def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_1D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# Stiffness matrix K doesn't change
oc.K = spsp.bmat([[-oc.L, -oc.Dz],
[oc.sigmaz*oc.Dz, oc.sigmaz]])
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
C = spsp.bmat([[oc.sigmaz*oc.m, None],
[None, oc.I]]) / self.dt
M = spsp.bmat([[oc.m, None],
[None, oc.empty]]) / self.dt**2
Stilde_inv = M+C
Stilde_inv.data = 1./Stilde_inv.data
self.A_k = Stilde_inv*(2*M - oc.K + C)
self.A_km1 = -1*Stilde_inv*(M)
self.A_f = Stilde_inv
def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_1D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# Stiffness matrix K doesn't change
oc.K = spsp.bmat([[ -oc.L, -oc.Dz],
[oc.sigmaz*oc.Dz, oc.sigmaz]])
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
C = spsp.bmat([[oc.sigmaz*oc.m, None],
[ None, oc.I]]) / self.dt
M = spsp.bmat([[oc.m, None],
[None, oc.empty]]) / self.dt**2
Stilde_inv = M+C
Stilde_inv.data = 1./Stilde_inv.data
self.A_k = Stilde_inv*(2*M - oc.K + C)
self.A_km1 = -1*Stilde_inv*(M)
self.A_f = Stilde_inv
def _build_helmholtz_operator(self, nu):
omega = 2 * np.pi * nu
# we build the right helmholtz operator compact or not
if self._local_support_spec['spatial_dimension'] == 1:
# 1D the compact is not implemented so we raise a warming and use the auxiliary field PML
return (-(omega**2) * self.M + omega * 1j * self.C +
self.K).tocsc()
elif self.mesh.x.lbc.type == 'pml':
if self.mesh.x.lbc.domain_bc.compact:
# building the compact operator
oc = self.operator_components
if self._local_support_spec['spatial_dimension'] == 2:
dx, dz = self.mesh.deltas
length_pml_x = dx * (self.mesh.x.lbc.n - 1)
length_pml_z = dz * (self.mesh.z.lbc.n - 1)
H1 = -(omega**2) * oc.M
H2 = make_diag_mtx(-1j / (omega * length_pml_x) * oc.sxp /
(1 - 1j / omega * oc.sx)**3).dot(oc.Dx)
H3 = make_diag_mtx(-1j / (omega * length_pml_z) * oc.szp /
(1 - 1j / omega * oc.sz)**3).dot(oc.Dz)
H4 = -make_diag_mtx(1.0 / (1 - 1j / omega * oc.sx)**2).dot(
oc.Dxx)
H5 = -make_diag_mtx(1.0 / (1 - 1j / omega * oc.sz)**2).dot(
oc.Dzz)
return (H1 + H2 + H3 + H4 + H5).tocsc(
) # csc is used for the sparse solvers right now
if self._local_support_spec['spatial_dimension'] == 3:
dx, dy, dz = self.mesh.deltas
length_pml_x = dx * (self.mesh.x.lbc.n - 1)
length_pml_y = dy * (self.mesh.y.lbc.n - 1)
length_pml_z = dz * (self.mesh.z.lbc.n - 1)
H1 = -(omega**2) * oc.M
H2 = make_diag_mtx(-1j / (omega * length_pml_x) * oc.sxp /
(1 - 1j / omega * oc.sx)**3).dot(oc.Dx)
H3 = make_diag_mtx(-1j / (omega * length_pml_y) * oc.syp /
(1 - 1j / omega * oc.sy)**3).dot(oc.Dy)
H4 = make_diag_mtx(-1j / (omega * length_pml_z) * oc.szp /
(1 - 1j / omega * oc.sz)**3).dot(oc.Dz)
H5 = -make_diag_mtx(1.0 / (1 - 1j / omega * oc.sx)**2).dot(
oc.Dxx)
H6 = -make_diag_mtx(1.0 / (1 - 1j / omega * oc.sy)**2).dot(
oc.Dyy)
H7 = -make_diag_mtx(1.0 / (1 - 1j / omega * oc.sz)**2).dot(
oc.Dzz)
return (H1 + H2 + H3 + H4 + H5 + H6 + H7).tocsc()
else:
return (-(omega**2) * self.M + omega * 1j * self.C +
self.K).tocsc()
else:
return (
-(omega**2) * self.M + omega * 1j * self.C +
self.K).tocsc() # csc is used for the sparse solvers right now
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.I = spsp.eye(dof, dof)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build other useful things
oc.empty = spsp.csr_matrix((dof, dof))
# Stiffness matrix K doesn't change
self.K = spsp.bmat([[-oc.L, -oc.Dz],
[oc.sigmaz*oc.Dz, oc.sigmaz]])
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
self.C = spsp.bmat([[oc.sigmaz*oc.m, None],
[None, oc.I]])
self.M = spsp.bmat([[oc.m, None],
[None, oc.empty]])
self.dM = oc.I
self.dC = oc.sigmaz
self.dK = 0
def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
nx = self.mesh.x.n
nz = self.mesh.z.n
#We actually don't need to correct the operators for the boundary nodes. But it will make the final matrix a little more sparse and therefore faster. Also, it makes is more clear that something needs to happen to the boundary. Any failure to do so can easily be seen by eye.
self.left_boundary_nodes = np.arange(
0, nz) #includes left corner dirichlet nodes
self.right_boundary_nodes = np.arange(
(nx - 1) * nz, nx * nz) #uncludes right corner dirichlet nodes
self.top_boundary_nodes = np.arange(
nz, (nx - 1) * nz,
nx) #does not include corner dirichlet nodes
self.bot_boundary_nodes = np.arange(
2 * nz - 1, nx * nz - 1,
nx) #does not include corner dirichlet nodes
self.all_boundary_nodes = np.concatenate([
self.left_boundary_nodes, self.right_boundary_nodes,
self.top_boundary_nodes, self.bot_boundary_nodes
])
# build laplacian
oc.L = build_derivative_matrix(self.mesh, 2,
self.spatial_accuracy_order)
#Set the rows corresponding to the boundary nodes in oc.L to zero (could be any junk value). This is not necessary, but will indicate that I will have to correct the boundary values at the end of the time increment.
