def getError(self):
#Test function
phi = lambda x: np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1])
j_funX = lambda x: -np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi *
x[:, 1])
j_funY = lambda x: -np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
jX_ana = j_funX(self.M.gridFx)
jY_ana = j_funY(self.M.gridFy)
j_ana = np.r_[jX_ana, jY_ana]
#TODO: Check where our boundary conditions are CCx or Nx
# fxm,fxp,fym,fyp = self.M.faceBoundaryInd
# gBFx = self.M.gridFx[(fxm|fxp),:]
# gBFy = self.M.gridFy[(fym|fyp),:]
fxm, fxp, fym, fyp = self.M.cellBoundaryInd
gBFx = self.M.gridCC[(fxm | fxp), :]
gBFy = self.M.gridCC[(fym | fyp), :]
bc = phi(np.r_[gBFx, gBFy])
# P = sp.csr_matrix(([-1,1],([0,self.M.nF-1],[0,1])), shape=(self.M.nF, 2))
P, Pin, Pout = self.M.getBCProjWF('dirichlet')
Mc = self.M.getFaceInnerProduct()
McI = Utils.sdInv(self.M.getFaceInnerProduct())
G = -self.M.faceDiv.T * Utils.sdiag(self.M.vol)
D = self.M.faceDiv
j = McI * (G * xc_ana + P * bc)
q = D * j
# self.M.plotImage(j, 'FxFy', showIt=True)
# Rearrange if we know q to solve for x
A = D * McI * G
rhs = q_ana - D * McI * P * bc
if self.myTest == 'j':
err = np.linalg.norm((j - j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q - q_ana), np.inf)
elif self.myTest == 'xc':
xc = Solver(A) * (rhs)
err = np.linalg.norm((xc - xc_ana), np.inf)
elif self.myTest == 'xcJ':
xc = Solver(A) * (rhs)
j = McI * (G * xc + P * bc)
err = np.linalg.norm((j - j_ana), np.inf)
return err
def MagneticsDiffSecondaryInv(mesh, model, data, **kwargs):
"""
Inversion module for MagneticsDiffSecondary
"""
from SimPEG import Optimization, Regularization, Parameters, ObjFunction, Inversion
prob = MagneticsDiffSecondary(mesh, model)
miter = kwargs.get('maxIter', 10)
if prob.ispaired:
prob.unpair()
if data.ispaired:
data.unpair()
prob.pair(data)
# Create an optimization program
opt = Optimization.InexactGaussNewton(maxIter=miter)
opt.bfgsH0 = Solver(sp.identity(model.nP), flag='D')
# Create a regularization program
reg = Regularization.Tikhonov(model)
# Create an objective function
beta = Parameters.BetaSchedule(beta0=1e0)
obj = ObjFunction.BaseObjFunction(data, reg, beta=beta)
# Create an inversion object
inv = Inversion.BaseInversion(obj, opt)
return inv, reg
def getError(self):
# Test function
def phi_fun(x):
return np.cos(np.pi * x)
def j_fun(x):
return np.pi * np.sin(np.pi * x)
def phi_deriv(x):
return -j_fun(x)
def q_fun(x):
return (np.pi**2) * np.cos(np.pi * x)
xc_ana = phi_fun(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
# Get boundary locations
vecN = self.M.vectorNx
vecC = self.M.vectorCCx
# Setup Mixed B.C (alpha, beta, gamma)
alpha_xm, alpha_xp = 1., 1.
beta_xm, beta_xp = 1., 1.
