def upwindSchemeMatrix(nx,cfl):
upwindMat=cdmath.SparseMatrixPetsc(nx,nx,2)
for i in range(nx):
upwindMat.setValue(i,i,1+cfl)
upwindMat.setValue(i,(i-1)%nx,-cfl)
return upwindMat
def centeredSchemeMatrix(nx,cfl):
centeredMat=cdmath.SparseMatrixPetsc(nx,nx,2)
for i in range(nx):
centeredMat.setValue(i,i,1+cfl)
centeredMat.setValue(i,(i-1)%nx,-cfl)
return centeredMat
def computeStaggeredDivergenceMatrix(a, b, nx, nbVoisinsMax, dt, scaling):
nbCells = nx
dx = (b - a) / nx
dim = 1
nbComp = dim + 1
normal = cdmath.Vector(dim)
implMat = cdmath.SparseMatrixPetsc(nbCells * nbComp, nbCells * nbComp,
(nbVoisinsMax + 1) * nbComp)
S1, S2 = staggeredMatrices(dt / dx, scaling)
for k in range(nbCells): #On parcourt les cellules
if (k == 0):
implMat.addValue(k * nbComp, (k + 1) * nbComp, S1)
implMat.addValue(k * nbComp, k * nbComp, S1 * (-1.))
elif (k == nbCells - 1):
implMat.addValue(k * nbComp, k * nbComp, S2)
implMat.addValue(k * nbComp, (k - 1) * nbComp, S2 * (-1.))
else:
implMat.addValue(k * nbComp, (k + 1) * nbComp, S1)
implMat.addValue(k * nbComp, k * nbComp, S1 * (-1.))
implMat.addValue(k * nbComp, k * nbComp, S2)
implMat.addValue(k * nbComp, (k - 1) * nbComp, S2 * (-1.))
return implMat
def computeUpwindDivergenceMatrix(a,b,nx,nbVoisinsMax,dt,scaling):
nbCells = nx
dx=(b-a)/nx
dim=1
nbComp=dim+1
normal=cdmath.Vector(dim)
implMat=cdmath.SparseMatrixPetsc(nbCells*nbComp,nbCells*nbComp,(nbVoisinsMax+1)*nbComp)
A,absA= jacobianMatrices(dt/dx,scaling)
for j in range(nbCells):#On parcourt les cellules
if ( j==0) :
implMat.addValue(j*nbComp,(j+1)*nbComp,(A-absA)/2)
implMat.addValue(j*nbComp, j*nbComp,(A-absA)/2*(-1.))
elif ( j==nbCells-1) :
implMat.addValue(j*nbComp, j*nbComp,(A+absA)/2)
implMat.addValue(j*nbComp,(j-1)*nbComp,(A+absA)/2*(-1.))
else :
implMat.addValue(j*nbComp,(j+1)*nbComp,(A-absA)/2)
implMat.addValue(j*nbComp, j*nbComp,(A-absA)/2*(-1.))
implMat.addValue(j*nbComp, j*nbComp,(A+absA)/2)
implMat.addValue(j*nbComp,(j-1)*nbComp,(A+absA)/2*(-1.))
return implMat
def computeDivergenceMatrix(my_mesh, nbVoisinsMax, dt, test_bc, velocity):
nbCells = my_mesh.getNumberOfCells()
dim = my_mesh.getMeshDimension()
nbComp = 1
normal = cdmath.Vector(dim)
implMat = cdmath.SparseMatrixPetsc(nbCells * nbComp, nbCells * nbComp,
(nbVoisinsMax + 1) * nbComp)
for j in range(nbCells): #On parcourt les cellules
Cj = my_mesh.getCell(j)
nbFaces = Cj.getNumberOfFaces()
for k in range(nbFaces):
indexFace = Cj.getFacesId()[k]
Fk = my_mesh.getFace(indexFace)
for i in range(dim):
normal[i] = Cj.getNormalVector(k, i)
#normale sortante
Am = upwinding_coeff(normal,
dt * Fk.getMeasure() / Cj.getMeasure(),
velocity)
cellAutre = -1
if (not Fk.isBorder()):
# hypothese: La cellule d'index indexC1 est la cellule courante index j */
if (Fk.getCellsId()[0] == j):
# hypothese verifiée
cellAutre = Fk.getCellsId()[1]
elif (Fk.getCellsId()[1] == j):
# hypothese non verifiée
cellAutre = Fk.getCellsId()[0]
else:
raise ValueError(
"computeFluxes: problem with mesh, unknown cel number")
implMat.addValue(j * nbComp, cellAutre * nbComp, Am)
implMat.addValue(j * nbComp, j * nbComp, Am * (-1.))
