class FEMNavierStokesModel2d_channel:
def __init__(self, pde, mesh, p, dt, T):
self.p = p
self.mesh = mesh
self.dt = dt
self.T = T
self.itype = self.mesh.itype
self.ftype = self.mesh.ftype
self.pde = pde
self.vspace = LagrangeFiniteElementSpace(mesh, p + 1)
self.pspace = LagrangeFiniteElementSpace(mesh, p)
self.vdof = self.vspace.dof
self.pdof = self.pspace.dof
self.cellmeasure = mesh.entity_measure('cell')
self.integralalg = FEMeshIntegralAlg(self.mesh,
p + 4,
cellmeasure=self.cellmeasure)
self.uh0 = self.vspace.function()
self.uh1 = self.vspace.function()
self.ph = self.pspace.function()
@timer
def NS_VC_Solver(self):
"""
The Navier-Stokes Velocity-Correction scheme solver.
"""
pde = self.pde
dt = self.dt
uh0 = self.uh0
uh1 = self.uh1
ph = self.ph
vspace = self.vspace
pspace = self.pspace
vdof = self.vdof
pdof = self.pdof
pface2dof = pdof.face_to_dof() # (NE,fldof)
pcell2dof = pdof.cell_to_dof() # (NC,cldof)
ucell2dof = vspace.cell_to_dof() # (NC,cldof)
idxDirEdge = self.set_Dirichlet_edge()
cellidxDir = self.mesh.ds.edge2cell[idxDirEdge, 0]
localidxDir = self.mesh.ds.edge2cell[idxDirEdge, 2]
Dir_face2dof = pface2dof[idxDirEdge, :] # (NDir,flodf)
Dir_cell2dof = pcell2dof[cellidxDir, :] # (NDir,cldof)
n_Dir = self.mesh.face_unit_normal(index=idxDirEdge) # (NDir,2)
dir_face_measure = self.mesh.entity_measure(
'face', index=idxDirEdge) # (NDir,2)
cell_measure = self.mesh.cell_area()
dir_cell_measure = self.mesh.cell_area(cellidxDir)
f_q = self.integralalg.faceintegrator
f_bcs, f_ws = f_q.get_quadrature_points_and_weights(
) # f_bcs.shape: (NQ,(GD-1)+1)
f_pp = self.mesh.bc_to_point(
f_bcs, index=idxDirEdge
) # f_pp.shape: (NQ,NDir,GD) the physical Gauss points
c_q = self.integralalg.cellintegrator
c_bcs, c_ws = c_q.get_quadrature_points_and_weights(
) # c_bcs.shape: (NQ,GD+1)
c_pp = self.mesh.bc_to_point(
c_bcs) # c_pp.shape: (NQ_cell,NC,GD) the physical Gauss points
last_uh0 = vspace.function()
last_uh1 = vspace.function()
next_ph = pspace.function()
# # t^{n+1}: Pressure-Left-StiffMatrix
plsm = self.pspace.stiff_matrix()
basis_int = pspace.integral_basis()
p_phi = pspace.face_basis(
f_bcs) # (NQ,1,fldof). 实际上这里可以直接用 pspace.basis(f_bcs), 两个函数的代码是相同的
p_gphi_f = pspace.edge_grad_basis(f_bcs, cellidxDir,
localidxDir) # (NDir,NQ,cldof,GD)
p_gphi_c = pspace.grad_basis(c_bcs) # (NQ_cell,NC,ldof,GD)
# # t^{n+1}: Velocity-Left-MassMatrix and -StiffMatrix
ulmm = self.vspace.mass_matrix()
ulsm = self.vspace.stiff_matrix()
u_phi = vspace.basis(c_bcs) # (NQ,1,cldof)
next_t = 0
dofLocalIdxInflow, localIdxInflow = self.set_velocity_inflow_dof()
idxOutFlowEdge = self.set_outflow_edge()
for nt in range(int(self.T / dt)):
curr_t = nt * dt
next_t = curr_t + dt
# ---------------------------------------
# 1st-step: get the p^{n+1}
# ---------------------------------------
# # Pressure-Right-Matrix
if curr_t == 0.:
# for Dirichlet-face-integration
last_gu_val0 = pde.grad_velocity0(f_pp,
0) # grad_u0: (NQ,NDir,GD)
last_gu_val1 = pde.grad_velocity1(f_pp,
0) # grad_u1: (NQ,NDir,GD)
# for cell-integration
last_u_val = pde.velocityInitialValue(c_pp) # (NQ,NC,GD)
last_u_val0 = last_u_val[..., 0] # (NQ,NC)
last_u_val1 = last_u_val[..., 1] # (NQ,NC)
last_nolinear_val0 = pde.NS_nolinearTerm_0(c_pp, 0) # (NQ,NC)
last_nolinear_val1 = pde.NS_nolinearTerm_1(c_pp, 0) # (NQ,NC)
else:
# for Dirichlet-face-integration
# last_gu_val0 = vspace.grad_value(last_uh0, f_bcs) # grad_u0: (NQ,NDir,GD)
# last_gu_val1 = vspace.grad_value(last_uh1, f_bcs) # grad_u1: (NQ,NDir,GD)
last_gu_val0 = self.uh_grad_value_at_faces(
