\( \usepackage{euler} % must be loaded before fontspec for the whole doc (below); this must be kept for pngmath, however \usepackage{hyperref} \usepackage{amsmath} \usepackage{amsbsy} %\usepackage{mathabx} \usepackage{underscore} \usepackage[all]{xy} % Metadata of the pdf output \hypersetup{pdftitle={Yade Documentation}} \hypersetup{pdfauthor={Smilauer, V., et al.}} \hypersetup{pdfkeywords={Discrete element method, granular materials, contact law, hydromechanical coupling}} % symbols \renewcommand{\mat}{\boldsymbol} \renewcommand{\vec}{\boldsymbol} \renewcommand{\tens}{\boldsymbol} %normalized and locfram here \def\normalized#1{\widehat{#1}} \def\locframe#1{\widetilde{#1}} % timestep \def\Dt{\Delta t} \def\Dtcr{\Dt_{\rm cr}} % algorithm complexity \def\bigO#1{{\mathcal{O}(#1)}} % variants for greek symbols \let\epsilon\varepsilon \let\theta\vartheta \let\phi\varphi % shorthands \let\sig\sigma \let\eps\epsilon % variables at different points of time \def\prev#1{#1^-} \def\pprev#1{#1^\ominus} \def\curr#1{#1^{\circ}} \def\nnext#1{#1^\oplus} \def\next#1{#1^+} % shorthands for geometry \def\currn{\curr{\vec{n}}} \def\currC{\curr{\vec{C}}} \def\uT{\vec{u}_T} \def\curruT{\curr{\vec{u}}_T} \def\prevuT{\prev{\vec{u}}_T} \def\currn{\curr{\vec{n}}} \def\prevn{\prev{\vec{n}}} % motion \def\pprevvel{\pprev{\dot{\vec{u}}}} \def\nnextvel{\nnext{\dot{\vec{u}}}} \def\curraccel{\curr{\ddot{\vec{u}}}} \def\prevpos{\prev{\vec{u}}} \def\currpos{\curr{\vec{u}}} \def\nextpos{\next{\vec{u}}} \def\curraaccel{\curr{\dot{\vec{\omega}}}} \def\pprevangvel{\pprev{\vec{\omega}}} \def\nnextangvel{\nnext{\vec{\omega}}} \def\loccurr#1{\curr{\locframe{#1}}} \def\numCPU{n_{\rm cpu}} \DeclareMathOperator{\Align}{Align} \DeclareMathOperator{\sign}{sgn} % sorting algorithms \def\isleq#1{\currelem{#1}\ar@/^/[ll]^{\leq}} \def\isnleq#1{\currelem{#1}\ar@/^/[ll]^{\not\leq}} \def\currelem#1{\fbox{$#1$}} \def\sortSep{||} \def\sortInv{\hbox{\phantom{||}}} \def\sortlines#1{\xymatrix@=3pt{#1}} \def\crossBound{||\mkern-18mu<} \)

# Source code for yade.pack

    # encoding: utf-8
    # 2009 © Václav Šmilauer <eudoxos@arcig.cz>
    """
    Creating packings and filling volumes defined by boundary representation or constructive solid geometry.

    For examples, see

    * :ysrc:`examples/gts-horse/gts-operators.py`
    * :ysrc:`examples/gts-horse/gts-random-pack-obb.py`
    * :ysrc:`examples/gts-horse/gts-random-pack.py`
    * :ysrc:`examples/test/pack-cloud.py`
    * :ysrc:`examples/test/pack-predicates.py`
    * :ysrc:`examples/packs/packs.py`
    * :ysrc:`examples/gts-horse/gts-horse.py`
    * :ysrc:`examples/WireMatPM/wirepackings.py`
    """
    import itertools, warnings
    from numpy import arange
    from math import sqrt
    from yade import utils

    from yade.wrapper import *

    from yade.minieigenHP import *

    ## compatibility hack for python 2.5 (21/8/2009)
    ## can be safely removed at some point
    if 'product' not in dir(itertools):

       def product(*args, **kwds):
          "http://docs.python.org/library/itertools.html#itertools.product"
          pools = list(map(tuple, args)) * kwds.get('repeat', 1)
          result = [[]]
          for pool in pools:
             result = [x + [y] for x in result for y in pool]
          for prod in result:
             yield tuple(prod)

       itertools.product = product

    # for now skip the import, but try in inGtsSurface constructor again, to give error if we really use it
    try:
       import gts
    except ImportError:
       pass

    from yade._packPredicates import *  ## imported in randomDensePack as well

    # import SpherePack
    from yade._packSpheres import *
    from yade._packObb import *

    ##
    # extend _packSphere.SpherePack c++ class by this method
    ##
    if (True):

       def SpherePack_toSimulation(self, rot=Matrix3.Identity, **kw):
          r"""Append spheres directly to the simulation. In addition calling :yref:`O.bodies.append<BodyContainer.append>`,
      this method also appropriately sets periodic cell information of the simulation.

