def __init__(self, cell: Cell, cell_basis_l: Basis, cell_basis_k: Basis, cell_basis_k1: Basis, unknown: Unknown):
"""
================================================================================================================
Class :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
================================================================================================================
Attributes :
================================================================================================================
"""
# --------------------------------------------------------------------------------------------------------------
# Building the cell
# --------------------------------------------------------------------------------------------------------------
centroid = cell.centroid
volume = cell.volume
quadrature_points = cell.quadrature_points
quadrature_weights = cell.quadrature_weights
# --------------------------------------------------------------------------------------------------------------
super().__init__(centroid, volume, diameter, quadrature_points, quadrature_weights)
m_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(cell, cell_basis_k, cell_basis_k)
m_phi_k_phi_k_inv = np.linalg.inv(m_phi_k_phi_k)
m_grad_phi_k1_grad_phi_k1_list = []
m_phi_k_grad_phi_l_list = []
m_grad_phi_k1_phi_l_list = []
for j in range(unknown.problem_dimension):
# ----------------------------------------------------------------------------------------------------------
m_grad_phi_k1_grad_phi_k1_j = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, j
)
m_grad_phi_k1_grad_phi_k1_list.append(m_grad_phi_k1_grad_phi_k1_j)
# ----------------------------------------------------------------------------------------------------------
m_phi_k_grad_phi_l_j = Integration.get_cell_advection_matrix_in_cell(cell, cell_basis_k, cell_basis_l, j)
m_phi_k_grad_phi_l_list.append(m_phi_k_grad_phi_l_j)
# ----------------------------------------------------------------------------------------------------------
m_phi_l_grad_phi_k1_j = Integration.get_cell_advection_matrix_in_cell(cell, cell_basis_l, cell_basis_k1, j)
m_grad_phi_k1_phi_l_j = m_phi_l_grad_phi_k1_j.T
m_grad_phi_k1_phi_l_list.append(m_grad_phi_k1_phi_l_j)
def __init__(
self,
face: Face,
face_basis: Basis,
face_reference_frame_transformation_matrix: Mat,
unknown: Unknown,
pressure: List[Callable] = None,
):
"""
================================================================================================================
Class :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
================================================================================================================
Attributes :
================================================================================================================
"""
if pressure is None:
pressure_vector = np.zeros(
(unknown.field_dimension * face_basis.basis_dimension, ))
else:
pressure_vector = np.zeros(
(unknown.field_dimension * face_basis.basis_dimension, ))
for i, pressure_component in enumerate(pressure):
if pressure_component is None:
pressure_vector_component = np.zeros(
(face_basis.basis_dimension, ))
else:
# face: Face, face_basis: Basis, face_reference_frame_transformation_matrix: Mat, pressure: Callable,
pressure_vector_component = Integration.get_face_pressure_vector_in_face(
face, face_basis,
face_reference_frame_transformation_matrix,
pressure_component)
pressure_vector[i * face_basis.basis_dimension:(i + 1) *
face_basis.
basis_dimension] += pressure_vector_component
self.pressure_vector = pressure_vector
def __init__(self, face_vertices_connectivity_matrix: Mat, face: Face, face_basis_k: Basis):
"""
================================================================================================================
Class :
================================================================================================================
The Mface class
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Attributes :
================================================================================================================
-
"""
# --------------------------------------------------------------------------------------------------------------
mat_F_psi_k_psi_k = Integration.get_face_mass_matrix_in_face(
face, face_basis_k, face_basis_k, face.reference_frame_transformation_matrix
)
self.mat_F_psi_k_psi_k_inv = np.linalg.inv(mat_F_psi_k_psi_k)
def compute_faces_matrices(self, faces: List[Face], face_basis_k: Basis):
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
mat_F_psi_k_psi_k_inv_list = []
for face in faces:
mat_F_psi_k_psi_k = Integration.get_face_mass_matrix_in_face(
face, face_basis_k, face_basis_k,
face.reference_frame_transformation_matrix)
mat_F_psi_k_psi_k_inv = np.linalg.inv(mat_F_psi_k_psi_k)
mat_F_psi_k_psi_k_inv_list.append(mat_F_psi_k_psi_k_inv)
return mat_F_psi_k_psi_k_inv_list
def get_local_mass_operator(
self,
cell: Cell,
faces: List[Face],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
face_basis_k: Basis,
unknown: Unknown,
) -> Mat:
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
local_problem_size = self.get_local_problem_size(
faces, cell_basis_l, face_basis_k, unknown)
local_mass_operator = np.zeros(
(local_problem_size, local_problem_size))
m_cell_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
directions = range(unknown.field_dimension)
for i in directions:
r0 = i * cell_basis_l.basis_dimension
r1 = (i + 1) * cell_basis_l.basis_dimension
local_mass_operator[r0:r1, r0:r1] += m_cell_phi_l_phi_l
return local_mass_operator
def __init__(self,
cell: Cell,
cell_basis: Basis,
unknown: Unknown,
load: List[Callable] = None):
"""
================================================================================================================
Class :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
================================================================================================================
Attributes :
================================================================================================================
"""
if load is None:
load_vector = np.zeros(
(unknown.field_dimension * cell_basis.basis_dimension, ))
else:
load_vector = np.zeros(
(unknown.field_dimension * cell_basis.basis_dimension, ))
for i, load_component in enumerate(load):
if load_component is None:
load_vector_component = np.zeros(
(cell_basis.basis_dimension, ))
else:
load_vector_component = Integration.get_cell_load_vector_in_cell(
cell, cell_basis, load_component)
load_vector[i * cell_basis.basis_dimension:(
i +
1) * cell_basis.basis_dimension] -= load_vector_component
self.load_vector = load_vector
def compute_hybrid_matrices(
self,
cell: Cell,
faces: List[Face],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
face_basis_k: Basis,
unknown: Unknown,
):
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR / STABILIZATION OPERATOR
# ==============================================================================================================
mat_F_phi_k_phi_l_list = []
mat_F_phi_k_psi_k_list = []
for mface in mfaces:
mat_F_phi_k_phi_l = Integration.get_cell_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k, cell_basis_l)
mat_F_phi_k_phi_l_list.append(mat_F_phi_k_phi_l_list)
# ----------------------------------------------------------------------------------------------------------
mat_F_phi_k_psi_k = Integration.get_hybrid_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k, face_basis_k,
mface.face.reference_frame_transformation_matrix)
mat_F_phi_k_psi_k_list.append(mat_F_phi_k_psi_k)
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
mat_F_grad_phi_k1_psi_k_list = []
mat_F_grad_phi_k1_phi_l_list = []
for mface in mfaces:
mat_F_grad_phi_k1_psi_k_j_list = []
mat_F_grad_phi_k1_phi_l_j_list = []
for j in range(len(unknown.problem_dimension)):
# ------------------------------------------------------------------------------------------------------
mat_F_grad_phi_k1_psi_k_j = Integration.get_hybrid_advection_matrix_in_face(
mcell.cell,
mface.face,
cell_basis_k1,
face_basis_k,
mface.face.reference_frame_transformation_matrix,
j,
)
mat_F_grad_phi_k1_psi_k_j_list.append(
mat_F_grad_phi_k1_psi_k_j)
