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<?php $root=""; ?><?php require($root."navigation.php"); ?><html><head> <title>libMesh Examples</title> <?php load_style($root); ?></head><body><?php make_navigation("examples",$root)?><div class="content"><h1>A Series of Example Programs</h1>The following series of example programs have beendesigned to get you started on the right foot.For the most part they are arranged in order ofincreasing complexity, and could be attempted inthat order. Click the links below, or use themenu on the left to navigate the examples.<ul><li><L1><a href="ex1.php">Creation of a Mesh Object</a></L1></li><!-- This is the first example program. It simply demonstrates --><!-- how to create a mesh object. A mesh is read from file, --><!-- information is printed to the screen, and the mesh is then --><!-- written. --><li><L1><a href="ex2.php">Defining a Simple System</a></L1></li><!-- The second example program demonstrates how to --><!-- create an equation system for a simple scalar system. This --><!-- example will also introduce some of the issues involved with using Petsc --><!-- in your application. --><li><L1><a href="ex3.php">Solving a 2D Poisson Problem</a></L1></li><!-- This is the third example program. It builds on --><!-- the second example program by showing how to solve a simple --><!-- Poisson system. This example also introduces the notion --><!-- of customized matrix assembly functions, working with an --><!-- exact solution, and using element iterators. --><!-- We will not comment on things that --><!-- were already explained in the second example. --><li><L1><a href="ex4.php">Solving a 2D or 3D Poisson Problem in Parallel</a></L1></li><!-- This is the fourth example program. It builds on --><!-- the third example program by showing how to formulate --><!-- the code in a dimension-independent way. Very minor --><!-- changes allow rhe example will allow the problem to be --><!-- solved in two or three dimensions and in parallel. --><li><L1><a href="ex0.php">Solving 1D PDE Using Adaptive Mesh Refinement</a></L1></li><!-- This example demonstrates how to solve a simple 1D problem using --><!-- adaptive mesh refinement. The PDE that is solved is: -epsilon*u''(x) + --><!-- u(x) = 1, on the domain [0,1] with boundary conditions u(0) = u(1) = 0 --><!-- and where epsilon << 1. --><!-- <br> --><!-- The approach used to solve 1D problems in libMesh is virtually --><!-- identical to solving 2D or 3D problems, so in this sense this example --><!-- represents a good starting point for new users. Note that many --><!-- concepts are used in this example which are explained more fully in --><!-- subsequent examples. --><li><L1><a href="ex5.php">Run-time Selection of Quadrature Rules</a></L1></li><!-- This example changes the previous example by enabling --><!-- run-time selection of quadrature rules. --><li><L1><a href="ex6.php">Infinite Elements for the Wave Equation</a></L1></li><!-- This example introduces "infinite elements" which may be used for --><!-- certain classes of applications. The wave equation is solved in this --><!-- example. --><i>For this example to work you must have configured thelibrary with the --enable-ifem option</i><li><L1><a href="ex7.php">Introduction to Complex Numbers and the "FrequencySystem"</a></L1></li><!-- This is the seventh example program. It builds on --><!-- the previous example programs, introduces complex --><!-- numbers and the FrequencySystem class to solve a --><!-- simple Helmholtz equation grad(p)*grad(p)+(omega/c)^2*p=0, --><!-- for multiple frequencies rather efficiently. --><!-- The FrequencySystem class offers two solution styles, --><!-- namely to solve large systems, or to solve --><!-- moderately-sized systems fast, for multiple frequencies. --><!-- The latter approach is implemented here. --><!-- For this example the library has to be compiled with --><!-- complex numbers enabled. --><li><L1><a href="ex8.php">The Newmark System and the Wave Equation</a></L1></li><!-- This example solves the wave equation in a hybrid-mesh pipe. The mesh --><!-- consists of <code>HEX8</code> and <code>PRISM6</code> element types. --><!-- The pressure at a point in the pipe is extracted and can be plotted as --><!-- a function of time. --><li><L1><a href="ex9.php">Solving a Transient Linear System in Parallel</a></L1></li><!-- This example shows how a simple, linear transient --><!-- system can be solved in parallel. The system is simple --><!-- scalar convection-diffusion with a specified external --><!-- velocity. The initial condition is given, and the --><!-- solution is advanced in time with a standard Crank-Nicholson --><!-- time-stepping strategy. --><li><L1><a href="ex10.php">Solving a Transient System with Adaptive Mesh Refinement</a></L1></li><!-- This example shows how a simple, linear transient --><!-- system can be solved in parallel. The system is simple --><!-- scalar convection-diffusion with a specified external --><!-- velocity. The initial condition is given, and the --><!-- solution is advanced in time with a standard Crank-Nicholson --><!-- time-stepping strategy. This example differs from the previous --><!-- example by employing adaptive mesh refinement (AMR) and the --><!-- Kelly et. al. error indicator. --><li><L1><a href="ex11.php">Solving a System of Equations</a></L1></li><!-- This example shows how to solve a simple, linear system of equations. The --><!-- familiar Stokes equations for incompressible fluid flow are solved. To satisfy --><!-- the LBB criterion different approximation spaces are used for the velocity and --><!-- pressure fields. --><li><L1><a href="ex12.php">Using the <code>MeshData</code> class</a></L1></li><!