def initial_conditions_wave_system(my_mesh):
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
if (dim != 1):
raise ValueError(
"initial_conditions_wave_system: Mesh dimension should be 1")
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, dim)
U = cdmath.Field("Conservative vector", cdmath.CELLS, my_mesh, dim + 1)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
pressure_field[i] = p0
if (x > 0.5):
velocity_field[i, 0] = 1
else:
velocity_field[i, 0] = -1
U[i, 0] = p0
U[i, 1] = rho0 * velocity_field[i, 0]
return U, pressure_field, velocity_field
def initial_conditions_wave_system(my_mesh):
test_desc["Initial_data"]="Constant pressure, divergence free velocity"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
y = my_mesh.getCell(i).y()
pressure_field[i] = p0
if(dim==1):
velocity_field[i,0] = 1
velocity_field[i,1] = 0
velocity_field[i,2] = 0
elif(dim==2):
velocity_field[i,0] = sin(pi*x)*cos(pi*y)
velocity_field[i,1] = -sin(pi*y)*cos(pi*x)
velocity_field[i,2] = 0
elif(dim==3):
z = my_mesh.getCell(i).z()
velocity_field[i,0] = sin(pi*x)*cos(pi*y)*cos(pi*z)
velocity_field[i,1] = sin(pi*y)*cos(pi*x)*cos(pi*z)
velocity_field[i,2] = -2*sin(pi*z)*cos(pi*x)*cos(pi*y)
return pressure_field, velocity_field
def initial_conditions_square_vortex(my_mesh):
print "Initial data : Square vortex (Constant pressure, divergence free velocity)"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
y = my_mesh.getCell(i).y()
z = my_mesh.getCell(i).z()
pressure_field[i] = p0
if(dim==1):
velocity_field[i,0] = 1
velocity_field[i,1] = 0
velocity_field[i,2] = 0
elif(dim==2):
velocity_field[i,0] = sin(pi*x)*cos(pi*y)
velocity_field[i,1] = -sin(pi*y)*cos(pi*x)
velocity_field[i,2] = 0
elif(dim==3):
velocity_field[i,0] = sin(pi*x)*cos(pi*y)*cos(pi*z)
velocity_field[i,1] = sin(pi*y)*cos(pi*x)*cos(pi*z)
velocity_field[i,2] = -2*sin(pi*z)*cos(pi*x)*cos(pi*y)
return pressure_field, velocity_field
def initial_conditions_RiemannProblem(my_mesh):
print "Initial data : Riemann problem"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
xcentre = 0
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
velocity_field[i, 0] = 0
velocity_field[i, 1] = 0
velocity_field[i, 2] = 0
if x < xcentre:
pressure_field[i] = p0
pass
else:
pressure_field[i] = p0 / 2
pass
pass
return pressure_field, velocity_field
def source_term_and_stat_solution_wave_system(my_mesh):
test_desc["Source_term"] = "True"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
source_vector = cdmath.Vector(nbCells * (dim + 1))
stat_pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
stat_momentum_field = cdmath.Field("Momentum", cdmath.CELLS, my_mesh, 3)
for k in range(nbCells):
x = my_mesh.getCell(k).x()
y = my_mesh.getCell(k).y()
z = my_mesh.getCell(k).z()
if (dim == 1):
source_vector[k * (dim + 1) + 0] = -2 * c0 * c0 * sign(x - 1. / 2)
source_vector[k * (dim + 1) + 1] = 0
stat_pressure_field[k] = abs(x - 1. / 2) * (x - 1. / 2)
stat_momentum_field[k, 0] = -2 * abs(x - 1. / 2)
stat_momentum_field[k, 1] = 0
stat_momentum_field[k, 2] = 0
elif (dim == 2):
source_vector[k * (dim + 1) + 0] = -2 * c0 * c0 * (
sign(x - 1. / 2) * (y - 1. / 2) * abs(y - 1. / 2) +
(x - 1. / 2) * abs(x - 1. / 2) * sign(y - 1. / 2))
source_vector[k * (dim + 1) + 1] = 0
source_vector[k * (dim + 1) + 2] = 0
stat_pressure_field[k] = (x - 1. / 2) * abs(x - 1. / 2) * (
y - 1. / 2) * abs(y - 1. / 2)
stat_momentum_field[
k, 0] = -2 * abs(x - 1. / 2) * (y - 1. / 2) * abs(y - 1. / 2)
stat_momentum_field[
k, 1] = -2 * (x - 1. / 2) * abs(x - 1. / 2) * abs(y - 1. / 2)
stat_momentum_field[k, 2] = 0
elif (dim == 3):
source_vector[k * (dim + 1) + 0] = -2 * c0 * c0 * (
sign(x - 1. / 2) * (y - 1. / 2) * abs(y - 1. / 2) *
(z - 1. / 2) * abs(z - 1. / 2) +
(x - 1. / 2) * abs(x - 1. / 2) * sign(y - 1. / 2) *
(z - 1. / 2) * abs(z - 1. / 2) +
(x - 1. / 2) * abs(x - 1. / 2) *
(y - 1. / 2) * abs(y - 1. / 2) * sign(z - 1. / 2))
source_vector[k * (dim + 1) + 1] = 0
source_vector[k * (dim + 1) + 2] = 0
source_vector[k * (dim + 1) + 3] = 0
stat_pressure_field[k] = (x - 1. / 2) * abs(x - 1. / 2) * (
y - 1. / 2) * abs(y - 1. / 2) * (z - 1. / 2) * abs(z - 1. / 2)
stat_momentum_field[k, 0] = -2 * abs(x - 1. / 2) * (
y - 1. / 2) * abs(y - 1. / 2) * (z - 1. / 2) * abs(z - 1. / 2)
stat_momentum_field[k, 1] = -2 * (x - 1. / 2) * abs(
x - 1. / 2) * abs(y - 1. / 2) * (z - 1. / 2) * abs(z - 1. / 2)
stat_momentum_field[k, 2] = -2 * (x - 1. / 2) * abs(x - 1. / 2) * (
y - 1. / 2) * abs(y - 1. / 2) * abs(z - 1. / 2)
return source_vector, stat_pressure_field, stat_momentum_field
def source_term_and_stat_solution_wave_system(my_mesh):
test_desc["Source_term"] = "True"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
source_vector = cdmath.Vector(nbCells * (dim + 1))
stat_pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
stat_velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for k in range(nbCells):
x = my_mesh.getCell(k).x()
y = my_mesh.getCell(k).y()
z = my_mesh.getCell(k).z()
if (dim == 1):
source_vector[k * (dim + 1) +
0] = c0 * c0 * 4 * pi * pi * sin(2 * pi * x)
source_vector[k * (dim + 1) + 1] = 0
stat_pressure_field[k] = sin(2 * pi * x)
stat_velocity_field[k, 0] = -2 * pi * cos(2 * pi * x) / rho0
stat_velocity_field[k, 1] = 0
stat_velocity_field[k, 2] = 0
elif (dim == 2):
source_vector[k * (dim + 1) + 0] = 2 * c0 * c0 * 4 * pi * pi * sin(
2 * pi * x) * sin(2 * pi * y)
source_vector[k * (dim + 1) + 1] = 0
source_vector[k * (dim + 1) + 2] = 0
stat_pressure_field[k] = sin(2 * pi * x) * sin(2 * pi * y)
stat_velocity_field[k, 0] = -2 * pi * cos(2 * pi * x) * sin(
2 * pi * y) / rho0
stat_velocity_field[k, 1] = -2 * pi * sin(2 * pi * x) * cos(
2 * pi * y) / rho0
stat_velocity_field[k, 2] = 0
elif (dim == 3):
source_vector[k * (dim + 1) + 0] = 3 * c0 * c0 * 4 * pi * pi * sin(
2 * pi * x) * sin(2 * pi * y) * sin(2 * pi * z)
source_vector[k * (dim + 1) + 1] = 0
source_vector[k * (dim + 1) + 2] = 0
source_vector[k * (dim + 1) + 3] = 0
stat_pressure_field[k] = sin(2 * pi * x) * sin(2 * pi * y) * sin(
2 * pi * z)
stat_velocity_field[k, 0] = -2 * pi * cos(2 * pi * x) * sin(
2 * pi * y) * sin(2 * pi * z) / rho0
stat_velocity_field[k, 1] = -2 * pi * sin(2 * pi * x) * cos(
2 * pi * y) * sin(2 * pi * z) / rho0
stat_velocity_field[k, 2] = -2 * pi * sin(2 * pi * x) * sin(
2 * pi * y) * cos(2 * pi * z) / rho0
return source_vector, stat_pressure_field, stat_velocity_field
def initial_conditions_wave_system_staggered(my_mesh):
test_desc["Initial_data"] = "Constant pressure, divergence free velocity"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
if (not my_mesh.isStructured()):
raise ValueError("WaveSystemStaggered: the mesh should be structured")
NxNyNz = my_mesh.getCellGridStructure()
DxDyDz = my_mesh.getDXYZ()
dx = DxDyDz[0]
if (dim >= 2):
dy = DxDyDz[1]
if (dim == 3):
dz = DxDyDz[2]
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
Ci = my_mesh.getCell(i)
x = Ci.x()
pressure_field[i] = p0
#We take only the normal component of the velocity on a cartesian grid
#We save the x component from the back face, the y component from the left face and the z component from the bottom face
#Warning : boundary values should be the same for left and right as well as top and down (front and back in 3D) boundaries
if (dim == 2):
y = Ci.y()
velocity_field[i, 0] = sin(pi * (x - 0.5 * dx)) * cos(
pi * y) # value on the left face
velocity_field[i, 1] = -sin(pi * (y - 0.5 * dy)) * cos(
pi * x) # value on the bottom face
velocity_field[i, 2] = 0
elif (dim == 3):
y = Ci.y()
z = Ci.z()
velocity_field[i,
0] = sin(pi * (x - 0.5 * dx)) * cos(pi * y) * cos(
pi * z) # value on the back face
velocity_field[i,
1] = sin(pi * (y - 0.5 * dy)) * cos(pi * x) * cos(
pi * z) # value on the left face
velocity_field[i, 2] = -2 * sin(pi * (z - 0.5 * dz)) * cos(
pi * x) * cos(pi * y) # value on the bottom face
else:
raise ValueError(
"initial_conditions_wave_system_staggered: the 1D mesh not yet implemented"
)
return pressure_field, velocity_field
def initial_conditions_shock(my_mesh, isCircle):
print "Initial data : Spherical wave"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
rayon = 0.15
if (not isCircle):
xcentre = 0.5
ycentre = 0.5
zcentre = 0.5
else:
xcentre = 0.
ycentre = 0.
zcentre = 0.
