degree=2) # no slip condition at the top
# Now we wire up
bcs_fluid = {
'velocity': [(facet_lookup['F_bottom'], uf_bottom),
((facet_lookup['I_bottom'], uf_top))],
'traction': [],
'pressure': [(facet_lookup['F_left'], pf_in),
(facet_lookup['F_right'], pf_out)]
}
# Get the initial conditions ------------------------------------- FIXME
# Solve fluid steady state
uf_out, pf_out = File('./output/poisseuille/uf.pvd'), File(
'./output/poisseuille/pf.pvd')
# Solve fluid problem
u_f, p_f = solve_fluid(Wf,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_fluid,
parameters=fluid_parameters)
u_f.rename("uf", "tmp")
p_f.rename("pf", "tmp")
uf_out << (u_f)
pf_out << (p_f)
def PVS_simulation(args):
""" Solve the flow and tracer transport in the PVS :
Outputs :
- a logfile with information about the simulation
- .pvd files with the u, p and c field at specified args.toutput time period
Return : u, p, c 1D array of the u, p, c fields on the middle line """
# output folder name
outputfolder=args.output_folder+'/'+args.job_name+'/'
if not os.path.exists(outputfolder):
os.makedirs(outputfolder)
if not os.path.exists(outputfolder+'/profiles'):
os.makedirs(outputfolder+'/profiles')
if not os.path.exists(outputfolder+'/fields'):
os.makedirs(outputfolder+'/fields')
# Create output files
#txt files
csv_p=open(outputfolder+'profiles'+'/pressure.txt', 'w')
csv_u=open(outputfolder+'profiles'+'/velocity.txt', 'w')
csv_c=open(outputfolder+'profiles'+'/concentration.txt', 'w')
csv_rv=open(outputfolder+'profiles'+'/radius.txt', 'w')
#pvd files
uf_out, pf_out= File(outputfolder+'fields'+'/uf.pvd'), File(outputfolder+'fields'+'/pf.pvd')
c_out= File(outputfolder+'fields'+'/c.pvd')
facets_out=File(outputfolder+'fields'+'/facets.pvd')
# Create logger
logger = logging.getLogger()
logger.setLevel(logging.INFO)
# log to a file
now = datetime.now().strftime("%Y%m%d_%H%M%S")
filename = os.path.join(outputfolder+'/', 'PVS_info.log')
file_handler = logging.FileHandler(filename,mode='w')
file_handler.setLevel(logging.INFO)
#formatter = logging.Formatter("%(asctime)s %(filename)s, %(lineno)d, %(funcName)s: %(message)s")
#file_handler.setFormatter(formatter)
logger.addHandler(file_handler)
# log to the console
console_handler = logging.StreamHandler()
level = logging.INFO
console_handler.setLevel(level)
logger.addHandler(console_handler)
# initialise logging
logging.info(title1("Simulation of the PVS flow and tracer transport using non steady solver and diffusion-advection solvers"))
logging.info("Date and time:"+datetime.now().strftime("%m/%d/%Y, %H:%M:%S"))
logging.info('Job name : '+args.job_name)
logging.debug('logging initialized')
# Set parameters
logging.info(title1("Parameters"))
# Geometry params
logging.info('\n * Geometry')
Rv = args.radius_vessel # centimeters
Rpvs =args.radius_pvs # centimeters
L = args.length # centimeters
logging.info('Vessel radius : %e cm'%Rv)
logging.info('PVS radius : %e cm'%Rpvs)
logging.info('PVS length : %e cm'%L)
#Mesh
logging.info('\n * Mesh')
#number of cells in the radial direction
Nr=args.N_radial
DR=(Rpvs-Rv)/Nr
#number of cells in the axial direction
if args.N_axial :
Nl=args.N_axial
else :
Nl=round(L/DR)
DY=L/Nl
logging.info('N axial : %i'%Nl)
logging.info('N radial : %e'%Nr)
#time parameters
logging.info('\n * Time')
toutput=args.toutput
tfinal=args.tend
dt=args.time_step
logging.info('final time: %e s'%tfinal)
logging.info('output period : %e s'%toutput)
logging.info('time step : %e s'%dt)
# approximate CFL for fluid solver : need to compute max velocity depending on the wall displacement...
# maybe just add a warning in computation with actual velocity
#Uapprox=500e-4 #upper limit for extected max velocity
#CFL_dt=0.25*DY/Uapprox
#if CFL_dt < dt :
# logging.warning('The specified time step of %.2e s does not fullfil the fluid CFL condition. New fluid time step : %.2e s'%(dt, CFL_dt))
#dt_fluid=min(dt,CFL_dt)
dt_fluid=dt
# approximate CFL for tracer solver
dt_advdiff=dt
# material parameters
logging.info('\n * Fluid properties')
mu=args.viscosity
rho=args.density
logging.info('density: %e g/cm3'%rho)
logging.info('dynamic viscosity : %e dyn s/cm2'%mu)
logging.info('\n* Tracer properties')
D=args.diffusion_coef
sigma_gauss=args.sigma
logging.info('Free diffusion coef: %e cm2/s'%D)
logging.info('STD of initial gaussian profile: %e '%sigma_gauss)
xi_gauss=args.initial_pos
logging.info('Initial position: %e cm2'%xi_gauss)
logging.info('\n * ALE')
kappa=args.ale_parameter
logging.info('ALE parameter: %e '%kappa)
logging.info('\n * Lateral BC')
resistance=args.resistance
logging.info('inner resistance: %e '%resistance)
if resistance == 0 :
lateral_bc='free'
logging.info('right BC will be set to the free assumption')
elif resistance < 0 :
lateral_bc='noflow'
logging.info('right BC will be set to the no flow assumption')
else :
lateral_bc='resistance'
logging.info('right BC will be set to the resistance assumption')
fluid_parameters = {'mu': mu, 'rho': rho, 'dt':dt_fluid}
tracer_parameters={'kappa': D, 'dt':dt_advdiff}
ale_parameters = {'kappa': kappa}
# Mesh
logging.info(title1('Meshing'))
logging.info('cell size : %e cm'%(np.sqrt(DR**2+DY**2)))
logging.info('nb cells: %i'%(Nl*Nr*2))
mesh_f= RectangleMesh(Point(0, Rv), Point(L, Rpvs), Nl, Nr)
fluid_bdries = MeshFunction("size_t", mesh_f, mesh_f.topology().dim()-1,0)
# Label facets
xy = mesh_f.coordinates().copy()
x, y = xy.T
xmin=x.min()
xmax=x.max()
ymin=y.min()
ymax=y.max()
tol=min(DR,DY)/2 #cm
class Boundary_left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0],xmin,tol) #left
class Boundary_right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0],xmax,tol)# right
class Boundary_bottom(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymin,tol) #bottom
class Boundary_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymax,tol) #top
btop= Boundary_top()
bbottom = Boundary_bottom()
bleft = Boundary_left()
bright = Boundary_right()
bbottom.mark(fluid_bdries, 2)
btop.mark(fluid_bdries, 4)
bleft.mark(fluid_bdries, 1)
bright.mark(fluid_bdries, 3)
facet_lookup = {'x_min': 1 ,'y_min': 2, 'x_max': 3, 'y_max': 4}
facets_out << fluid_bdries
#FEM space
logging.info(title1("Set FEM spaces"))
logging.info('\n * Fluid')
Vf_elm = VectorElement('Lagrange', triangle, 2)
Qf_elm = FiniteElement('Lagrange', triangle, 1)
Wf_elm = MixedElement([Vf_elm, Qf_elm])
Wf = FunctionSpace(mesh_f, Wf_elm)
logging.info('Velocity : "Lagrange", triangle, 2')
logging.info('Pressure : "Lagrange", triangle, 1')
logging.info('\n * Tracer')
Ct_elm = FiniteElement('Lagrange', triangle, 1)
Ct = FunctionSpace(mesh_f, Ct_elm)
logging.info('Concentration : "Lagrange", triangle, 1')
logging.info('\n * ALE')
Va_elm = VectorElement('Lagrange', triangle, 1)
Va = FunctionSpace(mesh_f, Va_elm)
logging.info('ALE displacement: "Lagrange", triangle, 1')
# Setup of boundary conditions
logging.info(title1("Boundary conditions"))
import sympy
tn = sympy.symbols("tn")
tnp1 = sympy.symbols("tnp1")
sin = sympy.sin
sqrt = sympy.sqrt
logging.info('\n * Cross section area parameters')
ai=args.ai
fi=args.fi
phii=args.phii
logging.info('ai (dimensionless): '+'%e '*len(ai)%tuple(ai))
logging.info('fi (Hz) : '+'%e '*len(fi)%tuple(fi))
logging.info('phii (rad) : '+'%e '*len(phii)%tuple(phii))
functionR = Rpvs-(Rpvs-Rv)*(1+sum([a*sin(2*pi*f*tn+phi) for a,f,phi in zip(ai,fi,phii)])) # displacement
R_vessel = sympy.printing.ccode(functionR)
functionV = sympy.diff(functionR,tn) # velocity
V_vessel = sympy.printing.ccode(functionV)
#Delta U for ALE. I dont really like this
functionUALE=-(Rpvs-Rv)*(1+sum([a*sin(2*pi*f*tnp1+phi) for a,f,phi in zip(ai,fi,phii)]))+(Rpvs-Rv)*(1+sum([a*sin(2*pi*f*tn+phi) for a,f,phi in zip(ai,fi,phii)]))
UALE_vessel = sympy.printing.ccode(functionUALE)
vf_bottom = Expression(('0',V_vessel ), tn = 0, degree=2) # no slip no gap condition at vessel wall
uale_bottom = Expression(('0',UALE_vessel ), tn = 0, tnp1=1, degree=2) # displacement for ALE at vessel wall
logging.info('\n * Lateral assumption')
logging.info(lateral_bc)
logging.info('\n * Fluid')
logging.info('Left : zero pressure')
if lateral_bc=='free' :
logging.info('Right : zero pressure')
elif lateral_bc=='resistance' :
logging.info('Right : resistance')
else :
logging.info('Right : no flow')
logging.info('Top : no slip no gap fixed wall')
logging.info('Bottom : no slip no gap moving wall')
logging.info('\n * Tracer concentration')
logging.info('Left : zero concentration')
if lateral_bc=='free' :
logging.info('Right : zero concentration')
else :
logging.info('Right : no flux')
logging.info('Top : no flux')
logging.info('Bottom : no flux')
logging.info('\n * ALE')
logging.info('Left : no flux')
logging.info('Right : no flux')
logging.info('Top : no displacement')
logging.info('Bottom : vessel displacement')
# Now we wire up
if lateral_bc=='free' :
bcs_fluid = {'velocity': [(facet_lookup['y_min'],vf_bottom),
(facet_lookup['y_max'], Constant((0,0)))],
'traction': [],
'pressure': [(facet_lookup['x_min'], Constant(0)),
(facet_lookup['x_max'], Constant(0))]}
elif lateral_bc=='resistance' :
