User’s Guide¶
Table of Contents
This manual provides reference documentation to SfePy from a user’s perspective.
Running a Simulation¶
The following should be run in the top-level directory of the SfePy source
tree after compiling the C extension files. See
Installation for full installation instructions info. The
$ indicates the command prompt of your terminal.
Basic Usage¶
$ ./simple.py examples/diffusion/poisson.py
- Creates
cylinder.vtk
- Creates
$ ./simple.py examples/navier_stokes/stokes.py
- Creates
channels_symm944t.vtk
- Creates
$ ./runTests.py
- See Running Tests
$ ./isfepy
- Follow the help information printed on startup
Surface Extraction¶
$ ./findSurf.py meshes/quantum/cube.node -
- Creates
surf_cube.mesh
- Creates
Applications¶
Phononic Materials
$ ./phonon.py -p examples/phononic/band_gaps.py
- see
examples/phononic/output/
- see
schroedinger.py(order is important below):
$ ./schroedinger.py --2d --create-mesh
$ ./schroedinger.py --2d --hydrogen
$ ./postproc.py mesh.vtk
Stand-Alone Examples¶
$ python examples/rs_correctors.py
$ python examples/compare_elastic_materials.py
$ python examples/live_plot.py
Running Tests¶
The tests are run by the runTests.py script:
$ ./runTests.py -h
Usage: runTests.py [options] [test_filename[ test_filename ...]]
Options:
--version show program's version number and exit
-h, --help show this help message and exit
--print-doc print the docstring of this file (howto write new
tests)
-d directory, --dir=directory
directory with tests [default: tests]
-o directory, --output=directory
directory for storing test results and temporary files
[default: output-tests]
--debug raise silenced exceptions to see what was wrong
--filter-none do not filter any messages
--filter-less filter output (suppress all except test messages)
--filter-more filter output (suppress all except test result
messages)
Common Tasks¶
Run all tests, filter output; result files related to the tests can be found in output-tests directory:
./runTests.py ./runTests.py --filter-more ./runTests.py --filter-less
Run a particular test file, filter output:
./runTests.py tests/test_input_le.py # Test if linear elasticity input file works.
Debug a failing test:
./runTests.py tests/test_input_le.py --debug
Computations and Examples¶
The example problems in the examples directory can be computed by the script
simple.py which is in the top-level directory of the SfePy distribution.
If it is run without arguments, a help message is printed:
$ ./simple.py
Usage: simple.py [options] filename_in
Options:
--version show program's version number and exit
-h, --help show this help message and exit
-c "key : value, ...", --conf="key : value, ..."
override problem description file items, written as
python dictionary without surrouding braces
-O "key : value, ...", --options="key : value, ..."
override options item of problem description, written
as python dictionary without surrouding braces
-o filename basename of output file(s) [default: <basename of
input file>]
--format=format output file format, one of: {vtk, h5, mesh} [default:
vtk]
--log=file log all messages to specified file (existing file will
be overwritten!)
-q, --quiet do not print any messages to screen
--save-ebc save problem state showing EBC (Dirichlet conditions)
--save-regions save problem regions as meshes
--save-regions-as-groups
save problem regions in a single mesh but mark them by
using different element/node group numbers
--save-field-meshes save meshes of problem fields (with extra DOF nodes)
--solve-not do not solve (use in connection with --save-*)
--list=what list data, what can be one of: {terms}
Additional (stand-alone) examples are in the examples/ directory, e.g.:
$ python examples/compare_elastic_materials.py
Parametric study example:
$ ./simple.py examples/diffusion/poisson_parametric_study.py
Common Tasks¶
Run a simulation:
./simple.py examples/diffusion/poisson.py ./simple.py examples/diffusion/poisson.py -o some_results # -> produces some_results.vtk
Print available terms:
./simple.py --list=terms
Run a simulation and also save Dirichlet boundary conditions:
./simple.py --save-ebc examples/diffusion/poisson.py # -> produces an additional .vtk file with BC visualization
Visualization of Results¶
The postproc.py script can be used for quick postprocessing and
visualization of the SfePy results. It requires mayavi2 installed on your
system. Running postproc.py without arguments produces:
$ ./postproc.py
Usage: postproc.py [options] filename
This is a script for quick Mayavi-based visualizations of finite element
computations results.
