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) 2006-2024 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 member functions for generic elements
 
#include <float.h>
 
// oomph-lib includes
#include "elements.h"
#include "timesteppers.h"
#include "integral.h"
#include "shape.h"
#include "oomph_definitions.h"
#include "element_with_external_element.h"
 
namespace oomph
{
  /// Static boolean to suppress warnings about any repeated
  /// data. Defaults to false.
  bool GeneralisedElement::Suppress_warning_about_any_repeated_data = false;
 
 
  /// Static boolean to suppress warnings about repeated internal
  /// data. Defaults to false
  bool GeneralisedElement::Suppress_warning_about_repeated_internal_data =
    false;
 
 
  /// Static boolean to suppress warnings about repeated external
  /// data. Defaults to true
  bool GeneralisedElement::Suppress_warning_about_repeated_external_data = true;
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  //  Functions for generalised elements
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Add a (pointer to an) internal data object to the element and
  /// return the index required to obtain it from the access
  /// function \c internal_data_pt()
  //=======================================================================
  unsigned GeneralisedElement::add_internal_data(Data* const& data_pt,
                                                 const bool& fd)
  {
    // Local cache of numbers of internal and external data
    const unsigned n_internal_data = Ninternal_data;
    const unsigned n_external_data = Nexternal_data;
 
    // Find out whether the data is already stored in the array
 
    // Loop over the number of internals
    // The internal data are stored at the beginning of the array
    for (unsigned i = 0; i < n_internal_data; i++)
    {
      // If the passed pointer is stored in the array
      if (internal_data_pt(i) == data_pt)
      {
#ifdef PARANOID
        if (!Suppress_warning_about_repeated_internal_data)
        {
          oomph_info << std::endl << std::endl;
          oomph_info
            << "---------------------------------------------------------------"
               "--"
            << std::endl
            << "Info: Data is already included in this element's internal "
               "storage."
            << std::endl
            << "It's stored as entry " << i << " and I'm not adding it again."
            << std::endl
            << std::endl
            << "Note: You can suppress this message by recompiling without"
            << "\n      PARANOID or setting the boolean \n"
            << "\n      "
               "GeneralisedElement::Suppress_warning_about_repeated_internal_"
               "data"
            << "\n\n      to true." << std::endl
            << "---------------------------------------------------------------"
               "--"
            << std::endl
            << std::endl;
        }
#endif
        // Return the index to the data object
        return i;
      }
    }
 
    // Allocate new storage for the pointers to data
    Data** new_data_pt = new Data*[n_internal_data + n_external_data + 1];
 
    // Copy the old internal values across to the beginning of the array
    for (unsigned i = 0; i < n_internal_data; i++)
    {
      new_data_pt[i] = Data_pt[i];
    }
 
    // Now add the new value to the end of the internal data
    new_data_pt[n_internal_data] = data_pt;
 
    // Copy the external values across
    for (unsigned i = 0; i < n_external_data; i++)
    {
      new_data_pt[n_internal_data + 1 + i] = Data_pt[n_internal_data + i];
    }
 
    // Delete the storage associated with the previous values
    delete[] Data_pt;
 
    // Set the pointer to the new storage
    Data_pt = new_data_pt;
 
    // Resize the array of boolean flags
    Data_fd.resize(n_internal_data + n_external_data + 1);
    // Shuffle the flags for the external data to the end of the array
    for (unsigned i = n_external_data; i > 0; i--)
    {
      Data_fd[n_internal_data + i] = Data_fd[n_internal_data + i - 1];
    }
    // Now add the new flag to the end of the internal data
    Data_fd[n_internal_data] = fd;
 
    // Increase the number of internals
    ++Ninternal_data;
 
    // Return the final index to the new internal data
    return n_internal_data;
  }
 
  //=======================================================================
  /// Add the contents of the queue global_eqn_numbers to
  /// the local storage for the equation numbers, which represents the
  /// local-to-global translation scheme. It is essential that the entries
  /// are added in order, i.e. from the front.
  //=======================================================================
  void GeneralisedElement::add_global_eqn_numbers(
    std::deque<unsigned long> const& global_eqn_numbers,
    std::deque<double*> const& global_dof_pt)
  {
    // Find the number of dofs
    const unsigned n_dof = Ndof;
    // Find the number of additional dofs
    const unsigned n_additional_dof = global_eqn_numbers.size();
    // If there are none, return immediately
    if (n_additional_dof == 0)
    {
      return;
    }
 
    // Find the new total number of equation numbers
    const unsigned new_n_dof = n_dof + n_additional_dof;
    // Create storage for all equations, initialised to NULL
    unsigned long* new_eqn_number = new unsigned long[new_n_dof];
 
    // Copy over the existing values to the start new storage
    for (unsigned i = 0; i < n_dof; i++)
    {
      new_eqn_number[i] = Eqn_number[i];
    }
 
    // Set an index to the next position in the new storage
    unsigned index = n_dof;
    // Loop over the queue and add it's entries to our new storage
    for (std::deque<unsigned long>::const_iterator it =
           global_eqn_numbers.begin();
         it != global_eqn_numbers.end();
         ++it)
    {
      // Add the value to the storage
      new_eqn_number[index] = *it;
      // Increase the array index
      ++index;
    }
 
 
    // If a non-empty dof deque has been passed then do stuff
    const unsigned n_additional_dof_pt = global_dof_pt.size();
    if (n_additional_dof_pt > 0)
    {
// If it's size is not the same as the equation numbers complain
#ifdef PARANOID
      if (n_additional_dof_pt != n_additional_dof)
      {
        std::ostringstream error_stream;
        error_stream
          << "global_dof_pt is non-empty, yet it does not have the same size\n"
          << "as global_eqn_numbers.\n"
          << "There are " << n_additional_dof << " equation numbers,\n"
          << "but " << n_additional_dof_pt << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Create storge for all dofs initialised to NULL
      double** new_dof_pt = new double*[new_n_dof];
      // Copy over the exisiting values to the start of new storage
      for (unsigned i = 0; i < n_dof; i++)
      {
        new_dof_pt[i] = Dof_pt[i];
      }
 
      // Set an index to the next position in the new storage
      unsigned index = n_dof;
      // Loop over the queue and add it's entries to our new storage
      for (std::deque<double*>::const_iterator it = global_dof_pt.begin();
           it != global_dof_pt.end();
           ++it)
      {
        // Add the value to the storage
        new_dof_pt[index] = *it;
        // Increase the array index
        ++index;
      }
 
      // Now delete the old storage
      delete[] Dof_pt;
      // Set the pointer to address the new storage
      Dof_pt = new_dof_pt;
    }
 
    // Now delete the old for the equation numbers storage
    delete[] Eqn_number;
    // Set the pointer to address the new storage
    Eqn_number = new_eqn_number;
    // Finally update the number of degrees of freedom
    Ndof = new_n_dof;
  }
 
  //========================================================================
  /// Empty dense matrix used as a dummy argument to combined
  /// residual and jacobian functions in the case when only the residuals
  /// are being assembled
  //========================================================================
  DenseMatrix<double> GeneralisedElement::Dummy_matrix;
 
  //========================================================================
  /// Static storage used when pointers to the dofs are being assembled by
  /// add_global_eqn_numbers()
  //========================================================================
  std::deque<double*> GeneralisedElement::Dof_pt_deque;
 
 
  //=========================================================================
  /// Default value used as the increment for finite difference calculations
  /// of the jacobian matrices
  //=========================================================================
  double GeneralisedElement::Default_fd_jacobian_step = 1.0e-8;
 
  //==========================================================================
  /// Destructor for generalised elements: Wipe internal data. Pointers
  /// to external data get NULLed  but are not deleted because they
  /// are (generally) shared by lots of elements.
  //==========================================================================
  GeneralisedElement::~GeneralisedElement()
  {
    // Delete each of the objects stored as internal data
    for (unsigned i = Ninternal_data; i > 0; i--)
    {
      // The objects are stored at the beginning of the Data_pt array
      delete Data_pt[i - 1];
      Data_pt[i - 1] = 0;
    }
 
    // Now delete the storage for internal and external data
    delete[] Data_pt;
 
    // Now if we have allocated storage for the local equation for
    // the internal and external data, delete it.
    if (Data_local_eqn)
    {
      delete[] Data_local_eqn[0];
      delete[] Data_local_eqn;
    }
 
    // Delete the storage for the global equation numbers
    delete[] Eqn_number;
  }
 
 
  //=======================================================================
  /// Add a (pointer to an) external data object to the element and
  /// return the index required to obtain it from the access
  /// function \c external_data_pt()
  //=======================================================================
  unsigned GeneralisedElement::add_external_data(Data* const& data_pt,
                                                 const bool& fd)
  {
    // Find the numbers of internal and external data
    const unsigned n_internal_data = Ninternal_data;
    const unsigned n_external_data = Nexternal_data;
    // Find out whether the data is already stored in the array
 
    // Loop over the number of externals
    // The external data are stored at the end of the array Data_pt
    for (unsigned i = 0; i < n_external_data; i++)
    {
      // If the passed pointer is stored in the array
      if (external_data_pt(i) == data_pt)
      {
#ifdef PARANOID
        if (!Suppress_warning_about_repeated_external_data)
        {
          oomph_info << std::endl << std::endl;
          oomph_info
            << "---------------------------------------------------------------"
               "--"
            << std::endl
            << "Info: Data is already included in this element's external "
               "storage."
            << std::endl
            << "It's stored as entry " << i << " and I'm not adding it again"
            << std::endl
            << std::endl
            << "Note: You can suppress this message by recompiling without"
            << "\n      PARANOID or setting the boolean \n"
            << "\n      "
               "GeneralisedElement::Suppress_warning_about_repeated_external_"
               "data"
            << "\n\n      to true." << std::endl
            << "---------------------------------------------------------------"
               "--"
            << std::endl
            << std::endl;
        }
#endif
        // Return the index to the data object
        return i;
      }
    }
 
    // Allocate new storage for the pointers to data
    Data** new_data_pt = new Data*[n_internal_data + n_external_data + 1];
 
    // Copy the old internal and external values across to the new array
    for (unsigned i = 0; i < (n_internal_data + n_external_data); i++)
    {
      new_data_pt[i] = Data_pt[i];
    }
 
    // Add the new data pointer to the end of the array
    new_data_pt[n_internal_data + n_external_data] = data_pt;
 
    // Delete the storage associated with the previous values
    delete[] Data_pt;
 
    // Set the pointer to the new storage
    Data_pt = new_data_pt;
 
    // Resize the array of boolean flags
    Data_fd.resize(n_internal_data + n_external_data + 1);
    // Now add the new flag to the end of the external data
    Data_fd[n_internal_data + n_external_data] = fd;
 
    // Increase the number of externals
    ++Nexternal_data;
 
    // Return the final index to the new external data
    return n_external_data;
  }
 
  //========================================================================
  /// Flush all the external data, i.e. clear the pointers to external
  /// data from the internal storage.
  //========================================================================
  void GeneralisedElement::flush_external_data()
  {
    // Get the numbers of internal and external data
    const unsigned n_external_data = Nexternal_data;
    // If there is external data
    if (n_external_data > 0)
    {
      // Storage for new data, initialised to NULL
      Data** new_data_pt = 0;
 
      // Find the number of internal data
      const unsigned n_internal_data = Ninternal_data;
      // If there is internal data resize Data_pt and copy over
      // the pointers
      if (n_internal_data > 0)
      {
        // The new data pointer should only be the size of the internal data
        new_data_pt = new Data*[n_internal_data];
        // Copy over the internal data only
        for (unsigned i = 0; i < n_internal_data; i++)
        {
          new_data_pt[i] = Data_pt[i];
        }
      }
 
      // Delete the old storage
      delete[] Data_pt;
      // Set the new storage, this will be NULL if there is no internal data
      Data_pt = new_data_pt;
      // Set the number of externals to zero
      Nexternal_data = 0;
 
      // Resize the array of boolean flags to the number of internals
      Data_fd.resize(n_internal_data);
    }
  }
 
 
  //=========================================================================
  /// Remove the object addressed by data_pt from the external data array
  /// Note that this could mess up the numbering of other external data
  //========================================================================
  void GeneralisedElement::flush_external_data(Data* const& data_pt)
  {
    // Get the current numbers of external data
    const unsigned n_external_data = Nexternal_data;
    // Index of the data to be removed
    // We initialise this to be n_external_data, and it will be smaller than
    // n_external_data if the data pointer is found in the array
    unsigned index = n_external_data;
    // Loop over the external data and find the argument
    for (unsigned i = 0; i < n_external_data; i++)
    {
      // If we have found the data pointer, set the index and break
      if (external_data_pt(i) == data_pt)
      {
        index = i;
        break;
      }
    }
 
    // If we have found an index less than Nexternal_data, then we have found
    // the data in the array
    if (index < n_external_data)
    {
      // Initialise a new array to NULL
      Data** new_data_pt = 0;
 
      // Find the number of internals
      const unsigned n_internal_data = Ninternal_data;
 
      // Find the new number of total data items (one fewer)
      const unsigned n_total_data = n_internal_data + n_external_data - 1;
 
      // Create a new array containing the data items, if we have any
      if (n_total_data > 0)
      {
        new_data_pt = new Data*[n_total_data];
      }
 
      // Copy over all the internal data
      for (unsigned i = 0; i < n_internal_data; i++)
      {
        new_data_pt[i] = Data_pt[i];
      }
 
      // Now copy over the un-flushed data
      unsigned counter = 0;
      for (unsigned i = 0; i < n_external_data; i++)
      {
        // If we are not at the deleted index
        if (i != index)
        {
          // Copy the undeleted entry into the next entry in our new
          // array of pointers to Data
          new_data_pt[n_internal_data + counter] = Data_pt[i];
          // Increase the counter
          ++counter;
        }
      }
 
      // Delete the storage associated with the previous values
      delete[] Data_pt;
 
      // Set pointers to the new storage, will be NULL if no data left
      Data_pt = new_data_pt;
 
      // Remove the entry from the array of boolean flags
      Data_fd.erase(Data_fd.begin() + n_internal_data + index);
 
      // Decrease the number of externals
      --Nexternal_data;
 
      // Issue a warning if there will be external data remaining
      if (Nexternal_data > 1)
      {
        std::ostringstream warning_stream;
        warning_stream
          << "Data removed from element's external data   " << std::endl
          << "You may have to update the indices for remaining data "
          << std::endl
          << "This can be achieved by using add_external_data()    "
          << std::endl;
        OomphLibWarning(warning_stream.str(),
                        "GeneralisedElement::flush_external_data()",
                        OOMPH_EXCEPTION_LOCATION);
      }
    }
  }
 
 
  //==========================================================================
  /// This function loops over the internal data of the element and assigns
  /// GLOBAL equation numbers to the data objects.
  ///
  /// Pass:
  /// - the current total number of dofs, global_number, which gets
  ///   incremented.
  /// - Dof_pt, the Vector of pointers to the global dofs
  ///   (to which any internal dofs are added).
  /// .
  //==========================================================================
  void GeneralisedElement::assign_internal_eqn_numbers(
    unsigned long& global_number, Vector<double*>& Dof_pt)
  {
    // Loop over the internal data and assign the equation numbers
    // The internal data are stored at the beginning of the Data_pt array
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->assign_eqn_numbers(global_number, Dof_pt);
    }
  }
 
 
  //==========================================================================
  /// Function to describe the 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(...)
  //==========================================================================
  void GeneralisedElement::describe_dofs(
    std::ostream& out, const std::string& current_string) const
  {
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      std::stringstream conversion;
      conversion << " of Internal Data " << i << current_string;
      std::string in(conversion.str());
      internal_data_pt(i)->describe_dofs(out, in);
    }
  }
 
  //==========================================================================
  /// 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(...)
  //==========================================================================
  void GeneralisedElement::describe_local_dofs(
    std::ostream& out, const std::string& current_string) const
  {
    // Find the number of internal and external data
    const unsigned n_internal_data = Ninternal_data;
    const unsigned n_external_data = Nexternal_data;
 
    // Now loop over the internal data and describe local equation numbers
    for (unsigned i = 0; i < n_internal_data; i++)
    {
      // Pointer to the internal data
      Data* const data_pt = internal_data_pt(i);
 
      std::stringstream conversion;
      conversion << " of Internal Data " << i << current_string;
      std::string in(conversion.str());
      data_pt->describe_dofs(out, in);
    }
 
 
    // Now loop over the external data and assign local equation numbers
    for (unsigned i = 0; i < n_external_data; i++)
    {
      // Pointer to the external data
      Data* const data_pt = external_data_pt(i);
 
      std::stringstream conversion;
      conversion << " of External Data " << i << current_string;
      std::string in(conversion.str());
      data_pt->describe_dofs(out, in);
    }
  }
 
  //==========================================================================
  /// This function loops over the internal data of the element and add
  /// pointers to their unconstrained values to a map indexed by the global
  /// equation number.
  //==========================================================================
  void GeneralisedElement::add_internal_value_pt_to_map(
    std::map<unsigned, double*>& map_of_value_pt)
  {
    // Loop over the internal data and add their data to the map
    // The internal data are stored at the beginning of the Data_pt array
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->add_value_pt_to_map(map_of_value_pt);
    }
  }
 
 
#ifdef OOMPH_HAS_MPI
  //=========================================================================
  /// Add all internal data and time history values to the vector in
  /// the internal storage order
  //=========================================================================
  void GeneralisedElement::add_internal_data_values_to_vector(
    Vector<double>& vector_of_values)
  {
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->add_values_to_vector(vector_of_values);
    }
  }
 
  //========================================================================
  /// Read all internal data and time history values from the vector
  /// starting from index. On return the index will be
  /// set to the value at the end of the data that has been read in
  //========================================================================
  void GeneralisedElement::read_internal_data_values_from_vector(
    const Vector<double>& vector_of_values, unsigned& index)
  {
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->read_values_from_vector(vector_of_values, index);
    }
  }
 
  //=========================================================================
  /// Add all equation numbers associated with internal data
  /// to the vector in the internal storage order
  //=========================================================================
  void GeneralisedElement::add_internal_eqn_numbers_to_vector(
    Vector<long>& vector_of_eqn_numbers)
  {
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->add_eqn_numbers_to_vector(vector_of_eqn_numbers);
    }
  }
 
  //=========================================================================
  ///  Read all equation numbers associated with internal data
  /// from the vector
  /// starting from index. On return the index will be
  /// set to the value at the end of the data that has been read in
  //=========================================================================
  void GeneralisedElement::read_internal_eqn_numbers_from_vector(
    const Vector<long>& vector_of_eqn_numbers, unsigned& index)
  {
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      internal_data_pt(i)->read_eqn_numbers_from_vector(vector_of_eqn_numbers,
                                                        index);
    }
  }
 
#endif
 
 
  //====================================================================
  /// Setup the arrays of local equation numbers for the element.
  /// In general, this function should not need to be overloaded. Instead
  /// the function assign_all_generic_local_eqn_numbers() will be overloaded
  /// by different types of Element.
  /// That said, the function is virtual so that it
  /// may be overloaded by the user if they *really* know what they are doing.
  //==========================================================================
  void GeneralisedElement::assign_local_eqn_numbers(
    const bool& store_local_dof_pt)
  {
    clear_global_eqn_numbers();
    assign_all_generic_local_eqn_numbers(store_local_dof_pt);
    assign_additional_local_eqn_numbers();
 
    // Check that no global equation numbers are repeated
#ifdef PARANOID
 
    std::ostringstream error_stream;
 
    // Loop over the array of equation numbers and add to set to assess
    // uniqueness
    std::map<unsigned, bool> is_repeated;
    std::set<unsigned long> set_of_global_eqn_numbers;
    unsigned old_ndof = 0;
    for (unsigned n = 0; n < Ndof; ++n)
    {
      set_of_global_eqn_numbers.insert(Eqn_number[n]);
      if (set_of_global_eqn_numbers.size() == old_ndof)
      {
        error_stream << "Repeated global eqn: " << Eqn_number[n] << std::endl;
        is_repeated[Eqn_number[n]] = true;
      }
      old_ndof = set_of_global_eqn_numbers.size();
    }
 
    // If the sizes do not match we have a repeat, throw an error
    if (set_of_global_eqn_numbers.size() != Ndof)
    {
#ifdef OOMPH_HAS_MPI
      error_stream << "Element is ";
      if (!is_halo()) error_stream << "not ";
      error_stream << "a halo element\n\n";
#endif
      error_stream << "\nLocal/lobal equation numbers: " << std::endl;
      for (unsigned n = 0; n < Ndof; ++n)
      {
        error_stream << "  " << n << "        " << Eqn_number[n] << std::endl;
      }
 
      // It's helpful for debugging purposes to output more about this element
      error_stream << std::endl << " element address is " << this << std::endl;
 
      // Check if the repeated dofs are among the internal Data values
      unsigned nint = this->ninternal_data();
      error_stream << "Number of internal data " << nint << std::endl;
      for (unsigned i = 0; i < nint; i++)
      {
        Data* data_pt = this->internal_data_pt(i);
        unsigned nval = data_pt->nvalue();
        for (unsigned j = 0; j < nval; j++)
        {
          int eqn_no = data_pt->eqn_number(j);
          error_stream << "Internal dof: " << eqn_no << std::endl;
          if (is_repeated[unsigned(eqn_no)])
          {
            error_stream << "Repeated internal dof: " << eqn_no << std::endl;
          }
        }
      }
 
      // Check if the repeated dofs are among the external Data values
      unsigned next = this->nexternal_data();
      error_stream << "Number of external data " << next << std::endl;
      for (unsigned i = 0; i < next; i++)
      {
        Data* data_pt = this->external_data_pt(i);
        unsigned nval = data_pt->nvalue();
        for (unsigned j = 0; j < nval; j++)
        {
          int eqn_no = data_pt->eqn_number(j);
          error_stream << "External dof: " << eqn_no << std::endl;
          if (is_repeated[unsigned(eqn_no)])
          {
            error_stream << "Repeated external dof: " << eqn_no;
            Node* nod_pt = dynamic_cast<Node*>(data_pt);
            if (nod_pt != 0)
            {
              error_stream << " (is a node at: ";
              unsigned ndim = nod_pt->ndim();
              for (unsigned ii = 0; ii < ndim; ii++)
              {
                error_stream << nod_pt->x(ii) << " ";
              }
            }
            error_stream << ")\n";
          }
        }
      }
 
 
      // If it's an element with external element check the associated
      // Data
      ElementWithExternalElement* e_el_pt =
        dynamic_cast<ElementWithExternalElement*>(this);
      if (e_el_pt != 0)
      {
        // Check if the repeated dofs are among the external Data values
        {
          unsigned next = e_el_pt->nexternal_interaction_field_data();
          error_stream << "Number of external element data " << next
                       << std::endl;
          Vector<Data*> data_pt(e_el_pt->external_interaction_field_data_pt());
          for (unsigned i = 0; i < next; i++)
          {
            unsigned nval = data_pt[i]->nvalue();
            for (unsigned j = 0; j < nval; j++)
            {
              int eqn_no = data_pt[i]->eqn_number(j);
              error_stream << "External element dof: " << eqn_no << std::endl;
              if (is_repeated[unsigned(eqn_no)])
              {
                error_stream << "Repeated external element dof: " << eqn_no;
                Node* nod_pt = dynamic_cast<Node*>(data_pt[i]);
                if (nod_pt != 0)
                {
                  error_stream << " (is a node at: ";
                  unsigned ndim = nod_pt->ndim();
                  for (unsigned ii = 0; ii < ndim; ii++)
                  {
                    error_stream << nod_pt->x(ii) << " ";
                  }
                }
                error_stream << ")\n";
              }
            }
          }
        }
 