Rpressure=Expression('R*Q+p0', R = resistance, Q=0, p0=0, degree=1) #
# Compute pressure to impose according to the flow at previous time step and resistance.
bcs_fluid = {'velocity': [(facet_lookup['y_min'],vf_bottom),
(facet_lookup['y_max'], Constant((0,0)))],
'traction': [],
'pressure': [(facet_lookup['x_min'], Constant(0)),
(facet_lookup['x_max'], Rpressure)]}
else :
bcs_fluid = {'velocity': [(facet_lookup['y_min'],vf_bottom),
(facet_lookup['y_max'], Constant((0,0))),
(facet_lookup['x_max'], Constant((0,0)))], # I would like only normal flow to be zero
'traction': [],
'pressure': [(facet_lookup['x_min'], Constant(0))]}
if lateral_bc=='free' :
bcs_tracer = {'concentration': [(facet_lookup['x_max'], Constant(0)),
(facet_lookup['x_min'], Constant(0))],
'flux': [(facet_lookup['y_max'], Constant(0)),
(facet_lookup['y_min'], Constant(0))]}
else :
bcs_tracer = {'concentration': [(facet_lookup['x_min'], Constant(0))],
'flux': [(facet_lookup['x_max'], Constant(0)),
(facet_lookup['y_max'], Constant(0)),
(facet_lookup['y_min'], Constant(0))]}
bcs_ale = {'dirichlet': [(facet_lookup['y_min'], uale_bottom),
(facet_lookup['y_max'], Constant((0, 0)))],
'neumann': [(facet_lookup['x_min'], Constant((0, 0))),
(facet_lookup['x_max'], Constant((0, 0)))]}
# We collect the time dependent BC for update
driving_expressions = (uale_bottom,vf_bottom)
# Initialisation :
logging.info(title1("Initialisation"))
logging.info("\n * Fluid")
logging.info("Velocity : zero field")
logging.info("Pressure : zero field")
uf_n = project(Constant((0, 0)), Wf.sub(0).collapse())
pf_n = project(Constant(0), Wf.sub(1).collapse())
logging.info("\n * Tracer")
logging.info("Concentration : Gaussian profile")
logging.info(" Centered at mid length")
logging.info(" STD parameter = %e"%sigma_gauss)
c_0 = Expression('exp(-a*pow(x[0]-b, 2)) ', degree=1, a=1/2/sigma_gauss**2, b=xi_gauss)
c_n = project(c_0,Ct)
# Save initial state
uf_n.rename("uf", "tmp")
pf_n.rename("pf", "tmp")
c_n.rename("c", "tmp")
uf_out << (uf_n, 0)
pf_out << (pf_n, 0)
c_out << (c_n, 0)
files=[csv_p,csv_u,csv_c]
fields=[pf_n,uf_n.sub(0),c_n]
slice_line = line([0,(Rpvs+Rv)/2],[L,(Rpvs+Rv)/2], 100)
for csv_file,field in zip(files,fields) :
#print the x scale
values=np.linspace(0,L,100)
row=[0]+list(values)
csv_file.write(('%e'+', %e'*len(values)+'\n')%tuple(row))
#print the initial 1D slice
values = line_sample(slice_line, field)
row=[0]+list(values)
csv_file.write(('%e'+', %e'*len(values)+'\n')%tuple(row))
############# RUN ############3
logging.info(title1("Run"))
# Time loop
time = 0.
timestep=0
# Here I dont know if there will be several dt for advdiff and fluid solver
while time < tfinal:
# Update boundary conditions
for expr in driving_expressions:
hasattr(expr, 'tn') and setattr(expr, 'tn', time)
hasattr(expr, 'tnp1') and setattr(expr, 'tnp1', time+dt)
if lateral_bc=='resistance' :
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
Flow=assemble(2*pi*r*dot(uf_n, n)*ds(facet_lookup['x_max']))
print('Right outflow : %e \n'%Flow)
setattr(Rpressure, 'Q', Flow)
#Solve ALE and move mesh
eta_f = solve_ale(Va, f=Constant((0, 0)), bdries=fluid_bdries, bcs=bcs_ale,
parameters=ale_parameters)
ALE.move(mesh_f, eta_f)
mesh_f.bounding_box_tree().build(mesh_f)
# Solve fluid problem
uf_, pf_ = solve_fluid(Wf, u_0=uf_n, f=Constant((0, 0)), bdries=fluid_bdries, bcs=bcs_fluid,
parameters=fluid_parameters)
tracer_parameters["T0"]=time
tracer_parameters["nsteps"]=1
tracer_parameters["dt"]=dt
# Solve tracer problem
c_, T0= solve_adv_diff(Ct, velocity=uf_-eta_f/Constant(dt), phi=Constant(1), f=Constant(0), c_0=c_n, phi_0=Constant(1),
bdries=fluid_bdries, bcs=bcs_tracer, parameters=tracer_parameters)
# Update current solution
uf_n.assign(uf_)
pf_n.assign(pf_)
c_n.assign(c_)
#Update time
time += dt
timestep+=1
# Save output
if(timestep % int(toutput/dt) == 0):
logging.info("\n*** save output time %e s"%time)
logging.info("number of time steps %i"%timestep)
# may report Courant number or other important values that indicate how is doing the run
uf_.rename("uf", "tmp")
pf_.rename("pf", "tmp")
c_.rename("c", "tmp")
uf_out << (uf_, time)
pf_out << (pf_, time)
c_out << (c_, time)
# Get the 1 D profiles at umax (to be changed in cyl coordinate)
mesh_points=mesh_f.coordinates()
x=mesh_points[:,0]
y=mesh_points[:,1]
xmin=min(x)
xmax=max(x)
ymin=min(y)
ymax=max(y)
#slice_line = line([xmin,(ymin+ymax)/2],[xmax,(ymin+ymax)/2], 100)
logging.info('Rpvs : %e'%ymax)
logging.info('Rvn : %e'%ymin)
files=[csv_p,csv_u,csv_c]
fields=[pf_n,uf_n.sub(0),c_n]
field_names=['pressure (dyn/cm2)','axial velocity (cm/s)','concentration']
for csv_file,field,name in zip(files,fields,field_names) :
#values = line_sample(slice_line, field)
values =profile(field,xmin,xmax,ymin,ymax)
logging.info('Max '+name+' : %.2e'%max(abs(values)))
#logging.info('Norm '+name+' : %.2e'%field.vector().norm('linf'))
row=[time]+list(values)
csv_file.write(('%e'+', %e'*len(values)+'\n')%tuple(row))
csv_rv.write('%e, %e\n'%(time,ymin))
def PVS_simulation(args):
""" Solve the flow and tracer transport in the PVS, assumption flat plates :
Outputs :
- a logfile with information about the simulation
- .pvd files with the u, p and c field at specified args.toutput time period
Return : u, p, c 1D array of the u, p, c fields on the middle line """
# output folder name
outputfolder = args.output_folder + '/' + args.job_name + '/'
if not os.path.exists(outputfolder):
os.makedirs(outputfolder)
if not os.path.exists(outputfolder + '/profiles'):
os.makedirs(outputfolder + '/profiles')
if not os.path.exists(outputfolder + '/fields'):
os.makedirs(outputfolder + '/fields')
# Create output files
#txt files
csv_p = open(outputfolder + 'profiles' + '/pressure.txt', 'w')
csv_u = open(outputfolder + 'profiles' + '/velocity.txt', 'w')
csv_c = open(outputfolder + 'profiles' + '/concentration.txt', 'w')
csv_rv = open(outputfolder + 'profiles' + '/radius.txt', 'w')
#pvd files
uf_out, pf_out = File(outputfolder + 'fields' +
'/uf.pvd'), File(outputfolder + 'fields' + '/pf.pvd')
c_out = File(outputfolder + 'fields' + '/c.pvd')
facets_out = File(outputfolder + 'fields' + '/facets.pvd')
# Create logger
logger = logging.getLogger()
logger.setLevel(logging.INFO)
# log to a file
now = datetime.now().strftime("%Y%m%d_%H%M%S")
filename = os.path.join(outputfolder + '/', 'PVS_info.log')
file_handler = logging.FileHandler(filename, mode='w')
file_handler.setLevel(logging.INFO)
#formatter = logging.Formatter("%(asctime)s %(filename)s, %(lineno)d, %(funcName)s: %(message)s")
#file_handler.setFormatter(formatter)
logger.addHandler(file_handler)
# log to the console
console_handler = logging.StreamHandler()
level = logging.INFO
console_handler.setLevel(level)
logger.addHandler(console_handler)
# initialise logging
logging.info(
' ______ ______ _ _ _ _ \n'
)
logging.info(
'| _ \ \ / / ___| ___(_)_ __ ___ _ _| | __ _| |_(_) ___ _ __ \n'
)
logging.info(
"| |_) \ \ / /\___ \ / __| | '_ ` _ \| | | | |/ _` | __| |/ _ \| '_ \ \n"
)
logging.info(
'| __/ \ V / ___) | \__ \ | | | | | | |_| | | (_| | |_| | (_) | | | |\n'
)
logging.info(
'|_| \_/ |____/ |___/_|_| |_| |_|\__,_|_|\__,_|\__|_|\___/|_| |_|\n\n'
)
logging.info(
title1(
"Simulation of the PVS flow and tracer transport using steady stokes solver and diffusion-advection solver. Flat plates geometry"
))
logging.info("Date and time:" +
datetime.now().strftime("%m/%d/%Y, %H:%M:%S"))
logging.info('Job name : ' + args.job_name)
logging.debug('logging initialized')
# Set parameters
logging.info(title1("Parameters"))
# Geometry params
logging.info('\n * Geometry')
Rv = args.radius_vessel # centimeters
Rpvs = args.radius_pvs # centimeters
L = args.length # centimeters
Rsas = args.radius_sas + Rv
Lsas = args.length_sas
logging.info('Vessel radius : %e cm' % Rv)
logging.info('PVS radius : %e cm' % Rpvs)
logging.info('PVS length : %e cm' % L)
logging.info('SAS length : %e cm' % Lsas)
logging.info('SAS radius : %e cm' % Rsas)
logging.info('\n * Cross section area deformation parameters')
ai = args.ai
fi = args.fi
phii = args.phii
logging.info('ai (dimensionless): ' + '%e ' * len(ai) % tuple(ai))
logging.info('fi (Hz) : ' + '%e ' * len(fi) % tuple(fi))
logging.info('phii (rad) : ' + '%e ' * len(phii) % tuple(phii))
#Mesh
logging.info('\n * Mesh')
#number of cells in the radial direction
Nr = args.N_radial
DR = (Rpvs - Rv) / Nr
logging.info('N radial : %e' % Nr)
#time parameters
logging.info('\n * Time')
toutput = args.toutput
tfinal = args.tend
dt = args.time_step
logging.info('final time: %e s' % tfinal)
logging.info('output period : %e s' % toutput)
logging.info('time step : %e s' % dt)