        >>> from yade import pack; from math import *
        >>> sp=pack.SpherePack()

      Create random periodic packing with 20 spheres:

        >>> sp.makeCloud((0,0,0),(5,5,5),rMean=.5,rRelFuzz=.5,periodic=True,num=20)
        20

      Virgin simulation is aperiodic:

        >>> O.reset()
        >>> O.periodic
        False

      Add generated packing to the simulation, rotated by 45° along +z

        >>> sp.toSimulation(rot=Quaternion((0,0,1),pi/4),color=(0,0,1))
        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

      Periodic properties are transferred to the simulation correctly, including rotation (this could be avoided by explicitly passing "hSize=O.cell.hSize" as an argument):

        >>> O.periodic
        True
        >>> O.cell.refSize
        Vector3(5,5,5)
        >>> O.cell.hSize # doctest: +SKIP
        Matrix3(3.53553,-3.53553,0, 3.53553,3.53553,0, 0,0,5)

      The current state (even if rotated) is taken as mechanically undeformed, i.e. with identity transformation:

        >>> O.cell.trsf
        Matrix3(1,0,0, 0,1,0, 0,0,1)

      :param Quaternion/Matrix3 rot: rotation of the packing, which will be applied on spheres and will be used to set :yref:`Cell.trsf` as well.
      :param \*\*kw: passed to :yref:`yade.utils.sphere`
      :return: list of body ids added (like :yref:`O.bodies.append<BodyContainer.append>`)
      """
          if isinstance(rot, Quaternion):
             rot = rot.toRotationMatrix()
          assert (isinstance(rot, Matrix3))
          if self.isPeriodic:
             O.periodic = True
          if self.cellSize != Vector3.Zero and self.isPeriodic:
             O.cell.hSize = rot * Matrix3(self.cellSize[0], 0, 0, 0, self.cellSize[1], 0, 0, 0, self.cellSize[2])
             O.cell.trsf = Matrix3.Identity
          if not self.hasClumps():
             return O.bodies.append([utils.sphere(rot * c, r, **kw) for c, r in self])
          else:
             standalone, clumps = self.getClumps()
             ids = O.bodies.append([utils.sphere(rot * c, r, **kw) for c, r in self])  # append all spheres first
             clumpIds = []
             userColor = 'color' in kw
             for clump in clumps:
                clumpIds.append(
                        O.bodies.clump([ids[i] for i in clump])
                )  # clump spheres with given ids together, creating the clump object as well
                # make all spheres within one clump a single color, unless color was specified by the user
                if not userColor:
                   for i in clump[1:]:
                      O.bodies[ids[i]].shape.color = O.bodies[ids[clump[0]]].shape.color
             return ids + clumpIds

       SpherePack.toSimulation = SpherePack_toSimulation

       class inGtsSurface_py(Predicate):
          """This class was re-implemented in c++, but should stay here to serve as reference for implementing
        Predicates in pure python code. C++ allows us to play dirty tricks in GTS which are not accessible
        through pygts itself; the performance penalty of pygts comes from fact that if constructs and destructs
        bb tree for the surface at every invocation of gts.Point().is_inside(). That is cached in the c++ code,
        provided that the surface is not manipulated with during lifetime of the object (user's responsibility).

        ---

        Predicate for GTS surfaces. Constructed using an already existing surfaces, which must be closed.

          import gts
          surf=gts.read(open('horse.gts'))
          inGtsSurface(surf)

        .. note::
          Padding is optionally supported by testing 6 points along the axes in the pad distance. This
          must be enabled in the ctor by saying doSlowPad=True. If it is not enabled and pad is not zero,
          warning is issued.
        """

          def __init__(self, surf, noPad=False):
             # call base class ctor; necessary for virtual methods to work as expected.
             # see comments in _packPredicates.cpp for struct PredicateWrap.
             super(inGtsSurface, self).__init__()
             if not surf.is_closed():
                raise RuntimeError("Surface for inGtsSurface predicate must be closed.")
             self.surf = surf
             self.noPad = noPad
             inf = float('inf')
             mn, mx = [inf, inf, inf], [-inf, -inf, -inf]
             for v in surf.vertices():
                c = v.coords()
                mn, mx = [min(mn[i], c[i]) for i in (0, 1, 2)], [max(mx[i], c[i]) for i in (0, 1, 2)]
             self.mn, self.mx = tuple(mn), tuple(mx)
             import gts

          def aabb(self):
             return self.mn, self.mx

          def __call__(self, _pt, pad=0.):
             p = gts.Point(*_pt)
             if self.noPad:
                if pad != 0:
                   warnings.warn("Padding disabled in ctor, using 0 instead.")
                return p.is_inside(self.surf)
             pp = [
                     gts.Point(_pt[0] - pad, _pt[1], _pt[2]),
                     gts.Point(_pt[0] + pad, _pt[1], _pt[2]),
                     gts.Point(_pt[0], _pt[1] - pad, _pt[2]),
                     gts.Point(_pt[0], _pt[1] + pad, _pt[2]),
                     gts.Point(_pt[0], _pt[1], _pt[2] - pad),
                     gts.Point(_pt[0], _pt[1], _pt[2] + pad)
             ]
             return p.is_inside(self.surf) and pp[0].is_inside(self.surf) and pp[1].is_inside(self.surf) and pp[2].is_inside(
                     self.surf
             ) and pp[3].is_inside(self.surf) and pp[4].is_inside(self.surf) and pp[5].is_inside(self.surf)

       class inSpace(Predicate):
          """Predicate returning True for any points, with infinite bounding box."""

          def __init__(self, _center=Vector3().Zero):
             self._center = _center

          def aabb(self):
             inf = float('inf')
             return Vector3(-inf, -inf, -inf), Vector3(inf, inf, inf)

          def center(self):
             return self._center

          def dim(self):
             inf = float('inf')
             return Vector3(inf, inf, inf)

          def __call__(self, pt, pad):
             return True

       class inHalfSpace(Predicate):
          """Predicate returning True  any points, with infinite bounding box."""

