Qspectral_elements.h

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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) 2006-2023 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
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// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header functions for classes that define Qelements
 
// Include guards to prevent multiple inclusion of the header
#ifndef OOMPH_QSPECTRAL_ELEMENT_HEADER
#define OOMPH_QSPECTRAL_ELEMENT_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// oomph-lib headers
#include "Qelements.h"
 
namespace oomph
{
  //=====================================================================
  /// Class that returns the shape functions associated with legendre
  //=====================================================================
  class OneDLegendreShapeParam : public Shape
  {
  public:
    static std::map<unsigned, Vector<double>> z;
 
    /// Static function used to populate the stored positions
    static inline void calculate_nodal_positions(const unsigned& order)
    {
      // If we haven't already calculated these
      if (z.find(order) == z.end())
      {
        Orthpoly::gll_nodes(order, z[order]);
      }
    }
 
    static inline double nodal_position(const unsigned& order,
                                        const unsigned& n)
    {
      return z[order][n];
    }
 
    /// Constructor
    OneDLegendreShapeParam(const unsigned& order, const double& s)
      : Shape(order)
    {
      using namespace Orthpoly;
 
      unsigned p = order - 1;
      // Now populate the shape function
      for (unsigned i = 0; i < order; i++)
      {
        // If we're at one of the nodes, the value must be 1.0
        if (std::fabs(s - z[order][i]) < Orthpoly::eps)
        {
          (*this)[i] = 1.0;
        }
        // Otherwise use the lagrangian interpolant
        else
        {
          (*this)[i] =
            (1.0 - s * s) * dlegendre(p, s) /
            (p * (p + 1) * legendre(p, z[order][i]) * (z[order][i] - s));
        }
      }
    }
  };
 
 
  class OneDLegendreDShapeParam : public Shape
  {
  public:
    // Constructor
    OneDLegendreDShapeParam(const unsigned& order, const double& s)
      : Shape(order)
    {
      unsigned p = order - 1;
      Vector<double> z = OneDLegendreShapeParam::z[order];
 
      bool root = false;
      for (unsigned i = 0; i < order; i++)
      {
        unsigned rootnum = 0;
        for (unsigned j = 0; j < order; j++)
        { // Loop over roots to check if
          if (std::fabs(s - z[j]) < 10.0 * Orthpoly::eps)
          { // s happens to be a root.
            root = true;
            break;
          }
          rootnum += 1;
        }
        if (root == true)
        {
          if (i == rootnum && i == 0)
          {
            (*this)[i] = -(1.0 + p) * p / 4.0;
          }
          else if (i == rootnum && i == p)
          {
            (*this)[i] = (1.0 + p) * p / 4.0;
          }
          else if (i == rootnum)
          {
            (*this)[i] = 0.0;
          }
          else
          {
            (*this)[i] = Orthpoly::legendre(p, z[rootnum]) /
                         Orthpoly::legendre(p, z[i]) / (z[rootnum] - z[i]);
          }
        }
        else
        {
          (*this)[i] =
            ((1 + s * (s - 2 * z[i])) / (s - z[i]) * Orthpoly::dlegendre(p, s) -
             (1 - s * s) * Orthpoly::ddlegendre(p, s)) /
            p / (p + 1.0) / Orthpoly::legendre(p, z[i]) / (s - z[i]);
        }
        root = false;
      }
    }
  };
 
 
  //========================================================================
  // A Base class for Spectral elements
  //========================================================================
  class SpectralElement : public virtual FiniteElement
  {
  protected:
    /// Additional Storage for shared spectral data
    Vector<Data*>* Spectral_data_pt;
 
    /// Vector that represents the spectral order in each dimension
    Vector<unsigned> Spectral_order;
 
    /// Vector that represents the nodal spectral order
    Vector<unsigned> Nodal_spectral_order;
 
    /// Local equation numbers for the spectral degrees of freedom
    DenseMatrix<int> Spectral_local_eqn;
 
  public:
    SpectralElement() : FiniteElement(), Spectral_data_pt(0) {}
 
    virtual ~SpectralElement()
    {
      if (Spectral_data_pt != 0)
      {
        delete Spectral_data_pt;
        Spectral_data_pt = 0;
      }
    }
 
 
    /// Return the i-th data object associated with the polynomials
    /// of order p. Note that i <= p.
    Data* spectral_data_pt(const unsigned& i) const
    {
      return (*Spectral_data_pt)[i];
    }
 
    /// Function to describe the local dofs of the element. The ostream
    /// specifies the output stream to which the description
    /// is written; the string stores the currently
    /// assembled output that is ultimately written to the
    /// output stream by Data::describe_dofs(...); it is typically
    /// built up incrementally as we descend through the
    /// call hierarchy of this function when called from
    /// Problem::describe_dofs(...)
    virtual void describe_local_dofs(std::ostream& out,
                                     const std::string& current_string) const
    {
      // Do the standard finite element stuff
      FiniteElement::describe_local_dofs(out, current_string);
      // Now get the number of spectral data.
      unsigned n_spectral = nspectral();
 
