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https://github.com/wassname/simpeg.git
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rowanc1
543 lines
21 KiB
Python
543 lines
21 KiB
Python
from scipy import sparse as sp
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from SimPEG.Utils import sub2ind, ndgrid, mkvc, getSubArray, sdiag, inv3X3BlockDiagonal, inv2X2BlockDiagonal, makePropertyTensor, invPropertyTensor, spzeros, isScalar
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import numpy as np
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class InnerProducts(object):
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"""
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This is a base for the SimPEG.Mesh classes. This mixIn creates the all the inner product matrices that you need!
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"""
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def __init__(self):
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raise Exception('InnerProducts is a base class providing inner product matrices for meshes and cannot run on its own. Inherit to your favorite Mesh class.')
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def getFaceInnerProduct(self, materialProperty=None, returnP=False,
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invertProperty=False, doFast=True):
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"""
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:param numpy.array materialProperty: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
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:param bool returnP: returns the projection matrices
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:param bool invertProperty: inverts the material property
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:param bool doFast: do a faster implementation if available.
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:rtype: scipy.csr_matrix
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:return: M, the inner product matrix (nF, nF)
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"""
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fast = None
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if returnP is False and hasattr(self, '_fastFaceInnerProduct') and doFast:
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fast = self._fastFaceInnerProduct(materialProperty=materialProperty, invertProperty=invertProperty)
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if fast is not None:
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return fast
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if invertProperty:
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materialProperty = invPropertyTensor(self, materialProperty)
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Mu = makePropertyTensor(self, materialProperty)
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d = self.dim
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# We will multiply by sqrt on each side to keep symmetry
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V = sp.kron(sp.identity(d), sdiag(np.sqrt((2**(-d))*self.vol)))
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if d == 1:
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fP = _getFacePx(self)
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P000 = V*fP('fXm')
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P100 = V*fP('fXp')
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elif d == 2:
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fP = _getFacePxx(self)
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P000 = V*fP('fXm', 'fYm')
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P100 = V*fP('fXp', 'fYm')
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P010 = V*fP('fXm', 'fYp')
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P110 = V*fP('fXp', 'fYp')
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elif d == 3:
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fP = _getFacePxxx(self)
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P000 = V*fP('fXm', 'fYm', 'fZm')
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P100 = V*fP('fXp', 'fYm', 'fZm')
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P010 = V*fP('fXm', 'fYp', 'fZm')
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P110 = V*fP('fXp', 'fYp', 'fZm')
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P001 = V*fP('fXm', 'fYm', 'fZp')
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P101 = V*fP('fXp', 'fYm', 'fZp')
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P011 = V*fP('fXm', 'fYp', 'fZp')
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P111 = V*fP('fXp', 'fYp', 'fZp')
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A = P000.T*Mu*P000 + P100.T*Mu*P100
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P = [P000, P100]
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if d > 1:
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A = A + P010.T*Mu*P010 + P110.T*Mu*P110
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P += [P010, P110]
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if d > 2:
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A = A + P001.T*Mu*P001 + P101.T*Mu*P101 + P011.T*Mu*P011 + P111.T*Mu*P111
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P += [P001, P101, P011, P111]
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if returnP:
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return A, P
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else:
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return A
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def getFaceInnerProductDeriv(self, materialProperty=None, v=None, P=None, doFast=True):
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"""
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:param numpy.array materialProperty: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
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:param numpy.array v: vector to multiply (required in the general implementation)
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:param list P: list of projection matrices
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:param bool doFast: do a faster implementation if available.
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:rtype: scipy.csr_matrix
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:return: dMdm, the derivative of the inner product matrix (nF, nC*nA)
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"""
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fast = None
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if hasattr(self, '_fastFaceInnerProductDeriv') and doFast:
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fast = self._fastFaceInnerProductDeriv(materialProperty=materialProperty, v=v)
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if fast is not None:
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return fast
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if P is None:
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M, P = self.getFaceInnerProduct(materialProperty=materialProperty, returnP=True)
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return self._getInnerProductDeriv(materialProperty, v, P, self.nF)
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def getEdgeInnerProduct(self, materialProperty=None, returnP=False,
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invertProperty=False, doFast=True):
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"""
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:param numpy.array materialProperty: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
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:param bool returnP: returns the projection matrices
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:param bool invertProperty: inverts the material property
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:param bool doFast: do a faster implementation if available.
