fourier_decomposed_helmholtz_elements.cc

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//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 1.0; svn revision $LastChangedRevision$
//LIC//
//LIC// $LastChangedDate$
//LIC// 
//LIC// Copyright (C) 2006-2016 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//====================================================================
//Non-inline functions for Helmholtz elements
#include "fourier_decomposed_helmholtz_elements.h"


namespace oomph
{


//========================================================================
/// Helper namespace for functions required for Helmholtz computations
//========================================================================
 namespace Legendre_functions_helper
 {

//========================================================================
// Factorial
//========================================================================
  double factorial(const unsigned& l)
  {
   if(l==0) return 1.0;
   return double(l*factorial(l-1));
  }


//========================================================================
/// Legendre polynomials depending on one parameter
//========================================================================
  double plgndr1(const unsigned& n, const double& x)
  {
   unsigned i;
   double pmm,pmm1;
   double pmm2=0;

#ifdef PARANOID
   // Shout if things went wrong  
   if (std::fabs(x) > 1.0)
    {
     std::ostringstream error_stream;    
     error_stream << "Bad arguments in routine plgndr1: x=" << x 
                  << " but should be less than 1 in absolute value.\n";  
     throw OomphLibError(error_stream.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
#endif

   // Compute pmm : if l=m it's finished
   pmm=1.0;
   if (n == 0) 
    {
     return pmm;
    }

   pmm1=x*1.0;
   if (n == 1) 
    {
     return pmm1;
    }
   else
    {
     for (i=2;i<=n;i++) 
      {
       pmm2=(x*(2*i-1)*pmm1-(i-1)*pmm)/i;
       pmm=pmm1;
       pmm1=pmm2;
      }
     return pmm2;
    }
  
  }//end of plgndr1



//========================================================================
// Legendre polynomials depending on two parameters
//========================================================================
  double plgndr2(const unsigned& l, const unsigned& m, const double& x)
  {
   unsigned i,ll;
   double fact,pmm,pmmp1,somx2;
   double pll=0.0;

#ifdef PARANOID
   // Shout if things went wrong  
   if (std::fabs(x) > 1.0)
    {
     std::ostringstream error_stream;    
     error_stream << "Bad arguments in routine plgndr2: x=" << x 
                  << " but should be less than 1 in absolute value.\n";  
     throw OomphLibError(error_stream.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
#endif

   // This one is easy...
   if (m > l)
    {
     return 0.0;
    }
  
   // Compute pmm : if l=m it's finished
   pmm=1.0;
   if (m > 0)
    {
     somx2=sqrt((1.0-x)*(1.0+x));
     fact=1.0;
     for (i=1;i<=m;i++) 
      {
       pmm *= -fact*somx2;
       fact += 2.0;
      }
    }
   if (l == m) return pmm;
  
   // Compute pmmp1 : if l=m+1 it's finished
   else 
    {
     pmmp1=x*(2*m+1)*pmm;
     if (l == (m+1))
      {
       return pmmp1;
      }
     // Compute pll : if l>m+1 it's finished
     else 
      {
       for (ll=m+2;ll<=l;ll++) 
        {
         pll=(x*(2*ll-1)*pmmp1-(ll+m-1)*pmm)/(ll-m);
         pmm=pmmp1;
         pmmp1=pll;
        }
       return pll;
      }
    }
  }//end of plgndr2

 } // end namespace



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


//======================================================================
/// Set the data for the number of Variables at each node, always two
/// (real and imag part) in every case
//======================================================================
 template<unsigned NNODE_1D>
 const unsigned QFourierDecomposedHelmholtzElement<NNODE_1D>::Initial_Nvalue=2;


//======================================================================
/// Compute element residual Vector and/or element Jacobian matrix 
/// 
/// flag=1: compute both
/// flag=0: compute only residual Vector
///
/// Pure version without hanging nodes
//======================================================================
 void  FourierDecomposedHelmholtzEquations::
 fill_in_generic_residual_contribution_fourier_decomposed_helmholtz(
  Vector<double> &residuals, 
  DenseMatrix<double> &jacobian, 
  const unsigned& flag) 
 {
  //Find out how many nodes there are
  const unsigned n_node = nnode();
  