      // Now loop over the nodes again and assign local equation numbers
      for (unsigned n = 0; n < n_spectral; n++)
      {
        // Can we upcast to a node
        Node* cast_node_pt = dynamic_cast<Node*>(spectral_data_pt(n));
        if (cast_node_pt)
        {
          std::stringstream conversion;
          conversion << " of Node " << n << current_string;
          std::string in(conversion.str());
          cast_node_pt->describe_dofs(out, in);
        }
        // Otherwise it is data.
        else
        {
          Data* data_pt = spectral_data_pt(n);
          std::stringstream conversion;
          conversion << " of Data " << n << current_string;
          std::string in(conversion.str());
          data_pt->describe_dofs(out, in);
        }
      }
    }
 
    /// Assign the local equation numbers. If the boolean argument is
    /// true then store degrees of freedom at Dof_pt
    virtual void assign_all_generic_local_eqn_numbers(
      const bool& store_local_dof_pt)
    {
      // Do the standard finite element stuff
      FiniteElement::assign_all_generic_local_eqn_numbers(store_local_dof_pt);
 
      // Now need to loop over the spectral data
      unsigned n_spectral = nspectral();
      if (n_spectral > 0)
      {
        // Find the number of local equations assigned so far
        unsigned local_eqn_number = ndof();
 
        // Find number of values stored at the first node
        unsigned max_n_value = spectral_data_pt(0)->nvalue();
        // Loop over the other nodes and find the maximum number of values
        // stored
        for (unsigned n = 1; n < n_spectral; n++)
        {
          unsigned n_value = spectral_data_pt(n)->nvalue();
          if (n_value > max_n_value)
          {
            max_n_value = n_value;
          }
        }
 
        // Resize the storage for the nodal local equation numbers
        // initially all local equations are unclassified
        Spectral_local_eqn.resize(
          n_spectral, max_n_value, Data::Is_unclassified);
 
        // Construct a set of pointers to the nodes
        std::set<Data*> set_of_node_pt;
        unsigned n_node = nnode();
        for (unsigned n = 0; n < n_node; n++)
        {
          set_of_node_pt.insert(node_pt(n));
        }
 
        // A local queue to store the global equation numbers
        std::deque<unsigned long> global_eqn_number_queue;
 
        // Now loop over the nodes again and assign local equation numbers
        for (unsigned n = 0; n < n_spectral; n++)
        {
          // Need to find whether the data is, in fact a node
          //(and hence already assigned)
 
          // Can we upcast to a node
          Node* cast_node_pt = dynamic_cast<Node*>(spectral_data_pt(n));
          if ((cast_node_pt) &&
              (set_of_node_pt.find(cast_node_pt) != set_of_node_pt.end()))
          {
            // Find the number of values
            unsigned n_value = cast_node_pt->nvalue();
            // Copy them across
            for (unsigned j = 0; j < n_value; j++)
            {
              Spectral_local_eqn(n, j) =
                nodal_local_eqn(get_node_number(cast_node_pt), j);
            }
            // Remove from the set
            set_of_node_pt.erase(cast_node_pt);
          }
          // Otherwise it's just data
          else
          {
            Data* const data_pt = spectral_data_pt(n);
            unsigned n_value = data_pt->nvalue();
            // Loop over the number of values
            for (unsigned j = 0; j < n_value; j++)
            {
              // Get the GLOBAL equation number
              long eqn_number = data_pt->eqn_number(j);
              // If the GLOBAL equation number is positive (a free variable)
              if (eqn_number >= 0)
              {
                // Add the GLOBAL equation number to the
                // local-to-global translation
                // scheme
                global_eqn_number_queue.push_back(eqn_number);
                // Add pointer to the dof to the queue if required
                if (store_local_dof_pt)
                {
                  GeneralisedElement::Dof_pt_deque.push_back(
                    data_pt->value_pt(j));
                }
                // Add the local equation number to the local scheme
                Spectral_local_eqn(n, j) = local_eqn_number;
                // Increase the local number
                local_eqn_number++;
              }
              else
              {
                // Set the local scheme to be pinned
                Spectral_local_eqn(n, j) = Data::Is_pinned;
              }
            }
          } // End of case when it's not a nodal dof
        }
 
        // Now add our global equations numbers to the internal element storage
        add_global_eqn_numbers(global_eqn_number_queue,
                               GeneralisedElement::Dof_pt_deque);
        // Clear the memory used in the deque
        if (store_local_dof_pt)
        {
          std::deque<double*>().swap(GeneralisedElement::Dof_pt_deque);
        }
 
      } // End of case when there are spectral degrees of freedom
    }
 
    unsigned nspectral() const
    {
      if (Spectral_data_pt == 0)
      {
        return 0;
      }
      else
      {
        return Spectral_data_pt->size();
      }
    }
  };
 
 
  //=======================================================================
  /// General QLegendreElement class
  ///
  /// Empty, just establishes the template parameters
  //=======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class QSpectralElement
  {
  };
 