# ------------------------------------------------------------------------------------------------------
mat_F_phi_l_grad_phi_k1_j = Integration.get_cell_advection_matrix_in_face(
mcell.cell, mface.face, cell_basis_l, cell_basis_k1, j)
# ------------------------------------------------------------------------------------------------------
mat_F_grad_phi_k1_phi_l_j = mat_F_phi_l_grad_phi_k1_j.T
mat_F_grad_phi_k1_phi_l_j_list.append(
mat_F_grad_phi_k1_phi_l_j)
# ----------------------------------------------------------------------------------------------------------
mat_F_grad_phi_k1_psi_k_list.append(mat_F_grad_phi_k1_psi_k_j_list)
mat_F_grad_phi_k1_phi_l_list.append(mat_F_grad_phi_k1_phi_l_j_list)
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
mat_F_psi_k_phi_k1_list = []
for mface in mfaces:
mat_F_phi_k1_psi_k = Integration.get_hybrid_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k1, face_basis_k,
mface.face.reference_frame_transformation_matrix)
mat_F_psi_k_phi_k1_list.append(mat_F_phi_k1_psi_k)
return (
mat_F_phi_k_phi_l_list,
mat_F_phi_k_psi_k_list,
mat_F_grad_phi_k1_psi_k_list,
mat_F_grad_phi_k1_phi_l_list,
mat_F_psi_k_phi_k1_list,
)
def compute_cell_matrices(self, cell: Cell, cell_basis_l: Basis,
cell_basis_k: Basis, cell_basis_k1: Basis,
unknown: Unknown):
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
mat_T_phi_k_phi_k_inv = np.linalg.inv(mat_T_phi_k_phi_k)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_grad_phi_l_list = []
for j in range(unknown.problem_dimension):
mat_T_phi_k_grad_phi_l = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, j)
mat_T_phi_k_grad_phi_l_list.append(mat_T_phi_k_grad_phi_l)
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_grad_phi_k1_grad_phi_k1_inv_list = []
for j in range(unknown.problem_dimension):
mat_T_grad_phi_k1_grad_phi_k1_j = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, j)
mat_T_grad_phi_k1_grad_phi_k1_inv_list.append(
mat_T_grad_phi_k1_grad_phi_k1_j)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_grad_phi_k1_phi_l_list = []
for j in range(unknown.problem_dimension):
mat_T_phi_l_grad_phi_k1_j = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1, j)
mat_T_grad_phi_k1_phi_l_j = mat_T_phi_l_grad_phi_k1_j.T
mat_T_grad_phi_k1_phi_l_list.append(mat_T_grad_phi_k1_phi_l_j)
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
mat_T_phi_k_phi_k1 = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k1)
proj_T_T_k1_k = mat_T_phi_k_phi_k_inv @ mat_T_phi_k_phi_k1
# ==============================================================================================================
# PROJECTION OPERATOR
# ==============================================================================================================
mat_T_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
mat_T_phi_l_phi_l_inv = np.linalg.inv(mat_T_phi_l_phi_l)
return (
mat_T_phi_k_phi_k_inv,
mat_T_phi_k_grad_phi_l_list,
mat_T_grad_phi_k1_grad_phi_k1_inv_list,
mat_T_grad_phi_k1_phi_l_list,
proj_T_T_k1_k,
)
def solve(
vertices: Mat,
faces: List[Face],
cells: List[Cell],
operators: List[Operator],
cells_faces_connectivity_matrix: Mat,
cells_vertices_connectivity_matrix: Mat,
faces_vertices_connectivity_matrix: Mat,
nsets: dict,
# flags: List[str],
nsets_faces: dict,
cell_basis_l: Basis,
cell_basis_k: Basis,
face_basis_k: Basis,
unknown: Unknown,
tangent_matrices: List[Mat],
stabilization_parameter: float,
boundary_conditions: dict,
load: List[Callable],
):
"""
====================================================================================================================
Description :
====================================================================================================================
====================================================================================================================
Parameters :
====================================================================================================================
====================================================================================================================
Exemple :
====================================================================================================================
"""
# ------------------------------------------------------------------------------------------------------------------
# Global matrix
# ------------------------------------------------------------------------------------------------------------------
number_of_faces = len(faces_vertices_connectivity_matrix)
number_of_cells = len(cells_vertices_connectivity_matrix)
total_system_size = number_of_faces * face_basis_k.basis_dimension * unknown.field_dimension
global_matrix = np.zeros((total_system_size, total_system_size))
global_vector = np.zeros((total_system_size, ))
stored_matrices = []
# ------------------------------------------------------------------------------------------------------------------
# Global matrix
# ------------------------------------------------------------------------------------------------------------------
cells_indices = range(len(cells))
for cell_index in cells_indices:
local_cell = cells[cell_index]
local_faces = [
faces[i] for i in cells_faces_connectivity_matrix[cell_index]
]
# --------------------------------------------------------------------------------------------------------------
# External forces
# --------------------------------------------------------------------------------------------------------------
number_of_local_faces = len(local_faces)
a = unknown.field_dimension * (
cell_basis_l.basis_dimension +
number_of_local_faces * face_basis_k.basis_dimension)
local_external_forces = np.zeros((a, ))
load_vector = Load(local_cell, cell_basis_l, unknown, load).load_vector
l0 = 0
l1 = unknown.field_dimension * cell_basis_l.basis_dimension
local_external_forces[l0:l1] += load_vector
connectivity = cells_faces_connectivity_matrix[cell_index]
local_faces_indices = cells_faces_connectivity_matrix[cell_index]
for local_face_index, global_face_index in enumerate(
local_faces_indices):
face = faces[global_face_index]
face_reference_frame_transformation_matrix = Operator.get_face_passmat(
local_cell, face)
# for boundary_name, nset in zip(nsets, nsets.values()):
for boundary_name, nset in zip(nsets_faces, nsets_faces.values()):
if connectivity[local_face_index] in nset:
pressure = boundary_conditions[boundary_name][1]
pressure_vector = Pressure(
face, face_basis_k,
face_reference_frame_transformation_matrix, unknown,
pressure).pressure_vector
else:
pressure_vector = Pressure(
face, face_basis_k,
face_reference_frame_transformation_matrix,
unknown).pressure_vector
l0 = local_face_index * unknown.field_dimension * face_basis_k.basis_dimension
l1 = (local_face_index +
1) * unknown.field_dimension * face_basis_k.basis_dimension
local_external_forces[l0:l1] += pressure_vector
# --------------------------------------------------------------------------------------------------------------
# Stffness matrix
# --------------------------------------------------------------------------------------------------------------
tangent_matrix = tangent_matrices[cell_index]
tangent_matrix_size = tangent_matrix.shape[0]
total_size = tangent_matrix_size * cell_basis_k.basis_dimension
local_mass_matrix = np.zeros((total_size, total_size))
m_phi_phi_cell = Integration.get_cell_mass_matrix_in_cell(
local_cell, cell_basis_k, cell_basis_k)
for i in range(tangent_matrix.shape[0]):
for j in range(tangent_matrix.shape[1]):
m = tangent_matrix[i, j] * m_phi_phi_cell
# ------------------------------------------------------------------------------------------------------
l0 = i * cell_basis_k.basis_dimension
l1 = (i + 1) * cell_basis_k.basis_dimension
c0 = j * cell_basis_k.basis_dimension
c1 = (j + 1) * cell_basis_k.basis_dimension
# ------------------------------------------------------------------------------------------------------
local_mass_matrix[l0:l1, c0:c1] += m
local_gradient_operator = operators[cell_index].local_gradient_operator
# print(local_gradient_operator.shape)
local_stiffness_form = local_gradient_operator.T @ local_mass_matrix @ local_gradient_operator
# --------------------------------------------------------------------------------------------------------------
# Stabilization matrix
# --------------------------------------------------------------------------------------------------------------
local_stabilization_operator = stabilization_parameter * operators[
