pml_time_harmonic_linear_elasticity_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// Copyright (C) 2006-2025 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 elements that solve the equations of linear
// elasticity in cartesian coordinates
 
#include "pml_time_harmonic_linear_elasticity_elements.h"
 
 
namespace oomph
{
  /// Static default value for square of frequency
  template<unsigned DIM>
  double
    PMLTimeHarmonicLinearElasticityEquationsBase<DIM>::Default_omega_sq_value =
      1.0;
 
 
  //////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// Compute the strain tensor at local coordinate s
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquationsBase<DIM>::get_strain(
    const Vector<double>& s, DenseMatrix<std::complex<double>>& strain) const
  {
#ifdef PARANOID
    if ((strain.ncol() != DIM) || (strain.nrow() != DIM))
    {
      std::ostringstream error_message;
      error_message << "Strain matrix is " << strain.ncol() << " x "
                    << strain.nrow() << ", but dimension of the equations is "
                    << DIM << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // Find out how many position types there are
    unsigned n_position_type = this->nnodal_position_type();
 
    if (n_position_type != 1)
    {
      std::ostringstream error_message;
      error_message << "PMLTimeHarmonicLinearElasticity is not yet "
                    << "implemented for more than one position type"
                    << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
 
    // Find out how many nodes there are in the element
    unsigned n_node = nnode();
 
    // Find the indices at which the local velocities are stored
    std::complex<unsigned> u_nodal_index[DIM];
    for (unsigned i = 0; i < DIM; i++)
    {
      u_nodal_index[i] = u_index_time_harmonic_linear_elasticity(i);
    }
 
    // Set up memory for the shape and derivative functions
    Shape psi(n_node);
    DShape dpsidx(n_node, DIM);
 
    // Call the derivatives of the shape functions
    (void)dshape_eulerian(s, psi, dpsidx);
 
    // Calculate interpolated values of the derivative of global position
    DenseMatrix<std::complex<double>> interpolated_dudx(
      DIM, DIM, std::complex<double>(0.0, 0.0));
 
    // Loop over nodes
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over velocity components
      for (unsigned i = 0; i < DIM; i++)
      {
        // Get the nodal value
        const std::complex<double> u_value =
          std::complex<double>(this->nodal_value(l, u_nodal_index[i].real()),
                               this->nodal_value(l, u_nodal_index[i].imag()));
 
        // Loop over derivative directions
        for (unsigned j = 0; j < DIM; j++)
        {
          interpolated_dudx(i, j) += u_value * dpsidx(l, j);
        }
      }
    }
 
    /// Now fill in the entries of the strain tensor
    for (unsigned i = 0; i < DIM; i++)
    {
      // Do upper half of matrix
      // Note that j must be signed here for the comparison test to work
      // Also i must be cast to an int
      for (int j = (DIM - 1); j >= static_cast<int>(i); j--)
      {
        // Off diagonal terms
        if (static_cast<int>(i) != j)
        {
          strain(i, j) =
            0.5 * (interpolated_dudx(i, j) + interpolated_dudx(j, i));
        }
        // Diagonal terms will including growth factor when it comes back in
        else
        {
          strain(i, i) = interpolated_dudx(i, i);
        }
      }
      // Matrix is symmetric so just copy lower half
      for (int j = (i - 1); j >= 0; j--)
      {
        strain(i, j) = strain(j, i);
      }
    }
  }
 
 
  //======================================================================
  /// Compute the Cauchy stress tensor at local coordinate s for
  /// displacement formulation.
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::get_stress(
    const Vector<double>& s, DenseMatrix<std::complex<double>>& stress) const
  {
#ifdef PARANOID
    if ((stress.ncol() != DIM) || (stress.nrow() != DIM))
    {
      std::ostringstream error_message;
      error_message << "Stress matrix is " << stress.ncol() << " x "
                    << stress.nrow() << ", but dimension of the equations is "
                    << DIM << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Get strain
    DenseMatrix<std::complex<double>> strain(DIM, DIM);
    this->get_strain(s, strain);
 
