complex_harmonic.cc

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//LIC// ====================================================================
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//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
//LIC// 
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//LIC//====================================================================
//Driver to solve the harmonic equation as a quadratic eigenvalue problem,
//by solving two first-order systems with homogeneous Dirichlet boundary
//conditions. The sign is chosen to that all the eigenvalues are imaginary
//when Mu is small (Mu=0 actually leads to a singular mass matrix so should 
//be avoided), but pairs merge to become real as Mu increases.
 
// Generic oomph-lib routines
#include "generic.h"
 
// Include the mesh
#include "meshes/one_d_mesh.h"
 
using namespace std;
 
using namespace oomph;
 
/// Namespace for the shift applied to the eigenproblem
namespace EigenproblemShift
{
 //Parameter chosen so that only one complex-conjugate pair has merged
 double Mu = 6.5; //Will lead to singular mass matrix if set to zero
}
 
 
//===================================================================
/// Function-type-object to perform comparison of complex data types
/// Needed to sort the complex eigenvalues into order based on the
/// size of the real part.
//==================================================================
template <class T>
class ComplexLess
{
public:
 
 /// Comparison in terms of magnitude of complex number
 bool operator()(const complex<T> &x, const complex<T> &y) const
  {
   //Return the order in terms of magnitude if they are not equal
   //Include a tolerance to avoid processor specific ordering
   if(std::abs(std::abs(x) - std::abs(y)) > 1.0e-10 )
    {return std::abs(x) < std::abs(y);}
   //Otherwise sort on real part, again with a tolerance
   else if(std::abs(x.real() - y.real()) > 1.0e-10)
    {return x.real() < y.real();}
   //Otherwise sort on imaginary part
   else 
    {return x.imag() < y.imag();}
  }
};
 
 
//=================================================================
/// A class for all elements that solve the eigenvalue problem
/// \f[ 
/// \frac{\partial w}{\partial x}  = \lambda u
/// \f] 
/// \f[
/// \frac{\partial u}{\partial x} = (\lambda - \mu) w
/// \f]
/// This class  contains the generic maths. Shape functions, geometric
/// mapping etc. must get implemented in derived class.
//================================================================
class ComplexHarmonicEquations : public virtual FiniteElement
{
 
public:
 /// Empty Constructor
 ComplexHarmonicEquations() {}
 
 /// Access function: First eigenfunction value at local node n
 /// Note that solving the eigenproblem does not assign values
 /// to this storage space. It is used for output purposes only.
 virtual inline double u(const unsigned& n) const 
  {return nodal_value(n,0);}
 
 /// Second eigenfunction value at local node n
 virtual inline double w(const unsigned& n) const 
  {return nodal_value(n,1);}
 
 /// Output the eigenfunction with default number of plot points
 void output(ostream &outfile) 
  {
   unsigned nplot=5;
   output(outfile,nplot);
  }
 
 /// Output FE representation of soln: x,y,u or x,y,z,u at 
 /// Nplot  plot points
 void output(ostream &outfile, const unsigned &nplot)
  {
   //Vector of local coordinates
   Vector<double> s(1);
 
   // 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);
     //Output the coordinate and the eigenfunction
     outfile << interpolated_x(s,0) << " " << interpolated_u(s)
             << " " << interpolated_w(s) << std::endl;   
    }
   // Write tecplot footer (e.g. FE connectivity lists)
   write_tecplot_zone_footer(outfile,nplot);
  }
 
 /// Assemble the contributions to the jacobian and mass
 /// matrices
 void fill_in_contribution_to_jacobian_and_mass_matrix(
  Vector<double> &residuals,
  DenseMatrix<double> &jacobian, DenseMatrix<double> &mass_matrix)
  {
   //Find out how many nodes there are
   unsigned n_node = nnode();
   
   //Set up memory for the shape functions and their derivatives
   Shape psi(n_node);
   DShape dpsidx(n_node,1);
 
   //Set the number of integration points
   unsigned n_intpt = integral_pt()->nweight();
   
   //Integers to store the local equation and unknown numbers
   int local_eqn=0, local_unknown=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_eulerian_at_knot(ipt,psi,dpsidx);
     
     //Premultiply the weights and the Jacobian
     double W = w*J;
 
