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
# Plot over slice circle
finiteElementsOnSphere_0vtu = pvs.XMLUnstructuredGridReader(
FileName=["FiniteElementsOnSphere" + '_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("./FiniteElementsOnSphere" + "_Slice" + '.png',
magnification=1,
quality=100,
view=renderView1)
plotOnSortedLines1 = pvs.PlotOnSortedLines(Input=slice1)
lineChartView2 = pvs.CreateView('XYChartView')
plotOnSortedLines1Display = pvs.Show(plotOnSortedLines1, lineChartView2)
plotOnSortedLines1Display.UseIndexForXAxis = 0
plotOnSortedLines1Display.XArrayName = 'arc_length'
plotOnSortedLines1Display.SeriesVisibility = ['ResultField (1)']
pvs.SaveScreenshot("./FiniteElementsOnSphere" + "_PlotOnSortedLine_" + '.png',
magnification=1,
quality=100,
view=lineChartView2)
pvs.Delete(lineChartView2)
print("Integral of the numerical solution", my_ResultField.integral(0))
print(
"Numerical solution of Poisson equation on a sphere using finite elements done"
)