    // Now fill in the entries of the stress tensor without exploiting
    // symmetry -- sorry too lazy. This fct is only used for
    // postprocessing anyway...
    for (unsigned i = 0; i < DIM; i++)
    {
      for (unsigned j = 0; j < DIM; j++)
      {
        stress(i, j) = 0.0;
        for (unsigned k = 0; k < DIM; k++)
        {
          for (unsigned l = 0; l < DIM; l++)
          {
            stress(i, j) += this->E(i, j, k, l) * strain(k, l);
          }
        }
      }
    }
  }
 
 
  //=======================================================================
  /// Compute the residuals for the linear elasticity equations in
  /// cartesian coordinates. Flag indicates if we want Jacobian too.
  //=======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::
    fill_in_generic_contribution_to_residuals_time_harmonic_linear_elasticity(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag)
  {
    // Find out how many nodes there are
    unsigned n_node = this->nnode();
 
#ifdef PARANOID
    // Find out how many positional dofs there are
    unsigned n_position_type = this->nnodal_position_type();
 
    if (n_position_type != 1)
    {
      std::ostringstream error_message;
      error_message << "PMLTimeHarmonicLinearElasticity is not yet "
                    << "implemented for more than one position type"
                    << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // Throw and error if an elasticity tensor has not been set
    if (this->Elasticity_tensor_pt == 0)
    {
      throw OomphLibError("No elasticity tensor set.",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Find the indices at which the local velocities are stored
    std::complex<unsigned> u_nodal_index[DIM];
    for (unsigned i = 0; i < DIM; i++)
    {
      u_nodal_index[i] = this->u_index_time_harmonic_linear_elasticity(i);
    }
 
 
    // Square of non-dimensional frequency
    const double omega_sq_local = this->omega_sq();
 
    // Set up memory for the shape functions
    Shape psi(n_node);
    DShape dpsidx(n_node, DIM);
 
    // Set the value of Nintpt -- the number of integration points
    unsigned n_intpt = this->integral_pt()->nweight();
 
    // Set the vector to hold the local coordinates in the element
    Vector<double> s(DIM);
 
    // Integers to store the local equation 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++)
    {
      // Assign the values of s
      for (unsigned i = 0; i < DIM; ++i)
      {
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Call the derivatives of the shape functions (and get Jacobian)
      double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
      // Storage for Eulerian coordinates (initialised to zero)
      Vector<double> interpolated_x(DIM, 0.0);
 
      // Displacement
      Vector<std::complex<double>> interpolated_u(
        DIM, std::complex<double>(0.0, 0.0));
 
      // Calculate interpolated values of the derivative of global position
      // wrt lagrangian coordinates
      DenseMatrix<std::complex<double>> interpolated_dudx(
        DIM, DIM, std::complex<double>(0.0, 0.0));
 
      // Calculate displacements and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over displacement components (deformed position)
        for (unsigned i = 0; i < DIM; i++)
        {
          // Calculate the Lagrangian coordinates and the accelerations
          interpolated_x[i] += this->raw_nodal_position(l, i) * psi(l);
 
 
          // Get the nodal displacements
          const std::complex<double> u_value = std::complex<double>(
            this->raw_nodal_value(l, u_nodal_index[i].real()),
            this->raw_nodal_value(l, u_nodal_index[i].imag()));
 
          interpolated_u[i] += u_value * psi(l);
 
          // Loop over derivative directions
          for (unsigned j = 0; j < DIM; j++)
          {
            interpolated_dudx(i, j) += u_value * dpsidx(l, j);
          }
        }
      }
 
      // Get body force
      Vector<std::complex<double>> body_force_vec(DIM);
      this->body_force(interpolated_x, body_force_vec);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
 
      /// All the PML weights that participate in the assemby process
      /// are computed here. pml_inverse_jacobian_diagonals are are used to
      /// transform derivatives in real x to derivatives in transformed space
      /// \f$\tilde x \f$.
      /// pml_jacobian_det allows us to transform volume integrals in
      /// transformed space to real space.
      /// If the PML is not enabled via enable_pml, both default to 1.0.
      Vector<std::complex<double>> pml_inverse_jacobian_diagonal(DIM);
      std::complex<double> pml_jacobian_det;
      this->compute_pml_coefficients(
        ipt, interpolated_x, pml_inverse_jacobian_diagonal, pml_jacobian_det);
 