cell_index].local_stabilization_operator
# --------------------------------------------------------------------------------------------------------------
# Local matrix
# --------------------------------------------------------------------------------------------------------------
local_matrix = local_stiffness_form + local_stabilization_operator
# --------------------------------------------------------------------------------------------------------------
# Static condensation
# --------------------------------------------------------------------------------------------------------------
(
m_cell_cell_inv,
m_cell_faces,
m_faces_cell,
m_faces_faces,
v_cell,
v_faces,
) = Condensation.get_system_decomposition(local_matrix,
local_external_forces,
unknown, cell_basis_l)
# --------------------------------------------------------------------------------------------------------------
# Static condensation
# --------------------------------------------------------------------------------------------------------------
m_cond, v_cond = Condensation.get_condensated_system(
m_cell_cell_inv,
m_cell_faces,
m_faces_cell,
m_faces_faces,
v_cell,
v_faces,
)
v_cell, m_cell_faces, m_cell_cell_inv
stored_matrices.append((v_cell, m_cell_faces, m_cell_cell_inv))
# --------------------------------------------------------------------------------------------------------------
# Assembly
# --------------------------------------------------------------------------------------------------------------
cell_faces_connectivity_matrix = cells_faces_connectivity_matrix[
cell_index]
for local_index_col, global_index_col in enumerate(
cell_faces_connectivity_matrix):
g0c = global_index_col * face_basis_k.basis_dimension * unknown.field_dimension
g1c = (global_index_col +
1) * face_basis_k.basis_dimension * unknown.field_dimension
l0c = local_index_col * face_basis_k.basis_dimension * unknown.field_dimension
l1c = (local_index_col +
1) * face_basis_k.basis_dimension * unknown.field_dimension
global_vector[g0c:g1c] += v_cond[l0c:l1c]
for local_index_row, global_index_row in enumerate(
cell_faces_connectivity_matrix):
g0r = global_index_row * face_basis_k.basis_dimension * unknown.field_dimension
g1r = (
global_index_row +
1) * face_basis_k.basis_dimension * unknown.field_dimension
l0r = local_index_row * face_basis_k.basis_dimension * unknown.field_dimension
l1r = (
local_index_row +
1) * face_basis_k.basis_dimension * unknown.field_dimension
global_matrix[g0r:g1r, g0c:g1c] += m_cond[l0r:l1r, l0c:l1c]
# ------------------------------------------------------------------------------------------------------------------
# Displacement enforcement through Lagrange multiplier
# ------------------------------------------------------------------------------------------------------------------
number_of_constrained_faces = 0
# for boundary_name, nset in zip(nsets, nsets.values()):
for boundary_name, nset in zip(nsets_faces, nsets_faces.values()):
# print(boundary_name, nset)
displacement = boundary_conditions[boundary_name][0]
for displacement_component in displacement:
if not displacement_component is None:
number_of_constrained_faces += len(nset)
lagrange_multiplyer_matrix = np.zeros((
number_of_constrained_faces * face_basis_k.basis_dimension,
number_of_faces * unknown.field_dimension *
face_basis_k.basis_dimension,
))
h_vector = np.zeros(
(number_of_constrained_faces * face_basis_k.basis_dimension, ))
iter_constrained_face = 0
# for boundary_name, nset in zip(nsets, nsets.values()):
for boundary_name, nset in zip(nsets_faces, nsets_faces.values()):
for face_global_index in nset:
face = faces[face_global_index]
face_reference_frame_transformation_matrix = face.reference_frame_transformation_matrix
# ----------------------------------------------------------------------------------------------------------
m_psi_psi_face = Integration.get_face_mass_matrix_in_face(
face, face_basis_k, face_basis_k,
face_reference_frame_transformation_matrix)
m_psi_psi_face_inv = np.linalg.inv(m_psi_psi_face)
# ----------------------------------------------------------------------------------------------------------
displacement = boundary_conditions[boundary_name][0]
for direction, displacement_component in enumerate(displacement):
if not displacement_component is None:
displacement_vector = Displacement(
face, face_basis_k,
face_reference_frame_transformation_matrix,
displacement_component).displacement_vector
# if False:
# print("hello")
# displacement_vector = np.zeros((face_basis.basis_dimension,))
# if displacement_component(0) != 0:
# displacement_vector[0] = 1.0
# print(displacement_vector)
# --------------------------------------------------------------------------------------------------
displacement_vector = m_psi_psi_face_inv @ displacement_vector
# --------------------------------------------------------------------------------------------------
l0 = iter_constrained_face * face_basis_k.basis_dimension
l1 = (iter_constrained_face +
1) * face_basis_k.basis_dimension
c0 = (face_global_index * unknown.field_dimension *
face_basis_k.basis_dimension +
direction * face_basis_k.basis_dimension)
c1 = (face_global_index * unknown.field_dimension *
face_basis_k.basis_dimension
) + (direction + 1) * face_basis_k.basis_dimension
# --------------------------------------------------------------------------------------------------
lagrange_multiplyer_matrix[l0:l1, c0:c1] += np.eye(
face_basis_k.basis_dimension)
# --------------------------------------------------------------------------------------------------
h_vector[l0:l1] += displacement_vector
iter_constrained_face += 1
# print("h_vector : \n{}".format(h_vector))
# print("lagrange_multiplyer_matrix : \n{}".format(lagrange_multiplyer_matrix))
# ------------------------------------------------------------------------------------------------------------------
double_lagrange = False
# ------------------------------------------------------------------------------------------------------------------
# If a single Lagrange multiplyier is used to enforce Dirichlet boundary conditions
# ------------------------------------------------------------------------------------------------------------------
if not double_lagrange:
global_vector_2 = np.zeros(
(total_system_size +
number_of_constrained_faces * face_basis_k.basis_dimension, ))
# --------------------------------------------------------------------------------------------------------------
global_vector_2[:total_system_size] += global_vector
global_vector_2[total_system_size:] += h_vector
# --------------------------------------------------------------------------------------------------------------
global_matrix_2 = np.zeros((
total_system_size +
number_of_constrained_faces * face_basis_k.basis_dimension,
total_system_size +
number_of_constrained_faces * face_basis_k.basis_dimension,
))
# --------------------------------------------------------------------------------------------------------------
global_matrix_2[:total_system_size, :
total_system_size] += global_matrix
global_matrix_2[:total_system_size,
total_system_size:] += lagrange_multiplyer_matrix.T
global_matrix_2[total_system_size:, :
total_system_size] += lagrange_multiplyer_matrix
# ------------------------------------------------------------------------------------------------------------------
# If double Lagrange multiplyiers are used to enforce Dirichlet boundary conditions
# ------------------------------------------------------------------------------------------------------------------
else:
global_vector_2 = np.zeros(
(total_system_size + 2 *
(number_of_constrained_faces * face_basis_k.basis_dimension), ))
# --------------------------------------------------------------------------------------------------------------
l0 = 0
l1 = total_system_size
global_vector_2[l0:l1] += global_vector
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size
l1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_vector_2[l0:l1] += h_vector
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
l1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_vector_2[l0:l1] += h_vector
