> import damask
        >>> g = damask.Grid(np.zeros([32]*3,int), np.ones(3)*1e-4)
        >>> g.mirror('xy',True)
        cells  a b c: 64 x 64 x 32
        size   x y z: 0.0002 x 0.0002 x 0.0001
        origin x y z: 0.0   0.0   0.0
        # materials: 1

        """
        valid = ['x','y','z']
        if not set(directions).issubset(valid):
            raise ValueError(f'invalid direction {set(directions).difference(valid)} specified')

        limits = [None,None] if reflect else [-2,0]
        mat = self.material.copy()

        if 'x' in directions:
            mat = np.concatenate([mat,mat[limits[0]:limits[1]:-1,:,:]],0)
        if 'y' in directions:
            mat = np.concatenate([mat,mat[:,limits[0]:limits[1]:-1,:]],1)
        if 'z' in directions:
            mat = np.concatenate([mat,mat[:,:,limits[0]:limits[1]:-1]],2)

        return Grid(material = mat,
                    size     = self.size/self.cells*np.asarray(mat.shape),
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','mirror')],
                   )


    def flip(self,directions):
        """
        Flip grid along given directions.

        Parameters
        ----------
        directions : iterable containing str
            Direction(s) along which the grid is flipped.
            Valid entries are 'x', 'y', 'z'.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        valid = ['x','y','z']
        if not set(directions).issubset(valid):
            raise ValueError(f'invalid direction {set(directions).difference(valid)} specified')

        mat = np.flip(self.material, (valid.index(d) for d in directions if d in valid))

        return Grid(material = mat,
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','flip')],
                   )


    def scale(self,cells,periodic=True):
        """
        Scale grid to new cells.

        Parameters
        ----------
        cells : numpy.ndarray of shape (3)
            Number of cells in x,y,z direction.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        Examples
        --------
        Double resolution.

        >>> import numpy as np
        >>> import damask
        >>> g = damask.Grid(np.zeros([32]*3,int),np.ones(3)*1e-4)
        >>> g.scale(g.cells*2)
        cells  a b c: 64 x 64 x 64
        size   x y z: 0.0001 x 0.0001 x 0.0001
        origin x y z: 0.0   0.0   0.0
        # materials: 1

        """
        return Grid(material = ndimage.interpolation.zoom(
                                                           self.material,
                                                           cells/self.cells,
                                                           output=self.material.dtype,
                                                           order=0,
                                                           mode=('wrap' if periodic else 'nearest'),
                                                           prefilter=False
                                                          ),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','scale')],
                   )


    def clean(self,stencil=3,selection=None,periodic=True):
        """
        Smooth grid by selecting most frequent material index within given stencil at each location.

        Parameters
        ----------
        stencil : int, optional
            Size of smoothing stencil.
        selection : list, optional
            Field values that can be altered. Defaults to all.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        def mostFrequent(arr,selection=None):
            me = arr[arr.size//2]
            if selection is None or me in selection:
                unique, inverse = np.unique(arr, return_inverse=True)
                return unique[np.argmax(np.bincount(inverse))]
            else:
                return me

        return Grid(material = ndimage.filters.generic_filter(
                                                               self.material,
                                                               mostFrequent,
                                                               size=(stencil if selection is None else stencil//2*2+1,)*3,
                                                               mode=('wrap' if periodic else 'nearest'),
                                                               extra_keywords=dict(selection=selection),
                                                              ).astype(self.material.dtype),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','clean')],
                   )


    def renumber(self):
        """
        Renumber sorted material indices as 0,...,N-1.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        _,renumbered = np.unique(self.material,return_inverse=True)

        return Grid(material = renumbered.reshape(self.cells),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','renumber')],
                   )


    def rotate(self,R,fill=None):
        """
        Rotate grid (pad if required).

        Parameters
        ----------
        R : damask.Rotation
            Rotation to apply to the grid.
        fill : int or float, optional
            Material index to fill the corners. Defaults to material.max() + 1.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        if fill is None: fill = np.nanmax(self.material) + 1
        dtype = float if isinstance(fill,float) or self.material.dtype in np.sctypes['float'] else int

        material = self.material
        # These rotations are always applied in the reference coordinate system, i.e. (z,x,z) not (z,x',z'')
        # see https://www.cs.utexas.edu/~theshark/courses/cs354/lectures/cs354-14.pdf
        for angle,axes in zip(R.as_Euler_angles(degrees=True)[::-1], [(0,1),(1,2),(0,1)]):
            material_temp = ndimage.rotate(material,angle,axes,order=0,prefilter=False,output=dtype,cval=fill)
            # avoid scipy interpolation errors for rotations close to multiples of 90°
            material = material_temp if np.prod(material_temp.shape) != np.prod(material.shape) else \
                       np.rot90(material,k=np.rint(angle/90.).astype(int),axes=axes)

        origin = self.origin-(np.asarray(material.shape)-self.cells)*.5 * self.size/self.cells

        return Grid(material = material,
                    size     = self.size/self.cells*np.asarray(material.shape),
                    origin   = origin,
                    comments = self.comments+[util.execution_stamp('Grid','rotate')],
                   )


    def canvas(self,cells=None,offset=None,fill=None):
        """
        Crop or enlarge/pad grid.

        Parameters
        ----------
        cells : numpy.ndarray of shape (3)
            Number of cells  x,y,z direction.
        offset : numpy.ndarray of shape (3)
            Offset (measured in cells) from old to new grid [0,0,0].
        fill : int or float, optional
            Material index to fill the background. Defaults to material.max() + 1.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        Examples
        --------
        Remove 1/2 of the microstructure in z-direction.