 
  //=======================================================================
  /// General QSpectralElement class specialised to one spatial dimension
  //=======================================================================
  template<unsigned NNODE_1D>
  class QSpectralElement<1, NNODE_1D> : public virtual SpectralElement,
                                        public virtual LineElementBase
  {
  private:
    /// Default integration rule: Gaussian integration of same 'order'
    /// as the element
    // This is sort of optimal, because it means that the integration is exact
    // for the shape functions. Can overwrite this in specific element
    // defintion.
    static GaussLobattoLegendre<1, NNODE_1D> integral;
 
  public:
    /// Constructor
    QSpectralElement()
    {
      // There are NNODE_1D nodes in this element
      this->set_n_node(NNODE_1D);
      Spectral_order.resize(1, NNODE_1D);
      Nodal_spectral_order.resize(1, NNODE_1D);
      // Set the elemental and nodal dimensions
      this->set_dimension(1);
      // Assign integral point
      this->set_integration_scheme(&integral);
      // Calculate the nodal positions for the shape functions
      OneDimensionalLegendreShape<NNODE_1D>::calculate_nodal_positions();
      // OneDLegendreShapeParam::calculate_nodal_positions(NNODE_1D);
    }
 
    /// Min. value of local coordinate
    double s_min() const
    {
      return -1.0;
    }
 
    /// Max. value of local coordinate
    double s_max() const
    {
      return 1.0;
    }
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      return 2;
    }
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      unsigned n_node_1d = nnode_1d();
      Node* nod_pt;
      switch (j)
      {
        case 0:
          nod_pt = this->node_pt(0);
          break;
        case 1:
          nod_pt = this->node_pt(n_node_1d - 1);
          break;
        default:
          std::ostringstream error_message;
          error_message << "Vertex node number is " << j
                        << " but must be from 0 to 1\n";
 
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
      }
      return nod_pt;
    }
 
    /// Get local coordinates of node j in the element; vector sets its own size
    void local_coordinate_of_node(const unsigned& n, Vector<double>& s) const
    {
      s.resize(1);
      s[0] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n);
    }
 
    /// Get the local fractino of node j in the element
    void local_fraction_of_node(const unsigned& n, Vector<double>& s_fraction)
    {
      this->local_coordinate_of_node(n, s_fraction);
      // Resize
      s_fraction[0] = 0.5 * (s_fraction[0] + 1.0);
    }
 
    /// The local one-d fraction is the same
    double local_one_d_fraction_of_node(const unsigned& n1d, const unsigned& i)
    {
      return 0.5 *
             (OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n1d) + 1.0);
    }
 
    /// Calculate the geometric shape functions at local coordinate s
    inline void shape(const Vector<double>& s, Shape& psi) const;
 
    /// Compute the  geometric shape functions and
    /// derivatives w.r.t. local coordinates at local coordinate s
    inline void dshape_local(const Vector<double>& s,
                             Shape& psi,
                             DShape& dpsids) const;
 
    /// Compute the geometric shape functions, derivatives and
    /// second derivatives w.r.t. local coordinates at local coordinate s
    /// d2psids(i,0) = \f$ d^2 \psi_j / d s^2 \f$
    inline void d2shape_local(const Vector<double>& s,
                              Shape& psi,
                              DShape& dpsids,
                              DShape& d2psids) const;
 
    /// Overload the template-free interface for the calculation of
    /// inverse jacobian matrix. This is a one-dimensional element, so
    /// use the 1D version.
    double invert_jacobian_mapping(const DenseMatrix<double>& jacobian,
                                   DenseMatrix<double>& inverse_jacobian) const
    {
      return FiniteElement::invert_jacobian<1>(jacobian, inverse_jacobian);
    }
 
 
    /// Number of nodes along each element edge
    unsigned nnode_1d() const
    {
      return NNODE_1D;
    }
 
    /// C-style output
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C_style output at n_plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
    /// Output
    void output(std::ostream& outfile);
 
    /// Output at n_plot points
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    ///  Get cector of local coordinates of plot point i (when plotting
    /// nplot points in each "coordinate direction).
    void get_s_plot(
      const unsigned& i,
      const unsigned& nplot,
      Vector<double>& s,
      const bool& use_equally_spaced_interior_sample_points = false) const
    {
      if (nplot > 1)
      {
        s[0] = -1.0 + 2.0 * double(i) / double(nplot - 1);
        if (use_equally_spaced_interior_sample_points)
        {
          double range = 2.0;
          double dx_new = range / double(nplot);
          double range_new = double(nplot - 1) * dx_new;
          s[0] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[0]) / range;
        }
      }
      else
      {
        s[0] = 0.0;
      }
    }
 
    /// Return string for tecplot zone header (when plotting
    /// nplot points in each "coordinate direction)
    std::string tecplot_zone_string(const unsigned& nplot) const
    {
      std::ostringstream header;
      header << "ZONE I=" << nplot << "\n";
      return header.str();
    }
 
    /// Return total number of plot points (when plotting
    /// nplot points in each "coordinate direction)
    unsigned nplot_points(const unsigned& nplot) const
    {
      unsigned DIM = 1;
      unsigned np = 1;
      for (unsigned i = 0; i < DIM; i++)
      {
        np *= nplot;
      }
      return np;
    }
 
