/
time_int_schemes.py
559 lines (411 loc) · 14.2 KB
/
time_int_schemes.py
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from dolfin import *
import numpy as np
import scipy.sparse as sps
import scipy.sparse.linalg as spsla
import krypy.linsys
from scipy.linalg import qr
from scipy.io import loadmat, savemat
import dolfin_to_nparrays as dtn
###
# solve M\dot v + K(v) -B'p = fv
# Bv = fpbc
###
def mass_fem_ip(q1,q2,M):
"""M^-1 inner product
"""
return np.dot(q1.T.conj(),krypy.linsys.cg(M,q2,tol=1e-12)['xk'])
def smamin_fem_ip(qqpq1,qqpq2,Mv,Mp,Nv,Npc):
""" M^-1 ip for the extended system
"""
return mass_fem_ip(qqpq1[:Nv,],qqpq2[:Nv,],Mv) + \
mass_fem_ip(qqpq1[Nv:-Npc,],qqpq2[Nv:-Npc,],Mp) + \
mass_fem_ip(qqpq1[-Npc:,],qqpq2[-Npc:,],Mp)
def halfexp_euler_smarminex(MSme,BSme,MP,FvbcSme,FpbcSme,B2BoolInv,PrP,TsP,vp_init,qqpq_init=None):
"""halfexplicit euler for the NSE in index 1 formulation
"""
N, Pdof = PrP.N, PrP.Pdof
Nts, t0, tE, dt, Nv, Np = init_time_stepping(PrP,TsP)
tcur = t0
Npc = Np-1
# remove the p - freedom
if Pdof == 0:
BSme = BSme[1:,:]
FpbcSmeC = FpbcSme[1:,]
MPc = MP[1:,:][:,1:]
else:
BSme = sps.vstack([BSme[:Pdof,:],BSme[Pdof+1:,:]])
raise Warning('TODO: Implement this')
B1Sme = BSme[:,:Nv-(Np-1)]
B2Sme = BSme[:,Nv-(Np-1):]
M1Sme = MSme[:,:Nv-(Np-1)]
M2Sme = MSme[:,Nv-(Np-1):]
#### The matrix to be solved in every time step
#
# 1/dt*M11 M12 -B2' 0 q1
# 1/dt*M21 M22 -B1' 0 tq2
# 1/dt*B1 B2 0 0 * p = rhs
# B1 0 0 B2 q2
#
# cf. preprint
#
MFac = 1
## Weights for the 'conti' eqns to balance the residuals
WC = 0.5
WCD = 0.5
PFac = 1#dt/WCD
PFacI = 1#WCD/dt
IterA1 = MFac * sps.hstack([sps.hstack([1.0/dt*M1Sme, M2Sme]),
-PFacI*BSme.T])
IterA2 = WCD * sps.hstack([sps.hstack([1.0/dt*B1Sme, B2Sme]),
sps.csr_matrix((Np-1, Np-1))])
IterASp = sps.vstack([IterA1,IterA2])
IterA3 = WC * sps.hstack([sps.hstack([B1Sme,sps.csr_matrix((Np-1,2*(Np-1)))]),
B2Sme])
IterA = sps.vstack([
sps.hstack([IterASp, sps.csr_matrix((Nv+Np-1,Np-1))]),
IterA3])