          def __init__(self, _center=Vector3().Zero, _dir=Vector3(1, 0, 0)):
             self._center = Vector3(_center)
             self._dir = Vector3(_dir)
             assert self._dir.norm() > 0., "Direction has to be nonzero vector"
             self._dir.normalize()
             self._d = -self._dir.dot(self._center)

          def aabb(self):
             d, c = self._dir, self._center
             inf = float('inf')
             min = Vector3(-inf, -inf, -inf)
             max = Vector3(+inf, +inf, +inf)
             for i in range(3):
                j = (i + 1) % 3
                k = (i + 2) % 3
                if d[i] == 0 and d[j] == 0:
                   if d[k] > 0:
                      min[k] = c[k]
                   else:
                      max[k] = c[k]
             return min, max

          def center(self):
             return self._center

          def dim(self):
             inf = float('inf')
             return Vector3(inf, inf, inf)

          def __call__(self, pt, pad):
             v = self._dir.dot(pt) + self._d
             return v > pad

       class inConvexPolyhedron(Predicate):

          def __init__(self, planes):
             self._inHalfSpaces = [inHalfSpace(c, d) for c, d in planes]
             self._min, self._max = self._computeAabb()

          def _computeAabb(self):
             try:
                import scipy.optimize
             except ImportError:
                raise ImportError("scipy (package python-scipy) needed for pack.inConvexPolyhedron")
             min, max = Vector3.Zero, Vector3.Zero
             A, b = [], []
             for h in self._inHalfSpaces:
                A.append(tuple(-h._dir))
                b.append(h._d)
             inf = float('inf')
             for i in range(3):
                c = Vector3.Zero
                #
                c[i] = 1
                opt = scipy.optimize.linprog(c, A_ub=A, b_ub=b, bounds=(-inf, inf))
                errmsg = "Something wrong in pack.inConvexPolyhedron defining planes.\nThe scipy.optimize.linprog output:\n{}\n"
                if not opt.success:
                   raise ValueError(errmsg.format(opt))
                min[i] = opt.x[i]
                #
                c[i] = -1
                opt = scipy.optimize.linprog(c, A_ub=A, b_ub=b, bounds=(-inf, inf))
                if not opt.success:
                   raise ValueError(errmsg.format(opt))
                max[i] = opt.x[i]
             return min, max

          def aabb(self):
             return Vector3(self._min), Vector3(self._max)

          def center(self):
             return .5 * (self._min + self._max)

          def dim(self):
             return self._max - self._min

          def __call__(self, pt, pad):
             for p in self._inHalfSpaces:
                if not p(pt, pad):
                   return False
             return True

    #####
    ## surface construction and manipulation
    #####

    def gtsSurface2Facets(surf, **kw):
       r"""Construct facets from given GTS surface. \*\*kw is passed to utils.facet."""
       import gts
       return [utils.facet([v.coords() for v in face.vertices()], **kw) for face in surf.faces()]

    def sweptPolylines2gtsSurface(pts, threshold=0, capStart=False, capEnd=False):
       """Create swept suface (as GTS triangulation) given same-length sequences of points (as 3-tuples).

    If threshold is given (>0), then

    * degenerate faces (with edges shorter than threshold) will not be created
    * gts.Surface().cleanup(threshold) will be called before returning, which merges vertices mutually closer than threshold. In case your pts are closed (last point concident with the first one) this will the surface strip of triangles. If you additionally have capStart==True and capEnd==True, the surface will be closed.

    .. note:: capStart and capEnd make the most naive polygon triangulation (diagonals) and will perhaps fail for non-convex sections.

    .. warning:: the algorithm connects points sequentially; if two polylines are mutually rotated or have inverse sense, the algorithm will not detect it and connect them regardless in their given order.
       """
       import gts  # will raise an exception in gts-less builds
       if not len(set([len(pts1) for pts1 in pts])) == 1:
          raise RuntimeError("Polylines must be all of the same length!")
       vtxs = [[gts.Vertex(x, y, z) for x, y, z in pts1] for pts1 in pts]
       sectEdges = [[gts.Edge(vtx[i], vtx[i + 1]) for i in range(0, len(vtx) - 1)] for vtx in vtxs]
       interSectEdges = [[] for i in range(0, len(vtxs) - 1)]
       for i in range(0, len(vtxs) - 1):
          for j in range(0, len(vtxs[i])):
             interSectEdges[i].append(gts.Edge(vtxs[i][j], vtxs[i + 1][j]))
             if j < len(vtxs[i]) - 1:
                interSectEdges[i].append(gts.Edge(vtxs[i][j], vtxs[i + 1][j + 1]))
       if threshold > 0:  # replace edges of zero length with None; their faces will be skipped

          def fixEdges(edges):
             for i, e in enumerate(edges):
                if (Vector3(e.v1.x, e.v1.y, e.v1.z) - Vector3(e.v2.x, e.v2.y, e.v2.z)).norm() < threshold:
                   edges[i] = None

          for ee in sectEdges:
             fixEdges(ee)
          for ee in interSectEdges:
             fixEdges(ee)
       surf = gts.Surface()
       for i in range(0, len(vtxs) - 1):
          for j in range(0, len(vtxs[i]) - 1):
             ee1 = interSectEdges[i][2 * j + 1], sectEdges[i + 1][j], interSectEdges[i][2 * j]
             ee2 = sectEdges[i][j], interSectEdges[i][2 * j + 2], interSectEdges[i][2 * j + 1]
             if None not in ee1:
                surf.add(gts.Face(interSectEdges[i][2 * j + 1], sectEdges[i + 1][j], interSectEdges[i][2 * j]))
             if None not in ee2:
                surf.add(gts.Face(sectEdges[i][j], interSectEdges[i][2 * j + 2], interSectEdges[i][2 * j + 1]))

