oomph-lib

In this example we discuss the SUPG-stabilised solution of the 2D advection-diffusion problem
For large values of $\alpha,$ this boundary data approaches a step, oriented at an angle $\Phi$ against the $x_1-$
axis.
$\mathbf{w}\left(x_1,x_2 \right) = \left( \begin{array}{c} \sin(6x_2) \\ \cos(6x_1) \end{array} \right), \ \ \ \ \ \ \ \ \ (4)$
The figures below show plots of the solution for $\Phi=45^\circ, \ \alpha=50$ and a Peclet number of $ Pe = 200$
, with and without SUPG stabilisation. The wire-mesh plot shows the solution computed on a 10x10 mesh, the shaded surface represents the solution obtained from an unstabilised solution on a 150x150 mesh. Note how SUPG stabilisation "suppresses the wiggles" on the relatively coarse mesh.
The driver code
To be written:
//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented, 
//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC//           Version 0.90. August 3, 2009.
//LIC// 
//LIC// Copyright (C) 2006-2009 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
//LIC// This library is distributed in the hope that it will be useful,
//LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
//LIC// 
//LIC// You should have received a copy of the GNU Lesser General Public
//LIC// License along with this library; if not, write to the Free Software
//LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
//LIC// 02110-1301  USA.
//LIC// 
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//Driver for a simple 2D adv diff problem with SUPG stabilisation

//Generic routines
#include "generic.h"

// The Poisson equations
#include "advection_diffusion.h"

// The mesh 
#include "meshes/rectangular_quadmesh.h"

using namespace std;

using namespace oomph;


 
//======start_of_namespace============================================
/// Namespace for global parameters: Unforced problem with
/// boundary values corresponding to a steep tanh step profile
/// oriented at 45 degrees across the domain.
//====================================================================
namespace GlobalPhysicalParameters
{

 /// Peclet number
 double Peclet=200.0;

 /// Parameter for steepness of step in boundary values
 double Alpha=50.0;

 /// Parameter for angle of step in boundary values: 45 degrees
 double TanPhi=1.0;

 /// Some "solution" for assignment of boundary values
 void get_boundary_values(const Vector<double>& x, Vector<double>& u)
 {
  u[0]=tanh(1.0-Alpha*(TanPhi*x[0]-x[1]));
 }

 /// Zero source function
 void source_function(const Vector<double>& x_vect, double& source)
 {
  source=0.0;
 }

 /// Wind
 void wind_function(const Vector<double>& x, Vector<double>& wind)
 {
  wind[0]=sin(6.0*x[1]);
  wind[1]=cos(6.0*x[0]);
 }

 
} // end of namespace

//////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////


//====== start_of_problem_class=======================================
/// 2D AdvectionDiffusion problem on rectangular domain, discretised 
/// with refineable 2D QAdvectionDiffusion elements. The specific type
/// of element is specified via the template parameter.
//====================================================================
template<class ELEMENT> 
class SUPGAdvectionDiffusionProblem : public Problem
{

public:

 /// \short Constructor: Pass pointer to source and wind functions, and
 /// flag to indicate if stabilisation is to be used.
 SUPGAdvectionDiffusionProblem(
  AdvectionDiffusionEquations<2>::AdvectionDiffusionSourceFctPt source_fct_pt,
  AdvectionDiffusionEquations<2>::AdvectionDiffusionWindFctPt wind_fct_pt,
  const bool& use_stabilisation);

 /// Destructor. Empty
 ~SUPGAdvectionDiffusionProblem(){}

 /// \short Update the problem specs before solve: Reset boundary conditions
 /// to the values from the tanh solution and compute stabilisation
 /// parameter.
 void actions_before_newton_solve();

 /// Update the problem after solve (empty)
 void actions_after_newton_solve(){}

 /// \short Doc the solution.
 void doc_solution();

 /// \short Overloaded version of the problem's access function to 
 /// the mesh. Recasts the pointer to the base Mesh object to 
 /// the actual mesh type.
 RectangularQuadMesh<ELEMENT>* mesh_pt() 
  {
   return dynamic_cast<RectangularQuadMesh<ELEMENT>*>(
    Problem::mesh_pt());
  }

private:

 /// DocInfo object
 DocInfo Doc_info;

 /// Pointer to source function
 AdvectionDiffusionEquations<2>::AdvectionDiffusionSourceFctPt Source_fct_pt;

 /// Pointer to wind function
 AdvectionDiffusionEquations<2>::AdvectionDiffusionWindFctPt Wind_fct_pt;

 /// Flag to indicate if stabilisation is to be used
 bool Use_stabilisation;

}; // end of problem class



//=====start_of_constructor===============================================
/// \short Constructor for AdvectionDiffusion problem: Pass pointer to 
/// source function and wind functions and flag to indicate 
/// if stabilisation is to be used.
//========================================================================
template<class ELEMENT>
SUPGAdvectionDiffusionProblem<ELEMENT>::SUPGAdvectionDiffusionProblem(
 AdvectionDiffusionEquations<2>::AdvectionDiffusionSourceFctPt source_fct_pt,
 AdvectionDiffusionEquations<2>::AdvectionDiffusionWindFctPt wind_fct_pt,
 const bool& use_stabilisation)
 :  Source_fct_pt(source_fct_pt), Wind_fct_pt(wind_fct_pt), 
    Use_stabilisation(use_stabilisation)
{ 

 // Set output directory
 if (use_stabilisation)
  {
   Doc_info.set_directory("RESLT_stabilised");
  }
 else
  {
   Doc_info.set_directory("RESLT_unstabilised");
  }