oc.L = self._zero_mat_rows(oc.L, self.all_boundary_nodes)
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
oc._numpy_components_built = True
C = self.model_parameters.C #just wavespeed. Don't confuse with the matrix C that used to be here for a while
oc.m = make_diag_mtx((C**-2).reshape(-1, ))
K = -oc.L
M = oc.m / self.dt**2
M_bc_zero = self._zero_mat_rows(
M, self.all_boundary_nodes
) #put boundary values to zero, not really necessary because boundary values will be overwritten anyway. But easier to see that something must happen when the algorithm will default them to zero so you know you need to correct them.
Stilde_inv = M
Stilde_inv.data = 1. / Stilde_inv.data
self.A_k = Stilde_inv * (2 * M_bc_zero - K)
self.A_km1 = -1 * Stilde_inv * (M_bc_zero)
self.A_f = Stilde_inv
def _turn_sparse_rows_to_identity(A, rows_to_keep, rows_to_change):
#Convenience function for removing some rows from the sparse laplacian
#Had some major performance problems by simply slicing in all the matrix formats I tried.
nr, nc = A.shape
if nr != nc:
raise Exception('assuming square matrix')
#Create diagonal matrix. When we multiply A by this matrix we can remove rows
rows_to_keep_diag = np.zeros(nr, dtype='int32')
rows_to_keep_diag[rows_to_keep] = 1
diag_mat_remove_rows = make_diag_mtx(rows_to_keep_diag)
#The matrix below has the rows we want to turn into identity turned to 0
A_with_rows_removed = diag_mat_remove_rows * A
#Make diag matrix that has diagonal entries in the rows we want to be identity
rows_to_change_diag = np.zeros(nr, dtype='int32')
rows_to_change_diag[rows_to_change] = 1
A_with_identity_rows = make_diag_mtx(rows_to_change_diag)
A_modified = A_with_rows_removed + A_with_identity_rows
return A_modified
def _turn_sparse_rows_to_identity(A, rows_to_keep, rows_to_change):
#Convenience function for removing some rows from the sparse laplacian
#Had some major performance problems by simply slicing in all the matrix formats I tried.
nr,nc = A.shape
if nr != nc:
raise Exception('assuming square matrix')
#Create diagonal matrix. When we multiply A by this matrix we can remove rows
rows_to_keep_diag = np.zeros(nr, dtype='int32')
rows_to_keep_diag[rows_to_keep] = 1
diag_mat_remove_rows = make_diag_mtx(rows_to_keep_diag)
#The matrix below has the rows we want to turn into identity turned to 0
A_with_rows_removed = diag_mat_remove_rows*A
#Make diag matrix that has diagonal entries in the rows we want to be identity
rows_to_change_diag = np.zeros(nr, dtype='int32')
rows_to_change_diag[rows_to_change] = 1
A_with_identity_rows = make_diag_mtx(rows_to_change_diag)
A_modified = A_with_rows_removed + A_with_identity_rows
return A_modified
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
# a more readable reference
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh, 2,
self.spatial_accuracy_order)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.minus_sigmax = make_diag_mtx(-sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.minus_sigmaz = make_diag_mtx(-sz)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx * sz)
oc.minus_sigma_xPz = oc.minus_sigmax + oc.minus_sigmaz
oc.sigma_zMx_Dx = make_diag_mtx(sz - sx) * oc.Dx
oc.sigma_xMz_Dz = make_diag_mtx(sx - sz) * oc.Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m_inv = make_diag_mtx((C**2).reshape(-1, ))
self.A = spsp.bmat(
[[oc.empty, oc.I, oc.empty, oc.empty],
[
oc.m_inv * oc.L - oc.sigma_xz, oc.minus_sigma_xPz,
oc.m_inv * oc.Dx, oc.m_inv * oc.Dz
], [oc.sigma_zMx_Dx, oc.empty, oc.minus_sigmax, oc.empty],
[oc.sigma_xMz_Dz, oc.empty, oc.empty, oc.minus_sigmaz]])
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmaz (stored as -1*sigmaz)
sz = build_sigma(self.mesh, self.mesh.z)
oc.minus_sigmaz = make_diag_mtx(-sz)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m_inv = make_diag_mtx((C**2).reshape(-1,))
self.A = spsp.bmat([[oc.empty, oc.I, oc.empty ],
[oc.m_inv*oc.L, oc.minus_sigmaz, oc.m_inv*oc.Dz ],
[oc.minus_sigmaz*oc.Dz, oc.empty, oc.minus_sigmaz]])
def _build_helmholtz_operator(self, nu):
omega = 2*np.pi*nu
# we build the right helmholtz operator compact or not
if self._local_support_spec['spatial_dimension']==1:
# 1D the compact is not implemented so we raise a warming and use the auxiliary field PML
return (-(omega**2)*self.M + omega*1j*self.C + self.K).tocsc()
elif self.mesh.x.lbc.type == 'pml':
if self.mesh.x.lbc.domain_bc.compact:
# building the compact operator
oc = self.operator_components
if self._local_support_spec['spatial_dimension']==2:
dx, dz = self.mesh.deltas
length_pml_x = dx * (self.mesh.x.lbc.n -1 )
length_pml_z = dz * (self.mesh.z.lbc.n -1 )
H1 = -(omega**2)*oc.M
H2 = make_diag_mtx(-1j/(omega*length_pml_x) * oc.sxp / (1 - 1j/omega * oc.sx)**3).dot(oc.Dx)
