from SimPEG import np, sp, Utils, Solver import matplotlib.pyplot as plt import matplotlib class ZCurve(object): """ The Z-order curve is generated by interleaving the bits of an offset. See: https://github.com/cortesi/scurve Aldo Cortesi """ def __init__(self, dimension, bits): """ dimension: Number of dimensions bits: The number of bits per co-ordinate. Total number of points is 2**(bits*dimension). """ self.dimension, self.bits = dimension, bits def bitrange(self, x, width, start, end): """ Extract a bit range as an integer. (start, end) is inclusive lower bound, exclusive upper bound. """ return x >> (width-end) & ((2**(end-start))-1) def index(self, p): p.reverse() idx = 0 iwidth = self.bits * self.dimension for i in range(iwidth): bitoff = self.bits-(i/self.dimension)-1 poff = self.dimension-(i%self.dimension)-1 b = self.bitrange(p[poff], self.bits, bitoff, bitoff+1) << i idx |= b return idx def point(self, idx): p = [0]*self.dimension iwidth = self.bits * self.dimension for i in range(iwidth): b = self.bitrange(idx, iwidth, i, i+1) << (iwidth-i-1)/self.dimension p[i%self.dimension] |= b p.reverse() return p def SortGrid(grid, offset=0): """ Sorts a grid by the x0 location. """ eps = 1e-7 def mycmp(c1,c2): c1 = grid[c1-offset] c2 = grid[c2-offset] if c1.size == 2: if np.abs(c1[1] - c2[1]) < eps: return c1[0] - c2[0] return c1[1] - c2[1] elif c1.size == 3: if np.abs(c1[2] - c2[2]) < eps: if np.abs(c1[1] - c2[1]) < eps: return c1[0] - c2[0] return c1[1] - c2[1] return c1[2] - c2[2] class K(object): def __init__(self, obj, *args): self.obj = obj def __lt__(self, other): return mycmp(self.obj, other.obj) < 0 def __gt__(self, other): return mycmp(self.obj, other.obj) > 0 def __eq__(self, other): return mycmp(self.obj, other.obj) == 0 def __le__(self, other): return mycmp(self.obj, other.obj) <= 0 def __ge__(self, other): return mycmp(self.obj, other.obj) >= 0 def __ne__(self, other): return mycmp(self.obj, other.obj) != 0 return sorted(range(offset,grid.shape[0]+offset), key=K) class Tree(object): def __init__(self, h_in, levels=3): assert type(h_in) is list, 'h_in must be a list' assert len(h_in) > 1, "len(h_in) must be greater than 1" h = range(len(h_in)) for i, h_i in enumerate(h_in): if type(h_i) in [int, long, float]: # This gives you something over the unit cube. h_i = np.ones(int(h_i))/int(h_i) assert isinstance(h_i, np.ndarray), ("h[%i] is not a numpy array." % i) assert len(h_i.shape) == 1, ("h[%i] must be a 1D numpy array." % i) assert len(h_i) == 2**levels, "must make h and levels match" h[i] = h_i[:] # make a copy. self.h = h self._levels = levels self._levelBits = int(np.ceil(np.sqrt(levels)))+1 self.__dirty__ = True #: The numbering is dirty! self._z = ZCurve(self.dim, 20) self._treeInds = set() self._treeInds.add(0) @property def levels(self): return self._levels @property def dim(self): return len(self.h) @property def nC(self): return len(self._treeInds) @property def nN(self): self.number() return self._nN @property def nF(self): self.number() return self._nF @property def nFx(self): self.number() return self._nFx @property def nFy(self): self.number() return self._nFy @property def nFz(self): self.number() return None if self.dim < 3 else