else:
if (
test_bc == "Periodic"
and Fk.getGroupName() != "Neumann"
): #Periodic boundary condition unless Wall/Neumann specified explicitly
indexFP = my_mesh.getIndexFacePeriodic(
indexFace,
my_mesh.getName() == "squareWithBrickWall",
my_mesh.getName() == "squareWithHexagons")
Fp = my_mesh.getFace(indexFP)
cellAutre = Fp.getCellsId()[0]
implMat.addValue(j * nbComp, cellAutre * nbComp, Am)
implMat.addValue(j * nbComp, j * nbComp, Am * (-1.))
elif (test_bc != "Neumann" and Fk.getGroupName() != "Neumann"
): #Nothing to do for Neumann boundary condition
print Fk.getGroupName()
raise ValueError(
"computeFluxes: Unknown boundary condition name")
return implMat
def computeDivergenceMatrix(my_mesh,nbVoisinsMax,dt,test_bc):
nbCells = my_mesh.getNumberOfCells()
dim=my_mesh.getMeshDimension()
nbComp=dim+1
normal=cdmath.Vector(dim)
implMat=cdmath.SparseMatrixPetsc(nbCells*nbComp,nbCells*nbComp,(nbVoisinsMax+1)*nbComp)
idMoinsJacCL=cdmath.Matrix(nbComp)
for j in range(nbCells):#On parcourt les cellules
Cj = my_mesh.getCell(j)
nbFaces = Cj.getNumberOfFaces();
for k in range(nbFaces) :
indexFace = Cj.getFacesId()[k];
Fk = my_mesh.getFace(indexFace);
for i in range(dim) :
normal[i] = Cj.getNormalVector(k, i);#normale sortante
Am=jacobianMatrices( normal,dt*Fk.getMeasure()/Cj.getMeasure());
cellAutre =-1
if ( not Fk.isBorder()) :
# hypothese: La cellule d'index indexC1 est la cellule courante index j */
if (Fk.getCellsId()[0] == j) :
# hypothese verifiée
cellAutre = Fk.getCellsId()[1];
elif(Fk.getCellsId()[1] == j) :
# hypothese non verifiée
cellAutre = Fk.getCellsId()[0];
else :
raise ValueError("computeFluxes: problem with mesh, unknown cel number")
implMat.addValue(j*nbComp,cellAutre*nbComp,Am)
implMat.addValue(j*nbComp, j*nbComp,Am*(-1.))
else :
if( test_bc=="Periodic" and Fk.getGroupName() != "Wall" and Fk.getGroupName() != "Paroi" and Fk.getGroupName() != "Neumann"):#Periodic boundary condition unless Wall/Neumann specified explicitly
test_desc["Boundary_conditions"]="Periodic"
indexFP = my_mesh.getIndexFacePeriodic(indexFace)
Fp = my_mesh.getFace(indexFP)
cellAutre = Fp.getCellsId()[0]
implMat.addValue(j*nbComp,cellAutre*nbComp,Am)
implMat.addValue(j*nbComp, j*nbComp,Am*(-1.))