# approximate CFL for fluid solver : need to compute max velocity depending on the wall displacement...
# maybe just add a warning in computation with actual velocity
#Uapprox=500e-4 #upper limit for extected max velocity
#CFL_dt=0.25*DY/Uapprox
#if CFL_dt < dt :
# logging.warning('The specified time step of %.2e s does not fullfil the fluid CFL condition. New fluid time step : %.2e s'%(dt, CFL_dt))
#dt_fluid=min(dt,CFL_dt)
dt_fluid = dt
# approximate CFL for tracer solver
dt_advdiff = dt
# material parameters
logging.info('\n * Fluid properties')
mu = args.viscosity
rho = args.density
logging.info('density: %e g/cm3' % rho)
logging.info('dynamic viscosity : %e dyn s/cm2' % mu)
logging.info('\n* Tracer properties')
D = args.diffusion_coef
sigma_gauss = args.sigma
logging.info('Free diffusion coef: %e cm2/s' % D)
logging.info('STD of initial gaussian profile: %e ' % sigma_gauss)
xi_gauss = args.initial_pos
logging.info('Initial position: %e cm2' % xi_gauss)
logging.info('\n * ALE')
kappa = args.ale_parameter
logging.info('ALE parameter: %e ' % kappa)
logging.info('\n * Lateral BC')
resistance = args.resistance
logging.info('inner resistance: %e ' % resistance)
if resistance == 0:
lateral_bc = 'free'
logging.info('right BC will be set to the free assumption')
elif resistance < 0:
lateral_bc = 'noflow'
logging.info('right BC will be set to the no flow assumption')
else:
lateral_bc = 'resistance'
logging.info('right BC will be set to the resistance assumption')
fluid_parameters = {'mu': mu, 'rho': rho, 'dt': dt_fluid}
tracer_parameters = {'kappa': D, 'dt': dt_advdiff}
ale_parameters = {'kappa': kappa}
# Mesh
logging.info(title1('Meshing'))
logging.info('cell size : %e cm' % (DR))
from sleep.mesh import mesh_model2d, load_mesh2d, set_mesh_size
import sys
gmsh.initialize(['', '-format', 'msh2'])
model = gmsh.model
import math
Apvs0 = math.pi * Rpvs**2
Av0 = math.pi * Rv**2
A0 = Apvs0 - Av0
# progressive mesh
factory = model.occ
a = factory.addPoint(-Lsas, Rv, 0)
b = factory.addPoint(L, Rv, 0)
c = factory.addPoint(L, Rpvs, 0)
d = factory.addPoint(0, Rpvs, 0)
e = factory.addPoint(0, Rsas, 0)
f = factory.addPoint(-Lsas, Rsas, 0)
fluid_lines = [
factory.addLine(*p)
for p in ((a, b), (b, c), (c, d), (d, e), (e, f), (f, a))
]
named_lines = dict(
zip(('bottom', 'pvs_right', 'pvs_top', 'brain_surf', 'sas_top',
'sas_left'), fluid_lines))
fluid_loop = factory.addCurveLoop(fluid_lines)
fluid = factory.addPlaneSurface([fluid_loop])
factory.synchronize()
model.addPhysicalGroup(2, [fluid], 1)
for name in named_lines:
tag = named_lines[name]
model.addPhysicalGroup(1, [tag], tag)
# boxes for mesh refinement
cell_size = DR * (Rpvs - Rv) / (Rpvs - Rv)
boxes = []
# add box on the PVS for mesh
field = model.mesh.field
fid = 1
field.add('Box', fid)
field.setNumber(fid, 'XMin', 0)
field.setNumber(fid, 'XMax', L)
field.setNumber(fid, 'YMin', Rv)
field.setNumber(fid, 'YMax', Rpvs)
field.setNumber(fid, 'VIn', cell_size)
field.setNumber(fid, 'VOut', DR * 50)
field.setNumber(fid, 'Thickness', Rsas)
boxes.append(fid)
# Combine
field.add('Min', 2)
field.setNumbers(2, 'FieldsList', boxes)
field.setAsBackgroundMesh(2)
model.occ.synchronize()
h5_filename = outputfolder + '/mesh.h5'
tags = {'cell': {'F': 1}, 'facet': {}}
mesh_model2d(model, tags, h5_filename)
mesh_f, markers, lookup = load_mesh2d(h5_filename)
gmsh.finalize()
# todo : how to know nb of cells ?
#logging.info('nb cells: %i'%(Nl*Nr*2))
fluid_bdries = MeshFunction("size_t", mesh_f,
mesh_f.topology().dim() - 1, 0)
# Label facets
xy = mesh_f.coordinates().copy()
x, y = xy.T
xmin = x.min()
xmax = x.max()
ymin = y.min()
ymax = y.max()
tol = cell_size / 2 #cm
class Boundary_sas_left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], -Lsas, tol)
# downstream
class Boundary_pvs_right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], L, tol)
class Boundary_sas_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], Rsas, tol) and (x[0] < tol)
class Boundary_pvs_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], Rpvs, tol) and (x[0] > -tol)
# brain
class Boundary_brainsurf(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], 0, tol) and (x[1] > Rpvs - tol)
class Boundary_bottom(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymin, tol)
btop_sas = Boundary_sas_top()
btop_pvs = Boundary_pvs_top()
bbottom = Boundary_bottom()
bvert_left = Boundary_sas_left()
bvert_brain = Boundary_brainsurf()
bvert_right = Boundary_pvs_right()
btop_sas.mark(fluid_bdries, 1)
btop_pvs.mark(fluid_bdries, 2)
bbottom.mark(fluid_bdries, 3)
bvert_left.mark(fluid_bdries, 4)
bvert_brain.mark(fluid_bdries, 5)
bvert_right.mark(fluid_bdries, 6)
facet_lookup = {
'sas_top': 1,
'pvs_top': 2,
'vessel': 3,
'sas_left': 4,
'brain_surf': 5,
'pvs_right': 6
}
facets_out << fluid_bdries
#FEM space
logging.info(title1("Set FEM spaces"))
logging.info('\n * Fluid')
Vf_elm = VectorElement('Lagrange', triangle, 2)
Qf_elm = FiniteElement('Lagrange', triangle, 1)
Wf_elm = MixedElement([Vf_elm, Qf_elm])
Wf = FunctionSpace(mesh_f, Wf_elm)
logging.info('Velocity : "Lagrange", triangle, 2')
logging.info('Pressure : "Lagrange", triangle, 1')
logging.info('\n * Tracer')
Ct_elm = FiniteElement('Lagrange', triangle, 1)
Ct = FunctionSpace(mesh_f, Ct_elm)
logging.info('Concentration : "Lagrange", triangle, 1')
logging.info('\n * ALE')
Va_elm = VectorElement('Lagrange', triangle, 1)
Va = FunctionSpace(mesh_f, Va_elm)
logging.info('ALE displacement: "Lagrange", triangle, 1')
# Setup of boundary conditions
logging.info(title1("Boundary conditions"))
import sympy
tn = sympy.symbols("tn")
tnp1 = sympy.symbols("tnp1")
sin = sympy.sin
sqrt = sympy.sqrt
functionR = Rpvs - (Rpvs - Rv) * (1 + sum(
[a * sin(2 * pi * f * tn + phi)
for a, f, phi in zip(ai, fi, phii)])) # displacement bottom plate
R_vessel = sympy.printing.ccode(functionR)
functionV = sympy.diff(functionR, tn) # velocity
V_vessel = sympy.printing.ccode(functionV)
#Delta U for ALE. I dont really like this
functionUALE = -(Rpvs - Rv) * (1 + sum([
a * sin(2 * pi * f * tnp1 + phi) for a, f, phi in zip(ai, fi, phii)
])) + (Rpvs - Rv) * (1 + sum(
[a * sin(2 * pi * f * tn + phi) for a, f, phi in zip(ai, fi, phii)]))
UALE_vessel = sympy.printing.ccode(functionUALE)
vf_bottom = Expression(('0', V_vessel), tn=0,
degree=2) # no slip no gap condition at vessel wall
uale_bottom = Expression(('0', UALE_vessel), tn=0, tnp1=1,
degree=2) # displacement for ALE at vessel wall
logging.info('\n * Lateral assumption')
logging.info(lateral_bc)
logging.info('\n * Fluid')
logging.info('Left : zero pressure')
if lateral_bc == 'free':
logging.info('Right : zero pressure')
elif lateral_bc == 'resistance':
logging.info('Right : resistance')
else:
logging.info('Right : no flow')
logging.info('Top : no slip no gap fixed wall')
logging.info('Bottom : no slip no gap moving wall')
logging.info('\n * Tracer concentration')
logging.info('Left : zero concentration')
if lateral_bc == 'free':
logging.info('Right : zero concentration')
else:
logging.info('Right : no flux')
logging.info('Top : no flux')
logging.info('Bottom : no flux')
logging.info('\n * ALE')
logging.info('Left : no flux')
logging.info('Right : no flux')
logging.info('Top : no displacement')
logging.info('Bottom : vessel displacement')
# Now we wire up
if lateral_bc == 'free':
bcs_fluid = {
'velocity': [(facet_lookup['vessel'], vf_bottom),
(facet_lookup['sas_left'], Constant((0, 0))),
(facet_lookup['brain_surf'], Constant((0, 0))),
(facet_lookup['pvs_top'], Constant((0, 0)))],
'traction': [],
'pressure': [(facet_lookup['sas_top'], Constant(0)),
(facet_lookup['pvs_right'], Constant(0))]
}
elif lateral_bc == 'resistance':
