def solve_newton_2(problem: Problem,
material: Material,
verbose: bool = False,
debug_mode: DebugMode = DebugMode.NONE):
time_step_index_count = 0
clean_res_dir(problem.res_folder_path)
problem.create_output(problem.res_folder_path)
_dx = problem.field.field_dimension
_fk = problem.finite_element.face_basis_k.dimension
_cl = problem.finite_element.cell_basis_l.dimension
external_forces_coefficient = 1.0
normalization_lagrange_coefficient = material.lagrange_parameter
# ----------------------------------------------------------------------------------------------------------
# SET SYSTEM SIZE
# ----------------------------------------------------------------------------------------------------------
_constrained_system_size, _system_size = problem.get_total_system_size()
faces_unknown_vector = np.zeros((_constrained_system_size), dtype=real)
faces_unknown_vector_previous_step = np.zeros((_constrained_system_size),
dtype=real)
residual_values = []
time_step_temp = problem.time_steps[0]
initial_time_steps = [ff for ff in problem.time_steps]
for time_step_index, time_step in enumerate(problem.time_steps):
local_time_steps = [time_step]
for local_time_step_index, local_time_step in enumerate(
local_time_steps):
print("0K")
# --- SET TEMPERATURE
material.set_temperature()
# --- PRINT DATA
print(
"----------------------------------------------------------------------------------------------------"
)
print("TIME_STEP : {} | LOAD_VALUE : {}".format(
time_step_index, time_step))
# --- WRITE RES FILES
# if time_step in initial_time_steps:
# file_suffix = "{}".format(time_step_index).zfill(6)
# problem.create_vertex_res_files(problem.res_folder_path, file_suffix)
# problem.create_quadrature_points_res_files(problem.res_folder_path, file_suffix, material)
for iteration in range(problem.number_of_iterations):
# --------------------------------------------------------------------------------------------------
# SET SYSTEM MATRIX AND VECTOR
# --------------------------------------------------------------------------------------------------
tangent_matrix = np.zeros(
(_constrained_system_size, _constrained_system_size),
dtype=real)
residual = np.zeros((_constrained_system_size), dtype=real)
# --------------------------------------------------------------------------------------------------
# SET TIME INCREMENT
# --------------------------------------------------------------------------------------------------
if time_step_index == 0:
_dt = time_step
else:
_dt = time_step - problem.time_steps[time_step_index - 1]
_dt = np.float64(_dt)
# _dt = 0.0
# _dt = np.float64(0.0)
# --------------------------------------------------------------------------------------------------
# FOR ELEMENT LOOP
# --------------------------------------------------------------------------------------------------
_qp = 0
stab_coef = 1.0
iter_face_constraint = 0
for _element_index, element in enumerate(problem.elements):
cell_quadrature_size = element.cell.get_quadrature_size(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
cell_quadrature_points = element.cell.get_quadrature_points(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
cell_quadrature_weights = element.cell.get_quadrature_weights(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
x_c = element.cell.get_centroid()
h_c = element.cell.get_diameter()
_nf = len(element.faces)
_c0_c = _dx * _cl
# --- INITIALIZE MATRIX AND VECTORS
element_stiffness_matrix = np.zeros(
(element.element_size, element.element_size), dtype=real)
element_internal_forces = np.zeros((element.element_size, ),
dtype=real)
element_external_forces = np.zeros((element.element_size, ),
dtype=real)
# --- RUN OVER EACH QUADRATURE POINT
for _qc in range(cell_quadrature_size):
_w_q_c = cell_quadrature_weights[_qc]
# _w_q_c = np.abs(cell_quadrature_weights[_qc])
_x_q_c = cell_quadrature_points[:, _qc]
# --- COMPUTE STRAINS AND SET THEM IN THE BEHAVIOUR LAW
transformation_gradient = element.get_transformation_gradient(
faces_unknown_vector, _qc)
material.mat_data.s1.gradients[
_qp] = transformation_gradient
# --- INTEGRATE BEHAVIOUR LAW
integ_res = mgis_bv.integrate(material.mat_data,
material.integration_type,
_dt, _qp, (_qp + 1))
# stored_energies', 'dissipated_energies', 'internal_state_variables
# print(material.mat_data.s1.internal_state_variables)
# --- VOLUMETRIC FORCES
v = problem.finite_element.cell_basis_l.evaluate_function(
_x_q_c, x_c, h_c)
for load in problem.loads:
vl = _w_q_c * v * load.function(time_step, _x_q_c)
_re0 = load.direction * _cl
_re1 = (load.direction + 1) * _cl
element_external_forces[_re0:_re1] += vl
# --- COMPUTE STIFFNESS MATRIX CONTRIBUTION AT QUADRATURE POINT
element_stiffness_matrix += _w_q_c * (
element.gradients_operators[_qc].T @ material.mat_data.
K[_qp] @ element.gradients_operators[_qc])
# --- COMPUTE STIFFNESS MATRIX CONTRIBUTION AT QUADRATURE POINT
element_internal_forces += _w_q_c * (
element.gradients_operators[_qc].T
@ material.mat_data.s1.thermodynamic_forces[_qp])
_qp += 1
if debug_mode == DebugMode.LIGHT:
print("ELEM : {} | INTERNAL_FORCES_BEFORE STAB : \n {}".
format(_element_index, element_internal_forces))
if verbose:
print("ELEM : {} | INTERNAL_FORCES_BEFORE STAB : \n {}".
