r""" Laplace equation with comments. Find :math:t such that: .. math:: \int_{\Omega} c \nabla s \cdot \nabla t = 0 \;, \quad \forall s \;. """ #! Poisson Equation #! ================ #$\centerline{Example input file, \today} #! Mesh #! ---- from sfepy import data_dir filename_mesh = data_dir + '/meshes/3d/cylinder.mesh' #! Materials #! --------- #$ Here we define just a constant coefficient $c$ of the Poisson equation, #$using the 'values' attribute. Other possible attribute is 'function', for #$ material coefficients computed/obtained at runtime. material_2 = { 'name' : 'coef', 'values' : {'val' : 1.0}, } #! Regions #! ------- 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)', } #! Fields #! ------ #! A field is used mainly to define the approximation on a (sub)domain, i.e. to #$define the discrete spaces$V_h$, where we seek the solution. #! #! The Poisson equation can be used to compute e.g. a temperature distribution, #! so let us call our field 'temperature'. On the region 'Omega' #! it will be approximated using P1 finite elements. field_1 = { 'name' : 'temperature', 'dtype' : 'real', 'shape' : (1,), 'region' : 'Omega', 'approx_order' : 1, } #! Variables #! --------- #! One field can be used to generate discrete degrees of freedom (DOFs) of #! several variables. Here the unknown variable (the temperature) is called #! 't', it's associated DOF name is 't.0' --- this will be referred to #! in the Dirichlet boundary section (ebc). The corresponding test variable of #! the weak formulation is called 's'. Notice that the 'dual' item of a test #! variable must specify the unknown it corresponds to. variable_1 = { 'name' : 't', 'kind' : 'unknown field', 'field' : 'temperature', 'order' : 0, # order in the global vector of unknowns } variable_2 = { 'name' : 's', 'kind' : 'test field', 'field' : 'temperature', 'dual' : 't', } #! Boundary Conditions #! ------------------- #! Essential (Dirichlet) boundary conditions can be specified as follows: ebc_1 = { 'name' : 't1', 'region' : 'Gamma_Left', 'dofs' : {'t.0' : 2.0}, } ebc_2 = { 'name' : 't2', 'region' : 'Gamma_Right', 'dofs' : {'t.0' : -2.0}, } #! Equations #! --------- #$ The weak formulation of the Poisson equation is: #$\begin{center} #$ Find $t \in V$, such that #\int_{\Omega} c\ \nabla t : \nabla s = f, \quad \forall s \in V_0$. #$ \end{center} #$The equation below directly corresponds to the discrete version of the #$ above, namely: #$\begin{center} #$ Find $\bm{t} \in V_h$, such that #\bm{s}^T (\int_{\Omega_h} c\ \bm{G}^T G) \bm{t} = 0, \quad \forall \bm{s} #$\in V_{h0}$, #$\end{center} #$ where $\nabla u \approx \bm{G} \bm{u}$. Below we use $f = 0$ (Laplace #\$ equation). #! We also define an integral here. integral_1 = { 'name' : 'i1', 'kind' : 'v', 'order' : 2, } equations = { 'Temperature' : """dw_laplace.i1.Omega( coef.val, s, t ) = 0""" } #! Linear solver parameters #! --------------------------- #! Use umfpack, if available, otherwise superlu. solver_0 = { 'name' : 'ls', 'kind' : 'ls.scipy_direct', 'method' : 'auto', } #! Nonlinear solver parameters #! --------------------------- #! Even linear problems are solved by a nonlinear solver (KISS rule) - only one #! iteration is needed and the final rezidual is obtained for free. 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) } #! Options #! ------- #! Use them for anything you like... Here we show how to tell which solvers #! should be used - reference solvers by their names. options = { 'nls' : 'newton', 'ls' : 'ls', }