      // Loop over the test functions, nodes of the element
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the displacement components
        for (unsigned a = 0; a < DIM; a++)
        {
          // Get real and imaginary equation numbers
          local_eqn_real = this->nodal_local_eqn(l, u_nodal_index[a].real());
          local_eqn_imag = this->nodal_local_eqn(l, u_nodal_index[a].imag());
 
          //===== EQUATIONS OF PML TIME HARMONIC LINEAR ELASTICITY ========
          // (This is where the maths happens)
 
          // Calculate jacobian and residual contributions from acceleration and
          // body force
          std::complex<double> mass_jacobian_contribution =
            -omega_sq_local * pml_jacobian_det;
          std::complex<double> mass_residual_contribution =
            (-omega_sq_local * interpolated_u[a] - body_force_vec[a]) *
            pml_jacobian_det;
 
          // Calculate jacobian and residual contributions from stress term
          std::complex<double> stress_jacobian_contributions[DIM][DIM][DIM];
          std::complex<double> stress_residual_contributions[DIM];
          for (unsigned b = 0; b < DIM; b++)
          {
            stress_residual_contributions[b] = 0.0;
            for (unsigned c = 0; c < DIM; c++)
            {
              for (unsigned d = 0; d < DIM; d++)
              {
                stress_jacobian_contributions[b][c][d] =
                  this->E(a, b, c, d) * pml_jacobian_det *
                  pml_inverse_jacobian_diagonal[b] *
                  pml_inverse_jacobian_diagonal[d];
 
                stress_residual_contributions[b] +=
                  stress_jacobian_contributions[b][c][d] *
                  interpolated_dudx(c, d);
              }
            }
          }
 
          //===== ADD CONTRIBUTIONS TO GLOBAL RESIDUALS AND JACOBIAN ========
 
          /*IF it's not a boundary condition*/
          if (local_eqn_real >= 0)
          {
            // Acceleration and body force
            residuals[local_eqn_real] +=
              mass_residual_contribution.real() * psi(l) * W;
 
            // Stress term
            for (unsigned b = 0; b < DIM; b++)
            {
              // Add the stress terms to the residuals
              residuals[local_eqn_real] +=
                stress_residual_contributions[b].real() * dpsidx(l, b) * W;
            }
 
            // Jacobian entries
            if (flag)
            {
              // Loop over the displacement basis functions again
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Loop over the displacement components again
                for (unsigned c = 0; c < DIM; c++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, u_nodal_index[c].real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, u_nodal_index[c].imag());
                  // If it's not pinned
                  if (local_unknown_real >= 0)
                  {
                    // Inertial term
                    if (a == c)
                    {
                      jacobian(local_eqn_real, local_unknown_real) +=
                        mass_jacobian_contribution.real() * psi(l) * psi(l2) *
                        W;
                    }
 
                    // Stress term
                    for (unsigned b = 0; b < DIM; b++)
                    {
                      for (unsigned d = 0; d < DIM; d++)
                      {
                        // Add the contribution to the Jacobian matrix
                        jacobian(local_eqn_real, local_unknown_real) +=
                          stress_jacobian_contributions[b][c][d].real() *
                          dpsidx(l2, d) * dpsidx(l, b) * W;
                      }
                    }
                  } // End of if not boundary condition
 
                  if (local_unknown_imag >= 0)
                  {
                    // Inertial term
                    if (a == c)
                    {
                      jacobian(local_eqn_real, local_unknown_imag) -=
                        mass_jacobian_contribution.imag() * psi(l) * psi(l2) *
                        W;
                    }
 