H3 = make_diag_mtx(-1j/(omega*length_pml_z) * oc.szp / (1 - 1j/omega * oc.sz)**3).dot(oc.Dz)
H4 = - make_diag_mtx(1.0/(1 - 1j/omega * oc.sx)**2).dot(oc.Dxx)
H5 = - make_diag_mtx(1.0/(1 - 1j/omega * oc.sz)**2).dot(oc.Dzz)
return (H1 + H2 + H3 + H4 + H5).tocsc() # csc is used for the sparse solvers right now
if self._local_support_spec['spatial_dimension']==3:
dx, dy, dz = self.mesh.deltas
length_pml_x = dx * (self.mesh.x.lbc.n -1 )
length_pml_y = dy * (self.mesh.y.lbc.n -1 )
length_pml_z = dz * (self.mesh.z.lbc.n -1 )
H1 = -(omega**2)*oc.M
H2 = make_diag_mtx(-1j/(omega*length_pml_x) * oc.sxp / (1 - 1j/omega * oc.sx)**3).dot(oc.Dx)
H3 = make_diag_mtx(-1j/(omega*length_pml_y) * oc.syp / (1 - 1j/omega * oc.sy)**3).dot(oc.Dy)
H4 = make_diag_mtx(-1j/(omega*length_pml_z) * oc.szp / (1 - 1j/omega * oc.sz)**3).dot(oc.Dz)
H5 = - make_diag_mtx(1.0/(1 - 1j/omega * oc.sx)**2).dot(oc.Dxx)
H6 = - make_diag_mtx(1.0/(1 - 1j/omega * oc.sy)**2).dot(oc.Dyy)
H7 = - make_diag_mtx(1.0/(1 - 1j/omega * oc.sz)**2).dot(oc.Dzz)
return (H1 + H2 + H3 + H4 + H5 + H6 + H7).tocsc()
else:
return (-(omega**2)*self.M + omega*1j*self.C + self.K).tocsc()
else:
return (-(omega**2)*self.M + omega*1j*self.C + self.K).tocsc() # csc is used for the sparse solvers right now
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmaz (stored as -1*sigmaz)
sz = build_sigma(self.mesh, self.mesh.z)
oc.minus_sigmaz = make_diag_mtx(-sz)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m_inv = make_diag_mtx((C**2).reshape(-1,))
self.A = spsp.bmat([[oc.empty, oc.I, oc.empty ],
[oc.m_inv*oc.L, oc.minus_sigmaz, oc.m_inv*oc.Dz ],
[oc.minus_sigmaz*oc.Dz, oc.empty, oc.minus_sigmaz]])
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
# a more readable reference
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.minus_sigmax = make_diag_mtx(-sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.minus_sigmaz = make_diag_mtx(-sz)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.minus_sigma_xPz = oc.minus_sigmax + oc.minus_sigmaz
oc.sigma_zMx_Dx = make_diag_mtx(sz-sx)*oc.Dx
oc.sigma_xMz_Dz = make_diag_mtx(sx-sz)*oc.Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m_inv = make_diag_mtx((C**2).reshape(-1,))
self.A = spsp.bmat([[oc.empty, oc.I, oc.empty, oc.empty ],
[oc.m_inv*oc.L-oc.sigma_xz, oc.minus_sigma_xPz, oc.m_inv*oc.Dx, oc.m_inv*oc.Dz ],
[oc.sigma_zMx_Dx, oc.empty, oc.minus_sigmax, oc.empty ],
[oc.sigma_xMz_Dz, oc.empty, oc.empty, oc.minus_sigmaz ]])
def _rebuild_operators(self):
if self.mesh.x.lbc.type == 'pml' and self.compact:
# build intermediates for the compact operator
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.M = make_diag_mtx(self.model_parameters.C.squeeze()**-2)
oc.I = spsp.eye(dof, dof)
# build the static components
if not built:
# build Dxx
oc.Dxx = build_derivative_matrix(
self.mesh,
2,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dzz
oc.Dzz = build_derivative_matrix(
self.mesh,
2,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dx
oc.Dx = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dz = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build sigma
oc.sx, oc.sz, oc.sxp, oc.szp = self._sigma_PML(self.mesh)
oc._numpy_components_built = True
self.dK = 0
self.dC = 0
self.dM = oc.I
else:
# build intermediates for operator with auxiliary fields
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
oc.I = spsp.eye(dof, dof)
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(
self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx * sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz - sx)) * oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx - sz)) * oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1, ))
self.K = spsp.bmat(
[[oc.m * oc.sigma_xz - oc.L, oc.minus_Dx, oc.minus_Dz],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
self.C = spsp.bmat([[oc.m * oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I]])
self.M = spsp.bmat([[oc.m, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
self.dK = oc.sigma_xz
self.dC = oc.sigma_xPz
self.dM = oc.I
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
self.K = spsp.bmat([[oc.m*oc.sigma_xz-oc.L, oc.minus_Dx, oc.minus_Dz ],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty ],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz ]])
self.C = spsp.bmat([[oc.m*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I ]])
self.M = spsp.bmat([[ oc.m, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
if self.mesh.x.lbc.type == 'pml' and self.compact:
# build intermediates for the compact operator
raise ValidationFunctionError(
" This solver is in construction and is not yet complete. For variable density and compact PML."