Examples
--------
The examples assume that runTests.py has been run successfully and the
resulting data files are present.
- view data in output-tests/test_navier_stokes.vtk
$ python postproc.py output-tests/test_navier_stokes.vtk
$ python postproc.py output-tests/test_navier_stokes.vtk --3d
- create animation (forces offscreen rendering) from
output-tests/test_time_poisson.*.vtk
$ python postproc.py output-tests/test_time_poisson.*.vtk -a mov
- create animation (forces offscreen rendering) from
output-tests/test_hyperelastic.*.vtk
The range specification for the displacements 'u' is required, as
output-tests/test_hyperelastic.00.vtk contains only zero
displacements which leads to invisible glyph size.
$ python postproc.py output-tests/test_hyperelastic.*.vtk --ranges=u,0,0.02 -a mov
- same as above, but slower frame rate
$ python postproc.py output-tests/test_hyperelastic.*.vtk --ranges=u,0,0.02 -a mov --ffmpeg-options="-r 2 -sameq"
Options:
--version show program's version number and exit
-h, --help show this help message and exit
-l, --list-ranges do not plot, only list names and ranges of all data
-n, --no-show do not call mlab.show()
--no-offscreen force no offscreen rendering for --no-show
--3d 3d plot mode
--view=angle,angle[,distance[,focal_point]]
camera azimuth, elevation angles, and optionally also
distance and focal point coordinates (without []) as
in `mlab.view()` [default: if --3d is True: "45,45",
else: "0,0"]
--roll=angle camera roll angle [default: 0.0]
--fgcolor=R,G,B foreground color, that is the color of all text
annotation labels (axes, orientation axes, scalar bar
labels) [default: 0.0,0.0,0.0]
--bgcolor=R,G,B background color [default: 1.0,1.0,1.0]
--layout=layout layout for multi-field plots, one of: rowcol, colrow,
row, col [default: rowcol]
--scalar-mode=mode mode for plotting scalars with --3d, one of:
cut_plane, iso_surface, both [default: iso_surface]
--vector-mode=mode mode for plotting vectors, one of: arrows, norm,
arrows_norm, warp_norm [default: arrows_norm]
-s scale, --scale-glyphs=scale
relative scaling of glyphs (vector field
visualization) [default: 0.05]
--clamping glyph clamping mode
--ranges=name1,min1,max1:name2,min2,max2:...
force data ranges [default: automatic from data]
-b, --scalar-bar show scalar bar for each data
--wireframe show wireframe of mesh surface for each data
--opacity=opacity global surface and wireframe opacity in [0.0, 1.0]
[default: 1.0]
--rel-text-width=width
relative text annotation width [default: 0.02]
-w, --watch watch the results file for changes (single file mode
only)
-o filename, --output=filename
view image file name [default: 'view.png']
--output-dir=directory
output directory for saving view images; ignored when
-o option is given, as the directory part of the
filename is taken instead [default: '.']
-a <ffmpeg-supported format>, --animation=<ffmpeg-supported format>
if set to a ffmpeg-supported format (e.g. mov, avi,
mpg), ffmpeg is installed and results of multiple time
steps are given, an animation is created in the same
directory as the view images
--ffmpeg-options="<ffmpeg options>"
ffmpeg animation encoding options (enclose in "")
[default: -r 10 -sameq]
-r resolution, --resolution=resolution
image resolution in NxN format [default: shorter axis:
600; depends on layout: for rowcol it is 800x600]
--all draw all data (normally, node_groups and mat_id are
omitted)
--only-names=list of names
draw only named data
--group-names=name1,...,nameN:...
superimpose plots of data in each group
--subdomains=mat_id_name,threshold_limits,single_color
superimpose surfaces of subdomains over each data;
example value: mat_id,0,None,True
--step=step set the time step [default: 0]
--anti-aliasing=value
value of anti-aliasing [default: mayavi2 default]
-d 'var_name0,function_name0,par0=val0,par1=val1,...:var_name1,...', --domain-specific='var_name0,function_name0,par0=val0,par1=val1,...:var_name1,...'
domain specific drawing functions and configurations
As a simple example, try:
$ ./simple.py examples/diffusion/poisson.py
$ ./postproc.py cylinder.vtk
The following window should display:
The -l switch lists information contained in a results file, e.g.:
$ ./postproc.py -l cylinder.vtk
sfepy: 0: cylinder.vtk
point scalars
"node_groups" (354,) range: 0 0 l2_norm_range: 0.0 0.0
"t" (354,) range: -2.0 2.0 l2_norm_range: 0.0106091 2.0
cell scalars
"mat_id" (1348,) range: 6 6 l2_norm_range: 6.0 6.0
Problem Description File¶
Here we discuss the basic items that users have to specify in their input
files. For complete examples, see the problem description files in the
examples/ directory of SfePy.
