elements.h

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// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
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// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
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// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for classes that define element objects
 
// Include guard to prevent multiple inclusions of the header
#ifndef OOMPH_GENERIC_ELEMENTS_HEADER
#define OOMPH_GENERIC_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include <map>
#include <deque>
#include <string>
#include <list>
 
// oomph-lib includes
#include "integral.h"
#include "nodes.h"
#include "geom_objects.h"
 
namespace oomph
{
  // Forward definition for time
  class Time;
 
 
  //========================================================================
  /// A Generalised Element class.
  ///
  /// The main components of a GeneralisedElement are:
  /// - pointers to its internal and external Data, which are stored together
  ///   in a single array so that we need only store one pointer.
  ///   The internal Data are placed at the beginning
  ///   of the array and the external Data are stored at the end.
  /// - a pointer to a global Time
  /// - a lookup table that establishes the relation between local
  ///   and global equation numbers.
  ///
  /// We also provide interfaces for functions that compute the
  /// element's Jacobian matrix and/or the Vector of residuals.
  /// In addition, an interface that returns a mass matrix --- the matrix
  /// of terms that multiply any time derivatives in the problem --- is
  /// also provided to permit explicit time-stepping and the solution
  /// of the generalised eigenproblems.
  //========================================================================
  class GeneralisedElement
  {
  private:
    /// Storage for the global equation numbers represented by the
    /// degrees of freedom in the element.
    unsigned long* Eqn_number;
 
    /// Storage for array of pointers to degrees of freedom ordered
    /// by local equation number. This information is not needed, except in
    /// continuation, bifurcation tracking and periodic orbit calculations.
    /// It is not set up unless the control flag
    /// Problem::Store_local_dof_pts_in_elements = true
    double** Dof_pt;
 
    /// Storage for pointers to internal and external data.
    /// The data is of the same type and so can be addressed by
    /// a single pointer. The idea is that the array is of
    /// total size Ninternal_data + Nexternal_data.
    /// The internal data are stored at the beginning of the array
    /// and the external data are stored at the end of the array.
    Data** Data_pt;
 
    /// Pointer to array storage for local equation numbers associated
    /// with internal and external data. Again, we save storage by using
    /// a single pointer to access this information. The first index of the
    /// array is of dimension Nineternal_data + Nexternal_data and the second
    /// index varies with the number of values stored at the data object.
    /// In the most general case, however, the scheme is as memory efficient
    /// as possible.
    int** Data_local_eqn;
 
    /// Number of degrees of freedom
    unsigned Ndof;
 
    /// Number of internal data
    unsigned Ninternal_data;
 
    /// Number of external data
    unsigned Nexternal_data;
 
    /// Storage for boolean flags of size Ninternal_data + Nexternal_data
    /// that correspond to the data used in the element. The flags
    /// indicate whether the particular
    /// internal or external data should be included in the general
    /// finite-difference loop in fill_in_jacobian_from_internal_by_fd() or
    /// fill_in_jacobian_from_external_by_fd(). The default is that all
    /// data WILL be included in the finite-difference loop, but in many
    /// circumstances it is possible to treat certain (external) data
    /// analytically. The use of an STL vector is optimal for memory
    /// use because the booleans are represented as single-bits.
    std::vector<bool> Data_fd;
 
  protected:
#ifdef OOMPH_HAS_MPI
 
    /// Non-halo processor ID for Data; -1 if it's not a halo.
    int Non_halo_proc_ID;
 
    /// Does this element need to be kept as a halo element during a distribute?
    bool Must_be_kept_as_halo;
 
#endif
 
    /// 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(). The boolean indicates whether
    /// the datum should be included in the general finite-difference loop
    /// when calculating the jacobian. The default value is true, i.e.
    /// the data will be included in the finite differencing.
    unsigned add_internal_data(Data* const& data_pt, const bool& fd = true);
 
 
    /// Return the status of the boolean flag indicating whether
    /// the internal data is included in the finite difference loop
    inline bool internal_data_fd(const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Return the value
      return Data_fd[i];
    }
 
 
    /// Set the boolean flag to exclude the internal datum from
    /// the finite difference loop when computing the jacobian matrix
    inline void exclude_internal_data_fd(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Set the value
      Data_fd[i] = false;
    }
 
    /// Set the boolean flag to include the internal datum in
    /// the finite difference loop when computing the jacobian matrix
    inline void include_internal_data_fd(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Set the value
      Data_fd[i] = true;
    }
 
    /// Clear the storage for the global equation numbers
    /// and pointers to dofs (if stored)
    void clear_global_eqn_numbers()
    {
      delete[] Eqn_number;
      Eqn_number = 0;
      delete[] Dof_pt;
      Dof_pt = 0;
      Ndof = 0;
    }
 
    /// Add the contents of the queue global_eqn_numbers
    /// to the local storage for the local-to-global translation scheme.
    /// It is essential that the entries in the queue are added IN ORDER
    /// i.e. from the front.
    void add_global_eqn_numbers(
      std::deque<unsigned long> const& global_eqn_numbers,
      std::deque<double*> const& global_dof_pt);
 
    /// Empty dense matrix used as a dummy argument to combined
    /// residual and jacobian functions in the case when only the residuals
    /// are being assembled
    static DenseMatrix<double> Dummy_matrix;
 
    /// Static storage for deque used to add_global_equation_numbers
    /// when pointers to the dofs in each element are not required
    static std::deque<double*> Dof_pt_deque;
 
    /// Assign the local equation numbers for the internal
    /// and external Data
    /// This must be called after the global equation numbers have all been
    /// assigned. It is virtual so that it can be overloaded by
    /// ElementWithExternalElements so that any external data from the
    /// external elements in included in the numbering scheme.
    /// If the boolean argument is true then pointers to the dofs will be
    /// stored in Dof_pt
    virtual void assign_internal_and_external_local_eqn_numbers(
      const bool& store_local_dof_pt);
 
    /// Assign all the local equation numbering schemes that can
    /// be applied generically for the element. In most cases, this is the
    /// function that will be overloaded by inherited classes. It is required
    /// to ensure that assign_additional_local_eqn_numbers() can always be
    /// called after ALL other local equation numbering has been performed.
    /// The default for the GeneralisedElement is simply to call internal
    /// and external local equation numbering.
    /// If the boolean argument is true then pointers to the dofs will be stored
    /// in Dof_pt
    virtual inline void assign_all_generic_local_eqn_numbers(
      const bool& store_local_dof_pt)
    {
      this->assign_internal_and_external_local_eqn_numbers(store_local_dof_pt);
    }
 
    /// Setup any additional look-up schemes for local equation numbers.
    /// Examples of use include using local storage to refer to explicit
    /// degrees of freedom. The additional memory cost of such storage may
    /// or may not be offset by fast local access.
    virtual void assign_additional_local_eqn_numbers() {}
 
    /// Return the local equation number corresponding to the j-th
    /// value stored at the i-th internal data
    inline int internal_local_eqn(const unsigned& i, const unsigned& j) const
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
      else
      {
        unsigned n_value = internal_data_pt(i)->nvalue();
        if (j >= n_value)
        {
          std::ostringstream error_message;
          error_message << "Range Error: value " << j << " at internal data "
                        << i << " is not in the range (0," << n_value - 1
                        << ")";
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
#endif
      // Check that the data has been allocated
#ifdef PARANOID
      if (Data_local_eqn == 0)
      {
        throw OomphLibError(
          "Internal local equation numbers have not been allocated",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Return the local equation number as stored in the Data_local_eqn array
      return Data_local_eqn[i][j];
    }
 
    /// Return the local equation number corresponding to the j-th
    /// value stored at the i-th external data
    inline int external_local_eqn(const unsigned& i, const unsigned& j)
    {
      // Check that the data has been allocated
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: External data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
      else
      {
        unsigned n_value = external_data_pt(i)->nvalue();
        if (j >= n_value)
        {
          std::ostringstream error_message;
          error_message << "Range Error: value " << j << " at internal data "
                        << i << " is not in the range (0," << n_value - 1
                        << ")";
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
#endif
#ifdef PARANOID
      if (Data_local_eqn == 0)
      {
        throw OomphLibError(
          "External local equation numbers have not been allocated",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Return the  local equation number stored as the j-th value of the
      // i-th external data object.
      return Data_local_eqn[Ninternal_data + i][j];
    }
 
    /// Add the elemental contribution to the residuals vector. Note that
    /// this function will NOT initialise the residuals vector. It must be
    /// called after the residuals vector has been initialised to zero.
    virtual void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      std::string error_message =
        "Empty fill_in_contribution_to_residuals() has been called.\n";
      error_message +=
        "This function is called from the default implementations of\n";
      error_message += "get_residuals() and get_jacobian();\n";
      error_message +=
        "and must calculate the residuals vector without initialising any of ";
      error_message += "its entries.\n\n";
 
      error_message +=
        "If you do not wish to use these defaults, you must overload both\n";
      error_message += "get_residuals() and get_jacobian(), which must "
                       "initialise the entries\n";
      error_message += "of the residuals vector and jacobian matrix to zero.\n";
 
      error_message += "N.B. the default get_jacobian() function employs "
                       "finite differencing\n";
 
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Calculate the contributions to the jacobian from the internal
    /// degrees of freedom using finite differences.
    /// This version of the function assumes that the residuals vector has
    /// already been calculated. 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 fill_in_jacobian_from_internal_by_fd(Vector<double>& residuals,
                                              DenseMatrix<double>& jacobian,
                                              const bool& fd_all_data = false);
 
    /// Calculate the contributions to the jacobian from the internal
    /// degrees of freedom using finite differences. This version computes
    /// the residuals vector before calculating the jacobian terms.
    /// 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 fill_in_jacobian_from_internal_by_fd(DenseMatrix<double>& jacobian,
                                              const bool& fd_all_data = false)
    {
      // Allocate storage for the residuals
      Vector<double> residuals(Ndof, 0.0);
      // Get the residuals for the entire element
      get_residuals(residuals);
      // Call the jacobian calculation
      fill_in_jacobian_from_internal_by_fd(residuals, jacobian, fd_all_data);
    }
 
 
    /// Calculate the contributions to the jacobian from the external
    /// degrees of freedom using finite differences.
    /// This version of the function assumes that the residuals vector has
    /// already been calculated.
    /// If the boolean argument is true, the finite
    /// differencing will be performed for all external 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 fill_in_jacobian_from_external_by_fd(Vector<double>& residuals,
                                              DenseMatrix<double>& jacobian,
                                              const bool& fd_all_data = false);
 
 
    /// Calculate the contributions to the jacobian from the external
    /// degrees of freedom using finite differences. This version computes
    /// the residuals vector before calculating the jacobian terms.
    /// 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 fill_in_jacobian_from_external_by_fd(DenseMatrix<double>& jacobian,
                                              const bool& fd_all_data = false)
    {
      // Allocate storage for a residuals vector
      Vector<double> residuals(Ndof);
      // Get the residuals for the entire element
      get_residuals(residuals);
      // Call the jacobian calculation
      fill_in_jacobian_from_external_by_fd(residuals, jacobian, fd_all_data);
    }
 
    /// Function that is called before the finite differencing of
    /// any internal data. This may be overloaded to update any dependent
    /// data before finite differencing takes place.
    virtual inline void update_before_internal_fd() {}
 
    /// Function that is call after the finite differencing of
    /// the internal data. This may be overloaded to reset any dependent
    /// variables that may have changed during the finite differencing.
    virtual inline void reset_after_internal_fd() {}
 
    /// Function called within the finite difference loop for
    /// internal data after a change in any values in the i-th
    /// internal data object.
    virtual inline void update_in_internal_fd(const unsigned& i) {}
 
    /// Function called within the finite difference loop for
    /// internal data after the values in the i-th external data object
    /// are reset. The default behaviour is to call the update function.
    virtual inline void reset_in_internal_fd(const unsigned& i)
    {
      update_in_internal_fd(i);
    }
 
 
    /// Function that is called before the finite differencing of
    /// any external data. This may be overloaded to update any dependent
    /// data before finite differencing takes place.
    virtual inline void update_before_external_fd() {}
 
    /// Function that is call after the finite differencing of
    /// the external data. This may be overloaded to reset any dependent
    /// variables that may have changed during the finite differencing.
    virtual inline void reset_after_external_fd() {}
 
    /// Function called within the finite difference loop for
    /// external data after a change in any values in the i-th
    /// external data object
    virtual inline void update_in_external_fd(const unsigned& i) {}
 
    /// Function called within the finite difference loop for
    /// external data after the values in the i-th external data object
    /// are reset. The default behaviour is to call the update function.
    virtual inline void reset_in_external_fd(const unsigned& i)
    {
      update_in_external_fd(i);
    }
 
    /// Add the elemental contribution to the jacobian matrix.
    /// and the residuals vector. Note that
    /// this function will NOT initialise the residuals vector or the jacobian
    /// matrix. It must be called after the residuals vector and
    /// jacobian matrix have been initialised to zero. The default
    /// is to use finite differences to calculate the jacobian
    virtual void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                  DenseMatrix<double>& jacobian)
    {
      // Add the contribution to the residuals
      fill_in_contribution_to_residuals(residuals);
      // Allocate storage for the full residuals (residuals of entire element)
      Vector<double> full_residuals(Ndof);
      // Get the residuals for the entire element
      get_residuals(full_residuals);
      // Calculate the contributions from the internal dofs
      //(finite-difference the lot by default)
      fill_in_jacobian_from_internal_by_fd(full_residuals, jacobian, true);
      // Calculate the contributions from the external dofs
      //(finite-difference the lot by default)
      fill_in_jacobian_from_external_by_fd(full_residuals, jacobian, true);
    }
 
    /// Add the elemental contribution to the mass matrix matrix.
    /// and the residuals vector. Note that
    /// this function should 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
    virtual void fill_in_contribution_to_mass_matrix(
      Vector<double>& residuals, DenseMatrix<double>& mass_matrix);
 
    /// 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.
    virtual void fill_in_contribution_to_jacobian_and_mass_matrix(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix);
 
    /// 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 to use finite differences to calculate
    /// the derivatives.
    virtual void fill_in_contribution_to_dresiduals_dparameter(
      double* const& parameter_pt, Vector<double>& dres_dparam);
 
 
    /// 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.
    virtual void fill_in_contribution_to_djacobian_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam);
 
    /// Add the elemental contribution to the derivative of the
    /// jacobian matrix,
    /// mass matrix and the residuals vector with respect to the
    /// passed 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.
    virtual void 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);
 
 
    /// 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}
    virtual void fill_in_contribution_to_hessian_vector_products(
      Vector<double> const& Y,
      DenseMatrix<double> const& C,
      DenseMatrix<double>& product);
 
    /// Fill in the contribution to the inner products between given
    /// pairs of history values
    virtual void fill_in_contribution_to_inner_products(
      Vector<std::pair<unsigned, unsigned>> const& history_index,
      Vector<double>& inner_product);
 
