from scipy import sparse as sp from SimPEG.Utils import sub2ind, ndgrid, mkvc, getSubArray, sdiag, inv3X3BlockDiagonal, inv2X2BlockDiagonal, makePropertyTensor import numpy as np class InnerProducts(object): """ Class creates the inner product matrices that you need! InnerProducts is a base class providing inner product matrices for meshes and cannot run on its own. Inherit to your favorite Mesh class. **Example problem for DC resistivity** .. math:: \sigma^{-1}\mathbf{J} = \\nabla \phi We can define in weak form by integrating with a general face function F: .. math:: \int_{\\text{cell}}{\sigma^{-1}\mathbf{J} \cdot \mathbf{F}} = \int_{\\text{cell}}{\\nabla \phi \cdot \mathbf{F}} \int_{\\text{cell}}{\sigma^{-1}\mathbf{J} \cdot \mathbf{F}} = \int_{\\text{cell}}{(\\nabla \cdot \mathbf{F}) \phi } + \int_{\partial \\text{cell}}{ \phi \mathbf{F} \cdot \mathbf{n}} We can then discretize for every cell: .. math:: v_{\\text{cell}} \sigma^{-1} (\mathbf{J}_x \mathbf{F}_x +\mathbf{J}_y \mathbf{F}_y + \mathbf{J}_z \mathbf{F}_z ) = -\phi^{\\top} v_{\\text{cell}} (\mathbf{D}_{\\text{cell}} \mathbf{F}) + \\text{BC} We can represent this in vector form (again this is for every cell), and will generalize for the case of anisotropic (tensor) sigma. .. math:: \mathbf{F}_c^{\\top} (\sqrt{v_{\\text{cell}}} \Sigma^{-1} \sqrt{v_{\\text{cell}}}) \mathbf{J}_c = -\phi^{\\top} v_{\\text{cell}}( v_\\text{cell}^{-1} \mathbf{D}_{\\text{cell}} \mathbf{A} \mathbf{F}) + \\text{BC} We multiply by volume on each side of the tensor conductivity to keep symmetry in the system. Here J_c is the Cartesian J (on the faces) and must be calculated differently depending on the mesh: .. math:: \mathbf{J}_c = \mathbf{Q}_{(i)}\mathbf{J}_\\text{TENSOR} = \mathbf{N}_{(i)}^{-1}\mathbf{Q}_{(i)}\mathbf{J}_\\text{LOM} Here the i index refers to where we choose to approximate this integral. We will approximate this relation at every node of the cell, there are 8 in 3D, using a projection matrix Q_i to pick the appropriate fluxes. We will then average to the cell center. For the TENSOR mesh, this looks like: .. math:: \mathbf{F}^{\\top} {1\over 8} \left(\sum_{i=1}^8 \mathbf{Q}_{(i)}^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma^{-1} \sqrt{v_{\\text{cell}}} \mathbf{Q}_{(i)} \\right) \mathbf{J} = -\mathbf{F}^{\\top} \mathbf{A} \mathbf{D}_{\\text{cell}}^{\\top} \phi + \\text{BC} \mathbf{M}(\Sigma^{-1}) \mathbf{J} = -\mathbf{A} \mathbf{D}_{\\text{cell}}^{\\top} \phi + \\text{BC} \mathbf{M}(\Sigma^{-1}) = {1\over 8} \left(\sum_{i=1}^8 \mathbf{Q}_{(i)}^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma^{-1} \sqrt{v_{\\text{cell}}} \mathbf{Q}_{(i)} \\right) The M is returned if mu is set equal to \Sigma^{-1}. If requested (returnP=True) the projection matricies are returned as well (ordered by nodes). Here each P (3*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates: .. math:: \mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)} Note that this is completed for each cell in the mesh at the same time. """ def __init__(self): 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.') def getFaceInnerProduct(M, mu=None, returnP=False): """ :param numpy.array mu: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6)) :param bool returnP: returns the projection matrices :rtype: scipy.csr_matrix :return: M, the inner product matrix (sum(nF), sum(nF)) Depending on the number of columns (either 1, 3, or 6) of mu, the material property is interpreted