def quadQuadFE(coord,Dmat):
nInt=3 # 2 quadrature points in each direction.
(pos,wei)=numericalIntegration.gaussQuad(nInt)
Kloc=np.zeros((2*len(coord),2*len(coord)))
for i in range(nInt):
for j in range(nInt):
_,Bmat,Jacobian=shapeFunction.shapeQuadQuadFE(coord,pos[i],pos[j])
Kloc+=np.matmul(Bmat.transpose(),np.matmul(Dmat,Bmat))*Jacobian*wei[i]*wei[j]
return Kloc
def ICMFE(coord,Dmat):
nInt=2 # 2 quadrature points in each direction.
(pos,wei)=numericalIntegration.gaussQuad(nInt)
Kloc=np.zeros((2*len(coord),2*len(coord)))
Hloc=np.zeros((4,4))
Eloc=np.zeros((2*len(coord),4))
for i in range(nInt):
for j in range(nInt):
Bmat,Gmat,Jacobian=shapeFunction.shapeICMFE(coord,pos[i],pos[j])
Kloc+=np.matmul(Bmat.transpose(),np.matmul(Dmat,Bmat))*Jacobian*wei[i]*wei[j]
Hloc+=np.matmul(Gmat.transpose(),np.matmul(Dmat,Gmat))*Jacobian*wei[i]*wei[j]
Eloc+=np.matmul(Bmat.transpose(),np.matmul(Dmat,Gmat))*Jacobian*wei[i]*wei[j]
[email protected](Hloc,Eloc.transpose())
return Kloc,Hloc,Eloc
def quadraticFE(inputData):
"""The solver for second order finite elements (9-node quads & 6-node triangles)."""
startTime=time.time()
parameters=inputData[0]
# Material matrices.
materialList=inputData[6]
materialMatrix=[None]*len(materialList)
for i in range(len(materialList)):
materialMatrix[i]=twoDMaterialMatrix(materialList[i],parameters[1])
# Assemble stiffness matrix.
coordinates=inputData[5]
meshes=inputData[3]
materialMeshList=inputData[4]
nodeElements=toDC.nodeElementList(coordinates,meshes)
Iglo=[]
Jglo=[]
Vglo=[]
for i in range(len(meshes[0])):
coord=AMORE.getCoord(coordinates,meshes[0][i])
if len(meshes[0][i])==6:
Kloc=stiffnessMatrix.triQuadFE(coord,materialMatrix[materialMeshList[0][i]])
elif len(meshes[0][i])==9:
Kloc=stiffnessMatrix.quadQuadFE(coord,materialMatrix[materialMeshList[0][i]])
else: raise ValueError("Wrong element numbering!")
Iloc,Jloc,Vloc=stiffnessMatrix.sparsifyElementMatrix(Kloc,meshes[0][i])
Iglo.extend(Iloc)
Jglo.extend(Jloc)
Vglo.extend(Vloc)
Iglo=np.array(Iglo,dtype=int)
Jglo=np.array(Jglo,dtype=int)
Vglo=np.array(Vglo,dtype='d')
Kglo=scipy.sparse.coo_matrix((Vglo,(Iglo,Jglo)),shape=(2*len(coordinates),2*len(coordinates))).tocsr()
print("Assembling stiffness matrix costs %s seconds."%(time.time()-startTime))
startTime=time.time()
# Force term.
indForce=inputData[-2]
if indForce[0]: # Body force is imposed.
pass
forceList=inputData[-1]
nInt=3
if indForce[1]: nInt+=2 # Customized boundary force.
