def check_input_opa(NU, femp=None):
if femp is None:
# from dolfin_navier_scipy.problem_setups import cyl_fems
# femp = cyl_fems(2)
from dolfin_navier_scipy.problem_setups import drivcav_fems
femp = drivcav_fems(20)
V = femp["V"]
Q = femp["Q"]
cdcoo = femp["cdcoo"]
# get the system matrices
stokesmats = dts.get_stokessysmats(V, Q)
# check the B
B1, Mu = cou.get_inp_opa(cdcoo=cdcoo, V=V, NU=NU, xcomp=0)
B2, Mu = cou.get_inp_opa(cdcoo=cdcoo, V=V, NU=NU, xcomp=1)
# get the rhs expression of Bu
Bu1 = spsla.spsolve(
stokesmats["M"],
B1 * np.vstack([np.linspace(0, 1, NU).reshape((NU, 1)), np.linspace(0, 1, NU).reshape((NU, 1))]),
)
Bu2 = spsla.spsolve(
stokesmats["M"],
B2 * np.vstack([np.linspace(0, 1, NU).reshape((NU, 1)), np.linspace(0, 1, NU).reshape((NU, 1))]),
)
Bu3 = spsla.spsolve(stokesmats["M"], B1 * np.vstack([1 * np.ones((NU, 1)), 0.2 * np.ones((NU, 1))]))
bu1 = dolfin.Function(V)
bu1.vector().set_local(Bu1)
bu1.vector()[2] = 1 # for scaling and illustration purposes
bu2 = dolfin.Function(V)
bu2.vector().set_local(Bu2)
bu2.vector()[2] = 1 # for scaling and illustration purposes
bu3 = dolfin.Function(V)
bu3.vector().set_local(Bu3)
bu3.vector()[2] = 1 # for scaling and illustration purposes
plt.figure(11)
dolfin.plot(bu1, title="plot of Bu - extending in x")
plt.figure(12)
dolfin.plot(bu2, title="plot of Bu - extending in y")
plt.figure(13)
dolfin.plot(bu3, title="plot of Bu - extending in y")
# dolfin.plot(V.mesh())
dolfin.interactive()
plt.show(block=False)
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 comp_exp_nsmats(problemname='drivencavity',
N=10,
Re=1e2,
nu=None,
linear_system=False,
refree=False,
bccontrol=False,
palpha=None,
use_old_data=False,
mddir='pathtodatastorage'):
"""compute and export the system matrices for Navier-Stokes equations
Parameters
---
refree : boolean, optional
whether to use `Re=1` (so that the `Re` number can be applied later by
scaling the corresponding matrices, defaults to `False`
linear_system : boolean, optional
whether to compute/return the linearized system, defaults to `False`
bccontrol : boolean, optional
whether to model boundary control at the cylinder via penalized robin
boundary conditions, defaults to `False`
palpha : float, optional
penalization parameter for the boundary control, defaults to `None`,
`palpha` is mandatory for `linear_system`
"""
if refree:
Re = 1
print 'For the Reynoldsnumber free mats, we set Re=1'
if problemname == 'drivencavity' and bccontrol:
raise NotImplementedError('boundary control for the driven cavity' +
' is not implemented yet')
if linear_system and bccontrol and palpha is None:
raise UserWarning('For the linear system a' +
' value for `palpha` is needed')
if not linear_system and bccontrol:
raise NotImplementedError('Nonlinear system with boundary control' +
' is not implemented yet')
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc = \
dnsps.get_sysmats(problem=problemname, bccontrol=bccontrol, N=N, Re=Re)
if linear_system and bccontrol:
Arob = stokesmatsc['A'] + 1. / palpha * stokesmatsc['Arob']
Brob = 1. / palpha * stokesmatsc['Brob']
elif linear_system:
Brob = 0
invinds = femp['invinds']
A, J = stokesmatsc['A'], stokesmatsc['J']
fvc, fpc = rhsd_vfrc['fvc'], rhsd_vfrc['fpr']
fv_stbc, fp_stbc = rhsd_stbc['fv'], rhsd_stbc['fp']
invinds = femp['invinds']
NV = invinds.shape[0]