)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.M = make_diag_mtx(self.model_parameters.C.squeeze()**-2)
# build the static components
if not built:
# build Dxx
oc.Dxx = build_derivative_matrix(
self.mesh,
2,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dzz
oc.Dzz = build_derivative_matrix(
self.mesh,
2,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dx
oc.Dx = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dz = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build sigma
oc.sx, oc.sz, oc.sxp, oc.szp = self._sigma_PML(self.mesh)
oc._numpy_components_built = True
else:
# build intermediates for operator with auxiliary fields
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(
self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx * sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz - sx)) * oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx - sz)) * oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1, ))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1, ))
# build heterogenous laplacian
sh = self.mesh.shape(include_bc=True, as_grid=True)
deltas = [self.mesh.x.delta, self.mesh.z.delta]
oc.L = build_derivative_matrix_VDA(self.mesh,
2,
self.spatial_accuracy_order,
alpha=rho**-1)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Currently the creation of oc.L is slow. This is because we have implemented a cenetered heterogenous laplacian.
# To speed up computation, we could compute a div(m2 grad) operator that is not centered by simply multiplying
# a divergence operator by oc.m2 by a gradient operator.
self.K = spsp.bmat([[
oc.m1 * oc.sigma_xz - oc.L, oc.minus_Dx * oc.m2,
oc.minus_Dz * oc.m2
], [oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
self.C = spsp.bmat([[oc.m1 * oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I]])
self.M = spsp.bmat([[oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1,))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1,))
# build heterogenous laplacian
sh = self.mesh.shape(include_bc=True,as_grid=True)
deltas = [self.mesh.x.delta,self.mesh.z.delta]
oc.L = build_heterogenous_laplacian(sh,rho**-1,deltas)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Currently the creation of oc.L is slow. This is because we have implemented a cenetered heterogenous laplacian.
# To speed up computation, we could compute a div(m2 grad) operator that is not centered by simply multiplying
# a divergence operator by oc.m2 by a gradient operator.
self.K = spsp.bmat([[oc.m1*oc.sigma_xz-oc.L, oc.minus_Dx*oc.m2, oc.minus_Dz*oc.m2 ],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty ],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz ]])
self.C = spsp.bmat([[oc.m1*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I ]])
self.M = spsp.bmat([[ oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
if self.mesh.x.lbc.type == 'pml' and self.compact:
# build intermediates for compact operator
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.M = make_diag_mtx(self.model_parameters.C.squeeze()**-2)
# build the static components
if not built:
# build Dxx
oc.Dxx = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dzz
oc.Dzz = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dyy
oc.Dyy = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='y',
use_shifted_differences=self.spatial_shifted_differences)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dy = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='y',
use_shifted_differences=self.spatial_shifted_differences)
# build sigma
oc.sx, oc.sy, oc.sz, oc.sxp, oc.syp, oc.szp = self._sigma_PML(self.mesh)
oc._numpy_components_built = True
else :
# build intermediates for operator with auxiliary fields
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmay
sy = build_sigma(self.mesh, self.mesh.y)
oc.sigmay = make_diag_mtx(sy)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dy
oc.minus_Dy = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='y',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dy.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.minus_I = -1*oc.I
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_sum_pair_prod = make_diag_mtx(sx*sy + sx*sz + sy*sz)
oc.sigma_sum = make_diag_mtx(sx+sy+sz)
oc.sigma_prod = make_diag_mtx(sx*sy*sz)
oc.minus_sigma_yPzMx_Dx = make_diag_mtx(sy+sz-sx)*oc.minus_Dx
oc.minus_sigma_xPzMy_Dy = make_diag_mtx(sx+sz-sy)*oc.minus_Dy
oc.minus_sigma_xPyMz_Dz = make_diag_mtx(sx+sy-sz)*oc.minus_Dz
oc.minus_sigma_yz_Dx = make_diag_mtx(sy*sz)*oc.minus_Dx
oc.minus_sigma_zx_Dy = make_diag_mtx(sz*sx)*oc.minus_Dy
oc.minus_sigma_xy_Dz = make_diag_mtx(sx*sy)*oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
self.K = spsp.bmat([[oc.m*oc.sigma_sum_pair_prod-oc.L, oc.m*oc.sigma_prod, oc.minus_Dx, oc.minus_Dy, oc.minus_Dz ],
[oc.minus_I, oc.empty, oc.empty, oc.empty, oc.empty ],
[oc.minus_sigma_yPzMx_Dx, oc.minus_sigma_yz_Dx, oc.sigmax, oc.empty, oc.empty ],
[oc.minus_sigma_xPzMy_Dy, oc.minus_sigma_zx_Dy, oc.empty, oc.sigmay, oc.empty ],
[oc.minus_sigma_xPyMz_Dz, oc.minus_sigma_xy_Dz, oc.empty, oc.empty, oc.sigmaz ]])
self.C = spsp.bmat([[oc.m*oc.sigma_sum, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.I, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.I ]]) / self.dt
self.M = spsp.bmat([[ oc.m, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
K = spsp.bmat([[oc.m*oc.sigma_xz-oc.L, oc.minus_Dx, oc.minus_Dz],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
C = spsp.bmat([[oc.m*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I]]) / self.dt
M = spsp.bmat([[oc.m, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]]) / self.dt**2
Stilde_inv = M+C
Stilde_inv.data = 1./Stilde_inv.data
self.A_k = Stilde_inv*(2*M - K + C)
self.A_km1 = -1*Stilde_inv*(M)
self.A_f = Stilde_inv
def _build_derivative_matrix_staggered_structured_cartesian(mesh,
derivative, order_accuracy,
dimension='all',
alpha = None,
return_1D_matrix=False,
**kwargs):
#Some of the operators could be cached the same way I did to make 'build_permutation_matrix' faster.
#Could be considered if the current speed is ever considered to be insufficient.
import time
tt = time.time()
if return_1D_matrix:
raise Exception('Not yet implemented')
if derivative < 1 or derivative > 2:
raise ValueError('Only defined for first and second order right now')
if derivative == 1 and dimension not in ['x', 'y', 'z']:
raise ValueError('First derivative requires a direciton')
sh = mesh.shape(include_bc = True, as_grid = True) #Will include PML padding
if len(sh) != 2: raise Exception('currently hardcoded 2D implementation, relatively straight-forward to change. Look at the function build_derivative_matrix to get a more general function.')
nx = sh[0]
nz = sh[-1]
#Currently I am working with density input on the regular grid.