pos,wei=numericalIntegration.gaussQuad(nInt)
fglo=np.zeros((2*len(coordinates),1))
for i in range(len(forceList)//2):
node1=forceList[2*i][0]
node2=forceList[2*i+1][0]
# length=lenEdge(coordinates[node1],coordinates[node2])
# Find the element.
elementPosition=(set(nodeElements[node1]) & set(nodeElements[node2])).pop()
numbering=meshes[elementPosition[0]][elementPosition[1]]
node3=findMidNode(numbering,node1,node2)
force1=np.array([forceList[2*i][1:3]]).transpose()
force2=np.array([forceList[2*i+1][1:3]]).transpose()
floc=np.zeros((6,1))
coord=np.array([coordinates[node1],coordinates[node2],[0.0,0.0]])
if coordinates[node3]: coord[2,:]=np.array(coordinates[node3])
else: coord[2,:]=0.5*(coord[0,:]+coord[1,:])
for j in range(nInt):
# Only support linear force distribution.
# Otherwise, use customized boundary force.
Nmat=shapeFunction.oneDLinear(pos[j])
force=Nmat[0,0]*force1+Nmat[0,2]*force2
quadNmat,Jacobian=shapeFunction.oneDQuadratic(pos[j],coord)
floc+=wei[j]*Jacobian*np.matmul(quadNmat.transpose(),force)
fglo[2*node1:2*node1+2,0]+=floc[0:2,0]
fglo[2*node2:2*node2+2,0]+=floc[2:4,0]
fglo[2*node3:2*node3+2,0]+=floc[4:6,0]
print("Calculating force term costs %s seconds."%(time.time()-startTime))
startTime=time.time()
# Impose constraints.
fixList=np.zeros((2*len(coordinates),1))
fixIndexList=np.zeros((2*len(coordinates),1),dtype=int)
constraintList=inputData[-3]
# Very important!!! Sort the constraints!!!
constraintList.sort(key=lambda item:item[0])
for i in constraintList:
if i[1]:
fixList[2*i[0]]=i[3]
fixIndexList[2*i[0]]=1
if i[2]:
fixList[2*i[0]+1]=i[4]
fixIndexList[2*i[0]+1]=1
# Solve.
fglo-=(Kglo.dot(fixList))
Kglo_complete=Kglo.copy()
Kglo=Kglo.tolil()
count=0
for i in constraintList:
if i[1]:
delete_row_lil(Kglo,2*i[0]-count)
fglo=np.delete(fglo,2*i[0]-count)
count+=1
if i[2]:
delete_row_lil(Kglo,2*i[0]+1-count)
fglo=np.delete(fglo,2*i[0]+1-count)
count+=1
Kglo=Kglo.transpose()
count=0
for i in constraintList:
if i[1]:
delete_row_lil(Kglo,2*i[0]-count)
count+=1
if i[2]:
delete_row_lil(Kglo,2*i[0]+1-count)
count+=1
print("Imposing constraints costs %s seconds."%(time.time()-startTime))
startTime=time.time()
Kglo=Kglo.tocsc()
# factor=cholesky(Kglo)
# disp=factor(fglo)
disp=spsolve(Kglo,fglo)
print("Solving the linear system costs %s seconds."%(time.time()-startTime))
# The complete displacement solution:
displacement=np.zeros((2*len(coordinates),1))
count=0
for i in range(2*len(coordinates)):
if fixIndexList[i]:
displacement[i]=fixList[i]
count+=1
else:
displacement[i]=disp[i-count]
energy=0.5*displacement.transpose()@Kglo_complete@displacement
return displacement,energy
def ICMFE(inputData):
"""The solver for (4-node) ICM finite elements. Warning: The code is only for squares.
For general quadrilaterals, the formulation needs to be modified to pass patch tests."""
startTime=time.time()
parameters=inputData[0]
# Material matrices.
materialList=inputData[6]
materialMatrix=[None]*len(materialList)
for i in range(len(materialList)):
materialMatrix[i]=twoDMaterialMatrix(materialList[i],parameters[1])
# Assemble stiffness matrix.
coordinates=inputData[5]
meshes=inputData[3]
materialMeshList=inputData[4]
Iglo=[]
Jglo=[]
Vglo=[]
for i in range(len(meshes[0])):
coord=AMORE.getCoord(coordinates,meshes[0][i])
if len(meshes[0][i])==4:
Kloc,_,_=stiffnessMatrix.ICMFE(coord,materialMatrix[materialMeshList[0][i]])
else: raise ValueError("Wrong element numbering!")
