netgen.py 47.4 KB
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# generation and planar cutting of periodic pore networks
#
# Daniel W. Meyer
# Institute of Fluid Dynamics, ETH Zurich
# January 2019
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import multiprocessing as mp
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from typing import List, Dict, Set, Tuple # for type hints in argument lists

LABELS = ('', 'in', 'out', 'cut') # don't change the order

class Pore:
    """Class of a pore connected to other pores via throats."""
    id = 0
    def __init__(self, pos: List[float], r: float, label: str = LABELS[0], \
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        throats: Set = None, index: int = 0, id: int = -1):
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        if id == -1: # no id given
            self.id = Pore.id; Pore.id = Pore.id+1
        else: # id provided
            self.id = id; Pore.id = max(id+1, Pore.id)
        self.pos = pos.copy() # position vector
        if throats is None: throats = set()
        self.throats = throats # throats set
        self.r = r # radius
        self.label = label # label string like '', 'in', or 'out'
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        # Identifier for pores in list used in generate_dendrogram
        self.index = index
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    def __repr__(self): return str(self.__class__) + ': ' + \
        str({k:self.__dict__[k] for k in self.__dict__ if k != 'throats'}) + \
        ' {0:d} throats'.format(len(self.throats)) # dont flush throat objects

class Throat:
    """Class of a throat connecting two pores.
    
    Periodic throats have a label 'X1 X2 X3' with Xc being element of {-1,0,1}.
    For Xc = 1, pore1 is at the right or upper domain bound in the c-direction
    and pore2 is at the left or lower bound. Vice versa for Xc = -1. With
    Xc = 0, the throat does not cross the bound periodically in c-direction.
    """
    id = 0
    def __init__(self, pore1: Pore, pore2: Pore, r: float, \
        label: str = LABELS[0], id: int = -1):
        if id == -1: # no id given
            self.id = Throat.id; Throat.id = Throat.id+1
        else: # id provided
            self.id = id; Throat.id = max(id+1, Throat.id)
        # pore objects
        self.pore1 = pore1
        self.pore2 = pore2
        # radius and label
        self.r = r
        self.label = label # label string like '' or periodicity label '1 0 -1'
    def __repr__(self): return str(self.__class__) + ': ' + str(self.__dict__)

class Network:
    """Network class with pore and throat lists.
    
    Lmax must be smaller than the smallest side length of the network ub-lb.
    """
    def __init__(self, lb: List[float] = [], ub: List[float] = [], \
        pores: List[Pore] = None, throats: List[Throat] = None, \
        Lmax: float = 0.0, label: str = None):
        # lower/upper bounds for network volume
        self.lb = lb.copy(); self.ub = ub.copy()
        # sets with pores and throats
        if pores is None: pores = set()
        if throats is None: throats = set()
        self.pores = pores.copy(); self.throats = throats.copy()
        # length of longest throat (distance between connected pores)
        self.Lmax = Lmax
        # network label or name
        from datetime import datetime
        if label is None:
            self.label = datetime.today().isoformat(sep=' ',timespec='seconds')
        else:
            self.label = label
    def __repr__(self):
        # count in/out resp. labelled pores
        k = sum([int(pore.label != LABELS[0]) for pore in self.pores])
        # report
        return 'Network \'' + self.label + '\' from ' + str(self.lb) + \
            ' to ' + str(self.ub) + \
            ' with {0:d}({1:d}) pores, {2:d} throats, Lmax = {3:e}'. \
            format(len(self.pores), k, len(self.throats), self.Lmax)
    def add_pore(self, pore: Pore):
        self.pores.add(pore)
    def remove_pore(self, pore: Pore):
        """Remove pore and all throats connected to it."""
        # disconnect throats from pores connected to pore
        for throat in pore.throats.copy():
            connected_pore = throat.pore1
            if (connected_pore == pore): connected_pore = throat.pore2
            connected_pore.throats.remove(throat)
            pore.throats.remove(throat)
            self.throats.remove(throat)
        # remove pore
        self.pores.remove(pore)
    def connect_pores(self, pore1: Pore, pore2: Pore, r: float, \
        throat_id: int = -1, label: str = LABELS[0]) -> Throat:
        """Establish a throat connection between two pores."""
        throat = Throat(pore1, pore2, r, label, throat_id)
        self.throats.add(throat)
        pore1.throats.add(throat); pore2.throats.add(throat)
        return throat

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class CellList:
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    """CellList class dividing physical domain into equally-sized cells
    
    find_candidates() assumes pore is located in interior of domain (non-periodic)
    """
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    def __init__(self, pores: List[Pore], domainSize: List[float], \
                         cellSize: float, Lmax: float):
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        from math import ceil
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        self.dim = len(domainSize)
        self.Lmax = Lmax
        # Number of cells in each dimension
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        self.nCells = [max(1, ceil(domainSize[i] / cellSize)) for i in range(self.dim)]
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        # Inverse of cell sizes in domain
        self.invCellSizes = [self.nCells[i] / domainSize[i] for i in range(self.dim)]
        # Total number of cells
        self.totalCells = prod(self.nCells)
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        # Pore positions 
        self.poresPos = [0.] * self.dim * len(pores)
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        # Sort pores according to cell membership
        self.poresSorted = [set() for _ in range(self.totalCells)]
        for pore in pores:
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            poreIdx = pore.index
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            cellIdx = self.pore_to_index(pore)
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            self.poresSorted[cellIdx].add(poreIdx)
            lb = self.dim * poreIdx
            ub = self.dim + lb
            self.poresPos[lb:ub] = pore.pos
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    # Helper for converting 3D index triple into 1D (flattened) index
    def flatten(self, i: int, j: int, k: int) -> int:
        return i + self.nCells[0] * (j + self.nCells[1] * k)
    
