ex9.php

一个用来实现偏微分方程中网格的计算库

<div class ="fragment">
<pre>
          assert (system_name == "Convection-Diffusion");
        
</pre>
</div>
<div class = "comment">
Get a reference to the Convection-Diffusion system object.
</div>

<div class ="fragment">
<pre>
          TransientLinearImplicitSystem & system =
            es.get_system&lt;TransientLinearImplicitSystem&gt;("Convection-Diffusion");
        
</pre>
</div>
<div class = "comment">
Project initial conditions at time 0
</div>

<div class ="fragment">
<pre>
          es.parameters.set&lt;Real&gt; ("time") = 0;
          
          system.project_solution(exact_value, NULL, es.parameters);
        }
        
        
        
</pre>
</div>
<div class = "comment">
Now we define the assemble function which will be used
by the EquationSystems object at each timestep to assemble
the linear system for solution.
</div>

<div class ="fragment">
<pre>
        void assemble_cd (EquationSystems& es,
        		  const std::string& system_name)
        {
        #ifdef ENABLE_AMR
</pre>
</div>
<div class = "comment">
It is a good idea to make sure we are assembling
the proper system.
</div>

<div class ="fragment">
<pre>
          assert (system_name == "Convection-Diffusion");
          
</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>
          TransientLinearImplicitSystem & system =
            es.get_system&lt;TransientLinearImplicitSystem&gt; ("Convection-Diffusion");
          
</pre>
</div>
<div class = "comment">
Get a constant reference to the Finite Element type
for the first (and only) variable in the system.
</div>

<div class ="fragment">
<pre>
          FEType fe_type = system.variable_type(0);
          
</pre>
</div>
<div class = "comment">
Build a Finite Element object of the specified type.  Since the
\p FEBase::build() member dynamically creates memory we will
store the object as an \p AutoPtr<FEBase>.  This can be thought
of as a pointer that will clean up after itself.
</div>

<div class ="fragment">
<pre>
          AutoPtr&lt;FEBase&gt; fe      (FEBase::build(dim, fe_type));
          AutoPtr&lt;FEBase&gt; fe_face (FEBase::build(dim, fe_type));
          
</pre>
</div>
<div class = "comment">
A Gauss quadrature rule for numerical integration.
Let the \p FEType object decide what order rule is appropriate.
</div>

<div class ="fragment">
<pre>
          QGauss qrule (dim,   fe_type.default_quadrature_order());
          QGauss qface (dim-1, fe_type.default_quadrature_order());
        
</pre>
</div>
<div class = "comment">
Tell the finite element object to use our quadrature rule.
</div>

<div class ="fragment">
<pre>
          fe-&gt;attach_quadrature_rule      (&qrule);
          fe_face-&gt;attach_quadrature_rule (&qface);
        
</pre>
</div>
<div class = "comment">
Here we define some references to cell-specific data that
will be used to assemble the linear system.  We will start
with the element Jacobian * quadrature weight at each integration point.   
</div>

<div class ="fragment">
<pre>
          const std::vector&lt;Real&gt;& JxW      = fe-&gt;get_JxW();
          const std::vector&lt;Real&gt;& JxW_face = fe_face-&gt;get_JxW();
          
</pre>
</div>
<div class = "comment">
The element shape functions evaluated at the quadrature points.
</div>

<div class ="fragment">
<pre>
          const std::vector&lt;std::vector&lt;Real&gt; &gt;& phi = fe-&gt;get_phi();
          const std::vector&lt;std::vector&lt;Real&gt; &gt;& psi = fe_face-&gt;get_phi();
        
</pre>
</div>
<div class = "comment">
The element shape function gradients evaluated at the quadrature
points.
</div>

<div class ="fragment">
<pre>
          const std::vector&lt;std::vector&lt;RealGradient&gt; &gt;& dphi = fe-&gt;get_dphi();
        
</pre>
</div>
<div class = "comment">
The XY locations of the quadrature points used for face integration
</div>

<div class ="fragment">
<pre>
          const std::vector&lt;Point&gt;& qface_points = fe_face-&gt;get_xyz();
          
</pre>
</div>
<div class = "comment">
A reference to the \p DofMap object for this system.  The \p DofMap
object handles the index translation from node and element numbers
to degree of freedom numbers.  We will talk more about the \p DofMap
in future examples.
</div>

<div class ="fragment">
<pre>
          const DofMap& dof_map = system.get_dof_map();
        
</pre>
</div>
<div class = "comment">
Define data structures to contain the element matrix
and right-hand-side vector contribution.  Following
basic finite element terminology we will denote these
"Ke" and "Fe".
</div>

<div class ="fragment">
<pre>
          DenseMatrix&lt;Number&gt; Ke;
          DenseVector&lt;Number&gt; Fe;
          
</pre>
</div>
<div class = "comment">
This vector will hold the degree of freedom indices for
the element.  These define where in the global system
the element degrees of freedom get mapped.
</div>

<div class ="fragment">
<pre>
          std::vector&lt;unsigned int&gt; dof_indices;
        
</pre>
</div>
<div class = "comment">
Here we extract the velocity & parameters that we put in the
EquationSystems object.
</div>

<div class ="fragment">
<pre>
          const RealVectorValue velocity =
            es.parameters.get&lt;RealVectorValue&gt; ("velocity");
        
          const Real dt = es.parameters.get&lt;Real&gt;   ("dt");
          const Real time = es.parameters.get&lt;Real&gt; ("time");
        
