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KTCFwd.py
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KTCFwd.py
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import numpy as np
import scipy as sp
from KTCMeshing import ELEMENT
class CMATRIX:# Compact presentation of a very sparse matrix
def __init__(self, mat, indi, indj=None):
self.mat = mat
self.i = indi
if indj != None:
self.j = indj
else:
self.j = indi
class EITFEM:
def __init__(self, Mesh2, Inj, Mpat=None, vincl=None, sigmamin=None, sigmamax=None):
self.Mesh2 = Mesh2
self.Inj = Inj
self.Mpat = Mpat
self.sigmamin = sigmamin if sigmamin is not None else 1e-9
self.sigmamax = sigmamax if sigmamax is not None else 1e9
self.zmin = 1e-6
self.Nel = len(Mesh2.elfaces)
self.nH2 = Mesh2.H.shape[0]
self.ng2 = Mesh2.g.shape[0]
self.vincl = vincl if vincl is not None else np.ones((len(Mesh2)*len(Mesh2)), dtype=bool)
self.mincl = np.array(vincl)
self.mincl.shape = (self.Nel-1,self.Inj.shape[1])
self.dA = None
self.C = np.matrix(np.vstack((np.ones((1, self.Nel-1)), -np.eye(self.Nel-1))))
def SolveForward(self, sigma, z):
# Forward solution function - solves the potential field and
# measured voltages given conductivity sigma and contact impedances
# z
# Project negative / too small sigma values to a given minimum
# Project too high sigma values to a given maximum
sigma[sigma < self.sigmamin] = self.sigmamin
sigma[sigma > self.sigmamax] = self.sigmamax
z[z < self.zmin] = self.zmin
# Compute the conductivity-dependent part of the FEM matrix
gN = max(np.shape(self.Mesh2.Node))
HN = max(np.shape(self.Mesh2.H)) # number of elements
k = 1
Arow = np.zeros((6 * HN, 6), dtype=np.uint32)
Acol = np.zeros((6 * HN, 6), dtype=np.uint32)
Aval = np.zeros((6 * HN, 6))
for ii in range(HN): # Go through all triangles
ind = self.Mesh2.H[ii, :]
gg = self.Mesh2.g[ind, :]
ss = sigma[ind[[0, 2, 4]]]
int = self.grinprod_gauss_quad_node(gg, ss)
Inds1 = np.tile(ind, (6, 1))
Acol[k-1:k+5, :] = Inds1
ind = ind.T
Inds2 = Inds1.T
Arow[k-1:k+5, :] = Inds2
Aval[k-1:k+5, :] = int
k = k + 6
A0 = sp.sparse.csr_matrix((Aval.flatten(), (Arow.flatten(), Acol.flatten())), shape=(self.ng2+self.Nel-1, self.ng2+self.Nel-1))
# Compute the rest of the FEM matrix
M = sp.sparse.csr_matrix((gN, self.Nel))
K = sp.sparse.csr_matrix((gN, gN))
s = np.zeros((self.Nel, 1))
g = self.Mesh2.g # Nodes
for ii in range(HN):
# Go through all triangles
ind = self.Mesh2.Element[ii].Topology # The indices to g of the ii'th triangle.
if self.Mesh2.Element[ii].Electrode: # Checks if the triangle ii is under an electrode
Ind = self.Mesh2.Element[ii].Electrode[1]
a = g[Ind[0], :]
b = g[Ind[1], :] # the 2nd order node
c = g[Ind[2], :]
InE = self.Mesh2.Element[ii].Electrode[0] # Electrode index.
s[InE] = s[InE] + 1 / z[InE] * self.electrlen(np.array([a, c])) # Assumes straight electrodes.