       def doCap(vtx, edg, start):
          ret = []
          eFan = [edg[0]] + [gts.Edge(vtx[i], vtx[0]) for i in range(2, len(vtx))]
          for i in range(1, len(edg)):
             ret += [gts.Face(eFan[i - 1], eFan[i], edg[i]) if start else gts.Face(eFan[i - 1], edg[i], eFan[i])]
          return ret

       caps = []
       if capStart:
          caps += doCap(vtxs[0], sectEdges[0], start=True)
       if capEnd:
          caps += doCap(vtxs[-1], sectEdges[-1], start=False)
       for cap in caps:
          surf.add(cap)
       if threshold > 0:
          surf.cleanup(threshold)
       return surf

    def gtsSurfaceBestFitOBB(surf):
       """Return (Vector3 center, Vector3 halfSize, Quaternion orientation) describing
       best-fit oriented bounding box (OBB) for the given surface. See cloudBestFitOBB
       for details."""
       import gts
       pts = [Vector3(v.x, v.y, v.z) for v in surf.vertices()]
       return cloudBestFitOBB(tuple(pts))

    def revolutionSurfaceMeridians(sects, angles, origin=Vector3().Zero, orientation=Quaternion().Identity):
       """Revolution surface given sequences of 2d points and sequence of corresponding angles,
       returning sequences of 3d points representing meridian sections of the revolution surface.
       The 2d sections are turned around z-axis, but they can be transformed
       using the origin and orientation arguments to give arbitrary orientation."""
       import math

       def toGlobal(x, y, z):
          return tuple(origin + orientation * (Vector3(x, y, z)))

       return [[toGlobal(x2d * math.cos(angles[i]), x2d * math.sin(angles[i]), y2d) for x2d, y2d in sects[i]] for i in range(0, len(sects))]

    ########
    ## packing generators
    ########

    def regularOrtho(predicate, radius, gap, **kw):
       """Return set of spheres in regular orthogonal grid, clipped inside solid given by predicate.
       Created spheres will have given radius and will be separated by gap space."""
       ret = []
       mn, mx = predicate.aabb()
       if (max([mx[i] - mn[i] for i in (0, 1, 2)]) == float('inf')):
          raise ValueError(
                  "Aabb of the predicate must not be infinite (didn't you use union | instead of intersection & for unbounded predicate such as notInNotch?"
          )
       xx, yy, zz = [arange(mn[i] + radius, mx[i] - radius, 2 * radius + gap) for i in (0, 1, 2)]
       for xyz in itertools.product(xx, yy, zz):
          if predicate(xyz, radius):
             ret += [utils.sphere(xyz, radius=radius, **kw)]
       if (len(ret) == 0):
          warnings.warn('No spheres are produced by regularOrtho-function', category=RuntimeWarning)
       return ret

    def regularHexa(predicate, radius, gap, **kw):
       """Return set of spheres in regular hexagonal grid, clipped inside solid given by predicate.
       Created spheres will have given radius and will be separated by gap space."""
       ret = []
       a = 2 * radius + gap
       hy, hz = a * sqrt(3) / 2., a * sqrt(6) / 3.
       mn, mx = predicate.aabb()
       dim = [mx[i] - mn[i] for i in (0, 1, 2)]
       if (max(dim) == float('inf')):
          raise ValueError(
                  "Aabb of the predicate must not be infinite (didn't you use union | instead of intersection & for unbounded predicate such as notInNotch?"
          )
       ii, jj, kk = [list(range(0, int(dim[0] / a) + 1)), list(range(0, int(dim[1] / hy) + 1)), list(range(0, int(dim[2] / hz) + 1))]
       for i, j, k in itertools.product(ii, jj, kk):
          #Simple HCP-lattice packing
          #http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres#Simple_hcp_lattice
          coordSph = Vector3((2 * i + ((j + k) % 2)), (sqrt(3.) * (j + 1. / 3. * (k % 2))), (2. * sqrt(6.) / 3. * k)) * (a / 2.0) + mn
          if predicate(coordSph, radius):
             ret += [utils.sphere(coordSph, radius=radius, **kw)]
       if (len(ret) == 0):
          warnings.warn('No spheres are produced by regularHexa-function', category=RuntimeWarning)
       return ret

    def filterSpherePack(predicate, spherePack, returnSpherePack=None, **kw):
       """Using given SpherePack instance, return spheres that satisfy predicate.
       It returns either a :yref:`yade._packSpheres.SpherePack` (if returnSpherePack) or a list.
       The packing will be recentered to match the predicate and warning is given if the predicate
       is larger than the packing."""
       if returnSpherePack == None:
          warnings.warn(
                  'The default behavior will change; specify returnSpherePack=True for the new behavior, and False to get rid of this warning (your code will break in the future, however). The returned SpherePack object can be added to the simulation using SpherePack.toSimulation()',
                  category=FutureWarning
          )
          returnSpherePack = False
       mn, mx = predicate.aabb()
       dimP, centP = predicate.dim(), predicate.center()
       dimS, centS = spherePack.dim(), spherePack.center()
       if dimP[0] > dimS[0] or dimP[1] > dimS[1] or dimP[2] > dimS[2]:
          warnings.warn("Packing's dimension (%s) doesn't fully contain dimension of the predicate (%s)." % (dimS, dimP))
       spherePack.translate(centP - centS)
       if returnSpherePack:
          ret = SpherePack()
          for c, r in spherePack:
             if predicate(c, r):
                ret.add(c, r)
          return ret
       else:
          # return particles to be added to O.bodies
          ret = []
          for s in spherePack:
             if predicate(s[0], s[1]):
                ret += [utils.sphere(s[0], radius=s[1], **kw)]
          return ret