    /// Fill in the contributions to the vectors that when taken
    /// as dot product with other history values give the inner product
    /// over the element
    virtual void fill_in_contribution_to_inner_product_vectors(
      Vector<unsigned> const& history_index,
      Vector<Vector<double>>& inner_product_vector);
 
  public:
    /// Constructor: Initialise all pointers and all values to zero
    GeneralisedElement()
      : Eqn_number(0),
        Dof_pt(0),
        Data_pt(0),
        Data_local_eqn(0),
        Ndof(0),
        Ninternal_data(0),
        Nexternal_data(0)
#ifdef OOMPH_HAS_MPI
        ,
        Non_halo_proc_ID(-1),
        Must_be_kept_as_halo(false)
#endif
    {
    }
 
    /// Virtual destructor to clean up any memory allocated by the object.
    virtual ~GeneralisedElement();
 
    /// Broken copy constructor
    GeneralisedElement(const GeneralisedElement&) = delete;
 
    /// Broken assignment operator
    void operator=(const GeneralisedElement&) = delete;
 
    /// Return a pointer to i-th internal data object.
    Data*& internal_data_pt(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Data_pt[i];
    }
 
    /// Return a pointer to i-th internal data object (const version)
    Data* const& internal_data_pt(const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (i >= Ninternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Ninternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Data_pt[i];
    }
 
 
    /// Return a pointer to i-th external data object.
    Data*& external_data_pt(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: External data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Data_pt[Ninternal_data + i];
    }
 
    /// Return a pointer to i-th external data object (const version)
    Data* const& external_data_pt(const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: External data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Data_pt[Ninternal_data + i];
    }
 
    /// Static boolean to suppress warnings about repeated internal
    /// data. Defaults to false.
    static bool Suppress_warning_about_repeated_internal_data;
 
    /// Static boolean to suppress warnings about repeated external
    /// data. Defaults to true.
    static bool Suppress_warning_about_repeated_external_data;
 
    /// Return the global equation number corresponding to the
    /// ieqn_local-th local equation number
    unsigned long eqn_number(const unsigned& ieqn_local) const
    {
#ifdef RANGE_CHECKING
      if (ieqn_local >= Ndof)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Equation number " << ieqn_local
                      << " is not in the range (0," << Ndof - 1 << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Eqn_number[ieqn_local];
    }
 
 
    /// Return the local equation number corresponding to the
    /// ieqn_global-th global equation number. Returns minus one (-1) if there
    /// is
    /// no local degree of freedom corresponding to the chosen global equation
    /// number
    int local_eqn_number(const unsigned long& ieqn_global) const
    {
      // Cache the number of degrees of freedom in the element
      const unsigned n_dof = this->Ndof;
      // Loop over the local equation numbers
      for (unsigned n = 0; n < n_dof; n++)
      {
        // If the global equation numbers match return
        if (ieqn_global == Eqn_number[n])
        {
          return n;
        }
      }
 
      // If we've got all the way to the end the number has not been found
      // return minus one.
      return -1;
    }
 
 
    /// Add a (pointer to an) external data object to the element and return its
    /// index (i.e. the index required to obtain it from
    /// the access function \c external_data_pt(...). The optional boolean
    /// flag indicates whether the data should be included in the
    /// general finite-difference loop when calculating the jacobian.
    /// The default value is true, i.e. the data will be included in the
    /// finite-differencing
    unsigned add_external_data(Data* const& data_pt, const bool& fd = true);
 
    /// Return the status of the boolean flag indicating whether
    /// the external data is included in the finite difference loop
    inline bool external_data_fd(const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Internal data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Return the value
      return Data_fd[Ninternal_data + i];
    }
 
 
    /// Set the boolean flag to exclude the external datum from the
    /// the finite difference loop when computing the jacobian matrix
    inline void exclude_external_data_fd(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: External data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Set the value
      Data_fd[Ninternal_data + i] = false;
    }
 
    /// Set the boolean flag to include the external datum in the
    /// the finite difference loop when computing the jacobian matrix
    inline void include_external_data_fd(const unsigned& i)
    {
#ifdef RANGE_CHECKING
      if (i >= Nexternal_data)
      {
        std::ostringstream error_message;
        error_message << "Range Error: External data " << i
                      << " is not in the range (0," << Nexternal_data - 1
                      << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Set the value
      Data_fd[Ninternal_data + i] = true;
    }
 
    /// Flush all external data
    void flush_external_data();
 
    /// Flush the object addressed by data_pt from the external data array
    void flush_external_data(Data* const& data_pt);
 
    /// Return the number of internal data objects.
    unsigned ninternal_data() const
    {
      return Ninternal_data;
    }
 
    /// Return the number of external data objects.
    unsigned nexternal_data() const
    {
      return Nexternal_data;
    }
 
    /// Return the number of equations/dofs in the element.
    unsigned ndof() const
    {
      return Ndof;
    }
 
    /// Return the vector of dof values at time level t
    void dof_vector(const unsigned& t, Vector<double>& dof)
    {
      // Check that the internal storage has been set up
#ifdef PARANOID
      if (Dof_pt == 0)
      {
        std::stringstream error_stream;
        error_stream << "Internal dof array not set up in element.\n"
                     << "In order to set it up you must call\n"
                     << "   Problem::enable_store_local_dof_in_elements()\n"
                     << "before the call to Problem::assign_eqn_numbers()\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Resize the vector
      dof.resize(this->Ndof);
      // Loop over the vector and fill in the desired values
      for (unsigned i = 0; i < this->Ndof; ++i)
      {
        dof[i] = Dof_pt[i][t];
      }
    }
 
    /// Return the vector of pointers to dof values
    void dof_pt_vector(Vector<double*>& dof_pt)
    {
      // Check that the internal storage has been set up
#ifdef PARANOID
      if (Dof_pt == 0)
      {
        std::stringstream error_stream;
        error_stream << "Internal dof array not set up in element.\n"
                     << "In order to set it up you must call\n"
                     << "   Problem::enable_store_local_dof_in_elements()\n"
                     << "before the call to Problem::assign_eqn_numbers()\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Resize the vector
      dof_pt.resize(this->Ndof);
      // Loop over the vector and fill in the desired values
      for (unsigned i = 0; i < this->Ndof; ++i)
      {
        dof_pt[i] = Dof_pt[i];
      }
    }
 
 
    /// Set the timestepper associated with the i-th internal data
    /// object
    void set_internal_data_time_stepper(const unsigned& i,
                                        TimeStepper* const& time_stepper_pt,
                                        const bool& preserve_existing_data)
    {
      this->internal_data_pt(i)->set_time_stepper(time_stepper_pt,
                                                  preserve_existing_data);
    }
 
    /// Assign the global equation numbers to the internal Data.
    /// The arguments are the current highest global equation number
    /// (which will be incremented) and a Vector of pointers to the
    /// global variables (to which any unpinned values in the internal Data are
    /// added).
    void assign_internal_eqn_numbers(unsigned long& global_number,
                                     Vector<double*>& 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 describe_dofs(std::ostream& out,
                       const std::string& current_string) const;
 
    /// Function to describe the local dofs of the element. The ostream
    /// specifies the output stream to which the description
    /// is written; the string stores the currently
    /// assembled output that is ultimately written to the
    /// output stream by Data::describe_dofs(...); it is typically
    /// built up incrementally as we descend through the
    /// call hierarchy of this function when called from
    /// Problem::describe_dofs(...)
    virtual void describe_local_dofs(std::ostream& out,
                                     const std::string& current_string) const;
 
    /// Add pointers to the internal data values to map indexed
    /// by the global equation number.
    void add_internal_value_pt_to_map(
      std::map<unsigned, double*>& 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 add_internal_data_values_to_vector(Vector<double>& 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 read_internal_data_values_from_vector(
      const Vector<double>& vector_of_values, unsigned& index);
 
    /// Add all equation numbers associated with internal data
    /// to the vector in the internal storage order
    void add_internal_eqn_numbers_to_vector(
      Vector<long>& 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 read_internal_eqn_numbers_from_vector(
      const Vector<long>& vector_of_eqn_numbers, unsigned& index);
 
#endif
 
    /// Setup the arrays of local equation numbers for the element.
    /// If the optional boolean argument is true, then pointers to the
    /// associated degrees of freedom are stored locally in the array Dof_pt
    virtual void assign_local_eqn_numbers(const bool& store_local_dof_pt);
 
    /// Complete the setup of any additional dependencies
    /// that the element may have. Empty virtual function that may be
    /// overloaded for specific derived elements. Used, e.g., for elements
    /// with algebraic node update functions to determine the "geometric
    /// Data", i.e. the Data that affects the element's shape.
    /// This function is called (for all elements) at the very beginning of the
    /// equation numbering procedure to ensure that all dependencies
    /// are accounted for.
    virtual void complete_setup_of_dependencies() {}
 
    /// Calculate the vector of residuals of the equations in the
    /// element. By default initialise the vector to zero and then call the
    /// fill_in_contribution_to_residuals() function. Note that this entire
    /// function can be overloaded if desired.
    virtual void get_residuals(Vector<double>& residuals)
    {
      // Zero the residuals vector
      residuals.initialise(0.0);
      // Add the elemental contribution to the residuals vector
      fill_in_contribution_to_residuals(residuals);
    }
 
    /// Calculate the elemental Jacobian matrix "d equation / d
    /// variable".
    virtual void get_jacobian(Vector<double>& residuals,
                              DenseMatrix<double>& jacobian)
    {
      // Zero the residuals vector
      residuals.initialise(0.0);
      // Zero the jacobian matrix
      jacobian.initialise(0.0);
      // Add the elemental contribution to the residuals vector
      fill_in_contribution_to_jacobian(residuals, jacobian);
    }
 
    /// Calculate the residuals and the elemental "mass" matrix, the
    /// matrix that multiplies the time derivative terms in a problem.
    virtual void get_mass_matrix(Vector<double>& residuals,
                                 DenseMatrix<double>& mass_matrix)
    {
      // Zero the residuals vector
      residuals.initialise(0.0);
      // Zero the mass matrix
      mass_matrix.initialise(0.0);
      // Add the elemental contribution to the vector and matrix
      fill_in_contribution_to_mass_matrix(residuals, mass_matrix);
    }
 
    /// Calculate the residuals and jacobian and elemental "mass" matrix,
    /// the matrix that multiplies the time derivative terms.
    virtual void get_jacobian_and_mass_matrix(Vector<double>& residuals,
                                              DenseMatrix<double>& jacobian,
                                              DenseMatrix<double>& mass_matrix)
    {
      // Zero the residuals vector
      residuals.initialise(0.0);
      // Zero the jacobian matrix
      jacobian.initialise(0.0);
      // Zero the mass matrix
      mass_matrix.initialise(0.0);
      // Add the elemental contribution to the vector and matrix
      fill_in_contribution_to_jacobian_and_mass_matrix(
        residuals, jacobian, mass_matrix);
    }
 
 
    /// Calculate the derivatives of the residuals with respect to
    /// a parameter
    virtual void get_dresiduals_dparameter(double* const& parameter_pt,
                                           Vector<double>& dres_dparam)
    {
      // Zero the dres_dparam vector
      dres_dparam.initialise(0.0);
      // Add the elemental contribution to the residuals vector
      this->fill_in_contribution_to_dresiduals_dparameter(parameter_pt,
                                                          dres_dparam);
    }
 
    /// Calculate the derivatives of the elemental Jacobian matrix
    /// and residuals with respect to a parameter
    virtual void get_djacobian_dparameter(double* const& parameter_pt,
                                          Vector<double>& dres_dparam,
                                          DenseMatrix<double>& djac_dparam)
    {
      // Zero the residuals vector
      dres_dparam.initialise(0.0);
      // Zero the jacobian matrix
      djac_dparam.initialise(0.0);
      // Add the elemental contribution to the residuals vector
      this->fill_in_contribution_to_djacobian_dparameter(
        parameter_pt, dres_dparam, djac_dparam);
    }
 
    /// Calculate the derivatives of the elemental Jacobian matrix
    /// mass matrix and residuals with respect to a parameter
    virtual void get_djacobian_and_dmass_matrix_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam)
    {
      // Zero the residuals derivative vector
      dres_dparam.initialise(0.0);
      // Zero the jacobian derivative  matrix
      djac_dparam.initialise(0.0);
      // Zero the mass matrix derivative
      dmass_matrix_dparam.initialise(0.0);
      // Add the elemental contribution to the residuals vector and matrices
      this->fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter(
        parameter_pt, dres_dparam, djac_dparam, dmass_matrix_dparam);
    }
 
 
    /// Calculate 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}
    virtual void get_hessian_vector_products(Vector<double> const& Y,
                                             DenseMatrix<double> const& C,
                                             DenseMatrix<double>& product)
    {
      // Initialise the product to zero
      product.initialise(0.0);
      /// Add the elemental contribution to the product
      this->fill_in_contribution_to_hessian_vector_products(Y, C, product);
    }
 
    /// Return the vector of inner product of the given pairs of
    /// history values
    virtual void get_inner_products(
      Vector<std::pair<unsigned, unsigned>> const& history_index,
      Vector<double>& inner_product)
    {
      inner_product.initialise(0.0);
      this->fill_in_contribution_to_inner_products(history_index,
                                                   inner_product);
    }
 
    ///  Compute the vectors that when taken as a dot product with
    /// other history values give the inner product over the element.
    virtual void get_inner_product_vectors(
      Vector<unsigned> const& history_index,
      Vector<Vector<double>>& inner_product_vector)
    {
      const unsigned n_inner_product = inner_product_vector.size();
      for (unsigned i = 0; i < n_inner_product; ++i)
      {
        inner_product_vector[i].initialise(0.0);
      }
      this->fill_in_contribution_to_inner_product_vectors(history_index,
                                                          inner_product_vector);
    }
 
 
    /// Self-test: Have all internal values been classified as
    /// pinned/unpinned? Return 0 if OK.
    virtual unsigned self_test();
 
 
    /// Compute norm of solution -- broken virtual can be overloaded
    /// by element writer to implement whatever norm is desired for
    /// the specific element
    virtual void compute_norm(Vector<double>& norm)
    {
      std::string error_message =
        "compute_norm(...) undefined for this element\n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Compute norm of solution -- broken virtual can be overloaded
    /// by element writer to implement whatever norm is desired for
    /// the specific element
    virtual void compute_norm(double& norm)
    {
      std::string error_message = "compute_norm undefined for this element \n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
#ifdef OOMPH_HAS_MPI
 
    /// Label the element as halo and specify processor that holds
    /// non-halo counterpart
    void set_halo(const unsigned& non_halo_proc_ID)
    {
      Non_halo_proc_ID = non_halo_proc_ID;
    }
 
    /// Label the element as not being a halo
    void set_nonhalo()
    {
      Non_halo_proc_ID = -1;
    }
 
    /// Is this element a halo?
    bool is_halo() const
    {
      return (Non_halo_proc_ID != -1);
    }
 
    /// ID of processor ID that holds non-halo counterpart
    /// of halo element; negative if not a halo.
    int non_halo_proc_ID()
    {
      return Non_halo_proc_ID;
    }
 