format(_element_index, element_internal_forces))
# --- STAB PARAMETER CHANGE
stab_param = stab_coef * material.stabilization_parameter
# --- ADDING STABILIZATION CONTRIBUTION AT THE ELEMENT LEVEL
element_stiffness_matrix += stab_param * element.stabilization_operator
# element_stiffness_matrix -= stab_param * element.stabilization_operator
# --- ADDING STABILIZATION CONTRIBUTION AT THE ELEMENT LEVEL
element_internal_forces += (
stab_param * element.stabilization_operator
@ element.get_element_unknown_vector(faces_unknown_vector))
# element_internal_forces -= (
# stab_param
# * element.stabilization_operator
# @ element.get_element_unknown_vector(problem.field, problem.finite_element, faces_unknown_vector)
# )
# if verbose:
# print(
# "ELEM : {} | INTERNAL_FORCES_AFTER STAB : \n {}".format(_element_index, element_internal_forces)
# )
# _iv0 = _element_index * len(element.cell.quadrature_weights)
# _iv1 = (_element_index + 1) * len(element.cell.quadrature_weights)
# print(
# "ELEM : {} | DISPLACEMENT S0 : \n {}".format(
# _element_index,
# element.get_element_unknown_vector(
# problem.field, problem.finite_element, faces_unknown_vector_previous_step
# ),
# )
# )
# print(
# "ELEM : {} | GRADIENTS S0 : \n {}".format(
# _element_index, material.mat_data.s0.gradients[_iv0:_iv1]
# )
# )
# print(
# "ELEM : {} | DISPLACEMENT S1 : \n {}".format(
# _element_index,
# element.get_element_unknown_vector(
# problem.field, problem.finite_element, faces_unknown_vector
# ),
# )
# )
# print(
# "ELEM : {} | GRADIENTS S1 : \n {}".format(
# _element_index, material.mat_data.s1.gradients[_iv0:_iv1]
# )
# )
# if not material.behaviour_name == "Elasticity":
# print(
# "ELEM : {} | INTERNAL_STATE_VARIABLES S0 : \n {}".format(
# _element_index, material.mat_data.s0.internal_state_variables[_iv0:_iv1]
# )
# )
# print(
# "ELEM : {} | INTERNAL_STATE_VARIABLES S1 : \n {}".format(
# _element_index, material.mat_data.s1.internal_state_variables[_iv0:_iv1]
# )
# )
# --- BOUNDARY CONDITIONS
for boundary_condition in problem.boundary_conditions:
# --- DISPLACEMENT CONDITIONS
if boundary_condition.boundary_type == BoundaryType.DISPLACEMENT:
for f_local, f_global in enumerate(
element.faces_indices):
if f_global in problem.mesh.faces_boundaries_connectivity[
boundary_condition.boundary_name]:
_l0 = _system_size + iter_face_constraint * _fk
_l1 = _system_size + (iter_face_constraint +
1) * _fk
_c0 = _cl * _dx + (
f_local * _dx *
_fk) + boundary_condition.direction * _fk
_c1 = _cl * _dx + (f_local * _dx * _fk) + (
boundary_condition.direction + 1) * _fk
_r0 = f_global * _fk * _dx + _fk * boundary_condition.direction
_r1 = f_global * _fk * _dx + _fk * (
boundary_condition.direction + 1)
# -------
# face_displacement = element_unknown_increment[_r0:_r1]
# face_displacement = np.copy(element_unknown_increment[_c0:_c1])
# face_displacement = element_unknown_increment[_c0:_c1]
face_lagrange = faces_unknown_vector[_l0:_l1]
face_displacement = faces_unknown_vector[
_r0:_r1]
_m_psi_psi_face = np.zeros((_fk, _fk),
dtype=real)
_v_face_imposed_displacement = np.zeros(
(_fk, ), dtype=real)
face = element.faces[f_local]
x_f = face.get_centroid()
h_f = face.get_diameter()
face_rot = face.get_rotation_matrix()
face_quadrature_size = face.get_quadrature_size(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_points = face.get_quadrature_points(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_weights = face.get_quadrature_weights(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
for _qf in range(face_quadrature_size):
# _h_f = element.faces[f_local].shape.diameter
# _x_f = element.faces[f_local].shape.centroid
_x_q_f = face_quadrature_points[:, _qf]
_w_q_f = face_quadrature_weights[_qf]
_s_q_f = (face_rot @ _x_q_f)[:-1]
_s_f = (face_rot @ x_f)[:-1]
# v = problem.finite_element.face_basis_k.evaluate_function(
# _x_q_f,
# element.faces[f_local].shape.centroid,
# element.faces[f_local].shape.diameter,
# )
v = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
_v_face_imposed_displacement += (
_w_q_f * v *
boundary_condition.function(
time_step, _x_q_f))
# _m_psi_psi_face += blocks.get_face_mass_matrix_in_face(
# element.faces[f_local],
# problem.finite_element.face_basis_k,
# problem.finite_element.face_basis_k,
# _x_q_f,
# _w_q_f,
# )
_psi_k = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
_m_psi_psi_face += _w_q_f * np.tensordot(
_psi_k, _psi_k, axes=0)
_m_psi_psi_face_inv = np.linalg.inv(
_m_psi_psi_face)
if debug_mode == 0:
print(
"FACE MASS MATRIX IN DIRICHLET BOUND COND :"
)
print("{}".format(
np.linalg.cond(_m_psi_psi_face)))
imposed_face_displacement = _m_psi_psi_face_inv @ _v_face_imposed_displacement
face_displacement_difference = face_displacement - imposed_face_displacement
# print(face_lagrange)
# --- LAGRANGE INTERNAL FORCES PART
element_internal_forces[
_c0:
_c1] += normalization_lagrange_coefficient * face_lagrange
# --- LAGRANGE MULTIPLIERS PART
# residual[_l0:_l1] += (
# normalization_lagrange_coefficient * face_displacement_difference
# )
residual[
_l0:
_l1] -= normalization_lagrange_coefficient * face_displacement_difference
# --- LAGRANGE MATRIX PART
tangent_matrix[
_l0:_l1, _r0:
_r1] += normalization_lagrange_coefficient * np.eye(
_fk, dtype=real)
tangent_matrix[
_r0:_r1, _l0:
