def mesh2triang(mesh: df.Mesh) -> tri.Triangulation:
"""Return an unstructured grid with the connectivity given by matplotlib
Code borrowed from Chris Richardson
"""
xy = mesh.coordinates()
return tri.Triangulation(xy[:, 0], xy[:, 1], mesh.cells())
def vtk_ug_to_dolfin_mesh(ug):
"""
Create a DOLFIN Mesh from a vtkUnstructuredGrid object
"""
if not isinstance(ug, vtk.vtkUnstructuredGrid):
raise TypeError("Expected a 'vtkUnstructuredGrid'")
# Get mesh data
num_cells = int(ug.GetNumberOfCells())
num_vertices = int(ug.GetNumberOfPoints())
# Get topological and geometrical dimensions
cell = ug.GetCell(0)
gdim = int(cell.GetCellDimension())
cell_type = cell.GetCellType()
if cell_type not in [vtk.VTK_TETRA, vtk.VTK_TRIANGLE]:
raise TypeError("DOLFIN only support meshes of triangles " + \
"and tetrahedrons.")
tdim = 3 if cell_type == vtk.VTK_TETRA else 2
# Create empty DOLFIN Mesh
mesh = Mesh()
editor = MeshEditor()
editor.open(mesh, tdim, gdim)
editor.init_cells(num_cells)
editor.init_vertices(num_vertices)
editor.close()
# Assign the cell and vertex informations directly from the vtk data
cells_array = array_handler.vtk2array(ug.GetCells().GetData())
# Get the assumed fixed size of indices and create an index array
cell_size = cell.GetPointIds().GetNumberOfIds()
cellinds = np.arange(len(cells_array))
# Each cell_ids_size:th data point need to be deleted from the
# index array
ind_delete = slice(0, len(cells_array), cell_size+1)
# Check that the removed value all have the same value which should
# be the size of the data
if not np.all(cells_array[ind_delete]==cell_size):
raise ValueError("Expected all cells to be of the same size")
cellinds = np.delete(cellinds, ind_delete)
# Get cell data from mesh and make it writeable (cell data is non
# writeable by default) and update the values
mesh_cells = mesh.cells()
mesh_cells.flags.writeable = True
mesh_cells[:] = np.reshape(cells_array[cellinds], \
(num_cells , cell_size))
# Set coordinates from vtk data
vertex_array = array_handler.vtk2array(ug.GetPoints().GetData())
if vertex_array.shape[1] != gdim:
vertex_array = vertex_array[:,:gdim]
mesh.coordinates()[:] = vertex_array
return mesh
def vtk_ug_to_dolfin_mesh(ug):
"""
Create a DOLFIN Mesh from a vtkUnstructuredGrid object
"""
if not isinstance(ug, vtk.vtkUnstructuredGrid):
raise TypeError("Expected a 'vtkUnstructuredGrid'")
# Get mesh data
num_cells = int(ug.GetNumberOfCells())
num_vertices = int(ug.GetNumberOfPoints())
# Get topological and geometrical dimensions
cell = ug.GetCell(0)
gdim = int(cell.GetCellDimension())
cell_type = cell.GetCellType()
if cell_type not in [vtk.VTK_TETRA, vtk.VTK_TRIANGLE]:
raise TypeError("DOLFIN only support meshes of triangles " + \
"and tetrahedrons.")