bb1 = self.bound_quad1(np.array([a, b, c]))
bb2 = self.bound_quad2(np.array([a, b, c]))
for il in range(6):
eind = np.where(self.Mesh2.Element[ii].Topology[il] == self.Mesh2.Element[ii].Electrode[1])[0]
if eind.size != 0:
M[ind[il], InE] = M[ind[il], InE] - 1 / z[InE] * bb1[eind[0]]
for im in range(6):
eind1 = np.where(self.Mesh2.Element[ii].Topology[il] == self.Mesh2.Element[ii].Electrode[1])[0]
eind2 = np.where(self.Mesh2.Element[ii].Topology[im] == self.Mesh2.Element[ii].Electrode[1])[0]
if eind1.size != 0 and eind2.size != 0:
K[ind[il], ind[im]] = K[ind[il], ind[im]] + 1 / z[InE] * bb2[eind1[0], eind2[0]]
tS = sp.sparse.csr_matrix(np.diag(s.flatten()))
S = sp.sparse.csr_matrix(self.C.T * tS * self.C)
M = M * self.C
#S0 = sp.sparse.csr_matrix(np.block([[K.toarray(), M.toarray()], [M.toarray().T, S.toarray()]]))
S0 = sp.sparse.bmat(
[
[K, M],
[M.T, S]
])
self.A = A0 + S0
self.b = np.concatenate((np.zeros((self.ng2, self.Inj.shape[1])), self.C.T * self.Inj), axis=0) # RHS vector
UU = sp.sparse.linalg.spsolve(self.A, self.b) # Solve the FEM system of equations
self.theta = UU # FEM solution vectors for each current injection
self.Pot = UU[0:self.ng2, :] # The electric potential field for each current injection
self.Imeas = self.Inj # Injected currents
self.Umeas = self.Mpat.T * self.C * self.theta[self.ng2:, :] # Measured voltages between electrodes
self.Umeas = self.Umeas.T[self.mincl.T].T
self.QC = np.block([[np.zeros((np.shape(self.Mpat)[1], self.ng2)), self.Mpat.T @ self.C]])
fsol = self.Umeas # Measured potentials, vectorized
return fsol
def Jacobian(self, sigma=None, z=None):
if sigma is not None and z is not None:
self.SolveForward(sigma, z)
else:
if self.theta is None:
raise ValueError("Jacobian cannot be computed without first computing a forward solution")
if self.dA is None:
temp_elements = []
for element in self.Mesh2.Element:
temp_element = ELEMENT(element.Topology[[0, 2, 4]], element.Electrode)
temp_elements.append(temp_element)
self.dA = self.jacob_nodesigma2nd3(self.Mesh2.Node, temp_elements, self.Mesh2.Node, self.Mesh2.Element)
m = len(sigma)
n = len(self.Umeas)
Jleft = sp.sparse.linalg.spsolve(self.A.T, self.QC.T)
Jright = self.theta
Js = np.zeros((n, m))
for ii in range(m):
Jid = self.dA[ii].i
Jtemp = -Jleft.T[:, Jid] @ self.dA[ii].mat @ Jright[Jid, :]
Js[:, ii] = Jtemp.T[self.mincl.T]
return Js
def Jacobianz(self, sigma=None, z=None):
if sigma is not None and z is not None:
self.SolveForward(sigma, z)
else:
if self.theta is None:
raise ValueError("Jacobian cannot be computed without first computing a forward solution")
dA_dz = self.ComputedA_dz( z)
m = len(z)
n = len(self.Umeas)
#Jleft = -self.QC / self.A
Jleft = -self.QC @ sp.sparse.linalg.inv(self.A)
Jright = self.theta
Jz = np.zeros((n, m))
for ii in range(m):
Jtemp = Jleft @ dA_dz[ii] @ Jright
Jz[:, ii] = Jtemp.T[self.mincl.T]
return Jz
def SetInvGamma(self, noise_percentage, noise_percentage2=0, meas_data=None):
if meas_data is not None:
meas = meas_data