Rpressure = Expression('R*Q+p0', R=resistance, Q=0, p0=0, degree=1) #
# Compute pressure to impose according to the flow at previous time step and resistance.
bcs_fluid = {
'velocity': [(facet_lookup['vessel'], vf_bottom),
(facet_lookup['sas_left'], Constant((0, 0))),
(facet_lookup['brain_surf'], Constant((0, 0))),
(facet_lookup['pvs_top'], Constant((0, 0)))],
'traction': [],
'pressure': [(facet_lookup['sas_top'], Constant(0)),
(facet_lookup['pvs_right'], Rpressure)]
}
else:
bcs_fluid = {
'velocity':
[(facet_lookup['vessel'], vf_bottom),
(facet_lookup['sas_left'], Constant((0, 0))),
(facet_lookup['brain_surf'], Constant((0, 0))),
(facet_lookup['pvs_top'], Constant((0, 0))),
(facet_lookup['pvs_right'], Constant(
(0, 0)))], # would be more correct to impose only on u x
'traction': [],
'pressure': [(facet_lookup['sas_top'], Constant(0))]
}
if lateral_bc == 'free':
bcs_tracer = {
'concentration': [(facet_lookup['sas_top'], Constant(0)),
(facet_lookup['pvs_right'], Constant(0))],
'flux': [(facet_lookup['vessel'], Constant(0)),
(facet_lookup['sas_left'], Constant(0)),
(facet_lookup['brain_surf'], Constant(0)),
(facet_lookup['pvs_top'], Constant(0))]
}
else:
bcs_tracer = {
'concentration': [(facet_lookup['sas_top'], Constant(0))],
'flux': [(facet_lookup['vessel'], Constant(0)),
(facet_lookup['sas_left'], Constant(0)),
(facet_lookup['brain_surf'], Constant(0)),
(facet_lookup['pvs_top'], Constant(0)),
(facet_lookup['pvs_right'], Constant(0))]
}
bcs_ale = {
'dirichlet': [(facet_lookup['vessel'], uale_bottom),
(facet_lookup['sas_top'], Constant((0, 0))),
(facet_lookup['pvs_top'], Constant((0, 0))),
(facet_lookup['brain_surf'], Constant((0, 0)))],
'neumann': [(facet_lookup['sas_left'], Constant((0, 0))),
(facet_lookup['pvs_right'], Constant((0, 0)))]
}
# We collect the time dependent BC for update
driving_expressions = (uale_bottom, vf_bottom)
# Initialisation :
logging.info(title1("Initialisation"))
# We start at a time shift
tshift = -1 / 4 / fi[0] ##todo : modify to be compatible with phii
# Update boundary conditions
for expr in driving_expressions:
hasattr(expr, 'tn') and setattr(expr, 'tn', 0)
hasattr(expr, 'tnp1') and setattr(expr, 'tnp1', tshift)
#Solve ALE and move mesh
eta_f = solve_ale(Va,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_ale,
parameters=ale_parameters)
ALE.move(mesh_f, eta_f)
mesh_f.bounding_box_tree().build(mesh_f)
logging.info("\n * Fluid")
logging.info("Velocity : zero field")
logging.info("Pressure : zero field")
uf_n = project(Constant((0, 0)), Wf.sub(0).collapse())
pf_n = project(Constant(0), Wf.sub(1).collapse())
logging.info("\n * Tracer")
logging.info("Concentration : Gaussian profile")
logging.info(" Centered at mid length")
logging.info(" STD parameter = %e" % sigma_gauss)
c_0 = Expression('exp(-a*pow(x[0]-b, 2)) ',
degree=1,
a=1 / 2 / sigma_gauss**2,
b=xi_gauss)
c_n = project(c_0, Ct)
# Save initial state
uf_n.rename("uf", "tmp")
pf_n.rename("pf", "tmp")
c_n.rename("c", "tmp")
uf_out << (uf_n, 0)
pf_out << (pf_n, 0)
c_out << (c_n, 0)
# Get the 1 D profiles at umax (to be changed in cyl coordinate)
mesh_points = mesh_f.coordinates()
x = mesh_points[:, 0]
y = mesh_points[:, 1]
xmin = 0
xmax = L
ymin = min(y)
ymax = Rpvs
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
field_names = [
'pressure (dyn/cm2)', 'axial velocity (cm/s)', 'concentration'
]
for csv_file, field, name in zip(files, fields, field_names):
#values = line_sample(slice_line, field)
values = profile(field, xmin, xmax, ymin, ymax)
logging.info('Max ' + name + ' : %.2e' % max(abs(values)))
#logging.info('Norm '+name+' : %.2e'%field.vector().norm('linf'))
row = [0] + list(values)
csv_file.write(('%e' + ', %e' * len(values) + '\n') % tuple(row))
csv_rv.write('%e, %e' % (0, ymin))
############# RUN ############3
logging.info(title1("Run"))
# Time loop
### We start at the quarter of the period so that velocity=0
print('tini = ', tshift)
time = tshift
timestep = 0
# Here I dont know if there will be several dt for advdiff and fluid solver
while time < tfinal + tshift:
# Update boundary conditions
for expr in driving_expressions:
hasattr(expr, 'tn') and setattr(expr, 'tn', time)
hasattr(expr, 'tnp1') and setattr(expr, 'tnp1', time + dt)
if lateral_bc == 'resistance':
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
Flow = assemble(2 * pi * r * dot(uf_n, n) *
ds(facet_lookup['x_max']))
print('Right outflow : %e \n' % Flow)
setattr(Rpressure, 'Q', Flow)
#Solve ALE and move mesh
eta_f = solve_ale(Va,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_ale,
parameters=ale_parameters)
ALE.move(mesh_f, eta_f)
mesh_f.bounding_box_tree().build(mesh_f)
# Solve fluid problem
uf_, pf_ = solve_fluid(Wf,
u_0=uf_n,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_fluid,
parameters=fluid_parameters)
tracer_parameters["T0"] = time
tracer_parameters["nsteps"] = 1
tracer_parameters["dt"] = dt
# Solve tracer problem
c_, T0 = solve_adv_diff(Ct,
velocity=uf_ - eta_f / Constant(dt),
phi=Constant(1),
f=Constant(0),
c_0=c_n,
phi_0=Constant(1),
bdries=fluid_bdries,
bcs=bcs_tracer,
parameters=tracer_parameters)
# Update current solution
uf_n.assign(uf_)
pf_n.assign(pf_)
c_n.assign(c_)
#Update time
time += dt
timestep += 1
# Save output
if (timestep % int(toutput / dt) == 0):
logging.info("\n*** save output time %e s" % (time - tshift))
logging.info("number of time steps %i" % timestep)
# may report Courant number or other important values that indicate how is doing the run
uf_.rename("uf", "tmp")
pf_.rename("pf", "tmp")
c_.rename("c", "tmp")
uf_out << (uf_, time - tshift)
pf_out << (pf_, time - tshift)
c_out << (c_, time - tshift)
# Get the 1 D profiles at umax (to be changed in cyl coordinate)
mesh_points = mesh_f.coordinates()
x = mesh_points[:, 0]
y = mesh_points[:, 1]
xmin = 0
xmax = L
ymin = min(y)
ymax = Rpvs
#slice_line = line([xmin,(ymin+ymax)/2],[xmax,(ymin+ymax)/2], 100)
logging.info('Rpvs : %e' % ymax)
logging.info('Rvn : %e' % ymin)
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
field_names = [
'pressure (dyn/cm2)', 'axial velocity (cm/s)', 'concentration'
]
for csv_file, field, name in zip(files, fields, field_names):
#values = line_sample(slice_line, field)
values = profile(field, xmin, xmax, ymin, ymax)
logging.info('Max ' + name + ' : %.2e' % max(abs(values)))
#logging.info('Norm '+name+' : %.2e'%field.vector().norm('linf'))
row = [time - tshift] + list(values)
csv_file.write(
('%e' + ', %e' * len(values) + '\n') % tuple(row))
csv_rv.write('%e, %e' % (time - tshift, ymin))
def SAS_simulation(args):
""" Solve the flow and tracer transport in the PVS :
Outputs :
- a logfile with information about the simulation
- .pvd files with the u, p and c field at specified args.toutput time period
Return : u, p, c 1D array of the u, p, c fields on the middle line """
# output folder name
outputfolder = args.output_folder + '/' + args.job_name + '/'
if not os.path.exists(outputfolder):
os.makedirs(outputfolder)
if not os.path.exists(outputfolder + '/profiles'):
os.makedirs(outputfolder + '/profiles')
if not os.path.exists(outputfolder + '/fields'):
os.makedirs(outputfolder + '/fields')
# Create output files
#txt files
csv_p = open(outputfolder + 'profiles' + '/pressure.txt', 'w')
csv_u = open(outputfolder + 'profiles' + '/velocity.txt', 'w')
csv_c = open(outputfolder + 'profiles' + '/concentration.txt', 'w')
csv_rv = open(outputfolder + 'profiles' + '/radius.txt', 'w')
#pvd files
uf_out, pf_out = File(outputfolder + 'fields' +
'/uf.pvd'), File(outputfolder + 'fields' + '/pf.pvd')
c_out = File(outputfolder + 'fields' + '/c.pvd')
facets_out = File(outputfolder + 'fields' + '/facets.pvd')
# Create logger
logger = logging.getLogger()
logger.setLevel(logging.INFO)
# log to a file
now = datetime.now().strftime("%Y%m%d_%H%M%S")
filename = os.path.join(outputfolder + '/', 'PVS_info.log')
file_handler = logging.FileHandler(filename, mode='w')
file_handler.setLevel(logging.INFO)
#formatter = logging.Formatter("%(asctime)s %(filename)s, %(lineno)d, %(funcName)s: %(message)s")
#file_handler.setFormatter(formatter)
logger.addHandler(file_handler)
# log to the console
console_handler = logging.StreamHandler()
level = logging.INFO
console_handler.setLevel(level)
logger.addHandler(console_handler)
# initialise logging
logging.info(
title1(
"Simulation of the PVS flow and tracer transport using non steady solver and diffusion-advection solvers"
))
logging.info("Date and time:" +
datetime.now().strftime("%m/%d/%Y, %H:%M:%S"))
logging.info('Job name : ' + args.job_name)
logging.debug('logging initialized')
# Set parameters
logging.info(title1("Parameters"))
# Geometry params
logging.info('\n * Geometry')
Rv = args.radius_inner # centimeters
Rpvs = args.radius_outer # centimeters
L = args.length # centimeters
logging.info('Inner radius : %e cm' % Rv)
logging.info('Outer radius : %e cm' % Rpvs)
logging.info('Length : %e cm' % L)
#Mesh
logging.info('\n * Mesh')
#number of cells in the radial direction
Nr = args.N_radial
DR = (Rpvs - Rv) / Nr
#number of cells in the axial direction
if args.N_axial:
Nl = args.N_axial
else:
Nl = round(L / DR)
DY = L / Nl
logging.info('N axial : %i' % Nl)
logging.info('N radial : %e' % Nr)
#time parameters
logging.info('\n * Time')
toutput = args.toutput
tfinal = args.tend
dt = args.time_step
logging.info('final time: %e s' % tfinal)
logging.info('output period : %e s' % toutput)
logging.info('time step : %e s' % dt)
# approximate CFL for fluid solver : need to compute max velocity depending on the wall displacement...
# maybe just add a warning in computation with actual velocity
#Uapprox=500e-4 #upper limit for extected max velocity
#CFL_dt=0.25*DY/Uapprox
#if CFL_dt < dt :
# logging.warning('The specified time step of %.2e s does not fullfil the fluid CFL condition. New fluid time step : %.2e s'%(dt, CFL_dt))
#dt_fluid=min(dt,CFL_dt)
dt_fluid = dt
# approximate CFL for tracer solver
dt_advdiff = dt
# material parameters
logging.info('\n * Fluid properties')
mu = args.viscosity
rho = args.density
logging.info('density: %e g/cm3' % rho)
logging.info('dynamic viscosity : %e dyn s/cm2' % mu)
logging.info('\n* Tracer properties')
D = args.diffusion_coef
sigma_gauss = args.sigma
logging.info('Free diffusion coef: %e cm2/s' % D)
logging.info('STD of initial gaussian profile: %e ' % sigma_gauss)
xi_gauss = args.initial_pos
logging.info('Initial position: %e cm2' % xi_gauss)
logging.info('\n * Lateral BC')
compliance = args.compliance
logging.info('outer compliance: %e ' % compliance)
if compliance <= 0:
lateral_bc = 'free'
logging.info('right BC will be set to the free assumption')
else:
lateral_bc = 'compliance'
logging.info('right BC will be set to the compliance assumption')
fluid_parameters = {'mu': mu, 'rho': rho, 'dt': dt_fluid}
tracer_parameters = {'kappa': D, 'dt': dt_advdiff}
# Mesh
logging.info(title1('Meshing'))
logging.info('cell size : %e cm' % (np.sqrt(DR**2 + DY**2)))
logging.info('nb cells: %i' % (Nl * Nr * 2))
mesh_f = RectangleMesh(Point(0, Rv), Point(L, Rpvs), Nl, Nr)
fluid_bdries = MeshFunction("size_t", mesh_f,
mesh_f.topology().dim() - 1, 0)
# Label facets
xy = mesh_f.coordinates().copy()
x, y = xy.T
xmin = x.min()
xmax = x.max()
ymin = y.min()
ymax = y.max()
tol = min(DR, DY) / 2 #cm
class Boundary_left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], xmin, tol) #left
class Boundary_right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], xmax, tol) # right
class Boundary_bottom(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymin, tol) #bottom
class Boundary_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymax, tol) #top
btop = Boundary_top()
bbottom = Boundary_bottom()