  //Set up memory for the shape and test functions
  Shape psi(n_node), test(n_node);
  DShape dpsidx(n_node,2), dtestdx(n_node,2);
  
  //Set the value of n_intpt
  const unsigned n_intpt = integral_pt()->nweight();
  
  //Integers to store the local equation and unknown numbers
  int local_eqn_real=0, local_unknown_real=0;
  int local_eqn_imag=0, local_unknown_imag=0;
  
  //Loop over the integration points
  for(unsigned ipt=0;ipt<n_intpt;ipt++)
   {
    //Get the integral weight
    double w = integral_pt()->weight(ipt);
    
    //Call the derivatives of the shape and test functions
    double J = dshape_and_dtest_eulerian_at_knot_fourier_decomposed_helmholtz
     (ipt,psi,dpsidx,test,dtestdx);
    
    //Premultiply the weights and the Jacobian
    double W = w*J;
    
    //Calculate local values of unknown
    //Allocate and initialise to zero
    std::complex<double> interpolated_u(0.0,0.0);
    Vector<double> interpolated_x(2,0.0);
    Vector< std::complex<double> > interpolated_dudx(2);
    
    //Calculate function value and derivatives:
    //-----------------------------------------
    // Loop over nodes
    for(unsigned l=0;l<n_node;l++) 
     {
      // Loop over directions
      for(unsigned j=0;j<2;j++)
       {
        interpolated_x[j] += raw_nodal_position(l,j)*psi(l);
       }
      
      //Get the nodal value of the helmholtz unknown
      const std::complex<double> u_value(
       raw_nodal_value(l,u_index_fourier_decomposed_helmholtz().real()),
       raw_nodal_value(l,u_index_fourier_decomposed_helmholtz().imag()));
      
      //Add to the interpolated value
      interpolated_u += u_value*psi(l);
      
      // Loop over directions
      for(unsigned j=0;j<2;j++)
       {
        interpolated_dudx[j] += u_value*dpsidx(l,j);
       }
     }
    
    //Get source function
    //-------------------
    std::complex<double> source(0.0,0.0);
    get_source_fourier_decomposed_helmholtz(ipt,interpolated_x,source);
    
    double r = interpolated_x[0];
    double rr = r*r;
    double n = (double)fourier_wavenumber();
    double n_squared = n*n;
     
    // Assemble residuals and Jacobian
    //--------------------------------
    
    // Loop over the test functions
    for(unsigned l=0;l<n_node;l++)
     {
      
      // first, compute the real part contribution 
      //-------------------------------------------
      
      //Get the local equation
      local_eqn_real = 
       nodal_local_eqn(l,u_index_fourier_decomposed_helmholtz().real());
      
      /*IF it's not a boundary condition*/
      if(local_eqn_real >= 0)
       {
        // Add body force/source term and Helmholtz bit
        
        residuals[local_eqn_real] += 
         (source.real()-((-n_squared/rr)+k_squared())*
          interpolated_u.real())*test(l)*r*W;
        
        // The Helmholtz bit itself
        for(unsigned k=0;k<2;k++)
         {
          residuals[local_eqn_real] += 
           interpolated_dudx[k].real()*dtestdx(l,k)*r*W;
         }
        
        // Calculate the jacobian
        //-----------------------
        if(flag)
         {
          //Loop over the velocity shape functions again
          for(unsigned l2=0;l2<n_node;l2++)
           { 
            local_unknown_real = 
             nodal_local_eqn(l2,u_index_fourier_decomposed_helmholtz().real());
            