elif( test_bc=="Wall" or Fk.getGroupName() == "Wall" or Fk.getGroupName() == "Paroi"):#Wall boundary condition
test_desc["Boundary_conditions"]="Wall"
v=cdmath.Vector(dim+1)
for i in range(dim) :
v[i+1]=normal[i]
idMoinsJacCL=v.tensProduct(v)*2
implMat.addValue(j*nbComp,j*nbComp,Am*(-1.)*idMoinsJacCL)
elif(test_bc!="Neumann" and Fk.getGroupName() != "Neumann"):#Nothing to do for Neumann boundary condition
print Fk.getGroupName()
raise ValueError("computeFluxes: Unknown boundary condition name");
return implMat
def implicitSchemeMatrix(nx, cfl):
implMat = cdmath.SparseMatrixPetsc(nx, nx, 3)
for i in range(nx):
implMat.setValue(i, (i + 1) % nx, -cfl)
implMat.setValue(i, i, 1. + 2 * cfl)
implMat.setValue(i, (i - 1) % nx, -cfl)
return implMat
def computeDivergenceMatrix(my_mesh, nbVoisinsMax, dt, scaling, test_bc):
nbCells = my_mesh.getNumberOfCells()
dim = my_mesh.getMeshDimension()
nbComp = dim + 1
normal = cdmath.Vector(dim)
implMat = cdmath.SparseMatrixPetsc(nbCells * nbComp, nbCells * nbComp,
(nbVoisinsMax + 1) * nbComp)
idMoinsJacCL = cdmath.Matrix(nbComp)
for j in range(nbCells): #On parcourt les cellules
Cj = my_mesh.getCell(j)
nbFaces = Cj.getNumberOfFaces()
for k in range(nbFaces):
indexFace = Cj.getFacesId()[k]
Fk = my_mesh.getFace(indexFace)
for i in range(dim):
normal[i] = Cj.getNormalVector(k, i)
#normale sortante
Am = jacobianMatrices(normal,
dt * Fk.getMeasure() / Cj.getMeasure(),
scaling)
cellAutre = -1
if (not Fk.isBorder()):
# hypothese: La cellule d'index indexC1 est la cellule courante index j */
if (Fk.getCellsId()[0] == j):
# hypothese verifiée
cellAutre = Fk.getCellsId()[1]
elif (Fk.getCellsId()[1] == j):
# hypothese non verifiée
cellAutre = Fk.getCellsId()[0]
else:
raise ValueError(
"computeFluxes: problem with mesh, unknown cell number"
)
implMat.addValue(j * nbComp, cellAutre * nbComp, Am)
implMat.addValue(j * nbComp, j * nbComp, Am * (-1.))
else:
indexFP = my_mesh.getIndexFacePeriodic(
indexFace,
my_mesh.getName() == "squareWithBrickWall",
my_mesh.getName() == "squareWithHexagons")
Fp = my_mesh.getFace(indexFP)
cellAutre = Fp.getCellsId()[0]
implMat.addValue(j * nbComp, cellAutre * nbComp, Am)
implMat.addValue(j * nbComp, j * nbComp, Am * (-1.))
return implMat
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
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
else: # Détection des noeuds intérieurs
interiorNodes.append(i)
nbInteriorNodes = nbInteriorNodes + 1
maxNbNeighbours = max(
1 + Ni.getNumberOfCells(), maxNbNeighbours
) #true only in 2D, otherwise use function Ni.getNumberOfEdges()
print("Right hand side discretisation done")
print("Number of interior nodes=", nbInteriorNodes)
print("Number of boundary nodes=", nbBoundaryNodes)
print("Maximum 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 max 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)
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
