from scipy import sparse as sp from SimPEG.utils import sub2ind, ndgrid, mkvc, getSubArray, sdiag, inv3X3BlockDiagonal, inv2X2BlockDiagonal 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(self, mu=None, returnP=False): """Wrapper function, :py:func:`SimPEG.mesh.InnerProducts.InnerProducts.getEdgeInnerProduct` :py:func:`SimPEG.mesh.InnerProducts.InnerProducts.getEdgeInnerProduct2D` """ if self.dim == 2: return getFaceInnerProduct2D(self, mu, returnP) elif self.dim == 3: return getFaceInnerProduct(self, mu, returnP) def getEdgeInnerProduct(self, sigma=None, returnP=False): """Wrapper function, :py:func:`SimPEG.mesh.InnerProducts.InnerProducts.getFaceInnerProduct` :py:func:`SimPEG.mesh.InnerProducts.InnerProducts.getFaceInnerProduct2D` """ if self.dim == 2: return getEdgeInnerProduct2D(self, sigma, returnP) elif self.dim == 3: return getEdgeInnerProduct(self, sigma, returnP) # ------------------------ 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 getFaceInnerProduct(mesh, 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, P001, P010, P011, P100, P101, P110, 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. """ if mu is None: # default is ones mu = np.ones((mesh.nC, 1)) m = np.array([mesh.nCx, mesh.nCy, mesh.nCz]) nc = mesh.nC i, j, k = np.int64(range(m[0])), np.int64(range(m[1])), np.int64(range(m[2])) iijjkk = ndgrid(i, j, k) ii, jj, kk = iijjkk[:, 0], iijjkk[:, 1], iijjkk[:, 2] if mesh._meshType == 'LOM': fN1 = mesh.r(mesh.normals, 'F', 'Fx', 'M') fN2 = mesh.r(mesh.normals, 'F', 'Fy', 'M') fN3 = mesh.r(mesh.normals, 'F', 'Fz', 'M') def Pxxx(pos): ind1 = sub2ind(mesh.nFx, np.c_[ii + pos[0][0], jj + pos[0][1], kk + pos[0][2]]) ind2 = sub2ind(mesh.nFy, np.c_[ii + pos[1][0], jj + pos[1][1], kk + pos[1][2]]) + mesh.nFv[0] ind3 = sub2ind(mesh.nFz, np.c_[ii + pos[2][0], jj + pos[2][1], kk + pos[2][2]]) + mesh.nFv[0] + mesh.nFv[1] IND = np.r_[ind1, ind2, ind3].flatten() PXXX = sp.coo_matrix((np.ones(3*nc), (range(3*nc), IND)), shape=(3*nc, np.sum(mesh.nF))).tocsr() if mesh._meshType == 'LOM': I3x3 = inv3X3BlockDiagonal(getSubArray(fN1[0], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(fN1[1], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(fN1[2], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(fN2[0], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(fN2[1], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(fN2[2], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(fN3[0], [i + pos[2][0], j + pos[2][1], k + pos[2][2]]), getSubArray(fN3[1], [i + pos[2][0], j + pos[2][1], k + pos[2][2]]), getSubArray(fN3[2], [i + pos[2][0], j + pos[2][1], k + pos[2][2]])) PXXX = I3x3 * PXXX return PXXX # 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 # Square root of cell volume multiplied by 1/8 v = np.sqrt(0.125*mesh.vol) V3 = sdiag(np.r_[v, v, v]) # We will multiply on each