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
velocity_field[i, 0] = 0
velocity_field[i, 1] = 0
velocity_field[i, 2] = 0
x = my_mesh.getCell(i).x()
valX = (x - xcentre) * (x - xcentre)
if (dim == 1):
val = sqrt(valX)
if (dim == 2):
y = my_mesh.getCell(i).y()
valY = (y - ycentre) * (y - ycentre)
val = sqrt(valX + valY)
if (dim == 3):
y = my_mesh.getCell(i).y()
z = my_mesh.getCell(i).z()
valY = (y - ycentre) * (y - ycentre)
valZ = (z - zcentre) * (z - zcentre)
val = sqrt(valX + valY + valZ)
if val < rayon:
pressure_field[i] = p0
pass
else:
pressure_field[i] = p0 / 2
pass
pass
return pressure_field, velocity_field
def source_term_and_stat_solution_transport_equation(my_mesh,velocity):
test_desc["Source_term"]="True"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
source_vector = cdmath.Vector(nbCells)
stat_field = cdmath.Field("Stationary field", cdmath.CELLS, my_mesh, 1)
for k in range(nbCells):
x = my_mesh.getCell(k).x()
y = my_mesh.getCell(k).y()
z = my_mesh.getCell(k).z()
if(dim==1):
source_vector[k] = 2*pi*velocity[0]*cos(2*pi*x)
stat_field[k] = sin(2*pi*x)
elif(dim==2):
source_vector[k*(dim+1)+0] = 2*pi*(velocity[0]*cos(2*pi*x)*sin(2*pi*y)+velocity[1]*sin(2*pi*x)*cos(2*pi*y))
stat_field[k] = sin(2*pi*x)*sin(2*pi*y)
elif(dim==3):
source_vector[k*(dim+1)+0] = 2*pi*(velocity[0]*cos(2*pi*x)*sin(2*pi*y)*sin(2*pi*z)+velocity[1]*sin(2*pi*x)*cos(2*pi*y)*sin(2*pi*z)+velocity[2]*sin(2*pi*x)*sin(2*pi*y)*cos(2*pi*z))
stat_field[k] = sin(2*pi*x)*sin(2*pi*y)*sin(2*pi*z)
return source_vector, stat_field
def initial_conditions_wave_system(my_mesh):
print "Spherical wave initial data"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
rayon = 0.15
xcentre = 0.5
ycentre = 0.5
zcentre = 0.5
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
U = cdmath.Field("Conservative vector", cdmath.CELLS, my_mesh, dim + 1)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
y = my_mesh.getCell(i).y()
velocity_field[i, 0] = 0
velocity_field[i, 1] = 0
velocity_field[i, 2] = 0
valX = (x - xcentre) * (x - xcentre)
valY = (y - ycentre) * (y - ycentre)
if (dim == 2):
val = sqrt(valX + valY)
if (dim == 3):
z = my_mesh.getCell(i).z()
valZ = (z - zcentre) * (z - zcentre)
val = sqrt(valX + valY + valZ)
if val < rayon:
pressure_field[i] = p0
pass
else:
pressure_field[i] = p0 / 2
pass
pass
U[i, 0] = pressure_field[i]
U[i, 1] = rho0 * velocity_field[i, 0]
U[i, 2] = rho0 * velocity_field[i, 1]
return U, pressure_field, velocity_field
def main():
a = -5.0
b = 5.0
nx = 1000
ntmax = 1000
dx = (b - a) / nx
pi = 3.1415927
# Transport velocity
cfl = 0.5
u = 3.
dt = cfl * dx / u
my_mesh = cdmath.Mesh(a, b, nx)
conc = cdmath.Field("Concentration", cdmath.CELLS, my_mesh, 1)
# Initial conditions
sigma = math.sqrt(0.2)
for i in xrange(my_mesh.getNumberOfCells()):
x = my_mesh.getCell(i).x()
conc[i] = 0.5 / (sigma * math.sqrt(2 * pi)) * math.exp(-0.5 * math.pow(
(x / sigma), 2))
pass
time = 0.
tmax = 3.0
it = 0
print("MED post-treatment of the solution at T=" + str(time) + "…")
output_filename = "EqTr1D"
conc.setTime(time, it)
conc.writeMED(output_filename)
conc.writeVTK(output_filename)
conc.writeCSV(output_filename)
output_freq = 10
# Time loop
while (it < ntmax and time <= tmax):
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " +
str(dt))
conc[0] = conc[0] - u * dt / dx * (
conc[0] - conc[my_mesh.getNumberOfCells() - 1])
for j in xrange(1, my_mesh.getNumberOfCells()):
conc[j] = conc[j] - u * dt / dx * (conc[j] - conc[j - 1])
pass
time += dt
it += 1
if (it % output_freq == 0):
conc.setTime(time, it)
conc.writeMED(output_filename, False)
conc.writeVTK(output_filename, False)
conc.writeCSV(output_filename)
pass
pass
print("CDMATH calculation done.")
return
def initial_conditions_disk_vortex(my_mesh):
print "Disk vortex initial data"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
if(dim!=2):
raise ValueError("Wave system on disk : mesh dimension should be 2")
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for i in range(nbCells):
x = my_mesh.getCell(i).x()
y = my_mesh.getCell(i).y()
pressure_field[i] = p0
velocity_field[i,0] = -y
velocity_field[i,1] = x
velocity_field[i,2] = 0
return pressure_field, velocity_field
def initial_conditions(my_mesh):
rayon = 0.15
xcentre = 0.25
ycentre = 0.25
y_field = cdmath.Field("Y field", cdmath.CELLS, my_mesh, 1)
nbCells = my_mesh.getNumberOfCells()
for j in range(nbCells):
x = my_mesh.getCell(j).x()
y = my_mesh.getCell(j).y()
valX = (x - xcentre) * (x - xcentre)
valY = (y - ycentre) * (y - ycentre)
val = math.sqrt(valX + valY)
if val < rayon:
y_field[j] = 1.0
pass
else:
y_field[j] = 0.0
pass
pass
return y_field
def initial_conditions_transport_equation(my_mesh):
print "Initial_data", "Shock"
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
initial_field = cdmath.Field("unknown", cdmath.CELLS, my_mesh, 1)
rayon = 0.15
xcentre = 0.5
ycentre = 0.5
zcentre = 0.5
for i in range(nbCells):
x = my_mesh.getCell(i).x()
valX = (x - xcentre) * (x - xcentre)
if (dim == 1):
val = sqrt(valX)
if (dim == 2):
y = my_mesh.getCell(i).y()
valY = (y - ycentre) * (y - ycentre)
val = sqrt(valX + valY)
if (dim == 3):
y = my_mesh.getCell(i).y()
z = my_mesh.getCell(i).z()
valY = (y - ycentre) * (y - ycentre)
valZ = (z - zcentre) * (z - zcentre)
val = sqrt(valX + valY + valZ)
if val < rayon:
initial_field[i] = 1
pass
else:
initial_field[i] = 0
pass
pass
return initial_field
my_mesh.setGroupAtPlan(1.,0,eps,"DirichletBorder") #Bord DROIT
my_mesh.setGroupAtPlan(0.,1,eps,"DirichletBorder") #Bord BAS
my_mesh.setGroupAtPlan(1.,1,eps,"DirichletBorder") #Bord HAUT
#my_mesh.setGroupAtPlan(0,2,eps,"DirichletBorder") #Bord AVANT si calcul 3D
#my_mesh.setGroupAtPlan(1,2,eps,"DirichletBorder") #Bord ARRIERE si calcul 3D
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
print("Fin chargement du maillage")
print("nb of nodes = ", nbNodes)
print("nb of cells = ", nbCells)
# Discrétisation du second membre et détermination des noeuds intérieurs
#=======================================================================
my_RHS_field = cdmath.Field("RHS field", cdmath.NODES, my_mesh, 1)
nbInteriorNodes = 0
nbBoundaryNodes = 0
maxNbNeighbours = 0#This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
interiorNodes = []
boundaryNodes = []
# Parcours des noeuds pour discrétisation du second membre et extraction 1) des noeuds intérieur 2) des noeuds frontière 3) du nb max voisins d'un noeud
for i in range(nbNodes):
Ni = my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
#z=Ni.z() si calcul 3D
my_RHS_field[i] = 2*pi*pi*sin(pi*x)*sin(pi*y)#mettre la fonction definie au second membre de l'edp
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
boundaryNodes.append(i)
def solve(filename,resolution,meshType, testColor):
start = time.time()
test_desc["Mesh_type"]=meshType
test_desc["Test_color"]=testColor
#Chargement du maillage triangulaire de la sphère
#=======================================================================================
my_mesh = cdmath.Mesh(filename+".med")
if(not my_mesh.isTriangular()) :
raise ValueError("Wrong cell types : mesh is not made of triangles")
if(my_mesh.getMeshDimension()!=2) :
raise ValueError("Wrong mesh dimension : expected a surface of dimension 2")
if(my_mesh.getSpaceDimension()!=3) :
raise ValueError("Wrong space dimension : expected a space of dimension 3")
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
test_desc["Space_dimension"]=my_mesh.getSpaceDimension()
test_desc["Mesh_dimension"]=my_mesh.getMeshDimension()
test_desc["Mesh_number_of_elements"]=my_mesh.getNumberOfNodes()
test_desc["Mesh_cell_type"]=my_mesh.getElementTypes()
print("Mesh building/loading done")
print("nb of nodes=", nbNodes)
print("nb of cells=", nbCells)
#Discrétisation du second membre et détermination des noeuds intérieurs
#======================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.NODES, my_mesh, 1)
maxNbNeighbours = 0#This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des noeuds pour discrétisation du second membre et extraction du nb max voisins d'un noeud
for i in range(nbNodes):
Ni=my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
z = Ni.z()
my_RHSfield[i]=12*y*(3*x*x-y*y)/pow(x*x+y*y+z*z,3/2)#vecteur propre du laplacien sur la sphère
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
raise ValueError("Mesh should not contain borders")
else:
maxNbNeighbours = max(1+Ni.getNumberOfCells(),maxNbNeighbours)
test_desc["Mesh_max_number_of_neighbours"]=maxNbNeighbours
print("Right hand side discretisation done")
print("Max nb of neighbours=", maxNbNeighbours)
print("Integral of the RHS", my_RHSfield.integral(0))
# Construction de la matrice de rigidité et du vecteur second membre du système linéaire
#=======================================================================================
Rigidite=cdmath.SparseMatrixPetsc(nbNodes,nbNodes,maxNbNeighbours)# warning : third argument is number of non zero coefficients per line
RHS=cdmath.Vector(nbNodes)
# Vecteurs gradient de la fonction de forme associée à chaque noeud d'un triangle
GradShapeFunc0=cdmath.Vector(3)
GradShapeFunc1=cdmath.Vector(3)
GradShapeFunc2=cdmath.Vector(3)
normalFace0=cdmath.Vector(3)
normalFace1=cdmath.Vector(3)
#On parcourt les triangles du domaine
for i in range(nbCells):
Ci=my_mesh.getCell(i)
#Contribution à la matrice de rigidité
nodeId0=Ci.getNodeId(0)
nodeId1=Ci.getNodeId(1)
nodeId2=Ci.getNodeId(2)
N0=my_mesh.getNode(nodeId0)
N1=my_mesh.getNode(nodeId1)
N2=my_mesh.getNode(nodeId2)
#Build normal to cell Ci
normalFace0[0]=Ci.getNormalVector(0,0)
normalFace0[1]=Ci.getNormalVector(0,1)
normalFace0[2]=Ci.getNormalVector(0,2)
normalFace1[0]=Ci.getNormalVector(1,0)
normalFace1[1]=Ci.getNormalVector(1,1)
normalFace1[2]=Ci.getNormalVector(1,2)
normalCell = normalFace0.crossProduct(normalFace1)
test = normalFace0.tensProduct(normalFace1)
normalCell = normalCell/normalCell.norm()
cellMat=cdmath.Matrix(4)
cellMat[0,0]=N0.x()
cellMat[0,1]=N0.y()
cellMat[0,2]=N0.z()
cellMat[1,0]=N1.x()
cellMat[1,1]=N1.y()
cellMat[1,2]=N1.z()
cellMat[2,0]=N2.x()
cellMat[2,1]=N2.y()
cellMat[2,2]=N2.z()
cellMat[3,0]=normalCell[0]
cellMat[3,1]=normalCell[1]