    /// Build the lower-dimensional FaceElement (an element of type
    /// QSpectralElement<0,NNODE_1D>). The face index takes two values
    /// corresponding to the two possible faces:
    /// -1 (Left)  s[0] = -1.0
    /// +1 (Right) s[0] =  1.0
    void build_face_element(const int& face_index,
                            FaceElement* face_element_pt);
  };
 
 
  //=======================================================================
  /// Shape function for specific QSpectralElement<1,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<1, NNODE_1D>::shape(const Vector<double>& s,
                                            Shape& psi) const
  {
    // Call the OneDimensional Shape functions
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]);
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]);
 
    // Now let's loop over the nodal points in the element
    // and copy the values back in
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      psi(i) = psi1[i];
    }
  }
 
  //=======================================================================
  /// Derivatives of shape functions for specific  QSpectralElement<1,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<1, NNODE_1D>::dshape_local(const Vector<double>& s,
                                                   Shape& psi,
                                                   DShape& dpsids) const
  {
    // Call the shape functions and derivatives
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]);
    OneDimensionalLegendreDShape<NNODE_1D> dpsi1ds(s[0]);
 
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]);
    // OneDLegendreDShapeParam dpsi1ds(NNODE_1D,s[0]);
 
    // Loop over shape functions in element and assign to psi
    for (unsigned l = 0; l < NNODE_1D; l++)
    {
      psi(l) = psi1[l];
      dpsids(l, 0) = dpsi1ds[l];
    }
  }
 
  //=======================================================================
  /// Second derivatives of shape functions for specific QSpectralElement<1,..>
  ///  d2psids(i,0) = \f$ d^2 \psi_j / d s^2 \f$
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<1, NNODE_1D>::d2shape_local(const Vector<double>& s,
                                                    Shape& psi,
                                                    DShape& dpsids,
                                                    DShape& d2psids) const
  {
    std::ostringstream error_message;
    error_message
      << "\nd2shpe_local currently not implemented for this element\n";
    throw OomphLibError(
      error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
 
    /*  //Call the shape functions and derivatives */
    /*  OneDimensionalShape<NNODE_1D> psi1(s[0]); */
    /*  OneDimensionalDShape<NNODE_1D> dpsi1ds(s[0]); */
    /*  OneDimensionalD2Shape<NNODE_1D> d2psi1ds(s[0]); */
 
    /*  //Loop over shape functions in element and assign to psi */
    /*  for(unsigned l=0;l<NNODE_1D;l++)  */
    /*   { */
    /*    psi[l] = psi1[l]; */
    /*    dpsids[l][0] = dpsi1ds[l]; */
    /*    d2psids[l][0] = d2psi1ds[l]; */
    /*   } */
  }
 
 
  //=======================================================================
  /// General QSpectralElement class specialised to two spatial dimensions
  //=======================================================================
  template<unsigned NNODE_1D>
  class QSpectralElement<2, NNODE_1D> : public virtual SpectralElement,
                                        public virtual QuadElementBase
  {
  private:
    /// Default integration rule: Gaussian integration of same 'order'
    /// as the element
    // This is sort of optimal, because it means that the integration is exact
    // for the shape functions. Can overwrite this in specific element
    // defintion.
    static GaussLobattoLegendre<2, NNODE_1D> integral;
 
  public:
    /// Constructor
    QSpectralElement()
    {
      // There are NNODE_1D^2 nodes in this element
      this->set_n_node(NNODE_1D * NNODE_1D);
      Spectral_order.resize(2, NNODE_1D);
      Nodal_spectral_order.resize(2, NNODE_1D);
      // Set the elemental and nodal dimensions
      this->set_dimension(2);
      // Assign integral pointer
      this->set_integration_scheme(&integral);
      // Calculate the nodal positions for the shape functions
      OneDimensionalLegendreShape<NNODE_1D>::calculate_nodal_positions();
      // OneDLegendreShapeParam::calculate_nodal_positions(NNODE_1D);
    }
 
    /// Min. value of local coordinate
    double s_min() const
    {
      return -1.0;
    }
 
    /// Max. value of local coordinate
    double s_max() const
    {
      return 1.0;
    }
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      return 4;
    }
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      unsigned n_node_1d = nnode_1d();
      Node* nod_pt;
      switch (j)
      {
        case 0:
          nod_pt = this->node_pt(0);
          break;
        case 1:
          nod_pt = this->node_pt(n_node_1d - 1);
          break;
        case 2:
          nod_pt = this->node_pt(n_node_1d * (n_node_1d - 1));
          break;
        case 3:
          nod_pt = this->node_pt(n_node_1d * n_node_1d - 1);
          break;
        default:
          std::ostringstream error_message;
          error_message << "Vertex node number is " << j
                        << " but must be from 0 to 3\n";
 
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
      }
      return nod_pt;
    }
 