            //If at a non-zero degree of freedom add in the entry
            if(local_unknown_real >= 0)
             {
              //Add contribution to elemental Matrix
              for(unsigned i=0;i<2;i++)
               {
                jacobian(local_eqn_real,local_unknown_real) 
                 += dpsidx(l2,i)*dtestdx(l,i)*r*W;
               }
              
              // Add the helmholtz contribution
              jacobian(local_eqn_real,local_unknown_real)
               -= ((-n_squared/rr)+k_squared())*psi(l2)*test(l)*r*W;
              
             }//end of local_unknown
           }
         }
       }
      
      // Second, compute the imaginary part contribution 
      //------------------------------------------------
      
      //Get the local equation
      local_eqn_imag = nodal_local_eqn
       (l,u_index_fourier_decomposed_helmholtz().imag());
      
      /*IF it's not a boundary condition*/
      if(local_eqn_imag >= 0)
       {
        // Add body force/source term and Helmholtz bit
        residuals[local_eqn_imag] += 
         (source.imag() - ( (-n_squared/rr) + k_squared() ) * 
          interpolated_u.imag() )*test(l)*r*W;
        
        // The Helmholtz bit itself
        for(unsigned k=0;k<2;k++)
         {
          residuals[local_eqn_imag] += interpolated_dudx[k].imag()*
           dtestdx(l,k)*r*W;
         }
        
        // Calculate the jacobian
        //-----------------------
        if(flag)
         {
          //Loop over the velocity shape functions again
          for(unsigned l2=0;l2<n_node;l2++)
           { 
            local_unknown_imag = nodal_local_eqn
             (l2,u_index_fourier_decomposed_helmholtz().imag());
            
            //If at a non-zero degree of freedom add in the entry
            if(local_unknown_imag >= 0)
             {
              //Add contribution to Elemental Matrix
              for(unsigned i=0;i<2;i++)
               {
                jacobian(local_eqn_imag,local_unknown_imag) 
                 += dpsidx(l2,i)*dtestdx(l,i)*r*W;
               }
              // Add the helmholtz contribution
              jacobian(local_eqn_imag,local_unknown_imag)
               -= ((-n_squared/rr)+k_squared())*psi(l2)*test(l)*r*W;
             }
           }
         }
       }
      
     }
   } // End of loop over integration points
 }   
 
 
//======================================================================
/// Self-test:  Return 0 for OK
//======================================================================
 unsigned  FourierDecomposedHelmholtzEquations::self_test()
 {
  
  bool passed=true;
  
  // Check lower-level stuff
  if (FiniteElement::self_test()!=0)
   {
    passed=false;
   }
  
  // Return verdict
  if (passed)
   {
    return 0;
   }
  else
   {
    return 1;
   }
 }
 
 
 
//======================================================================
/// Output function:
///
///   r,z,u_re,u_imag  
///
/// nplot points in each coordinate direction
//======================================================================
 void  FourierDecomposedHelmholtzEquations::output(std::ostream &outfile, 
                                                   const unsigned &nplot)
 {
  
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Tecplot header info
  outfile << tecplot_zone_string(nplot);
  
  // Loop over plot points
  unsigned num_plot_points=nplot_points(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    std::complex<double> u(interpolated_u_fourier_decomposed_helmholtz(s));
    for(unsigned i=0;i<2;i++) 
     {
      outfile << interpolated_x(s,i) << " ";
     }
    outfile << u.real() << " " << u.imag() << std::endl;   
    
   }
  
  // Write tecplot footer (e.g. FE connectivity lists)
  write_tecplot_zone_footer(outfile,nplot);
  
 }
 



//======================================================================
/// Output function for real part of full time-dependent solution
///
///  u = Re( (u_r +i u_i) exp(-i omega t)
///
/// at phase angle omega t = phi.
///
///  r,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
 void  FourierDecomposedHelmholtzEquations::output_real(std::ostream &outfile, 
                                                        const double& phi,
                                                        const unsigned &nplot)
 {
  
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Tecplot header info
  outfile << tecplot_zone_string(nplot);
  