FE Mesh¶
A FE mesh defining a domain geometry can be stored in several formats:
- legacy VTK (
.vtk) - custom HDF5 file (
.h5) - medit mesh file (
.mesh) - tetgen mesh files (
.node,.ele) - comsol text mesh file (
.txt) - abaqus text mesh file (
.inp) - avs-ucd text mesh file (
.inp) - hypermesh text mesh file (
.hmascii) - hermes3d mesh file (
.mesh3d) - nastran text mesh file (
.bdf) - gambit neutral text mesh file (
.neu) - salome/pythonocc med binary mesh file (
.med)
Example:
filename_mesh = 'meshes/3d/cylinder.vtk'
The VTK and HDF5 formats can be used for storing the results. The format can be selected in options, see Miscellaneous.
The following geometry elements are supported:
Regions¶
Regions serve to select a certain part of the computational domain (= selection of nodes and elements of a FE mesh). They are used to define the boundary conditions, the domains of terms and materials etc.
Region selection syntax
- Entity selections
allnodes of surfacenodes of group <integer>nodes of group <str>(if mesh format supports reading boundary condition nodes)nodes in <expr>nodes by <function>node <id>[, <id>, ...]elements of group <integer>elements by <efunction>element <id>[, <id>, ...]assumes group 0 (ig = 0)element (<ig>, <id>)[, (<ig>, <id>), ...]r.<name of another region>
- Notation
<expr>is a logical expression like(y <= 0.00001) & (x < 0.11)<function>is e.g.,afunction( x, y, z, otherArgs )<efunction>is e.g.,efunction( domain )
- Region operations
- Node-wise:
+n,-n,*n(union, set difference, intersection) - Element-wise:
+e,-e,*e(union, set difference, intersection)
- Node-wise:
- Additional specification:
- ‘forbid’ : ‘group <integer>’ - forbid elements of listed groups
- ‘can_cells’ : <boolean> - determines whether a region can have cells (volume in 3D)
- Entity selections
Region definition syntax
Long syntax: a region is defined by the following Python dictionary (denote optional keys):
region_<number> = { 'name' : <name>, 'select' : <selection>, ['forbid'] : group <integer>[, <integer>], ['can_cells'] : <boolean>, }
Example definitions:
region_20 = { 'name' : 'Left', 'select' : 'nodes in (x < -0.499)' } region_21 = { 'name' : 'Right', 'select' : 'nodes in (x > 0.499)' } region_31 = { 'name' : 'Gamma1', 'select' : """(elements of group 1 *n elements of group 4) +n (elements of group 2 *n elements of group 4) +n ((r.Left +n r.Right) *n elements of group 4) """, 'forbid' : 'group 1 2' }
Short syntax:
regions = { <name> : ( <selection>, {[<additional spec.>]} ) }
Example definitions:
regions = { 'Left' : ('nodes in (x < -0.499)', {}), 'Right' : ('nodes in (x > 0.499)', {}), 'Gamma1' : ("""(elements of group 1 *n elements of group 4) +n (elements of group 2 *n elements of group 4) +n ((r.Left +n r.Right) *n elements of group 4)""", {'forbid' : 'group 1 2'}), }
Fields¶
Fields correspond to FE spaces
Long syntax:
field_<number> = { 'name' : <name>, 'dtype' : <data_type>, 'shape' : <shape>, 'region' : <region_name>, 'approx_order' : <approx_order> }
- where
- <data_type> is a numpy type (float64 or complex128) or ‘real’ or ‘complex’
- <shape> is the number of DOFs per node: 1 or (1,) or ‘scalar’, space dimension (2, or (2,) or 3 or (3,)) or ‘vector’; it can be other positive integer than just 1, 2, or 3
- <region_name> is the name of region where the field is defined
- <approx_order> is the FE approximation order, e.g. 0, 1, 2, ‘1B’ (1 with bubble)
Example: scalar P1 elements in 2D on a region Omega:
field_1 = { 'name' : 'temperature', 'dtype' : 'real', 'shape' : 'scalar', 'region' : 'Omega', 'approx_order' : 1 }
Short syntax:
fields = { <name> : (<data_type>, <shape>, <region_name>, <approx_order>) }
Example: scalar P1 elements in 2D on a region Omega:
fields = { 'temperature' : ('real', 1, 'Omega', 1), }
The following approximation orders can be used:
- simplex elements: 1, 2, ‘1B’, ‘2B’
- tensor product elements: 0, 1, ‘1B’
Optional bubble function enrichment is marked by ‘B’.