# --------------------------------------------------------------------------------------------------------------
global_matrix_2 = np.zeros((
total_system_size + 2 *
(number_of_constrained_faces * face_basis_k.basis_dimension),
total_system_size + 2 *
(number_of_constrained_faces * face_basis_k.basis_dimension),
))
# --------------------------------------------------------------------------------------------------------------
l0 = 0
l1 = total_system_size
c0 = 0
c1 = total_system_size
global_matrix_2[l0:l1, c0:c1] += global_matrix
# --------------------------------------------------------------------------------------------------------------
l0 = 0
l1 = total_system_size
c0 = total_system_size
c1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] += lagrange_multiplyer_matrix.T
# --------------------------------------------------------------------------------------------------------------
l0 = 0
l1 = total_system_size
c0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] += lagrange_multiplyer_matrix.T
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size
l1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = 0
c1 = total_system_size
global_matrix_2[l0:l1, c0:c1] += lagrange_multiplyer_matrix
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
l1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = 0
c1 = total_system_size
global_matrix_2[l0:l1, c0:c1] += lagrange_multiplyer_matrix
# --------------------------------------------------------------------------------------------------------------
id_lag = np.eye(number_of_constrained_faces *
face_basis_k.basis_dimension)
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size
l1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = total_system_size
c1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] += id_lag
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size
l1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] -= id_lag
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
l1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = total_system_size
c1 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] -= id_lag
# --------------------------------------------------------------------------------------------------------------
l0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
l1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
c0 = total_system_size + (number_of_constrained_faces *
face_basis_k.basis_dimension)
c1 = total_system_size + 2 * (number_of_constrained_faces *
face_basis_k.basis_dimension)
global_matrix_2[l0:l1, c0:c1] += id_lag
# print("global_matrix_2 : \n{}".format(global_matrix_2))
# print("global_vector_2 : \n{}".format(global_vector_2))
# ------------------------------------------------------------------------------------------------------------------
# Solving the global system for faces unknowns
# ------------------------------------------------------------------------------------------------------------------
# print("global_matrix : \n{}".format(global_matrix_2))
global_solution = np.linalg.solve(global_matrix_2, global_vector_2)
# print("global_solution : \n{}".format(global_solution))
# ------------------------------------------------------------------------------------------------------------------
# Getting the number of vertices
# ------------------------------------------------------------------------------------------------------------------
number_of_quadrature_points = 0
for cell in cells:
number_of_quadrature_points_in_cell = cell.quadrature_points.shape[0]
number_of_quadrature_points += number_of_quadrature_points_in_cell
number_of_vertices = vertices.shape[0]
# ------------------------------------------------------------------------------------------------------------------
quadrature_points = np.zeros(
(number_of_quadrature_points, unknown.problem_dimension))
quadrature_weights = np.zeros((number_of_quadrature_points, ))
unknowns_at_vertices = [
np.zeros((number_of_vertices, ))
for i in range(unknown.field_dimension)
]
f_unknowns_at_vertices = [
np.zeros((number_of_vertices, ))
for i in range(unknown.field_dimension)
]
unknowns_at_quadrature_points = [
np.zeros((number_of_quadrature_points, ))
for i in range(unknown.field_dimension)
]
vertices_weights = np.zeros((number_of_vertices, ))
f_vertices_weights = np.zeros((number_of_vertices, ))
# ------------------------------------------------------------------------------------------------------------------
# Desassembly
# ------------------------------------------------------------------------------------------------------------------
quadrature_point_count = 0
# ==================================================================================================================
# ==================================================================================================================
faces_indices = range(number_of_faces)
for face_index in faces_indices:
face_vertices_connectivity_matrix = faces_vertices_connectivity_matrix[
face_index]
face_vertices = vertices[face_vertices_connectivity_matrix]
face = faces[face_index]
l0 = face_index * unknown.field_dimension * face_basis_k.basis_dimension
l1 = (face_index +
1) * unknown.field_dimension * face_basis_k.basis_dimension
x_face = global_solution[l0:l1]
for i, vertex in enumerate(face_vertices):
vertex_in_face = Face.get_points_in_face_reference_frame(
vertex, face.reference_frame_transformation_matrix)
centroid_in_face = Face.get_points_in_face_reference_frame(
face.centroid, face.reference_frame_transformation_matrix)
v = face_basis_k.get_phi_vector(vertex_in_face, centroid_in_face,
face.diameter)
for direction in range(unknown.field_dimension):
l0 = direction * face_basis_k.basis_dimension
l1 = (direction + 1) * face_basis_k.basis_dimension
vertex_value_vector = v * x_face[l0:l1]
global_index = face_vertices_connectivity_matrix[i]
l0 = global_index
l1 = global_index + 1
f_unknowns_at_vertices[direction][l0:l1] += np.sum(
vertex_value_vector)
f_vertices_weights[l0:l1] += 1.0
# ==================================================================================================================
# ==================================================================================================================
x_cell_list, x_faces_list = [], []
for cell_index in cells_indices:
# --------------------------------------------------------------------------------------------------------------
# Getting faces unknowns
# --------------------------------------------------------------------------------------------------------------
local_cell = cells[cell_index]
cell_faces_connectivity_matrix = cells_faces_connectivity_matrix[
cell_index]
local_faces = [faces[i] for i in cell_faces_connectivity_matrix]
faces_unknown_dimension = len(
local_faces
) * face_basis_k.basis_dimension * unknown.field_dimension
x_faces = np.zeros((faces_unknown_dimension, ))
for local_index_col, global_index_col in enumerate(
cell_faces_connectivity_matrix):
g0c = global_index_col * face_basis_k.basis_dimension * unknown.field_dimension
g1c = (global_index_col +
1) * face_basis_k.basis_dimension * unknown.field_dimension
l0c = local_index_col * face_basis_k.basis_dimension * unknown.field_dimension
l1c = (local_index_col +
1) * face_basis_k.basis_dimension * unknown.field_dimension
x_faces[l0c:l1c] += global_solution[g0c:g1c]
# --------------------------------------------------------------------------------------------------------------
# Decondensation : getting cell unknowns
# --------------------------------------------------------------------------------------------------------------
(v_cell, m_cell_faces, m_cell_cell_inv) = stored_matrices[cell_index]
x_cell = Condensation.get_cell_unknown(m_cell_cell_inv, m_cell_faces,
v_cell, x_faces)
x_cell_list.append(x_cell)
x_faces_list.append(x_faces)
x_element = np.zeros((len(x_cell) + len(x_faces), ))
p1 = cell_basis_l.basis_dimension * unknown.field_dimension
x_element[:p1] += x_cell
x_element[p1:] += x_faces
# operators[cell_index]
# --------------------------------------------------------------------------------------------------------------
cell_vertices_connectivity_matrix = cells_vertices_connectivity_matrix[
cell_index]
local_vertices = vertices[cell_vertices_connectivity_matrix]