                    // Stress term
                    for (unsigned b = 0; b < DIM; b++)
                    {
                      for (unsigned d = 0; d < DIM; d++)
                      {
                        // Add the contribution to the Jacobian matrix
                        jacobian(local_eqn_real, local_unknown_imag) -=
                          stress_jacobian_contributions[b][c][d].imag() *
                          dpsidx(l2, d) * dpsidx(l, b) * W;
                      }
                    }
                  } // End of if not boundary condition
                }
              }
            } // End of jacobian calculation
 
          } // End of if not boundary condition for real eqn
 
 
          /*IF it's not a boundary condition*/
          if (local_eqn_imag >= 0)
          {
            // Acceleration and body force
            residuals[local_eqn_imag] +=
              mass_residual_contribution.imag() * psi(l) * W;
 
            // Stress term
            for (unsigned b = 0; b < DIM; b++)
            {
              // Add the stress terms to the residuals
              residuals[local_eqn_imag] +=
                stress_residual_contributions[b].imag() * dpsidx(l, b) * W;
            }
 
            // Jacobian entries
            if (flag)
            {
              // Loop over the displacement basis functions again
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Loop over the displacement components again
                for (unsigned c = 0; c < DIM; c++)
                {
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, u_nodal_index[c].imag());
                  local_unknown_real =
                    this->nodal_local_eqn(l2, u_nodal_index[c].real());
                  // If it's not pinned
                  if (local_unknown_imag >= 0)
                  {
                    // Inertial term
                    if (a == c)
                    {
                      jacobian(local_eqn_imag, local_unknown_imag) +=
                        mass_jacobian_contribution.real() * psi(l) * psi(l2) *
                        W;
                    }
 
                    // Stress term
                    for (unsigned b = 0; b < DIM; b++)
                    {
                      for (unsigned d = 0; d < DIM; d++)
                      {
                        // Add the contribution to the Jacobian matrix
                        jacobian(local_eqn_imag, local_unknown_imag) +=
                          stress_jacobian_contributions[b][c][d].real() *
                          dpsidx(l2, d) * dpsidx(l, b) * W;
                      }
                    }
                  } // End of if not boundary condition
 
                  if (local_unknown_real >= 0)
                  {
                    // Inertial term
                    if (a == c)
                    {
                      jacobian(local_eqn_imag, local_unknown_real) +=
                        mass_jacobian_contribution.imag() * psi(l) * psi(l2) *
                        W;
                    }
 
                    // Stress term
                    for (unsigned b = 0; b < DIM; b++)
                    {
                      for (unsigned d = 0; d < DIM; d++)
                      {
                        // Add the contribution to the Jacobian matrix
                        jacobian(local_eqn_imag, local_unknown_real) +=
                          stress_jacobian_contributions[b][c][d].imag() *
                          dpsidx(l2, d) * dpsidx(l, b) * W;
                      }
                    }
                  } // End of if not boundary condition
                }
              }
            } // End of jacobian calculation
 
          } // End of if not boundary condition for imag eqn
 
        } // End of loop over coordinate directions
      } // End of loop over shape functions
    } // End of loop over integration points
  }
 
  //=======================================================================
  /// Output exact solution x,y,[z],u_r,v_r,[w_r],u_i,v_i,[w_i]
  //=======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output_fct(
    std::ostream& outfile,
    const unsigned& nplot,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
  {
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Vector for coordintes
    Vector<double> x(DIM);
 
    // Tecplot header info
    outfile << this->tecplot_zone_string(nplot);
 
    // Exact solution Vector
    Vector<double> exact_soln(2 * DIM);
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Get x position as Vector
      this->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 < DIM; i++)
      {
        outfile << x[i] << " ";
      }
      for (unsigned i = 0; i < 2 * DIM; i++)
      {
        outfile << exact_soln[i] << " ";
      }
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //=======================================================================
  /// Output: x,y,[z],u,v,[w]
  //=======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output(
    std::ostream& outfile, const unsigned& nplot)
  {
    // Initialise local coord, global coord and solution vectors
    Vector<double> s(DIM);
    Vector<double> x(DIM);
    Vector<std::complex<double>> u(DIM);
 
    // Tecplot header info
    outfile << this->tecplot_zone_string(nplot);
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Get Eulerian coordinates and displacements
      this->interpolated_x(s, x);
      this->interpolated_u_time_harmonic_linear_elasticity(s, u);
 