bleft = Boundary_left()
bright = Boundary_right()
bbottom.mark(fluid_bdries, 2)
btop.mark(fluid_bdries, 4)
bleft.mark(fluid_bdries, 1)
bright.mark(fluid_bdries, 3)
facet_lookup = {'x_min': 1, 'y_min': 2, 'x_max': 3, 'y_max': 4}
facets_out << fluid_bdries
#FEM space
logging.info(title1("Set FEM spaces"))
logging.info('\n * Fluid')
Vf_elm = VectorElement('Lagrange', triangle, 2)
Qf_elm = FiniteElement('Lagrange', triangle, 1)
Wf_elm = MixedElement([Vf_elm, Qf_elm])
Wf = FunctionSpace(mesh_f, Wf_elm)
logging.info('Velocity : "Lagrange", triangle, 2')
logging.info('Pressure : "Lagrange", triangle, 1')
logging.info('\n * Tracer')
Ct_elm = FiniteElement('Lagrange', triangle, 1)
Ct = FunctionSpace(mesh_f, Ct_elm)
logging.info('Concentration : "Lagrange", triangle, 1')
# Setup of boundary conditions
logging.info(title1("Boundary conditions"))
import sympy
tn = sympy.symbols("tn")
sin = sympy.sin
cos = sympy.cos
sqrt = sympy.sqrt
logging.info('\n * inner face flow parameters')
ai = args.ai
fi = args.fi
phii = args.phii
logging.info('ai (dimensionless): ' + '%e ' * len(ai) % tuple(ai))
logging.info('fi (Hz) : ' + '%e ' * len(fi) % tuple(fi))
logging.info('phii (rad) : ' + '%e ' * len(phii) % tuple(phii))
#Brain outflow due to vessel contraction/dilation
#blood_vol=12.5e-3#*1.5 #cm3 total cerebral blood volume
brainvolume = pi * Rv**2 * L
brainsurface = 2 * pi * Rv * L
blood_vol = 0.035 * brainvolume
pc_arterioles = 0.01 #pc of total blood volume in arterioles
V0_arterioles = pc_arterioles * blood_vol
Asas = pi * (Rpvs**2) - pi * Rv**2
#CSF production
Qprod = 6e-6 #cm3/s
#Steady velocity due to production
Uprod = Qprod / Asas
#Concentration in the brain
cbrain = Expression('c0*(1-tn)', c0=1, tn=0, degree=2)
logging.info('brain volume: %e cm3 ' % brainvolume)
logging.info('brain surface: %e cm2 ' % brainsurface)
logging.info('blood volume: %e cm3 ' % blood_vol)
logging.info('pourcentage of arterioles: %e pc ' % (pc_arterioles * 100))
functionV = V0_arterioles * (1 + sum(
[a * sin(2 * pi * f * tn + phi)
for a, f, phi in zip(ai, fi, phii)])) # blood volume
functionU = sympy.diff(functionV, tn) / brainsurface # velocity
#functionU=V0_arterioles*(sum([a*2*pi*f*cos(2*pi*f*tn+phi) for a,f,phi in zip(ai,fi,phii)]))/brainsurface
ubottom = sympy.printing.ccode(functionU)
vf_bottom = Expression(('0', ubottom), tn=0,
degree=2) # no slip no gap condition at vessel wall
logging.info('\n * Lateral assumption')
logging.info(lateral_bc)
logging.info('\n * Fluid')
logging.info('Left : zero pressure')
if lateral_bc == 'free':
logging.info('Right : zero pressure')
else:
logging.info('Right : compliance')
logging.info('Top : no slip no gap fixed wall')
logging.info('Bottom : no slip and brain inflow')
logging.info('\n * Tracer concentration')
logging.info('Left : zero concentration')
if lateral_bc == 'free':
logging.info('Right : zero concentration')
else:
logging.info('Right : no flux')
logging.info('Top : no flux')
logging.info('Bottom : no flux')
# Now we wire up
if lateral_bc == 'free':
bcs_fluid = {
'velocity': [(facet_lookup['y_min'], vf_bottom),
(facet_lookup['y_max'], Constant((0, 0))),
(facet_lookup['x_max'], Constant((-Uprod, 0)))],
'traction': [],
'pressure': [(facet_lookup['x_min'], Constant(0))]
}
else:
E = 100
P0 = 5330
Pn = 5330
Pout = Expression('Pn', Pn=Pn, degree=0) #
# Compute pressure to impose according to the flow at previous time step and compliance
bcs_fluid = {
'velocity': [(facet_lookup['y_min'], vf_bottom),
(facet_lookup['y_max'], Constant((0, 0))),
(facet_lookup['x_max'], Constant((-Uprod, 0)))],
'traction': [],
'pressure': [(facet_lookup['x_min'], Pout)]
}
if lateral_bc == 'free':
bcs_tracer = {
'concentration': [(facet_lookup['x_max'], Constant(0)),
(facet_lookup['x_min'], Constant(0)),
(facet_lookup['y_min'], cbrain)],
'flux': [(facet_lookup['y_max'], Constant(0))]
}
else:
bcs_tracer = {
'concentration': [(facet_lookup['x_min'], Constant(0)),
(facet_lookup['y_min'], cbrain)],
'flux': [(facet_lookup['x_max'], Constant(0)),
(facet_lookup['y_max'], Constant(0))]
}
# We collect the time dependent BC for update
driving_expressions = (vf_bottom)
# Initialisation :
logging.info(title1("Initialisation"))
logging.info("\n * Fluid")
logging.info("Velocity : zero field")
logging.info("Pressure : zero field")
uf_n = project(Constant((0, 0)), Wf.sub(0).collapse())
pf_n = project(Constant(0), Wf.sub(1).collapse())
#logging.info("\n * Tracer")
#logging.info("Concentration : Gaussian profile")
#logging.info(" Centered at mid length")
#logging.info(" STD parameter = %e"%sigma_gauss)
#c_0 = Expression('exp(-a*pow(x[0]-b, 2)) ', degree=1, a=1/2/sigma_gauss**2, b=xi_gauss)
c_n = project(Constant(0), Ct)
# Save initial state
uf_n.rename("uf", "tmp")
pf_n.rename("pf", "tmp")
c_n.rename("c", "tmp")
uf_out << (uf_n, 0)
pf_out << (pf_n, 0)
c_out << (c_n, 0)
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
slice_line = line([0, (Rpvs + Rv) / 2], [L, (Rpvs + Rv) / 2], 100)
for csv_file, field in zip(files, fields):
#print the x scale
values = np.linspace(0, L, 100)
row = [0] + list(values)
csv_file.write(('%e' + ', %e' * len(values) + '\n') % tuple(row))
#print the initial 1D slice
values = line_sample(slice_line, field)
row = [0] + list(values)
csv_file.write(('%e' + ', %e' * len(values) + '\n') % tuple(row))
############# RUN ############3
logging.info(title1("Run"))
# Time loop
time = 0.
timestep = 0
intQout = 0
# Here I dont know if there will be several dt for advdiff and fluid solver
while time < tfinal:
# Update boundary conditions
for expr in driving_expressions:
hasattr(expr, 'tn') and setattr(expr, 'tn', time)
vf_bottom.tn = time
if lateral_bc == 'compliance':
# Compliance law dP/dt=dP/dV.dV/dt=dP/dV.qout=E*P*qout
#qout=2 pi int_Rv^Rpvs u r dr
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
Qout = assemble(2 * pi * r * dot(uf_n, n) *
ds(facet_lookup['x_max']))
#Pn+1=pn+dt*(E*P*qoutn)
#Pout.Pn=Pn+dt*E*Pn*Qout
# dP/dt=dP/dV.dV/dt=dP/dV.qout=E*P*qout
# ln(p)-ln(p0)=E int_0^t Qout
#p=p0exp(E int_0^t Qout)
intQout += Qout * dt
Pout.Pn = P0 * exp(E * intQout)
print('Right pressure : %e mmHg \n' % (Pout.Pn / 1333))
# Solve fluid problem
uf_, pf_ = solve_fluid(Wf,
u_0=uf_n,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_fluid,
parameters=fluid_parameters)
tracer_parameters["T0"] = time
tracer_parameters["nsteps"] = 1
tracer_parameters["dt"] = dt
# Solve tracer problem
c_, T0 = solve_adv_diff(Ct,
velocity=uf_,
mesh_displacement=Constant((0, 0)),
f=Constant(0),
phi_0=c_n,
bdries=fluid_bdries,
bcs=bcs_tracer,
parameters=tracer_parameters)
# Update current solution
uf_n.assign(uf_)
pf_n.assign(pf_)
c_n.assign(c_)
#Update time
time += dt
timestep += 1
# Save output
if (timestep % int(toutput / dt) == 0):
logging.info("\n*** save output time %e s" % time)
logging.info("number of time steps %i" % timestep)
# may report Courant number or other important values that indicate how is doing the run
uf_.rename("uf", "tmp")
pf_.rename("pf", "tmp")
c_.rename("c", "tmp")
uf_out << (uf_, time)
pf_out << (pf_, time)
c_out << (c_, time)
# Get the 1 D profiles at umax (to be changed in cyl coordinate)
mesh_points = mesh_f.coordinates()
x = mesh_points[:, 0]
y = mesh_points[:, 1]
xmin = min(x)
xmax = max(x)
ymin = min(y)
ymax = max(y)
#slice_line = line([xmin,(ymin+ymax)/2],[xmax,(ymin+ymax)/2], 100)
logging.info('Rpvs : %e' % ymax)
logging.info('Rvn : %e' % ymin)
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
field_names = [
'pressure (dyn/cm2)', 'axial velocity (cm/s)', 'concentration'
]
for csv_file, field, name in zip(files, fields, field_names):
#values = line_sample(slice_line, field)
values = profile_cyl(field, xmin, xmax, ymin, ymax)
logging.info('Max ' + name + ' : %.2e' % max(abs(values)))
#logging.info('Norm '+name+' : %.2e'%field.vector().norm('linf'))
row = [time] + list(values)
csv_file.write(
('%e' + ', %e' * len(values) + '\n') % tuple(row))
csv_rv.write('%e, %e' % (time, ymin))
def PVS_simulation(args):
""" Solve the flow and tracer transport in the PVS :
Outputs :
- a logfile with information about the simulation
- .pvd files with the u, p and c field at specified args.toutput time period
Return : u, p, c 1D array of the u, p, c fields on the middle line """
# output folder name
outputfolder = args.output_folder + '/' + args.job_name + '/'
if not os.path.exists(outputfolder):
os.makedirs(outputfolder)
#if not os.path.exists(outputfolder+'profiles'):
# os.makedirs(outputfolder+'profiles')
if not os.path.exists(outputfolder + 'fields'):
os.makedirs(outputfolder + 'fields')
# Create output files
#txt files
csv_p = open(args.output_folder + '/' + args.job_name + '_pressure.txt',
'w')
csv_u = open(args.output_folder + '/' + args.job_name + '_velocity.txt',
'w')
csv_c = open(
args.output_folder + '/' + args.job_name + '_concentration.txt', 'w')
csv_rv = open(args.output_folder + '/' + args.job_name + '_radius.txt',
'w')
csv_mass = open(args.output_folder + '/' + args.job_name + '_mass.txt',
'w')
#pvd files
uf_out, pf_out = File(outputfolder + 'fields' +
'/uf.pvd'), File(outputfolder + 'fields' + '/pf.pvd')
c_out = File(outputfolder + 'fields' + '/c.pvd')
facets_out = File(outputfolder + 'fields' + '/facets.pvd')
# Create logger
logger = logging.getLogger()
logger.setLevel(logging.INFO)
# log to a file
now = datetime.now().strftime("%Y%m%d_%H%M%S")
filename = args.output_folder + '/' + args.job_name + '_PVSinfo.log'
file_handler = logging.FileHandler(filename, mode='w')
file_handler.setLevel(logging.INFO)
#formatter = logging.Formatter("%(asctime)s %(filename)s, %(lineno)d, %(funcName)s: %(message)s")
#file_handler.setFormatter(formatter)
logger.addHandler(file_handler)
# log to the console
console_handler = logging.StreamHandler()
level = logging.INFO
console_handler.setLevel(level)
logger.addHandler(console_handler)
# initialise logging
# initialise logging
logging.info(
' ______ ______ _ _ _ _ '
)
logging.info(
'| _ \ \ / / ___| ___(_)_ __ ___ _ _| | __ _| |_(_) ___ _ __ '
)
logging.info(
"| |_) \ \ / /\___ \ / __| | '_ ` _ \| | | | |/ _` | __| |/ _ \| '_ \ "
)
logging.info(
'| __/ \ V / ___) | \__ \ | | | | | | |_| | | (_| | |_| | (_) | | | |'
)
logging.info(
'|_| \_/ |____/ |___/_|_| |_| |_|\__,_|_|\__,_|\__|_|\___/|_| |_|\n'
)
logging.info(
title1(
"Simulation of the PVS flow and tracer transport using stoke solver and diffusion-advection solver"
))
logging.info("Date and time:" +
datetime.now().strftime("%m/%d/%Y, %H:%M:%S"))
logging.info('Job name : ' + args.job_name)
logging.debug('logging initialized')
# Set parameters
logging.info(title1("Parameters"))
# Geometry params
logging.info('\n * Geometry')
Rv = args.radius_vessel # centimeters
Rpvs = args.radius_pvs # centimeters
L = args.length # centimeters
logging.info('Vessel radius : %e cm' % Rv)
logging.info('PVS radius : %e cm' % Rpvs)
logging.info('PVS length : %e cm' % L)
# test presence of the SAS compartment on the mesh
isSAS = args.SAS
if isSAS:
logging.info('Add a SAS compartment on the left')
Rsas = args.radius_sas + Rv
Lsas = args.length_sas
logging.info('SAS length : %e cm' % Lsas)
logging.info('SAS radius : %e cm' % Rsas)
#Mesh
logging.info('\n * Mesh')
#number of cells in the radial direction
Nr = args.N_radial
DR = (Rpvs - Rv) / Nr
#number of cells in the axial direction
if args.N_axial:
Nl = args.N_axial
else:
Nl = round(L / DR)
DY = L / Nl
logging.info('N axial : %i' % Nl)
logging.info('N radial : %e' % Nr)
#time parameters
logging.info('\n * Time')
toutput = args.toutput
tfinal = args.tend
dt = args.time_step
logging.info('final time: %e s' % tfinal)
logging.info('output period : %e s' % toutput)
logging.info('time step : %e s' % dt)