        >>> import numpy as np
        >>> import damask
        >>> g = damask.Grid(np.zeros([32]*3,int),np.ones(3)*1e-4)
        >>> g.canvas(np.array([32,32,16],int))
        cells  a b c: 33 x 32 x 16
        size   x y z: 0.0001 x 0.0001 x 5e-05
        origin x y z: 0.0   0.0   0.0
        # materials: 1

        """
        if offset is None: offset = 0
        if fill is None: fill = np.nanmax(self.material) + 1
        dtype = float if int(fill) != fill or self.material.dtype in np.sctypes['float'] else int

        canvas = np.full(self.cells if cells is None else cells,fill,dtype)

        LL = np.clip( offset,           0,np.minimum(self.cells,     cells+offset))
        UR = np.clip( offset+cells,     0,np.minimum(self.cells,     cells+offset))
        ll = np.clip(-offset,           0,np.minimum(     cells,self.cells-offset))
        ur = np.clip(-offset+self.cells,0,np.minimum(     cells,self.cells-offset))

        canvas[ll[0]:ur[0],ll[1]:ur[1],ll[2]:ur[2]] = self.material[LL[0]:UR[0],LL[1]:UR[1],LL[2]:UR[2]]

        return Grid(material = canvas,
                    size     = self.size/self.cells*np.asarray(canvas.shape),
                    origin   = self.origin+offset*self.size/self.cells,
                    comments = self.comments+[util.execution_stamp('Grid','canvas')],
                   )


    def substitute(self,from_material,to_material):
        """
        Substitute material indices.

        Parameters
        ----------
        from_material : iterable of ints
            Material indices to be substituted.
        to_material : iterable of ints
            New material indices.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        def mp(entry,mapper):
            return mapper[entry] if entry in mapper else entry

        mp = np.vectorize(mp)
        mapper = dict(zip(from_material,to_material))

        return Grid(material = mp(self.material,mapper).reshape(self.cells),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','substitute')],
                   )


    def sort(self):
        """
        Sort material indices such that min(material) is located at (0,0,0).

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        a = self.material.flatten(order='F')
        from_ma = pd.unique(a)
        sort_idx = np.argsort(from_ma)
        ma = np.unique(a)[sort_idx][np.searchsorted(from_ma,a,sorter = sort_idx)]

        return Grid(material = ma.reshape(self.cells,order='F'),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','sort')],
                   )


    def vicinity_offset(self,vicinity=1,offset=None,trigger=[],periodic=True):
        """
        Offset material index of points in the vicinity of xxx.

        Different from themselves (or listed as triggers) within a given (cubic) vicinity,
        i.e. within the region close to a grain/phase boundary.
        ToDo: use include/exclude as in seeds.from_grid

        Parameters
        ----------
        vicinity : int, optional
            Voxel distance checked for presence of other materials.
            Defaults to 1.
        offset : int, optional
            Offset (positive or negative) to tag material indices,
            defaults to material.max()+1.
        trigger : list of ints, optional
            List of material indices that trigger a change.
            Defaults to [], meaning that any different neighbor triggers a change.
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.

        Returns
        -------
        updated : damask.Grid
            Updated grid-based geometry.

        """
        def tainted_neighborhood(stencil,trigger):

            me = stencil[stencil.shape[0]//2]
            return np.any(stencil != me if len(trigger) == 0 else
                          np.in1d(stencil,np.array(list(set(trigger) - {me}))))

        offset_ = np.nanmax(self.material)+1 if offset is None else offset
        mask = ndimage.filters.generic_filter(self.material,
                                              tainted_neighborhood,
                                              size=1+2*vicinity,
                                              mode='wrap' if periodic else 'nearest',
                                              extra_keywords={'trigger':trigger})

        return Grid(material = np.where(mask, self.material + offset_,self.material),
                    size     = self.size,
                    origin   = self.origin,
                    comments = self.comments+[util.execution_stamp('Grid','vicinity_offset')],
                   )


    def get_grain_boundaries(self,periodic=True,directions='xyz'):
        """
        Create VTK unstructured grid containing grain boundaries.

        Parameters
        ----------
        periodic : Boolean, optional
            Assume grid to be periodic. Defaults to True.
        directions : iterable containing str, optional
            Direction(s) along which the boundaries are determined.
            Valid entries are 'x', 'y', 'z'. Defaults to 'xyz'.

        Returns
        -------
        grain_boundaries : damask.VTK
            VTK-based geometry of grain boundary network.

        """
        valid = ['x','y','z']
        if not set(directions).issubset(valid):
            raise ValueError(f'invalid direction {set(directions).difference(valid)} specified')

        o = [[0, self.cells[0]+1,           np.prod(self.cells[:2]+1)+self.cells[0]+1, np.prod(self.cells[:2]+1)],
             [0, np.prod(self.cells[:2]+1), np.prod(self.cells[:2]+1)+1,               1],
             [0, 1,                         self.cells[0]+1+1,                         self.cells[0]+1]] # offset for connectivity

        connectivity = []
        for i,d in enumerate(['x','y','z']):
            if d not in directions: continue
            mask = self.material != np.roll(self.material,1,i)
            for j in [0,1,2]:
                mask = np.concatenate((mask,np.take(mask,[0],j)*(i==j)),j)
            if i == 0 and not periodic: mask[0,:,:] = mask[-1,:,:] = False
            if i == 1 and not periodic: mask[:,0,:] = mask[:,-1,:] = False
            if i == 2 and not periodic: mask[:,:,0] = mask[:,:,-1] = False

            base_nodes = np.argwhere(mask.flatten(order='F')).reshape(-1,1)
            connectivity.append(np.block([base_nodes + o[i][k] for k in range(4)]))

        coords = grid_filters.coordinates0_node(self.cells,self.size,self.origin).reshape(-1,3,order='F')
        return VTK.from_unstructured_grid(coords,np.vstack(connectivity),'QUAD')