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:rtype: scipy.csr_matrix
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:return: M, the inner product matrix (nE, nE)
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"""
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fast = None
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if returnP is False and hasattr(self, '_fastEdgeInnerProduct') and doFast:
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fast = self._fastEdgeInnerProduct(materialProperty=materialProperty, invertProperty=invertProperty)
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if fast is not None:
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return fast
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if invertProperty:
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materialProperty = invPropertyTensor(self, materialProperty)
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Mu = makePropertyTensor(self, materialProperty)
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d = self.dim
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# We will multiply by sqrt on each side to keep symmetry
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V = sp.kron(sp.identity(d), sdiag(np.sqrt((2**(-d))*self.vol)))
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if d == 1:
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raise NotImplementedError('getEdgeInnerProduct not implemented for 1D')
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elif d == 2:
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eP = _getEdgePxx(self)
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P000 = V*eP('eX0', 'eY0')
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P100 = V*eP('eX0', 'eY1')
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P010 = V*eP('eX1', 'eY0')
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P110 = V*eP('eX1', 'eY1')
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elif d == 3:
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eP = _getEdgePxxx(self)
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P000 = V*eP('eX0', 'eY0', 'eZ0')
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P100 = V*eP('eX0', 'eY1', 'eZ1')
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P010 = V*eP('eX1', 'eY0', 'eZ2')
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P110 = V*eP('eX1', 'eY1', 'eZ3')
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P001 = V*eP('eX2', 'eY2', 'eZ0')
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P101 = V*eP('eX2', 'eY3', 'eZ1')
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P011 = V*eP('eX3', 'eY2', 'eZ2')
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P111 = V*eP('eX3', 'eY3', 'eZ3')
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Mu = makePropertyTensor(self, materialProperty)
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A = P000.T*Mu*P000 + P100.T*Mu*P100 + P010.T*Mu*P010 + P110.T*Mu*P110
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P = [P000, P100, P010, P110]
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if d == 3:
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A = A + P001.T*Mu*P001 + P101.T*Mu*P101 + P011.T*Mu*P011 + P111.T*Mu*P111
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P += [P001, P101, P011, P111]
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if returnP:
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return A, P
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else:
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return A
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def getEdgeInnerProductDeriv(self, materialProperty=None, v=None, P=None, doFast=True):
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"""
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:param numpy.array materialProperty: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
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:param numpy.array v: vector to multiply (required in the general implementation)
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:param list P: list of projection matrices
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:param bool doFast: do a faster implementation if available.
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:rtype: scipy.csr_matrix
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:return: dMdm, the derivative of the inner product matrix (nE, nC*nA)
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"""
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fast = None
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if hasattr(self, '_fastEdgeInnerProductDeriv') and doFast:
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fast = self._fastEdgeInnerProductDeriv(materialProperty=materialProperty, v=v)
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if fast is not None:
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return fast
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if P is None:
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M, P = self.getEdgeInnerProduct(materialProperty=materialProperty, returnP=True)
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return self._getInnerProductDeriv(materialProperty, v, P, self.nE)
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def _getInnerProductDeriv(self, materialProperty, v, P, n):
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"""
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:param numpy.array materialProperty: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
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:param numpy.array v: vector to multiply (required in the general implementation)
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:param list P: list of projection matrices
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:param int n: nF or nE
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:rtype: scipy.csr_matrix
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:return: dMdm, the derivative of the inner product matrix (n, nC*nA)
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"""
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if materialProperty is None:
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return None
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if v is None:
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raise Exception('v must be supplied for this implementation.')