 // Setup mesh

 // # of elements in x-direction
 unsigned n_x=40;

 // # of elements in y-direction
 unsigned n_y=40;

 // Domain length in x-direction
 double l_x=1.0;

 // Domain length in y-direction
 double l_y=2.0;

 // Build and assign mesh
 Problem::mesh_pt() = 
  new RectangularQuadMesh<ELEMENT>(n_x,n_y,l_x,l_y);

  
 // Set the boundary conditions for this problem: All nodes are
 // free by default -- only need to pin the ones that have Dirichlet 
 // conditions here
 unsigned num_bound = mesh_pt()->nboundary();
 for(unsigned ibound=0;ibound<num_bound;ibound++)
  {
   unsigned num_nod= mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     mesh_pt()->boundary_node_pt(ibound,inod)->pin(0); 
    }
  } // end loop over boundaries
 
 // Complete the build of all elements so they are fully functional 

 // Loop over the elements to set up element-specific 
 // things that cannot be handled by the (argument-free!) ELEMENT 
 // constructor: Pass pointer to source function
 unsigned n_element = mesh_pt()->nelement();
 for(unsigned i=0;i<n_element;i++)
  {
   // Upcast from GeneralsedElement to the present element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(mesh_pt()->element_pt(i));

   //Set the source function pointer
   el_pt->source_fct_pt() = Source_fct_pt;

   //Set the wind function pointer
   el_pt->wind_fct_pt() = Wind_fct_pt;

   // Set the Peclet number
   el_pt->pe_pt() = &GlobalPhysicalParameters::Peclet;
  }

 // Setup equation numbering scheme
 cout <<"Number of equations: " << assign_eqn_numbers() << std::endl; 

} // end of constructor


//========================================start_of_actions_before_newton_solve===
/// Update the problem specs before solve: (Re-)set boundary conditions
//========================================================================
template<class ELEMENT>
void SUPGAdvectionDiffusionProblem<ELEMENT>::actions_before_newton_solve()
{
 // How many boundaries are there?
 unsigned num_bound = mesh_pt()->nboundary();
 
 //Loop over the boundaries
 for(unsigned ibound=0;ibound<num_bound;ibound++)
  {
   // How many nodes are there on this boundary?
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);

   // Loop over the nodes on boundary
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Get pointer to node
     Node* nod_pt=mesh_pt()->boundary_node_pt(ibound,inod);

     // Extract nodal coordinates from node:
     Vector<double> x(2);
     x[0]=nod_pt->x(0);
     x[1]=nod_pt->x(1);

     // Get boundary value
     Vector<double> u(1);
     GlobalPhysicalParameters::get_boundary_values(x,u);

     // Assign the value to the one (and only) nodal value at this node
     nod_pt->set_value(0,u[0]);
    }
  } 

 // Now loop over all elements and set the stabilisation parameter
 unsigned n_element = mesh_pt()->nelement();
 for(unsigned i=0;i<n_element;i++)
  {
   // Upcast from GeneralsedElement to the present element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(mesh_pt()->element_pt(i));
   

   // Use stabilisation?
   if (Use_stabilisation)
    {
     //Compute stabilisation parameter
     el_pt->compute_stabilisation_parameter();
    }
   else
    {
     //Compute stabilisation parameter
     el_pt->switch_off_stabilisation();
    }

  }

}  // end of actions before solve



//===============start_of_doc=============================================
/// Doc the solution
//========================================================================
template<class ELEMENT>
void SUPGAdvectionDiffusionProblem<ELEMENT>::doc_solution()
{ 

 ofstream some_file;
 char filename[100];

 // Number of plot points: npts x npts
 unsigned npts=5;

 // Output solution 
 //-----------------
 sprintf(filename,"%s/soln%i.dat",Doc_info.directory().c_str(),
         Doc_info.number());
 some_file.open(filename);
 mesh_pt()->output(some_file,npts);
 some_file.close();

} // end of doc


//===== start_of_main=====================================================
/// Driver code for 2D AdvectionDiffusion problem
//========================================================================
int main()
{

 //Set up the problem with stabilisation
 {
  bool use_stabilisation=true;
  
  // Create the problem with 2D nine-node elements from the
  // QAdvectionDiffusionElement family. Pass pointer to 
  // source and wind function. 
  SUPGAdvectionDiffusionProblem<QSUPGAdvectionDiffusionElement<2,3> > 
   problem(&GlobalPhysicalParameters::source_function,
           &GlobalPhysicalParameters::wind_function,
           use_stabilisation);
  
  // Solve the problem
  problem.newton_solve();
  
  //Output the solution
  problem.doc_solution();
  
 }
 
 
 
 //Set up the problem without stabilisation
 {
  
  bool use_stabilisation=false;
  
  // Create the problem with 2D nine-node elements from the
  // QAdvectionDiffusionElement family. Pass pointer to 
  // source and wind function. 
  SUPGAdvectionDiffusionProblem<QSUPGAdvectionDiffusionElement<2,3> > 
   problem(&GlobalPhysicalParameters::source_function,
           &GlobalPhysicalParameters::wind_function,
           use_stabilisation);
  
  // Solve the problem
  problem.newton_solve();
  
  //Output the solution
  problem.doc_solution();
  
 }


 
} // end of main
Example problem: SUPG-stabilised solution of the 2D advection diffusion equation

The driver code

To be written:

Source files for this tutorial

PDF file