#In the derivation of the variable density solver we only require density at the stagger points
#For now I am just interpolating density defined on regular points towards the stagger points and use that as 'density model'.
#Later it is probably better to define the density directly on the stagger points (and evaluate density gradient there to update directly at these points?)
if type(alpha) == None: #If no alpha is given, we set it to a uniform vector. The result should be the homogeneous Laplacian.
alpha = np.ones(nx*nz)
alpha = alpha.flatten() #make 1D
dx = mesh.x.delta
dz = mesh.z.delta
#Get 1D linear interpolation matrices
Jx_1d = build_linear_interpolation_matrix_part(nx)
Jz_1d = build_linear_interpolation_matrix_part(nz)
#Get 1D derivative matrix for first spatial derivative using the desired order of accuracy
lbc_x = _set_bc(mesh.x.lbc)
rbc_x = _set_bc(mesh.x.rbc)
lbc_z = _set_bc(mesh.z.lbc)
rbc_z = _set_bc(mesh.z.rbc)
Dx_1d = _build_staggered_first_derivative_matrix_part(nx, order_accuracy, h=dx, lbc = lbc_x, rbc = rbc_x)
Dz_1d = _build_staggered_first_derivative_matrix_part(nz, order_accuracy, h=dz, lbc = lbc_z, rbc = rbc_z)
#Some empty matrices of the right shape so we can use kronsum to get the proper 2D matrices for the operations we want.
#The same is used in the homogeneous 'build_derivative_matrix' function.
Ix = spsp.eye(nx)
Iz = spsp.eye(nz)
Dx_2d = spsp.kron(Dx_1d, Iz, format='csr')
if dimension == 'x' and derivative == 1:
return Dx_2d
Dz_2d = spsp.kron(Ix, Dz_1d, format='csr')
if dimension == 'z' and derivative == 1:
return Dz_2d
#If we are evaluating this we want to make the heterogeneous Laplacian
Jx_2d = spsp.kron(Jx_1d, Iz, format='csr')
Jz_2d = spsp.kron(Ix, Jz_1d, format='csr')
#alpha interpolated to x stagger points. Make diag mat
diag_alpha_x = make_diag_mtx(Jx_2d*alpha)
#alpha interpolated to z stagger points. Make diag mat
diag_alpha_z = make_diag_mtx(Jz_2d*alpha)
#Create laplacian components
#The negative transpose of Dx and Dz takes care of the divergence term of the heterogeneous laplacian
Dxx_2d = -Dx_2d.T*diag_alpha_x*Dx_2d
Dzz_2d = -Dz_2d.T*diag_alpha_z*Dz_2d
#Correct the Laplacian around the boundary. This is also done in the homogeneous Laplacian
#I want the heterogeneous Laplacian to be the same as the homogeneous Laplacian when alpha is uniform
#This is the only part of the Laplacian that deviates from symmetry, just as in the homogeneous case.
#But because of these conditions on the dirichlet boundary the wavefield will always equal 0 there and this deviation from symmetry is fine.
#For indexing, get list of all boundary node numbers
left_node_nrs = np.arange(nz)
right_node_nrs = np.arange((nx-1)*nz,nx*nz)
top_node_nrs = np.arange(nz,(nx-1)*nz,nz) #does not include left and right top node
bot_node_nrs = top_node_nrs + nz - 1 #does not include left and right top node
all_boundary_node_nrs = np.concatenate((left_node_nrs, right_node_nrs, top_node_nrs, bot_node_nrs))
nb = all_boundary_node_nrs.size
L = Dxx_2d + Dzz_2d
all_node_numbers = np.arange(0,(nx*nz), dtype='int32')
internal_node_numbers = list(set(all_node_numbers) - set(all_boundary_node_nrs))
L = L.tocsr() #so we can extract rows efficiently
#Operation below fixes the boundary rows quite efficiently.
L_fixed = _turn_sparse_rows_to_identity(L, internal_node_numbers, all_boundary_node_nrs)
return L_fixed.tocsr()
def _rebuild_operators(self):
VariableDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
oc.minus_Dx = copy.deepcopy(oc.Dx) #more storage, but less computations
oc.minus_Dx.data *= -1
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
oc.minus_Dz = copy.deepcopy(oc.Dz) #more storage, but less computations
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1,))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1,))
oc.L = build_derivative_matrix_VDA(self.mesh,
2,
self.spatial_accuracy_order,
alpha = rho**-1
)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Ian's implementation used the regular Dx operators in the PML even though the heterogeneous Laplacian with staggered derivative operators is used in the physical domain.
# I (Bram) did not change this or investigate if this causes some problems but am noting it here for completeness.
K = spsp.bmat([[oc.m1*oc.sigma_xz-oc.L, oc.minus_Dx*oc.m2, oc.minus_Dz*oc.m2 ],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty ],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz ]])
C = spsp.bmat([[oc.m1*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I ]]) / self.dt
M = spsp.bmat([[ oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]]) / self.dt**2
Stilde_inv = M+C
Stilde_inv.data = 1./Stilde_inv.data
self.A_k = Stilde_inv*(2*M - K + C)
self.A_km1 = -1*Stilde_inv*(M)
self.A_f = Stilde_inv
def _rebuild_operators(self):
VariableDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
oc.minus_Dx = copy.deepcopy(
oc.Dx) #more storage, but less computations
oc.minus_Dx.data *= -1
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
oc.minus_Dz = copy.deepcopy(
oc.Dz) #more storage, but less computations
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx * sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz - sx)) * oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx - sz)) * oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1, ))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1, ))
oc.L = build_derivative_matrix_VDA(self.mesh,
2,
self.spatial_accuracy_order,
alpha=rho**-1)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Ian's implementation used the regular Dx operators in the PML even though the heterogeneous Laplacian with staggered derivative operators is used in the physical domain.