## Preconditioning ...
#
if TsP.SadPtPrec:
MLump = np.atleast_2d(MSme.diagonal()).T
MLump2 = MLump[-(Np-1):,]
MLumpI = 1./MLump
MLumpI1 = MLumpI[:-(Np-1),]
MLumpI2 = MLumpI[-(Np-1):,]
def PrecByB2(qqpq):
qq = MLumpI*qqpq[:Nv,]
p = qqpq[Nv:-(Np-1),]
p = spsla.spsolve(B2Sme, p)
p = MLump2*np.atleast_2d(p).T
p = spsla.spsolve(B2Sme.T,p)
p = np.atleast_2d(p).T
q2 = qqpq[-(Np-1):,]
q2 = spsla.spsolve(B2Sme,q2)
q2 = np.atleast_2d(q2).T
return np.vstack([np.vstack([qq, -p]), q2])
MGmr = spsla.LinearOperator( (Nv+2*(Np-1),Nv+2*(Np-1)), matvec=PrecByB2, dtype=np.float32 )
TsP.Ml = MGmr
def smamin_ip(qqpq1, qqpq2):
"""inner product that 'inverts' the preconditioning
for better comparability of the residuals, i.e. the tolerances
"""
def _inv_prec(qqpq):
qq = MLump*qqpq[:Nv,]
p = qqpq[Nv:-(Np-1),]
p = B2Sme.T*p
p = MLumpI2*p
p = B2Sme*p
q2 = qqpq[-(Np-1):,]
q2 = B2Sme*q2
return qq, p, q2
if TsP.SadPtPrec:
qq1,p1,q21 = _inv_prec(qqpq1)
qq2,p2,q22 = _inv_prec(qqpq2)
else:
qq1,p1,q21 = qqpq1[:Nv,], qqpq1[Nv:-(Np-1),], qqpq1[-(Np-1):,]
qq2,p2,q22 = qqpq2[:Nv,], qqpq2[Nv:-(Np-1),], qqpq2[-(Np-1):,]
return mass_fem_ip(qq1,qq2,MSme) + mass_fem_ip(p1,p2,MPc) + mass_fem_ip(q21,q22,MPc)
def smamin_prec_fem_ip(qqpq1,qqpq2):
""" M ip for the preconditioned residuals
"""
return np.dot(qqpq1[:Nv,].T.conj(),MSme*qqpq2[:Nv,]) + \
np.dot(qqpq1[Nv:-Npc,].T.conj(),MPc*qqpq2[Nv:-Npc,]) + \
np.dot(qqpq1[-Npc:,].T.conj(),MPc*qqpq2[-Npc:,])
v, p = expand_vp_dolfunc(PrP, vp=vp_init, vc=None, pc=None, pdof=None)
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
vp_old = np.copy(vp_init)
q1_old = vp_init[~B2BoolInv,]
q2_old = vp_init[B2BoolInv,]
if qqpq_init is None:
# initial value for tq2
ConV, CurFv = get_conv_curfv_rearr(v,PrP,tcur,B2BoolInv)
tq2_old = spsla.spsolve(M2Sme[-(Np-1):,:], CurFv[-(Np-1):,])
#tq2_old = MLumpI2*CurFv[-(Np-1):,]
tq2_old = np.atleast_2d(tq2_old).T
# state vector of the smaminex system : [ q1^+, tq2^c, p^c, q2^+]
qqpq_old = np.zeros((Nv+2*(Np-1),1))
qqpq_old[:Nv-(Np-1),] = q1_old
qqpq_old[Nv-(Np-1):Nv,] = tq2_old
qqpq_old[Nv:Nv+Np-1,] = PFac * vp_old[Nv:,]
qqpq_old[Nv+Np-1:,] = q2_old
else:
qqpq_old = qqpq_init
ContiRes, VelEr, PEr, TolCorL = [], [], [], []
for etap in range(1,TsP.NOutPutPts +1 ):
for i in range(Nts/TsP.NOutPutPts):
ConV, CurFv = get_conv_curfv_rearr(v,PrP,tcur,B2BoolInv)
gdot = np.zeros((Np-1,1)) # TODO: implement \dot g
Iterrhs = 1.0/dt * np.vstack([MFac * M1Sme*q1_old, WCD*B1Sme*q1_old]) \
+ np.vstack([MFac*(FvbcSme+CurFv-ConV),WCD*gdot])
Iterrhs = np.vstack([Iterrhs,FpbcSmeC])
#Norm of rhs of index-1 formulation
if TsP.TolCorB:
NormRhsInd1 = np.sqrt(smamin_fem_ip(Iterrhs,Iterrhs,MSme,MPc,Nv,Npc))[0][0]
TolCor = 1.0 / np.max([NormRhsInd1,1])
else:
TolCor = 1.0
if TsP.linatol == 0:
q1_tq2_p_q2_new = spsla.spsolve(IterA,Iterrhs)
qqpq_old = np.atleast_2d(q1_tq2_p_q2_new).T
else:
# Values from previous calculations to initialize gmres
if TsP.UsePreTStps:
dname = 'ValSmaMinNts%dN%dtcur%e' % (Nts, N, tcur)
try:
IniV = loadmat(dname)
qqpq_old = IniV['qqpq_old']
except IOError:
pass
q1_tq2_p_q2_new = krypy.linsys.gmres(IterA, Iterrhs,
x0=qqpq_old, Ml=TsP.Ml, Mr=TsP.Mr,
inner_product=smamin_prec_fem_ip,
tol=TolCor*TsP.linatol, maxiter=TsP.MaxIter,
max_restarts=20)
qqpq_old = np.atleast_2d(q1_tq2_p_q2_new['xk'])
if TsP.SaveTStps:
from scipy.io import savemat
dname = 'ValSmaMinNts%dN%dtcur%e' % (Nts, N, tcur)
savemat(dname, { 'qqpq_old': qqpq_old })
if q1_tq2_p_q2_new['info'] != 0:
print q1_tq2_p_q2_new['relresvec'][-5:]
raise Warning('no convergence')
print 'Needed %d of max %r iterations: final relres = %e\n TolCor was %e' %(len(q1_tq2_p_q2_new['relresvec']), TsP.MaxIter, q1_tq2_p_q2_new['relresvec'][-1], TolCor )
q1_old = qqpq_old[:Nv-(Np-1),]
q2_old = qqpq_old[-Npc:,]
# Extract the 'actual' velocity and pressure
vc = np.zeros((Nv,1))
vc[~B2BoolInv,] = q1_old
vc[B2BoolInv,] = q2_old
print np.linalg.norm(vc)
pc = PFacI * qqpq_old[Nv:Nv+Np-1,]
v, p = expand_vp_dolfunc(PrP, vp=None, vc=vc, pc=pc, pdof = Pdof)
tcur += dt
# the errors and residuals
vCur, pCur = PrP.v, PrP.p
vCur.t = tcur
pCur.t = tcur - dt
ContiRes.append(comp_cont_error(v,FpbcSme,PrP.Q))
VelEr.append(errornorm(vCur,v))
PEr.append(errornorm(pCur,p))
TolCorL.append(TolCor)
if i + etap == 1 and TsP.SaveIniVal:
from scipy.io import savemat
dname = 'IniValSmaMinN%s' % N
savemat(dname, { 'qqpq_old': qqpq_old })
print '%d of %d time steps completed ' % (etap*Nts/TsP.NOutPutPts, Nts)
if TsP.ParaviewOutput:
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
TsP.Residuals.ContiRes.append(ContiRes)
TsP.Residuals.VelEr.append(VelEr)
TsP.Residuals.PEr.append(PEr)
TsP.TolCor.append(TolCorL)
return
def halfexp_euler_nseind2(Mc,MP,Ac,BTc,Bc,fvbc,fpbc,vp_init,PrP,TsP):
"""halfexplicit euler for the NSE in index 2 formulation
"""
####
#
# Basic Eqn:
#
# 1/dt*M -B.T q+ 1/dt*M*qc - K(qc) + fc
# B * pc = g
#
########
Nts, t0, tE, dt, Nv, Np = init_time_stepping(PrP,TsP)
tcur = t0
MFac = 1
CFac = 1 #/dt
PFac = -1 #-1 for symmetry (if CFac==1)
PFacI = -1
v, p = expand_vp_dolfunc(PrP, vp=vp_init, vc=None, pc=None)
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
IterAv = MFac*sps.hstack([1.0/dt*Mc,PFacI*(-1)*BTc[:,:-1]])
IterAp = CFac*sps.hstack([Bc[:-1,:],sps.csr_matrix((Np-1,Np-1))])
IterA = sps.vstack([IterAv,IterAp])
MPc = MP[:-1,:][:,:-1]
vp_old = vp_init
ContiRes, VelEr, PEr, TolCorL = [], [], [], []
# M matrix for the minres routine
# M accounts for the FEM discretization
def _MInv(vp):
v, p = vp[:Nv,], vp[Nv:,]
Mv = krypy.linsys.cg(Mc,v,tol=1e-14)['xk']
Mp = krypy.linsys.cg(MPc,p,tol=1e-14)['xk']
return np.vstack([Mv,Mp])
MInv = spsla.LinearOperator( (Nv+Np-1,Nv+Np-1), matvec=_MInv, dtype=np.float32 )
def ind2_ip(vp1,vp2):
"""
for applying the fem inner product
