def comp_evcs_evls(compall=False, nevs=None, linsysstr=None, bigamat=None,
bigmmatshift=None, bigmmat=None, retinstevcs=False):
whichevecs = '_leftevs' if leftevecs else '_rightevs'
bcstr = '_palpha{0}'.format(palpha) if bccontrol else ''
levstr = linsysstr + bcstr + whichevecs
try:
if debug:
raise IOError()
evls = dou.load_npa(levstr)
print 'loaded the eigenvalues of the linearized system: \n' + levstr
if retinstevcs:
evcs = dou.load_npa(levstr+'_instableevcs')
return evls, evcs
except IOError:
print 'computing the eigenvalues of the linearized system: \n' + levstr
if compall:
A = bigamat.todense()
M = bigmmat.todense() if not shiftall else bigmmatshift.todense()
evls = spla.eigvals(A, M, overwrite_a=True, check_finite=False)
else:
if leftevecs:
bigamat = bigamat.T
# TODO: we assume that M is symmetric
msplu = spsla.splu(bigmmatshift)
def minvavec(vvec):
return msplu.solve(bigamat*vvec)
miaop = spsla.LinearOperator(bigamat.shape, matvec=minvavec,
dtype=bigamat.dtype)
shiftamm = bigamat - sigma*bigmmatshift
sammsplu = spsla.splu(shiftamm)
def samminvmvec(vvec):
return sammsplu.solve(bigmmatshift*vvec)
samminvop = spsla.LinearOperator(bigamat.shape, matvec=samminvmvec,
dtype=bigamat.dtype)
evls, evcs = spsla.eigs(miaop, k=nevs, sigma=sigma,
OPinv=samminvop, maxiter=100,
# sigma=None, which='SM',
return_eigenvectors=retinstevcs)
dou.save_npa(evls, levstr)
if retinstevcs:
instevlsindx = np.real(evls) > 0
instevcs = evcs[:, instevlsindx]
dou.save_npa(instevcs, levstr+'_instableevcs')
else:
instevcs = None
return evls, instevcs
def compcong():
try:
zwc = dou.load_npa(fdstr + '__zwc')
print 'at least __zwc is there'
except IOError:
zwc = get_gramians(mmat=mmat, amat=f_mat,
jmat=stokesmatsc['J'],
bmat=b_mat, wmat=c_mat_reg.T,
nwtn_adi_dict=nap,
z0=zinic)['zfac']
dou.save_npa(zwc, fdstr + '__zwc')
def compcong():
try:
zwc = dou.load_npa(fdstr + '__zwc')
print('yeyeyeah, __zwc is there')
except IOError:
# XXX: why here bmat*Rmhalf and in zwo not?
zwc = get_ricadifacs(mmat=mmat, amat=fmat, jmat=jmat,
bmat=bmat, wmat=cmat.T,
z0=zinic, **adidict)['zfac']
dou.save_npa(zwc, fdstr + '__zwc')
return
def compobsg():
try:
zwo = dou.load_npa(fdstr + '__zwo')
print 'at least __zwo is there'
except IOError:
zwo = get_gramians(mmat=mmat.T, amat=f_mat.T,
jmat=stokesmatsc['J'],
bmat=c_mat_reg.T, wmat=b_mat,
nwtn_adi_dict=nap,
z0=zinio)['zfac']
dou.save_npa(zwo, fdstr + '__zwo')
def compobsg():
if zwconly:
print('we only compute zwc')
return
try:
zwo = dou.load_npa(fdstr + '__zwo')
print('yeyeyeah, __zwo is there')
except IOError:
zwo = get_ricadifacs(mmat=mmat.T, amat=fmat.T, jmat=jmat,
bmat=cmat.T, wmat=bmat,
z0=zinio, **adidict)['zfac']
dou.save_npa(zwo, fdstr + '__zwo')
return
def rothe_time_int(problem='cylinderwake', nu=None, Re=None,
Nts=256, t0=0.0, tE=0.2, Nll=[2],
viniv=None, piniv=None, Nini=None,
scheme=None, dtstrdct={}, method=2):
trange = np.linspace(t0, tE, Nts+1)
t = trange[0]
dtstrdct.update(dict(t=0, N=Nll[0]))
cdatstr = get_dtstr(**dtstrdct)
dou.save_npa(viniv, cdatstr + '__vel')
curvdict = {t: cdatstr + '__vel'}
dou.save_npa(piniv, cdatstr + '__p')
logger.info('v/p saved to ' + cdatstr + '__v/__p')
curpdict = {t: cdatstr + '__p'}
smaminex = True if method == 1 else False
vcurvec = viniv
logger.debug(' t={0}, |v|={1}'.format(t, npla.norm(vcurvec)))
curmeshdict = get_curmeshdict(problem=problem, N=Nll[0], nu=nu, Re=Re,
scheme=scheme, smaminex=smaminex)
curmeshdict.update(coefalu=None)
Vc = curmeshdict['V']
for tk, t in enumerate(trange[1:]):
cts = t - trange[tk]
if not Nll[tk+1] == Nll[tk]:
curmeshdict = get_curmeshdict(problem=problem, N=Nll[tk+1], nu=nu,
Re=Re, scheme=scheme,
smaminex=smaminex)
curmeshdict.update(coefalu=None)
logger.info('changed the mesh from N={0} to N={1} at t={2}'.
format(Nll[tk], Nll[tk+1], t))
# change in the mesh
Nvc = curmeshdict['A'].shape[0]
logger.debug("t={0}, dim V={1}".format(t, curmeshdict['V'].dim()))
if smaminex:
vpcur, coefalu = \
roth_upd_smmx(vvec=vcurvec, cts=cts,
Vc=Vc, diribcsc=curmeshdict['diribcs'],
nmd=curmeshdict, returnalu=True)
else: # index 2
vpcur, coefalu = \
roth_upd_ind2(vvec=vcurvec, cts=cts,
Vc=Vc, diribcsc=curmeshdict['diribcs'],
nmd=curmeshdict, returnalu=True)
dtstrdct.update(dict(t=t, N=Nll[tk+1]))
cdatstr = get_dtstr(**dtstrdct)
# add the boundary values to the velocity
vcurvec = dts.append_bcs_vec(vpcur[:Nvc], **curmeshdict)
logger.debug(' t={0}, |v|={1}'.format(t, npla.norm(vcurvec)))
dou.save_npa(vcurvec, cdatstr+'__vel')
curvdict.update({t: cdatstr+'__vel'})
dou.save_npa(vpcur[Nvc:, :], cdatstr+'__p')
curpdict.update({t: cdatstr+'__p'})
curmeshdict.update(dict(coefalu=coefalu))
Vc = curmeshdict['V']
return curvdict, curpdict
def optcon_nse(problemname='drivencavity',
N=10,
Nts=10,
nu=1e-2,
clearprvveldata=False,
ini_vel_stokes=False,
stst_control=False,
closed_loop=True,
outernwtnstps=1,
t0=None,
tE=None,
use_ric_ini_nu=None,
alphau=1e-9,
gamma=1e-3,
spec_tip_dict=None,
nwtn_adi_dict=None,
linearized_nse=False,
stokes_flow=False,
ystar=None):
tip = time_int_params(Nts, t0=t0, tE=tE)
if spec_tip_dict is not None:
tip.update(spec_tip_dict)
if nwtn_adi_dict is not None:
tip['nwtn_adi_dict'] = nwtn_adi_dict
problemdict = dict(drivencavity=dnsps.drivcav_fems,
cylinderwake=dnsps.cyl_fems)
problemfem = problemdict[problemname]
femp = problemfem(N)
# output
ddir = 'data/'
try:
os.chdir(ddir)
except OSError:
raise Warning('need "' + ddir + '" subdir for storing the data')
os.chdir('..')
if linearized_nse and not outernwtnstps == 1:
raise Warning('Linearized problem can have only one Newton step')
if closed_loop:
if stst_control:
data_prfx = ddir + 'stst_' + problemname + '__'
else:
data_prfx = ddir + 'tdst_' + problemname + '__'
else:
data_prfx = ddir + problemname + '__'
if stokes_flow:
data_prfx = data_prfx + 'stokes__'
# specify in what spatial direction Bu changes. The remaining is constant
if problemname == 'drivencavity':
uspacedep = 0
elif problemname == 'cylinderwake':
uspacedep = 1
stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'], femp['fv'], femp['fp'], t=0)
# remove the freedom in the pressure
stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]
# reduce the matrices by resolving the BCs
(stokesmatsc, rhsd_stbc, invinds, bcinds,
bcvals) = dts.condense_sysmatsbybcs(stokesmats, femp['diribcs'])
print 'Dimension of the div matrix: ', stokesmatsc['J'].shape
# pressure freedom and dirichlet reduced rhs
rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])
# add the info on boundary and inner nodes
bcdata = {'bcinds': bcinds, 'bcvals': bcvals, 'invinds': invinds}
femp.update(bcdata)
# casting some parameters
NV = len(femp['invinds'])
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
# soldict.update(rhsd_vfrc) # adding fvc, fpr
soldict.update(fv=rhsd_stbc['fv'] + rhsd_vfrc['fvc'],
fp=rhsd_stbc['fp'] + rhsd_vfrc['fpr'],
N=N,
nu=nu,
trange=tip['tmesh'],
get_datastring=get_datastr,
data_prfx=data_prfx,
clearprvdata=clearprvveldata,
paraviewoutput=tip['ParaviewOutput'],
vfileprfx=tip['proutdir'] + 'vel_',
pfileprfx=tip['proutdir'] + 'p_')
#
# Prepare for control
#
contp = ContParams(femp['odcoo'], ystar=ystar, alphau=alphau, gamma=gamma)
# casting some parameters
NY, NU = contp.NY, contp.NU
contsetupstr = problemname + '__NV{0}NU{1}NY{2}'.format(NV, NU, NY)
# get the control and observation operators
try:
b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
print 'loaded `b_mat`'
except IOError:
print 'computing `b_mat`...'
b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'],
V=femp['V'],
NU=NU,
xcomp=uspacedep)
dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
try:
mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
print 'loaded `c_mat`'
except IOError:
print 'computing `c_mat`...'
mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
V=femp['V'],
NY=NY)
dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')
# restrict the operators to the inner nodes
mc_mat = mc_mat[:, invinds][:, :]
b_mat = b_mat[invinds, :][:, :]
# for further use:
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
if contp.ystarx is None:
c_mat = c_mat[NY:, :][:, :] # TODO: Do this right
mc_mat = mc_mat[NY:, :][:, :] # TODO: Do this right
y_masmat = y_masmat[:NY, :][:, :NY] # TODO: Do this right
mct_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=mc_mat.T,
transposedprj=True)
# set the weighing matrices
contp.R = contp.alphau * u_masmat
#
# solve the differential-alg. Riccati eqn for the feedback gain X
# via computing factors Z, such that X = -Z*Z.T
#
# at the same time we solve for the affine-linear correction w
#
# tilde B = BR^{-1/2}
tb_mat = lau.apply_invsqrt_fromright(contp.R, b_mat, output='sparse')
# tb_dense = np.array(tb_mat.todense())
trct_mat = lau.apply_invsqrt_fromright(y_masmat,
mct_mat_reg,
output='dense')
if closed_loop:
cntpstr = 'NV{3}NY{0}NU{1}alphau{2}gamma{4}'.\
format(contp.NU, contp.NY, contp.alphau, NV, contp.gamma)
else:
cntpstr = ''
# we gonna use this quite often
M, A = stokesmatsc['M'], stokesmatsc['A']
datastrdict = dict(time=None, meshp=N, nu=nu, Nts=Nts, data_prfx=data_prfx)
# compute the uncontrolled steady state (Navier-)Stokes solution
# as initial value
if ini_vel_stokes:
# compute the uncontrolled steady state Stokes solution
ini_vel, newtonnorms = snu.solve_steadystate_nse(vel_nwtn_stps=0,
vel_pcrd_stps=0,
**soldict)
soldict.update(dict(iniv=ini_vel))
else:
ini_vel, newtonnorms = snu.solve_steadystate_nse(**soldict)
soldict.update(dict(iniv=ini_vel))
if closed_loop:
if stst_control:
if stokes_flow:
convc_mat = sps.csr_matrix((NV, NV))
rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
lin_point = None
else:
lin_point, newtonnorms = snu.solve_steadystate_nse(**soldict)
(convc_mat, rhs_con,
rhsv_conbc) = snu.get_v_conv_conts(prev_v=lin_point,
invinds=invinds,
V=femp['V'],
diribcs=femp['diribcs'])
# infinite control horizon, steady target state
cdatstr = get_datastr(time=None,
meshp=N,
nu=nu,
Nts=None,
data_prfx=data_prfx)
try:
Z = dou.load_npa(cdatstr + cntpstr + '__Z')
print 'loaded ' + cdatstr + cntpstr + '__Z'
except IOError:
if use_ric_ini_nu is not None:
cdatstr = get_datastr(nwtn=None,
time=None,
meshp=N,
nu=use_ric_ini_nu,
Nts=None,
data_prfx=data_prfx)
try:
zini = dou.load_npa(ddir + cdatstr + cntpstr + '__Z')
print 'Initialize Newton ADI by Z from ' + cdatstr
except IOError:
raise Warning('No data for initialization of '
' Newton ADI -- need ' + cdatstr + '__Z')
cdatstr = get_datastr(meshp=N, nu=nu, data_prfx=data_prfx)
else:
zini = None
parnadi = pru.proj_alg_ric_newtonadi
Z = parnadi(mmat=M,
amat=-A - convc_mat,
jmat=stokesmatsc['J'],
bmat=tb_mat,
wmat=trct_mat,
z0=zini,
nwtn_adi_dict=tip['nwtn_adi_dict'])['zfac']
dou.save_npa(Z, fstring=cdatstr + cntpstr + '__Z')
print 'saved ' + cdatstr + cntpstr + '__Z'
if tip['compress_z']:
Zc = pru.compress_Zsvd(Z,
thresh=tip['comprz_thresh'],
k=tip['comprz_maxc'])
Z = Zc
fvnstst = rhs_con + rhsv_conbc + rhsd_stbc['fv'] + rhsd_vfrc['fvc']
# X = -ZZ.T
mtxtb_stst = -pru.get_mTzzTtb(M.T, Z, tb_mat)
mtxfv_stst = -pru.get_mTzzTtb(M.T, Z, fvnstst)
fl = mc_mat.T * contp.ystarvec(0)
wft = lau.solve_sadpnt_smw(amat=A.T + convc_mat.T,
jmat=stokesmatsc['J'],
rhsv=fl + mtxfv_stst,
umat=mtxtb_stst,
vmat=tb_mat.T)[:NV]
auxstrg = cdatstr + cntpstr
dou.save_npa(wft, fstring=cdatstr + cntpstr + '__w')
dou.save_npa(mtxtb_stst, fstring=cdatstr + cntpstr + '__mtxtb')
feedbackthroughdict = {
None: dict(w=auxstrg + '__w', mtxtb=auxstrg + '__mtxtb')
}
cns = 0
soldict.update(data_prfx=data_prfx + '_cns{0}'.format(cns))
if linearized_nse:
soldict.update(vel_pcrd_stps=0,
vel_nwtn_stps=1,
lin_vel_point={None: lin_point})
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True,
static_feedback=True,
tb_mat=tb_mat,
stokes_flow=stokes_flow,
clearprvveldata=True,
feedbackthroughdict=feedbackthroughdict, **soldict)
else: # time dep closed loop
cns_data_prfx = 'data/cnsvars'
invd = init_nwtnstps_value_dict
curnwtnsdict = invd(tmesh=tip['tmesh'], data_prfx=cns_data_prfx)
# initialization: compute the forward solution
if stokes_flow:
dictofvels = None
else:
dictofvels = snu.solve_nse(return_dictofvelstrs=True,
stokes_flow=stokes_flow,
**soldict)
# dbs.plot_vel_norms(tip['tmesh'], dictofvels)
# function for the time depending parts
# -- to be passed to the solver
def get_tdpart(time=None,
dictofvalues=None,
feedback=False,
V=None,
invinds=None,
diribcs=None,
**kw):
if stokes_flow:
convc_mat = sps.csr_matrix((NV, NV))
rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
curvel = dou.load_npa(dictofvalues[time])
convc_mat, rhs_con, rhsv_conbc = \
snu.get_v_conv_conts(prev_v=curvel, invinds=invinds,
V=V, diribcs=diribcs)
return convc_mat, rhsv_conbc + rhs_con
gttdprtargs = dict(dictofvalues=dictofvels,
V=femp['V'],
diribcs=femp['diribcs'],
invinds=invinds)
# old version rhs
# ftilde = rhs_con + rhsv_conbc + rhsd_stbc['fv']
for cns in range(outernwtnstps):
datastrdict.update(data_prfx=data_prfx + cntpstr +
'_cns{0}'.format(cns))
soldict.update(data_prfx=data_prfx + cntpstr +
'_cns{0}'.format(cns))
sfd = sdr.solve_flow_daeric
feedbackthroughdict = \
sfd(mmat=M, amat=A, jmat=stokesmatsc['J'],
bmat=b_mat,
# cmat=ct_mat_reg.T,
mcmat=mct_mat_reg.T,
v_is_my=True, rmat=contp.alphau*u_masmat,
vmat=y_masmat, rhsv=rhsd_stbc['fv'],
gamma=contp.gamma,
rhsp=None,
tmesh=tip['tmesh'], ystarvec=contp.ystarvec,
nwtn_adi_dict=tip['nwtn_adi_dict'],
comprz_thresh=tip['comprz_thresh'],
comprz_maxc=tip['comprz_maxc'], save_full_z=False,
get_tdpart=get_tdpart, gttdprtargs=gttdprtargs,
curnwtnsdict=curnwtnsdict,
get_datastr=get_datastr, gtdtstrargs=datastrdict)
# for t in tip['tmesh']: # feedbackthroughdict.keys():
# curw = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
# print cns, t, np.linalg.norm(curw)
cdatstr = get_datastr(time='all',
meshp=N,
nu=nu,
Nts=None,
data_prfx=data_prfx)
if linearized_nse:
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True, tb_mat=tb_mat,
lin_vel_point=dictofvels,
feedbackthroughdict=feedbackthroughdict,
vel_nwtn_stps=1,
vel_pcrd_stps=0,
**soldict)
else:
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True, tb_mat=tb_mat,
stokes_flow=stokes_flow,
feedbackthroughdict=feedbackthroughdict,
vel_pcrd_stps=1,
vel_nwtn_stps=2,
**soldict)
# for t in dictofvels.keys():
# curw = dou.load_npa(dictofvels[t])
# print cns, t, np.linalg.norm(curw)
gttdprtargs.update(dictofvalues=dictofvels)
else:
# no control
feedbackthroughdict = None
tb_mat = None
cdatstr = get_datastr(meshp=N,
nu=nu,
time='all',
Nts=Nts,
data_prfx=data_prfx)
soldict.update(clearprvdata=True)
dictofvels = snu.solve_nse(feedbackthroughdict=feedbackthroughdict,
tb_mat=tb_mat,
closed_loop=closed_loop,
stokes_flow=stokes_flow,
return_dictofvelstrs=True,
static_feedback=stst_control,
**soldict)
(yscomplist, ystarlist) = dou.extract_output(dictofpaths=dictofvels,
tmesh=tip['tmesh'],
c_mat=c_mat,
ystarvec=contp.ystarvec)
save_output_json(yscomplist,
tip['tmesh'].tolist(),
ystar=ystarlist,
fstring=cdatstr + cntpstr + '__sigout')
costfunval = eval_costfunc(W=y_masmat,
V=contp.gamma * y_masmat,
R=None,
tbmat=tb_mat,
cmat=c_mat,
ystar=contp.ystarvec,
tmesh=tip['tmesh'],
veldict=dictofvels,
fbftdict=feedbackthroughdict)
print 'Value of cost functional: ', costfunval
costfunval = eval_costfunc(W=y_masmat,
V=contp.gamma * y_masmat,
R=None,
tbmat=tb_mat,
cmat=c_mat,
ystar=contp.ystarvec,
penau=False,
tmesh=tip['tmesh'],
veldict=dictofvels,
fbftdict=feedbackthroughdict)
print 'Value of cost functional not considering `u`: ', costfunval
print 'dim of v :', femp['V'].dim()
charlene = .15 if problemname == 'cylinderwake' else 1.0
print 'Re = charL / nu = {0}'.format(charlene / nu)
def solve_steadystate_nse(A=None, J=None, JT=None, M=None,
fv=None, fp=None,
V=None, Q=None, invinds=None, diribcs=None,
dbcvals=None, dbcinds=None,
diricontbcinds=None, diricontbcvals=None,
diricontfuncs=None,
return_vp=False, ppin=-1,
return_nwtnupd_norms=False,
N=None, nu=None,
vel_pcrd_stps=10, vel_pcrd_tol=1e-4,
vel_nwtn_stps=20, vel_nwtn_tol=5e-15,
clearprvdata=False,
useolddata=False,
vel_start_nwtn=None,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
save_data=True,
save_intermediate_steps=False,
vfileprfx='', pfileprfx='',
verbose=True,
**kw):
"""
Solution of the steady state nonlinear NSE Problem
using Newton's scheme. If no starting value is provided, the iteration
is started with the steady state Stokes solution.