-- This example describes the use of the <code>MeshData</code> class. --><!-- More on this later. --><li><L1><a href="ex13.php">Unsteady Navier-Stokes Equations - Unsteady Nonlinear System</a></L1></li><!-- This example shows how a simple, unsteady, nonlinear system of equations --><!-- can be solved in parallel. The system of equations are the familiar --><!-- Navier-Stokes equations for low-speed incompressible fluid flow. This --><!-- example introduces the concept of the inner nonlinear loop for each --><!-- timestep, and requires a good deal of linear algebra number-crunching --><!-- at each step. If you have the General Mesh Viewer (GMV) installed, --><!-- the script movie.sh in this directory will also take appropriate screen --><!-- shots of each of the solution files in the time sequence. These rgb files --><!-- can then be animated with the "animate" utility of ImageMagick if it is --><!-- installed on your system. On a PIII 1GHz machine in debug mode, this --><!-- example takes a little over a minute to run. If you would like to see --><!-- a more detailed time history, or compute more timesteps, that is certainly --><!-- possible by changing the n_timesteps and dt variables below. --><li><L1><a href="ex14.php">Laplace's Equation in an L-Shaped Domain</a></L1></li><!-- This example solves the Laplace equation on the classic "L-shaped" --><!-- domain with adaptive mesh refinement. In this case, the exact --><!-- solution is u(r,\theta) = r^{2/3} * \sin ( (2/3) * \theta), but --><!-- the standard Kelley error indicator is used to estimate the error. --><!-- The initial mesh contains three QUAD9 elements which represent the --><!-- standard quadrants I, II, and III of the domain [-1,1]x[-1,1], --><!-- i.e. --><!-- Element 0: [-1,0]x[ 0,1] --><!-- Element 1: [ 0,1]x[ 0,1] --><!-- Element 2: [-1,0]x[-1,0] --><!-- The mesh is provided in the standard libMesh ASCII format file --><!-- named "lshaped.xda". In addition, an input file named "ex14.in" --><!-- is provided which allows the user to set several parameters for --><!-- the solution so that the problem can be re-run without a --><!-- re-compile. The solution technique employed is to have a --><!-- refinement loop with a linear solve inside followed by a --><!-- refinement of the grid and projection of the solution to the new grid --><!-- In the final loop iteration, there is no additional --><!-- refinement after the solve. In the input file "ex14.in", the variable --><!-- "max_r_steps" controls the number of refinement steps, --><!-- "max_r_level" controls the maximum element refinement level, and --><!-- "refine_percentage" / "coarsen_percentage" determine the number of --><!-- elements which will be refined / coarsened at each step. --><!-- <li><L1><a href="ex15.php">Example 15</a> - Biharmonic Equation</L1></li> --><!-- This example solves the Biharmonic equation on a square domain using a --><!-- Galerkin formulation with C1 elements approximating the H^2_0 function --><!-- space. The initial mesh contains two TRI6 elements. The mesh is --><!-- provided in the standard libMesh ASCII format file named "domain.xda". --><!-- In addition, an input file named "ex15.in" is provided which allows --><!-- the user to set several parameters for the solution so that the --><!-- problem can be re-run without a re-compile. The solution technique --><!-- employed is to have a refinement loop with a linear solve inside --><!-- followed by a refinement of the grid and projection of the solution to --><!-- the new grid In the final loop iteration, there is no additional --><!-- refinement after the solve. In the input file "ex15.in", the variable --><!-- "max_r_steps" controls the number of refinement steps, and --><!-- "max_r_level" controls the maximum element refinement level. --><li><L1><a href="ex16.php">Solving an Eigen Problem</a></L1></li><!-- This example introduces the EigenSystem and shows how libMesh can be --><!-- used for eigenvalue analysis. For solving eigen problems, libMesh --><!-- interfaces SLEPc --><!-- (<a href="http://www.grycap.upv.es/slepc/">www.grycap.upv.es/slepc/</a>) --><!-- which again is based on PETSc. Hence, this example will only work if --><!-- the library is compiled with SLEPc support enabled. In this example --><!-- some eigenvalues for a standard symmetric eigenvalue problem --><!-- A*x=lambda*x are computed, where the matrix A is assembled according --><!-- to a mass matrix. --><li><L1><a href="ex17.php">Solving a generalized Eigen Problem</a></L1></li><!-- This example shows how the previous EigenSolver example --><!-- can be adapted to solve generailzed eigenvalue problems. --><!-- For solving eigen problems, libMesh interfaces --><!-- SLEPc --><!-- (<a href="http://www.grycap.upv.es/slepc/">www.grycap.upv.es/slepc/</a>) --><!-- which again is based on PETSc. --><!-- Hence, this example will only work if the library is compiled --><!-- with SLEPc support enabled. --><!-- In this example some eigenvalues for a generalized symmetric --><!-- eigenvalue problem A*x=lambda*B*x are computed, where the --><!-- matrices A and B are assembled according to stiffness and --><!-- mass matrix, respectively. --> <li><L1><a href="ex18.php">Unsteady Navier-Stokes Equations with DiffSystem</a></L1></li><!-- This example shows how the transient nonlinear problem from --><!-- example 13 can be solved using the DiffSystem class framework to --><!-- simplify the user-implemented equations. --></ul></div><!--<div id="navBeta"></div>--><?php make_footer() ?></body></html><?php if (0) { ?># Local Variables:# mode: html# End:<?php } ?>⌨️ 快捷键说明
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