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    def pore_to_triplet(self, poreIdx: int) -> List[int]:
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        from math import floor
        # Shift pore position to be positive first (buffer layer)
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        porePos = self.fetch_pos(poreIdx)
        return [floor( (porePos[i] + self.Lmax) * self.invCellSizes[i]) \
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                        for i in range(self.dim)]
    # Compute index of given pore in cell list based on position
    def pore_to_index(self, pore: Pore) -> int:
        from math import floor
        # Shift pore position to be positive first (buffer layer)
        cellIdx = self.flatten(*[floor( (pore.pos[i] + self.Lmax) * self.invCellSizes[i]) \
                        for i in range(self.dim)])
        return cellIdx
    
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    def is_valid_index(self, idx: int) -> bool:
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        return 0 <= idx < self.totalCells
        
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    # Compute dictionary of all nbor candidates of given pore
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    def find_candidates(self, poreIdx: int) -> Dict[int, float]:    
        candidates  = {}
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        porePos = self.fetch_pos(poreIdx)
        cellIdxTrip = self.pore_to_triplet(poreIdx)
        
        nearestCellIdx = self.flatten(*cellIdxTrip)
        for nborIdx in self.poresSorted[nearestCellIdx]:
            if (nborIdx == poreIdx): continue
            l = distance(self.fetch_pos(nborIdx), porePos)
            if (l < self.Lmax):
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                candidates[nborIdx] = l
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        # TODO: Speed up finding of suitable candidates (e.g. multiprocessing)
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        for n in range(self.dim):
            for i in [-1, 1]:
                cellIdx = self.flatten(cellIdxTrip[0] + i * (n == 0), \
                                       cellIdxTrip[1] + i * (n == 1), \
                                       cellIdxTrip[2] + i * (n == 2))
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                # Check if valid cell index
                #assert(self.is_valid_index(nborIdx))
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                # Check all pores within current neighbor-cell
                for nborIdx in self.poresSorted[cellIdx]:
                    l = distance(self.fetch_pos(nborIdx), porePos)
                    if (l < self.Lmax):
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                        candidates[nborIdx] = l
        
        return candidates
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    def fetch_pos(self, poreIdx: int) -> List[float]:
        lb = self.dim * poreIdx
        ub = self.dim + lb
        return self.poresPos[lb:ub]
        
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    def par_find_candidates(self, in_q: mp.SimpleQueue, out_q: mp.SimpleQueue):
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        while True:
            # Retrieve payload from in-queue
            payload = in_q.get()
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            if payload is None:
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                break  # Terminate process
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            poreIdx     = payload[0]
            neighbors   = payload[1]
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            porePos = self.fetch_pos(poreIdx)
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            candidates = {}
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            for nborIdx in neighbors:
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                if (nborIdx == poreIdx): continue
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                l = distance(porePos, self.fetch_pos(nborIdx))
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                if (l < self.Lmax):
                    candidates[nborIdx] = l
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            # Put result in out-queue
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            out_q.put(candidates)
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    # Only consider subset of all neighboring cells (closest 7)
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    def nbor_indices(self, poreIdx: int) -> List[int]:
        cellIdx = self.pore_to_triplet(poreIdx)
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        # i, j, k; i +- 1, j, k; i, j +- 1, k; i, j, k +- 1;
        neighborhood = [self.flatten(*cellIdx)]  # Cell containing the pore
        neighborhood += [self.flatten(cellIdx[0] + i * (n == 0), \
                                      cellIdx[1] + i * (n == 1), \
                                      cellIdx[2] + i * (n == 2))
                        for n in range(self.dim) for i in [-1, 1]]
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        return neighborhood
        
    def nbor_cell_counts(self, neighborhood: List[int]) -> List[int]:
        return [len(self.poresSorted[i]) for i in neighborhood]
        
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    def cell_stats(self):
        cellCounts = [len(self.poresSorted[i]) for i in range(self.totalCells)]
        n    = sum(cellCounts)
        max_ = max(cellCounts)
        min_ = min(cellCounts)
        avg  = n / self.totalCells
        
        """
        print("Cell statistics:")
        print("Number of cells (%d, %d, %d)" % (self.nCells[0], self.nCells[1], self.nCells[2]))
        print("Max. pores per cell: %d (%f%%)" % (max_, max_ / n * 100.))
        print("Min. pores per cell: %d (%f%%)" % (min_, min_ / n * 100.))
        print("Avg. pores per cell: %.1f (%f%%)" % (avg, avg / n * 100.))
        """
        return (max_, min_, avg)

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    # Remove pore from sorted pores
    def expel(self, pore: Pore):
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        poreCellIdx = self.pore_to_index(pore)
        self.poresSorted[poreCellIdx].remove(pore.index)
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def distance(p1: List[float], p2: List[float]):
    """Distance between points p1 and p2."""
    mag = 0.0
    for c1, c2 in zip(p1,p2): mag += (c1-c2)**2
    return mag**0.5


def prod(v):
    """Cumulative product of vector components."""
    from functools import reduce
    return(reduce(lambda a,b: a*b, v))


def random_direction(d: int) -> List[float]:
    """Unity vector with uniformly distributed orientation."""
    from math import pi, sin, cos, acos
    from random import random
    phi = 2*pi * random()
    if (d == 2):
        return([cos(phi), sin(phi)])
    elif (d == 3):
        theta = acos(2*random() - 1)
        return([sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)])
    else:
        raise ValueError("random_direction supports d = 2 or 3")


def throat_ends(throat: Throat, L: List[float]) -> Tuple[List[float]]:
    """Provide the correct relative end points of a possibly periodic throat."""
    x1 = throat.pore1.pos.copy(); x2 = throat.pore2.pos.copy()
    # account for periodic throats
    if len(throat.label) != 0:
        direct = [int(k) for k in throat.label.split()] # periodicity
        x2 = [x2[k] + direct[k] * L[k] for k in range(len(L))]
    return (x1, x2)



def imperial_read(net_file_pref: str) -> Network:
    """Read network data in Imperial College format.
    