    /// Insist that this element be kept as a halo element during a distribute?
    void set_must_be_kept_as_halo()
    {
      Must_be_kept_as_halo = true;
    }
 
    /// Do not insist that this element be kept as a halo element during
    /// distribution
    void unset_must_be_kept_as_halo()
    {
      Must_be_kept_as_halo = false;
    }
 
    /// Test whether the element must be kept as a halo element
    bool must_be_kept_as_halo() const
    {
      return Must_be_kept_as_halo;
    }
 
#endif
 
    /// Double used for the default finite difference step in elemental
    /// jacobian calculations
    static double Default_fd_jacobian_step;
 
    /// The number of types of degrees of freedom in this element
    /// are sub-divided into
    virtual unsigned ndof_types() const
    {
      // error message stream
      std::ostringstream error_message;
      error_message << "ndof_types() const has not been implemented for this \n"
                    << "element\n"
                    << std::endl;
      // throw error
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Create a list of pairs for the unknowns that this element
    /// is "in charge of" -- ignore any unknowns associated with external
    /// \c Data. The first entry in each pair must contain the global equation
    /// number of the unknown, while the second one contains the number
    /// of the DOF type that this unknown is associated with.
    /// (The function can obviously only be called if the equation numbering
    /// scheme has been set up.)
    virtual void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const
    {
      // error message stream
      std::ostringstream error_message;
      error_message << "get_dof_numbers_for_unknowns() const has not been \n"
                    << " implemented for this element\n"
                    << std::endl;
      // throw error
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
  };
 
  /// Enumeration a finite element's geometry "type". Either "Q" (square,
  /// cubeoid like) or "T" (triangle, tetrahedron).
  namespace ElementGeometry
  {
    enum ElementGeometry
    {
      Q,
      T
    };
  }
 
  // Forward class definitions that are used in FiniteElements
  class FaceElement;
  class MacroElement;
  class TimeStepper;
  class Shape;
  class DShape;
  class Integral;
 
  //===========================================================================
  /// Helper namespace for tolerances and iterations within the Newton
  /// method used in the locate_zeta function in FiniteElements.
  //===========================================================================
  namespace Locate_zeta_helpers
  {
    /// Convergence tolerance for the Newton solver
    extern double Newton_tolerance;
 
    /// Maximum number of Newton iterations
    extern unsigned Max_newton_iterations;
 
    /// Multiplier for (zeta-based) outer radius of element used for
    /// deciding that point is outside element. Set to negative value
    /// to suppress test.
    extern double Radius_multiplier_for_fast_exit_from_locate_zeta;
 
    /// 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 for locate_zeta)
    extern unsigned N_local_points;
 
  } // namespace Locate_zeta_helpers
 
 
  /// Typedef for the function that translates the face coordinate
  /// to the coordinate in the bulk element
  typedef void (*CoordinateMappingFctPt)(const Vector<double>& s,
                                         Vector<double>& s_bulk);
 
  /// Typedef for the function that returns the partial derivative
  /// of the local coordinates in the bulk element
  /// with respect to the coordinates along the face.
  /// In addition this function returns an index of one of the
  /// bulk local coordinates that varies away from the edge
  typedef void (*BulkCoordinateDerivativesFctPt)(
    const Vector<double>& s,
    DenseMatrix<double>& ds_bulk_dsface,
    unsigned& interior_direction);
 
 
  //========================================================================
  /// A general Finite Element class.
  ///
  /// The main components of a FiniteElement are:
  /// - pointers to its Nodes
  /// - pointers to its internal Data (inherited from GeneralisedElement)
  /// - pointers to its external Data (inherited from GeneralisedElement)
  /// - a pointer to a spatial integration scheme
  /// - a pointer to the global  Time object (inherited from GeneralisedElement)
  /// - a lookup table which establishes the relation between local
  ///   and global equation numbers (inherited from GeneralisedElement)
  ///
  /// We also provide interfaces for functions that compute the
  /// element's Jacobian matrix and/or the Vector of residuals
  /// (inherited from GeneralisedElement) plus various output routines.
  //========================================================================
  class FiniteElement : public virtual GeneralisedElement, public GeomObject
  {
  private:
    /// Pointer to the spatial integration scheme
    Integral* Integral_pt;
 
    /// Storage for pointers to the nodes in the element
    Node** Node_pt;
 
    /// Storage for the local equation numbers associated with
    /// the values stored at the nodes
    int** Nodal_local_eqn;
 
    /// Number of nodes in the element
    unsigned Nnode;
 
    /// The spatial dimension of the element, i.e. the number
    /// of local coordinates used to parametrize it.
    unsigned Elemental_dimension;
 
    /// The spatial dimension of the nodes in the element.
    /// We assume that nodes have the same spatial dimension, because
    /// we cannot think of any "real" problems for which that would not
    /// be the case.
    unsigned Nodal_dimension;
 
    /// The number of coordinate types required to interpolate
    /// the element's geometry between the nodes. For Lagrange elements
    /// it is 1 (the default). It must be over-ridden by using
    /// the set_nposition_type() function in the constructors of elements
    /// that use generalised coordinate, e.g. for 1D Hermite elements
    /// Nnodal_position_types =2.
    unsigned Nnodal_position_type;
 
  protected:
    /// Assemble the jacobian matrix for the mapping from local
    /// to Eulerian coordinates, given the derivatives of the shape function
    /// w.r.t the local coordinates.
    virtual void assemble_local_to_eulerian_jacobian(
      const DShape& dpsids, DenseMatrix<double>& jacobian) const;
 
    /// Assemble the the "jacobian" matrix of second derivatives of the
    /// mapping from local to Eulerian coordinates, given
    /// the second derivatives of the shape functions w.r.t. local coordinates.
    virtual void assemble_local_to_eulerian_jacobian2(
      const DShape& d2psids, DenseMatrix<double>& jacobian2) const;
 
    /// Assemble the covariant Eulerian base vectors, assuming that
    /// the derivatives of the shape functions with respect to the local
    /// coordinates have already been constructed.
    virtual void assemble_eulerian_base_vectors(
      const DShape& dpsids, DenseMatrix<double>& interpolated_G) const;
 
    /// Default return value for required_nvalue(n) which gives the
    /// number of "data" values required by the element at node n; for example,
    /// solving a Poisson equation would required only one "data" value at each
    /// node. The defaults is set to zero, because a general element is
    /// problem-less.
    static const unsigned Default_Initial_Nvalue;
 
    /// Default value for the tolerance to be used when locating nodes
    /// via local coordinates
    static const double Node_location_tolerance;
 
  public:
    /// Set the dimension of the element and initially set
    /// the dimension of the nodes to be the same as the dimension of the
    /// element.
    void set_dimension(const unsigned& dim)
    {
      Elemental_dimension = dim;
      Nodal_dimension = dim;
    }
 
    /// Set the dimension of the nodes in the element. This will
    /// typically only be required when constructing FaceElements or
    /// in beam and shell type elements where a lower dimensional surface
    /// is embedded in a higher dimensional space.
    void set_nodal_dimension(const unsigned& nodal_dim)
    {
      Nodal_dimension = nodal_dim;
    }
 
    /// Set the number of types required to interpolate the coordinate
    void set_nnodal_position_type(const unsigned& nposition_type)
    {
      Nnodal_position_type = nposition_type;
    }
 
 
    /// Set the number of nodes in the element to n, by resizing
    /// the storage for pointers to the Node objects.
    void set_n_node(const unsigned& n)
    {
      // This should only be done once, in a Node constructor
      //#ifdef PARANOID
      // if(Node_pt)
      // {
      //  OomphLibWarning(
      //   "You are resetting the number of nodes in an element -- Be careful",
      //   "FiniteElement::set_n_node()",
      //   OOMPH_EXCEPTION_LOCATION);
      // }
      //#endif
      // Delete any previous storage to avoid memory leaks
      // This will only happen in very special cases
      delete[] Node_pt;
      // Set the number of nodes
      Nnode = n;
      // Allocate the storage
      Node_pt = new Node*[n];
      // Initialise all the pointers to NULL
      for (unsigned i = 0; i < n; i++)
      {
        Node_pt[i] = 0;
      }
    }
 
    /// Return the local equation number corresponding to the i-th
    /// value at the n-th local node.
    inline int nodal_local_eqn(const unsigned& n, const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (n >= Nnode)
      {
        std::ostringstream error_message;
        error_message << "Range Error: Node number " << n
                      << " is not in the range (0," << Nnode - 1 << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
      else
      {
        unsigned n_value = node_pt(n)->nvalue();
        if (i >= n_value)
        {
          std::ostringstream error_message;
          error_message << "Range Error: value " << i << " at node " << n
                        << " is not in the range (0," << n_value - 1 << ")";
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
#endif
#ifdef PARANOID
      // Check that the equations have been allocated
      if (Nodal_local_eqn == 0)
      {
        throw OomphLibError(
          "Nodal local equation numbers have not been allocated",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Nodal_local_eqn[n][i];
    }
 
 
    /// 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 dJ_eulerian_at_knot(const unsigned& ipt,
                               Shape& psi,
                               DenseMatrix<double>& djacobian_dX) const;
 
  protected:
    /// Static array that holds the number of second derivatives
    /// as a function of the dimension of the element
    static const unsigned N2deriv[];
 
    /// Take the matrix passed as jacobian and return its inverse in
    /// inverse_jacobian. This function is templated by the dimension of the
    /// element because matrix inversion cannot be written efficiently in a
    /// generic manner.
    template<unsigned DIM>
    double invert_jacobian(const DenseMatrix<double>& jacobian,
                           DenseMatrix<double>& inverse_jacobian) const;
 
 
    /// A template-free interface that takes the matrix passed as
    /// jacobian and return its inverse in inverse_jacobian. By default the
    /// function will use the dimension of the element to call the correct
    /// invert_jacobian(..) function. This should be overloaded for efficiency
    /// (removal of a switch statement) in specific elements.
    virtual double invert_jacobian_mapping(
      const DenseMatrix<double>& jacobian,
      DenseMatrix<double>& inverse_jacobian) const;
 
 
    /// Calculate the mapping from local to Eulerian coordinates,
    /// given the derivatives of the shape functions w.r.t. local coordinates.
    /// Returns the determinant of the jacobian, the jacobian and inverse
    /// jacobian
    virtual double local_to_eulerian_mapping(
      const DShape& dpsids,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& inverse_jacobian) const
    {
      // Assemble the jacobian
      assemble_local_to_eulerian_jacobian(dpsids, jacobian);
      // Invert the jacobian (use the template-free interface)
      return invert_jacobian_mapping(jacobian, inverse_jacobian);
    }
 
 
    /// Calculate the mapping from local to Eulerian coordinates,
    /// given the derivatives of the shape functions w.r.t. local coordinates,
    /// Return only the determinant of the jacobian and the inverse of the
    /// mapping (ds/dx).
    double local_to_eulerian_mapping(
      const DShape& dpsids, DenseMatrix<double>& inverse_jacobian) const
    {
      // Find the dimension of the element
      const unsigned el_dim = dim();
      // Assign memory for the jacobian
      DenseMatrix<double> jacobian(el_dim);
      // Calculate the jacobian and inverse
      return local_to_eulerian_mapping(dpsids, jacobian, inverse_jacobian);
    }
 
    /// Calculate the mapping from local to Eulerian coordinates given
    /// the derivatives of the shape functions w.r.t the local 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.
    /// This function returns the determinant of the jacobian, the jacobian
    /// and the inverse jacobian.
    virtual double local_to_eulerian_mapping_diagonal(
      const DShape& dpsids,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& inverse_jacobian) const;
 
    /// A template-free interface that calculates the derivative of the
    /// jacobian of a mapping with respect to the nodal coordinates X_ij.
    /// To do this it requires the jacobian matrix and the derivatives of the
    /// shape functions w.r.t. the local coordinates. By default the function
    /// will use the dimension of the element to call the correct
    /// dJ_eulerian_dnodal_coordinates_templated_helper(..) function. This
    /// should be overloaded for efficiency (removal of a switch statement)
    /// in specific elements.
    virtual void dJ_eulerian_dnodal_coordinates(
      const DenseMatrix<double>& jacobian,
      const DShape& dpsids,
      DenseMatrix<double>& djacobian_dX) const;
 
    /// Calculate the derivative of the jacobian of a mapping with
    /// respect to the nodal coordinates X_ij using the jacobian matrix
    /// and the derivatives of the shape functions w.r.t. the local
    /// coordinates. This function is templated by the dimension of the
    /// element.
    template<unsigned DIM>
    void dJ_eulerian_dnodal_coordinates_templated_helper(
      const DenseMatrix<double>& jacobian,
      const DShape& dpsids,
      DenseMatrix<double>& djacobian_dX) const;
 
    /// A template-free interface that calculates 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$
    /// By default the function will use the dimension of the element to call
    /// the correct d_dshape_eulerian_dnodal_coordinates_templated_helper(..)
    /// function. This should be overloaded for efficiency (removal of a
    /// switch statement) in specific elements.
    virtual void 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;
 
    /// 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$, using the determinant of
    /// the jacobian mapping, its derivative w.r.t. the nodal coordinates
    /// \f$ 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 templated by the dimension of the element.
    template<unsigned DIM>
    void d_dshape_eulerian_dnodal_coordinates_templated_helper(
      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;
 
    /// Convert derivative w.r.t.local coordinates to derivatives
    /// w.r.t the coordinates used to assemble the inverse_jacobian passed
    /// in the mapping. On entry, dbasis must contain the basis function
    /// derivatives w.r.t. the local coordinates; it will contain the
    /// derivatives w.r.t. the new coordinates on exit.
    /// This is virtual so that it may be overloaded if desired
    /// for efficiency reasons.
    virtual void transform_derivatives(
      const DenseMatrix<double>& inverse_jacobian, DShape& dbasis) const;
 
    /// Convert derivative w.r.t local coordinates to derivatives
    /// w.r.t the coordinates used to assemble the inverse jacobian passed
    /// in the mapping, assuming that the coordinates are aligned in the
    /// direction of the local coordinates. On entry dbasis must contain
    /// the derivatives of the basis functions w.r.t. the local coordinates;
    /// it will contain the derivatives w.r.t. the new coordinates.
    /// are converted into the new  using the mapping inverse_jacobian.
    void transform_derivatives_diagonal(
      const DenseMatrix<double>& inverse_jacobian, DShape& dbasis) const;
 
    /// 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. By default this function will call
    /// transform_second_derivatives_template<>(...) using the dimension of
    /// the element as the template parameter. It is virtual so that it can
    /// be overloaded by a specific element to save using a switch statement.
    /// Optionally, the element writer may wish to use the
    /// transform_second_derivatives_diagonal<>(...) function
    /// On entry dbasis and d2basis must contain the derivatives w.r.t. the
    /// local coordinates; on exit they will be the derivatives w.r.t. the
    /// transformed coordinates.
    virtual void transform_second_derivatives(
      const DenseMatrix<double>& jacobian,
      const DenseMatrix<double>& inverse_jacobian,
      const DenseMatrix<double>& jacobian2,
      DShape& dbasis,
      DShape& d2basis) const;
 