side to keep symmetry P000 = V3*Pxxx([[0, 0, 0], [0, 0, 0], [0, 0, 0]]) P100 = V3*Pxxx([[1, 0, 0], [0, 0, 0], [0, 0, 0]]) P010 = V3*Pxxx([[0, 0, 0], [0, 1, 0], [0, 0, 0]]) P110 = V3*Pxxx([[1, 0, 0], [0, 1, 0], [0, 0, 0]]) P001 = V3*Pxxx([[0, 0, 0], [0, 0, 0], [0, 0, 1]]) P101 = V3*Pxxx([[1, 0, 0], [0, 0, 0], [0, 0, 1]]) P011 = V3*Pxxx([[0, 0, 0], [0, 1, 0], [0, 0, 1]]) P111 = V3*Pxxx([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) if mu.size == mesh.nC: # Isotropic! mu = mkvc(mu) # ensure it is a vector. Mu = sdiag(np.r_[mu, mu, mu]) elif mu.shape[1] == 3: # Diagonal tensor Mu = sdiag(np.r_[mu[:, 0], mu[:, 1], mu[:, 2]]) elif mu.shape[1] == 6: # Fully anisotropic row1 = sp.hstack((sdiag(mu[:, 0]), sdiag(mu[:, 3]), sdiag(mu[:, 4]))) row2 = sp.hstack((sdiag(mu[:, 3]), sdiag(mu[:, 1]), sdiag(mu[:, 5]))) row3 = sp.hstack((sdiag(mu[:, 4]), sdiag(mu[:, 5]), sdiag(mu[:, 2]))) Mu = sp.vstack((row1, row2, row3)) A = P000.T*Mu*P000 + P001.T*Mu*P001 + P010.T*Mu*P010 + P011.T*Mu*P011 + P100.T*Mu*P100 + P101.T*Mu*P101 + P110.T*Mu*P110 + P111.T*Mu*P111 P = [P000, P001, P010, P011, P100, P101, P110, P111] if returnP: return A, P else: return A def getFaceInnerProduct2D(mesh, mu=None, returnP=False): """ :param numpy.array mu: material property (tensor properties are possible) at each cell center (nC, (1, 2, or 3)) :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, 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 mu is None: # default is ones mu = np.ones((mesh.nC, 1)) m = np.array([mesh.nCx, mesh.nCy]) nc = mesh.nC i, j = np.int64(range(m[0])), np.int64(range(m[1])) iijj = ndgrid(i, j) ii, jj = iijj[:, 0], iijj[:, 1] if mesh._meshType == 'LOM': fN1 = mesh.r(mesh.normals, 'F', 'Fx', 'M') fN2 = mesh.r(mesh.normals, 'F', 'Fy', 'M') def Pxx(pos): ind1 = sub2ind(mesh.nFx, np.c_[ii + pos[0][0], jj + pos[0][1]]) ind2 = sub2ind(mesh.nFy, np.c_[ii + pos[1][0], jj + pos[1][1]]) + mesh.nFv[0] IND = np.r_[ind1, ind2].flatten() PXX = sp.coo_matrix((np.ones(2*nc), (range(2*nc), IND)), shape=(2*nc, np.sum(mesh.nF))).tocsr() if mesh._meshType == 'LOM': I2x2 = inv2X2BlockDiagonal(getSubArray(fN1[0], [i + pos[0][0], j + pos[0][1]]), getSubArray(fN1[1], [i + pos[0][0], j + pos[0][1]]), getSubArray(fN2[0], [i + pos[1][0], j + pos[1][1]]), getSubArray(fN2[1], [i + pos[1][0], j + pos[1][1]])) PXX = I2x2 * PXX return PXX # 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 # Square root of cell volume multiplied by 1/4 v = np.sqrt(0.25*mesh.vol) V2 = sdiag(np.r_[v, v]) # We will multiply on each side to keep symmetry P00 = V2*Pxx([[0, 0], [0, 0]]) P10 = V2*Pxx([[1, 0], [0, 0]]) P01 = V2*Pxx([[0, 0], [0, 1]]) P11 = V2*Pxx([[1, 0], [0, 1]]) if mu.size == mesh.nC: # Isotropic! mu = mkvc(mu) # ensure it is a vector. Mu = sdiag(np.r_[mu, mu]) elif mu.shape[1] == 2: # Diagonal tensor Mu = sdiag(np.r_[mu[:, 0], mu[:, 1]]) elif