.. include:: links.inc .. _sec-tutorial: Tutorial ======== .. contents:: Table of Contents :local: :backlinks: top *SfePy* can be used in two basic ways: #. a black-box partial differential equation (PDE) solver, #. a Python package to build custom applications involving solving PDEs by the finite element (FE) method. 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 :doc:primer which focuses on a particular problem in detail. Notes on solving PDEs by the Finite Element Method -------------------------------------------------- The Finite Element Method (FEM) is the numerical method for solving Partial Differential Equations (PDEs). FEM was developed in the middle of XX. century and now it is widely used in different areas of science and engineering, including mechanical and structural design, biomedicine, electrical and power design, fluid dynamics and other. FEM is based on a very elegant mathematical theory of weak solution of PDEs. In this section we will briefly discuss basic ideas underlying FEM. .. _poisson-weak-form-tutorial: Strong form of Poisson's equation and its integration ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Let us start our discussion about FEM with the strong form of Poisson's equation .. math:: :label: eq_spoisson \Delta T = f(x), \quad x \in \Omega, .. math:: :label: eq_spoisson_dbc T = u(x), \quad x \in \Gamma_D, .. math:: :label: eq_spoisson_nbc \nabla T \cdot \mathbf{n} = g(x), \quad x \in \Gamma_N, where :math:\Omega \subset \mathbb{R}^n is the solution domain with the boundary :math:\partial \Omega, :math:\Gamma_D is the part of the boundary where Dirichlet boundary conditions are given, :math:\Gamma_N is the part of the boundary where Neumann boundary conditions are given, :math:T(x) is the unknown function to be found, :math:f(x), u(x), g(x) are known functions. FEM is based on a weak formulation. The weak form of the equation :eq:eq_spoisson is .. math:: \int\limits_{\Omega} (\Delta T - f) \cdot s \, \mathrm{d}\Omega = 0, where :math:s is a **test** function. Integrating this equation by parts .. math:: 0 = \int\limits_{\Omega} (\Delta T - f) \cdot s \, \mathrm{d}\Omega = \int\limits_{\Omega} \nabla \cdot (\nabla T) \cdot s \, \mathrm{d}\Omega - \int_{\Omega} f \cdot s \, \mathrm{d}\Omega = .. math:: = - \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega + \int\limits_{\Omega} \nabla \cdot (\nabla T \cdot s) \, \mathrm{d}\Omega - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega and applying Gauss theorem we obtain: .. math:: 0 = - \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega + \int\limits_{\Gamma_D \cup \Gamma_N} \!\!\!\! s \cdot (\nabla T \cdot \mathbf{n}) \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega or .. math:: \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega = \int\limits_{\Gamma_D \cup \Gamma_N} \!\!\!\! s \cdot (\nabla T \cdot \mathbf{n}) \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega. The surface integral term can be split into two integrals, one over the Dirichlet part of the surface and second over the Neumann part .. math:: :label: eq_wpoisson_full \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega = \int\limits_{\Gamma_D} s \cdot (\nabla T \cdot \mathbf{n}) \, \mathrm{d}\Gamma + \int\limits_{\Gamma_N} s \cdot (\nabla T \cdot \mathbf{n}) \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega. The equation :eq:eq_wpoisson_full is the initial weak form of the Poisson's problem :eq:eq_spoisson--:eq:eq_spoisson_nbc. But we can not work with it without applying the boundary conditions. So it is time to talk about the boundary conditions. Dirichlet Boundary Conditions """"""""""""""""""""""""""""" On the Dirichlet part of the surface we have two restrictions. One is the Dirichlet boundary conditions :math:T(x) = u(x) as they are, and the second is the integral term over :math:\Gamma_D in equation :eq:eq_wpoisson_full. To be consistent we have to use only the Dirichlet conditions and avoid the integral term. To implement this we can take the function :math:T \in V(\Omega) and the test function :math:s \in V_0(\Omega), where .. math:: V(\Omega) = \{f(x) \in H^1(\Omega)\}, .. math:: V_0(\Omega) = \{f(x) \in H^1(\Omega); f(x) = 0, x \in \Gamma_D\}. In other words the unknown function :math:T must be continuous together with its gradient in the domain. In contrast the test function :math:s must be also continuous together with its gradient in the domain but it should be zero on the surface :math:\Gamma_D. With this requirement the integral term over Dirichlet part of the surface is vanishing and the weak form of the Poisson equation for :math:T \in V(\Omega) and :math:s \in V_0(\Omega) becomes .. math:: \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega = \int\limits_{\Gamma_N} s \cdot (\nabla T \cdot \mathbf{n}) \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega, T(x) = u(x), \quad x \in \Gamma_D. That is why Dirichlet conditions in FEM terminology are called **Essential Boundary Conditions**. These conditions are not a part of the weak form and they are used as they are. Neumann Boundary Conditions """"""""""""""""""""""""""" The Neumann boundary conditions correspond to the known flux :math:g(x) = \nabla T \cdot \mathbf{n}. The integral term over the Neumann surface in the equation :eq:eq_wpoisson_full contains exactly the same flux. So we can use the known function :math:g(x) in the integral term: .. math:: \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega = \int\limits_{\Gamma_N} g \cdot s \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega, where test function :math:s also belongs to the space :math:V_0. That is why Neumann conditions in FEM terminology are called **Natural Boundary Conditions**. These conditions are a part of weak form terms. The weak form of the Poisson's equation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Now we can write the resulting weak form for the Poisson's problem :eq:eq_spoisson--:eq:eq_spoisson_nbc. For any test function :math:s \in V_0(\Omega) find :math:T \in V(\Omega) such that .. math:: :label: eq_wpoisson_final \boxed { \begin{split} \int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega & = \int\limits_{\Gamma_N} g \cdot s \, \mathrm{d}\Gamma - \int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega, \quad \mbox{and}\\ T(x) & = u(x), \quad x \in \Gamma_D. \end{split} } Discussion of discretization and meshing ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ It is planned to have an example of the discretization based on the Poisson's equation weak form :eq:eq_wpoisson_final. For now, please refer to the wikipedia page Finite Element Method_ for a basic description of the disretization and meshing. Numerical solution of the problem ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To solve numerically given problem based on the weak form :eq:eq_wpoisson_final we have to go through 5 steps: #. Define geometry of the domain :math:\Omega and surfaces :math:\Gamma_D and :math:\Gamma_N. #. Define the known functions :math:f, :math:u and :math:g. #. Define the unknown function :math:T and the test functions :math:s. #. Define essential boundary conditions (Dirichlet conditions) :math:T(x) = u(x), x \in \Gamma_D. #. Define equation and natural boundary conditions (Neumann conditions) as the set of all integral terms :math:\int\limits_{\Omega} \nabla T \cdot \nabla s \, \mathrm{d}\Omega, :math:\int\limits_{\Gamma_N} g \cdot s \, \mathrm{d}\Gamma, :math:\int\limits_{\Omega} f \cdot s \, \mathrm{d}\Omega. In the next section we will discuss how to define all these things in SfePy. Basic notions ------------- The simplest way of using *SfePy* is to solve a system of PDEs defined in a **problem description file**, also referred to as **input file**. In such a file, the problem is described using several keywords that allow one to define the equations, variables, finite element approximations, solvers, solution domain and subdomains etc., see :ref:sec-problem-description-file for a full list of those keywords. The syntax of the problem description file is very simple yet powerful, as the file itself is just a regular Python module that can be normally imported - no special parsing is necessary. The keywords mentioned above are regular Python variables (usually of the dict type) with special names. Historically, the keywords exist in two flavors: * **long syntax** is the original one - it is longer to type, but the individual fields are named, so it might be easier/understand to read for newcomers. * **short syntax** was added later to offer brevity for "expert" use. Below we show: #. how to solve a problem given by a problem description file, and #. explain the elements of the file on several examples. But let us begin with a slight detour... Sneak peek: what is going on under the hood ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ #. A top-level script (usually simple.py, as in this tutorial) reads in an input file. #. Following the contents of the input file, a :class:Problem  instance is created - this is the input file coming to life. Let us call the instance problem. * The problem sets up its domain, regions (various sub-domains), fields (the FE approximations), the equations and the solvers. The equations determine the materials and variables in use - only those are fully instantiated, so the input file can safely contain definitions of items that are not used actually. #. Prior to solution, problem.time_update() function