data_prfx = problemname + '__N{0}Re{1}'.format(N, Re)
if bccontrol:
data_prfx = data_prfx + '_penarob'
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
soldict.update(rhsd_vfrc) # adding fvc, fpr
fv = rhsd_vfrc['fvc'] + rhsd_stbc['fv']
fp = rhsd_vfrc['fpr'] + rhsd_stbc['fp']
# print 'get expmats: ||fv|| = {0}'.format(np.linalg.norm(fv))
# print 'get expmats: ||fp|| = {0}'.format(np.linalg.norm(fp))
# import scipy.sparse.linalg as spsla
# print 'get expmats: ||A|| = {0}'.format(spsla.norm(A))
# print 'get expmats: ||Arob|| = {0}'.format(spsla.norm(Arob))
# print 'get expmats: ||A|| = {0}'.format(spsla.norm(stokesmatsc['A']))
# raise Warning('TODO: debug')
soldict.update(fv=fv,
fp=fp,
N=N,
nu=nu,
clearprvdata=~use_old_data,
get_datastring=None,
data_prfx=ddir + data_prfx + '_stst',
paraviewoutput=False)
if bccontrol and linear_system:
soldict.update(A=Arob)
# compute the uncontrolled steady state Navier-Stokes solution
vp_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(return_vp=True,
**soldict)
v_ss_nse, p_ss_nse = vp_ss_nse[:NV], vp_ss_nse[NV:]
# specify in what spatial direction Bu changes. The remaining is constant
if problemname == 'drivencavity':
uspacedep = 0
elif problemname == 'cylinderwake':
uspacedep = 1
#
# Control mats
#
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, :][:, :]
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
# TODO: right choice of norms for y
# and necessity of regularization here
# by now, we go on number save
# the pressure observation mean over a small domain
if problemname == 'cylinderwake':
podcoo = dict(xmin=0.6, xmax=0.64, ymin=0.18, ymax=0.22)
elif problemname == 'drivencavity':
podcoo = dict(xmin=0.45, xmax=0.55, ymin=0.7, ymax=0.8)
else:
podcoo = femp['odcoo']
# description of the control and observation domains
dmd = femp['cdcoo']
xmin, xmax, ymin, ymax = dmd['xmin'], dmd['xmax'], dmd['ymin'], dmd['ymax']
velcondomstr = 'vel control domain: [{0}, {1}]x[{2}, {3}]\n'.\
format(xmin, xmax, ymin, ymax)
dmd = femp['odcoo']
xmin, xmax, ymin, ymax = dmd['xmin'], dmd['xmax'], dmd['ymin'], dmd['ymax']
velobsdomstr = 'vel observation domain: [{0}, {1}]x[{2}, {3}]\n'.\
format(xmin, xmax, ymin, ymax)
dmd = podcoo
xmin, xmax, ymin, ymax = dmd['xmin'], dmd['xmax'], dmd['ymin'], dmd['ymax']
pobsdomstr = 'pressure observation domain: [{0}, {1}]x[{2}, {3}]\n'.\
format(xmin, xmax, ymin, ymax)
pcmat = cou.get_pavrg_onsubd(odcoo=podcoo, Q=femp['Q'], ppin=None)
cdatstr = snu.get_datastr_snu(time=None, meshp=N, nu=nu, Nts=None)
(coors, xinds, yinds, corfunvec) = dts.get_dof_coors(femp['V'],
invinds=invinds)
ctrl_visu_str = \
' the (distributed) control setup is as follows \n' +\
' B maps into the domain of control -' +\
velcondomstr +\
' the first half of the columns' +\
'actuate in x-direction, the second in y direction \n' +\
' Cv measures averaged velocities in the domain of observation' +\
velobsdomstr +\
' Cp measures the averaged pressure' +\
' in the domain of pressure observation: ' +\
pobsdomstr +\
' the first components are in x, the last in y-direction \n\n' +\
' Visualization: \n\n' +\
' `coors` -- array of (x,y) coordinates in ' +\
' the same order as v[xinds] or v[yinds] \n' +\
' `xinds`, `yinds` -- indices of x and y components' +\
' of v = [vx, vy] -- note that indexing starts with 0\n' +\