_l1] += normalization_lagrange_coefficient * np.eye(
_fk, dtype=real)
# --- SET EXTERNAL FORCES COEFFICIENT
lagrange_external_forces = (
normalization_lagrange_coefficient *
imposed_face_displacement)
if np.max(np.abs(lagrange_external_forces)
) > external_forces_coefficient:
external_forces_coefficient = np.max(
np.abs(lagrange_external_forces))
iter_face_constraint += 1
elif boundary_condition.boundary_type == BoundaryType.PRESSURE:
for f_local, f_global in enumerate(
element.faces_indices):
if f_global in problem.mesh.faces_boundaries_connectivity[
boundary_condition.boundary_name]:
face = element.faces[f_local]
x_f = face.get_centroid()
h_f = face.get_diameter()
face_rot = face.get_rotation_matrix()
face_quadrature_size = face.get_quadrature_size(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_points = face.get_quadrature_points(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_weights = face.get_quadrature_weights(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
for _qf in range(face_quadrature_size):
# _h_f = element.faces[f_local].shape.diameter
# _x_f = element.faces[f_local].shape.centroid
_x_q_f = face_quadrature_points[:, _qf]
_w_q_f = face_quadrature_weights[_qf]
_s_q_f = (face_rot @ _x_q_f)[:-1]
_s_f = (face_rot @ x_f)[:-1]
# for qf in range(len(element.faces[f_local].quadrature_weights)):
# # _h_f = element.faces[f_local].shape.diameter
# # _x_f = element.faces[f_local].shape.centroid
# _x_q_f = face_quadrature_points[:, _qf]
# _w_q_f = face_quadrature_weights[_qf]
# _s_q_f = (element.faces[f_local].mapping_matrix @ _x_q_f)[:-1]
# _s_f = (element.faces[f_local].mapping_matrix @ _x_f)[:-1]
# v = problem.finite_element.face_basis_k.evaluate_function(
# _x_q_f,
# element.faces[f_local].shape.centroid,
# element.faces[f_local].shape.diameter,
# )
v = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
# _x_q_f = element.faces[f_local].quadrature_points[:, qf]
# _w_q_f = element.faces[f_local].quadrature_weights[qf]
# v = problem.finite_element.face_basis_k.evaluate_function(
# _x_q_f,
# element.faces[f_local].shape.centroid,
# element.faces[f_local].shape.diameter,
# )
vf = _w_q_f * v * boundary_condition.function(
time_step, _x_q_f)
_c0 = _dx * _cl + f_local * _dx * _fk + boundary_condition.direction * _fk
_c1 = _dx * _cl + f_local * _dx * _fk + (
boundary_condition.direction + 1) * _fk
# _r0 = f_global * _fk * _dx + _fk * boundary_condition.direction
# _r1 = f_global * _fk * _dx + _fk * (boundary_condition.direction + 1)
element_external_forces[_c0:_c1] += vf
# --- COMPUTE RESIDUAL AFTER VOLUMETRIC CONTRIBUTION
element_residual = element_internal_forces - element_external_forces
if verbose:
print("ELEM : {} | INTERNAL_FORCES_END : \n {}".format(
_element_index, element_internal_forces))
print("ELEM : {} | ELEMENT_RESIDUAL : \n {}".format(
_element_index, element_residual))
# --------------------------------------------------------------------------------------------------
# CONDENSATION
# --------------------------------------------------------------------------------------------------
m_cell_cell = element_stiffness_matrix[:_c0_c, :_c0_c]
m_cell_faces = element_stiffness_matrix[:_c0_c, _c0_c:]
m_faces_cell = element_stiffness_matrix[_c0_c:, :_c0_c]
m_faces_faces = element_stiffness_matrix[_c0_c:, _c0_c:]
# v_cell = element_residual[:_c0_c]
# v_faces = element_residual[_c0_c:]
v_cell = -element_residual[:_c0_c]
v_faces = -element_residual[_c0_c:]
_condtest = np.linalg.cond(m_cell_cell)
m_cell_cell_inv = np.linalg.inv(m_cell_cell)
if debug_mode == 0:
print("TANGENT MATRIX IN CONDENSATION COND :")
print("{}".format(np.linalg.cond(m_cell_cell)))
# ge = m_faces_cell @ m_cell_cell_inv
# gd = (m_faces_cell @ m_cell_cell_inv) @ m_cell_faces
K_cond = m_faces_faces - (
(m_faces_cell @ m_cell_cell_inv) @ m_cell_faces)
R_cond = v_faces - (m_faces_cell @ m_cell_cell_inv) @ v_cell
# --- SET CONDENSATION/DECONDENSATION MATRICES
element.m_cell_cell_inv = m_cell_cell_inv
element.m_cell_faces = m_cell_faces
element.v_cell = v_cell
# --- ASSEMBLY
for _i_local, _i_global in enumerate(element.faces_indices):
_rg0 = _i_global * (_fk * _dx)
_rg1 = (_i_global + 1) * (_fk * _dx)
_re0 = _i_local * (_fk * _dx)
_re1 = (_i_local + 1) * (_fk * _dx)
residual[_rg0:_rg1] += R_cond[_re0:_re1]
for _j_local, _j_global in enumerate(
element.faces_indices):
_cg0 = _j_global * (_fk * _dx)
_cg1 = (_j_global + 1) * (_fk * _dx)
_ce0 = _j_local * (_fk * _dx)
_ce1 = (_j_local + 1) * (_fk * _dx)
tangent_matrix[_rg0:_rg1,
_cg0:_cg1] += K_cond[_re0:_re1,
_ce0:_ce1]
# --- SET EXTERNAL FORCES COEFFICIENT
if np.max(np.abs(element_external_forces)
) > external_forces_coefficient:
external_forces_coefficient = np.max(
np.abs(element_external_forces))
# --------------------------------------------------------------------------------------------------
# RESIDUAL EVALUATION
# --------------------------------------------------------------------------------------------------
if external_forces_coefficient == 0.0:
external_forces_coefficient = 1.0
residual_evaluation = np.max(
np.abs(residual)) / external_forces_coefficient
print("ITER : {} | RES_MAX : {}".format(
str(iteration).zfill(4), residual_evaluation))