cellMat[3,2]=normalCell[2]
cellMat[0,3]=1
cellMat[1,3]=1
cellMat[2,3]=1
cellMat[3,3]=0
#Formule des gradients voir EF P1 -> calcul déterminants
GradShapeFunc0[0]= cellMat.partMatrix(0,0).determinant()/2
GradShapeFunc0[1]=-cellMat.partMatrix(0,1).determinant()/2
GradShapeFunc0[2]= cellMat.partMatrix(0,2).determinant()/2
GradShapeFunc1[0]=-cellMat.partMatrix(1,0).determinant()/2
GradShapeFunc1[1]= cellMat.partMatrix(1,1).determinant()/2
GradShapeFunc1[2]=-cellMat.partMatrix(1,2).determinant()/2
GradShapeFunc2[0]= cellMat.partMatrix(2,0).determinant()/2
GradShapeFunc2[1]=-cellMat.partMatrix(2,1).determinant()/2
GradShapeFunc2[2]= cellMat.partMatrix(2,2).determinant()/2
#Création d'un tableau (numéro du noeud, gradient de la fonction de forme
GradShapeFuncs={nodeId0 : GradShapeFunc0}
GradShapeFuncs[nodeId1]=GradShapeFunc1
GradShapeFuncs[nodeId2]=GradShapeFunc2
# Remplissage de la matrice de rigidité et du second membre
for j in [nodeId0,nodeId1,nodeId2] :
#Ajout de la contribution de la cellule triangulaire i au second membre du noeud j
RHS[j]=Ci.getMeasure()/3*my_RHSfield[j]+RHS[j] # intégrale dans le triangle du produit f x fonction de base
#Contribution de la cellule triangulaire i à la ligne j du système linéaire
for k in [nodeId0,nodeId1,nodeId2] :
Rigidite.addValue(j,k,GradShapeFuncs[j]*GradShapeFuncs[k]/Ci.getMeasure())
print("Linear system matrix building done")
# Résolution du système linéaire
#=================================
LS=cdmath.LinearSolver(Rigidite,RHS,100,1.E-2,"CG","ILU")#Remplacer CG par CHOLESKY pour solveur direct
LS.isSingular()#En raison de l'absence de bord
LS.setComputeConditionNumber()
SolSyst=LS.solve()
print "Preconditioner used : ", LS.getNameOfPc()
print "Number of iterations used : ", LS.getNumberOfIter()
print "Final residual : ", LS.getResidu()
print("Linear system solved")
test_desc["Linear_solver_algorithm"]=LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"]=LS.getNameOfPc()
test_desc["Linear_solver_precision"]=LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"]=LS.getNumberMaxOfIter()
test_desc["Linear_system_max_actual_iterations_number"]=LS.getNumberOfIter()
test_desc["Linear_system_max_actual_error"]=LS.getResidu()
test_desc["Linear_system_max_actual_condition number"]=LS.getConditionNumber()
# Création du champ résultat
#===========================
my_ResultField = cdmath.Field("ResultField", cdmath.NODES, my_mesh, 1)
for j in range(nbNodes):
my_ResultField[j]=SolSyst[j];#remplissage des valeurs pour les noeuds intérieurs
#sauvegarde sur le disque dur du résultat dans un fichier paraview
my_ResultField.writeVTK("FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes))
end = time.time()
print("Integral of the numerical solution", my_ResultField.integral(0))
print("Numerical solution of poisson equation on a sphere using finite elements done")
#Calcul de l'erreur commise par rapport à la solution exacte
#===========================================================
#The following formulas use the fact that the exact solution is equal the right hand side divided by 12
max_abs_sol_exacte=0
erreur_abs=0
max_sol_num=0
min_sol_num=0
for i in range(nbNodes) :
if max_abs_sol_exacte < abs(my_RHSfield[i]) :
max_abs_sol_exacte = abs(my_RHSfield[i])
if erreur_abs < abs(my_RHSfield[i]/12 - my_ResultField[i]) :
erreur_abs = abs(my_RHSfield[i]/12 - my_ResultField[i])
if max_sol_num < my_ResultField[i] :
max_sol_num = my_ResultField[i]
if min_sol_num > my_ResultField[i] :
min_sol_num = my_ResultField[i]
max_abs_sol_exacte = max_abs_sol_exacte/12
print("Absolute error = max(| exact solution - numerical solution |) = ",erreur_abs )
print("Relative error = max(| exact solution - numerical solution |)/max(| exact solution |) = ",erreur_abs/max_abs_sol_exacte)
print ("Maximum numerical solution = ", max_sol_num, " Minimum numerical solution = ", min_sol_num)
test_desc["Computational_time_taken_by_run"]=end-start
test_desc["Absolute_error"]=erreur_abs
test_desc["Relative_error"]=erreur_abs/max_abs_sol_exacte
#Postprocessing :
#================
# save 3D picture
PV_routines.Save_PV_data_to_picture_file("FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes)+'_0.vtu',"ResultField",'NODES',"FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes))
# save 3D clip
VTK_routines.Clip_VTK_data_to_VTK("FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes)+'_0.vtu',"Clip_VTK_data_to_VTK_"+ "FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes)+'_0.vtu',[0.25,0.25,0.25], [-0.5,-0.5,-0.5],resolution )
PV_routines.Save_PV_data_to_picture_file("Clip_VTK_data_to_VTK_"+"FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes)+'_0.vtu',"ResultField",'NODES',"Clip_VTK_data_to_VTK_"+"FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes))
# save plot around circumference
finiteElementsOnSphere_0vtu = pvs.XMLUnstructuredGridReader(FileName=["FiniteElementsOnSpherePoisson_"+meshType+str(nbNodes)+'_0.vtu'])
slice1 = pvs.Slice(Input=finiteElementsOnSphere_0vtu)
slice1.SliceType.Normal = [0.5, 0.5, 0.5]
renderView1 = pvs.GetActiveViewOrCreate('RenderView')
finiteElementsOnSphere_0vtuDisplay = pvs.Show(finiteElementsOnSphere_0vtu, renderView1)
pvs.ColorBy(finiteElementsOnSphere_0vtuDisplay, ('POINTS', 'ResultField'))
slice1Display = pvs.Show(slice1, renderView1)
pvs.SaveScreenshot("./FiniteElementsOnSpherePoisson"+"_Slice_"+meshType+str(nbNodes)+'.png', magnification=1, quality=100, view=renderView1)
plotOnSortedLines1 = pvs.PlotOnSortedLines(Input=slice1)
pvs.SaveData('./FiniteElementsOnSpherePoisson_PlotOnSortedLines'+meshType+str(nbNodes)+'.csv', proxy=plotOnSortedLines1)
lineChartView2 = pvs.CreateView('XYChartView')
plotOnSortedLines1Display = pvs.Show(plotOnSortedLines1, lineChartView2)
plotOnSortedLines1Display.UseIndexForXAxis = 0
plotOnSortedLines1Display.XArrayName = 'arc_length'
plotOnSortedLines1Display.SeriesVisibility = ['ResultField (1)']
pvs.SaveScreenshot("./FiniteElementsOnSpherePoisson"+"_PlotOnSortedLine_"+meshType+str(nbNodes)+'.png', magnification=1, quality=100, view=lineChartView2)
pvs.Delete(lineChartView2)
with open('test_Poisson'+str(my_mesh.getMeshDimension())+'D_EF_'+meshType+str(nbCells)+ "Cells.json", 'w') as outfile:
json.dump(test_desc, outfile)
return erreur_abs/max_abs_sol_exacte, nbNodes, min_sol_num, max_sol_num, end - start
def sigma_flux(VitesseX, VitesseY, cfl, y_field, indexFacesPerio):
# Fluxes calculation #
SumFlux = cdmath.Field("Fluxes", cdmath.CELLS, y_field.getMesh(), 1)
my_mesh = y_field.getMesh()
nbCells = my_mesh.getNumberOfCells()
normU = math.sqrt(VitesseX * VitesseX + VitesseY * VitesseY)
for j in range(nbCells):
Cj = my_mesh.getCell(j)
nbFace = Cj.getNumberOfFaces()
SumF = 0.0
minlengthFk = 1.E30
for k in range(nbFace):
indexFace = Cj.getFacesId()[k]
Fk = my_mesh.getFace(indexFace)
NormalX = Cj.getNormalVector(k, 0)
NormalY = Cj.getNormalVector(k, 1)
LengthFk = Fk.getMeasure()
UN = VitesseX * NormalX + VitesseY * NormalY
minlengthFk = min(minlengthFk, LengthFk / abs(UN))
minlengthFk = min(minlengthFk, LengthFk / abs(VitesseX))
minlengthFk = min(minlengthFk, LengthFk / abs(VitesseY))
conc = 0.0
cellCourante = j
cellAutre = -1
if (not Fk.isBorder()):
indexC1 = Fk.getCellsId()[0]
indexC2 = Fk.getCellsId()[1]
# hypothese: La cellule d'index indexC1 est la cellule courante index j #
if (indexC1 == j):
# hypothese verifie #
cellCourante = indexC1
cellAutre = indexC2
pass
elif (indexC2 == j):
# hypothese non verifie #
cellCourante = indexC2
cellAutre = indexC1
pass
# definir la cellule gauche et droite par le prduit vitesse * normale sortante
# si u*n>0 : rien a faire sinon inverser la gauche et la droite
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = y_field[cellAutre]
pass
pass
else:
# conditions aux limites neumann homogene #
if (Fk.getGroupName() == "LeftEdge" or Fk.getGroupName() == "RightEdge"):
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = 0.0
pass
pass
# conditions aux limites periodiques #
if (Fk.getGroupName() == "BottomEdge" or Fk.getGroupName() == "TopEdge"):
indexFP = indexFacesPerio[indexFace]
# une autre manière de recuperer l'index de la face periodique #
# int indexFP=my_mesh.getIndexFacePeriodic(indexFace);
Fp = my_mesh.getFace(indexFP)
indexCp = Fp.getCellsId()[0]
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = y_field[indexCp]
pass
pass
pass
SumF = SumF + UN * LengthFk * conc
pass
dt = cfl * minlengthFk / normU
SumFlux[j] = dt * SumF / Cj.getMeasure()
pass
return dt, SumFlux
def WaveSystem1DVF(ntmax, tmax, cfl, my_mesh, output_freq, resolution):
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
dt = 0.
time = 0.
it = 0
isStationary = False
SumFluxes = cdmath.Field("Fluxes", cdmath.CELLS, my_mesh, dim + 1)
# Initial conditions #
print("Construction of the initial condition …")
U, pressure_field, velocity_field = initial_conditions_wave_system(my_mesh)
dx_min = my_mesh.minRatioVolSurf()
dt = cfl * dx_min / c0
print(
"Starting computation of the linear wave system with an UPWIND explicit scheme …"
)
# Starting time loop
while (it < ntmax and time <= tmax and not isStationary):
computeFluxes(U, SumFluxes)
SumFluxes *= dt
maxVector = SumFluxes.normMax()
isStationary = maxVector[0] / p0 < precision and maxVector[
1] / rho0 < precision
U -= SumFluxes
time = time + dt
it = it + 1
#Sauvegardes
if (it % output_freq == 0 or it >= ntmax or isStationary
or time >= tmax):
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " +
str(dt))
print "Variation temporelle relative : pressure ", maxVector[
0] / p0, ", velocity x", maxVector[1] / rho0
print
for k in range(nbCells):
pressure_field[k] = U[k, 0]
velocity_field[k, 0] = U[k, 1] / rho0
pressure_field.setTime(time, it)
pressure_field.writeCSV("WaveSystem1DUpwind_pressure")
velocity_field.setTime(time, it)
velocity_field.writeCSV("WaveSystem1DUpwind_velocity")
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " + str(dt))
print "|| Un+1 - Un || : pressure ", maxVector[
0] / p0, ", velocity x", maxVector[1] / rho0
print
if (it >= ntmax):
print "Nombre de pas de temps maximum ntmax= ", ntmax, " atteint"
raise ValueError(
"Maximum number of time steps reached : Stationary state not found !!!!!!!"
