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mpfa3d.py
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mpfa3d.py
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import numpy as np
from tpfa import tpfaScheme
from mdot import mdot
from scipy.sparse import csc_matrix
from scipy.sparse.linalg import spsolve
class mpfa3dScheme(tpfaScheme):
def __init__(self,mesh):
super().__init__(mesh)
self.w, self.fw = self.weights()
self.dji, self.djk, self.keq, self.dirichletT, self.dirichletQ, self.neumann = self.preparacao()
self.T, self.Q = self.assembly_mpfa3d()
self.p = spsolve(self.T,self.Q)
def assembly_mpfa3d(self):
noI = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 0]
noJ = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 1]
noK = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 2]
area, N = self.calc_area(self.mesh.faces.internal)
veri = self.calc_veri(self.mesh.faces.internal, N)
left = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 0]
right = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 1]
left[veri], right[veri] = right[veri], left[veri]
gi = 0.5 * self.djk
gj = 0.5 * self.dji - 0.5 * self.djk
gk = -0.5 * self.dji
Gijk = self.keq * gi * self.node_value[noI] \
+ self.keq * gj * self.node_value[noJ] \
+ self.keq * gk * self.node_value[noK]
Fijk = self.keq * gi * self.fw[noI] \
+ self.keq * gj * self.fw[noJ] \
+ self.keq * gk * self.fw[noK]
rowT = np.concatenate([left, left, right, right])
colT = np.concatenate([left, right, right, left])
dataT = np.concatenate([self.keq, -self.keq, self.keq, -self.keq])
rowQ = np.concatenate([left, right, left, right])
colQ = np.zeros(2*len(left)+2*len(right))
dataQ = np.concatenate([Gijk, -Gijk, Fijk, -Fijk])
esurn = self.mesh.nodes.bridge_adjacencies(self.mesh.nodes.all, 0, 3)
for i in range(len(self.mesh.faces.internal)):
rowT = np.append(rowT,np.concatenate([left[i]*np.ones(len(esurn[noI[i]])), right[i]*np.ones(len(esurn[noI[i]])), \
left[i]*np.ones(len(esurn[noJ[i]])), right[i]*np.ones(len(esurn[noJ[i]])), \
left[i]*np.ones(len(esurn[noK[i]])), right[i]*np.ones(len(esurn[noK[i]]))]))
colT = np.append(colT,np.concatenate([esurn[noI[i]], esurn[noI[i]], \
esurn[noJ[i]], esurn[noJ[i]], \
esurn[noK[i]], esurn[noK[i]]]))
dataT = np.append(dataT,np.concatenate([-self.keq[i] * gi[i] * self.w[noI[i]], self.keq[i] * gi[i] * self.w[noI[i]], \
-self.keq[i] * gj[i] * self.w[noJ[i]], self.keq[i] * gj[i] * self.w[noJ[i]], \
-self.keq[i] * gk[i] * self.w[noK[i]], self.keq[i] * gk[i] * self.w[noK[i]]]))
leftb = self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 2, 3)[:, 0]
rowT = np.append(rowT, leftb)
colT = np.append(colT, leftb)
dataT = np.append(dataT, self.dirichletT)
rowQ = np.append(rowQ,[leftb, leftb])
colQ = np.append(colQ,[np.zeros(len(leftb)), np.zeros(len(leftb))])
dataQ = np.append(dataQ,[-self.dirichletQ, -self.neumann])