# print("ITER : {} | =====================================================================".format(str(iteration).zfill(4)))
# residual_values.append(residual_evaluation)
# if residual_evaluation < problem.tolerance:
if residual_evaluation < problem.tolerance:
# ----------------------------------------------------------------------------------------------
# UPDATE INTERNAL VARIABLES
# ----------------------------------------------------------------------------------------------
mgis_bv.update(material.mat_data)
print("ITERATIONS : {}".format(iteration + 1))
file_suffix = "{}".format(time_step_index).zfill(6)
problem.create_vertex_res_files(problem.res_folder_path,
file_suffix)
problem.create_quadrature_points_res_files(
problem.res_folder_path, file_suffix, material)
problem.write_vertex_res_files(problem.res_folder_path,
file_suffix,
faces_unknown_vector)
problem.write_quadrature_points_res_files(
problem.res_folder_path, file_suffix, material,
faces_unknown_vector)
# problem.create_output(problem.res_folder_path)
problem.fill_quadrature_stress_output(problem.res_folder_path,
"CAUCHY_STRESS",
time_step_index_count,
time_step, material)
problem.fill_quadrature_strain_output(problem.res_folder_path,
"STRAIN",
time_step_index_count,
time_step, material)
# problem.close_output(problem.res_folder_path)
time_step_index_count += 1
faces_unknown_vector_previous_step = np.copy(
faces_unknown_vector)
for element in problem.elements:
element.cell_unknown_vector_backup = np.copy(
element.cell_unknown_vector)
residual_values.append(residual_evaluation)
time_step_temp = time_step + 0.
break
elif iteration == problem.number_of_iterations - 1:
problem.time_steps.insert(time_step_index + 1,
(time_step + time_step_temp) / 2.)
faces_unknown_vector = np.copy(
faces_unknown_vector_previous_step)
for element in problem.elements:
# element.cell_unknown_vector = np.zeros((_cl * _dx,))
element.cell_unknown_vector = np.copy(
element.cell_unknown_vector_backup)
break
else:
# ----------------------------------------------------------------------------------------------
# SOLVE SYSTEM
# ----------------------------------------------------------------------------------------------
if debug_mode == 0:
print("K GLOBAL COND")
print("{}".format(np.linalg.cond(tangent_matrix)))
# sparse_global_matrix = csr_matrix(-tangent_matrix)
sparse_global_matrix = csr_matrix(tangent_matrix)
correction = spsolve(sparse_global_matrix, residual)
# print("CORRECTION :")
# print(correction)
# correction = np.linalg.solve(-tangent_matrix, residual)
faces_unknown_vector += correction
if verbose:
print(
"R_K_RES : \n {}".format(residual -
tangent_matrix @ correction))
print("R_K_RES_NORMALIZED : \n {}".format(
(residual - tangent_matrix @ correction) /
external_forces_coefficient))
# ----------------------------------------------------------------------------------------------
# DECONDENSATION
# ----------------------------------------------------------------------------------------------
for element in problem.elements:
_nf = len(element.faces)
# _c0_c = _dx * _cl
# --- DECONDENSATION - GET ELEMENT UNKNOWN CORRECTION
face_correction = np.zeros((_nf * _fk * _dx), dtype=real)
# element_correction = np.zeros((element.element_size,))
for _i_local, _i_global in enumerate(
element.faces_indices):
_c0_fg = _i_global * (_fk * _dx)
_c1_fg = (_i_global + 1) * (_fk * _dx)
# _c0_fl = (_dx * _cl) + _i_local * (_fk * _dx)
# _c1_fl = (_dx * _cl) + (_i_local + 1) * (_fk * _dx)
_c0_fl = _i_local * (_fk * _dx)
_c1_fl = (_i_local + 1) * (_fk * _dx)
# element_correction[_c0_fl:_c1_fl] += correction[_c0_fg:_c1_fg]
face_correction[_c0_fl:_c1_fl] += correction[
_c0_fg:_c1_fg]
# element_correction[:_c0_c] = element.m_cell_cell_inv @ (
# element.v_cell - element.m_cell_faces @ element_correction[_c0_c:]
# )
cell_correction = element.m_cell_cell_inv @ (
element.v_cell -
element.m_cell_faces @ face_correction)
# --- ADDING CORRECTION TO CURRENT DISPLACEMENT
element.cell_unknown_vector += cell_correction
plt.plot(range(len(residual_values)), residual_values, label="residual")
plt.plot(range(len(residual_values)),
[problem.tolerance for local_i in range(len(residual_values))],
label="tolerance")
plt.ylabel("resiudal after convergence")
plt.xlabel("time step index")
plt.title("evolution of the residual")
plt.legend()
plt.gcf().subplots_adjust(left=0.15)
plt.show()
return
def solve_newton_exact(problem: Problem,
material: Material,
verbose: bool = False,
debug_mode: DebugMode = DebugMode.NONE):
"""
Args:
material:
Returns:
"""
clean_res_dir(problem.res_folder_path)
_dx = problem.field.field_dimension
_fk = problem.finite_element.face_basis_k.dimension
_cl = problem.finite_element.cell_basis_l.dimension
external_forces_coefficient = 1.0
normalization_lagrange_coefficient = material.lagrange_parameter
# ----------------------------------------------------------------------------------------------------------
# SET SYSTEM SIZE
# ----------------------------------------------------------------------------------------------------------
_constrained_system_size, _system_size = problem.get_total_system_size()
faces_unknown_vector = np.zeros((_constrained_system_size), dtype=real)
faces_unknown_vector_previous_step = np.zeros((_constrained_system_size),
dtype=real)
residual_values = []
correction = np.zeros((_constrained_system_size), dtype=real)
