pml_fourier_decomposed_helmholtz_elements.cc

Go to the documentation of this file.

// 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// Copyright (C) 2006-2024 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 "pml_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
 
  } // namespace Legendre_functions_helper
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// 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
    QPMLFourierDecomposedHelmholtzElement<NNODE_1D>::Initial_Nvalue = 2;
 
  /// PML Helmholtz equations static data, so that by default we can point to a
  /// 0
  double
    PMLFourierDecomposedHelmholtzEquations::Default_Physical_Constant_Value =
      0.0;
 
  //======================================================================
  /// 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 PMLFourierDecomposedHelmholtzEquations::
    fill_in_generic_residual_contribution_pml_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_pml_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_pml_fourier_decomposed_helmholtz().real()),
          raw_nodal_value(l,
                          u_index_pml_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_pml_fourier_decomposed_helmholtz(ipt, interpolated_x, source);
 
      double n = (double)pml_fourier_wavenumber();
      double n_squared = n * n;
 
 
      // Declare a vector of complex numbers for pml weights on the Laplace bit
      Vector<std::complex<double>> pml_laplace_factor(2);
      // Declare a complex number for pml weights on the mass matrix bit
      std::complex<double> pml_k_squared_factor =
        std::complex<double>(1.0, 0.0);
 
      //  All the PML weights that participate in the assemby process
      // are computed here. pml_laplace_factor will contain the entries
      // for the Laplace bit, while pml_k_squared_factor contains the
      // contributions to the Helmholtz bit. Both default to 1.0, should the PML
      // not be enabled via enable_pml.
      compute_pml_coefficients(
        ipt, interpolated_x, pml_laplace_factor, pml_k_squared_factor);
 
 
      // Determine the complex r variable.  The variable is
      // only complex once it enters the right pml domain or either
      // of the two corner pml domains, otherwise it acts like the
      // variable r.
      std::complex<double> complex_r = std::complex<double>(1.0, 0.0);
      compute_complex_r(ipt, interpolated_x, complex_r);
 
      // Calculate Jacobian
      // ------------------
      std::complex<double> pml_k_squared_jacobian =
        -pml_k_squared_factor *
        (k_squared() - n_squared / (complex_r * complex_r)) * complex_r * W;
 
      // Term from the Laplace operator
      Vector<std::complex<double>> pml_laplace_jacobian(2);
      for (unsigned k = 0; k < 2; k++)
      {
        pml_laplace_jacobian[k] = pml_laplace_factor[k] * complex_r * W;
      }
 
      // Calculate residuals
      // -------------------
      // Note: it is a linear equation so residual = jacobian * u
      std::complex<double> pml_k_squared_residual =
        pml_k_squared_jacobian * interpolated_u;
 
      // Term from the Laplace operator
      Vector<std::complex<double>> pml_laplace_residual(2);
      for (unsigned k = 0; k < 2; k++)
      {
        pml_laplace_residual[k] =
          pml_laplace_jacobian[k] * interpolated_dudx[k];
      }
 
      // Assemble residuals and Jacobian
      //--------------------------------
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the equation numbers
        local_eqn_real =
          nodal_local_eqn(l, u_index_pml_fourier_decomposed_helmholtz().real());
        local_eqn_imag =
          nodal_local_eqn(l, u_index_pml_fourier_decomposed_helmholtz().imag());
 
        // first, add the real contributions
        //-------------------------------------------
 
        /*IF it's not a boundary condition*/
        if (local_eqn_real >= 0)
        {
          // Add k squared term of equation
          residuals[local_eqn_real] += pml_k_squared_residual.real() * test(l);
 
          // Add the term from the Laplace operator
          for (unsigned k = 0; k < 2; k++)
          {
            residuals[local_eqn_real] +=
              pml_laplace_residual[k].real() * dtestdx(l, k);
          }
 
          // Add contributions to the jacobian
          //----------------------------------
          if (flag)
          {
            // Loop over the velocity shape functions again
            for (unsigned l2 = 0; l2 < n_node; l2++)
            {
              // Get the unknown numbers
              local_unknown_real = nodal_local_eqn(
                l2, u_index_pml_fourier_decomposed_helmholtz().real());
              local_unknown_imag = nodal_local_eqn(
                l2, u_index_pml_fourier_decomposed_helmholtz().imag());
 