# I (Bram) did not change this or investigate if this causes some problems but am noting it here for completeness.
K = spsp.bmat([[
oc.m1 * oc.sigma_xz - oc.L, oc.minus_Dx * oc.m2,
oc.minus_Dz * oc.m2
], [oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
C = spsp.bmat([[oc.m1 * oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty], [oc.empty, oc.empty, oc.I]
]) / self.dt
M = spsp.bmat([[oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]]) / self.dt**2
Stilde_inv = M + C
Stilde_inv.data = 1. / Stilde_inv.data
self.A_k = Stilde_inv * (2 * M - K + C)
self.A_km1 = -1 * Stilde_inv * (M)
self.A_f = Stilde_inv
def _build_derivative_matrix_staggered_structured_cartesian(
mesh,
derivative,
order_accuracy,
dimension='all',
alpha=None,
return_1D_matrix=False,
**kwargs):
#Some of the operators could be cached the same way I did to make 'build_permutation_matrix' faster.
#Could be considered if the current speed is ever considered to be insufficient.
import time
tt = time.time()
if return_1D_matrix:
raise Exception('Not yet implemented')
if derivative < 1 or derivative > 2:
raise ValueError('Only defined for first and second order right now')
if derivative == 1 and dimension not in ['x', 'y', 'z']:
raise ValueError('First derivative requires a direciton')
sh = mesh.shape(include_bc=True, as_grid=True) #Will include PML padding
if len(sh) != 2:
raise Exception(
'currently hardcoded 2D implementation, relatively straight-forward to change. Look at the function build_derivative_matrix to get a more general function.'
)
nx = sh[0]
nz = sh[-1]
#Currently I am working with density input on the regular grid.
#In the derivation of the variable density solver we only require density at the stagger points
#For now I am just interpolating density defined on regular points towards the stagger points and use that as 'density model'.
#Later it is probably better to define the density directly on the stagger points (and evaluate density gradient there to update directly at these points?)
if type(
alpha
) == None: #If no alpha is given, we set it to a uniform vector. The result should be the homogeneous Laplacian.
alpha = np.ones(nx * nz)
alpha = alpha.flatten() #make 1D
dx = mesh.x.delta
dz = mesh.z.delta
#Get 1D linear interpolation matrices
Jx_1d = build_linear_interpolation_matrix_part(nx)
Jz_1d = build_linear_interpolation_matrix_part(nz)
#Get 1D derivative matrix for first spatial derivative using the desired order of accuracy
lbc_x = _set_bc(mesh.x.lbc)
rbc_x = _set_bc(mesh.x.rbc)
lbc_z = _set_bc(mesh.z.lbc)
rbc_z = _set_bc(mesh.z.rbc)
Dx_1d = _build_staggered_first_derivative_matrix_part(nx,
order_accuracy,
h=dx,
lbc=lbc_x,
rbc=rbc_x)
Dz_1d = _build_staggered_first_derivative_matrix_part(nz,
order_accuracy,
h=dz,
lbc=lbc_z,
rbc=rbc_z)
#Some empty matrices of the right shape so we can use kronsum to get the proper 2D matrices for the operations we want.
#The same is used in the homogeneous 'build_derivative_matrix' function.
Ix = spsp.eye(nx)
Iz = spsp.eye(nz)
Dx_2d = spsp.kron(Dx_1d, Iz, format='csr')
if dimension == 'x' and derivative == 1:
return Dx_2d
Dz_2d = spsp.kron(Ix, Dz_1d, format='csr')
if dimension == 'z' and derivative == 1:
return Dz_2d
#If we are evaluating this we want to make the heterogeneous Laplacian
Jx_2d = spsp.kron(Jx_1d, Iz, format='csr')
Jz_2d = spsp.kron(Ix, Jz_1d, format='csr')
#alpha interpolated to x stagger points. Make diag mat
diag_alpha_x = make_diag_mtx(Jx_2d * alpha)
#alpha interpolated to z stagger points. Make diag mat
diag_alpha_z = make_diag_mtx(Jz_2d * alpha)
#Create laplacian components
#The negative transpose of Dx and Dz takes care of the divergence term of the heterogeneous laplacian
Dxx_2d = -Dx_2d.T * diag_alpha_x * Dx_2d
Dzz_2d = -Dz_2d.T * diag_alpha_z * Dz_2d
#Correct the Laplacian around the boundary. This is also done in the homogeneous Laplacian
#I want the heterogeneous Laplacian to be the same as the homogeneous Laplacian when alpha is uniform
#This is the only part of the Laplacian that deviates from symmetry, just as in the homogeneous case.
#But because of these conditions on the dirichlet boundary the wavefield will always equal 0 there and this deviation from symmetry is fine.
#For indexing, get list of all boundary node numbers
left_node_nrs = np.arange(nz)
right_node_nrs = np.arange((nx - 1) * nz, nx * nz)
top_node_nrs = np.arange(nz, (nx - 1) * nz,
nz) #does not include left and right top node
bot_node_nrs = top_node_nrs + nz - 1 #does not include left and right top node
all_boundary_node_nrs = np.concatenate(
(left_node_nrs, right_node_nrs, top_node_nrs, bot_node_nrs))
nb = all_boundary_node_nrs.size
L = Dxx_2d + Dzz_2d
all_node_numbers = np.arange(0, (nx * nz), dtype='int32')
internal_node_numbers = list(
set(all_node_numbers) - set(all_boundary_node_nrs))
L = L.tocsr() #so we can extract rows efficiently
#Operation below fixes the boundary rows quite efficiently.