Parameters
----------
A : (N,N) sparse matrix
stiffness matrix aka discrete Laplacian, note the sign!
M : (N,N) sparse matrix
mass matrix
J : (M,N) sparse matrix
discrete divergence operator
JT : (N,M) sparse matrix, optional
discrete gradient operator, set to J.T if not provided
fv, fp : (N,1), (M,1) ndarrays
right hand sides restricted via removing the boundary nodes in the
momentum and the pressure freedom in the continuity equation
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
dbcinds: list, optional
indices of the Dirichlet boundary conditions
dbcvals: list, optional
values of the Dirichlet boundary conditions (as listed in `dbcinds`)
diricontbcinds: list, optional
list of dirichlet indices that are to be controlled
diricontbcvals: list, optional
list of the vals corresponding to `diricontbcinds`
diricontfuncs: list, optional
list like `[ufunc]` where `ufunc: (t, v) -> u` where `u` is used to
scale the corresponding `diricontbcvals`
return_vp : boolean, optional
whether to return also the pressure, defaults to `False`
vel_pcrd_stps : int, optional
Number of Picard iterations when computing a starting value for the
Newton scheme, cf. Elman, Silvester, Wathen: *FEM and fast iterative
solvers*, 2005, defaults to `100`
vel_pcrd_tol : real, optional
tolerance for the size of the Picard update, defaults to `1e-4`
vel_nwtn_stps : int, optional
Number of Newton iterations, defaults to `20`
vel_nwtn_tol : real, optional
tolerance for the size of the Newton update, defaults to `5e-15`
Returns:
---
vel_k : (N, 1) ndarray
the velocity vector, if not `return_vp`, else
(v, p) : tuple
of the velocity and the pressure vector
norm_nwtnupd_list : list, on demand
list of the newton upd errors
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if get_datastring is None:
get_datastring = get_datastr_snu
if JT is None:
JT = J.T
NV = J.shape[1]
#
# Compute or load the uncontrolled steady state Navier-Stokes solution
#
norm_nwtnupd_list = []
# a dict to be passed to the get_datastring function
datastrdict = dict(time=None, meshp=N, nu=nu,
Nts=None, data_prfx=data_prfx)
if clearprvdata:
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '*__vel*'):
os.remove(fname)
if useolddata:
try:
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = dou.load_npa(cdatstr + '__norm_nwtnupd')
vel_k = dou.load_npa(cdatstr + '__vel')
norm_nwtnupd_list.append(norm_nwtnupd)
if verbose:
print('found vel files')
print('norm of last Nwtn update: {0}'.format(norm_nwtnupd))
print('... loaded from ' + cdatstr)
if np.atleast_1d(norm_nwtnupd)[0] is None:
norm_nwtnupd = None
pass # nothing useful found
elif norm_nwtnupd < vel_nwtn_tol:
if not return_vp:
return vel_k, norm_nwtnupd_list
else:
pfv = get_pfromv(v=vel_k[:NV, :], V=V,
M=M, A=A, J=J, fv=fv,
dbcinds=dbcinds, dbcvals=dbcvals,
invinds=invinds, diribcs=diribcs)
return (np.vstack([vel_k, pfv]), norm_nwtnupd_list)
except IOError:
if verbose:
print('no old velocity data found')
norm_nwtnupd = None
else:
# we start from scratch
norm_nwtnupd = None
if paraviewoutput:
cdatstr = get_datastring(**datastrdict)
vfile = dolfin.File(vfileprfx+'__steadystates.pvd')
pfile = dolfin.File(pfileprfx+'__steadystates.pvd')
prvoutdict = dict(V=V, Q=Q, vfile=vfile, pfile=pfile,
invinds=invinds, diribcs=diribcs, ppin=ppin,
dbcinds=dbcinds, dbcvals=dbcvals,
vp=None, t=None, writeoutput=True)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' later
NV = A.shape[0]
if vel_start_nwtn is None:
loccntbcinds, cntrlldbcvals, glbcntbcinds = [], [], []
if diricontbcinds is None or diricontbcinds == []:
cmmat, camat, cj, cjt, cfv, cfp = M, A, J, JT, fv, fp
cnv = NV
dbcntinvinds = invinds
else:
def _localizecdbinds(cdbinds):
""" find the local indices of the control dirichlet boundaries
the given control dirichlet boundaries were indexed w.r.t. the
full space `V`. Here, in the matrices, we have already
resolved the constant Dirichlet bcs
"""
allinds = np.arange(V.dim())
redcdallinds = allinds[invinds]
# now: find the positions of the control dbcs in the reduced
# index vector
lclinds = np.searchsorted(redcdallinds, cdbinds, side='left')
return lclinds
for k, cdbidbv in enumerate(diricontbcinds):
ccntrlfunc = diricontfuncs[k]
# no time at steady state, no starting value
cntrlval = ccntrlfunc(None, None)
localbcinds = (_localizecdbinds(cdbidbv)).tolist()
loccntbcinds.extend(localbcinds) # adding the boundary inds
glbcntbcinds.extend(cdbidbv)
ccntrlldbcvals = [cntrlval*bcvl for bcvl in diricontbcvals[k]]
# adding the scaled boundary values
cntrlldbcvals.extend(ccntrlldbcvals)
dbcntinvinds = np.setdiff1d(invinds, glbcntbcinds).astype(np.int32)
matdict = dict(M=M, A=A, J=J, JT=JT, MP=None)
cmmat, camat, cjt, cj, _, cfv, cfp, _ = dts.\
condense_sysmatsbybcs(matdict, dbcinds=loccntbcinds,
dbcvals=cntrlldbcvals, mergerhs=True,
rhsdict=dict(fv=fv, fp=fp),
ret_unrolled=True)
cnv = cmmat.shape[0]
vp_stokes = lau.solve_sadpnt_smw(amat=camat, jmat=cj, jmatT=cjt,
rhsv=cfv, rhsp=cfp)
vp_stokes[cnv:] = -vp_stokes[cnv:]
# pressure was flipped for symmetry
# save the data
cdatstr = get_datastring(**datastrdict)
if save_data:
dou.save_npa(vp_stokes[:cnv, ], fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_stokes,
dbcinds=[dbcinds, glbcntbcinds],
dbcvals=[dbcvals, cntrlldbcvals],
invinds=dbcntinvinds))
dou.output_paraview(**prvoutdict)
# Stokes solution as starting value
vp_k = vp_stokes
vel_k = vp_stokes[:cnv, ]
else:
vel_k = vel_start_nwtn
matdict = dict(M=M, A=A, J=J, JT=JT, MP=None)
rhsdict = dict(fv=fv, fp=fp)
cndnsmtsdct = dict(dbcinds=loccntbcinds, mergerhs=True,
ret_unrolled=True)
# Picard iterations for a good starting value for Newton
for k in range(vel_pcrd_stps):
cntrlldbcvals = _unroll_cntrl_dbcs(diricontbcvals, diricontfuncs,
time=None, vel=vel_k)
(convc_mat,
rhs_con, rhsv_conbc) = \
get_v_conv_conts(prev_v=vel_k, V=V, diribcs=diribcs,
invinds=dbcntinvinds,
dbcinds=[dbcinds, glbcntbcinds],
dbcvals=[dbcvals, cntrlldbcvals], Picard=True)
_, _, _, _, _, cfv, cfp, _ = dts.\
condense_sysmatsbybcs(matdict, dbcvals=cntrlldbcvals,
rhsdict=rhsdict, **cndnsmtsdct)
vp_k = lau.solve_sadpnt_smw(amat=camat+convc_mat, jmat=cj, jmatT=cjt,
rhsv=cfv+rhsv_conbc, rhsp=cfp)
# vp_k = lau.solve_sadpnt_smw(amat=A+convc_mat, jmat=J, jmatT=JT,
# rhsv=fv+rhsv_conbc,
# rhsp=fp)
normpicupd = np.sqrt(m_innerproduct(cmmat, vel_k-vp_k[:cnv, ]))[0]
if verbose:
print('Picard iteration: {0} -- norm of update: {1}'.
format(k+1, normpicupd))
vel_k = vp_k[:cnv, ]
vp_k[cnv:] = -vp_k[cnv:]
# pressure was flipped for symmetry
if normpicupd < vel_pcrd_tol:
break
# Newton iteration
for vel_newtk, k in enumerate(range(vel_nwtn_stps)):
cdatstr = get_datastring(**datastrdict)
cntrlldbcvals = _unroll_cntrl_dbcs(diricontbcvals, diricontfuncs,
time=None, vel=vel_k)
_, _, _, _, _, cfv, cfp, _ = dts.\
condense_sysmatsbybcs(matdict, dbcvals=cntrlldbcvals,
rhsdict=rhsdict, **cndnsmtsdct)
(convc_mat, rhs_con, rhsv_conbc) = \
get_v_conv_conts(prev_v=vel_k, V=V, diribcs=diribcs,
invinds=dbcntinvinds,
dbcinds=[dbcinds, glbcntbcinds],
dbcvals=[dbcvals, cntrlldbcvals])
vp_k = lau.solve_sadpnt_smw(amat=camat+convc_mat, jmat=cj, jmatT=cjt,
rhsv=cfv+rhs_con+rhsv_conbc,
rhsp=cfp)
norm_nwtnupd = np.sqrt(m_innerproduct(cmmat, vel_k - vp_k[:cnv, :]))[0]
vel_k = vp_k[:cnv, ]
vp_k[cnv:] = -vp_k[cnv:]
# pressure was flipped for symmetry
if verbose:
print('Steady State NSE: Newton iteration: {0}'.format(vel_newtk) +
'-- norm of update: {0}'.format(norm_nwtnupd))
if save_data:
dou.save_npa(vel_k, fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_k, dbcvals=[dbcvals, cntrlldbcvals]))
dou.output_paraview(**prvoutdict)
if norm_nwtnupd < vel_nwtn_tol:
break
else:
if vel_nwtn_stps == 0:
print('No Newton steps = steady state probably not well converged')
else:
raise UserWarning('Steady State NSE: Newton has not converged')
if save_data:
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
prvoutdict.update(dict(vp=vp_k, dbcvals=[dbcvals, cntrlldbcvals]))
dou.output_paraview(**prvoutdict)
# savetomatlab = True
# if savetomatlab:
# export_mats_to_matlab(E=None, A=None, matfname='matexport')
vwc = _attach_cntbcvals(vel_k.flatten(), globbcinvinds=dbcntinvinds,
globbcinds=glbcntbcinds, dbcvals=cntrlldbcvals,
invinds=invinds, NV=V.dim())
if return_vp:
retthing = (vwc.reshape((NV, 1)), vp_k[cnv:, :])
else:
retthing = vwc.reshape((NV, 1))
if return_nwtnupd_norms:
return retthing, norm_nwtnupd_list
else:
return retthing
def optcon_nse(problemname='drivencavity',
N=10, Nts=10, nu=1e-2, clearprvveldata=False,
ini_vel_stokes=False, stst_control=False,
closed_loop=True,
outernwtnstps=1,
t0=None, tE=None,
use_ric_ini_nu=None,
alphau=1e-9, gamma=1e-3,
spec_tip_dict=None,
nwtn_adi_dict=None,
linearized_nse=False,
stokes_flow=False,
ystar=None):
tip = time_int_params(Nts, t0=t0, tE=tE)
if spec_tip_dict is not None:
tip.update(spec_tip_dict)
if nwtn_adi_dict is not None:
tip['nwtn_adi_dict'] = nwtn_adi_dict
problemdict = dict(drivencavity=dnsps.drivcav_fems,
cylinderwake=dnsps.cyl_fems)
problemfem = problemdict[problemname]
femp = problemfem(N)
# output
ddir = 'data/'
try:
os.chdir(ddir)
except OSError:
raise Warning('need "' + ddir + '" subdir for storing the data')
os.chdir('..')
if linearized_nse and not outernwtnstps == 1:
raise Warning('Linearized problem can have only one Newton step')
if closed_loop:
if stst_control:
data_prfx = ddir + 'stst_' + problemname + '__'
else:
data_prfx = ddir + 'tdst_' + problemname + '__'
else:
data_prfx = ddir + problemname + '__'
if stokes_flow:
data_prfx = data_prfx + 'stokes__'
# specify in what spatial direction Bu changes. The remaining is constant
if problemname == 'drivencavity':
uspacedep = 0
elif problemname == 'cylinderwake':
uspacedep = 1
stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'],
femp['fv'], femp['fp'], t=0)
# remove the freedom in the pressure
stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]
# reduce the matrices by resolving the BCs
(stokesmatsc, rhsd_stbc,
invinds, bcinds, bcvals) = dts.condense_sysmatsbybcs(stokesmats,
femp['diribcs'])
print 'Dimension of the div matrix: ', stokesmatsc['J'].shape
# pressure freedom and dirichlet reduced rhs
rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])
# add the info on boundary and inner nodes
bcdata = {'bcinds': bcinds, 'bcvals': bcvals, 'invinds': invinds}
femp.update(bcdata)
# casting some parameters
NV = len(femp['invinds'])
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
# soldict.update(rhsd_vfrc) # adding fvc, fpr
soldict.update(fv=rhsd_stbc['fv']+rhsd_vfrc['fvc'],
fp=rhsd_stbc['fp']+rhsd_vfrc['fpr'],
N=N, nu=nu,
trange=tip['tmesh'],
get_datastring=get_datastr,
data_prfx=data_prfx,
clearprvdata=clearprvveldata,
paraviewoutput=tip['ParaviewOutput'],
vfileprfx=tip['proutdir']+'vel_',
pfileprfx=tip['proutdir']+'p_')
#
# Prepare for control
#
contp = ContParams(femp['odcoo'], ystar=ystar, alphau=alphau, gamma=gamma)
# casting some parameters
NY, NU = contp.NY, contp.NU
contsetupstr = problemname + '__NV{0}NU{1}NY{2}'.format(NV, NU, NY)
# get the control and observation operators
try:
b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
print 'loaded `b_mat`'
except IOError:
print 'computing `b_mat`...'
b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'], V=femp['V'],
NU=NU, xcomp=uspacedep)
dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
try:
mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
print 'loaded `c_mat`'
except IOError:
print 'computing `c_mat`...'
mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
V=femp['V'], NY=NY)
dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')
# restrict the operators to the inner nodes
mc_mat = mc_mat[:, invinds][:, :]
b_mat = b_mat[invinds, :][:, :]
# for further use:
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
if contp.ystarx is None:
c_mat = c_mat[NY:, :][:, :] # TODO: Do this right
mc_mat = mc_mat[NY:, :][:, :] # TODO: Do this right
y_masmat = y_masmat[:NY, :][:, :NY] # TODO: Do this right
mct_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=mc_mat.T,
transposedprj=True)
# set the weighing matrices
contp.R = contp.alphau * u_masmat
#
# solve the differential-alg. Riccati eqn for the feedback gain X
# via computing factors Z, such that X = -Z*Z.T
#
# at the same time we solve for the affine-linear correction w
#
# tilde B = BR^{-1/2}
tb_mat = lau.apply_invsqrt_fromright(contp.R, b_mat, output='sparse')
# tb_dense = np.array(tb_mat.todense())
trct_mat = lau.apply_invsqrt_fromright(y_masmat,
mct_mat_reg, output='dense')
if closed_loop:
cntpstr = 'NV{3}NY{0}NU{1}alphau{2}gamma{4}'.\
format(contp.NU, contp.NY, contp.alphau, NV, contp.gamma)
else:
cntpstr = ''
# we gonna use this quite often
M, A = stokesmatsc['M'], stokesmatsc['A']
datastrdict = dict(time=None, meshp=N, nu=nu, Nts=Nts,
data_prfx=data_prfx)
# compute the uncontrolled steady state (Navier-)Stokes solution
# as initial value
if ini_vel_stokes:
# compute the uncontrolled steady state Stokes solution
ini_vel, newtonnorms = snu.solve_steadystate_nse(vel_nwtn_stps=0,
vel_pcrd_stps=0,
**soldict)
soldict.update(dict(iniv=ini_vel))
else:
ini_vel, newtonnorms = snu.solve_steadystate_nse(**soldict)
soldict.update(dict(iniv=ini_vel))
if closed_loop:
if stst_control:
if stokes_flow:
convc_mat = sps.csr_matrix((NV, NV))
rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
lin_point = None
else:
lin_point, newtonnorms = snu.solve_steadystate_nse(**soldict)
(convc_mat, rhs_con,
rhsv_conbc) = snu.get_v_conv_conts(prev_v=lin_point,
invinds=invinds,
V=femp['V'],
diribcs=femp['diribcs'])
# infinite control horizon, steady target state
cdatstr = get_datastr(time=None, meshp=N, nu=nu,
Nts=None, data_prfx=data_prfx)
try:
Z = dou.load_npa(cdatstr + cntpstr + '__Z')
print 'loaded ' + cdatstr + cntpstr + '__Z'
except IOError:
if use_ric_ini_nu is not None:
cdatstr = get_datastr(nwtn=None, time=None, meshp=N,
nu=use_ric_ini_nu, Nts=None,
data_prfx=data_prfx)
try:
zini = dou.load_npa(ddir + cdatstr
+ cntpstr + '__Z')
print 'Initialize Newton ADI by Z from ' + cdatstr
except IOError:
raise Warning('No data for initialization of '
' Newton ADI -- need ' + cdatstr
+ '__Z')
cdatstr = get_datastr(meshp=N, nu=nu,
data_prfx=data_prfx)
else:
zini = None
parnadi = pru.proj_alg_ric_newtonadi
Z = parnadi(mmat=M, amat=-A-convc_mat,
jmat=stokesmatsc['J'],
bmat=tb_mat, wmat=trct_mat, z0=zini,
nwtn_adi_dict=tip['nwtn_adi_dict'])['zfac']
dou.save_npa(Z, fstring=cdatstr + cntpstr + '__Z')
print 'saved ' + cdatstr + cntpstr + '__Z'
if tip['compress_z']:
Zc = pru.compress_Zsvd(Z, thresh=tip['comprz_thresh'],
k=tip['comprz_maxc'])
Z = Zc
fvnstst = rhs_con + rhsv_conbc + rhsd_stbc['fv'] + rhsd_vfrc['fvc']
# X = -ZZ.T
mtxtb_stst = -pru.get_mTzzTtb(M.T, Z, tb_mat)
mtxfv_stst = -pru.get_mTzzTtb(M.T, Z, fvnstst)
fl = mc_mat.T * contp.ystarvec(0)
wft = lau.solve_sadpnt_smw(amat=A.T+convc_mat.T,
jmat=stokesmatsc['J'],
rhsv=fl+mtxfv_stst,
umat=mtxtb_stst,
vmat=tb_mat.T)[:NV]
auxstrg = cdatstr + cntpstr
dou.save_npa(wft, fstring=cdatstr + cntpstr + '__w')
dou.save_npa(mtxtb_stst, fstring=cdatstr + cntpstr + '__mtxtb')
feedbackthroughdict = {None:
dict(w=auxstrg + '__w',
mtxtb=auxstrg + '__mtxtb')}
cns = 0
soldict.update(data_prfx=data_prfx+'_cns{0}'.format(cns))
if linearized_nse:
soldict.update(vel_pcrd_stps=0,
vel_nwtn_stps=1,
lin_vel_point={None: lin_point})
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True,
static_feedback=True,
tb_mat=tb_mat,
stokes_flow=stokes_flow,
clearprvveldata=True,
feedbackthroughdict=feedbackthroughdict, **soldict)
else: # time dep closed loop
cns_data_prfx = 'data/cnsvars'
invd = init_nwtnstps_value_dict
curnwtnsdict = invd(tmesh=tip['tmesh'],
data_prfx=cns_data_prfx)
# initialization: compute the forward solution
if stokes_flow:
dictofvels = None
else:
dictofvels = snu.solve_nse(return_dictofvelstrs=True,
stokes_flow=stokes_flow,
**soldict)
# dbs.plot_vel_norms(tip['tmesh'], dictofvels)
# function for the time depending parts
# -- to be passed to the solver
def get_tdpart(time=None, dictofvalues=None, feedback=False,
V=None, invinds=None, diribcs=None, **kw):
if stokes_flow:
convc_mat = sps.csr_matrix((NV, NV))
rhs_con, rhsv_conbc = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
curvel = dou.load_npa(dictofvalues[time])
convc_mat, rhs_con, rhsv_conbc = \
snu.get_v_conv_conts(prev_v=curvel, invinds=invinds,
V=V, diribcs=diribcs)
return convc_mat, rhsv_conbc+rhs_con
gttdprtargs = dict(dictofvalues=dictofvels,
V=femp['V'],
diribcs=femp['diribcs'],
invinds=invinds)
# old version rhs
# ftilde = rhs_con + rhsv_conbc + rhsd_stbc['fv']
for cns in range(outernwtnstps):
datastrdict.update(data_prfx=data_prfx+cntpstr+'_cns{0}'.
format(cns))
soldict.update(data_prfx=data_prfx+cntpstr+'_cns{0}'.
format(cns))
sfd = sdr.solve_flow_daeric
feedbackthroughdict = \
sfd(mmat=M, amat=A, jmat=stokesmatsc['J'],
bmat=b_mat,
# cmat=ct_mat_reg.T,
mcmat=mct_mat_reg.T,
v_is_my=True, rmat=contp.alphau*u_masmat,
vmat=y_masmat, rhsv=rhsd_stbc['fv'],
gamma=contp.gamma,
rhsp=None,
tmesh=tip['tmesh'], ystarvec=contp.ystarvec,
nwtn_adi_dict=tip['nwtn_adi_dict'],
comprz_thresh=tip['comprz_thresh'],
comprz_maxc=tip['comprz_maxc'], save_full_z=False,
get_tdpart=get_tdpart, gttdprtargs=gttdprtargs,
curnwtnsdict=curnwtnsdict,
get_datastr=get_datastr, gtdtstrargs=datastrdict)
# for t in tip['tmesh']: # feedbackthroughdict.keys():
# curw = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
# print cns, t, np.linalg.norm(curw)
cdatstr = get_datastr(time='all', meshp=N, nu=nu,
Nts=None, data_prfx=data_prfx)
if linearized_nse:
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True, tb_mat=tb_mat,
lin_vel_point=dictofvels,
feedbackthroughdict=feedbackthroughdict,
vel_nwtn_stps=1,
vel_pcrd_stps=0,
**soldict)
else:
dictofvels = snu.\
solve_nse(return_dictofvelstrs=True,
closed_loop=True, tb_mat=tb_mat,
stokes_flow=stokes_flow,
feedbackthroughdict=feedbackthroughdict,
vel_pcrd_stps=1,
vel_nwtn_stps=2,
**soldict)
# for t in dictofvels.keys():
# curw = dou.load_npa(dictofvels[t])
# print cns, t, np.linalg.norm(curw)
gttdprtargs.update(dictofvalues=dictofvels)
else:
# no control
feedbackthroughdict = None
tb_mat = None
cdatstr = get_datastr(meshp=N, nu=nu, time='all',
Nts=Nts, data_prfx=data_prfx)
soldict.update(clearprvdata=True)
dictofvels = snu.solve_nse(feedbackthroughdict=feedbackthroughdict,
tb_mat=tb_mat, closed_loop=closed_loop,
stokes_flow=stokes_flow,
return_dictofvelstrs=True,
static_feedback=stst_control,
**soldict)
(yscomplist,
ystarlist) = dou.extract_output(dictofpaths=dictofvels,
tmesh=tip['tmesh'],
c_mat=c_mat, ystarvec=contp.ystarvec)
save_output_json(yscomplist, tip['tmesh'].tolist(), ystar=ystarlist,
fstring=cdatstr + cntpstr + '__sigout')
costfunval = eval_costfunc(W=y_masmat, V=contp.gamma*y_masmat,
R=None, tbmat=tb_mat, cmat=c_mat,
ystar=contp.ystarvec,
tmesh=tip['tmesh'], veldict=dictofvels,
fbftdict=feedbackthroughdict)
print 'Value of cost functional: ', costfunval
costfunval = eval_costfunc(W=y_masmat, V=contp.gamma*y_masmat,
R=None, tbmat=tb_mat, cmat=c_mat,
ystar=contp.ystarvec, penau=False,
tmesh=tip['tmesh'], veldict=dictofvels,
fbftdict=feedbackthroughdict)
print 'Value of cost functional not considering `u`: ', costfunval
print 'dim of v :', femp['V'].dim()
charlene = .15 if problemname == 'cylinderwake' else 1.0
print 'Re = charL / nu = {0}'.format(charlene/nu)
def solve_nse(A=None, M=None, J=None, JT=None,
fv=None, fp=None,
fvc=None, fpc=None, # TODO: this is to catch deprecated calls
fv_tmdp=None, fv_tmdp_params={},
fv_tmdp_memory=None,
iniv=None, lin_vel_point=None,
stokes_flow=False,
trange=None,
t0=None, tE=None, Nts=None,
V=None, Q=None, invinds=None, diribcs=None,
output_includes_bcs=False,
N=None, nu=None,
ppin=-1,
closed_loop=False, static_feedback=False,
feedbackthroughdict=None,
return_vp=False,
tb_mat=None, c_mat=None,
vel_nwtn_stps=20, vel_nwtn_tol=5e-15,
vel_pcrd_stps=4,
krylov=None, krpslvprms={}, krplsprms={},
clearprvdata=False,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
vfileprfx='', pfileprfx='',
return_dictofvelstrs=False,
return_dictofpstrs=False,
dictkeysstr=False,
comp_nonl_semexp=False,
return_as_list=False,
start_ssstokes=False,
**kw):
"""
solution of the time-dependent nonlinear Navier-Stokes equation
.. math::
M\\dot v + Av + N(v)v + J^Tp = f \n
Jv =g
using a Newton scheme in function space, i.e. given :math:`v_k`,
we solve for the update like
.. math::
M\\dot v + Av + N(v_k)v + N(v)v_k + J^Tp = N(v_k)v_k + f,
and trapezoidal rule in time. To solve an *Oseen* system (linearization
about a steady state) or a *Stokes* system, set the number of Newton
steps to one and provide a linearization point and an initial value.
Parameters
----------
lin_vel_point : dictionary, optional
contains the linearization point for the first Newton iteration
* Steady State: {{`None`: 'path_to_nparray'}, {'None': nparray}}
* Newton: {`t`: 'path_to_nparray'}
defaults to `None`
dictkeysstr : boolean, optional
whether the `keys` of the result dictionaries are strings instead \
of floats, defaults to `False`
fv_tmdp : callable f(t, v, dict), optional
time-dependent part of the right-hand side, set to zero if None
fv_tmdp_params : dictionary, optional
dictionary of parameters to be passed to `fv_tmdp`, defaults to `{}`
fv_tmdp_memory : dictionary, optional
memory of the function
output_includes_bcs : boolean, optional
whether append the boundary nodes to the computed and stored \
velocities, defaults to `False`
krylov : {None, 'gmres'}, optional
whether or not to use an iterative solver, defaults to `None`
krpslvprms : dictionary, optional
to specify parameters of the linear solver for use in Krypy, e.g.,
* initial guess
* tolerance
* number of iterations
defaults to `None`
krplsprms : dictionary, optional
parameters to define the linear system like
* preconditioner
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
stokes_flow : boolean, optional
whether to consider the Stokes linearization, defaults to `False`
start_ssstokes : boolean, optional
for your convenience, compute and use the steady state stokes solution
as initial value, defaults to `False`
Returns
-------
dictofvelstrs : dictionary, on demand
dictionary with time `t` as keys and path to velocity files as values
dictofpstrs : dictionary, on demand
dictionary with time `t` as keys and path to pressure files as values
vellist : list, on demand
list of the velocity solutions
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if fvc is not None or fpc is not None: # TODO: this is for catching calls
raise UserWarning('deprecated use of `rhsd_vfrc`, use only `fv`, `fp`')
if get_datastring is None:
get_datastring = get_datastr_snu
if paraviewoutput:
prvoutdict = dict(V=V, Q=Q, invinds=invinds, diribcs=diribcs,
vp=None, t=None, writeoutput=True, ppin=ppin)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' here
if trange is None:
trange = np.linspace(t0, tE, Nts+1)
if comp_nonl_semexp and lin_vel_point is not None:
raise UserWarning('I am not sure what you want! ' +
'set either `lin_vel_point=None` ' +
'or `comp_nonl_semexp=False`! \n' +
'as it is I will compute a linear case')
if return_dictofpstrs:
gpfvd = dict(V=V, M=M, A=A, J=J,
fv=fv, fp=fp,
diribcs=diribcs, invinds=invinds)
NV, NP = A.shape[0], J.shape[0]
if fv_tmdp is None:
def fv_tmdp(time=None, curvel=None, **kw):
return np.zeros((NV, 1)), None
if iniv is None:
if start_ssstokes:
# Stokes solution as starting value
(fv_tmdp_cont,
fv_tmdp_memory) = fv_tmdp(time=0, **fv_tmdp_params)
vp_stokes =\
lau.solve_sadpnt_smw(amat=A, jmat=J, jmatT=JT,
rhsv=fv + fv_tmdp_cont,
krylov=krylov, krpslvprms=krpslvprms,
krplsprms=krplsprms, rhsp=fp)
iniv = vp_stokes[:NV]
else:
raise ValueError('No initial value given')
datastrdict = dict(time=None, meshp=N, nu=nu,
Nts=trange.size-1, data_prfx=data_prfx)
if return_as_list:
clearprvdata = True # we want the results at hand
if clearprvdata:
datastrdict['time'] = '*'
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '__vel*'):
os.remove(fname)
for fname in glob.glob(cdatstr + '__p*'):
os.remove(fname)
def _atdct(cdict, t, thing):
if dictkeysstr:
cdict.update({'{0}'.format(t): thing})
else:
cdict.update({t: thing})
def _gfdct(cdict, t):
if dictkeysstr:
return cdict['{0}'.format(t)]
else:
return cdict[t]
if stokes_flow:
vel_nwtn_stps = 1
vel_pcrd_stps = 0
print 'Stokes Flow!'
elif lin_vel_point is None:
comp_nonl_semexp_inig = True
if not comp_nonl_semexp:
print('No linearization point given - explicit' +
' treatment of the nonlinearity in the first Iteration')
else:
cur_linvel_point = lin_vel_point
comp_nonl_semexp_inig = False
newtk, norm_nwtnupd, norm_nwtnupd_list = 0, 1, []
# check for previously computed velocities
if lin_vel_point is None and not stokes_flow:
try:
datastrdict.update(dict(time=trange[-1]))
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = dou.load_npa(cdatstr + '__norm_nwtnupd')
v_old = dou.load_npa(cdatstr + '__vel')
norm_nwtnupd_list.append(norm_nwtnupd)
print 'found vel files'
print 'norm of last Nwtn update: {0}'.format(norm_nwtnupd)
if norm_nwtnupd < vel_nwtn_tol and not return_dictofvelstrs:
return
elif norm_nwtnupd < vel_nwtn_tol:
datastrdict.update(dict(time=trange[0]))
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
try:
p_old = dou.load_npa(cdatstr + '__p')
dictofpstrs = {}
_atdct(dictofpstrs, trange[0], cdatstr+'__p')
except:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
_atdct(dictofpstrs, trange[0], cdatstr+'__p')
for t in trange[1:]:
# test if the vels are there
v_old = dou.load_npa(cdatstr + '__vel')
# update the dict
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
_atdct(dictofvelstrs, t, cdatstr + '__vel')
try:
p_old = dou.load_npa(cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
except:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
comp_nonl_semexp = False
except IOError:
norm_nwtnupd = 2
print 'no old velocity data found'
def _append_bcs_ornot(vvec):
if output_includes_bcs: # make the switch here for better readibility
vwbcs = dts.append_bcs_vec(vvec, vdim=V.dim(),
invinds=invinds, diribcs=diribcs)
return vwbcs
else:
return vvec
def _get_mats_rhs_ts(mmat=None, dt=None, var_c=None,
coeffmat_c=None,
coeffmat_n=None,
fv_c=None, fv_n=None,
umat_c=None, vmat_c=None,
umat_n=None, vmat_n=None,
impeul=False):
""" to be tweaked for different int schemes
"""
solvmat = M + 0.5*dt*coeffmat_n
rhs = M*var_c + 0.5*dt*(fv_n + fv_c - coeffmat_c*var_c)
if umat_n is not None:
matvec = lau.mm_dnssps
umat = 0.5*dt*umat_n
vmat = vmat_n
# TODO: do we really need a PLUS here??'
rhs = rhs + 0.5*dt*matvec(umat_c, matvec(vmat_c, var_c))
else:
umat, vmat = umat_n, vmat_n
return solvmat, rhs, umat, vmat
v_old = iniv # start vector for time integration in every Newtonit
datastrdict['time'] = trange[0]
cdatstr = get_datastring(**datastrdict)
dou.save_npa(_append_bcs_ornot(v_old), fstring=cdatstr + '__vel')
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
dictofpstrs = {}
_atdct(dictofpstrs, trange[0], cdatstr+'__p')
if return_as_list:
vellist = []
vellist.append(_append_bcs_ornot(v_old))
while (newtk < vel_nwtn_stps and norm_nwtnupd > vel_nwtn_tol):
if stokes_flow:
newtk = vel_nwtn_stps
elif comp_nonl_semexp_inig and not comp_nonl_semexp:
pcrd_anyone = False
print 'explicit treatment of nonl. for initial guess'
elif vel_pcrd_stps > 0 and not comp_nonl_semexp:
vel_pcrd_stps -= 1
pcrd_anyone = True
print 'Picard iterations for initial value -- {0} left'.\
format(vel_pcrd_stps)
elif comp_nonl_semexp:
pcrd_anyone = False
newtk = vel_nwtn_stps
print 'No Newton iterations - explicit treatment of the nonlin.'