for time_step_index, time_step in enumerate(problem.time_steps):
# --- SET TEMPERATURE
material.set_temperature()
# --- PRINT DATA
print(
"----------------------------------------------------------------------------------------------------"
)
print("TIME_STEP : {} | LOAD_VALUE : {}".format(
time_step_index, time_step))
# --- WRITE RES FILES
file_suffix = "{}".format(time_step_index).zfill(6)
problem.create_vertex_res_files(problem.res_folder_path, file_suffix)
problem.create_quadrature_points_res_files(problem.res_folder_path,
file_suffix, material)
# correction = np.zeros((_constrained_system_size),dtype=real)
# --- RESET DISPLACEMENT
reset_displacement = False
if reset_displacement:
faces_unknown_vector = np.zeros((_constrained_system_size),
dtype=real)
for element in problem.elements:
element.cell_unknown_vector = np.zeros((_dx * _cl, ),
dtype=real)
for iteration in range(problem.number_of_iterations):
# --------------------------------------------------------------------------------------------------
# SET SYSTEM MATRIX AND VECTOR
# --------------------------------------------------------------------------------------------------
tangent_matrix = np.zeros(
(_constrained_system_size, _constrained_system_size),
dtype=real)
residual = np.zeros((_constrained_system_size), dtype=real)
# --------------------------------------------------------------------------------------------------
# SET TIME INCREMENT
# --------------------------------------------------------------------------------------------------
if time_step_index == 0:
_dt = time_step
else:
_dt = time_step - problem.time_steps[time_step_index - 1]
_dt = np.float64(_dt)
# --------------------------------------------------------------------------------------------------
# FOR ELEMENT LOOP
# --------------------------------------------------------------------------------------------------
_qp = 0
stab_coef = 1.0
stab_inf = np.inf
iter_face_constraint = 0
if False:
for element in problem.elements:
for _qc in range(len(element.cell.quadrature_weights)):
_w_q_c = element.cell.quadrature_weights[_qc]
_x_q_c = element.cell.quadrature_points[:, _qc]
# --- COMPUTE STRAINS AND SET THEM IN THE BEHAVIOUR LAW
transformation_gradient = element.get_transformation_gradient(
problem.field, problem.finite_element,
faces_unknown_vector, _qc)
material.mat_data.s1.gradients[
_qp] = transformation_gradient
# --- INTEGRATE BEHAVIOUR LAW
integ_res = mgis_bv.integrate(
material.mat_data, material.integration_type, _dt,
_qp, (_qp + 1))
eigen_values = np.linalg.eig(
material.mat_data.K[_qp])[0]
stab_elem = np.min(
eigen_values) / material.stabilization_parameter
if stab_elem < stab_inf and stab_elem > 1.0:
stab_inf = stab_elem
stab_coef = stab_elem
_qp += 1
mgis_bv.revert(material.mat_data)
_qp = 0
for _element_index, element in enumerate(problem.elements):
cell_quadrature_size = element.cell.get_quadrature_size(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
cell_quadrature_points = element.cell.get_quadrature_points(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
cell_quadrature_weights = element.cell.get_quadrature_weights(
problem.finite_element.construction_integration_order,
quadrature_type=problem.quadrature_type)
x_c = element.cell.get_centroid()
h_c = element.cell.get_diameter()
print("ELEMENT : {}".format(_element_index))
# --- GET FACES LOCAL CORRECTION
local_faces_correction = np.zeros(
(len(element.faces) * _dx * _fk), dtype=real)
for _f_local, _f_global in enumerate(element.faces_indices):
local_faces_correction[_f_local * _dx *
_fk:(_f_local + 1) * _dx * _fk] += (
correction[_f_global * _dx *
_fk:(_f_global + 1) *
_dx * _fk])
# --- LOCAL NEWTON
number_of_local_iterations = 20
# local_tolerance = problem.tolerance
local_tolerance = 1.e-4
R_cell_value_previous = np.inf
for _local_iteration in range(number_of_local_iterations):
_nf = len(element.faces)
_c0_c = _dx * _cl
# --- INITIALIZE MATRIX AND VECTORS
element_stiffness_matrix = np.zeros(
(element.element_size, element.element_size),
dtype=real)
element_internal_forces = np.zeros(
(element.element_size, ), dtype=real)
element_external_forces = np.zeros(
(element.element_size, ), dtype=real)
# --- RUN OVER EACH QUADRATURE POINT
_local_qp = 0
for _qc in range(cell_quadrature_size):
_w_q_c = cell_quadrature_weights[_qc]
_x_q_c = cell_quadrature_points[:, _qc]
# --- COMPUTE STRAINS AND SET THEM IN THE BEHAVIOUR LAW
transformation_gradient = element.get_transformation_gradient(
faces_unknown_vector, _qc)
material.mat_data.s1.gradients[
_qp] = transformation_gradient
# --- INTEGRATE BEHAVIOUR LAW
integ_res = mgis_bv.integrate(
material.mat_data, material.integration_type, _dt,
_qp, (_qp + 1))
# stored_energies', 'dissipated_energies', 'internal_state_variables
# print(material.mat_data.s1.internal_state_variables)
# --- VOLUMETRIC FORCES
v = problem.finite_element.cell_basis_l.evaluate_function(
_x_q_c, x_c, h_c)
for load in problem.loads:
vl = _w_q_c * v * load.function(time_step, _x_q_c)
_re0 = load.direction * _cl
_re1 = (load.direction + 1) * _cl
element_external_forces[_re0:_re1] += vl
# --- COMPUTE STIFFNESS MATRIX CONTRIBUTION AT QUADRATURE POINT
element_stiffness_matrix += _w_q_c * (
element.gradients_operators[_qc].T @ material.
