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perf_BDB_mic.py
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perf_BDB_mic.py
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# This example implements 1st-level homogenization of Biot-Darcy-Brinkman model of flow in deformable
# double porous media.
# The mathematical model is described in:
#
#ROHAN E., TURJANICOVÁ J., LUKEŠ V.
#The Biot-Darcy-Brinkman model of flow in deformable double porous media; homogenization and numerical modelling.
# Computers and Mathematics with applications, 78(9):3044-3066, 2019,
# https://doi.org/10.1016/j.camwa.2019.04.004
#
# Run calculation of homogenized coefficients:
#
# ./homogen.py example_perfusion_BDB/perf_BDB_mic.py
#
# The results are stored in `example_perfusion_BDB/results/micro` directory.
#
import numpy as nm
from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson
from sfepy.homogenization.utils import define_box_regions
from sfepy.discrete.fem.mesh import Mesh
import sfepy.discrete.fem.periodic as per
import sfepy.homogenization.coefs_base as cb
import os.path as osp
data_dir = 'example_perfusion_BDB'
#Definition of periodic boundary conditions
def get_periodic_bc(var_tab, dim=3, dim_tab=None):
if dim_tab is None:
dim_tab = {'x': ['left', 'right'],
'z': ['bottom', 'top'],
'y': ['near', 'far']}
periodic = {}
epbcs = {}
for ivar, reg in var_tab:
periodic['per_%s' % ivar] = pers = []
for idim in 'xyz'[0:dim]:
key = 'per_%s_%s' % (ivar, idim)
regs = ['%s_%s' % (reg, ii) for ii in dim_tab[idim]]
epbcs[key] = (regs, {'%s.all' % ivar: '%s.all' % ivar},
'match_%s_plane' % idim)
pers.append(key)
return epbcs, periodic
# define homogenized coefficients and subproblems for correctors
def define(filename_mesh=None):
eps0 = 0.01 # given scale parameter
if filename_mesh is None:
filename_mesh = osp.join(data_dir, 'micro_perf_puc.vtk')
mesh = Mesh.from_file(filename_mesh)
dim = 3
sym = (dim + 1) * dim // 2
sym_eye = 'nm.array([1,1,0])' if dim == 2 else 'nm.array([1,1,1,0,0,0])'
bbox = mesh.get_bounding_box()
regions = define_box_regions(mesh.dim, bbox[0], bbox[1], eps=1e-3)
regions.update({
'Y': 'all',
'Gamma_Y': ('vertices of surface', 'facet'),
# solid matrix
'Ys': 'cells of group 1',
'Ys_left': ('r.Ys *v r.Left', 'vertex'),
'Ys_right': ('r.Ys *v r.Right', 'vertex'),
'Ys_bottom': ('r.Ys *v r.Bottom', 'vertex'),
'Ys_top': ('r.Ys *v r.Top', 'vertex'),
'Gamma_Ysf': ('r.Ys *v r.Yf', 'facet', 'Ys'),
# channel
'Yf': 'cells of group 2',
'Yf0': ('r.Yf -v r.Gamma_Yfs', 'vertex'),
'Yf_left': ('r.Yf0 *v r.Left', 'vertex'),
'Yf_right': ('r.Yf0 *v r.Right', 'vertex'),
'Yf_bottom': ('r.Yf0 *v r.Bottom', 'vertex'),
'Yf_top': ('r.Yf0 *v r.Top', 'vertex'),
'Gamma_Yfs': ('r.Ys *v r.Yf', 'facet', 'Yf'),
})
if dim == 3:
regions.update({
'Ys_far': ('r.Ys *v r.Far', 'vertex'),
'Ys_near': ('r.Ys *v r.Near', 'vertex'),
'Yf_far': ('r.Yf0 *v r.Far', 'vertex'),
'Yf_near': ('r.Yf0 *v r.Near', 'vertex'),
})
fields = {
'volume': ('real', 'scalar', 'Y', 1),
'displacement': ('real', 'vector', 'Ys', 1),
'pressure': ('real', 'scalar', 'Yf', 1),
'velocity': ('real', 'vector', 'Yf', 2),
}
variables = {
# displacement
'u': ('unknown field', 'displacement'),
'v': ('test field', 'displacement', 'u'),
'Pi_u': ('parameter field', 'displacement', 'u'),
'U1': ('parameter field', 'displacement', '(set-to-None)'),
'U2': ('parameter field', 'displacement', '(set-to-None)'),
# velocity
'w': ('unknown field', 'velocity'),
'z': ('test field', 'velocity', 'w'),
'Pi_w': ('parameter field', 'velocity', 'w'),
'W1': ('parameter field', 'velocity', '(set-to-None)'),
'W2': ('parameter field', 'velocity', '(set-to-None)'),
# pressure
'p': ('unknown field', 'pressure'),
'q': ('test field', 'pressure', 'p'),
# volume
'volume': ('parameter field', 'volume', '(set-to-None)'),
}
functions = {
'match_x_plane': (per.match_x_plane,),
'match_y_plane': (per.match_y_plane,),
'match_z_plane': (per.match_z_plane,),
}
materials = {
'matrix': ({'D': stiffness_from_youngpoisson(dim, 1e3, 0.49)},),#Soft tissue
'fluid': ({
'eta_p': 3.6e-3 / eps0**2,#Rescaled blood viscosity