)
elif (isStationary):
print "Régime stationnaire atteint au pas de temps ", it, ", t= ", time
for k in range(nbCells):
pressure_field[k] = U[k, 0]
velocity_field[k, 0] = U[k, 1] / rho0
pressure_field.setTime(time, 0)
pressure_field.writeCSV("WaveSystem1DUpwind_pressure_Stat")
velocity_field.setTime(time, 0)
velocity_field.writeCSV("WaveSystem1DUpwind_velocity_Stat")
else:
print "Temps maximum Tmax= ", tmax, " atteint"
raise ValueError(
"Maximum time reached : Stationary state not found !!!!!!!")
def solve(my_mesh, filename, resolution, meshType, testColor):
start = time.time()
test_desc["Mesh_type"] = meshType
test_desc["Test_color"] = testColor
nbCells = my_mesh.getNumberOfCells()
if (my_mesh.getSpaceDimension() != 2 or my_mesh.getMeshDimension() != 2):
raise ValueError(
"Wrong space or mesh dimension : space and mesh dimensions should be 2"
)
test_desc["Space_dimension"] = my_mesh.getSpaceDimension()
test_desc["Mesh_dimension"] = my_mesh.getMeshDimension()
test_desc["Mesh_number_of_elements"] = my_mesh.getNumberOfCells()
test_desc["Mesh_cell_type"] = my_mesh.getElementTypesNames()
print("Mesh groups done")
print("Number of cells = ", nbCells)
#Discrétisation du second membre et extraction du nb max de voisins d'une cellule
#================================================================================
my_ExactSol = cdmath.Field("Exact_field", cdmath.CELLS, my_mesh, 1)
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
eps = 1e-6 #For coarse meshes
#parcours des cellules pour discrétisation du second membre et extraction du nb max de voisins d'une cellule
for i in range(nbCells):
Ci = my_mesh.getCell(i)
x = Ci.x()
y = Ci.y()
#Robust calculation of atan(2x/(x**2+y**2-1)
if x**2 + y**2 - 1 > eps:
print("!!! Warning Mesh ", meshType,
" !!! Cell is not in the unit disk.", ", eps=", eps,
", x**2+y**2 - 1=", x**2 + y**2 - 1)
#raise ValueError("Exact solution computation !!! Domain should be the unit disk.")
if x**2 + y**2 - 1 < -eps:
my_ExactSol[i] = atan(2 * x / (x**2 + y**2 - 1))
elif x > 0: #x**2+y**2-1>=0
my_ExactSol[i] = -pi / 2
elif x < 0: #x**2+y**2-1>=0
my_ExactSol[i] = pi / 2
else: #x=0
my_ExactSol[i] = 0
# compute maximum number of neighbours
maxNbNeighbours = max(1 + Ci.getNumberOfFaces(), maxNbNeighbours)
test_desc["Mesh_max_number_of_neighbours"] = maxNbNeighbours
print("Right hand side discretisation done")
print("Max nb of neighbours=", maxNbNeighbours)
# Construction de la matrice et du vecteur second membre du système linéaire
#===========================================================================
Rigidite = cdmath.SparseMatrixPetsc(
nbCells, nbCells, maxNbNeighbours
) # warning : third argument is maximum number of non zero coefficients per line of the matrix
RHS = cdmath.Vector(nbCells)
#Parcours des cellules du domaine
for i in range(nbCells):
Ci = my_mesh.getCell(i)
for j in range(Ci.getNumberOfFaces()): # parcours des faces voisinnes
Fj = my_mesh.getFace(Ci.getFaceId(j))
if not Fj.isBorder():
k = Fj.getCellId(0)
if k == i:
k = Fj.getCellId(1)
Ck = my_mesh.getCell(k)
distance = Ci.getBarryCenter().distance(Ck.getBarryCenter())
coeff = Fj.getMeasure() / Ci.getMeasure() / distance
Rigidite.setValue(i, k, -coeff) # terme extradiagonal
else:
coeff = Fj.getMeasure() / Ci.getMeasure() / Ci.getBarryCenter(
).distance(Fj.getBarryCenter())
#For the particular case where the mesh boundary does not coincide with the domain boundary
x = Fj.getBarryCenter().x()
y = Fj.getBarryCenter().y()
if x**2 + y**2 - 1 > eps:
print("!!! Warning Mesh ", meshType,
" !!! Face is not in the unit disk.", ", eps=", eps,
", x**2+y**2 - 1=", x**2 + y**2 - 1)
#raise ValueError("!!! Domain should be the unit disk.")
if x**2 + y**2 - 1 < -eps:
RHS[i] += coeff * atan(2 * x / (x**2 + y**2 - 1))
elif x > 0: #x**2+y**2-1>=0
RHS[i] += coeff * (-pi / 2)
elif x < 0: #x**2+y**2-1>=0
RHS[i] += coeff * pi / 2
else: #x=0
RHS[i] += 0
Rigidite.addValue(i, i, coeff) # terme diagonal
print("Linear system matrix building done")
# Résolution du système linéaire
#=================================
LS = cdmath.LinearSolver(Rigidite, RHS, 500, 1.E-6, "CG", "LU")
LS.setComputeConditionNumber()
SolSyst = LS.solve()
print("Preconditioner used : ", LS.getNameOfPc())
print("Number of iterations used : ", LS.getNumberOfIter())
print("Linear system solved")
test_desc["Linear_solver_algorithm"] = LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"] = LS.getNameOfPc()
test_desc["Linear_solver_precision"] = LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"] = LS.getNumberMaxOfIter()
test_desc[
"Linear_system_max_actual_iterations_number"] = LS.getNumberOfIter()
test_desc["Linear_system_max_actual_error"] = LS.getResidu()
test_desc[
"Linear_system_max_actual_condition number"] = LS.getConditionNumber()
# Création du champ résultat
#===========================
my_ResultField = cdmath.Field("ResultField", cdmath.CELLS, my_mesh, 1)
for i in range(nbCells):
my_ResultField[i] = SolSyst[i]
#sauvegarde sur le disque dur du résultat dans un fichier paraview
my_ResultField.writeVTK("FiniteVolumes2DPoissonStiffBC_DISK_" + meshType +
str(nbCells))
my_ExactSol.writeVTK("ExactSol2DPoissonStiffBC_DISK_" + meshType +
str(nbCells))
print(
"Numerical solution of 2D Poisson equation on a disk using finite volumes done"
)
end = time.time()
#Calcul de l'erreur commise par rapport à la solution exacte
#===========================================================
l2_norm_sol_exacte = my_ExactSol.normL2()[0]
l2_error = (my_ExactSol - my_ResultField).normL2()[0]
print(
"L2 relative error = norm( exact solution - numerical solution )/norm( exact solution ) = ",
l2_error / l2_norm_sol_exacte)
print("Maximum numerical solution = ", my_ResultField.max(),
" Minimum numerical solution = ", my_ResultField.min())
print("Maximum exact solution = ", my_ExactSol.max(),
" Minimum exact solution = ", my_ExactSol.min())
#Postprocessing :
#================
# Extraction of the diagonal data
diag_data = VTK_routines.Extract_field_data_over_line_to_numpyArray(
my_ResultField, [0, -1, 0], [0, 1, 0], resolution)
# save 2D picture
PV_routines.Save_PV_data_to_picture_file(
"FiniteVolumes2DPoissonStiffBC_DISK_" + meshType + str(nbCells) +
'_0.vtu', "ResultField", 'CELLS',
"FiniteVolumes2DPoissonStiffBC_DISK_" + meshType + str(nbCells))
PV_routines.Save_PV_data_to_picture_file(
"ExactSol2DPoissonStiffBC_DISK_" + meshType + str(nbCells) + '_0.vtu',
"Exact_field", 'CELLS',
"ExactSol2DPoissonStiffBC_DISK_" + meshType + str(nbCells))
test_desc["Computational_time_taken_by_run"] = end - start
test_desc["Absolute_error"] = l2_error
test_desc["Relative_error"] = l2_error / l2_norm_sol_exacte
with open(
'test_PoissonStiffBC' + str(my_mesh.getMeshDimension()) + 'D_VF_' +
"DISK_" + str(nbCells) + "Cells.json", 'w') as outfile:
json.dump(test_desc, outfile)
return l2_error / l2_norm_sol_exacte, nbCells, diag_data, my_ResultField.min(
), my_ResultField.max(), end - start
def WaveSystemVF(ntmax, tmax, cfl, my_mesh, output_freq, meshName, resolution,
scaling, test_bc):
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
dt = 0.
time = 0.