    For format specifics see PhD thesis of Taha Sochi from 2007 at Imperial.
    The names of the network files starts with the string net_file_pref.
    A Network object is returned.
    """
    # initialize network
    from os import sep
    network = Network(label = net_file_pref.split(sep)[-1])
    
    # read pore data
    with open(net_file_pref + '_node1.dat', 'r') as file:
        line = file.readline()
        size = [float(n) for n in line.replace('\t',' ').split()][1:]
        d = len(size) # dimensionality
        f = lambda words: (
            int(words[0]), # id
            [float(n) for n in words[1:(d+1)]], # pos
            [int(throat) for throat in words[(d+4+int(words[d+1])):]], # throats
            LABELS[int(words[d+2+int(words[d+1])]) + \
            2*int(words[d+3+int(words[d+1])])]) # label
        pores = [f(line.replace('\t',' ').split()) for line in file]
    pores.sort(key=lambda pore: pore[0])
    pores = [Pore(pore[1], 0.0, pore[3]) for pore in pores]
    # read pore radii
    with open(net_file_pref + '_node2.dat', 'r') as file:
        f = lambda words: (int(words[0]), float(words[2]))
        radii = [f(line.replace('\t',' ').split()) for line in file]
    radii.sort(key=lambda pore: pore[0])
    for k, pore in enumerate(pores): pore.r = radii[k][1]
    # add pores to network
    for pore in pores: network.add_pore(pore)

    # read throat data
    with open(net_file_pref + '_link1.dat', 'r') as file:
        n = int(file.readline().strip('\n ')) # number of throats
        f = lambda words: (
            int(words[0]), int(words[1])-1, int(words[2])-1, float(words[3]))
        throats = [f(line.replace('\t',' ').split()) for line in file]
    throats.sort(key=lambda throat: throat[0])
    # connect pores
    for throat in throats:
        if (throat[1] >= 0) and (throat[2] >= 0): # no in-/outflow throats
            network.connect_pores(pores[throat[1]], pores[throat[2]], throat[3])

    # determine maximum throat length
    Lmax = 0.0
    for throat in network.throats:
        L = distance(throat.pore1.pos, throat.pore2.pos)
        Lmax = max(L,Lmax)
    network.Lmax = Lmax

    network.lb = [0 for i in range(len(size))]
    network.ub = size
    return network


def plot_network(network: Network, labels: bool = False):
    """Plot pore network.
    
    Returns a figure object. If labels == True, pore and throat ids are shown.
    Pores with label == '' are plotted in black. Pores with labels starting
    with 'in', 'out', 'cut' are plotted in red, blue, magenta, respectively.
    """
    # setup plot
    import matplotlib.pyplot as plt
    from mpl_toolkits.mplot3d import Axes3D
    plt.ion()
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    
    # plot throats
    for throat in network.throats:
        if throat.label == LABELS[0]: col = 'black'
        else: col = 'grey'
        p1 = throat.pore1.pos; p2 = throat.pore2.pos
        ax.plot([p1[0], p2[0]], [p1[1], p2[1]], [p1[2], p2[2]],
            linewidth=0.5, color=col)
        if labels:
            ax.text3D((p1[0]+p2[0])/2, (p1[1]+p2[1])/2, (p1[2]+p2[2])/2,
                str(throat.id), fontsize=5)

    # plotting function for pores
    def plot_pores(label, size, color):
        pnts = [pore.pos for pore in network.pores \
            if (pore.label[:max(1,len(label))] == label)]
        if len(pnts) == 0: return # in case of empty point list
        pnts = list(map(list, zip(*pnts))) # transpose double list
        ax.scatter(*pnts, s=size, marker='.', c=color)
        if labels:
            for pore in network.pores:
                if (pore.label[:max(1,len(label))] == label):
                    ax.text3D(*pore.pos, str(pore.id), fontsize=7, color=color)
    # plot ordinary pores
    plot_pores(LABELS[0], 4, 'black')
    # plot inflow pores
    plot_pores(LABELS[1], 7, 'red')
    # plot outflow pores
    plot_pores(LABELS[2], 7, 'blue')
    # plot cut pores
    plot_pores(LABELS[3], 7, 'magenta')

    #ax.axis('equal')
    return fig


def generate_simple_net(n_pores: int, targetsize: List[float], \
    r_pore: float, r_throat: float, coordinatnumb: int, Lmax: float, \
    sd: int = None) -> Network:
    """Generate a spatially periodic network with uniform pore distribution."""
    d = len(targetsize) # number of spatial dimensions
    density = n_pores / prod(targetsize)
    origin = [0.0 for k in range(d)]
    Lb = (1/density)**(1/d)
    basenet = Network(origin, [Lb for k in range(d)], \
        [Pore(origin, r_pore, throats = {k for k in range(coordinatnumb)})], \
        [Throat(None, None, r_throat)], Lmax)
    basenet.label = 'simplenet_' + basenet.label
    return generate_imperial(basenet, targetsize, sd, False)


def generate_imperial(basenet: Network, targetsize: List[float], \
    sd: int = None, correlated: bool = True) -> Network:
    """Generate a new spatially periodic network.