alpha = np.r_[alpha_xm, alpha_xp]
beta = np.r_[beta_xm, beta_xp]
vecN = self.M.vectorNx
vecC = self.M.vectorCCx
phi_bc = phi_fun(vecN[[0, -1]])
phi_deriv_bc = phi_deriv(vecN[[0, -1]])
gamma = alpha * phi_bc + beta * phi_deriv_bc
x_BC, y_BC = getxBCyBC_CC(self.M, alpha, beta, gamma)
sigma = np.ones(self.M.nC)
Mfrho = self.M.getFaceInnerProduct(1. / sigma)
MfrhoI = self.M.getFaceInnerProduct(1. / sigma, invMat=True)
V = Utils.sdiag(self.M.vol)
Div = V * self.M.faceDiv
P_BC, B = self.M.getBCProjWF_simple()
q = q_fun(self.M.gridCC)
M = B * self.M.aveCC2F
G = Div.T - P_BC * Utils.sdiag(y_BC) * M
# Mrhoj = D.T V phi + P_BC*Utils.sdiag(y_BC)*M phi - P_BC*x_BC
rhs = V * q + Div * MfrhoI * P_BC * x_BC
A = Div * MfrhoI * G
if self.myTest == 'xc':
# TODO: fix the null space
Ainv = Solver(A)
xc = Ainv * rhs
err = np.linalg.norm((xc - xc_ana), np.inf)
else:
NotImplementedError
return err
def getError(self):
#Test function
phi = lambda x: np.cos(np.pi * x)
j_fun = lambda x: -np.pi * np.sin(np.pi * x)
q_fun = lambda x: -(np.pi**2) * np.cos(np.pi * x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
#TODO: Check where our boundary conditions are CCx or Nx
# vec = self.M.vectorNx
vec = self.M.vectorCCx
phi_bc = phi(vec[[0, -1]])
j_bc = j_fun(vec[[0, -1]])
P, Pin, Pout = self.M.getBCProjWF([['dirichlet', 'dirichlet']])
Mc = self.M.getFaceInnerProduct()
McI = Utils.sdInv(self.M.getFaceInnerProduct())
V = Utils.sdiag(self.M.vol)
G = -Pin.T * Pin * self.M.faceDiv.T * V
D = self.M.faceDiv
j = McI * (G * xc_ana + P * phi_bc)
q = V * D * Pin.T * Pin * j + V * D * Pout.T * j_bc
# Rearrange if we know q to solve for x
A = V * D * Pin.T * Pin * McI * G
rhs = V * q_ana - V * D * Pin.T * Pin * McI * P * phi_bc - V * D * Pout.T * j_bc
# A = D*McI*G
# rhs = q_ana - D*McI*P*phi_bc
if self.myTest == 'j':
err = np.linalg.norm((j - j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q - V * q_ana), np.inf)
elif self.myTest == 'xc':
#TODO: fix the null space
solver = SolverCG(A, maxiter=1000)
xc = solver * (rhs)
print 'ACCURACY', np.linalg.norm(Utils.mkvc(A * xc) - rhs)
err = np.linalg.norm((xc - xc_ana), np.inf)
elif self.myTest == 'xcJ':
#TODO: fix the null space
xc = Solver(A) * (rhs)
print np.linalg.norm(Utils.mkvc(A * xc) - rhs)
j = McI * (G * xc + P * phi_bc)
err = np.linalg.norm((j - j_ana), np.inf)
return err
def getError(self):
# Create some functions to integrate
fun = lambda x: np.sin(2 * np.pi * x[:, 0]) * np.sin(
2 * np.pi * x[:, 1]) * np.sin(2 * np.pi * x[:, 2])
sol = lambda x: -3. * ((2 * np.pi)**2) * fun(x)
self.M.setCellGradBC('dirichlet')
D = self.M.faceDiv
G = self.M.cellGrad
if self.forward:
sA = sol(self.M.gridCC)
sN = D * G * fun(self.M.gridCC)
err = np.linalg.norm((sA - sN), np.inf)
else:
fA = fun(self.M.gridCC)
fN = Solver(D * G) * (sol(self.M.gridCC))
err = np.linalg.norm((fA - fN), np.inf)
return err
def MagneticsDiffSecondaryInv(mesh, model, data, **kwargs):
"""
Inversion module for MagneticsDiffSecondary
"""
from SimPEG import Optimization, Regularization, Parameters, ObjFunction, Inversion