mu.shape[1] == 3: # Fully anisotropic row1 = sp.hstack((sdiag(mu[:, 0]), sdiag(mu[:, 2]))) row2 = sp.hstack((sdiag(mu[:, 2]), sdiag(mu[:, 1]))) Mu = sp.vstack((row1, row2)) A = P00.T*Mu*P00 + P10.T*Mu*P10 + P01.T*Mu*P01 + P11.T*Mu*P11 P = [P00, P10, P01, P11] if returnP: return A, P else: return A def getEdgeInnerProduct(mesh, 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, P001, P010, P011, P100, P101, P110, 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. """ if sigma is None: # default is ones sigma = np.ones((mesh.nC, 1)) m = np.array([mesh.nCx, mesh.nCy, mesh.nCz]) nc = mesh.nC i, j, k = np.int64(range(m[0])), np.int64(range(m[1])), np.int64(range(m[2])) iijjkk = ndgrid(i, j, k) ii, jj, kk = iijjkk[:, 0], iijjkk[:, 1], iijjkk[:, 2] if mesh._meshType == 'LOM': eT1 = mesh.r(mesh.tangents, 'E', 'Ex', 'M') eT2 = mesh.r(mesh.tangents, 'E', 'Ey', 'M') eT3 = mesh.r(mesh.tangents, 'E', 'Ez', 'M') def Pxxx(pos): ind1 = sub2ind(mesh.nEx, np.c_[ii + pos[0][0], jj + pos[0][1], kk + pos[0][2]]) ind2 = sub2ind(mesh.nEy, np.c_[ii + pos[1][0], jj + pos[1][1], kk + pos[1][2]]) + mesh.nEv[0] ind3 = sub2ind(mesh.nEz, np.c_[ii + pos[2][0], jj + pos[2][1], kk + pos[2][2]]) + mesh.nEv[0] + mesh.nEv[1] IND = np.r_[ind1, ind2, ind3].flatten() PXXX = sp.coo_matrix((np.ones(3*nc), (range(3*nc), IND)), shape=(3*nc, np.sum(mesh.nE))).tocsr() if mesh._meshType == 'LOM': I3x3 = inv3X3BlockDiagonal(getSubArray(eT1[0], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(eT1[1], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(eT1[2], [i + pos[0][0], j + pos[0][1], k + pos[0][2]]), getSubArray(eT2[0], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(eT2[1], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(eT2[2], [i + pos[1][0], j + pos[1][1], k + pos[1][2]]), getSubArray(eT3[0], [i + pos[2][0], j + pos[2][1], k + pos[2][2]]), getSubArray(eT3[1], [i + pos[2][0], j + pos[2][1], k + pos[2][2]]), getSubArray(eT3[2], [i + pos[2][0], j + pos[2][1], k + pos[2][2]])) PXXX = I3x3 * PXXX return PXXX # 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 # Square root of cell volume multiplied by 1/8 v = np.sqrt(0.125*mesh.vol) V3 = sdiag(np.r_[v, v, v]) # We will multiply on each side to keep symmetry P000 = V3*Pxxx([[0, 0, 0], [0, 0, 0], [0, 0, 0]]) P100 = V3*Pxxx([[0, 0, 0], [1, 0, 0], [1, 0, 0]]) P010 = V3*Pxxx([[0, 1, 0], [0, 0, 0], [0, 1, 0]]) P110 = V3*Pxxx([[0, 1, 0], [1, 0, 0], [1, 1, 0]]) P001 = V3*Pxxx([[0, 0, 1], [0, 0, 1], [0, 0, 0]]) P101 = V3*Pxxx([[0, 0, 1], [1, 0, 1], [1, 0, 0]]) P011 = V3*Pxxx([[0, 1, 1], [0, 0, 1], [0, 1, 0]]) P111 = V3*Pxxx([[0, 1, 1], [1, 0, 1], [1, 1, 0]]) if sigma.size == mesh.nC: # Isotropic! sigma = mkvc(sigma) # ensure it is a vector. Sigma = sdiag(np.r_[sigma, sigma, sigma]) elif sigma.shape[1] == 3: # Diagonal tensor Sigma = sdiag(np.r_[sigma[:, 0], sigma[:, 1], sigma[:, 