 
        // Check if the repeated dofs are among the external geom Data values
        {
          unsigned next = e_el_pt->nexternal_interaction_geometric_data();
          error_stream << "Number of external element geom data " << next
                       << std::endl;
          Vector<Data*> data_pt(
            e_el_pt->external_interaction_geometric_data_pt());
          for (unsigned i = 0; i < next; i++)
          {
            unsigned nval = data_pt[i]->nvalue();
            for (unsigned j = 0; j < nval; j++)
            {
              int eqn_no = data_pt[i]->eqn_number(j);
              error_stream << "External element geometric dof: " << eqn_no
                           << std::endl;
              if (is_repeated[unsigned(eqn_no)])
              {
                error_stream
                  << "Repeated external element geometric dof: " << eqn_no
                  << " " << data_pt[i]->value(j);
                Node* nod_pt = dynamic_cast<Node*>(data_pt[i]);
                if (nod_pt != 0)
                {
                  error_stream << " (is a node at: ";
                  unsigned ndim = nod_pt->ndim();
                  for (unsigned ii = 0; ii < ndim; ii++)
                  {
                    error_stream << nod_pt->x(i) << " ";
                  }
                  error_stream << ")";
                }
                error_stream << "\n";
              }
            }
          }
        }
      }
 
      // If it's a FiniteElement then output its nodes
      FiniteElement* f_el_pt = dynamic_cast<FiniteElement*>(this);
      if (f_el_pt != 0)
      {
        unsigned n_node = f_el_pt->nnode();
        for (unsigned n = 0; n < n_node; n++)
        {
          Node* nod_pt = f_el_pt->node_pt(n);
          unsigned nval = nod_pt->nvalue();
          for (unsigned j = 0; j < nval; j++)
          {
            int eqn_no = nod_pt->eqn_number(j);
            error_stream << "Node " << n << ": Nodal dof: " << eqn_no;
            if (eqn_no >= 0)
            {
              if (is_repeated[unsigned(eqn_no)])
              {
                error_stream << "Node " << n
                             << ": Repeated nodal dof: " << eqn_no;
                if (j >= f_el_pt->required_nvalue(n))
                {
                  error_stream << " (resized)";
                }
                error_stream << std::endl;
              }
            }
          }
          SolidNode* solid_nod_pt = dynamic_cast<SolidNode*>(nod_pt);
          if (solid_nod_pt != 0)
          {
            Data* data_pt = solid_nod_pt->variable_position_pt();
            unsigned nval = data_pt->nvalue();
            for (unsigned j = 0; j < nval; j++)
            {
              int eqn_no = data_pt->eqn_number(j);
              error_stream << "Node " << n << ": Positional dof: " << eqn_no
                           << std::endl;
              if (is_repeated[unsigned(eqn_no)])
              {
                error_stream << "Repeated positional dof: " << eqn_no << " "
                             << data_pt->value(j) << std::endl;
              }
            }
          }
        }
 
        // Output nodal coordinates of element
        n_node = f_el_pt->nnode();
        for (unsigned n = 0; n < n_node; n++)
        {
          Node* nod_pt = f_el_pt->node_pt(n);
          unsigned n_dim = nod_pt->ndim();
          error_stream << "Node " << n << " at ( ";
          for (unsigned i = 0; i < n_dim; i++)
          {
            error_stream << nod_pt->x(i) << " ";
          }
          error_stream << ")" << std::endl;
        }
      }
 
 
      // Shout?
      if (!Suppress_warning_about_any_repeated_data)
      {
        error_stream << std::endl << std::endl;
        error_stream
          << "---------------------------------------------------------------"
             "--"
          << std::endl
          << "Note: You can suppress this warning by recompiling without"
          << "\n      PARANOID or setting the boolean \n"
          << "\n      "
             "GeneralisedElement::Suppress_warning_about_any_repeated_data"
          << "\n\n      to true." << std::endl
          << std::endl
          << "Only do this if you know what you're doing; repeated equation\n"
          << "numbers are usually a sign of trouble...\n"
          << "---------------------------------------------------------------"
             "--"
          << std::endl
          << std::endl;
 
        // Issue warning
        OomphLibWarning(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
#endif
  }
 
 
  //==========================================================================
  /// This function loops over the internal and external data of the element,
  /// adds the GLOBAL equation numbers to the local-to-global look-up scheme and
  /// fills in the look-up schemes for the local equation
  /// numbers.
  /// If the boolean argument is true then pointers to the dofs will be
  /// stored in Dof_pt
  //==========================================================================
  void GeneralisedElement::assign_internal_and_external_local_eqn_numbers(
    const bool& store_local_dof_pt)
  {
    // Find the number of internal and external data
    const unsigned n_internal_data = Ninternal_data;
    const unsigned n_external_data = Nexternal_data;
    // Find the total number of data
    const unsigned n_total_data = n_internal_data + n_external_data;
 
    // If there is data
    if (n_total_data > 0)
    {
      // Find the number of local equations assigned so far
      unsigned local_eqn_number = ndof();
 
      // We need to find the total number of values stored in all the
      // internal and external data
      // Initialise to the number stored in the first data object
      unsigned n_total_values = Data_pt[0]->nvalue();
      // Loop over the other data and add the number of values stored
      for (unsigned i = 1; i < n_total_data; ++i)
      {
        n_total_values += Data_pt[i]->nvalue();
      }
 
      // If allocated delete the old storage
      if (Data_local_eqn)
      {
        delete[] Data_local_eqn[0];
        delete[] Data_local_eqn;
      }
 
      // If there are no values then we are done, null out the storage and
      // return
      if (n_total_values == 0)
      {
        Data_local_eqn = 0;
        return;
      }
 
      // Allocate the storage for the local equation numbers
      // The idea is that we store internal equation numbers followed by
      // external equation numbers
 
      // Firstly allocate pointers to the rows for each data item
      Data_local_eqn = new int*[n_total_data];
      // Now allocate storage for all the equation numbers
      Data_local_eqn[0] = new int[n_total_values];
      // Set all values to be unclassified
      for (unsigned i = 0; i < n_total_values; ++i)
      {
        Data_local_eqn[0][i] = Data::Is_unclassified;
      }
 
      // Loop over the remaining rows and set their pointers
      for (unsigned i = 1; i < n_total_data; ++i)
      {
        // Initially set the pointer to the i-th row to the pointer
        // to the i-1th row
        Data_local_eqn[i] = Data_local_eqn[i - 1];
        // Now increase the row pointer by the number of values
        // stored at the i-1th data object
        Data_local_eqn[i] += Data_pt[i - 1]->nvalue();
      }
 
      // A local queue to store the global equation numbers
      std::deque<unsigned long> global_eqn_number_queue;
 
      // Now loop over the internal data and assign local equation numbers
      for (unsigned i = 0; i < n_internal_data; i++)
      {
        // Pointer to the internal data
        Data* const data_pt = internal_data_pt(i);
        // Find the number of values stored at the internal data
        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 queue
            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 storage scheme
            Data_local_eqn[i][j] = local_eqn_number;
            // Increase the local number
            local_eqn_number++;
          }
          else
          {
            // Set the local scheme to be pinned
            Data_local_eqn[i][j] = Data::Is_pinned;
          }
        }
      } // End of loop over internal data
 
 
      // Now loop over the external data and assign local equation numbers
      for (unsigned i = 0; i < n_external_data; i++)
      {
        // Pointer to the external data
        Data* const data_pt = external_data_pt(i);
        // Find the number of values stored at the external data
        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 queue
            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
            Data_local_eqn[n_internal_data + i][j] = local_eqn_number;
            // Increase the local number
            local_eqn_number++;
          }
          else
          {
            // Set the local scheme to be pinned
            Data_local_eqn[n_internal_data + i][j] = Data::Is_pinned;
          }
        }
      }
 
      // 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);
      }
    }
  }
 
 
  //============================================================================
  /// This function calculates the entries of Jacobian matrix, used in
  /// the Newton method, associated with the internal degrees of freedom.
  /// It does this using finite differences,
  /// rather than an analytical formulation, so can be done in total generality.
  /// If the boolean argument is true, the finite
  /// differencing will be performed for all internal data, irrespective of
  /// the information in Data_fd. The default value (false)
  /// uses the information in Data_fd to selectively difference only
  /// certain data.
  //==========================================================================
  void GeneralisedElement::fill_in_jacobian_from_internal_by_fd(
    Vector<double>& residuals,
    DenseMatrix<double>& jacobian,
    const bool& fd_all_data)
  {
    // Locally cache the number of internal data
    const unsigned n_internal_data = Ninternal_data;
 
    // If there aren't any internal data, then return straight away
    if (n_internal_data == 0)
    {
      return;
    }
 
    // Call the update function to ensure that the element is in
    // a consistent state before finite differencing starts
    update_before_internal_fd();
 
    // Find the number of dofs in the element
    const unsigned n_dof = ndof();
 
    // Create newres vector
    Vector<double> newres(n_dof);
 
    // Integer storage for local unknown
    int local_unknown = 0;
 
    // Use the default finite difference step
    const double fd_step = Default_fd_jacobian_step;
 
    // Loop over the internal data
    for (unsigned i = 0; i < n_internal_data; i++)
    {
      // If we are doing all finite differences or
      // the variable is included in the finite difference loop, do it
      if (fd_all_data || internal_data_fd(i))
      {
        // Get the number of value at the internal data
        const unsigned n_value = internal_data_pt(i)->nvalue();
 
        // Loop over the number of values
        for (unsigned j = 0; j < n_value; j++)
        {
          // Get the local equation number
          local_unknown = internal_local_eqn(i, j);
          // If it's not pinned
          if (local_unknown >= 0)
          {
            // Get a pointer to the value of the internal data
            double* const value_pt = internal_data_pt(i)->value_pt(j);
 
            // Save the old value of the Internal data
            const double old_var = *value_pt;
 
            // Increment the value of the Internal data
            *value_pt += fd_step;
 
            // Now update any dependent variables
            update_in_internal_fd(i);
 
            // Calculate the new residuals
            get_residuals(newres);
 
            // Do finite differences
            for (unsigned m = 0; m < n_dof; m++)
            {
              double sum = (newres[m] - residuals[m]) / fd_step;
              // Stick the entry into the Jacobian matrix
              jacobian(m, local_unknown) = sum;
            }
 
            // Reset the Internal data
            *value_pt = old_var;
 
            // Reset any dependent variables
            reset_in_internal_fd(i);
          }
        }
      } // End of finite-differencing for datum (if block)
    }
 
    // End of finite difference loop
    // Final reset of any dependent data
    reset_after_internal_fd();
  }
 
  //============================================================================
  /// This function calculates the entries of Jacobian matrix, used in
  /// the Newton method, associated with the external degrees of freedom.
  /// It does this using finite differences,
  /// rather than an analytical formulation, so can be done in total generality.
  /// If the boolean argument is true, the finite
  /// differencing will be performed for all internal data, irrespective of
  /// the information in Data_fd. The default value (false)
  /// uses the information in Data_fd to selectively difference only
  /// certain data.
  //==========================================================================
  void GeneralisedElement::fill_in_jacobian_from_external_by_fd(
    Vector<double>& residuals,
    DenseMatrix<double>& jacobian,
    const bool& fd_all_data)
  {
    // Locally cache the number of external data
    const unsigned n_external_data = Nexternal_data;
    // If there aren't any external data, then return straight away
    if (n_external_data == 0)
    {
      return;
    }
 
    // Call the update function to ensure that the element is in
    // a consistent state before finite differencing starts
    update_before_external_fd();
 
    // Find the number of dofs in the element
    const unsigned n_dof = ndof();
 
    // Create newres vector
    Vector<double> newres(n_dof);
 
    // Integer storage for local unknown
    int local_unknown = 0;
 
    // Use the default finite difference step
    const double fd_step = Default_fd_jacobian_step;
 
    // Loop over the external data
    for (unsigned i = 0; i < n_external_data; i++)
    {
      // If we are doing all finite differences or
      // the variable is included in the finite difference loop, do it
      if (fd_all_data || external_data_fd(i))
      {
        // Get the number of value at the external data
        const unsigned n_value = external_data_pt(i)->nvalue();
 
        // Loop over the number of values
        for (unsigned j = 0; j < n_value; j++)
        {
          // Get the local equation number
          local_unknown = external_local_eqn(i, j);
          // If it's not pinned
          if (local_unknown >= 0)
          {
            // Get a pointer to the External data value
            double* const value_pt = external_data_pt(i)->value_pt(j);
 
            // Save the old value of the External data
            const double old_var = *value_pt;
 
            // Increment the value of the External data
            *value_pt += fd_step;
 
            // Now update any dependent variables
            update_in_external_fd(i);
 
            // Calculate the new residuals
            get_residuals(newres);
 
            // Do finite differences
            for (unsigned m = 0; m < n_dof; m++)
            {
              double sum = (newres[m] - residuals[m]) / fd_step;
              // Stick the entry into the Jacobian matrix
              jacobian(m, local_unknown) = sum;
            }
 
            // Reset the External data
            *value_pt = old_var;
 
            // Reset any dependent variables
            reset_in_external_fd(i);
          }
        }
      } // End of finite differencing for datum (if block)
    }
 
    // End of finite difference loop
    // Final reset of any dependent data
    reset_after_external_fd();
  }
 
  //=====================================================================
  /// Add the elemental contribution to the mass matrix
  /// and the residuals vector. Note that
  /// this function will NOT initialise the residuals vector or the mass
  /// matrix. It must be called after the residuals vector and
  /// jacobian matrix have been initialised to zero. The default
  /// is deliberately broken.
  //=====================================================================
  void GeneralisedElement::fill_in_contribution_to_mass_matrix(
    Vector<double>& residuals, DenseMatrix<double>& mass_matrix)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_mass_matrix() has been called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_mass_matrix();\n";
    error_message += "and must calculate the residuals vector and mass matrix ";
    error_message += "without initialising any of their entries.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message += "get_mass_matrix(), which must initialise the entries\n";
    error_message += "of the residuals vector and mass matrix to zero.\n";
 
    throw OomphLibError(
      error_message,
      "GeneralisedElement::fill_in_contribution_to_mass_matrix()",
      OOMPH_EXCEPTION_LOCATION);
  }
 
  //=====================================================================
  /// Add the elemental contribution to the jacobian matrix,
  /// mass matrix and the residuals vector. Note that
  /// this function should NOT initialise any entries.
  /// It must be called after the residuals vector and
  /// matrices have been initialised to zero. The default is deliberately
  /// broken.
  //=====================================================================
  void GeneralisedElement::fill_in_contribution_to_jacobian_and_mass_matrix(
    Vector<double>& residuals,
    DenseMatrix<double>& jacobian,
    DenseMatrix<double>& mass_matrix)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_jacobian_and_mass_matrix() has been ";
    error_message += "called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_jacobian_and_mass_matrix();\n";
    error_message +=
      "and must calculate the residuals vector and mass and jacobian matrices ";
    error_message += "without initialising any of their entries.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_jacobian_and_mass_matrix(), which must initialise the entries\n";
    error_message +=
      "of the residuals vector, jacobian and mass matrix to zero.\n";
 
    throw OomphLibError(
      error_message,
      "GeneralisedElement::fill_in_contribution_to_jacobian_and_mass_matrix()",
      OOMPH_EXCEPTION_LOCATION);
  }
 
 
  //=====================================================================
  /// Add the elemental contribution to the derivatives of
  /// the residuals with respect to a parameter. This function should
  /// NOT initialise any entries and must be called after the entries
  /// have been initialised to zero
  /// The default implementation is deliberately broken
  //========================================================================
  void GeneralisedElement::fill_in_contribution_to_dresiduals_dparameter(
    double* const& parameter_pt, Vector<double>& dres_dparam)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_dresiduals_dparameter() has been ";
    error_message += "called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_dresiduals_dparameter();\n";
    error_message += "and must calculate the derivatives of the residuals "
                     "vector with respect\n";
    error_message += "to the passed parameter ";
    error_message += "without initialising any values.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_dresiduals_dparameter(), which must initialise the entries\n";
    error_message += "of the returned vector to zero.\n";
 
    error_message +=
      "This function is intended for use instead of the default (global) \n";
    error_message +=
      "finite-difference routine when analytic expressions are to be used\n";
    error_message += "in continuation and bifurcation tracking problems.\n\n";
    error_message += "This function is only called when the function\n";
    error_message +=
      "Problem::set_analytic_dparameter() has been called in the driver code\n";
 
    throw OomphLibError(
      error_message,
      "GeneralisedElement::fill_in_contribution_to_dresiduals_dparameter()",
      OOMPH_EXCEPTION_LOCATION);
  }
 
  //======================================================================
  /// Add the elemental contribution to the derivatives of
  /// the elemental Jacobian matrix
  /// and residuals with respect to a parameter. This function should
  /// NOT initialise any entries and must be called after the entries
  /// have been initialised to zero
  /// The default implementation is to use finite differences to calculate
  /// the derivatives.
  //========================================================================
  void GeneralisedElement::fill_in_contribution_to_djacobian_dparameter(
    double* const& parameter_pt,
    Vector<double>& dres_dparam,
    DenseMatrix<double>& djac_dparam)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_djacobian_dparameter() has been ";
    error_message += "called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_djacobian_dparameter();\n";
    error_message +=
      "and must calculate the derivatives of residuals vector and jacobian ";
    error_message += "matrix\n";
    error_message += "with respect to the passed parameter ";
    error_message += "without initialising any values.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_djacobian_dparameter(), which must initialise the entries\n";
    error_message += "of the returned vector and matrix to zero.\n\n";
 
    error_message +=
      "This function is intended for use instead of the default (global) \n";
    error_message +=
      "finite-difference routine when analytic expressions are to be used\n";
    error_message += "in continuation and bifurcation tracking problems.\n\n";
    error_message += "This function is only called when the function\n";
    error_message +=
      "Problem::set_analytic_dparameter() has been called in the driver code\n";
 
 
    throw OomphLibError(
      error_message,
      "GeneralisedElement::fill_in_contribution_to_djacobian_dparameter()",
      OOMPH_EXCEPTION_LOCATION);
  }
 
 
  //=====================================================================
  /// Add the elemental contribution to the derivative of the
  /// jacobian matrix,
  /// mass matrix and the residuals vector with respect to a parameter.
  /// Note that
  /// this function should NOT initialise any entries.
  /// It must be called after the residuals vector and
  /// matrices have been initialised to zero. The default is deliberately
  /// broken.
  //=====================================================================
  void GeneralisedElement::
    fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter() "
      "has";
    error_message += " been called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_djacobian_and_dmass_matrix_dparameter();\n";
    error_message +=
      "and must calculate the residuals vector and mass and jacobian matrices ";
    error_message += "without initialising any of their entries.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message += "get_djacobian_and_dmass_matrix_dparameter(), which must "
                     "initialise the\n";
    error_message += "entries of the returned vector and  matrices to zero.\n";
 
 
    error_message +=
      "This function is intended for use instead of the default (global) \n";
    error_message +=
      "finite-difference routine when analytic expressions are to be used\n";
    error_message += "in continuation and bifurcation tracking problems.\n\n";
    error_message += "This function is only called when the function\n";
    error_message +=
      "Problem::set_analytic_dparameter() has been called in the driver code\n";
 
 
    throw OomphLibError(error_message,
                        "GeneralisedElement::fill_in_contribution_to_djacobian_"
                        "and_dmass_matrix_dparameter()",
                        OOMPH_EXCEPTION_LOCATION);
  }
 
 
  //========================================================================
  /// Fill in contribution to the product of the Hessian
  /// (derivative of Jacobian with
  /// respect to all variables) an eigenvector, Y, and
  /// other specified vectors, C
  /// (d(J_{ij})/d u_{k}) Y_{j} C_{k}
  /// At the moment the dof pointer is passed in to enable
  /// easy calculation of finite difference default
  //==========================================================================
  void GeneralisedElement::fill_in_contribution_to_hessian_vector_products(
    Vector<double> const& Y,
    DenseMatrix<double> const& C,
    DenseMatrix<double>& product)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_hessian_vector_products() has been ";
    error_message += "called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_hessian_vector_products(); ";
    error_message += " and must calculate the products \n";
    error_message += "of the hessian matrix with the (global) ";
    error_message += "vectors Y and C\n";
    error_message += "without initialising any values.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_hessian_vector_products(), which must initialise the entries\n";
    error_message += "of the returned vector to zero.\n\n";
 
    error_message +=
      "This function is intended for use instead of the default (global) \n";
    error_message +=
      "finite-difference routine when analytic expressions are to be used.\n";
    error_message += "This function is only called when the function\n";
    error_message += "Problem::set_analytic_hessian_products() has been called "
                     "in the driver code\n";
 
    throw OomphLibError(
      error_message,
      "GeneralisedElement::fill_in_contribution_to_hessian_vector_product()",
      OOMPH_EXCEPTION_LOCATION);
  }
 
  //========================================================================
  /// Fill in the contribution to the inner products between given
  /// pairs of history values
  //==========================================================================
  void GeneralisedElement::fill_in_contribution_to_inner_products(
    Vector<std::pair<unsigned, unsigned>> const& history_index,
    Vector<double>& inner_product)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_inner_products() has been called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_inner_products();\n ";
    error_message += "It must calculate the inner products over the element\n";
    error_message += "of given pairs of history values\n";
    error_message += "without initialision of the output vector.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_inner_products(), which must initialise the entries\n";
    error_message += "of the returned vector to zero.\n\n";
 
    throw OomphLibError(
      error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
  }
 
  //========================================================================
  /// Fill in the contributions to the vectors that when taken
  /// as dot product with other history values give the inner product
  /// over the element
  //==========================================================================
  void GeneralisedElement::fill_in_contribution_to_inner_product_vectors(
    Vector<unsigned> const& history_index,
    Vector<Vector<double>>& inner_product_vector)
  {
    std::string error_message =
      "Empty fill_in_contribution_to_inner_product_vectors() has been "
      "called.\n";
    error_message +=
      "This function is called from the default implementation of\n";
    error_message += "get_inner_product_vectors(); ";
    error_message += " and must calculate the vectors \n";
    error_message += "corresponding to the input history values\n ";
    error_message +=
      "that when multiplied by other vectors of history values\n";
    error_message += "return the appropriate dot products.\n\n";
    error_message += "The output vectors must not be initialised.\n\n";
 
    error_message +=
      "If you do not wish to use these defaults, you must overload\n";
    error_message +=
      "get_inner_products(), which must initialise the entries\n";
    error_message += "of the returned vector to zero.\n\n";
 
    throw OomphLibError(
      error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
  }
 
 
  //==========================================================================
  /// Self-test: Have all internal values been classified as
  /// pinned/unpinned? Return 0 if OK.
  //==========================================================================
  unsigned GeneralisedElement::self_test()
  {
    // Initialise
    bool passed = true;
 
    // Loop over internal Data
    for (unsigned i = 0; i < Ninternal_data; i++)
    {
      if (internal_data_pt(i)->self_test() != 0)
      {
        passed = false;
        oomph_info << "\n ERROR: Failed GeneralisedElement::self_test()!"
                   << std::endl;
        oomph_info << "for internal data object number: " << i << std::endl;
      }
    }
 