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d = self.dim
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Z = spzeros(self.nC, self.nC)
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if isScalar(materialProperty):
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dMdm = spzeros(n, 1)
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for i, p in enumerate(P):
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dMdm = dMdm + sp.csr_matrix((p.T * (p * v), (range(n), np.zeros(n))), shape=(n,1))
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if d == 1:
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if materialProperty.size == self.nC:
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dMdm = spzeros(n, self.nC)
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for i, p in enumerate(P):
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dMdm = dMdm + p.T * sdiag( p * v )
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elif d == 2:
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if materialProperty.size == self.nC:
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dMdm = spzeros(n, self.nC)
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:]
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dMdm = dMdm + p.T * sp.vstack((sdiag( y1 ), sdiag( y2 )))
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elif materialProperty.size == self.nC*2:
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dMdms = [spzeros(n, self.nC) for _ in range(2)]
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:]
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dMdms[0] = dMdms[0] + p.T * sp.vstack(( sdiag( y1 ), Z))
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dMdms[1] = dMdms[1] + p.T * sp.vstack(( Z, sdiag( y2 )))
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dMdm = sp.hstack(dMdms)
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elif materialProperty.size == self.nC*3:
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dMdms = [spzeros(n, self.nC) for _ in range(3)]
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:]
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dMdms[0] = dMdms[0] + p.T * sp.vstack(( sdiag( y1 ), Z))
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dMdms[1] = dMdms[1] + p.T * sp.vstack(( Z, sdiag( y2 )))
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dMdms[2] = dMdms[2] + p.T * sp.vstack(( sdiag( y2 ), sdiag( y1 )))
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dMdm = sp.hstack(dMdms)
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elif d == 3:
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if materialProperty.size == self.nC:
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dMdm = spzeros(n, self.nC)
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:self.nC*2]
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y3 = Y[self.nC*2:]
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dMdm = dMdm + p.T * sp.vstack((sdiag( y1 ), sdiag( y2 ), sdiag( y3 )))
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elif materialProperty.size == self.nC*3:
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dMdms = [spzeros(n, self.nC) for _ in range(3)]
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:self.nC*2]
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y3 = Y[self.nC*2:]
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dMdms[0] = dMdms[0] + p.T * sp.vstack(( sdiag( y1 ), Z, Z))
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dMdms[1] = dMdms[1] + p.T * sp.vstack(( Z, sdiag( y2 ), Z))
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dMdms[2] = dMdms[2] + p.T * sp.vstack(( Z, Z, sdiag( y3 )))
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dMdm = sp.hstack(dMdms)
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elif materialProperty.size == self.nC*6:
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dMdms = [spzeros(n, self.nC) for _ in range(6)]
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for i, p in enumerate(P):
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Y = p * v
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y1 = Y[:self.nC]
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y2 = Y[self.nC:self.nC*2]
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y3 = Y[self.nC*2:]
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dMdms[0] = dMdms[0] + p.T * sp.vstack(( sdiag( y1 ), Z, Z))
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dMdms[1] = dMdms[1] + p.T * sp.vstack(( Z, sdiag( y2 ), Z))
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dMdms[2] = dMdms[2] + p.T * sp.vstack(( Z, Z, sdiag( y3 )))
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dMdms[3] = dMdms[3] + p.T * sp.vstack(( sdiag( y2 ), sdiag( y1 ), Z))
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dMdms[4] = dMdms[4] + p.T * sp.vstack(( sdiag( y3 ), Z, sdiag( y1 )))
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dMdms[5] = dMdms[5] + p.T * sp.vstack(( Z, sdiag( y3 ), sdiag( y2 )))
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dMdm = sp.hstack(dMdms)
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return dMdm
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# ------------------------ Geometries ------------------------------
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#
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#
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# node(i,j,k+1) ------ edge2(i,j,k+1) ----- node(i,j+1,k+1)
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# / /
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# / / |
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# edge3(i,j,k) face1(i,j,k) edge3(i,j+1,k)
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# / / |
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# / / |
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# node(i,j,k) ------ edge2(i,j,k) ----- node(i,j+1,k)
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# | | |
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# | | node(i+1,j+1,k+1)
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# | | /
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# edge1(i,j,k) face3(i,j,k) edge1(i,j+1,k)
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# | | /
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# | | /
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# | |/
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# node(i+1,j,k) ------ edge2(i+1,j,k) ----- node(i+1,j+1,k)
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def _getFacePx(M):
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assert M._meshType == 'TENSOR', 'Only supported for a tensor mesh'
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return _getFacePx_Rectangular(M)
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def _getFacePxx(M):
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if M._meshType == 'TREE':
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return M._getFacePxx
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return _getFacePxx_Rectangular(M)
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def _getFacePxxx(M):
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if M._meshType == 'TREE':
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return M._getFacePxxx
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return _getFacePxxx_Rectangular(M)
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def _getEdgePxx(M):
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if M._meshType == 'TREE':
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return M._getEdgePxx
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return _getEdgePxx_Rectangular(M)
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def _getEdgePxxx(M):
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if M._meshType == 'TREE':
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return M._getEdgePxxx
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return _getEdgePxxx_Rectangular(M)
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def _getFacePx_Rectangular(M):
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"""Returns a function for creating projection matrices
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"""
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ii = np.int64(range(M.nCx))
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def Px(xFace):
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"""
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xFace is 'fXp' or 'fXm'
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"""
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posFx = 0 if xFace == 'fXm' else 1
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IND = ii + posFx
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PX = sp.csr_matrix((np.ones(M.nC), (range(M.nC), IND)), shape=(M.nC, M.nF))
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return PX
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return Px
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def _getFacePxx_Rectangular(M):
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"""returns a function for creating projection matrices
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Mats takes you from faces a subset of all faces on only the
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faces that you ask for.