    /// Convert derivatives and second derivatives w.r.t. local
    /// coordinates to derivatives and second derivatives w.r.t. the coordinates
    /// used to asssmble the jacobian, inverse jacobian and jacobian2 passed in
    /// the mapping. This is templated by dimension because the method of
    /// calculation varies significantly with the dimension. On entry dbasis and
    /// d2basis must contain the derivatives w.r.t. the local coordinates; on
    /// exit they will be the derivatives w.r.t. the transformed coordinates.
    template<unsigned DIM>
    void transform_second_derivatives_template(
      const DenseMatrix<double>& jacobian,
      const DenseMatrix<double>& inverse_jacobian,
      const DenseMatrix<double>& jacobian2,
      DShape& dbasis,
      DShape& d2basis) const;
 
    /// Convert derivatives and second derivatives w.r.t. local
    /// coordinates to derivatives and second derivatives w.r.t. the coordinates
    /// used to asssmble the jacobian, inverse jacobian and jacobian2 passed in
    /// the mapping. This version of the function assumes that the local
    /// coordinates are aligned with the global coordinates, i.e. the jacobians
    /// are diagonal On entry dbasis and d2basis must contain the derivatives
    /// w.r.t. the local coordinates; on exit they will be the derivatives
    /// w.r.t. the transformed coordinates.
    template<unsigned DIM>
    void transform_second_derivatives_diagonal(
      const DenseMatrix<double>& jacobian,
      const DenseMatrix<double>& inverse_jacobian,
      const DenseMatrix<double>& jacobian2,
      DShape& dbasis,
      DShape& d2basis) const;
 
    /// Pointer to the element's macro element (NULL by default)
    MacroElement* Macro_elem_pt;
 
    /// Calculate the contributions to the jacobian from the nodal
    /// degrees of freedom using finite differences.
    /// This version of the function assumes that the residuals vector has
    /// already been calculated
    virtual void fill_in_jacobian_from_nodal_by_fd(
      Vector<double>& residuals, DenseMatrix<double>& jacobian);
 
    /// Calculate the contributions to the jacobian from the nodal
    /// degrees of freedom using finite differences. This version computes
    /// the residuals vector before calculating the jacobian terms.
    void fill_in_jacobian_from_nodal_by_fd(DenseMatrix<double>& jacobian)
    {
      // Allocate storage for a residuals vector and initialise to zero
      unsigned n_dof = ndof();
      Vector<double> residuals(n_dof, 0.0);
      // Get the residuals for the entire element
      get_residuals(residuals);
      // Call the jacobian calculation
      fill_in_jacobian_from_nodal_by_fd(residuals, jacobian);
    }
 
    /// Function that is called before the finite differencing of
    /// any nodal data. This may be overloaded to update any dependent
    /// data before finite differencing takes place.
    virtual inline void update_before_nodal_fd() {}
 
    /// Function that is call after the finite differencing of
    /// the nodal data. This may be overloaded to reset any dependent
    /// variables that may have changed during the finite differencing.
    virtual inline void reset_after_nodal_fd() {}
 
    /// Function called within the finite difference loop for
    /// nodal data after a change in the i-th nodal value.
    virtual inline void update_in_nodal_fd(const unsigned& i) {}
 
    /// Function called within the finite difference loop for
    /// nodal data after the i-th nodal values is reset.
    /// The default behaviour is to call the update function.
    virtual inline void reset_in_nodal_fd(const unsigned& i)
    {
      update_in_nodal_fd(i);
    }
 
 
    /// Add the elemental contribution to the jacobian matrix.
    /// and the residuals vector. Note that
    /// this function will NOT initialise the residuals vector or the jacobian
    /// matrix. It must be called after the residuals vector and
    /// jacobian matrix have been initialised to zero. The default
    /// is to use finite differences to calculate the jacobian
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Add the contribution to the residuals
      fill_in_contribution_to_residuals(residuals);
      // Allocate storage for the full residuals (residuals of entire element)
      unsigned n_dof = ndof();
      Vector<double> full_residuals(n_dof);
      // Get the residuals for the entire element
      get_residuals(full_residuals);
      // Calculate the contributions from the internal dofs
      //(finite-difference the lot by default)
      fill_in_jacobian_from_internal_by_fd(full_residuals, jacobian, true);
      // Calculate the contributions from the external dofs
      //(finite-difference the lot by default)
      fill_in_jacobian_from_external_by_fd(full_residuals, jacobian, true);
      // Calculate the contributions from the nodal dofs
      fill_in_jacobian_from_nodal_by_fd(full_residuals, jacobian);
    }
 
  public:
    /// Function pointer for function that computes vector-valued
    /// steady "exact solution" \f$ {\bf f}({\bf x}) \f$
    /// as \f$ \mbox{\tt fct}({\bf x}, {\bf f}) \f$.
    typedef void (*SteadyExactSolutionFctPt)(const Vector<double>&,
                                             Vector<double>&);
 
    /// Function pointer for function that computes Vector-valued
    /// time-dependent function \f$ {\bf f}(t,{\bf x}) \f$
    /// as \f$ \mbox{\tt fct}(t, {\bf x}, {\bf f}) \f$.
    typedef void (*UnsteadyExactSolutionFctPt)(const double&,
                                               const Vector<double>&,
                                               Vector<double>&);
 
    /// Tolerance below which the jacobian is considered singular
    static double Tolerance_for_singular_jacobian;
 
    /// Boolean that if set to true allows a negative jacobian
    /// in the transform between global and local coordinates (negative surface
    /// area = left-handed coordinate system).
    static bool Accept_negative_jacobian;
 
    /// Static boolean to suppress output while checking
    /// for inverted elements
    static bool Suppress_output_while_checking_for_inverted_elements;
 
    /// Constructor
    FiniteElement()
      : GeneralisedElement(),
        Integral_pt(0),
        Node_pt(0),
        Nodal_local_eqn(0),
        Nnode(0),
        Elemental_dimension(0),
        Nodal_dimension(0),
        Nnodal_position_type(1),
        Macro_elem_pt(0)
    {
    }
 
    /// The destructor cleans up the static memory allocated
    /// for shape function storage. Internal and external data get
    /// wiped by the GeneralisedElement destructor; nodes get
    /// killed in mesh destructor.
    virtual ~FiniteElement();
 
    /// Broken copy constructor
    FiniteElement(const FiniteElement&) = delete;
 
    /// Broken assignment operator
    // Commented out broken assignment operator because this can lead to a
    // conflict warning when used in the virtual inheritence hierarchy.
    // Essentially the compiler doesn't realise that two separate
    // implementations of the broken function are the same and so, quite
    // rightly, it shouts.
    /*void operator=(const FiniteElement&) = delete;*/
 
    /// Check whether the local coordinate are valid or not
    virtual bool local_coord_is_valid(const Vector<double>& s)
    {
      throw OomphLibError(
        "local_coord_is_valid is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
      return true;
    }
 
    /// Adjust local coordinates so that they're located inside
    /// the element
    virtual void move_local_coord_back_into_element(Vector<double>& s) const
    {
      throw OomphLibError("move_local_coords_back_into_element() is not "
                          "implemented for this element\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// 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
    void get_centre_of_gravity_and_max_radius_in_terms_of_zeta(
      Vector<double>& cog, double& max_radius) const;
 
    /// Get local coordinates of node j in the element; vector
    /// sets its own size (broken virtual)
    virtual void local_coordinate_of_node(const unsigned& j,
                                          Vector<double>& s) const
    {
      throw OomphLibError(
        "local_coordinate_of_node(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Get the local fraction of the node j in the element
    /// A dumb, but correct default implementation is provided.
    virtual void local_fraction_of_node(const unsigned& j,
                                        Vector<double>& s_fraction);
 
    /// Get the local fraction of any node in the n-th position
    /// in a one dimensional expansion along the i-th local coordinate
    virtual double local_one_d_fraction_of_node(const unsigned& n1d,
                                                const unsigned& i)
    {
      std::string error_message =
        "local one_d_fraction_of_node is not implemented for this element\n";
      error_message +=
        "It only makes sense for elements that use tensor-product expansions\n";
 
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Set pointer to macro element -- can be overloaded in derived
    /// elements to perform additional tasks
    virtual void set_macro_elem_pt(MacroElement* macro_elem_pt)
    {
      Macro_elem_pt = macro_elem_pt;
    }
 
    /// Access function to pointer to macro element
    MacroElement* macro_elem_pt()
    {
      return Macro_elem_pt;
    }
 
    /// Global coordinates as function of local coordinates.
    /// Either via FE representation or via macro-element (if Macro_elem_pt!=0)
    void get_x(const Vector<double>& s, Vector<double>& x) const
    {
      // If there is no macro element then return interpolated x
      if (Macro_elem_pt == 0)
      {
        interpolated_x(s, x);
      }
      // Otherwise call the macro element representation
      else
      {
        get_x_from_macro_element(s, x);
      }
    }
 
 
    /// Global coordinates as function of local coordinates
    /// at previous time "level" t (t=0: present; t>0: previous).
    /// Either via FE representation of QElement or
    /// via macro-element (if Macro_elem_pt!=0).
    void get_x(const unsigned& t, const Vector<double>& s, Vector<double>& x)
    {
      // Get timestepper from first node
      TimeStepper* time_stepper_pt = node_pt(0)->time_stepper_pt();
 
      // Number of previous values
      const unsigned nprev = time_stepper_pt->nprev_values();
 
      // If t > nprev_values(), we're not dealing with a previous value
      // but a generalised history value -- this cannot be recovered from
      // macro element but must be determined by finite element interpolation
 
      // If there is no macro element, or we're dealing with a generalised
      // history value then use the FE representation
      if ((Macro_elem_pt == 0) || (t > nprev))
      {
        interpolated_x(t, s, x);
      }
      // Otherwise use the macro element representation
      else
      {
        get_x_from_macro_element(t, s, x);
      }
    }
 
 
    /// Global coordinates as function of local coordinates
    /// using macro element representation.
    /// (Broken virtual --- this  must be overloaded in specific geometric
    /// element classes)
    virtual void get_x_from_macro_element(const Vector<double>& s,
                                          Vector<double>& x) const
    {
      throw OomphLibError(
        "get_x_from_macro_element(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Global coordinates as function of local coordinates
    /// at previous time "level" t (t=0: present; t>0: previous).
    /// using macro element representation
    /// (Broken virtual -- overload in specific geometric element class
    /// if you want to use this functionality.)
    virtual void get_x_from_macro_element(const unsigned& t,
                                          const Vector<double>& s,
                                          Vector<double>& x)
    {
      throw OomphLibError(
        "get_x_from_macro_element(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Set the spatial integration scheme
    virtual void set_integration_scheme(Integral* const& integral_pt);
 
    /// Return the pointer to the integration scheme (const version)
    Integral* const& integral_pt() const
    {
      return Integral_pt;
    }
 
    /// Calculate the geometric shape functions
    /// at local coordinate s. This function must be overloaded for each
    /// specific geometric element.
    virtual void shape(const Vector<double>& s, Shape& psi) const = 0;
 
    /// Return the geometric shape function at the ipt-th integration
    /// point
    virtual void shape_at_knot(const unsigned& ipt, Shape& psi) const;
 
    /// Function to compute the geometric shape functions and
    /// derivatives w.r.t. local coordinates at local coordinate s.
    /// This function must be overloaded for each specific geometric element.
    /// (Broken virtual function --- specifies the interface)
    virtual void dshape_local(const Vector<double>& s,
                              Shape& psi,
                              DShape& dpsids) const
    {
      throw OomphLibError(
        "dshape_local() is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return the geometric shape function and its derivative w.r.t.
    /// the local coordinates at the ipt-th integration point.
    virtual void dshape_local_at_knot(const unsigned& ipt,
                                      Shape& psi,
                                      DShape& dpsids) const;
 
    /// Function to compute the geometric shape functions and also
    /// first and second derivatives w.r.t.
    /// local coordinates at local coordinate s. This function must
    /// be overloaded for each specific geometric element (if required).
    /// (Broken virtual function --- specifies the interface).
    ///  Numbering:
    ///  \b 1D:
    /// d2psids(i,0) = \f$ d^2 \psi_j / ds^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$
    virtual void d2shape_local(const Vector<double>& s,
                               Shape& psi,
                               DShape& dpsids,
                               DShape& d2psids) const
    {
      throw OomphLibError(
        "d2shape_local() is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return the geometric shape function and its first and
    /// second derivatives w.r.t.
    /// the local coordinates at the ipt-th integration point.
    ///  Numbering:
    ///  \b 1D:
    /// d2psids(i,0) = \f$ d^2 \psi_j / ds^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$
    virtual void d2shape_local_at_knot(const unsigned& ipt,
                                       Shape& psi,
                                       DShape& dpsids,
                                       DShape& d2psids) const;
 
    /// Return the Jacobian of mapping from local to global
    /// coordinates at local position s.
    virtual double J_eulerian(const Vector<double>& s) const;
 
    /// Return the Jacobian of the mapping from local to global
    /// coordinates at the ipt-th integration point
    virtual double J_eulerian_at_knot(const unsigned& ipt) const;
 
    /// Check that Jacobian of mapping between local and Eulerian
    /// coordinates at all integration points is positive.
    void check_J_eulerian_at_knots(bool& passed) const;
 
    /// Helper function used to check for singular or negative
    /// Jacobians in the transform from local to global or Lagrangian
    /// coordinates.
    void check_jacobian(const double& jacobian) const;
 
    /// 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.
    double dshape_eulerian(const Vector<double>& s,
                           Shape& psi,
                           DShape& dpsidx) const;
 
    /// Return the geometric shape functions and also first
    /// derivatives w.r.t. global coordinates at the ipt-th integration point.
    virtual double dshape_eulerian_at_knot(const unsigned& ipt,
                                           Shape& psi,
                                           DShape& dpsidx) const;
 
    /// Compute the geometric shape functions (psi) and first derivatives
    /// w.r.t. global coordinates (dpsidx) at the ipt-th integration point.
    /// 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.
    virtual double dshape_eulerian_at_knot(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsi,
      DenseMatrix<double>& djacobian_dX,
      RankFourTensor<double>& d_dpsidx_dX) const;
 
    /// Compute the geometric shape functions and also first
    /// and second derivatives w.r.t. global coordinates at local coordinate s;
    /// 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 d2shape_eulerian(const Vector<double>& s,
                            Shape& psi,
                            DShape& dpsidx,
                            DShape& d2psidx) const;
 
 
    /// Return the geometric shape functions and also first
    /// and second derivatives w.r.t. global coordinates at ipt-th integration
    /// point.
    ///  Numbering:
    ///  \b 1D:
    /// d2psidx(i,0) = \f$ d^2 \psi_j / d s^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$
    virtual double d2shape_eulerian_at_knot(const unsigned& ipt,
                                            Shape& psi,
                                            DShape& dpsidx,
                                            DShape& d2psidx) const;
 