mat_data.K[_qp] @ element.gradients_operators[_qc])
# --- COMPUTE STIFFNESS MATRIX CONTRIBUTION AT QUADRATURE POINT
element_internal_forces += _w_q_c * (
element.gradients_operators[_qc].T
@ material.mat_data.s1.thermodynamic_forces[_qp])
_qp += 1
_local_qp += 1
if verbose:
print(
"ELEM : {} | INTERNAL_FORCES_BEFORE STAB : \n {}".
format(_element_index, element_internal_forces))
# --- STAB PARAMETER CHANGE
stab_param = stab_coef * material.stabilization_parameter
# if check and iteration > 0:
# numerical_element_stiffness_matrix = (
# numerical_forces_bs_0[_element_index] -
# numerical_forces_bs_1[_element_index]
# ) / (2.0 * noise)
# check_num = np.max(
# np.abs(numerical_element_stiffness_matrix - element_stiffness_matrix)
# ) / np.max(np.abs(element_stiffness_matrix))
# if check_num > noise_tolerance:
# print(
# "AT ELEM : {} | CHECK_BEFORE_STAB : {}".format(str(_element_index).zfill(6), check_num)
# )
# --- ADDING STABILIZATION CONTRIBUTION AT THE ELEMENT LEVEL
element_stiffness_matrix += stab_param * element.stabilization_operator
# element_stiffness_matrix -= stab_param * element.stabilization_operator
# --- ADDING STABILIZATION CONTRIBUTION AT THE ELEMENT LEVEL
element_internal_forces += (
stab_param * element.stabilization_operator @ element.
get_element_unknown_vector(faces_unknown_vector))
# --- MATRIX VECTOR DECOMPOSITION
K_cc = element_stiffness_matrix[:_c0_c, :_c0_c]
K_cf = element_stiffness_matrix[:_c0_c, _c0_c:]
# local_external_forces_coefficient = np.max(np.abs(element_external_forces))
element_residual = element_internal_forces - element_external_forces
# if local_external_forces_coefficient == 0.:
# local_external_forces_coefficient = 1.0
# local_external_forces_coefficient = normalization_lagrange_coefficient
# local_external_forces_coefficient = 1./element.cell.shape.volume
R_cc = element_residual[:_c0_c]
if iteration == 0:
local_external_forces_coefficient = 1.0
cell_correction = np.ones((_cl * _dx, ), dtype=real)
if _local_iteration == 0 and iteration > 0:
cell_correction = np.ones((_cl * _dx, ), dtype=real)
local_external_forces_coefficient = np.max(
np.abs(R_cc))
# local_external_forces_coefficient = 1.
# if local_external_forces_coefficient == 0.0 or local_external_forces_coefficient < local_tolerance:
if local_external_forces_coefficient == 0.0:
local_external_forces_coefficient = 1.0
# R_cell = R_cc - K_cf @ local_faces_correction
# print("LOCAL_EXTERNAL_FORCES_COEF : {}".format(local_external_forces_coefficient))
R_cell = R_cc
local_residual_evaluation = np.max(
np.abs(R_cell) / local_external_forces_coefficient)
# local_residual_evaluation = np.max(np.abs(cell_correction))
# if local_residual_evaluation > R_cell_value_previous:
# print("!!!! RESIUDAL INCREASE :\n {}".format(element.cell.quadrature_points))
# R_cell_value_previous = local_residual_evaluation
print("LOCAL ITER : {} | RES MAX : {}".format(
_local_iteration, local_residual_evaluation))
if local_residual_evaluation < local_tolerance:
break
else:
cell_correction = np.linalg.solve(-K_cc, R_cell)
element.cell_unknown_vector += cell_correction
if _local_iteration != number_of_local_iterations - 1:
_qp -= _local_qp
# element_internal_forces -= (
# stab_param
# * element.stabilization_operator
# @ element.get_element_unknown_vector(problem.field, problem.finite_element, faces_unknown_vector)
# )
# --------------------------------------------------------------------------------------------------
# CONDENSATION
# --------------------------------------------------------------------------------------------------
m_cell_cell = element_stiffness_matrix[:_c0_c, :_c0_c]
m_cell_faces = element_stiffness_matrix[:_c0_c, _c0_c:]
m_faces_cell = element_stiffness_matrix[_c0_c:, :_c0_c]
m_faces_faces = element_stiffness_matrix[_c0_c:, _c0_c:]
# v_cell = element_residual[:_c0_c]
# v_faces = element_residual[_c0_c:]
v_cell = -element_residual[:_c0_c]
v_faces = -element_residual[_c0_c:]
m_cell_cell_inv = np.linalg.inv(m_cell_cell)
if debug_mode == 0:
print("TANGENT MATRIX IN CONDENSATION COND :")
print("{}".format(np.linalg.cond(m_cell_cell)))
# ge = m_faces_cell @ m_cell_cell_inv
# gd = (m_faces_cell @ m_cell_cell_inv) @ m_cell_faces
K_cond = m_faces_faces - (
(m_faces_cell @ m_cell_cell_inv) @ m_cell_faces)
R_cond = v_faces - (m_faces_cell @ m_cell_cell_inv) @ v_cell
# --- SET CONDENSATION/DECONDENSATION MATRICES
# element.m_cell_cell_inv = m_cell_cell_inv
# element.m_cell_faces = m_cell_faces
# element.v_cell = v_cell
# --- ASSEMBLY
for _i_local, _i_global in enumerate(element.faces_indices):
_rg0 = _i_global * (_fk * _dx)
_rg1 = (_i_global + 1) * (_fk * _dx)
_re0 = _i_local * (_fk * _dx)
_re1 = (_i_local + 1) * (_fk * _dx)
residual[_rg0:_rg1] += R_cond[_re0:_re1]
for _j_local, _j_global in enumerate(
element.faces_indices):