    // Loop over external Data
    for (unsigned i = 0; i < Nexternal_data; i++)
    {
      if (external_data_pt(i)->self_test() != 0)
      {
        passed = false;
        oomph_info << "\n ERROR: Failed GeneralisedElement::self_test()!"
                   << std::endl;
        oomph_info << "for external data object number: " << i << std::endl;
      }
    }
 
    // Return verdict
    if (passed)
    {
      return 0;
    }
    else
    {
      return 1;
    }
  }
 
 
  //======================================================================
  /// Helper namespace for tolerances, number of iterations, etc
  /// used in the locate_zeta function in FiniteElement
  //======================================================================
  namespace Locate_zeta_helpers
  {
    /// Convergence tolerance for the newton solver
    double Newton_tolerance = 1.0e-7;
 
    /// Maximum number of newton iterations
    unsigned Max_newton_iterations = 10;
 
    /// Multiplier for (zeta-based) outer radius of element used for
    /// deciding that point is outside element. Set to negative value
    /// to suppress test.
    double Radius_multiplier_for_fast_exit_from_locate_zeta = 1.5;
 
    /// Number of points along one dimension of each element used
    /// to create coordinates within the element in order to see
    /// which has the smallest initial residual (and is therefore used
    /// as the initial guess in the Newton method when locating coordinate)
    unsigned N_local_points = 5;
  } // namespace Locate_zeta_helpers
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  //  Functions for finite elements
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// Return the i-th coordinate at local node n. Do not use
  /// the hanging node representation.
  /// NOTE: Moved to cc file because of a possible compiler bug
  /// in gcc (yes, really!). The move to the cc file avoids inlining
  /// which appears to cause problems (only) when compiled with gcc
  /// and -O3; offensive "illegal read" is in optimised-out section
  /// of code and data that is allegedly illegal is readily readable
  /// (by other means) just before this function is called so I can't
  /// really see how we could possibly be responsible for this...
  //======================================================================
  double FiniteElement::raw_nodal_position(const unsigned& n,
                                           const unsigned& i) const
  {
    /* oomph_info << "n: "<< n << std::endl; */
    /* oomph_info << "i: "<< i << std::endl; */
    /* oomph_info << "node_pt(n): "<< node_pt(n) << std::endl; */
    /* oomph_info << "node_pt(n)->x(i): "<< node_pt(n)->x(i) << std::endl; */
    // double tmp=node_pt(n)->x(i);
    // oomph_info << "tmp: "<< tmp << std::endl;
    return node_pt(n)->x(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(...)
  //======================================================================
  void FiniteElement::describe_local_dofs(
    std::ostream& out, const std::string& current_string) const
  {
    // Call the standard finite element classification.
    GeneralisedElement::describe_local_dofs(out, current_string);
    describe_nodal_local_dofs(out, current_string);
  }
 
  //======================================================================
  // 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(...)
  //======================================================================
  void FiniteElement::describe_nodal_local_dofs(
    std::ostream& out, const std::string& current_string) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    for (unsigned n = 0; n < n_node; n++)
    {
      // Pointer to node
      Node* const nod_pt = node_pt(n);
 
      std::stringstream conversion;
      conversion << " of Node " << n << current_string;
      std::string in(conversion.str());
      nod_pt->describe_dofs(out, in);
    } // End if for n_node
  } // End describe_nodal_local_dofs
 
  //========================================================================
  /// Internal function used to check for singular or negative values
  /// of the determinant of the Jacobian of the mapping between local and
  /// global or lagrangian coordinates. Negative jacobians are allowed if the
  /// Accept_negative_jacobian flag is set to true.
  //========================================================================
  void FiniteElement::check_jacobian(const double& jacobian) const
  {
    // First check for a zero jacobian
    if (std::fabs(jacobian) < Tolerance_for_singular_jacobian)
    {
      if (FiniteElement::Suppress_output_while_checking_for_inverted_elements)
      {
        throw OomphLibQuietException();
      }
      else
      {
        std::ostringstream error_stream;
        error_stream
          << "Determinant of Jacobian matrix is zero --- "
          << "singular mapping!\nThe determinant of the "
          << "jacobian is " << std::fabs(jacobian)
          << " which is smaller than the treshold "
          << Tolerance_for_singular_jacobian << "\n"
          << "You can change this treshold, by specifying "
          << "FiniteElement::Tolerance_for_singular_jacobian \n"
          << "Here are the nodal coordinates of the inverted element\n"
          << "in the form \n\n       x,y[,z], hang_status\n\n"
          << "where hang_status = 1 or 2 for non-hanging or hanging\n"
          << "nodes respectively (useful for automatic sizing of\n"
          << "tecplot markers to identify the hanging nodes). \n\n";
        unsigned n_dim_nod = node_pt(0)->ndim();
        unsigned n_nod = nnode();
        unsigned hang_count = 0;
        for (unsigned j = 0; j < n_nod; j++)
        {
          for (unsigned i = 0; i < n_dim_nod; i++)
          {
            error_stream << node_pt(j)->x(i) << " ";
          }
          if (node_pt(j)->is_hanging())
          {
            error_stream << " 2";
            hang_count++;
          }
          else
          {
            error_stream << " 1";
          }
          error_stream << std::endl;
        }
        error_stream << std::endl << std::endl;
        if ((Macro_elem_pt != 0) && (0 != hang_count))
        {
          error_stream
            << "NOTE:  Offending element is associated with a MacroElement\n"
            << "       AND the element has hanging nodes! \n"
            << "       If an element is thin and highly curved, the \n"
            << "       constraints imposed by\n \n"
            << "       (1) inter-element continuity (imposed by the hanging\n"
            << "           node constraints) and \n\n"
            << "       (2) the need to respect curvilinear domain boundaries\n"
            << "           during mesh refinement (imposed by the element's\n"
            << "           macro element mapping)\n\n"
            << "       may be irreconcilable!  \n \n"
            << "       You may have to re-design your base mesh to avoid \n"
            << "       the creation of thin, highly curved elements during\n"
            << "       the refinement process.\n"
            << std::endl;
        }
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
 
 
    // Now check for negative jacobians, if we're not allowing them (default)
    if ((Accept_negative_jacobian == false) && (jacobian < 0.0))
    {
      if (FiniteElement::Suppress_output_while_checking_for_inverted_elements)
      {
        throw OomphLibQuietException();
      }
      else
      {
        std::ostringstream error_stream;
        error_stream << "Negative Jacobian in transform from "
                     << "local to global coordinates" << std::endl
                     << "       You have an inverted coordinate system!"
                     << std::endl
                     << std::endl;
        error_stream
          << "Here are the nodal coordinates of the inverted element\n"
          << "in the form \n\n       x,y[,z], hang_status\n\n"
          << "where hang_status = 1 or 2 for non-hanging or hanging\n"
          << "nodes respectively (useful for automatic sizing of\n"
          << "tecplot markers to identify the hanging nodes). \n\n";
        unsigned n_dim_nod = node_pt(0)->ndim();
        unsigned n_nod = nnode();
        unsigned hang_count = 0;
        for (unsigned j = 0; j < n_nod; j++)
        {
          for (unsigned i = 0; i < n_dim_nod; i++)
          {
            error_stream << node_pt(j)->x(i) << " ";
          }
          if (node_pt(j)->is_hanging())
          {
            error_stream << " 2";
            hang_count++;
          }
          else
          {
            error_stream << " 1";
          }
          error_stream << std::endl;
        }
        error_stream << std::endl << std::endl;
        if ((Macro_elem_pt != 0) && (0 != hang_count))
        {
          error_stream
            << "NOTE: The inverted element is associated with a MacroElement\n"
            << "       AND the element has hanging nodes! \n"
            << "       If an element is thin and highly curved, the \n"
            << "       constraints imposed by\n \n"
            << "       (1) inter-element continuity (imposed by the hanging\n"
            << "           node constraints) and \n\n"
            << "       (2) the need to respect curvilinear domain boundaries\n"
            << "           during mesh refinement (imposed by the element's\n"
            << "           macro element mapping)\n\n"
            << "       may be irreconcilable!  \n \n"
            << "       You may have to re-design your base mesh to avoid \n"
            << "       the creation of thin, highly curved elements during\n"
            << "       the refinement process.\n"
            << std::endl;
        }
 
        error_stream
          << std::endl
          << std::endl
          << "If you believe that inverted elements do not cause any\n"
          << "problems in your specific application you can \n "
          << "suppress this test by: " << std::endl
          << "  i) setting the (static) flag "
          << "FiniteElement::Accept_negative_jacobian to be true" << std::endl;
        error_stream << " ii) switching OFF the PARANOID flag" << std::endl
                     << std::endl;
 
        /// Throw an inverted error so it can be caught by the adaptive
        /// timestepper and arc length continuation for example.
        throw InvertedElementError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
  }
 
  //=========================================================================
  /// Internal function that is used to assemble the jacobian of the mapping
  /// from local coordinates (s) to the eulerian coordinates (x), given the
  /// derivatives of the shape functions. The entire jacobian matrix is
  /// constructed and this function will only work if there are the same number
  /// of local coordinates as global coordinates (i.e. for "bulk" elements).
  //=========================================================================
  void FiniteElement::assemble_local_to_eulerian_jacobian(
    const DShape& dpsids, DenseMatrix<double>& jacobian) const
  {
    // Locally cache the elemental dimension
    const unsigned el_dim = dim();
    // The number of shape functions must be equal to the number
    // of nodes (by definition)
    const unsigned n_shape = nnode();
    // The number of shape function types must be equal to the number
    // of nodal position types (by definition)
    const unsigned n_shape_type = nnodal_position_type();
 
#ifdef PARANOID
    // Check for dimensional compatibility
    if (Elemental_dimension != Nodal_dimension)
    {
      std::ostringstream error_message;
      error_message << "Dimension mismatch" << std::endl;
      error_message << "The elemental dimension: " << Elemental_dimension
                    << " must equal the nodal dimension: " << Nodal_dimension
                    << " for the jacobian of the mapping to be well-defined"
                    << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
 
    // Loop over the rows of the jacobian
    for (unsigned i = 0; i < el_dim; i++)
    {
      // Loop over the columns of the jacobian
      for (unsigned j = 0; j < el_dim; j++)
      {
        // Zero the entry
        jacobian(i, j) = 0.0;
        // Loop over the shape functions
        for (unsigned l = 0; l < n_shape; l++)
        {
          for (unsigned k = 0; k < n_shape_type; k++)
          {
            // Jacobian is dx_j/ds_i, which is represented by the sum
            // over the dpsi/ds_i of the nodal points X j
            // Call the Non-hanging version of positions
            // This is overloaded in refineable elements
            jacobian(i, j) += raw_nodal_position_gen(l, k, j) * dpsids(l, k, i);
          }
        }
      }
    }
  }
 
  //=========================================================================
  /// Internal function that is used to assemble the jacobian of second
  /// derivatives of the mapping from local coordinates (s) to the
  /// eulerian coordinates (x), given the second derivatives of the
  /// shape functions.
  //=========================================================================
  void FiniteElement::assemble_local_to_eulerian_jacobian2(
    const DShape& d2psids, DenseMatrix<double>& jacobian2) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Find the number of shape functions and shape functions types
    // Must be equal to the number of nodes and their position types
    // by the definition of the shape function.
    const unsigned n_shape = nnode();
    const unsigned n_shape_type = nnodal_position_type();
    // Find the number of second derivatives
    const unsigned n_row = N2deriv[el_dim];
 
    // Assemble the "jacobian" (d^2 x_j/ds_i^2) for second derivatives of
    // shape functions
    // Loop over the rows (number of second derivatives)
    for (unsigned i = 0; i < n_row; i++)
    {
      // Loop over the columns (element dimension
      for (unsigned j = 0; j < el_dim; j++)
      {
        // Zero the entry
        jacobian2(i, j) = 0.0;
        // Loop over the shape functions
        for (unsigned l = 0; l < n_shape; l++)
        {
          // Loop over the shape function types
          for (unsigned k = 0; k < n_shape_type; k++)
          {
            // Add the terms to the jacobian entry
            // Call the Non-hanging version of positions
            // This is overloaded in refineable elements
            jacobian2(i, j) +=
              raw_nodal_position_gen(l, k, j) * d2psids(l, k, i);
          }
        }
      }
    }
  }
 
  //=====================================================================
  /// Assemble the covariant Eulerian base vectors and return them in the
  /// matrix interpolated_G. The derivatives of the shape functions with
  /// respect to the local coordinate should already have been calculated
  /// before calling this function
  //=====================================================================
  void FiniteElement::assemble_eulerian_base_vectors(
    const DShape& dpsids, DenseMatrix<double>& interpolated_G) const
  {
    // Find the number of nodes and position types
    const unsigned n_node = nnode();
    const unsigned n_position_type = nnodal_position_type();
    // Find the dimension of the node and element
    const unsigned n_dim_node = nodal_dimension();
    const unsigned n_dim_element = dim();
 
    // Loop over the dimensions and compute the entries of the
    // base vector matrix
    for (unsigned i = 0; i < n_dim_element; i++)
    {
      for (unsigned j = 0; j < n_dim_node; j++)
      {
        // Initialise the j-th component of the i-th base vector to zero
        interpolated_G(i, j) = 0.0;
        for (unsigned l = 0; l < n_node; l++)
        {
          for (unsigned k = 0; k < n_position_type; k++)
          {
            interpolated_G(i, j) +=
              raw_nodal_position_gen(l, k, j) * dpsids(l, k, i);
          }
        }
      }
    }
  }
 
 
  //============================================================================
  /// Zero-d specialisation of function to calculate inverse of jacobian mapping
  //============================================================================
  template<>
  double FiniteElement::invert_jacobian<0>(
    const DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Issue a warning
    oomph_info << "\nWarning: You are trying to invert the jacobian for "
               << "a 'point' element" << std::endl
               << "This makes no sense and is almost certainly an error"
               << std::endl
               << std::endl;
 
    // Dummy return
    return (1.0);
  }
 
 
  //===========================================================================
  /// One-d specialisation of function to calculate inverse of jacobian mapping
  //===========================================================================
  template<>
  double FiniteElement::invert_jacobian<1>(
    const DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Calculate the determinant of the matrix
    const double det = jacobian(0, 0);
 
// Report if Matrix is singular or negative
#ifdef PARANOID
    check_jacobian(det);
#endif
 
    // Calculate the inverse --- trivial in 1D
    inverse_jacobian(0, 0) = 1.0 / jacobian(0, 0);
 
    // Return the determinant
    return (det);
  }
 
  //===========================================================================
  /// Two-d specialisation of function to calculate inverse of jacobian mapping
  //===========================================================================
  template<>
  double FiniteElement::invert_jacobian<2>(
    const DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Calculate the determinant of the matrix
    const double det =
      jacobian(0, 0) * jacobian(1, 1) - jacobian(0, 1) * jacobian(1, 0);
 
// Report if Matrix is singular or negative
#ifdef PARANOID
    check_jacobian(det);
#endif
 
    // Calculate the inverset of the 2x2 matrix
    inverse_jacobian(0, 0) = jacobian(1, 1) / det;
    inverse_jacobian(0, 1) = -jacobian(0, 1) / det;
    inverse_jacobian(1, 0) = -jacobian(1, 0) / det;
    inverse_jacobian(1, 1) = jacobian(0, 0) / det;
 
    // Return the jacobian
    return (det);
  }
 
  //=============================================================================
  /// Three-d specialisation of function to calculate inverse of jacobian
  /// mapping
  //=============================================================================
  template<>
  double FiniteElement::invert_jacobian<3>(
    const DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Calculate the determinant of the matrix
    const double det = jacobian(0, 0) * jacobian(1, 1) * jacobian(2, 2) +
                       jacobian(0, 1) * jacobian(1, 2) * jacobian(2, 0) +
                       jacobian(0, 2) * jacobian(1, 0) * jacobian(2, 1) -
                       jacobian(0, 0) * jacobian(1, 2) * jacobian(2, 1) -
                       jacobian(0, 1) * jacobian(1, 0) * jacobian(2, 2) -
                       jacobian(0, 2) * jacobian(1, 1) * jacobian(2, 0);
 
    // Report if Matrix is singular or negative
#ifdef PARANOID
    check_jacobian(det);
#endif
 
    // Calculate the inverse of the 3x3 matrix
    inverse_jacobian(0, 0) =
      (jacobian(1, 1) * jacobian(2, 2) - jacobian(1, 2) * jacobian(2, 1)) / det;
    inverse_jacobian(0, 1) =
      -(jacobian(0, 1) * jacobian(2, 2) - jacobian(0, 2) * jacobian(2, 1)) /
      det;
    inverse_jacobian(0, 2) =
      (jacobian(0, 1) * jacobian(1, 2) - jacobian(0, 2) * jacobian(1, 1)) / det;
    inverse_jacobian(1, 0) =
      -(jacobian(1, 0) * jacobian(2, 2) - jacobian(1, 2) * jacobian(2, 0)) /
      det;
    inverse_jacobian(1, 1) =
      (jacobian(0, 0) * jacobian(2, 2) - jacobian(0, 2) * jacobian(2, 0)) / det;
    inverse_jacobian(1, 2) =
      -(jacobian(0, 0) * jacobian(1, 2) - jacobian(0, 2) * jacobian(1, 0)) /
      det;
    inverse_jacobian(2, 0) =
      (jacobian(1, 0) * jacobian(2, 1) - jacobian(1, 1) * jacobian(2, 0)) / det;
    inverse_jacobian(2, 1) =
      -(jacobian(0, 0) * jacobian(2, 1) - jacobian(0, 1) * jacobian(2, 0)) /
      det;
    inverse_jacobian(2, 2) =
      (jacobian(0, 0) * jacobian(1, 1) - jacobian(0, 1) * jacobian(1, 0)) / det;
 
    // Return the determinant
    return (det);
  }
 
  //========================================================================
  /// Template-free interface for inversion of the jacobian of a mapping.
  /// This is slightly inefficient, given that it uses a switch statement.
  /// It can always be overloaded in specific geometric elements,
  /// for efficiency reasons.
  //========================================================================
  double FiniteElement::invert_jacobian_mapping(
    const DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Find the spatial dimension of the element
    const unsigned el_dim = dim();
    // Call the appropriate templated function, depending on the
    // element dimension
    switch (el_dim)
    {
      case 0:
        return invert_jacobian<0>(jacobian, inverse_jacobian);
        break;
      case 1:
        return invert_jacobian<1>(jacobian, inverse_jacobian);
        break;
      case 2:
        return invert_jacobian<2>(jacobian, inverse_jacobian);
        break;
      case 3:
        return invert_jacobian<3>(jacobian, inverse_jacobian);
        break;
        // Catch-all default case: issue warning and die
      default:
        std::ostringstream error_stream;
        error_stream << "Dimension of the element must be 0,1,2 or 3, not "
                     << el_dim << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
    // Dummy return for Intel compiler
    return 1.0;
  }
 
  //============================================================================
  /// Zero-d specialisation of function to calculate the derivative of the
  /// jacobian of a mapping with respect to the nodal coordinates X_ij.
  //============================================================================
  template<>
  void FiniteElement::dJ_eulerian_dnodal_coordinates_templated_helper<0>(
    const DenseMatrix<double>& jacobian,
    const DShape& dpsids,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Issue a warning
    oomph_info << "\nWarning: You are trying to calculate derivatives of "
               << "a jacobian w.r.t. nodal coordinates for a 'point' "
               << "element." << std::endl
               << "This makes no sense and is almost certainly an error."
               << std::endl
               << std::endl;
  }
 
  //===========================================================================
  /// One-d specialisation of function to calculate the derivative of the
  /// jacobian of a mapping with respect to the nodal coordinates X_ij.
  //===========================================================================
  template<>
  void FiniteElement::dJ_eulerian_dnodal_coordinates_templated_helper<1>(
    const DenseMatrix<double>& jacobian,
    const DShape& dpsids,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over nodes
    for (unsigned j = 0; j < n_node; j++)
    {
      djacobian_dX(0, j) = dpsids(j, 0);
    }
  }
 
  //===========================================================================
  /// Two-d specialisation of function to calculate the derivative of the
  /// jacobian of a mapping with respect to the nodal coordinates X_ij.
  //===========================================================================
  template<>
  void FiniteElement::dJ_eulerian_dnodal_coordinates_templated_helper<2>(
    const DenseMatrix<double>& jacobian,
    const DShape& dpsids,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over nodes
    for (unsigned j = 0; j < n_node; j++)
    {
      // i=0
      djacobian_dX(0, j) =
        dpsids(j, 0) * jacobian(1, 1) - dpsids(j, 1) * jacobian(0, 1);
 
      // i=1
      djacobian_dX(1, j) =
        dpsids(j, 1) * jacobian(0, 0) - dpsids(j, 0) * jacobian(1, 0);
    }
  }
 
  //=============================================================================
  /// Three-d specialisation of function to calculate the derivative of the
  /// jacobian of a mapping with respect to the nodal coordinates X_ij.
  //=============================================================================
  template<>
  void FiniteElement::dJ_eulerian_dnodal_coordinates_templated_helper<3>(
    const DenseMatrix<double>& jacobian,
    const DShape& dpsids,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over nodes
    for (unsigned j = 0; j < n_node; j++)
    {
      // i=0
      djacobian_dX(0, j) =
        dpsids(j, 0) *
          (jacobian(1, 1) * jacobian(2, 2) - jacobian(1, 2) * jacobian(2, 1)) +
        dpsids(j, 1) *
          (jacobian(0, 2) * jacobian(2, 1) - jacobian(0, 1) * jacobian(2, 2)) +
        dpsids(j, 2) *
          (jacobian(0, 1) * jacobian(1, 2) - jacobian(0, 2) * jacobian(1, 1));
 
      // i=1
      djacobian_dX(1, j) =
        dpsids(j, 0) *
          (jacobian(1, 2) * jacobian(2, 0) - jacobian(1, 0) * jacobian(2, 2)) +
        dpsids(j, 1) *
          (jacobian(0, 0) * jacobian(2, 2) - jacobian(0, 2) * jacobian(2, 0)) +
        dpsids(j, 2) *
          (jacobian(0, 2) * jacobian(1, 0) - jacobian(0, 0) * jacobian(1, 2));
 
      // i=2
      djacobian_dX(2, j) =
        dpsids(j, 0) *
          (jacobian(1, 0) * jacobian(2, 1) - jacobian(1, 1) * jacobian(2, 0)) +
        dpsids(j, 1) *
          (jacobian(0, 1) * jacobian(2, 0) - jacobian(0, 0) * jacobian(2, 1)) +
        dpsids(j, 2) *
          (jacobian(0, 0) * jacobian(1, 1) - jacobian(0, 1) * jacobian(1, 0));
    }
  }
 
  //============================================================================
  /// Zero-d specialisation of function to calculate the derivative w.r.t. the
  /// nodal coordinates \f$ X_{pq} \f$ of the derivative of the shape functions
  /// w.r.t. the global eulerian coordinates \f$ x_i \f$.
  //============================================================================
  template<>
  void FiniteElement::d_dshape_eulerian_dnodal_coordinates_templated_helper<0>(
    const double& det_jacobian,
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& djacobian_dX,
    const DenseMatrix<double>& inverse_jacobian,
    const DShape& dpsids,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Issue a warning
    oomph_info << "\nWarning: You are trying to calculate derivatives of "
               << "eulerian derivatives of shape functions w.r.t. nodal "
               << "coordinates for a 'point' element." << std::endl
               << "This makes no sense and is almost certainly an error."
               << std::endl
               << std::endl;
  }
 