      // Output the x,y,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
 
      // Output u,v,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << u[i].real() << " ";
      }
 
      // Output u,v,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << u[i].imag() << " ";
      }
 
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //======================================================================
  /// Output function for real part of full time-dependent solution
  /// constructed by adding the scattered field
  ///
  ///  u = Re( (u_r +i u_i) exp(-i omega t)
  ///
  /// at phase angle omega t = phi computed here, to the corresponding
  /// incoming wave specified via the function pointer.
  ///
  ///   x,y,u   or    x,y,z,u
  ///
  /// Output at nplot points in each coordinate direction
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output_total_real(
    std::ostream& outfile,
    FiniteElement::SteadyExactSolutionFctPt incoming_wave_fct_pt,
    const double& phi,
    const unsigned& nplot)
  {
    // Initialise local coord, global coord and solution vectors
    Vector<double> s(DIM);
    Vector<double> x(DIM);
    Vector<std::complex<double>> u(DIM);
 
    // Real and imag part of incoming wave
    Vector<double> incoming_soln(2 * DIM);
 
    // Tecplot header info
    outfile << this->tecplot_zone_string(nplot);
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Get Eulerian coordinates and displacements
      this->interpolated_x(s, x);
      this->interpolated_u_time_harmonic_linear_elasticity(s, u);
 
      // Get exact solution at this point
      (*incoming_wave_fct_pt)(x, incoming_soln);
 
      // Output the x,y,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
 
      // Output u,v,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << (u[i].real() + incoming_soln[2 * i]) * cos(phi) +
                     (u[i].imag() + incoming_soln[2 * i + 1]) * sin(phi)
                << " ";
      }
 
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(outfile, nplot);
  }
 
  //======================================================================
  /// Output function for imaginary part of full time-dependent solution
  ///
  ///  u = Im( (u_r +i u_i) exp(-i omega t))
  ///
  /// at phase angle omega t = phi.
  ///
  ///   x,y,u   or    x,y,z,u
  ///
  /// Output at nplot points in each coordinate direction
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output_real(
    std::ostream& outfile, const double& phi, const unsigned& nplot)
  {
    // Initialise local coord, global coord and solution vectors
    Vector<double> s(DIM);
    Vector<double> x(DIM);
    Vector<std::complex<double>> u(DIM);
 
    // Tecplot header info
    outfile << this->tecplot_zone_string(nplot);
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Get Eulerian coordinates and displacements
      this->interpolated_x(s, x);
      this->interpolated_u_time_harmonic_linear_elasticity(s, u);
 
      // Output the x,y,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
 
      // Output u,v,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << u[i].real() * cos(phi) + u[i].imag() * sin(phi) << " ";
      }
 
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(outfile, nplot);
  }
 
  //======================================================================
  /// Output function for imaginary part of full time-dependent solution
  ///
  ///  u = Im( (u_r +i u_i) exp(-i omega t))
  ///
  /// at phase angle omega t = phi.
  ///
  ///   x,y,u   or    x,y,z,u
  ///
  /// Output at nplot points in each coordinate direction
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output_imag(
    std::ostream& outfile, const double& phi, const unsigned& nplot)
  {
    // Initialise local coord, global coord and solution vectors
    Vector<double> s(DIM);
    Vector<double> x(DIM);
    Vector<std::complex<double>> u(DIM);
 
    // Tecplot header info
    outfile << this->tecplot_zone_string(nplot);
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Get Eulerian coordinates and displacements
      this->interpolated_x(s, x);
      this->interpolated_u_time_harmonic_linear_elasticity(s, u);
 
      // Output the x,y,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
 
      // Output u,v,..
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << u[i].imag() * cos(phi) - u[i].real() * sin(phi) << " ";
      }
 
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //=======================================================================
  /// C-style output: x,y,[z],u,v,[w]
  //=======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::output(
    FILE* file_pt, const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Tecplot header info
    fprintf(file_pt, "%s", this->tecplot_zone_string(nplot).c_str());
 