# --------------------------------------------------------------------------------------------------------------
for i, vertex in enumerate(local_vertices):
v = cell_basis_l.get_phi_vector(vertex, local_cell.centroid,
local_cell.diameter)
for direction in range(unknown.field_dimension):
l0 = direction * cell_basis_l.basis_dimension
l1 = (direction + 1) * cell_basis_l.basis_dimension
vertex_value_vector = v * x_cell[l0:l1]
global_index = cell_vertices_connectivity_matrix[i]
l0 = global_index
l1 = global_index + 1
unknowns_at_vertices[direction][l0:l1] += np.sum(
vertex_value_vector)
vertices_weights[l0:l1] += 1.0
# --------------------------------------------------------------------------------------------------------------
for i, quadrature_point in enumerate(local_cell.quadrature_points):
v = cell_basis_l.get_phi_vector(quadrature_point,
local_cell.centroid,
local_cell.diameter)
for direction in range(unknown.field_dimension):
l0 = direction * cell_basis_l.basis_dimension
l1 = (direction + 1) * cell_basis_l.basis_dimension
vertex_value_vector = v * x_cell[l0:l1]
l0 = quadrature_point_count
l1 = quadrature_point_count + 1
unknowns_at_quadrature_points[direction][l0:l1] += np.sum(
vertex_value_vector)
quadrature_points[quadrature_point_count] += quadrature_point
quadrature_weights[
quadrature_point_count] += local_cell.quadrature_weights[i]
quadrature_point_count += 1
# ------------------------------------------------------------------------------------------------------------------
# Global matrix
# ------------------------------------------------------------------------------------------------------------------
for direction in range(unknown.field_dimension):
unknowns_at_vertices[
direction] = unknowns_at_vertices[direction] / vertices_weights
f_unknowns_at_vertices[
direction] = f_unknowns_at_vertices[direction] / f_vertices_weights
return (
(vertices, unknowns_at_vertices),
(quadrature_points, unknowns_at_quadrature_points, quadrature_weights),
(vertices, f_unknowns_at_vertices),
(x_cell_list, x_faces_list),
# unknowns_at_faces
)
def __init__(
self,
cell_vertices_connectivity_matrix: Mat,
cell: Cell,
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
):
"""
================================================================================================================
Class :
================================================================================================================
The Mcell class
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Attributes :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
mat_T_phi_k_phi_k_inv = np.linalg.inv(mat_T_phi_k_phi_k)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_grad_phi_l_x = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, 0)
mat_T_phi_k_grad_phi_l_y = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, 1)
mat_T_phi_k_grad_phi_l_list = [
mat_T_phi_k_grad_phi_l_x, mat_T_phi_k_grad_phi_l_y
]
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_grad_phi_k1_grad_phi_k1_x = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, 0)
mat_T_grad_phi_k1_grad_phi_k1_y = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, 1)
mat_T_grad_phi_k1_grad_phi_k1_inv_list = [
mat_T_grad_phi_k1_grad_phi_k1_x, mat_T_grad_phi_k1_grad_phi_k1_y
]
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_l_grad_phi_k1_x = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1, 0)
mat_T_phi_l_grad_phi_k1_y = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1, 1)
mat_T_grad_phi_k1_phi_l_list = [
mat_T_phi_l_grad_phi_k1_x, mat_T_phi_l_grad_phi_k1_y
]
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
mat_T_phi_k_phi_k1 = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k1)
proj_T_T_k1_k = mat_T_phi_k_phi_k_inv @ mat_T_phi_k_phi_k1
# ==============================================================================================================
# PROJECTION OPERATOR
# ==============================================================================================================
mat_T_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
mat_T_phi_l_phi_l_inv = np.linalg.inv(mat_T_phi_l_phi_l)
def get_local_gradient_operator(
self,
cell: Cell,
faces: List[Face],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
face_basis_k: Basis,
unknown: Unknown,
) -> Mat:
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# --------------------------------------------------------------------------------------------------------------
# Getting the local problem size depending on the problem dimension and the field dimension
# --------------------------------------------------------------------------------------------------------------
local_problem_size = self.get_local_problem_size(
faces, cell_basis_l, face_basis_k, unknown)
# --------------------------------------------------------------------------------------------------------------
# Initializing the local gradient operator matrix
# --------------------------------------------------------------------------------------------------------------
local_gradient_operator = np.zeros(
(len(unknown.indices) * cell_basis_k.basis_dimension,
local_problem_size))
# --------------------------------------------------------------------------------------------------------------
m_cell_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
m_cell_phi_k_phi_k_inv = np.linalg.inv(m_cell_phi_k_phi_k)
# --------------------------------------------------------------------------------------------------------------
derivative_directions = range(unknown.problem_dimension)
directions = range(unknown.field_dimension)
for j in derivative_directions:
# ----------------------------------------------------------------------------------------------------------
# Compting matrices 1
# ----------------------------------------------------------------------------------------------------------
m_cell_phi_k_grad_phi_l = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, j)
for f, face in enumerate(faces):
passmat = Operator.get_face_passmat(cell, face)
normal_vector_component = passmat[-1, j]
# m_face_phi_l_psi_k = Integration.get_hybrid_mass_matrix_in_face(
# cell, face, cell_basis_l, face_basis_k, passmat
# )
m_face_phi_k_psi_k = Integration.get_hybrid_mass_matrix_in_face(
cell, face, cell_basis_k, face_basis_k, passmat)
m_face_phi_k_phi_l = Integration.get_cell_mass_matrix_in_face(
cell, face, cell_basis_k, cell_basis_l)
for i in directions:
# --------------------------------------------------------------------------------------------------
# bkbjkj
# --------------------------------------------------------------------------------------------------
line = self.get_line_from_indices(i, j, unknown)
r0 = line * cell_basis_k.basis_dimension
r1 = (line + 1) * cell_basis_k.basis_dimension
# --------------------------------------------------------------------------------------------------
c0_T = i * cell_basis_l.basis_dimension
c1_T = (i + 1) * cell_basis_l.basis_dimension
# --------------------------------------------------------------------------------------------------
# Getting the indices corresponding to the face_index-th face for the field on the ith axis
# --------------------------------------------------------------------------------------------------
c0_F = (unknown.field_dimension *
cell_basis_l.basis_dimension +
f * unknown.field_dimension *
face_basis_k.basis_dimension +
i * face_basis_k.basis_dimension)
c1_F = (unknown.field_dimension *
cell_basis_l.basis_dimension +
f * unknown.field_dimension *
face_basis_k.basis_dimension +
(i + 1) * face_basis_k.basis_dimension)
# --------------------------------------------------------------------------------------------------
# grad
# --------------------------------------------------------------------------------------------------
# local_gradient_operator[r0:r1, c0_T:c1_T] += m_cell_phi_k_grad_phi_l
local_gradient_operator[
r0:r1, c0_F:
c1_F] += m_face_phi_k_psi_k * normal_vector_component
local_gradient_operator[
r0:r1, c0_T:
c1_T] -= m_face_phi_k_phi_l * normal_vector_component
for i in directions:
line = self.get_line_from_indices(i, j, unknown)
r0 = line * cell_basis_k.basis_dimension
r1 = (line + 1) * cell_basis_k.basis_dimension