tdim = 3 if cell_type == vtk.VTK_TETRA else 2
# Create empty DOLFIN Mesh
mesh = Mesh()
editor = MeshEditor()
editor.open(mesh, tdim, gdim)
editor.init_cells(num_cells)
editor.init_vertices(num_vertices)
editor.close()
# Assign the cell and vertex informations directly from the vtk data
cells_array = array_handler.vtk2array(ug.GetCells().GetData())
# Get the assumed fixed size of indices and create an index array
cell_size = cell.GetPointIds().GetNumberOfIds()
cellinds = np.arange(len(cells_array))
# Each cell_ids_size:th data point need to be deleted from the
# index array
ind_delete = slice(0, len(cells_array), cell_size + 1)
# Check that the removed value all have the same value which should
# be the size of the data
if not np.all(cells_array[ind_delete] == cell_size):
raise ValueError("Expected all cells to be of the same size")
cellinds = np.delete(cellinds, ind_delete)
# Get cell data from mesh and make it writeable (cell data is non
# writeable by default) and update the values
mesh_cells = mesh.cells()
mesh_cells.flags.writeable = True
mesh_cells[:] = np.reshape(cells_array[cellinds], \
(num_cells , cell_size))
# Set coordinates from vtk data
vertex_array = array_handler.vtk2array(ug.GetPoints().GetData())
if vertex_array.shape[1] != gdim:
vertex_array = vertex_array[:, :gdim]
mesh.coordinates()[:] = vertex_array
return mesh
import numpy
from dolfin import Mesh
mesh = Mesh("mesh.xml.gz")
E = mesh.cells()
M = numpy.fromfile('materials.np')
I = -numpy.ones(mesh.num_vertices())
for i in range(6):
edx = numpy.nonzero((M > 10 * (i + 1)) * (M < 10 * (i + 2)))[0]
idx = (numpy.unique(E[edx, 0:3])).astype(int)
I[idx] = i * 0.2
edx = numpy.nonzero(M == 7)[0]
idx = (numpy.unique(E[edx, 0:3])).astype(int)
I[idx] = -2
from viper import Viper
pv = Viper(mesh, I)
pv.interactive()
class Mesh():
def __init__(self, dimension, robot_radius):
self.robot_radius = robot_radius
# Correct the map size by the robot_radius
self.dimension = list(map(lambda x: x - robot_radius, dimension))
self._cache_path = '/tmp/mesh.xml'
# Define the base rectangle.
self._map = mshr.Rectangle(Point(robot_radius, robot_radius),
Point(self.dimension[0], self.dimension[1]))
self._mesh2d = None
def add_circle_obstacle(self, p, radius, mirror=False, accuracy=10):
points = [Point(p[0], p[1])]
# Replicate the circle on the other edge of the map.
if mirror:
points.append(
Point(self.dimension[0] + self.robot_radius - p[0], p[1]))
for point in points:
self._map -= mshr.Circle(point, radius + self.robot_radius,
accuracy)
def add_rectangle_obstacle(self, p1, p2, mirror=False):
self._map -= mshr.Rectangle(
self.__correct_point(Point(p1[0], p1[1]), inverse=-1),
self.__correct_point(Point(p2[0], p2[1])))
# Replicate the data the other side of the map.
if mirror:
p1bis = Point(
self.dimension[0] + self.robot_radius - p1[0] +
self.robot_radius, p1[1] - self.robot_radius)
p2bis = Point(
self.dimension[0] + self.robot_radius - p2[0] -
self.robot_radius, p2[1] + self.robot_radius)
self._map -= mshr.Rectangle(p2bis, p1bis)
# As dolfin and mshr don't support leaning rectangle, we build the
# rectangle giving him the point with the minimum y data,
# the absolute y size, the width, the angle with the 0 starting
# with the x axis.
# The accuracy give the number of rectangles we want to cut the
# leaning rectangle.
def add_leaning_rectangle_obstacle(self,
p,
height,
width,
angle,
mirror=False,
accuracy=10):
height_chunck_size = height / (accuracy)
radian_angle = math.radians(angle)
for chunck in range(accuracy):