</pre>
</div>
<div class = "comment">
Now we will loop over all the elements in the mesh that
live on the local processor. We will compute the element
matrix and right-hand-side contribution.  Since the mesh
will be refined we want to only consider the ACTIVE elements,
hence we use a variant of the \p active_elem_iterator.
const_active_local_elem_iterator           el (mesh.elements_begin());
const const_active_local_elem_iterator end_el (mesh.elements_end());


<br><br></div>

<div class ="fragment">
<pre>
          MeshBase::const_element_iterator       el     = mesh.active_local_elements_begin();
          const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end(); 
        
          for ( ; el != end_el; ++el)
            {    
</pre>
</div>
<div class = "comment">
Store a pointer to the element we are currently
working on.  This allows for nicer syntax later.
</div>

<div class ="fragment">
<pre>
              const Elem* elem = *el;
              
</pre>
</div>
<div class = "comment">
Get the degree of freedom indices for the
current element.  These define where in the global
matrix and right-hand-side this element will
contribute to.
</div>

<div class ="fragment">
<pre>
              dof_map.dof_indices (elem, dof_indices);
              
</pre>
</div>
<div class = "comment">
Compute the element-specific data for the current
element.  This involves computing the location of the
quadrature points (q_point) and the shape functions
(phi, dphi) for the current element.
</div>

<div class ="fragment">
<pre>
              fe-&gt;reinit (elem);
              
</pre>
</div>
<div class = "comment">
Zero the element matrix and right-hand side before
summing them.  We use the resize member here because
the number of degrees of freedom might have changed from
the last element.  Note that this will be the case if the
element type is different (i.e. the last element was a
triangle, now we are on a quadrilateral).
</div>

<div class ="fragment">
<pre>
              Ke.resize (dof_indices.size(),
        		 dof_indices.size());
        
              Fe.resize (dof_indices.size());
              
</pre>
</div>
<div class = "comment">
Now we will build the element matrix and right-hand-side.
Constructing the RHS requires the solution and its
gradient from the previous timestep.  This myst be
calculated at each quadrature point by summing the
solution degree-of-freedom values by the appropriate
weight functions.
</div>

<div class ="fragment">
<pre>
              for (unsigned int qp=0; qp&lt;qrule.n_points(); qp++)
        	{
</pre>
</div>
<div class = "comment">
Values to hold the old solution & its gradient.
</div>

<div class ="fragment">
<pre>
                  Number   u_old = 0.;
        	  Gradient grad_u_old;
        	  
</pre>
</div>
<div class = "comment">
Compute the old solution & its gradient.
</div>

<div class ="fragment">
<pre>
                  for (unsigned int l=0; l&lt;phi.size(); l++)
        	    {
        	      u_old      += phi[l][qp]*system.old_solution  (dof_indices[l]);
        	      
</pre>
</div>
<div class = "comment">
This will work,
grad_u_old += dphi[l][qp]*system.old_solution (dof_indices[l]);
but we can do it without creating a temporary like this:
</div>

<div class ="fragment">
<pre>
                      grad_u_old.add_scaled (dphi[l][qp],system.old_solution (dof_indices[l]));
        	    }
        	  
</pre>
</div>
<div class = "comment">
Now compute the element matrix and RHS contributions.
</div>

<div class ="fragment">
<pre>
                  for (unsigned int i=0; i&lt;phi.size(); i++)
        	    {
</pre>
</div>
<div class = "comment">
The RHS contribution
</div>

<div class ="fragment">
<pre>
                      Fe(i) += JxW[qp]*(
</pre>
</div>
<div class = "comment">
Mass matrix term
</div>

<div class ="fragment">
<pre>
                                        u_old*phi[i][qp] +
        				-.5*dt*(
</pre>
</div>
<div class = "comment">
Convection term
(grad_u_old may be complex, so the
order here is important!)
</div>

<div class ="fragment">
<pre>
                                                (grad_u_old*velocity)*phi[i][qp] +
        					
</pre>
</div>
<div class = "comment">
Diffusion term
</div>

<div class ="fragment">
<pre>
                                                0.01*(grad_u_old*dphi[i][qp]))     
        				);
        	      
        	      for (unsigned int j=0; j&lt;phi.size(); j++)
        		{
</pre>
</div>
<div class = "comment">
The matrix contribution
</div>

<div class ="fragment">
<pre>
                          Ke(i,j) += JxW[qp]*(
</pre>
</div>
<div class = "comment">
Mass-matrix
</div>

<div class ="fragment">
<pre>
                                              phi[i][qp]*phi[j][qp] + 
        
        				      .5*dt*(
</pre>
</div>
<div class = "comment">
Convection term
</div>

<div class ="fragment">
<pre>
                                                     (velocity*dphi[j][qp])*phi[i][qp] +
        					     
</pre>
</div>
<div class = "comment">
Diffusion term
</div>

<div class ="fragment">
<pre>
                                                     0.01*(dphi[i][qp]*dphi[j][qp]))      
        				      );
        		} 
        	    } 
        	} 
        
</pre>
</div>
<div class = "comment">
At this point the interior element integration has
been completed.  However, we have not yet addressed
boundary conditions.  For this example we will only
consider simple Dirichlet boundary conditions imposed
via the penalty method. 

<br><br>The following loops over the sides of the element.
If the element has no neighbor on a side then that
side MUST live on a boundary of the domain.