# ------------------------------------------------------------------------------------------------------
c0_T = i * cell_basis_l.basis_dimension
c1_T = (i + 1) * cell_basis_l.basis_dimension
local_gradient_operator[r0:r1,
c0_T:c1_T] += m_cell_phi_k_grad_phi_l
local_gradient_operator[
r0:
r1, :] = m_cell_phi_k_phi_k_inv @ local_gradient_operator[
r0:r1, :]
return local_gradient_operator
def get_stabilization_operator(
self,
cell: Cell,
faces: List[Face],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
face_basis_k: Basis,
unknown: Unknown,
) -> Mat:
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# --------------------------------------------------------------------------------------------------------------
# rec
# --------------------------------------------------------------------------------------------------------------
local_reconstruction_operator = self.get_local_reconstruction_operator(
cell, faces, cell_basis_l, cell_basis_k, cell_basis_k1,
face_basis_k, unknown)
# --------------------------------------------------------------------------------------------------------------
# Initializing the local stabilization form matrix
# --------------------------------------------------------------------------------------------------------------
local_problem_size = self.get_local_problem_size(
faces, cell_basis_l, face_basis_k, unknown)
local_stabilization_operator = np.zeros(
(local_problem_size, local_problem_size))
derivative_directions = range(unknown.problem_dimension)
directions = range(unknown.field_dimension)
for i in directions:
# # ------------------------------------------------------------------------------------------------------
# r0_r = i * cell_basis_k1.basis_dimension
# r1_r = (i + 1) * cell_basis_k1.basis_dimension
# ------------------------------------------------------------------------------------------------------
c0_c = i * cell_basis_l.basis_dimension
c1_c = (i + 1) * cell_basis_l.basis_dimension
for f, face in enumerate(faces):
passmat = Operator.get_face_passmat(cell, face)
# normal_vector_component = passmat[-1, j]
# --------------------------------------------------------------------------------------------------
# Getting the indices corresponding to the face_indexth face for the field on the ith axis
# --------------------------------------------------------------------------------------------------
c0_f = (
unknown.field_dimension * cell_basis_l.basis_dimension +
f * unknown.field_dimension * face_basis_k.basis_dimension
+ i * face_basis_k.basis_dimension)
c1_f = (
unknown.field_dimension * cell_basis_l.basis_dimension +
f * unknown.field_dimension * face_basis_k.basis_dimension
+ (i + 1) * face_basis_k.basis_dimension)
# ------------------------------------------------------------------------------------------------------
# Compting matrices 1
# ------------------------------------------------------------------------------------------------------
m_face_id = np.eye(face_basis_k.basis_dimension)
m_face_psi_k_psi_k = Integration.get_face_mass_matrix_in_face(
face, face_basis_k, face_basis_k, passmat)
m_face_psi_k_psi_k_inv = np.linalg.inv(m_face_psi_k_psi_k)
# ------------------------------------------------------------------------------------------------------
m_cell_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
m_cell_phi_k_phi_k_inv = np.linalg.inv(m_cell_phi_k_phi_k)
m_cell_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
m_cell_phi_l_phi_l_inv = np.linalg.inv(m_cell_phi_l_phi_l)
# ------------------------------------------------------------------------------------------------------
# proj T -> F, l -> k
# ------------------------------------------------------------------------------------------------------
m_face_phi_l_psi_k = Integration.get_hybrid_mass_matrix_in_face(
cell, face, cell_basis_l, face_basis_k, passmat)
m_face_psi_k_phi_l = m_face_phi_l_psi_k.T
pi_c_f_l_k = m_face_psi_k_psi_k_inv @ m_face_psi_k_phi_l
# ------------------------------------------------------------------------------------------------------
# proj T -> F, k -> k
# ------------------------------------------------------------------------------------------------------
m_face_phi_k_psi_k = Integration.get_hybrid_mass_matrix_in_face(
cell, face, cell_basis_k, face_basis_k, passmat)
m_face_psi_k_phi_k = m_face_phi_k_psi_k.T
pi_c_f_k_k = m_face_psi_k_psi_k_inv @ m_face_psi_k_phi_k
# ------------------------------------------------------------------------------------------------------
# proj T -> F, k1 -> k
# ------------------------------------------------------------------------------------------------------
m_face_phi_k1_psi_k = Integration.get_hybrid_mass_matrix_in_face(
cell, face, cell_basis_k1, face_basis_k, passmat)
m_face_psi_k_phi_k1 = m_face_phi_k1_psi_k.T
pi_c_f_k1_k = m_face_psi_k_psi_k_inv @ m_face_psi_k_phi_k1
# ------------------------------------------------------------------------------------------------------
# proj T -> T, k1 -> l
# ------------------------------------------------------------------------------------------------------
m_cell_phi_l_phi_k1 = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1)
pi_c_c_k1_l = m_cell_phi_l_phi_l_inv @ m_cell_phi_l_phi_k1
# ------------------------------------------------------------------------------------------------------
m_stab_face_jump = np.zeros(
(face_basis_k.basis_dimension, local_problem_size))
m_stab_face_jump[:, c0_c:c1_c] -= pi_c_f_l_k
m_stab_face_jump[:, c0_f:c1_f] += m_face_id
# ------------------------------------------------------------------------------------------------------
c0_r = i * cell_basis_k1.basis_dimension
c1_r = (i + 1) * cell_basis_k1.basis_dimension
m_stab_face_jump -= pi_c_f_k1_k @ local_reconstruction_operator[
c0_r:c1_r, :]
m_stab_face_jump += pi_c_f_l_k @ (
pi_c_c_k1_l @ local_reconstruction_operator[c0_r:c1_r, :])
h_f = 1.0 / face.diameter
local_stabilization_operator += h_f * m_stab_face_jump.T @ m_face_psi_k_psi_k_inv @ m_stab_face_jump
return local_stabilization_operator
def get_local_reconstruction_operator(
self,
cell: Cell,
faces: List[Face],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
face_basis_k: Basis,
unknown: Unknown,
) -> Mat:
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# --------------------------------------------------------------------------------------------------------------
# Getting the local problem size depending on the problem dimension and the field dimension
# --------------------------------------------------------------------------------------------------------------
local_problem_size = self.get_local_problem_size(
faces, cell_basis_l, face_basis_k, unknown)
# --------------------------------------------------------------------------------------------------------------
# Initializing the local gradient operator matrix
# --------------------------------------------------------------------------------------------------------------
local_reconstruction_operator = np.zeros(
((unknown.field_dimension) * cell_basis_k1.basis_dimension,
local_problem_size))
# --------------------------------------------------------------------------------------------------------------
derivative_directions = range(unknown.problem_dimension)
directions = range(unknown.field_dimension)
m_cell_grad_phi_k1_grad_phi_k1_sum = np.zeros(
(cell_basis_k1.basis_dimension, cell_basis_k1.basis_dimension))
m_cell_grad_phi_k1_grad_phi_l_sum = np.zeros(
(cell_basis_k1.basis_dimension, cell_basis_l.basis_dimension))
for j in derivative_directions:
m_cell_grad_phi_k1_grad_phi_k1 = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, j)
m_cell_grad_phi_k1_grad_phi_k1_sum += m_cell_grad_phi_k1_grad_phi_k1
m_cell_grad_phi_k1_grad_phi_l = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_l, j)
m_cell_grad_phi_k1_grad_phi_l_sum += m_cell_grad_phi_k1_grad_phi_l
for f, face in enumerate(faces):
passmat = Operator.get_face_passmat(cell, face)
normal_vector_component = passmat[-1, j]
m_face_grad_phi_k1_phi_l_sum = np.zeros(
(cell_basis_k1.basis_dimension, cell_basis_l.basis_dimension))
m_face_grad_phi_k1_psi_k_sum = np.zeros(
(cell_basis_k1.basis_dimension, face_basis_k.basis_dimension))
for j in derivative_directions:
m_face_grad_phi_k1_psi_k = Integration.get_hybrid_advection_matrix_in_face(
