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<?php $root=""; ?><?php require($root."navigation.php"); ?><html><head> <?php load_style($root); ?></head> <body> <?php make_navigation("ex11",$root)?> <div class="content"><a name="comments"></a> <div class = "comment"><h1>Example 11 - Stokes Equations - Systems of Equations</h1><br><br>This example shows how a simple, linear system of equationscan be solved in parallel. The system of equations are the familiarStokes equations for low-speed incompressible fluid flow.<br><br>C++ include files that we need</div><div class ="fragment"><pre> #include <iostream> #include <algorithm> #include <math.h> </pre></div><div class = "comment">Basic include file needed for the mesh functionality.</div><div class ="fragment"><pre> #include "libmesh.h" #include "mesh.h" #include "mesh_generation.h" #include "gmv_io.h" #include "equation_systems.h" #include "fe.h" #include "quadrature_gauss.h" #include "dof_map.h" #include "sparse_matrix.h" #include "numeric_vector.h" #include "dense_matrix.h" #include "dense_vector.h" #include "linear_implicit_system.h" </pre></div><div class = "comment">For systems of equations the \p DenseSubMatrixand \p DenseSubVector provide convenient ways forassembling the element matrix and vector on acomponent-by-component basis.</div><div class ="fragment"><pre> #include "dense_submatrix.h" #include "dense_subvector.h" </pre></div><div class = "comment">The definition of a geometric element</div><div class ="fragment"><pre> #include "elem.h" </pre></div><div class = "comment">Function prototype. This function will assemble the systemmatrix and right-hand-side.</div><div class ="fragment"><pre> void assemble_stokes (EquationSystems& es, const std::string& system_name); </pre></div><div class = "comment">The main program.</div><div class ="fragment"><pre> int main (int argc, char** argv) {</pre></div><div class = "comment">Initialize libMesh.</div><div class ="fragment"><pre> libMesh::init (argc, argv); { </pre></div><div class = "comment">Set the dimensionality of the mesh = 2</div><div class ="fragment"><pre> const unsigned int dim = 2; </pre></div><div class = "comment">Create a two-dimensional mesh.</div><div class ="fragment"><pre> Mesh mesh (dim); </pre></div><div class = "comment">Use the MeshTools::Generation mesh generator to create a uniformgrid on the square [-1,1]^D. We instruct the mesh generatorto build a mesh of 8x8 \p Quad9 elements in 2D, or \p Hex27elements in 3D. Building these higher-order elements allowsus to use higher-order approximation, as in example 3.</div><div class ="fragment"><pre> MeshTools::Generation::build_square (mesh, 15, 15, 0., 1., 0., 1., QUAD9); </pre></div><div class = "comment">Print information about the mesh to the screen.</div><div class ="fragment"><pre> mesh.print_info(); </pre></div><div class = "comment">Create an equation systems object.</div><div class ="fragment"><pre> EquationSystems equation_systems (mesh); </pre></div><div class = "comment">Declare the system and its variables.</div><div class ="fragment"><pre> {</pre></div><div class = "comment">Creates a transient system named "Convection-Diffusion"</div><div class ="fragment"><pre> LinearImplicitSystem & system = equation_systems.add_system<LinearImplicitSystem> ("Stokes"); </pre></div><div class = "comment">Add the variables "u" & "v" to "Stokes". Theywill be approximated using second-order approximation.</div><div class ="fragment"><pre> system.add_variable ("u", SECOND); system.add_variable ("v", SECOND); </pre></div><div class = "comment">Add the variable "p" to "Stokes". This willbe approximated with a first-order basis,providing an LBB-stable pressure-velocity pair.</div><div class ="fragment"><pre> system.add_variable ("p", FIRST); </pre></div><div class = "comment">Give the system a pointer to the matrix assemblyfunction.</div><div class ="fragment"><pre> system.attach_assemble_function (assemble_stokes); </pre></div><div class = "comment">Initialize the data structures for the equation system.</div><div class ="fragment"><pre> equation_systems.init (); equation_systems.parameters.set<unsigned int>("linear solver maximum iterations") = 250; equation_systems.parameters.set<Real> ("linear solver tolerance") = TOLERANCE; </pre></div><div class = "comment">Prints information about the system to the screen.</div><div class ="fragment"><pre> equation_systems.print_info(); } </pre></div><div class = "comment">Assemble & solve the linear system,then write the solution.</div><div class ="fragment"><pre> equation_systems.get_system("Stokes").solve(); GMVIO(mesh).write_equation_systems ("out.gmv", equation_systems); } </pre></div><div class = "comment">All done. </div><div class ="fragment"><pre> return libMesh::close (); } void assemble_stokes (EquationSystems& es, const std::string& system_name) {</pre></div><div class = "comment">It is a good idea to make sure we are assemblingthe proper system.</div><div class ="fragment"><pre> assert (system_name == "Stokes"); </pre></div><div class = "comment">Get a constant reference to the mesh object.</div><div class ="fragment"><pre> const Mesh& mesh = es.get_mesh(); </pre></div><div class = "comment">The dimension that we are running</div><div class ="fragment"><pre> const unsigned int dim = mesh.mesh_dimension(); </pre></div><div class = "comment">Get a reference to the Convection-Diffusion system object.</div><div class ="fragment"><pre> LinearImplicitSystem & system = es.get_system<LinearImplicitSystem> ("Stokes"); </pre></div><div class = "comment">Numeric ids corresponding to each variable in the system</div><div class ="fragment"><pre> const unsigned int u_var = system.variable_number ("u"); const unsigned int v_var = system.variable_number ("v"); const unsigned int p_var = system.variable_number ("p"); </pre></div><div class = "comment">Get the Finite Element type for "u". Note this will bethe same as the type for "v".</div><div class ="fragment"><pre> FEType fe_vel_type = system.variable_type(u_var); </pre></div><div class = "comment">Get the Finite Element type for "p".</div><div class ="fragment"><pre> FEType fe_pres_type = system.variable_type(p_var); </pre></div><div class = "comment">Build a Finite Element object of the specified type forthe velocity variables.</div><div class ="fragment"><pre> AutoPtr<FEBase> fe_vel (FEBase::build(dim, fe_vel_type)); </pre></div><div class = "comment">Build a Finite Element object of the specified type forthe pressure variables.</div><div class ="fragment"><pre> AutoPtr<FEBase> fe_pres (FEBase::build(dim, fe_pres_type)); </pre></div><div class = "comment">A Gauss quadrature rule for numerical integration.Let the \p FEType object decide what order rule is appropriate.⌨️ 快捷键说明
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