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These are centered around a single nodes.
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For example, if this was your entire mesh:
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f3(Yp)
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2_______________3
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f0(Xm) | x | f1(Xp)
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|_______________|
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0 1
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f2(Ym)
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Pxx('fXm','fYm') = | 1, 0, 0, 0 |
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| 0, 0, 1, 0 |
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Pxx('fXp','fYm') = | 0, 1, 0, 0 |
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| 0, 0, 1, 0 |
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"""
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i, j = np.int64(range(M.nCx)), np.int64(range(M.nCy))
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iijj = ndgrid(i, j)
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ii, jj = iijj[:, 0], iijj[:, 1]
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if M._meshType == 'LRM':
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fN1 = M.r(M.normals, 'F', 'Fx', 'M')
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fN2 = M.r(M.normals, 'F', 'Fy', 'M')
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def Pxx(xFace, yFace):
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"""
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xFace is 'fXp' or 'fXm'
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yFace is 'fYp' or 'fYm'
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"""
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# no | node | f1 | f2
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# 00 | i ,j | i , j | i, j
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# 10 | i+1,j | i+1, j | i, j
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# 01 | i ,j+1 | i , j | i, j+1
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# 11 | i+1,j+1 | i+1, j | i, j+1
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posFx = 0 if xFace == 'fXm' else 1
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posFy = 0 if yFace == 'fYm' else 1
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ind1 = sub2ind(M.vnFx, np.c_[ii + posFx, jj])
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ind2 = sub2ind(M.vnFy, np.c_[ii, jj + posFy]) + M.nFx
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IND = np.r_[ind1, ind2].flatten()
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PXX = sp.csr_matrix((np.ones(2*M.nC), (range(2*M.nC), IND)), shape=(2*M.nC, M.nF))
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if M._meshType == 'LRM':
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I2x2 = inv2X2BlockDiagonal(getSubArray(fN1[0], [i + posFx, j]), getSubArray(fN1[1], [i + posFx, j]),
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getSubArray(fN2[0], [i, j + posFy]), getSubArray(fN2[1], [i, j + posFy]))
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PXX = I2x2 * PXX
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return PXX
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return Pxx
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def _getFacePxxx_Rectangular(M):
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"""returns a function for creating projection matrices
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Mats takes you from faces a subset of all faces on only the
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faces that you ask for.
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These are centered around a single nodes.