"""
v1, v2 = vp1[:Nv,], vp2[:Nv,]
p1, p2 = vp1[Nv:,], vp2[Nv:,]
return mass_fem_ip(v1,v2,Mc) + mass_fem_ip(p1,p2,MPc)
for etap in range(1,TsP.NOutPutPts + 1 ):
for i in range(Nts/TsP.NOutPutPts):
ConV = dtn.get_convvec(v, PrP.V)
CurFv = dtn.get_curfv(PrP.V, PrP.fv, PrP.invinds, tcur)
Iterrhs = np.vstack([MFac*1.0/dt*Mc*vp_old[:Nv,],np.zeros((Np-1,1))]) \
+ np.vstack([MFac*(fvbc+CurFv-ConV[PrP.invinds,]),
CFac*fpbc[:-1,]])
if TsP.linatol == 0:
vp_new = spsla.spsolve(IterA,Iterrhs)#,vp_old,tol=TsP.linatol)
vp_old = np.atleast_2d(vp_new).T
else:
if TsP.TolCorB:
NormRhsInd2 = np.sqrt(ind2_ip(Iterrhs,Iterrhs))[0][0]
TolCor = 1.0 / np.max([NormRhsInd2,1])
else:
TolCor = 1.0
ret = krypy.linsys.minres(IterA, Iterrhs,
x0=vp_old, tol=TolCor*TsP.linatol,
M=MInv)
vp_old = ret['xk']
print 'Needed %d iterations -- final relres = %e' %(len(ret['relresvec']), ret['relresvec'][-1] )
print 'TolCor was %e' % TolCor
vc = vp_old[:Nv,]
pc = PFacI*vp_old[Nv:,]
v, p = expand_vp_dolfunc(PrP, vp=None, vc=vc, pc=pc)
tcur += dt
# the errors
vCur, pCur = PrP.v, PrP.p
vCur.t = tcur
pCur.t = tcur - dt
ContiRes.append(comp_cont_error(v,fpbc,PrP.Q))
VelEr.append(errornorm(vCur,v))
PEr.append(errornorm(pCur,p))
TolCorL.append(TolCor)
print '%d of %d time steps completed ' % (etap*Nts/TsP.NOutPutPts, Nts)
if TsP.ParaviewOutput:
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
TsP.Residuals.ContiRes.append(ContiRes)
TsP.Residuals.VelEr.append(VelEr)
TsP.Residuals.PEr.append(PEr)
TsP.TolCor.append(TolCorL)
return
def qr_impeuler(Mc,Ac,BTc,Bc,fvbc,fpbc,vp_init,PrP,TsP=None):
""" with BTc[:-1,:] = Q*[R ; 0]
we define ~M = Q*M*Q.T , ~A = Q*A*Q.T , ~V = Q*v
and condense the system accordingly """
Nts, t0, tE, dt, Nv, Np = init_time_stepping(PrP,TsP)
tcur = t0
v, p = expand_vp_dolfunc(PrP, vp=vp_init, vc=None, pc=None)
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
Qm, Rm = qr(BTc.todense()[:,:-1],mode='full')
Qm_1 = Qm[:Np-1,] #first Np-1 rows of Q
Qm_2 = Qm[Np-1:,] #last Nv-(Np-1) rows of Q
TM_11 = np.dot(Qm_1,np.dot(Mc.todense(),Qm_1.T))
TA_11 = np.dot(Qm_1,np.dot(Ac.todense(),Qm_1.T))
TM_21 = np.dot(Qm_2,np.dot(Mc.todense(),Qm_1.T))
TA_21 = np.dot(Qm_2,np.dot(Ac.todense(),Qm_1.T))
TM_12 = np.dot(Qm_1,np.dot(Mc.todense(),Qm_2.T))
TA_12 = np.dot(Qm_1,np.dot(Ac.todense(),Qm_2.T))
TM_22 = np.dot(Qm_2,np.dot(Mc.todense(),Qm_2.T))
TA_22 = np.dot(Qm_2,np.dot(Ac.todense(),Qm_2.T))
Tv1 = np.linalg.solve(Rm[:Np-1,].T, fpbc[:-1,])
Tv2_old = np.dot(Qm_2, vp_init[:Nv,])
Tfv2 = np.dot(Qm_2, fvbc) + np.dot(TA_21, Tv1)
IterA2 = TM_22+dt*TA_22
for etap in range(1,11):
for i in range(Nts/10):
tcur = tcur + dt
Iter2rhs = np.dot(TM_22, Tv2_old) + dt*Tfv2
Tv2_new = np.linalg.solve(IterA2, Iter2rhs)
Tv2_old = Tv2_new
# Retransformation v = Q.T*Tv
vc_new = np.dot(Qm.T, np.vstack([Tv1, Tv2_new]))
RhsTv2dot = Tfv2 - np.dot(TA_22, Tv2_new)
Tv2dot = np.linalg.solve(TM_22, RhsTv2dot)