pos = chunck * height_chunck_size + height_chunck_size / 2
# Get the center of the little rectangle.
center_point = (pos * math.cos(radian_angle),
pos * math.sin(radian_angle))
self.__build_rectangle_by_center(p, center_point,
height_chunck_size, width, mirror)
def __build_rectangle_by_center(self,
base_point,
p,
height,
width,
mirror=False):
p1 = (base_point[0] + p[0] + width / 2,
base_point[1] + p[1] + height / 2)
p2 = (base_point[0] + p[0] - width / 2,
base_point[1] + p[1] - height / 2)
self.add_rectangle_obstacle(p1, p2, mirror)
def __correct_point(self, p, inverse=1):
return Point(p.x() + self.robot_radius * inverse,
p.y() + self.robot_radius * inverse)
def build(self, accuracy=5, cache=False):
if cache and os.path.exists(self._cache_path):
self._mesh2d = DolfinMesh(self._cache_path)
return
self._mesh2d = mshr.generate_mesh(self._map, accuracy)
if cache:
f = File(self._cache_path)
f << self._mesh2d
def get_connectivity_cells(self):
return self._mesh2d.cells()
def get_nodes(self):
return self._mesh2d.coordinates()
def display(self, accuracy=10):
if self._mesh2d is None:
self.build(accuracy)
plot(self._mesh2d, interactive=True)
import numpy
from dolfin import Mesh
mesh = Mesh("mesh.xml.gz")
E = mesh.cells()
M = numpy.fromfile('materials.np');
I = -numpy.ones(mesh.num_vertices())
for i in range(6):
edx = numpy.nonzero((M>10*(i+1))*(M<10*(i+2)))[0]
idx = (numpy.unique(E[edx,0:3])).astype(int)
I[idx] = i*0.2
edx = numpy.nonzero(M==7)[0]
idx = (numpy.unique(E[edx,0:3])).astype(int)
I[idx] = -2;
from viper import Viper
pv = Viper(mesh, I)
pv.interactive()
class FBVP:
def __init__(self, mesh_name, parameters, alpha_range=None):
# LOAD MESH AND PARAMETERS
self.parameters = parameters
self.mesh_name = mesh_name
if not alpha_range:
self.alpha_range = parameters.alpha_range
else:
self.alpha_range = alpha_range
mesh_path = self.parameters.get_mesh_path()
self.mesh = Mesh()
with XDMFFile(mesh_path) as f:
f.read(self.mesh)
print(f'Mesh size= {self.mesh.hmax()}')
# dimension of approximation space
self.dim = len(self.mesh.coordinates())
print(f'Dimension of solution space is {self.dim}')
self.V = FunctionSpace(self.mesh, 'CG', 1) # CG = P1
self.coords = self.mesh.coordinates()[dof_to_vertex_map(self.V)]
self.T = self.parameters.T # final time
self.w = TrialFunction(self.V)
self.u = TestFunction(self.V)
#######################################################################
# CONTROL SET CREATION
self.control_set = np.linspace(self.alpha_range[0],
self.alpha_range[1],
self.parameters.control_set_size)
self.control_set_size = len(self.control_set)
print(f'Discretized control set has size {self.control_set_size}')
#######################################################################
# BOUNDARY CONDITIONS
parameters.set_boundary_conditions(self.mesh)
self.boundary_markers = MeshFunction('size_t', self.mesh, 1)
self.boundary_markers.set_all(4) # pylint: disable=no-member
for i, omega in self.parameters.omegas.items():
omega.mark(self.boundary_markers, i)
self.ds = Measure('ds',
domain=self.mesh,
subdomain_data=self.boundary_markers)
self.dirichlet_bcs = [
DirichletBC(self.V, parameters.RHS_bound[j], self.boundary_markers,
j) for j in self.parameters.regions["Dirichlet"]
]
# Get indices of dirichlet and robin dofs
self.dirichlet_nodes_list = set()
self.dirichlet_nodes_dict = {}
for j in self.parameters.regions["Dirichlet"]:
bc = DirichletBC(self.V, Constant(0), self.boundary_markers, j)
self.dirichlet_nodes_list |= set(bc.get_boundary_values().keys())
self.dirichlet_nodes_dict[j] = list(
bc.get_boundary_values().keys())
self.robin_nodes_list = set()
self.robin_nodes_dict = {}
for j in self.parameters.regions["Robin"]:
bc = DirichletBC(self.V, Constant(0), self.boundary_markers, j)