else:
pcrd_anyone = False
newtk += 1
print 'Computing Newton Iteration {0}'.format(newtk)
v_old = iniv # start vector for time integration in every Newtonit
try:
if krpslvprms['krylovini'] == 'old':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
elif krpslvprms['krylovini'] == 'upd':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
vp_new = vp_old
cts_old = trange[1] - trange[0]
except (TypeError, KeyError):
pass # no inival for krylov solver required
vfile = dolfin.File(vfileprfx+'__timestep.pvd')
pfile = dolfin.File(pfileprfx+'__timestep.pvd')
prvoutdict.update(dict(vp=None, vc=iniv, t=trange[0],
pfile=pfile, vfile=vfile))
dou.output_paraview(**prvoutdict)
# ## current values_c for application of trap rule
if stokes_flow:
convc_mat_c = sps.csr_matrix((NV, NV))
rhs_con_c, rhsv_conbc_c = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
if comp_nonl_semexp or comp_nonl_semexp_inig:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point, trange[0]))
except KeyError:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point, None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_c, rhs_con_c, rhsv_conbc_c = \
get_v_conv_conts(prev_v=iniv, invinds=invinds,
V=V, diribcs=diribcs, Picard=pcrd_anyone)
(fv_tmdp_cont,
fv_tmdp_memory) = fv_tmdp(time=0,
curvel=v_old,
memory=fv_tmdp_memory,
**fv_tmdp_params)
fvn_c = fv + rhsv_conbc_c + rhs_con_c + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_c = dou.load_npa(feedbackthroughdict[None]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[None]['w'])
# print '\nnorm of feedback: ', np.linalg.norm(mtxtb_c)
else:
mtxtb_c = dou.load_npa(feedbackthroughdict[0]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[0]['w'])
fvn_c = fvn_c + tb_mat * (tb_mat.T * w_c)
vmat_c = mtxtb_c.T
try:
umat_c = np.array(tb_mat.todense())
except AttributeError:
umat_c = tb_mat
else:
vmat_c = None
umat_c = None
norm_nwtnupd = 0
print 'time to go',
for tk, t in enumerate(trange[1:]):
cts = t - trange[tk]
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
sys.stdout.write("\rEnd: {1} -- now: {0:f}".format(t, trange[-1]))
sys.stdout.flush()
# prv_datastrdict = copy.deepcopy(datastrdict)
# coeffs and rhs at next time instance
if stokes_flow:
convc_mat_n = sps.csr_matrix((NV, NV))
rhs_con_n, rhsv_conbc_n = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
if comp_nonl_semexp or comp_nonl_semexp_inig:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point, t))
except KeyError:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point,
None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_n, rhs_con_n, rhsv_conbc_n = \
get_v_conv_conts(prev_v=prev_v, invinds=invinds,
V=V, diribcs=diribcs, Picard=pcrd_anyone)
(fv_tmdp_cont,
fv_tmdp_memory) = fv_tmdp(time=t,
curvel=v_old,
memory=fv_tmdp_memory,
**fv_tmdp_params)
fvn_n = fv + rhsv_conbc_n + rhs_con_n + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_n = dou.load_npa(feedbackthroughdict[None]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[None]['w'])
# fb = np.dot(tb_mat*mtxtb_n.T, v_old)
# print '\nnorm of feedback: ', np.linalg.norm(fb)
# print '\nnorm of v_old: ', np.linalg.norm(v_old)
# print '\nnorm of feedthrough: ', np.linalg.norm(next_w)
else:
mtxtb_n = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[t]['w'])
fvn_n = fvn_n + tb_mat * (tb_mat.T * w_n)
vmat_n = mtxtb_n.T
try:
umat_n = np.array(tb_mat.todense())
except AttributeError:
umat_n = tb_mat
else:
vmat_n = None
umat_n = None
(solvmat, rhsv, umat,
vmat) = _get_mats_rhs_ts(mmat=M, dt=cts, var_c=v_old,
coeffmat_c=A + convc_mat_c,
coeffmat_n=A + convc_mat_n,
fv_c=fvn_c, fv_n=fvn_n,
umat_c=umat_c, vmat_c=vmat_c,
umat_n=umat_n, vmat_n=vmat_n)
try:
if krpslvprms['krylovini'] == 'old':
krpslvprms['x0'] = vp_old
elif krpslvprms['krylovini'] == 'upd':
vp_oldold = vp_old
vp_old = vp_new
krpslvprms['x0'] = vp_old + \
cts*(vp_old - vp_oldold)/cts_old
cts_old = cts
except (TypeError, KeyError):
pass # no inival for krylov solver required
vp_new = lau.solve_sadpnt_smw(amat=solvmat,
jmat=J, jmatT=JT,
rhsv=rhsv,
rhsp=fp,
krylov=krylov, krpslvprms=krpslvprms,
krplsprms=krplsprms,
umat=umat, vmat=vmat)
v_old = vp_new[:NV, ]
(umat_c, vmat_c, fvn_c,
convc_mat_c) = umat_n, vmat_n, fvn_n, convc_mat_n
dou.save_npa(_append_bcs_ornot(v_old), fstring=cdatstr + '__vel')
_atdct(dictofvelstrs, t, cdatstr + '__vel')
if return_dictofpstrs:
p_new = -1/cts*vp_new[NV:, ]
# p was flipped and scaled for symmetry
dou.save_npa(p_new, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_as_list:
vellist.append(_append_bcs_ornot(v_old))
prvoutdict.update(dict(vp=vp_new, t=t))
dou.output_paraview(**prvoutdict)
# integrate the Newton error
if stokes_flow or comp_nonl_semexp or comp_nonl_semexp_inig:
norm_nwtnupd = [None]
else:
if len(prev_v) > len(invinds):
prev_v = prev_v[invinds, :]
norm_nwtnupd += cts * m_innerproduct(M, v_old - prev_v)
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
norm_nwtnupd_list.append(norm_nwtnupd[0])
print '\nnorm of current Newton update: {}'.format(norm_nwtnupd)
comp_nonl_semexp = False
comp_nonl_semexp_inig = False
cur_linvel_point = dictofvelstrs
if return_dictofvelstrs:
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
elif return_as_list:
return vellist
else:
return
def solve_steadystate_nse(A=None, J=None, JT=None, M=None,
fv=None, fp=None,
V=None, Q=None, invinds=None, diribcs=None,
return_vp=False, ppin=-1,
N=None, nu=None,
vel_pcrd_stps=10, vel_pcrd_tol=1e-4,
vel_nwtn_stps=20, vel_nwtn_tol=5e-15,
clearprvdata=False,
vel_start_nwtn=None,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
save_intermediate_steps=False,
vfileprfx='', pfileprfx='',
**kw):
"""
Solution of the steady state nonlinear NSE Problem
using Newton's scheme. If no starting value is provided, the iteration
is started with the steady state Stokes solution.
Parameters
----------
A : (N,N) sparse matrix
stiffness matrix aka discrete Laplacian, note the sign!
M : (N,N) sparse matrix
mass matrix
J : (M,N) sparse matrix
discrete divergence operator
JT : (N,M) sparse matrix, optional
discrete gradient operator, set to J.T if not provided
fv, fp : (N,1), (M,1) ndarrays
right hand sides restricted via removing the boundary nodes in the
momentum and the pressure freedom in the continuity equation
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
return_vp : boolean, optional
whether to return also the pressure, defaults to `False`
vel_pcrd_stps : int, optional
Number of Picard iterations when computing a starting value for the
Newton scheme, cf. Elman, Silvester, Wathen: *FEM and fast iterative
solvers*, 2005, defaults to `100`
vel_pcrd_tol : real, optional
tolerance for the size of the Picard update, defaults to `1e-4`
vel_nwtn_stps : int, optional
Number of Newton iterations, defaults to `20`
vel_nwtn_tol : real, optional
tolerance for the size of the Newton update, defaults to `5e-15`
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if get_datastring is None:
get_datastring = get_datastr_snu
if JT is None:
JT = J.T
NV = J.shape[1]
#
# Compute or load the uncontrolled steady state Navier-Stokes solution
#
norm_nwtnupd_list = []
vel_newtk, norm_nwtnupd = 0, 1
# a dict to be passed to the get_datastring function
datastrdict = dict(time=None, meshp=N, nu=nu,
Nts=None, data_prfx=data_prfx)
if clearprvdata:
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '*__vel*'):
os.remove(fname)
try:
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = dou.load_npa(cdatstr + '__norm_nwtnupd')
vel_k = dou.load_npa(cdatstr + '__vel')
norm_nwtnupd_list.append(norm_nwtnupd)
print 'found vel files'
print 'norm of last Nwtn update: {0}'.format(norm_nwtnupd)
if norm_nwtnupd < vel_nwtn_tol:
if not return_vp:
return vel_k, norm_nwtnupd_list
else:
pfv = get_pfromv(v=vel_k[:NV, :], V=V, M=M, A=A, J=J, fv=fv,
invinds=invinds, diribcs=diribcs)
return (np.vstack([vel_k, pfv]), norm_nwtnupd_list)
except IOError:
print 'no old velocity data found'
norm_nwtnupd = None
if paraviewoutput:
cdatstr = get_datastring(**datastrdict)
vfile = dolfin.File(vfileprfx+'__steadystates.pvd')
pfile = dolfin.File(pfileprfx+'__steadystates.pvd')
prvoutdict = dict(V=V, Q=Q, vfile=vfile, pfile=pfile,
invinds=invinds, diribcs=diribcs, ppin=ppin,
vp=None, t=None, writeoutput=True)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' here
NV = A.shape[0]
if vel_start_nwtn is None:
vp_stokes = lau.solve_sadpnt_smw(amat=A, jmat=J, jmatT=JT,
rhsv=fv, rhsp=fp)
vp_stokes[NV:] = -vp_stokes[NV:]
# pressure was flipped for symmetry
# save the data
cdatstr = get_datastring(**datastrdict)
dou.save_npa(vp_stokes[:NV, ], fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_stokes))
dou.output_paraview(**prvoutdict)
# Stokes solution as starting value
vp_k = vp_stokes
vel_k = vp_stokes[:NV, ]
else:
vel_k = vel_start_nwtn
# Picard iterations for a good starting value for Newton
for k in range(vel_pcrd_stps):
(convc_mat,
rhs_con, rhsv_conbc) = get_v_conv_conts(prev_v=vel_k, invinds=invinds,
V=V, diribcs=diribcs,
Picard=True)
vp_k = lau.solve_sadpnt_smw(amat=A+convc_mat, jmat=J, jmatT=JT,
rhsv=fv+rhs_con+rhsv_conbc,
rhsp=fp)
normpicupd = np.sqrt(m_innerproduct(M, vel_k-vp_k[:NV, :]))[0]
print 'Picard iteration: {0} -- norm of update: {1}'\
.format(k+1, normpicupd)
vel_k = vp_k[:NV, ]
vp_k[NV:] = -vp_k[NV:]
# pressure was flipped for symmetry
if normpicupd < vel_pcrd_tol:
break
# Newton iteration
while (vel_newtk < vel_nwtn_stps
and (norm_nwtnupd is None or
norm_nwtnupd > vel_nwtn_tol or
norm_nwtnupd == np.array(None))):
vel_newtk += 1
cdatstr = get_datastring(**datastrdict)
(convc_mat,
rhs_con, rhsv_conbc) = get_v_conv_conts(vel_k, invinds=invinds,
V=V, diribcs=diribcs)
vp_k = lau.solve_sadpnt_smw(amat=A+convc_mat, jmat=J, jmatT=JT,
rhsv=fv+rhs_con+rhsv_conbc,
rhsp=fp)
norm_nwtnupd = np.sqrt(m_innerproduct(M, vel_k - vp_k[:NV, :]))[0]
vel_k = vp_k[:NV, ]
vp_k[NV:] = -vp_k[NV:]
# pressure was flipped for symmetry
print 'Steady State NSE: Newton iteration: {0} -- norm of update: {1}'\
.format(vel_newtk, norm_nwtnupd)
dou.save_npa(vel_k, fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_k))
dou.output_paraview(**prvoutdict)
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
dou.output_paraview(**prvoutdict)
# savetomatlab = True
# if savetomatlab:
# export_mats_to_matlab(E=None, A=None, matfname='matexport')
if return_vp:
return vp_k, norm_nwtnupd_list
else:
return vel_k, norm_nwtnupd_list
def solve_nse(A=None, M=None, J=None, JT=None,
fv=None, fp=None,
fvc=None, fpc=None, # TODO: this is to catch deprecated calls
fv_tmdp=None, fv_tmdp_params={},
fv_tmdp_memory=None,
iniv=None, lin_vel_point=None,
stokes_flow=False,
trange=None,
t0=None, tE=None, Nts=None,
V=None, Q=None, invinds=None, diribcs=None,
dbcinds=None, dbcvals=None,
output_includes_bcs=False,
N=None, nu=None,
ppin=-1,
closed_loop=False, static_feedback=False,
feedbackthroughdict=None,
return_vp=False,
tb_mat=None, cv_mat=None,
vel_nwtn_stps=20, vel_nwtn_tol=5e-15,
nsects=1, loc_nwtn_tol=5e-15, loc_pcrd_stps=True,
addfullsweep=False,
vel_pcrd_stps=4,
krylov=None, krpslvprms={}, krplsprms={},
clearprvdata=False,
useolddata=False,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
plttrange=None,
vfileprfx='', pfileprfx='',
return_dictofvelstrs=False,
return_dictofpstrs=False,
dictkeysstr=False,
treat_nonl_explct=False,
return_as_list=False,
verbose=True,
start_ssstokes=False,
**kw):
"""
solution of the time-dependent nonlinear Navier-Stokes equation
.. math::
M\\dot v + Av + N(v)v + J^Tp = f \n
Jv =g
using a Newton scheme in function space, i.e. given :math:`v_k`,
we solve for the update like
.. math::
M\\dot v + Av + N(v_k)v + N(v)v_k + J^Tp = N(v_k)v_k + f,
and trapezoidal rule in time. To solve an *Oseen* system (linearization
about a steady state) or a *Stokes* system, set the number of Newton
steps to one and provide a linearization point and an initial value.
Parameters
----------
lin_vel_point : dictionary, optional
contains the linearization point for the first Newton iteration
* Steady State: {{`None`: 'path_to_nparray'}, {'None': nparray}}
* Newton: {`t`: 'path_to_nparray'}
defaults to `None`
dictkeysstr : boolean, optional
whether the `keys` of the result dictionaries are strings instead \
of floats, defaults to `False`
fv_tmdp : callable f(t, v, dict), optional
time-dependent part of the right-hand side, set to zero if None
fv_tmdp_params : dictionary, optional
dictionary of parameters to be passed to `fv_tmdp`, defaults to `{}`
fv_tmdp_memory : dictionary, optional
memory of the function
output_includes_bcs : boolean, optional
whether append the boundary nodes to the computed and stored \
velocities, defaults to `False`
krylov : {None, 'gmres'}, optional
whether or not to use an iterative solver, defaults to `None`
krpslvprms : dictionary, optional
to specify parameters of the linear solver for use in Krypy, e.g.,
* initial guess
* tolerance
* number of iterations
defaults to `None`
krplsprms : dictionary, optional
parameters to define the linear system like
* preconditioner
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
stokes_flow : boolean, optional
whether to consider the Stokes linearization, defaults to `False`
start_ssstokes : boolean, optional
for your convenience, compute and use the steady state stokes solution
as initial value, defaults to `False`
treat_nonl_explct= string, optional
whether to treat the nonlinearity explicitly, defaults to `False`
nsects: int, optional
in how many segments the trange is split up. (The newton iteration
will be confined to the segments and, probably, converge faster than
the global iteration), defaults to `1`
loc_nwtn_tol: float, optional
tolerance for the newton iteration on the segments,
defaults to `1e-15`
loc_pcrd_stps: boolean, optional
whether to init with `vel_pcrd_stps` Picard steps on every section,
if `False`, Picard iterations are performed only on the first section,
defaults to `True`
addfullsweep: boolean, optional
whether to compute the newton iteration on the full `trange`,
useful to check and for the plots, defaults to `False`
cv_mat: (Ny, Nv) sparse array, optional
output matrix for velocity outputs, needed, e.g., for output dependent
feedback control, defaults to `None`
Returns
-------
dictofvelstrs : dictionary, on demand
dictionary with time `t` as keys and path to velocity files as values
dictofpstrs : dictionary, on demand
dictionary with time `t` as keys and path to pressure files as values
vellist : list, on demand
list of the velocity solutions
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if fvc is not None or fpc is not None: # TODO: this is for catching calls
raise UserWarning('deprecated use of `rhsd_vfrc`, use only `fv`, `fp`')
if get_datastring is None:
get_datastring = get_datastr_snu
if paraviewoutput:
prvoutdict = dict(V=V, Q=Q,
invinds=invinds, diribcs=diribcs, ppin=ppin,
vp=None, t=None,
tfilter=plttrange, writeoutput=True)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' here
if trange is None:
trange = np.linspace(t0, tE, Nts+1)
if treat_nonl_explct and lin_vel_point is not None:
raise UserWarning('cant use `lin_vel_point` ' +
'and explicit treatment of the nonlinearity')
if return_dictofpstrs:
gpfvd = dict(V=V, M=M, A=A, J=J,
fv=fv, fp=fp,
dbcinds=dbcinds, dbcvals=dbcvals,
diribcs=diribcs, invinds=invinds)
NV, NP = A.shape[0], J.shape[0]
fv = np.zeros((NV, 1)) if fv is None else fv
fp = np.zeros((NP, 1)) if fp is None else fp
if fv_tmdp is None:
def fv_tmdp(time=None, curvel=None, **kw):
return np.zeros((NV, 1)), None
if iniv is None:
if start_ssstokes:
# Stokes solution as starting value
vp_stokes =\
lau.solve_sadpnt_smw(amat=A, jmat=J, jmatT=JT,
rhsv=fv, # + fv_tmdp_cont,
krylov=krylov, krpslvprms=krpslvprms,
krplsprms=krplsprms, rhsp=fp)
iniv = vp_stokes[:NV]
else:
raise ValueError('No initial value given')
datastrdict = dict(time=None, meshp=N, nu=nu,
Nts=trange.size-1, data_prfx=data_prfx,
semiexpl=treat_nonl_explct)
if return_as_list:
clearprvdata = True # we want the results at hand
if clearprvdata:
datastrdict['time'] = '*'
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '__vel*'):
os.remove(fname)
for fname in glob.glob(cdatstr + '__p*'):
os.remove(fname)
def _atdct(cdict, t, thing):
if dictkeysstr:
cdict.update({'{0}'.format(t): thing})
else:
cdict.update({t: thing})
def _gfdct(cdict, t):
if dictkeysstr:
return cdict['{0}'.format(t)]
else:
return cdict[t]
if stokes_flow:
vel_nwtn_stps = 1
vel_pcrd_stps = 0
print('Stokes Flow!')
elif lin_vel_point is None:
comp_nonl_semexp_inig = True
if not treat_nonl_explct:
print(('No linearization point given - explicit' +
' treatment of the nonlinearity in the first Iteration'))
else:
cur_linvel_point = lin_vel_point
comp_nonl_semexp_inig = False
newtk, norm_nwtnupd = 0, 1
# check for previously computed velocities
if useolddata and lin_vel_point is None and not stokes_flow:
try:
datastrdict.update(dict(time=trange[-1]))
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = (dou.load_npa(cdatstr + '__norm_nwtnupd')).flatten()
try:
if norm_nwtnupd[0] is None:
norm_nwtnupd = 1.
except IndexError:
norm_nwtnupd = 1.
dou.load_npa(cdatstr + '__vel')
print('found vel files')
print('norm of last Nwtn update: {0}'.format(norm_nwtnupd))
print('... loaded from ' + cdatstr)
if norm_nwtnupd < vel_nwtn_tol and not return_dictofvelstrs:
return
elif norm_nwtnupd < vel_nwtn_tol or treat_nonl_explct:
# looks like converged / or semi-expl
# -- check if all values are there
# t0:
datastrdict.update(dict(time=trange[0]))
cdatstr = get_datastring(**datastrdict)
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
dictofpstrs = {}
for t in trange:
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
# test if the vels are there
v_old = dou.load_npa(cdatstr + '__vel')
# update the dict
_atdct(dictofvelstrs, t, cdatstr + '__vel')
if return_dictofpstrs:
try:
p_old = dou.load_npa(cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
except:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
# comp_nonl_semexp = False
except IOError:
norm_nwtnupd = 2
print('no old velocity data found')
def _append_bcs_ornot(vvec):
if output_includes_bcs: # make the switch here for better readibility
vwbcs = dts.append_bcs_vec(vvec, vdim=V.dim(),
invinds=invinds, diribcs=diribcs)
return vwbcs
else:
return vvec
def _get_mats_rhs_ts(mmat=None, dt=None, var_c=None,
coeffmat_c=None,
coeffmat_n=None,
fv_c=None, fv_n=None,
umat_c=None, vmat_c=None,
umat_n=None, vmat_n=None,
impeul=False):
""" to be tweaked for different int schemes
"""
solvmat = M + 0.5*dt*coeffmat_n
rhs = M*var_c + 0.5*dt*(fv_n + fv_c - coeffmat_c*var_c)
if umat_n is not None:
matvec = lau.mm_dnssps
umat = 0.5*dt*umat_n
vmat = vmat_n
# TODO: do we really need a PLUS here??'
rhs = rhs + 0.5*dt*matvec(umat_c, matvec(vmat_c, var_c))
else:
umat, vmat = umat_n, vmat_n
return solvmat, rhs, umat, vmat
v_old = iniv # start vector for time integration in every Newtonit
datastrdict['time'] = trange[0]
cdatstr = get_datastring(**datastrdict)
dou.save_npa(_append_bcs_ornot(v_old), fstring=cdatstr + '__vel')
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
dictofpstrs = {}
_atdct(dictofpstrs, trange[0], cdatstr+'__p')
else:
p_old = None
if return_as_list:
vellist = []
vellist.append(_append_bcs_ornot(v_old))
lensect = np.int(np.floor(trange.size/nsects))
loctrngs = []
for k in np.arange(nsects-1):
loctrngs.append(trange[k*lensect: (k+1)*lensect+1])
loctrngs.append(trange[(nsects-1)*lensect:])
if addfullsweep:
loctrngs.append(trange)
realiniv = np.copy(iniv)
if nsects == 1:
loc_nwtn_tol = vel_nwtn_tol
addfullsweep = False
loctrngs = [trange]
if loc_pcrd_stps:
vel_loc_pcrd_steps = vel_pcrd_stps
for loctrng in loctrngs:
dtvec = np.array(loctrng)[1:] - np.array(loctrng)[1:]
dotdtvec = dtvec[1:] - dtvec[:-1]
uniformgrid = np.allclose(np.linalg.norm(dotdtvec), 0)
coeffmatlu = None
while (newtk < vel_nwtn_stps and norm_nwtnupd > loc_nwtn_tol):
print('solve the NSE on the interval [{0}, {1}]'.