 
    /// Get local coordinates of node j in the element; vector sets its own size
    void local_coordinate_of_node(const unsigned& n, Vector<double>& s) const
    {
      s.resize(2);
      unsigned n0 = n % NNODE_1D;
      unsigned n1 = unsigned(double(n) / double(NNODE_1D));
      s[0] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n0);
      s[1] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n1);
    }
 
    /// Get the local fractino of node j in the element
    void local_fraction_of_node(const unsigned& n, Vector<double>& s_fraction)
    {
      this->local_coordinate_of_node(n, s_fraction);
      // Resize
      s_fraction[0] = 0.5 * (s_fraction[0] + 1.0);
      s_fraction[1] = 0.5 * (s_fraction[1] + 1.0);
    }
 
    /// The local one-d fraction is the same
    double local_one_d_fraction_of_node(const unsigned& n1d, const unsigned& i)
    {
      return 0.5 *
             (OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n1d) + 1.0);
    }
 
    /// Calculate the geometric shape functions at local coordinate s
    inline void shape(const Vector<double>& s, Shape& psi) const;
 
    /// Compute the  geometric shape functions and
    /// derivatives w.r.t. local coordinates at local coordinate s
    inline void dshape_local(const Vector<double>& s,
                             Shape& psi,
                             DShape& dpsids) const;
 
    /// Compute the geometric shape functions, derivatives and
    /// second derivatives w.r.t. local coordinates at local coordinate s
    /// d2psids(i,0) = \f$ \partial ^2 \psi_j / \partial s_0^2 \f$
    /// d2psids(i,1) = \f$ \partial ^2 \psi_j / \partial s_1^2 \f$
    /// d2psids(i,2) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_1 \f$
    inline void d2shape_local(const Vector<double>& s,
                              Shape& psi,
                              DShape& dpsids,
                              DShape& d2psids) const;
 
    /// Overload the template-free interface for the calculation of
    /// inverse jacobian matrix. This is a one-dimensional element, so
    /// use the 1D version.
    double invert_jacobian_mapping(const DenseMatrix<double>& jacobian,
                                   DenseMatrix<double>& inverse_jacobian) const
    {
      return FiniteElement::invert_jacobian<2>(jacobian, inverse_jacobian);
    }
 
 
    /// Number of nodes along each element edge
    unsigned nnode_1d() const
    {
      return NNODE_1D;
    }
 
    /// C-style output
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C_style output at n_plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
    /// Output
    void output(std::ostream& outfile);
 
    /// Output at n_plot points
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    ///  Get cector of local coordinates of plot point i (when plotting
    /// nplot points in each "coordinate direction).
    void get_s_plot(
      const unsigned& i,
      const unsigned& nplot,
      Vector<double>& s,
      const bool& use_equally_spaced_interior_sample_points = false) const
    {
      if (nplot > 1)
      {
        unsigned i0 = i % nplot;
        unsigned i1 = (i - i0) / nplot;
 
        s[0] = -1.0 + 2.0 * double(i0) / double(nplot - 1);
        s[1] = -1.0 + 2.0 * double(i1) / double(nplot - 1);
        if (use_equally_spaced_interior_sample_points)
        {
          double range = 2.0;
          double dx_new = range / double(nplot);
          double range_new = double(nplot - 1) * dx_new;
          s[0] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[0]) / range;
          s[1] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[1]) / range;
        }
      }
      else
      {
        s[0] = 0.0;
        s[1] = 0.0;
      }
    }
 
 
    /// Return string for tecplot zone header (when plotting
    /// nplot points in each "coordinate direction)
    std::string tecplot_zone_string(const unsigned& nplot) const
    {
      std::ostringstream header;
      header << "ZONE I=" << nplot << ", J=" << nplot << "\n";
      return header.str();
    }
 
    /// Return total number of plot points (when plotting
    /// nplot points in each "coordinate direction)
    unsigned nplot_points(const unsigned& nplot) const
    {
      unsigned DIM = 2;
      unsigned np = 1;
      for (unsigned i = 0; i < DIM; i++)
      {
        np *= nplot;
      }
      return np;
    }
 
    /// Build the lower-dimensional FaceElement (an element of type
    /// QSpectralElement<1,NNODE_1D>). The face index takes four values
    /// corresponding to the four possible faces:
    /// -1 (West)  s[0] = -1.0
    /// +1 (East)  s[0] =  1.0
    /// -2 (South) s[1] = -1.0
    /// +2 (North) s[1] =  1.0
    void build_face_element(const int& face_index,
                            FaceElement* face_element_pt);
  };
 
 
  //=======================================================================
  /// Shape function for specific QSpectralElement<2,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<2, NNODE_1D>::shape(const Vector<double>& s,
                                            Shape& psi) const
  {
    // Call the OneDimensional Shape functions
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]), psi2(s[1]);
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]), psi2(NNODE_1D,s[1]);
 
    // Now let's loop over the nodal points in the element
    // and copy the values back in
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        psi(NNODE_1D * i + j) = psi2[i] * psi1[j];
      }
    }
  }
 