L_fixed = _turn_sparse_rows_to_identity(L, internal_node_numbers,
all_boundary_node_nrs)
return L_fixed.tocsr()
def _rebuild_operators(self):
VariableDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1,))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1,))
# build heterogenous laplacian
sh = self.mesh.shape(include_bc=True,as_grid=True)
deltas = [self.mesh.x.delta,self.mesh.z.delta]
oc.L = build_heterogenous_laplacian(sh,rho**-1,deltas)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Currently the creation of oc.L is slow. This is because we have implemented a cenetered heterogenous laplacian.
# To speed up computation, we could compute a div(m2 grad) operator that is not centered by simply multiplying
# a divergence operator by oc.m2 by a gradient operator.
K = spsp.bmat([[oc.m1*oc.sigma_xz-oc.L, oc.minus_Dx*oc.m2, oc.minus_Dz*oc.m2 ],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty ],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz ]])
C = spsp.bmat([[oc.m1*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I ]]) / self.dt
M = spsp.bmat([[ oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]]) / self.dt**2
Stilde_inv = M+C
Stilde_inv.data = 1./Stilde_inv.data
self.A_k = Stilde_inv*(2*M - K + C)
self.A_km1 = -1*Stilde_inv*(M)
self.A_f = Stilde_inv
def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_3D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get("_numpy_components_built", False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh, 2, self.spatial_accuracy_order)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmay
sy = build_sigma(self.mesh, self.mesh.y)
oc.sigmay = make_diag_mtx(sy)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh, 1, self.spatial_accuracy_order, dimension="x")
oc.minus_Dx.data *= -1
# build Dy
oc.minus_Dy = build_derivative_matrix(self.mesh, 1, self.spatial_accuracy_order, dimension="y")
oc.minus_Dy.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh, 1, self.spatial_accuracy_order, dimension="z")
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.minus_I = -1 * oc.I
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_sum_pair_prod = make_diag_mtx(sx * sy + sx * sz + sy * sz)
oc.sigma_sum = make_diag_mtx(sx + sy + sz)
oc.sigma_prod = make_diag_mtx(sx * sy * sz)
oc.minus_sigma_yPzMx_Dx = make_diag_mtx(sy + sz - sx) * oc.minus_Dx
oc.minus_sigma_xPzMy_Dy = make_diag_mtx(sx + sz - sy) * oc.minus_Dy
oc.minus_sigma_xPyMz_Dz = make_diag_mtx(sx + sy - sz) * oc.minus_Dz
oc.minus_sigma_yz_Dx = make_diag_mtx(sy * sz) * oc.minus_Dx
oc.minus_sigma_zx_Dy = make_diag_mtx(sz * sx) * oc.minus_Dy
oc.minus_sigma_xy_Dz = make_diag_mtx(sx * sy) * oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C ** -2).reshape(-1))
K = spsp.bmat(
[
[oc.m * oc.sigma_sum_pair_prod - oc.L, oc.m * oc.sigma_prod, oc.minus_Dx, oc.minus_Dy, oc.minus_Dz],
[oc.minus_I, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.minus_sigma_yPzMx_Dx, oc.minus_sigma_yz_Dx, oc.sigmax, oc.empty, oc.empty],
[oc.minus_sigma_xPzMy_Dy, oc.minus_sigma_zx_Dy, oc.empty, oc.sigmay, oc.empty],
[oc.minus_sigma_xPyMz_Dz, oc.minus_sigma_xy_Dz, oc.empty, oc.empty, oc.sigmaz],
]
)
C = (
spsp.bmat(
[
[oc.m * oc.sigma_sum, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.I, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.I],
]
)
/ self.dt
)
M = (
spsp.bmat(
[
[oc.m, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
]
)
/ self.dt ** 2
)
Stilde_inv = M + C
Stilde_inv.data = 1.0 / Stilde_inv.data
self.A_k = Stilde_inv * (2 * M - K + C)
self.A_km1 = -1 * Stilde_inv * (M)
self.A_f = Stilde_inv
def _rebuild_operators(self):
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmay
sy = build_sigma(self.mesh, self.mesh.y)
oc.sigmay = make_diag_mtx(sy)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dy
oc.minus_Dy = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='y',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dy.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.minus_I = -1*oc.I
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_sum_pair_prod = make_diag_mtx(sx*sy + sx*sz + sy*sz)
oc.sigma_sum = make_diag_mtx(sx+sy+sz)
oc.sigma_prod = make_diag_mtx(sx*sy*sz)
oc.minus_sigma_yPzMx_Dx = make_diag_mtx(sy+sz-sx)*oc.minus_Dx
oc.minus_sigma_xPzMy_Dy = make_diag_mtx(sx+sz-sy)*oc.minus_Dy
oc.minus_sigma_xPyMz_Dz = make_diag_mtx(sx+sy-sz)*oc.minus_Dz
oc.minus_sigma_yz_Dx = make_diag_mtx(sy*sz)*oc.minus_Dx
oc.minus_sigma_zx_Dy = make_diag_mtx(sz*sx)*oc.minus_Dy
oc.minus_sigma_xy_Dz = make_diag_mtx(sx*sy)*oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
self.K = spsp.bmat([[oc.m*oc.sigma_sum_pair_prod-oc.L, oc.m*oc.sigma_prod, oc.minus_Dx, oc.minus_Dy, oc.minus_Dz ],
[oc.minus_I, oc.empty, oc.empty, oc.empty, oc.empty ],
[oc.minus_sigma_yPzMx_Dx, oc.minus_sigma_yz_Dx, oc.sigmax, oc.empty, oc.empty ],
[oc.minus_sigma_xPzMy_Dy, oc.minus_sigma_zx_Dy, oc.empty, oc.sigmay, oc.empty ],
[oc.minus_sigma_xPyMz_Dz, oc.minus_sigma_xy_Dz, oc.empty, oc.empty, oc.sigmaz ]])
self.C = spsp.bmat([[oc.m*oc.sigma_sum, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.I, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.I ]]) / self.dt
self.M = spsp.bmat([[ oc.m, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
if self.mesh.x.lbc.type == 'pml' and self.compact:
# build intermediates for the compact operator
raise ValidationFunctionError(" This solver is in construction and is not yet complete. For variable density and compact PML.")