' for testing use corfunvec wich is the interpolant of\n' +\
' f(x,y) = [x, y] on the grid \n\n' +\
'Created in `get_exp_nsmats.py` ' +\
'(see https://github.com/highlando/dolfin_navier_scipy) at\n' +\
datetime.datetime.now().strftime("%I:%M%p on %B %d, %Y")
if bccontrol and problemname == 'cylinderwake' and linear_system:
ctrl_visu_str = \
('the boundary control is realized via penalized robin \n' +
'boundary conditions, cf. e.g. [Hou/Ravindran `98], \n' +
'with predefined shape functions for the cylinder wake \n' +
'and the penalization parameter `palpha`={0}.').format(palpha) +\
ctrl_visu_str
if linear_system:
convc_mat, rhs_con, rhsv_conbc = \
snu.get_v_conv_conts(prev_v=v_ss_nse, invinds=invinds,
V=femp['V'], diribcs=femp['diribcs'])
# TODO: omg
if bccontrol:
f_mat = -Arob - convc_mat
else:
f_mat = -stokesmatsc['A'] - convc_mat
infostr = 'These are the coefficient matrices of the linearized ' +\
'Navier-Stokes Equations \n for the ' +\
problemname + ' to be used as \n\n' +\
' $M \\dot v = Av + J^Tp + Bu$ and $Jv = 0$ \n\n' +\
' the Reynoldsnumber is computed as L/nu \n' +\
' Note this is the reduced system for the velocity update\n' +\
' caused by the control, i.e., no boundary conditions\n' +\
' or inhomogeneities here. To get the actual flow, superpose \n' +\
' the steadystate velocity solution `v_ss_nse` \n\n' +\
ctrl_visu_str
matstr = (mddir + problemname + '__mats_N{0}_Re{1}').format(NV, Re)
if bccontrol:
matstr = matstr + '__penarob_palpha{0}'.format(palpha)
scipy.io.savemat(
matstr,
dict(A=f_mat,
M=stokesmatsc['M'],
nu=femp['nu'],
Re=femp['Re'],
J=stokesmatsc['J'],
B=b_mat,
C=c_mat,
Cp=pcmat,
Brob=Brob,
v_ss_nse=v_ss_nse,
info=infostr,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec))
print('matrices saved to ' + matstr)
elif refree:
hstr = ddir + problemname + '_N{0}_hmat'.format(N)
try:
hmat = dou.load_spa(hstr)
print 'loaded `hmat`'
except IOError:
print 'assembling hmat ...'
hmat = dts.ass_convmat_asmatquad(W=femp['V'], invindsw=invinds)
dou.save_spa(hmat, hstr)
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)
# diff_mat = stokesmatsc['A']
# bcconv_mat = bc_conv
# fv_bcdiff = fv_stbc
# fv_bcconv = - bc_rhs_conv
fv = fvc
fp = fpc
# fp_bc = fp_stbc
infostr = 'These are the coefficient matrices of the quadratic ' +\
'formulation of the Navier-Stokes Equations \n for the ' +\
problemname + ' to be used as \n\n' +\
' $M \\dot v + Av + H*kron(v,v) + J^Tp = Bu + fv$ \n' +\
' and $Jv = fp$ \n\n' +\
' the Reynoldsnumber is computed as L/nu \n' +\
' note that `A` contains the diffusion and the linear term \n' +\
' that comes from the dirichlet boundary values \n' +\
' as initial value one can use the provided steady state \n' +\
' Stokes solution \n' +\
' see https://github.com/highlando/dolfin_navier_scipy/blob/' +\
' master/tests/solve_nse_quadraticterm.py for appl example\n' +\
ctrl_visu_str
scipy.io.savemat(
mddir + problemname + 'quadform__mats_N{0}_Re{1}'.format(NV, Re),
dict(
A=f_mat,
M=stokesmatsc['M'],
H=-hmat,
fv=fv,
fp=fp,
nu=femp['nu'],
Re=femp['Re'],
J=stokesmatsc['J'],
B=b_mat,
Cv=c_mat,
Cp=pcmat,
info=infostr,
# ss_stokes=old_v,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec))
else:
hstr = ddir + problemname + '_N{0}_hmat'.format(N)
try:
hmat = dou.load_spa(hstr)
print 'loaded `hmat`'
except IOError:
print 'assembling hmat ...'