T = csc_matrix((dataT, (rowT, colT)), shape=(len(self.mesh.volumes), len(self.mesh.volumes)))
Q = csc_matrix((dataQ, (rowQ, colQ)), shape=(len(self.mesh.volumes), 1))
return T, Q
def preparacao(self):
permeabilities = self.vol_permeabilities()
# PARA FACES INTERNAS: ----------------------------------------------------------------------------------------
area, N = self.calc_area(self.mesh.faces.internal)
veri = self.calc_veri(self.mesh.faces.internal, N)
L = self.mesh.volumes.center(self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 0])
R = self.mesh.volumes.center(self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 1])
L[veri], R[veri] = R[veri], L[veri]
left = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 0]
right = self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 2, 3)[:, 1]
left[veri], right[veri] = right[veri], left[veri]
heights = self.calc_heights(self.mesh.faces.internal, N, L, R, 2); hL = heights[0]; hR = heights[1]
KL = permeabilities[left]; KR = permeabilities[right]
keq, dji, djk = self.calc_keq_dji_djk(N, L, R, KL, KR, hL, hR, area)
# FIM DO TRECHO PARA FACES INTERNAS ---------------------------------------------------------------------------
# PARA FACES DE CONTORNO: -------------------------------------------------------------------------------------
area, N = self.calc_area(self.mesh.faces.boundary)
veri = self.calc_veri(self.mesh.faces.boundary, N)
N[veri] = -N[veri]
L = self.mesh.volumes.center(self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 2, 3)[:,0])
hL = self.calc_heights(self.mesh.faces.boundary, N, L, R, 1)
KL = permeabilities[self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 2, 3)[:, 0]]
dirichletT, dirichletQ, neumann = self.calc_boundary_contribution(N, L, KL, hL, area)
# FIM DO TRECHO PARA FACES DE CONTORNO: -----------------------------------------------------------------------
return dji, djk, keq, dirichletT, dirichletQ, neumann
def vol_permeabilities(self):
region_flag = np.zeros(len(self.mesh.volumes))
for key in self.mesh.volumes.flag:
for i in self.mesh.volumes.flag[key]: region_flag[i] = key
permeabilities = []
for i in range(len(region_flag)): permeabilities.append(self.permeabilities[region_flag[i]])
permeabilities = np.array(permeabilities)
return permeabilities
def calc_area(self,faces):
I = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(faces, 0, 0)[:, 0])
J = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(faces, 0, 0)[:, 1])
K = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(faces, 0, 0)[:, 2])
N = 0.5 * np.cross(I - J, K - J)
area = np.linalg.norm(N, axis=1)
return area, N
def calc_veri(self, faces, N):
tdot = lambda x, y: (x * y).sum(axis=1)
L = self.mesh.volumes.center(self.mesh.faces.bridge_adjacencies(faces, 2, 3)[:, 0])
I = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(faces, 0, 0)[:, 0])