  //===========================================================================
  /// One-d specialisation of function to calculate the derivative w.r.t. the
  /// nodal coordinates \f$ X_{pq} \f$ of the derivative of the shape functions
  /// w.r.t. the global eulerian coordinates \f$ x_i \f$.
  //===========================================================================
  template<>
  void FiniteElement::d_dshape_eulerian_dnodal_coordinates_templated_helper<1>(
    const double& det_jacobian,
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& djacobian_dX,
    const DenseMatrix<double>& inverse_jacobian,
    const DShape& dpsids,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Find inverse of determinant of jacobian of mapping
    const double inv_det_jac = 1.0 / det_jacobian;
 
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over the shape functions
    for (unsigned q = 0; q < n_node; q++)
    {
      // Loop over the shape functions
      for (unsigned j = 0; j < n_node; j++)
      {
        d_dpsidx_dX(0, q, j, 0) =
          -djacobian_dX(0, q) * dpsids(j, 0) * inv_det_jac * inv_det_jac;
      }
    }
  }
 
  //===========================================================================
  /// Two-d specialisation of function to calculate the derivative w.r.t. the
  /// nodal coordinates \f$ X_{pq} \f$ of the derivative of the shape functions
  /// w.r.t. the global eulerian coordinates \f$ x_i \f$.
  //===========================================================================
  template<>
  void FiniteElement::d_dshape_eulerian_dnodal_coordinates_templated_helper<2>(
    const double& det_jacobian,
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& djacobian_dX,
    const DenseMatrix<double>& inverse_jacobian,
    const DShape& dpsids,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Find inverse of determinant of jacobian of mapping
    const double inv_det_jac = 1.0 / det_jacobian;
 
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over the spatial dimension (this must be 2)
    for (unsigned p = 0; p < 2; p++)
    {
      // Loop over the shape functions
      for (unsigned q = 0; q < n_node; q++)
      {
        // Loop over the shape functions
        for (unsigned j = 0; j < n_node; j++)
        {
          // i=0
          d_dpsidx_dX(p, q, j, 0) =
            -djacobian_dX(p, q) * (inverse_jacobian(0, 0) * dpsids(j, 0) +
                                   inverse_jacobian(0, 1) * dpsids(j, 1));
 
          if (p == 1)
          {
            d_dpsidx_dX(p, q, j, 0) +=
              dpsids(j, 0) * dpsids(q, 1) - dpsids(j, 1) * dpsids(q, 0);
          }
          d_dpsidx_dX(p, q, j, 0) *= inv_det_jac;
 
          // i=1
          d_dpsidx_dX(p, q, j, 1) =
            -djacobian_dX(p, q) * (inverse_jacobian(1, 1) * dpsids(j, 1) +
                                   inverse_jacobian(1, 0) * dpsids(j, 0));
 
          if (p == 0)
          {
            d_dpsidx_dX(p, q, j, 1) +=
              dpsids(j, 1) * dpsids(q, 0) - dpsids(j, 0) * dpsids(q, 1);
          }
          d_dpsidx_dX(p, q, j, 1) *= inv_det_jac;
        }
      }
    }
  }
 
  //=============================================================================
  /// Three-d specialisation of function to calculate the derivative w.r.t. the
  /// nodal coordinates \f$ X_{pq} \f$ of the derivative of the shape functions
  /// w.r.t. the global eulerian coordinates \f$ x_i \f$.
  //=============================================================================
  template<>
  void FiniteElement::d_dshape_eulerian_dnodal_coordinates_templated_helper<3>(
    const double& det_jacobian,
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& djacobian_dX,
    const DenseMatrix<double>& inverse_jacobian,
    const DShape& dpsids,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Find inverse of determinant of jacobian of mapping
    const double inv_det_jac = 1.0 / det_jacobian;
 
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Loop over the spatial dimension (this must be 3)
    for (unsigned p = 0; p < 3; p++)
    {
      // Loop over the shape functions
      for (unsigned q = 0; q < n_node; q++)
      {
        // Loop over the shape functions
        for (unsigned j = 0; j < n_node; j++)
        {
          // Terms not multiplied by delta function
          for (unsigned i = 0; i < 3; i++)
          {
            d_dpsidx_dX(p, q, j, i) =
              -djacobian_dX(p, q) * (inverse_jacobian(i, 0) * dpsids(j, 0) +
                                     inverse_jacobian(i, 1) * dpsids(j, 1) +
                                     inverse_jacobian(i, 2) * dpsids(j, 2));
          }
 
          // Delta function terms
          switch (p)
          {
            case 0:
              d_dpsidx_dX(p, q, j, 1) += ((dpsids(q, 2) * jacobian(1, 2) -
                                           dpsids(q, 1) * jacobian(2, 2)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 0) * jacobian(2, 2) -
                                           dpsids(q, 2) * jacobian(0, 2)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 1) * jacobian(0, 2) -
                                           dpsids(q, 0) * jacobian(1, 2)) *
                                            dpsids(j, 2));
 
              d_dpsidx_dX(p, q, j, 2) += ((dpsids(q, 1) * jacobian(2, 1) -
                                           dpsids(q, 2) * jacobian(1, 1)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 2) * jacobian(0, 1) -
                                           dpsids(q, 0) * jacobian(2, 1)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 0) * jacobian(1, 1) -
                                           dpsids(q, 1) * jacobian(0, 1)) *
                                            dpsids(j, 2));
              break;
 
            case 1:
 
              d_dpsidx_dX(p, q, j, 0) += ((dpsids(q, 1) * jacobian(2, 2) -
                                           dpsids(q, 2) * jacobian(1, 2)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 2) * jacobian(0, 2) -
                                           dpsids(q, 0) * jacobian(2, 2)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 0) * jacobian(1, 2) -
                                           dpsids(q, 1) * jacobian(0, 2)) *
                                            dpsids(j, 2));
 
              d_dpsidx_dX(p, q, j, 2) += ((dpsids(q, 2) * jacobian(1, 0) -
                                           dpsids(q, 1) * jacobian(2, 0)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 0) * jacobian(2, 0) -
                                           dpsids(q, 2) * jacobian(0, 0)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 1) * jacobian(0, 0) -
                                           dpsids(q, 0) * jacobian(1, 0)) *
                                            dpsids(j, 2));
              break;
 
            case 2:
 
              d_dpsidx_dX(p, q, j, 0) += ((dpsids(q, 2) * jacobian(1, 1) -
                                           dpsids(q, 1) * jacobian(2, 1)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 0) * jacobian(2, 1) -
                                           dpsids(q, 2) * jacobian(0, 1)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 1) * jacobian(0, 1) -
                                           dpsids(q, 0) * jacobian(1, 1)) *
                                            dpsids(j, 2));
 
              d_dpsidx_dX(p, q, j, 1) += ((dpsids(q, 1) * jacobian(2, 0) -
                                           dpsids(q, 2) * jacobian(1, 0)) *
                                            dpsids(j, 0) +
                                          (dpsids(q, 2) * jacobian(0, 0) -
                                           dpsids(q, 0) * jacobian(2, 0)) *
                                            dpsids(j, 1) +
                                          (dpsids(q, 0) * jacobian(1, 0) -
                                           dpsids(q, 1) * jacobian(0, 0)) *
                                            dpsids(j, 2));
              break;
          }
 
          // Divide through by the determinant of the Jacobian mapping
          for (unsigned i = 0; i < 3; i++)
          {
            d_dpsidx_dX(p, q, j, i) *= inv_det_jac;
          }
        }
      }
    }
  }
 
  //=======================================================================
  /// Default value for the number of values at a node
  //=======================================================================
  const unsigned FiniteElement::Default_Initial_Nvalue = 0;
 
  //======================================================================
  /// Default value that is used for the tolerance required when
  /// locating nodes via local coordinates
  const double FiniteElement::Node_location_tolerance = 1.0e-14;
 
  //=======================================================================
  /// Set the default tolerance for a singular jacobian
  //=======================================================================
  double FiniteElement::Tolerance_for_singular_jacobian = 1.0e-16;
 
  //======================================================================
  /// Set the default value of the Accept_negative_jacobian flag to be
  /// false
  //======================================================================
  bool FiniteElement::Accept_negative_jacobian = false;
 
 
  //======================================================================
  ///  Set default for static boolean to suppress output while checking
  /// for inverted elements
  //======================================================================
  bool FiniteElement::Suppress_output_while_checking_for_inverted_elements =
    false;
 
  //========================================================================
  /// Static array that holds the number of rows in the second derivative
  /// matrix as a function of spatial dimension. In one-dimension, there is
  /// only one possible second derivative. In two-dimensions, there are three,
  /// the two second derivatives and the mixed derivatives. In three
  /// dimensions there are six.
  //=========================================================================
  const unsigned FiniteElement::N2deriv[4] = {0, 1, 3, 6};
 
  //==========================================================================
  /// Calculate the mapping from local to eulerian coordinates
  /// assuming that the coordinates are aligned in the direction of the local
  /// coordinates, i.e. there are no cross terms and the jacobian is diagonal.
  /// The local derivatives are passed as dpsids and the jacobian and
  /// inverse jacobian are returned.
  //==========================================================================
  double FiniteElement::local_to_eulerian_mapping_diagonal(
    const DShape& dpsids,
    DenseMatrix<double>& jacobian,
    DenseMatrix<double>& inverse_jacobian) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Find the number of shape functions and shape functions types
    // Equal to the number of nodes and their position types by definition
    const unsigned n_shape = nnode();
    const unsigned n_shape_type = nnodal_position_type();
 
#ifdef PARANOID
    // Check for dimension compatibility
    if (Elemental_dimension != Nodal_dimension)
    {
      std::ostringstream error_message;
      error_message << "Dimension mismatch" << std::endl;
      error_message << "The elemental dimension: " << Elemental_dimension
                    << " must equal the nodal dimension: " << Nodal_dimension
                    << " for the jacobian of the mapping to be well-defined"
                    << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // In this case we assume that there are no cross terms, that is
    // global coordinate 0 is always in the direction of local coordinate 0
 
    // Loop over the coordinates
    for (unsigned i = 0; i < el_dim; i++)
    {
      // Zero the jacobian and inverse jacobian entries
      for (unsigned j = 0; j < el_dim; j++)
      {
        jacobian(i, j) = 0.0;
        inverse_jacobian(i, j) = 0.0;
      }
 
      // Loop over the shape functions
      for (unsigned l = 0; l < n_shape; l++)
      {
        // Loop over the types of dof
        for (unsigned k = 0; k < n_shape_type; k++)
        {
          // Derivatives are always dx_{i}/ds_{i}
          jacobian(i, i) += raw_nodal_position_gen(l, k, i) * dpsids(l, k, i);
        }
      }
    }
 
    // Now calculate the determinant of the matrix
    double det = 1.0;
    for (unsigned i = 0; i < el_dim; i++)
    {
      det *= jacobian(i, i);
    }
 
// Report if Matrix is singular, or negative
#ifdef PARANOID
    check_jacobian(det);
#endif
 
    // Calculate the inverse mapping (trivial in this case)
    for (unsigned i = 0; i < el_dim; i++)
    {
      inverse_jacobian(i, i) = 1.0 / jacobian(i, i);
    }
 
    // Return the value of the Jacobian
    return (det);
  }
 
  //========================================================================
  /// Template-free interface calculating the derivative of the jacobian
  /// of a mapping with respect to the nodal coordinates X_ij. This is
  /// slightly inefficient, given that it uses a switch statement. It can
  /// always be overloaded in specific geometric elements, for efficiency
  /// reasons.
  //========================================================================
  void FiniteElement::dJ_eulerian_dnodal_coordinates(
    const DenseMatrix<double>& jacobian,
    const DShape& dpsids,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Determine the spatial dimension of the element
    const unsigned el_dim = dim();
 
#ifdef PARANOID
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Check that djacobian_dX has the correct number of rows (= el_dim)
    if (djacobian_dX.nrow() != el_dim)
    {
      std::ostringstream error_message;
      error_message << "djacobian_dX must have the same number of rows as the"
                    << "\nspatial dimension of the element.";
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
    // Check that djacobian_dX has the correct number of columns (= n_node)
    if (djacobian_dX.ncol() != n_node)
    {
      std::ostringstream error_message;
      error_message
        << "djacobian_dX must have the same number of columns as the"
        << "\nnumber of nodes in the element.";
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Call the appropriate templated function, depending on the
    // element dimension
    switch (el_dim)
    {
      case 0:
        dJ_eulerian_dnodal_coordinates_templated_helper<0>(
          jacobian, dpsids, djacobian_dX);
        break;
      case 1:
        dJ_eulerian_dnodal_coordinates_templated_helper<1>(
          jacobian, dpsids, djacobian_dX);
        break;
      case 2:
        dJ_eulerian_dnodal_coordinates_templated_helper<2>(
          jacobian, dpsids, djacobian_dX);
        break;
      case 3:
        dJ_eulerian_dnodal_coordinates_templated_helper<3>(
          jacobian, dpsids, djacobian_dX);
        break;
        // Catch-all default case: issue warning and die
      default:
        std::ostringstream error_stream;
        error_stream << "Dimension of the element must be 0,1,2 or 3, not "
                     << el_dim << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
  }
 
  //========================================================================
  /// Template-free interface calculating the derivative w.r.t. the nodal
  /// coordinates \f$ X_{pq} \f$ of the derivative of the shape functions
  /// \f$ \psi_j \f$ w.r.t. the global eulerian coordinates \f$ x_i \f$.
  /// I.e. this function calculates
  /// \f[ \frac{\partial}{\partial X_{pq}} \left( \frac{\partial \psi_j}{\partial x_i} \right). \f]
  /// To do this it requires the determinant of the jacobian mapping, its
  /// derivative w.r.t. the nodal coordinates \f$ X_{pq} \f$, the inverse
  /// jacobian and the derivatives of the shape functions w.r.t. the local
  /// coordinates. The result is returned as a tensor of rank four.
  ///  Numbering:
  /// d_dpsidx_dX(p,q,j,i) = \f$ \frac{\partial}{\partial X_{pq}} \left( \frac{\partial \psi_j}{\partial x_i} \right) \f$
  /// This function is slightly inefficient, given that it uses a switch
  /// statement. It can always be overloaded in specific geometric elements,
  /// for efficiency reasons.
  //========================================================================
  void FiniteElement::d_dshape_eulerian_dnodal_coordinates(
    const double& det_jacobian,
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& djacobian_dX,
    const DenseMatrix<double>& inverse_jacobian,
    const DShape& dpsids,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Determine the spatial dimension of the element
    const unsigned el_dim = dim();
 
#ifdef PARANOID
    // Determine the number of nodes in the element
    const unsigned n_node = nnode();
 
    // Check that d_dpsidx_dX is of the correct size
    if (d_dpsidx_dX.nindex1() != el_dim || d_dpsidx_dX.nindex2() != n_node ||
        d_dpsidx_dX.nindex3() != n_node || d_dpsidx_dX.nindex4() != el_dim)
    {
      std::ostringstream error_message;
      error_message << "d_dpsidx_dX must be of the following dimensions:"
                    << "\nd_dpsidx_dX(el_dim,n_node,n_node,el_dim)";
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Call the appropriate templated function, depending on the
    // element dimension
    switch (el_dim)
    {
      case 0:
        d_dshape_eulerian_dnodal_coordinates_templated_helper<0>(
          det_jacobian,
          jacobian,
          djacobian_dX,
          inverse_jacobian,
          dpsids,
          d_dpsidx_dX);
        break;
      case 1:
        d_dshape_eulerian_dnodal_coordinates_templated_helper<1>(
          det_jacobian,
          jacobian,
          djacobian_dX,
          inverse_jacobian,
          dpsids,
          d_dpsidx_dX);
        break;
      case 2:
        d_dshape_eulerian_dnodal_coordinates_templated_helper<2>(
          det_jacobian,
          jacobian,
          djacobian_dX,
          inverse_jacobian,
          dpsids,
          d_dpsidx_dX);
        break;
      case 3:
        d_dshape_eulerian_dnodal_coordinates_templated_helper<3>(
          det_jacobian,
          jacobian,
          djacobian_dX,
          inverse_jacobian,
          dpsids,
          d_dpsidx_dX);
        break;
        // Catch-all default case: issue warning and die
      default:
        std::ostringstream error_stream;
        error_stream << "Dimension of the element must be 0,1,2 or 3, not "
                     << el_dim << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
  }
 
  //=====================================================================
  /// Convert derivatives w.r.t local coordinates to derivatives w.r.t the
  /// coordinates used to assemble the inverse jacobian mapping passed as
  /// inverse_jacobian. The derivatives passed in dbasis will be
  /// modified in this function from dbasisds to dbasisdX.
  //======================================================================
  void FiniteElement::transform_derivatives(
    const DenseMatrix<double>& inverse_jacobian, DShape& dbasis) const
  {
    // Find the number of basis functions and basis function types
    const unsigned n_basis = dbasis.nindex1();
    const unsigned n_basis_type = dbasis.nindex2();
    // Find the dimension of the array (Must be the elemental dimension)
    const unsigned n_dim = dim();
 
    // Allocate temporary (stack) storage of the dimension of the element
    double new_derivatives[n_dim];
 
    // Loop over the number of basis functions
    for (unsigned l = 0; l < n_basis; l++)
    {
      // Loop over type of basis functions
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        // Loop over the coordinates
        for (unsigned j = 0; j < n_dim; j++)
        {
          // Zero the new transformed derivatives
          new_derivatives[j] = 0.0;
          // Do premultiplication by inverse jacobian
          for (unsigned i = 0; i < n_dim; i++)
          {
            new_derivatives[j] += inverse_jacobian(j, i) * dbasis(l, k, i);
          }
        }
        // We now copy the new derivatives into the shape functions
        for (unsigned j = 0; j < n_dim; j++)
        {
          dbasis(l, k, j) = new_derivatives[j];
        }
      }
    }
  }
 
  //=====================================================================
  /// Convert derivatives w.r.t local coordinates to derivatives w.r.t the
  /// coordinates used to assemble the inverse jacobian mapping passed as
  /// inverse_jacobian, assuming that the mapping is diagonal. This merely
  /// saves a few loops, but is probably worth it.
  //======================================================================
  void FiniteElement::transform_derivatives_diagonal(
    const DenseMatrix<double>& inverse_jacobian, DShape& dbasis) const
  {
    // Find the number of basis functions and basis function types
    const unsigned n_basis = dbasis.nindex1();
    const unsigned n_basis_type = dbasis.nindex2();
    // Find the dimension of the array (must be the elemental dimension)
    const unsigned n_dim = dim();
 
    // Loop over the number of basis functions
    for (unsigned l = 0; l < n_basis; l++)
    {
      // Loop over type of basis functions
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        // Loop over the coordinates
        for (unsigned j = 0; j < n_dim; j++)
        {
          dbasis(l, k, j) *= inverse_jacobian(j, j);
        }
      }
    }
  }
 
  //=======================================================================
  /// Convert derivatives and second derivatives w.r.t local coordinates to
  /// derivatives w.r.t. the coordinates used to assemble the jacobian,
  /// inverse_jacobian and jacobian 2 passed. This must be specialised
  /// for each dimension, otherwise it gets very ugly
  /// Specialisation to one dimension.
  //=======================================================================
  template<>
  void FiniteElement::transform_second_derivatives_template<1>(
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& inverse_jacobian,
    const DenseMatrix<double>& jacobian2,
    DShape& dbasis,
    DShape& d2basis) const
  {
    // Find the number of basis functions and basis function types
    const unsigned n_basis = dbasis.nindex1();
    const unsigned n_basis_type = dbasis.nindex2();
 
    // The second derivatives are easy, because there can be no mixed terms
    // Loop over number of basis functions
    for (unsigned l = 0; l < n_basis; l++)
    {
      // Loop over number of basis function types
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        d2basis(l, k, 0) =
          d2basis(l, k, 0) / (jacobian(0, 0) * jacobian(0, 0))
          // Second term comes from taking d/dx of (dpsi/ds / dx/ds)
          - dbasis(l, k, 0) * jacobian2(0, 0) /
              (jacobian(0, 0) * jacobian(0, 0) * jacobian(0, 0));
      }
    }
 
    // Assemble the first derivatives (do this last so that we don't
    // overwrite the dphids before we use it in the above)
    transform_derivatives(inverse_jacobian, dbasis);
  }
 
  //=======================================================================
  /// Convert derivatives and second derivatives w.r.t local coordinates to
  /// derivatives w.r.t. the coordinates used to assemble the jacobian,
  /// inverse_jacobian and jacobian 2 passed. This must be specialised
  /// for each dimension, otherwise it gets very ugly.
  /// Specialisation to two spatial dimensions
  //=======================================================================
  template<>
  void FiniteElement::transform_second_derivatives_template<2>(
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& inverse_jacobian,
    const DenseMatrix<double>& jacobian2,
    DShape& dbasis,
    DShape& d2basis) const
  {
    // Find the number of basis functions and basis function types
    const unsigned n_basis = dbasis.nindex1();
    const unsigned n_basis_type = dbasis.nindex2();
 
    // Calculate the determinant
    const double det =
      jacobian(0, 0) * jacobian(1, 1) - jacobian(0, 1) * jacobian(1, 0);
 
    // Second derivatives ... the approach taken here is to construct
    // dphidX/ds which can then be used to calculate the second derivatives
    // using the relationship d/dX = inverse_jacobian*d/ds
 
    double ddetds[2];
 
    ddetds[0] =
      jacobian2(0, 0) * jacobian(1, 1) + jacobian(0, 0) * jacobian2(2, 1) -
      jacobian2(0, 1) * jacobian(1, 0) - jacobian(0, 1) * jacobian2(2, 0);
    ddetds[1] =
      jacobian2(2, 0) * jacobian(1, 1) + jacobian(0, 0) * jacobian2(1, 1) -
      jacobian2(2, 1) * jacobian(1, 0) - jacobian(0, 1) * jacobian2(1, 0);
 
    // Calculate the derivatives of the inverse jacobian wrt the local
    // coordinates
    double dinverse_jacobiands[2][2][2];
 
    dinverse_jacobiands[0][0][0] =
      jacobian2(2, 1) / det - jacobian(1, 1) * ddetds[0] / (det * det);
    dinverse_jacobiands[0][1][0] =
      -jacobian2(0, 1) / det + jacobian(0, 1) * ddetds[0] / (det * det);
    dinverse_jacobiands[1][0][0] =
      -jacobian2(2, 0) / det + jacobian(1, 0) * ddetds[0] / (det * det);
    dinverse_jacobiands[1][1][0] =
      jacobian2(0, 0) / det - jacobian(0, 0) * ddetds[0] / (det * det);
 
    dinverse_jacobiands[0][0][1] =
      jacobian2(1, 1) / det - jacobian(1, 1) * ddetds[1] / (det * det);
    dinverse_jacobiands[0][1][1] =
      -jacobian2(2, 1) / det + jacobian(0, 1) * ddetds[1] / (det * det);
    dinverse_jacobiands[1][0][1] =
      -jacobian2(1, 0) / det + jacobian(1, 0) * ddetds[1] / (det * det);
    dinverse_jacobiands[1][1][1] =
      jacobian2(2, 0) / det - jacobian(0, 0) * ddetds[1] / (det * det);
 
