This tutorial focuses on the first way and introduces the basic concepts
and nomenclature used in the following parts of the documentation. Check
also the Primer which focuses on a particular problem in detail.
Here we discuss the contents of the examples/diffusion/poisson.py problem
description file. For additional examples, see the problem description files in
the examples/ directory of SfePy.
The problem at hand is the following:
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where
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is a subset of the domain
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boundary. For simplicity, we set
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, but we still work with
the material constant
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even though it has no influence on the
solution in this case. We also assume zero fluxes over
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, i.e.
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there. The
particular boundary conditions used below are
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on the left side
of the cylindrical domain depicted in the previous section and
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on the right side.
The first step to do is to write (6) in weak
formulation (5). The
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,
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. So only one term in weak form
(5) remains:
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Comparing the above integral term with the
long table in Term Overview, we can see that SfePy contains
this term under name dw_laplace. We are now ready to proceed to the
actual problem definition.
The example uses long syntax of the keywords. In next subsection, we
show the same example written in short syntax.
Open the examples/diffusion/poisson.py file in your favorite text
editor. Note that the file is a regular python source code.
from sfepy import data_dir
filename_mesh = data_dir + '/meshes/3d/cylinder.mesh'
The filename_mesh variable points to the file containing the mesh for the
particular problem. SfePy supports a variety of mesh formats.
material_2 = {
'name' : 'coef',
'values' : {'val' : 1.0},
}
Here we define just a constant coefficient
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of the Poisson
equation, using the
'values' attribute. Other possible attribute is
'function', for material coefficients computed/obtained at runtime.
Many finite element problems require the definition of material parameters.
These can be handled in SfePy with material variables which associate the
material parameters with the corresponding region of the mesh.
region_1000 = {
'name' : 'Omega',
'select' : 'elements of group 6',
}
region_03 = {
'name' : 'Gamma_Left',
'select' : 'nodes in (x < 0.00001)',
}
region_4 = {
'name' : 'Gamma_Right',
'select' : 'nodes in (x > 0.099999)',
}
Regions assign names to various parts of the finite element mesh. The region
names can later be referred to, for example when specifying portions of the mesh
to apply boundary conditions to. Regions can be specified in a variety of ways,
including by element or by node. Here, Omega is the elemental domain over which
the PDE is solved and Gamma_Left and Gamma_Right define surfaces upon which the
boundary conditions will be applied.
field_1 = {
'name' : 'temperature',
'dtype' : 'real',
'shape' : (1,),
'region' : 'Omega',
'approx_order' : 1,
}
A field is used mainly to define the approximation on a (sub)domain, i.e. to
define the discrete spaces
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, where we seek the solution.
The Poisson equation can be used to compute e.g. a temperature distribution,
so let us call our field 'temperature'. On the region 'Omega'
it will be approximated using linear finite elements.
A field in a given region defines the finite element approximation.
Several variables can use the same field, see below.
variable_1 = {
'name' : 't',
'kind' : 'unknown field',
'field' : 'temperature',
'order' : 0, # order in the global vector of unknowns
}
variable_2 = {
'name' : 's',
'kind' : 'test field',
'field' : 'temperature',
'dual' : 't',
}
One field can be used to generate discrete degrees of freedom (DOFs) of
several variables. Here the unknown variable (the temperature) is called
't', it’s associated DOF name is 't.0' — this will be referred
to in the Dirichlet boundary section (ebc). The corresponding test
variable of the weak formulation is called 's'. Notice that the
'dual' item of a test variable must specify the unknown it
corresponds to.
For each unknown (or state) variable there has to be a test (or virtual)
variable defined, as usual in weak formulation of PDEs.
ebc_1 = {
'name' : 't1',
'region' : 'Gamma_Left',
'dofs' : {'t.0' : 2.0},
}
ebc_2 = {
'name' : 't2',
'region' : 'Gamma_Right',
'dofs' : {'t.0' : -2.0},
}
Essential (Dirichlet) boundary conditions can be specified as above.