     //Assemble the contributions to the mass matrix
     //Loop over the test functions
     for(unsigned l=0;l<n_node;l++)
      {
       //Get the local equation number
       local_eqn = u_local_eqn(l,0);
       //If it's not a boundary condition
       if(local_eqn >= 0)
        {
         //Loop over the shape functions
         for(unsigned l2=0;l2<n_node;l2++)
          { 
           local_unknown = u_local_eqn(l2,0);
           //If at a non-zero degree of freedom add in the entry
           if(local_unknown >= 0)
            {
             //This corresponds to the top left B block
             mass_matrix(local_eqn, local_unknown) += psi(l2)*psi(l)*W;
            }
           local_unknown = u_local_eqn(l2,1);
           //If at a non-zero degree of freedom add in the entry
           if(local_unknown >= 0)
            {
             //This corresponds to the top right A block
             jacobian(local_eqn,local_unknown) += dpsidx(l2,0)*psi(l)*W;
            }
          }
        }
 
       //Get the local equation number
       local_eqn = u_local_eqn(l,1);
       //IF it's not a boundary condition
       if(local_eqn >= 0)
        {
         //Loop over the shape functions
         for(unsigned l2=0;l2<n_node;l2++)
          { 
           local_unknown = u_local_eqn(l2,0);
           //If at a non-zero degree of freedom add in the entry
           if(local_unknown >= 0)
            {
             //This corresponds to the lower left A block
             jacobian(local_eqn,local_unknown) += dpsidx(l2,0)*psi(l)*W;
            }
           local_unknown = u_local_eqn(l2,1);
           //If at a non-zero degree of freedom add in the entry
           if(local_unknown >= 0)
            {
             //This corresponds to the lower right B block
             mass_matrix(local_eqn, local_unknown) += psi(l2)*psi(l)*W;
             //This corresponds to the lower right \mu B block
             jacobian(local_eqn,local_unknown) += 
              EigenproblemShift::Mu*psi(l2)*psi(l)*W;
            }
          }
        }
      }
    }
 } //end_of_fill_in_contribution_to_jacobian_and_mass_matrix
 
 /// Return FE representation of function value u(s) at local coordinate s
 inline double interpolated_u(const Vector<double> &s) const
  {
   unsigned n_node = nnode();
 
   //Local shape functions
   Shape psi(n_node);
 
   //Find values of basis function
   this->shape(s,psi);
 
   //Initialise value of u
   double interpolated_u = 0.0;
 
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_node;l++)  {interpolated_u+=u(l)*psi[l];}
 
   //Return the interpolated value of the eigenfunction
   return(interpolated_u);
  }
 
 /// Return FE representation of function value w(s) at local coordinate s
 inline double interpolated_w(const Vector<double> &s) const
  {
   unsigned n_node = nnode();
 
   //Local shape functions
   Shape psi(n_node);
 
   //Find values of basis function
   this->shape(s,psi);
 
   //Initialise value of u
   double interpolated_u = 0.0;
 
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_node;l++)  {interpolated_u+=w(l)*psi[l];}
 
   //Return the interpolated value of the eigenfunction
   return(interpolated_u);
  }
 
 
protected:
 
 /// Shape/test functions and derivs w.r.t. to global coords at 
 /// local coord. s; return  Jacobian of mapping
 virtual double dshape_eulerian(const Vector<double> &s, 
                                Shape &psi, 
                                DShape &dpsidx) const=0;
 
 /// Shape/test functions and derivs w.r.t. to global coords at 
 /// integration point ipt; return  Jacobian of mapping
 virtual double dshape_eulerian_at_knot(const unsigned &ipt, 
                                        Shape &psi, 
                                        DShape &dpsidx) const=0;
 
 /// Access function that returns the local equation number
 /// of the unknown in the problem. Default is to assume that it is the
 /// first (only) value stored at the node.
 virtual inline int u_local_eqn(const unsigned &n, const unsigned &i)
  {return nodal_local_eqn(n,i);}
 
    private:
 
};
 
 
 
//======================================================================
/// QComplexHarmonicElement<NNODE_1D> elements are 1D  Elements with 
/// NNODE_1D nodal points that are used to solve the ComplexHarmonic eigenvalue
/// Problem described by ComplexHarmonicEquations.
//======================================================================
template <unsigned NNODE_1D>
class QComplexHarmonicElement : public virtual QElement<1,NNODE_1D>, 
                         public ComplexHarmonicEquations
{
 
  public:
 
 /// Constructor: Call constructors for QElement and 
 /// Poisson equations
 QComplexHarmonicElement() : QElement<1,NNODE_1D>(), 
                             ComplexHarmonicEquations() {}
 