self.robin_nodes_list |= set(bc.get_boundary_values().keys())
self.robin_nodes_dict[j] = list(bc.get_boundary_values().keys())
bc = DirichletBC(self.V, Constant(0), 'on_boundary')
self.boundary_nodes_list = bc.get_boundary_values().keys()
self.robint_nodes_list = set()
self.robint_nodes_dict = {}
for j in self.parameters.regions["RobinTime"]:
bc = DirichletBC(self.V, Constant(0), self.boundary_markers, j)
self.robint_nodes_list |= set(bc.get_boundary_values().keys())
self.robint_nodes_dict[j] = list(bc.get_boundary_values().keys())
#######################################################################
# ASSEMBLY
time_start = time.process_time()
self.assemble_diagonal_matrix() # auxilliary generic diagonal matrix
# used for vector*matrix multiplication of dolfin matrices
self.assemble_lumpedmm() # lumped mass matrix
# which serves the role of identity operator
self.assemble_laplacian() # discrete laplacian
self.ad_data_path = f'out/{self.parameters.experiment}'
Path(self.ad_data_path).mkdir(parents=True, exist_ok=True)
self.timesteps = self.parameters.get_number_of_timesteps()
self.assemble_HJBe() # assembly of explicit operators
self.assemble_HJBi() # assembly of implicit operators
self.assemble_RHS() # assembly of forcing term
print('Final time assembly complete')
print(f'Assembly took {time.process_time() - time_start} seconds')
print('===========================================================')
def assemble_diagonal_matrix(self):
print("Assembling auxilliary diagonal matrix")
""" Operator assembled to get the right sparameterssity pattern
(non-zero terms can only exist on diagonal)
"""
mesh3 = UnitIntervalMesh(self.dim)
V3 = FunctionSpace(mesh3, "DG", 0)
wid = TrialFunction(V3)
uid = TestFunction(V3)
self.diag_matrix = assemble(uid * wid * dx)
self.diag_matrix.zero()
def assemble_lumpedmm(self):
""" Assembly lumped mass matrix - equal to 0 on robin boundary since
there is no time derivative there.
"""
print("Assembling lumped mass matrix")
mass_form = self.w * self.u * dx
mass_action_form = action(mass_form, Constant(1))
self.MM_terms = assemble(mass_action_form)
for n in self.robint_nodes_list:
self.MM_terms[n] = 1.0
for n in self.robin_nodes_list:
self.MM_terms[n] = 0.0
self.mass_matrix = assemble(mass_form)
self.mass_matrix.zero()
self.mass_matrix.set_diagonal(self.MM_terms)
self.scipy_mass_matrix = toscipy(self.mass_matrix)
def assemble_laplacian(self):
print("Assembling laplacians")
# laplacian discretisation
self.laplacian = assemble(dot(grad(self.w), grad(self.u)) * dx)
def assemble_HJBe(self, t=None):
""" Assembly explicit operator for every control in the control set,
then apply artificial diffusion to all of them. Note that artificial
diffusion is calculated differently on the boundary nodes.
"""
self.explicit_matrices = np.empty(self.control_set_size, dtype=object)
self.explicit_diffusion = np.empty([self.control_set_size, self.dim])
diagonal_vector = Vector(self.mesh.mpi_comm(), self.dim)
global_min_timestep = float('inf')
# Create explicit operator for each control in control set
for k, alpha in enumerate(self.control_set):
if k % 10 == 0:
print(f'Assembling explicit matrix under control {k}'
f' out of {self.control_set_size}')
# coefficients in the interior
advection_x = self.parameters.adv_x(alpha)
advection_y = self.parameters.adv_y(alpha)
reaction = self.parameters.lin(alpha)
if t is not None:
advection_x.t = t
advection_y.t = t
reaction.t = t
# Discretize PDE (set values on boundary rows to zero)
b = np.array([advection_x, advection_y])
E_interior_form = (np.dot(b, grad(self.w)) +
reaction * self.w) * self.u * dx
explicit_matrix = assemble(E_interior_form)
set_zero_rows(explicit_matrix, self.boundary_nodes_list)