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
my_RHSfield[i] = 9*sin(3*phi)*cos(3*theta+ phi)/(r*r) + (10*sin(3*phi)*cos(3*theta+ phi) + 6*cos(3*phi)*sin(3*theta+ phi))/((R+r*cos(theta))*(R+r*cos(theta))) - 3*sin(theta)*sin(3*phi)*sin(3*theta+ phi)/(r*(R+r*cos(theta))) #for the right hand side we use the function given in the article of Olshanskii, Reusken 2009, page 19
if my_mesh.isBorderNode(i): # Détection des noeuds frontière
raise ValueError("Mesh should not contain borders")
else:
maxNbNeighbours = max(1+Ni.getNumberOfCells(),maxNbNeighbours)#true only for planar cells, otherwise use function Ni.getNumberOfEdges()
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)
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)
def computeDivergenceMatrix(my_mesh, nbVoisinsMax, dt, scaling):
nbCells = my_mesh.getNumberOfCells()
dim = my_mesh.getMeshDimension()
nbComp = dim + 1
if (not my_mesh.isStructured()):
raise ValueError("WaveSystemStaggered: the mesh should be structured")
NxNyNz = my_mesh.getCellGridStructure()
DxDyDz = my_mesh.getDXYZ()
implMat = cdmath.SparseMatrixPetsc(nbCells * nbComp, nbCells * nbComp,
(nbVoisinsMax + 1) * nbComp)
idMoinsJacCL = cdmath.Matrix(nbComp)
if (dim == 1):
nx = NxNyNz[0]
dx = DxDyDz[0]
if (scaling == 0):
for k in range(nbCells):
implMat.addValue(k, 1 * nbCells + k, -c0 * c0 * dt / dx)
implMat.addValue(k, 1 * nbCells + (k + 1) % nx,
c0 * c0 * dt / dx)
implMat.addValue(1 * nbCells + k, k, dt / dx)
implMat.addValue(1 * nbCells + (k + 1) % nx, k, -dt / dx)
else: # scaling >0
for k in range(nbCells):
implMat.addValue(k, 1 * nbCells + k, -c0 * dt / dx)
implMat.addValue(k, 1 * nbCells + (k + 1) % nx, c0 * dt / dx)
implMat.addValue(1 * nbCells + k, k, c0 * dt / dx)
implMat.addValue(1 * nbCells + (k + 1) % nx, k, -c0 * dt / dx)
elif (dim == 2): # k = j*nx+i
nx = NxNyNz[0]
ny = NxNyNz[1]
dx = DxDyDz[0]
dy = DxDyDz[1]
if (scaling == 0):
for k in range(nbCells):
i = k % nx
j = k // nx
implMat.addValue(k, 1 * nbCells + j * nx + i,
-c0 * c0 * dt / dx)
implMat.addValue(k, 1 * nbCells + j * nx + (i + 1) % nx,
c0 * c0 * dt / dx)
implMat.addValue(k, 2 * nbCells + j * nx + i,
-c0 * c0 * dt / dy)
implMat.addValue(k, 2 * nbCells + ((j + 1) % ny) * nx + i,
c0 * c0 * dt / dy)
implMat.addValue(1 * nbCells + j * nx + i, k, dt / dx)
implMat.addValue(1 * nbCells + j * nx + (i + 1) % nx, k,
-dt / dx)
implMat.addValue(2 * nbCells + j * nx + i, k, dt / dy)
implMat.addValue(2 * nbCells + ((j + 1) % ny) * nx + i, k,
-dt / dy)
else: # scaling >0
for k in range(nbCells):
i = k % nx
j = k // nx
implMat.addValue(k, 1 * nbCells + j * nx + i, -c0 * dt / dx)
implMat.addValue(k, 1 * nbCells + j * nx + (i + 1) % nx,
c0 * dt / dx)
implMat.addValue(k, 2 * nbCells + j * nx + i, -c0 * dt / dy)
implMat.addValue(k, 2 * nbCells + ((j + 1) % ny) * nx + i,
c0 * dt / dy)
implMat.addValue(1 * nbCells + j * nx + i, k, c0 * dt / dx)
implMat.addValue(1 * nbCells + j * nx + (i + 1) % nx, k,