self._nFz @property def nE(self): self.number() if self.dim == 2: return self.nF elif self.dim == 3: return len(self.edges) @property def nEx(self): self.number() if self.dim == 2: return self._nFy elif self.dim == 3: return self._nEx @property def nEy(self): self.number() if self.dim == 2: return self._nFx elif self.dim == 3: return self._nEy @property def nEz(self): self.number() return None if self.dim < 3 else self._nEz @property def vol(self): self.number() return self._vol @property def area(self): self.number() return self._area @property def edge(self): self.number() if self.dim == 2: return np.r_[self._area[self.nFx:], self._area[:self.nFx]] @property def _sortedInds(self): if getattr(self, '__sortedInds', None) is None: self.__sortedInds = sorted(self._treeInds) return self.__sortedInds @property def permuteCC(self): #TODO: cache these? P = SortGrid(self.gridCC) return sp.identity(self.nC).tocsr()[P,:] @property def permuteF(self): #TODO: cache these? P = SortGrid(self.gridFx) P += SortGrid(self.gridFy, offset=self.nFx) if self.dim == 3: P += SortGrid(self.gridFz, offset=self.nFx+self.nFy) return sp.identity(self.nF).tocsr()[P,:] @property def permuteE(self): #TODO: cache these? if self.dim == 2: P = SortGrid(self.gridFy) P += SortGrid(self.gridFx, offset=self.nEx) return sp.identity(self.nE).tocsr()[P,:] if self.dim == 3: raise Exception() def _structureChange(self): if self.__dirty__: return deleteThese = ['__sortedInds', '_gridCC', '_gridFx'] for p in deleteThese: if hasattr(self, p): delattr(self, p) self.__dirty__ = True def _index(self, pointer): assert len(pointer) is self.dim+1 assert pointer[-1] <= self.levels x = self._z.index([p for p in pointer[:-1]]) # copy return (x << self._levelBits) + pointer[-1] def _pointer(self, index): assert type(index) in [int, long] n = index & (2**self._levelBits-1) p = self._z.point(index >> self._levelBits) return p + [n] def __contains__(self, v): if type(v) in [int, long]: return v in self._treeInds return self._index(v) in self._treeInds def refine(self, function=None, recursive=True, cells=None): cells = cells if cells is not None else sorted(self._treeInds) recurse = [] for cell in cells: p = self._pointer(cell) do = function(self._cellC(cell)) > p[-1] if do: recurse += self._refineCell(cell) if recursive and len(recurse) > 0: self.refine(function=function, recursive=True, cells=recurse) return recurse def _refineCell(self, pointer): self._structureChange() pointer = self._asPointer(pointer) ind = self._asIndex(pointer) assert ind in self h = self._levelWidth(pointer[-1])/2 # halfWidth nL = pointer[-1] + 1 # new level add = lambda p:p[0]+p[1] added = [] def addCell(p): i = self._index(p+[nL]) self._treeInds.add(i) added.append(i) addCell(map(add, zip(pointer[:-1], [0,0,0][:self.dim]))) addCell(map(add, zip(pointer[:-1], [h,0,0][:self.dim]))) addCell(map(add, zip(pointer[:-1], [0,h,0][:self.dim]))) addCell(map(add, zip(pointer[:-1], [h,h,0][:self.dim]))) if self.dim == 3: addCell(map(add, zip(pointer[:-1], [0,0,h]))) addCell(map(add, zip(pointer[:-1], [h,0,h]))) addCell(map(add, zip(pointer[:-1], [0,h,h]))) addCell(map(add, zip(pointer[:-1], [h,h,h]))) self._treeInds.remove(ind) return added def _corsenCell(self, pointer): self._structureChange() raise Exception('Not yet implemented') def _asPointer(self, ind): if type(ind) in [int, long]: return self._pointer(ind) if type(ind) is list: return ind if isinstance(ind, np.ndarray): return ind.tolist() raise Exception def _asIndex(self, pointer): if type(pointer) in [int, long]: return pointer if type(pointer) is list: return self._index(pointer) raise Exception def _parentPointer(self, pointer): mod = self._levelWidth(pointer[-1]-1) return [p - (p % mod) for p in pointer[:-1]] + [pointer[-1]-1] def _cellN(self, p): p = self._asPointer(p) return [hi[:p[ii]].sum() for ii, hi in enumerate(self.h)] def _cellH(self, p): p = self._asPointer(p) w = self._levelWidth(p[-1]) return [hi[p[ii]:p[ii]+w].sum() for ii, hi in enumerate(self.h)] def _cellC(self, p): return (np.array(self._cellH(p))/2.0 + self._cellN(p)).tolist() def _levelWidth(self, level): return 2**(self.levels - level) def _isInsideMesh(self, pointer): inside = True for p in pointer[:-1]: inside = inside and p >= 0 and p < 2**self.levels return inside def _getNextCell(self, ind, direction=0, positive=True): """ Returns a None, int, list, or nested list The int is the cell number. """ pointer = self._asPointer(ind) step = (1 if positive else -1) * self._levelWidth(pointer[-1]) nextCell = [p if ii is not direction else p + step for ii, p in enumerate(pointer)] if not self._isInsideMesh(nextCell): return None # it might be the same size as me? if nextCell in self: return self._index(nextCell) # it might be smaller than me? if nextCell[-1] + 1 <= self.levels: # if I am not the smallest. nextCell[-1] += 1 if not positive: nextCell[direction] -= step/2 # Get the closer one if nextCell in self: # there is at least one hw = self._levelWidth(pointer[-1]) / 2 nextCell = np.array([p if ii is not direction else p + (step/2 if positive else 0) for ii, p in enumerate(pointer)]) if self.dim == 2: if direction == 0: children = [0,0,1], [0,hw,1] if direction == 1: children = [0,0,1], [hw,0,1] elif self.dim == 3: if direction == 0: children = [0,0,0,1], [0,hw,0,1], [0,0,hw,1], [0,hw,hw,1] if direction == 1: children = [0,0,0,1], [hw,0,0,1], [0,0,hw,1], [hw,0,hw,1] if direction == 2: children = [0,0,0,1], [hw,0,0,1], [0,hw,0,1], [hw,hw,0,1] nextCells = [] for child in children: nextCells.append(self._getNextCell(nextCell + child, direction=direction,positive=positive)) return nextCells # it might be bigger than me? return self._getNextCell(self._parentPointer(pointer), direction=direction, positive=positive) @property def gridCC(self): if getattr(self, '_gridCC', None) is None: self._gridCC = np.zeros((len(self._treeInds),self.dim)) for ii, ind in enumerate(self._sortedInds): p = self._asPointer(ind) self._gridCC[ii, :] = self._cellC(p) return self._gridCC @property def gridFx(self): if getattr(self, '_gridFx', None) is None: self.number() return self._gridFx @property def gridFy(self): if getattr(self, '_gridFy', None) is None: self.number() return self._gridFy @property def gridFz(self): if self.dim < 3: return None if getattr(self, '_gridFz', None) is None: self.number() return self._gridFz def _onSameLevel(self, i0, i1): p0 = self._asPointer(i0) p1 = self._asPointer(i1) return p0[-1] == p1[-1] def number(self, force=False): if not self.