Variables¶
Variables use the FE approximation given by the specified field:
Long syntax:
variables_<number> = { 'name' : <name>, 'kind' : <kind>, 'field' : <field_name>, ['order' : <order>,] ['dual' : <variable_name>,] ['history' : <history_size>,] }
- where
- <kind> - ‘unknown field’, ‘test field’ or ‘parameter field’
- <order> - primary variable - order in the global vector of unknowns
- <history_size> - number of time steps to remember prior to current step
Example, long syntax:
variable_1 = { 'name' : 't', 'kind' : 'unknown field', 'field' : 'temperature', 'order' : 0, # order in the global vector of unknowns 'history' : 1, } variable_2 = { 'name' : 's', 'kind' : 'test field', 'field' : 'temperature', 'dual' : 't', }
Short syntax:
variables = { <name> : (<kind>, <field_name>, <spec.>, [<history>]) }
where
<spec> - in case of: primary variable - order in the global vector of unknowns, dual variable - name of primary variable
Example, short syntax:
variables = { 't' : ('unknown field', 'temperature', 0, 1), 's' : ('test field', 'temperature', 't'), }
Integrals¶
Define the integral type and quadrature rule. This keyword is optional, as the integration orders can be specified directly in equations, see below.
Long syntax:
integral_<number> = { 'name' : <name>, 'kind' : <kind>, 'order' : <order>, }
where
- <name> - the integral name - it has to begin with ‘i’!
- <kind> - volume ‘v’ or surface ‘s’ integral
- <order> - the order of polynomials to integrate, or ‘custom’ for integrals with explicitly given values and weights
Example, long syntax:
integral_1 = { 'name' : 'i1', 'kind' : 'v', 'order' : 2, } import numpy as nm N = 2 integral_2 = { 'name' : 'i2', 'kind' : 'v', 'order' : 'custom', 'vals' : zip(nm.linspace( 1e-10, 0.5, N ), nm.linspace( 1e-10, 0.5, N )), 'weights' : [1./N] * N, }
Short syntax:
integrals = { <name> : (<kind>, <order>) }
Example, short syntax:
import numpy as nm N = 2 integrals = { 'i1' : ('v', 2), 'i2' : ('v', 'custom', zip(nm.linspace( 1e-10, 0.5, N ), nm.linspace( 1e-10, 0.5, N )), [1./N] * N), }
Boundary Conditions¶
The boundary conditions apply in a given region given by its name, and, optionally, in selected times. The times can be given either using a list of tuples (t0, t1) making the condition active for t0 <= t < t1, or by a name of a function taking the time argument and returning True or False depending on whether the condition is active at the given time or not.
Dirichlet (essential) boundary conditions, long syntax:
ebc_<number> = { 'name' : <name>, 'region' : <region_name>, ['times' : <times_specification>,] 'dofs' : {<dof_specification> : <value>[, <dof_specification> : <value>, ...]} }
Example:
ebc_1 = { 'name' : 'ZeroSurface', 'region' : 'Surface', 'times' : [(0.5, 1.0), (2.3, 5)], 'dofs' : {'u.all' : 0.0, 'phi.all' : 0.0}, }
Dirichlet (essential) boundary conditions, short syntax:
ebcs = { <name> : (<region_name>, [<times_specification>,] {<dof_specification> : <value>[, <dof_specification> : <value>, ...]},...) }
Example:
ebcs = { 'u1' : ('Left', {'u.all' : 0.0}), 'u2' : ('Right', [(0.0, 1.0)], {'u.0' : 0.1}), 'phi' : ('Surface', {'phi.all' : 0.0}), }
Initial Conditions¶
Initial conditions are applied prior to the boundary conditions - no special care must be used for the boundary dofs.