    /// Assign the local equation numbers for Data stored at the nodes
    /// Virtual so that it can be overloaded by RefineableFiniteElements.
    /// If the boolean is true then the pointers to the degrees of freedom
    /// associated with each equation number are stored in Dof_pt
    virtual void assign_nodal_local_eqn_numbers(const bool& store_local_dof_pt);
 
    /// Function to describe the local dofs of the element[s]. The
    /// ostream specifies the output stream to which the description is written;
    /// the string stores the currently assembled output that is ultimately
    /// written to the output stream by Data::describe_dofs(...); it is
    /// typically built up incrementally as we descend through the call
    /// hierarchy of this function when called from Problem::describe_dofs(...)
    virtual void describe_local_dofs(std::ostream& out,
                                     const std::string& current_string) const;
 
    /// Function to describe the local dofs of the element[s]. The
    /// ostream specifies the output stream to which the description is written;
    /// the string stores the currently assembled output that is ultimately
    /// written to the output stream by Data::describe_dofs(...); it is
    /// typically built up incrementally as we descend through the call
    /// hierarchy of this function when called from Problem::describe_dofs(...)
    virtual void describe_nodal_local_dofs(
      std::ostream& out, const std::string& current_string) const;
 
    /// Overloaded version of the calculation of the local equation
    /// numbers. If the boolean argument is true then pointers to the degrees
    /// of freedom associated with each equation number are stored locally
    /// in the array Dof_pt.
    virtual inline void assign_all_generic_local_eqn_numbers(
      const bool& store_local_dof_pt)
    {
      // GeneralisedElement's version assigns internal and external data
      GeneralisedElement::assign_all_generic_local_eqn_numbers(
        store_local_dof_pt);
      // Need simply to add the nodal numbering scheme
      assign_nodal_local_eqn_numbers(store_local_dof_pt);
    }
 
    /// Return a pointer to the local node n
    Node*& node_pt(const unsigned& n)
    {
#ifdef RANGE_CHECKING
      if (n >= Nnode)
      {
        std::ostringstream error_message;
        error_message << "Range Error: " << n << " is not in the range (0,"
                      << Nnode - 1 << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return Node_pt[n];
    }
 
    /// Return a pointer to the local node n (const version)
    Node* const& node_pt(const unsigned& n) const
    {
#ifdef RANGE_CHECKING
      if (n >= Nnode)
      {
        std::ostringstream error_message;
        error_message << "Range Error: " << n << " is not in the range (0,"
                      << Nnode - 1 << ")";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      return Node_pt[n];
    }
 
    /// Return the number of nodes
    unsigned nnode() const
    {
      return Nnode;
    }
 
    /// Return the number of nodes along one edge of the element
    /// Default is to return zero --- must be overloaded by geometric
    /// elements
    virtual unsigned nnode_1d() const
    {
      return 0;
    }
 
    /// 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 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 tmp; // node_pt(n)->x(i); */
    /* } */
 
    /// Return the i-th coordinate at local node n, at time level t
    /// (t=0: present; t>0: previous time level).
    /// Do not use the hanging node representation.
    double raw_nodal_position(const unsigned& t,
                              const unsigned& n,
                              const unsigned& i) const
    {
      return node_pt(n)->x(t, i);
    }
 
    /// Return the i-th component of nodal velocity: dx/dt at local node
    /// n. Do not use the hanging node representation.
    double raw_dnodal_position_dt(const unsigned& n, const unsigned& i) const
    {
      return node_pt(n)->dx_dt(i);
    }
 
    /// Return the i-th component of j-th derivative of nodal position:
    /// d^jx/dt^j at node n. Do not use the hanging node representation.
    double raw_dnodal_position_dt(const unsigned& n,
                                  const unsigned& j,
                                  const unsigned& i) const
    {
      return node_pt(n)->dx_dt(j, i);
    }
 
    /// Return the value of the k-th type of the i-th positional variable
    /// at the local node n. Do not use the hanging node representation.
    double raw_nodal_position_gen(const unsigned& n,
                                  const unsigned& k,
                                  const unsigned& i) const
    {
      return node_pt(n)->x_gen(k, i);
    }
 
    /// Return the generalised nodal position (type k, i-th variable)
    /// at previous timesteps at local node n. Do not use the hanging node
    /// representation.
    double raw_nodal_position_gen(const unsigned& t,
                                  const unsigned& n,
                                  const unsigned& k,
                                  const unsigned& i) const
    {
      return node_pt(n)->x_gen(t, k, i);
    }
 
    /// i-th component of time derivative (velocity) of the
    /// generalised position, dx(k,i)/dt at local node n.
    /// `Type': k; Coordinate direction: i. Do not use the hanging node
    /// representation.
    double raw_dnodal_position_gen_dt(const unsigned& n,
                                      const unsigned& k,
                                      const unsigned& i) const
    {
      return node_pt(n)->dx_gen_dt(k, i);
    }
 
    /// i-th component of j-th time derivative  of the
    /// generalised position, dx(k,i)/dt at local node n.
    /// `Type': k; Coordinate direction: i. Do not use the hanging node
    /// representation.
    double raw_dnodal_position_gen_dt(const unsigned& j,
                                      const unsigned& n,
                                      const unsigned& k,
                                      const unsigned& i) const
    {
      return node_pt(n)->dx_gen_dt(j, k, i);
    }
 
 
    /// Return the i-th coordinate at local node n. If the
    /// node is hanging, the appropriate interpolation is handled by the
    /// position function in the Node class.
    double nodal_position(const unsigned& n, const unsigned& i) const
    {
      return node_pt(n)->position(i);
    }
 
    /// Return the i-th coordinate at local node n, at time level t
    /// (t=0: present; t>0: previous time level)
    /// Returns suitably interpolated version for hanging nodes.
    double nodal_position(const unsigned& t,
                          const unsigned& n,
                          const unsigned& i) const
    {
      return node_pt(n)->position(t, i);
    }
 
    /// Return the i-th component of nodal velocity: dx/dt at local node n.
    double dnodal_position_dt(const unsigned& n, const unsigned& i) const
    {
      return node_pt(n)->dposition_dt(i);
    }
 
    /// Return the i-th component of j-th derivative of nodal position:
    /// d^jx/dt^j at node n.
    double dnodal_position_dt(const unsigned& n,
                              const unsigned& j,
                              const unsigned& i) const
    {
      return node_pt(n)->dposition_dt(j, i);
    }
 
    /// Return the value of the k-th type of the i-th positional variable
    /// at the local node n.
    double nodal_position_gen(const unsigned& n,
                              const unsigned& k,
                              const unsigned& i) const
    {
      return node_pt(n)->position_gen(k, i);
    }
 
    /// Return the generalised nodal position (type k, i-th variable)
    /// at previous timesteps at local node n
    double nodal_position_gen(const unsigned& t,
                              const unsigned& n,
                              const unsigned& k,
                              const unsigned& i) const
    {
      return node_pt(n)->position_gen(t, k, i);
    }
 
    /// i-th component of time derivative (velocity) of the
    /// generalised position, dx(k,i)/dt at local node n.
    /// `Type': k; Coordinate direction: i.
    double dnodal_position_gen_dt(const unsigned& n,
                                  const unsigned& k,
                                  const unsigned& i) const
    {
      return node_pt(n)->dposition_gen_dt(k, i);
    }
 
    /// i-th component of j-th time derivative  of the
    /// generalised position, dx(k,i)/dt at local node n.
    /// `Type': k; Coordinate direction: i.
    double dnodal_position_gen_dt(const unsigned& j,
                                  const unsigned& n,
                                  const unsigned& k,
                                  const unsigned& i) const
    {
      return node_pt(n)->dposition_gen_dt(j, k, i);
    }
 
 
    /// 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}
    virtual void get_dresidual_dnodal_coordinates(
      RankThreeTensor<double>& dresidual_dnodal_coordinates);
 
    /// This is an empty function that establishes a uniform
    /// interface for all (derived) elements that involve time-derivatives.
    /// Such elements are/should be implemented in ALE form to allow
    /// mesh motions. The additional expense associated with the
    /// computation of the mesh velocities is, of course, superfluous
    /// if the elements are used in problems in which the mesh is
    /// stationary. This function should therefore be overloaded
    /// in all derived elements that are formulated in ALE form
    /// to suppress the computation of the mesh velocities.
    /// The user disables the ALE functionality at his/her own risk!
    /// If the mesh does move after all, then the results will be wrong.
    /// Here we simply issue a warning message stating that the empty
    /// function has been called.
    virtual void disable_ALE()
    {
      std::ostringstream warn_message;
      warn_message
        << "Warning: You have just called the default (empty) function \n\n"
        << "    FiniteElement::disable_ALE() \n\n"
        << "This suggests that you've either tried to call it for an element\n"
        << "that \n"
        << "(1) does not involve time-derivatives (e.g. a Poisson element) \n"
        << "(2) an element for which the time-derivatives aren't implemented \n"
        << "    in ALE form \n"
        << "(3) an element for which the ALE form of the equations can't be \n"
        << "    be disabled (yet).\n";
      OomphLibWarning(
        warn_message.str(), "Problem::disable_ALE()", OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// (Re-)enable ALE, i.e. take possible mesh motion into account
    /// when evaluating the time-derivative. This function is empty
    /// and simply establishes a common interface for all derived
    /// elements that are formulated in ALE form.
    virtual void enable_ALE()
    {
      std::ostringstream warn_message;
      warn_message
        << "Warning: You have just called the default (empty) function \n\n"
        << "    FiniteElement::enable_ALE() \n\n"
        << "This suggests that you've either tried to call it for an element\n"
        << "that \n"
        << "(1) does not involve time-derivatives (e.g. a Poisson element) \n"
        << "(2) an element for which the time-derivatives aren't implemented \n"
        << "    in ALE form \n"
        << "(3) an element for which the ALE form of the equations can't be \n"
        << "    be disabled (yet)\n"
        << "(4) an element for which this function has not been (properly) \n "
        << "    implemented. This is likely to be a bug!\n ";
      OomphLibWarning(
        warn_message.str(), "Problem::enable_ALE()", OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Number of values that must be stored at local node n by
    /// the element. The default is 0, until over-ridden by a particular
    /// element. For example, a Poisson equation requires only one value to be
    /// stored at each node; 2D Navier--Stokes equations require two values
    /// (velocity components) to be stored at each Node (provided that the
    /// pressure interpolation is discontinuous).
    virtual unsigned required_nvalue(const unsigned& n) const
    {
      return Default_Initial_Nvalue;
    }
 
    /// Return the number of coordinate types that the element requires
    /// to interpolate the geometry between
    /// the nodes. For Lagrange elements it is 1.
    unsigned nnodal_position_type() const
    {
      return Nnodal_position_type;
    }
 
    /// Return boolean to indicate if any of the element's nodes
    /// are geometrically hanging.
    bool has_hanging_nodes() const
    {
      unsigned nnod = nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        if (node_pt(j)->is_hanging())
        {
          return true;
        }
      }
      return false;
    }
 
    /// Return the required Eulerian dimension of the nodes in this element
    unsigned nodal_dimension() const
    {
      return Nodal_dimension;
    }
 
    /// Return the number of vertex nodes in this element. Broken virtual
    /// function in "pure" finite elements.
    virtual unsigned nvertex_node() const
    {
      std::string error_msg = "Not implemented for FiniteElement.";
      throw OomphLibError(
        error_msg, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Pointer to the j-th vertex node in the element. Broken virtual
    /// function in "pure" finite elements.
    virtual Node* vertex_node_pt(const unsigned& j) const
    {
      std::string error_msg = "Not implemented for FiniteElement.";
      throw OomphLibError(
        error_msg, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Construct the local node n and return a pointer to the newly
    /// created node object.
    virtual Node* construct_node(const unsigned& n)
    {
      // Create a node and assign it to the local node pointer
      // The dimension and number of values are taken from internal element data
      node_pt(n) =
        new Node(Nodal_dimension, Nnodal_position_type, required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can use it
      return node_pt(n);
    }
 
    /// Construct the local node n, including
    /// storage for history values required by timestepper, and return a pointer
    /// to the newly created node object.
    virtual Node* construct_node(const unsigned& n,
                                 TimeStepper* const& time_stepper_pt)
    {
      // Create a node and assign it to the local node pointer.
      // The dimension and number of values are taken from internal element data
      node_pt(n) = new Node(time_stepper_pt,
                            Nodal_dimension,
                            Nnodal_position_type,
                            required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Construct the local node n as a boundary node;
    /// that is a node that MAY be placed on a mesh boundary
    /// and return a pointer to the newly created node object.
    virtual Node* construct_boundary_node(const unsigned& n)
    {
      // Create a node and assign it to the local node pointer
      // The dimension and number of values are taken from internal element data
      node_pt(n) = new BoundaryNode<Node>(
        Nodal_dimension, Nnodal_position_type, required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can use it
      return node_pt(n);
    }
 
    /// Construct the local node n, including
    /// storage for history values required by timestepper,
    /// as a boundary node; that is a node that MAY be placed on a mesh
    /// boundary and return a pointer to the newly created node object.
    virtual Node* construct_boundary_node(const unsigned& n,
                                          TimeStepper* const& time_stepper_pt)
    {
      // Create a node and assign it to the local node pointer.
      // The dimension and number of values are taken from internal element data
      node_pt(n) = new BoundaryNode<Node>(time_stepper_pt,
                                          Nodal_dimension,
                                          Nnodal_position_type,
                                          required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Return the number of the node *node_pt if this node
    /// is in the element, else return -1;
    int get_node_number(Node* const& node_pt) const;
 
 
    /// If there is a node at this local coordinate, return the pointer
    /// to the node
    virtual Node* get_node_at_local_coordinate(const Vector<double>& s) const;
 
    /// Return the i-th value stored at local node n
    /// but do NOT take hanging nodes into account
    double raw_nodal_value(const unsigned& n, const unsigned& i) const
    {
      return node_pt(n)->raw_value(i);
    }
 
    /// Return the i-th value stored at local node n, at time level t
    /// (t=0: present; t>0 previous timesteps), but do NOT take hanging nodes
    /// into account
    double raw_nodal_value(const unsigned& t,
                           const unsigned& n,
                           const unsigned& i) const
    {
      return node_pt(n)->raw_value(t, i);
    }
 
    /// Return the i-th value stored at local node n.
    /// Produces suitably interpolated values for hanging nodes.
    double nodal_value(const unsigned& n, const unsigned& i) const
    {
      return node_pt(n)->value(i);
    }
 
    /// Return the i-th value stored at local node n, at time level t
    /// (t=0: present; t>0 previous timesteps). Produces suitably interpolated
    /// values for hanging nodes.
    double nodal_value(const unsigned& t,
                       const unsigned& n,
                       const unsigned& i) const
    {
      return node_pt(n)->value(t, i);
    }
 
    /// Return the spatial dimension of the element, i.e.
    /// the number of local coordinates required to parametrise its
    /// geometry.
    unsigned dim() const
    {
      return Elemental_dimension;
    }
 
    /// Return the geometry type of the element (either Q or T usually).
    virtual ElementGeometry::ElementGeometry element_geometry() const
    {
      std::string err = "Broken virtual function.";
      throw OomphLibError(
        err, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return FE interpolated coordinate x[i] at local coordinate s
    virtual double interpolated_x(const Vector<double>& s,
                                  const unsigned& i) const;
 