    def _memoizePacking(memoizeDb, sp, radius, rRelFuzz, wantPeri, fullDim, noPrint=False):
       import sys
       if not memoizeDb:
          return
       import pickle, sqlite3, time, os
       if os.path.exists(memoizeDb):
          conn = sqlite3.connect(memoizeDb)
       else:
          conn = sqlite3.connect(memoizeDb)
          c = conn.cursor()
          c.execute(
                  'create table packings (radius real, rRelFuzz real, dimx real, dimy real, dimz real, N integer, timestamp real, periodic integer, pack blob)'
          )
       c = conn.cursor()
       packBlob = memoryview(pickle.dumps(sp.toList(), pickle.HIGHEST_PROTOCOL))
       packDim = sp.cellSize if wantPeri else fullDim
       c.execute(
               'insert into packings values (?,?,?,?,?,?,?,?,?)', (
                       radius,
                       rRelFuzz,
                       packDim[0],
                       packDim[1],
                       packDim[2],
                       len(sp),
                       time.time(),
                       wantPeri,
                       packBlob,
               )
       )
       c.close()
       conn.commit()
       if not noPrint:
          print("Packing saved to the database", memoizeDb)

    def _getMemoizedPacking(memoizeDb, radius, rRelFuzz, x1, y1, z1, fullDim, wantPeri, fillPeriodic, spheresInCell, memoDbg=False, noPrint=False):
       """Return suitable SpherePack read from *memoizeDb* if found, None otherwise.

          :param fillPeriodic: whether to fill fullDim by repeating periodic packing
          :param wantPeri: only consider periodic packings
       """
       import os, os.path, sqlite3, time, pickle, sys
       if memoDbg and not noPrint:

          def memoDbgMsg(s):
             print(s)
       else:

          def memoDbgMsg(s):
             pass

       if not memoizeDb or not os.path.exists(memoizeDb):
          if memoizeDb:
             memoDbgMsg("Database %s does not exist." % memoizeDb)
          return None
       # find suitable packing and return it directly
       conn = sqlite3.connect(memoizeDb)
       c = conn.cursor()
       try:
          c.execute('select radius,rRelFuzz,dimx,dimy,dimz,N,timestamp,periodic from packings order by N')
       except sqlite3.OperationalError:
          raise RuntimeError(
                  "ERROR: database `" + memoizeDb +
                  "' not compatible with randomDensePack (corrupt, deprecated format or not a db created by randomDensePack)"
          )
       for row in c:
          R, rDev, X, Y, Z, NN, timestamp, isPeri = row[0:8]
          scale = radius / R
          rDev *= scale
          X *= scale
          Y *= scale
          Z *= scale
          memoDbgMsg(
                  "Considering packing (radius=%g±%g,N=%g,dim=%g×%g×%g,%s,scale=%g), created %s" %
                  (R, .5 * rDev, NN, X, Y, Z, "periodic" if isPeri else "non-periodic", scale, time.asctime(time.gmtime(timestamp)))
          )
          if not isPeri and wantPeri:
             memoDbgMsg("REJECT: is not periodic, which is requested.")
             continue
          if wantPeri and (X / x1 > 0.9 or X / x1 < 0.6):
             memoDbgMsg("REJECT: initSize differs too much from scaled packing size.")
             continue
          if (rRelFuzz == 0 and rDev != 0) or (rRelFuzz != 0 and rDev == 0) or (rRelFuzz != 0 and abs((rDev - rRelFuzz) / rRelFuzz) > 1e-2):
             memoDbgMsg("REJECT: radius fuzz differs too much (%g, %g desired)" % (rDev, rRelFuzz))
             continue  # radius fuzz differs too much
          if isPeri and wantPeri:
             if spheresInCell > NN and spheresInCell > 0:
                memoDbgMsg("REJECT: Number of spheres in the packing too small")
                continue
             if abs((y1 / x1) / (Y / X) - 1) > 0.3 or abs((z1 / x1) / (Z / X) - 1) > 0.3:
                memoDbgMsg(
                        "REJECT: proportions (y/x=%g, z/x=%g) differ too much from what is desired (%g, %g)." %
                        (Y / X, Z / X, y1 / x1, z1 / x1)
                )
                continue
          else:
             if (X < fullDim[0] or Y < fullDim[1] or Z < fullDim[2]):
                memoDbgMsg("REJECT: not large enough")
                continue  # not large enough
          memoDbgMsg("ACCEPTED")
          if not noPrint:
             print(
                     "Found suitable packing in %s (radius=%g±%g,N=%g,dim=%g×%g×%g,%s,scale=%g), created %s" %
                     (memoizeDb, R, rDev, NN, X, Y, Z, "periodic" if isPeri else "non-periodic", scale, time.asctime(time.gmtime(timestamp)))
             )
          c.execute('select pack from packings where timestamp=?', (timestamp,))
          sp = SpherePack(pickle.loads(c.fetchone()[0]))
          sp.scale(scale)
          if isPeri and wantPeri:
             sp.isPeriodic = True
             sp.cellSize = (X, Y, Z)
             if fillPeriodic:
                sp.cellFill(Vector3(fullDim[0], fullDim[1], fullDim[2]))
          #sp.cellSize=(0,0,0) # resetting cellSize avoids warning when rotating
          return sp
          #if orientation: sp.rotate(*orientation.toAxisAngle())
          #return filterSpherePack(predicate,sp,material=material)
       #print "No suitable packing in database found, running",'PERIODIC compression' if wantPeri else 'triaxial'
       #sys.stdout.flush()