    // Set up derivatives of dpsidx wrt local coordinates
    DShape dphidXds0(n_basis, n_basis_type, 2),
      dphidXds1(n_basis, n_basis_type, 2);
 
    for (unsigned l = 0; l < n_basis; l++)
    {
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        for (unsigned j = 0; j < 2; j++)
        {
          // Note that we can't have an inner loop because of the
          // convention I've chosen for the mixed derivatives!
          dphidXds0(l, k, j) = dinverse_jacobiands[j][0][0] * dbasis(l, k, 0) +
                               dinverse_jacobiands[j][1][0] * dbasis(l, k, 1) +
                               inverse_jacobian(j, 0) * d2basis(l, k, 0) +
                               inverse_jacobian(j, 1) * d2basis(l, k, 2);
 
          dphidXds1(l, k, j) = dinverse_jacobiands[j][0][1] * dbasis(l, k, 0) +
                               dinverse_jacobiands[j][1][1] * dbasis(l, k, 1) +
                               inverse_jacobian(j, 0) * d2basis(l, k, 2) +
                               inverse_jacobian(j, 1) * d2basis(l, k, 1);
        }
      }
    }
 
    // Now calculate the DShape d2phidX
    for (unsigned l = 0; l < n_basis; l++)
    {
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        // Zero dpsidx
        for (unsigned j = 0; j < 3; j++)
        {
          d2basis(l, k, j) = 0.0;
        }
 
        // Do premultiplication by inverse jacobian
        for (unsigned i = 0; i < 2; i++)
        {
          d2basis(l, k, i) = inverse_jacobian(i, 0) * dphidXds0(l, k, i) +
                             inverse_jacobian(i, 1) * dphidXds1(l, k, i);
        }
        // Calculate mixed derivative term
        d2basis(l, k, 2) += inverse_jacobian(0, 0) * dphidXds0(l, k, 1) +
                            inverse_jacobian(0, 1) * dphidXds1(l, k, 1);
      }
    }
 
    // Assemble the first derivatives second, so that we don't
    // overwrite dphids
    transform_derivatives(inverse_jacobian, dbasis);
  }
 
 
  //=======================================================================
  /// Convert derivatives and second derivatives w.r.t local coordinates to
  /// derivatives w.r.t. the coordinates used to assemble the jacobian,
  /// inverse_jacobian and jacobian 2 passed. This must be specialised
  /// for each dimension, otherwise it gets very ugly
  /// Specialisation to one dimension.
  //=======================================================================
  template<>
  void FiniteElement::transform_second_derivatives_diagonal<1>(
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& inverse_jacobian,
    const DenseMatrix<double>& jacobian2,
    DShape& dbasis,
    DShape& d2basis) const
  {
    FiniteElement::transform_second_derivatives_template<1>(
      jacobian, inverse_jacobian, jacobian2, dbasis, d2basis);
  }
 
 
  //=========================================================================
  /// Convert second derivatives w.r.t. local coordinates to
  /// second derivatives w.r.t. the coordinates passed in the tensor
  /// coordinate. Specialised to two spatial dimension
  //=========================================================================
  template<>
  void FiniteElement::transform_second_derivatives_diagonal<2>(
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& inverse_jacobian,
    const DenseMatrix<double>& jacobian2,
    DShape& dbasis,
    DShape& d2basis) const
  {
    // Find the number of basis functions and basis function types
    const unsigned n_basis = dbasis.nindex1();
    const unsigned n_basis_type = dbasis.nindex2();
 
    // Again we assume that there are no cross terms and that  coordinate
    // i depends only upon local coordinate i
 
    // Now calculate the DShape d2phidx
    for (unsigned l = 0; l < n_basis; l++)
    {
      for (unsigned k = 0; k < n_basis_type; k++)
      {
        // Second derivatives
        d2basis(l, k, 0) =
          d2basis(l, k, 0) / (jacobian(0, 0) * jacobian(0, 0)) -
          dbasis(l, k, 0) * jacobian2(0, 0) /
            (jacobian(0, 0) * jacobian(0, 0) * jacobian(0, 0));
 
        d2basis(l, k, 1) =
          d2basis(l, k, 1) / (jacobian(1, 1) * jacobian(1, 1)) -
          dbasis(l, k, 1) * jacobian2(1, 1) /
            (jacobian(1, 1) * jacobian(1, 1) * jacobian(1, 1));
 
        d2basis(l, k, 2) = d2basis(l, k, 2) / (jacobian(0, 0) * jacobian(1, 1));
      }
    }
 
 
    // Assemble the first derivatives
    transform_derivatives_diagonal(inverse_jacobian, dbasis);
  }
 
 
  //=============================================================================
  /// Convert derivatives and second derivatives w.r.t. local coordiantes
  /// to derivatives and second derivatives w.r.t. the coordinates used to
  /// assemble the jacobian, inverse jacobian and jacobian2 passed to the
  /// function. This is a template-free general interface, that should be
  /// overloaded for efficiency
  //============================================================================
  void FiniteElement::transform_second_derivatives(
    const DenseMatrix<double>& jacobian,
    const DenseMatrix<double>& inverse_jacobian,
    const DenseMatrix<double>& jacobian2,
    DShape& dbasis,
    DShape& d2basis) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Choose the appropriate function based on the dimension of the element
    switch (el_dim)
    {
      case 1:
        transform_second_derivatives_template<1>(
          jacobian, inverse_jacobian, jacobian2, dbasis, d2basis);
        break;
      case 2:
        transform_second_derivatives_template<2>(
          jacobian, inverse_jacobian, jacobian2, dbasis, d2basis);
        break;
 
      case 3:
        throw OomphLibError("Not implemented yet ... maybe one day",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
 
        // transform_second_derivatives_template<3>(dphids,d2phids,jacobian,
        //                                         inverse_jacobian,jacobian2,
        //                                         dphidX,d2phidX);
        break;
        // Catch-all default case: issue warning and die
      default:
        std::ostringstream error_stream;
        error_stream << "Dimension of the element must be 1,2 or 3, not "
                     << el_dim << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
  }
 
 
  //======================================================================
  /// The destructor cleans up the memory allocated
  /// for storage of pointers to nodes. Internal and external data get
  /// wiped by the GeneralisedElement destructor; nodes get
  /// killed in mesh destructor.
  //=======================================================================
  FiniteElement::~FiniteElement()
  {
    // Delete the storage allocated for the pointers to the loca nodes
    delete[] Node_pt;
 
    // Delete the storage allocated for the nodal numbering schemes
    if (Nodal_local_eqn)
    {
      delete[] Nodal_local_eqn[0];
      delete[] Nodal_local_eqn;
    }
  }
 
 
  //==============================================================
  /// Get the local fraction of the node j in the element;
  /// vector sets its own size
  //==============================================================
  void FiniteElement::local_fraction_of_node(const unsigned& j,
                                             Vector<double>& s_fraction)
  {
    // Default implementation is rather dumb
    // Get the local coordinate and scale by local coordinate range
    local_coordinate_of_node(j, s_fraction);
    unsigned n_coordinates = s_fraction.size();
    for (unsigned i = 0; i < n_coordinates; i++)
    {
      s_fraction[i] = (s_fraction[i] - s_min()) / (s_max() - s_min());
    }
  }
 
  //=======================================================================
  /// Set the spatial integration scheme and also calculate the values of the
  /// shape functions and their derivatives w.r.t. the local coordinates,
  /// placing the values into storage so that they may be re-used,
  /// without recalculation
  //=======================================================================
  void FiniteElement::set_integration_scheme(Integral* const& integral_pt)
  {
    // Assign the integration scheme
    Integral_pt = integral_pt;
  }
 
  //=========================================================================
  /// Return the shape function stored at the ipt-th integration
  /// point.
  //=========================================================================
  void FiniteElement::shape_at_knot(const unsigned& ipt, Shape& psi) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Storage for the local coordinates of the integration point
    Vector<double> s(el_dim);
    // Set the local coordinate
    for (unsigned i = 0; i < el_dim; i++)
    {
      s[i] = integral_pt()->knot(ipt, i);
    }
    // Get the shape function
    shape(s, psi);
  }
 
  //=========================================================================
  /// Return the shape function and its derivatives w.r.t. the local
  /// coordinates at the ipt-th integration point.
  //=========================================================================
  void FiniteElement::dshape_local_at_knot(const unsigned& ipt,
                                           Shape& psi,
                                           DShape& dpsids) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Storage for the local coordinates of the integration point
    Vector<double> s(el_dim);
    // Set the local coordinate
    for (unsigned i = 0; i < el_dim; i++)
    {
      s[i] = integral_pt()->knot(ipt, i);
    }
    // Get the shape function and derivatives
    dshape_local(s, psi, dpsids);
  }
 
  //=========================================================================
  /// Calculate the shape function and its first and second derivatives
  /// w.r.t. local coordinates at the ipt-th integration point.
  /// Numbering:
  /// \b 1D:
  /// d2psids(i,0) = \f$ d^2 \psi_j / d s^2 \f$
  /// \b 2D:
  /// 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$
  /// \b 3D:
  /// 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$
  //=========================================================================
  void FiniteElement::d2shape_local_at_knot(const unsigned& ipt,
                                            Shape& psi,
                                            DShape& dpsids,
                                            DShape& d2psids) const
  {
    // Find the dimension of the element
    const unsigned el_dim = dim();
    // Storage for the local coordinates of the integration point
    Vector<double> s(el_dim);
    // Set the local coordinate
    for (unsigned i = 0; i < el_dim; i++)
    {
      s[i] = integral_pt()->knot(ipt, i);
    }
    // Get the shape function and first and second derivatives
    d2shape_local(s, psi, dpsids, d2psids);
  }
 
  //=========================================================================
  /// Compute the geometric shape functions and also
  /// first derivatives w.r.t. global coordinates at local coordinate s;
  /// Returns Jacobian of mapping from global to local coordinates.
  /// Most general form of the function, but may be over-loaded, if desired
  //=========================================================================
  double FiniteElement::dshape_eulerian(const Vector<double>& s,
                                        Shape& psi,
                                        DShape& dpsi) const
  {
    // Find the element dimension
    const unsigned el_dim = dim();
 
    // Get the values of the shape functions and their local derivatives
    // Temporarily stored in dpsi
    dshape_local(s, psi, dpsi);
 
    // Allocate memory for the inverse jacobian
    DenseMatrix<double> inverse_jacobian(el_dim);
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to be dpsidx
    transform_derivatives(inverse_jacobian, dpsi);
    // Return the determinant of the jacobian
    return det;
  }
 
  //========================================================================
  /// Compute the geometric shape functions and also first
  /// derivatives w.r.t. global coordinates at integration point ipt.
  /// Most general form of function, but may be over-loaded if desired
  //========================================================================
  double FiniteElement::dshape_eulerian_at_knot(const unsigned& ipt,
                                                Shape& psi,
                                                DShape& dpsi) const
  {
    // Find the element dimension
    const unsigned el_dim = dim();
 
    // Get the values of the shape function and local derivatives
    // Temporarily store it in dpsi
    dshape_local_at_knot(ipt, psi, dpsi);
 
    // Allocate memory for the inverse jacobian
    DenseMatrix<double> inverse_jacobian(el_dim);
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to dpsidx
    transform_derivatives(inverse_jacobian, dpsi);
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //========================================================================
  /// Compute the geometric shape functions (psi) at integration point
  /// ipt. Return the determinant of the jacobian of the mapping (detJ).
  /// Additionally calculate the derivatives of "detJ" w.r.t. the
  /// nodal coordinates.
  //========================================================================
  double FiniteElement::dJ_eulerian_at_knot(
    const unsigned& ipt, Shape& psi, DenseMatrix<double>& djacobian_dX) const
  {
    // Find the element dimension
    const unsigned el_dim = dim();
 
    // Get the values of the shape function and local derivatives
    unsigned nnod = nnode();
    DShape dpsi(nnod, el_dim);
    dshape_local_at_knot(ipt, psi, dpsi);
 
    // Allocate memory for the jacobian and the inverse of the jacobian
    DenseMatrix<double> jacobian(el_dim), inverse_jacobian(el_dim);
 
    // Now calculate the inverse jacobian
    const double det =
      local_to_eulerian_mapping(dpsi, jacobian, inverse_jacobian);
 
    // Calculate the derivative of the jacobian w.r.t. nodal coordinates
    // Note: must call this before "transform_derivatives(...)" since this
    // function requires dpsids rather than dpsidx
    dJ_eulerian_dnodal_coordinates(jacobian, dpsi, djacobian_dX);
 
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //========================================================================
  /// Compute the geometric shape functions (psi) and first
  /// derivatives w.r.t. global coordinates (dpsidx) at integration point
  /// ipt. Return the determinant of the jacobian of the mapping (detJ).
  /// Additionally calculate the derivatives of both "detJ" and "dpsidx"
  /// w.r.t. the nodal coordinates.
  /// Most general form of function, but may be over-loaded if desired.
  //========================================================================
  double FiniteElement::dshape_eulerian_at_knot(
    const unsigned& ipt,
    Shape& psi,
    DShape& dpsi,
    DenseMatrix<double>& djacobian_dX,
    RankFourTensor<double>& d_dpsidx_dX) const
  {
    // Find the element dimension
    const unsigned el_dim = dim();
 
    // Get the values of the shape function and local derivatives
    // Temporarily store in dpsi
    dshape_local_at_knot(ipt, psi, dpsi);
 
    // Allocate memory for the jacobian and the inverse of the jacobian
    DenseMatrix<double> jacobian(el_dim), inverse_jacobian(el_dim);
 
    // Now calculate the inverse jacobian
    const double det =
      local_to_eulerian_mapping(dpsi, jacobian, inverse_jacobian);
 
    // Calculate the derivative of the jacobian w.r.t. nodal coordinates
    // Note: must call this before "transform_derivatives(...)" since this
    // function requires dpsids rather than dpsidx
    dJ_eulerian_dnodal_coordinates(jacobian, dpsi, djacobian_dX);
 
    // Calculate the derivative of dpsidx w.r.t. nodal coordinates
    // Note: this function also requires dpsids rather than dpsidx
    d_dshape_eulerian_dnodal_coordinates(
      det, jacobian, djacobian_dX, inverse_jacobian, dpsi, d_dpsidx_dX);
 
    // Now set the values of the derivatives to dpsidx
    transform_derivatives(inverse_jacobian, dpsi);
 
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //===========================================================================
  /// Compute the geometric shape functions and also first
  /// and second derivatives w.r.t. global coordinates at local coordinate s;
  /// Also returns Jacobian of mapping from global to local coordinates.
  /// Numbering:
  /// \b 1D:
  /// d2psidx(i,0) = \f$ d^2 \psi_j / d x^2 \f$
  /// \b 2D:
  /// d2psidx(i,0) = \f$ \partial^2 \psi_j / \partial x_0^2 \f$
  /// d2psidx(i,1) = \f$ \partial^2 \psi_j / \partial x_1^2 \f$
  /// d2psidx(i,2) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_1 \f$
  /// \b 3D:
  /// d2psidx(i,0) = \f$ \partial^2 \psi_j / \partial x_0^2 \f$
  /// d2psidx(i,1) = \f$ \partial^2 \psi_j / \partial x_1^2 \f$
  /// d2psidx(i,2) = \f$ \partial^2 \psi_j / \partial x_2^2 \f$
  /// d2psidx(i,3) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_1 \f$
  /// d2psidx(i,4) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_2 \f$
  /// d2psidx(i,5) = \f$ \partial^2 \psi_j / \partial x_1 \partial x_2 \f$
  //===========================================================================
  double FiniteElement::d2shape_eulerian(const Vector<double>& s,
                                         Shape& psi,
                                         DShape& dpsi,
                                         DShape& d2psi) const
  {
    // Find the values of the indices of the shape functions
    // Locally cached.
    // Find the element dimension
    const unsigned el_dim = dim();
    // Find the number of second derivatives required
    const unsigned n_deriv = N2deriv[el_dim];
 
    // Get the values of the shape function and local derivatives
    d2shape_local(s, psi, dpsi, d2psi);
 
    // Allocate memory for the jacobian and inverse jacobian
    DenseMatrix<double> jacobian(el_dim), inverse_jacobian(el_dim);
    // Calculate the jacobian and inverse jacobian
    const double det =
      local_to_eulerian_mapping(dpsi, jacobian, inverse_jacobian);
 
    // Allocate memory for the jacobian of second derivatives
    DenseMatrix<double> jacobian2(n_deriv, el_dim);
    // Assemble the jacobian of second derivatives
    assemble_local_to_eulerian_jacobian2(d2psi, jacobian2);
 
    // Now set the value of the derivatives
    transform_second_derivatives(
      jacobian, inverse_jacobian, jacobian2, dpsi, d2psi);
    // Return the determinant of the mapping
    return det;
  }
 
  //===========================================================================
  /// Compute the geometric shape functions and also first
  /// and second derivatives w.r.t. global coordinates at ipt-th integration
  /// point
  /// Returns Jacobian of mapping from global to local coordinates.
  /// This is the most general version, may be overloaded, if desired.
  /// Numbering:
  /// \b 1D:
  /// d2psidx(i,0) = \f$ d^2 \psi_j / d x^2 \f$
  /// \b 2D:
  /// d2psidx(i,0) = \f$ \partial^2 \psi_j / \partial x_0^2 \f$
  /// d2psidx(i,1) = \f$ \partial^2 \psi_j / \partial x_1^2 \f$
  /// d2psidx(i,2) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_1 \f$
  /// \b 3D:
  /// d2psidx(i,0) = \f$ \partial^2 \psi_j / \partial x_0^2 \f$
  /// d2psidx(i,1) = \f$ \partial^2 \psi_j / \partial x_1^2 \f$
  /// d2psidx(i,2) = \f$ \partial^2 \psi_j / \partial x_2^2 \f$
  /// d2psidx(i,3) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_1 \f$
  /// d2psidx(i,4) = \f$ \partial^2 \psi_j / \partial x_0 \partial x_2 \f$
  /// d2psidx(i,5) = \f$ \partial^2 \psi_j / \partial x_1 \partial x_2 \f$
  //==========================================================================
  double FiniteElement::d2shape_eulerian_at_knot(const unsigned& ipt,
                                                 Shape& psi,
                                                 DShape& dpsi,
                                                 DShape& d2psi) const
  {
    // Find the values of the indices of the shape functions
    // Locally cached
    // Find the element dimension
    const unsigned el_dim = dim();
    // Find the number of second derivatives required
    const unsigned n_deriv = N2deriv[el_dim];
 
    // Get the values of the shape function and local derivatives
    d2shape_local_at_knot(ipt, psi, dpsi, d2psi);
 
    // Allocate memory for the jacobian and inverse jacobian
    DenseMatrix<double> jacobian(el_dim), inverse_jacobian(el_dim);
    // Calculate the jacobian and inverse jacobian
    const double det =
      local_to_eulerian_mapping(dpsi, jacobian, inverse_jacobian);
 
    // Allocate memory for the jacobian of second derivatives
    DenseMatrix<double> jacobian2(n_deriv, el_dim);
    // Assemble the jacobian of second derivatives
    assemble_local_to_eulerian_jacobian2(d2psi, jacobian2);
 
    // Now set the value of the derivatives
    transform_second_derivatives(
      jacobian, inverse_jacobian, jacobian2, dpsi, d2psi);
    // Return the determinant of the mapping
    return det;
  }
 
 
  //==========================================================================
  /// This function loops over the nodal data of the element, adds the
  /// GLOBAL equation numbers to the local-to-global look-up scheme and
  /// fills in the Nodal_local_eqn look-up scheme for the local equation
  /// numbers
  /// If the boolean argument is true then pointers to the dofs will be
  /// stored in Dof_pt
  //==========================================================================
  void FiniteElement::assign_nodal_local_eqn_numbers(
    const bool& store_local_dof_pt)
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // If there are nodes
    if (n_node > 0)
    {
      // Find the number of local equations assigned so far
      unsigned local_eqn_number = ndof();
 
      // We need to find the total number of values stored at the node
      // Initialise to the number of values stored at the first node
      unsigned n_total_values = node_pt(0)->nvalue();
      // Loop over the other nodes and add the values stored
      for (unsigned n = 1; n < n_node; n++)
      {
        n_total_values += node_pt(n)->nvalue();
      }
 
      // If allocated delete the old storage
      if (Nodal_local_eqn)
      {
        delete[] Nodal_local_eqn[0];
        delete[] Nodal_local_eqn;
      }
 
      // If there are no values, we are done, null out the storage and
      // return
      if (n_total_values == 0)
      {
        Nodal_local_eqn = 0;
        return;
      }
 
      // Resize the storage for the nodal local equation numbers
      // Firstly allocate pointers to rows for each node
      Nodal_local_eqn = new int*[n_node];
      // Now allocate storage for the equation numbers
      Nodal_local_eqn[0] = new int[n_total_values];
      // initially all local equations are unclassified
      for (unsigned i = 0; i < n_total_values; i++)
      {
        Nodal_local_eqn[0][i] = Data::Is_unclassified;
      }
 
      // Loop over the remaining rows and set their pointers
      for (unsigned n = 1; n < n_node; ++n)
      {
        // Initially set the pointer to the i-th row to the pointer
        // to the i-1th row
        Nodal_local_eqn[n] = Nodal_local_eqn[n - 1];
        // Now increase the row pointer by the number of values
        // stored at the i-1th node
        Nodal_local_eqn[n] += Node_pt[n - 1]->nvalue();
      }
 
 
      // 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_node; n++)
      {
        // Pointer to node
        Node* const nod_pt = node_pt(n);
 
        // Find the number of values stored at the node
        unsigned n_value = nod_pt->nvalue();
 
        // Loop over the number of values
        for (unsigned j = 0; j < n_value; j++)
        {
          // Get the GLOBAL equation number
          long eqn_number = nod_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 queue
            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(nod_pt->value_pt(j));
            }
            // Add the local equation number to the local scheme
            Nodal_local_eqn[n][j] = local_eqn_number;
            // Increase the local number
            local_eqn_number++;
          }
          else
          {
            // Set the local scheme to be pinned
            Nodal_local_eqn[n][j] = Data::Is_pinned;
          }
        }
      }
 
      // 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);
      }
    }
  }
 
 
  //============================================================================
  /// This function calculates the entries of Jacobian matrix, used in
  /// the Newton method, associated with the nodal degrees of freedom.
  /// It does this using finite differences,
  /// rather than an analytical formulation, so can be done in total generality.
  //==========================================================================
  void FiniteElement::fill_in_jacobian_from_nodal_by_fd(
    Vector<double>& residuals, DenseMatrix<double>& jacobian)
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // If there aren't any nodes, then return straight awayy
    if (n_node == 0)
    {
      return;
    }
 
    // Call the update function to ensure that the element is in
    // a consistent state before finite differencing starts
    update_before_nodal_fd();
 
    // Find the number of dofs in the element
    const unsigned n_dof = ndof();
    // Create newres vector
    Vector<double> newres(n_dof);
 
    // Integer storage for local unknown
    int local_unknown = 0;
 
    // Use the default finite difference step
    const double fd_step = Default_fd_jacobian_step;
 
    // Loop over the nodes
    for (unsigned n = 0; n < n_node; n++)
    {
      // Get the number of values stored at the node
      const unsigned n_value = node_pt(n)->nvalue();
 