Long syntax:
ic_<number> = { 'name' : <name>, 'region' : <region_name>, 'dofs' : {<dof_specification> : <value>[, <dof_specification> : <value>, ...]} }
Example:
ic_1 = { 'name' : 'ic', 'region' : 'Omega', 'dofs' : {'T.0' : 5.0}, }
Short syntax:
ics = { <name> : (<region_name>, {<dof_specification> : <value>[, <dof_specification> : <value>, ...]},...) }
Example:
ics = { 'ic' : ('Omega', {'T.0' : 5.0}), }
Materials¶
Materials are used to define constitutive parameters (e.g. stiffness, permeability, or viscosity), and other non-field arguments of terms (e.g. known traction or volume forces). Depending on a particular term, the parameters can be constants, functions defined over FE mesh nodes, functions defined in the elements, etc.
Example, long syntax:
material_10 = { 'name' : 'm', 'values' : { # This gets tiled to all physical QPs (constant function) 'val' : [0.0, -1.0, 0.0], # This does not - '.' denotes a special value, e.g. a flag. '.val0' : [0.0, 0.1, 0.0], }, } material_3 = { 'name' : 'm2', 'function' : 'some_function', } def some_function(ts, coor, region, ig, mode=None): out = {} if mode == 'qp': # <array of shape (coor.shape[0], n_row, n_col)> out['val'] = nm.ones((coor.shape[0], 1, 1), dtype=nm.float64) else: # special mode out['val0'] = True
Example, short syntax:
material = { 'm' : ({'val' : [0.0, -1.0, 0.0]},), 'm2' : 'some_function', 'm3' : (None, 'some_function'), # Same as the above line. }
Example, short syntax, different material parameters in regions ‘Yc’, ‘Ym’:
from sfepy.mechanics.matcoefs import stiffness_from_youngpoisson dim = 3 materials = { 'mat' : ({'D' : { 'Ym': stiffness_from_youngpoisson(dim, 7.0e9, 0.4), 'Yc': stiffness_from_youngpoisson(dim, 70.0e9, 0.2)} },), }
Equations and Terms¶
Equations can be built by combining terms listed in Term Table.
Examples¶
Laplace equation, named integral:
equations = { 'Temperature' : """dw_laplace.i1.Omega( coef.val, s, t ) = 0""" }
Laplace equation, simplified integral given by order:
equations = { 'Temperature' : """dw_laplace.2.Omega( coef.val, s, t ) = 0""" }
Laplace equation, automatic integration order (not implemented yet!):
equations = { 'Temperature' : """dw_laplace.a.Omega( coef.val, s, t ) = 0""" }
Navier-Stokes equations:
equations = { 'balance' : """+ dw_div_grad.i2.Omega( fluid.viscosity, v, u ) + dw_convect.i2.Omega( v, u ) - dw_stokes.i1.Omega( v, p ) = 0""", 'incompressibility' : """dw_stokes.i1.Omega( u, q ) = 0""", }
Configuring Solvers¶
In SfePy, a non-linear solver has to be specified even when solving a linear problem. The linear problem is/should be then solved in one iteration of the nonlinear solver.
Linear solver, long syntax:
solver_0 = { 'name' : 'ls', 'kind' : 'ls.umfpack', }
Nonlinear solver, long syntax:
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) }
Solvers, short syntax:
solvers = { 'ls' : ('ls.scipy_direct', {}), 'newton' : ('nls.newton', {'i_max' : 1, 'problem' : 'nonlinear'}), }
Solver selection:
options = { 'nls' : 'newton', 'ls' : 'ls', }
Functions¶
Functions are a way of customizing SfePy behavior. They make it possible to define material properties, boundary conditions, parametric sweeps, and other items in an arbitrary manner. Functions are normal Python functions declared in the Problem Definition file, so they can invoke the full power of Python. In order for SfePy to make use of the functions, they must be declared using the function keyword. See the examples below.
Defining Material Parameters¶
The functions for defining material parameters can work in two modes, distinguished by the mode argument. The two modes are ‘qp’ and ‘special’. The first mode is used for usual functions that define parameters in quadrature points (hence ‘qp’), while the second one can be used for special values like various flags.