prob = Simulation3DDifferential(mesh, survey=data, mu=model)
miter = kwargs.get("maxIter", 10)
# Create an optimization program
opt = Optimization.InexactGaussNewton(maxIter=miter)
opt.bfgsH0 = Solver(sp.identity(model.nP), flag="D")
# Create a regularization program
reg = Regularization.Tikhonov(model)
# Create an objective function
beta = Parameters.BetaSchedule(beta0=1e0)
obj = ObjFunction.BaseObjFunction(prob, reg, beta=beta)
# Create an inversion object
inv = Inversion.BaseInversion(obj, opt)
return inv, reg
def run(plotIt=True, n=60):
np.random.seed(5)
# Here we are going to rearrange the equations:
# (phi_ - phi)/dt = A*(d2fdphi2*(phi_ - phi) + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*(d2fdphi2*phi_ - d2fdphi2*phi + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*d2fdphi2*phi_ + A*( - d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - phi = dt*A*d2fdphi2*phi_ + dt*A*(- d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - dt*A*d2fdphi2 * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*dfdphi - dt*A*d2fdphi2*phi - dt*A*L*phi_ + phi
# (dt*A*d2fdphi2 - I) * phi_ = dt*A*d2fdphi2*phi + dt*A*L*phi_ - phi - dt*A*dfdphi
# (dt*A*d2fdphi2 - I - dt*A*L) * phi_ = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
h = [(0.25, n)]
M = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 0.
dexp = -5
phi = np.random.normal(loc=0.5, scale=0.01, size=M.nC)
ii, jj = 0, 0
PHIS = []
capture = np.logspace(-1, np.log10(duration), 8)
while elapsed < duration:
dt = min(100, np.exp(dexp))
elapsed += dt
dexp += 0.05
dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
d2fdphi2 = Utils.sdiag(a**2 * 2 * (1 - 6 * phi * (1 - phi)))
MAT = (dt*A*d2fdphi2 - I - dt*A*L)
rhs = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
phi = Solver(MAT)*rhs
if elapsed > capture[jj]:
PHIS += [(elapsed, phi.copy())]
jj += 1
if ii % 10 == 0:
print(ii, elapsed)
ii += 1
if plotIt:
fig, axes = plt.subplots(2, 4, figsize=(14, 6))
axes = np.array(axes).flatten().tolist()
for ii, ax in zip(np.linspace(0, len(PHIS)-1, len(axes)), axes):
ii = int(ii)
M.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))
def simulateMT(mesh, sigma, frequency, rtype="app_res"):
"""
Compute apparent resistivity and phase at each frequency.
Return apparent resistivity and phase for rtype="app_res",
or impedance for rtype="impedance"
"""
# Angular frequency (rad/s)
def omega(freq):
return 2*np.pi*freq
# make sure we are working with numpy arrays
if type(frequency) is float:
frequency = np.r_[frequency] # make it a list to loop over later if it is just a scalar
elif type(frequency) is list:
frequency = np.array(frequency)
# Grad
mesh.setCellGradBC([['dirichlet', 'dirichlet']]) # Setup boundary conditions
Grad = mesh.cellGrad # Gradient matrix
# MfMu
mu = np.ones(mesh.nC)*mu_0 # magnetic permeability values for all cells
Mmu = Utils.sdiag(mesh.aveCC2F * mu)
# Mccsigma
sigmahat = sigma # quasi-static assumption
Msighat = Utils.sdiag(sigmahat)
# Div
Div = mesh.faceDiv # Divergence matrix