it = 0
isStationary = False
nbVoisinsMax = my_mesh.getMaxNbNeighbours(cdmath.CELLS)
iterGMRESMax = 50
#iteration vectors
Un = cdmath.Vector(nbCells * (dim + 1))
dUn = cdmath.Vector(nbCells * (dim + 1))
# Initial conditions #
print("Construction of the initial condition …")
if (meshName.find("square") == -1 and meshName.find("Square") == -1
and meshName.find("cube") == -1 and meshName.find("Cube") == -1):
print "Mesh name : ", meshName
raise ValueError(
"Mesh name should contain substring square or cube to use wave system with smooth source term"
)
pressure_field = cdmath.Field("Pressure", cdmath.CELLS, my_mesh, 1)
velocity_field = cdmath.Field("Velocity", cdmath.CELLS, my_mesh, 3)
for k in range(nbCells): # fields initialisation
pressure_field[k] = 0
velocity_field[k, 0] = 0
velocity_field[k, 1] = 0
velocity_field[k, 2] = 0
S, stat_pressure, stat_velocity = source_term_and_stat_solution_wave_system(
my_mesh)
for k in range(nbCells):
Un[k * (dim + 1) + 0] = pressure_field[k]
Un[k * (dim + 1) + 1] = rho0 * velocity_field[k, 0]
Un[k * (dim + 1) + 2] = rho0 * velocity_field[k, 1]
if (dim == 3):
Un[k * (dim + 1) + 3] = rho0 * velocity_field[k, 2]
if (scaling == 1):
Vn = Un.deepCopy()
for k in range(nbCells):
Vn[k * (dim + 1) + 0] = Vn[k * (dim + 1) + 0] / (c0 * c0)
S[k * (dim + 1) + 0] = S[k * (dim + 1) + 0] / (c0 * c0)
elif (scaling == 2):
Vn = Un.deepCopy()
for k in range(nbCells):
Vn[k * (dim + 1) + 0] = Vn[k * (dim + 1) + 0] / c0
S[k * (dim + 1) + 0] = S[k * (dim + 1) + 0] / c0
#sauvegarde de la donnée initiale
pressure_field.setTime(time, it)
pressure_field.writeVTK("WaveSystem" + str(dim) + "DCentered" + meshName +
"_pressure")
velocity_field.setTime(time, it)
velocity_field.writeVTK("WaveSystem" + str(dim) + "DCentered" + meshName +
"_velocity")
#Postprocessing : save 2D picture
PV_routines.Save_PV_data_to_picture_file(
"WaveSystem" + str(dim) + "DCentered" + meshName + "_pressure" +
'_0.vtu', "Pressure", 'CELLS',
"WaveSystem" + str(dim) + "DCentered" + meshName + "_pressure_initial")
PV_routines.Save_PV_data_to_picture_file(
"WaveSystem" + str(dim) + "DCentered" + meshName + "_velocity" +
'_0.vtu', "Velocity", 'CELLS',
"WaveSystem" + str(dim) + "DCentered" + meshName + "_velocity_initial")
total_pressure_initial = pressure_field.integral(
) #For conservation test later
total_velocity_initial = velocity_field.integral(
) #For conservation test later
dx_min = my_mesh.minRatioVolSurf()
dt = cfl * dx_min / c0
divMat = computeDivergenceMatrix(my_mesh, nbVoisinsMax, dt, scaling,
test_bc)
#Adding the momentum friction term
for j in range(nbCells):
for i in range(dim):
divMat.addValue(j * (dim + 1) + 1 + i, j * (dim + 1) + 1 + i, dt)
#Add the identity matrix on the diagonal
if (scaling == 0 or scaling == 2):
divMat.diagonalShift(1) #only after filling all coefficients
else:
for j in range(nbCells):
divMat.addValue(j * (dim + 1), j * (dim + 1),
1 / (c0 * c0)) #/(c0*c0)
for i in range(dim):
divMat.addValue(j * (dim + 1) + 1 + i,
j * (dim + 1) + 1 + i, 1)
if (scaling == 0):
LS = cdmath.LinearSolver(divMat, Un + S * dt, iterGMRESMax, precision,
"GMRES", "ILU")
else:
LS = cdmath.LinearSolver(divMat, Vn + S * dt, iterGMRESMax, precision,
"GMRES", "ILU")
LS.setComputeConditionNumber()
test_desc["Linear_solver_algorithm"] = LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"] = LS.getNameOfPc()
test_desc["Linear_solver_precision"] = LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"] = LS.getNumberMaxOfIter()
test_desc["Numerical_parameter_space_step"] = dx_min
test_desc["Numerical_parameter_time_step"] = dt
test_desc["Linear_solver_with_scaling"] = scaling
test_desc['Linear_system_max_actual_iterations_number'] = 0
test_desc["Linear_system_max_actual_error"] = 0
test_desc["Linear_system_max_actual_condition number"] = 0
print(
"Starting computation of the linear wave system with a centered scheme …"
)
# Starting time loop
while (it < ntmax and time <= tmax and not isStationary):
dUn = Un.deepCopy()
if (scaling == 0):
LS.setSndMember(Un + S * dt)
else:
LS.setSndMember(Vn + S * dt)
if (scaling < 2):
Un = LS.solve()
if (scaling == 1):
Vn = Un.deepCopy()
for k in range(nbCells):
Vn[k * (dim + 1) + 0] = Vn[k * (dim + 1) + 0] / (c0 * c0)
else: #( scaling==2)
Vn = LS.solve()
Un = Vn.deepCopy()
for k in range(nbCells):
Un[k * (dim + 1) + 0] = c0 * Vn[k * (dim + 1) + 0]
if (not LS.getStatus()):
print "Linear system did not converge ", iterGMRES, " GMRES iterations"
raise ValueError("Pas de convergence du système linéaire")
dUn -= Un
test_desc["Linear_system_max_actual_iterations_number"] = max(
LS.getNumberOfIter(),
test_desc["Linear_system_max_actual_iterations_number"])
test_desc["Linear_system_max_actual_error"] = max(
LS.getResidu(), test_desc["Linear_system_max_actual_error"])
test_desc["Linear_system_max_actual_condition number"] = max(
LS.getConditionNumber(),
test_desc["Linear_system_max_actual_condition number"])
maxVector = dUn.maxVector(dim + 1)
isStationary = maxVector[0] < precision and maxVector[
1] / rho0 < precision and maxVector[2] / rho0 < precision
if (dim == 3):
isStationary = isStationary and maxVector[3] / rho0 < precision
time = time + dt
it = it + 1
#Sauvegardes
if (it == 1 or it % output_freq == 0 or it >= ntmax or isStationary
or time >= tmax):
print "-- Iter: " + str(it) + ", Time: " + str(
time) + ", dt: " + str(dt)
print "Variation temporelle relative : pressure ", maxVector[
0], ", velocity x", maxVector[
1] / rho0, ", velocity y", maxVector[2] / rho0
print "Linear system converged in ", LS.getNumberOfIter(
), " GMRES iterations"
delta_press = 0
delta_v = cdmath.Vector(dim)
for k in range(nbCells):
pressure_field[k] = Un[k * (dim + 1) + 0]
velocity_field[k, 0] = Un[k * (dim + 1) + 1] / rho0
if (dim > 1):
velocity_field[k, 1] = Un[k * (dim + 1) + 2] / rho0
if (dim > 2):
velocity_field[k, 2] = Un[k * (dim + 1) + 3] / rho0
if (abs(stat_pressure[k] - pressure_field[k]) > delta_press):
delta_press = abs(stat_pressure[k] - pressure_field[k])
if (abs(stat_velocity[k, 0] - velocity_field[k, 0]) >
delta_v[0]):
delta_v[0] = abs(stat_velocity[k, 0] -
velocity_field[k, 0])
if (abs(stat_velocity[k, 1] - velocity_field[k, 1]) >
delta_v[1]):
delta_v[1] = abs(stat_velocity[k, 1] -
velocity_field[k, 1])
if (dim == 3):
if (abs(stat_velocity[k, 2] - velocity_field[k, 2]) >
delta_v[2]):
delta_v[2] = abs(stat_velocity[k, 2] -
velocity_field[k, 2])
pressure_field.setTime(time, it)
pressure_field.writeVTK(
"WaveSystem" + str(dim) + "DCentered" + meshName + "_pressure",
False)
velocity_field.setTime(time, it)
velocity_field.writeVTK(
"WaveSystem" + str(dim) + "DCentered" + meshName + "_velocity",
False)
print "Ecart au stationnaire exact : error_p= ", delta_press, " error_||u||= ", delta_v.maxVector(
)[0]
print
print "-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " + str(dt)
print "Variation temporelle relative : pressure ", maxVector[
0], ", velocity x", maxVector[1] / rho0, ", velocity y", maxVector[
2] / rho0
print
if (it >= ntmax):
print "Nombre de pas de temps maximum ntmax= ", ntmax, " atteint"
raise ValueError(
"Maximum number of time steps reached : Stationary state not found !!!!!!!"
)
elif (isStationary):
print "Régime stationnaire atteint au pas de temps ", it, ", t= ", time
print "------------------------------------------------------------------------------------"
pressure_field.setTime(time, 0)
pressure_field.writeVTK("WaveSystem" + str(dim) + "DCentered" +
meshName + "_pressure_Stat")
velocity_field.setTime(time, 0)
velocity_field.writeVTK("WaveSystem" + str(dim) + "DCentered" +
meshName + "_velocity_Stat")
#Postprocessing : Extraction of the diagonal data
if (dim == 2):
diag_data_press = VTK_routines.Extract_field_data_over_line_to_numpyArray(
pressure_field, [0, 1, 0], [1, 0, 0], resolution)
diag_data_vel = VTK_routines.Extract_field_data_over_line_to_numpyArray(
velocity_field, [0, 1, 0], [1, 0, 0], resolution)
elif (dim == 3):
diag_data_press = VTK_routines.Extract_field_data_over_line_to_numpyArray(
pressure_field, [0, 0, 0], [1, 1, 1], resolution)
diag_data_vel = VTK_routines.Extract_field_data_over_line_to_numpyArray(
velocity_field, [0, 0, 0], [1, 1, 1], resolution)
#Postprocessing : save 2D picture
PV_routines.Save_PV_data_to_picture_file(
"WaveSystem" + str(dim) + "DCentered" + meshName +
"_pressure_Stat" + '_0.vtu', "Pressure", 'CELLS', "WaveSystem" +
str(dim) + "DCentered" + meshName + "_pressure_Stat")
PV_routines.Save_PV_data_to_picture_file(
"WaveSystem" + str(dim) + "DCentered" + meshName +
"_velocity_Stat" + '_0.vtu', "Velocity", 'CELLS', "WaveSystem" +
str(dim) + "DCentered" + meshName + "_velocity_Stat")
return delta_press, delta_v.maxVector(
)[0], nbCells, time, it, velocity_field.getNormEuclidean().max(
), diag_data_press, diag_data_vel, test_desc[
"Linear_system_max_actual_condition number"]
else:
print "Temps maximum Tmax= ", tmax, " atteint"
raise ValueError(
"Maximum time reached : Stationary state not found !!!!!!!")
def WaveSystemVF(ntmax, tmax, cfl, my_mesh, output_freq, filename,resolution):
dim=my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
meshName=my_mesh.getName()
dt = 0.
time = 0.
it=0;
isStationary=False;
SumFluxes = cdmath.Field("Fluxes", cdmath.CELLS, my_mesh, dim+1)
# Initial conditions #
print("Construction of the initial condition …")
if(filename.find("square")>-1 or filename.find("Square")>-1 or filename.find("cube")>-1 or filename.find("Cube")>-1):
pressure_field, velocity_field = initial_conditions_square_shock(my_mesh)
elif(filename.find("disk")>-1 or filename.find("Disk")>-1):
pressure_field, velocity_field = initial_conditions_disk_vortex(my_mesh)
else:
print "Mesh name : ", filename
raise ValueError("Mesh name should contain substring square, cube or disk")
U = cdmath.Field("Conservative vector", cdmath.CELLS, my_mesh, dim+1)
for i in range(nbCells):
U[i,0] = pressure_field[i]
U[i,1] = rho0*velocity_field[i,0]
U[i,2] = rho0*velocity_field[i,1]
#sauvegarde de la donnée initiale
pressure_field.setTime(time,it);
pressure_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_pressure");
velocity_field.setTime(time,it);
velocity_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_velocity");
dx_min=my_mesh.minRatioVolSurf()
dt = cfl * dx_min / c0
print("Starting computation of the linear wave system with an explicit UPWIND scheme …")
# Starting time loop
while (it<ntmax and time <= tmax and not isStationary):
computeFluxes(U,SumFluxes);
SumFluxes*=dt;
maxVector=SumFluxes.normMax()
isStationary= maxVector[0]/p0<precision and maxVector[1]/rho0<precision and maxVector[2]/rho0<precision;
U-=SumFluxes;
time=time+dt;
it=it+1;
#Sauvegardes
if(it%output_freq==0 or it>=ntmax or isStationary or time >=tmax):