has to be called to setup boundary conditions, material parameters and other potentially time-dependent data. This holds also for stationary problems with a single "time step". #. The solution is then obtained by calling problem.solve() function. #. Finally, the solution can be stored using problem.save_state() The above last three steps are essentially repeated for each time step. So that is it - using the code a black-box PDE solver shields the user from having to create the :class:Problem  instance by hand. But note that this is possible, and often necessary when the flexibility of the default solvers is not enough. At the end of the tutorial an example demonstrating the interactive creation of the problem is shown, see :ref:sec-interactive-example-linear-elasticity. Now let us continue with running a simulation. Running a simulation -------------------- The following commands should be run in the top-level directory of the *SfePy* source tree after compiling the C extension files. See :ref:introduction_installation for full installation instructions. .. _invoking_command_line: Invoking *SfePy* from the command line ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This section introduces the basics of running *SfePy* on the command line. The $ indicates the command prompt of your terminal. * The script simple.py is the most basic starting point in *SfePy*. It is invoked as follows::$ ./simple.py examples/diffusion/poisson.py * examples/diffusion/poisson.py is the *SfePy* *problem description* file, which defines the problem to be solved in terms *SfePy* can understand * Running the above command creates the output file cylinder.vtk in the *SfePy* top-level directory * *SfePy* can also be invoked interactively using IPython_ with custom imports, as described in :ref:using-ipython. In the SfePy top-level directory, run:: $ipython --profile=sfepy See :ref:sec-interactive-example-linear-elasticity for more information. .. _postprocessing: Postprocessing the results ^^^^^^^^^^^^^^^^^^^^^^^^^^ * The postproc.py script can be used for quick postprocessing and visualization of the *SfePy* output files. It requires mayavi2 installed on your system. * As a simple example, try::$ ./postproc.py cylinder.vtk * The following interactive 3D window should display: .. image:: images/postproc_simple.png :width: 70 % :align: center * The left mouse button by itself orbits the 3D view * Holding shift and the left mouse button pans the view * Holding control and the left mouse button rotates about the screen normal axis * The right mouse button controls the zoom .. _poisson-example-tutorial: Example problem description file -------------------------------- Here we discuss the contents of the :download:examples/diffusion/poisson.py <../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: .. math:: :label: eq_laplace_static c \Delta T = f \mbox{ in }\Omega,\quad T(t) = \bar{T}(t) \mbox{ on } \Gamma \;, where :math:\Gamma \subseteq \Omega is a subset of the domain :math:\Omega boundary. For simplicity, we set :math:f \equiv 0, but we still work with the material constant :math:c even though it has no influence on the solution in this case. We also assume zero fluxes over :math:\partial \Omega \setminus \Gamma, i.e. :math:\pdiff{T}{\ul{n}} = 0 there. The particular boundary conditions used below are :math:T = 2 on the left side of the cylindrical domain depicted in the previous section and :math:T = -2 on the right side. The first step to do is to write :eq:eq_laplace_static in *weak formulation* :eq:eq_wpoisson_final. The :math:f = 0, :math:g = \pdiff{T}{\ul{n}} = 0. So only one term in weak form :eq:eq_wpoisson_final remains: .. math:: :label: eq_wlaplace_static \int_{\Omega} c\ \nabla T \cdot \nabla s = 0, \quad \forall s \in V_0 \;. Comparing the above integral term with the long table in :ref: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. Long syntax of keywords ^^^^^^^^^^^^^^^^^^^^^^^ The example uses **long syntax** of the keywords. In next subsection, we show the same example written in **short syntax**. Open the :download:examples/diffusion/poisson.py <../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 :math:c 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' : 'cells of group 6', } region_03 = { 'name' : 'Gamma_Left', 'select' : 'vertices in (x < 0.00001)', 'kind' : 'facet', } region_4 = { 'name' : 'Gamma_Right', 'select' : 'vertices in (x > 0.099999)', 'kind' : 'facet', } 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 :math:V_h, 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' : 'i', '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.i.Omega( coef.val, s, t ) = 0""" } The equation above directly corresponds to the discrete version of :eq:eq_wlaplace_static, namely: Find :math:\bm{t} \in V_h, such that .. math:: \bm{s}^T (\int_{\Omega_h} c\ \bm{G}^T G) \bm{t} = 0, \quad \forall \bm{s} \in V_{h0} \;, where :math:\nabla u \approx \bm{G} \bm{u}. 