# Right Hand Side
B = mesh.cellGradBC # a matrix for boundary conditions
Exbc = np.r_[0., 1.] # boundary values for Ex
# Right-hand side
rhs = np.r_[
-B*Exbc,
np.zeros(mesh.nC)
]
# loop over frequencies
Zxy = []
for freq in frequency:
# A-matrix
A = sp.vstack([
sp.hstack([Grad, 1j*omega(freq)*Mmu]), # Top row of A matrix
sp.hstack((Msighat, Div)) # Bottom row of A matrix
])
Ainv = Solver(A) # Factorize A matrix
sol = Ainv*rhs # Solve A^-1 rhs = sol
Ex = sol[:mesh.nC] # Extract Ex from solution vector u
Hy = sol[mesh.nC:mesh.nC+mesh.nN] # Extract Hy from solution vector u
Zxy.append(- 1./Hy[-1]) # Impedance at the surface
# turn it into an array
Zxy = np.array(Zxy)
if rtype.lower() == "impedance":
return Zxy
elif rtype.lower() == "app_res":
app_res = abs(Zxy)**2 / (mu_0*omega(frequency))
app_phase = np.rad2deg(np.arctan(Zxy.imag / Zxy.real))
return app_res, app_phase
else:
raise Exception("rtype must be 'impedance' or 'app_res', not {}".format(rtype.lower()))
def dc_resistivity(
log_sigma_background=1., # Conductivity of the background, S/m
log_sigma_block=2, # Conductivity of the block, S/m
plot_type='potential' # "conductivity", "potential", or "current"
):
from pylab import rcParams
rcParams['figure.figsize'] = 10, 10
# Define a unit-cell mesh
mesh = Mesh.TensorMesh([100, 100]) # setup a mesh on which to solve
# model parameters
sigma_background = 10**log_sigma_background
sigma_block = 10**log_sigma_block
# add a block to our model
x_block = np.r_[0.4, 0.6]
y_block = np.r_[0.4, 0.6]
# assign them on the mesh
# create a physical property model
sigma = sigma_background * np.ones(mesh.nC)
block_indices = ((mesh.gridCC[:, 0] >= x_block[0]) & # left boundary
(mesh.gridCC[:, 0] <= x_block[1]) & # right boundary
(mesh.gridCC[:, 1] >= y_block[0]) & # bottom boundary
(mesh.gridCC[:, 1] <= y_block[1])) # top boundary
# add the block to the physical property model
sigma[block_indices] = sigma_block
# Define a source
a_loc, b_loc = np.r_[0.2, 0.5], np.r_[0.8, 0.5]
source_locs = [a_loc, b_loc]
# locate it on the mesh
source_loc_inds = Utils.closestPoints(mesh, source_locs)
a_loc_mesh = mesh.gridCC[source_loc_inds[0], :]
b_loc_mesh = mesh.gridCC[source_loc_inds[1], :]
if plot_type == 'conductivity':
plt.colorbar(mesh.plotImage(sigma)[0])
plt.plot(a_loc_mesh[0], a_loc_mesh[1], 'wv', markersize=8)
plt.plot(b_loc_mesh[0], b_loc_mesh[1], 'w^', markersize=8)
plt.title('electrical conductivity, $\sigma$')
return
# Assemble and solve the DC resistivity problem
Div = mesh.faceDiv
Sigma = mesh.getFaceInnerProduct(sigma, invProp=True, invMat=True)
Vol = Utils.sdiag(mesh.vol)
# assemble the system matrix
A = Vol * Div * Sigma * Div.T * Vol
# right hand side
q = np.zeros(mesh.nC)
q[source_loc_inds] = np.r_[+1, -1]
# solve the DC resistivity problem
Ainv = Solver(A) # create a matrix that behaves like A inverse
phi = Ainv * q
if plot_type == 'potential':
plt.colorbar(mesh.plotImage(phi)[0])
plt.title('Electric Potential, $\phi$')
return
if plot_type == 'current':
j = Sigma * mesh.faceDiv.T * Utils.sdiag(mesh.vol) * phi
plt.colorbar(