Boundary conditions place restrictions on the finite element formulation and
create a unique solution to the problem. Here, we specify that a temperature of
+2 is applied to the left surface of the mesh and a temperature of -2 is applied
to the right surface.
integral_1 = {
'name' : 'i1',
'kind' : 'v',
'order' : 2,
}
Integrals specify which numerical scheme to use. Here we are using a 2nd order
quadrature over a 3 dimensional space.
equations = {
'Temperature' : """dw_laplace.i1.Omega( coef.val, s, t ) = 0"""
}
The equation above directly corresponds to the discrete version of
(7), namely: Find
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, such that
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where
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.
The equations block is the heart of the SfePy problem definition file. Here,
we are specifying that the Laplacian of the temperature (in the weak
formulation) is 0, where coef.val is a material constant. We are using the
i1 integral defined previously, over the domain specified by the region
Omega.
The above syntax is useful for defining custom integrals with
user-defined quadrature points and weights, see Integrals. The
above uniform integration can be more easily achieved by:
equations = {
'Temperature' : """dw_laplace.2.Omega( coef.val, s, t ) = 0"""
}
The integration order is specified directly in place of the integral
name. The integral definition is superfluous in this case.
solver_0 = {
'name' : 'ls',
'kind' : 'ls.scipy_direct',
'method' : 'auto',
}
Here, we specify which kind of solver to use for linear equations.
solver_1 = {
'name' : 'newton',
'kind' : 'nls.newton',
'i_max' : 1,
'eps_a' : 1e-10,
'eps_r' : 1.0,
'macheps' : 1e-16,
'lin_red' : 1e-2, # Linear system error < (eps_a * lin_red).
'ls_red' : 0.1,
'ls_red_warp' : 0.001,
'ls_on' : 1.1,
'ls_min' : 1e-5,
'check' : 0,
'delta' : 1e-6,
'is_plot' : False,
'problem' : 'nonlinear', # 'nonlinear' or 'linear' (ignore i_max)
}
Here, we specify the nonlinear solver kind and options. The convergence
parameters can be adjusted if necessary, otherwise leave the default.
Even linear problems are solved by a nonlinear solver (KISS rule) - only one
iteration is needed and the final rezidual is obtained for free.
options = {
'nls' : 'newton',
'ls' : 'ls',
}
The solvers to use are specified in the options block. We can define multiple
solvers with different convergence parameters if necessary.
That’s it! Now it is possible to proceed as described in
Invoking SfePy from the command line.
The same diffusion equation example as above in short syntax reads,
see examples/diffusion/poisson_short_syntax.py, as follows:
1
2
3
4
5
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9
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60 | r"""
Laplace equation.
The same example as poisson.py, but using the short syntax of keywords.
Find :math:`t` such that:
.. math::
\int_{\Omega} c \nabla s \cdot \nabla t
= 0
\;, \quad \forall s \;.
"""
from sfepy import data_dir
filename_mesh = data_dir + '/meshes/3d/cylinder.mesh'
materials = {
'coef' : ({'val' : 1.0},),
}
regions = {
'Omega' : ('all', {}), # or 'elements of group 6'
'Gamma_Left' : ('nodes in (x < 0.00001)', {}),
'Gamma_Right' : ('nodes in (x > 0.099999)', {}),
}
fields = {
'temperature' : ('real', 1, 'Omega', 1),
}
variables = {
't' : ('unknown field', 'temperature', 0),
's' : ('test field', 'temperature', 't'),
}
ebcs = {
't1' : ('Gamma_Left', {'t.0' : 2.0}),
't2' : ('Gamma_Right', {'t.0' : -2.0}),
}
integrals = {
'i1' : ('v', 2),
}
equations = {
'Temperature' : """dw_laplace.i1.Omega( coef.val, s, t ) = 0"""
}
solvers = {
'ls' : ('ls.scipy_direct', {}),
'newton' : ('nls.newton',
{'i_max' : 1,
'eps_a' : 1e-10,
}),
}
options = {
'nls' : 'newton',
'ls' : 'ls',
}
|
This example shows how to use SfePy interactively, but also how to
make a custom simulation script. We will use isfepy for the
explanation, but regular Python shell, or IPython would do as well,
provided the proper modules are imported (see the help information
printed on startup of isfepy).