-c0 * dt / dx)
implMat.addValue(2 * nbCells + j * nx + i, k, c0 * dt / dy)
implMat.addValue(2 * nbCells + ((j + 1) % ny) * nx + i, k,
-c0 * dt / dy)
elif (dim == 3): # k = l*nx*ny+j*nx+i
nx = NxNyNz[0]
ny = NxNyNz[1]
nz = NxNyNz[2]
dx = DxDyDz[0]
dy = DxDyDz[1]
dz = DxDyDz[2]
if (scaling == 0):
for k in range(nbCells):
i = k % nx
j = (k // nx) % ny
l = k // (nx * ny)
implMat.addValue(k, 1 * nbCells + l * nx * ny + j * nx + i,
-c0 * c0 * dt / dx)
implMat.addValue(
k, 1 * nbCells + l * nx * ny + j * nx + (i + 1) % nx,
c0 * c0 * dt / dx)
implMat.addValue(k, 2 * nbCells + l * nx * ny + j * nx + i,
-c0 * c0 * dt / dy)
implMat.addValue(
k, 2 * nbCells + l * nx * ny + ((j + 1) % ny) * nx + i,
c0 * c0 * dt / dy)
implMat.addValue(k, 3 * nbCells + l * nx * ny + j * nx + i,
-c0 * c0 * dt / dz)
implMat.addValue(
k, 3 * nbCells + ((l + 1) % nz) * nx * ny + j * nx + i,
c0 * c0 * dt / dz)
implMat.addValue(1 * nbCells + l * nx * ny + j * nx + i, k,
dt / dx)
implMat.addValue(
1 * nbCells + l * nx * ny + j * nx + (i + 1) % nx, k,
-dt / dx)
implMat.addValue(2 * nbCells + l * nx * ny + j * nx + i, k,
dt / dy)
implMat.addValue(
2 * nbCells + l * nx * ny + ((j + 1) % ny) * nx + i, k,
-dt / dy)
implMat.addValue(3 * nbCells + l * nx * ny + j * nx + i, k,
dt / dz)
implMat.addValue(
3 * nbCells + ((l + 1) % nz) * nx * ny + j * nx + i, k,
-dt / dz)
else: # scaling >0
for k in range(nbCells):
i = k % nx
j = (k // nx) % ny
l = k // (nx * ny)
implMat.addValue(k, 1 * nbCells + l * nx * ny + j * nx + i,
-c0 * dt / dx)
implMat.addValue(
k, 1 * nbCells + l * nx * ny + j * nx + (i + 1) % nx,
c0 * dt / dx)
implMat.addValue(k, 2 * nbCells + l * nx * ny + j * nx + i,
-c0 * dt / dy)
implMat.addValue(
k, 2 * nbCells + l * nx * ny + ((j + 1) % ny) * nx + i,
c0 * dt / dy)
implMat.addValue(k, 3 * nbCells + l * nx * ny + j * nx + i,
-c0 * dt / dz)
implMat.addValue(
k, 3 * nbCells + ((l + 1) % nz) * nx * ny + j * nx + i,
c0 * dt / dz)
implMat.addValue(1 * nbCells + l * nx * ny + j * nx + i, k,
c0 * dt / dx)
implMat.addValue(
1 * nbCells + l * nx * ny + j * nx + (i + 1) % nx, k,
-c0 * dt / dx)
implMat.addValue(2 * nbCells + l * nx * ny + j * nx + i, k,
c0 * dt / dy)
implMat.addValue(
2 * nbCells + l * nx * ny + ((j + 1) % ny) * nx + i, k,
-c0 * dt / dy)
implMat.addValue(3 * nbCells + l * nx * ny + j * nx + i, k,
c0 * dt / dz)
implMat.addValue(
3 * nbCells + ((l + 1) % nz) * nx * ny + j * nx + i, k,
-c0 * dt / dz)
#Add the identity matrix on the diagonal
#divMat.diagonalShift(1)#only after filling all coefficients
for j in range(nbCells * nbComp):
implMat.addValue(j, j, 1) #/(c0*c0)
return implMat
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 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 EulerSystemVF(ntmax, tmax, cfl, my_mesh, output_freq, filename, resolution,
isImplicit):
dim = my_mesh.getMeshDimension()
nbCells = my_mesh.getNumberOfCells()
meshName = my_mesh.getName()
dt = 0.
time = 0.