last_uh0, f_bcs, cellidxDir,
localidxDir) # grad_u0: (NQ,NDir,GD)
last_gu_val1 = self.uh_grad_value_at_faces(
last_uh1, f_bcs, cellidxDir,
localidxDir) # grad_u0: (NQ,NDir,GD)
# for cell-integration
last_u_val0 = vspace.value(last_uh0, c_bcs) # (NQ,NC)
last_u_val1 = vspace.value(last_uh1, c_bcs)
last_nolinear_val = self.NSNolinearTerm(
last_uh0, last_uh1,
c_bcs) # last_nolinear_val.shape: (NQ,NC,GD)
last_nolinear_val0 = last_nolinear_val[..., 0] # (NQ,NC)
last_nolinear_val1 = last_nolinear_val[..., 1] # (NQ,NC)
uDir_val = pde.dirichlet(
f_pp, next_t, bd_threshold=localIdxInflow) # (NQ,NDir,GD)
f_val = pde.source(c_pp, next_t) # (NQ,NC,GD)
# # --- to update the pressure value --- # #
# # to get the Pressure's Right-hand Vector
prv = np.zeros((pdof.number_of_global_dofs(), ),
dtype=self.ftype) # (Npdof,)
# for Dirichlet faces integration
dir_int0 = -1 / dt * np.einsum('i, ijk, jk, ijn, j->jn', f_ws,
uDir_val, n_Dir, p_phi,
dir_face_measure) # (NDir,fldof)
dir_int1 = -pde.nu * (
np.einsum('i, j, ij, jin, j->jn', f_ws, n_Dir[:, 1],
last_gu_val1[..., 0] - last_gu_val0[..., 1],
p_gphi_f[..., 0], dir_face_measure) +
np.einsum('i, j, ij, jin, j->jn', f_ws, -n_Dir[:, 0],
last_gu_val1[..., 0] - last_gu_val0[..., 1],
p_gphi_f[..., 1], dir_face_measure)) # (NDir,cldof)
# for cell integration
cell_int0 = 1 / dt * (
np.einsum('i, ij, ijk, j->jk', c_ws, last_u_val0,
p_gphi_c[..., 0], cell_measure) +
np.einsum('i, ij, ijk, j->jk', c_ws, last_u_val1,
p_gphi_c[..., 1], cell_measure)) # (NC,cldof)
cell_int1 = -(
np.einsum('i, ij, ijk, j->jk', c_ws, last_nolinear_val0,
p_gphi_c[..., 0], cell_measure) +
np.einsum('i, ij, ijk, j->jk', c_ws, last_nolinear_val1,
p_gphi_c[..., 1], cell_measure)) # (NC,cldof)
cell_int2 = (np.einsum('i, ij, ijk, j->jk', c_ws, f_val[..., 0],
p_gphi_c[..., 0], cell_measure) +
np.einsum('i, ij, ijk, j->jk', c_ws, f_val[..., 1],
p_gphi_c[..., 1], cell_measure)
) # (NC,cldof)
np.add.at(prv, Dir_face2dof, dir_int0)
np.add.at(prv, Dir_cell2dof, dir_int1)
np.add.at(prv, pcell2dof, cell_int0 + cell_int1 + cell_int2)
# # Method I: The following code is right! Pressure satisfies \int_\Omega p = 0
# plsm_temp = bmat([[plsm, basis_int.reshape(-1, 1)], [basis_int, None]], format='csr')
# prv = np.r_[prv, 0]
# next_ph[:] = spsolve(plsm_temp, prv)[:-1] # we have added one addtional dof
# # Method II: Using the Dirichlet boundary of pressure
def dir_pressure(p):
return pde.pressure_dirichlet(p, next_t)
bc = DirichletBC(pspace, dir_pressure, threshold=idxOutFlowEdge)
plsm_temp, prv = bc.apply(plsm.copy(), prv)
next_ph[:] = spsolve(plsm_temp, prv).reshape(-1)
# # --- to update the velocity value --- # #
next_gradph = pspace.grad_value(next_ph, c_bcs) # (NQ,NC,2)
# the velocity u's Left-Matrix
ulm0 = 1 / dt * ulmm + pde.nu * ulsm
ulm1 = 1 / dt * ulmm + pde.nu * ulsm
# # to get the u's Right-hand Vector
def dir_u0(p):
return pde.dirichlet(p, next_t,
bd_threshold=dofLocalIdxInflow)[..., 0]
def dir_u1(p):
return pde.dirichlet(p, next_t,
bd_threshold=dofLocalIdxInflow)[..., 1]
# for the first-component of velocity
urv0 = np.zeros((vdof.number_of_global_dofs(), ),
dtype=self.ftype) # (Nvdof,)
urv0_temp = np.einsum('i, ij, ijk, j->jk', c_ws,
last_u_val0 / dt - next_gradph[..., 0] -
last_nolinear_val0 + f_val[..., 0], u_phi,
cell_measure) # (NC,clodf)
np.add.at(urv0, ucell2dof, urv0_temp)
u0_bc = DirichletBC(vspace, dir_u0, threshold=idxDirEdge)
ulm0, urv0 = u0_bc.apply(ulm0, urv0)
last_uh0[:] = spsolve(ulm0, urv0).reshape(-1)
# for the second-component of velocity
urv1 = np.zeros((vdof.number_of_global_dofs(), ),
