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https://github.com/wassname/simpeg.git
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rowanc1
572 lines
23 KiB
Python
572 lines
23 KiB
Python
from scipy import sparse as sp
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from SimPEG.Utils import sub2ind, ndgrid, mkvc, getSubArray, sdiag, inv3X3BlockDiagonal, inv2X2BlockDiagonal, makePropertyTensor
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import numpy as np
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class InnerProducts(object):
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"""
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Class creates the inner product matrices that you need!
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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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**Example problem for DC resistivity**
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.. math::
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\sigma^{-1}\mathbf{J} = \\nabla \phi
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We can define in weak form by integrating with a general face function F:
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.. math::
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\int_{\\text{cell}}{\sigma^{-1}\mathbf{J} \cdot \mathbf{F}} = \int_{\\text{cell}}{\\nabla \phi \cdot \mathbf{F}}
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\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}}
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We can then discretize for every cell:
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.. math::
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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}
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We can represent this in vector form (again this is for every cell), and will generalize for the case of anisotropic (tensor) sigma.
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.. math::
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\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}
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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:
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.. math::
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\mathbf{J}_c = \mathbf{Q}_{(i)}\mathbf{J}_\\text{TENSOR} = \mathbf{N}_{(i)}^{-1}\mathbf{Q}_{(i)}\mathbf{J}_\\text{LOM}
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Here the i index refers to where we choose to approximate this integral.
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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.
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We will then average to the cell center. For the TENSOR mesh, this looks like:
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.. math::
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\mathbf{F}^{\\top}
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{1\over 8}
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\left(\sum_{i=1}^8
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\mathbf{Q}_{(i)}^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma^{-1} \sqrt{v_{\\text{cell}}} \mathbf{Q}_{(i)}
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\\right)
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\mathbf{J}
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=
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-\mathbf{F}^{\\top} \mathbf{A} \mathbf{D}_{\\text{cell}}^{\\top} \phi + \\text{BC}
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\mathbf{M}(\Sigma^{-1}) \mathbf{J}
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=
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-\mathbf{A} \mathbf{D}_{\\text{cell}}^{\\top} \phi + \\text{BC}
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\mathbf{M}(\Sigma^{-1}) = {1\over 8}
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\left(\sum_{i=1}^8
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\mathbf{Q}_{(i)}^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma^{-1} \sqrt{v_{\\text{cell}}} \mathbf{Q}_{(i)}
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\\right)
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The M is returned if mu is set equal to \Sigma^{-1}.
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If requested (returnP=True) the projection matricies are returned as well (ordered by nodes).
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Here each P (3*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates:
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.. math::
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\mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)}
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Note that this is completed for each cell in the mesh at the same time.
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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(M, mu=None, returnP=False):
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"""
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:param numpy.array mu: 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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:rtype: scipy.csr_matrix
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:return: M, the inner product matrix (sum(nF), sum(nF))
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Depending on the number of columns (either 1, 3, or 6) of mu, the material property is interpreted as follows:
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.. math::
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\\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 & 0 \\\\ 0 & \mu_{1} & 0 \\\\ 0 & 0 & \mu_{1} \end{matrix}\\right]
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\\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 & 0 \\\\ 0 & \mu_{2} & 0 \\\\ 0 & 0 & \mu_{3} \end{matrix}\\right]
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\\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]
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\mathbf{M}(\\vec{\mu}) = {1\over 8}
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\left(\sum_{i=1}^8
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\mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \\vec{\mu} \sqrt{v_{\\text{cell}}} \mathbf{J}_c
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\\right)
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If requested (returnP=True) the projection matricies are returned as well (ordered by nodes)::
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P = [P000, P100, P010, P110, P001, P101, P011, P111]
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Here each P (3*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates:
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.. math::
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\mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)}
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Note that this is completed for each cell in the mesh at the same time.
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**For 2D:**
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Depending on the number of columns (either 1, 2, or 3) of mu, the material property is interpreted as follows:
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.. math::
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\\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 \\\\ 0 & \mu_{1} \end{matrix}\\right]
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\\vec{\mu} = \left[\\begin{matrix} \mu_{1} & 0 \\\\ 0 & \mu_{2} \end{matrix}\\right]
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\\vec{\mu} = \left[\\begin{matrix} \mu_{1} & \mu_{3} \\\\ \mu_{3} & \mu_{2} \end{matrix}\\right]
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.. math::
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\mathbf{M}(\\vec{\mu}) = {1\over 4}
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\left(\sum_{i=1}^4
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\mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \\vec{\mu} \sqrt{v_{\\text{cell}}} \mathbf{J}_c
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\\right)
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If requested (returnP=True) the projection matricies are returned as well (ordered by nodes)::
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P = [P00, P10, P01, P11]
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Here each P (2*nC, sum(nF)) is a combination of the projection, volume, and any normalization to Cartesian coordinates:
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.. math::
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\mathbf{P}_{(i)} = \sqrt{ {1\over 4} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)}
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Note that this is completed for each cell in the mesh at the same time.