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 i 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 :ref:ug_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, '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 :ref:invoking_command_line. Short syntax of keywords ^^^^^^^^^^^^^^^^^^^^^^^^ The same diffusion equation example as above in **short syntax** reads, see :download:examples/diffusion/poisson_short_syntax.py <../examples/diffusion/poisson_short_syntax.py>, as follows: .. literalinclude:: ../examples/diffusion/poisson_short_syntax.py :linenos: .. _sec-interactive-example-linear-elasticity: Interactive Example: Linear Elasticity -------------------------------------- This example shows how to use *SfePy* interactively, but also how to make a custom simulation script. We will use IPython_ with custom imports, as described in :ref:using-ipython, for the explanation, but regular Python shell would do as well, provided the proper modules are imported. We wish to solve the following linear elasticity problem: .. math:: :label: eq_linear_elasticity - \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 \;, where the stress is defined as :math:\sigma_{ij} = 2 \mu e_{ij} + \lambda e_{kk} \delta_{ij}, :math:\lambda, :math:\mu are the LamÃ©'s constants, the strain is :math:e_{ij}(\ul{u}) = \frac{1}{2}(\pdiff{u_i}{x_j} + \pdiff{u_j}{x_i}) and :math:\ul{f} are volume forces. This can be written in general form as :math:\sigma_{ij}(\ul{u}) = D_{ijkl} e_{kl}(\ul{u}), where in our case :math:D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl}. In the weak form the equation :eq:eq_linear_elasticity is .. math:: :label: eq_wlinear_elasticity \int_{\Omega} D_{ijkl} e_{kl}(\ul{u}) e_{ij}(\ul{v}) + \int_{\Omega} f_i v_i = 0 \;, where :math:\ul{v} is the test function, and both :math:\ul{u}, :math:\ul{v} 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 :ref:tutorial_interactive_source. In the SfePy top-level directory, run:: \$ ipython --profile=sfepy Read a finite element mesh, that defines the domain :math:\Omega. .. sourcecode:: ipython In [1]: mesh = Mesh.from_file('meshes/2d/rectangle_tri.mesh') Create a domain. The domain allows defining regions or subdomains. .. sourcecode:: ipython In [2]: domain = FEDomain('domain', mesh) Define the regions - the whole domain :math:\Omega, where the solution is sought, and :math:\Gamma_1, :math:\Gamma_2, 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. .. sourcecode:: ipython 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', ...: 'vertices in x < %.10f' % (min_x + eps), ...: 'facet') In [7]: gamma2 = domain.create_region('Gamma2', ...: 'vertices in x > %.10f' % (max_x - eps), ...: 'facet') Next we define the actual finite element approximation using the :class:Field class. .. sourcecode:: ipython In [8]: field = Field.from_args('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 :math:\ub and the test variable :math:\vb. .. sourcecode:: ipython In [9]: u = FieldVariable('u', 'unknown', field) In [10]: v = FieldVariable('v', 'test', field, 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 LamÃ© constants :math:\lambda = 1, :math:\mu = 1. The volume forces will be defined also as a material, as a constant (column) vector :math:[0.02, 0.01]^T. .. sourcecode:: ipython 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. .. sourcecode:: ipython In [14]: integral = Integral('i', order=3) Now we are ready to define the two terms and build the equations. .. sourcecode:: ipython 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]) The equations have to be completed by boundary conditions. Let us clamp the left edge :math:\Gamma_1, and shift the right edge :math:\Gamma_2 in the :math:x direction a bit, depending on the :math:y coordinate. .. sourcecode:: ipython In [20]: from sfepy.discrete.