  //=======================================================================
  /// Derivatives of shape functions for specific  QSpectralElement<2,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<2, NNODE_1D>::dshape_local(const Vector<double>& s,
                                                   Shape& psi,
                                                   DShape& dpsids) const
  {
    // Call the shape functions and derivatives
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]), psi2(s[1]);
    OneDimensionalLegendreDShape<NNODE_1D> dpsi1ds(s[0]), dpsi2ds(s[1]);
 
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]), psi2(NNODE_1D,s[1]);
    // OneDLegendreDShapeParam dpsi1ds(NNODE_1D,s[0]), dpsi2ds(NNODE_1D,s[1]);
 
    // Index for the shape functions
    unsigned index = 0;
    // Loop over shape functions in element
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        // Assign the values
        dpsids(index, 0) = psi2[i] * dpsi1ds[j];
        dpsids(index, 1) = dpsi2ds[i] * psi1[j];
        psi[index] = psi2[i] * psi1[j];
        // Increment the index
        ++index;
      }
    }
  }
 
 
  //=======================================================================
  /// Second derivatives of shape functions for specific  QSpectralElement<2,..>
  /// d2psids(i,0) = \f$ \partial ^2 \psi_j / \partial s_0^2 \f$
  /// d2psids(i,1) = \f$ \partial ^2 \psi_j / \partial s_1^2 \f$
  /// d2psids(i,2) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_1 \f$
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<2, NNODE_1D>::d2shape_local(const Vector<double>& s,
                                                    Shape& psi,
                                                    DShape& dpsids,
                                                    DShape& d2psids) const
  {
    std::ostringstream error_message;
    error_message
      << "\nd2shpe_local currently not implemented for this element\n";
    throw OomphLibError(
      error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
 
    /*  //Call the shape functions and derivatives */
    /*  OneDimensionalShape<NNODE_1D> psi1(s[0]); */
    /*  OneDimensionalDShape<NNODE_1D> dpsi1ds(s[0]); */
    /*  OneDimensionalD2Shape<NNODE_1D> d2psi1ds(s[0]); */
 
    /*  //Loop over shape functions in element and assign to psi */
    /*  for(unsigned l=0;l<NNODE_1D;l++)  */
    /*   { */
    /*    psi[l] = psi1[l]; */
    /*    dpsids[l][0] = dpsi1ds[l]; */
    /*    d2psids[l][0] = d2psi1ds[l]; */
    /*   } */
  }
 
 
  //=======================================================================
  /// General QSpectralElement class specialised to three spatial dimensions
  //=======================================================================
  template<unsigned NNODE_1D>
  class QSpectralElement<3, NNODE_1D> : public virtual SpectralElement,
                                        public virtual BrickElementBase
  {
  private:
    /// Default integration rule: Gaussian integration of same 'order'
    /// as the element
    // This is sort of optimal, because it means that the integration is exact
    // for the shape functions. Can overwrite this in specific element
    // defintion.
    static GaussLobattoLegendre<3, NNODE_1D> integral;
 
  public:
    /// Constructor
    QSpectralElement()
    {
      // There are NNODE_1D^3 nodes in this element
      this->set_n_node(NNODE_1D * NNODE_1D * NNODE_1D);
      Spectral_order.resize(3, NNODE_1D);
      Nodal_spectral_order.resize(3, NNODE_1D);
      // Set the elemental and nodal dimensions
      this->set_dimension(3);
      // Assign integral point
      this->set_integration_scheme(&integral);
      // Calculate the nodal positions for the shape functions
      OneDimensionalLegendreShape<NNODE_1D>::calculate_nodal_positions();
      // OneDLegendreShapeParam::calculate_nodal_positions(NNODE_1D);
    }
 
    /// Min. value of local coordinate
    double s_min() const
    {
      return -1.0;
    }
 
    /// Max. value of local coordinate
    double s_max() const
    {
      return 1.0;
    }
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      return 8;
    }
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      unsigned n_node_1d = nnode_1d();
      Node* nod_pt;
      switch (j)
      {
        case 0:
          nod_pt = this->node_pt(0);
          break;
        case 1:
          nod_pt = this->node_pt(n_node_1d - 1);
          break;
        case 2:
          nod_pt = this->node_pt(n_node_1d * (n_node_1d - 1));
          break;
        case 3:
          nod_pt = this->node_pt(n_node_1d * n_node_1d - 1);
          break;
        case 4:
          nod_pt = this->node_pt(n_node_1d * n_node_1d * (n_node_1d - 1));
          break;
        case 5:
          nod_pt = this->node_pt(n_node_1d * n_node_1d * (n_node_1d - 1) +
                                 (n_node_1d - 1));
          break;
        case 6:
          nod_pt = this->node_pt(n_node_1d * n_node_1d * n_node_1d - n_node_1d);
          break;
        case 7:
          nod_pt = this->node_pt(n_node_1d * n_node_1d * n_node_1d - 1);
          break;
        default:
          std::ostringstream error_message;
          error_message << "Vertex node number is " << j
                        << " but must be from 0 to 7\n";
 
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
      }
      return nod_pt;
    }
 