cell, face, cell_basis_k1, face_basis_k, passmat, j)
m_face_grad_phi_k1_psi_k_sum += m_face_grad_phi_k1_psi_k
m_face_phi_l_grad_phi_k1 = Integration.get_cell_advection_matrix_in_face(
cell, face, cell_basis_l, cell_basis_k1, j)
m_face_grad_phi_k1_phi_l = m_face_phi_l_grad_phi_k1.T
m_face_grad_phi_k1_phi_l_sum += m_face_grad_phi_k1_phi_l
for i in directions:
# --------------------------------------------------------------------------------------------------
# bkbjkj
# --------------------------------------------------------------------------------------------------
r0 = i * cell_basis_k1.basis_dimension
r1 = (i + 1) * cell_basis_k1.basis_dimension
# --------------------------------------------------------------------------------------------------
# Getting the indices corresponding to the face_index-th face for the field on the ith axis
# --------------------------------------------------------------------------------------------------
c0_F = (
unknown.field_dimension * cell_basis_l.basis_dimension +
f * unknown.field_dimension * face_basis_k.basis_dimension
+ i * face_basis_k.basis_dimension)
c1_F = (
unknown.field_dimension * cell_basis_l.basis_dimension +
f * unknown.field_dimension * face_basis_k.basis_dimension
+ (i + 1) * face_basis_k.basis_dimension)
c0_T = i * cell_basis_l.basis_dimension
c1_T = (i + 1) * cell_basis_l.basis_dimension
# --------------------------------------------------------------------------------------------------
# rec
# --------------------------------------------------------------------------------------------------
local_reconstruction_operator[r0:r1, c0_F:c1_F] += (
m_face_grad_phi_k1_psi_k_sum * normal_vector_component)
local_reconstruction_operator[r0:r1, c0_T:c1_T] -= (
m_face_grad_phi_k1_phi_l_sum * normal_vector_component)
for i in directions:
r0 = i * cell_basis_k1.basis_dimension
r1 = (i + 1) * cell_basis_k1.basis_dimension
m_cell_grad_phi_k1_grad_phi_k1_sum_inv = np.linalg.inv(
m_cell_grad_phi_k1_grad_phi_k1_sum[1:, 1:])
local_reconstruction_operator[r0 + 1:r1, :] = (
m_cell_grad_phi_k1_grad_phi_k1_sum_inv
@ local_reconstruction_operator[r0 + 1:r1, :])
# ------------------------------------------------------------------------------------------------------
cell_integration_vector_k1 = Integration.get_cell_integration_vector(
cell, cell_basis_k1)
cell_integration_vector_l = Integration.get_cell_integration_vector(
cell, cell_basis_l)
# ------------------------------------------------------------------------------------------------------
m_mat = np.zeros(
(cell_basis_l.basis_dimension, local_problem_size))
c0_m = i * cell_basis_l.basis_dimension
c1_m = (i + 1) * cell_basis_l.basis_dimension
m_mat[:, c0_m:c1_m] += np.eye(cell_basis_l.basis_dimension)
# ------------------------------------------------------------------------------------------------------
a = (cell_integration_vector_l @ m_mat -
cell_integration_vector_k1[1:]
@ local_reconstruction_operator[r0 + 1:r1, :])
a = (1.0 / cell_integration_vector_k1[0]) * a
local_reconstruction_operator[r0:r0 + 1, :] = a
# for i in directions:
# r0 = i * cell_basis_k1.basis_dimension
# r1 = (i + 1) * cell_basis_k1.basis_dimension
# c0_T = i * cell_basis_l.basis_dimension
# c1_T = (i + 1) * cell_basis_l.basis_dimension
# #
# local_reconstruction_operator[r0:r1, c0_T:c1_T] += m_cell_grad_phi_k1_grad_phi_l_sum
# local_reconstruction_operator[r0:r1, :] = local_reconstruction_operator[r0:r1, :]
# for j in derivative_directions:
# # ------------------------------------------------------------------------------------------------------
# # Compting matrices 1
# # ------------------------------------------------------------------------------------------------------
# m_cell_grad_phi_k1_grad_phi_k1 = Integration.get_cell_stiffness_matrix_in_cell(
# cell, cell_basis_k1, cell_basis_k1, j
# )
# m_cell_grad_phi_k1_grad_phi_k1_sum += m_cell_grad_phi_k1_grad_phi_k1
# m_cell_grad_phi_k1_grad_phi_l = Integration.get_cell_stiffness_matrix_in_cell(
# cell, cell_basis_k1, cell_basis_l, j
# )
# for f, face in enumerate(faces):
# passmat = Operator.get_face_passmat(cell, face)
# normal_vector_component = passmat[-1, j]
# m_face_grad_phi_k1_psi_k
# m_face_grad_phi_k1_psi_k = Integration.get_hybrid_advection_matrix_in_face(
# cell, face, cell_basis_k1, face_basis_k, passmat, j
# )
# # m_face_grad_phi_k1_phi_l = Integration.get_cell_advection_matrix_in_face(
# # cell, face, cell_basis_k1, cell_basis_l, j
# # )
# m_face_phi_l_grad_phi_k1 = Integration.get_cell_advection_matrix_in_face(
# cell, face, cell_basis_l, cell_basis_k1, j
# )
# m_face_grad_phi_k1_phi_l = m_face_phi_l_grad_phi_k1.T
# for i in directions:
# # --------------------------------------------------------------------------------------------------
# # bkbjkj
# # --------------------------------------------------------------------------------------------------
# r0 = i * cell_basis_k1.basis_dimension
# r1 = (i + 1) * cell_basis_k1.basis_dimension
# # --------------------------------------------------------------------------------------------------
# c0_T = i * cell_basis_l.basis_dimension
# c1_T = (i + 1) * cell_basis_l.basis_dimension
# # --------------------------------------------------------------------------------------------------
# # Getting the indices corresponding to the face_index-th face for the field on the ith axis
# # --------------------------------------------------------------------------------------------------
# c0_F = (
# unknown.field_dimension * cell_basis_l.basis_dimension
# + f * unknown.field_dimension * face_basis_k.basis_dimension
# + i * face_basis_k.basis_dimension
# )
# c1_F = (
# unknown.field_dimension * cell_basis_l.basis_dimension
# + f * unknown.field_dimension * face_basis_k.basis_dimension
# + (i + 1) * face_basis_k.basis_dimension
# )
# # --------------------------------------------------------------------------------------------------
# # rec
# # --------------------------------------------------------------------------------------------------
# local_reconstruction_operator[r0:r1, c0_F:c1_F] += (
# m_face_grad_phi_k1_psi_k * normal_vector_component
# )
# local_reconstruction_operator[r0:r1, c0_T:c1_T] -= (
# m_face_grad_phi_k1_phi_l * normal_vector_component
# )
# for i in directions:
# r0 = i * cell_basis_k1.basis_dimension
# r1 = (i + 1) * cell_basis_k1.basis_dimension
# # ------------------------------------------------------------------------------------------------------
# c0_T = i * cell_basis_l.basis_dimension
# c1_T = (i + 1) * cell_basis_l.basis_dimension
# local_reconstruction_operator[r0:r1, c0_T:c1_T] += m_cell_grad_phi_k1_grad_phi_l
# # --------------------------------------------------------------------------------------------------------------
# for i in directions:
# # a_sum = np.zeros((local_problem_size,))
# # for i_sum in directions:
# r0 = i * cell_basis_k1.basis_dimension
# r1 = (i + 1) * cell_basis_k1.basis_dimension
# # print("grad_matrx : \n{}".format(m_cell_grad_phi_k1_grad_phi_k1_sum[1:, 1:]))
# m_cell_grad_phi_k1_grad_phi_k1_sum_inv = np.linalg.inv(m_cell_grad_phi_k1_grad_phi_k1_sum[1:, 1:])
# local_reconstruction_operator[r0 + 1 : r1, :] = (
# m_cell_grad_phi_k1_grad_phi_k1_sum_inv @ local_reconstruction_operator[r0 + 1 : r1, :]
# )
# # ------------------------------------------------------------------------------------------------------
# cell_integration_vector_k1 = Integration.get_cell_integration_vector(cell, cell_basis_k1)
# cell_integration_vector_l = Integration.get_cell_integration_vector(cell, cell_basis_l)
# # ------------------------------------------------------------------------------------------------------
# m_mat = np.zeros((cell_basis_l.basis_dimension, local_problem_size))
# c0_m = i * cell_basis_l.basis_dimension
# c1_m = (i + 1) * cell_basis_l.basis_dimension
# m_mat[:, c0_m:c1_m] += np.eye(cell_basis_l.basis_dimension)
# # ------------------------------------------------------------------------------------------------------
# a = (
# cell_integration_vector_l @ m_mat
# - cell_integration_vector_k1[1:] @ local_reconstruction_operator[r0 + 1 : r1, :]
# )
# a = (1.0 / cell_integration_vector_k1[0]) * a
# local_reconstruction_operator[r0 : r0 + 1, :] = a
return local_reconstruction_operator
def __init__(
self,
cell_faces_connectivity_matrix: Mat,