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown_real >= 0)
              {
                // Add the helmholtz contribution to the jacobian
                jacobian(local_eqn_real, local_unknown_real) +=
                  pml_k_squared_jacobian.real() * psi(l2) * test(l);
 
                // Add the term from the Laplace operator to the jacobian
                for (unsigned k = 0; k < 2; k++)
                {
                  jacobian(local_eqn_real, local_unknown_real) +=
                    pml_laplace_jacobian[k].real() * dpsidx(l2, k) *
                    dtestdx(l, k);
                }
              }
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown_imag >= 0)
              {
                // Add k squared term to jacobian
                jacobian(local_eqn_real, local_unknown_imag) +=
                  -pml_k_squared_jacobian.imag() * psi(l2) * test(l);
 
                // Add contribution to elemental Matrix
                for (unsigned k = 0; k < 2; k++)
                {
                  jacobian(local_eqn_real, local_unknown_imag) +=
                    -pml_laplace_jacobian[k].imag() * dpsidx(l2, k) *
                    dtestdx(l, k);
                }
              }
 
            } // end of loop over velocity shape functions
          } // end of if(flag)
        }
 
        // Second, add the imaginary contribution
        //------------------------------------------------
 
        // IF it's not a boundary condition
        if (local_eqn_imag >= 0)
        {
          // Add k squared term of equation
          residuals[local_eqn_imag] += pml_k_squared_residual.imag() * test(l);
 
          // Add the term from the Laplace operator
          for (unsigned k = 0; k < 2; k++)
          {
            residuals[local_eqn_imag] +=
              pml_laplace_residual[k].imag() * dtestdx(l, k);
          }
 
          // Add the contribution to 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_pml_fourier_decomposed_helmholtz().imag());
              local_unknown_real = nodal_local_eqn(
                l2, u_index_pml_fourier_decomposed_helmholtz().real());
 
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown_imag >= 0)
              {
                // Add the helmholtz contribution
                jacobian(local_eqn_imag, local_unknown_imag) +=
                  pml_k_squared_jacobian.real() * psi(l2) * test(l);
 
                // Add contribution to elemental Matrix
                for (unsigned k = 0; k < 2; k++)
                {
                  jacobian(local_eqn_imag, local_unknown_imag) +=
                    pml_laplace_jacobian[k].real() * dpsidx(l2, k) *
                    dtestdx(l, k);
                }
              }
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown_real >= 0)
              {
                // Add the helmholtz contribution
                jacobian(local_eqn_imag, local_unknown_real) +=
                  pml_k_squared_jacobian.imag() * psi(l2) * test(l);
 
                // Add contribution to elemental Matrix
                for (unsigned k = 0; k < 2; k++)
                {
                  jacobian(local_eqn_imag, local_unknown_real) +=
                    pml_laplace_jacobian[k].imag() * dpsidx(l2, k) *
                    dtestdx(l, k);
                }
              }
            }
          }
        }
      }
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Self-test:  Return 0 for OK
  //======================================================================
  unsigned PMLFourierDecomposedHelmholtzEquations::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 PMLFourierDecomposedHelmholtzEquations::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_pml_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 PMLFourierDecomposedHelmholtzEquations::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_pml_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 PMLFourierDecomposedHelmholtzEquations::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_pml_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 PMLFourierDecomposedHelmholtzEquations::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 PMLFourierDecomposedHelmholtzEquations::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 PMLFourierDecomposedHelmholtzEquations::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_pml_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 PMLFourierDecomposedHelmholtzEquations::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_pml_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 QPMLFourierDecomposedHelmholtzElement<2>;
  template class QPMLFourierDecomposedHelmholtzElement<3>;
  template class QPMLFourierDecomposedHelmholtzElement<4>;
 
  BermudezPMLMappingAndTransformedCoordinate
    PMLFourierDecomposedHelmholtzEquations::
      Default_pml_mapping_and_transformed_coordinate;
 
} // namespace oomph