format(loctrng[0], loctrng[-1]))
if stokes_flow:
pcrd_anyone = False
newtk = vel_nwtn_stps
elif comp_nonl_semexp_inig and not treat_nonl_explct:
pcrd_anyone = False
loc_treat_nonl_explct = True
print('explicit treatment of nonl. for initial guess!')
elif treat_nonl_explct:
pcrd_anyone = False
loc_treat_nonl_explct = True
newtk = vel_nwtn_stps
print('No Newton iterations - explicit treatment ' +
'of the nonlinearity')
if not comp_nonl_semexp_inig and not treat_nonl_explct:
if vel_pcrd_stps > 0:
vel_pcrd_stps -= 1
pcrd_anyone = True
print('Picard iterations for initial value -- {0} left'.
format(vel_pcrd_stps))
else:
pcrd_anyone = False
newtk += 1
print('Computing Newton Iteration {0}'.format(newtk))
v_old = iniv # start vector for time integration in every Newtonit
try:
if krpslvprms['krylovini'] == 'old':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
elif krpslvprms['krylovini'] == 'upd':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
vp_new = vp_old
cts_old = loctrng[1] - loctrng[0]
except (TypeError, KeyError):
pass # no inival for krylov solver required
vfile = dolfin.File(vfileprfx+'__timestep.pvd')
pfile = dolfin.File(pfileprfx+'__timestep.pvd')
prvoutdict.update(dict(vp=None, vc=iniv, pc=p_old, t=loctrng[0],
dbcinds=dbcinds, dbcvals=dbcvals,
pfile=pfile, vfile=vfile))
dou.output_paraview(**prvoutdict)
# ## current values_c for application of trap rule
if stokes_flow:
convc_mat_c = sps.csr_matrix((NV, NV))
rhs_con_c, rhsv_conbc_c = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
if loc_treat_nonl_explct is not None:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point,
loctrng[0]))
except KeyError:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point,
None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_c, rhs_con_c, rhsv_conbc_c = \
get_v_conv_conts(prev_v=iniv, invinds=invinds,
semi_explicit=loc_treat_nonl_explct,
dbcinds=dbcinds, dbcvals=dbcvals,
V=V, diribcs=diribcs, Picard=pcrd_anyone)
cury = None if cv_mat is None else cv_mat.dot(v_old)
(fv_tmdp_cont,
fv_tmdp_memory) = fv_tmdp(time=0,
curvel=v_old,
cury=cury,
memory=fv_tmdp_memory,
**fv_tmdp_params)
_rhsconvc = 0. if pcrd_anyone else rhs_con_c
fvn_c = fv + rhsv_conbc_c + _rhsconvc + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_c = dou.load_npa(feedbackthroughdict[None]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[None]['w'])
else:
mtxtb_c = dou.load_npa(feedbackthroughdict[0]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[0]['w'])
fvn_c = fvn_c + tb_mat * (tb_mat.T * w_c)
vmat_c = mtxtb_c.T
try:
umat_c = np.array(tb_mat.todense())
except AttributeError:
umat_c = tb_mat
else:
vmat_c = None
umat_c = None
norm_nwtnupd = 0
if verbose:
# define at which points of time the progress is reported
nouts = 10 # number of output points
locnts = loctrng.size # TODO: trange may be a list...
filtert = np.arange(0, locnts,
np.int(np.floor(locnts/nouts)))
loctinstances = loctrng[filtert]
loctinstances[0] = loctrng[1]
loctinstances = loctinstances.tolist()
print('doing the time integration...')
for tk, t in enumerate(loctrng[1:]):
cts = t - loctrng[tk]
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
try:
if verbose and t == loctinstances[0]:
curtinst = loctinstances.pop(0)
# print("runtime: {0} -- t: {1} -- tE: {2:f}".
# format(time.clock(), curtinst, loctrng[-1]))
print("runtime: {0:.1f} - t/tE: {1:.2f} - t: {2:.4f}".
format(time.clock(), curtinst/loctrng[-1],
curtinst))
except IndexError:
pass # if something goes wrong, don't stop
# coeffs and rhs at next time instance
if stokes_flow:
convc_mat_n = sps.csr_matrix((NV, NV))
rhs_con_n = np.zeros((NV, 1))
rhsv_conbc_n = np.zeros((NV, 1))
else:
if loc_treat_nonl_explct:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point, t))
except KeyError:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point,
None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_n, rhs_con_n, rhsv_conbc_n = \
get_v_conv_conts(prev_v=prev_v, invinds=invinds, V=V,
semi_explicit=loc_treat_nonl_explct,
dbcinds=dbcinds, dbcvals=dbcvals,
diribcs=diribcs, Picard=pcrd_anyone)
cury = None if cv_mat is None else cv_mat.dot(v_old)
(fv_tmdp_cont,
fv_tmdp_memory) = fv_tmdp(time=t,
curvel=v_old,
cury=cury,
memory=fv_tmdp_memory,
**fv_tmdp_params)
_rhsconvn = 0. if pcrd_anyone else rhs_con_n
fvn_n = fv + rhsv_conbc_n + _rhsconvn + fv_tmdp_cont
if loc_treat_nonl_explct and not closed_loop:
fvn_c = fv + rhsv_conbc_n + _rhsconvn + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_n = dou.\
load_npa(feedbackthroughdict[None]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[None]['w'])
# fb = np.dot(tb_mat*mtxtb_n.T, v_old)
# print '\nnorm of feedback: ', np.linalg.norm(fb)
# print '\nnorm of v_old: ', np.linalg.norm(v_old)
else:
mtxtb_n = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[t]['w'])
fvn_n = fvn_n + tb_mat * (tb_mat.T * w_n)
vmat_n = mtxtb_n.T
try:
umat_n = np.array(tb_mat.todense())
except AttributeError:
umat_n = tb_mat
else:
vmat_n = None
umat_n = None
(solvmat, rhsv, umat,
vmat) = _get_mats_rhs_ts(mmat=M, dt=cts, var_c=v_old,
coeffmat_c=A + convc_mat_c,
coeffmat_n=A + convc_mat_n,
fv_c=fvn_c, fv_n=fvn_n,
umat_c=umat_c, vmat_c=vmat_c,
umat_n=umat_n, vmat_n=vmat_n)
try:
if krpslvprms['krylovini'] == 'old':
krpslvprms['x0'] = vp_old
elif krpslvprms['krylovini'] == 'upd':
vp_oldold = vp_old
vp_old = vp_new
krpslvprms['x0'] = vp_old + \
cts*(vp_old - vp_oldold)/cts_old
cts_old = cts
except (TypeError, KeyError):
pass # no inival for krylov solver required
if loc_treat_nonl_explct and uniformgrid and not krylov:
if coeffmatlu is None:
print('gonna compute an LU of the coefficient ' +
'matrix \n and reuse it in the time stepping')
vp_new, coeffmatlu = \
lau.solve_sadpnt_smw(amat=solvmat, jmat=J, jmatT=JT,
rhsv=rhsv, rhsp=fp,
sadlu=coeffmatlu,
return_alu=True,
umat=umat, vmat=vmat)
else:
vp_new = lau.solve_sadpnt_smw(amat=solvmat,
jmat=J, jmatT=JT,
rhsv=rhsv,
rhsp=fp,
krylov=krylov,
krpslvprms=krpslvprms,
krplsprms=krplsprms,
umat=umat, vmat=vmat)
v_old = vp_new[:NV, ]
(umat_c, vmat_c, fvn_c,
convc_mat_c) = umat_n, vmat_n, fvn_n, convc_mat_n
dou.save_npa(_append_bcs_ornot(v_old),
fstring=cdatstr + '__vel')
_atdct(dictofvelstrs, t, cdatstr + '__vel')
p_new = -1/cts*vp_new[NV:, ]
# p was flipped and scaled for symmetry
if return_dictofpstrs:
dou.save_npa(p_new, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_as_list:
vellist.append(_append_bcs_ornot(v_old))
# integrate the Newton error
if stokes_flow or treat_nonl_explct:
norm_nwtnupd = None
elif comp_nonl_semexp_inig:
norm_nwtnupd = 1.
else:
if len(prev_v) > len(invinds):
prev_v = prev_v[invinds, :]
addtonwtnupd = cts * m_innerproduct(M, v_old - prev_v)
norm_nwtnupd += np.float(addtonwtnupd.flatten()[0])
if newtk == vel_nwtn_stps or norm_nwtnupd < loc_nwtn_tol:
# paraviewoutput in the (probably) last newton sweep
prvoutdict.update(dict(vc=v_old, pc=p_new, t=t))
dou.output_paraview(**prvoutdict)
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
print('\nnorm of current Newton update: {}'.format(norm_nwtnupd))
# print('\nsaved `norm_nwtnupd(={0})'.format(norm_nwtnupd) +
# ' to ' + cdatstr)
loc_treat_nonl_explct = False
comp_nonl_semexp_inig = False
cur_linvel_point = dictofvelstrs
iniv = v_old
if not treat_nonl_explct and lin_vel_point is None:
comp_nonl_semexp_inig = True
if addfullsweep and loctrng is loctrngs[-2]:
comp_nonl_semexp_inig = False
iniv = realiniv
loc_nwtn_tol = vel_nwtn_tol
elif loc_pcrd_stps:
vel_pcrd_stps = vel_loc_pcrd_steps
norm_nwtnupd = 1.
newtk = 0
if return_dictofvelstrs:
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
elif return_as_list:
return vellist
else:
return
def lqgbt(problemname='drivencavity',
N=10, Re=1e2, plain_bt=False,
use_ric_ini=None, t0=0.0, tE=1.0, Nts=11,
NU=3, NY=3,
bccontrol=True, palpha=1e-5,
paraoutput=True,
trunc_lqgbtcv=1e-6,
nwtn_adi_dict=None,
comp_freqresp=False, comp_stepresp='nonlinear',
closed_loop=False, multiproc=False,
perturbpara=1e-3):
"""Main routine for LQGBT
Parameters
----------
problemname : string, optional
what problem to be solved, 'cylinderwake' or 'drivencavity'
N : int, optional
parameter for the dimension of the space discretization
Re : real, optional
Reynolds number, defaults to `1e2`
plain_bt : boolean, optional
whether to try simple *balanced truncation*, defaults to False
use_ric_ini : real, optional
use the solution with this Re number as stabilizing initial guess,
defaults to `None`
t0, tE, Nts : real, real, int, optional
starting and endpoint of the considered time interval, number of
time instancses, default to `0.0, 1.0, 11`
bccontrol : boolean, optional
whether to apply boundary control via penalized robin conditions,
defaults to `False`
NU, NY : int, optional
dimensions of components of in and output space (will double because
there are two components), default to `3, 3`
comp_freqresp : boolean, optional
whether to compute and compare the frequency responses,
defaults to `False`
comp_stepresp : {'nonlinear', False, None}
whether to compute and compare the step responses
| if False -> no step response
| if == 'nonlinear' -> compare linear reduced to nonlinear full model
| else -> linear reduced versus linear full model
defaults to `False`
trunc_lqgbtcv : real, optional
threshold at what the lqgbt characteristiv values are truncated,
defaults to `1e-6`
closed_loop : {'full_state_fb', 'red_output_fb', False, None}
how to do the closed loop simulation:
| if False -> no simulation
| if == 'full_state_fb' -> full state feedback
| if == 'red_output_fb' -> reduced output feedback
| else -> no control is applied
defaults to `False`
"""
typprb = 'BT' if plain_bt else 'LQG-BT'
print '\n ### We solve the {0} problem for the {1} at Re={2} ###\n'.\
format(typprb, problemname, Re)
if nwtn_adi_dict is not None:
nap = nwtn_adi_dict
else:
nap = nwtn_adi_params()['nwtn_adi_dict']
# output
ddir = 'data/'
try:
os.chdir(ddir)
except OSError:
raise Warning('need "' + ddir + '" subdir for storing the data')
os.chdir('..')
# stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
# rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'],
# femp['fv'], femp['fp'], t=0)
# # remove the freedom in the pressure
# stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
# stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
# rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]
# # reduce the matrices by resolving the BCs
# (stokesmatsc,
# rhsd_stbc,
# invinds,
# bcinds,
# bcvals) = dts.condense_sysmatsbybcs(stokesmats,
# femp['diribcs'])
# # pressure freedom and dirichlet reduced rhs
# rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])
# # add the info on boundary and inner nodes
# bcdata = {'bcinds': bcinds,
# 'bcvals': bcvals,
# 'invinds': invinds}
# femp.update(bcdata)
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc \
= dnsps.get_sysmats(problem=problemname, N=N, Re=Re,
bccontrol=bccontrol, scheme='TH')
nu = femp['charlen']/Re
# specify in what spatial direction Bu changes. The remaining is constant
uspacedep = femp['uspacedep']
# casting some parameters
invinds, NV = femp['invinds'], len(femp['invinds'])
prbstr = '_bt' if plain_bt else '_lqgbt'
# contsetupstr = 'NV{0}NU{1}NY{2}alphau{3}'.format(NV, NU, NY, alphau)
if bccontrol:
import scipy.sparse as sps
contsetupstr = 'NV{0}_bcc_NY{1}'.format(NV, NY)
stokesmatsc['A'] = stokesmatsc['A'] + 1./palpha*stokesmatsc['Arob']
b_mat = 1./palpha*stokesmatsc['Brob']
u_masmat = sps.eye(b_mat.shape[1], format='csr')
else:
contsetupstr = 'NV{0}NU{1}NY{2}'.format(NV, NU, NY)
def get_fdstr(Re):
return ddir + problemname + '_Re{0}_'.format(Re) + \
contsetupstr + prbstr
fdstr = get_fdstr(Re)
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
# soldict.update(rhsd_vfrc) # adding fvc, fpr
soldict.update(fv=rhsd_stbc['fv']+rhsd_vfrc['fvc'],
fp=rhsd_stbc['fp']+rhsd_vfrc['fpr'],
N=N, nu=nu, data_prfx=fdstr)
#
# Prepare for control
#
# get the control and observation operators
if not bccontrol:
try:
b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
print 'loaded `b_mat`'
except IOError:
print 'computing `b_mat`...'
b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'], V=femp['V'],
NU=NU, xcomp=uspacedep)
dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
b_mat = b_mat[invinds, :][:, :]
# tb_mat = 1./np.sqrt(alphau)
try:
mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
print 'loaded `c_mat`'
except IOError:
print 'computing `c_mat`...'
mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
V=femp['V'], NY=NY)
dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
# restrict the operators to the inner nodes
mc_mat = mc_mat[:, invinds][:, :]
c_mat = c_mat[:, invinds][:, :]
c_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=c_mat.T,
transposedprj=True).T
# c_mat_reg = np.array(c_mat.todense())
# TODO: right choice of norms for y
# and necessity of regularization here
# by now, we go on number save
#
# setup the system for the correction
#
#
# compute the uncontrolled steady state Stokes solution
#
v_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(**soldict)
(convc_mat, rhs_con,
rhsv_conbc) = snu.get_v_conv_conts(prev_v=v_ss_nse, invinds=invinds,
V=femp['V'], diribcs=femp['diribcs'])
f_mat = - stokesmatsc['A'] - convc_mat
mmat = stokesmatsc['M']
# ssv_rhs = rhsv_conbc + rhsv_conbc + rhsd_vfrc['fvc'] + rhsd_stbc['fv']
if plain_bt:
get_gramians = pru.solve_proj_lyap_stein
else:
get_gramians = pru.proj_alg_ric_newtonadi
truncstr = '__lqgbtcv{0}'.format(trunc_lqgbtcv)
try:
tl = dou.load_npa(fdstr + '__tl' + truncstr)
tr = dou.load_npa(fdstr + '__tr' + truncstr)
print 'loaded the left and right transformations: \n' + \
fdstr + '__tl/__tr' + truncstr
except IOError:
print 'computing the left and right transformations' + \
' and saving to: \n' + fdstr + '__tl/__tr' + truncstr
try:
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
print 'loaded factor of the Gramians: \n\t' + \
fdstr + '__zwc/__zwo'
except IOError:
zinic, zinio = None, None
if use_ric_ini is not None:
fdstr = get_fdstr(use_ric_ini)
try:
zinic = dou.load_npa(fdstr + '__zwc')
zinio = dou.load_npa(fdstr + '__zwo')
print 'Initialize Newton ADI by zwc/zwo from ' + fdstr
except IOError:
raise UserWarning('No initial guess with Re={0}'.