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.M = make_diag_mtx(self.model_parameters.C.squeeze()**-2)
# build the static components
if not built:
# build Dxx
oc.Dxx = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dzz
oc.Dzz = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build sigma
oc.sx, oc.sz, oc.sxp, oc.szp = self._sigma_PML(self.mesh)
oc._numpy_components_built = True
else:
# build intermediates for operator with auxiliary fields
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
kappa = self.model_parameters.kappa
rho = self.model_parameters.rho
oc.m1 = make_diag_mtx((kappa**-1).reshape(-1,))
oc.m2 = make_diag_mtx((rho**-1).reshape(-1,))
# build heterogenous laplacian
sh = self.mesh.shape(include_bc=True,as_grid=True)
deltas = [self.mesh.x.delta,self.mesh.z.delta]
oc.L = build_heterogenous_laplacian(sh,rho**-1,deltas)
# oc.L is a heterogenous laplacian operator. It computes div(m2 grad), where m2 = 1/rho.
# Currently the creation of oc.L is slow. This is because we have implemented a cenetered heterogenous laplacian.
# To speed up computation, we could compute a div(m2 grad) operator that is not centered by simply multiplying
# a divergence operator by oc.m2 by a gradient operator.
self.K = spsp.bmat([[oc.m1*oc.sigma_xz-oc.L, oc.minus_Dx*oc.m2, oc.minus_Dz*oc.m2 ],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty ],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz ]])
self.C = spsp.bmat([[oc.m1*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I ]])
self.M = spsp.bmat([[ oc.m1, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
def _rebuild_operators(self):
ConstantDensityAcousticTimeScalar_2D._rebuild_operators(self)
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh, 2,
self.spatial_accuracy_order)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x')
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z')
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx * sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz - sx)) * oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx - sz)) * oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1, ))
K = spsp.bmat([[oc.m * oc.sigma_xz - oc.L, oc.minus_Dx, oc.minus_Dz],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
C = spsp.bmat([[oc.m * oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty], [oc.empty, oc.empty, oc.I]
]) / self.dt
M = spsp.bmat([[oc.m, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]]) / self.dt**2
Stilde_inv = M + C
Stilde_inv.data = 1. / Stilde_inv.data
self.A_k = Stilde_inv * (2 * M - K + C)
self.A_km1 = -1 * Stilde_inv * (M)
self.A_f = Stilde_inv
def _rebuild_operators(self):
if self.mesh.x.lbc.type == 'pml' and self.compact:
# build intermediates for the compact operator
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
built = oc.get('_numpy_components_built', False)
oc.M = make_diag_mtx(self.model_parameters.C.squeeze()**-2)
oc.I = spsp.eye(dof, dof)
# build the static components
if not built:
# build Dxx
oc.Dxx = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dzz
oc.Dzz = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build Dx
oc.Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
# build Dz
oc.Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
# build sigma
oc.sx, oc.sz, oc.sxp, oc.szp = self._sigma_PML(self.mesh)
oc._numpy_components_built = True
self.dK = 0
self.dC = 0
self.dM = oc.I
else:
# build intermediates for operator with auxiliary fields
dof = self.mesh.dof(include_bc=True)
oc = self.operator_components
oc.I = spsp.eye(dof, dof)
built = oc.get('_numpy_components_built', False)
# build the static components
if not built:
# build laplacian
oc.L = build_derivative_matrix(self.mesh,
2,
self.spatial_accuracy_order,
use_shifted_differences=self.spatial_shifted_differences)
# build sigmax
sx = build_sigma(self.mesh, self.mesh.x)
oc.sigmax = make_diag_mtx(sx)
# build sigmaz
sz = build_sigma(self.mesh, self.mesh.z)
oc.sigmaz = make_diag_mtx(sz)
# build Dx
oc.minus_Dx = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='x',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dx.data *= -1
# build Dz
oc.minus_Dz = build_derivative_matrix(self.mesh,
1,
self.spatial_accuracy_order,
dimension='z',
use_shifted_differences=self.spatial_shifted_differences)
oc.minus_Dz.data *= -1
# build other useful things
oc.I = spsp.eye(dof, dof)
oc.empty = spsp.csr_matrix((dof, dof))
# useful intermediates
oc.sigma_xz = make_diag_mtx(sx*sz)
oc.sigma_xPz = oc.sigmax + oc.sigmaz
oc.minus_sigma_zMx_Dx = make_diag_mtx((sz-sx))*oc.minus_Dx
oc.minus_sigma_xMz_Dz = make_diag_mtx((sx-sz))*oc.minus_Dz
oc._numpy_components_built = True
C = self.model_parameters.C
oc.m = make_diag_mtx((C**-2).reshape(-1,))
self.K = spsp.bmat([[oc.m*oc.sigma_xz-oc.L, oc.minus_Dx, oc.minus_Dz],
[oc.minus_sigma_zMx_Dx, oc.sigmax, oc.empty],
[oc.minus_sigma_xMz_Dz, oc.empty, oc.sigmaz]])
self.C = spsp.bmat([[oc.m*oc.sigma_xPz, oc.empty, oc.empty],
[oc.empty, oc.I, oc.empty],
[oc.empty, oc.empty, oc.I]])
self.M = spsp.bmat([[oc.m, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty],
[oc.empty, oc.empty, oc.empty]])
self.dK = oc.sigma_xz
self.dC = oc.sigma_xPz
self.dM = oc.I