    Based on an existing network basenet, generate a new network of size
    targetsize by using the algorithm outlined on p.54 of the PhD thesis of
    Nasiru Abiodun Idowu from 2009 at Imperial College.
    Pores are uniformly distribute with pore number density as in basenet.
    Pore radii and number of throat connections are sampled from basenet.
    Closest pores are connected. Throat radii are taken from basenet such
    that throat radius and radius-sum of the connected pores are correlated.
    Periodic throats are marked with a periodicity label, e.g., '1 0 -1' in 3d.
    """
    from random import seed, random, randint
    seed(sd)
    d = len(targetsize) # number of spatial dimensions
    # check target size (must be > basenet.Lmax)
    if any([L < basenet.Lmax for L in targetsize]):
        raise NameError('targetsize must be > Lmax of basenet!')

    # pores, uniformly distributed
    # number of pores in new network
    import math
    basesize = list(map(lambda a,b: a-b, basenet.ub, basenet.lb))
    n = math.ceil(prod(targetsize) * len(basenet.pores) / prod(basesize))
    # distribute pores uniformly
    basepores = [pore for pore in basenet.pores] # make indexable
    pores = [Pore(pos=[random()*L for L in targetsize], # pos
        # radius, label, number of throat connections
        r=basepores[idx].r, label=LABELS[0],
        throats=len(basepores[idx].throats)) for j, idx in \
        enumerate([randint(0,len(basepores)-1) for k in range(n)])]

    # add pore buffer layers for spatial periodicity
    n = len(pores)
    _ = __add_buffer_layers(pores, targetsize, basenet.Lmax)

    # throats, connect pores
    # triangulation
    import numpy as np
    from scipy.spatial import Delaunay
    points = np.array([pore.pos for pore in pores])
    tri = Delaunay(points) # possibility for missing (coplanar) points!
    # return neighbors of point pnt based on triangulation tri
    def neighbors_of(pnt, tri):
        return tri.vertex_neighbor_vertices[1][ \
            tri.vertex_neighbor_vertices[0][pnt]: \
            tri.vertex_neighbor_vertices[0][pnt+1]]
    # connect pores
    throats = []
    for k, pore in enumerate(pores[:n]):
        # initialize neighbor lists
        nbors = neighbors_of(k, tri)
        nbors = dict(zip(nbors,
            [distance(pore.pos, pores[j].pos) for j in nbors]))
        oldies = {k:0.0}
        # establish throat connections between pore and its neighbors
        while (pore.throats > 0) \
            and (len(nbors) > 0) and (min(nbors.values()) <= basenet.Lmax):
            # find closest neighbor
            for nbor in nbors:
                if nbors[nbor] == min(nbors.values()): break
            # if nbor is periodic copy, get original pore and periodicity label
            if (pores[nbor].label != LABELS[0]):
                nbor_o, lbl = pores[nbor].label.split(' ',1)
                nbor_o = int(nbor_o)
            else:
                nbor_o = nbor; lbl = LABELS[0]
            # is nbor a valid pore to connect to?
            if not ((nbor_o <= k) or (nbor in oldies) or \
                (pores[nbor_o].throats == 0)):
                # yes -> add connection to pore nbor_o
                throats.append([k, nbor_o, lbl])
                pore.throats = pore.throats - 1
                pores[nbor_o].throats = pores[nbor_o].throats - 1
            # nbor has been dealt with
            oldies[nbor] = nbors[nbor]
            # expand search neighborhood
            for j in neighbors_of(nbor, tri):
                nbors[j] = distance(pores[k].pos, pores[j].pos)
            for oldie in oldies.keys():
                if oldie in nbors: del nbors[oldie]
    
    # initialize throat sets of pores
    k = 0 # count of unrealised throats
    for pore in pores[:n]:
        k = k + pore.throats
        pore.throats = set() # initialize throats set

    # set throat radii
    # sort branch weights (branch weight is prop. to sum of pore radii)
    bw = [pores[throat[0]].r + pores[throat[1]].r for throat in throats]
    bw = list(enumerate(bw))
    bw.sort(key=lambda w: w[1])
    # sample throat radii from basenet
    basethroats = [throat for throat in basenet.throats] # make indexable
    r = [randint(0,len(basethroats)-1) for k in range(len(bw))]
    r = [basethroats[k].r for k in r]
    if correlated: r = sorted(r)
    # assign radii
    for k in range(len(r)): throats[bw[k][0]].append(r[k])

    # assemble and return network
    network = Network(lb=[0.0 for k in range(d)],
        ub=targetsize, Lmax=basenet.Lmax, label='from_' + basenet.label)
    for pore in pores[:n]: network.add_pore(pore)
    for throat in throats: network.connect_pores(pore1=pores[throat[0]],
        pore2=pores[throat[1]], label=throat[2], r=throat[3])
    return network
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def cell_list_scaling(basenet: Network, targetsize: List[int], \
    cutoff: float = float('inf'), sd: int = None, mute: bool = False):
    from random import seed, random, randint
    from itertools import product
    seed(sd)
    d = len(targetsize) # number of spatial dimensions
    # check target size multiplicator (must be >= 1)
    if any([i < 1 for i in targetsize]):
        raise NameError('targetsize must be >= 1!')
    # size of new network
    L = list(map(lambda lb,ub,i: (ub-lb)*i, basenet.lb, basenet.ub, targetsize))
    # check target size (must be > basenet.Lmax)
    if any([Lc < basenet.Lmax for Lc in L]):
        raise NameError('targetsize leads network < Lmax of basenet!')