dtype=self.ftype) # (Nvdof,)
urv1_temp = np.einsum('i, ij, ijk, j->jk', c_ws,
last_u_val1 / dt - next_gradph[..., 1] -
last_nolinear_val1 + f_val[..., 1], u_phi,
cell_measure) # (NC,clodf)
np.add.at(urv1, ucell2dof, urv1_temp)
u1_bc = DirichletBC(vspace, dir_u1, threshold=idxDirEdge)
ulm1, urv1 = u1_bc.apply(ulm1, urv1)
last_uh1[:] = spsolve(ulm1, urv1).reshape(-1)
if nt % 100 == 0:
print('# ------------ logging the circle info ------------ #')
print('current t = ', curr_t)
p_l2err, u0_l2err, u1_l2err = self.currt_error(
next_ph, last_uh0, last_uh1, next_t)
print('p_l2err = %e, u0_l2err = %e, u1_l2err = %e' %
(p_l2err, u0_l2err, u1_l2err))
print(
'# ------------------------------------------------- # \n')
if np.isnan(p_l2err) | np.isnan(u0_l2err) | np.isnan(u1_l2err):
print('Some error is nan: breaking the program')
break
# print('end of current time')
# print('# ------------ the end error ------------ #')
# p_l2err, u0_l2err, u1_l2err = self.currt_error(next_ph, last_uh0, last_uh1, next_t)
# print('p_l2err = %e, u0_l2err = %e, u1_l2err = %e' % (p_l2err, u0_l2err, u1_l2err))
return last_uh0, last_uh1, next_ph
def currt_error(self, ph, uh0, uh1, t):
pde = self.pde
def currt_pressure(p):
return pde.pressure(p, t)
p_l2err = self.pspace.integralalg.L2_error(currt_pressure, ph)
# print('p_l2err = %e' % p_l2err)
def currt_u0(p):
return pde.velocity(p, t)[..., 0]
u0_l2err = self.vspace.integralalg.L2_error(currt_u0, uh0)
# print('u0_l2err = %e' % u0_l2err)
def currt_u1(p):
return pde.velocity(p, t)[..., 1]
u1_l2err = self.vspace.integralalg.L2_error(currt_u1, uh1)
# print('u1_l2err = %e' % u1_l2err)
return p_l2err, u0_l2err, u1_l2err
def uh_grad_value_at_faces(self, vh, f_bcs, cellidx, localidx):
cell2dof = self.vdof.cell2dof
f_gphi = self.vspace.edge_grad_basis(f_bcs, cellidx,
localidx) # (NE,NQ,cldof,GD)
# val.shape: (NQ,NE,GD)
# vh.shape: (v_gdof,)
# vh[cell2dof[cellidx]].shape: (Ncellidx,cldof)
val = np.einsum('ik, ijkm->jim', vh[cell2dof[cellidx]], f_gphi)
return val
def NSNolinearTerm(self, uh0, uh1, bcs):
vspace = self.vspace
val0 = vspace.value(uh0, bcs) # val0.shape: (NQ,NC)
val1 = vspace.value(uh1, bcs) # val1.shape: (NQ,NC)
gval0 = vspace.grad_value(uh0, bcs) # gval0.shape: (NQ,NC,2)
gval1 = vspace.grad_value(uh1, bcs)
NSNolinear = np.empty(gval0.shape,
dtype=self.ftype) # NSNolinear.shape: (NQ,NC,2)
NSNolinear[..., 0] = val0 * gval0[..., 0] + val1 * gval0[..., 1]
NSNolinear[..., 1] = val0 * gval1[..., 0] + val1 * gval1[..., 1]
return NSNolinear
def set_Dirichlet_edge(self, idxDirEdge=None):
if idxDirEdge is not None:
return idxDirEdge
mesh = self.mesh
node = mesh.node # (NV,2)
edge2cell = mesh.ds.edge_to_cell()
isBdEdge = (edge2cell[:, 0] == edge2cell[:, 1]
) # (NE,), the bool vars, to get the boundary edges
idxBdEdge, = np.nonzero(isBdEdge)
edge2node = mesh.ds.edge_to_node() # (NE,2)
mid_coor = (node[edge2node[:, 0], :] +
node[edge2node[:, 1], :]) / 2 # (NE,2)
bd_mid = mid_coor[isBdEdge, :]
isOutflow = np.abs(bd_mid[:, 0] - 1.0) < 1e-6
isDirEdge = ~isOutflow # here, we set all the boundary but the right edges are Dir edges
idxDirEdge = idxBdEdge[isDirEdge] # (NE_Dir,)
return idxDirEdge
def set_outflow_edge(self, idxOutEdge=None):
if idxOutEdge is not None:
return idxOutEdge
mesh = self.mesh
node = mesh.node # (NV,2)
edge2cell = mesh.ds.edge_to_cell()
isBdEdge = (edge2cell[:, 0] == edge2cell[:, 1]
) # (NE,), the bool vars, to get the boundary edges
idxBdEdge, = np.nonzero(isBdEdge)
edge2node = mesh.ds.edge_to_node() # (NE,2)