veri = np.nonzero((tdot(I - L, N) - np.absolute(tdot(I - L, N))) / np.absolute(tdot(I - L, N)))
return veri
def calc_heights(self, faces, N, L, R, i):
tdot = lambda x, y: (x * y).sum(axis=1)
normal = np.transpose(N.T/np.linalg.norm(N, axis=1))
I = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(faces, 0, 0)[:, 0])
if i==2:
heights = np.array([np.absolute(tdot(I - L, normal)), np.absolute(tdot(I - R, normal))])
else:
heights = np.array(np.absolute(tdot(I - L, normal)))
return heights
def calc_keq_dji_djk(self, N, L, R, KL, KR, hL, hR, area):
tdot = lambda x, y: (x * y).sum(axis=1)
I = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 0])
J = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 1])
K = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.internal, 0, 0)[:, 2])
JI = I - J; JK = K - J; LR = R - L; TJI = np.cross(N, JI); TJK = np.cross(N, JK)
KnL = mdot(N, KL, N) / (area * area)
KnR = mdot(N, KR, N) / (area * area)
keq = (area * KnL * KnR) / (hL * KnR + hR * KnL)
KTJIL = mdot(N, KL, TJI) / (area * area)
KTJKR = mdot(N, KR, TJK) / (area * area)
KTJIR = mdot(N, KR, TJI) / (area * area)
KTJKL = mdot(N, KL, TJK) / (area * area)
dji = (tdot(TJI, LR) / (area * area)) - (1 / area) * (hL * (KTJIL / KnL) + hR * (KTJIR / KnR))
djk = (tdot(TJK, LR) / (area * area)) - (1 / area) * (hL * (KTJKL / KnL) + hR * (KTJKR / KnR))
return keq, dji, djk
def calc_boundary_contribution(self, N, L, KL, hL, area):
tdot = lambda x, y: (x * y).sum(axis=1)
noI = self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 0]
noJ = self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 1]
noK = self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 2]
I = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 0])
J = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 1])
K = self.mesh.nodes.coords(self.mesh.faces.bridge_adjacencies(self.mesh.faces.boundary, 0, 0)[:, 2])
JI = I - J; JK = K - J; LJ = J - L
veri = np.nonzero((tdot(np.cross(JI, JK), N) - np.absolute(tdot(np.cross(JI, JK), N))) / np.absolute(tdot(np.cross(JI, JK), N)))
I[veri], K[veri] = K[veri], I[veri]; noI[veri], noK[veri] = noK[veri], noI[veri]
JI = I - J; JK = K - J
TJI = np.cross(N, JI); TJK = np.cross(N, JK)
pI = self.node_value[noI]; pJ = self.node_value[noJ]; pK = self.node_value[noK]
KnL = mdot(N, KL, N) / (area * area)
KTJIL = mdot(N, KL, TJI) / (area * area)
KTJKL = mdot(N, KL, TJK) / (area * area)
dirichletQ = (1 / (2 * hL * area)) * ((tdot(-TJK, LJ) * KnL + hL * area * KTJKL) * (pI-pJ) - \
2 * (area * area) * KnL * pJ + (tdot(-TJI, LJ) * KnL + hL * area * KTJIL) * (pJ-pK))
dirichletT = (1 / (2 * hL * area)) * (2 * (area * area) * KnL)
neumann = np.zeros(len(self.mesh.faces.boundary))
for key in self.mesh.faces.flag:
for i in self.mesh.faces.flag[key]:
neumann[i] = area[i] * self.flag_value[key] * (key > 200)
dirichletQ[i] *= (key < 200)