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, problem=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. .. sourcecode:: ipython 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 :class:Problem 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. .. sourcecode:: ipython In [30]: pb = Problem('elasticity', equations=eqs, nls=nls, ls=ls) The :class:Problem has several handy methods for debugging. Let us try saving the regions into a VTK file. .. sourcecode:: ipython In [31]: pb.save_regions_as_groups('regions') And view them. .. sourcecode:: ipython In [32]: view = Viewer('regions.vtk') In [33]: view() You should see this: .. image:: images/linear_elasticity_regions.png :width: 70 % :align: center Finally, we apply the boundary conditions, solve the problem, save and view the results. .. sourcecode:: ipython In [34]: pb.time_update(ebcs=Conditions([fix_u, shift_u])) In [35]: vec = pb.solve() In [36]: print nls_status In [37]: pb.save_state('linear_elasticity.vtk', vec) In [38]: view = Viewer('linear_elasticity.vtk') In [39]: view() This is the resulting image: .. image:: images/linear_elasticity_solution1.png :width: 70 % :align: center The default view is not very fancy. Let us show the displacements by shifting the mesh. Close the previous window and do: .. sourcecode:: ipython In [56]: view(vector_mode='warp_norm', rel_scaling=2, ....: is_scalar_bar=True, is_wireframe=True) And the result is: .. image:: images/linear_elasticity_solution2.png :width: 70 % :align: center See the docstring of view() and play with its options. .. _tutorial_interactive_source: Complete Example as a Script ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The source code: :download:linear_elasticity.py <../examples/standalone/interactive/linear_elasticity.py>. It should be run from the *SfePy* source directory so that it finds the mesh file. .. literalinclude:: ../examples/standalone/interactive/linear_elasticity.py :linenos: .. _isogeometric_analysis: Isogeometric Analysis --------------------- Isogeometric analysis_ (IGA) is a recently developed computational approach that allows using the NURBS-based domain description from CAD design tools also for approximation purposes similar to the finite element method. The implementation is SfePy is based on Bezier extraction of NURBS as developed in [1]_. This approach allows reusing the existing finite element assembling routines, as still the evaluation of weak forms occurs locally in "elements" and the local contributions are then assembled to the global system. Current Implementation ^^^^^^^^^^^^^^^^^^^^^^ The IGA code is still very preliminary and some crucial components are missing. The current implementation is also very slow, as it is in pure Python. The following already works: - single patch tensor product domain support in 2D and 3D - region selection based on topological Bezier mesh, see below - Dirichlet boundary conditions constant on entire sides of a patch - both scalar and vector volume terms work - term integration over the whole domain as well as a volume subdomain - simple linearization (output file generation) based on sampling the results with uniform parametric vectors - basic domain generation with script/gen_iga_patch.py based on igakit_ The following is not implemented yet: - tests - theoretical convergence rate verification - fast basis evaluation - surface terms - general Dirichlet boundary conditions (non-constant and/or on a subset of a side) - other boundary conditions - evaluation in arbitrary point in the physical domain - proper (adaptive) linearization for post-processing - support for multiple NURBS patches Domain Description """""""""""""""""" The domain description is in custom HDF5-based files with .iga extension. Such a file contains: - NURBS patch data (knots, degrees, control points and weights). Those can either be generated using igakit, created manually or imported from other tools. - Bezier extraction operators and corresponding DOF connectivity (computed by SfePy). - Bezier mesh control points, weights and connectivity (computed by SfePy). The Bezier mesh is used to create a topological Bezier mesh - a subset of the Bezier mesh containing the Bezier element corner vertices only. Those vertices are interpolatory (are on the exact geometry) and so can be used for region selections. Region Selection """""""""""""""" The domain description files contain vertex sets for regions corresponding to the patch sides, named 'xiIJ', where I is the parametric axis (0, 1, or 2) and J is 0 or 1 for the beginning and end of the axis knot span. Other regions can be defined in the usual way, using the topological Bezier mesh entities. Examples ^^^^^^^^ The examples demonstrating the use of IGA in SfePy are: - :ref:diffusion-poisson_iga - :ref:linear_elasticity-linear_elastic_iga Their problem description files are almost the same as their FEM equivalents, with the following differences: - There is filename_domain instead of filename_mesh. - Fields are defined as follows:: fields = { 'temperature' : ('real', 1, 'Omega', None, 'H1', 'iga'), } The approximation order is None as it is given by the NURBS degrees in the domain description. .. [1] Michael J. Borden, Michael A. Scott, John A. Evans, Thomas J. R. Hughes: Isogeometric finite element data structures based on Bezier extraction of NURBS, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, March 2010.