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"""
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if M.dim == 1:
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v = np.sqrt(0.5*M.vol)
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V1 = sdiag(v) # We will multiply on each side to keep symmetry
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Px = _getFacePx(M)
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P000 = V1*Px('fXm')
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P100 = V1*Px('fXp')
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elif M.dim == 2:
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# Square root of cell volume multiplied by 1/4
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v = np.sqrt(0.25*M.vol)
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V2 = sdiag(np.r_[v, v]) # We will multiply on each side to keep symmetry
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Pxx = _getFacePxx(M)
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P000 = V2*Pxx('fXm', 'fYm')
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P100 = V2*Pxx('fXp', 'fYm')
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P010 = V2*Pxx('fXm', 'fYp')
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P110 = V2*Pxx('fXp', 'fYp')
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elif M.dim == 3:
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# Square root of cell volume multiplied by 1/8
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v = np.sqrt(0.125*M.vol)
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V3 = sdiag(np.r_[v, v, v]) # We will multiply on each side to keep symmetry
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Pxxx = _getFacePxxx(M)
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P000 = V3*Pxxx('fXm', 'fYm', 'fZm')
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P100 = V3*Pxxx('fXp', 'fYm', 'fZm')
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P010 = V3*Pxxx('fXm', 'fYp', 'fZm')
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P110 = V3*Pxxx('fXp', 'fYp', 'fZm')
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P001 = V3*Pxxx('fXm', 'fYm', 'fZp')
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P101 = V3*Pxxx('fXp', 'fYm', 'fZp')
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P011 = V3*Pxxx('fXm', 'fYp', 'fZp')
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P111 = V3*Pxxx('fXp', 'fYp', 'fZp')
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Mu = makePropertyTensor(M, mu)
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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 M.dim > 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 M.dim > 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 getEdgeInnerProduct(M, sigma=None, returnP=False):
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"""
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:param numpy.array sigma: 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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:rtype: scipy.csr_matrix
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:return: M, the inner product matrix (sum(nE), sum(nE))
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Depending on the number of columns (either 1, 3, or 6) of sigma, the material property is interpreted as follows:
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.. math::
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\Sigma = \left[\\begin{matrix} \sigma_{1} & 0 & 0 \\\\ 0 & \sigma_{1} & 0 \\\\ 0 & 0 & \sigma_{1} \end{matrix}\\right]
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\Sigma = \left[\\begin{matrix} \sigma_{1} & 0 & 0 \\\\ 0 & \sigma_{2} & 0 \\\\ 0 & 0 & \sigma_{3} \end{matrix}\\right]
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\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]
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What is returned:
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.. math::
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\mathbf{M}(\Sigma) = {1\over 8}
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\left(\sum_{i=1}^8
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\mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma \sqrt{v_{\\text{cell}}} \mathbf{J}_c
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\\right)
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If requested (returnP=True) the projection matricies are returned as well (ordered by nodes)::
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P = [P000, P100, P010, P110, P001, P101, P011, P111]
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Here each P (3*nC, sum(nE)) is a combination of the projection, volume, and any normalization to Cartesian coordinates:
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.. math::
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\mathbf{P}_{(i)} = \sqrt{ {1\over 8} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)}
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Note that this is completed for each cell in the mesh at the same time.
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**For 2D:**
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Depending on the number of columns (either 1, 2, or 3) of sigma, the material property is interpreted as follows:
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.. math::
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\Sigma = \left[\\begin{matrix} \sigma_{1} & 0 \\\\ 0 & \sigma_{1} \end{matrix}\\right]
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\Sigma = \left[\\begin{matrix} \sigma_{1} & 0 \\\\ 0 & \sigma_{2} \end{matrix}\\right]
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\Sigma = \left[\\begin{matrix} \sigma_{1} & \sigma_{3} \\\\ \sigma_{3} & \sigma_{2} \end{matrix}\\right]
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.. math::
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\mathbf{M}(\Sigma) = {1\over 4}
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\left(\sum_{i=1}^4
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\mathbf{J}_c^{-\\top} \sqrt{v_{\\text{cell}}} \Sigma \sqrt{v_{\\text{cell}}} \mathbf{J}_c
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\\right)
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If requested (returnP=True) the projection matricies are returned as well (ordered by nodes)::
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P = [P00, P10, P01, P11]
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Here each P (2*nC, sum(nE)) is a combination of the projection, volume, and any normalization to Cartesian coordinates:
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.. math::
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\mathbf{P}_{(i)} = \sqrt{ {1\over 4} v_{\\text{cell}}} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\\text{LOM only}} \mathbf{Q}_{(i)}
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Note that this is completed for each cell in the mesh at the same time.
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"""
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if M.dim == 1:
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raise NotImplementedError('getEdgeInnerProduct not implemented for 1D')
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# We will multiply by V on each side to keep symmetry
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elif M.dim == 2:
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# Square root of cell volume multiplied by 1/4
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v = np.sqrt(0.25*M.vol)
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V = sdiag(np.r_[v, v])
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eP = _getEdgePxx(M)
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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 M.dim == 3:
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# Square root of cell volume multiplied by 1/8
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v = np.sqrt(0.125*M.vol)
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V = sdiag(np.r_[v, v, v])
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eP = _getEdgePxxx(M)
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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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Sigma = makePropertyTensor(M, sigma)
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A = P000.T*Sigma*P000 + P100.T*Sigma*P100 + P010.T*Sigma*P010 + P110.T*Sigma*P110
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P = [P000, P100, P010, P110]
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if M.dim == 3:
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A = A + P001.T*Sigma*P001 + P101.T*Sigma*P101 + P011.T*Sigma*P011 + P111.T*Sigma*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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# ------------------------ 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('m','m') = | 1, 0, 0, 0 |
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| 0, 0, 1, 0 |
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Pxx('p','m') = | 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 == 'LOM':
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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 == 'LOM':
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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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|
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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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|
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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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|
|
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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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|
|
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if M._meshType == 'LOM':
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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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|
|
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def Pxxx(xFace, yFace, zFace):
|
|
"""
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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
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|
# 100 | i+1,j ,k | i+1, j, k | i, j , k | i, j, k
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|
# 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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|
|
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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 == '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)
|