We wish to solve the following linear elasticity problem:
System Message: WARNING/2 ( - \pdiff{\sigma_{ij}(\ul{u})}{x_j} + f_i = 0 \mbox{ in }\Omega,
\quad \ul{u} = 0 \mbox{ on } \Gamma_1,
\quad u_1 = \bar{u}_1 \mbox{ on } \Gamma_2 \;,)
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where the stress is defined as
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,
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,
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are the Lamé’s
constants, the strain is
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and
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are
volume forces. This can be written in general form as
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, where in our case
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.
In the weak form the equation (8) is
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f_i v_i = 0 \;,)
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where
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is the test function, and both
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,
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belong to a suitable function space.
Hint: Whenever you create a new object (e.g. a Mesh instance, see
below), try to print it using the print statement - it will give you
insight about the object internals.
The whole example summarized in a script is below in
Complete Example as a Script.
Run the isfepy script:
The output should look like this:
Python 2.6.5 console for SfePy 2010.2-git-11cfd34 (8c4664610ed4b85851966326aaa7ce36e560ce7a)
These commands were executed:
>>> from sfepy.base.base import *
>>> from sfepy.fem import *
>>> from sfepy.applications import solve_pde
>>> from sfepy.postprocess import Viewer
When in SfePy source directory, try:
>>> pb, vec, data = solve_pde('examples/diffusion/poisson.py')
>>> view = Viewer(pb.get_output_name())
>>> view()
When in another directory (and SfePy is installed), try:
>>> from sfepy import data_dir
>>> pb, vec, data = solve_pde(data_dir + '/examples/diffusion/poisson.py')
>>> view = Viewer(pb.get_output_name())
>>> view()
Documentation can be found at http://sfepy.org
In [1]:
Read a finite element mesh, that defines the domain
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.
In [1]: mesh = Mesh.from_file('meshes/2d/rectangle_tri.mesh')
Create a domain. The domain allows defining regions or subdomains.
In [2]: domain = Domain('domain', mesh)
sfepy: setting up domain edges...
sfepy: ...done in 0.01 s
Define the regions - the whole domain
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, where the solution
is sought, and
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,
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, where the boundary
conditions will be applied. As the domain is rectangular, we first get a
bounding box to get correct bounds for selecting the boundary edges.
In [3]: min_x, max_x = domain.get_mesh_bounding_box()[:,0]
In [4]: eps = 1e-8 * (max_x - min_x)
In [5]: omega = domain.create_region('Omega', 'all')
In [6]: gamma1 = domain.create_region('Gamma1',
...: 'nodes in x < %.10f' % (min_x + eps))
In [7]: gamma2 = domain.create_region('Gamma2',
...: 'nodes in x > %.10f' % (max_x - eps))
Next we define the actual finite element approximation using the
Field class.
In [8]: field = Field('fu', nm.float64, 'vector', omega,
...: space='H1', poly_space_base='lagrange', approx_order=2)
Using the field fu, we can define both the unknown variable
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and
the test variable
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.
In [9]: u = FieldVariable('u', 'unknown', field, mesh.dim)
In [10]: v = FieldVariable('v', 'test', field, mesh.dim,
....: primary_var_name='u')
Before we can define the terms to build the equation of linear
elasticity, we have to create also the materials, i.e. define the
(constitutive) parameters. The linear elastic material m will be
defined using the two Lamé constants
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,
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1)
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. The volume forces will be defined also as a material, as a constant
(column) vector
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.
In [11]: m = Material('m', lam=1.0, mu=1.0)
In [12]: f = Material('f', val=[[0.02], [0.01]])
One more thing needs to be defined - the numerical quadrature that will
be used to integrate each term over its domain.
In [14]: integral = Integral('i', order=3)
Now we are ready to define the two terms and build the equations.