mesh.plotImage(j, vType='F', view='vec', streamOpts={'color':
'w'})[0])
plt.title('Current, $j$')
return
if showIt: plt.show()
return [scalarMap]
if __name__ == '__main__':
import numpy as np
from SimPEG import Mesh, Utils, Solver
hx = [(5, 2, -1.3), (2, 4), (5, 2, 1.3)]
hy = [(2, 2, -1.3), (2, 6), (2, 2, 1.3)]
hz = [(2, 2, -1.3), (2, 6), (2, 2, 1.3)]
M = Mesh.TensorMesh([hx, hy, hz], x0=[10, 20, 14])
q = np.zeros(M.vnC)
q[[4, 4], [4, 4], [2, 6]] = [-1, 1]
q = Utils.mkvc(q)
A = M.faceDiv * M.cellGrad
b = Solver(A) * (q)
M.plotSlice(M.cellGrad * b,
'F',
view='vec',
grid=True,
pcolorOpts={'alpha': 0.8})
M2 = Mesh.TensorMesh([10, 20], x0=[10, 5])
f = np.r_[np.sin(M2.gridFx[:, 0] * 2 * np.pi),
np.sin(M2.gridFy[:, 1] * 2 * np.pi)]
M2.plotImage(f, 'F', view='vec', grid=True, pcolorOpts={'alpha': 0.8})
M2.plotImage(f, 'Fx')
f = np.r_[np.sin(M2.gridEx[:, 0] * 2 * np.pi),
np.sin(M2.gridEy[:, 1] * 2 * np.pi)]
M2.plotImage(f, 'E', view='vec', grid=True, pcolorOpts={'alpha': 0.8})
def getError(self):
# Test function
def phi_fun(x):
return (np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1]) *
np.cos(np.pi * x[:, 2]))
def j_funX(x):
return (np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1]) *
np.cos(np.pi * x[:, 2]))
def j_funY(x):
return (np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi * x[:, 1]) *
np.cos(np.pi * x[:, 2]))
def j_funZ(x):
return (np.pi * np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1]) *
np.sin(np.pi * x[:, 2]))
def phideriv_funX(x):
return -j_funX(x)
def phideriv_funY(x):
return -j_funY(x)
def phideriv_funZ(x):
return -j_funZ(x)
def q_fun(x):
return 3 * (np.pi**2) * phi_fun(x)
xc_ana = phi_fun(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
jX_ana = j_funX(self.M.gridFx)
jY_ana = j_funY(self.M.gridFy)
j_ana = np.r_[jX_ana, jY_ana, jY_ana]
# Get boundary locations
fxm, fxp, fym, fyp, fzm, fzp = self.M.faceBoundaryInd
gBFxm = self.M.gridFx[fxm, :]
gBFxp = self.M.gridFx[fxp, :]
gBFym = self.M.gridFy[fym, :]
gBFyp = self.M.gridFy[fyp, :]
gBFzm = self.M.gridFz[fzm, :]
gBFzp = self.M.gridFz[fzp, :]
# Setup Mixed B.C (alpha, beta, gamma)
alpha_xm = np.ones_like(gBFxm[:, 0])
alpha_xp = np.ones_like(gBFxp[:, 0])
beta_xm = np.ones_like(gBFxm[:, 0])
beta_xp = np.ones_like(gBFxp[:, 0])
alpha_ym = np.ones_like(gBFym[:, 1])
alpha_yp = np.ones_like(gBFyp[:, 1])
beta_ym = np.ones_like(gBFym[:, 1])
beta_yp = np.ones_like(gBFyp[:, 1])
alpha_zm = np.ones_like(gBFzm[:, 2])
alpha_zp = np.ones_like(gBFzp[:, 2])
beta_zm = np.ones_like(gBFzm[:, 2])
beta_zp = np.ones_like(gBFzp[:, 2])
phi_bc_xm, phi_bc_xp = phi_fun(gBFxm), phi_fun(gBFxp)
phi_bc_ym, phi_bc_yp = phi_fun(gBFym), phi_fun(gBFyp)
phi_bc_zm, phi_bc_zp = phi_fun(gBFzm), phi_fun(gBFzp)
phiderivX_bc_xm = phideriv_funX(gBFxm)
phiderivX_bc_xp = phideriv_funX(gBFxp)
phiderivY_bc_ym = phideriv_funY(gBFym)
phiderivY_bc_yp = phideriv_funY(gBFyp)
phiderivY_bc_zm = phideriv_funZ(gBFzm)
phiderivY_bc_zp = phideriv_funZ(gBFzp)
def gamma_fun(alpha, beta, phi, phi_deriv):
return alpha * phi + beta * phi_deriv