    /// Return FE interpolated coordinate x[i] at local coordinate s
    /// at previous timestep t (t=0: present; t>0: previous timestep)
    virtual double interpolated_x(const unsigned& t,
                                  const Vector<double>& s,
                                  const unsigned& i) const;
 
    /// Return FE interpolated position x[] at local coordinate s as Vector
    virtual void interpolated_x(const Vector<double>& s,
                                Vector<double>& x) const;
 
    /// Return FE interpolated position x[] at local coordinate s
    /// at previous timestep t as Vector (t=0: present; t>0: previous timestep)
    virtual void interpolated_x(const unsigned& t,
                                const Vector<double>& s,
                                Vector<double>& x) const;
 
    /// Return t-th time-derivative of the
    /// i-th FE-interpolated Eulerian coordinate at
    /// local coordinate s.
    virtual double interpolated_dxdt(const Vector<double>& s,
                                     const unsigned& i,
                                     const unsigned& t);
 
    /// Compte t-th time-derivative of the
    /// FE-interpolated Eulerian coordinate vector at
    /// local coordinate s.
    virtual void interpolated_dxdt(const Vector<double>& s,
                                   const unsigned& t,
                                   Vector<double>& dxdt);
 
    /// A standard FiniteElement is fixed, so there are no geometric
    /// data when viewed in its GeomObject incarnation
    inline unsigned ngeom_data() const
    {
      return 0;
    }
 
    /// A standard FiniteElement is fixed, so there are no geometric
    /// data when viewed in its GeomObject incarnation
    inline Data* geom_data_pt(const unsigned& j)
    {
      return 0;
    }
 
    /// Return the parametrised position of the FiniteElement in
    /// its incarnation as a GeomObject, r(zeta).
    /// The position is given by the Eulerian coordinate and the intrinsic
    /// coordinate (zeta) is the local coordinate of the element (s).
    void position(const Vector<double>& zeta, Vector<double>& r) const
    {
      this->interpolated_x(zeta, r);
    }
 
    /// Return the parametrised position of the FiniteElement
    /// in its GeomObject incarnation:
    /// r(zeta). The position is given by the Eulerian coordinate and the
    /// intrinsic coordinate (zeta) is the local coordinate of the element (s)
    /// This version of the function returns the position as a function
    /// of time t=0: current time; t>0: previous
    /// timestep. Works for t=0 but needs to be overloaded
    /// if genuine time-dependence is required.
    void position(const unsigned& t,
                  const Vector<double>& zeta,
                  Vector<double>& r) const
    {
      this->interpolated_x(t, zeta, r);
    }
 
    /// Return the t-th time derivative of the
    /// parametrised position of the FiniteElement
    /// in its GeomObject incarnation:
    /// \f$ \frac{d^{t} dr(zeta)}{d t^{t}} \f$.
    /// Call the t-th time derivative of the FE-interpolated Eulerian coordinate
    void dposition_dt(const Vector<double>& zeta,
                      const unsigned& t,
                      Vector<double>& drdt)
    {
      this->interpolated_dxdt(zeta, t, drdt);
    }
 
    /// Specify the values of the "global" intrinsic coordinate, zeta,
    /// of a compound geometric object (a mesh of elements) when
    /// the element is viewied as a sub-geometric object.
    /// The default assumption is that the element will be
    /// treated as a sub-geometric object in a bulk Mesh of other elements
    /// (geometric objects). The "global" coordinate of the compound geometric
    /// object is simply the Eulerian coordinate, x.
    /// The second default assumption is that the coordinate zeta will
    /// be stored at the nodes and interpolated using the shape functions
    /// of the element. This function returns the value of zeta stored at
    /// local node n, where k is the type of coordinate and i is the
    /// coordinate direction. The function is virtual so that it can
    /// be overloaded by different types of element: FaceElements
    /// and SolidFiniteElements
    virtual double zeta_nodal(const unsigned& n,
                              const unsigned& k,
                              const unsigned& i) const
    {
      // By default return the value for nodal_position_gen
      return nodal_position_gen(n, k, i);
    }
 
 
    /// 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 interpolated_zeta(const Vector<double>& s, Vector<double>& zeta) const;
 
    /// 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.
    /// 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.
    /// By default don't use any value passed in to the local coordinate s
    /// as the initial guess in the Newton method
    void locate_zeta(const Vector<double>& zeta,
                     GeomObject*& geom_object_pt,
                     Vector<double>& s,
                     const bool& use_coordinate_as_initial_guess = false);
 
 
    /// Update the positions of all nodes in the element using
    /// each node update function. The default implementation may
    /// be overloaded so that more efficient versions can be written
    virtual void 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
    ///  - the pointer to a Data object
    /// and
    /// - 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. The geometric data, which includes the positions
    /// of SolidNodes, is treated separately by the function
    /// \c identify_geometric_data()
    virtual void identify_field_data_for_interactions(
      std::set<std::pair<Data*, unsigned>>& paired_field_data);
 
 
    /// The purpose of this
    /// function is to identify all \c Data objects that affect the elements'
    /// geometry. This function is implemented as an empty virtual
    /// function since it can only be implemented in conjunction
    /// with a node-update strategy. A specific implementation
    /// is provided in the ElementWithMovingNodes class.
    virtual void identify_geometric_data(std::set<Data*>& geometric_data_pt) {}
 
 
    /// Min value of local coordinate
    virtual double s_min() const
    {
      throw OomphLibError("s_min() isn't implemented for this element\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
      // Dummy return
      return 0.0;
    }
 
    /// Max. value of local coordinate
    virtual double s_max() const
    {
      throw OomphLibError("s_max() isn't implemented for this element\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
      // Dummy return
      return 0.0;
    }
 
    /// Calculate the size of the element (length, area, volume,...)
    /// 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 size() const;
 
 
    /// Broken virtual
    /// 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.
    virtual double compute_physical_size() const
    {
      throw OomphLibError(
        "compute_physical_size() isn't implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
      // Dummy return
      return 0.0;
    }
 
    /// Virtual function to write the double precision numbers that
    /// appear in a single line of output into the data vector. Empty virtual,
    /// can be overloaded for specific elements; used e.g. by LineVisualiser.
    virtual void point_output_data(const Vector<double>& s,
                                   Vector<double>& data)
    {
    }
 
    /// Output solution (as defined by point_output_data())
    /// at local cordinates s
    void point_output(std::ostream& outfile, const Vector<double>& s)
    {
      // Get point data
      Vector<double> data;
      this->point_output_data(s, data);
 
      // Output
      unsigned n = data.size();
      for (unsigned i = 0; i < n; i++)
      {
        outfile << data[i] << " ";
      }
    }
 
    /// Return the number of actual plot points for paraview
    /// plot with parameter nplot. Broken virtual; can be overloaded
    /// in specific elements.
    virtual unsigned nplot_points_paraview(const unsigned& nplot) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
 
      // Dummy unsigned
      return 0;
    }
 
    /// Return the number of local sub-elements for paraview plot with
    /// parameter nplot. Broken virtual; can be overloaded
    /// in specific elements.
    virtual unsigned nsub_elements_paraview(const unsigned& nplot) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
 
      // Dummy unsigned
      return 0;
    }
 
    /// Paraview output -- this outputs the coordinates at the plot
    /// points (for parameter nplot) to specified output file.
    void output_paraview(std::ofstream& file_out, const unsigned& nplot) const
    {
      // Decide the dimensions of the nodes
      unsigned nnod = nnode();
      if (nnod == 0) return;
      unsigned n = node_pt(0)->ndim();
 
      // Vector for local coordinates
      Vector<double> s(n, 0.0);
 
      // Vector for cartesian coordinates
      Vector<double> x(n, 0.0);
 
      // Determine the total number of plotpoints
      unsigned plot = nplot_points_paraview(nplot);
 
      // Loop over the local points
      for (unsigned j = 0; j < plot; j++)
      {
        // Determine where in the local element the point is
        this->get_s_plot(j, nplot, s);
 
        // Update the cartesian coordinates vector
        this->interpolated_x(s, x);
 
        // Print the global coordinates. Note: no whitespace after last
        // coordinate or paraview is very unhappy.
        for (unsigned i = 0; i < n - 1; i++)
        {
          file_out << x[i] << " ";
        }
        file_out << x[n - 1];
 
        // Since unstructured grid always needs 3 components for each
        // point, output 0's by default
        switch (n)
        {
          case 1:
            file_out << " 0"
                     << " 0" << std::endl;
            break;
 
          case 2:
            file_out << " 0" << std::endl;
            break;
 
          case 3:
            file_out << std::endl;
            break;
 
          // Paraview can't handle more than 3 dimensions, output error
          default:
            throw OomphLibError(
              "Printing PlotPoint to .vtu failed; it has >3 dimensions.",
              OOMPH_CURRENT_FUNCTION,
              OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
 
    /// Fill in the offset information for paraview plot. Broken virtual.
    /// Needs to be implemented for each new geometric element type; see
    /// http://www.vtk.org/VTK/img/file-formats.pdf
    virtual void write_paraview_output_offset_information(
      std::ofstream& file_out, const unsigned& nplot, unsigned& counter) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return the paraview element type.  Broken virtual.
    /// Needs to be implemented for each new geometric element type; see
    /// http://www.vtk.org/VTK/img/file-formats.pdf
    virtual void write_paraview_type(std::ofstream& file_out,
                                     const unsigned& nplot) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return the offsets for the paraview sub-elements. Broken
    /// virtual. Needs to be implemented for each new geometric element type;
    /// see http://www.vtk.org/VTK/img/file-formats.pdf
    virtual void write_paraview_offsets(std::ofstream& file_out,
                                        const unsigned& nplot,
                                        unsigned& offset_sum) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Number of scalars/fields output by this element. Broken
    /// virtual. Needs to be implemented for each new specific element type.
    virtual unsigned nscalar_paraview() const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
 
      // Dummy unsigned
      return 0;
    }
 
    /// Write values of the i-th scalar field at the plot points. Broken
    /// virtual. Needs to be implemented for each new specific element type.
    virtual void scalar_value_paraview(std::ofstream& file_out,
                                       const unsigned& i,
                                       const unsigned& nplot) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Write values of the i-th scalar field at the plot points. Broken
    /// virtual. Needs to be implemented for each new specific element type.
    virtual void scalar_value_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Write values of the i-th scalar field at the plot points. Broken
    /// virtual. Needs to be implemented for each new specific element type.
    virtual void scalar_value_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt) const
    {
      throw OomphLibError(
        "This function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Name of the i-th scalar field. Default implementation
    /// returns V1 for the first one, V2 for the second etc. Can (should!) be
    /// overloaded with more meaningful names in specific elements.
    virtual std::string scalar_name_paraview(const unsigned& i) const
    {
      return "V" + StringConversion::to_string(i);
    }
 
    /// Output the element data --- typically the values at the
    /// nodes in a format suitable for post-processing.
    virtual void output(std::ostream& outfile)
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output the element data --- pass (some measure of)
    /// the number of plot points per element
    virtual void output(std::ostream& outfile, const unsigned& n_plot)
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output the element data at time step t. This is const because
    /// it is newly added and so can be done easily. Really all the
    /// output(...) functions should be const!
    virtual void output(const unsigned& t,
                        std::ostream& outfile,
                        const unsigned& n_plot) const
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output the element data --- typically the values at the
    /// nodes in a format suitable for post-processing.
    /// (C style output)
    virtual void output(FILE* file_pt)
    {
      throw OomphLibError("C-style otput function function hasn't been "
                          "implemented for this element",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output the element data --- pass (some measure of)
    /// the number of plot points per element
    /// (C style output)
    virtual void output(FILE* file_pt, const unsigned& n_plot)
    {
      throw OomphLibError("C-style output function function hasn't been "
                          "implemented for this element",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output an exact solution over the element.
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& n_plot,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for exact solution",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Output a time-dependent exact solution over the element.
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& n_plot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for exact solution",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Output a time-dependent exact solution over the element.
    virtual void output_fct(std::ostream& outfile,
                            const unsigned& n_plot,
                            const double& time,
                            const SolutionFunctorBase& exact_soln) const
    {
      throw OomphLibError(
        "Output function function hasn't been implemented for exact solution",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    ///  Get cector of local coordinates of plot point i (when plotting
    /// nplot points in each "coordinate direction").
    /// Generally these plot points will be uniformly spaced across the element.
    /// The optional final boolean flag (default: false) allows them to
    /// be shifted inwards to avoid duplication of plot point points between
    /// elements -- useful when they are used in locate_zeta, say.
    virtual void get_s_plot(const unsigned& i,
                            const unsigned& nplot,
                            Vector<double>& s,
                            const bool& shifted_to_interior = false) const
    {
      throw OomphLibError(
        "get_s_plot(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    };
 
    /// Return string for tecplot zone header (when plotting
    /// nplot points in each "coordinate direction")
    virtual std::string tecplot_zone_string(const unsigned& nplot) const
    {
      throw OomphLibError(
        "tecplot_zone_string(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
      return "dummy return";
    }
 
    /// Add tecplot zone "footer" to output stream (when plotting
    /// nplot points in each "coordinate direction").
    /// Empty by default -- can be used, e.g., to add FE connectivity
    /// lists to elements that need it.
    virtual void write_tecplot_zone_footer(std::ostream& outfile,
                                           const unsigned& nplot) const {};
 
    /// Add tecplot zone "footer" to C-style output. (when plotting
    /// nplot points in each "coordinate direction").
    /// Empty by default -- can be used, e.g., to add FE connectivity
    /// lists to elements that need it.
    virtual void write_tecplot_zone_footer(FILE* file_pt,
                                           const unsigned& nplot) const {};
 
    /// Return total number of plot points (when plotting
    /// nplot points in each "coordinate direction")
    virtual unsigned nplot_points(const unsigned& nplot) const
    {
      throw OomphLibError(
        "nplot_points(...) is not implemented for this element",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
      // Dummy return
      return 0;
    }
 
 
    /// Calculate the norm of the error and that of the exact solution.
    virtual void compute_error(
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      double& error,
      double& norm)
    {
      std::string error_message = "compute_error undefined for this element \n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Calculate the norm of the error and that of the exact solution.
    virtual void compute_error(
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
      const double& time,
      double& error,
      double& norm)
    {
      std::string error_message = "time-dependent compute_error ";
      error_message += "undefined for this element \n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Given the exact solution \f$ {\bf f}({\bf x}) \f$ this function
    /// calculates the norm of the error and that of the exact solution.
    /// Version with vectors of norms and errors so that different variables'
    /// norms and errors can be returned individually
    virtual void compute_error(
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      Vector<double>& error,
      Vector<double>& norm)
    {
      std::string error_message = "compute_error undefined for this element \n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Given the exact solution \f$ {\bf f}({\bf x}) \f$ this function
    /// calculates the norm of the error and that of the exact solution.
    /// Version with vectors of norms and errors so that different variables'
    /// norms and errors can be returned individually
    virtual void compute_error(
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
      const double& time,
      Vector<double>& error,
      Vector<double>& norm)
    {
      std::string error_message = "time-dependent compute_error ";
      error_message += "undefined for this element \n";
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Plot the error when compared
    /// against a given exact solution \f$ {\bf f}({\bf x}) \f$.
    /// Also calculates the norm of the
    /// error and that of the exact solution.
    virtual void compute_error(
      std::ostream& outfile,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      double& error,
      double& norm)
    {
      std::string error_message = "compute_error undefined for this element \n";
      outfile << error_message;
 