RhsP = np.dot(Qm_1,fvbc) - np.dot(TA_11,Tv1) \
- np.dot(TA_12,Tv2_new) - np.dot(TM_12,Tv2dot)
pc_new = - np.linalg.solve(Rm[:Np-1,],RhsP)
v, p = expand_vp_dolfunc(PrP, vp=None, vc=vc_new, pc=pc_new)
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
print '%d of %d time steps completed ' % (etap*Nts/10,Nts)
return
def plain_impeuler(Mc,Ac,BTc,Bc,fvbc,fpbc,vp_init,PrP,TsP):
Nts, t0, tE, dt, Nv, Np = init_time_stepping(PrP,TsP)
v, p = expand_vp_dolfunc(PrP, vp=vp_init, vc=None, pc=None)
tcur = t0
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
IterAv = sps.hstack([Mc+dt*Ac,-dt*BTc])
IterAp = sps.hstack([-dt*Bc,sps.csr_matrix((Np,Np))])
IterA = sps.vstack([IterAv,IterAp]).todense()[:-1,:-1]
vp_old = vp_init
for etap in range(1,11):
for i in range(Nts/10):
tcur = tcur + dt
Iterrhs = np.vstack([Mc*vp_old[:Nv,],np.zeros((Np-1,1))]) \
+ dt*np.vstack([fvbc,fpbc[:-1,]])
vp_new = np.linalg.solve(IterA,Iterrhs)
vp_old = vp_new
print '%d of %d time steps completed ' % (etap*Nts/10,Nts)
v, p = expand_vp_dolfunc(PrP, vp=vp_new, vc=None, pc=None)
TsP.UpFiles.u_file << v, tcur
TsP.UpFiles.p_file << p, tcur
return
def comp_cont_error(v,fpbc,Q):
"""Compute the L2 norm of the residual of the continuity equation
Bv = g
"""
q = TrialFunction(Q)
divv = assemble(q*div(v)*dx)
conRhs = Function(Q)
conRhs.vector().set_local(fpbc)
#raise Warning('debugggg')
ContEr = norm(conRhs.vector()-divv)
return ContEr
def expand_vp_dolfunc(PrP, vp=None, vc=None, pc=None, pdof=None):
"""expand v and p to the dolfin function representation
pdof = pressure dof that was set zero
"""
v = Function(PrP.V)
p = Function(PrP.Q)
if vp is not None:
if vp.ndim == 1:
vc = vp[:len(PrP.invinds)].reshape(len(PrP.invinds),1)
pc = vp[len(PrP.invinds):].reshape(PrP.Q.dim()-1,1)
else:
vc = vp[:len(PrP.invinds),:]
pc = vp[len(PrP.invinds):,:]
ve = np.zeros((PrP.V.dim(),1))
# fill in the boundary values
for bc in PrP.velbcs:
bcdict = bc.get_boundary_values()
ve[bcdict.keys(),0] = bcdict.values()
ve[PrP.invinds] = vc
if pdof is None:
pe = np.vstack([pc,[0]])
elif pdof == 0:
pe = np.vstack([[0],pc])
elif pdof == -1:
pe = pc
else:
pe = np.vstack([pc[:pdof],np.vstack([[0.02],pc[pdof:]])])
v.vector().set_local(ve)
p.vector().set_local(pe)
return v, p
def get_conv_curfv_rearr(v,PrP,tcur,B2BoolInv):
ConV = dtn.get_convvec(v, PrP.V)
ConV = ConV[PrP.invinds,]
ConV = np.vstack([ConV[~B2BoolInv],ConV[B2BoolInv]])
CurFv = dtn.get_curfv(PrP.V, PrP.fv, PrP.invinds, tcur)
if len(CurFv) != len(PrP.invinds):
raise Warning('Need fv at innernodes here')
CurFv = np.vstack([CurFv[~B2BoolInv],CurFv[B2BoolInv]])
return ConV, CurFv
def init_time_stepping(PrP,TsP):
"""what every method starts with """
Nts, t0, tE = TsP.Nts, TsP.t0, TsP.tE
dt = (tE-t0)/Nts
Nv = len(PrP.invinds)
Np = PrP.Q.dim()
if Nts % TsP.NOutPutPts != 0:
TsP.NOutPutPts = 1
return Nts, t0, tE, dt, Nv, Np