gamma_xm = gamma_fun(alpha_xm, beta_xm, phi_bc_xm, phiderivX_bc_xm)
gamma_xp = gamma_fun(alpha_xp, beta_xp, phi_bc_xp, phiderivX_bc_xp)
gamma_ym = gamma_fun(alpha_ym, beta_ym, phi_bc_ym, phiderivY_bc_ym)
gamma_yp = gamma_fun(alpha_yp, beta_yp, phi_bc_yp, phiderivY_bc_yp)
gamma_zm = gamma_fun(alpha_zm, beta_zm, phi_bc_zm, phiderivY_bc_zm)
gamma_zp = gamma_fun(alpha_zp, beta_zp, phi_bc_zp, phiderivY_bc_zp)
alpha = [alpha_xm, alpha_xp, alpha_ym, alpha_yp, alpha_zm, alpha_zp]
beta = [beta_xm, beta_xp, beta_ym, beta_yp, beta_zm, beta_zp]
gamma = [gamma_xm, gamma_xp, gamma_ym, gamma_yp, gamma_zm, gamma_zp]
x_BC, y_BC = getxBCyBC_CC(self.M, alpha, beta, gamma)
sigma = np.ones(self.M.nC)
Mfrho = self.M.getFaceInnerProduct(1. / sigma)
MfrhoI = self.M.getFaceInnerProduct(1. / sigma, invMat=True)
V = Utils.sdiag(self.M.vol)
Div = V * self.M.faceDiv
P_BC, B = self.M.getBCProjWF_simple()
q = q_fun(self.M.gridCC)
M = B * self.M.aveCC2F
G = Div.T - P_BC * Utils.sdiag(y_BC) * M
rhs = V * q + Div * MfrhoI * P_BC * x_BC
A = Div * MfrhoI * G
if self.myTest == 'xc':
# TODO: fix the null space
Ainv = Solver(A)
xc = Ainv * rhs
err = np.linalg.norm((xc - xc_ana), np.inf)
else:
NotImplementedError
return err
def run(plotIt=True, n=60):
np.random.seed(5)
# Here we are going to rearrange the equations:
# (phi_ - phi)/dt = A*(d2fdphi2*(phi_ - phi) + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*(d2fdphi2*phi_ - d2fdphi2*phi + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*d2fdphi2*phi_ + A*( - d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - phi = dt*A*d2fdphi2*phi_ + dt*A*(- d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - dt*A*d2fdphi2 * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*dfdphi - dt*A*d2fdphi2*phi - dt*A*L*phi_ + phi
# (dt*A*d2fdphi2 - I) * phi_ = dt*A*d2fdphi2*phi + dt*A*L*phi_ - phi - dt*A*dfdphi
# (dt*A*d2fdphi2 - I - dt*A*L) * phi_ = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
h = [(0.25, n)]
M = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 0.
dexp = -5
phi = np.random.normal(loc=0.5, scale=0.01, size=M.nC)
ii, jj = 0, 0
PHIS = []
capture = np.logspace(-1, np.log10(duration), 8)
while elapsed < duration:
dt = min(100, np.exp(dexp))
elapsed += dt
dexp += 0.05
dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
d2fdphi2 = Utils.sdiag(a**2 * 2 * (1 - 6 * phi * (1 - phi)))
MAT = (dt * A * d2fdphi2 - I - dt * A * L)
rhs = (dt * A * d2fdphi2 - I) * phi - dt * A * dfdphi
phi = Solver(MAT) * rhs
if elapsed > capture[jj]:
PHIS += [(elapsed, phi.copy())]
jj += 1
if ii % 10 == 0:
print(ii, elapsed)
ii += 1
if plotIt:
fig, axes = plt.subplots(2, 4, figsize=(14, 6))
axes = np.array(axes).flatten().tolist()
for ii, ax in zip(np.linspace(0, len(PHIS) - 1, len(axes)), axes):
ii = int(ii)
M.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))
def run(plotIt=True, n=60):
"""
Mesh: Operators: Cahn Hilliard
==============================
This example is based on the example in the FiPy_ library.
Please see their documentation for more information about the
Cahn-Hilliard equation.