 
    /// Get local coordinates of node j in the element; vector sets its own size
    void local_coordinate_of_node(const unsigned& n, Vector<double>& s) const
    {
      s.resize(3);
      unsigned n0 = n % NNODE_1D;
      unsigned n1 = unsigned(double(n) / double(NNODE_1D)) % NNODE_1D;
      unsigned n2 = unsigned(double(n) / double(NNODE_1D * NNODE_1D));
      s[0] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n0);
      s[1] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n1);
      s[2] = OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n2);
    }
 
    /// Get the local fractino of node j in the element
    void local_fraction_of_node(const unsigned& n, Vector<double>& s_fraction)
    {
      this->local_coordinate_of_node(n, s_fraction);
      // Resize
      s_fraction[0] = 0.5 * (s_fraction[0] + 1.0);
      s_fraction[1] = 0.5 * (s_fraction[1] + 1.0);
      s_fraction[2] = 0.5 * (s_fraction[2] + 1.0);
    }
 
    /// The local one-d fraction is the same
    double local_one_d_fraction_of_node(const unsigned& n1d, const unsigned& i)
    {
      return 0.5 *
             (OneDimensionalLegendreShape<NNODE_1D>::nodal_position(n1d) + 1.0);
    }
 
    /// Calculate the geometric shape functions at local coordinate s
    inline void shape(const Vector<double>& s, Shape& psi) const;
 
    /// Compute the  geometric shape functions and
    /// derivatives w.r.t. local coordinates at local coordinate s
    inline void dshape_local(const Vector<double>& s,
                             Shape& psi,
                             DShape& dpsids) const;
 
    /// Compute the geometric shape functions, derivatives and
    /// second derivatives w.r.t. local coordinates at local coordinate s
    /// d2psids(i,0) = \f$ \partial ^2 \psi_j / \partial s_0^2 \f$
    /// d2psids(i,1) = \f$ \partial ^2 \psi_j / \partial s_1^2 \f$
    /// d2psids(i,2) = \f$ \partial ^2 \psi_j / \partial s_2^2 \f$
    /// d2psids(i,3) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_1 \f$
    /// d2psids(i,4) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_2 \f$
    /// d2psids(i,5) = \f$ \partial ^2 \psi_j / \partial s_1 \partial s_2 \f$
    inline void d2shape_local(const Vector<double>& s,
                              Shape& psi,
                              DShape& dpsids,
                              DShape& d2psids) const;
 
 
    /// Overload the template-free interface for the calculation of
    /// inverse jacobian matrix. This is a one-dimensional element, so
    /// use the 3D version.
    double invert_jacobian_mapping(const DenseMatrix<double>& jacobian,
                                   DenseMatrix<double>& inverse_jacobian) const
    {
      return FiniteElement::invert_jacobian<3>(jacobian, inverse_jacobian);
    }
 
    /// Number of nodes along each element edge
    unsigned nnode_1d() const
    {
      return NNODE_1D;
    }
 
    /// C-style output
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C_style output at n_plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
    /// Output
    void output(std::ostream& outfile);
 
    /// Output at nplot points
    void output(std::ostream& outfile, const unsigned& nplot);
 
    ///  Get cector of local coordinates of plot point i (when plotting
    /// nplot points in each "coordinate direction).
    void get_s_plot(
      const unsigned& i,
      const unsigned& nplot,
      Vector<double>& s,
      const bool& use_equally_spaced_interior_sample_points = false) const
    {
      if (nplot > 1)
      {
        unsigned i01 = i % (nplot * nplot);
        unsigned i0 = i01 % nplot;
        unsigned i1 = (i01 - i0) / nplot;
        unsigned i2 = (i - i01) / (nplot * nplot);
 
        s[0] = -1.0 + 2.0 * double(i0) / double(nplot - 1);
        s[1] = -1.0 + 2.0 * double(i1) / double(nplot - 1);
        s[2] = -1.0 + 2.0 * double(i2) / double(nplot - 1);
        if (use_equally_spaced_interior_sample_points)
        {
          double range = 2.0;
          double dx_new = range / double(nplot);
          double range_new = double(nplot - 1) * dx_new;
          s[0] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[0]) / range;
          s[1] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[1]) / range;
          s[2] = -1.0 + 0.5 * dx_new + range_new * (1.0 + s[2]) / range;
        }
      }
      else
      {
        s[0] = 0.0;
        s[1] = 0.0;
        s[2] = 0.0;
      }
    }
 
 
    /// Return string for tecplot zone header (when plotting
    /// nplot points in each "coordinate direction)
    std::string tecplot_zone_string(const unsigned& nplot) const
    {
      std::ostringstream header;
      header << "ZONE I=" << nplot << ", J=" << nplot << ", K=" << nplot
             << "\n";
      return header.str();
    }
 
    /// Return total number of plot points (when plotting
    /// nplot points in each "coordinate direction)
    unsigned nplot_points(const unsigned& nplot) const
    {
      unsigned DIM = 3;
      unsigned np = 1;
      for (unsigned i = 0; i < DIM; i++)
      {
        np *= nplot;
      }
      return np;
    }
 