hmat = dts.ass_convmat_asmatquad(W=femp['V'], invindsw=invinds)
dou.save_spa(hmat, hstr)
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)
f_mat = -stokesmatsc['A'] - bc_conv
l_mat = -bc_conv
fv = fv_stbc + fvc - bc_rhs_conv
fp = fp_stbc + fpc
vp_stokes = lau.solve_sadpnt_smw(amat=A,
jmat=J,
rhsv=fv_stbc + fvc,
rhsp=fp_stbc + fpc)
old_v = vp_stokes[:NV]
p_stokes = -vp_stokes[NV:] # the pressure was flipped for symmetry
infostr = 'These are the coefficient matrices of the quadratic ' +\
'formulation of the Navier-Stokes Equations \n for the ' +\
problemname + ' to be used as \n\n' +\
' $M \\dot v = Av + H*kron(v,v) + J^Tp + Bu + fv$ \n' +\
' and $Jv = fp$ \n\n' +\
' the Reynoldsnumber is computed as L/nu \n' +\
' note that `A` contains the diffusion and the linear term `L`\n' +\
' that comes from the dirichlet boundary values \n' +\
' for linearizations it might be necessary to consider `A-L` \n' +\
' as initial value one can use the provided steady state \n' +\
' Stokes solution \n' +\
' see https://github.com/highlando/dolfin_navier_scipy/blob/' +\
' master/tests/solve_nse_quadraticterm.py for appl example\n' +\
ctrl_visu_str
scipy.io.savemat(
mddir + problemname + 'quadform__mats_N{0}_Re{1}'.format(NV, Re),
dict(A=f_mat,
M=stokesmatsc['M'],
H=-hmat,
fv=fv,
fp=fp,
L=l_mat,
nu=femp['nu'],
Re=femp['Re'],
J=stokesmatsc['J'],
B=b_mat,
Cv=c_mat,
Cp=pcmat,
info=infostr,
p_ss_stokes=p_stokes,
p_ss_nse=p_ss_nse,
v_ss_stokes=old_v,
v_ss_nse=v_ss_nse,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec))
def comp_exp_nsmats(
problemname="drivencavity",
N=10,
Re=1e2,
nu=None,
linear_system=False,
refree=False,
bccontrol=False,
palpha=None,
use_old_data=False,
mddir="pathtodatastorage",
):
"""compute and export the system matrices for Navier-Stokes equations
Parameters
---
refree : boolean, optional
whether to use `Re=1` (so that the `Re` number can be applied later by
scaling the corresponding matrices, defaults to `False`
linear_system : boolean, optional
whether to compute/return the linearized system, defaults to `False`
bccontrol : boolean, optional
whether to model boundary control at the cylinder via penalized robin
boundary conditions, defaults to `False`
palpha : float, optional
penalization parameter for the boundary control, defaults to `None`,
`palpha` is mandatory for `linear_system`
"""
if refree:
Re = 1
print "For the Reynoldsnumber free mats, we set Re=1"
if problemname == "drivencavity" and bccontrol:
raise NotImplementedError("boundary control for the driven cavity" + " is not implemented yet")
if linear_system and bccontrol and palpha is None:
raise UserWarning("For the linear system a" + " value for `palpha` is needed")
if not linear_system and bccontrol:
raise NotImplementedError("Nonlinear system with boundary control" + " is not implemented yet")
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc = dnsps.get_sysmats(problem=problemname, bccontrol=bccontrol, N=N, Re=Re)
if linear_system and bccontrol:
Arob = stokesmatsc["A"] + 1.0 / palpha * stokesmatsc["Arob"]
Brob = 1.0 / palpha * stokesmatsc["Brob"]
elif linear_system:
Brob = 0
invinds = femp["invinds"]
A, J = stokesmatsc["A"], stokesmatsc["J"]
fvc, fpc = rhsd_vfrc["fvc"], rhsd_vfrc["fpr"]
fv_stbc, fp_stbc = rhsd_stbc["fv"], rhsd_stbc["fp"]
invinds = femp["invinds"]
NV = invinds.shape[0]
data_prfx = problemname + "__N{0}Re{1}".format(N, Re)
if bccontrol:
data_prfx = data_prfx + "_penarob"
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
soldict.update(rhsd_vfrc) # adding fvc, fpr
fv = rhsd_vfrc["fvc"] + rhsd_stbc["fv"]
fp = rhsd_vfrc["fpr"] + rhsd_stbc["fp"]