The "Cahn-Hilliard" equation separates a field \\\\( \\\\phi \\\\)
into 0 and 1 with smooth transitions.
.. math::
\\frac{\partial \phi}{\partial t} = \\nabla \cdot D \\nabla \left( \\frac{\partial f}{\partial \phi} - \epsilon^2 \\nabla^2 \phi \\right)
Where \\\\( f \\\\) is the energy function \\\\( f = ( a^2 / 2 )\\\\phi^2(1 - \\\\phi)^2 \\\\)
which drives \\\\( \\\\phi \\\\) towards either 0 or 1, this competes with the term
\\\\(\\\\epsilon^2 \\\\nabla^2 \\\\phi \\\\) which is a diffusion term that creates smooth changes in \\\\( \\\\phi \\\\).
The equation can be factored:
.. math::
\\frac{\partial \phi}{\partial t} = \\nabla \cdot D \\nabla \psi \\\\
\psi = \\frac{\partial^2 f}{\partial \phi^2} (\phi - \phi^{\\text{old}}) + \\frac{\partial f}{\partial \phi} - \epsilon^2 \\nabla^2 \phi
Here we will need the derivatives of \\\\( f \\\\):
.. math::
\\frac{\partial f}{\partial \phi} = (a^2/2)2\phi(1-\phi)(1-2\phi)
\\frac{\partial^2 f}{\partial \phi^2} = (a^2/2)2[1-6\phi(1-\phi)]
The implementation below uses backwards Euler in time with an
exponentially increasing time step. The initial \\\\( \\\\phi \\\\)
is a normally distributed field with a standard deviation of 0.1 and
mean of 0.5. The grid is 60x60 and takes a few seconds to solve ~130
times. The results are seen below, and you can see the field separating
as the time increases.
.. _FiPy: http://www.ctcms.nist.gov/fipy/examples/cahnHilliard/generated/examples.cahnHilliard.mesh2DCoupled.html
"""
np.random.seed(5)
# Here we are going to rearrange the equations:
# (phi_ - phi)/dt = A*(d2fdphi2*(phi_ - phi) + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*(d2fdphi2*phi_ - d2fdphi2*phi + dfdphi - L*phi_)
# (phi_ - phi)/dt = A*d2fdphi2*phi_ + A*( - d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - phi = dt*A*d2fdphi2*phi_ + dt*A*(- d2fdphi2*phi + dfdphi - L*phi_)
# phi_ - dt*A*d2fdphi2 * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
# (I - dt*A*d2fdphi2) * phi_ = dt*A*dfdphi - dt*A*d2fdphi2*phi - dt*A*L*phi_ + phi
# (dt*A*d2fdphi2 - I) * phi_ = dt*A*d2fdphi2*phi + dt*A*L*phi_ - phi - dt*A*dfdphi
# (dt*A*d2fdphi2 - I - dt*A*L) * phi_ = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
h = [(0.25, n)]
M = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 0.
dexp = -5
phi = np.random.normal(loc=0.5, scale=0.01, size=M.nC)
ii, jj = 0, 0
PHIS = []
capture = np.logspace(-1, np.log10(duration), 8)
while elapsed < duration:
dt = min(100, np.exp(dexp))
elapsed += dt
dexp += 0.05
dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
d2fdphi2 = Utils.sdiag(a**2 * 2 * (1 - 6 * phi * (1 - phi)))
MAT = (dt * A * d2fdphi2 - I - dt * A * L)
rhs = (dt * A * d2fdphi2 - I) * phi - dt * A * dfdphi
phi = Solver(MAT) * rhs
if elapsed > capture[jj]:
PHIS += [(elapsed, phi.copy())]
jj += 1
if ii % 10 == 0:
print(ii, elapsed)
ii += 1
if plotIt:
fig, axes = plt.subplots(2, 4, figsize=(14, 6))
axes = np.array(axes).flatten().tolist()
for ii, ax in zip(np.linspace(0, len(PHIS) - 1, len(axes)), axes):
ii = int(ii)
M.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))