as follows: .. math:: \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 & 0 \\\\ 0 & \mu_{1} & 0 \\\\ 0 & 0 & \mu_{1} \end{matrix}\\right] \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 & 0 \\\\ 0 & \mu_{2} & 0 \\\\ 0 & 0 & \mu_{3} \end{matrix}\\right] \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & \mu_{4} & \mu_{5} \\\\ \mu_{4} & \mu_{2} & \mu_{6} \\\\ \mu_{5} & \mu_{6} & \mu_{3} \end{matrix}\\right] \mathbf{M}(\\vec{\mu}) = {1\over 8} \left(\sum_{i=1}^8 \mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \\vec{\mu} \sqrt{v_{\\text{cell}}} \mathbf{J}_c \\right) If requested (returnP=True) the projection matricies are returned as well (ordered by nodes):: P = [P000, P100, P010, P110, P001, P101, P011, P111] Here each P (3*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates: .. math:: \mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)} Note that this is completed for each cell in the mesh at the same time. **For 2D:** Depending on the number of columns (either 1, 2, or 3) of mu, the material property is interpreted as follows: .. math:: \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 \\\\ 0 & \mu_{1} \end{matrix}\\right] \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 \\\\ 0 & \mu_{2} \end{matrix}\\right] \\vec{\mu} = \left[\\begin{matrix} \mu_{1} & \mu_{3} \\\\ \mu_{3} & \mu_{2} \end{matrix}\\right] .. math:: \mathbf{M}(\\vec{\mu}) = {1\over 4} \left(\sum_{i=1}^4 \mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \\vec{\mu} \sqrt{v_{\\text{cell}}} \mathbf{J}_c \\right) If requested (returnP=True) the projection matricies are returned as well (ordered by nodes):: P = [P00, P10, P01, P11] Here each P (2*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates: .. math:: \mathbf{P}_{(i)} = \sqrt{ {1\over 4} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)} Note that this is completed for each cell in the mesh at the same time. """ if M.dim == 1: v = np.sqrt(0.5*M.vol) V1 = sdiag(v) # We will multiply on each side to keep symmetry Px = _getFacePx(M) P000 = V1*Px('fXm') P100 = V1*Px('fXp') elif M.dim == 2: # Square root of cell volume multiplied by 1/4 v = np.sqrt(0.25*M.vol) V2 = sdiag(np.r_[v, v]) # We will multiply on each side to keep symmetry Pxx = _getFacePxx(M) P000 = V2*Pxx('fXm', 'fYm') P100 = V2*Pxx('fXp', 'fYm') P010 = V2*Pxx('fXm', 'fYp') P110 = V2*Pxx('fXp', 'fYp') elif M.dim == 3: # Square root of cell volume multiplied by 1/8 v = np.sqrt(0.125*M.vol) V3 = sdiag(np.r_[v, v, v]) # We will multiply on each side to keep symmetry Pxxx = _getFacePxxx(M) P000 = V3*Pxxx('fXm', 'fYm', 'fZm') P100 = V3*Pxxx('fXp', 'fYm', 'fZm') P010 = V3*Pxxx('fXm', 'fYp', 'fZm') P110 = V3*Pxxx('fXp', 'fYp', 'fZm') P001 = V3*Pxxx('fXm', 'fYm', 'fZp') P101 = V3*Pxxx('fXp', 'fYm', 'fZp') P011 = V3*Pxxx('fXm', 'fYp', 'fZp') P111 = V3*Pxxx('fXp', 'fYp', 'fZp') Mu = makePropertyTensor(M, mu) A = P000.T*Mu*P000 + P100.T*Mu*P100 P = [P000, P100] if M.dim > 1: A = A + P010.T*Mu*P010 + P110.T*Mu*P110 P += [P010, P110] if M.dim > 2: A = A + P001.T*Mu*P001 + P101.T*Mu*P101 + P011.T*Mu*P011 + P111.T*Mu*P111 P += [P001, P101, P011, P111] if returnP: return A, P else: return A def getEdgeInnerProduct(M, sigma=None, returnP=False): """ :param numpy.array sigma: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6)) :param bool returnP: returns the projection matrices :rtype: scipy.csr_matrix :return: M, the inner product matrix (sum(nE), sum(nE)) Depending on the number of