format(use_ric_ini))
fdstr = get_fdstr(Re)
print 'computing factors of Gramians: \n\t' + \
fdstr + '__zwc/__zwo'
def compobsg():
try:
zwo = dou.load_npa(fdstr + '__zwo')
print 'at least __zwo is there'
except IOError:
zwo = get_gramians(mmat=mmat.T, amat=f_mat.T,
jmat=stokesmatsc['J'],
bmat=c_mat_reg.T, wmat=b_mat,
nwtn_adi_dict=nap,
z0=zinio)['zfac']
dou.save_npa(zwo, fdstr + '__zwo')
def compcong():
try:
zwc = dou.load_npa(fdstr + '__zwc')
print 'at least __zwc is there'
except IOError:
zwc = get_gramians(mmat=mmat, amat=f_mat,
jmat=stokesmatsc['J'],
bmat=b_mat, wmat=c_mat_reg.T,
nwtn_adi_dict=nap,
z0=zinic)['zfac']
dou.save_npa(zwc, fdstr + '__zwc')
if multiproc:
print '\n ### multithread start - ' +\
'output might be intermangled'
p1 = multiprocessing.Process(target=compobsg)
p2 = multiprocessing.Process(target=compcong)
p1.start()
p2.start()
p1.join()
p2.join()
print '### multithread end'
else:
compobsg()
compcong()
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
print 'computing the left and right transformations' + \
' and saving to:\n' + fdstr + '__tr/__tl' + truncstr
tl, tr = btu.\
compute_lrbt_transfos(zfc=zwc, zfo=zwo,
mmat=stokesmatsc['M'],
trunck={'threshh': trunc_lqgbtcv})
dou.save_npa(tl, fdstr + '__tl' + truncstr)
dou.save_npa(tr, fdstr + '__tr' + truncstr)
print 'NV = {0}, NP = {2}, k = {1}'.\
format(tl.shape[0], tl.shape[1], stokesmatsc['J'].shape[0])
if comp_freqresp:
btu.compare_freqresp(mmat=stokesmatsc['M'], amat=f_mat,
jmat=stokesmatsc['J'], bmat=b_mat,
cmat=c_mat, tr=tr, tl=tl,
plot=True, datastr=fdstr + '__tl' + truncstr)
if comp_stepresp is not False:
if comp_stepresp == 'nonlinear':
stp_rsp_nwtn = 3
stp_rsp_dtpr = 'nonl_stepresp_'
else:
stp_rsp_nwtn = 1
stp_rsp_dtpr = 'stepresp_'
def fullstepresp_lnse(bcol=None, trange=None, ini_vel=None,
cmat=None, soldict=None):
soldict.update(fv_stbc=rhsd_stbc['fv']+bcol,
vel_nwtn_stps=stp_rsp_nwtn, trange=trange,
iniv=ini_vel, lin_vel_point=ini_vel,
clearprvdata=True, data_prfx=stp_rsp_dtpr,
return_dictofvelstrs=True)
dictofvelstrs = snu.solve_nse(**soldict)
return cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
c_mat=cmat, load_data=dou.load_npa)
# differences in the initial vector
# print np.dot(c_mat_reg, v_ss_nse)
# print np.dot(np.dot(c_mat_reg, tr),
# np.dot(tl.T, stokesmatsc['M']*v_ss_nse))
jsonstr = fdstr + stp_rsp_dtpr + '_Nred{0}_t0tENts{1}{2}{3}.json'.\
format(tl.shape[1], t0, tE, Nts)
btu.compare_stepresp(tmesh=np.linspace(t0, tE, Nts),
a_mat=f_mat, c_mat=c_mat_reg, b_mat=b_mat,
m_mat=stokesmatsc['M'],
tr=tr, tl=tl, iniv=v_ss_nse,
# ss_rhs=ssv_rhs,
fullresp=fullstepresp_lnse, fsr_soldict=soldict,
plot=True, jsonstr=jsonstr)
# compute the regulated system
trange = np.linspace(t0, tE, Nts)
if closed_loop is False:
return
elif closed_loop == 'full_state_fb':
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
mtxtb = pru.get_mTzzTtb(stokesmatsc['M'].T, zwc, b_mat)
def fv_tmdp_fullstatefb(time=None, curvel=None,
linv=None, tb_mat=None, tbxm_mat=None, **kw):
"""realizes a full state static feedback as a function
that can be passed to a solution routine for the
unsteady Navier-Stokes equations
Parameters
----------
time : real
current time
curvel : (N,1) nparray
current velocity
linv : (N,1) nparray
linearization point for the linear model
tb_mat : (N,K) nparray
input matrix containing the input weighting
tbxm_mat : (N,K) nparray
`tb_mat * gain * mass`
Returns
-------
actua : (N,1) nparray
current contribution to the right-hand side
, : dictionary
dummy `{}` for consistency
"""
actua = -lau.comp_uvz_spdns(tb_mat, tbxm_mat, curvel-linv)
# actua = 0*curvel
print '\nnorm of deviation', np.linalg.norm(curvel-linv)
# print 'norm of actuation {0}'.format(np.linalg.norm(actua))
return actua, {}
tmdp_fsfb_dict = dict(linv=v_ss_nse, tb_mat=b_mat, tbxm_mat=mtxtb.T)
fv_tmdp = fv_tmdp_fullstatefb
fv_tmdp_params = tmdp_fsfb_dict
fv_tmdp_memory = None
elif closed_loop == 'red_output_fb':
try:
xok = dou.load_npa(fdstr+truncstr+'__xok')
xck = dou.load_npa(fdstr+truncstr+'__xck')
ak_mat = dou.load_npa(fdstr+truncstr+'__ak_mat')
ck_mat = dou.load_npa(fdstr+truncstr+'__ck_mat')
bk_mat = dou.load_npa(fdstr+truncstr+'__bk_mat')
except IOError:
print 'couldn"t load the red system - compute it'
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
ak_mat = np.dot(tl.T, f_mat*tr)
ck_mat = lau.mm_dnssps(c_mat_reg, tr)
bk_mat = lau.mm_dnssps(tl.T, b_mat)
tltm, trtm = tl.T*stokesmatsc['M'], tr.T*stokesmatsc['M']
xok = np.dot(np.dot(tltm, zwo), np.dot(zwo.T, tltm.T))
xck = np.dot(np.dot(trtm, zwc), np.dot(zwc.T, trtm.T))
dou.save_npa(xok, fdstr+truncstr+'__xok')
dou.save_npa(xck, fdstr+truncstr+'__xck')
dou.save_npa(ak_mat, fdstr+truncstr+'__ak_mat')
dou.save_npa(ck_mat, fdstr+truncstr+'__ck_mat')
dou.save_npa(bk_mat, fdstr+truncstr+'__bk_mat')
obs_bk = np.dot(xok, ck_mat.T)
DT = (tE - t0)/(Nts-1)
sysmatk_inv = np.linalg.inv(np.eye(ak_mat.shape[1]) - DT*(ak_mat -
np.dot(np.dot(xok, ck_mat.T), ck_mat) -
np.dot(bk_mat, np.dot(bk_mat.T, xck))))
def fv_tmdp_redoutpfb(time=None, curvel=None, memory=None,
linvel=None,
ipsysk_mat_inv=None,
obs_bk=None, cts=None,
b_mat=None, c_mat=None,
xck=None, bk_mat=None,
**kw):
"""realizes a reduced static output feedback as a function
that can be passed to a solution routine for the
unsteady Navier-Stokes equations
For convinience the
Parameters
----------
time : real
current time
curvel : (N,1) nparray
current velocity. For consistency, the full state is taken
as input. However, internally, we only use the observation
`y = c_mat*curvel`
memory : dictionary
contains values from previous call, in particular the
previous state estimate
linvel : (N,1) nparray
linearization point for the linear model
ipsysk_mat_inv : (K,K) nparray
inverse of the system matrix that defines the update
of the state estimate
obs_bk : (K,NU) nparray
input matrix in the observer
cts : real
time step length
b_mat : (N,NU) sparse matrix
input matrix of the full system
c_mat=None,
c_mat : (NY,N) sparse matrix
output matrix of the full system
xck : (K,K) nparray
reduced solution of the CARE
bk_mat : (K,NU) nparray
reduced input matrix
Returns
-------
actua : (N,1) nparray
the current actuation
memory : dictionary
to be passed back in the next timestep
"""
xk_old = memory['xk_old']
buk = cts*np.dot(obs_bk,
lau.mm_dnssps(c_mat, (curvel-linvel)))
xk_old = np.dot(ipsysk_mat_inv, xk_old + buk)
# cts*np.dot(obs_bk,
# lau.mm_dnssps(c_mat, (curvel-linvel))))
memory['xk_old'] = xk_old
actua = -lau.mm_dnssps(b_mat,
np.dot(bk_mat.T, np.dot(xck, xk_old)))
print '\nnorm of deviation', np.linalg.norm(curvel-linvel)
print 'norm of actuation {0}'.format(np.linalg.norm(actua))
return actua, memory
fv_rofb_dict = dict(cts=DT, linvel=v_ss_nse, b_mat=b_mat,
c_mat=c_mat_reg, obs_bk=obs_bk, bk_mat=bk_mat,
ipsysk_mat_inv=sysmatk_inv, xck=xck)
fv_tmdp = fv_tmdp_redoutpfb
fv_tmdp_params = fv_rofb_dict
fv_tmdp_memory = dict(xk_old=np.zeros((tl.shape[1], 1)))
else:
fv_tmdp = None
fv_tmdp_params = {}
fv_tmdp_memory = {}
perturbini = perturbpara*np.ones((NV, 1))
reg_pertubini = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=perturbini)
soldict.update(fv_stbc=rhsd_stbc['fv'],
trange=trange,
iniv=v_ss_nse + reg_pertubini,
lin_vel_point=None,
clearprvdata=True, data_prfx=fdstr + truncstr,
fv_tmdp=fv_tmdp,
comp_nonl_semexp=True,
fv_tmdp_params=fv_tmdp_params,
fv_tmdp_memory=fv_tmdp_memory,
return_dictofvelstrs=True)
outstr = truncstr + '{0}'.format(closed_loop) \
+ 't0{0}tE{1}Nts{2}N{3}Re{4}'.format(t0, tE, Nts, N, Re)
if paraoutput:
soldict.update(paraviewoutput=True,
vfileprfx='results/vel_'+outstr,
pfileprfx='results/p_'+outstr)
dictofvelstrs = snu.solve_nse(**soldict)
yscomplist = cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
c_mat=c_mat, load_data=dou.load_npa)
dou.save_output_json(dict(tmesh=trange.tolist(), outsig=yscomplist),
fstring=fdstr + truncstr + '{0}'.format(closed_loop) +
't0{0}tE{1}Nts{2}'.format(t0, tE, Nts) +
'inipert{0}'.format(perturbpara))
dou.plot_outp_sig(tmesh=trange, outsig=yscomplist)
format(NN, Re, compall)
try:
if debug:
raise IOError()
levs = dou.load_npa(levstr)
print 'loaded the eigenvalues of the linearized system'
except IOError:
print 'computing the eigenvalues of the linearized system'
if compall:
A = asysmat.todense()
M = msysmat.todense() if not shiftall else mshiftsmat.todense()
levs = spla.eigvals(A, M, overwrite_a=True, check_finite=False)
else:
levs = spsla.eigs(asysmat, M=mshiftsmat, sigma=1, k=10, which='LR',
return_eigenvectors=False)
dou.save_npa(levs, levstr)
plt.figure(1)
# plt.xlim((-20, 15))
# plt.ylim((-100, 100))
plt.plot(np.real(levs), np.imag(levs), '+')
plt.show(block=False)
projsys = False
if projsys:
mfac = spsla.splu(M)
mijtl = []
for kcol in range(J.shape[0]):
curcol = np.array(J.T[:, kcol].todense())
micc = np.array(mfac.solve(curcol.flatten()))
mijtl.append(micc.reshape((NV, 1)))
def set_inival(whichinival='sstokes', soldict=None, perturbpara=None,
v_ss_nse=None, trange=None, tpp=None, fdstr=None,
retvssnse=False):
''' compute the wanted initial value and set it in the soldict
'''
if (retvssnse or whichinival == 'sstate+d') and v_ss_nse is None:
ret_v_ss_nse = snu.solve_steadystate_nse(**soldict)
elif v_ss_nse is not None:
ret_v_ss_nse = v_ss_nse
else:
ret_v_ss_nse is None
if whichinival == 'sstokes':
print('we start with Stokes -- `perturbpara` is not considered')
soldict.update(dict(iniv=None, start_ssstokes=True))
shortinivstr = 'sks'
return shortinivstr, ret_v_ss_nse
if (whichinival == 'sstate+d' or
whichinival == 'snse+d++' or whichinival == 'sstate+du'):
perturbini = perturbpara*np.ones((soldict['M'].shape[0], 1))
reg_pertubini = lau.app_prj_via_sadpnt(amat=soldict['M'],
jmat=soldict['J'],
rhsv=perturbini)
if whichinival == 'sstate+d':
soldict.update(dict(iniv=ret_v_ss_nse + reg_pertubini))
shortinivstr = 'ssd{0}'.format(perturbpara)
return shortinivstr, ret_v_ss_nse
if whichinival == 'sstokes++' or whichinival == 'snse+d++':
lctrng = (trange[trange < tpp]).tolist()
lctrng.append(tpp)
stksppdtstr = fdstr + 't0{0:.1f}tE{1:.4f}'.\
format(trange[0], tpp) + whichinival
try:
sstokspp = dou.load_npa(stksppdtstr)
print('loaded ' + stksppdtstr + ' for inival')
except IOError:
inivsoldict = {}
inivsoldict.update(soldict) # containing A, J, JT
inivsoldict['fv_tmdp'] = None # don't want control here
# import ipdb; ipdb.set_trace()
inivsoldict.update(trange=np.array(lctrng),
comp_nonl_semexp=True,
return_dictofvelstrs=True)
if whichinival == 'sstokes++':
print('solving for `stokespp({0})` as inival'.format(tpp))
inivsoldict.update(iniv=None, start_ssstokes=True)
else:
inivsoldict.update(iniv=ret_v_ss_nse+reg_pertubini)
print('solving for `nse+d+pp({0})` as inival'.format(tpp))
dcvlstrs = snu.solve_nse(**inivsoldict)
sstokspp = dou.load_npa(dcvlstrs[tpp])
dou.save_npa(sstokspp, stksppdtstr)
soldict.update(dict(iniv=sstokspp))
shortinivstr = 'sk{0}'.format(tpp) if whichinival == 'sstokes++' \
else 'nsk{0}'.format(tpp)
return shortinivstr, ret_v_ss_nse
def test_qbdae_ass(problemname='cylinderwake',
N=1,
Re=None,
nu=3e-2,
t0=0.0,
tE=1.0,
Nts=100,
use_saved_mats=None):
trange = np.linspace(t0, tE, Nts + 1)
DT = (tE - t0) / Nts
rdir = 'results/'
ddir = 'data/'
if use_saved_mats is None:
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc = \
dns.problem_setups.get_sysmats(problem=problemname, N=N, Re=Re)
invinds = femp['invinds']
A, J, M = stokesmatsc['A'], stokesmatsc['J'], stokesmatsc['M']
L = 0 * A
fvc, fpc = rhsd_vfrc['fvc'], rhsd_vfrc['fpr']
fv_stbc, fp_stbc = rhsd_stbc['fv'], rhsd_stbc['fp']
hstr = ddir + problemname + '_N{0}_hmat'.format(N)
try:
hmat = dou.load_spa(hstr)
print('loaded `hmat`')
except IOError:
print('assembling hmat ...')
hmat = dnsts.ass_convmat_asmatquad(W=femp['V'], invindsw=invinds)
dou.save_spa(hmat, hstr)
invinds = femp['invinds']
NV, NP = invinds.shape[0], J.shape[0]
zerv = np.zeros((NV, 1))
bc_conv, bc_rhs_conv, rhsbc_convbc = \
snu.get_v_conv_conts(prev_v=zerv, V=femp['V'], invinds=invinds,
diribcs=femp['diribcs'], Picard=False)
fp = fp_stbc + fpc
fv = fv_stbc + fvc - bc_rhs_conv
if linnsesol:
vp_nse, _ = snu.\
solve_steadystate_nse(A=A, J=J, JT=None, M=M,
fv=fv_stbc + fvc,
fp=fp_stbc + fpc,
V=femp['V'], Q=femp['Q'],
invinds=invinds,
diribcs=femp['diribcs'],
return_vp=False, ppin=-1,
N=N, nu=nu,
clearprvdata=False)
old_v = vp_nse[:NV]
else:
import sadptprj_riclyap_adi.lin_alg_utils as lau
# Stokes solution as initial value
vp_stokes = lau.solve_sadpnt_smw(amat=A,
jmat=J,
rhsv=fv_stbc + fvc,
rhsp=fp_stbc + fpc)
old_v = vp_stokes[:NV]
sysmat = sps.vstack([
sps.hstack([M + DT * (A + bc_conv), J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])
])
if use_saved_mats is not None:
# if saved as in ../get_exp_mats
import scipy.io
mats = scipy.io.loadmat(use_saved_mats)
A = -mats['A']
L = -mats['L']
Re = mats['Re']
N = A.shape[0]
M = mats['M']
J = mats['J']
hmat = -mats['H']
fv = mats['fv']
fp = mats['fp']
NV, NP = fv.shape[0], fp.shape[0]
old_v = mats['ss_stokes']
sysmat = sps.vstack([
sps.hstack([M + DT * A, J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])
])
if compevs:
import matplotlib.pyplot as plt
import scipy.linalg as spla
hlstr = ddir + problemname + '_N{0}_Re{1}Nse{2}_hlmat'.\
format(N, Re, linnsesol)
HL = linearzd_quadterm(hmat, old_v, hlstr=hlstr)
print(HL.shape)
asysmat = sps.vstack([
sps.hstack([-(A - L + HL), J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])
])
msysmat = sps.vstack([
sps.hstack([M, sps.csc_matrix((NV, NP))]),
sps.hstack([sps.csc_matrix((NP, NV)),
sps.csc_matrix((NP, NP))])
])
levstr = ddir + problemname + '_N{0}Re{1}Nse{2}_levs'.\
format(N, Re, linnsesol)
try:
levs = dou.load_npa(levstr)
if debug:
raise IOError()
print('loaded the eigenvalues of the linearized system')
except IOError:
print('computing the eigenvalues of the linearized system')
A = asysmat.todense()
M = msysmat.todense()
levs = spla.eigvals(A, M, overwrite_a=True, check_finite=False)
dou.save_npa(levs, levstr)
plt.figure(1)
# plt.xlim((-25, 15))
# plt.ylim((-50, 50))
plt.plot(np.real(levs), np.imag(levs), '+')
plt.show(block=False)
if timeint:
print('computing LU once...')
sysmati = spsla.factorized(sysmat)
vfile = dolfin.File(rdir + problemname + 'qdae__vel.pvd')
pfile = dolfin.File(rdir + problemname + 'qdae__p.pvd')
prvoutdict = dict(V=femp['V'],
Q=femp['Q'],
vfile=vfile,
pfile=pfile,
invinds=invinds,
diribcs=femp['diribcs'],
vp=None,
t=None,
writeoutput=True)
print('doing the time loop...')
for t in trange:
crhsv = M * old_v + DT * (fv - hmat * np.kron(old_v, old_v))
crhs = np.vstack([crhsv, fp])
vp_new = np.atleast_2d(sysmati(crhs.flatten())).T
prvoutdict.update(dict(vp=vp_new, t=t))
dou.output_paraview(**prvoutdict)
old_v = vp_new[:NV]
print(t, np.linalg.norm(old_v))
def test_qbdae_ass(problemname='cylinderwake', N=1, Re=None, nu=3e-2,
t0=0.0, tE=1.0, Nts=100, use_saved_mats=None):
trange = np.linspace(t0, tE, Nts+1)
DT = (tE-t0)/Nts
rdir = 'results/'
ddir = 'data/'
if use_saved_mats is None:
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc = \
dns.problem_setups.get_sysmats(problem=problemname, N=N, Re=Re)
invinds = femp['invinds']
A, J, M = stokesmatsc['A'], stokesmatsc['J'], stokesmatsc['M']
L = 0*A
fvc, fpc = rhsd_vfrc['fvc'], rhsd_vfrc['fpr']
fv_stbc, fp_stbc = rhsd_stbc['fv'], rhsd_stbc['fp']
hstr = ddir + problemname + '_N{0}_hmat'.format(N)
try:
hmat = dou.load_spa(hstr)
print('loaded `hmat`')
except IOError:
print('assembling hmat ...')