The shape and type of data returned in the ‘special’ mode can be arbitrary (depending on the term used). On the other hand, in the ‘qp’ mode all the data have to be numpy float64 arrays with shape (n_coor, n_row, n_col), where n_coor is the number of quadrature points given by the coors argument, n_coor = coors.shape[0], and (n_row, n_col) is the shape of a material parameter in each quadrature point. For example, for scalar parameters, the shape is (n_coor, 1, 1).
Examples¶
See examples/diffusion/poisson_functions.py for a complete problem
description file demonstrating how to use different kinds of functions.
functions for defining regions:
def get_circle(coors, domain=None): r = nm.sqrt(coors[:,0]**2.0 + coors[:,1]**2.0) return nm.where(r < 0.2)[0] functions = { 'get_circle' : (get_circle,), }
functions for defining boundary conditions:
def get_p_edge(ts, coors, bc=None, problem=None): if bc.name == 'p_left': return nm.sin(nm.pi * coors[:,1]) else: return nm.cos(nm.pi * coors[:,1]) functions = { 'get_p_edge' : (get_p_edge,), } ebcs = { 'p' : ('Gamma', {'p.0' : 'get_p_edge'}), }
The values can be given by a function of time, coordinates and possibly other data, for example:
ebcs = { 'f1' : ('Gamma1', {'u.0' : 'get_ebc_x'}), 'f2' : ('Gamma2', {'u.all' : 'get_ebc_all'}), } def get_ebc_x(coors, amplitude): z = coors[:, 2] val = amplitude * nm.sin(z * 2.0 * nm.pi) return val def get_ebc_all(ts, coors): x, y, z = coors[:, 0], coors[:, 1], coors[:, 2] val = ts.step * nm.r_[x, y, z] return val functions = { 'get_ebc_x' : (lambda ts, coors, bc, problem, **kwargs: get_ebc_x(coors, 5.0),), 'get_ebc_all' : (lambda ts, coors, bc, problem, **kwargs: get_ebc_all(ts, coors),), }
Note that when setting more than one component as in get_ebc_all() above, the function should return a single one-dimensional vector with all values of the first component, then of the second one etc. concatenated together.
function for defining usual material parameters:
def get_pars(ts, coors, mode=None, region=None, ig=None): if mode == 'qp': val = coors[:,0] val.shape = (coors.shape[0], 1, 1) return {'x_coor' : val} functions = { 'get_pars' : (get_pars,), }
function for defining special material parameters, with an extra argument:
def get_pars_special(ts, coors, mode=None, region=None, ig=None, extra_arg=None): if mode == 'special': if extra_arg == 'hello!': ic = 0 else: ic = 1 return {('x_%s' % ic) : coors[:,ic]} functions = { 'get_pars1' : (lambda ts, coors, mode=None, region=None, ig=None: get_pars_special(ts, coors, mode, region, ig, extra_arg='hello!'),), } # Just another way of adding a function, besides 'functions' keyword. function_1 = { 'name' : 'get_pars2', 'function' : lambda ts, coors,mode=None, region=None, ig=None: get_pars_special(ts, coors, mode, region, ig, extra_arg='hi!'), }
function combining both kinds of material parameters:
def get_pars_both(ts, coors, mode=None, region=None, ig=None): out = {} if mode == 'special': out['flag'] = coors.max() > 1.0 elif mode == 'qp': val = coors[:,1] val.shape = (coors.shape[0], 1, 1) out['y_coor'] = val return out functions = { 'get_pars_both' : (get_pars_both,), }
function for setting values of a parameter variable:
variable_1 = { 'name' : 'p', 'kind' : 'parameter field', 'field' : 'temperature', 'like' : None, 'special' : {'setter' : 'get_load_variable'}, } def get_load_variable(ts, coors, region=None): y = coors[:,1] val = 5e5 * y return val functions = { 'get_load_variable' : (get_load_variable,) }
Miscellaneous¶
The options can be used to select solvers, output file format, output directory, to register functions to be called at various phases of the solution (the hooks), and for other settings.