      // Loop over the number of values
      for (unsigned i = 0; i < n_value; i++)
      {
        // Get the local equation number
        local_unknown = nodal_local_eqn(n, i);
        // If it's not pinned
        if (local_unknown >= 0)
        {
          // Store a pointer to the nodal data value
          double* const value_pt = node_pt(n)->value_pt(i);
 
          // Save the old value of the Nodal data
          const double old_var = *value_pt;
 
          // Increment the value of the Nodal data
          *value_pt += fd_step;
 
          // Now update any dependent variables
          update_in_nodal_fd(i);
 
          // Calculate the new residuals
          get_residuals(newres);
 
          // Do finite differences
          for (unsigned m = 0; m < n_dof; m++)
          {
            double sum = (newres[m] - residuals[m]) / fd_step;
            // Stick the entry into the Jacobian matrix
            jacobian(m, local_unknown) = sum;
          }
 
          // Reset the Nodal data
          *value_pt = old_var;
 
          // Reset any dependent variables
          reset_in_nodal_fd(i);
        }
      }
    }
 
    // End of finite difference loop
    // Final reset of any dependent data
    reset_after_nodal_fd();
  }
 
 
  //=======================================================================
  /// Compute derivatives of elemental residual vector with respect
  /// to nodal coordinates. Default implementation by FD can be overwritten
  /// for specific elements.
  /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
  /// /=======================================================================
  void FiniteElement::get_dresidual_dnodal_coordinates(
    RankThreeTensor<double>& dresidual_dnodal_coordinates)
  {
    // Number of nodes
    unsigned n_nod = nnode();
 
    // If the element has no nodes (why??!!) return straightaway
    if (n_nod == 0) return;
 
    // Get dimension from first node
    unsigned dim_nod = node_pt(0)->ndim();
 
    // Number of dofs
    unsigned n_dof = ndof();
 
    // Get reference residual
    Vector<double> res(n_dof);
    Vector<double> res_pls(n_dof);
    get_residuals(res);
 
    // FD step
    double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
 
    // Do FD loop
    for (unsigned j = 0; j < n_nod; j++)
    {
      // Get node
      Node* nod_pt = node_pt(j);
 
      // Loop over coordinate directions
      for (unsigned i = 0; i < dim_nod; i++)
      {
        // Make backup
        double backup = nod_pt->x(i);
 
        // Do FD step. No node update required as we're
        // attacking the coordinate directly...
        nod_pt->x(i) += eps_fd;
 
        // Perform auxiliary node update function
        nod_pt->perform_auxiliary_node_update_fct();
 
        // Get advanced residual
        get_residuals(res_pls);
 
        // Fill in FD entries [Loop order is "wrong" here as l is the
        // slow index but this is in a function that's costly anyway
        // and gives us the fastest loop outside where these tensor
        // is actually used.]
        for (unsigned l = 0; l < n_dof; l++)
        {
          dresidual_dnodal_coordinates(l, i, j) =
            (res_pls[l] - res[l]) / eps_fd;
        }
 
        // Reset coordinate. No node update required as we're
        // attacking the coordinate directly...
        nod_pt->x(i) = backup;
 
        // Perform auxiliary node update function
        nod_pt->perform_auxiliary_node_update_fct();
      }
    }
  }
 
 
  //===============================================================
  /// Return the number of the node located at *node_pt
  /// if this node is in the element, else return \f$ -1 \f$
  //===============================================================
  int FiniteElement::get_node_number(Node* const& global_node_pt) const
  {
    // Initialise the number to -1
    int number = -1;
    // Find the number of nodes
    unsigned n_node = nnode();
#ifdef PARANOID
    {
      // Error check that node does not appear in element more than once
      unsigned count = 0;
      // Storage for the local node numbers of the element
      std::vector<int> local_node_number;
      // Loop over the nodes
      for (unsigned i = 0; i < n_node; i++)
      {
        // If the node is present increase the counter
        // and store the local node number
        if (node_pt(i) == global_node_pt)
        {
          ++count;
          local_node_number.push_back(i);
        }
      }
 
      // If the node appears more than once, complain
      if (count > 1)
      {
        std::ostringstream error_stream;
        error_stream << "Node " << global_node_pt << " appears " << count
                     << " times in an element." << std::endl
                     << "In positions: ";
        for (std::vector<int>::iterator it = local_node_number.begin();
             it != local_node_number.end();
             ++it)
        {
          error_stream << *it << " ";
        }
        error_stream << std::endl << "That seems very odd." << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
#endif
 
    // Loop over the nodes
    for (unsigned i = 0; i < n_node; i++)
    {
      // If the passed node pointer is present in the element
      // set number to be its local node number
      if (node_pt(i) == global_node_pt)
      {
        number = i;
        break;
      }
    }
 
    // Return the node number
    return number;
  }
 
 
  //==========================================================================
  /// If there is a node at the local coordinate, s, return the pointer
  /// to  the node. If not return 0. Note that this is a default, brute
  /// force implementation, can almost certainly be made more efficient for
  /// specific elements.
  //==========================================================================
  Node* FiniteElement::get_node_at_local_coordinate(
    const Vector<double>& s) const
  {
    // Locally cache the tolerance
    const double tol = Node_location_tolerance;
    Vector<double> s_node;
    // Locally cache the member data
    const unsigned el_dim = Elemental_dimension;
    const unsigned n_node = Nnode;
    // Loop over the nodes
    for (unsigned n = 0; n < n_node; n++)
    {
      bool Match = true;
      // Find the local coordinate of the node
      local_coordinate_of_node(n, s_node);
      for (unsigned i = 0; i < el_dim; i++)
      {
        // Calculate the difference between coordinates
        // and if it's bigger than our tolerance
        // break out of the (inner)loop
        if (std::fabs(s[i] - s_node[i]) > tol)
        {
          Match = false;
          break;
        }
      }
      // If we haven't complained then we have a match
      if (Match)
      {
        return node_pt(n);
      }
    }
    // If we get here, we have no match
    return 0;
  }
 
  //======================================================================
  /// Compute centre of gravity of all nodes and radius of node that
  /// is furthest from it. Used to assess approximately if a point
  /// is likely to be contained with an element in locate_zeta-like
  /// operations. NOTE: All computed in terms of zeta!
  //======================================================================
  void FiniteElement::get_centre_of_gravity_and_max_radius_in_terms_of_zeta(
    Vector<double>& cog, double& max_radius) const
  {
    // Initialise
    cog.resize(Elemental_dimension);
    max_radius = 0.0;
 
    // Get cog
    unsigned nnod = nnode();
    for (unsigned j = 0; j < nnod; j++)
    {
      for (unsigned i = 0; i < Elemental_dimension; i++)
      {
        cog[i] += zeta_nodal(j, 0, i);
      }
    }
    for (unsigned i = 0; i < Elemental_dimension; i++)
    {
      cog[i] /= double(nnod);
    }
 
    // Get max distance
    for (unsigned j = 0; j < nnod; j++)
    {
      double dist_squared = 0.0;
      for (unsigned i = 0; i < Elemental_dimension; i++)
      {
        dist_squared +=
          (cog[i] - zeta_nodal(j, 0, i)) * (cog[i] - zeta_nodal(j, 0, i));
      }
      if (dist_squared > max_radius) max_radius = dist_squared;
    }
    max_radius = sqrt(max_radius);
  }
 
  //======================================================================
  /// Return FE interpolated coordinate x[i] at local coordinate s
  //======================================================================
  double FiniteElement::interpolated_x(const Vector<double>& s,
                                       const unsigned& i) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // Find the number of positional types
    const unsigned n_position_type = nnodal_position_type();
    // Assign storage for the local shape function
    Shape psi(n_node, n_position_type);
    // Find the values of shape function
    shape(s, psi);
 
    // Initialise value of x
    double interpolated_x = 0.0;
    // Loop over the local nodes
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over the number of dofs
      for (unsigned k = 0; k < n_position_type; k++)
      {
        interpolated_x += nodal_position_gen(l, k, i) * psi(l, k);
      }
    }
 
    return (interpolated_x);
  }
 
  //=========================================================================
  /// Return FE interpolated coordinate x[i] at local coordinate s
  /// at previous timestep t (t=0: present; t>0: previous timestep)
  //========================================================================
  double FiniteElement::interpolated_x(const unsigned& t,
                                       const Vector<double>& s,
                                       const unsigned& i) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // Find the number of positional types
    const unsigned n_position_type = nnodal_position_type();
 
    // Assign storage for the local shape function
    Shape psi(n_node, n_position_type);
    // Find the values of shape function
    shape(s, psi);
 
    // Initialise value of x
    double interpolated_x = 0.0;
    // Loop over the local nodes
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over the number of dofs
      for (unsigned k = 0; k < n_position_type; k++)
      {
        interpolated_x += nodal_position_gen(t, l, k, i) * psi(l, k);
      }
    }
 
    return (interpolated_x);
  }
 
  //=======================================================================
  /// Return FE interpolated position x[] at local coordinate s as Vector
  //=======================================================================
  void FiniteElement::interpolated_x(const Vector<double>& s,
                                     Vector<double>& x) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // Find the number of positional types
    const unsigned n_position_type = nnodal_position_type();
    // Find the dimension stored in the node
    const unsigned nodal_dim = nodal_dimension();
 
    // Assign storage for the local shape function
    Shape psi(n_node, n_position_type);
    // Find the values of shape function
    shape(s, psi);
 
    // Loop over the dimensions
    for (unsigned i = 0; i < nodal_dim; i++)
    {
      // Initilialise value of x[i] to zero
      x[i] = 0.0;
      // Loop over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the number of dofs
        for (unsigned k = 0; k < n_position_type; k++)
        {
          x[i] += nodal_position_gen(l, k, i) * psi(l, k);
        }
      }
    }
  }
 
  //==========================================================================
  /// Return FE interpolated position x[] at local coordinate s
  /// at previous timestep t as Vector (t=0: present; t>0: previous timestep)
  //==========================================================================
  void FiniteElement::interpolated_x(const unsigned& t,
                                     const Vector<double>& s,
                                     Vector<double>& x) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // Find the number of positional types
    const unsigned n_position_type = nnodal_position_type();
    // Find the dimensions of the nodes
    const unsigned nodal_dim = nodal_dimension();
 
    // Assign storage for the local shape function
    Shape psi(n_node, n_position_type);
    // Find the values of shape function
    shape(s, psi);
 
    // Loop over the dimensions
    for (unsigned i = 0; i < nodal_dim; i++)
    {
      // Initilialise value of x[i] to zero
      x[i] = 0.0;
      // Loop over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the number of dofs
        for (unsigned k = 0; k < n_position_type; k++)
        {
          x[i] += nodal_position_gen(t, l, k, i) * psi(l, k);
        }
      }
    }
  }
 
  //========================================================================
  /// Calculate the determinant of the
  /// Jacobian of the mapping between local and global
  /// coordinates at the position. Works directly from the base vectors
  /// without assuming that coordinates match spatial dimension. Will
  /// be overloaded in FaceElements, in which the elemental dimension does
  /// not match the spatial dimension. WARNING: this is always positive
  /// and cannot be used to check if the element is inverted, say!
  //========================================================================
  double FiniteElement::J_eulerian(const Vector<double>& s) const
  {
    // Find the number of nodes and position types
    const unsigned n_node = nnode();
    const unsigned n_position_type = nnodal_position_type();
    // Find the dimension of the node and element
    const unsigned n_dim_node = nodal_dimension();
    const unsigned n_dim_element = dim();
 
    // Set up dummy memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, n_dim_element);
    // Get the shape functions and local derivatives
    dshape_local(s, psi, dpsids);
 
    // Right calculate the base vectors
    DenseMatrix<double> interpolated_G(n_dim_element, n_dim_node);
    assemble_eulerian_base_vectors(dpsids, interpolated_G);
 
    // Calculate the metric tensor of the element
    DenseMatrix<double> G(n_dim_element, n_dim_element, 0.0);
    for (unsigned i = 0; i < n_dim_element; i++)
    {
      for (unsigned j = 0; j < n_dim_element; j++)
      {
        for (unsigned k = 0; k < n_dim_node; k++)
        {
          G(i, j) += interpolated_G(i, k) * interpolated_G(j, k);
        }
      }
    }
 
    // Calculate the determinant of the metric tensor
    double det = 0.0;
    switch (n_dim_element)
    {
      case 0:
        throw OomphLibError("Cannot calculate J_eulerian() for point element\n",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
        break;
      case 1:
        det = G(0, 0);
        break;
      case 2:
        det = G(0, 0) * G(1, 1) - G(0, 1) * G(1, 0);
        break;
      case 3:
        det = G(0, 0) * G(1, 1) * G(2, 2) + G(0, 1) * G(1, 2) * G(2, 0) +
              G(0, 2) * G(1, 0) * G(2, 1) - G(0, 0) * G(1, 2) * G(2, 1) -
              G(0, 1) * G(1, 0) * G(2, 2) - G(0, 2) * G(1, 1) * G(2, 0);
        break;
      default:
        oomph_info << "More than 3 dimensions in J_eulerian()" << std::endl;
        break;
    }
 
    // Return the Jacobian (square-root of the determinant of the metric tensor)
    return sqrt(det);
  }
 
  //========================================================================
  /// Compute the Jacobian of the mapping between the local and global
  /// coordinates at the ipt-th integration point
  //========================================================================
  double FiniteElement::J_eulerian_at_knot(const unsigned& ipt) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
    // Find the number of position types
    const unsigned n_position_type = nnodal_position_type();
    // Find the dimension of the node and element
    const unsigned n_dim_node = nodal_dimension();
    const unsigned n_dim_element = dim();
 
    // Set up dummy memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, n_dim_element);
    // Get the shape functions and local derivatives at the knot
    // This call may use the stored versions, which is why this entire
    // function doesn't just call J_eulerian(s), after reading out s from
    // the knots.
    dshape_local_at_knot(ipt, psi, dpsids);
 
    // Right calculate the base vectors
    DenseMatrix<double> interpolated_G(n_dim_element, n_dim_node);
    assemble_eulerian_base_vectors(dpsids, interpolated_G);
 
    // Calculate the metric tensor of the element
    DenseMatrix<double> G(n_dim_element, n_dim_element, 0.0);
    for (unsigned i = 0; i < n_dim_element; i++)
    {
      for (unsigned j = 0; j < n_dim_element; j++)
      {
        for (unsigned k = 0; k < n_dim_node; k++)
        {
          G(i, j) += interpolated_G(i, k) * interpolated_G(j, k);
        }
      }
    }
 
    // Calculate the determinant of the metric tensor
    double det = 0.0;
    switch (n_dim_element)
    {
      case 0:
        throw OomphLibError("Cannot calculate J_eulerian() for point element\n",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
        break;
      case 1:
        det = G(0, 0);
        break;
      case 2:
        det = G(0, 0) * G(1, 1) - G(0, 1) * G(1, 0);
        break;
      case 3:
        det = G(0, 0) * G(1, 1) * G(2, 2) + G(0, 1) * G(1, 2) * G(2, 0) +
              G(0, 2) * G(1, 0) * G(2, 1) - G(0, 0) * G(1, 2) * G(2, 1) -
              G(0, 1) * G(1, 0) * G(2, 2) - G(0, 2) * G(1, 1) * G(2, 0);
        break;
      default:
        oomph_info << "More than 3 dimensions in J_eulerian()" << std::endl;
        break;
    }
 
    // Return the Jacobian (square-root of the determinant of the metric tensor)
    return sqrt(det);
  }
 
  //========================================================================
  /// Check that Jacobian of mapping between local and Eulerian
  /// coordinates at all integration points is positive.
  //========================================================================
  void FiniteElement::check_J_eulerian_at_knots(bool& passed) const
  {
    // Bypass check deep down in the guts...
    bool backup = FiniteElement::Accept_negative_jacobian;
    FiniteElement::Accept_negative_jacobian = true;
 
    // Let's be optimistic...
    passed = true;
 
    // Find the number of nodes
    const unsigned n_node = nnode();
 
    // Find the number of position types
    const unsigned n_position_type = nnodal_position_type();
 
    // DRAIG: Unused variable
    // const unsigned n_dim_node = nodal_dimension();
 
    // Find the dimension of the node and element
    const unsigned n_dim_element = dim();
 
    // Set up dummy memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsi(n_node, n_dim_element);
 
    unsigned nintpt = integral_pt()->nweight();
    for (unsigned ipt = 0; ipt < nintpt; ipt++)
    {
      double jac = dshape_eulerian_at_knot(ipt, psi, dpsi);
 
      // Are we dead yet?
      if (jac <= 0.0)
      {
        passed = false;
 
        // Reset
        FiniteElement::Accept_negative_jacobian = backup;
 
        return;
      }
    }
 
    // Reset
    FiniteElement::Accept_negative_jacobian = backup;
  }
 
  //====================================================================
  /// Calculate the size of the element (in Eulerian computational
  /// coordinates. Use suitably overloaded compute_physical_size()
  /// function to compute the actual size (taking into account
  /// factors such as 2pi or radii the integrand). Such function
  /// can only be implemented on an equation-by-equation basis.
  //====================================================================
  double FiniteElement::size() const
  {
    // Initialise the sum to zero
    double sum = 0.0;
 
    // Loop over the integration points
    const unsigned n_intpt = integral_pt()->nweight();
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
      // Get the value of the Jacobian of the mapping to global coordinates
      double J = J_eulerian_at_knot(ipt);
 
      // Add the product to the sum
      sum += w * J;
    }
 
    // Return the answer
    return (sum);
  }
 
  //==================================================================
  /// Integrate Vector-valued time-dep function over element
  //==================================================================
  void FiniteElement::integrate_fct(
    FiniteElement::UnsteadyExactSolutionFctPt integrand_fct_pt,
    const double& time,
    Vector<double>& integral)
  {
    // Initialise all components of integral Vector and setup integrand vector
    const unsigned ncomponents = integral.size();
    Vector<double> integrand(ncomponents);
    for (unsigned i = 0; i < ncomponents; i++)
    {
      integral[i] = 0.0;
    }
 
    // Figure out the global (Eulerian) spatial dimension of the
    // element
    const unsigned n_dim_eulerian = nodal_dimension();
 
    // Allocate Vector of global Eulerian coordinates
    Vector<double> x(n_dim_eulerian);
 
    // Set the value of n_intpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Vector of local coordinates
    const unsigned n_dim = dim();
    Vector<double> s(n_dim);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign the values of s
      for (unsigned i = 0; i < n_dim; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Assign the values of the global Eulerian coordinates
      for (unsigned i = 0; i < n_dim_eulerian; i++)
      {
        x[i] = interpolated_x(s, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Get Jacobian of mapping
      double J = J_eulerian(s);
 
      // Evaluate the integrand at the knot points
      integrand_fct_pt(time, x, integrand);
 
      // Add to components of integral Vector
      for (unsigned i = 0; i < ncomponents; i++)
      {
        integral[i] += integrand[i] * w * J;
      }
    }
  }
 
  //==================================================================
  /// Integrate Vector-valued function over element
  //==================================================================
  void FiniteElement::integrate_fct(
    FiniteElement::SteadyExactSolutionFctPt integrand_fct_pt,
    Vector<double>& integral)
  {
    // Initialise all components of integral Vector
    const unsigned ncomponents = integral.size();
    Vector<double> integrand(ncomponents);
    for (unsigned i = 0; i < ncomponents; i++)
    {
      integral[i] = 0.0;
    }
 
    // Figure out the global (Eulerian) spatial dimension of the
    // element by checking the Eulerian dimension of the nodes
    const unsigned n_dim_eulerian = nodal_dimension();
 
    // Allocate Vector of global Eulerian coordinates
    Vector<double> x(n_dim_eulerian);
 
    // Set the value of n_intpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Vector of local coordinates
    const unsigned n_dim = dim();
    Vector<double> s(n_dim);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign the values of s
      for (unsigned i = 0; i < n_dim; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Assign the values of the global Eulerian coordinates
      for (unsigned i = 0; i < n_dim_eulerian; i++)
      {
        x[i] = interpolated_x(s, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Get Jacobian of mapping
      double J = J_eulerian(s);
 
      // Evaluate the integrand at the knot points
      integrand_fct_pt(x, integrand);
 
      // Add to components of integral Vector
      for (unsigned i = 0; i < ncomponents; i++)
      {
        integral[i] += integrand[i] * w * J;
      }
    }
  }
 
  //==========================================================================
  /// Self-test: Have all internal values been classified as
  /// pinned/unpinned? Has pointer to spatial integration scheme
  /// been set? Return 0 if OK.
  //==========================================================================
  unsigned FiniteElement::self_test()
  {
    // Initialise
    bool passed = true;
 
    if (GeneralisedElement::self_test() != 0)
    {
      passed = false;
    }
 
    // Check that pointer to spatial integration scheme has been set
    if (integral_pt() == 0)
    {
      passed = false;
 
      OomphLibWarning("Pointer to spatial integration scheme has not been set.",
                      "FiniteElement::self_test()",
                      OOMPH_EXCEPTION_LOCATION);
    }
 
    // If the dimension of the element is zero (point element), there
    // is not jacobian
    const unsigned dim_el = dim();
 
    if (dim_el > 0)
    {
      // Loop over integration points to check sign of Jacobian
      //-------------------------------------------------------
 
      // Set the value of n_intpt
      const unsigned n_intpt = integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s(dim_el);
 
      // Find the number of local nodes
      const unsigned n_node = nnode();
      const unsigned n_position_type = nnodal_position_type();
      // Set up memory for the shape and test functions
      Shape psi(n_node, n_position_type);
      DShape dpsidx(n_node, dim_el);
 
      // Jacobian
      double jacobian;
 
 
      // Two ways of testing for negative Jacobian for non-FaceElements
      unsigned ntest = 1;
 
      // For FaceElements checking the Jacobian via dpsidx doesn't
      // make sense
      FiniteElement* tmp_pt = const_cast<FiniteElement*>(this);
      FaceElement* face_el_pt = dynamic_cast<FaceElement*>(tmp_pt);
      if (face_el_pt == 0)
      {
        ntest = 2;
      }
 
      // For now overwrite -- the stuff above fails for Bretherton.
      // Not sure why.
      ntest = 1;
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < dim_el; i++)
        {
          s[i] = integral_pt()->knot(ipt, i);
        }
 
 
        // Do tests
        for (unsigned test = 0; test < ntest; test++)
        {
          switch (test)
          {
            case 0:
 
              // Get the jacobian from the mapping between local and Eulerian
              // coordinates
              jacobian = J_eulerian(s);
 
              break;
 
            case 1:
 
              // Call the geometrical shape functions and derivatives
              // This also computes the Jacobian by a slightly different
              // method
              jacobian = dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
              break;
 
            default:
 
              throw OomphLibError("Never get here",
                                  OOMPH_CURRENT_FUNCTION,
                                  OOMPH_EXCEPTION_LOCATION);
          }
 
 
          // Check for a singular jacobian
          if (std::fabs(jacobian) < 1.0e-16)
          {
            std::ostringstream warning_stream;
            warning_stream << "Determinant of Jacobian matrix is zero at ipt "
                           << ipt << std::endl;
            OomphLibWarning(warning_stream.str(),
                            "FiniteElement::self_test()",
                            OOMPH_EXCEPTION_LOCATION);
            passed = false;
            // Skip the next test
            continue;
          }
 