In [15]: from sfepy.terms import Term
In [16]: t1 = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral, omega, m=m, v=v, u=u)
In [17]: t2 = Term.new('dw_volume_lvf(f.val, v)', integral, omega, f=f, v=v)
In [18]: eq = Equation('balance', t1 + t2)
In [19]: eqs = Equations([eq])
sfepy: setting up dof connectivities...
sfepy: ...done in 0.00 s
sfepy: describing geometries...
sfepy: ...done in 0.00 s
The equations have to be completed by boundary conditions. Let us clamp
the left edge
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, and shift the right edge
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in the
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direction a bit, depending on the
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coordinate.
In [20]: from sfepy.fem.conditions import Conditions, EssentialBC
In [21]: fix_u = EssentialBC('fix_u', gamma1, {'u.all' : 0.0})
In [22]: def shift_u_fun(ts, coors, bc=None, shift=0.0):
....: val = shift * coors[:,1]**2
....: return val
....:
In [23]: bc_fun = Function('shift_u_fun', shift_u_fun,
....: extra_args={'shift' : 0.01})
In [24]: shift_u = EssentialBC('shift_u', gamma2, {'u.0' : bc_fun})
The last thing to define before building the problem are the
solvers. Here we just use a sparse direct SciPy solver and the SfePy
Newton solver with default parameters. We also wish to store the
convergence statistics of the Newton solver. As the problem is linear,
it should converge in one iteration.
In [25]: from sfepy.solvers.ls import ScipyDirect
In [26]: from sfepy.solvers.nls import Newton
In [27]: ls = ScipyDirect({})
In [28]: nls_status = IndexedStruct()
In [29]: nls = Newton({}, lin_solver=ls, status=nls_status)
Now we are ready to create a ProblemDefinition instance. Note
that the step above is not really necessary - the above solvers are
constructed by default. We did them to get the nls_status.
In [30]: pb = ProblemDefinition('elasticity', equations=eqs, nls=nls, ls=ls)
The ProblemDefinition has several handy methods for
debugging. Let us try saving the regions into a VTK file.
In [31]: pb.save_regions_as_groups('regions')
sfepy: saving regions as groups...
sfepy: Omega
sfepy: Gamma1
sfepy: Gamma2
sfepy: Gamma1
sfepy: ...done
And view them.
In [32]: view = Viewer('regions.vtk')
In [33]: view()
sfepy: point scalars Gamma1 [ 0. 0. 0.]
sfepy: point scalars Gamma2 [ 11. 0. 0.]
sfepy: point scalars Omega [ 22. 0. 0.]
Out[33]: <sfepy.postprocess.viewer.ViewerGUI object at 0x93ea5f0>
You should see this:
Finally, we apply the boundary conditions, solve the problem, save and
view the results.
In [34]: pb.time_update(ebcs=Conditions([fix_u, shift_u]))
sfepy: updating materials...
sfepy: m
sfepy: f
sfepy: ...done in 0.01 s
sfepy: updating variables...
sfepy: ...done
sfepy: matrix shape: (1815, 1815)
sfepy: assembling matrix graph...
sfepy: ...done in 0.00 s
sfepy: matrix structural nonzeros: 39145 (1.19e-02% fill)
In [35]: vec = pb.solve()
sfepy: nls: iter: 0, residual: 1.343114e+01 (rel: 1.000000e+00)
sfepy: rezidual: 0.00 [s]
sfepy: solve: 0.01 [s]
sfepy: matrix: 0.00 [s]
sfepy: nls: iter: 1, residual: 2.567997e-14 (rel: 1.911972e-15)
In [36]: print nls_status
-------> print(nls_status)
IndexedStruct
condition:
0
err:
2.56799662867e-14
err0:
13.4311385972
time_stats:
{'rezidual': 0.0, 'solve': 0.010000000000001563, 'matrix': 0.0}
In [37]: pb.save_state('linear_elasticity.vtk', vec)
In [38]: view = Viewer('linear_elasticity.vtk')
In [39]: view()
sfepy: point vectors u [ 0. 0. 0.]