 ///  Required  # of `values' (pinned or dofs) 
 /// at node n. Here there are two (u and w)
 inline unsigned required_nvalue(const unsigned &n) const {return 2;}
 
 /// Output function overloaded from ComplexHarmonicEquations
 void output(ostream &outfile) 
  {ComplexHarmonicEquations::output(outfile);}
 
 ///  Output function overloaded from ComplexHarmonicEquations
 void output(ostream &outfile, const unsigned &Nplot) 
  {ComplexHarmonicEquations::output(outfile,Nplot);}
 
 
protected:
 
/// Shape, test functions & derivs. w.r.t. to global coords. Return Jacobian.
 inline double dshape_eulerian(const Vector<double> &s, 
                               Shape &psi, 
                               DShape &dpsidx) const
  {return QElement<1,NNODE_1D>::dshape_eulerian(s,psi,dpsidx);}
 
 
 /// Shape, test functions & derivs. w.r.t. to global coords. at
 /// integration point ipt. Return Jacobian.
 inline double dshape_eulerian_at_knot(const unsigned& ipt,
                                       Shape &psi, 
                                       DShape &dpsidx) const
  {return QElement<1,NNODE_1D>::dshape_eulerian_at_knot(ipt,psi,dpsidx);}
 
}; //end_of_QComplexHarmonic_class_definition
 
 
//==start_of_problem_class============================================
/// 1D ComplexHarmonic problem in unit interval.
//====================================================================
template<class ELEMENT,class EIGEN_SOLVER> 
class ComplexHarmonicProblem : public Problem
{
public:
 
 /// Constructor: Pass number of elements and pointer to source function
 ComplexHarmonicProblem(const unsigned& n_element);
 
 /// Destructor. Clean up the mesh and solver
 ~ComplexHarmonicProblem(){delete this->mesh_pt(); 
  delete this->eigen_solver_pt();}
 
 /// Solve the problem
 void solve(const unsigned &label);
 
 /// Doc the solution, pass the number of the case considered,
 /// so that output files can be distinguished.
 void doc_solution(const unsigned& label);
 
}; // end of problem class
 
 
 
//=====start_of_constructor===============================================
/// Constructor for 1D ComplexHarmonic problem in unit interval.
/// Discretise the 1D domain with n_element elements of type ELEMENT.
/// Specify function pointer to source function. 
//========================================================================
template<class ELEMENT,class EIGEN_SOLVER>
ComplexHarmonicProblem<ELEMENT,EIGEN_SOLVER>::ComplexHarmonicProblem(
 const unsigned& n_element)
{ 
 //Create the eigen solver
 this->eigen_solver_pt() = new EIGEN_SOLVER;
 
 //Set domain length 
 double L=1.0;
 
 // Build mesh and store pointer in Problem
 Problem::mesh_pt() = new OneDMesh<ELEMENT>(n_element,L);
 
 // Set the boundary conditions for this problem: By default, all nodal
 // values are free -- we only need to pin the ones that have 
 // Dirichlet conditions. 
 
 // Pin the single nodal value at the single node on mesh 
 // boundary 0 (= the left domain boundary at x=0)
 mesh_pt()->boundary_node_pt(0,0)->pin(0);
 
 // Pin the single nodal value at the single node on mesh 
 // boundary 1 (= the right domain boundary at x=1)
 mesh_pt()->boundary_node_pt(1,0)->pin(0);
 
 // Setup equation numbering scheme
 assign_eqn_numbers();
 
} // end of constructor
 
 
 
//===start_of_doc=========================================================
/// Doc the solution in tecplot format. Label files with label.
//========================================================================
template<class ELEMENT,class EIGEN_SOLVER>
void ComplexHarmonicProblem<ELEMENT,EIGEN_SOLVER>::doc_solution(
 const unsigned& label)
{ 
 
 ofstream some_file;
 char filename[100];
 
 // Number of plot points
 unsigned npts;
 npts=5; 
 
 // Output solution with specified number of plot points per element
 sprintf(filename,"soln%i.dat",label);
 some_file.open(filename);
 mesh_pt()->output(some_file,npts);
 some_file.close();
 