# approximate CFL for fluid solver : need to compute max velocity depending on the wall displacement...
# maybe just add a warning in computation with actual velocity
#Uapprox=500e-4 #upper limit for extected max velocity
#CFL_dt=0.25*DY/Uapprox
#if CFL_dt < dt :
# logging.warning('The specified time step of %.2e s does not fullfil the fluid CFL condition. New fluid time step : %.2e s'%(dt, CFL_dt))
#dt_fluid=min(dt,CFL_dt)
dt_fluid = dt
# approximate CFL for tracer solver
dt_advdiff = dt
# material parameters
logging.info('\n * Fluid properties')
mu = args.viscosity
rho = args.density
logging.info('density: %e g/cm3' % rho)
logging.info('dynamic viscosity : %e dyn s/cm2' % mu)
logging.info('\n* Tracer properties')
D = args.diffusion_coef
sigma_gauss = args.sigma
logging.info('Free diffusion coef: %e cm2/s' % D)
logging.info('STD of initial gaussian profile: %e ' % sigma_gauss)
xi_gauss = args.initial_pos
logging.info('Initial position: %e cm2' % xi_gauss)
logging.info('\n * ALE')
kappa = args.ale_parameter
logging.info('ALE parameter: %e ' % kappa)
logging.info('\n * Lateral BC')
resistance = args.resistance
logging.info('inner resistance: %e ' % resistance)
if resistance == 0:
lateral_bc = 'free'
logging.info('right BC will be set to the free assumption')
elif resistance < 0:
lateral_bc = 'noflow'
logging.info('right BC will be set to the no flow assumption')
else:
lateral_bc = 'resistance'
logging.info('right BC will be set to the resistance assumption')
fluid_parameters = {'mu': mu, 'rho': rho, 'dt': dt_fluid}
tracer_parameters = {'kappa': D, 'dt': dt_advdiff}
ale_parameters = {'kappa': kappa}
# Setup of boundary conditions
logging.info(title1("Boundary conditions"))
logging.info('\n * Cross section area parameters')
if args.cycle:
logging.info('frequency and amplitude data from cycle ' + args.cycle)
timedependentfa = True
else:
timedependentfa = False
ai = args.ai
fi = args.fi
phii = args.phii
logging.info('ai (dimensionless): ' + '%e ' * len(ai) % tuple(ai))
logging.info('fi (Hz) : ' + '%e ' * len(fi) % tuple(fi))
logging.info('phii (rad) : ' + '%e ' * len(phii) % tuple(phii))
if timedependentfa:
##
logging.info('creation of cycle')
cycleObj = ReadCycle('../stages/cycles.yml', args.cycle)
totalcycletime = np.sum(cycleObj.durations)
spantime, listspana, listspanf, spanRv, spanh0, spanRpvs = cycleObj.generatedata(
int(tfinal / totalcycletime) + 1)
logging.info('*** Simulation of %s cycle ' % cycleObj.name)
# adjust last time in order to be able to interpolate
#spantime[-1]=max(tfinal+2*dt,spantime[-1])
from scipy.interpolate import interp1d
varflist = {}
varalist = {}
for freq in listspanf:
varflist[freq] = interp1d(spantime, listspanf[freq])
varalist[freq] = interp1d(spantime, listspana[freq])
varh0 = interp1d(spantime, spanh0)
varRpvs = interp1d(spantime, spanRpvs)
#varda=interp1d(spantime,dadt,kind="previous")
fs = 1 / dt # test if same result if multiplying by 10 ?
time = 0 + np.arange(
int((tfinal + 2 * dt) * fs)
) / fs # longer than the simulation time because we need tn+1 for the U ALE computation
modulation = {}
for freq in listspanf:
modulation[freq] = varalist[freq](time) * np.sin(
2 * np.pi * np.cumsum(varflist[freq](time)) / fs)
OuterRadius = varRpvs(time)
InnerRadius = varRpvs(time) - varh0(time) * (
1 + modulation['cardiac'] + modulation['resp'] + modulation['LF'] +
modulation['VLF'])
# define the thickness interpolation function
interph0 = interp1d(
time,
varh0(time) * (1 + modulation['cardiac'] + modulation['resp'] +
modulation['LF'] + modulation['VLF']))
## More easy here to take the numerical derivative of the radius
##dadt= np.array(list(np.diff(vara(time))/np.diff(time))+[0.0])
##dRadiusdt= -(Rpvs-Rv)*(dadt*np.sin(2*np.pi*np.cumsum(varf(time))/fs)+vara(time)*np.cos((2*np.pi*np.cumsum(varf(time))/fs))*(2*np.pi*varf(time)))
douterRadiusdt = np.array(
list(np.diff(OuterRadius) / np.diff(time)) + [0.0])
dinnerRadiusdt = np.array(
list(np.diff(InnerRadius) / np.diff(time)) + [0.0])
# define an expression for the radius and the derivative
class Interp(UserExpression):
def __init__(self, x, y, **kwargs):
super().__init__(self, **kwargs)
self.interp = interp1d(x, y)
self.tn = 0
def eval(self, values, x):
values[0] = 0
values[1] = self.interp(self.tn)
def value_shape(self):
return (2, )
class Interpdiff(UserExpression):
def __init__(self, x, y, **kwargs):
super().__init__(self, **kwargs)
self.interp = interp1d(x, y)
self.tn = 0
self.tnp1 = dt
def eval(self, values, x):
values[0] = 0
values[1] = self.interp(self.tnp1) - self.interp(self.tn)
def value_shape(self):
return (2, )
interpRpvs = Interp(time, OuterRadius, degree=1)
interpRv = Interp(time, InnerRadius, degree=1)
#initial values for Rv and Rpvs
uale_top = Interpdiff(time, OuterRadius, degree=1)
uale_bottom = Interpdiff(time, InnerRadius, degree=1)
vf_top = Interp(time, douterRadiusdt, degree=1)
vf_bottom = Interp(time, dinnerRadiusdt, degree=1)
Rvfunction = interp1d(time, InnerRadius)
Rpvsfunction = interp1d(time, OuterRadius)
dRvdtfunction = interp1d(time, dinnerRadiusdt)
else:
import sympy
tn = sympy.symbols("tn")
tnp1 = sympy.symbols("tnp1")
sin = sympy.sin
sqrt = sympy.sqrt
functionR = Rpvs - (Rpvs - Rv) * (1 + sum([
a * sin(2 * pi * f * tn + phi) for a, f, phi in zip(ai, fi, phii)
])) # displacement
R_vessel = sympy.printing.ccode(functionR)
functionV = sympy.diff(functionR, tn) # velocity
V_vessel = sympy.printing.ccode(functionV)
#Delta U for ALE. I dont really like this
functionUALE = -(Rpvs - Rv) * (1 + sum([
a * sin(2 * pi * f * tnp1 + phi)
for a, f, phi in zip(ai, fi, phii)
])) + (Rpvs - Rv) * (1 + sum([
a * sin(2 * pi * f * tn + phi) for a, f, phi in zip(ai, fi, phii)
]))
UALE_vessel = sympy.printing.ccode(functionUALE)
vf_bottom = Expression(
('0', V_vessel), tn=0,
degree=2) # no slip no gap condition at vessel wall
uale_bottom = Expression(
('0', UALE_vessel), tn=0, tnp1=1,
degree=2) # displacement for ALE at vessel wall
vf_top = Constant((0, 0))
uale_top = Constant((0, 0))