    # pores, distributed based on dendrogram of basenet
    if (not mute): print("distributing pores...", end="", flush=True)
    # make indexable and discard in-/outflow pores
    basepores = [pore for pore in basenet.pores \
        if ((pore.label[:len(LABELS[1])] != LABELS[1]) \
        and (pore.label[:len(LABELS[2])] != LABELS[2]))]
    # dendrogram-based or uniform pore distribution
    pores = []
    if (cutoff != cutoff): # uniform
        print("\b"*21 + "uniform pore distribution")
        # loop over network sections & add uniformly distributed pores drawn from basenet
        for s in product(*[range(k) for k in targetsize]):
            for k, pore in enumerate(basepores):
                pos = [random()*l for l in L]
                pores.append(Pore(pos=pos, r=pore.r, label=LABELS[0], 
                    throats=pore.throats.copy()))
    else: # dendrogram-based
        # extract cluster hierarchy from basenet
        centroids = [pore.pos for pore in basepores]
        from scipy.cluster import hierarchy
        clustree = hierarchy.linkage(centroids, method = 'centroid')
        # determine cluster centroids and weights
        weights = len(basepores)*[1] # pores have weight = 1
        for cluster in clustree:
            p1 = int(cluster[0]); p2 = int(cluster[1])
            w1 = weights[p1]; w2 = weights[p2]
            centroids.append(list(map(lambda x1,x2: (w1*x1 + w2*x2)/(w1 + w2),
                centroids[p1], centroids[p2])))
            weights.append(w1 + w2)
        # loop over network sections (twisted basenet copies)
        for s in product(*[range(k) for k in targetsize]):
            # twist centers of sufficiently small cluster
            touched = [False]*len(centroids)
            # positions of pores based on rotations of linked cluster-pairs
            for k in range(len(centroids)-1,len(basepores)-1,-1):
                i = k-len(basepores) # index of cluster in clustree
                ctr = centroids[k] # center of rotation
                dist = clustree[i,2] # cluster distance
                # check if cluster/pore needs to be twisted
                if ((dist < cutoff) or touched[k]):
                    p1 = int(clustree[i,0]); p2 = int(clustree[i,1])
                    w1 = weights[p1]; w2 = weights[p2]
                    vec = random_direction(d)
                    centroids[p1] = \
                        [ctr[j] + w2*dist/(w1+w2)*vec[j] for j in range(d)]
                    centroids[p2] = \
                        [ctr[j] - w1*dist/(w1+w2)*vec[j] for j in range(d)]
                    touched[p1] = touched[p2] = True
            # add section pores in random order to pores of new network
            for k, pore in enumerate(basepores):
                pos = [c-lb + si*(ub-lb) \
                    for c,si,lb,ub in zip(centroids[k],s,basenet.lb,basenet.ub)]
                pores.insert(randint(0,len(pores)), Pore(pos=pos,
                    r=pore.r, label=LABELS[0], throats=pore.throats.copy()))
        print("\b"*21 + "left {0:d} of {1:d} clusters (incl. {2:d} pores) untouched".\
            format(sum([int(not j) for j in touched]), len(touched), len(basepores)))
        # flip pores outside back into domain
        for pore in pores:
            for k in range(d):
                if pore.pos[k] < 0:
                    pore.pos[k] = L[k] + pore.pos[k]
                elif pore.pos[k] >= L[k]:
                    pore.pos[k] = pore.pos[k] - L[k]

    # add pore buffer layers for spatial periodicity
    n = len(pores)
    copies = __add_buffer_layers(pores, L, basenet.Lmax)
    
    # Domain size including periodic buffer layers on all sides
    trueDomainSize = [L[i] + 2 * basenet.Lmax for i in range(d)]
    # Initialize cell-list; Place each pore in respective cell-set
    cellList = CellList(pores, trueDomainSize, basenet.Lmax, basenet.Lmax)
    # Print cell stats to console
    max_, min_, avg = cellList.cell_stats()
    return (max_, min_, avg)
    
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def generate_dendrogram(basenet: Network, targetsize: List[int], \
    cutoff: float = float('inf'), sd: int = None, mute: bool = False) \
    -> Network:
    """Generate a new spatially periodic network.

    Based on an existing network basenet, generate a new network of size
    targetsize by using a dendrogram-based algorithm that accounts for the
    spatial distribution/clustering of pores.
    The pore-clustering dendrogram is taken from basenet. Pore radii and number
    of throats are sampled from basenet. Pores are connected based on the
    throat lengths from the basenet data. Throat radii are taken from basenet.
    Periodic throats are marked with a periodicity label, e.g., '1 0 -1' in 3d.
    If cutoff == float('nan'), pores are uniformly distributed.
    """
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    import multiprocessing as mp
    from random import seed, random, randint
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    from itertools import product
    seed(sd)
    d = len(targetsize) # number of spatial dimensions
    # check target size multiplicator (must be >= 1)
    if any([i < 1 for i in targetsize]):
        raise NameError('targetsize must be >= 1!')
    # size of new network
    L = list(map(lambda lb,ub,i: (ub-lb)*i, basenet.lb, basenet.ub, targetsize))
    # check target size (must be > basenet.Lmax)
    if any([Lc < basenet.Lmax for Lc in L]):
        raise NameError('targetsize leads network < Lmax of basenet!')