    // Loop over plot points
    unsigned num_plot_points = this->nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      this->get_s_plot(iplot, nplot, s);
 
      // Coordinates
      for (unsigned i = 0; i < DIM; i++)
      {
        fprintf(file_pt, "%g ", this->interpolated_x(s, i));
      }
 
      // Displacement
      for (unsigned i = 0; i < DIM; i++)
      {
        fprintf(
          file_pt,
          "%g ",
          this->interpolated_u_time_harmonic_linear_elasticity(s, i).real());
      }
      for (unsigned i = 0; i < DIM; i++)
      {
        fprintf(
          file_pt,
          "%g ",
          this->interpolated_u_time_harmonic_linear_elasticity(s, i).imag());
      }
    }
    fprintf(file_pt, "\n");
 
    // Write tecplot footer (e.g. FE connectivity lists)
    this->write_tecplot_zone_footer(file_pt, nplot);
  }
 
 
  //=======================================================================
  /// Compute norm of the solution
  //=======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::compute_norm(double& norm)
  {
    // Initialise
    norm = 0.0;
 
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Vector for coordintes
    Vector<double> x(DIM);
 
    // Displacement vector
    Vector<std::complex<double>> disp(DIM);
 
    // Find out how many nodes there are in the element
    unsigned n_node = this->nnode();
 
    Shape psi(n_node);
 
    // Set the value of n_intpt
    unsigned n_intpt = this->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 < DIM; i++)
      {
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Get jacobian of mapping
      double J = this->J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get FE function value
      this->interpolated_u_time_harmonic_linear_elasticity(s, disp);
 
      // Add to norm
      for (unsigned ii = 0; ii < DIM; ii++)
      {
        norm += (disp[ii].real() * disp[ii].real() +
                 disp[ii].imag() * disp[ii].imag()) *
                W;
      }
    }
  }
 
  //======================================================================
  /// Validate against exact solution
  ///
  /// Solution is provided via function pointer.
  ///
  //======================================================================
  template<unsigned DIM>
  void PMLTimeHarmonicLinearElasticityEquations<DIM>::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(DIM);
 
    // Vector for coordintes
    Vector<double> x(DIM);
 
    // Displacement vector
    Vector<std::complex<double>> disp(DIM);
 
    // Find out how many nodes there are in the element
    unsigned n_node = this->nnode();
 
    Shape psi(n_node);
 
    // Set the value of n_intpt
    unsigned n_intpt = this->integral_pt()->nweight();
 
    // Exact solution Vector
    Vector<double> exact_soln(2 * DIM);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Get jacobian of mapping
      double J = this->J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get x position as Vector
      this->interpolated_x(s, x);
 
      // Get exact solution at this point
      (*exact_soln_pt)(x, exact_soln);
 
      // Get FE function value
      this->interpolated_u_time_harmonic_linear_elasticity(s, disp);
 
      // Add to  norm
      for (unsigned ii = 0; ii < DIM; ii++)
      {
        // Add to error and norm
        error += ((exact_soln[ii] - disp[ii].real()) *
                    (exact_soln[ii] - disp[ii].real()) +
                  (exact_soln[ii + DIM] - disp[ii].imag()) *
                    (exact_soln[ii + DIM] - disp[ii].imag())) *
                 W;
        norm += (disp[ii].real() * disp[ii].real() +
                 disp[ii].imag() * disp[ii].imag()) *
                W;
      }
    }
  }
 
 
  // Instantiate the required elements
  template class PMLTimeHarmonicLinearElasticityEquationsBase<2>;
  template class PMLTimeHarmonicLinearElasticityEquations<2>;
 
  template class QPMLTimeHarmonicLinearElasticityElement<3, 3>;
  template class PMLTimeHarmonicLinearElasticityEquationsBase<3>;
  template class PMLTimeHarmonicLinearElasticityEquations<3>;
 
  template<unsigned DIM>
  ContinuousBermudezPMLMapping
    PMLTimeHarmonicLinearElasticityEquationsBase<DIM>::Default_pml_mapping;
 
 
} // namespace oomph