## Is there a better way to do that ?
Rvfunction = lambda t: functionR.subs(tn, t).evalf()
Rpvsfunction = lambda t: Rpvs
dRvdtfunction = lambda t: functionV.subs(tn, t).evalf()
if isSAS:
import sympy
tn = sympy.symbols("tn")
sin = sympy.sin
# We add the possibility for rigid moion of the brain
a_rigid = args.arigid #cm
f_rigid = args.frigid #Hz
logging.info('ridig motion of the brain, amplitude : %.2e um' %
a_rigid)
logging.info('ridig motion of the brain, frequency : %.2e Hz' %
f_rigid)
functionY = a_rigid * sin(2 * pi * f_rigid * tn)
functionVbone = sympy.diff(functionY, tn) # velocity
V_bone = sympy.printing.ccode(functionVbone)
vf_bone = Expression(('0', V_bone), tn=0, degree=2)
logging.info('\n * Lateral assumption')
logging.info(lateral_bc)
logging.info('\n * Fluid')
logging.info('Left : zero pressure')
if lateral_bc == 'free':
logging.info('Right : zero pressure')
elif lateral_bc == 'resistance':
logging.info('Right : resistance')
else:
logging.info('Right : no flow')
logging.info('Top : no slip no gap fixed wall')
logging.info('Bottom : no slip no gap moving wall')
logging.info('\n * Tracer concentration')
sas_bc = args.sasbc
init_concentration_type = args.c0init
init_concentration_value = args.c0valuePVS
logging.info('Left BC scenario :', sas_bc)
if lateral_bc == 'free':
logging.info('Right : zero concentration')
else:
productionrate = args.productionrate
if productionrate:
logging.info(
'Right : imposed solute production rate : %e ([c]/s)' %
args.productionrate)
else:
logging.info('Right : no flux')
logging.info('Top : no flux')
logging.info('Bottom : no flux')
logging.info('\n * ALE')
logging.info('Left : no flux')
logging.info('Right : no flux')
logging.info('Top : no displacement')
logging.info('Bottom : vessel displacement')
# Mesh
logging.info(title1('Meshing'))
if isSAS:
Rv = Rvfunction(0)
Rpvs = Rpvsfunction(0)
DR = (Rpvs - Rv) / Nr
logging.info('cell size : %e cm' % (DR))
from sleep.mesh import mesh_model2d, load_mesh2d, set_mesh_size
import gmsh
gmsh.initialize(['', '-format', 'msh2'])
model = gmsh.model
import math
Apvs0 = math.pi * Rpvs**2
Av0 = math.pi * Rv**2
A0 = Apvs0 - Av0
# progressive mesh
factory = model.occ
a = factory.addPoint(-Lsas, Rv, 0)
b = factory.addPoint(L, Rv, 0)
c = factory.addPoint(L, Rpvs, 0)
d = factory.addPoint(0, Rpvs, 0)
e = factory.addPoint(0, Rsas, 0)
f = factory.addPoint(-Lsas, Rsas, 0)
fluid_lines = [
factory.addLine(*p)
for p in ((a, b), (b, c), (c, d), (d, e), (e, f), (f, a))
]
named_lines = dict(
zip(('bottom', 'pvs_right', 'pvs_top', 'brain_surf', 'sas_top',
'sas_left'), fluid_lines))
fluid_loop = factory.addCurveLoop(fluid_lines)
fluid = factory.addPlaneSurface([fluid_loop])
factory.synchronize()
model.addPhysicalGroup(2, [fluid], 1)
for name in named_lines:
tag = named_lines[name]
model.addPhysicalGroup(1, [tag], tag)
# boxes for mesh refinement
boxes = []
# add box on the PVS for mesh
field = model.mesh.field
fid = 1
field.add('Box', fid)
field.setNumber(fid, 'XMin', 0)
field.setNumber(fid, 'XMax', L)
field.setNumber(fid, 'YMin', Rvfunction(0))
field.setNumber(fid, 'YMax', Rpvsfunction(0))
field.setNumber(fid, 'VIn', DR)
field.setNumber(fid, 'VOut', DR * 50)
field.setNumber(fid, 'Thickness', Rsas)
boxes.append(fid)
# Combine
field.add('Min', 2)
field.setNumbers(2, 'FieldsList', boxes)
field.setAsBackgroundMesh(2)
model.occ.synchronize()
h5_filename = outputfolder + '/mesh.h5'
tags = {'cell': {'F': 1}, 'facet': {}}
mesh_model2d(model, tags, h5_filename)
mesh_f, markers, lookup = load_mesh2d(h5_filename)
gmsh.finalize()
else:
# simple PVS mesh
logging.info('cell size : %e cm' % (np.sqrt(DR**2 + DY**2)))
logging.info('nb cells: %i' % (Nl * Nr * 2))
mesh_f = RectangleMesh(Point(0, Rvfunction(0)),
Point(L, Rpvsfunction(0)), Nl, Nr)
## Refinement at the SAS boundary
if args.refineleft:
x = mesh_f.coordinates()[:, 0]
y = mesh_f.coordinates()[:, 1]
#Deformation of the mesh
def deform_mesh(x, y):
x = L * (x / L)**2.5
return [x, y]
x_bar, y_bar = deform_mesh(x, y)
xy_bar_coor = np.array([x_bar, y_bar]).transpose()
mesh_f.coordinates()[:] = xy_bar_coor
mesh_f.bounding_box_tree().build(mesh_f)
fluid_bdries = MeshFunction("size_t", mesh_f,
mesh_f.topology().dim() - 1, 0)
# Label facets
xy = mesh_f.coordinates().copy()
x, y = xy.T
xmin = x.min()
xmax = x.max()
ymin = y.min()
ymax = y.max()
tol = min(DR, DY) / 2 #cm
if isSAS:
class Boundary_sas_left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], -Lsas, tol)
# downstream
class Boundary_pvs_right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], L, tol)
class Boundary_sas_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], Rsas, tol) and (x[0] < tol)
class Boundary_pvs_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], Rpvs, tol) and (x[0] > -tol)
# brain
class Boundary_brainsurf(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], 0,
tol) and (x[1] > Rpvs - tol)
class Boundary_bottom(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymin, tol)
btop_sas = Boundary_sas_top()
btop_pvs = Boundary_pvs_top()
bbottom = Boundary_bottom()
bvert_left = Boundary_sas_left()
bvert_brain = Boundary_brainsurf()
bvert_right = Boundary_pvs_right()
btop_sas.mark(fluid_bdries, 1)
btop_pvs.mark(fluid_bdries, 2)
bbottom.mark(fluid_bdries, 3)
bvert_left.mark(fluid_bdries, 4)
bvert_brain.mark(fluid_bdries, 5)
bvert_right.mark(fluid_bdries, 6)
facet_lookup = {
'sas_out': 1,
'pvs_tissue': 2,
'vessel': 3,
'sas_bone': 4,
'sas_tissue': 5,
'pvs_end': 6
}
else:
# simple PVS mesh
class Boundary_left(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], xmin, tol) #left
class Boundary_right(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], xmax, tol) # right
class Boundary_bottom(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymin, tol) #bottom
class Boundary_top(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[1], ymax, tol) #top
btop = Boundary_top()
bbottom = Boundary_bottom()
bleft = Boundary_left()
bright = Boundary_right()
bbottom.mark(fluid_bdries, 2)
btop.mark(fluid_bdries, 4)
bleft.mark(fluid_bdries, 1)
bright.mark(fluid_bdries, 3)
facet_lookup = {
'sas_out': 1,
'vessel': 2,
'pvs_end': 3,
'pvs_tissue': 4
}
facets_out << fluid_bdries
# Now we wire up
rate_prod = Expression('rate/surface',
rate=productionrate,
surface=1,
degree=1)
if lateral_bc == 'free':
bcs_fluid = {
'velocity': [(facet_lookup['vessel'], vf_bottom),
(facet_lookup['pvs_tissue'], vf_top)],
'traction': [],
'pressure': [(facet_lookup['sas_out'], Constant(0)),
(facet_lookup['pvs_end'], Constant(0))]
}
elif lateral_bc == 'resistance':
Rpressure = Expression('R*Q+p0', R=resistance, Q=0, p0=0, degree=1) #
# Compute pressure to impose according to the flow at previous time step and resistance.
bcs_fluid = {
'velocity': [(facet_lookup['vessel'], vf_bottom),
(facet_lookup['pvs_tissue'], vf_top)],
'traction': [],
'pressure': [(facet_lookup['sas_out'], Constant(0)),
(facet_lookup['pvs_end'], Rpressure)]
}
else:
bcs_fluid = {
'velocity':
[(facet_lookup['vessel'], vf_bottom),
(facet_lookup['pvs_tissue'], vf_top),
(facet_lookup['pvs_end'], Constant(
(0, 0)))], # I would like only normal flow to be zero
'traction': [],
'pressure': [(facet_lookup['sas_out'], Constant(0))]
}
#### This is overwritten later depending on the scenario
bcs_tracer_in = {
'concentration': [(facet_lookup['sas_out'], 0)],
'flux': [(facet_lookup['pvs_end'], rate_prod),
(facet_lookup['pvs_tissue'], Constant(0)),
(facet_lookup['vessel'], Constant(0))]
}
bcs_tracer_out = {
'concentration': [],
'flux': [(facet_lookup['sas_out'], Constant(0)),
(facet_lookup['pvs_end'], rate_prod),
(facet_lookup['pvs_tissue'], Constant(0)),
(facet_lookup['vessel'], Constant(0))]
}
# todo : add possibility to have other BC at pvs_end
#if lateral_bc=='free' :
# bcs_tracer = {'concentration': [(facet_lookup['pvs_end'], Constant(0)),
# (facet_lookup['sas_out'], c_SAS)],
# 'flux': [(facet_lookup['pvs_tissue'], Constant(0)),
# (facet_lookup['vessel'], Constant(0))]}
# add BC on the extra boundary in the mesh of the SAS
if isSAS:
bcs_fluid['velocity'].append((facet_lookup['sas_bone'], vf_bone))
bcs_fluid['velocity'].append(
(facet_lookup['sas_tissue'], Constant((0, 0))))
bcs_tracer_in['flux'].append((facet_lookup['sas_bone'], Constant(0)))
bcs_tracer_in['flux'].append((facet_lookup['sas_tissue'], Constant(0)))
bcs_tracer_out['flux'].append((facet_lookup['sas_bone'], Constant(0)))
bcs_tracer_out['flux'].append(
(facet_lookup['sas_tissue'], Constant(0)))
# BC for ALE (not used anymore)
bcs_ale = {
'dirichlet': [(facet_lookup['vessel'], uale_bottom),
(facet_lookup['pvs_tissue'], uale_top)],
'neumann': [(facet_lookup['sas_out'], Constant((0, 0))),
(facet_lookup['pvs_end'], Constant((0, 0)))]
}
# We collect the time dependent BC for update
driving_expressions = [uale_bottom, vf_bottom, uale_top, vf_top]
if isSAS:
driving_expressions.append(vf_bone)
#FEM space
logging.info(title1("Set FEM spaces"))
logging.info('\n * Fluid')
Vf_elm = VectorElement('Lagrange', triangle, 2)
Qf_elm = FiniteElement('Lagrange', triangle, 1)
Wf_elm = MixedElement([Vf_elm, Qf_elm])
Wf = FunctionSpace(mesh_f, Wf_elm)
logging.info('Velocity : "Lagrange", triangle, 2')
logging.info('Pressure : "Lagrange", triangle, 1')
logging.info('\n * Tracer')
Ct_elm = FiniteElement('Lagrange', triangle, 1)
Ct = FunctionSpace(mesh_f, Ct_elm)
logging.info('Concentration : "Lagrange", triangle, 1')
logging.info('\n * ALE')
Va_elm = VectorElement('Lagrange', triangle, 1)
Va = FunctionSpace(mesh_f, Va_elm)
logging.info('ALE displacement: "Lagrange", triangle, 1')
# Initialisation :
logging.info(title1("Initialisation"))
c_SAS = Expression('m/VCSF', m=0, VCSF=40e-3, degree=2)
#initial concentration in SAS
if sas_bc == 'scenarioA':
cSAS = 0
else:
cSAS = args.c0valueSAS
# number of vessels used for mass balance
Nvessels = 6090
# initial volume of CSF in PVS
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
# volume of pvs
VPVS = 2 * np.pi * assemble(Constant(1.0) * r * dx(mesh_f))
# initial volume of CSF in SAS : assumed to be 10 times larger than volume in PVS
VCSF = 10 * VPVS #40e-3
# initial pressure of the CSF
PCSF = 4 # mmHg
# initial volume of arterial blood
Vblood = 4e-3 # ml
# equivalent vessel length used for the compliance function and assessement of ICP
leq = Vblood / (np.pi * Rvfunction(0)**2)
# initial tracer mass in the CSF
m = cSAS * VCSF
# constant production of CSF
Qprod = 6e-6 # ml/s
# Outflow resistance
Rcsf = 5 / 1.7e-5 # mmHg/(ml/s)
# CSF compliance
Ccsf = 1e-3 #ml/mmHg
if sas_bc == 'scenarioA':
logging.info('Left : zero concentration')
# initial outflow of CSF (not used, just for output file)
Qout = 0
elif sas_bc == 'scenarioB':
logging.info('Left : mass conservation, no CSF outflow')
# initial outflow of CSF
Qout = 0
elif sas_bc == 'scenarioC':
logging.info('Left : mass conservation, constant CSF outflow')
# initial outflow of CSF
Qout = Qprod
elif sas_bc == 'scenarioD':
logging.info(
'Left : mass conservation, pressure dependent CSF outflow')
# initial outflow of CSF
Qout = Qprod
# venous pressure
Pss = PCSF - Qout * Rcsf
logging.info("\n * Fluid")
logging.info("Velocity : zero field")
logging.info("Pressure : zero field")
uf_n = project(Constant((0, 0)), Wf.sub(0).collapse())
pf_n = project(Constant(0), Wf.sub(1).collapse())
logging.info("\n * Tracer")
if init_concentration_type == 'gaussian':
logging.info("Concentration : Gaussian profile")
logging.info(" Centered at xi = %e" % xi_gauss)
logging.info(" STD parameter = %e" % sigma_gauss)
logging.info(" Max value=%e" % init_concentration_value)
c_0 = Expression('c0*exp(-a*pow(x[0]-b, 2)) ',
degree=1,
a=1 / 2 / sigma_gauss**2,
b=xi_gauss,
c0=init_concentration_value)
c_n = project(c_0, Ct)
elif init_concentration_type == 'constant':
logging.info("Concentration : Uniform profile")
if isSAS:
logging.info("Value in PVS=%e" % init_concentration_value)
logging.info("Value in SAS=%e" % cSAS)
c_0 = Expression('x[0]>0 ? cPVS : cSAS ',
degree=2,
cPVS=init_concentration_value,
cSAS=cSAS)
c_n = project(c_0, Ct)
else:
logging.info("Value=%e" % init_concentration_value)
c_n = project(Constant(init_concentration_value), Ct)
elif init_concentration_type == 'null':
logging.info("Concentration : zero in the vessel")
c_n = project(Constant(0), Ct)
else:
logging.info("Concentration : Uniform profile (default)")
logging.info("Value=%e" % 0)
## Initialisation
c_n = project(Constant(0), Ct)
# Save initial state
uf_n.rename("uf", "tmp")
pf_n.rename("pf", "tmp")
c_n.rename("c", "tmp")
uf_out << (uf_n, 0)
pf_out << (pf_n, 0)
c_out << (c_n, 0)
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
slice_line = line([0, (Rpvs + Rv) / 2], [L, (Rpvs + Rv) / 2], 100)
for csv_file, field in zip(files, fields):
#print the x scale
values = np.linspace(0, L, 100)
row = [0] + list(values)
csv_file.write(('%e' + ', %e' * len(values) + '\n') % tuple(row))
#print the initial 1D slice
values = line_sample(slice_line, field)
row = [0] + list(values)
csv_file.write(('%e' + ', %e' * len(values) + '\n') % tuple(row))
############# RUN ############
logging.info(title1("Run"))
# Time loop
time = 0.
