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 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 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