# print 'get expmats: ||fv|| = {0}'.format(np.linalg.norm(fv))
# print 'get expmats: ||fp|| = {0}'.format(np.linalg.norm(fp))
# import scipy.sparse.linalg as spsla
# print 'get expmats: ||A|| = {0}'.format(spsla.norm(A))
# print 'get expmats: ||Arob|| = {0}'.format(spsla.norm(Arob))
# print 'get expmats: ||A|| = {0}'.format(spsla.norm(stokesmatsc['A']))
# raise Warning('TODO: debug')
soldict.update(
fv=fv,
fp=fp,
N=N,
nu=nu,
clearprvdata=~use_old_data,
get_datastring=None,
data_prfx=ddir + data_prfx + "_stst",
paraviewoutput=False,
)
if bccontrol and linear_system:
soldict.update(A=Arob)
# compute the uncontrolled steady state Navier-Stokes solution
vp_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(return_vp=True, **soldict)
v_ss_nse, p_ss_nse = vp_ss_nse[:NV], vp_ss_nse[NV:]
# specify in what spatial direction Bu changes. The remaining is constant
if problemname == "drivencavity":
uspacedep = 0
elif problemname == "cylinderwake":
uspacedep = 1
#
# Control mats
#
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, :][:, :]
c_mat = lau.apply_massinv(y_masmat, mc_mat, output="sparse")
# TODO: right choice of norms for y
# and necessity of regularization here
# by now, we go on number save
# the pressure observation mean over a small domain
if problemname == "cylinderwake":
podcoo = dict(xmin=0.6, xmax=0.64, ymin=0.18, ymax=0.22)
elif problemname == "drivencavity":
podcoo = dict(xmin=0.45, xmax=0.55, ymin=0.7, ymax=0.8)
else:
podcoo = femp["odcoo"]
# description of the control and observation domains
dmd = femp["cdcoo"]
xmin, xmax, ymin, ymax = dmd["xmin"], dmd["xmax"], dmd["ymin"], dmd["ymax"]
velcondomstr = "vel control domain: [{0}, {1}]x[{2}, {3}]\n".format(xmin, xmax, ymin, ymax)
dmd = femp["odcoo"]
xmin, xmax, ymin, ymax = dmd["xmin"], dmd["xmax"], dmd["ymin"], dmd["ymax"]
velobsdomstr = "vel observation domain: [{0}, {1}]x[{2}, {3}]\n".format(xmin, xmax, ymin, ymax)
dmd = podcoo
xmin, xmax, ymin, ymax = dmd["xmin"], dmd["xmax"], dmd["ymin"], dmd["ymax"]
pobsdomstr = "pressure observation domain: [{0}, {1}]x[{2}, {3}]\n".format(xmin, xmax, ymin, ymax)
pcmat = cou.get_pavrg_onsubd(odcoo=podcoo, Q=femp["Q"], ppin=None)
cdatstr = snu.get_datastr_snu(time=None, meshp=N, nu=nu, Nts=None)
(coors, xinds, yinds, corfunvec) = dts.get_dof_coors(femp["V"], invinds=invinds)
ctrl_visu_str = (
" the (distributed) control setup is as follows \n"
+ " B maps into the domain of control -"
+ velcondomstr
+ " the first half of the columns"
+ "actuate in x-direction, the second in y direction \n"
+ " Cv measures averaged velocities in the domain of observation"
+ velobsdomstr
+ " Cp measures the averaged pressure"
+ " in the domain of pressure observation: "
+ pobsdomstr
+ " the first components are in x, the last in y-direction \n\n"
+ " Visualization: \n\n"
+ " `coors` -- array of (x,y) coordinates in "
+ " the same order as v[xinds] or v[yinds] \n"
+ " `xinds`, `yinds` -- indices of x and y components"
+ " of v = [vx, vy] -- note that indexing starts with 0\n"
+ " for testing use corfunvec wich is the interpolant of\n"
+ " f(x,y) = [x, y] on the grid \n\n"
+ "Created in `get_exp_nsmats.py` "
+ "(see https://github.com/highlando/dolfin_navier_scipy) at\n"
+ datetime.datetime.now().strftime("%I:%M%p on %B %d, %Y")
)
if bccontrol and problemname == "cylinderwake" and linear_system:
ctrl_visu_str = (
"the boundary control is realized via penalized robin \n"
+ "boundary conditions, cf. e.g. [Hou/Ravindran `98], \n"
+ "with predefined shape functions for the cylinder wake \n"
+ "and the penalization parameter `palpha`={0}."