    /// Build the lower-dimensional FaceElement (an element of type
    /// QSpectralElement<2,NNODE_1D>). The face index takes six values
    /// corresponding to the six possible faces:
    /// -1 (Left)   s[0] = -1.0
    /// +1 (Right)  s[0] =  1.0
    /// -2 (Down)   s[1] = -1.0
    /// +2 (Up)     s[1] =  1.0
    /// -3 (Back)   s[2] = -1.0
    /// +3 (Front)  s[2] =  1.0
    void build_face_element(const int& face_index,
                            FaceElement* face_element_pt);
  };
 
 
  //=======================================================================
  /// Shape function for specific QSpectralElement<3,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<3, NNODE_1D>::shape(const Vector<double>& s,
                                            Shape& psi) const
  {
    // Call the OneDimensional Shape functions
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]), psi2(s[1]), psi3(s[2]);
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]), psi2(NNODE_1D,s[1]),
    // psi3(NNODE_1D,s[2]);
 
    // Now let's loop over the nodal points in the element
    // and copy the values back in
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        for (unsigned k = 0; k < NNODE_1D; k++)
        {
          psi(NNODE_1D * NNODE_1D * i + NNODE_1D * j + k) =
            psi3[i] * psi2[j] * psi1[k];
        }
      }
    }
  }
 
  //=======================================================================
  /// Derivatives of shape functions for specific  QSpectralElement<3,..>
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<3, NNODE_1D>::dshape_local(const Vector<double>& s,
                                                   Shape& psi,
                                                   DShape& dpsids) const
  {
    // Call the shape functions and derivatives
    // OneDLegendreShapeParam psi1(NNODE_1D,s[0]), psi2(NNODE_1D,s[1]),
    // psi3(NNODE_1D,s[2]);
    // OneDLegendreDShapeParam dpsi1ds(NNODE_1D,s[0]), dpsi2ds(NNODE_1D,s[1]),
    // dpsi3ds(NNODE_1D,s[2]);
    OneDimensionalLegendreShape<NNODE_1D> psi1(s[0]), psi2(s[1]), psi3(s[2]);
    OneDimensionalLegendreDShape<NNODE_1D> dpsi1ds(s[0]), dpsi2ds(s[1]),
      dpsi3ds(s[2]);
 
 
    // Index for the shape functions
    unsigned index = 0;
 
    // Loop over shape functions in element
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        for (unsigned k = 0; k < NNODE_1D; k++)
        {
          // Assign the values
          dpsids(index, 0) = psi3[i] * psi2[j] * dpsi1ds[k];
          dpsids(index, 1) = psi3[i] * dpsi2ds[j] * psi1[k];
          dpsids(index, 2) = dpsi3ds[i] * psi2[j] * psi1[k];
          psi[index] = psi3[i] * psi2[j] * psi1[k];
          // Increment the index
          ++index;
        }
      }
    }
  }
 
 
  //=======================================================================
  /// Second derivatives of shape functions for specific QSpectralElement<3,..>
  /// d2psids(i,0) = \f$ \partial ^2 \psi_j / \partial s_0^2 \f$
  /// d2psids(i,1) = \f$ \partial ^2 \psi_j / \partial s_1^2 \f$
  /// d2psids(i,2) = \f$ \partial ^2 \psi_j / \partial s_2^2 \f$
  /// d2psids(i,3) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_1 \f$
  /// d2psids(i,4) = \f$ \partial ^2 \psi_j / \partial s_0 \partial s_2 \f$
  /// d2psids(i,5) = \f$ \partial ^2 \psi_j / \partial s_1 \partial s_2 \f$
  //=======================================================================
  template<unsigned NNODE_1D>
  void QSpectralElement<3, NNODE_1D>::d2shape_local(const Vector<double>& s,
                                                    Shape& psi,
                                                    DShape& dpsids,
                                                    DShape& d2psids) const
  {
    std::ostringstream error_message;
    error_message
      << "\nd2shpe_local currently not implemented for this element\n";
    throw OomphLibError(
      error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
 
    /*  //Call the shape functions and derivatives */
    /*  OneDimensionalShape<NNODE_1D> psi1(s[0]); */
    /*  OneDimensionalDShape<NNODE_1D> dpsi1ds(s[0]); */
    /*  OneDimensionalD2Shape<NNODE_1D> d2psi1ds(s[0]); */
 
    /*  //Loop over shape functions in element and assign to psi */
    /*  for(unsigned l=0;l<NNODE_1D;l++)  */
    /*   { */
    /*    psi[l] = psi1[l]; */
    /*    dpsids[l][0] = dpsi1ds[l]; */
    /*    d2psids[l][0] = d2psi1ds[l]; */
    /*   } */
  }
 
  //==============================================================
  /// A class that is used to template the refineable Q spectral elements
  /// by dimension. It's really nothing more than a policy class
  //=============================================================
  template<unsigned DIM>
  class RefineableQSpectralElement
  {
  public:
    /// Empty constuctor
    RefineableQSpectralElement() {}
  };
 
} // namespace oomph
 
#endif