mid_coor = (node[edge2node[:, 0], :] +
node[edge2node[:, 1], :]) / 2 # (NE,2)
bd_mid = mid_coor[isBdEdge, :]
isOutflow = np.abs(bd_mid[:, 0] - 1.0) < 1e-6
idxOutEdge = idxBdEdge[isOutflow] # (NE_Dir,)
return idxOutEdge
def set_velocity_inflow_dof(self):
mesh = self.mesh
node = mesh.node # (NV,2)
edge2node = mesh.ds.edge_to_node() # (NE,2)
idxDirEdge = self.set_Dirichlet_edge()
dirIndicator = np.full(idxDirEdge.shape, False)
mid_coor = (node[edge2node[:, 0], :] +
node[edge2node[:, 1], :]) / 2 # (NE,2)
bd_mid = mid_coor[idxDirEdge, :]
# --- set inflow edges --- #
inflow_x = 0.0
isInflow = np.abs(bd_mid[:, 0] - inflow_x) < 1e-6
localIdxInflow = np.nonzero(isInflow)
idxInflow = idxDirEdge[isInflow]
# --- set inflow dof --- #
edge2dof = self.vdof.edge_to_dof()
dir_dof = np.unique(edge2dof[idxDirEdge].flatten())
inflow_dof = np.unique(edge2dof[idxInflow].flatten())
dofLocalIdxInflow = np.searchsorted(dir_dof, inflow_dof)
return dofLocalIdxInflow, localIdxInflow
def set_Neumann_edge(self, idxNeuEdge=None):
if idxNeuEdge is not None:
return idxNeuEdge
mesh = self.mesh
edge2cell = mesh.ds.edge_to_cell()
bdEdge = (edge2cell[:, 0] == edge2cell[:, 1]
) # the bool vars, to get the boundary edges
isNeuEdge = bdEdge # here, we first set all the boundary edges are Neu edges
issetNeuEdge = 'no'
if issetNeuEdge == 'no':
isNeuEdge = None
idxNeuEdge, = np.nonzero(isNeuEdge) # (NE_Dir,)
return idxNeuEdge
class FEMCahnHilliardModel2d:
def __init__(self, pde, mesh, p, dt):
self.pde = pde
self.p = p
self.mesh = mesh
self.timemesh, self.dt = self.pde.time_mesh(dt)
self.itype = self.mesh.itype
self.ftype = self.mesh.ftype
self.pde = pde
self.space = LagrangeFiniteElementSpace(mesh, p)
self.dof = self.space.dof
self.cellmeasure = mesh.entity_measure('cell')
self.integralalg = FEMeshIntegralAlg(self.mesh, p + 4, cellmeasure=self.cellmeasure)
self.uh = self.space.function()
self.wh = self.space.function()
self.StiffMatrix = self.space.stiff_matrix()
self.MassMatrix = self.space.mass_matrix()
def setCoefficient_T1stOrder(self, dt_minimum=None):
pde = self.pde
dt_min = self.dt if dt_minimum is None else dt_minimum
m = pde.m
epsilon = pde.epsilon
s = np.sqrt(4 * epsilon / (m * dt_min))
alpha = 1. / (2 * epsilon) * (-s + np.sqrt(abs(s ** 2 - 4 * epsilon / (m * self.dt))))
return s, alpha
def setCoefficient_T2ndOrder(self, dt_minimum=None):
pde = self.pde
dt_min = self.dt if dt_minimum is None else dt_minimum
m = pde.m
epsilon = pde.epsilon
s = np.sqrt(4 * (3/2) * epsilon / (m * dt_min))
alpha = 1. / (2 * epsilon) * (-s + np.sqrt(abs(s ** 2 - 4 * (3/2) * epsilon / (m * self.dt))))
return s, alpha
def uh_grad_value_at_faces(self, vh, f_bcs, cellidx, localidx):
cell2dof = self.dof.cell2dof
f_gphi = self.space.edge_grad_basis(f_bcs, cellidx, localidx) # (NE,NQ,cldof,GD)
val = np.einsum('ik, ijkm->jim', vh[cell2dof[cellidx]], f_gphi) # (NQ,NE,GD)
return val
def grad_free_energy_at_faces(self, uh, f_bcs, idxBdEdge, cellidx, localidx):
"""
1. Compute the grad of free energy at FACE Gauss-integration points (barycentric coordinates).
2. In this function, the free energy has NO coefficients.
-------
:param uh:
:param f_bcs: f_bcs.shape: (NQ,(GD-1)+1)
:return:
"""
uh_val = self.space.value(uh, f_bcs)[..., idxBdEdge] # (NQ,NBE)
guh_val = self.uh_grad_value_at_faces(uh, f_bcs, cellidx, localidx) # (NQ,NBE,GD)
guh_val[..., 0] = 3 * uh_val ** 2 * guh_val[..., 0] - guh_val[..., 0]
guh_val[..., 1] = 3 * uh_val ** 2 * guh_val[..., 1] - guh_val[..., 1]
return guh_val # (NQ,NBE,2)
def grad_free_energy_at_cells(self, uh, c_bcs):
"""
