diffusion/poisson_periodic_boundary_condition.pyΒΆ
Description
This example is using a mesh generated by gmsh. Both the .geo script used by gmsh to generate the file and the .mesh file can be found in meshes.
The mesh is suitable for periodic boundary conditions. It consists of a cylinder enclosed by a box in the x and y directions.
The cylinder will act as a power source.
Transient Laplace equation in time interval
with a localized power source and periodic boundary conditions:Find
for such that:
r"""
This example is using a mesh generated by gmsh. Both the
.geo script used by gmsh to generate the file and the .mesh
file can be found in meshes.
The mesh is suitable for periodic boundary conditions. It consists
of a cylinder enclosed by a box in the x and y directions.
The cylinder will act as a power source.
Transient Laplace equation in time interval
:math:`t \in [0, t_{\rm final}]` with a localized power source and
periodic boundary conditions:
Find :math:`T(t)` for :math:`t \in [0, t_{\rm final}]` such that:
.. math::
\int_{\Omega}c s \pdiff{T}{t}
+ \int_{\Omega} \sigma_2 \nabla s \cdot \nabla T
= \int_{\Omega_2} P_3 T
\;, \quad \forall s \;.
"""
from sfepy import data_dir
import numpy as nm
import sfepy.fem.periodic as per
filename_mesh = data_dir + '/meshes/3d/cylinder_in_box.mesh'
t0 = 0.0
t1 = 1.
n_step = 11
power_per_volume =1.e2 # Heating power per volume of the cylinder
capacity_cylinder = 1. # Heat capacity of cylinder
capacity_fill = 1. # Heat capacity of filling material
conductivity_cylinder = 1. # Heat conductivity of cylinder
conductivity_fill = 1. # Heat conductivity of filling material
def cylinder_material_func(ts, coors, problem, mode=None, **kwargs):
"""
Returns the thermal conductivity, the thermal mass, and the power of the
material in the cylinder.
"""
if mode == 'qp':
shape = (coors.shape[0], 1, 1)
power = nm.empty(shape, dtype=nm.float64)
if ts.step < 5:
# The power is turned on in the first 5 steps only.
power.fill(power_per_volume)
else:
power.fill(0.0)
conductivity = nm.ones(shape) * conductivity_cylinder
capacity = nm.ones(shape) * capacity_cylinder
return {'power' : power, 'capacity' : capacity,
'conductivity' : conductivity}
materials = {
'cylinder' : 'cylinder_material_func',
'fill' : ({'capacity' : capacity_fill,
'conductivity' : conductivity_fill,},),
}
fields = {
'temperature' : ('real', 1, 'Omega', 1),
}
variables = {
'T' : ('unknown field', 'temperature', 1, 1),
's' : ('test field', 'temperature', 'T'),
}
regions = {
'Omega' : ('all', {}),
'cylinder' : ('elements of group 444', {}),
'fill' : ('elements of group 555', {}),
'Gamma_Left' : ('nodes in (x < -2.4999)', {}),
'y+' : ('nodes in (y >2.4999)', {}),
'y-' : ('nodes in (y <-2.4999)', {}),
'z+' : ('nodes in (z >0.4999)', {}),
'z-' : ('nodes in (z <-0.4999)', {}),
}
ebcs = {
'T1' : ('Gamma_Left', {'T.0' : 0.0}),
}
# The matching functions link the elements on each side with that on the
# opposing side.
functions = {
'cylinder_material_func' : (cylinder_material_func,),
"match_y_plane" : (per.match_y_plane,),
"match_z_plane" : (per.match_z_plane,),
}
epbcs = {
# In the y-direction
'periodic_y' : (['y+', 'y-'], {'T.0' : 'T.0'}, 'match_y_plane'),
# and in the z-direction. Due to the symmetry of the problem, this periodic
# boundary condition is actually not necessary, but we include it anyway.
'periodic_z' : (['z+', 'z-'], {'T.0' : 'T.0'}, 'match_z_plane'),
}
ics = {
'ic' : ('Omega', {'T.0' : 0.0}),
}
integrals = {
'i1' : ('v', 1),
}
equations = {
'Temperature' :
"""dw_volume_dot.i1.cylinder( cylinder.capacity, ds/dt, dT/dt )
dw_volume_dot.i1.fill( fill.capacity, ds/dt, dT/dt )
dw_laplace.i1.cylinder( cylinder.conductivity, s, T )
dw_laplace.i1.fill( fill.conductivity, s, T )
= dw_volume_integrate.i1.cylinder( cylinder.power, s )"""
}
solvers = {
'ls' : ('ls.scipy_direct', {}),
'newton' : ('nls.newton', {
'i_max' : 1,
'eps_a' : 1e-10,
'eps_r' : 1.0,
'problem' : 'nonlinear'
}),
'ts' : ('ts.simple', {
't0' : t0,
't1' : t1,
'dt' : None,
'n_step' : n_step, # has precedence over dt!
'quasistatic' : False,
}),
}
options = {
'nls' : 'newton',
'ls' : 'ls',
'ts' : 'ts',
'output_dir' : 'output',
'save_steps' : -1,
}