# Calculate diffusion necessary to make explicit operator monotone
min_diffusion = np.zeros(explicit_matrix.size(0))
for rows, row_num in getrows(explicit_matrix,
self.laplacian,
ignore=self.boundary_nodes_list):
min_diffusion[row_num] = calc_ad(rows, row_num)
self.explicit_diffusion[k] = min_diffusion
discrete_diffusion = apply_ad(self.laplacian, min_diffusion)
explicit_matrix += discrete_diffusion
for j in self.parameters.regions['RobinTime']:
self.set_directional_derivative(
explicit_matrix,
region=j,
nodes=self.robint_nodes_dict[j],
control=alpha,
time=t)
explicit_matrix.get_diagonal(diagonal_vector)
current_min_timestep = get_min_timestep(
diagonal_vector, self.MM_terms,
self.dirichlet_nodes_list | self.robin_nodes_list)
global_min_timestep = min(current_min_timestep,
global_min_timestep)
self.explicit_matrices[k] = toscipy(explicit_matrix)
#######################################################################
min_timesteps = int(self.T / global_min_timestep) + 1
if not self.timesteps or self.timesteps < min_timesteps:
self.timesteps = min_timesteps
try:
filename = (f'meshes/{self.parameters.domain}/'
f'Mvalues-{self.parameters.experiment}.json')
with open(filename, 'r') as f:
min_timesteps_dict = json.load(f)
except FileNotFoundError:
min_timesteps_dict = {}
min_timesteps_dict[self.parameters.mesh_name] = self.timesteps
with open(filename, 'w') as f:
json.dump(min_timesteps_dict, f)
self.timestep_size = self.T / self.timesteps # time step size
self.parameters.calculate_save_interval()
self.explicit_matrices = self.timestep_size * self.explicit_matrices - \
np.repeat(self.scipy_mass_matrix, self.control_set_size)
print('Checking if the explicit operators satisfy'
' monotonicity conditions')
for explicit_matrix in self.explicit_matrices:
explicit_check(explicit_matrix, self.dirichlet_nodes_list)
def assemble_RHS(self, t=None):
"""Assemble right hand side of the FBVP
"""
print('Assembling RHS')
self.forcing_terms = np.empty([self.control_set_size, self.dim])
for i, alpha in enumerate(self.control_set):
rhs = self.parameters.RHSt(alpha)
if t is not None:
rhs.t = t
# Initialise forcing term
F = np.array(assemble(rhs * self.u * dx)[:])
for j in self.parameters.regions['RobinTime']:
rhs = self.parameters.RHS_bound[j](alpha)
if t is not None:
rhs.t = t
bc = DirichletBC(self.V, rhs, self.boundary_markers, j)
vals = bc.get_boundary_values()
F[list(vals.keys())] = list(vals.values())
for j in self.parameters.regions['Robin']:
rhs = self.parameters.RHS_bound[j](alpha)
if t is not None:
rhs.t = t
bc = DirichletBC(self.V, rhs, self.boundary_markers, j)
vals = bc.get_boundary_values()
F[list(vals.keys())] = list(vals.values())
self.forcing_terms[i] = self.timestep_size * F
def assemble_HJBi(self, t=None):
""" Assembly matrix discretizing second order terms after diffusion
moved to the explicit operator is subtracted. Whenever amount of
diffusion used to make some row of an explicit operator monotonic
exceeds the amount of natural diffusion at that node we call it
artificial diffusion. In such case, this row of the implicit operator
is multiplied by zero
"""
print('Assembling implicit operators')
self.implicit_matrices = []
remaining_diffusion = Function(self.V)
max_art_dif = 0.0
max_art_dif_loc = None
diffusion_matrix = self.diag_matrix.copy()
for explicit_diffusion, alpha in \
zip(self.explicit_diffusion, self.control_set):
diffusion = self.parameters.diffusion(alpha)
if t is not None:
diffusion.t = t
diff = interpolate(diffusion, self.V).vector()
if not np.all(diff >= 0):
raise Exception("Choose non-negative diffusion")
diff_vec = np.array([
diff[i] if i not in self.boundary_nodes_list else 0.0
for i in range(self.dim)
])
artificial_diffusion = explicit_diffusion - diff_vec
if np.amax(artificial_diffusion) > max_art_dif:
max_art_dif = np.amax(artificial_diffusion)
max_art_dif_loc = self.coords[np.argmax(artificial_diffusion)]
# discretise second order terms
remaining_diffusion.vector()[:] = np.maximum(
-artificial_diffusion, [0] * self.dim)
diffusion_matrix.set_diagonal(remaining_diffusion.vector())
implicit_matrix = matmult(diffusion_matrix, self.laplacian)
for j in self.parameters.regions['Robin']:
self.set_directional_derivative(implicit_matrix,
region=j,
nodes=self.robin_nodes_dict[j],
control=alpha,
time=t)
self.implicit_matrices.append(self.timestep_size *
implicit_matrix + self.mass_matrix)
self.scipy_implicit_matrices = [
toscipy(mat) for mat in self.implicit_matrices
]
print('Checking if the implicit operators satisfy'
' monotonicity conditions')
for implicit_matrix in self.scipy_implicit_matrices:
implicit_check(implicit_matrix)
with open(self.ad_data_path + '/ad.txt', 'a') as f:
time_str = f' at time {t}\n' if t is not None else '\n'
f.write(f'For mesh {self.mesh_name} max value of artificial'
' diffusion coefficient was'
f' {max_art_dif} at {max_art_dif_loc}' + time_str)
def set_directional_derivative(self,
operator,
region,
nodes,
control,
time=None):
if region in self.parameters.regions['Robin']:
adv_x = self.parameters.robin_adv_x[region](control)
adv_y = self.parameters.robin_adv_y[region](control)
lin = self.parameters.robin_lin[region](control)
elif region in self.parameters.regions['RobinTime']:
adv_x = self.parameters.robint_adv_x[region](control)
adv_y = self.parameters.robint_adv_y[region](control)
lin = self.parameters.robint_lin[region](control)
if time is not None:
adv_x.t = time
adv_y.t = time
lin.t = time
b = (interpolate(adv_x, self.V), interpolate(adv_y, self.V))
c = interpolate(lin, self.V)
for n in nodes:
# node coordinates
x = self.coords[n]
# evaluate advection at robin node
b_x = np.array([b[0].vector()[n], b[1].vector()[n]])
# denominator used to calculate directional derivative
if np.linalg.norm(b_x) > 1:
lamb = 0.1 * self.mesh.hmin() / np.linalg.norm(b_x)
else:
lamb = 0.1 * self.mesh.hmin()
# position of first node of the stencil
x_prev = x - lamb * b_x
# Find cell containing first node of stencil and get its
# dof/vertex coordinates
try:
cell_ind = self.mesh.bounding_box_tree(
).compute_entity_collisions(Point(x_prev))[0]
except IndexError:
i = 16
while i > 2:
# sometimes Fenics does not detect nodes if boundary is
# parallel to the boundary advection due to rounding errors
# so try different precisions just to be sure
try:
cell_ind = self.mesh.bounding_box_tree(
).compute_entity_collisions(Point(np.round(x_prev,
i)))[0]
break
except IndexError:
i -= 1
else:
raise Exception(
"Boundary advection outside tangential cone")
cell_vertices = self.mesh.cells()[cell_ind]
cell_dofs = vertex_to_dof_map(self.V)[cell_vertices]
cell_coords = self.mesh.coordinates()[cell_vertices]
# calculate weigth of each vertex in the cell (using
# barycentric coordinates)
A = np.vstack((cell_coords.T, np.ones(3)))
rhs = np.append(x_prev, np.ones(1))
weights = np.linalg.solve(A, rhs)
weights = [w if w > 1e-14 else 0 for w in weights]
dof_to_weight = dict(zip(cell_dofs, weights))
# calculate directional derivative at each node using
# weights to interpolate value of numerical solution at
# x_prev
row = operator.getrow(n)
indices = row[0]
data = row[1]
for dof in cell_dofs:
pos = np.where(indices == dof)[0][0]
if dof != n:
data[pos] = -dof_to_weight[dof] / lamb
else:
c_n = c.vector()[dof]
# make sure reaction term is positive adding artificial
# constant if necessary
if region in self.parameters.regions['Robin']:
c_n = max(c_n, min(lamb, 1E-8))
data[pos] = (1 - dof_to_weight[dof]) / lamb + c_n
operator.set([data], [n], indices)
operator.apply('insert')