else:
meas = self.Umeas
var_meas = np.power(((noise_percentage / 100) * (np.abs(meas))),2)
var_meas = var_meas + np.power((noise_percentage2 / 100) * np.max(np.abs(meas)),2)
Gamma_n = np.diag(np.array(var_meas).flatten())
InvGamma_n = np.linalg.inv(Gamma_n)
self.InvGamma_n = sp.sparse.csr_matrix(InvGamma_n)
self.Ln = sp.sparse.csr_matrix(np.linalg.cholesky(InvGamma_n))
self.InvLn = sp.sparse.csr_matrix(np.linalg.cholesky(Gamma_n))
q = 0
def grinprod_gauss_quad_node(self, g, sigma):
w = np.array([1/6, 1/6, 1/6])
ip = np.array([[1/2, 0], [1/2, 1/2], [0, 1/2]])
int_sum = 0
for ii in range(3):
S = np.array([1 - ip[ii, 0] - ip[ii, 1], ip[ii, 0], ip[ii, 1]])
L = np.array([[4 * (ip[ii, 0] + ip[ii, 1]) - 3, -8 * ip[ii, 0] - 4 * ip[ii, 1] + 4, 4 * ip[ii, 0] - 1, 4 * ip[ii, 1], 0, -4 * ip[ii, 1]],
[4 * (ip[ii, 0] + ip[ii, 1]) - 3, -4 * ip[ii, 0], 0, 4 * ip[ii, 0], 4 * ip[ii, 1] - 1, -8 * ip[ii, 1] - 4 * ip[ii, 0] + 4]])
Jt = L @ g
iJt = np.linalg.inv(Jt)
dJt = np.abs(np.linalg.det(Jt))
G = iJt @ L
int_sum += w[ii] * (S.T @ sigma) * G.T @ G * dJt
return int_sum
def bound_quad1(self, g):
w = np.array([1/2, 1/2])
ip = np.array([1/2 - 1/6 * np.sqrt(3), 1/2 + 1/6 * np.sqrt(3)])
int_sum = 0
for ii in range(2):
S = np.array([2 * ip[ii]**2 - 3 * ip[ii] + 1,
-4 * ip[ii]**2 + 4 * ip[ii],
2 * ip[ii]**2 - ip[ii]])
dJt = np.sqrt((g[0, 0] * (4 * ip[ii] - 3) + g[1, 0] * (4 - 8 * ip[ii]) + g[2, 0] * (4 * ip[ii] - 1))**2 +
(g[0, 1] * (4 * ip[ii] - 3) + g[1, 1] * (4 - 8 * ip[ii]) + g[2, 1] * (4 * ip[ii] - 1))**2)
int_sum += w[ii] * S * dJt
return int_sum
def bound_quad2(self, g):
w = np.array([5/18, 8/18, 5/18])
ip = np.array([1/2 - 1/10 * np.sqrt(15), 1/2, 1/2 + 1/10 * np.sqrt(15)])
int_sum = np.zeros((3,3))
for ii in range(3):
S = np.array([2 * ip[ii]**2 - 3 * ip[ii] + 1,
-4 * ip[ii]**2 + 4 * ip[ii],
2 * ip[ii]**2 - ip[ii]])
dJt = np.sqrt((g[0, 0] * (4 * ip[ii] - 3) + g[1, 0] * (4 - 8 * ip[ii]) + g[2, 0] * (4 * ip[ii] - 1))**2 +
(g[0, 1] * (4 * ip[ii] - 3) + g[1, 1] * (4 - 8 * ip[ii]) + g[2, 1] * (4 * ip[ii] - 1))**2)
temp = np.outer(S,S)
int_sum += w[ii] * np.outer(S,S) * dJt
return int_sum
def electrlen(self, g):
dJt = np.sqrt((g[1, 0] - g[0, 0])**2 + (g[1, 1] - g[0, 1])**2)
return dJt
def jacob_nodesigma2nd3(self, Nodesig, Elementsig, Node, Element):# Returns a CMATRIX list representing the derivative of the stiffness matrix w.r.t. coefficients of to the conductivity
gNs = len(Nodesig)
gN = len(Node)
Agrads = []
for jj in range(gNs):
nzi = np.array([],dtype= np.uint32)
nzj = np.array([],dtype= np.uint32)
nzv = np.array([],dtype= np.float64)
indexmap = np.zeros((gN,1),dtype= np.uint32)
Aa = sp.sparse.csr_matrix((gN, gN))
El = Nodesig[jj].ElementConnection
for ii in range(len(El)):
indsig = Elementsig[El[ii]].Topology
ind = Element[El[ii]].Topology
gg = np.array([Node[i].Coordinate for i in ind]).reshape(6, 2)
I = np.where(jj == np.array(indsig))
if len(I[0]) > 0:
anis = self.grinprodgaus2ndnode3(gg, I[0][0])
tmpind = np.tile(ind, (6,1))