  // Loop over plot points
  unsigned num_plot_points=nplot_points(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    std::complex<double> u(interpolated_u_fourier_decomposed_helmholtz(s));
    for(unsigned i=0;i<2;i++) 
     {
      outfile << interpolated_x(s,i) << " ";
     }
    outfile << u.real()*cos(phi)+u.imag()*sin(phi) << std::endl;   
    
   }
  
  // Write tecplot footer (e.g. FE connectivity lists)
  write_tecplot_zone_footer(outfile,nplot);
  
 }
 
 
//======================================================================
/// C-style output function:
///
///   r,z,u 
///
/// nplot points in each coordinate direction
//======================================================================
 void  FourierDecomposedHelmholtzEquations::output(FILE* file_pt,
                                                   const unsigned &nplot)
 {
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Tecplot header info
  fprintf(file_pt,"%s",tecplot_zone_string(nplot).c_str());
  
  // Loop over plot points
  unsigned num_plot_points=nplot_points(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    std::complex<double> u(interpolated_u_fourier_decomposed_helmholtz(s));
    
    for(unsigned i=0;i<2;i++) 
     {
      fprintf(file_pt,"%g ",interpolated_x(s,i));
     }
    
    for(unsigned i=0;i<2;i++) 
     {
      fprintf(file_pt,"%g ",interpolated_x(s,i));
     }
    fprintf(file_pt,"%g ",u.real());
    fprintf(file_pt,"%g \n",u.imag());
    
   }
  
  // Write tecplot footer (e.g. FE connectivity lists)
  write_tecplot_zone_footer(file_pt,nplot);
 }
 
 
 
//======================================================================
 /// Output exact solution
 /// 
 /// Solution is provided via function pointer.
 /// Plot at a given number of plot points.
 ///
 ///   r,z,u_exact
//======================================================================
 void FourierDecomposedHelmholtzEquations::output_fct(
  std::ostream &outfile, 
  const unsigned &nplot, 
  FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
 {
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Vector for coordintes
  Vector<double> x(2);
  
  // Tecplot header info
  outfile << tecplot_zone_string(nplot);
  
  // Exact solution Vector 
  Vector<double> exact_soln(2);
  
  // Loop over plot points
  unsigned num_plot_points=nplot_points(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    
    // Get x position as Vector
    interpolated_x(s,x);
    
    // Get exact solution at this point
    (*exact_soln_pt)(x,exact_soln);
    
    //Output r,z,u_exact
    for(unsigned i=0;i<2;i++)
     {
      outfile << x[i] << " ";
     }
    outfile << exact_soln[0] << " " <<  exact_soln[1] << std::endl;  
   }
  
  // Write tecplot footer (e.g. FE connectivity lists)
  write_tecplot_zone_footer(outfile,nplot);
 }
 
 
 
//======================================================================
/// Output function for real part of full time-dependent fct
///
///  u = Re( (u_r +i u_i) exp(-i omega t)
///
/// at phase angle omega t = phi.
///
///   r,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
 void FourierDecomposedHelmholtzEquations::output_real_fct(
  std::ostream &outfile, 
  const double& phi,
  const unsigned &nplot, 
  FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
 {
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Vector for coordintes
  Vector<double> x(2);
  
  // Tecplot header info
  outfile << tecplot_zone_string(nplot);
  
  // Exact solution Vector 
  Vector<double> exact_soln(2);
  
  // Loop over plot points
  unsigned num_plot_points=nplot_points(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    
    // Get x position as Vector
    interpolated_x(s,x);
    
    // Get exact solution at this point
    (*exact_soln_pt)(x,exact_soln);
    
    //Output x,y,...,u_exact
    for(unsigned i=0;i<2;i++)
     {
      outfile << x[i] << " ";
     }
    outfile << exact_soln[0]*cos(phi)+ exact_soln[1]*sin(phi) << std::endl;  
   }
  
  // Write tecplot footer (e.g. FE connectivity lists)
  write_tecplot_zone_footer(outfile,nplot);
 }
 