    # pores, distributed based on dendrogram of basenet
    if (not mute): print("distributing pores...", end="", flush=True)
    # make indexable and discard in-/outflow pores
    basepores = [pore for pore in basenet.pores \
        if ((pore.label[:len(LABELS[1])] != LABELS[1]) \
        and (pore.label[:len(LABELS[2])] != LABELS[2]))]
    # dendrogram-based or uniform pore distribution
    pores = []
    if (cutoff != cutoff): # uniform
        print("\b"*21 + "uniform pore distribution")
        # loop over network sections & add uniformly distributed pores drawn from basenet
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        runningIndex = 0
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        for s in product(*[range(k) for k in targetsize]):
            for k, pore in enumerate(basepores):
                pos = [random()*l for l in L]
                pores.append(Pore(pos=pos, r=pore.r, label=LABELS[0], 
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                    throats=pore.throats.copy(), index=runningIndex))
                runningIndex += 1
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    else: # dendrogram-based
        # extract cluster hierarchy from basenet
        centroids = [pore.pos for pore in basepores]
        from scipy.cluster import hierarchy
        clustree = hierarchy.linkage(centroids, method = 'centroid')
        # determine cluster centroids and weights
        weights = len(basepores)*[1] # pores have weight = 1
        for cluster in clustree:
            p1 = int(cluster[0]); p2 = int(cluster[1])
            w1 = weights[p1]; w2 = weights[p2]
            centroids.append(list(map(lambda x1,x2: (w1*x1 + w2*x2)/(w1 + w2),
                centroids[p1], centroids[p2])))
            weights.append(w1 + w2)
        # loop over network sections (twisted basenet copies)
        for s in product(*[range(k) for k in targetsize]):
            # twist centers of sufficiently small cluster
            touched = [False]*len(centroids)
            # positions of pores based on rotations of linked cluster-pairs
            for k in range(len(centroids)-1,len(basepores)-1,-1):
                i = k-len(basepores) # index of cluster in clustree
                ctr = centroids[k] # center of rotation
                dist = clustree[i,2] # cluster distance
                # check if cluster/pore needs to be twisted
                if ((dist < cutoff) or touched[k]):
                    p1 = int(clustree[i,0]); p2 = int(clustree[i,1])
                    w1 = weights[p1]; w2 = weights[p2]
                    vec = random_direction(d)
                    centroids[p1] = \
                        [ctr[j] + w2*dist/(w1+w2)*vec[j] for j in range(d)]
                    centroids[p2] = \
                        [ctr[j] - w1*dist/(w1+w2)*vec[j] for j in range(d)]
                    touched[p1] = touched[p2] = True
            # add section pores in random order to pores of new network
            for k, pore in enumerate(basepores):
                pos = [c-lb + si*(ub-lb) \
                    for c,si,lb,ub in zip(centroids[k],s,basenet.lb,basenet.ub)]
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                pores.insert(randint(0,len(pores)), Pore(pos=pos,
                    r=pore.r, label=LABELS[0], throats=pore.throats.copy()))
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        print("\b"*21 + "left {0:d} of {1:d} clusters (incl. {2:d} pores) untouched".\
            format(sum([int(not j) for j in touched]), len(touched), len(basepores)))
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        # flip pores outside back into domain and set respective index of all pores
        for i, pore in enumerate(pores):
            pore.index = i
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            for k in range(d):
                if pore.pos[k] < 0:
                    pore.pos[k] = L[k] + pore.pos[k]
                elif pore.pos[k] >= L[k]:
                    pore.pos[k] = pore.pos[k] - L[k]

    # add pore buffer layers for spatial periodicity
    n = len(pores)
    copies = __add_buffer_layers(pores, L, basenet.Lmax)
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    # Flag for handling the case where original nbor pore and its periodic
    # copies are potential candidates of a given pore at the same time 
    # (necessary only if target size is 1 in at least one dimension)
    isPeriodicCheck = 1 in targetsize
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    # Domain size including periodic buffer layers on all sides
    trueDomainSize = [L[i] + 2 * basenet.Lmax for i in range(d)]
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    # Initialize cell-list; Place each pore in respective cell-set
    cellList = CellList(pores, trueDomainSize, basenet.Lmax, basenet.Lmax)
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    # throats, connect pores
    if (not mute): print("\b"*21 + "connecting")
    throats = []
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    k = 0 # count of unrealised throats
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    nthreads = 4
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    # Spawn worker threads
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    in_q  = mp.SimpleQueue()
    out_q = mp.SimpleQueue()
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    workers = []
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    for _ in range(nthreads):
        worker = mp.Process(target=cellList.par_find_candidates, args=(in_q, out_q))
        worker.start()
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        workers.append(worker)
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    avg_throat_diff = 0.
    count = 0
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    for poreIdx, pore in enumerate(pores[:n]):
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        if (not mute):
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            print(f"progress {((poreIdx+1) / n)*100.:.1f}%", end="\r", flush=True)
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        if (len(pore.throats) == 0): continue # no further connection needed
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        # TODO: Search neighboring cells in parallel
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        #candidates = cellList.find_candidates(poreIdx)
        neighborhood = cellList.nbor_indices(poreIdx)
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        for nborCellIdx in neighborhood:
            in_q.put((poreIdx, cellList.poresSorted[nborCellIdx]))
        
        candidates = {}
        for _ in range(len(neighborhood)):
            cellCandidates = out_q.get()
            # Add cell candidates to global candidates
            candidates = {**candidates, **cellCandidates}
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        # establish throat connections between pore and best candidate
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        # TODO: Perform this for all throats simultaneously
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        #conpores = set()
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        while (len(pore.throats) > 0):
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            throat = pore.throats.pop()
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            lt = distance(*throat_ends(throat,
                [ub-lb for lb,ub in zip(basenet.lb,basenet.ub)])) # target len.
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            if (len(candidates) == 0):
                k = k+1; continue  # no candidates left
            nborIdx, ltc = min(candidates.items(), key=lambda pl: abs(pl[1] - lt))
            nbor = pores[nborIdx]
            