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Plot the error when compared against a given time-dependent exact
    /// solution \f$ {\bf f}(t,{\bf x}) \f$. Also calculates the norm of the
    /// error and that of the exact solution.
    virtual void compute_error(
      std::ostream& outfile,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
      const double& time,
      double& error,
      double& norm)
    {
      std::string error_message =
        "time-dependent compute_error undefined for this element \n";
      outfile << error_message;
 
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Plot the error when compared against a given exact solution
    /// \f$ {\bf f}({\bf x}) \f$. Also calculates the norm of the error and
    /// that of the exact solution.
    /// The version with vectors of norms and errors so that different
    /// variables' norms and errors can be returned individually
    virtual void compute_error(
      std::ostream& outfile,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      Vector<double>& error,
      Vector<double>& norm)
    {
      std::string error_message = "compute_error undefined for this element \n";
      outfile << error_message;
 
      throw OomphLibError(error_message,
                          "FiniteElement::compute_error()",
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Plot the error when compared against a given time-dependent exact
    /// solution \f$ {\bf f}(t,{\bf x}) \f$. Also calculates the norm of the
    /// error and that of the exact solution.
    /// The version with vectors of norms and errors so that different
    /// variables' norms and errors can be returned individually
    virtual void compute_error(
      std::ostream& outfile,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
      const double& time,
      Vector<double>& error,
      Vector<double>& norm)
    {
      std::string error_message =
        "time-dependent compute_error undefined for this element \n";
      outfile << error_message;
 
      throw OomphLibError(error_message,
                          "FiniteElement::compute_error()",
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Plot the error when compared
    /// against a given exact solution \f$ {\bf f}({\bf x}) \f$.
    /// Also calculates the maximum absolute error
    virtual void compute_abs_error(
      std::ostream& outfile,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      double& error)
    {
      std::string error_message =
        "compute_abs_error undefined for this element \n";
      outfile << error_message;
 
      throw OomphLibError(
        error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Evaluate integral of a Vector-valued function
    /// \f$ {\bf f}({\bf x}) \f$ over the element.
    void integrate_fct(FiniteElement::SteadyExactSolutionFctPt integrand_fct_pt,
                       Vector<double>& integral);
 
 
    /// Evaluate integral of a Vector-valued, time-dependent function
    /// \f$ {\bf f}(t,{\bf x}) \f$ over the element.
    void integrate_fct(
      FiniteElement::UnsteadyExactSolutionFctPt integrand_fct_pt,
      const double& time,
      Vector<double>& integral);
 
    /// Function for building a lower dimensional FaceElement
    /// on the specified face of the FiniteElement.
    /// The arguments are the index of the face, an integer whose value
    /// depends on the particular element type, and a pointer to the
    /// FaceElement.
    virtual void build_face_element(const int& face_index,
                                    FaceElement* face_element_pt);
 
 
    /// Self-test: Check inversion of element & do self-test for
    /// GeneralisedElement. Return 0 if OK.
    virtual unsigned self_test();
 
    /// Get the number of the ith node on face face_index (in the bulk node
    /// vector).
    virtual unsigned get_bulk_node_number(const int& face_index,
                                          const unsigned& i) const
    {
      std::string err = "Not implemented for this element.";
      throw OomphLibError(
        err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
    }
 
    /// Get the sign of the outer unit normal on the face given by face_index.
    virtual int face_outer_unit_normal_sign(const int& face_index) const
    {
      std::string err = "Not implemented for this element.";
      throw OomphLibError(
        err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
    }
 
    virtual unsigned nnode_on_face() const
    {
      std::string err = "Not implemented for this element.";
      throw OomphLibError(
        err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
    }
 
    /// Range check for face node numbers
    void face_node_number_error_check(const unsigned& i) const
    {
#ifdef RANGE_CHECKING
      if (i > nnode_on_face())
      {
        std::string err = "Face node index i out of range on face.";
        throw OomphLibError(
          err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
      }
#endif
    }
 
    /// Get a pointer to the function mapping face coordinates to bulk
    /// coordinates
    virtual CoordinateMappingFctPt face_to_bulk_coordinate_fct_pt(
      const int& face_index) const
    {
      std::string err = "Not implemented for this element.";
      throw OomphLibError(
        err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
    }
 
    /// Get a pointer to the derivative of the mapping from face to bulk
    /// coordinates.
    virtual BulkCoordinateDerivativesFctPt bulk_coordinate_derivatives_fct_pt(
      const int& face_index) const
    {
      std::string err = "Not implemented for this element.";
      throw OomphLibError(
        err, OOMPH_EXCEPTION_LOCATION, OOMPH_CURRENT_FUNCTION);
    }
  };
 
 
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Point element has just a single node and a single shape function
  /// which is identically equal to one.
  //=======================================================================
  class PointElement : public virtual FiniteElement
  {
  public:
    /// Constructor
    PointElement()
    {
      /// Allocate storage for the pointers to the nodes,
      /// There is only one nodes
      this->set_n_node(1);
 
      // Set the integration scheme
      this->set_integration_scheme(&Default_integration_scheme);
    }
 
    /// Broken copy constructor
    PointElement(const PointElement&) = delete;
 
    /// Broken assignment operator
    /*void operator=(const PointElement&) = delete;*/
 
    /// Calculate the geometric shape functions at local coordinate s
    void shape(const Vector<double>& s, Shape& psi) const;
 
    /// Get local coordinates of node j in the element; vector sets its own size
    void local_coordinate_of_node(const unsigned& j, Vector<double>& s) const
    {
      s.resize(0);
    }
 
    ///  Return spatial dimension of element (=number of local
    /// coordinates needed to parametrise the element)
    // unsigned dim() const {return 0;}
 
  private:
    /// Default integration scheme
    static PointIntegral Default_integration_scheme;
  };
 
 
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
 
 
  class GeomObject;
 
  //=======================================================================
  /// A class to specify the initial conditions for a solid body.
  /// Solid bodies are often discretised with
  /// Hermite-type elements, for which the assignment
  /// of the generalised nodal values is nontrivial since they represent
  /// derivatives  w.r.t. to the local coordinates. A SolidInitialCondition
  /// object specifies initial position (i.e. shape), velocity and acceleration
  /// of the structure with a geometric object.
  /// An integer specifies which time-derivative derivative is currently
  /// assigned. See example codes for a demonstration of its use.
  //=======================================================================
  class SolidInitialCondition
  {
  public:
    /// Constructor: Pass geometric object; initialise time deriv to 0
    SolidInitialCondition(GeomObject* geom_object_pt)
      : Geom_object_pt(geom_object_pt), IC_time_deriv(0){};
 
 
    /// Broken copy constructor
    SolidInitialCondition(const SolidInitialCondition&) = delete;
 
    /// Broken assignment operator
    void operator=(const SolidInitialCondition&) = delete;
 
    /// (Reference to) pointer to geom object that specifies the initial
    /// condition
    GeomObject*& geom_object_pt()
    {
      return Geom_object_pt;
    }
 
    /// Which time derivative are we currently assigning?
    unsigned& ic_time_deriv()
    {
      return IC_time_deriv;
    }
 
 
  private:
    /// Pointer to the GeomObject that specifies the initial
    /// condition (shape, veloc and accel)
    GeomObject* Geom_object_pt;
 
    /// Which time derivative (0,1,2) are we currently assigning
    unsigned IC_time_deriv;
  };
 
 
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
 
 
  //============================================================================
  /// SolidFiniteElement class.
  ///
  /// Solid elements are elements whose nodal positions are unknowns
  /// in the problem -- their nodes are SolidNodes. In such elements,
  /// the nodes not only have a variable (Eulerian) but also a fixed
  /// (Lagrangian) position. The positional variables have their own
  /// local equation numbering scheme which is set up with
  /// \code assign_solid_local_eqn_numbers () \endcode
  /// The derivatives of the
  /// `solid equations' (i.e. the equations that determine the
  /// nodal positions) with respect to the nodal positions, required
  /// in the Jacobian matrix, are determined by finite differencing.
  ///
  /// In the present form, the SolidFiniteElement represents
  /// a fully functional base class for `proper' solid mechanics elements,
  /// but it can also be combined
  /// (via inheritance) with elements that solve additional equations.
  /// This is particularly useful in cases where the solid equations
  /// are merely used to update the nodal positions in a moving mesh
  /// problem, although this can prove costly in practice.
  //============================================================================
  class SolidFiniteElement : public virtual FiniteElement
  {
  public:
    /// Set the lagrangian dimension of the element --- the number
    /// of lagrangian coordinates stored at the nodes in the element.
    void set_lagrangian_dimension(const unsigned& lagrangian_dimension)
    {
      Lagrangian_dimension = lagrangian_dimension;
    }
 
    /// Return whether there is internal solid data (e.g. discontinuous
    /// solid pressure). At present, this is used to report an error in
    /// setting initial conditions for ElasticProblems which can't
    /// handle such cases. The default is false.
    virtual bool has_internal_solid_data()
    {
      return false;
    }
 
    /// Pointer to function that computes the "multiplier" for the
    /// inertia terms in the consistent determination of the initial
    /// conditions for Newmark timestepping.
    typedef double (*MultiplierFctPt)(const Vector<double>& xi);
 
    /// Constructor: Set defaults
    SolidFiniteElement()
      : FiniteElement(),
        Undeformed_macro_elem_pt(0),
        Solid_ic_pt(0),
        Multiplier_fct_pt(0),
        Position_local_eqn(0),
        Lagrangian_dimension(0),
        Nnodal_lagrangian_type(1),
        Solve_for_consistent_newmark_accel_flag(false)
    {
    }
 
    /// Destructor to clean up any allocated memory
    virtual ~SolidFiniteElement();
 
    /// Broken copy constructor
    SolidFiniteElement(const SolidFiniteElement&) = delete;
 
    /// Broken assignment operator
    /*void operator=(const SolidFiniteElement&) = delete;*/
 
    /// The number of geometric data affecting a SolidFiniteElemnet is
    /// the same as the number of nodes (one variable position data per node)
    inline unsigned ngeom_data() const
    {
      return nnode();
    }
 
    /// Return pointer to the j-th Data item that the object's
    /// shape depends on. (Redirects to the nodes' positional Data).
    inline Data* geom_data_pt(const unsigned& j)
    {
      return static_cast<SolidNode*>(node_pt(j))->variable_position_pt();
    }
 
    /// Specify Data that affects the geometry of the element
    /// by adding the position Data to the set that's passed in.
    /// (This functionality is required in FSI problems; set is used to
    /// avoid double counting).
    void identify_geometric_data(std::set<Data*>& geometric_data_pt)
    {
      // Loop over the node update data and add to the set
      const unsigned n_node = this->nnode();
      for (unsigned n = 0; n < n_node; n++)
      {
        geometric_data_pt.insert(
          dynamic_cast<SolidNode*>(this->node_pt(n))->variable_position_pt());
      }
    }
 
 
    /// In a SolidFiniteElement, the "global" intrinsic coordinate
    /// of the element when viewed as part of a compound geometric
    /// object (a Mesh) is, by default, the Lagrangian coordinate
    /// Note the assumption here is that we are always using isoparameteric
    /// elements in which the lagrangian coordinate is interpolated by the
    /// same shape functions as the eulerian coordinate.
    inline double zeta_nodal(const unsigned& n,
                             const unsigned& k,
                             const unsigned& i) const
    {
      return lagrangian_position_gen(n, k, i);
    }
 
    /// Eulerian and Lagrangian coordinates as function of the
    /// local coordinates: The Eulerian position is returned in
    /// FE-interpolated form (\c x_fe) and then in the form obtained
    /// from the "current" MacroElement representation (if it exists -- if not,
    /// \c x is the same as \c x_fe). This allows the Domain/MacroElement-
    /// based representation to be used to apply displacement boundary
    /// conditions exactly. Ditto for the Lagrangian coordinates returned
    /// in xi_fe and xi.
    /// (Broken virtual -- overload in specific geometric element class
    /// if you want to use this functionality.)
    virtual void get_x_and_xi(const Vector<double>& s,
                              Vector<double>& x_fe,
                              Vector<double>& x,
                              Vector<double>& xi_fe,
                              Vector<double>& xi) const
    {
      throw OomphLibError(
        "get_x_and_xi(...) is not implemented for this element\n",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Set pointer to MacroElement -- overloads generic version
    /// and uses the MacroElement
    /// also as the default for the "undeformed" configuration.
    /// This assignment must be overwritten with
    /// set_undeformed_macro_elem_pt(...) if the deformation of
    /// the solid body is driven by a deformation of the
    /// "current" Domain/MacroElement representation of it's boundary.
    /// Can be overloaded in derived classes to perform additional
    /// tasks
    virtual void set_macro_elem_pt(MacroElement* macro_elem_pt)
    {
      Macro_elem_pt = macro_elem_pt;
      Undeformed_macro_elem_pt = macro_elem_pt;
    }
 
    /// Set pointers to "current" and "undeformed" MacroElements.
    /// Can be overloaded in derived classes to perform additional
    /// tasks
    virtual void set_macro_elem_pt(MacroElement* macro_elem_pt,
                                   MacroElement* undeformed_macro_elem_pt)
    {
      Macro_elem_pt = macro_elem_pt;
      Undeformed_macro_elem_pt = undeformed_macro_elem_pt;
    }
 
    /// Set pointer to "undeformed" macro element.
    /// Can be overloaded in derived classes to perform additional
    /// tasks.
    void set_undeformed_macro_elem_pt(MacroElement* undeformed_macro_elem_pt)
    {
      Undeformed_macro_elem_pt = undeformed_macro_elem_pt;
    }
 
 
    /// Access function to pointer to "undeformed" macro element
    MacroElement* undeformed_macro_elem_pt()
    {
      return Undeformed_macro_elem_pt;
    }
 
    /// Calculate shape functions and derivatives w.r.t. Lagrangian
    /// coordinates at local coordinate s. Returns the Jacobian of the mapping
    /// from Lagrangian to local coordinates.
    double dshape_lagrangian(const Vector<double>& s,
                             Shape& psi,
                             DShape& dpsidxi) const;
 
    /// Return the geometric shape functions and also first
    /// derivatives w.r.t. Lagrangian coordinates at ipt-th integration point.
    virtual double dshape_lagrangian_at_knot(const unsigned& ipt,
                                             Shape& psi,
                                             DShape& dpsidxi) const;
 