).format(palpha) + ctrl_visu_str
if linear_system:
convc_mat, rhs_con, rhsv_conbc = snu.get_v_conv_conts(
prev_v=v_ss_nse, invinds=invinds, V=femp["V"], diribcs=femp["diribcs"]
)
# TODO: omg
if bccontrol:
f_mat = -Arob - convc_mat
else:
f_mat = -stokesmatsc["A"] - convc_mat
infostr = (
"These are the coefficient matrices of the linearized "
+ "Navier-Stokes Equations \n for the "
+ problemname
+ " to be used as \n\n"
+ " $M \\dot v = Av + J^Tp + Bu$ and $Jv = 0$ \n\n"
+ " the Reynoldsnumber is computed as L/nu \n"
+ " Note this is the reduced system for the velocity update\n"
+ " caused by the control, i.e., no boundary conditions\n"
+ " or inhomogeneities here. To get the actual flow, superpose \n"
+ " the steadystate velocity solution `v_ss_nse` \n\n"
+ ctrl_visu_str
)
matstr = (mddir + problemname + "__mats_N{0}_Re{1}").format(NV, Re)
if bccontrol:
matstr = matstr + "__penarob_palpha{0}".format(palpha)
scipy.io.savemat(
matstr,
dict(
A=f_mat,
M=stokesmatsc["M"],
nu=femp["nu"],
Re=femp["Re"],
J=stokesmatsc["J"],
B=b_mat,
C=c_mat,
Cp=pcmat,
Brob=Brob,
v_ss_nse=v_ss_nse,
info=infostr,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec,
),
)
print ("matrices saved to " + matstr)
elif refree:
hstr = ddir + problemname + "_N{0}_hmat".format(N)
try:
hmat = dou.load_spa(hstr)
print "loaded `hmat`"
except IOError:
print "assembling hmat ..."
hmat = dts.ass_convmat_asmatquad(W=femp["V"], invindsw=invinds)
dou.save_spa(hmat, hstr)
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
)
# diff_mat = stokesmatsc['A']
# bcconv_mat = bc_conv
# fv_bcdiff = fv_stbc
# fv_bcconv = - bc_rhs_conv
fv = fvc
fp = fpc
# fp_bc = fp_stbc
infostr = (
"These are the coefficient matrices of the quadratic "
+ "formulation of the Navier-Stokes Equations \n for the "
+ problemname
+ " to be used as \n\n"
+ " $M \\dot v + Av + H*kron(v,v) + J^Tp = Bu + fv$ \n"
+ " and $Jv = fp$ \n\n"
+ " the Reynoldsnumber is computed as L/nu \n"
+ " note that `A` contains the diffusion and the linear term \n"
+ " that comes from the dirichlet boundary values \n"
+ " as initial value one can use the provided steady state \n"
+ " Stokes solution \n"
+ " see https://github.com/highlando/dolfin_navier_scipy/blob/"
+ " master/tests/solve_nse_quadraticterm.py for appl example\n"
+ ctrl_visu_str
)
scipy.io.savemat(
mddir + problemname + "quadform__mats_N{0}_Re{1}".format(NV, Re),
dict(
A=f_mat,
M=stokesmatsc["M"],
H=-hmat,
fv=fv,
fp=fp,
nu=femp["nu"],
Re=femp["Re"],
J=stokesmatsc["J"],
B=b_mat,
Cv=c_mat,
Cp=pcmat,
info=infostr,
# ss_stokes=old_v,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec,
),
)
else:
hstr = ddir + problemname + "_N{0}_hmat".format(N)
try:
hmat = dou.load_spa(hstr)
print "loaded `hmat`"
except IOError:
print "assembling hmat ..."
hmat = dts.ass_convmat_asmatquad(W=femp["V"], invindsw=invinds)
dou.save_spa(hmat, hstr)
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
)
f_mat = -stokesmatsc["A"] - bc_conv
l_mat = -bc_conv
fv = fv_stbc + fvc - bc_rhs_conv
fp = fp_stbc + fpc
vp_stokes = lau.solve_sadpnt_smw(amat=A, jmat=J, rhsv=fv_stbc + fvc, rhsp=fp_stbc + fpc)
old_v = vp_stokes[:NV]
p_stokes = -vp_stokes[NV:] # the pressure was flipped for symmetry
infostr = (
"These are the coefficient matrices of the quadratic "
+ "formulation of the Navier-Stokes Equations \n for the "
+ problemname
+ " to be used as \n\n"
+ " $M \\dot v = Av + H*kron(v,v) + J^Tp + Bu + fv$ \n"
+ " and $Jv = fp$ \n\n"
+ " the Reynoldsnumber is computed as L/nu \n"
+ " note that `A` contains the diffusion and the linear term `L`\n"
+ " that comes from the dirichlet boundary values \n"
+ " for linearizations it might be necessary to consider `A-L` \n"
+ " as initial value one can use the provided steady state \n"
+ " Stokes solution \n"
+ " see https://github.com/highlando/dolfin_navier_scipy/blob/"
+ " master/tests/solve_nse_quadraticterm.py for appl example\n"
+ ctrl_visu_str
)
scipy.io.savemat(
mddir + problemname + "quadform__mats_N{0}_Re{1}".format(NV, Re),
dict(
A=f_mat,
M=stokesmatsc["M"],
H=-hmat,
fv=fv,
fp=fp,
L=l_mat,
nu=femp["nu"],
Re=femp["Re"],
J=stokesmatsc["J"],
B=b_mat,
Cv=c_mat,
Cp=pcmat,
info=infostr,
p_ss_stokes=p_stokes,
p_ss_nse=p_ss_nse,
v_ss_stokes=old_v,
v_ss_nse=v_ss_nse,
contsetupstr=contsetupstr,
datastr=cdatstr,
coors=coors,
xinds=xinds,
yinds=yinds,
corfunvec=corfunvec,
),
)
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)
def gopod(problemname='drivencavity',
N=10, Re=1e2, t0=0.0, tE=1.0, Nts=11, NU=3, NY=3,
paraoutput=True, multiproc=False,
krylov=None, krpslvprms={}, krplsprms={}):
"""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`
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`
NU, NY : int, optional
dimensions of components of in and output space (will double because
there are two components), default to `3, 3`
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
"""
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc, data_prfx, ddir, proutdir = \
dnsps.get_sysmats(problem=problemname, N=N, Re=Re)
# specify in what spatial direction Bu changes. The remaining is constant
uspacedep = femp['uspacedep']
# output
ddir = 'data/'
try:
os.chdir(ddir)
except OSError:
raise Warning('need "' + ddir + '" subdir for storing the data')
os.chdir('..')