1. Compute the grad of free energy at CELL Gauss-integration points (barycentric coordinates).
2. In this function, the free energy has NO coefficients.
-------
:param uh:
:param c_bcs: c_bcs.shape: (NQ,GD+1)
:return:
"""
uh_val = self.space.value(uh, c_bcs) # (NQ,NC)
guh_val = self.space.grad_value(uh, c_bcs) # (NQ,NC,2)
guh_val[..., 0] = 3 * uh_val ** 2 * guh_val[..., 0] - guh_val[..., 0]
guh_val[..., 1] = 3 * uh_val ** 2 * guh_val[..., 1] - guh_val[..., 1]
return guh_val # (NQ,NC,2)
@timer
def CH_Solver_T1stOrder(self):
pde = self.pde
timemesh = self.timemesh
NT = len(timemesh)
dt = self.dt
uh = self.uh
wh = self.wh
sm = self.StiffMatrix
mm = self.MassMatrix
space = self.space
dof = self.dof
face2dof = dof.face_to_dof() # (NE,fldof)
cell2dof = dof.cell_to_dof() # (NC,cldof)
dt_min = pde.dt_min if hasattr(pde, 'dt_min') else dt
s, alpha = self.setCoefficient_T1stOrder(dt_minimum=dt_min)
m = pde.m
epsilon = pde.epsilon
eta = pde.eta
print(' # #################################### #')
print(' Time 1st-order scheme')
print(' # #################################### #')
print(' # ------------ parameters ------------ #')
print(' s = %.4e, alpha = %.4e, m = %.4e, epsilon = %.4e, eta = %.4e' % (s, alpha, m, epsilon, eta))
print(' t0 = %.4e, T = %.4e, dt = %.4e' % (timemesh[0], timemesh[-1], dt))
print(' ')
idxNeuEdge = self.set_Neumann_edge()
nBd = self.mesh.face_unit_normal(index=idxNeuEdge) # (NBE,2)
NeuCellIdx = self.mesh.ds.edge2cell[idxNeuEdge, 0]
NeuLocalIdx = self.mesh.ds.edge2cell[idxNeuEdge, 2]
neu_face_measure = self.mesh.entity_measure('face', index=idxNeuEdge) # (Nneu,2)
cell_measure = self.mesh.cell_area()
f_q = self.integralalg.faceintegrator
f_bcs, f_ws = f_q.get_quadrature_points_and_weights() # f_bcs.shape: (NQ,(GD-1)+1)
f_pp = self.mesh.bc_to_point(f_bcs, index=idxNeuEdge) # f_pp.shape: (NQ,NBE,GD) the physical Gauss points
c_q = self.integralalg.cellintegrator
c_bcs, c_ws = c_q.get_quadrature_points_and_weights() # c_bcs.shape: (NQ,GD+1)
c_pp = self.mesh.bc_to_point(c_bcs) # c_pp.shape: (NQ_cell,NC,GD) the physical Gauss points
phi_f = space.face_basis(f_bcs) # (NQ,1,fldof). 实际上这里可以直接用 pspace.basis(f_bcs), 两个函数的代码是相同的
phi_c = space.basis(c_bcs) # (NQ,NC,clodf)
gphi_c = space.grad_basis(c_bcs) # (NQ,NC,cldof,GD)
# # time-looping
print(' # ------------ begin the time-looping ------------ #')
for nt in range(NT-1):
currt_t = timemesh[nt]
next_t = currt_t + dt
if nt % max([int(NT / 10), 1]) == 0:
print(' currt_t = %.4e' % currt_t)
if nt == 0:
# the initial value setting
u_c = pde.solution(c_pp, pde.t0) # (NQC,NC)
gu_c = pde.gradient(c_pp, pde.t0) # (NQC,NC,2)
u_f = pde.solution(f_pp, pde.t0) # (NQF,NBE)
gu_f = pde.gradient(f_pp, pde.t0) # (NQF,NBE,2)
grad_free_energy_c = epsilon / eta ** 2 * (3 * np.repeat(u_c[..., np.newaxis], 2, axis=2) ** 2 * gu_c - gu_c)
grad_free_energy_f = epsilon / eta ** 2 * (3 * np.repeat(u_f[..., np.newaxis], 2, axis=2) ** 2 * gu_f - gu_f)
uh_val = u_c # (NQ,NC)
guh_val_c = gu_c # (NQ,NC,2)
guh_val_f = gu_f # (NQ,NE,2)
# gu_c_test = gu_c.copy()
# gu_c_test[..., 0] = epsilon / eta ** 2 * (3 * u_c ** 2 * gu_c_test[..., 0] - gu_c_test[..., 0])
# gu_c_test[..., 1] = epsilon / eta ** 2 * (3 * u_c ** 2 * gu_c_test[..., 1] - gu_c_test[..., 1])
# gu_f_test = gu_f.copy()
# gu_f_test[..., 0] = epsilon / eta ** 2 * (3 * u_f ** 2 * gu_f_test[..., 0] - gu_f_test[..., 0])
# gu_f_test[..., 1] = epsilon / eta ** 2 * (3 * u_f ** 2 * gu_f_test[..., 1] - gu_f_test[..., 1])
# print(np.allclose(gu_c_test, grad_free_energy_c))
# print(np.allclose(gu_f_test, grad_free_energy_f))
else:
grad_free_energy_c = epsilon / eta ** 2 * self.grad_free_energy_at_cells(uh, c_bcs) # (NQ,NC,2)
grad_free_energy_f = epsilon / eta ** 2 * self.grad_free_energy_at_faces(uh, f_bcs, idxNeuEdge, NeuCellIdx,