_cg0 = _j_global * (_fk * _dx)
_cg1 = (_j_global + 1) * (_fk * _dx)
_ce0 = _j_local * (_fk * _dx)
_ce1 = (_j_local + 1) * (_fk * _dx)
tangent_matrix[_rg0:_rg1,
_cg0:_cg1] += K_cond[_re0:_re1,
_ce0:_ce1]
# --- SET EXTERNAL FORCES COEFFICIENT
if np.max(np.abs(element_external_forces)
) > external_forces_coefficient:
external_forces_coefficient = np.max(
np.abs(element_external_forces))
# if check and iteration > 0:
# numerical_element_stiffness_matrix = (
# numerical_forces_as_0[_element_index] -
# numerical_forces_as_1[_element_index]
# ) / (2.0 * noise)
# check_num = np.max(
# np.abs(numerical_element_stiffness_matrix - element_stiffness_matrix)
# ) / np.max(np.abs(element_stiffness_matrix))
# if check_num > noise_tolerance:
# print(
# "AT ELEM : {} | CHECK_BEFORE_STAB : {}".format(str(_element_index).zfill(6), check_num)
# )
if verbose:
pass
# print("ELEM : {} | INTERNAL_FORCES_AFTER STAB : \n {}".format(_element_index,
# element_internal_forces))
# _iv0 = _element_index * len(element.cell.quadrature_weights)
# _iv1 = (_element_index + 1) * len(element.cell.quadrature_weights)
# print("ELEM : {} | DISPLACEMENT S0 : \n {}".format(_element_index,
# element.get_element_unknown_vector(problem.field,
# problem.finite_element,
# faces_unknown_vector_previous_step)))
# print("ELEM : {} | GRADIENTS S0 : \n {}".format(_element_index,
# material.mat_data.s0.gradients[_iv0:_iv1]))
# print("ELEM : {} | DISPLACEMENT S1 : \n {}".format(_element_index,
# element.get_element_unknown_vector(problem.field,
# problem.finite_element,
# faces_unknown_vector)))
# print("ELEM : {} | GRADIENTS S1 : \n {}".format(_element_index,
# material.mat_data.s1.gradients[_iv0:_iv1]))
# if not material.behaviour_name == "Elasticity":
# print("ELEM : {} | INTERNAL_STATE_VARIABLES S0 : \n {}".format(_element_index,
# material.mat_data.s0.internal_state_variables[
# _iv0:_iv1]))
# print("ELEM : {} | INTERNAL_STATE_VARIABLES S1 : \n {}".format(_element_index,
# material.mat_data.s1.internal_state_variables[
# _iv0:_iv1]))
# --- BOUNDARY CONDITIONS
for boundary_condition in problem.boundary_conditions:
# --- DISPLACEMENT CONDITIONS
if boundary_condition.boundary_type == BoundaryType.DISPLACEMENT:
for f_local, f_global in enumerate(
element.faces_indices):
if (f_global in
problem.mesh.faces_boundaries_connectivity[
boundary_condition.boundary_name]):
_l0 = _system_size + iter_face_constraint * _fk
_l1 = _system_size + (iter_face_constraint +
1) * _fk
_c0 = _cl * _dx + (
f_local * _dx *
_fk) + boundary_condition.direction * _fk
_c1 = _cl * _dx + (f_local * _dx * _fk) + (
boundary_condition.direction + 1) * _fk
_r0 = f_global * _fk * _dx + _fk * boundary_condition.direction
_r1 = f_global * _fk * _dx + _fk * (
boundary_condition.direction + 1)
# -------
# face_displacement = element_unknown_increment[_r0:_r1]
# face_displacement = np.copy(element_unknown_increment[_c0:_c1])
# face_displacement = element_unknown_increment[_c0:_c1]
face_lagrange = faces_unknown_vector[_l0:_l1]
face_displacement = faces_unknown_vector[
_r0:_r1]
_m_psi_psi_face = np.zeros((_fk, _fk),
dtype=real)
_v_face_imposed_displacement = np.zeros(
(_fk, ), dtype=real)
face = element.faces[f_local]
x_f = face.get_centroid()
h_f = face.get_diameter()
face_rot = face.get_rotation_matrix()
face_quadrature_size = face.get_quadrature_size(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_points = face.get_quadrature_points(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_weights = face.get_quadrature_weights(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
for _qf in range(face_quadrature_size):
_x_q_f = face_quadrature_points[:, _qf]
_w_q_f = face_quadrature_weights[_qf]
_s_q_f = (face_rot @ _x_q_f)[:-1]
_s_f = (face_rot @ x_f)[:-1]
v = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
_v_face_imposed_displacement += (
_w_q_f * v *
boundary_condition.function(
time_step, _x_q_f))
_psi_k = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
_m_psi_psi_face += _w_q_f * np.tensordot(
_psi_k, _psi_k, axes=0)
_m_psi_psi_face_inv = np.linalg.inv(
_m_psi_psi_face)
if debug_mode == 0:
print(
"FACE MASS MATRIX IN DIRICHLET BOUND COND :"
)
print("{}".format(
np.linalg.cond(_m_psi_psi_face)))
imposed_face_displacement = _m_psi_psi_face_inv @ _v_face_imposed_displacement
face_displacement_difference = face_displacement - imposed_face_displacement
# --- LAGRANGE INTERNAL FORCES PART
# element_internal_forces[_c0:_c1] += (
# normalization_lagrange_coefficient * face_lagrange
# )
# residual[_r0:_r1] += (
# normalization_lagrange_coefficient * face_lagrange
# )
residual[_r0:_r1] -= (
normalization_lagrange_coefficient *
face_lagrange)
# --- LAGRANGE MULTIPLIERS PART
# residual[_l0:_l1] += (
# normalization_lagrange_coefficient * face_displacement_difference
# )
residual[_l0:_l1] -= (