} // end of doc
 
//=======================start_of_solve==============================
/// Solve the eigenproblem 
//===================================================================
template<class ELEMENT,class EIGEN_SOLVER>
void ComplexHarmonicProblem<ELEMENT,EIGEN_SOLVER>::
solve(const unsigned& label)
{ 
 //Set external storage for the eigenvalues
 Vector<complex<double> > eigenvalues;
 //Set external storage for the eigenvectors
 Vector<DoubleVector> eigenvector_real;
 Vector<DoubleVector> eigenvector_imag;
 //Desired number eigenvalues enough to include the first two real
 //eigenvalues and then the complex conjugate pair. If the number is set
 //to four then the original iterative solver prefers the degenerate (real) eigenvalues Mu, Mu.
 unsigned n_eval=7;
 
 //Solve the eigenproblem
 this->solve_eigenproblem(n_eval,eigenvalues,eigenvector_real,eigenvector_imag);
 
 //We now need to sort the output based on the size of the real part
 //of the eigenvalues.
 //This is because the solver does not necessarily sort the eigenvalues
 Vector<complex<double> > sorted_eigenvalues = eigenvalues;
 sort(sorted_eigenvalues.begin(),sorted_eigenvalues.end(),
      ComplexLess<double>());
 
 //Read out the smallest eigenvalue 
 complex<double> temp_evalue = sorted_eigenvalues[0];
 unsigned smallest_index=0;
 //Loop over the unsorted eigenvalues and find the entry that corresponds
 //to our second smallest eigenvalue.
 for(unsigned i=0;i<eigenvalues.size();i++)
  {
   //Note that equality tests for doubles are bad, but it was just
   //sorted data, so should be fine
   if(eigenvalues[i] == temp_evalue) {smallest_index=i; break;}
  }
 
 //Normalise the eigenvector 
 {
  //Get the dimension of the eigenvector
  unsigned dim = eigenvector_real[smallest_index].nrow();
  double length=0.0;
  //Loop over all the entries
  for(unsigned i=0;i<dim;i++)
   {
    //Add the contribution to the length
    length += std::pow(eigenvector_real[smallest_index][i],2.0);
   }
  //Now take the magnitude
  length = sqrt(length);
  //Fix the sign
  if(eigenvector_real[smallest_index][0] < 0) {length *= -1.0;}
  //Finally normalise
  for(unsigned i=0;i<dim;i++)
   {
    eigenvector_real[smallest_index][i] /= length;
   }
 }
 
 //Now assign the second eigenvector to the dofs of the problem
 this->assign_eigenvector_to_dofs(eigenvector_real[smallest_index]);
 //Output solution for this case (label output files with "1")
 this->doc_solution(label);
 
 char filename[100];
 sprintf(filename,"eigenvalues%i.dat",label);
 
 //Open an output file for the sorted eigenvalues
 ofstream evalues(filename);
 for(unsigned i=0;i<n_eval;i++)
  {
   //Print to screen
   cout << sorted_eigenvalues[i].real() << " " 
        << sorted_eigenvalues[i].imag() << std::endl;
   //Send to file
   evalues << sorted_eigenvalues[i].real() << " " 
           << sorted_eigenvalues[i].imag() << std::endl;
  }
 
 evalues.close();
} //end_of_solve
 
 
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
 
 
//======start_of_main==================================================
/// Driver for 1D Poisson problem
//=====================================================================
int main(int argc, char **argv)
{
//Want to test Trilinos if we have it, so we must initialise MPI
//if we have compiled with it
#ifdef OOMPH_HAS_MPI
 MPI_Helpers::init(argc,argv);
#endif
 
 // Set up the problem: 
 unsigned n_element=100; //Number of elements
 
 clock_t t_start1 = clock();
 //Solve with LAPACK_QZ
 {
 
  ComplexHarmonicProblem<QComplexHarmonicElement<3>,LAPACK_QZ> 
   problem(n_element);
  
  problem.solve(1);
 }
 clock_t t_end1 = clock();
 
#ifdef OOMPH_HAS_TRILINOS
 clock_t t_start2 = clock();
//Solve with Anasazi
 {
  ComplexHarmonicProblem<QComplexHarmonicElement<3>,ANASAZI> problem(n_element);
  problem.solve(2);
 }
 clock_t t_end2 = clock();
#endif
 
 std::cout << "LAPACK TIME: " << (double)(t_end1 - t_start1)/CLOCKS_PER_SEC
           << std::endl;
 
#ifdef OOMPH_HAS_TRILINOS
  std::cout << "ANASAZI TIME: " << (double)(t_end2 - t_start2)/CLOCKS_PER_SEC
           << std::endl;
#endif
 
#ifdef OOMPH_HAS_MPI
 MPI_Helpers::finalize();
#endif
 
} // end of main