__dirty__ and not force: return facesX, facesY, facesZ = [], [], [] areaX, areaY, areaZ = [], [], [] hangingFacesX, hangingFacesY, hangingFacesZ = [], [], [] faceXCount, faceYCount, faceZCount = -1, -1, -1 fXm,fXp,fYm,fYp,fZm,fZp = range(6) vol, nodes = [], [] def addXFace(count, p, positive=True): n = self._cellN(p) w = self._cellH(p) areaX.append(w[1] if self.dim == 2 else w[1]*w[2]) if self.dim == 2: facesX.append([n[0] + (w[0] if positive else 0), n[1] + w[1]/2.0]) elif self.dim == 3: facesX.append([n[0] + (w[0] if positive else 0), n[1] + w[1]/2.0, n[2] + w[2]/2.0]) return count + 1 def addYFace(count, p, positive=True): n = self._cellN(p) w = self._cellH(p) areaY.append(w[0] if self.dim == 2 else w[0]*w[2]) if self.dim == 2: facesY.append([n[0] + w[0]/2.0, n[1] + (w[1] if positive else 0)]) elif self.dim == 3: facesY.append([n[0] + w[0]/2.0, n[1] + (w[1] if positive else 0), n[2] + w[2]/2.0]) return count + 1 def addZFace(count, p, positive=True): n = self._cellN(p) w = self._cellH(p) areaZ.append(w[0]*w[1]) facesZ.append([n[0] + w[0]/2.0, n[1] + w[1]/2.0, n[2] + (w[2] if positive else 0)]) return count + 1 # c2cn = dict() c2f = dict() def gc2f(ind): if ind in c2f: return c2f[ind] c2f_ind = [list() for _ in xrange(2*self.dim)] c2f[ind] = c2f_ind return c2f_ind def processCell(ind, faceCount, addFace, hangingFaces, DIR=0): fM,fP=(0,1) if DIR == 0 else (2,3) if DIR == 1 else (4,5) p = self._asPointer(ind) if self._getNextCell(p, direction=DIR, positive=False) is None: faceCount = addFace(faceCount, p, positive=False) gc2f(ind)[fM] += [faceCount] nextCell = self._getNextCell(p, direction=DIR) # Add the next Xface if nextCell is None: # on the boundary faceCount = addFace(faceCount, p) gc2f(ind)[fP] += [faceCount] elif type(nextCell) in [int, long] and self._onSameLevel(p,nextCell): # same sized cell faceCount = addFace(faceCount, p) gc2f(ind)[fP] += [faceCount] gc2f(nextCell)[fM] += [faceCount] elif type(nextCell) in [int, long] and not self._onSameLevel(p,nextCell): # the cell is bigger than me faceCount = addFace(faceCount, p) gc2f(ind)[fP] += [faceCount] gc2f(nextCell)[fM] += [faceCount] hangingFaces.append(faceCount) elif type(nextCell) is list: # the cell is smaller than me # TODO: ensure that things are balanced. p0 = self._pointer(nextCell[0]) p1 = self._pointer(nextCell[1]) faceCount = addFace(faceCount, p0, positive=False) gc2f(nextCell[0])[fM] += [faceCount] faceCount = addFace(faceCount, p1, positive=False) gc2f(nextCell[1])[fM] += [faceCount] gc2f(ind)[fP] += [faceCount-1,faceCount] hangingFaces += [faceCount-1, faceCount] return faceCount for ii, ind in enumerate(self._sortedInds): # c2cn[ind] = ii vol.append(np.prod(self._cellH(ind))) faceXCount = processCell(ind, faceXCount, addXFace, hangingFacesX, DIR=0) faceYCount = processCell(ind, faceYCount, addYFace, hangingFacesY, DIR=1) if self.dim == 3: faceZCount = processCell(ind, faceZCount, addZFace, hangingFacesZ, DIR=2) self._c2f = c2f self._area = np.array(areaX + areaY + (areaZ if self.dim == 3 else [])) self._vol = np.array(vol) self._gridFx = np.array(facesX) self._gridFy = np.array(facesY) self._hangingFacesX = hangingFacesX self._hangingFacesY = hangingFacesY if self.dim == 3: self._gridFz = np.array(facesZ) self._nFz = self._gridFz.shape[0] self._hangingFacesZ = hangingFacesZ self._nC = len(self._sortedInds) self._nFx = self._gridFx.shape[0] self._nFy = self._gridFy.shape[0] self._nF = self._nFx + self._nFy + (self._nFz if self.dim == 3 else 0) self.