it = 0
isStationary = False
nbVoisinsMax = my_mesh.getMaxNbNeighbours(cdmath.CELLS)
# Initial conditions #
print("Construction of the initial condition …")
if (dim == 1 or 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_shock(
my_mesh, False)
elif (filename.find("disk") > -1 or filename.find("Disk") > -1):
pressure_field, velocity_field = initial_conditions_shock(
my_mesh, True)
else:
print "Mesh name : ", filename
raise ValueError(
"Mesh name should contain substring square, cube or disk")
#iteration vectors
Un = cdmath.Vector(nbCells * (dim + 1))
dUn = cdmath.Vector(nbCells * (dim + 1))
for k in range(nbCells):
Un[k * (dim + 1) + 0] = pressure_field[k]
Un[k * (dim + 1) + 1] = rho0 * velocity_field[k, 0]
if (dim >= 2):
Un[k + 2 *
nbCells] = rho0 * velocity_field[k,
1] # value on the bottom face
if (dim == 3):
Un[k + 3 * nbCells] = rho0 * initial_velocity[k, 2]
#sauvegarde de la donnée initiale
pressure_field.setTime(time, it)
pressure_field.writeVTK("EulerSystem" + str(dim) + "DUpwind" +
"_isImplicit" + str(isImplicit) + meshName +
"_pressure")
velocity_field.setTime(time, it)
velocity_field.writeVTK("EulerSystem" + str(dim) + "DUpwind" +
"_isImplicit" + str(isImplicit) + meshName +
"_velocity")
dx_min = my_mesh.minRatioVolSurf()
divMat = cdmath.SparseMatrixPetsc(nbCells * (1 + dim), nbCells * (1 + dim),
(nbVoisinsMax + 1) * (1 + dim))
if (isImplicit):
iterGMRESMax = 50
LS = cdmath.LinearSolver(divMat, Un, iterGMRESMax, precision, "GMRES",
"ILU")
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):
divMat.zeroEntries() #sets the matrix coefficients to zero
vp_max = computeDivergenceMatrix(my_mesh, divMat,
Un) #To update at each time step
dt = cfl * dx_min / vp_max #To update at each time step
if (isImplicit):
#Adding the identity matrix on the diagonal
divMat.diagonalShift(1) #only after filling all coefficients
dUn = Un.deepCopy()
LS.setSndMember(Un)
Un = LS.solve()
if (not LS.getStatus()):
print "Linear system did not converge ", LS.getNumberOfIter(
), " GMRES iterations"
raise ValueError("Pas de convergence du système linéaire")
dUn -= Un
else:
dUn = divMat * Un
Un -= dUn
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))
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
pressure_field.setTime(time, it)
pressure_field.writeVTK(
"EulerSystem" + str(dim) + "DUpwind" + "_isImplicit" +
str(isImplicit) + meshName + "_pressure", False)
velocity_field.setTime(time, it)
velocity_field.writeVTK(
"EulerSystem" + str(dim) + "DUpwind" + "_isImplicit" +
str(isImplicit) + meshName + "_velocity", False)
print("-- Iter: " + str(it) + ", Time: " + str(time) + ", dt: " + str(dt))
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("EulerSystem" + str(dim) + "DUpwind" +
"_isImplicit" + str(isImplicit) + meshName +
"_pressure_Stat")
velocity_field.setTime(time, 0)
velocity_field.writeVTK("EulerSystem" + str(dim) + "DUpwind" +
"_isImplicit" + str(isImplicit) + 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(
"EulerSystem" + str(dim) + "DUpwind" + "_isImplicit" +
str(isImplicit) + meshName + "_pressure_Stat" + '_0.vtu',
"Pressure", 'CELLS',
"EulerSystem" + str(dim) + "DUpwind" + meshName + "_pressure_Stat")
PV_routines.Save_PV_data_to_picture_file(
"EulerSystem" + str(dim) + "DUpwind" + "_isImplicit" +
str(isImplicit) + meshName + "_velocity_Stat" + '_0.vtu',
"Velocity", 'CELLS',
"EulerSystem" + str(dim) + "DUpwind" + meshName + "_velocity_Stat")
else:
print "Temps maximum Tmax= ", tmax, " atteint"