    def randomDensePack(
            predicate,
            radius,
            material=-1,
            dim=None,
            cropLayers=0,
            rRelFuzz=0.,
            spheresInCell=0,
            memoizeDb=None,
            useOBB=False,
            memoDbg=False,
            color=None,
            returnSpherePack=None,
            seed=-1
    ):
       """Generator of random dense packing with given geometry properties, using TriaxialTest (aperiodic)
       or PeriIsoCompressor (periodic). The periodicity depens on whether   the spheresInCell parameter is given.

       *O.switchScene()* magic is used to have clean simulation for TriaxialTest without deleting the original simulation.
       This function therefore should never run in parallel with some code accessing your simulation.

       :param predicate: solid-defining predicate for which we generate packing
       :param spheresInCell: if given, the packing will be periodic, with given number of spheres in the periodic cell.
       :param radius: mean radius of spheres
       :param rRelFuzz: relative fuzz of the radius -- e.g. radius=10, rRelFuzz=.2, then spheres will have radii 10 ± (10*.2)), with an uniform distribution.
          0 by default, meaning all spheres will have exactly the same radius.
       :param cropLayers: (aperiodic only) how many layers of spheres will be added to the computed dimension of the box so that there no
          (or not so much, at least) boundary effects at the boundaries of the predicate.
       :param dim: dimension of the packing, to override dimensions of the predicate (if it is infinite, for instance)
       :param memoizeDb: name of sqlite database (existent or nonexistent) to find an already generated packing or to store
          the packing that will be generated, if not found (the technique of caching results of expensive computations
          is known as memoization). Fuzzy matching is used to select suitable candidate -- packing will be scaled, rRelFuzz
          and dimensions compared. Packing that are too small are dictarded. From the remaining candidate, the one with the
          least number spheres will be loaded and returned.
       :param useOBB: effective only if a inGtsSurface predicate is given. If true (not default), oriented bounding box will be
          computed first; it can reduce substantially number of spheres for the triaxial compression (like 10× depending on
          how much asymmetric the body is), see examples/gts-horse/gts-random-pack-obb.py
       :param memoDbg: show packings that are considered and reasons why they are rejected/accepted
       :param returnSpherePack: see the corresponding argument in :yref:`yade.pack.filterSpherePack`