dirichletT[i] *= (key < 200)
return dirichletT, dirichletQ, neumann
def weights(self):
esurn = self.mesh.nodes.bridge_adjacencies(self.mesh.nodes.all, 0, 3)
w = []; fw = []; node = 0
for i in esurn:
Q = self.mesh.nodes.coords(node)
noI, noJ, noK, ocI, ocJ, ocK, ofI, ofJ, ofK = self.calc_nos_viz(i, node)
I = self.mesh.nodes.coords(noI)
J = self.mesh.nodes.coords(noJ)
K = self.mesh.nodes.coords(noK)
T = np.array([0.5 * (Q + I), (1 /3) * (Q + I + K), 0.5 * (Q + K), (1 /3) * (Q + J + K), 0.5 * (Q + J), ( 1 /3) * (Q + I + J)])
t = np.array([[0, 5, 1],[1, 3, 2],[3, 5, 4],[1, 5, 3]])
csi = self.calc_csi(Q, T, t, i)
neta, sigma, neubf = self.calc_neta_sigma(node, Q, T, ocI, ocJ, ocK, ofI, ofJ, ofK, i)
lam, gam = self.calc_lam(node, csi, neta, sigma, neubf, ocI, ocJ, ocK, i)
S = sum(lam) if any(lam!=0) else 1
w.append(lam/S)
fw.append(gam/S)
node = node + 1
w = np.array(w)
fw = np.array(fw)
return w, fw
def calc_nos_viz(self, esurn, node):
tdot = lambda x, y: (x * y).sum(axis=1)
othernodes = self.mesh.volumes.bridge_adjacencies(esurn, 3, 0)[self.mesh.volumes.bridge_adjacencies(esurn, 3, 0) != node].reshape((len(esurn), 3))
elemfaceelem = self.mesh.volumes.bridge_adjacencies(esurn, 2, 3)
noI = othernodes[:, 0]; noJ = othernodes[:, 1]; noK = othernodes[:, 2]
I = self.mesh.nodes.coords(noI); J = self.mesh.nodes.coords(noJ); K = self.mesh.nodes.coords(noK)
N = 0.5 * np.cross(I - J, K - J); C = self.mesh.volumes.center(esurn)
veri = np.nonzero((tdot(I - C, N) - np.absolute(tdot(I - C, N))) / np.absolute(tdot(I - C, N)))
noJ[veri], noK[veri] = noK[veri], noJ[veri]
elemsurnoI = self.mesh.nodes.bridge_adjacencies(noI, 0, 3)
elemsurnoJ = self.mesh.nodes.bridge_adjacencies(noJ, 0, 3)
elemsurnoK = self.mesh.nodes.bridge_adjacencies(noK, 0, 3)
ocI = -1 * np.ones(len(elemsurnoI)); ocJ = -1 * np.ones(len(elemsurnoJ)); ocK = -1 * np.ones(len(elemsurnoK))
for j in range(len(elemsurnoI)):
for e in elemfaceelem[j][:]:
if len([d for d, x in enumerate(e == elemsurnoI[j][:]) if x])==0: ocI[j] = e
for j in range(len(elemsurnoJ)):
for e in elemfaceelem[j][:]:
if len([d for d, x in enumerate(e == elemsurnoJ[j][:]) if x])==0: ocJ[j] = e
for j in range(len(elemsurnoK)):
for e in elemfaceelem[j][:]:
if len([d for d, x in enumerate(e == elemsurnoK[j][:]) if x])==0: ocK[j] = e
faceelem = self.mesh.volumes.bridge_adjacencies(esurn, 2, 2)
facesurnoI = self.mesh.nodes.bridge_adjacencies(noI, 0, 2)
facesurnoJ = self.mesh.nodes.bridge_adjacencies(noJ, 0, 2)
facesurnoK = self.mesh.nodes.bridge_adjacencies(noK, 0, 2)
ofI = -1 * np.ones(len(facesurnoI)); ofJ = -1 * np.ones(len(facesurnoJ)); ofK = -1 * np.ones(len(facesurnoK))
for j in range(len(facesurnoI)):
for e in faceelem[j][:]:
if len([d for d, x in enumerate(e == facesurnoI[j][:]) if x]) == 0: ofI[j] = e
for j in range(len(facesurnoJ)):
for e in faceelem[j][:]:
if len([d for d, x in enumerate(e == facesurnoJ[j][:]) if x]) == 0: ofJ[j] = e