data_prfx = ddir + data_prfx
# casting some parameters
NV = len(femp['invinds'])
# contsetupstr = 'NV{0}NU{1}NY{2}alphau{3}'.format(NV, NU, NY, alphau)
contsetupstr = 'NV{0}NU{1}NY{2}Re{3}'.format(NV, NU, NY, 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_stbc=rhsd_stbc['fv'], fp_stbc=rhsd_stbc['fp'],
N=N, nu=femp['nu'], data_prfx=data_prfx)
soldict.update(paraviewoutput=paraoutput)
soldict.update(krylov=krylov, krplsprms=krplsprms, krpslvprms=krpslvprms)
#
# Prepare for control
#
# 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
invinds = femp['invinds']
mc_mat = mc_mat[:, invinds][:, :]
b_mat = b_mat[invinds, :][:, :]
# tb_mat = 1./np.sqrt(alphau)
# setup the system for the correction
#
# # compute the uncontrolled steady state Stokes solution
#
v_ss_stokes, list_norm_nwtnupd = \
snu.solve_steadystate_nse(vel_pcrd_stps=0, vel_nwtn_stps=0,
clearprvdata=True, **soldict)
tmesh = np.linspace(t0, tE, Nts)
soldict.update(trange=tmesh,
iniv=v_ss_stokes,
lin_vel_point=v_ss_stokes,
clearprvdata=True,
vel_nwtn_stps=1,
return_dictofvelstrs=False,
paraviewoutput=True,
vfileprfx='results/fullvel',
pfileprfx='results/fullp')
convc_mat_n, rhs_con_n, rhsv_conbc_n = \
snu.get_v_conv_conts(prev_v=v_ss_stokes, invinds=invinds,
V=femp['V'], diribcs=femp['diribcs'],
Picard=False)
convc_mat_z, rhs_con_z, rhsv_conbc_z = \
snu.get_v_conv_conts(prev_v=0*v_ss_stokes, invinds=invinds,
V=femp['V'], diribcs=femp['diribcs'],
Picard=False)
vellist = snu.solve_nse(return_as_list=True, **soldict)
velar = np.array(vellist)[:, :, 0].T
rhsv = soldict['fv_stbc'] + soldict['fvc'] + rhsv_conbc_n + rhs_con_n
rhsp = soldict['fp_stbc'] + soldict['fpr']
# print 'fvstbc', np.linalg.norm(soldict['fv_stbc'])
# print 'fvc', np.linalg.norm(soldict['fvc'])
# print 'rhsvconbc', np.linalg.norm(rhsv_conbc_n)
# print 'rhscon', np.linalg.norm(rhs_con_n)
print 'velarshape :', velar.shape
checkreturns = False
if checkreturns:
inivel = velar[:, 0:1]
ylist = snu.solve_nse(A=soldict['A']+convc_mat_n-convc_mat_z,
M=soldict['M'],
J=soldict['J'], fvc=rhsv, fpr=rhsp,
iniv=inivel,
fv_stbc=0*rhsv - rhsv_conbc_z - rhs_con_z,
fp_stbc=0*rhsp,
lin_vel_point=0*inivel, trange=tmesh,
V=femp['V'], Q=femp['Q'],
invinds=femp['invinds'],
diribcs=femp['diribcs'], N=soldict['N'],
nu=soldict['nu'],
vel_nwtn_stps=1,
return_as_list=True)
velarcheck = np.array(ylist)[:, :, 0].T
print np.linalg.norm(velarcheck - velar)
# print np.linalg.norm(velarcheck[:, 1] - velar[:, 1])
return (soldict['M'], soldict['A']+convc_mat_n, velar,
rhsv, b_mat, tmesh, soldict['J'])
def lqgbt(problemname='drivencavity',
N=10, Nts=10, nu=1e-2, plain_bt=True,
savetomatfiles=False):
tip = time_int_params(Nts, nu)
problemdict = dict(drivencavity=dnsps.drivcav_fems,
cylinderwake=dnsps.cyl_fems)
problemfem = problemdict[problemname]
femp = problemfem(N)
data_prfx = problemname + '__'
NU, NY = 3, 4
# specify in what spatial direction Bu changes. The remaining is constant
if problemname == 'drivencavity':
uspacedep = 0
elif problemname == 'cylinderwake':
uspacedep = 1
