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', show_it=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 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)
mesh = self.M
phi_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
phi_bc = phi(mesh.boundary_faces)
MfI = mesh.get_face_inner_product(invert_matrix=True)
M_bf = mesh.boundary_face_scalar_integral
V = utils.sdiag(mesh.cell_volumes)
G = -mesh.face_divergence.T * V
D = mesh.face_divergence
# Sinc the xc_bc is a known, move it to the RHS!
A = V @ D @ MfI @ G
rhs = V @ q_ana - V @ D @ MfI @ M_bf @ phi_bc
phi_test = Solver(A) * rhs
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (phi_test - phi_ana))
else:
err = np.linalg.norm(phi_test - phi_ana) / np.sqrt(mesh.n_cells)
return err
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 = discretize.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 * discretize.utils.sdiag(y_BC) * M
# Mrhoj = D.T V phi + P_BC*discretize.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.sin(np.pi * x[:, 0]) * np.sin(np.pi * x[:, 1])
j_funX = lambda x: np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
j_funY = lambda x: np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi *
x[:, 1])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
mesh = self.M
phi_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
phi_bc = phi(mesh.boundary_faces)
jx_bc = j_funX(mesh.boundary_faces)
jy_bc = j_funY(mesh.boundary_faces)
j_bc = np.c_[jx_bc, jy_bc]
j_bc_dot_n = np.sum(j_bc * mesh.boundary_face_outward_normals, axis=-1)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
# get indices of x0 boundary and y0 boundary
n_bounary_faces = len(j_bc_dot_n)
dirichlet_locs = np.any(mesh.boundary_faces == 0.0, axis=1)
alpha = np.zeros(n_bounary_faces)
alpha[dirichlet_locs] = 1.0
beta = np.zeros(n_bounary_faces)
beta[~dirichlet_locs] = 1.0
gamma = alpha * phi_bc + beta * j_bc_dot_n
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
phi_test = Solver(A) * rhs
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (phi_test - phi_ana))
else:
err = np.linalg.norm((phi_test - phi_ana)) / np.sqrt(mesh.n_cells)
return err
def getError(self):
# Test function
phi = lambda x: np.cos(0.5 * np.pi * x)
j_fun = lambda x: -0.5 * np.pi * np.sin(0.5 * np.pi * x)
q_fun = lambda x: -0.25 * (np.pi**2) * np.cos(0.5 * np.pi * x)
mesh = self.M
xc_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
j_ana = j_fun(mesh.faces_x)
phi_bc = phi(mesh.boundary_faces)
j_bc = j_fun(mesh.boundary_faces)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
alpha = np.r_[1.0, 0.0]
beta = np.r_[0.0, 1.0]
gamma = alpha * phi_bc + beta * j_bc * mesh.boundary_face_outward_normals
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
if self.myTest == "xc":
xc = Solver(A) * rhs
err = np.linalg.norm(xc - xc_ana) / np.sqrt(mesh.n_cells)
elif self.myTest == "xcJ":
xc = Solver(A) * rhs
j = MfI @ ((-G + B_bc) @ xc + b_bc)
err = np.linalg.norm(j - j_ana, np.inf)
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):
# Test function
phi = lambda x: np.sin(np.pi * x)
j_fun = lambda x: np.pi * np.cos(np.pi * x)
q_fun = lambda x: -1 * (np.pi**2) * phi(x)
mesh = self.M
mesh.origin = [
-0.25,
]
phi_ana = phi(mesh.nodes)
j_ana = j_fun(mesh.edges_x)
q_ana = q_fun(mesh.nodes)
phi_bc = phi(mesh.boundary_nodes)
j_bc = j_fun(mesh.boundary_nodes)
# construct matrix with robin operator
beta = 1.0
if self.boundary_type == "Robin":
alpha = 1.0
elif self.boundary_type == "Mixed":
alpha = np.r_[1.0, 0.0]
else:
alpha = 0.0
gamma = alpha * phi_bc + beta * j_bc * mesh.boundary_face_outward_normals
Me = mesh.get_edge_inner_product()
Mn = sp.diags(mesh.average_node_to_cell.T @ mesh.cell_volumes)
G = mesh.nodal_gradient
B_bc, b_bc = mesh.edge_divergence_weak_form_robin(alpha, beta, gamma)
A = -G.T @ Me @ G + B_bc
rhs = Mn @ q_ana - b_bc
if self.boundary_type == "Nuemann":
A[0, :] = 0.0
A[0, 0] = 1.0
rhs[0] = phi_ana[0]
phi_test = Solver(A) * rhs
err = np.linalg.norm(phi_test - phi_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 getError(self):
# Test function
phi = lambda x: np.sin(np.pi * x[:, 0]) * np.sin(np.pi * x[:, 1])
j_funX = lambda x: np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
j_funY = lambda x: np.pi * np.cos(np.pi * x[:, 1]) * np.sin(np.pi *
x[:, 0])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
mesh = self.M
if self._meshType == "rotateCurv":
nodes_x, nodes_y = mesh.node_list
nodes_x -= 0.25
nodes_y -= 0.25
mesh = discretize.CurvilinearMesh([nodes_x, nodes_y])
else:
mesh.origin = np.r_[-0.25, -0.25]
phi_ana = phi(mesh.nodes)
q_ana = q_fun(mesh.nodes)
if self.boundary_type == "Nuemann":
# Nuemann with J defined at boundary nodes
jx_bc = j_funX(mesh.boundary_nodes)
jy_bc = j_funY(mesh.boundary_nodes)
j_bc = np.c_[jx_bc, jy_bc]
M_bn = mesh.boundary_node_vector_integral
B_bc = sp.csr_matrix((mesh.n_nodes, mesh.n_nodes))
b_bc = M_bn @ (j_bc.reshape(-1, order='F'))
else:
phi_bc = phi(mesh.boundary_faces)
jx_bc = j_funX(mesh.boundary_faces)
jy_bc = j_funY(mesh.boundary_faces)
j_bc = np.c_[jx_bc, jy_bc]
j_bc_dot_n = np.sum(j_bc * mesh.boundary_face_outward_normals,
axis=-1)
# construct matrix with robin operator
if self.boundary_type == "Robin":
alpha = 1.0
else:
# get indices of x0 boundary and y0 boundary
n_boundary_faces = len(j_bc_dot_n)
robin_locs = np.any(mesh.boundary_faces == -0.25, axis=1)
alpha = np.zeros(n_boundary_faces)
alpha[robin_locs] = 1.0
beta = 1.0
gamma = alpha * phi_bc + beta * j_bc_dot_n
B_bc, b_bc = mesh.edge_divergence_weak_form_robin(
alpha, beta, gamma)
Me = mesh.get_edge_inner_product()
Mn = sp.diags(mesh.average_node_to_cell.T @ mesh.cell_volumes)
G = mesh.nodal_gradient
A = -G.T @ Me @ G + B_bc
rhs = Mn @ q_ana - b_bc
if self.boundary_type == "Nuemann":
A[0, :] = 0.0
A[0, 0] = 1.0
rhs[0] = phi_ana[0]
phi_test = Solver(A) * rhs
err = np.linalg.norm(phi_test - phi_ana, np.inf)
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 = discretize.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = discretize.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 = discretize.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 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 = discretize.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 * discretize.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