hmat = dnsts.ass_convmat_asmatquad(W=femp['V'], invindsw=invinds)
dou.save_spa(hmat, hstr)
invinds = femp['invinds']
NV, NP = invinds.shape[0], J.shape[0]
zerv = np.zeros((NV, 1))
bc_conv, bc_rhs_conv, rhsbc_convbc = \
snu.get_v_conv_conts(prev_v=zerv, V=femp['V'], invinds=invinds,
diribcs=femp['diribcs'], Picard=False)
fp = fp_stbc + fpc
fv = fv_stbc + fvc - bc_rhs_conv
if linnsesol:
vp_nse, _ = snu.\
solve_steadystate_nse(A=A, J=J, JT=None, M=M,
fv=fv_stbc + fvc,
fp=fp_stbc + fpc,
V=femp['V'], Q=femp['Q'],
invinds=invinds,
diribcs=femp['diribcs'],
return_vp=False, ppin=-1,
N=N, nu=nu,
clearprvdata=False)
old_v = vp_nse[:NV]
else:
import sadptprj_riclyap_adi.lin_alg_utils as lau
# Stokes solution as initial value
vp_stokes = lau.solve_sadpnt_smw(amat=A, jmat=J,
rhsv=fv_stbc + fvc,
rhsp=fp_stbc + fpc)
old_v = vp_stokes[:NV]
sysmat = sps.vstack([sps.hstack([M+DT*(A+bc_conv), J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])])
if use_saved_mats is not None:
# if saved as in ../get_exp_mats
import scipy.io
mats = scipy.io.loadmat(use_saved_mats)
A = - mats['A']
L = - mats['L']
Re = mats['Re']
N = A.shape[0]
M = mats['M']
J = mats['J']
hmat = -mats['H']
fv = mats['fv']
fp = mats['fp']
NV, NP = fv.shape[0], fp.shape[0]
old_v = mats['ss_stokes']
sysmat = sps.vstack([sps.hstack([M+DT*A, J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])])
if compevs:
import matplotlib.pyplot as plt
import scipy.linalg as spla
hlstr = ddir + problemname + '_N{0}_Re{1}Nse{2}_hlmat'.\
format(N, Re, linnsesol)
HL = linearzd_quadterm(hmat, old_v, hlstr=hlstr)
print(HL.shape)
asysmat = sps.vstack([sps.hstack([-(A-L+HL), J.T]),
sps.hstack([J, sps.csc_matrix((NP, NP))])])
msysmat = sps.vstack([sps.hstack([M, sps.csc_matrix((NV, NP))]),
sps.hstack([sps.csc_matrix((NP, NV)),
sps.csc_matrix((NP, NP))])])
levstr = ddir + problemname + '_N{0}Re{1}Nse{2}_levs'.\
format(N, Re, linnsesol)
try:
levs = dou.load_npa(levstr)
if debug:
raise IOError()
print('loaded the eigenvalues of the linearized system')
except IOError:
print('computing the eigenvalues of the linearized system')
A = asysmat.todense()
M = msysmat.todense()
levs = spla.eigvals(A, M, overwrite_a=True, check_finite=False)
dou.save_npa(levs, levstr)
plt.figure(1)
# plt.xlim((-25, 15))
# plt.ylim((-50, 50))
plt.plot(np.real(levs), np.imag(levs), '+')
plt.show(block=False)
if timeint:
print('computing LU once...')
sysmati = spsla.factorized(sysmat)
vfile = dolfin.File(rdir + problemname + 'qdae__vel.pvd')
pfile = dolfin.File(rdir + problemname + 'qdae__p.pvd')
prvoutdict = dict(V=femp['V'], Q=femp['Q'], vfile=vfile, pfile=pfile,
invinds=invinds, diribcs=femp['diribcs'],
vp=None, t=None, writeoutput=True)
print('doing the time loop...')
for t in trange:
crhsv = M*old_v + DT*(fv - hmat*np.kron(old_v, old_v))
crhs = np.vstack([crhsv, fp])
vp_new = np.atleast_2d(sysmati(crhs.flatten())).T
prvoutdict.update(dict(vp=vp_new, t=t))
dou.output_paraview(**prvoutdict)
old_v = vp_new[:NV]
print(t, np.linalg.norm(old_v))
def solve_steadystate_nse(A=None,
J=None,
JT=None,
M=None,
fv=None,
fp=None,
V=None,
Q=None,
invinds=None,
diribcs=None,
return_vp=False,
ppin=-1,
N=None,
nu=None,
vel_pcrd_stps=10,
vel_pcrd_tol=1e-4,
vel_nwtn_stps=20,
vel_nwtn_tol=5e-15,
clearprvdata=False,
vel_start_nwtn=None,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
save_intermediate_steps=False,
vfileprfx='',
pfileprfx='',
**kw):
"""
Solution of the steady state nonlinear NSE Problem
using Newton's scheme. If no starting value is provided, the iteration
is started with the steady state Stokes solution.
Parameters
----------
A : (N,N) sparse matrix
stiffness matrix aka discrete Laplacian, note the sign!
M : (N,N) sparse matrix
mass matrix
J : (M,N) sparse matrix
discrete divergence operator
JT : (N,M) sparse matrix, optional
discrete gradient operator, set to J.T if not provided
fv, fp : (N,1), (M,1) ndarrays
right hand sides restricted via removing the boundary nodes in the
momentum and the pressure freedom in the continuity equation
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
return_vp : boolean, optional
whether to return also the pressure, defaults to `False`
vel_pcrd_stps : int, optional
Number of Picard iterations when computing a starting value for the
Newton scheme, cf. Elman, Silvester, Wathen: *FEM and fast iterative
solvers*, 2005, defaults to `100`
vel_pcrd_tol : real, optional
tolerance for the size of the Picard update, defaults to `1e-4`
vel_nwtn_stps : int, optional
Number of Newton iterations, defaults to `20`
vel_nwtn_tol : real, optional
tolerance for the size of the Newton update, defaults to `5e-15`
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if get_datastring is None:
get_datastring = get_datastr_snu
if JT is None:
JT = J.T
NV = J.shape[1]
#
# Compute or load the uncontrolled steady state Navier-Stokes solution
#
norm_nwtnupd_list = []
vel_newtk, norm_nwtnupd = 0, 1
# a dict to be passed to the get_datastring function
datastrdict = dict(time=None,
meshp=N,
nu=nu,
Nts=None,
data_prfx=data_prfx)
if clearprvdata:
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '*__vel*'):
os.remove(fname)
try:
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = dou.load_npa(cdatstr + '__norm_nwtnupd')
vel_k = dou.load_npa(cdatstr + '__vel')
norm_nwtnupd_list.append(norm_nwtnupd)
print('found vel files')
print('norm of last Nwtn update: {0}'.format(norm_nwtnupd))
print('... loaded from ' + cdatstr)
if np.atleast_1d(norm_nwtnupd)[0] is None:
norm_nwtnupd = None
pass # nothing useful found
elif norm_nwtnupd < vel_nwtn_tol:
if not return_vp:
return vel_k, norm_nwtnupd_list
else:
pfv = get_pfromv(v=vel_k[:NV, :],
V=V,
M=M,
A=A,
J=J,
fv=fv,
invinds=invinds,
diribcs=diribcs)
return (np.vstack([vel_k, pfv]), norm_nwtnupd_list)
except IOError:
print('no old velocity data found')
norm_nwtnupd = None
if paraviewoutput:
cdatstr = get_datastring(**datastrdict)
vfile = dolfin.File(vfileprfx + '__steadystates.pvd')
pfile = dolfin.File(pfileprfx + '__steadystates.pvd')
prvoutdict = dict(V=V,
Q=Q,
vfile=vfile,
pfile=pfile,
invinds=invinds,
diribcs=diribcs,
ppin=ppin,
vp=None,
t=None,
writeoutput=True)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' here
NV = A.shape[0]
if vel_start_nwtn is None:
vp_stokes = lau.solve_sadpnt_smw(amat=A,
jmat=J,
jmatT=JT,
rhsv=fv,
rhsp=fp)
vp_stokes[NV:] = -vp_stokes[NV:]
# pressure was flipped for symmetry
# save the data
cdatstr = get_datastring(**datastrdict)
dou.save_npa(vp_stokes[:NV, ], fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_stokes))
dou.output_paraview(**prvoutdict)
# Stokes solution as starting value
vp_k = vp_stokes
vel_k = vp_stokes[:NV, ]
else:
vel_k = vel_start_nwtn
# Picard iterations for a good starting value for Newton
for k in range(vel_pcrd_stps):
(convc_mat, rhs_con, rhsv_conbc) = get_v_conv_conts(prev_v=vel_k,
invinds=invinds,
V=V,
diribcs=diribcs,
Picard=True)
vp_k = lau.solve_sadpnt_smw(amat=A + convc_mat,
jmat=J,
jmatT=JT,
rhsv=fv + rhs_con + rhsv_conbc,
rhsp=fp)
normpicupd = np.sqrt(m_innerproduct(M, vel_k - vp_k[:NV, :]))[0]
print('Picard iteration: {0} -- norm of update: {1}'.format(
k + 1, normpicupd))
vel_k = vp_k[:NV, ]
vp_k[NV:] = -vp_k[NV:]
# pressure was flipped for symmetry
if normpicupd < vel_pcrd_tol:
break
# Newton iteration
while (vel_newtk < vel_nwtn_stps
and (norm_nwtnupd is None or norm_nwtnupd > vel_nwtn_tol
or norm_nwtnupd == np.array(None))):
vel_newtk += 1
cdatstr = get_datastring(**datastrdict)
(convc_mat, rhs_con, rhsv_conbc) = get_v_conv_conts(vel_k,
invinds=invinds,
V=V,
diribcs=diribcs)
vp_k = lau.solve_sadpnt_smw(amat=A + convc_mat,
jmat=J,
jmatT=JT,
rhsv=fv + rhs_con + rhsv_conbc,
rhsp=fp)
norm_nwtnupd = np.sqrt(m_innerproduct(M, vel_k - vp_k[:NV, :]))[0]
vel_k = vp_k[:NV, ]
vp_k[NV:] = -vp_k[NV:]
# pressure was flipped for symmetry
print('Steady State NSE: Newton iteration: {0} -- norm of update: {1}'.
format(vel_newtk, norm_nwtnupd))
dou.save_npa(vel_k, fstring=cdatstr + '__vel')
prvoutdict.update(dict(vp=vp_k))
dou.output_paraview(**prvoutdict)
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
dou.output_paraview(**prvoutdict)
# savetomatlab = True
# if savetomatlab:
# export_mats_to_matlab(E=None, A=None, matfname='matexport')
if return_vp:
return vp_k, norm_nwtnupd_list
else:
return vel_k, norm_nwtnupd_list
def solve_nse(
A=None,
M=None,
J=None,
JT=None,
fv=None,
fp=None,
fvc=None,
fpc=None, # TODO: this is to catch deprecated calls
fv_tmdp=None,
fv_tmdp_params={},
fv_tmdp_memory=None,
iniv=None,
lin_vel_point=None,
stokes_flow=False,
trange=None,
t0=None,
tE=None,
Nts=None,
V=None,
Q=None,
invinds=None,
diribcs=None,
output_includes_bcs=False,
N=None,
nu=None,
ppin=-1,
closed_loop=False,
static_feedback=False,
feedbackthroughdict=None,
return_vp=False,
tb_mat=None,
c_mat=None,
vel_nwtn_stps=20,
vel_nwtn_tol=5e-15,
nsects=1,
loc_nwtn_tol=5e-15,
loc_pcrd_stps=True,
addfullsweep=False,
vel_pcrd_stps=4,
krylov=None,
krpslvprms={},
krplsprms={},
clearprvdata=False,
get_datastring=None,
data_prfx='',
paraviewoutput=False,
vfileprfx='',
pfileprfx='',
return_dictofvelstrs=False,
return_dictofpstrs=False,
dictkeysstr=False,
comp_nonl_semexp=False,
return_as_list=False,
verbose=True,
start_ssstokes=False,
**kw):
"""
solution of the time-dependent nonlinear Navier-Stokes equation
.. math::
M\\dot v + Av + N(v)v + J^Tp = f \n
Jv =g
using a Newton scheme in function space, i.e. given :math:`v_k`,
we solve for the update like
.. math::
M\\dot v + Av + N(v_k)v + N(v)v_k + J^Tp = N(v_k)v_k + f,
and trapezoidal rule in time. To solve an *Oseen* system (linearization
about a steady state) or a *Stokes* system, set the number of Newton
steps to one and provide a linearization point and an initial value.
Parameters
----------
lin_vel_point : dictionary, optional
contains the linearization point for the first Newton iteration
* Steady State: {{`None`: 'path_to_nparray'}, {'None': nparray}}
* Newton: {`t`: 'path_to_nparray'}
defaults to `None`
dictkeysstr : boolean, optional
whether the `keys` of the result dictionaries are strings instead \
of floats, defaults to `False`
fv_tmdp : callable f(t, v, dict), optional
time-dependent part of the right-hand side, set to zero if None
fv_tmdp_params : dictionary, optional
dictionary of parameters to be passed to `fv_tmdp`, defaults to `{}`
fv_tmdp_memory : dictionary, optional
memory of the function
output_includes_bcs : boolean, optional
whether append the boundary nodes to the computed and stored \
velocities, defaults to `False`
krylov : {None, 'gmres'}, optional
whether or not to use an iterative solver, defaults to `None`
krpslvprms : dictionary, optional
to specify parameters of the linear solver for use in Krypy, e.g.,
* initial guess
* tolerance
* number of iterations
defaults to `None`
krplsprms : dictionary, optional
parameters to define the linear system like
* preconditioner
ppin : {int, None}, optional
which dof of `p` is used to pin the pressure, defaults to `-1`
stokes_flow : boolean, optional
whether to consider the Stokes linearization, defaults to `False`
start_ssstokes : boolean, optional
for your convenience, compute and use the steady state stokes solution
as initial value, defaults to `False`
nsects: int, optional
in how many segments the trange is split up. (The newton iteration
will be confined to the segments and, probably, converge faster than
the global iteration), defaults to `1`
loc_nwtn_tol: float, optional
tolerance for the newton iteration on the segments,
defaults to `1e-15`
loc_pcrd_stps: boolean, optional
whether to init with `vel_pcrd_stps` Picard steps on every section,
if `False`, Picard iterations are performed only on the first section,
defaults to `True`
addfullsweep: boolean, optional
whether to compute the newton iteration on the full `trange`,
useful to check and for the plots, defaults to `False`
Returns
-------
dictofvelstrs : dictionary, on demand
dictionary with time `t` as keys and path to velocity files as values
dictofpstrs : dictionary, on demand
dictionary with time `t` as keys and path to pressure files as values
vellist : list, on demand
list of the velocity solutions
"""
import sadptprj_riclyap_adi.lin_alg_utils as lau
if fvc is not None or fpc is not None: # TODO: this is for catching calls
raise UserWarning('deprecated use of `rhsd_vfrc`, use only `fv`, `fp`')
if get_datastring is None:
get_datastring = get_datastr_snu
if paraviewoutput:
prvoutdict = dict(V=V,
Q=Q,
invinds=invinds,
diribcs=diribcs,
vp=None,
t=None,
writeoutput=True,
ppin=ppin)
else:
prvoutdict = dict(writeoutput=False) # save 'if statements' here
if trange is None:
trange = np.linspace(t0, tE, Nts + 1)
if comp_nonl_semexp and lin_vel_point is not None:
raise UserWarning('I am not sure what you want! ' +
'set either `lin_vel_point=None` ' +
'or `comp_nonl_semexp=False`! \n' +
'as it is, I will compute a linear case')
if return_dictofpstrs:
gpfvd = dict(V=V,
M=M,
A=A,
J=J,
fv=fv,
fp=fp,
diribcs=diribcs,
invinds=invinds)
NV, NP = A.shape[0], J.shape[0]
if fv_tmdp is None:
def fv_tmdp(time=None, curvel=None, **kw):
return np.zeros((NV, 1)), None
if iniv is None:
if start_ssstokes:
# Stokes solution as starting value
(fv_tmdp_cont, fv_tmdp_memory) = fv_tmdp(time=0, **fv_tmdp_params)
vp_stokes =\
lau.solve_sadpnt_smw(amat=A, jmat=J, jmatT=JT,
rhsv=fv + fv_tmdp_cont,
krylov=krylov, krpslvprms=krpslvprms,
krplsprms=krplsprms, rhsp=fp)
iniv = vp_stokes[:NV]
else:
raise ValueError('No initial value given')
datastrdict = dict(time=None,
meshp=N,
nu=nu,
Nts=trange.size - 1,
data_prfx=data_prfx)
if return_as_list:
clearprvdata = True # we want the results at hand
if clearprvdata:
datastrdict['time'] = '*'
cdatstr = get_datastring(**datastrdict)
for fname in glob.glob(cdatstr + '__vel*'):
os.remove(fname)
for fname in glob.glob(cdatstr + '__p*'):
os.remove(fname)
def _atdct(cdict, t, thing):
if dictkeysstr:
cdict.update({'{0}'.format(t): thing})
else:
cdict.update({t: thing})
def _gfdct(cdict, t):
if dictkeysstr:
return cdict['{0}'.format(t)]
else:
return cdict[t]
if stokes_flow:
vel_nwtn_stps = 1
vel_pcrd_stps = 0
print('Stokes Flow!')
elif lin_vel_point is None:
comp_nonl_semexp_inig = True
if not comp_nonl_semexp:
print(('No linearization point given - explicit' +
' treatment of the nonlinearity in the first Iteration'))
else:
cur_linvel_point = lin_vel_point
comp_nonl_semexp_inig = False
newtk, norm_nwtnupd = 0, 1
# check for previously computed velocities
if lin_vel_point is None and not stokes_flow:
try:
datastrdict.update(dict(time=trange[-1]))
cdatstr = get_datastring(**datastrdict)
norm_nwtnupd = (dou.load_npa(cdatstr + '__norm_nwtnupd')).flatten()
try:
if norm_nwtnupd[0] is None:
norm_nwtnupd = 1.
except IndexError:
norm_nwtnupd = 1.