'aux_compress': 1e-18},),#Auxillary compressibility
}
ebcs = {
'fixed_u': ('Corners', {'u.all': 0.0}),
'fixed_w': ('Gamma_Yfs', {'w.all': 0.0}),
}
epbcs, periodic = get_periodic_bc([('u', 'Ys'), ('p', 'Yf'), ('w', 'Yf')])
integrals = {
'i': 4,
}
options = {
'coefs': 'coefs',
'coefs_filename': 'coefs_micro',
'requirements': 'requirements',
'volume': {
'variables': ['u', 'p'],
'expression': 'd_volume.i.Ys(u) + d_volume.i.Yf(p)',
},
'output_dir': data_dir+'/results/micro',
'ls': 'ls',
'file_per_var': True,
'absolute_mesh_path': True,
'multiprocessing': True,
'output_prefix': 'micro:',
}
#Definition of used solvers
solvers = {
'ls': ('ls.mumps', {}),
'ns_em9': ('nls.newton', {
'i_max': 1,
'eps_a': 1e-9,
'eps_r': 1e-3,
'problem': 'nonlinear'}),
'ns_em12': ('nls.newton', {
'i_max': 1,
'eps_a': 1e-12,
'eps_r': 1e-3,
'problem': 'nonlinear'}),
}
#Definition of homogenized coefficients, see (22) and (23)
coefs = {
#Elasticity coefficient
'A': {
'requires': ['pis_u', 'corrs_omega_ij'],
'expression': 'dw_lin_elastic.i.Ys(matrix.D, U1, U2)',
'set_variables': [('U1', ('corrs_omega_ij', 'pis_u'), 'u'),
('U2', ('corrs_omega_ij', 'pis_u'), 'u')],
'class': cb.CoefSymSym,
},
#Biot coefficient
'hat_B': {
'status': 'auxiliary',
'requires': ['corrs_omega_ij'],
'expression': '- ev_div.i.Ys(U1)',
'set_variables': [('U1', 'corrs_omega_ij', 'u')],
'class': cb.CoefSym,
},
'B': {
'requires': ['c.phi_f', 'c.hat_B'],
'expression': 'c.hat_B + c.phi_f * %s' % sym_eye,
'class': cb.CoefEval,
},
'M': {
'requires': ['corrs_omega_p'],
'expression': 'dw_lin_elastic.i.Ys(matrix.D, U1, U2)',
'set_variables': [('U1', 'corrs_omega_p', 'u'),
('U2', 'corrs_omega_p', 'u')],
'class': cb.CoefOne,
},
#Permeability
'K': {
'requires': ['corrs_psi_i'],
'expression': 'dw_div_grad.i.Yf(W1, W2)', # !!!
'set_variables': [('W1', 'corrs_psi_i', 'w'),
('W2', 'corrs_psi_i', 'w')],
'class': cb.CoefDimDim,
},
#Volume fraction of fluid part
'phi_f': {
'requires': ['c.vol'],
'expression': 'c.vol["fraction_Yf"]',
'class': cb.CoefEval,
},
#Coefficient for storing viscosity
'eta_p': {
'expression': '%e' % materials['fluid'][0]['eta_p'],
'class': cb.CoefEval,
},
#Volume fractions
'vol': {
'regions': ['Ys', 'Yf'],
'expression': 'd_volume.i.%s(volume)',
'class': cb.VolumeFractions,
},
#Surface volume fractions
'surf_vol': {
'regions': ['Ys', 'Yf'],
'expression': 'd_surface.i.%s(volume)',
'class': cb.VolumeFractions,
},
'filenames': {},
}
#Definition of microscopic corrector problems
requirements = {
#Definition of \Pi^{ij}_k
'pis_u': {
'variables': ['u'],
'class': cb.ShapeDimDim,
},
#Correcotr like class returning ones
'pis_w': {
'variables': ['w'],
'class': cb.OnesDim,
},
#Corrector problem related to elasticity, see (17)
'corrs_omega_ij': {
'requires': ['pis_u'],
'ebcs': ['fixed_u'],
'epbcs': periodic['per_u'],
'is_linear': True,
'equations': {
'balance_of_forces':
"""dw_lin_elastic.i.Ys(matrix.D, v, u)
= - dw_lin_elastic.i.Ys(matrix.D, v, Pi_u)"""
},
'set_variables': [('Pi_u', 'pis_u', 'u')],
'class': cb.CorrDimDim,
'save_name': 'corrs_omega_ij',
'dump_variables': ['u'],
'solvers': {'ls': 'ls', 'nls': 'ns_em9'},
},
# Corrector problem related to elasticity, see (18)
'corrs_omega_p': {
'requires': [],
'ebcs': ['fixed_u'],
'epbcs': periodic['per_u'],
'equations': {
'balance_of_forces':
"""dw_lin_elastic.i.Ys(matrix.D, v, u)
= -dw_surface_ltr.i.Gamma_Ysf(v)"""
},
'class': cb.CorrOne,
'save_name': 'corrs_omega_p',
'dump_variables': ['u'],
'solvers': {'ls': 'ls', 'nls': 'ns_em3'},
},
#Corrector problem related to velocity, see (19)
'corrs_psi_i': {
'requires': ['pis_w'],
'ebcs': ['fixed_w'],
'epbcs': periodic['per_w'] + periodic['per_p'],
'is_linear': True,
'equations': {
'balance_of_forces': # !!!
"""dw_div_grad.i.Yf(fluid.eta_p,z, w)
- dw_stokes.i.Yf(z, p)
= dw_volume_dot.i.Yf(z, Pi_w)""",
'incompressibility':
"""dw_stokes.i.Yf(w, q)
+ dw_volume_dot.i.Yf(fluid.aux_compress, q, p)
= 0""",#
},
'set_variables': [('Pi_w', 'pis_w', 'w')],
'class': cb.CorrDim,
'save_name': 'corrs_psi_i',
'dump_variables': ['w', 'p'],
'solvers': {'ls': 'ls', 'nls': 'ns_em12'},
},
}
return locals()