Out[39]: <sfepy.postprocess.viewer.ViewerGUI object at 0xad61bf0>
This is the resulting image:
The default view is not very fancy. Let us show the displacements by
shifting the mesh. Close the previous window and do:
In [56]: view(vector_mode='warp_norm', rel_scaling=2,
....: is_scalar_bar=True, is_wireframe=True)
sfepy: point vectors u [ 0. 0. 0.]
Out[56]: <sfepy.postprocess.viewer.ViewerGUI object at 0xad61bf0>
And the result is:
See the docstring of view() and play with its options.
The source code: linear_elasticity.py. It should be
run from the SfePy source directory so that it finds the mesh file.
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94 | #!/usr/bin/env python
from optparse import OptionParser
import numpy as nm
import sys
sys.path.append('.')
from sfepy.base.base import IndexedStruct
from sfepy.fem import (Mesh, Domain, H1NodalVolumeField, FieldVariable,
Material, Integral, Function, Equation, Equations,
ProblemDefinition)
from sfepy.terms import Term
from sfepy.fem.conditions import Conditions, EssentialBC
from sfepy.solvers.ls import ScipyDirect
from sfepy.solvers.nls import Newton
from sfepy.postprocess import Viewer
def shift_u_fun(ts, coors, bc=None, problem=None, shift=0.0):
"""
Define a displacement depending on the y coordinate.
"""
val = shift * coors[:,1]**2
return val
usage = """%prog [options]"""
help = {
'show' : 'show the results figure',
}
def main():
from sfepy import data_dir
parser = OptionParser(usage=usage, version='%prog')
parser.add_option('-s', '--show',
action="store_true", dest='show',
default=False, help=help['show'])
options, args = parser.parse_args()
mesh = Mesh.from_file(data_dir + '/meshes/2d/rectangle_tri.mesh')
domain = Domain('domain', mesh)
min_x, max_x = domain.get_mesh_bounding_box()[:,0]
eps = 1e-8 * (max_x - min_x)
omega = domain.create_region('Omega', 'all')
gamma1 = domain.create_region('Gamma1',
'nodes in x < %.10f' % (min_x + eps))
gamma2 = domain.create_region('Gamma2',
'nodes in x > %.10f' % (max_x - eps))
field = H1NodalVolumeField('fu', nm.float64, 'vector', omega,
approx_order=2)
u = FieldVariable('u', 'unknown', field, mesh.dim)
v = FieldVariable('v', 'test', field, mesh.dim, primary_var_name='u')
m = Material('m', lam=1.0, mu=1.0)
f = Material('f', val=[[0.02], [0.01]])
integral = Integral('i', order=3)
t1 = Term.new('dw_lin_elastic_iso(m.lam, m.mu, v, u)',
integral, omega, m=m, v=v, u=u)
t2 = Term.new('dw_volume_lvf(f.val, v)', integral, omega, f=f, v=v)
eq = Equation('balance', t1 + t2)
eqs = Equations([eq])
fix_u = EssentialBC('fix_u', gamma1, {'u.all' : 0.0})
bc_fun = Function('shift_u_fun', shift_u_fun, extra_args={'shift' : 0.01})
shift_u = EssentialBC('shift_u', gamma2, {'u.0' : bc_fun})
ls = ScipyDirect({})
nls_status = IndexedStruct()
nls = Newton({}, lin_solver=ls, status=nls_status)
pb = ProblemDefinition('elasticity', equations=eqs, nls=nls, ls=ls)
pb.save_regions_as_groups('regions')
pb.time_update(ebcs=Conditions([fix_u, shift_u]))
vec = pb.solve()
print nls_status
pb.save_state('linear_elasticity.vtk', vec)
if options.show:
view = Viewer('linear_elasticity.vtk')
view(vector_mode='warp_norm', rel_scaling=2,
is_scalar_bar=True, is_wireframe=True)
if __name__ == '__main__':
main()
|