for j in range(len(facesurnoK)):
for e in faceelem[j][:]:
if len([d for d, x in enumerate(e == facesurnoK[j][:]) if x]) == 0: ofK[j] = e
return noI, noJ, noK, ocI, ocJ, ocK, ofI, ofJ, ofK
def calc_csi(self,Q,T,t,esurn):
tdot = lambda x, y: (x * y).sum(axis=1)
permeabilities = self.vol_permeabilities(); Kk = permeabilities[esurn]
NQ = 0.5 * np.cross(T[t[:,0],:,:] - T[t[:,1],:,:], T[t[:,2],:,:] - T[t[:,1],:,:])
QI = T[t[:, 0], :, :] - Q; QJ = T[t[:, 1], :, :] - Q; QK = T[t[:, 2], :, :] - Q
Nti = 0.5 * np.cross(QJ, QK); Ntj = 0.5 * np.cross(QK, QI); Ntk = 0.5 * np.cross(QI, QJ)
csi = []
for i in range(len(esurn)):
csi.append([0.0]*6)
csi = np.array(csi)
for i in range(len(NQ)):
vol_t = abs((1/6) * tdot(2*Ntk[i],QK[i]))
csi[:,t[i,0]] += mdot(NQ[i], Kk, Nti[i]) / (3 * vol_t)
csi[:,t[i,1]] += mdot(NQ[i], Kk, Ntj[i]) / (3 * vol_t)
csi[:,t[i,2]] += mdot(NQ[i], Kk, Ntk[i]) / (3 * vol_t)
csi = csi.T
return csi
def calc_neta_sigma(self, node, Q, T, ocI, ocJ, ocK, ofI, ofJ, ofK, esurn):
tdot = lambda x, y: (x * y).sum(axis=1)
permeabilities = self.vol_permeabilities(); Kk = permeabilities[esurn]; eps = 1e-9
KkoI = []; KkoJ = []; KkoK = []; CoI = []; CoJ = []; CoK = []
KK = np.array([[0.0] * len(esurn) for y in range(6)]); area = np.zeros_like(KK); height = np.zeros_like(KK)
Nf = np.zeros_like(T); t = list(range(6)); u = [1,2,3,4,5,0]; C = self.mesh.volumes.center(esurn)
C += eps * np.random.rand(*C.shape); valueopI = []; valueopJ = []; valueopK = []
for i in range(len(esurn)):
KkoI.append(permeabilities[int(ocI[i])]) if ocI[i]>=0 else KkoI.append([[0]*3 for y in range(3)])
KkoJ.append(permeabilities[int(ocJ[i])]) if ocJ[i]>=0 else KkoJ.append([[0]*3 for y in range(3)])
KkoK.append(permeabilities[int(ocK[i])]) if ocK[i]>=0 else KkoK.append([[0]*3 for y in range(3)])
CoI.append(self.mesh.volumes.center(int(ocI[i]))) if ocI[i]>=0 else CoI.append(Q)
CoJ.append(self.mesh.volumes.center(int(ocJ[i]))) if ocJ[i]>=0 else CoJ.append(Q)
CoK.append(self.mesh.volumes.center(int(ocK[i]))) if ocK[i]>=0 else CoK.append(Q)
valueopI.append(self.flag_value[self.face_flag[int(ofI[i])]])
valueopJ.append(self.flag_value[self.face_flag[int(ofJ[i])]])
valueopK.append(self.flag_value[self.face_flag[int(ofK[i])]])
KkoI = np.array(KkoI); KkoJ = np.array(KkoJ); KkoK = np.array(KkoK)
CoI = np.array(CoI); CoJ = np.array(CoJ); CoK = np.array(CoK)
for i in range(6):
Nf[i] = 0.5 * np.cross(T[u[i]] - Q, T[t[i]] - Q)
area[i] = np.linalg.norm(Nf[i], axis=1)
height[i] = tdot(T[t[i]] - C, Nf[i]) / np.linalg.norm(Nf[i], axis=1)
hoI = eps + tdot(CoI - Q, 0.5 * np.cross(T[4, :, :] - Q, T[2, :, :] - Q)) /\
np.linalg.norm(0.5 * np.cross(T[4, :, :] - Q, T[2, :, :] - Q), axis=1)
hoJ = eps + tdot(CoJ - Q, 0.5 * np.cross(T[2, :, :] - Q, T[0, :, :] - Q)) /\
np.linalg.norm(0.5 * np.cross(T[2, :, :] - Q, T[0, :, :] - Q), axis=1)