mcell: Mcell,
mfaces: List[Mface],
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
unknown: Unknown,
):
"""
================================================================================================================
Class :
================================================================================================================
The Mbox class
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Attributes :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR / STABILIZATION OPERATOR
# ==============================================================================================================
self.mat_F_phi_k_phi_l_list = []
self.mat_F_phi_k_psi_k_list = []
for mface in mfaces:
mat_F_phi_k_phi_l = Integration.get_cell_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k, cell_basis_l)
self.mat_F_phi_k_phi_l_list.append(mat_F_phi_k_phi_l_list)
# ----------------------------------------------------------------------------------------------------------
mat_F_phi_k_psi_k = Integration.get_hybrid_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k, face_basis_k,
mface.face.reference_frame_transformation_matrix)
self.mat_F_phi_k_psi_k_list.append(mat_F_phi_k_psi_k)
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
self.mat_F_grad_phi_k1_psi_k_list = []
self.mat_F_grad_phi_k1_phi_l_list = []
for mface in mfaces:
mat_F_grad_phi_k1_psi_k_j_list = []
mat_F_grad_phi_k1_phi_l_j_list = []
for j in range(len(unknown.problem_dimension)):
# ------------------------------------------------------------------------------------------------------
mat_F_grad_phi_k1_psi_k_j = Integration.get_hybrid_advection_matrix_in_face(
mcell.cell,
mface.face,
cell_basis_k1,
face_basis_k,
mface.face.reference_frame_transformation_matrix,
j,
)
mat_F_grad_phi_k1_psi_k_j_list.append(
mat_F_grad_phi_k1_psi_k_j)
# ------------------------------------------------------------------------------------------------------
mat_F_phi_l_grad_phi_k1_j = Integration.get_cell_advection_matrix_in_face(
mcell.cell, mface.face, cell_basis_l, cell_basis_k1, j)
# ------------------------------------------------------------------------------------------------------
mat_F_grad_phi_k1_phi_l_j = mat_F_phi_l_grad_phi_k1_j.T
mat_F_grad_phi_k1_phi_l_j_list.append(
mat_F_grad_phi_k1_phi_l_j)
# ----------------------------------------------------------------------------------------------------------
self.mat_F_grad_phi_k1_psi_k_list.append(
mat_F_grad_phi_k1_psi_k_j_list)
self.mat_F_grad_phi_k1_phi_l_list.append(
mat_F_grad_phi_k1_phi_l_j_list)
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
self.mat_F_psi_k_phi_k1_list = []
for mface in mfaces:
mat_F_phi_k1_psi_k = Integration.get_hybrid_mass_matrix_in_face(
mcell.cell, mface.face, cell_basis_k1, face_basis_k,
mface.face.reference_frame_transformation_matrix)
self.mat_F_psi_k_phi_k1_list.append(mat_F_phi_k1_psi_k)
def __init__(
self,
cell_vertices_connectivity_matrix: Mat,
cell: Cell,
cell_basis_l: Basis,
cell_basis_k: Basis,
cell_basis_k1: Basis,
unknown: Unknown,
):
"""
================================================================================================================
Class :
================================================================================================================
The Mcell class
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Attributes :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
self.mat_T_phi_k_phi_k_inv = np.linalg.inv(mat_T_phi_k_phi_k)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
self.mat_T_phi_k_grad_phi_l_list = []
for j in range(unknown.problem_dimension):
mat_T_phi_k_grad_phi_l = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, j)
self.mat_T_phi_k_grad_phi_l_list.append(mat_T_phi_k_grad_phi_l)
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
self.mat_T_grad_phi_k1_grad_phi_k1_inv_list = []
for j in range(unknown.problem_dimension):
mat_T_grad_phi_k1_grad_phi_k1_j = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, j)
self.mat_T_grad_phi_k1_grad_phi_k1_inv_list.append(
mat_T_grad_phi_k1_grad_phi_k1_j)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
self.mat_T_grad_phi_k1_phi_l_list = []
for j in range(unknown.problem_dimension):
mat_T_phi_l_grad_phi_k1_j = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1, j)
mat_T_grad_phi_k1_phi_l_j = mat_T_phi_l_grad_phi_k1_j.T
self.mat_T_grad_phi_k1_phi_l_list.append(mat_T_grad_phi_k1_phi_l_j)
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
mat_T_phi_k_phi_k1 = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k1)
self.proj_T_T_k1_k = mat_T_phi_k_phi_k_inv @ mat_T_phi_k_phi_k1
# ==============================================================================================================
# PROJECTION OPERATOR
# ==============================================================================================================
mat_T_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
self.mat_T_phi_l_phi_l_inv = np.linalg.inv(mat_T_phi_l_phi_l)
def compute_cell_matrices(self, cell: Cell, cell_basis_l: Basis,
cell_basis_k: Basis, cell_basis_k1: Basis):
"""
================================================================================================================
Method :
================================================================================================================
================================================================================================================
Parameters :
================================================================================================================
-
================================================================================================================
Returns :
================================================================================================================
-
"""
# ==============================================================================================================
# GRADIENT OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_phi_k = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k)
mat_T_phi_k_phi_k_inv = np.linalg.inv(mat_T_phi_k_phi_k)
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_k_grad_phi_l_x = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_k, cell_basis_l, 0)
mat_T_phi_k_grad_phi_l_list = [mat_T_phi_k_grad_phi_l_x]
# ==============================================================================================================
# RECONSTRUCTION OPERATOR
# ==============================================================================================================
# left-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_grad_phi_k1_grad_phi_k1_x = Integration.get_cell_stiffness_matrix_in_cell(
cell, cell_basis_k1, cell_basis_k1, 0)
mat_T_grad_phi_k1_grad_phi_k1_inv_list = [
mat_T_grad_phi_k1_grad_phi_k1_x
]
# --------------------------------------------------------------------------------------------------------------
# right-hand side
# --------------------------------------------------------------------------------------------------------------
mat_T_phi_l_grad_phi_k1_x = Integration.get_cell_advection_matrix_in_cell(
cell, cell_basis_l, cell_basis_k1, 0)
mat_T_grad_phi_k1_phi_l_x = mat_T_phi_l_grad_phi_k1_x.T
mat_T_grad_phi_k1_phi_l_list = [mat_T_grad_phi_k1_phi_l_x]
# ==============================================================================================================
# STABILIZATION OPERATOR
# ==============================================================================================================
mat_T_phi_k_phi_k1 = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_k, cell_basis_k1)
proj_T_T_k1_k = mat_T_phi_k_phi_k_inv @ mat_T_phi_k_phi_k1
# ==============================================================================================================
# PROJECTION OPERATOR
# ==============================================================================================================
mat_T_phi_l_phi_l = Integration.get_cell_mass_matrix_in_cell(
cell, cell_basis_l, cell_basis_l)
mat_T_phi_l_phi_l_inv = np.linalg.inv(mat_T_phi_l_phi_l)
return (
mat_T_phi_k_phi_k_inv,
mat_T_phi_k_grad_phi_l_list,
mat_T_grad_phi_k1_grad_phi_k1_inv_list,
mat_T_grad_phi_k1_phi_l_list,
proj_T_T_k1_k,
)