NeuLocalIdx) # (NQ,NE,2)
uh_val = space.value(uh, c_bcs) # (NQ,NC)
guh_val_c = space.grad_value(uh, c_bcs) # (NQ,NC,2)
guh_val_f = self.uh_grad_value_at_faces(uh, f_bcs, NeuCellIdx, NeuLocalIdx) # (NQ,NE,2)
# # # ---------------------------------------- yc test --------------------------------------------------- # #
# if nt == 0:
# def init_solution(p):
# return pde.solution(p, 0)
# uh[:] = space.interpolation(init_solution)
# grad_free_energy_c = epsilon / eta ** 2 * self.grad_free_energy_at_cells(uh, c_bcs) # (NQ,NC,2)
# grad_free_energy_f = epsilon / eta ** 2 * self.grad_free_energy_at_faces(uh, f_bcs, idxNeuEdge, NeuCellIdx,
# NeuLocalIdx) # (NQ,NE,2)
# uh_val = space.value(uh, c_bcs) # (NQ,NC)
# guh_val_c = space.grad_value(uh, c_bcs) # (NQ,NC,2)
# guh_val_f = self.uh_grad_value_at_faces(uh, f_bcs, NeuCellIdx, NeuLocalIdx) # (NQ,NE,2)
# # # --------------------------------------- end test ---------------------------------------------------- # #
Neumann = pde.neumann(f_pp, next_t, nBd) # (NQ,NE)
LaplaceNeumann = pde.laplace_neumann(f_pp, next_t, nBd) # (NQ,NE)
f_val = pde.source(c_pp, next_t, m, epsilon, eta) # (NQ,NC)
# # get the auxiliary equation Right-hand-side-Vector
aux_rv = np.zeros((dof.number_of_global_dofs(),), dtype=self.ftype) # (Ndof,)
# # aux_rhs_c_0: -1. / (epsilon * m * dt) * (uh^n,phi)_\Omega
aux_rhs_c_0 = -1. / (epsilon * m) * (1/dt * np.einsum('i, ij, ijk, j->jk', c_ws, uh_val, phi_c, cell_measure) +
np.einsum('i, ij, ijk, j->jk', c_ws, f_val, phi_c, cell_measure)) # (NC,cldof)
# # aux_rhs_c_1: -s / epsilon * (\nabla uh^n, \nabla phi)_\Omega
aux_rhs_c_1 = -s / epsilon * (
np.einsum('i, ij, ijk, j->jk', c_ws, guh_val_c[..., 0], gphi_c[..., 0], cell_measure)
+ np.einsum('i, ij, ijk, j->jk', c_ws, guh_val_c[..., 1], gphi_c[..., 1], cell_measure)) # (NC,cldof)
# # aux_rhs_c_2: 1 / epsilon * (\nabla h(uh^n), \nabla phi)_\Omega
aux_rhs_c_2 = 1. / epsilon * (
np.einsum('i, ij, ijk, j->jk', c_ws, grad_free_energy_c[..., 0], gphi_c[..., 0], cell_measure)
+ np.einsum('i, ij, ijk, j->jk', c_ws, grad_free_energy_c[..., 1], gphi_c[..., 1], cell_measure)) # (NC,cldof)
# # aux_rhs_f_0: (\nabla wh^{n+1}\cdot n, phi)_\Gamma, wh is the solution of auxiliary equation
aux_rhs_f_0 = alpha * np.einsum('i, ij, ijn, j->jn', f_ws, Neumann, phi_f, neu_face_measure) \
+ np.einsum('i, ij, ijn, j->jn', f_ws, LaplaceNeumann, phi_f, neu_face_measure) # (Nneu,fldof)
# # aux_rhs_f_1: s / epsilon * (\nabla uh^n \cdot n, phi)_\Gamma
aux_rhs_f_1 = s / epsilon * np.einsum('i, ijk, jk, ijn, j->jn', f_ws, guh_val_f, nBd, phi_f,
neu_face_measure) # (Nneu,fldof)
# # aux_rhs_f_2: -1 / epsilon * (\nabla h(uh^n) \cdot n, phi)_\Gamma
aux_rhs_f_2 = -1. / epsilon * np.einsum('i, ijk, jk, ijn, j->jn', f_ws, grad_free_energy_f, nBd, phi_f,
neu_face_measure) # (Nneu,fldof)
np.add.at(aux_rv, cell2dof, aux_rhs_c_0 + aux_rhs_c_1 + aux_rhs_c_2)
np.add.at(aux_rv, face2dof[idxNeuEdge, :], aux_rhs_f_0 + aux_rhs_f_1 + aux_rhs_f_2)
# # update the solution of auxiliary equation
wh[:] = spsolve(sm + (alpha + s / epsilon) * mm, aux_rv)
# # update the original solution
orig_rv = np.zeros((dof.number_of_global_dofs(),), dtype=self.ftype) # (Ndof,)
wh_val = space.value(wh, c_bcs) # (NQ,NC)
orig_rhs_c = - np.einsum('i, ij, ijk, j->jk', c_ws, wh_val, phi_c, cell_measure) # (NC,cldof)
orig_rhs_f = np.einsum('i, ij, ijn, j->jn', f_ws, Neumann, phi_f, neu_face_measure)
np.add.at(orig_rv, cell2dof, orig_rhs_c)
np.add.at(orig_rv, face2dof[idxNeuEdge, :], orig_rhs_f)
uh[:] = spsolve(sm - alpha * mm, orig_rv)
# currt_l2err = self.currt_error(uh, next_t)
# print('max(uh) = ', max(uh))
# if max(uh[:]) > 1e5:
# break
print(' # ------------ end the time-looping ------------ #\n')