__dirty__ = False @property def faceDiv(self): # print self._c2f if getattr(self, '_faceDiv', None) is None: self.number() # TODO: Preallocate! I, J, V = [], [], [] PM = [-1,1]*self.dim # plus / minus offset = [0,0,self.nFx,self.nFx,self.nFx+self.nFy,self.nFx+self.nFy] for ii, ind in enumerate(self._sortedInds): faces = self._c2f[ind] for off, pm, face in zip(offset,PM,faces): j = [_ + off for _ in face] I += [ii]*len(j) J += j V += [pm]*len(j) VOL = self.vol D = sp.csr_matrix((V,(I,J)), shape=(self.nC, self.nF)) S = self.area self._faceDiv = Utils.sdiag(1.0/VOL)*D*Utils.sdiag(S) return self._faceDiv def plotGrid(self, ax=None, showIt=False): axOpts = {'projection':'3d'} if self.dim == 3 else {} if ax is None: ax = plt.subplot(111, **axOpts) else: assert isinstance(ax,matplotlib.axes.Axes), "ax must be an Axes!" fig = ax.figure for ind in self._sortedInds: p = self._asPointer(ind) n = self._cellN(p) h = self._cellH(p) x = [n[0] , n[0] + h[0], n[0] + h[0], n[0] , n[0]] y = [n[1] , n[1] , n[1] + h[1], n[1] + h[1], n[1]] z = [n[2] , n[2] , n[2] , n[2] , n[2]] ax.plot(x,y, 'b-', zs=None if self.dim == 2 else z) if self.dim == 3: z = [n[2] + h[2], n[2] + h[2], n[2] + h[2], n[2] + h[2], n[2] + h[2]] ax.plot(x,y, 'b-', zs=z) sides = [0,0], [h[0],0], [0,h[1]], [h[0],h[1]] for s in sides: x = [n[0] + s[0], n[0] + s[0]] y = [n[1] + s[1], n[1] + s[1]] z = [n[2] , n[2] + h[2]] ax.plot(x,y, 'b-', zs=z) ax.plot(self.gridCC[[0,-1],0], self.gridCC[[0,-1],1], 'ro', zs=None if self.dim == 2 else self.gridCC[[0,-1],2]) ax.plot(self.gridCC[:,0], self.gridCC[:,1], 'r.', zs=None if self.dim == 2 else self.gridCC[:,2]) ax.plot(self.gridCC[:,0], self.gridCC[:,1], 'r:', zs=None if self.dim == 2 else self.gridCC[:,2]) # ax.plot(self.gridFx[self._hangingFacesX,0], self.gridFx[self._hangingFacesX,1], 'gs', ms=10, mfc='none', mec='green', zs=None if self.dim == 2 else self.gridFx[self._hangingFacesX,2]) # ax.plot(self.gridFx[:,0], self.gridFx[:,1], 'g>', zs=None if self.dim == 2 else self.gridFx[:,2]) # ax.plot(self.gridFy[self._hangingFacesY,0], self.gridFy[self._hangingFacesY,1], 'gs', ms=10, mfc='none', mec='green', zs=None if self.dim == 2 else self.gridFy[self._hangingFacesY,2]) # ax.plot(self.gridFy[:,0], self.gridFy[:,1], 'g^', zs=None if self.dim == 2 else self.gridFy[:,2]) if self.dim == 3: ax.plot(self.gridFz[self._hangingFacesZ,0], self.gridFz[self._hangingFacesZ,1], 'gs', ms=10, mfc='none', mec='green', zs=self.gridFz[self._hangingFacesZ,2]) ax.plot(self.gridFz[:,0], self.gridFz[:,1], 'g^', zs=self.gridFz[:,2]) if showIt:plt.show() if __name__ == '__main__': def function(xc): r = xc - np.r_[0.5,0.5] dist = np.sqrt(r.dot(r)) # if dist < 0.05: # return 5 if dist < 0.1: return 4 if dist < 0.3: return 3 if dist < 1.0: return 2 else: return 0 T = Tree([4,4,4],levels=2) T.refine(lambda xc:1) T._refineCell([0,0,0,1]) T.plotGrid(showIt=True)