          // Check sign of Jacobian
          if ((Accept_negative_jacobian == false) && (jacobian < 0.0))
          {
            std::ostringstream warning_stream;
            warning_stream << "Jacobian negative at integration point ipt="
                           << ipt << std::endl;
            warning_stream
              << "If you think that this is what you want you may: "
              << std::endl
              << "set the (static) flag "
              << "FiniteElement::Accept_negative_jacobian to be true"
              << std::endl;
 
            OomphLibWarning(warning_stream.str(),
                            "FiniteElement::self_test()",
                            OOMPH_EXCEPTION_LOCATION);
            passed = false;
          }
 
        } // end of loop over two tests
      }
    } // End of non-zero dimension check
 
    // Return verdict
    if (passed)
    {
      return 0;
    }
    else
    {
      return 1;
    }
  }
 
 
  //=======================================================================
  /// Return the t-th time-derivative of the
  /// i-th FE-interpolated Eulerian coordinate at
  /// local coordinate s.
  //=======================================================================
  double FiniteElement::interpolated_dxdt(const Vector<double>& s,
                                          const unsigned& i,
                                          const unsigned& t_deriv)
  {
    // Find the number of nodes and positions (locally cached)
    const unsigned n_node = nnode();
    const unsigned n_position_type = nnodal_position_type();
    // Get shape functions: specify # of nodes, # of positional dofs
    Shape psi(n_node, n_position_type);
    shape(s, psi);
 
    // Initialise
    double drdt = 0.0;
 
    // Assemble time derivative
    // Loop over nodes
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over types of dof
      for (unsigned k = 0; k < n_position_type; k++)
      {
        drdt += dnodal_position_gen_dt(t_deriv, l, k, i) * psi(l, k);
      }
    }
    return drdt;
  }
 
 
  //=======================================================================
  /// Compute t-th time-derivative of the
  /// FE-interpolated Eulerian coordinate vector at
  /// local coordinate s.
  //=======================================================================
  void FiniteElement::interpolated_dxdt(const Vector<double>& s,
                                        const unsigned& t_deriv,
                                        Vector<double>& dxdt)
  {
    // Find the number of nodes and positions (locally cached)
    const unsigned n_node = nnode();
    const unsigned n_position_type = nnodal_position_type();
    const unsigned nodal_dim = nodal_dimension();
 
    // Get shape functions: specify # of nodes, # of positional dofs
    Shape psi(n_node, n_position_type);
    shape(s, psi);
 
    // Loop over directions
    for (unsigned i = 0; i < nodal_dim; i++)
    {
      // Initialise
      dxdt[i] = 0.0;
 
      // Assemble time derivative
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over types of dof
        for (unsigned k = 0; k < n_position_type; k++)
        {
          dxdt[i] += dnodal_position_gen_dt(t_deriv, l, k, i) * psi(l, k);
        }
      }
    }
  }
 
  //============================================================================
  /// Calculate the interpolated value of zeta, the intrinsic coordinate
  /// of the element when viewed as a compound geometric object within a Mesh
  /// as a function of the local coordinate of the element, s.
  ///  The default
  /// assumption is the zeta is interpolated using the shape functions of
  /// the element with the values given by zeta_nodal().
  /// A MacroElement representation of the intrinsic coordinate parametrised
  /// by the local coordinate s is used if available.
  /// Choosing the MacroElement representation of zeta (Eulerian x by default)
  /// allows a correspondence to be established between elements on different
  /// Meshes covering the same curvilinear domain in cases where one element
  /// is much coarser than the other.
  //==========================================================================
  void FiniteElement::interpolated_zeta(const Vector<double>& s,
                                        Vector<double>& zeta) const
  {
    // If there is a macro element use it
    if (Macro_elem_pt != 0)
    {
      this->get_x_from_macro_element(s, zeta);
    }
    // Otherwise interpolate zeta_nodal using the shape functions
    else
    {
      // Find the number of nodes
      const unsigned n_node = this->nnode();
      // Find the number of positional types
      const unsigned n_position_type = this->nnodal_position_type();
      // Storage for the shape functions
      Shape psi(n_node, n_position_type);
      // Get the values of the shape functions at the local coordinate s
      this->shape(s, psi);
 
      // Find the number of coordinates
      const unsigned ncoord = this->dim();
      // Initialise the value of zeta to zero
      for (unsigned i = 0; i < ncoord; i++)
      {
        zeta[i] = 0.0;
      }
 
      // Add the contributions from each nodal dof to the interpolated value
      // of zeta.
      for (unsigned l = 0; l < n_node; l++)
      {
        for (unsigned k = 0; k < n_position_type; k++)
        {
          // Locally cache the value of the shape function
          const double psi_ = psi(l, k);
          for (unsigned i = 0; i < ncoord; i++)
          {
            zeta[i] += this->zeta_nodal(l, k, i) * psi_;
          }
        }
      }
    }
  }
 
  //==========================================================================
  /// For a given value of zeta, the "global" intrinsic coordinate of
  /// a mesh of FiniteElements represented as a compound geometric object,
  /// find the local coordinate in this element that corresponds to the
  /// requested value of zeta.
  /// This is achieved in generality by using Newton's method to find the value
  /// of the local coordinate, s, such that
  /// interpolated_zeta(s) is equal to the requested value of zeta.
  /// If zeta cannot be located in this element, geom_object_pt is set
  /// to NULL. If zeta is located in this element, we return its "this"
  /// pointer.
  /// Setting the optional bool argument to true means that the coordinate
  /// argument "s" is used as the initial guess. (Default is false).
  //=========================================================================
  void FiniteElement::locate_zeta(const Vector<double>& zeta,
                                  GeomObject*& geom_object_pt,
                                  Vector<double>& s,
                                  const bool& use_coordinate_as_initial_guess)
  {
    // Find the number of coordinates, the dimension of the element
    // This must be the same for the local and intrinsic global coordinate
    unsigned ncoord = this->dim();
 
    // Fast return based on centre of gravity and max. radius of any nodal
    // point
    if (Locate_zeta_helpers::Radius_multiplier_for_fast_exit_from_locate_zeta >
        0.0)
    {
      Vector<double> cog(ncoord);
      double max_radius = 0.0;
      get_centre_of_gravity_and_max_radius_in_terms_of_zeta(cog, max_radius);
 
      // Get radius
      double radius = 0.0;
      for (unsigned i = 0; i < ncoord; i++)
      {
        radius += (cog[i] - zeta[i]) * (cog[i] - zeta[i]);
      }
      radius = sqrt(radius);
      if (radius > Locate_zeta_helpers::
                       Radius_multiplier_for_fast_exit_from_locate_zeta *
                     max_radius)
      {
        geom_object_pt = 0;
        return;
      }
    }
 
    // Assign storage for the vector and matrix used in Newton's method
    Vector<double> dx(ncoord, 0.0);
    DenseDoubleMatrix jacobian(ncoord, ncoord, 0.0);
 
    // // Make a list of (equally-spaced) local coordinates inside the element
    // unsigned n_list=Locate_zeta_helpers::N_local_points;
 
    // double list_space=(1.0/(double(n_list)-1.0))*(s_max()-s_min());
 
    // Possible initial guesses for Newton's method
    Vector<Vector<double>> s_list;
 
    // If the boolean argument use_coordinate_as_initial_guess was set
    // to true then we don't need to initialise s
    if (!use_coordinate_as_initial_guess)
    {
      // Vector of local coordinates
      Vector<double> s_c(ncoord);
 
      // Loop over plot points
      unsigned num_plot_points =
        nplot_points(Locate_zeta_helpers::N_local_points);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        get_s_plot(iplot, Locate_zeta_helpers::N_local_points, s_c);
        s_list.push_back(s_c);
      }
    }
 
    // Counter for the number of Newton steps
    unsigned count = 0;
 
    // Control flag for the Newton loop
    bool keep_going = true;
 
    // Storage for the interpolated value of x
    Vector<double> inter_x(ncoord);
 
    // Ifwe have specified the coordinate already
    if (use_coordinate_as_initial_guess)
    {
      // Get the value of x at the initial guess
      this->interpolated_zeta(s, inter_x);
 
      // Set up the residuals
      for (unsigned i = 0; i < ncoord; i++)
      {
        dx[i] = zeta[i] - inter_x[i];
      }
    }
    else
    {
      // Find the smallest residual from the list of coordinates made earlier
      double my_min_resid = DBL_MAX;
      Vector<double> s_local(ncoord);
      Vector<double> work_x(ncoord);
      Vector<double> work_dx(ncoord);
 
      unsigned n_list_coord = s_list.size();
 
      for (unsigned i_coord = 0; i_coord < n_list_coord; i_coord++)
      {
        for (unsigned i = 0; i < ncoord; i++)
        {
          s_local[i] = s_list[i_coord][i];
        }
        // get_x for this coordinate
        this->interpolated_zeta(s_local, work_x);
 
        // calculate residuals
        for (unsigned i = 0; i < ncoord; i++)
        {
          work_dx[i] = zeta[i] - work_x[i];
        }
 
        double maxres = std::fabs(
          *std::max_element(work_dx.begin(), work_dx.end(), AbsCmp<double>()));
 
        // test against previous residuals
        if (maxres < my_min_resid)
        {
          my_min_resid = maxres;
          dx = work_dx;
          inter_x = work_x;
          s = s_local;
        }
      }
    }
 
    // Main Newton Loop
    do // start of do while loop
    {
      // Increase loop counter
      count++;
 
      // Bail out if necessary (without an error for now...)
      if (count > Locate_zeta_helpers::Max_newton_iterations)
      {
        keep_going = false;
        continue;
      }
 
      // If it's the first time round the loop, check the initial residuals
      if (count == 1)
      {
        double maxres =
          std::fabs(*std::max_element(dx.begin(), dx.end(), AbsCmp<double>()));
 
        // If it's small enough exit
        if (maxres < Locate_zeta_helpers::Newton_tolerance)
        {
          keep_going = false;
          continue;
        }
      }
 
      // Is there a macro element? If so, assemble the Jacobian by FD-ing
      if (macro_elem_pt() != 0)
      {
        // Assemble jacobian on the fly by finite differencing
        Vector<double> work_s = s;
        Vector<double> r = inter_x; // i.e. the result of previous call to get_x
 
        // Finite difference step
        double fd_step = GeneralisedElement::Default_fd_jacobian_step;
 
        // Storage for calculated r from incremented s
        Vector<double> work_r(ncoord, 0.0);
 
        // Loop over s coordinates
        for (unsigned i = 0; i < ncoord; i++)
        {
          // Increment work_s by a small amount
          work_s[i] += fd_step;
 
          // Calculate work_r from macro element
          this->interpolated_zeta(work_s, work_r);
 
          // Loop over r to fill Jacobian
          for (unsigned j = 0; j < ncoord; j++)
          {
            jacobian(j, i) = -(work_r[j] - r[j]) / fd_step;
          }
 
          // Reset work_s
          work_s[i] = s[i];
        }
      }
      else // no macro element, so compute Jacobian with shape functions etc.
      {
        // Compute the entries of the Jacobian matrix
        unsigned n_node = this->nnode();
        unsigned n_position_type = this->nnodal_position_type();
        Shape psi(n_node, n_position_type);
        DShape dpsids(n_node, n_position_type, ncoord);
 
        // Get the local shape functions and their derivatives
        dshape_local(s, psi, dpsids);
 
        // Calculate the values of dxds
        DenseMatrix<double> interpolated_dxds(ncoord, ncoord, 0.0);
 
        // MH: No longer needed
        //          //This implementation will only work for n_position_type=1
        //          //since the function nodal_position_gen does not yet exist
        // #ifdef PARANOID
        //          if (n_position_type!=1)
        //           {
        //            std::ostringstream error_stream;
        //            error_stream << "This implementation does not exist
        //            yet;\n"
        //                         << "it currently uses
        //                         raw_nodal_position_gen\n"
        //                         << "which does not take hangingness into
        //                         account\n"
        //                         << "It will work if n_position_type=1\n";
        //            throw OomphLibError(error_stream.str(),
        //         OOMPH_CURRENT_FUNCTION,
        //                                OOMPH_EXCEPTION_LOCATION);
        //           }
        // #endif
 
        // Loop over the nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over position type even though it should be 1; the
          // functionality for n_position_type>1 will exist in the future
          for (unsigned k = 0; k < n_position_type; k++)
          {
            // Add the contribution from the nodal coordinates to the matrix
            for (unsigned i = 0; i < ncoord; i++)
            {
              for (unsigned j = 0; j < ncoord; j++)
              {
                interpolated_dxds(i, j) +=
                  this->zeta_nodal(l, k, i) * dpsids(l, k, j);
              }
            }
          }
        }
 
        // The entries of the Jacobian matrix are merely dresiduals/ds
        // i.e. \f$ -dx/ds \f$
        for (unsigned i = 0; i < ncoord; i++)
        {
          for (unsigned j = 0; j < ncoord; j++)
          {
            jacobian(i, j) = -interpolated_dxds(i, j);
          }
        }
      }
 
      // Now solve the damn thing
      try
      {
        jacobian.solve(dx);
      }
      catch (OomphLibError& error)
      {
        // I've caught the error so shut up!
        error.disable_error_message();
#ifdef PARANOID
        oomph_info << "Error in linear solve for "
                   << "FiniteElement::locate_zeta()" << std::endl;
        oomph_info << "Should not affect the result!" << std::endl;
#endif
      }
 
      // Add the correction to the local coordinates
      for (unsigned i = 0; i < ncoord; i++)
      {
        s[i] -= dx[i];
      }
 
      // Get the new residuals
      this->interpolated_zeta(s, inter_x);
      for (unsigned i = 0; i < ncoord; i++)
      {
        dx[i] = zeta[i] - inter_x[i];
      }
 
      // Get the maximum residuals
      double maxres =
        std::fabs(*std::max_element(dx.begin(), dx.end(), AbsCmp<double>()));
 
      // If we have converged jump straight to the test at the end of the loop
      if (maxres < Locate_zeta_helpers::Newton_tolerance)
      {
        keep_going = false;
        continue;
      }
    } while (keep_going);
 
    // Test whether the local coordinates are valid or not
    bool valid = local_coord_is_valid(s);
 
    // If not valid, experimentally push back into element
    // and see if the result is still valid (within the Newton tolerance)
    if (!valid)
    {
      move_local_coord_back_into_element(s);
 
      // Check residuals again
      this->interpolated_zeta(s, inter_x);
      for (unsigned i = 0; i < ncoord; i++)
      {
        dx[i] = zeta[i] - inter_x[i];
      }
 
      // Get the maximum residuals
      double maxres =
        std::fabs(*std::max_element(dx.begin(), dx.end(), AbsCmp<double>()));
 
      // Are we still OK?
      if (maxres > Locate_zeta_helpers::Newton_tolerance)
      {
        // oomph_info
        // << "Pushing back inside has violated the Newton tolerance: max_res =
        // "
        // << maxres << std::endl;
        geom_object_pt = 0;
        return;
      }
    }
 
    // It is also possible now that it may not have converged "correctly",
    // i.e. count is greater than Max_newton_iterations
    if (count > Locate_zeta_helpers::Max_newton_iterations)
    {
      // Don't trust the current answer, return null
      geom_object_pt = 0;
      return;
    }
 
    // Otherwise the required point is located in "this" element:
    geom_object_pt = this;
  }
 
 
  //=======================================================================
  /// Loop over all nodes in the element and update their positions
  /// using each node's (algebraic) update function
  //=======================================================================
  void FiniteElement::node_update()
  {
    const unsigned n_node = nnode();
    for (unsigned n = 0; n < n_node; n++)
    {
      node_pt(n)->node_update();
    }
  }
 
  //======================================================================
  /// The purpose of this function is to identify all possible
  /// Data that can affect the fields interpolated by the FiniteElement.
  /// The information will typically be used in interaction problems in
  /// which the FiniteElement provides a forcing term for an
  /// ElementWithExternalElement. The Data must be provided as
  /// \c paired_load data containing (a) the pointer to a Data object
  /// and (b) the index of the value in that Data object.
  /// The generic implementation (should be overloaded in more specific
  /// applications) is to include all nodal and internal Data stored in
  /// the FiniteElement. Note that the geometric data,
  /// which includes the positions
  /// of SolidNodes, is treated separately by the function
  /// \c identify_geometric_data()
  //======================================================================
  void FiniteElement::identify_field_data_for_interactions(
    std::set<std::pair<Data*, unsigned>>& paired_field_data)
  {
    // Loop over all internal data
    const unsigned n_internal = this->ninternal_data();
    for (unsigned n = 0; n < n_internal; n++)
    {
      // Cache the data pointer
      Data* const dat_pt = this->internal_data_pt(n);
      // Find the number of data values stored in the data object
      const unsigned n_value = dat_pt->nvalue();
      // Add the index of each data value and the pointer to the set
      // of pairs
      for (unsigned i = 0; i < n_value; i++)
      {
        paired_field_data.insert(std::make_pair(dat_pt, i));
      }
    }
 
    // Loop over all the nodes
    const unsigned n_node = this->nnode();
    for (unsigned n = 0; n < n_node; n++)
    {
      // Cache the node pointer
      Node* const nod_pt = this->node_pt(n);
      // Find the number of values stored at the node
      const unsigned n_value = nod_pt->nvalue();
      // Add the index of each data value and the pointer to the set
      // of pairs
      for (unsigned i = 0; i < n_value; i++)
      {
        paired_field_data.insert(std::make_pair(nod_pt, i));
      }
    }
  }
 
  void FiniteElement::build_face_element(const int& face_index,
                                         FaceElement* face_element_pt)
  {
    // Set the nodal dimension
    face_element_pt->set_nodal_dimension(nodal_dimension());
 
    // Set the pointer to the orginal "bulk" element
    face_element_pt->bulk_element_pt() = this;
 
#ifdef OOMPH_HAS_MPI
    // Pass on non-halo proc ID
    face_element_pt->set_halo(Non_halo_proc_ID);
#endif
 
    // Set the face index
    face_element_pt->face_index() = face_index;
 
    // Get number of bulk nodes on a face of this element
    const unsigned nnode_face = nnode_on_face();
 
    // Set the function pointer for face coordinate to bulk coordinate
    // mapping
    face_element_pt->face_to_bulk_coordinate_fct_pt() =
      face_to_bulk_coordinate_fct_pt(face_index);
 
    // Set the function pointer for the derivative of the face coordinate to
    // bulk coordinate mapping
    face_element_pt->bulk_coordinate_derivatives_fct_pt() =
      bulk_coordinate_derivatives_fct_pt(face_index);
 
    // Resize storage for the number of values originally stored each of the
    // face element's nodes.
    face_element_pt->nbulk_value_resize(nnode_face);
 
    // Resize storage for the bulk node numbers corresponding to the face
    // element's nodes.
    face_element_pt->bulk_node_number_resize(nnode_face);
 
    // Copy bulk_node_numbers and nbulk_values
    for (unsigned i = 0; i < nnode_face; i++)
    {
      // Find the corresponding bulk node's number
      unsigned bulk_number = get_bulk_node_number(face_index, i);
 
      // Assign the pointer and number into the face element
      face_element_pt->node_pt(i) = node_pt(bulk_number);
      face_element_pt->bulk_node_number(i) = bulk_number;
 
      // Set the number of values originally stored at this node
      face_element_pt->nbulk_value(i) = required_nvalue(bulk_number);
    }
 
    // Set the outer unit normal sign
    face_element_pt->normal_sign() = face_outer_unit_normal_sign(face_index);
  }
 
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
 
  // Initialise the static variable.
  // Do not ignore warning for discontinuous tangent vectors.
  bool FaceElement::Ignore_discontinuous_tangent_warning = false;
 
 
  //========================================================================
  /// Output boundary coordinate zeta
  //========================================================================
  void FaceElement::output_zeta(std::ostream& outfile, const unsigned& nplot)
  {
    // Vector of local coordinates
    unsigned n_dim = dim();
    Vector<double> s(n_dim);
 
    // 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);
 
      // Spatial coordinates are one higher
      for (unsigned i = 0; i < n_dim + 1; i++)
      {
        outfile << interpolated_x(s, i) << " ";
      }
 
      // Boundary coordinate
      Vector<double> zeta(n_dim);
      interpolated_zeta(s, zeta);
      for (unsigned i = 0; i < n_dim; i++)
      {
        outfile << zeta[i] << " ";
      }
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //========================================================================
  /// Calculate the determinant of the
  /// Jacobian of the mapping between local and global
  /// coordinates at the position s. Overloaded from FiniteElement.
  //========================================================================
  double FaceElement::J_eulerian(const Vector<double>& s) const
  {
    // Find out the sptial dimension of the element
    unsigned n_dim_el = this->dim();
 
    // Bail out if we're in a point element -- not sure what
    // J_eulerian actually is, but this is harmless
    if (n_dim_el == 0) return 1.0;
 
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Find out how many positional dofs there are
    unsigned n_position_type = this->nnodal_position_type();
 
    // Find out the dimension of the node
    unsigned n_dim = this->nodal_dimension();
 
    // Set up memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, n_dim_el);
 
    // Only need to call the local derivatives
    dshape_local(s, psi, dpsids);
 
    // Also calculate the surface Vectors (derivatives wrt local coordinates)
    DenseMatrix<double> interpolated_A(n_dim_el, n_dim, 0.0);
 
    // Calculate positions and derivatives
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over positional dofs
      for (unsigned k = 0; k < n_position_type; k++)
      {
        // Loop over coordinates
        for (unsigned i = 0; i < n_dim; i++)
        {
          // Loop over LOCAL derivative directions, to calculate the tangent(s)
          for (unsigned j = 0; j < n_dim_el; j++)
          {
            interpolated_A(j, i) +=
              nodal_position_gen(l, bulk_position_type(k), i) * dpsids(l, k, j);
          }
        }
      }
    }
    // Now find the local deformed metric tensor from the tangent Vectors
    DenseMatrix<double> A(n_dim_el, n_dim_el, 0.0);
    for (unsigned i = 0; i < n_dim_el; i++)
    {
      for (unsigned j = 0; j < n_dim_el; j++)
      {
        // Take the dot product
        for (unsigned k = 0; k < n_dim; k++)
        {
          A(i, j) += interpolated_A(i, k) * interpolated_A(j, k);
        }
      }
    }
 
    // Find the determinant of the metric tensor
    double Adet = 0.0;
    switch (n_dim_el)
    {
      case 1:
        Adet = A(0, 0);
        break;
      case 2:
        Adet = A(0, 0) * A(1, 1) - A(0, 1) * A(1, 0);
        break;
      default:
        throw OomphLibError("Wrong dimension in FaceElement",
                            "FaceElement::J_eulerian()",
                            OOMPH_EXCEPTION_LOCATION);
    }
 
    // Return
    return sqrt(Adet);
  }
 
 
  //========================================================================
  /// Compute the Jacobian of the mapping between the local and global
  /// coordinates at the ipt-th integration point. Overloaded from
  /// FiniteElement.
  //========================================================================
  double FaceElement::J_eulerian_at_knot(const unsigned& ipt) const
  {
    // Find the number of nodes
    const unsigned n_node = nnode();
 
    // Find the number of position types
    const unsigned n_position_type = nnodal_position_type();
 
    // Find the dimension of the node and element
    const unsigned n_dim = nodal_dimension();
    const unsigned n_dim_el = dim();
 