l2err, h1err = self.currt_error(uh, timemesh[-1])
print(' # ------------ the last errors ------------ #')
print(' l2err = %.4e, h1err = %.4e' % (l2err, h1err))
return l2err, h1err
def CH_Solver_T2ndOrder(self):
pde = self.pde
timemesh = self.timemesh
NT = len(timemesh)
space = self.space
dt = self.dt
uh = self.uh
last_uh = space.function()
wh = self.wh
sm = self.StiffMatrix
mm = self.MassMatrix
dof = self.dof
face2dof = dof.face_to_dof() # (NE,fldof)
cell2dof = dof.cell_to_dof() # (NC,cldof)
dt_min = pde.dt_min if hasattr(pde, 'dt_min') else dt
s, alpha = self.setCoefficient_T2ndOrder(dt_minimum=dt_min)
m = pde.m
epsilon = pde.epsilon
eta = pde.eta
print(' # #################################### #')
print(' Time 2nd-order scheme')
print(' # #################################### #')
print(' # ------------ parameters ------------ #')
print(' s = %.4e, alpha = %.4e, m = %.4e, epsilon = %.4e, eta = %.4e' % (s, alpha, m, epsilon, eta))
print(' t0 = %.4e, T = %.4e, dt = %.4e' % (timemesh[0], timemesh[-1], dt))
print(' ')
idxNeuEdge = self.set_Neumann_edge()
nBd = self.mesh.face_unit_normal(index=idxNeuEdge) # (NBE,2)
NeuCellIdx = self.mesh.ds.edge2cell[idxNeuEdge, 0]
NeuLocalIdx = self.mesh.ds.edge2cell[idxNeuEdge, 2]
neu_face_measure = self.mesh.entity_measure('face', index=idxNeuEdge) # (Nneu,2)
cell_measure = self.mesh.cell_area()
f_q = self.integralalg.faceintegrator
f_bcs, f_ws = f_q.get_quadrature_points_and_weights() # f_bcs.shape: (NQ,(GD-1)+1)
f_pp = self.mesh.bc_to_point(f_bcs, index=idxNeuEdge) # f_pp.shape: (NQ,NBE,GD) the physical Gauss points
c_q = self.integralalg.cellintegrator
c_bcs, c_ws = c_q.get_quadrature_points_and_weights() # c_bcs.shape: (NQ,GD+1)
c_pp = self.mesh.bc_to_point(c_bcs) # c_pp.shape: (NQ_cell,NC,GD) the physical Gauss points
phi_f = space.face_basis(f_bcs) # (NQ,1,fldof). 实际上这里可以直接用 pspace.basis(f_bcs), 两个函数的代码是相同的
phi_c = space.basis(c_bcs) # (NQ,NC,clodf)
gphi_c = space.grad_basis(c_bcs) # (NQ,NC,cldof,GD)
# # time-looping
print(' # ------------ begin the time-looping ------------ #')
def init_solution(p):
return pde.solution(p, 0)
last_uh[:] = space.interpolation(init_solution)
for nt in range(NT-1):
currt_t = timemesh[nt]
next_t = currt_t + dt
if nt % max([int(NT/10), 1]) == 0:
print(' currt_t = %.4e' % currt_t)
if nt == 0:
# the initial value setting
s, alpha = self.setCoefficient_T1stOrder(dt_minimum=dt_min)
u_c = pde.solution(c_pp, pde.t0) # (NQC,NC)
gu_c = pde.gradient(c_pp, pde.t0) # (NQC,NC,2)
u_f = pde.solution(f_pp, pde.t0) # (NQF,NBE)
gu_f = pde.gradient(f_pp, pde.t0) # (NQF,NBE,2)
grad_free_energy_c = epsilon / eta ** 2 * (3 * np.repeat(u_c[..., np.newaxis], 2, axis=2) ** 2 * gu_c - gu_c)
grad_free_energy_f = epsilon / eta ** 2 * (3 * np.repeat(u_f[..., np.newaxis], 2, axis=2) ** 2 * gu_f - gu_f)
uh_val = u_c # (NQ,NC)
guh_val_c = gu_c # (NQ,NC,2)
guh_val_f = gu_f # (NQ,NE,2)
else:
s, alpha = self.setCoefficient_T2ndOrder(dt_minimum=dt_min)
grad_free_energy_c = epsilon / eta ** 2 * self.grad_free_energy_at_cells(2*uh - last_uh, c_bcs) # (NQ,NC,2)
grad_free_energy_f = epsilon / eta ** 2 * self.grad_free_energy_at_faces(2*uh - last_uh, f_bcs, idxNeuEdge, NeuCellIdx, NeuLocalIdx) # (NQ,NE,2)
uh_val = space.value(2 * uh - 1/2 * last_uh, c_bcs) # (NQ,NC)
guh_val_c = space.grad_value(2*uh - last_uh, c_bcs) # (NQ,NC,2)
guh_val_f = self.uh_grad_value_at_faces(2*uh - last_uh, f_bcs, NeuCellIdx, NeuLocalIdx) # (NQ,NE,2)
last_uh[:] = uh[:] # update the last_uh
Neumann = pde.neumann(f_pp, next_t, nBd) # (NQ,NE)
LaplaceNeumann = pde.laplace_neumann(f_pp, next_t, nBd) # (NQ,NE)
f_val = pde.source(c_pp, next_t, m, epsilon, eta) # (NQ,NC)
# # get the auxiliary equation Right-hand-side-Vector
aux_rv = np.zeros((dof.number_of_global_dofs(),), dtype=self.ftype) # (Ndof,)