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"""
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i, j, k = np.int64(range(M.nCx)), np.int64(range(M.nCy)), np.int64(range(M.nCz))
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iijjkk = ndgrid(i, j, k)
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ii, jj, kk = iijjkk[:, 0], iijjkk[:, 1], iijjkk[:, 2]
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if M._meshType == 'LRM':
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fN1 = M.r(M.normals, 'F', 'Fx', 'M')
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fN2 = M.r(M.normals, 'F', 'Fy', 'M')
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fN3 = M.r(M.normals, 'F', 'Fz', 'M')
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def Pxxx(xFace, yFace, zFace):
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"""
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xFace is 'fXp' or 'fXm'
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yFace is 'fYp' or 'fYm'
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zFace is 'fZp' or 'fZm'
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"""
|
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# no | node | f1 | f2 | f3
|
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# 000 | i ,j ,k | i , j, k | i, j , k | i, j, k
|
|
# 100 | i+1,j ,k | i+1, j, k | i, j , k | i, j, k
|
|
# 010 | i ,j+1,k | i , j, k | i, j+1, k | i, j, k
|
|
# 110 | i+1,j+1,k | i+1, j, k | i, j+1, k | i, j, k
|
|
# 001 | i ,j ,k+1 | i , j, k | i, j , k | i, j, k+1
|
|
# 101 | i+1,j ,k+1 | i+1, j, k | i, j , k | i, j, k+1
|
|
# 011 | i ,j+1,k+1 | i , j, k | i, j+1, k | i, j, k+1
|
|
# 111 | i+1,j+1,k+1 | i+1, j, k | i, j+1, k | i, j, k+1
|
|
|
|
posX = 0 if xFace == 'fXm' else 1
|
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posY = 0 if yFace == 'fYm' else 1
|
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posZ = 0 if zFace == 'fZm' else 1
|
|
|
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ind1 = sub2ind(M.vnFx, np.c_[ii + posX, jj, kk])
|
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ind2 = sub2ind(M.vnFy, np.c_[ii, jj + posY, kk]) + M.nFx
|
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ind3 = sub2ind(M.vnFz, np.c_[ii, jj, kk + posZ]) + M.nFx + M.nFy
|
|
|
|
IND = np.r_[ind1, ind2, ind3].flatten()
|
|
|
|
PXXX = sp.coo_matrix((np.ones(3*M.nC), (range(3*M.nC), IND)), shape=(3*M.nC, M.nF)).tocsr()
|
|
|
|
if M._meshType == 'LRM':
|
|
I3x3 = inv3X3BlockDiagonal(getSubArray(fN1[0], [i + posX, j, k]), getSubArray(fN1[1], [i + posX, j, k]), getSubArray(fN1[2], [i + posX, j, k]),
|
|
getSubArray(fN2[0], [i, j + posY, k]), getSubArray(fN2[1], [i, j + posY, k]), getSubArray(fN2[2], [i, j + posY, k]),
|
|
getSubArray(fN3[0], [i, j, k + posZ]), getSubArray(fN3[1], [i, j, k + posZ]), getSubArray(fN3[2], [i, j, k + posZ]))
|
|
PXXX = I3x3 * PXXX
|
|
|
|
return PXXX
|
|
return Pxxx
|
|
|
|
def _getEdgePxx_Rectangular(M):
|
|
i, j = np.int64(range(M.nCx)), np.int64(range(M.nCy))
|
|
|
|
iijj = ndgrid(i, j)
|
|
ii, jj = iijj[:, 0], iijj[:, 1]
|
|
|
|
if M._meshType == 'LRM':
|
|
eT1 = M.r(M.tangents, 'E', 'Ex', 'M')
|
|
eT2 = M.r(M.tangents, 'E', 'Ey', 'M')
|
|
|
|
def Pxx(xEdge, yEdge):
|
|