Additional options (including solver selection):
options = { # string, output directory 'output_dir' : 'output/<output_dir>', # 'vtk' or 'h5', output file (results) format 'output_format' : 'h5', # string, nonlinear solver name 'nls' : 'newton', # string, linear solver name 'ls' : 'ls', # string, time stepping solver name 'ts' : 'ts', # int, number of time steps when results should be saved (spaced # regularly from 0 to n_step), or -1 for all time steps 'save_steps' : -1, # string, a function to be called after each time step 'step_hook' : '<step_hook_function>', # string, a function to be called after each time step, used to # update the results to be saved 'post_process_hook' : '<post_process_hook_function>', # string, as above, at the end of simulation 'post_process_hook_final' : '<post_process_hook_final_function>', # string, a function to generate probe instances 'gen_probes' : '<gen_probes_function>', # string, a function to probe data 'probe_hook' : '<probe_hook_function>', # string, a function to modify problem definition parameters 'parametric_hook' : '<parametric_hook_function>', }
post_process_hookenables computing derived quantities, like stress or strain, from the primary unknown variables. See the examples inexamples/large_deformation/directory.parametric_hookmakes it possible to run parametric studies by modifying the problem description programmatically. Seeexamples/diffusion/poisson_parametric_study.pyfor an example.output_dirredirects output files to specified directory
Building Equations in SfePy¶
Equations in SfePy are built using terms, which correspond directly to the integral forms of weak formulation of a problem to be solved. As an example, let us consider the Laplace equation in time interval
:The weak formulation of (1) is: Find
, such thatwhere we assume no fluxes over
. In the syntax used in SfePy input files, this can be written as:dw_mass_scalar.i1.Omega( s, dT/dt ) + dw_laplace.i1.Omega( coef, s, T) = 0
which directly corresponds to the discrete version of (2): Find
, such thatwhere
, for . The integrals over the discrete domain are approximated by a numerical quadrature, that is named in our case.Syntax of Terms in Equations¶
The terms in equations are written in form:
<term_name>.<i>.<r>( <arg1>, <arg2>, ... )
where <i> denotes an integral name (i.e. a name of numerical quadrature to
use) and <r> marks a region (domain of the integral). In the following,
<virtual> corresponds to a test function, <state> to a unknown function
and <parameter> to a known function arguments.
When solving, the individual terms in equations are evaluated in the ‘weak’ mode. The evaluation modes are described in the next section.
Term Evaluation¶
Terms can be evaluated in two ways:
- implicitly by using them in equations;
- explicitly using
ProblemDefinition.evaluate(). This way is mostly used in the postprocessing.
Each term supports one or more evaluation modes:
- ‘weak’ : Assemble (in the finite element sense) either the vector or matrix
depending on diff_var argument (the name of variable to differentiate with
respect to) of
Term.evaluate(). This mode is usually used implicitly when building the linear system corresponding to given equations. - ‘eval’ : The evaluation mode integrates the term (= integral) over a region. The result has the same dimension as the quantity being integrated. This mode can be used, for example, to compute some global quantities during postprocessing such as fluxes or total values of extensive quantities (mass, volume, energy, ...).
- ‘el_avg’ : The element average mode results in an array of a quantity averaged in each element of a region. This is the mode for postprocessing.
- ‘el’ : The element integral value mode results in an array of a quantity integrated over each element of a region. This mode is supported only by some special terms.
- ‘qp’ : The quadrature points mode results in an array of a quantity interpolated into quadrature points of each element in a region. This mode is used when further point-wise calculations with the result are needed. The same element type and number of quadrature points in each element are assumed.
Not all terms support all the modes - consult the documentation of the individual terms. There are, however, certain naming conventions:
- ‘dw_*’ terms support ‘weak’ mode
- ‘dq_*’ terms support ‘qp’ mode
- ‘d_*’, ‘di_*’ terms support ‘eval’ mode
- ‘ev_*’ terms support ‘eval’, ‘el_avg’ and ‘qp’ modes
Note that the naming prefixes are due to history when the mode argument to
ProblemDefinition.evaluate() and Term.evaluate() was not available. Now they are often
redundant, but are kept around to indicate the evaluation purpose of each term.
Several examples of using the ProblemDefinition.evaluate() function are shown below.
Solution Postprocessing¶
A solution to equations given in a problem description file is given by the variables of the ‘unknown field’ kind, that are set in the solution procedure. By default, those are the only values that are stored into a results file. The solution postprocessing allows computing additional, derived, quantities, based on the primary variables values, as well as any other quantities to be stored in the results.