# 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'],
tip['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)
# casting some parameters
NV, DT, INVINDS = len(femp['invinds']), tip['dt'], 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_stbc=rhsd_stbc['fv'], fp_stbc=rhsd_stbc['fp'],
N=N, nu=tip['nu'],
nnewtsteps=tip['nnewtsteps'],
vel_nwtn_tol=tip['vel_nwtn_tol'],
ddir=ddir, get_datastring=None,
data_prfx=data_prfx,
paraviewoutput=tip['ParaviewOutput'],
vfileprfx=tip['proutdir']+'vel_',
pfileprfx=tip['proutdir']+'p_')
#
# compute the uncontrolled steady state Stokes solution
#
v_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(**soldict)
#
# Prepare for control
#
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, :][:, :]
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
# 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
#
(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
cdatstr = snu.get_datastr_snu(time=None, meshp=N, nu=tip['nu'],
Nts=None, dt=None)
if savetomatfiles:
import datetime
import scipy.io
(coors, xinds,
yinds, corfunvec) = dts.get_dof_coors(femp['V'], invinds=invinds)
infostr = 'These are the coefficient matrices of the linearized ' +\
'Navier-Stokes Equations \n for the ' +\
problemname + ' to be used as \n\n' +\
' $M \\dot v = Av + J^Tp + Bu$ and $Jv = 0$ \n\n' +\
' the Reynoldsnumber is computed as L/nu \n' +\
' Note that this is the reduced system for the velocity update\n' +\
' caused by the control, i.e., no boundary conditions\n' +\
' or inhomogeneities here. To get the actual flow, superpose \n' +\
' the steadystate velocity solution `v_ss_nse` \n\n' +\
' the control setup is as follows \n' +\
' B maps into the domain of control - the first half of the colums' +\
'actuate in x-direction, the second in y direction \n' +\
' C measures averaged velocities in the domain of observation' +\
' the first components are in x, the last in y-direction \n\n' +\
' Visualization: \n\n' +\
' `coors` -- array of (x,y) coordinates in ' +\
' the same order as v[xinds] or v[yinds] \n' +\
' `xinds`, `yinds` -- indices of x and y components' +\
' of v = [vx, vy] -- note that indexing starts with 0\n' +\
' for testing use corfunvec wich is the interpolant of\n' +\
' f(x,y) = [x, y] on the grid \n\n' +\
'Created in `exp_cylinder_mats.py` ' +\
'(see https://github.com/highlando/lqgbt-oseen) at\n' +\
datetime.datetime.now().strftime("%I:%M%p on %B %d, %Y")
mddir = '/afs/mpi-magdeburg.mpg.de/data/csc/projects/qbdae-nse/data/'
if problemname == 'cylinderwake':
charlen = 0.15 # diameter of the cylinder
Re = charlen/nu
elif problemname == 'drivencavity':
Re = nu
else:
Re = nu
scipy.io.savemat(mddir + problemname +
'__mats_N{0}_Re{1}'.format(NV, Re),
dict(A=f_mat, M=stokesmatsc['M'], nu=nu, Re=Re,
J=stokesmatsc['J'], B=b_mat, C=c_mat,
v_ss_nse=v_ss_nse, info=infostr,
contsetupstr=contsetupstr, datastr=cdatstr,
coors=coors, xinds=xinds, yinds=yinds,
corfunvec=corfunvec))
return
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)