2]]) elif sigma.shape[1] == 6: # Fully anisotropic row1 = sp.hstack((sdiag(sigma[:, 0]), sdiag(sigma[:, 3]), sdiag(sigma[:, 4]))) row2 = sp.hstack((sdiag(sigma[:, 3]), sdiag(sigma[:, 1]), sdiag(sigma[:, 5]))) row3 = sp.hstack((sdiag(sigma[:, 4]), sdiag(sigma[:, 5]), sdiag(sigma[:, 2]))) Sigma = sp.vstack((row1, row2, row3)) A = P000.T*Sigma*P000 + P001.T*Sigma*P001 + P010.T*Sigma*P010 + P011.T*Sigma*P011 + P100.T*Sigma*P100 + P101.T*Sigma*P101 + P110.T*Sigma*P110 + P111.T*Sigma*P111 P = [P000, P001, P010, P011, P100, P101, P110, P111] if returnP: return A, P else: return A def getEdgeInnerProduct2D(mesh, sigma=None, returnP=False): """ :param numpy.array sigma: material property (tensor properties are possible) at each cell center (nC, (1, 2, or 3)) :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, 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 sigma is None: # default is ones sigma = np.ones((mesh.nC, 1)) m = np.array([mesh.nCx, mesh.nCy]) nc = mesh.nC i, j = np.int64(range(m[0])), np.int64(range(m[1])) iijj = ndgrid(i, j) ii, jj = iijj[:, 0], iijj[:, 1] if mesh._meshType == 'LOM': eT1 = mesh.r(mesh.tangents, 'E', 'Ex', 'M') eT2 = mesh.r(mesh.tangents, 'E', 'Ey', 'M') def Pxx(pos): ind1 = sub2ind(mesh.nEx, np.c_[ii + pos[0][0], jj + pos[0][1]]) ind2 = sub2ind(mesh.nEy, np.c_[ii + pos[1][0], jj + pos[1][1]]) + mesh.nEv[0] IND = np.r_[ind1, ind2].flatten() PXX = sp.coo_matrix((np.ones(2*nc), (range(2*nc), IND)), shape=(2*nc, np.sum(mesh.nE))).tocsr() if mesh._meshType == 'LOM': I2x2 = inv2X2BlockDiagonal(getSubArray(eT1[0], [i + pos[0][0], j + pos[0][1]]), getSubArray(eT1[1], [i + pos[0][0], j + pos[0][1]]), getSubArray(eT2[0], [i + pos[1][0], j + pos[1][1]]), getSubArray(eT2[1], [i + pos[1][0], j + pos[1][1]])) PXX = I2x2 * PXX return PXX # 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 # Square root of cell volume multiplied by 1/4 v = np.sqrt(0.25*mesh.vol) V2 = sdiag(np.r_[v, v]) # We will multiply on each side to keep symmetry P00 = V2*Pxx([[0, 0], [0, 0]]) P10 = V2*Pxx([[0, 0], [1, 0]]) P01 = V2*Pxx([[0, 1], [0, 0]]) P11 = V2*Pxx([[0, 1], [1, 0]]) if sigma.size == mesh.nC: # Isotropic! sigma = mkvc(sigma) # ensure it is a vector. Sigma = sdiag(np.r_[sigma, sigma]) elif sigma.shape[1] == 2: # Diagonal tensor Sigma = sdiag(np.r_[sigma[:, 0], sigma[:, 1]]) elif sigma.shape[1] == 3: # Fully anisotropic row1 = sp.hstack((sdiag(sigma[:, 0]), sdiag(sigma[:, 2]))) row2 = sp.hstack((sdiag(sigma[:, 2]), sdiag(sigma[:, 1]))) Sigma = sp.vstack((row1, row2)) A = P00.T*Sigma*P00 + P10.T*Sigma*P10 + P01.T*Sigma*P01 + P11.T*Sigma*P11 P = [P00, P10, P01, P11] if returnP: return A, P else: return A 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])] mesh = TensorMesh(h) mu = np.ones((mesh.nC, 6)) A, P = mesh.getFaceInnerProduct(mu, returnP=True) B, P = mesh.getEdgeInnerProduct(mu, returnP=True)