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            diff = abs(lt - ltc)
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            avg_throat_diff += diff
            count += 1
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            lt = ltc
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            nborc = nbor # memorize potential periodic copy
            # find original of buffer layer pore
            if (nbor.label != LABELS[0]):
                j, lbl = nbor.label.split(' ',1)
                nbor = pores[int(j)]
            else:
                lbl = LABELS[0]
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            # Remove nbor from candidates as well as all periodic copies
            del candidates[nborc.index]
            # Remove original non-periodic nbor as well as all its
            # periodic copies. Only applies if targetsize[i] = 1 for some i
            if isPeriodicCheck:
                if nbor.index in candidates:
                    del candidates[nbor.index]
                    
                for cpore in copies[nbor]:
                    if cpore.index in candidates:
                        del candidates[cpore.index]
            
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            # connect pore to nbor
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            throats.append([pore, nbor, lbl, throat.r])
            # remove most similar throat of nbor
            ranking = [(distance(*throat_ends(nthroat,
                [ub-lb for lb,ub in zip(basenet.lb,basenet.ub)])),
                nthroat) for nthroat in nbor.throats]
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            _, nthroat = min(ranking, key=lambda lth: abs(lth[0] - lt))
            nbor.throats.remove(nthroat)
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            # remove fully connected nbor from search neighborhood
            if (len(nbor.throats) == 0):
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                cellList.expel(nbor)
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                for cpore in copies[nbor]:
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                    cellList.expel(cpore)
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        # remove fully connected pore from search neighborhood
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        cellList.expel(pore)
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        for cpore in copies[pore]:
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            cellList.expel(cpore)
    
    # Terminate worker threads
    for _ in range(nthreads):
        in_q.put(None)
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    for worker in workers:
        worker.join()
        
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    percent = k / (k + len(throats)) * 100.
    print("\b"*24 + "{0:d} throats in total, {1:d} unrealised ({2:.1f}%)".\
        format(len(throats), k, percent))
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    avg_throat_diff /= count
    print("Avg. throat length difference: %e" % avg_throat_diff)
    print("Relative to Lmax: %.1f%%" % (avg_throat_diff / basenet.Lmax * 100.))
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    # assemble and return network
    network = Network(lb=[0.0 for k in range(d)],
        ub=L, Lmax=basenet.Lmax, label='from_' + basenet.label)
    for pore in pores[:n]: network.add_pore(pore)
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    for throat in throats: network.connect_pores(pore1=throat[0],
        pore2=throat[1], label=throat[2], r=throat[3])
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    return network


def __add_buffer_layers(pores: List[Pore], targetsize: List[float],
    Lbuffer: float) -> Dict[Pore,Set[Pore]]:
    """Add pore buffer layers for spatial periodicity.
    
    Returns a dict where each key is a pore and the corresponding value
    is a set containing the copies of that pore.
    """
    n = len(pores) # number of pores inside domain
    d = len(targetsize) # number of spatial dimensions
    # label = pore index + periodicity label (-1,0,1) in each dim.
    for k, pore in enumerate(pores): pore.label = str(k) + d * ' 0'
    # add pore buffer layers
    copies = {pore:set() for pore in pores} # {pore:{set of periodic copies}}
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    runningIndex = n
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    for k in range(d):
        # left bound
        pores_layer = [pore for pore in pores \
            if targetsize[k]-Lbuffer <= pore.pos[k] < targetsize[k]]
        for pore in pores_layer:
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            pcopy = Pore(pore.pos, pore.r, throats=pore.throats, index=runningIndex, id=pore.id)
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            pcopy.pos[k] = pcopy.pos[k]-targetsize[k]
            lbl = pore.label.split(); lbl[k + 1] = '-1'
            pcopy.label = ' '.join(lbl) # periodicity label
            pores.append(pcopy)
            copies[pores[int(lbl[0])]].add(pcopy)
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            runningIndex += 1
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        # right bound
        pores_layer = [pore for pore in pores \
            if 0 <= pore.pos[k] < Lbuffer]
        for pore in pores_layer:
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            pcopy = Pore(pore.pos, pore.r, throats=pore.throats, index=runningIndex, id=pore.id)
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            pcopy.pos[k] = pcopy.pos[k]+targetsize[k]
            lbl = pore.label.split(); lbl[k + 1] = '1'
            pcopy.label = ' '.join(lbl) # periodicity label
            pores.append(pcopy)
            copies[pores[int(lbl[0])]].add(pcopy)
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            runningIndex += 1
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    # reset labels of pores inside domain (without buffer layers)
    for pore in pores[:n]: pore.label = LABELS[0]
    return copies


def open_periodic_network(network: Network, c: int, inouts: Dict = {}):
    """Open periodic throats in the spatial direction c>0 of a network.
    