columns (either 1, 3, or 6) of sigma, the material property is interpreted as follows: .. math:: \Sigma = \left[\\begin{matrix} \sigma_{1} & 0 & 0 \\\\ 0 & \sigma_{1} & 0 \\\\ 0 & 0 & \sigma_{1} \end{matrix}\\right] \Sigma = \left[\\begin{matrix} \sigma_{1} & 0 & 0 \\\\ 0 & \sigma_{2} & 0 \\\\ 0 & 0 & \sigma_{3} \end{matrix}\\right] \Sigma = \left[\\begin{matrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\\\ \sigma_{4} & \sigma_{2} & \sigma_{6} \\\\ \sigma_{5} & \sigma_{6} & \sigma_{3} \end{matrix}\\right] What is returned: .. math:: \mathbf{M}(\Sigma) = {1\over 8} \left(\sum_{i=1}^8 \mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma \sqrt{v_{\\text{cell}}} \mathbf{J}_c \\right) If requested (returnP=True) the projection matricies are returned as well (ordered by nodes):: P = [P000, P100, P010, P110, P001, P101, P011, P111] Here each P (3*nC, sum(nE)) is a combination of the projection, volume, and any normalization to Cartesian coordinates: .. math:: \mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)} Note that this is completed for each cell in the mesh at the same time. **For 2D:** Depending on the number of columns (either 1, 2, or 3) of sigma, the material property is interpreted as follows: .. math:: \Sigma = \left[\\begin{matrix} \sigma_{1} & 0 \\\\ 0 & \sigma_{1} \end{matrix}\\right] \Sigma = \left[\\begin{matrix} \sigma_{1} & 0 \\\\ 0 & \sigma_{2} \end{matrix}\\right] \Sigma = \left[\\begin{matrix} \sigma_{1} & \sigma_{3} \\\\ \sigma_{3} & \sigma_{2} \end{matrix}\\right] .. math:: \mathbf{M}(\Sigma) = {1\over 4} \left(\sum_{i=1}^4 \mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma \sqrt{v_{\\text{cell}}} \mathbf{J}_c \\right) If requested (returnP=True) the projection matricies are returned as well (ordered by nodes):: P = [P00, P10, P01, P11] Here each P (2*nC, sum(nE)) is a combination of the projection, volume, and any normalization to Cartesian coordinates: .. math:: \mathbf{P}_{(i)} = \sqrt{ {1\over 4} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)} Note that this is completed for each cell in the mesh at the same time. """ if M.dim == 1: raise NotImplementedError('getEdgeInnerProduct not implemented for 1D') # We will multiply by V on each side to keep symmetry elif M.dim == 2: # Square root of cell volume multiplied by 1/4 v = np.sqrt(0.25*M.vol) V = sdiag(np.r_[v, v]) eP = _getEdgePxx(M) P000 = V*eP('eX0', 'eY0') P100 = V*eP('eX0', 'eY1') P010 = V*eP('eX1', 'eY0') P110 = V*eP('eX1', 'eY1') elif M.dim == 3: # Square root of cell volume multiplied by 1/8 v = np.sqrt(0.125*M.vol) V = sdiag(np.r_[v, v, v]) eP = _getEdgePxxx(M) P000 = V*eP('eX0', 'eY0', 'eZ0') P100 = V*eP('eX0', 'eY1', 'eZ1') P010 = V*eP('eX1', 'eY0', 'eZ2') P110 = V*eP('eX1', 'eY1', 'eZ3') P001 = V*eP('eX2', 'eY2', 'eZ0') P101 = V*eP('eX2', 'eY3', 'eZ1') P011 = V*eP('eX3', 'eY2', 'eZ2') P111 = V*eP('eX3', 'eY3', 'eZ3') Sigma = makePropertyTensor(M, sigma) A = P000.T*Sigma*P000 + P100.T*Sigma*P100 + P010.T*Sigma*P010 + P110.T*Sigma*P110 P = [P000, P100, P010, P110] if M.dim == 3: A = A + P001.T*Sigma*P001 + P101.T*Sigma*P101 + P011.T*Sigma*P011 + P111.T*Sigma*P111 P += [P001, P101, P011, P111] if returnP: return A, P else: return A # ------------------------ Geometries ------------------------------ # # # node(i,j,k+1) ------ edge2(i,j,k+1) ----- node(i,j+1,k+1) # / / # / / | # edge3(i,j,k) face1(i,j,k) edge3(i,j+1,k) # / / | # / / | # node(i,j,k) ------ edge2(i,j,k) ----- node(i,j+1,k) # | | | # | | node(i+1,j+1,k+1) # | | / # edge1(i,j,k) face3(i,j,k) edge1(i,j+1,k) # | | / # | | / # | |/ # node(i+1,j,k) ------ edge2(i+1,j,k) ----- node(i+1,j+1,k) def _getFacePx(M): assert M._meshType == 'TENSOR', 'Only supported for a tensor mesh' return _getFacePx_Rectangular(M) def _getFacePxx(M): if M._meshType == 'TREE': return M._getFacePxx return _getFacePxx_Rectangular(M) def _getFacePxxx(M): if M._meshType == 'TREE': return M._getFacePxxx return _getFacePxxx_Rectangular(M) def _getEdgePxx(M): if M._meshType == 'TREE': return M._getEdgePxx return _getEdgePxx_Rectangular(M) def _getEdgePxxx(M): if M._meshType == 'TREE': return M._getEdgePxxx return _getEdgePxxx_Rectangular(M) def _getFacePx_Rectangular(M): """Returns a function for creating projection matrices """ ii = np.int64(range(M.nCx)) def Px(xFace): """ xFace is 'fXp' or 'fXm' """ posFx = 0 if xFace == 'fXm' else 1 IND = ii + posFx PX = sp.csr_matrix((np.ones(M.nC), (range(M.nC), IND)), shape=(M.nC, M.nF)) return PX return Px def _getFacePxx_Rectangular(M): """returns a function for creating projection matrices Mats takes you from faces a subset of all faces on only the faces that you ask for. These are centered around a single nodes. For example, if this was your entire mesh: f3(Yp) 2_______________3 | | | | | | f0(Xm) | x | f1(Xp) | | | | |_______________| 0 1 f2(Ym) Pxx('m','m') = | 1, 0, 0, 0 | | 0, 0, 1, 0 | Pxx('p','m') = | 0, 1, 0, 0 | | 0, 0, 1, 0 | """ 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 == 'LOM': fN1 = M.r(M.normals, 'F', 'Fx', 'M') fN2 = M.r(M.normals, 'F', 'Fy', 'M') def Pxx(xFace, yFace): """ xFace is 'fXp' or 'fXm' yFace is 'fYp' or 'fYm' """ # no | node | f1 | f2 # 00 | i ,j | i , j | i, j # 10 | i+1,j | i+1, j | i, j # 01 | i ,j+1 | i , j | i, j+1 # 11 | i+1,j+1 | i+1, j | i, j+1 posFx = 0 if xFace == 'fXm' else 1 posFy = 0 if yFace == 'fYm' else 1 ind1 = sub2ind(M.vnFx, np.c_[ii + posFx, jj]) ind2 = sub2ind(M.vnFy, np.c_[ii, jj + posFy]) + M.nFx IND = np.r_[ind1, ind2].flatten() PXX = sp.csr_matrix((np.ones(2*M.nC), (range(2*M.nC), IND)), shape=(2*M.nC, M.nF)) if M._meshType == 'LOM': I2x2 = inv2X2BlockDiagonal(getSubArray(fN1[0], [i + posFx, j]), getSubArray(fN1[1], [i + posFx, j]), getSubArray(fN2[0], [i, j + posFy]), getSubArray(fN2[1], [i, j + posFy])) PXX = I2x2 * PXX return PXX return Pxx def _getFacePxxx_Rectangular(M): """returns a function for creating projection matrices Mats takes you from faces a subset of all faces on only the faces that you ask for. These are centered around a single nodes. """ 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 == 'LOM': fN1 = M.r(M.normals, 'F', 'Fx', 'M') fN2 = M.r(M.normals, 'F', 'Fy', 'M') fN3 = M.r(M.normals, 'F', 'Fz', 'M') def Pxxx(xFace, yFace, zFace): """ xFace is 'fXp' or 'fXm' yFace is 'fYp' or 'fYm' zFace is 'fZp' or 'fZm' """ # no | node | f1 | f2 | f3 # 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 posY = 0 if yFace == 'fYm' else 1 posZ = 0 if zFace == 'fZm' else 1 ind1 = sub2ind(M.vnFx, np.c_[ii + posX, jj, kk]) ind2 = sub2ind(M.vnFy, np.c_[ii, jj + posY, kk]) + M.nFx 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 == 'LOM': 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 == 'LOM': 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 == 'LOM': 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 == 'LOM': 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 == 'LOM': 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)