timestep = 0
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
# volume of pvs
volume = 2 * np.pi * assemble(Constant(1.0) * r * dx(mesh_f))
# integral of concentration
intc = 2 * np.pi * assemble(r * c_n * dx(mesh_f))
# tracer mass out of the system
mout = 0
csv_mass.write('%s, %s, %s, %s, %s, %s, %s, %s, %s\n' %
('time', 'mass PVS', 'mass CSF', 'mass out', 'Total mass',
'PVS volume', 'CSF volume', 'P csf', 'Q out'))
csv_mass.write('%e, %e, %e, %e, %e, %e, %e, %e, %e\n' %
(time, Nvessels * intc, m, mout, Nvessels * intc + m + mout,
Nvessels * volume, VCSF, PCSF, Qout))
# ALE deformation function
expressionDeformation = Expression((
"0",
"x[1]<=rpvs ? (x[1]-rpvs)/(rpvs-rvessel)*htarget+rpvstarget-x[1]:rpvstarget-rpvs"
),
rvessel=0,
rpvs=1,
rpvstarget=1,
htarget=1,
degree=1)
# Extend normal to 3d as GradAxisym(scalar) is 3-vector
normal = as_vector((Constant(-1), Constant(0), Constant(0)))
# Here I dont know if there will be several dt for advdiff and fluid solver
while time < tfinal:
for expr in driving_expressions:
hasattr(expr, 'tn') and setattr(expr, 'tn', time)
hasattr(expr, 'tnp1') and setattr(expr, 'tnp1', time + dt)
if lateral_bc == 'resistance':
Flow = assemble(2 * pi * r * dot(uf_n, n) *
ds(facet_lookup['pvs_end']))
setattr(Rpressure, 'Q', Flow)
#Solve ALE and move mesh
#eta_f = solve_ale(Va, f=Constant((0, 0)), bdries=fluid_bdries, bcs=bcs_ale, parameters=ale_parameters)
# Or just compute the deformation
xy = mesh_f.coordinates()
x, y = xy.T
expressionDeformation.rpvstarget = Rpvsfunction(time)
expressionDeformation.htarget = Rpvsfunction(time) - Rvfunction(time)
expressionDeformation.rvessel = min(
y[x > 0]) # We look only in the PVS (x>0) not the SAS
expressionDeformation.rpvs = max(y[x > 0]) #
#eta_f = interpolate(expressionDeformation,VectorFunctionSpace(mesh_f,"CG",1))
eta_f = project(expressionDeformation, Va)
ALE.move(mesh_f, eta_f)
mesh_f.bounding_box_tree().build(mesh_f)
# update the coordinates
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
# Solve fluid problem
uf_, pf_ = solve_fluid(Wf,
u_0=uf_n,
f=Constant((0, 0)),
bdries=fluid_bdries,
bcs=bcs_fluid,
parameters=fluid_parameters)
# Solve tracer problem
tracer_parameters["T0"] = time
tracer_parameters["nsteps"] = 1
tracer_parameters["dt"] = dt
# If the fluid is exiting the PVS we compute the amount of mass entering the SAS. The tracer left BC is free.
# If the fluid is entering the PVS then we impose the concentration in the SAS at the left BC.
#Fluid flow at the BC
FluidFlow = assemble(2 * pi * r * dot(uf_, n) *
ds(facet_lookup['sas_out']))
# n is directed in the outward direction : is it ?
print('fluid flow : ', FluidFlow)
if FluidFlow > 0:
# then the fluid is going out and we impose natural BC for concentration
bcs_tracer = bcs_tracer_out
else:
# then the fluid is going in and we impose the SAS concentration
cmean = assemble(
2 * pi * r * c_n * ds(facet_lookup['sas_out'])) / assemble(
2 * pi * r * Constant(1) * ds(facet_lookup['sas_out']))
# we allow the possibility to use a relaxation here
alpha = 0. # 0 means no relaxation
c_imposed = (1 - alpha) * cSAS + alpha * cmean
c_imposed = max(c_imposed, 0)
bcs_tracer = bcs_tracer_in
bcs_tracer['concentration'] = [(facet_lookup['sas_out'],
Constant(c_imposed))]
c_, T0 = solve_adv_diff(Ct,
velocity=uf_ - eta_f / Constant(dt),
phi=Constant(1),
f=Constant(0),
c_0=c_n,
phi_0=Constant(1),
bdries=fluid_bdries,
bcs=bcs_tracer,
parameters=tracer_parameters)
Massflow = assemble(2 * pi * r * dot(uf_ - eta_f / Constant(dt), n) *
c_ * ds(facet_lookup['sas_out']))
Massdiffusion = tracer_parameters["kappa"] * assemble(2 * pi * r * dot(
cyl.GradAxisym(c_), normal) * ds(facet_lookup['sas_out']))
if sas_bc == 'scenarioD':
# update CSF outflow
Qout = max((PCSF - Pss) / Rcsf, 0) # valve
# update CSF pressure
PCSF += dt / Ccsf * (Qprod - Qout) + np.pi * leq * (
Rvfunction(time + dt)**2 - Rvfunction(time)**2) / Ccsf
# link between leq and Nvessels ?
if sas_bc == 'scenarioA':
if FluidFlow >= 0:
# mainly advection
mout += dt * Nvessels * Massflow
# lost mass in the PVS due to diffusion
mout += -dt * Nvessels * Massdiffusion
else:
# Advected mass
m += dt * Nvessels * Massflow - dt * Qout * cSAS
if FluidFlow >= 0: # when in-flow we impose c sas at the boundary so no c gradient
# Adding diffusion
m += -dt * Nvessels * Massdiffusion
mout += Qout * cSAS * dt
# update the volume of CSF
#VCSF+=dt*Nvessels*FluidFlow
# should correspond to the volume change due to vessel dilation
#VCSF+=np.pi*leq*(Rvfunction(time+dt)**2-Rvfunction(time)**2)
# update tracer concentration in SAS
cSAS = m / VCSF
rate_prod.surface = assemble(2 * pi * r * ds(facet_lookup['pvs_end']))
# Update current solution
uf_n.assign(uf_)
pf_n.assign(pf_)
c_n.assign(c_)
#Update time
time += dt
timestep += 1
# Save output
if (timestep % int(toutput / dt) == 0):
logging.info("\n*** save output time %e s" % time)
logging.info("number of time steps %i" % timestep)
# may report Courant number or other important values that indicate how is doing the run
uf_.rename("uf", "tmp")
pf_.rename("pf", "tmp")
c_.rename("c", "tmp")
uf_out << (uf_, time)
pf_out << (pf_, time)
c_out << (c_, time)
# Get the 1 D profiles at umax (to be changed in cyl coordinate)
mesh_points = mesh_f.coordinates()
x = mesh_points[:, 0]
y = mesh_points[:, 1]
xmin = min(x)
xmax = max(x)
ymin = min(y[x > 0])
ymax = max(y[x > 0])
# update the coordinates
z, r = SpatialCoordinate(mesh_f)
ds = Measure('ds', domain=mesh_f, subdomain_data=fluid_bdries)
n = FacetNormal(mesh_f)
#slice_line = line([xmin,(ymin+ymax)/2],[xmax,(ymin+ymax)/2], 100)
logging.info('Rpvs : %e' % ymax)
logging.info('Rvn : %e' % ymin)
files = [csv_p, csv_u, csv_c]
fields = [pf_n, uf_n.sub(0), c_n]
field_names = [
'pressure (dyn/cm2)', 'axial velocity (cm/s)', 'concentration'
]
for csv_file, field, name in zip(files, fields, field_names):
#values = line_sample(slice_line, field)
values = profile(field, xmin, xmax, ymin, ymax)
logging.info('Max ' + name + ' : %.2e' % max(abs(values)))
#logging.info('Norm '+name+' : %.2e'%field.vector().norm('linf'))
row = [time] + list(values)
csv_file.write(
('%e' + ', %e' * len(values) + '\n') % tuple(row))
csv_file.flush()
csv_rv.write(('%e, %e\n') % (time, ymin))
csv_rv.flush()
# volume of pvs
volume = 2 * np.pi * assemble(Constant(1.0) * r * dx(mesh_f))
# integral of concentration
intc = 2 * np.pi * assemble(r * c_ * dx(mesh_f))
csv_mass.write('%e, %e, %e, %e, %e, %e, %e, %e, %e\n' %
(time, Nvessels * intc, m, mout, Nvessels * intc +
m + mout, Nvessels * volume, VCSF, PCSF, Qout))
csv_mass.flush()
ds(facet_lookup['out']))
#Pn+1=pn+dt*(E*P*qoutn)
#Pout.Pn=Pn+dt*E*Pn*Qout
# dP/dt=dP/dV.dV/dt=dP/dV.qout=E*P*qout
# ln(p)-ln(p0)=E int_0^t Qout
#p=p0exp(E int_0^t Qout)
intQout += Qout * dt
Pout.Pn = P0 * math.exp(E * intQout)
print('Qout', Qout)
print('integral Qout', intQout)
# Solve fluid problem
uf_, pf_ = solve_fluid(Wf,
u_0=uf_n,
f=df.Constant((0, 0)),
bdries=facet_f,
bcs=bcs_fluid,
parameters=fluid_parameters)
tracer_parameters["T0"] = time
# Solve tracer problem
c_, T0 = solve_adv_diff(Ct,
velocity=uf_,
mesh_displacement=uf_ * 0,
f=df.Constant(0),
phi_0=c_n,
bdries=facet_f,
bcs=bcs_tracer,
parameters=tracer_parameters)