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " + str(dt))
print "Variation temporelle relative : pressure ", maxVector[0]/p0 ,", velocity x", maxVector[1]/rho0 ,", velocity y", maxVector[2]/rho0
for k in range(nbCells):
pressure_field[k]=U[k,0]
velocity_field[k,0]=U[k,1]/rho0
if(dim>1):
velocity_field[k,1]=U[k,2]/rho0
if(dim>2):
velocity_field[k,2]=U[k,3]/rho0
pressure_field.setTime(time,it);
pressure_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_pressure",False);
velocity_field.setTime(time,it);
velocity_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_velocity",False);
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " + str(dt))
print "Variation temporelle relative : pressure ", maxVector[0]/p0 ,", velocity x", maxVector[1]/rho0 ,", velocity y", maxVector[2]/rho0
print
if(it>=ntmax):
print "Nombre de pas de temps maximum ntmax= ", ntmax, " atteint"
elif(isStationary):
print "Régime stationnaire atteint au pas de temps ", it, ", t= ", time
pressure_field.setTime(time,0);
pressure_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_pressure_Stat");
velocity_field.setTime(time,0);
velocity_field.writeVTK("WaveSystem"+str(dim)+"DUpwind"+meshName+"_velocity_Stat");
#Postprocessing : Extraction of the diagonal data
diag_data_press=VTK_routines.Extract_field_data_over_line_to_numpyArray(pressure_field,[0,1,0],[1,0,0], resolution)
diag_data_vel =VTK_routines.Extract_field_data_over_line_to_numpyArray(velocity_field,[0,1,0],[1,0,0], resolution)
#Postprocessing : save 2D picture
PV_routines.Save_PV_data_to_picture_file("WaveSystem"+str(dim)+"DUpwind"+meshName+"_pressure_Stat"+'_0.vtu',"Pressure",'CELLS',"WaveSystem"+str(dim)+"DUpwind"+meshName+"_pressure_Stat")
PV_routines.Save_PV_data_to_picture_file("WaveSystem"+str(dim)+"DUpwind"+meshName+"_velocity_Stat"+'_0.vtu',"Velocity",'CELLS',"WaveSystem"+str(dim)+"DUpwind"+meshName+"_velocity_Stat")
else:
print "Temps maximum Tmax= ", tmax, " atteint"
eps=1e-6
my_mesh.setGroupAtPlan(0,0,eps,"DirichletBorder")#Bord GAUCHE
my_mesh.setGroupAtPlan(1,0,eps,"DirichletBorder")#Bord DROIT
my_mesh.setGroupAtPlan(0,1,eps,"DirichletBorder")#Bord BAS
my_mesh.setGroupAtPlan(1,1,eps,"DirichletBorder")#Bord HAUT
my_mesh.setGroupAtPlan(0,2,eps,"DirichletBorder")#Bord AVANT
my_mesh.setGroupAtPlan(1,2,eps,"DirichletBorder")#Bord ARRIERE
nbCells = my_mesh.getNumberOfCells()
print("Mesh loading done")
print("Number of cells ", nbCells)
#Discrétisation du second membre et extraction du nb max de voisins d'une cellule
#================================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.CELLS, my_mesh, 1)
my_ExactSol = cdmath.Field("EXACT_SOL", cdmath.CELLS, my_mesh, 1)
maxNbNeighbours=0#This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des cellules pour discrétisation du second membre et extraction du nb max de voisins d'une cellule
for i in range(nbCells):
Ci = my_mesh.getCell(i)
x = Ci.x()
y = Ci.y()
z = Ci.z()
r=sqrt(x*x+y*y+z*z)
phi=atan2(y,x)
theta=atan2(y,z*sin(phi))
my_RHSfield[i]=6*r*sin(theta)**2*cos(phi)+3*(r-1)*cos(phi)#mettre la fonction definie au second membre de l'edp
def sigma_flux(VitesseX, VitesseY, cfl, y_field, indexFacesPerio):
# Calculation of fluxes #
SumFlux = cdmath.Field("Fluxes", cdmath.CELLS, y_field.getMesh(), 1)
my_mesh = y_field.getMesh()
nbCells = my_mesh.getNumberOfCells()
normU = math.sqrt(VitesseX * VitesseX + VitesseY * VitesseY)
for j in range(nbCells):
Cj = my_mesh.getCell(j)
nbFace = Cj.getNumberOfFaces()
SumF = 0.0
minlengthFk = 1.E30
for k in range(nbFace):
indexFace = Cj.getFacesId()[k]
Fk = my_mesh.getFace(indexFace)
NormalX = Cj.getNormalVector(k, 0)
NormalY = Cj.getNormalVector(k, 1)
LengthFk = Fk.getMeasure()
UN = VitesseX * NormalX + VitesseY * NormalY
minlengthFk = min(minlengthFk, LengthFk / abs(UN))
minlengthFk = min(minlengthFk, LengthFk / abs(VitesseX))
minlengthFk = min(minlengthFk, LengthFk / abs(VitesseY))
conc = 0.0
cellCourante = j
cellAutre = -1
if (not Fk.isBorder()):
indexC1 = Fk.getCellsId()[0]
indexC2 = Fk.getCellsId()[1]
# hypothesis: the cell of index indexC1 is the current cell of index j #
if (indexC1 == j):
# hypothese is verified #
cellCourante = indexC1
cellAutre = indexC2
pass
elif (indexC2 == j):
# hypothesis is not verified #
cellCourante = indexC2
cellAutre = indexC1
pass
# define left and right cell with the product of velocity * outward normal vector
# if u*n>0: nothing to do, else invert left and right
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = y_field[cellAutre]
pass
pass
else:
# homogeneous Neumann boundary conditions #
if (Fk.getGroupName() == "GAUCHE" or Fk.getGroupName() == "DROITE"):
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = 0.0
pass
pass
# periodical boundary conditions #
if (Fk.getGroupName() == "BAS" or Fk.getGroupName() == "HAUT"):
indexFP = indexFacesPerio[indexFace]
# another way to get the index of the periodical face #
# int indexFP=my_mesh.getIndexFacePeriodic(indexFace);
Fp = my_mesh.getFace(indexFP)
indexCp = Fp.getCellsId()[0]
if (UN > 1.E-15):
conc = y_field[cellCourante]
pass
else:
conc = y_field[indexCp]
pass
pass
pass
SumF = SumF + UN * LengthFk * conc
pass
dt = cfl * minlengthFk / normU
SumFlux[j] = dt * SumF / Cj.getMeasure()
pass
return dt, SumFlux
def solve(filename, resolution, meshName, testColor):
start = time.time()
test_desc["Mesh_name"] = meshName
test_desc["Test_color"] = testColor
#Chargement du maillage triangulaire du disque unité D(0,1)
#=======================================================================================
my_mesh = cdmath.Mesh(filename + ".med")
if (my_mesh.getSpaceDimension() != 2 or my_mesh.getMeshDimension() != 2):
raise ValueError(
"Wrong space or mesh dimension : space and mesh dimensions should be 2"
)
if (not my_mesh.isTriangular()):
raise ValueError("Wrong cell types : mesh is not made of triangles")
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
test_desc["Space_dimension"] = my_mesh.getSpaceDimension()
test_desc["Mesh_dimension"] = my_mesh.getMeshDimension()
test_desc["Mesh_number_of_elements"] = my_mesh.getNumberOfNodes()
test_desc["Mesh_cell_type"] = my_mesh.getElementTypesNames()
print("Mesh loading done")
print("Number of nodes=", nbNodes)
print("Number of cells=", nbCells)
#Détermination des noeuds intérieurs
#======================================================================
my_ExactSol = cdmath.Field("Exact_field", cdmath.NODES, my_mesh, 1)
nbInteriorNodes = 0
nbBoundaryNodes = 0
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
interiorNodes = []
boundaryNodes = []
eps = 1e-6
#parcours des noeuds pour discrétisation du second membre et extraction 1) des noeuds intérieur 2) des noeuds frontière 3) du nb max voisins d'un noeud
for i in range(nbNodes):
Ni = my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
#Robust calculation of atan(2x/(x**2+y**2-1)
#my_ExactSol[i]=atan2(2*x*sign(x**2+y**2-1),abs(x**2+y**2-1))#mettre la solution exacte de l'edp
if x**2 + y**2 - 1 > eps:
print("!!! Warning Mesh ", meshName,
" !!! Node is not in the unit disk.", ", eps=", eps,
", x**2+y**2-1=", x**2 + y**2 - 1)
#raise ValueError("!!! Domain should be the unit disk.")
if x**2 + y**2 - 1 < -eps:
my_ExactSol[i] = atan(2 * x / (x**2 + y**2 - 1))
elif x > 0: #x**2+y**2-1>=0
my_ExactSol[i] = -pi / 2
elif x < 0: #x**2+y**2-1>=0
my_ExactSol[i] = pi / 2
else: #x=0
my_ExactSol[i] = 0
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
boundaryNodes.append(i)
nbBoundaryNodes = nbBoundaryNodes + 1
else: # Détection des noeuds intérieurs
interiorNodes.append(i)
nbInteriorNodes = nbInteriorNodes + 1
maxNbNeighbours = max(
1 + Ni.getNumberOfCells(), maxNbNeighbours
) #true only in 2D, need a function Ni.getNumberOfNeighbourNodes()
test_desc["Mesh_max_number_of_neighbours"] = maxNbNeighbours
print("Right hand side discretisation done")
print("Number of interior nodes=", nbInteriorNodes)
print("Number of boundary nodes=", nbBoundaryNodes)
print("Max number of neighbours=", maxNbNeighbours)
# Construction de la matrice de rigidité et du vecteur second membre du système linéaire
#=======================================================================================
Rigidite = cdmath.SparseMatrixPetsc(
nbInteriorNodes, nbInteriorNodes, maxNbNeighbours
) # warning : third argument is maximum number of non zero coefficients per line of the matrix
RHS = cdmath.Vector(nbInteriorNodes)
M = cdmath.Matrix(3, 3)
# Vecteurs gradient de la fonction de forme associée à chaque noeud d'un triangle (hypothèse 2D)
GradShapeFunc0 = cdmath.Vector(2)
GradShapeFunc1 = cdmath.Vector(2)
GradShapeFunc2 = cdmath.Vector(2)
#On parcourt les triangles du domaine
for i in range(nbCells):
Ci = my_mesh.getCell(i)
#Contribution à la matrice de rigidité
nodeId0 = Ci.getNodeId(0)
nodeId1 = Ci.getNodeId(1)
nodeId2 = Ci.getNodeId(2)
N0 = my_mesh.getNode(nodeId0)
N1 = my_mesh.getNode(nodeId1)
N2 = my_mesh.getNode(nodeId2)
M[0, 0] = N0.x()
M[0, 1] = N0.y()
M[0, 2] = 1
M[1, 0] = N1.x()
M[1, 1] = N1.y()
M[1, 2] = 1
M[2, 0] = N2.x()
M[2, 1] = N2.y()
M[2, 2] = 1
#Values of each shape function at each node
values0 = [1, 0, 0]
values1 = [0, 1, 0]
values2 = [0, 0, 1]
GradShapeFunc0 = gradientNodal(M, values0) / 2
GradShapeFunc1 = gradientNodal(M, values1) / 2
GradShapeFunc2 = gradientNodal(M, values2) / 2
#Création d'un tableau (numéro du noeud, gradient de la fonction de forme)
GradShapeFuncs = {nodeId0: GradShapeFunc0}
GradShapeFuncs[nodeId1] = GradShapeFunc1
GradShapeFuncs[nodeId2] = GradShapeFunc2
# Remplissage de la matrice de rigidité et du second membre
for j in [nodeId0, nodeId1, nodeId2]:
if boundaryNodes.count(
j
) == 0: #seuls les noeuds intérieurs contribuent au système linéaire (matrice de rigidité et second membre)
j_int = interiorNodes.index(
j) #indice du noeud j en tant que noeud intérieur
#Pas de contribution au second membre car pas de terme source
boundaryContributionAdded = False #Needed in case j is a border cell
#Contribution de la cellule triangulaire i à la ligne j_int du système linéaire
for k in [nodeId0, nodeId1, nodeId2]:
if boundaryNodes.count(
k
) == 0: #seuls les noeuds intérieurs contribuent à la matrice du système linéaire
k_int = interiorNodes.index(
k) #indice du noeud k en tant que noeud intérieur
Rigidite.addValue(
j_int, k_int, GradShapeFuncs[j] *
GradShapeFuncs[k] / Ci.getMeasure())
elif boundaryContributionAdded == False: #si condition limite non nulle au bord (ou maillage non recouvrant), ajouter la contribution du bord au second membre de la cellule j
if boundaryNodes.count(nodeId0) != 0:
u0 = my_ExactSol[nodeId0]
else:
u0 = 0
if boundaryNodes.count(nodeId1) != 0:
u1 = my_ExactSol[nodeId1]
else:
u1 = 0
if boundaryNodes.count(nodeId2) != 0:
u2 = my_ExactSol[nodeId2]
else:
u2 = 0
boundaryContributionAdded = True #Contribution from the boundary to matrix line j is done in one step
GradGh = gradientNodal(M, [u0, u1, u2]) / 2
RHS[j_int] += -(GradGh *
GradShapeFuncs[j]) / Ci.getMeasure()
print("Linear system matrix building done")
# Résolution du système linéaire
#=================================
LS = cdmath.LinearSolver(
Rigidite, RHS, 100, 1.E-6, "CG",
"LU") #Remplacer CG par CHOLESKY pour solveur direct
LS.setComputeConditionNumber()
SolSyst = LS.solve()
print("Preconditioner used : ", LS.getNameOfPc())
print("Number of iterations used : ", LS.getNumberOfIter())
print("Linear system solved")
test_desc["Linear_solver_algorithm"] = LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"] = LS.getNameOfPc()
test_desc["Linear_solver_precision"] = LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"] = LS.getNumberMaxOfIter()
test_desc[
"Linear_system_max_actual_iterations_number"] = LS.getNumberOfIter()
test_desc["Linear_system_max_actual_error"] = LS.getResidu()
test_desc[
"Linear_system_max_actual_condition number"] = LS.getConditionNumber()
# Création du champ résultat
#===========================
my_ResultField = cdmath.Field("ResultField", cdmath.NODES, my_mesh, 1)
for j in range(nbInteriorNodes):
my_ResultField[interiorNodes[j]] = SolSyst[j]
#remplissage des valeurs pour les noeuds intérieurs
for j in range(nbBoundaryNodes):
my_ResultField[boundaryNodes[j]] = my_ExactSol[boundaryNodes[j]]
#remplissage des valeurs pour les noeuds frontière (condition limite)
#sauvegarde sur le disque dur du résultat dans un fichier paraview
my_ResultField.writeVTK("FiniteElements2DPoissonStiffBC_DISK_" + meshName +
str(nbNodes))
print(
"Numerical solution of 2D Poisson equation on a disk using finite elements done"
)
end = time.time()
#Calcul de l'erreur commise par rapport à la solution exacte
#===========================================================
l2_norm_sol_exacte = my_ExactSol.normL2()[0]
l2_error = (my_ExactSol - my_ResultField).normL2()[0]
print(
"L2 relative error = norm( exact solution - numerical solution )/norm( exact solution ) = ",
l2_error / l2_norm_sol_exacte)
print("Maximum numerical solution = ", my_ResultField.max(),
" Minimum numerical solution = ", my_ResultField.min())
print("Maximum exact solution = ", my_ExactSol.max(),
" Minimum exact solution = ", my_ExactSol.min())
#Postprocessing :
#================
# Extraction of the diagonal data
diag_data = VTK_routines.Extract_field_data_over_line_to_numpyArray(
my_ResultField, [0, -1, 0], [0, 1, 0], resolution)
# save 2D picture
PV_routines.Save_PV_data_to_picture_file(
"FiniteElements2DPoissonStiffBC_DISK_" + meshName + str(nbNodes) +
'_0.vtu', "ResultField", 'NODES',
"FiniteElements2DPoissonStiffBC_DISK_" + meshName + str(nbNodes))
test_desc["Computational_time_taken_by_run"] = end - start
test_desc["Absolute_error"] = l2_error
test_desc["Relative_error"] = l2_error / l2_norm_sol_exacte
with open(
'test_PoissonStiffBC' + str(my_mesh.getMeshDimension()) + 'D_EF_' +
"DISK_" + meshName + str(nbCells) + "Cells.json", 'w') as outfile:
json.dump(test_desc, outfile)
return l2_error / l2_norm_sol_exacte, nbNodes, diag_data, my_ResultField.min(
), my_ResultField.max(), end - start
raise ValueError(
"Wrong space or mesh dimension : space and mesh dimensions should be 2"
)
if (not my_mesh.isTriangular()):
raise ValueError("Wrong cell types : mesh is not made of triangles")
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
print("Mesh loading done")
print("Number of nodes=", nbNodes)
print("Number of cells=", nbCells)
#Discrétisation du second membre et détermination des noeuds intérieurs
#======================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.NODES, my_mesh, 1)
my_ExactSol = cdmath.Field("Exact_field", cdmath.NODES, my_mesh, 1)
nbInteriorNodes = 0
nbBoundaryNodes = 0
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
interiorNodes = []
boundaryNodes = []
#parcours des noeuds pour discrétisation du second membre et extraction 1) des noeuds intérieur 2) des noeuds frontière 3) du nb max voisins d'un noeud
for i in range(nbNodes):
Ni = my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
r = sqrt(x * x + y * y)
theta = atan2(y, x)
if (my_mesh.getMeshDimension() != 2):
raise ValueError(
"Wrong mesh dimension : expected a surface of dimension 2")
if (my_mesh.getSpaceDimension() != 3):
raise ValueError("Wrong space dimension : expected a space of dimension 3")
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
print("Mesh building/loading done")
print("nb of nodes=", nbNodes)
print("nb of cells=", nbCells)
#Discrétisation du second membre et détermination des noeuds intérieurs
#======================================================================
my_RHSfield = cdmath.Field("RHS field", cdmath.NODES, my_mesh, 1)
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des noeuds pour discrétisation du second membre et extraction du nb max voisins d'un noeud
for i in range(nbNodes):
Ni = my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
z = Ni.z()
my_RHSfield[i] = 12 * y * (3 * x * x - y * y) / pow(
x * x + y * y + z * z,
3 / 2) #vecteur propre du laplacien sur la sphère
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
raise ValueError("Mesh should not contain borders")
else:
eps = 1e-6
my_mesh.setGroupAtPlan(0, 0, eps, "DirichletBorder") #Bord GAUCHE
my_mesh.setGroupAtPlan(1, 0, eps, "DirichletBorder") #Bord DROIT
my_mesh.setGroupAtPlan(0, 1, eps, "DirichletBorder") #Bord BAS
my_mesh.setGroupAtPlan(1, 1, eps, "DirichletBorder") #Bord HAUT
my_mesh.setGroupAtPlan(0, 2, eps, "DirichletBorder") #Bord AVANT
my_mesh.setGroupAtPlan(1, 2, eps, "DirichletBorder") #Bord ARRIERE
nbCells = my_mesh.getNumberOfCells()
print("Mesh loading done")
print("Number of cells ", nbCells)
#Discrétisation du second membre et extraction du nb max de voisins d'une cellule
#================================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.CELLS, my_mesh, 1)
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des cellules pour discrétisation du second membre et extraction du nb max de voisins d'une cellule
for i in range(nbCells):
Ci = my_mesh.getCell(i)
x = Ci.x()
y = Ci.y()
z = Ci.z()
my_RHSfield[i] = 3 * pi * pi * sin(pi * x) * sin(pi * y) * sin(
pi * z) #mettre la fonction definie au second membre de l edp
# compute maximum number of neighbours
maxNbNeighbours = max(1 + Ci.getNumberOfFaces(), maxNbNeighbours)
print("Right hand side discretisation done")
print("Maximum number of neighbours=", maxNbNeighbours)
raise ValueError("Wrong space dimension : expected a space of dimension 3")
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
print("Mesh building/loading done")
print("nb of nodes=", nbNodes)
print("nb of cells=", nbCells)
# Torus radii (calculation will fail if the mesh is not correct)
R=1 #Grand rayon
r=0.6 #Petit rayon
#Discrétisation du second membre, de la solution exacte et détermination des noeuds intérieurs
#======================================================================
my_RHSfield = cdmath.Field("RHS field", cdmath.NODES, my_mesh, 1)
exactSolField = cdmath.Field("Exact solution field", cdmath.NODES, my_mesh, 1)
maxNbNeighbours = 0#This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des noeuds pour discrétisation du second membre et extraction du nb max voisins d'un noeud
for i in range(nbNodes):
Ni=my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
z = Ni.z()
theta=atan2(z,sqrt(x*x+y*y)-R)
phi=atan2(y,x)
exactSolField[i] = sin(3*phi)*cos(3*theta+ phi) # for the exact solution we use the funtion given in the article of Olshanskii, Reusken 2009, page 19
def solve(filename,resolution,meshType, testColor):
start = time.time()
test_desc["Mesh_type"]=meshType
test_desc["Test_color"]=testColor
#Chargement du maillage triangulaire du domaine carré [0,1]x[0,1], définition des bords
#=======================================================================================
my_mesh = cdmath.Mesh(filename+".med")
if( my_mesh.getSpaceDimension()!=2 or my_mesh.getMeshDimension()!=2) :
raise ValueError("Wrong space or mesh dimension : space and mesh dimensions should be 2")
if(not my_mesh.isTriangular()) :
raise ValueError("Wrong cell types : mesh is not made of triangles")
eps=1e-6
my_mesh.setGroupAtPlan(0.,0,eps,"DirichletBorder")#Bord GAUCHE
my_mesh.setGroupAtPlan(1.,0,eps,"DirichletBorder")#Bord DROIT
my_mesh.setGroupAtPlan(0.,1,eps,"DirichletBorder")#Bord BAS
my_mesh.setGroupAtPlan(1.,1,eps,"DirichletBorder")#Bord HAUT
nbNodes = my_mesh.getNumberOfNodes()
nbCells = my_mesh.getNumberOfCells()
test_desc["Space_dimension"]=my_mesh.getSpaceDimension()
test_desc["Mesh_dimension"]=my_mesh.getMeshDimension()
test_desc["Mesh_number_of_elements"]=my_mesh.getNumberOfNodes()
test_desc["Mesh_cell_type"]=my_mesh.getElementTypes()
print("Mesh loading done")
print("Number of nodes=", nbNodes)
print("Number of cells=", nbCells)
#Discrétisation du second membre et détermination des noeuds intérieurs
#======================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.NODES, my_mesh, 1)
nbInteriorNodes = 0
nbBoundaryNodes = 0
maxNbNeighbours = 0#This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
interiorNodes=[]
boundaryNodes=[]
#parcours des noeuds pour discrétisation du second membre et extraction 1) des noeuds intérieur 2) des noeuds frontière 3) du nb max voisins d'un noeud
for i in range(nbNodes):
Ni=my_mesh.getNode(i)
x = Ni.x()
y = Ni.y()
my_RHSfield[i]=2*pi*pi*sin(pi*x)*sin(pi*y)#mettre la fonction definie au second membre de l'edp
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
boundaryNodes.append(i)
nbBoundaryNodes=nbBoundaryNodes+1
else: # Détection des noeuds intérieurs
interiorNodes.append(i)
nbInteriorNodes=nbInteriorNodes+1
maxNbNeighbours= max(1+Ni.getNumberOfCells(),maxNbNeighbours) #true only in 2D, need a function Ni.getNumberOfNeighbourNodes()
test_desc["Mesh_max_number_of_neighbours"]=maxNbNeighbours
print("Right hand side discretisation done")
print("Number of interior nodes=", nbInteriorNodes)
print("Number of boundary nodes=", nbBoundaryNodes)
print("Max number of neighbours=", maxNbNeighbours)
# Construction de la matrice de rigidité et du vecteur second membre du système linéaire
#=======================================================================================
Rigidite=cdmath.SparseMatrixPetsc(nbInteriorNodes,nbInteriorNodes,maxNbNeighbours) # warning : third argument is maximum number of non zero coefficients per line of the matrix
RHS=cdmath.Vector(nbInteriorNodes)
# Vecteurs gradient de la fonction de forme associée à chaque noeud d'un triangle (hypothèse 2D)
GradShapeFunc0=cdmath.Vector(2)