    To avoid identical in/out pores resulting from different periodic throats
    connecting to the same original pore, one can keep track of newly generated
    in/out pores through the inouts pore dict {orig1:{"0 0 1":copy1, ...}, ...}.
    """
    # loop over (periodic) throats
    for throat in network.throats.copy(): # shallow copy (iteration changes set)
        # is throat a periodic connection in direction c?
        if len(throat.label) == 0: continue
        direct = [int(c) for c in throat.label.split()]
        if direct[c-1] == 0: continue
        # open periodic throat connection
        __open_periodic_throat(network, throat, direct, c, inouts)


def __open_periodic_throat(network: Network,
    throat: Throat, direct: List[int], c: int, inouts: Set[Pore]) \
    -> Set[Throat]:
    """Open a periodic throat connection.
    
    direct is the encoded periodicity label of the throat and c>0 is the
    spatial direction used for the in-/outflow labeling. The inouts dict
    is used to avoid the formation of pores with identical positions.
    """
    d = len(network.lb) # number of spatial dimensions
    p1 = throat.pore1; p2 = throat.pore2; r = throat.r
    # identify existing in/out pore or construct new one
    # needed in case of many periodic throats connecting to the same pore
    def get_inout_pore(pore_o, direct, pos, label, inouts) -> Pore:
        dirkey = str([abs(d) for d in direct]) # periodicity direction key
        # find existing inout pore copy (or periodic original)
        if pore_o in inouts: # is there an inout copy of the original pore?
            if dirkey in inouts[pore_o]: # is there copy in right direct?
                return inouts[pore_o][dirkey] # yes, return inout copy
        # if no inout copy exists, create a new one
        pore = Pore(pos, pore_o.r, label)
        if pore_o in inouts: # add new direct
            inout = inouts[pore_o] # dict dirkey:inout-pore
            inout[dirkey] = pore
        else: inouts[pore_o] = {dirkey:pore} # add original to inout dict
        return pore
    # pore2
    pos = [0.0 for j in range(d)] # init new position object
    # copy and shift pore2
    for j in range(d):
        pos[j] = p2.pos[j] + direct[j] * (network.ub[j]-network.lb[j])
    p2c = get_inout_pore(p2, direct, pos,
        LABELS[1+round((direct[c-1]+1)/2)] + str(c), inouts)
    network.add_pore(p2c)
    t1 = network.connect_pores(p1, p2c, throat.r)
    # pore1
    pos = [0.0 for j in range(d)] # init new position object
    # copy and shift pore1
    for j in range(d):
        pos[j] = p1.pos[j] - direct[j] * (network.ub[j]-network.lb[j])
    p1c = get_inout_pore(p1, direct, pos,
        LABELS[1+round((-direct[c-1]+1)/2)] + str(c), inouts)
    network.add_pore(p1c)
    t2 = network.connect_pores(p2, p1c, throat.r)
    # remove periodic throat
    network.throats.remove(throat)
    p1.throats.remove(throat); p2.throats.remove(throat)
    return {t1, t2}


def cut_network(network: Network, x: float, c: int, label: str = '',
    inouts: Dict = {}):
    """Cut a (periodic) network at position x_c with c>0.

    New pores are labelled and introduced at throat intersection points
    with the cutting plane. Before cutting, periodic throat connections
    in the c-direction are opened by calling open_periodic_network,
    which makes use of the inouts pore dictionary.
    """
    d = len(network.lb) # number of spatial dimensions
    # open periodic pores in direction c
    open_periodic_network(network, c, inouts)
    # cut throats incl. cut periodic ones being periodic normal to direction c
    throats = network.throats.copy() # shallow copy (iteration changes set)
    while len(throats) > 0:
        throats_next = set() # cut possible periodic pores in a 2nd round
        # cut throats ...
        for throat in throats:
            p1 = throat.pore1; p2 = throat.pore2
            # ... intersecting with cutting plane at x_c
            if (p1.pos[c-1] - x) * (x - p2.pos[c-1]) > 0:
                # treat periodic pores
                if len(throat.label) != 0:
                    direct = \
                        [int(k) for k in throat.label.split()] # periodicity
                    for k, cn in enumerate(direct):
                        if cn != 0: break # find first +/-1 periodic component
                    throats_periodic = __open_periodic_throat( \
                        network, throat, direct, k+1, inouts)
                    throats_next = throats_next | throats_periodic
                    continue
                # cut position
                f = (x - p1.pos[c-1]) / (p2.pos[c-1] - p1.pos[c-1])
                pos = [(p2.pos[j]-p1.pos[j])*f + p1.pos[j] for j in range(d)]
                pos[c-1] = x # to avoid tiny rounding errors
                # introduce and connect pore
                pore = Pore(pos, r=0.0, label=LABELS[3] + str(c) + label)
                network.add_pore(pore)
                # throat p1 - pore (reconnect throat from p2 to p1)
                throat.pore2 = pore; pore.throats.add(throat)
                # throat pore - p2 (new throat object)
                p2.throats.remove(throat)
                network.connect_pores(pore, p2, throat.r)
        # cut newly opened periodic throats
        throats = throats_next


def erase_network(network: Network, x: float, c: int, direct: bool, label: str):
    """Remove pores and connected throats from a network.
    
    Pores with positions < or > x_c for direct = True or False, respectively,
    are removed (c>0). At the same time, pores at x_c are labelled and network
    bounds updated.
    """
    for pore in network.pores.copy():
        # remove pores (and connected throats)
        if ((pore.pos[c-1] < x) and direct) or \
            ((x < pore.pos[c-1]) and not direct): network.remove_pore(pore)
        # relabel pores on cutting plane x_c
        elif pore.pos[c-1] == x: pore.label = label