//======================================================================
 /// Validate against exact solution
 /// 
 /// Solution is provided via function pointer.
 /// Plot error at a given number of plot points.
 ///
//======================================================================
 void FourierDecomposedHelmholtzEquations::compute_error(
  std::ostream &outfile, 
  FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
  double& error, double& norm)
 { 
  
  // Initialise
  error=0.0;
  norm=0.0;
  
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Vector for coordintes
  Vector<double> x(2);
  
  //Find out how many nodes there are in the element
  unsigned n_node = nnode();
  
  Shape psi(n_node);
  
  //Set the value of n_intpt
  unsigned n_intpt = integral_pt()->nweight();
  
  // Tecplot 
  outfile << "ZONE" << std::endl;
  
  // Exact solution Vector 
  Vector<double> exact_soln(2);
  
  //Loop over the integration points
  for(unsigned ipt=0;ipt<n_intpt;ipt++)
   {
    
    //Assign values of s
    for(unsigned i=0;i<2;i++)
     {
      s[i] = integral_pt()->knot(ipt,i);
     }
    
    //Get the integral weight
    double w = integral_pt()->weight(ipt);
    
    // Get jacobian of mapping
    double J=J_eulerian(s);
    
    //Premultiply the weights and the Jacobian
    double W = w*J;
    
    // Get x position as Vector
    interpolated_x(s,x);
    
    // Get FE function value
    std::complex<double> u_fe=interpolated_u_fourier_decomposed_helmholtz(s);
    
    // Get exact solution at this point
    (*exact_soln_pt)(x,exact_soln);
    
    //Output r,z,error
    for(unsigned i=0;i<2;i++)
     {
      outfile << x[i] << " ";
     }
    outfile << exact_soln[0] << " "  << exact_soln[1] << " " 
            << exact_soln[0]-u_fe.real() << " " << exact_soln[1]-u_fe.imag() 
            << std::endl;  
    
    // Add to error and norm
    norm+=(exact_soln[0]*exact_soln[0]+exact_soln[1]*exact_soln[1])*W;
    error+=( (exact_soln[0]-u_fe.real())*(exact_soln[0]-u_fe.real())+
             (exact_soln[1]-u_fe.imag())*(exact_soln[1]-u_fe.imag()) )*W;
    
   }
 }
 
 

//======================================================================
 /// Compute norm of fe solution
//======================================================================
 void FourierDecomposedHelmholtzEquations::compute_norm(double& norm)
 { 
  
  // Initialise
  norm=0.0;
  
  //Vector of local coordinates
  Vector<double> s(2);
  
  // Vector for coordintes
  Vector<double> x(2);
  
  //Find out how many nodes there are in the element
  unsigned n_node = nnode();
  
  Shape psi(n_node);
  
  //Set the value of n_intpt
  unsigned n_intpt = integral_pt()->nweight();
  
  //Loop over the integration points
  for(unsigned ipt=0;ipt<n_intpt;ipt++)
   {
    
    //Assign values of s
    for(unsigned i=0;i<2;i++)
     {
      s[i] = integral_pt()->knot(ipt,i);
     }
    
    //Get the integral weight
    double w = integral_pt()->weight(ipt);
    
    // Get jacobian of mapping
    double J=J_eulerian(s);
    
    //Premultiply the weights and the Jacobian
    double W = w*J;
    
    // Get FE function value
    std::complex<double> u_fe=interpolated_u_fourier_decomposed_helmholtz(s);
    
    // Add to  norm
    norm+=(u_fe.real()*u_fe.real()+u_fe.imag()*u_fe.imag())*W;
    
   }
 }
 
 
 
 
 
//====================================================================
// Force build of templates
//====================================================================
template class QFourierDecomposedHelmholtzElement<2>;
template class QFourierDecomposedHelmholtzElement<3>;
template class QFourierDecomposedHelmholtzElement<4>;

}