       :return: SpherePack object with spheres, filtered by the predicate.
       """
       import sqlite3, os.path, pickle, time, sys, numpy
       from math import pi
       from yade import _packPredicates
       wantPeri = (spheresInCell > 0)
       if 'inGtsSurface' in dir(_packPredicates) and type(predicate) == inGtsSurface and useOBB:
          center, dim, orientation = gtsSurfaceBestFitOBB(predicate.surf)
          print("Best-fit oriented-bounding-box computed for GTS surface, orientation is", orientation)
          dim *= 2  # gtsSurfaceBestFitOBB returns halfSize
       else:
          if not dim:
             dim = predicate.dim()
          if max(dim) == float('inf'):
             raise RuntimeError("Infinite predicate and no dimension of packing requested.")
          center = predicate.center()
          orientation = None
       if not wantPeri:
          fullDim = tuple([dim[i] + 4 * cropLayers * radius for i in (0, 1, 2)])
       else:
          # compute cell dimensions now, as they will be compared to ones stored in the db
          # they have to be adjusted to not make the cell to small WRT particle radius
          fullDim = dim
          cloudPorosity = 0.25  # assume this number for the initial cloud (can be underestimated)
          beta, gamma = fullDim[1] / fullDim[0], fullDim[2] / fullDim[0]  # ratios β=y₀/x₀, γ=z₀/x₀
          N100 = spheresInCell / cloudPorosity  # number of spheres for cell being filled by spheres without porosity
          x1 = radius * (1 / (beta * gamma) * N100 * (4 / 3.) * pi)**(1 / 3.)
          y1, z1 = beta * x1, gamma * x1
          vol0 = x1 * y1 * z1
          maxR = radius * (1 + rRelFuzz)
          x1 = max(x1, 8 * maxR)
          y1 = max(y1, 8 * maxR)
          z1 = max(z1, 8 * maxR)
          vol1 = x1 * y1 * z1
          N100 *= vol1 / vol0  # volume might have been increased, increase number of spheres to keep porosity the same
          sp = _getMemoizedPacking(
                  memoizeDb, radius, rRelFuzz, x1, y1, z1, fullDim, wantPeri, fillPeriodic=True, spheresInCell=spheresInCell, memoDbg=False
          )
          if sp:
             if orientation:
                sp.cellSize = (0, 0, 0)  # resetting cellSize avoids warning when rotating
                sp.rotate(*orientation.toAxisAngle())
             return filterSpherePack(predicate, sp, material=material, returnSpherePack=returnSpherePack)
          else:
             print("No suitable packing in database found, running", 'PERIODIC compression' if wantPeri else 'triaxial')
          sys.stdout.flush()
       O.switchScene()
       O.resetThisScene()  ### !!
       if wantPeri:
          # x1,y1,z1 already computed above
          sp = SpherePack()
          O.periodic = True
          #O.cell.refSize=(x1,y1,z1)
          O.cell.setBox((x1, y1, z1))
          #print cloudPorosity,beta,gamma,N100,x1,y1,z1,O.cell.refSize
          #print x1,y1,z1,radius,rRelFuzz
          O.materials.append(FrictMat(young=3e10, density=2400))
          num = sp.makeCloud(Vector3().Zero, O.cell.refSize, radius, rRelFuzz, spheresInCell, True, seed=seed)
          O.engines = [
                  ForceResetter(),
                  InsertionSortCollider([Bo1_Sphere_Aabb()], verletDist=.05 * radius),
                  InteractionLoop([Ig2_Sphere_Sphere_ScGeom()], [Ip2_FrictMat_FrictMat_FrictPhys()], [Law2_ScGeom_FrictPhys_CundallStrack()]),
                  PeriIsoCompressor(
                          charLen=2 * radius, stresses=[-100e9, -1e8], maxUnbalanced=1e-2, doneHook='O.pause();', globalUpdateInt=5, keepProportions=True
                  ),
                  NewtonIntegrator(damping=.6)
          ]
          O.materials.append(FrictMat(young=30e9, frictionAngle=.5, poisson=.3, density=1e3))
          for s in sp:
             O.bodies.append(utils.sphere(s[0], s[1]))
          O.dt = utils.PWaveTimeStep()
          O.run()
          O.wait()
          sp = SpherePack()
          sp.fromSimulation()
          #print 'Resulting cellSize',sp.cellSize,'proportions',sp.cellSize[1]/sp.cellSize[0],sp.cellSize[2]/sp.cellSize[0]
          # repetition to the required cell size will be done below, after memoizing the result
       else:
          assumedFinalDensity = 0.6
          V = (4.0 / 3.0) * pi * radius**3.0
          N = assumedFinalDensity * fullDim[0] * fullDim[1] * fullDim[2] / V
          TriaxialTest(
                  numberOfGrains=int(N),
                  radiusMean=radius,
                  radiusStdDev=rRelFuzz,
                  # upperCorner is just size ratio, if radiusMean is specified
                  upperCorner=fullDim,
                  seed=seed,
                  ## no need to touch any the following
                  noFiles=True,
                  lowerCorner=[0, 0, 0],
                  sigmaIsoCompaction=-4e4,
                  sigmaLateralConfinement=-5e2,
                  compactionFrictionDeg=1,
                  StabilityCriterion=.02,
                  strainRate=.2,
                  thickness=0,
                  maxWallVelocity=.1,
                  wallOversizeFactor=1.5,
                  autoUnload=True,
                  autoCompressionActivation=False,
                  internalCompaction=True
          ).load()
          while (numpy.isnan(utils.unbalancedForce()) or utils.unbalancedForce() > 0.005):
             O.run(500, True)
          sp = SpherePack()
          sp.fromSimulation()
       O.switchScene()  ### !!
       _memoizePacking(memoizeDb, sp, radius, rRelFuzz, wantPeri, fullDim)
       if wantPeri:
          sp.cellFill(Vector3(fullDim[0], fullDim[1], fullDim[2]))
       if orientation:
          sp.cellSize = (0, 0, 0)
          # reset periodicity to avoid warning when rotating periodic packing
          sp.rotate(*orientation.toAxisAngle())
       return filterSpherePack(predicate, sp, material=material, color=color, returnSpherePack=returnSpherePack)

    def randomPeriPack(radius, initSize, rRelFuzz=0.0, memoizeDb=None, noPrint=False, seed=-1):
       """Generate periodic dense packing.

       A cell of initSize is stuffed with as many spheres as possible, then we run periodic compression with PeriIsoCompressor, just like with
       randomDensePack.

       :param radius: mean sphere radius
       :param rRelFuzz: relative fuzz of sphere radius (equal distribution); see the same param for randomDensePack.
       :param initSize: initial size of the periodic cell.

       :return: SpherePack object, which also contains periodicity information.
       """
       from math import pi
       sp = _getMemoizedPacking(
               memoizeDb,
               radius,
               rRelFuzz,
               initSize[0],
               initSize[1],
               initSize[2],
               fullDim=Vector3(0, 0, 0),
               wantPeri=True,
               fillPeriodic=False,
               spheresInCell=-1,
               memoDbg=True,
               noPrint=noPrint
       )
       if sp:
          return sp
       O.switchScene()
       O.resetThisScene()
       sp = SpherePack()
       O.periodic = True
       #O.cell.refSize=initSize
       O.cell.setBox(initSize)
       sp.makeCloud(Vector3().Zero, O.cell.refSize, radius, rRelFuzz, -1, True, seed=seed)
       O.engines = [
               ForceResetter(),
               InsertionSortCollider([Bo1_Sphere_Aabb()], verletDist=.05 * radius),
               InteractionLoop([Ig2_Sphere_Sphere_ScGeom()], [Ip2_FrictMat_FrictMat_FrictPhys()], [Law2_ScGeom_FrictPhys_CundallStrack()]),
               PeriIsoCompressor(
                       charLen=2 * radius, stresses=[-100e9, -1e8], maxUnbalanced=1e-2, doneHook='O.pause();', globalUpdateInt=20, keepProportions=True
               ),
               NewtonIntegrator(damping=.8)
       ]
       O.materials.append(FrictMat(young=30e9, frictionAngle=.1, poisson=.3, density=1e3))
       for s in sp:
          O.bodies.append(utils.sphere(s[0], s[1]))
       O.dt = utils.PWaveTimeStep()
       O.timingEnabled = True
       O.run()
       O.wait()
       ret = SpherePack()
       ret.fromSimulation()
       _memoizePacking(memoizeDb, ret, radius, rRelFuzz, wantPeri=True, fullDim=Vector3(0, 0, 0), noPrint=noPrint)  # fullDim unused
       O.switchScene()
       return ret