GradShapeFunc1=cdmath.Vector(2)
GradShapeFunc2=cdmath.Vector(2)
#On parcourt les triangles du domaine
for i in range(nbCells):
Ci=my_mesh.getCell(i)
#Contribution à la matrice de rigidité
nodeId0=Ci.getNodeId(0)
nodeId1=Ci.getNodeId(1)
nodeId2=Ci.getNodeId(2)
N0=my_mesh.getNode(nodeId0)
N1=my_mesh.getNode(nodeId1)
N2=my_mesh.getNode(nodeId2)
#Formule des gradients voir EF P1 -> calcul déterminants
GradShapeFunc0[0]= (N1.y()-N2.y())/2
GradShapeFunc0[1]=-(N1.x()-N2.x())/2
GradShapeFunc1[0]=-(N0.y()-N2.y())/2
GradShapeFunc1[1]= (N0.x()-N2.x())/2
GradShapeFunc2[0]= (N0.y()-N1.y())/2
GradShapeFunc2[1]=-(N0.x()-N1.x())/2
#Création d'un tableau (numéro du noeud, gradient de la fonction de forme
GradShapeFuncs={nodeId0 : GradShapeFunc0}
GradShapeFuncs[nodeId1]=GradShapeFunc1
GradShapeFuncs[nodeId2]=GradShapeFunc2
# Remplissage de la matrice de rigidité et du second membre
for j in [nodeId0,nodeId1,nodeId2] :
if boundaryNodes.count(j)==0 : #seuls les noeuds intérieurs contribuent au système linéaire (matrice de rigidité et second membre)
j_int=interiorNodes.index(j)#indice du noeud j en tant que noeud intérieur
#Ajout de la contribution de la cellule triangulaire i au second membre du noeud j
RHS[j_int]=Ci.getMeasure()/3*my_RHSfield[j]+RHS[j_int] # intégrale dans le triangle du produit f x fonction de base
#Contribution de la cellule triangulaire i à la ligne j_int du système linéaire
for k in [nodeId0,nodeId1,nodeId2] :
if boundaryNodes.count(k)==0 : #seuls les noeuds intérieurs contribuent à la matrice du système linéaire
k_int=interiorNodes.index(k)#indice du noeud k en tant que noeud intérieur
Rigidite.addValue(j_int,k_int,GradShapeFuncs[j]*GradShapeFuncs[k]/Ci.getMeasure())
#else: si condition limite non nulle au bord, ajouter la contribution du bord au second membre de la cellule j
print("Linear system matrix building done")
# Résolution du système linéaire
#=================================
LS=cdmath.LinearSolver(Rigidite,RHS,100,1.E-6,"CG","ILU")#Remplacer CG par CHOLESKY pour solveur direct
LS.setComputeConditionNumber()
SolSyst=LS.solve()
print "Preconditioner used : ", LS.getNameOfPc()
print "Number of iterations used : ", LS.getNumberOfIter()
print("Linear system solved")
test_desc["Linear_solver_algorithm"]=LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"]=LS.getNameOfPc()
test_desc["Linear_solver_precision"]=LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"]=LS.getNumberMaxOfIter()
test_desc["Linear_system_max_actual_iterations_number"]=LS.getNumberOfIter()
test_desc["Linear_system_max_actual_error"]=LS.getResidu()
test_desc["Linear_system_max_actual_condition number"]=LS.getConditionNumber()
# Création du champ résultat
#===========================
my_ResultField = cdmath.Field("ResultField", cdmath.NODES, my_mesh, 1)
for j in range(nbInteriorNodes):
my_ResultField[interiorNodes[j]]=SolSyst[j];#remplissage des valeurs pour les noeuds intérieurs
for j in range(nbBoundaryNodes):
my_ResultField[boundaryNodes[j]]=0;#remplissage des valeurs pour les noeuds frontière (condition limite)
#sauvegarde sur le disque dur du résultat dans un fichier paraview
my_ResultField.writeVTK("FiniteElements2DPoisson_SQUARE_"+meshType+str(nbNodes))
print("Numerical solution of 2D Poisson equation on a square using finite elements done")
end = time.time()
#Calcul de l'erreur commise par rapport à la solution exacte
#===========================================================
#The following formulas use the fact that the exact solution is equal the right hand side divided by 2*pi*pi
max_abs_sol_exacte=max(my_RHSfield.max(),-my_RHSfield.min())/(2*pi*pi)
max_sol_num=my_ResultField.max()
min_sol_num=my_ResultField.min()
erreur_abs=0
for i in range(nbNodes) :
if erreur_abs < abs(my_RHSfield[i]/(2*pi*pi) - my_ResultField[i]) :
erreur_abs = abs(my_RHSfield[i]/(2*pi*pi) - my_ResultField[i])
print("Relative error = max(| exact solution - numerical solution |)/max(| exact solution |) = ",erreur_abs/max_abs_sol_exacte)
print ("Maximum numerical solution = ", max_sol_num, " Minimum numerical solution = ", min_sol_num)
#Postprocessing :
#================
# Extraction of the diagonal data
diag_data=VTK_routines.Extract_field_data_over_line_to_numpyArray(my_ResultField,[0,1,0],[1,0,0], resolution)
# save 2D picture
PV_routines.Save_PV_data_to_picture_file("FiniteElements2DPoisson_SQUARE_"+meshType+str(nbNodes)+'_0.vtu',"ResultField",'NODES',"FiniteElements2DPoisson_SQUARE_"+meshType+str(nbNodes))
test_desc["Computational_time_taken_by_run"]=end-start
test_desc["Absolute_error"]=erreur_abs
test_desc["Relative_error"]=erreur_abs/max_abs_sol_exacte
with open('test_Poisson'+str(my_mesh.getMeshDimension())+'D_EF_'+"SQUARE_"+meshType+str(nbCells)+ "Cells.json", 'w') as outfile:
json.dump(test_desc, outfile)
return erreur_abs/max_abs_sol_exacte, nbNodes, diag_data, min_sol_num, max_sol_num, end - start
def solve(my_mesh, filename, resolution, meshType, testColor):
start = time.time()
test_desc["Mesh_type"] = meshType
test_desc["Test_color"] = testColor
nbCells = my_mesh.getNumberOfCells()
if (my_mesh.getSpaceDimension() != 2 or my_mesh.getMeshDimension() != 2):
raise ValueError(
"Wrong space or mesh dimension : space and mesh dimensions should be 2"
)
test_desc["Space_dimension"] = my_mesh.getSpaceDimension()
test_desc["Mesh_dimension"] = my_mesh.getMeshDimension()
test_desc["Mesh_number_of_elements"] = my_mesh.getNumberOfCells()
test_desc["Mesh_cell_type"] = my_mesh.getElementTypesNames()
print("Mesh groups done")
print("Number of cells = ", nbCells)
#Discrétisation du second membre et extraction du nb max de voisins d'une cellule
#================================================================================
my_RHSfield = cdmath.Field("RHS_field", cdmath.CELLS, my_mesh, 1)
maxNbNeighbours = 0 #This is to determine the number of non zero coefficients in the sparse finite element rigidity matrix
#parcours des cellules pour discrétisation du second membre et extraction du nb max de voisins d'une cellule
for i in range(nbCells):
Ci = my_mesh.getCell(i)
x = Ci.x()
y = Ci.y()
my_RHSfield[i] = 2 * pi * pi * sin(pi * x) * sin(
pi * y) #mettre la fonction definie au second membre de l edp
# compute maximum number of neighbours
maxNbNeighbours = max(1 + Ci.getNumberOfFaces(), maxNbNeighbours)
test_desc["Mesh_max_number_of_neighbours"] = maxNbNeighbours
print("Right hand side discretisation done")
print("Max nb of neighbours=", maxNbNeighbours)
# Construction de la matrice et du vecteur second membre du système linéaire
#===========================================================================
Rigidite = cdmath.SparseMatrixPetsc(
nbCells, nbCells, maxNbNeighbours
) # warning : third argument is maximum number of non zero coefficients per line of the matrix
RHS = cdmath.Vector(nbCells)
#Parcours des cellules du domaine
for i in range(nbCells):
RHS[i] = my_RHSfield[
i] #la valeur moyenne du second membre f dans la cellule i
Ci = my_mesh.getCell(i)
for j in range(Ci.getNumberOfFaces()): # parcours des faces voisinnes
Fj = my_mesh.getFace(Ci.getFaceId(j))
if not Fj.isBorder():
k = Fj.getCellId(0)
if k == i:
k = Fj.getCellId(1)
Ck = my_mesh.getCell(k)
distance = Ci.getBarryCenter().distance(Ck.getBarryCenter())
coeff = Fj.getMeasure() / Ci.getMeasure() / distance
Rigidite.setValue(i, k, -coeff) # terme extradiagonal
else:
coeff = Fj.getMeasure() / Ci.getMeasure() / Ci.getBarryCenter(
).distance(Fj.getBarryCenter())
#For the particular case where the mesh boundary does not coincide with the domain boundary
x = Fj.getBarryCenter().x()
y = Fj.getBarryCenter().y()
RHS[i] += coeff * sin(pi * x) * sin(
pi * y
) #mettre ici la condition limite du problème de Dirichlet
Rigidite.addValue(i, i, coeff) # terme diagonal
print("Linear system matrix building done")
# Résolution du système linéaire
#=================================
LS = cdmath.LinearSolver(Rigidite, RHS, 500, 1.E-6, "CG", "LU")
LS.setComputeConditionNumber()
SolSyst = LS.solve()
print "Preconditioner used : ", LS.getNameOfPc()
print "Number of iterations used : ", LS.getNumberOfIter()
print("Linear system solved")
test_desc["Linear_solver_algorithm"] = LS.getNameOfMethod()
test_desc["Linear_solver_preconditioner"] = LS.getNameOfPc()
test_desc["Linear_solver_precision"] = LS.getTolerance()
test_desc["Linear_solver_maximum_iterations"] = LS.getNumberMaxOfIter()
test_desc[
"Linear_system_max_actual_iterations_number"] = LS.getNumberOfIter()
test_desc["Linear_system_max_actual_error"] = LS.getResidu()
test_desc[
"Linear_system_max_actual_condition number"] = LS.getConditionNumber()
# Création du champ résultat
#===========================
my_ResultField = cdmath.Field("ResultField", cdmath.CELLS, my_mesh, 1)
for i in range(nbCells):
my_ResultField[i] = SolSyst[i]
#sauvegarde sur le disque dur du résultat dans un fichier paraview
my_ResultField.writeVTK("FiniteVolumes2DPoisson_SQUARE_" + meshType +
str(nbCells))
print(
"Numerical solution of 2D Poisson equation on a square using finite elements done"
)
end = time.time()
#Calcul de l'erreur commise par rapport à la solution exacte
#===========================================================
#The following formulas use the fact that the exact solution is equal the right hand side divided by 2*pi*pi
max_abs_sol_exacte = max(my_RHSfield.max(),
-my_RHSfield.min()) / (2 * pi * pi)
max_sol_num = my_ResultField.max()
min_sol_num = my_ResultField.min()
erreur_abs = 0
for i in range(nbCells):
if erreur_abs < abs(my_RHSfield[i] /
(2 * pi * pi) - my_ResultField[i]):
erreur_abs = abs(my_RHSfield[i] / (2 * pi * pi) -
my_ResultField[i])
print(
"Relative error = max(| exact solution - numerical solution |)/max(| exact solution |) = ",
erreur_abs / max_abs_sol_exacte)
print("Maximum numerical solution = ", max_sol_num,
" Minimum numerical solution = ", min_sol_num)
#Postprocessing :
#================
# Extraction of the diagonal data
diag_data = VTK_routines.Extract_field_data_over_line_to_numpyArray(
my_ResultField, [0, 1, 0], [1, 0, 0], resolution)
# save 2D picture
PV_routines.Save_PV_data_to_picture_file(
"FiniteVolumes2DPoisson_SQUARE_" + meshType + str(nbCells) + '_0.vtu',
"ResultField", 'CELLS',
"FiniteVolumes2DPoisson_SQUARE_" + meshType + str(nbCells))
test_desc["Computational_time_taken_by_run"] = end - start
test_desc["Absolute_error"] = erreur_abs
test_desc["Relative_error"] = erreur_abs / max_abs_sol_exacte
with open(
'test_Poisson' + str(my_mesh.getMeshDimension()) + 'D_VF_' +
str(nbCells) + "Cells.json", 'w') as outfile:
json.dump(test_desc, outfile)
return erreur_abs / max_abs_sol_exacte, nbCells, diag_data, min_sol_num, max_sol_num, end - start