aux_rv_temp = np.zeros((dof.number_of_global_dofs(),), dtype=self.ftype)
# # aux_rhs_c_0: -1. / (epsilon * m * dt) * (uh^n,phi)_\Omega
aux_rhs_c_0 = -1. / (epsilon * m) * (1/dt * np.einsum('i, ij, ijk, j->jk', c_ws, uh_val, phi_c, cell_measure) +
np.einsum('i, ij, ijk, j->jk', c_ws, f_val, phi_c, cell_measure)) # (NC,cldof)
# # aux_rhs_c_1: -s / epsilon * (\nabla uh^n, \nabla phi)_\Omega
aux_rhs_c_1 = -s / epsilon * (
np.einsum('i, ij, ijk, j->jk', c_ws, guh_val_c[..., 0], gphi_c[..., 0], cell_measure)
+ np.einsum('i, ij, ijk, j->jk', c_ws, guh_val_c[..., 1], gphi_c[..., 1], cell_measure)) # (NC,cldof)
# # aux_rhs_c_2: 1 / epsilon * (\nabla h(uh^n), \nabla phi)_\Omega
aux_rhs_c_2 = 1. / epsilon * (
np.einsum('i, ij, ijk, j->jk', c_ws, grad_free_energy_c[..., 0], gphi_c[..., 0], cell_measure)
+ np.einsum('i, ij, ijk, j->jk', c_ws, grad_free_energy_c[..., 1], gphi_c[..., 1], cell_measure)) # (NC,cldof)
# # aux_rhs_f_0: (\nabla wh^{n+1}\cdot n, phi)_\Gamma, wh is the solution of auxiliary equation
aux_rhs_f_0 = alpha * np.einsum('i, ij, ijn, j->jn', f_ws, Neumann, phi_f, neu_face_measure) \
+ np.einsum('i, ij, ijn, j->jn', f_ws, LaplaceNeumann, phi_f, neu_face_measure) # (Nneu,fldof)
# # aux_rhs_f_1: s / epsilon * (\nabla uh^n \cdot n, phi)_\Gamma
aux_rhs_f_1 = s / epsilon * np.einsum('i, ijk, jk, ijn, j->jn', f_ws, guh_val_f, nBd, phi_f,
neu_face_measure) # (Nneu,fldof)
# # aux_rhs_f_2: -1 / epsilon * (\nabla h(uh^n) \cdot n, phi)_\Gamma
aux_rhs_f_2 = -1. / epsilon * np.einsum('i, ijk, jk, ijn, j->jn', f_ws, grad_free_energy_f, nBd, phi_f,
neu_face_measure) # (Nneu,fldof)
np.add.at(aux_rv_temp, cell2dof, aux_rhs_c_0)
np.add.at(aux_rv, cell2dof, aux_rhs_c_0 + aux_rhs_c_1 + aux_rhs_c_2)
np.add.at(aux_rv, face2dof[idxNeuEdge, :], aux_rhs_f_0 + aux_rhs_f_1 + aux_rhs_f_2)
# # update the solution of auxiliary equation
wh[:] = spsolve(sm + (alpha + s / epsilon) * mm, aux_rv)
# # update the original solution
orig_rv = np.zeros((dof.number_of_global_dofs(),), dtype=self.ftype) # (Ndof,)
wh_val = space.value(wh, c_bcs) # (NQ,NC)
orig_rhs_c = - np.einsum('i, ij, ijk, j->jk', c_ws, wh_val, phi_c, cell_measure) # (NC,cldof)
orig_rhs_f = np.einsum('i, ij, ijn, j->jn', f_ws, Neumann, phi_f, neu_face_measure)
np.add.at(orig_rv, cell2dof, orig_rhs_c)
np.add.at(orig_rv, face2dof[idxNeuEdge, :], orig_rhs_f)
uh[:] = spsolve(sm - alpha * mm, orig_rv)
print(' # ------------ end the time-looping ------------ #\n')
l2err, h1err = self.currt_error(uh, timemesh[-1])
print(' # ------------ the last errors ------------ #')
print(' l2err = %.4e, h1err = %.4e' % (l2err, h1err))
return l2err, h1err
def currt_error(self, uh, t):
pde = self.pde
def currt_solution(p):
return pde.solution(p, t)
l2err = self.space.integralalg.L2_error(currt_solution, uh)
def currt_grad_solution(p):
return pde.gradient(p, t)
h1err = self.space.integralalg.L2_error(currt_grad_solution, uh.grad_value)
return l2err, h1err
def set_Dirichlet_edge(self, idxDirEdge=None):
if idxDirEdge is not None:
return idxDirEdge
mesh = self.mesh
edge2cell = mesh.ds.edge_to_cell()
isBdEdge = (edge2cell[:, 0] == edge2cell[:, 1]) # (NE,), the bool vars, to get the boundary edges
isDirEdge = isBdEdge # here, we set all the boundary edges are Dir edges
idxDirEdge, = np.nonzero(isDirEdge) # (NE_Dir,)
return idxDirEdge
def set_Neumann_edge(self, idxNeuEdge=None):
if idxNeuEdge is not None:
return idxNeuEdge
mesh = self.mesh
edge2cell = mesh.ds.edge_to_cell()
bdEdge = (edge2cell[:, 0] == edge2cell[:, 1]) # the bool vars, to get the boundary edges
isNeuEdge = bdEdge # here, we first set all the boundary edges are Neu edges
idxNeuEdge, = np.nonzero(isNeuEdge) # (NE_Dir,)
return idxNeuEdge