nzi = np.append(nzi,tmpind.T.flatten())
nzj = np.append(nzj,tmpind.flatten())
nzv = np.append(nzv,anis.flatten())
unzi = np.unique(nzi)
nunz = len(unzi)
tmat = np.zeros((nunz,nunz))
t = np.array(range(nunz))
t.shape = (nunz,1)
indexmap[unzi] = t
for ii in range(len(nzi)):
ti = indexmap[nzi[ii]]
tj = indexmap[nzj[ii]]
tmat[ti,tj] += nzv[ii]
Agrads.append(CMATRIX(tmat,unzi))
#Agrad[:, jj] = np.array(Aa).flatten()
J = Agrads
return J
def grinprodgaus2ndnode3(self, g, I):
w = np.array([1/40] * 3 + [1/15] * 3 + [27/120])
ip = np.array([[0, 0],
[1, 0],
[0, 1],
[1/2, 0],
[1/2, 1/2],
[0, 1/2],
[1/3, 1/3]])
int_sum = 0
for ii in range(7):
S = np.array([1 - ip[ii, 0] - ip[ii, 1], ip[ii, 0], ip[ii, 1]])
L = np.array([[4 * (ip[ii, 0] + ip[ii, 1]) - 3, -8 * ip[ii, 0] - 4 * ip[ii, 1] + 4,
4 * ip[ii, 0] - 1, 4 * ip[ii, 1], 0, -4 * ip[ii, 1]],
[4 * (ip[ii, 0] + ip[ii, 1]) - 3, -4 * ip[ii, 0],
0, 4 * ip[ii, 0], 4 * ip[ii, 1] - 1, -8 * ip[ii, 1] - 4 * ip[ii, 0] + 4]])
Jt = L @ g
iJt = np.linalg.inv(Jt)
dJt = abs(np.linalg.det(Jt))
G = iJt @ L
int_sum += w[ii] * S[I] * G.T @ G * dJt
return int_sum
def regroupAgrad2cell(self, Agrad, ng=None, ngs=None):
if ng is None and ngs is None:
ng = int(np.sqrt(Agrad.shape[0]))
ngs = Agrad.shape[1]
arow = [None] * ng
acol = [None] * ng
aval = [None] * ng
for k in range(ngs):
S = np.array(Agrad[:, k]).reshape(ng, ng)
I, J = np.nonzero(S)
I = np.unique(I)
J = np.unique(J)
arow[k] = I
acol[k] = J
aval[k] = S[np.ix_(I, J)]
return arow, acol, aval
def ComputedA_dz(self, z):
dA_dz = [None] * self.Nel
gN = len(self.Mesh2.Node)
HN = len(self.Mesh2.H)
M = sp.sparse.lil_matrix((gN, self.Nel))
K = [sp.sparse.lil_matrix((gN, gN)) for _ in range(self.Nel)]
s = np.zeros(self.Nel)
g = np.array([node.Coordinate for node in self.Mesh2.Node]).reshape(gN, 2)
for ii in range(HN):
ind = self.Mesh2.Element[ii].Topology
if len(self.Mesh2.Element[ii].Electrode) > 0:
Ind = self.Mesh2.Element[ii].Electrode[1]
a = g[Ind[0]]
b = g[Ind[1]]
c = g[Ind[2]]
InE = self.Mesh2.Element[ii].Electrode[0]
s[InE] -= (1 / (z[InE]**2)) * self.electrlen(np.array([a, c]))
bb1 = self.bound_quad1(np.array([a, b, c]))
bb2 = self.bound_quad2(np.array([a, b, c]))
for il in range(6):
eind = np.where(self.Mesh2.Element[ii].Topology[il] == self.Mesh2.Element[ii].Electrode[1])[0]
if eind.size > 0:
M[ind[il], InE] += (1 / (z[InE]**2)) * bb1[eind[0]]
for im in range(6):
eind1 = np.where(self.Mesh2.Element[ii].Topology[il] == self.Mesh2.Element[ii].Electrode[1])[0]
eind2 = np.where(self.Mesh2.Element[ii].Topology[im] == self.Mesh2.Element[ii].Electrode[1])[0]
if eind1.size > 0 and eind2.size > 0:
K[InE][ind[il], ind[im]] -= (1 / (z[InE]**2)) * bb2[eind1[0], eind2[0]]
for ii in range(self.Nel):
tmp = np.zeros(self.Nel)
tmp[ii] = s[ii]
S = sp.sparse.diags(tmp)
S = self.C.T @ S @ self.C
#Mtmp = np.zeros_like(M)
Mtmp = sp.sparse.lil_matrix((gN, self.Nel))
Mtmp[:, ii] = M[:, ii]
Mtmp = Mtmp @ self.C
dA_dz[ii] = np.block([[K[ii].toarray(), Mtmp],
[Mtmp.T, S]])
return dA_dz