    def hexaNet(radius, cornerCoord=[0, 0, 0], xLength=1., yLength=0.5, mos=0.08, a=0.04, b=0.04, startAtCorner=True, isSymmetric=False, **kw):
       """Definition of the particles for a hexagonal wire net in the x-y-plane for the WireMatPM.

       :param radius: radius of the particle
       :param cornerCoord: coordinates of the lower left corner of the net
       :param xLenght: net length in x-direction
       :param yLenght: net length in y-direction
       :param mos: mesh opening size (horizontal distance between the double twists)
       :param a: length of double-twist
       :param b: height of single wire section
       :param startAtCorner: if true the generation starts with a double-twist at the lower left corner
       :param isSymmetric: defines if the net is symmetric with respect to the y-axis

       :return: set of spheres which defines the net (net) and exact dimensions of the net (lx,ly).

       .. note:: This packing works for the WireMatPM only. The particles at the corner are always generated first. For examples on how to use this packing see examples/WireMatPM. In order to create the proper interactions for the net the interaction radius has to be adapted in the simulation.
       """
       # check input dimension
       if (xLength < mos):
          raise ValueError("xLength must be greater than mos!")
       if (yLength < 2 * a + b):
          raise ValueError("yLength must be greater than 2*a+b!")
       xstart = cornerCoord[0]
       ystart = cornerCoord[1]
       z = cornerCoord[2]
       ab = a + b
       # number of double twisted sections in y-direction and real length ly
       ny = int((yLength - a) / ab) + 1
       ly = ny * a + (ny - 1) * b
       jump = 0
       # number of sections in x-direction and real length lx
       if isSymmetric:
          nx = int(xLength / mos) + 1
          lx = (nx - 1) * mos
          if not startAtCorner:
             nx += -1
       else:
          nx = int((xLength - 0.5 * mos) / mos) + 1
          lx = (nx - 1) * mos + 0.5 * mos
       net = []
       # generate corner particles
       if startAtCorner:
          if (ny % 2 == 0):  # if ny even no symmetry in y-direction
             net += [utils.sphere((xstart, ystart + ly, z), radius=radius, **kw)]  # upper left corner
             if isSymmetric:
                net += [utils.sphere((xstart + lx, ystart + ly, z), radius=radius, **kw)]  # upper right corner
             else:
                net += [utils.sphere((xstart + lx, ystart, z), radius=radius, **kw)]  # lower right corner
          else:  # if ny odd symmetry in y-direction
             if not isSymmetric:
                net += [utils.sphere((xstart + lx, ystart, z), radius=radius, **kw)]  # lower right corner
                net += [utils.sphere((xstart + lx, ystart + ly, z), radius=radius, **kw)]  # upper right corner
          jump = 1
       else:  # do not start at corner
          if (ny % 2 == 0):  # if ny even no symmetry in y-direction
             net += [utils.sphere((xstart, ystart, z), radius=radius, **kw)]  # lower left corner
             if isSymmetric:
                net += [utils.sphere((xstart + lx, ystart, z), radius=radius, **kw)]  # lower right corner
             else:
                net += [utils.sphere((xstart + lx, ystart + ly, z), radius=radius, **kw)]  # upper right corner
          else:  # if ny odd symmetry in y-direction
             net += [utils.sphere((xstart, ystart, z), radius=radius, **kw)]  # lower left corner
             net += [utils.sphere((xstart, ystart + ly, z), radius=radius, **kw)]  # upper left corner
             if isSymmetric:
                net += [utils.sphere((xstart + lx, ystart, z), radius=radius, **kw)]  # lower right corner
                net += [utils.sphere((xstart + lx, ystart + ly, z), radius=radius, **kw)]  # upper right corner
          xstart += 0.5 * mos
       # generate other particles
       if isSymmetric:
          for i in range(ny):
             y = ystart + i * ab
             for j in range(nx):
                x = xstart + j * mos
                # add two particles of one vertical section (double-twist)
                net += [utils.sphere((x, y, z), radius=radius, **kw)]
                net += [utils.sphere((x, y + a, z), radius=radius, **kw)]
             # set values for next section
             xstart = xstart - 0.5 * mos * pow(-1, i + jump)
             nx = int(nx + 1 * pow(-1, i + jump))
       else:
          for i in range(ny):
             y = ystart + i * ab
             for j in range(nx):
                x = xstart + j * mos
                # add two particles of one vertical section (double-twist)
                net += [utils.sphere((x, y, z), radius=radius, **kw)]
                net += [utils.sphere((x, y + a, z), radius=radius, **kw)]
             # set values for next section
             xstart = xstart - 0.5 * mos * pow(-1, i + jump)
       return [net, lx, ly]