# no | node | e1 | e2
|
|
# 00 | i ,j | i ,j | i ,j
|
|
# 10 | i+1,j | i ,j | i+1,j
|
|
# 01 | i ,j+1 | i ,j+1 | i ,j
|
|
# 11 | i+1,j+1 | i ,j+1 | i+1,j
|
|
posX = 0 if xEdge == 'eX0' else 1
|
|
posY = 0 if yEdge == 'eY0' else 1
|
|
|
|
ind1 = sub2ind(M.vnEx, np.c_[ii, jj + posX])
|
|
ind2 = sub2ind(M.vnEy, np.c_[ii + posY, jj]) + M.nEx
|
|
|
|
IND = np.r_[ind1, ind2].flatten()
|
|
|
|
PXX = sp.coo_matrix((np.ones(2*M.nC), (range(2*M.nC), IND)), shape=(2*M.nC, M.nE)).tocsr()
|
|
|
|
if M._meshType == 'LRM':
|
|
I2x2 = inv2X2BlockDiagonal(getSubArray(eT1[0], [i, j + posX]), getSubArray(eT1[1], [i, j + posX]),
|
|
getSubArray(eT2[0], [i + posY, j]), getSubArray(eT2[1], [i + posY, j]))
|
|
PXX = I2x2 * PXX
|
|
|
|
return PXX
|
|
return Pxx
|
|
|
|
def _getEdgePxxx_Rectangular(M):
|
|
i, j, k = np.int64(range(M.nCx)), np.int64(range(M.nCy)), np.int64(range(M.nCz))
|
|
|
|
iijjkk = ndgrid(i, j, k)
|
|
ii, jj, kk = iijjkk[:, 0], iijjkk[:, 1], iijjkk[:, 2]
|
|
|
|
if M._meshType == 'LRM':
|
|
eT1 = M.r(M.tangents, 'E', 'Ex', 'M')
|
|
eT2 = M.r(M.tangents, 'E', 'Ey', 'M')
|
|
eT3 = M.r(M.tangents, 'E', 'Ez', 'M')
|
|
|
|
def Pxxx(xEdge, yEdge, zEdge):
|
|
|
|
# no | node | e1 | e2 | e3
|
|
# 000 | i ,j ,k | i ,j ,k | i ,j ,k | i ,j ,k
|
|
# 100 | i+1,j ,k | i ,j ,k | i+1,j ,k | i+1,j ,k
|
|
# 010 | i ,j+1,k | i ,j+1,k | i ,j ,k | i ,j+1,k
|
|
# 110 | i+1,j+1,k | i ,j+1,k | i+1,j ,k | i+1,j+1,k
|
|
# 001 | i ,j ,k+1 | i ,j ,k+1 | i ,j ,k+1 | i ,j ,k
|
|
# 101 | i+1,j ,k+1 | i ,j ,k+1 | i+1,j ,k+1 | i+1,j ,k
|
|
# 011 | i ,j+1,k+1 | i ,j+1,k+1 | i ,j ,k+1 | i ,j+1,k
|
|
# 111 | i+1,j+1,k+1 | i ,j+1,k+1 | i+1,j ,k+1 | i+1,j+1,k
|
|
|
|
posX = [0,0] if xEdge == 'eX0' else [1, 0] if xEdge == 'eX1' else [0,1] if xEdge == 'eX2' else [1,1]
|
|
posY = [0,0] if yEdge == 'eY0' else [1, 0] if yEdge == 'eY1' else [0,1] if yEdge == 'eY2' else [1,1]
|
|
posZ = [0,0] if zEdge == 'eZ0' else [1, 0] if zEdge == 'eZ1' else [0,1] if zEdge == 'eZ2' else [1,1]
|
|
|
|
ind1 = sub2ind(M.vnEx, np.c_[ii, jj + posX[0], kk + posX[1]])
|
|
ind2 = sub2ind(M.vnEy, np.c_[ii + posY[0], jj, kk + posY[1]]) + M.nEx
|
|
ind3 = sub2ind(M.vnEz, np.c_[ii + posZ[0], jj + posZ[1], kk]) + M.nEx + M.nEy
|
|
|
|
IND = np.r_[ind1, ind2, ind3].flatten()
|
|
|
|
PXXX = sp.coo_matrix((np.ones(3*M.nC), (range(3*M.nC), IND)), shape=(3*M.nC, M.nE)).tocsr()
|
|
|
|
if M._meshType == 'LRM':
|
|
I3x3 = inv3X3BlockDiagonal(getSubArray(eT1[0], [i, j + posX[0], k + posX[1]]), getSubArray(eT1[1], [i, j + posX[0], k + posX[1]]), getSubArray(eT1[2], [i, j + posX[0], k + posX[1]]),
|
|
getSubArray(eT2[0], [i + posY[0], j, k + posY[1]]), getSubArray(eT2[1], [i + posY[0], j, k + posY[1]]), getSubArray(eT2[2], [i + posY[0], j, k + posY[1]]),
|
|
getSubArray(eT3[0], [i + posZ[0], j + posZ[1], k]), getSubArray(eT3[1], [i + posZ[0], j + posZ[1], k]), getSubArray(eT3[2], [i + posZ[0], j + posZ[1], k]))
|
|
PXXX = I3x3 * PXXX
|
|
|
|
return PXXX
|
|
return Pxxx
|
|
|
|
if __name__ == '__main__':
|
|
from TensorMesh import TensorMesh
|
|
h = [np.array([1, 2, 3, 4]), np.array([1, 2, 1, 4, 2]), np.array([1, 1, 4, 1])]
|
|
M = TensorMesh(h)
|
|
mu = np.ones((M.nC, 6))
|
|
A, P = M.getFaceInnerProduct(mu, returnP=True)
|
|
B, P = M.getEdgeInnerProduct(mu, returnP=True)
|