dou.load_npa(cdatstr + '__vel')
print('found vel files')
print('norm of last Nwtn update: {0}'.format(norm_nwtnupd))
print('... loaded from ' + cdatstr)
if norm_nwtnupd < vel_nwtn_tol and not return_dictofvelstrs:
return
elif norm_nwtnupd < vel_nwtn_tol:
# looks like converged -- check if all values are there
# t0:
datastrdict.update(dict(time=trange[0]))
cdatstr = get_datastring(**datastrdict)
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
dictofpstrs = {}
# try:
# p_old = dou.load_npa(cdatstr + '__p')
# dictofpstrs = {}
# _atdct(dictofpstrs, trange[0], cdatstr+'__p')
# except:
# p_old = get_pfromv(v=v_old, **gpfvd)
# dou.save_npa(p_old, fstring=cdatstr + '__p')
# _atdct(dictofpstrs, trange[0], cdatstr+'__p')
for t in trange:
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
# test if the vels are there
v_old = dou.load_npa(cdatstr + '__vel')
# update the dict
_atdct(dictofvelstrs, t, cdatstr + '__vel')
try:
p_old = dou.load_npa(cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
except:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
comp_nonl_semexp = False
except IOError:
norm_nwtnupd = 2
print('no old velocity data found')
def _append_bcs_ornot(vvec):
if output_includes_bcs: # make the switch here for better readibility
vwbcs = dts.append_bcs_vec(vvec,
vdim=V.dim(),
invinds=invinds,
diribcs=diribcs)
return vwbcs
else:
return vvec
def _get_mats_rhs_ts(mmat=None,
dt=None,
var_c=None,
coeffmat_c=None,
coeffmat_n=None,
fv_c=None,
fv_n=None,
umat_c=None,
vmat_c=None,
umat_n=None,
vmat_n=None,
impeul=False):
""" to be tweaked for different int schemes
"""
solvmat = M + 0.5 * dt * coeffmat_n
rhs = M * var_c + 0.5 * dt * (fv_n + fv_c - coeffmat_c * var_c)
if umat_n is not None:
matvec = lau.mm_dnssps
umat = 0.5 * dt * umat_n
vmat = vmat_n
# TODO: do we really need a PLUS here??'
rhs = rhs + 0.5 * dt * matvec(umat_c, matvec(vmat_c, var_c))
else:
umat, vmat = umat_n, vmat_n
return solvmat, rhs, umat, vmat
v_old = iniv # start vector for time integration in every Newtonit
datastrdict['time'] = trange[0]
cdatstr = get_datastring(**datastrdict)
dou.save_npa(_append_bcs_ornot(v_old), fstring=cdatstr + '__vel')
dictofvelstrs = {}
_atdct(dictofvelstrs, trange[0], cdatstr + '__vel')
if return_dictofpstrs:
p_old = get_pfromv(v=v_old, **gpfvd)
dou.save_npa(p_old, fstring=cdatstr + '__p')
dictofpstrs = {}
_atdct(dictofpstrs, trange[0], cdatstr + '__p')
else:
p_old = None
if return_as_list:
vellist = []
vellist.append(_append_bcs_ornot(v_old))
lensect = np.int(np.floor(trange.size / nsects))
loctrngs = []
for k in np.arange(nsects - 1):
loctrngs.append(trange[k * lensect:(k + 1) * lensect + 1])
loctrngs.append(trange[(nsects - 1) * lensect:])
if addfullsweep:
loctrngs.append(trange)
realiniv = np.copy(iniv)
if nsects == 1:
loc_nwtn_tol = vel_nwtn_tol
addfullsweep = False
loctrngs = [trange]
if loc_pcrd_stps:
vel_loc_pcrd_steps = vel_pcrd_stps
for loctrng in loctrngs:
while (newtk < vel_nwtn_stps and norm_nwtnupd > loc_nwtn_tol):
print('solve the NSE on the interval [{0}, {1}]'.format(
loctrng[0], loctrng[-1]))
if stokes_flow:
newtk = vel_nwtn_stps
elif comp_nonl_semexp_inig and not comp_nonl_semexp:
pcrd_anyone = False
print('explicit treatment of nonl. for initial guess')
elif vel_pcrd_stps > 0 and not comp_nonl_semexp:
vel_pcrd_stps -= 1
pcrd_anyone = True
print('Picard iterations for initial value -- {0} left'.format(
vel_pcrd_stps))
elif comp_nonl_semexp:
pcrd_anyone = False
newtk = vel_nwtn_stps
print('No Newton iterations - explicit treatment' +
'of the nonlinerity')
else:
pcrd_anyone = False
newtk += 1
print('Computing Newton Iteration {0}'.format(newtk))
v_old = iniv # start vector for time integration in every Newtonit
try:
if krpslvprms['krylovini'] == 'old':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
elif krpslvprms['krylovini'] == 'upd':
vp_old = np.vstack([v_old, np.zeros((NP, 1))])
vp_new = vp_old
cts_old = loctrng[1] - loctrng[0]
except (TypeError, KeyError):
pass # no inival for krylov solver required
vfile = dolfin.File(vfileprfx + '__timestep.pvd')
pfile = dolfin.File(pfileprfx + '__timestep.pvd')
prvoutdict.update(
dict(vp=None,
vc=iniv,
pc=p_old,
t=loctrng[0],
pfile=pfile,
vfile=vfile))
dou.output_paraview(**prvoutdict)
# ## current values_c for application of trap rule
if stokes_flow:
convc_mat_c = sps.csr_matrix((NV, NV))
rhs_con_c, rhsv_conbc_c = np.zeros((NV, 1)), np.zeros((NV, 1))
else:
if comp_nonl_semexp or comp_nonl_semexp_inig:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(
_gfdct(cur_linvel_point, loctrng[0]))
except KeyError:
try:
prev_v = dou.load_npa(
_gfdct(cur_linvel_point, None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_c, rhs_con_c, rhsv_conbc_c = \
get_v_conv_conts(prev_v=iniv, invinds=invinds,
V=V, diribcs=diribcs, Picard=pcrd_anyone)
(fv_tmdp_cont, fv_tmdp_memory) = fv_tmdp(time=0,
curvel=v_old,
memory=fv_tmdp_memory,
**fv_tmdp_params)
fvn_c = fv + rhsv_conbc_c + rhs_con_c + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_c = dou.load_npa(feedbackthroughdict[None]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[None]['w'])
# print '\nnorm of feedback: ', np.linalg.norm(mtxtb_c)
else:
mtxtb_c = dou.load_npa(feedbackthroughdict[0]['mtxtb'])
w_c = dou.load_npa(feedbackthroughdict[0]['w'])
fvn_c = fvn_c + tb_mat * (tb_mat.T * w_c)
vmat_c = mtxtb_c.T
try:
umat_c = np.array(tb_mat.todense())
except AttributeError:
umat_c = tb_mat
else:
vmat_c = None
umat_c = None
norm_nwtnupd = 0
if verbose:
print('time to go')
for tk, t in enumerate(loctrng[1:]):
cts = t - loctrng[tk]
datastrdict.update(dict(time=t))
cdatstr = get_datastring(**datastrdict)
if verbose:
sys.stdout.write("\rEnd: {1} -- now: {0:f}".format(
t, loctrng[-1]))
sys.stdout.flush()
# prv_datastrdict = copy.deepcopy(datastrdict)
# coeffs and rhs at next time instance
if stokes_flow:
convc_mat_n = sps.csr_matrix((NV, NV))
rhs_con_n = np.zeros((NV, 1))
rhsv_conbc_n = np.zeros((NV, 1))
else:
if comp_nonl_semexp or comp_nonl_semexp_inig:
prev_v = v_old
else:
try:
prev_v = dou.load_npa(_gfdct(cur_linvel_point, t))
except KeyError:
try:
prev_v = dou.load_npa(
_gfdct(cur_linvel_point, None))
except TypeError:
prev_v = cur_linvel_point[None]
convc_mat_n, rhs_con_n, rhsv_conbc_n = \
get_v_conv_conts(prev_v=prev_v, invinds=invinds, V=V,
diribcs=diribcs, Picard=pcrd_anyone)
(fv_tmdp_cont, fv_tmdp_memory) = fv_tmdp(time=t,
curvel=v_old,
memory=fv_tmdp_memory,
**fv_tmdp_params)
fvn_n = fv + rhsv_conbc_n + rhs_con_n + fv_tmdp_cont
if closed_loop:
if static_feedback:
mtxtb_n = dou.\
load_npa(feedbackthroughdict[None]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[None]['w'])
# fb = np.dot(tb_mat*mtxtb_n.T, v_old)
# print '\nnorm of feedback: ', np.linalg.norm(fb)
# print '\nnorm of v_old: ', np.linalg.norm(v_old)
else:
mtxtb_n = dou.load_npa(feedbackthroughdict[t]['mtxtb'])
w_n = dou.load_npa(feedbackthroughdict[t]['w'])
fvn_n = fvn_n + tb_mat * (tb_mat.T * w_n)
vmat_n = mtxtb_n.T
try:
umat_n = np.array(tb_mat.todense())
except AttributeError:
umat_n = tb_mat
else:
vmat_n = None
umat_n = None
(solvmat, rhsv, umat,
vmat) = _get_mats_rhs_ts(mmat=M,
dt=cts,
var_c=v_old,
coeffmat_c=A + convc_mat_c,
coeffmat_n=A + convc_mat_n,
fv_c=fvn_c,
fv_n=fvn_n,
umat_c=umat_c,
vmat_c=vmat_c,
umat_n=umat_n,
vmat_n=vmat_n)
try:
if krpslvprms['krylovini'] == 'old':
krpslvprms['x0'] = vp_old
elif krpslvprms['krylovini'] == 'upd':
vp_oldold = vp_old
vp_old = vp_new
krpslvprms['x0'] = vp_old + \
cts*(vp_old - vp_oldold)/cts_old
cts_old = cts
except (TypeError, KeyError):
pass # no inival for krylov solver required
vp_new = lau.solve_sadpnt_smw(amat=solvmat,
jmat=J,
jmatT=JT,
rhsv=rhsv,
rhsp=fp,
krylov=krylov,
krpslvprms=krpslvprms,
krplsprms=krplsprms,
umat=umat,
vmat=vmat)
v_old = vp_new[:NV, ]
(umat_c, vmat_c, fvn_c,
convc_mat_c) = umat_n, vmat_n, fvn_n, convc_mat_n
dou.save_npa(_append_bcs_ornot(v_old),
fstring=cdatstr + '__vel')
_atdct(dictofvelstrs, t, cdatstr + '__vel')
p_new = -1 / cts * vp_new[NV:, ]
# p was flipped and scaled for symmetry
if return_dictofpstrs:
dou.save_npa(p_new, fstring=cdatstr + '__p')
_atdct(dictofpstrs, t, cdatstr + '__p')
if return_as_list:
vellist.append(_append_bcs_ornot(v_old))
prvoutdict.update(dict(vc=v_old, pc=p_new, t=t))
dou.output_paraview(**prvoutdict)
# integrate the Newton error
if stokes_flow or comp_nonl_semexp:
norm_nwtnupd = None
elif comp_nonl_semexp_inig:
norm_nwtnupd = 1.
else:
if len(prev_v) > len(invinds):
prev_v = prev_v[invinds, :]
addtonwtnupd = cts * m_innerproduct(M, v_old - prev_v)
norm_nwtnupd += np.float(addtonwtnupd.flatten()[0])
dou.save_npa(norm_nwtnupd, cdatstr + '__norm_nwtnupd')
print('\nnorm of current Newton update: {}'.format(norm_nwtnupd))
# print('\nsaved `norm_nwtnupd(={0})'.format(norm_nwtnupd) +
# ' to ' + cdatstr)
comp_nonl_semexp = False
comp_nonl_semexp_inig = False
cur_linvel_point = dictofvelstrs
iniv = v_old
if not comp_nonl_semexp:
comp_nonl_semexp_inig = True
if addfullsweep and loctrng is loctrngs[-2]:
comp_nonl_semexp_inig = False
iniv = realiniv
loc_nwtn_tol = vel_nwtn_tol
elif loc_pcrd_stps:
vel_pcrd_stps = vel_loc_pcrd_steps
norm_nwtnupd = 1.
newtk = 0
if return_dictofvelstrs:
if return_dictofpstrs:
return dictofvelstrs, dictofpstrs
else:
return dictofvelstrs
elif return_as_list:
return vellist
else:
return
def solve_flow_daeric(mmat=None, amat=None, jmat=None, bmat=None,
cmat=None, rhsv=None, rhsp=None,
mcmat=None, v_is_my=False,
rmat=None, vmat=None,
gamma=1.0,
tmesh=None, ystarvec=None,
nwtn_adi_dict=None,
curnwtnsdict=None,
comprz_thresh=None, comprz_maxc=None, save_full_z=False,
get_tdpart=None, gttdprtargs=None,
get_datastr=None, gtdtstrargs=None,
check_c_consist=True):
"""
Routine for the solution of the DAE Riccati
.. math::
\\dot{MXM^T} + F^TXM + M^TXM + M^TXGXM + L(Y) = W \\\\
JXM = 0 \\quad \\text{and} \\quad M^TXJ = 0 \\\\
M^TX(T)M = W
where :math:`F=A+N(t)`,
where :math:`W:=C^T V C`, :math:`G:=B R^{-1} B^T`,
and where :math:`L(Y)` is the Lagrange multiplier term.
Simultaneously we solve for the feedforward term :math:`w`:
.. math::
\\dot{M^Tw} - [M^TXG+F^T]w - J^Tv = C^T V y^* + M^T[Xf + Yg] \\\\
Jw = 0 \\\\
M^Tw(T) = C^T V y^*(T)
Note that :math:`V=M_y` if the norm of :math:`Y` is used
in the cost function.
Parameters
----------
cmat : (NY, NV) array
the (regularized aka projected) output matrix
mcmat : (NY, NV) array
output matrix times the mass matrix in the output space
gamma : float, optional
weighting parameter for penalization of the terminal value,
TODO: rather provide the right weighting matrix V,
defaults to `1.0`
v_is_my : boolean
whether the weighting matrix is the same as the mass matrix, \
defaults to `False`
get_tdpart : callable f(t)
returns the `mattd, rhstd` -- time dependent coefficients matrices
and right hand side at time `t`
gtdtstrargs : dictionary
**kwargs to the current data string
gttdprtargs : dictionary
`**kwargs` to get_tdpart
Returns
-------
feedbackthroughdict : dictionary
with time instances as keys and
| `w` -- the current feedthrough value
| `mtxtb` -- the current feedback gain part `(R.-1/2 * B.T * X * M).T`
as values
"""
if check_c_consist:
if v_is_my and mcmat is not None:
mic = lau.apply_massinv(mmat.T, mcmat.T)
if np.linalg.norm(jmat*mic) > 1e-12:
raise Warning('mcmat.T needs to be in the kernel of J*M.-1')
elif cmat is not None:
mic = lau.apply_massinv(mmat.T, cmat.T)
if np.linalg.norm(jmat*mic) > 1e-12:
raise Warning('cmat.T needs to be in the kernel of J*M.-1')
MT, AT, NV = mmat.T, amat.T, amat.shape[0]
gtdtstrargs.update(time=tmesh[-1])
cdatstr = get_datastr(**gtdtstrargs)
# set/compute the terminal values aka starting point
if v_is_my and mcmat is not None:
tct_mat = lau.apply_invsqrt_fromright(vmat, mcmat.T, output='dense')
else:
tct_mat = lau.apply_sqrt_fromright(vmat, cmat.T, output='dense')
# TODO: good handling of bmat and umasmat
tb_mat = lau.apply_invsqrt_fromright(rmat, bmat, output='sparse')
# bmat_rpmo = bmat * np.linalg.inv(np.array(rmat.todense()))
Zc = np.sqrt(gamma)*lau.apply_massinv(mmat, tct_mat)
mtxtb = -pru.get_mTzzTtb(mmat.T, Zc, tb_mat)
# mtxbrm = pru.get_mTzzTtb(mmat.T, Zc, bmat_rpmo)
dou.save_npa(Zc, fstring=cdatstr + '__Z')
dou.save_npa(mtxtb, fstring=cdatstr + '__mtxtb')
if ystarvec is not None:
wc = lau.apply_massinv(MT, gamma*np.dot(mcmat.T, ystarvec(tmesh[-1])))
dou.save_npa(wc, fstring=cdatstr + '__w')
else:
wc = None
feedbackthroughdict = {tmesh[-1]: dict(w=cdatstr + '__w',
mtxtb=cdatstr + '__mtxtb')}
# save the end values
if curnwtnsdict is not None:
dou.save_npa(wc, fstring=curnwtnsdict[tmesh[-1]]['w'])
dou.save_npa(mtxtb, fstring=curnwtnsdict[tmesh[-1]]['mtxtb'])
# time integration
for tk, t in reversed(list(enumerate(tmesh[:-1]))):
cts = tmesh[tk+1] - t
print 'Time is {0}, timestep is {1}'.\
format(t, cts)
# get the previous time time-dep matrices
gtdtstrargs.update(time=t)
cdatstr = get_datastr(**gtdtstrargs)
nmattd, rhsvtd = get_tdpart(time=t, **gttdprtargs)
# get the feedback from the current newton step
if curnwtnsdict is not None:
try:
cnsw = dou.load_npa(curnwtnsdict[t]['w'])
cnsmtxtb = dou.load_npa(curnwtnsdict[t]['mtxtb'])
except IOError:
cnsw, cnsmtxtb = None, None
else:
cnsw, cnsmtxtb = None, None
try:
Zc = dou.load_npa(cdatstr + '__Z')
except IOError:
# coeffmat for nwtn adi
ft_mat = -(0.5*MT + cts*(AT + nmattd.T))
# rhs for nwtn adi
w_mat = np.hstack([MT*Zc, np.sqrt(cts)*tct_mat])
# feedback from a previous Newton step
mtxb = np.sqrt(cts)*cnsmtxtb if cnsmtxtb is not None else None
Zp = pru.proj_alg_ric_newtonadi(mmat=MT,
amat=ft_mat, transposed=True,
mtxoldb=mtxb,
jmat=jmat,
bmat=np.sqrt(cts)*tb_mat,
wmat=w_mat, z0=Zc,
nwtn_adi_dict=nwtn_adi_dict
)['zfac']
if comprz_maxc is not None or comprz_thresh is not None:
Zc = pru.compress_Zsvd(Zp, thresh=comprz_thresh,
k=comprz_maxc)
else:
Zc = Zp
if save_full_z:
dou.save_npa(Zp, fstring=cdatstr + '__Z')
else:
dou.save_npa(Zc, fstring=cdatstr + '__Z')
# and the affine correction
at_mat = MT + cts*(AT + nmattd.T)
# current rhs
ftilde = rhsvtd + rhsv
# apply the feedback and through
if cnsw is not None:
ftilde = rhsvtd + rhsv + cnsw
cnsmtxtb = cnsmtxtb + mtxtb if cnsmtxtb is not None else mtxtb
mtxft = pru.get_mTzzTtb(MT, Zc, ftilde)
fl1 = np.dot(mcmat.T, ystarvec(t))
rhswc = MT*wc + cts*(fl1 - mtxft)
mtxtb = -pru.get_mTzzTtb(MT, Zc, tb_mat)
# mtxtbrm = pru.get_mTzzTtb(MT, Zc, bmat_rpmo)
wc = lau.solve_sadpnt_smw(amat=at_mat, jmat=jmat,
umat=cts*cnsmtxtb, vmat=tb_mat.T,
rhsv=rhswc)[:NV]
# wc = lau.solve_sadpnt_smw(amat=at_mat, jmat=jmat,
# umat=-cts*mtxbrm, vmat=bmat.T,
# rhsv=rhswc)[:NV]
# update the feedback in Newton
if curnwtnsdict is not None:
cnsw = cnsw + wc if cnsw is not None else wc
cnsmtxtb = cnsmtxtb + mtxtb if cnsmtxtb is not None else mtxtb
dou.save_npa(cnsw, fstring=curnwtnsdict[t]['w'])
dou.save_npa(cnsmtxtb, fstring=curnwtnsdict[t]['mtxtb'])
dou.save_npa(wc, fstring=cdatstr + '__w')
dou.save_npa(mtxtb, fstring=cdatstr + '__mtxtb')
# dou.save_npa(mtxbrm, fstring=cdatstr + '__mtxbrm')
feedbackthroughdict.update({t: dict(w=cdatstr + '__w',
# mtxbrm=cdatstr + '__mtxbrm')})
mtxtb=cdatstr + '__mtxtb')})
return feedbackthroughdict