Iloc,Jloc,Vloc=stiffnessMatrix.sparsifyElementMatrix(Kloc,meshes[0][i])
Iglo.extend(Iloc)
Jglo.extend(Jloc)
Vglo.extend(Vloc)
Iglo=np.array(Iglo,dtype=int)
Jglo=np.array(Jglo,dtype=int)
Vglo=np.array(Vglo,dtype='d')
Kglo=scipy.sparse.coo_matrix((Vglo,(Iglo,Jglo)),shape=(2*len(coordinates),2*len(coordinates))).tocsr()
print("Assembling stiffness matrix costs %s seconds."%(time.time()-startTime))
startTime=time.time()
# Force term.
indForce=inputData[-2]
if indForce[0]: # Body force is imposed.
pass
forceList=inputData[-1]
nInt=2
if indForce[1]: nInt+=3 # Customized boundary force.
pos,wei=numericalIntegration.gaussQuad(nInt)
fglo=np.zeros((2*len(coordinates),1))
for i in range(len(forceList)//2):
node1=forceList[2*i][0]
node2=forceList[2*i+1][0]
length=lenEdge(coordinates[node1],coordinates[node2])
force1=np.array([forceList[2*i][1:3]]).transpose()
force2=np.array([forceList[2*i+1][1:3]]).transpose()
floc=np.zeros((4,1))
for j in range(nInt):
Nmat=shapeFunction.oneDLinear(pos[j])
force=Nmat[0,0]*force1+Nmat[0,2]*force2
floc+=0.5*wei[j]*length*np.matmul(Nmat.transpose(),force)
fglo[2*node1:2*node1+2,0]+=floc[0:2,0]
fglo[2*node2:2*node2+2,0]+=floc[2:4,0]
print("Calculating force term costs %s seconds."%(time.time()-startTime))
startTime=time.time()
# Impose constraints.
fixList=np.zeros((2*len(coordinates),1))
fixIndexList=np.zeros((2*len(coordinates),1),dtype=int)
constraintList=inputData[-3]
# Very important!!! Sort the constraints!!!
constraintList.sort(key=lambda item:item[0])
for i in constraintList:
if i[1]:
fixList[2*i[0]]=i[3]
fixIndexList[2*i[0]]=1
if i[2]:
fixList[2*i[0]+1]=i[4]
fixIndexList[2*i[0]+1]=1
# Solve.
fglo-=(Kglo.dot(fixList))
Kglo_complete=Kglo.copy()
Kglo=Kglo.tolil()
count=0
for i in constraintList:
if i[1]:
delete_row_lil(Kglo,2*i[0]-count)
fglo=np.delete(fglo,2*i[0]-count)
count+=1
if i[2]:
delete_row_lil(Kglo,2*i[0]+1-count)
fglo=np.delete(fglo,2*i[0]+1-count)
count+=1
Kglo=Kglo.transpose()
count=0
for i in constraintList:
if i[1]:
delete_row_lil(Kglo,2*i[0]-count)
count+=1
if i[2]:
delete_row_lil(Kglo,2*i[0]+1-count)
count+=1
print("Imposing constraints costs %s seconds."%(time.time()-startTime))
startTime=time.time()
Kglo=Kglo.tocsc()
print("Number of non-zero sparse matrix entries = %s."%Kglo.count_nonzero())
# factor=cholesky(Kglo)
# disp=factor(fglo)
disp=spsolve(Kglo,fglo)
print("Solving the linear system costs %s seconds."%(time.time()-startTime))
# The complete displacement solution:
displacement=np.zeros((2*len(coordinates),1))
count=0
for i in range(2*len(coordinates)):
if fixIndexList[i]:
displacement[i]=fixList[i]
count+=1
else:
displacement[i]=disp[i-count]
energy=0.5*displacement.transpose()@Kglo_complete@displacement
return displacement,energy