Let us illustrate this using several typical examples. Let us assume that the postprocessing function is called ‘post_process()’, and is added to options as discussed in Miscellaneous, see ‘post_process_hook’ and ‘post_process_hook_final’. Then:
compute stress and strain given the displacements (variable u):
def post_process(out, problem, state, extend=False): """ This will be called after the problem is solved. Parameters ---------- out : dict The output dictionary, where this function will store additional data. problem : ProblemDefinition instance The current ProblemDefinition instance. state : State instance The computed state, containing FE coefficients of all the unknown variables. extend : bool The flag indicating whether to extend the output data to the whole domain. It can be ignored if the problem is solved on the whole domain already. Returns ------- out : dict The updated output dictionary. """ from sfepy.base.base import Struct # Cauchy strain averaged in elements. strain = problem.evaluate('ev_cauchy_strain.i1.Omega(u)', mode='el_avg') out['cauchy_strain'] = Struct(name='output_data', mode='cell', data=strain, dofs=None) # Cauchy stress averaged in elements. stress = problem.evaluate('ev_cauchy_stress.i1.Omega(solid.D, u)', mode='el_avg') out['cauchy_stress'] = Struct(name='output_data', mode='cell', data=stress, dofs=None) return out
The full example is linear_elasticity/linear_elastic_probes.py.
compute diffusion velocity given the pressure:
def post_process(out, pb, state, extend=False): from sfepy.base.base import Struct dvel = pb.evaluate('ev_diffusion_velocity.i1.Omega(m.K, p)', mode='el_avg') out['dvel'] = Struct(name='output_data', mode='cell', data=dvel, dofs=None) return out
The full example is biot/biot_npbc.py.
store values of a non-homogeneous material parameter:
def post_process(out, pb, state, extend=False): from sfepy.base.base import Struct mu = pb.evaluate('ev_integrate_mat.2.Omega(nonlinear.mu, u)', mode='el_avg', copy_materials=False, verbose=False) out['mu'] = Struct(name='mu', mode='cell', data=mu, dofs=None) return out
The full example is linear_elasticity/material_nonlinearity.py.
compute volume of a region (u is any variable defined in the region Omega):
volume = problem.evaluate('d_volume.2.Omega(u)')
Probing¶
Probing applies interpolation to output the solution along specified paths. As
mentioned in Miscellaneous, it relies on defining two additional
functions, namely the ‘gen_probes’ function, that should create the required
probes (see sfepy.fem.probes), and the ‘probe_hook’ function that
performs the actual probing of the results for each of the probes. This
function can return the probing results, as well as a handle to a corresponding
matplotlib figure. See Primer for additional explanation.
Available Solvers¶
This Section describes solvers available in SfePy from user’s perspective. There internal/external solvers include linear, nonlinear, eigenvalue, optimization and time stepping solvers.
Nonlinear Solvers¶
Almost every problem, even linear, is solved in SfePy using a nonlinear solver that calls a linear solver in each iteration. This approach unifies treatment of linear and non-linear problems, and simplifies application of Dirichlet (essential) boundary conditions, as the linear system computes not a solution, but a solution increment, i.e., it always has zero boundary conditions.
The following solvers are available:
- ‘nls.newton’: Newton solver with backtracking line-search - this is the default solver, that is used for almost all examples.
- ‘nls.oseen’: Oseen problem solver tailored for stabilized Navier-Stokes equations (see navier_stokes/stabilized_navier_stokes.py).
- ‘nls.scipy_broyden_like’: interface to Broyden and Anderson solvers from scipy.optimize.
- ‘nls.semismooth_newton’: Semismooth Newton method for contact/friction problems.
Linear Solvers¶
A good linear solver is key to solving efficiently stationary as well as transient PDEs with implicit time-stepping. The following solvers are available:
- ‘ls.scipy_direct’: direct solver from SciPy - this is the default solver for all examples. It is strongly recommended to install umfpack and its SciPy wrappers to get good performance.
- ‘ls.umfpack’: alias to ‘ls.scipy_direct’.
- ‘ls.scipy_iterative’: Interface to SciPy iterative solvers.
- ‘ls.pyamg’: Interface to PyAMG solvers.
- ‘ls.petsc’: Interface to Krylov subspace solvers of PETSc.
- ‘ls.petsc_parallel’: Interface to Krylov subspace solvers of PETSc able to run in parallel by storing the system to disk and running a separate script via mpiexec.
- ‘ls.schur_complement’: Schur complement problem solver.