hoK = eps + tdot(CoK - Q, 0.5 * np.cross(T[0, :, :] - Q, T[4, :, :] - Q)) /\
np.linalg.norm(0.5 * np.cross(T[0, :, :] - Q, T[4, :, :] - Q), axis=1)
Kn = np.zeros_like(KK); Kt = np.zeros_like(KK); Kto = np.zeros_like(KK); Kno = np.zeros_like(KK)
Do1 = np.zeros_like(KK); D1 = np.zeros_like(KK); D2 = np.zeros_like(KK); Do2 = np.zeros_like(KK)
sigmao = np.zeros_like(KK); neubf = np.zeros_like(KK)
for i in range(6):
tau1 = np.cross(Nf[i], T[t[i]] - Q)
tau2 = np.cross(Nf[i], T[u[i]] - Q)
Kn[i, :] = mdot(Nf[i], Kk, Nf[i]) / (area[i] * area[i])
Kt[i, :] = mdot(Nf[i], Kk, tau1) / (area[i] * area[i])
D1[i, :] = tdot(-tau1, Q - C) / (2 * height[i] * area[i])
D2[i, :] = tdot(-tau2, Q - C) / (2 * height[i] * area[i])
if i == 0 or i == 1: Kto[i, :] = mdot(Nf[i], KkoJ, tau1) / (area[i] * area[i]); \
Kno[i, :] = mdot(Nf[i], KkoJ, Nf[i]) / (area[i] * area[i]); \
Do1[i, :] = tdot(-tau1, CoJ - Q) / (2 * hoJ * area[i]); \
Do2[i, :] = tdot(-tau2, CoJ - Q) / (2 * hoJ * area[i]); \
sigmao[i, :] = (area[i] / hoJ) * (mdot(Nf[i], KkoJ, Nf[i]) / (area[i] * area[i])); \
neubf[i, :] = area[i] * valueopJ * (self.node_flag[node] > 200) * (ocJ<0)
if i == 2 or i == 3: Kto[i, :] = mdot(Nf[i], KkoI, tau1) / (area[i] * area[i]); \
Kno[i, :] = mdot(Nf[i], KkoI, Nf[i]) / (area[i] * area[i]); \
Do1[i, :] = tdot(-tau1, CoI - Q) / (2 * hoI * area[i]); \
Do2[i, :] = tdot(-tau2, CoI - Q) / (2 * hoI * area[i]); \
sigmao[i, :] = (area[i] / hoI) * (mdot(Nf[i], KkoI, Nf[i]) / (area[i] * area[i])); \
neubf[i, :] = area[i] * valueopI * (self.node_flag[node] > 200) * (ocI<0)
if i == 4 or i == 5: Kto[i, :] = mdot(Nf[i], KkoK, tau1) / (area[i] * area[i]); \
Kno[i, :] = mdot(Nf[i], KkoK, Nf[i]) / (area[i] * area[i]); \
Do1[i, :] = tdot(-tau1, CoK - Q) / (2 * hoK * area[i]); \
Do2[i, :] = tdot(-tau2, CoK - Q) / (2 * hoK * area[i]); \
sigmao[i, :] = (area[i] / hoK) * (mdot(Nf[i], KkoK, Nf[i]) / (area[i] * area[i])); \
neubf[i, :] = area[i] * valueopK * (self.node_flag[node] > 200) * (ocK<0)
sigma = np.array([(area / height) * Kn, sigmao])
neta = np.array([D1 * Kn - Do1 * Kno + 0.5 * Kt - 0.5 * Kto, D2 * Kn - Do2 * Kno + 0.5 * Kt - 0.5 * Kto])
return neta, sigma, neubf
def calc_lam(self, node, csi, neta, sigma, neubf, ocI, ocJ, ocK, esurn):
lam = np.array([0.0]*len(esurn))
gam = 0
t = list(range(6)); u = [1,2,3,4,5,0]
if self.node_flag[node] > 200:
for i in range(len(esurn)):
H = np.array([[0.0] * 6 for y in range(6)])
C = np.array([[0.0] * 4 for y in range(6)])
H[t, t] = -neta[1, :, i]
H[t, u] = neta[0, :, i]
B = np.linalg.inv(H)
A = csi[:,i]
C[:,0] = -sigma[0,:,i]; C[0:2,1] = -sigma[1,0:2,i]; C[2:4,2] = -sigma[1,2:4,i]; C[4:6,3] = -sigma[1,4:6,i]
L = (A@B)@C
lam[esurn == esurn[i]] += L[0]
lam[esurn == ocJ[i]] += L[1]
lam[esurn == ocI[i]] += L[2]
lam[esurn == ocK[i]] += L[3]
gam += sum((A@B)*neubf[:,i]) + sum(neubf[:,i])
return lam, gam
# import pdb; pdb.set_trace()