normalization_lagrange_coefficient *
face_displacement_difference)
# --- LAGRANGE MATRIX PART
tangent_matrix[
_l0:_l1, _r0:
_r1] += normalization_lagrange_coefficient * np.eye(
_fk, dtype=real)
tangent_matrix[
_r0:_r1, _l0:
_l1] += normalization_lagrange_coefficient * np.eye(
_fk, dtype=real)
# --- SET EXTERNAL FORCES COEFFICIENT
lagrange_external_forces = (
normalization_lagrange_coefficient *
imposed_face_displacement)
if np.max(np.abs(lagrange_external_forces)
) > external_forces_coefficient:
external_forces_coefficient = np.max(
np.abs(lagrange_external_forces))
iter_face_constraint += 1
elif boundary_condition.boundary_type == BoundaryType.PRESSURE:
for f_local, f_global in enumerate(
element.faces_indices):
if (f_global in
problem.mesh.faces_boundaries_connectivity[
boundary_condition.boundary_name]):
face = element.faces[f_local]
x_f = face.get_centroid()
h_f = face.get_diameter()
face_rot = face.get_rotation_matrix()
face_quadrature_size = face.get_quadrature_size(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_points = face.get_quadrature_points(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
face_quadrature_weights = face.get_quadrature_weights(
problem.finite_element.
construction_integration_order,
quadrature_type=problem.quadrature_type,
)
for _qf in range(face_quadrature_size):
# _h_f = element.faces[f_local].shape.diameter
# _x_f = element.faces[f_local].shape.centroid
_x_q_f = face_quadrature_points[:, _qf]
_w_q_f = face_quadrature_weights[_qf]
_s_q_f = (face_rot @ _x_q_f)[:-1]
_s_f = (face_rot @ x_f)[:-1]
v = problem.finite_element.face_basis_k.evaluate_function(
_s_q_f, _s_f, h_f)
vf = _w_q_f * v * boundary_condition.function(
time_step, _x_q_f)
# _c0 = _dx * _cl + f_local * _dx * _fk + boundary_condition.direction * _fk
# _c1 = _dx * _cl + f_local * _dx * _fk + (boundary_condition.direction + 1) * _fk
_r0 = f_global * _fk * _dx + _fk * boundary_condition.direction
_r1 = f_global * _fk * _dx + _fk * (
boundary_condition.direction + 1)
# element_external_forces[_c0:_c1] += vf
# residual[_r0:_r1] -= vf
residual[_r0:_r1] += vf
if np.max(np.abs(
vf)) > external_forces_coefficient:
external_forces_coefficient = np.max(
np.abs(vf))
# --------------------------------------------------------------------------------------------------
# RESIDUAL EVALUATION
# --------------------------------------------------------------------------------------------------
if external_forces_coefficient == 0.0:
external_forces_coefficient = 1.0
residual_evaluation = np.max(
np.abs(residual)) / external_forces_coefficient
print("ITER : {} | RES_MAX : {}".format(
str(iteration).zfill(4), residual_evaluation))
# print("ITER : {} | =====================================================================".format(str(iteration).zfill(4)))
# residual_values.append(residual_evaluation)
# if residual_evaluation < problem.tolerance:
if residual_evaluation < problem.tolerance or iteration == problem.number_of_iterations - 1:
# ----------------------------------------------------------------------------------------------
# UPDATE INTERNAL VARIABLES
# ----------------------------------------------------------------------------------------------
mgis_bv.update(material.mat_data)
print("ITERATIONS : {}".format(iteration + 1))
problem.write_vertex_res_files(problem.res_folder_path,
file_suffix,
faces_unknown_vector)
problem.write_quadrature_points_res_files(
problem.res_folder_path, file_suffix, material,
faces_unknown_vector)
faces_unknown_vector_previous_step += faces_unknown_vector
residual_values.append(residual_evaluation)
break
else:
# ----------------------------------------------------------------------------------------------
# SOLVE SYSTEM
# ----------------------------------------------------------------------------------------------
if debug_mode == 0:
print("K GLOBAL COND")
print("{}".format(np.linalg.cond(tangent_matrix)))
# sparse_global_matrix = csr_matrix(-tangent_matrix)
sparse_global_matrix = csr_matrix(tangent_matrix)
correction = spsolve(sparse_global_matrix, residual)
print("CORRECTION :")
# print(correction)
# correction = np.linalg.solve(-tangent_matrix, residual)
faces_unknown_vector += correction
if verbose:
print(
"R_K_RES : \n {}".format(residual -
tangent_matrix @ correction))
print("R_K_RES_NORMALIZED : \n {}".format(
(residual - tangent_matrix @ correction) /
external_forces_coefficient))
plt.plot(range(len(residual_values)), residual_values, label="residual")
plt.plot(range(len(residual_values)),
[problem.tolerance for local_i in range(len(residual_values))],
label="tolerance")
plt.ylabel("resiudal after convergence")
plt.xlabel("time step index")
plt.title("evolution of the residual")
plt.legend()
plt.gcf().subplots_adjust(left=0.15)
plt.show()
return