    // Set up dummy memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, n_dim_el);
 
    // Get the shape functions and local derivatives at the knot
    // This call may use the stored versions, which is why this entire
    // function doesn't just call J_eulerian(s), after reading out s from
    // the knots.
    dshape_local_at_knot(ipt, psi, dpsids);
 
    // Also calculate the surface Vectors (derivatives wrt local coordinates)
    DenseMatrix<double> interpolated_A(n_dim_el, n_dim, 0.0);
 
    // Calculate positions and derivatives
    for (unsigned l = 0; l < n_node; l++)
    {
      // Loop over positional dofs
      for (unsigned k = 0; k < n_position_type; k++)
      {
        // Loop over coordinates
        for (unsigned i = 0; i < n_dim; i++)
        {
          // Loop over LOCAL derivative directions, to calculate the tangent(s)
          for (unsigned j = 0; j < n_dim_el; j++)
          {
            interpolated_A(j, i) +=
              nodal_position_gen(l, bulk_position_type(k), i) * dpsids(l, k, j);
          }
        }
      }
    }
 
    // Now find the local deformed metric tensor from the tangent Vectors
    DenseMatrix<double> A(n_dim_el, n_dim_el, 0.0);
    for (unsigned i = 0; i < n_dim_el; i++)
    {
      for (unsigned j = 0; j < n_dim_el; j++)
      {
        // Take the dot product
        for (unsigned k = 0; k < n_dim; k++)
        {
          A(i, j) += interpolated_A(i, k) * interpolated_A(j, k);
        }
      }
    }
 
    // Find the determinant of the metric tensor
    double Adet = 0.0;
    switch (n_dim_el)
    {
      case 1:
        Adet = A(0, 0);
        break;
      case 2:
        Adet = A(0, 0) * A(1, 1) - A(0, 1) * A(1, 0);
        break;
      default:
        throw OomphLibError("Wrong dimension in FaceElement",
                            "FaceElement::J_eulerian_at_knot()",
                            OOMPH_EXCEPTION_LOCATION);
    }
 
    // Return
    return sqrt(Adet);
  }
 
  //========================================================================
  /// Check that Jacobian of mapping between local and Eulerian
  /// coordinates at all integration points is positive.
  //========================================================================
  void FaceElement::check_J_eulerian_at_knots(bool& passed) const
  {
    // Let's be optimistic...
    passed = true;
 
    // Find the number of nodes
    const unsigned n_node = nnode();
 
    // Find the number of position types
    const unsigned n_position_type = nnodal_position_type();
 
    // Find the dimension of the node and element
    const unsigned n_dim = nodal_dimension();
    const unsigned n_dim_el = dim();
 
    // Set up dummy memory for the shape functions
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, n_dim_el);
 
 
    unsigned nintpt = integral_pt()->nweight();
    for (unsigned ipt = 0; ipt < nintpt; ipt++)
    {
      // Get the shape functions and local derivatives at the knot
      // This call may use the stored versions, which is why this entire
      // function doesn't just call J_eulerian(s), after reading out s from
      // the knots.
      dshape_local_at_knot(ipt, psi, dpsids);
 
      // Also calculate the surface Vectors (derivatives wrt local coordinates)
      DenseMatrix<double> interpolated_A(n_dim_el, n_dim, 0.0);
 
      // Calculate positions and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over positional dofs
        for (unsigned k = 0; k < n_position_type; k++)
        {
          // Loop over coordinates
          for (unsigned i = 0; i < n_dim; i++)
          {
            // Loop over LOCAL derivative directions, to calculate the
            // tangent(s)
            for (unsigned j = 0; j < n_dim_el; j++)
            {
              interpolated_A(j, i) +=
                nodal_position_gen(l, bulk_position_type(k), i) *
                dpsids(l, k, j);
            }
          }
        }
      }
 
      // Now find the local deformed metric tensor from the tangent Vectors
      DenseMatrix<double> A(n_dim_el, n_dim_el, 0.0);
      for (unsigned i = 0; i < n_dim_el; i++)
      {
        for (unsigned j = 0; j < n_dim_el; j++)
        {
          // Take the dot product
          for (unsigned k = 0; k < n_dim; k++)
          {
            A(i, j) += interpolated_A(i, k) * interpolated_A(j, k);
          }
        }
      }
 
      // Find the determinant of the metric tensor
      double Adet = 0.0;
      switch (n_dim_el)
      {
        case 1:
          Adet = A(0, 0);
          break;
        case 2:
          Adet = A(0, 0) * A(1, 1) - A(0, 1) * A(1, 0);
          break;
        default:
          throw OomphLibError("Wrong dimension in FaceElement",
                              "FaceElement::J_eulerian_at_knot()",
                              OOMPH_EXCEPTION_LOCATION);
      }
 
 
      // Are we dead yet?
      if (Adet <= 0.0)
      {
        passed = false;
        return;
      }
    }
  }
 
  //=======================================================================
  /// Compute the tangent vector(s) and the outer unit normal
  /// vector at the specified local coordinate.
  /// In two spatial dimensions, a "tangent direction" is not required.
  /// In three spatial dimensions, a tangent direction is required
  /// (set via set_tangent_direction(...)), and we project the tanent direction
  /// on to the surface. The second tangent vector is taken to be the cross
  /// product of the projection and the unit normal.
  //=======================================================================
  void FaceElement::continuous_tangent_and_outer_unit_normal(
    const Vector<double>& s,
    Vector<Vector<double>>& tang_vec,
    Vector<double>& unit_normal) const
  {
    // Find the spatial dimension of the FaceElement
    const unsigned element_dim = dim();
 
    // Find the overall dimension of the problem
    //(assume that it's the same for all nodes)
    const unsigned spatial_dim = nodal_dimension();
 
#ifdef PARANOID
    // Check the number of local coordinates passed
    if (s.size() != element_dim)
    {
      std::ostringstream error_stream;
      error_stream
        << "Local coordinate s passed to outer_unit_normal() has dimension "
        << s.size() << std::endl
        << "but element dimension is " << element_dim << std::endl;
 
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // Check that if the Tangent_direction_pt is set, then
    // it is the same length as the spatial dimension.
    if (Tangent_direction_pt != 0 &&
        Tangent_direction_pt->size() != spatial_dim)
    {
      std::ostringstream error_stream;
      error_stream << "Tangent direction vector has dimension "
                   << Tangent_direction_pt->size() << std::endl
                   << "but spatial dimension is " << spatial_dim << std::endl;
 
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // Check the dimension of the normal vector
    if (unit_normal.size() != spatial_dim)
    {
      std::ostringstream error_stream;
      error_stream << "Unit normal passed to outer_unit_normal() has dimension "
                   << unit_normal.size() << std::endl
                   << "but spatial dimension is " << spatial_dim << std::endl;
 
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
 
    // The number of tangent vectors.
    unsigned ntangent_vec = tang_vec.size();
 
    // For the tangent vector,
    // if element_dim = 2, tang_vec is a 2D Vector of length 3.
    // if element_dim = 1, tang_vec is a 1D Vector of length 2.
    // if element_dim = 0, tang_vec is a 1D Vector of length 1.
    switch (element_dim)
    {
        // Point element, derived from a 1D element.
      case 0:
      {
        // Check that tang_vec is a 1D vector.
        if (ntangent_vec != 1)
        {
          std::ostringstream error_stream;
          error_stream
            << "This is a 0D FaceElement, we need one tangent vector.\n"
            << "You have given me " << tang_vec.size() << " vector(s).\n";
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
      break;
 
      // Line element, derived from a 2D element.
      case 1:
      {
        // Check that tang_vec is a 1D vector.
        if (ntangent_vec != 1)
        {
          std::ostringstream error_stream;
          error_stream
            << "This is a 1D FaceElement, we need one tangent vector.\n"
            << "You have given me " << tang_vec.size() << " vector(s).\n";
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
      break;
 
      // Plane element, derived from 3D element.
      case 2:
      {
        // Check that tang_vec is a 2D vector.
        if (ntangent_vec != 2)
        {
          std::ostringstream error_stream;
          error_stream
            << "This is a 2D FaceElement, we need two tangent vectors.\n"
            << "You have given me " << tang_vec.size() << " vector(s).\n";
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
      break;
      // There are no other elements!
      default:
 
        throw OomphLibError(
          "Cannot have a FaceElement with dimension higher than 2",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
        break;
    }
 
    // Check the lengths of each sub vectors.
    for (unsigned vec_i = 0; vec_i < ntangent_vec; vec_i++)
    {
      if (tang_vec[vec_i].size() != spatial_dim)
      {
        std::ostringstream error_stream;
        error_stream << "This problem has " << spatial_dim
                     << " spatial dimensions.\n"
                     << "But the " << vec_i << " tangent vector has size "
                     << tang_vec[vec_i].size() << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
 
#endif
 
 
    // Now let's consider the different element dimensions
    switch (element_dim)
    {
        // Point element, derived from a 1D element.
      case 0:
      {
        std::ostringstream error_stream;
        error_stream
          << "It is unclear how to calculate a tangent and normal vectors for "
          << "a point element.\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
      break;
 
      // Line element, derived from a 2D element, in this case
      // the normal is a mess of cross products
      // We need an interior direction, so we must find the local
      // derivatives in the BULK element
      case 1:
      {
        // Find the number of nodes in the bulk element
        const unsigned n_node_bulk = Bulk_element_pt->nnode();
        // Find the number of position types in the bulk element
        const unsigned n_position_type_bulk =
          Bulk_element_pt->nnodal_position_type();
 
        // Construct the local coordinate in the bulk element
        Vector<double> s_bulk(2);
        // Get the local coordinates in the bulk element
        get_local_coordinate_in_bulk(s, s_bulk);
 
        // Allocate storage for the shape functions and their derivatives wrt
        // local coordinates
        Shape psi(n_node_bulk, n_position_type_bulk);
        DShape dpsids(n_node_bulk, n_position_type_bulk, 2);
        // Get the value of the shape functions at the given local coordinate
        Bulk_element_pt->dshape_local(s_bulk, psi, dpsids);
 
        // Calculate all derivatives of the spatial coordinates
        // wrt local coordinates
        DenseMatrix<double> interpolated_dxds(2, spatial_dim, 0.0);
 
        // Loop over all parent nodes
        for (unsigned l = 0; l < n_node_bulk; l++)
        {
          // Loop over all position types in the bulk
          for (unsigned k = 0; k < n_position_type_bulk; k++)
          {
            // Loop over derivative direction
            for (unsigned j = 0; j < 2; j++)
            {
              // Loop over coordinate directions
              for (unsigned i = 0; i < spatial_dim; i++)
              {
                // Compute the spatial derivative
                interpolated_dxds(j, i) +=
                  Bulk_element_pt->nodal_position_gen(l, k, i) *
                  dpsids(l, k, j);
              }
            }
          }
        }
 
        // Initialise the tangent, interior tangent and normal vectors to zero
        // The idea is that even if the element is in a two-dimensional space,
        // the normal cannot be calculated without embedding the element in
        // three dimensions, in which case, the tangent and interior tangent
        // will have zero z-components.
        Vector<double> tangent(3, 0.0), interior_tangent(3, 0.0),
          normal(3, 0.0);
 
        // We must get the relationship between the coordinate along the face
        // and the local coordinates in the bulk element
        // We must also find an interior direction
        DenseMatrix<double> dsbulk_dsface(2, 1, 0.0);
        unsigned interior_direction = 0;
        get_ds_bulk_ds_face(s, dsbulk_dsface, interior_direction);
        // Load in the values for the tangents
        for (unsigned i = 0; i < spatial_dim; i++)
        {
          // Tangent to the face is the derivative wrt to the face coordinate
          // which is calculated using dsbulk_dsface and the chain rule
          tangent[i] = interpolated_dxds(0, i) * dsbulk_dsface(0, 0) +
                       interpolated_dxds(1, i) * dsbulk_dsface(1, 0);
          // Interior tangent to the face is the derivative in the interior
          // direction
          interior_tangent[i] = interpolated_dxds(interior_direction, i);
        }
 
        // Now the (3D) normal to the element is the interior tangent
        // crossed with the tangent
        VectorHelpers::cross(interior_tangent, tangent, normal);
 
        // We find the line normal by crossing the element normal with the
        // tangent
        Vector<double> full_normal(3);
        VectorHelpers::cross(normal, tangent, full_normal);
 
        // Copy the appropriate entries into the unit normal
        // Two or Three depending upon the spatial dimension of the system
        for (unsigned i = 0; i < spatial_dim; i++)
        {
          unit_normal[i] = full_normal[i];
        }
 
        // Finally normalise unit normal and multiply by the Normal_sign
        double length = VectorHelpers::magnitude(unit_normal);
        for (unsigned i = 0; i < spatial_dim; i++)
        {
          unit_normal[i] *= Normal_sign / length;
        }
 
        // Create the tangent vector
        tang_vec[0][0] = -unit_normal[1];
        tang_vec[0][1] = unit_normal[0];
      }
      break;
 
      // Plane element, derived from 3D element, in this case the normal
      // is just the cross product of the two surface tangents
      // We assume, therefore, that we have three spatial coordinates
      // and two surface coordinates
      // Then we need only to get the derivatives wrt the local coordinates
      // in this face element
      case 2:
      {
#ifdef PARANOID
        // Check that we actually have three spatial dimensions
        if (spatial_dim != 3)
        {
          std::ostringstream error_stream;
          error_stream << "There are only " << spatial_dim
                       << "coordinates at the nodes of this 2D FaceElement,\n"
                       << "which must have come from a 3D Bulk element\n";
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
#endif
 
        // Find the number of nodes in the element
        const unsigned n_node = this->nnode();
        // Find the number of position types
        const unsigned n_position_type = this->nnodal_position_type();
 
        // Allocate storage for the shape functions and their derivatives wrt
        // local coordinates
        Shape psi(n_node, n_position_type);
        DShape dpsids(n_node, n_position_type, 2);
        // Get the value of the shape functions at the given local coordinate
        this->dshape_local(s, psi, dpsids);
 
        // Calculate all derivatives of the spatial coordinates
        // wrt local coordinates
        Vector<Vector<double>> interpolated_dxds(2, Vector<double>(3, 0));
 
        // Loop over all nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over all position types
          for (unsigned k = 0; k < n_position_type; k++)
          {
            // Loop over derivative directions
            for (unsigned j = 0; j < 2; j++)
            {
              // Loop over coordinate directions
              for (unsigned i = 0; i < 3; i++)
              {
                // Compute the spatial derivative
                // Remember that we need to translate the position type
                // to its location in the bulk node
                interpolated_dxds[j][i] +=
                  this->nodal_position_gen(l, bulk_position_type(k), i) *
                  dpsids(l, k, j);
              }
            }
          }
        }
 
        // We now take the cross product of the two normal vectors
        VectorHelpers::cross(
          interpolated_dxds[0], interpolated_dxds[1], unit_normal);
 
        // Finally normalise unit normal
        double normal_length = VectorHelpers::magnitude(unit_normal);
 
        for (unsigned i = 0; i < spatial_dim; i++)
        {
          unit_normal[i] *= Normal_sign / normal_length;
        }
 
        // Next we create the continuous tangent vectors!
        if (Tangent_direction_pt != 0)
        // There is a general direction that the first tangent vector should
        // follow.
        {
          // Project the Tangent direction vector onto the surface.
          // Project Tangent_direction_pt D onto the plane P defined by
          // T1 and T2:
          // proj_P(D) = proj_T1(D) + proj_T2(D), where D is
          // Tangent_direction_pt, recall that proj_u(v) = (u.v)/(u.u) * u
 
          // Get the direction vector. The vector is NOT copied! :)
          Vector<double>& direction_vector = *Tangent_direction_pt;
 
#ifdef PARANOID
          // Check that the angle between the direction vector and the normal
          // is not less than 10 degrees
          if (VectorHelpers::angle(direction_vector, unit_normal) <
              (10.0 * MathematicalConstants::Pi / 180.0))
          {
            std::ostringstream err_stream;
            err_stream << "The angle between the unit normal and the \n"
                       << "direction vector is less than 10 degrees.";
            throw OomphLibError(err_stream.str(),
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
#endif
 
          // Calculate the two scalings, (u.v) / (u.u)
          double t1_scaling =
            VectorHelpers::dot(interpolated_dxds[0], direction_vector) /
            VectorHelpers::dot(interpolated_dxds[0], interpolated_dxds[0]);
 
          double t2_scaling =
            VectorHelpers::dot(interpolated_dxds[1], direction_vector) /
            VectorHelpers::dot(interpolated_dxds[1], interpolated_dxds[1]);
 
          // Finish off the projection.
          tang_vec[0][0] = t1_scaling * interpolated_dxds[0][0] +
                           t2_scaling * interpolated_dxds[1][0];
          tang_vec[0][1] = t1_scaling * interpolated_dxds[0][1] +
                           t2_scaling * interpolated_dxds[1][1];
          tang_vec[0][2] = t1_scaling * interpolated_dxds[0][2] +
                           t2_scaling * interpolated_dxds[1][2];
 
          // The second tangent vector is the cross product of
          // tang_vec[0] and the normal vector N.
          VectorHelpers::cross(tang_vec[0], unit_normal, tang_vec[1]);
 
          // Normalise the tangent vectors
          for (unsigned vec_i = 0; vec_i < 2; vec_i++)
          {
            // Get the length...
            double tang_length = VectorHelpers::magnitude(tang_vec[vec_i]);
 
            for (unsigned dim_i = 0; dim_i < spatial_dim; dim_i++)
            {
              tang_vec[vec_i][dim_i] /= tang_length;
            }
          }
        }
        else
        // A direction for the first tangent vector is not supplied, we do our
        // best to keep it continuous. But if the normal vector is aligned with
        // the z axis, then the tangent vector may "flip", even within elements.
        // This is because we check this by comparing doubles.
        // The code is copied from impose_parallel_outflow_element.h,
        // NO modification is done except for a few variable name changes.
        {
          double a, b, c;
          a = unit_normal[0];
          b = unit_normal[1];
          c = unit_normal[2];
 
          if (a != 0.0 || b != 0.0)
          {
            double a_sq = a * a;
            double b_sq = b * b;
            double c_sq = c * c;
 
            tang_vec[0][0] = -b / sqrt(a_sq + b_sq);
            tang_vec[0][1] = a / sqrt(a_sq + b_sq);
            tang_vec[0][2] = 0;
 
            double z = (a_sq + b_sq) / sqrt(a_sq * c_sq + b_sq * c_sq +
                                            (a_sq + b_sq) * (a_sq + b_sq));
 
            tang_vec[1][0] = -(a * c * z) / (a_sq + b_sq);
            tang_vec[1][1] = -(b * c * z) / (a_sq + b_sq);
            tang_vec[1][2] = z;
            // NB : we didn't use the fact that N is normalized,
            // that's why we have these unsimplified formulas
          }
          else if (c != 0.0)
          {
            if (!FaceElement::Ignore_discontinuous_tangent_warning)
            {
              std::ostringstream warn_stream;
              warn_stream
                << "I have detected that your normal vector is [0,0,1].\n"
                << "Since this function compares against 0.0, you may\n"
                << "get discontinuous tangent vectors.";
              OomphLibWarning(warn_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
            }
 
            tang_vec[0][0] = 1.0;
            tang_vec[0][1] = 0.0;
            tang_vec[0][2] = 0.0;
 
            tang_vec[1][0] = 0.0;
            tang_vec[1][1] = 1.0;
            tang_vec[1][2] = 0.0;
          }
          else
          {
            throw OomphLibError("You have a zero normal vector!! ",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
        }
      }
      break;
 
      default:
 
        throw OomphLibError(
          "Cannot have a FaceElement with dimension higher than 2",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
        break;
    }
  }
 
  //=======================================================================
  /// Compute the tangent vector(s) and the outer unit normal
  /// vector at the ipt-th integration point. This is a wrapper around
  /// continuous_tangent_and_outer_unit_normal(...) with the integration points
  /// converted into local coordinates.
  //=======================================================================
  void FaceElement::continuous_tangent_and_outer_unit_normal(
    const unsigned& ipt,
    Vector<Vector<double>>& tang_vec,
    Vector<double>& unit_normal) const
  {
    // Find the dimension of the element
    const unsigned element_dim = dim();
    // Find the local coordinates of the ipt-th integration point
    Vector<double> s(element_dim);
    for (unsigned i = 0; i < element_dim; i++)
    {
      s[i] = integral_pt()->knot(ipt, i);
    }
    // Call the outer unit normal function
    continuous_tangent_and_outer_unit_normal(s, tang_vec, unit_normal);
  }
 
  //=======================================================================
  /// Compute the outer unit normal at the specified local coordinate
  //=======================================================================
  void FaceElement::outer_unit_normal(const Vector<double>& s,
                                      Vector<double>& unit_normal) const
  {
    // Find the spatial dimension of the FaceElement
    const unsigned element_dim = dim();
 
    // Find the overall dimension of the problem
    //(assume that it's the same for all nodes)
    const unsigned spatial_dim = nodal_dimension();
 
#ifdef PARANOID
    // Check the number of local coordinates passed
    if (s.size() != element_dim)
    {
      std::ostringstream error_stream;
      error_stream
        << "Local coordinate s passed to outer_unit_normal() has dimension "
        << s.size() << std::endl
        << "but element dimension is " << element_dim << std::endl;
 
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // Check the dimension of the normal vector
    if (unit_normal.size() != spatial_dim)
    {
      std::ostringstream error_stream;
      error_stream << "Unit normal passed to outer_unit_normal() has dimension "
                   << unit_normal.size() << std::endl
                   << "but spatial dimension is " << spatial_dim << std::endl;
 
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    /*  //The spatial dimension of the bulk element will be element_dim+1
      const unsigned bulk_dim = element_dim + 1;
 
      //Find the number of nodes in the bulk element
      const unsigned n_node_bulk = Bulk_element_pt->nnode();
      //Find the number of position types in the bulk element
      const unsigned n_position_type_bulk =
       Bulk_element_pt->nnodal_position_type();
 
      //Construct the local coordinate in the bulk element
      Vector<double> s_bulk(bulk_dim);
      //Set the value of the bulk coordinate that is fixed on the face
      //s_bulk[s_fixed_index()] = s_fixed_value();
 
      //Set the other bulk coordinates
      //for(unsigned i=0;i<element_dim;i++) {s_bulk[bulk_s_index(i)] = s[i];}
 
      get_local_coordinate_in_bulk(s,s_bulk);
 
      //Allocate storage for the shape functions and their derivatives wrt
      //local coordinates
      Shape psi(n_node_bulk,n_position_type_bulk);
      DShape dpsids(n_node_bulk,n_position_type_bulk,bulk_dim);
      //Get the value of the shape functions at the given local coordinate
      Bulk_element_pt->dshape_local(s_bulk,psi,dpsids);
 
      //Calculate all derivatives of the spatial coordinates wrt local
      coordinates DenseMatrix<double> interpolated_dxds(bulk_dim,spatial_dim);
      //Initialise to zero
      for(unsigned j=0;j<bulk_dim;j++)
       {for(unsigned i=0;i<spatial_dim;i++) {interpolated_dxds(j,i) = 0.0;}}
 
      //Loop over all parent nodes
      for(unsigned l=0;l<n_node_bulk;l++)
       {
        //Loop over all position types in the bulk
        for(unsigned k=0;k<n_position_type_bulk;k++)
         {