    /// Compute the geometric shape functions and also first
    /// and second derivatives w.r.t. Lagrangian coordinates at
    /// local coordinate s;
    /// Returns Jacobian of mapping from Lagrangian to local coordinates.
    ///  Numbering:
    ///  \b 1D:
    /// d2pidxi(i,0) = \f$ d^2 \psi_j / d \xi^2 \f$
    ///  \b 2D:
    /// d2psidxi(i,0) = \f$ \partial^2 \psi_j / \partial \xi_0^2 \f$
    /// d2psidxi(i,1) = \f$ \partial^2 \psi_j / \partial \xi_1^2 \f$
    /// d2psidxi(i,2) = \f$ \partial^2 \psi_j / \partial \xi_0 \partial \xi_1 \f$
    ///  \b 3D:
    /// d2psidxi(i,0) = \f$ \partial^2 \psi_j / \partial \xi_0^2 \f$
    /// d2psidxi(i,1) = \f$ \partial^2 \psi_j / \partial \xi_1^2 \f$
    /// d2psidxi(i,2) = \f$ \partial^2 \psi_j / \partial \xi_2^2 \f$
    /// d2psidxi(i,3) = \f$ \partial^2 \psi_j/\partial \xi_0 \partial \xi_1 \f$
    /// d2psidxi(i,4) = \f$ \partial^2 \psi_j/\partial \xi_0 \partial \xi_2 \f$
    /// d2psidxi(i,5) = \f$ \partial^2 \psi_j/\partial \xi_1 \partial \xi_2 \f$
    double d2shape_lagrangian(const Vector<double>& s,
                              Shape& psi,
                              DShape& dpsidxi,
                              DShape& d2psidxi) const;
 
    /// Return  the geometric shape functions and also first
    /// and second derivatives w.r.t. Lagrangian coordinates at
    /// the ipt-th integration point.
    /// Returns Jacobian of mapping from Lagrangian to local coordinates.
    ///  Numbering:
    ///  \b 1D:
    /// d2pidxi(i,0) = \f$ d^2 \psi_j / d^2 \xi^2 \f$
    ///  \b 2D:
    /// d2psidxi(i,0) = \f$ \partial^2 \psi_j/\partial^2 \xi_0^2 \f$
    /// d2psidxi(i,1) = \f$ \partial^2 \psi_j/\partial^2 \xi_1^2 \f$
    /// d2psidxi(i,2) = \f$ \partial^2 \psi_j/\partial \xi_0 \partial \xi_1 \f$
    ///  \b 3D:
    /// d2psidxi(i,0) = \f$ \partial^2 \psi_j / \partial^2 \xi_0^2 \f$
    /// d2psidxi(i,1) = \f$ \partial^2 \psi_j / \partial^2 \xi_1^2 \f$
    /// d2psidxi(i,2) = \f$ \partial^2 \psi_j / \partial^2 \xi_2^2 \f$
    /// d2psidxi(i,3) = \f$ \partial^2 \psi_j / \partial \xi_0 \partial \xi_1 \f$
    /// d2psidxi(i,4) = \f$ \partial^2 \psi_j / \partial \xi_0 \partial \xi_2 \f$
    /// d2psidxi(i,5) = \f$ \partial^2 \psi_j / \partial \xi_1 \partial \xi_2 \f$
    virtual double d2shape_lagrangian_at_knot(const unsigned& ipt,
                                              Shape& psi,
                                              DShape& dpsidxi,
                                              DShape& d2psidxi) const;
 
    /// Return the number of Lagrangian coordinates that the
    /// element requires at all nodes.
    /// This is by default the elemental dimension. If we ever need any
    /// other case, it can be implemented.
    unsigned lagrangian_dimension() const
    {
      return Lagrangian_dimension;
    }
 
    /// Return the number of types of (generalised)
    /// nodal Lagrangian coordinates required to
    /// interpolate the Lagrangian coordinates in the element. (E.g.
    /// 1 for Lagrange-type elements; 2 for Hermite beam elements;
    /// 4 for Hermite shell elements). Default value is 1. Needs to
    /// be overloaded for any other element.
    unsigned nnodal_lagrangian_type() const
    {
      return Nnodal_lagrangian_type;
    }
 
    /// Construct the local node n and return a pointer to it.
    Node* construct_node(const unsigned& n)
    {
      // Construct a solid node and assign it to the local node pointer vector.
      // The dimension and number of values are taken from internal element data
      // The number of solid pressure dofs are also taken from internal data
      // The number of timesteps to be stored comes from the problem!
      node_pt(n) = new SolidNode(lagrangian_dimension(),
                                 nnodal_lagrangian_type(),
                                 nodal_dimension(),
                                 nnodal_position_type(),
                                 required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Construct the local node n and return
    /// a pointer to it. Additionally, create storage for `history'
    /// values as required by timestepper
    Node* construct_node(const unsigned& n, TimeStepper* const& time_stepper_pt)
    {
      // Construct a solid node and assign it to the local node pointer vector
      // The dimension and number of values are taken from internal element data
      // The number of solid pressure dofs are also taken from internal data
      node_pt(n) = new SolidNode(time_stepper_pt,
                                 lagrangian_dimension(),
                                 nnodal_lagrangian_type(),
                                 nodal_dimension(),
                                 nnodal_position_type(),
                                 required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Construct the local node n and return a pointer to it.
    /// in the case when it is a boundary node; that is it MAY be
    /// located on a Mesh boundary
    Node* construct_boundary_node(const unsigned& n)
    {
      // Construct a solid node and assign it to the local node pointer vector.
      // The dimension and number of values are taken from internal element data
      // The number of solid pressure dofs are also taken from internal data
      // The number of timesteps to be stored comes from the problem!
      node_pt(n) = new BoundaryNode<SolidNode>(lagrangian_dimension(),
                                               nnodal_lagrangian_type(),
                                               nodal_dimension(),
                                               nnodal_position_type(),
                                               required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Construct the local node n and return
    /// a pointer to it, in the case when the node MAY be located
    /// on a boundary. Additionally, create storage for `history'
    /// values as required by timestepper
    Node* construct_boundary_node(const unsigned& n,
                                  TimeStepper* const& time_stepper_pt)
    {
      // Construct a solid node and assign it to the local node pointer vector
      // The dimension and number of values are taken from internal element data
      // The number of solid pressure dofs are also taken from internal data
      node_pt(n) = new BoundaryNode<SolidNode>(time_stepper_pt,
                                               lagrangian_dimension(),
                                               nnodal_lagrangian_type(),
                                               nodal_dimension(),
                                               nnodal_position_type(),
                                               required_nvalue(n));
      // Now return a pointer to the node, so that the mesh can find it
      return node_pt(n);
    }
 
    /// Overload assign_all_generic_local_equation numbers to
    /// include the data associated with solid dofs.
    /// It remains virtual so that it can be overloaded
    /// by RefineableSolidElements. If the boolean argument is true
    /// then the degrees of freedom are stored in Dof_pt
    virtual inline void assign_all_generic_local_eqn_numbers(
      const bool& store_local_dof_pt)
    {
      // Call the standard finite element equation numbering
      //(internal, external and nodal data).
      FiniteElement::assign_all_generic_local_eqn_numbers(store_local_dof_pt);
      // Assign the numbering for the solid dofs
      assign_solid_local_eqn_numbers(store_local_dof_pt);
    }
 
    /// 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 describe_local_dofs(std::ostream& out,
                             const std::string& current_string) const;
 
    /// Return i-th Lagrangian coordinate at local node n without using
    /// the hanging representation
    double raw_lagrangian_position(const unsigned& n, const unsigned& i) const
    {
      return static_cast<SolidNode*>(node_pt(n))->xi(i);
    }
 
    /// Return Generalised Lagrangian coordinate at local node n.
    /// `Direction' i, `Type' k. Does not use the hanging node representation
    double raw_lagrangian_position_gen(const unsigned& n,
                                       const unsigned& k,
                                       const unsigned& i) const
    {
      return static_cast<SolidNode*>(node_pt(n))->xi_gen(k, i);
    }
 
    /// Return i-th Lagrangian coordinate at local node n
    double lagrangian_position(const unsigned& n, const unsigned& i) const
    {
      return static_cast<SolidNode*>(node_pt(n))->lagrangian_position(i);
    }
 
    /// Return Generalised Lagrangian coordinate at local node n.
    /// `Direction' i, `Type' k.
    double lagrangian_position_gen(const unsigned& n,
                                   const unsigned& k,
                                   const unsigned& i) const
    {
      return static_cast<SolidNode*>(node_pt(n))->lagrangian_position_gen(k, i);
    }
 
    /// Return i-th FE-interpolated Lagrangian coordinate xi[i] at
    /// local coordinate s.
    virtual double interpolated_xi(const Vector<double>& s,
                                   const unsigned& i) const;
 
    /// Compute FE interpolated Lagrangian coordinate vector xi[] at
    /// local coordinate s as Vector
    virtual void interpolated_xi(const Vector<double>& s,
                                 Vector<double>& xi) const;
 
    /// Compute derivatives of FE-interpolated Lagrangian coordinates xi
    /// with respect to local coordinates: dxids[i][j]=dxi_i/ds_j.
    virtual void interpolated_dxids(const Vector<double>& s,
                                    DenseMatrix<double>& dxids) const;
 
    /// Return the Jacobian of mapping from local to Lagrangian
    /// coordinates at local position s.  NOT YET IMPLEMENTED
    virtual void J_lagrangian(const Vector<double>& s) const
    {
      // Must be implemented and overloaded in FaceElements
      throw OomphLibError("Function not implemented yet",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Return the Jacobian of the mapping from local to Lagrangian
    /// coordinates at the ipt-th integration point. NOT YET IMPLEMENTED
    virtual double J_lagrangian_at_knot(const unsigned& ipt) const
    {
      // Must be implemented and overloaded in FaceElements
      throw OomphLibError("Function not implemented yet",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Pointer to object that describes the initial condition.
    SolidInitialCondition*& solid_ic_pt()
    {
      return Solid_ic_pt;
    }
 
    /// Set to alter the problem being solved when
    /// assigning the initial conditions for time-dependent problems:
    /// solve for the history value that
    /// corresponds to the acceleration in the Newmark scheme by demanding
    /// that the PDE is satisifed at the initial time.  In this case
    /// the Jacobian is replaced by the mass matrix.
    void enable_solve_for_consistent_newmark_accel()
    {
      Solve_for_consistent_newmark_accel_flag = true;
    }
 
    /// Set to reset the problem being solved to be the standard problem
    void disable_solve_for_consistent_newmark_accel()
    {
      Solve_for_consistent_newmark_accel_flag = false;
    }
 
    /// Access function: Pointer to multiplicator function for assignment
    /// of consistent assignement of initial conditions for Newmark scheme
    MultiplierFctPt& multiplier_fct_pt()
    {
      return Multiplier_fct_pt;
    }
 
 
    /// Access function: Pointer to multiplicator function for assignment
    /// of consistent assignement of initial conditions for Newmark scheme
    /// (const version)
    MultiplierFctPt multiplier_fct_pt() const
    {
      return Multiplier_fct_pt;
    }
 
 
    /// Compute the residuals for the setup of an initial condition.
    /// The global equations are:
    /// \f[ 0 = \int \left( \sum_{j=1}^N \sum_{k=1}^K X_{ijk} \psi_{jk}(\xi_n) - \frac{\partial^D R^{(IC)}_i(\xi_n)}{\partial t^D} \right) \psi_{lm}(\xi_n) \ dv \mbox{ \ \ \ \ for \ \ \ $l=1,...,N, \ \ m=1,...,K$} \f]
    /// where \f$ N \f$ is the number of nodes in the mesh and \f$ K \f$
    /// the number of generalised nodal coordinates. The initial shape
    /// of the solid body, \f$ {\bf R}^{(IC)},\f$ and its time-derivatives
    /// are specified via the \c GeomObject that is stored in the
    /// \c SolidFiniteElement::SolidInitialCondition object. The latter also
    /// stores the order of the time-derivative \f$ D \f$ to be assigned.
    virtual void get_residuals_for_solid_ic(Vector<double>& residuals)
    {
      residuals.initialise(0.0);
      fill_in_residuals_for_solid_ic(residuals);
    }
 
    /// Fill in the residuals for the setup of an initial condition.
    /// The global equations are:
    /// \f[ 0 = \int \left( \sum_{j=1}^N \sum_{k=1}^K X_{ijk} \psi_{jk}(\xi_n) - \frac{\partial^D R^{(IC)}_i(\xi_n)}{\partial t^D} \right) \psi_{lm}(\xi_n) \ dv \mbox{ \ \ \ \ for \ \ \ $l=1,...,N, \ \ m=1,...,K$} \f]
    /// where \f$ N \f$ is the number of nodes in the mesh and \f$ K \f$
    /// the number of generalised nodal coordinates. The initial shape
    /// of the solid body, \f$ {\bf R}^{(IC)},\f$ and its time-derivatives
    /// are specified via the \c GeomObject that is stored in the
    /// \c SolidFiniteElement::SolidInitialCondition object. The latter also
    /// stores the order of the time-derivative \f$ D \f$ to be assigned.
    void fill_in_residuals_for_solid_ic(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using a dummy matrix argument
      fill_in_generic_jacobian_for_solid_ic(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// Fill in the residuals and Jacobian for the setup of an
    /// initial condition. The global equations are:
    /// \f[ 0 = \int \left( \sum_{j=1}^N \sum_{k=1}^K X_{ijk} \psi_{jk}(\xi_n) - \frac{\partial^D R^{(IC)}_i(\xi_n)}{\partial t^D} \right) \psi_{lm}(\xi_n) \ dv \mbox{ \ \ \ \ for \ \ \ $l=1,...,N, \ \ m=1,...,K$} \f]
    /// where \f$ N \f$ is the number of nodes in the mesh and \f$ K \f$
    /// the number of generalised nodal coordinates. The initial shape
    /// of the solid body, \f$ {\bf R}^{(IC)},\f$ and its time-derivatives
    /// are specified via the \c GeomObject that is stored in the
    /// \c SolidFiniteElement::SolidInitialCondition object. The latter also
    /// stores the order of the time-derivative \f$ D \f$ to be assigned.
    void fill_in_jacobian_for_solid_ic(Vector<double>& residuals,
                                       DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_jacobian_for_solid_ic(residuals, jacobian, 1);
    }
 
 
    /// Fill in the contributions of the Jacobian matrix
    /// for the consistent assignment of the initial "accelerations" in
    /// Newmark scheme. In this case the Jacobian is the mass matrix.
    void fill_in_jacobian_for_newmark_accel(DenseMatrix<double>& jacobian);
 
 
    /// Calculate the L2 norm of the displacement u=R-r to overload the
    /// compute_norm function in the GeneralisedElement base class
    void compute_norm(double& el_norm);
 
  protected:
    /// Helper function to fill in the residuals and (if flag==1) the
    /// Jacobian for the setup of an initial condition. The global equations
    /// are:
    /// \f[ 0 = \int \left( \sum_{j=1}^N \sum_{k=1}^K X_{ijk} \psi_{jk}(\xi_n) - \frac{\partial^D R^{(IC)}_i(\xi_n)}{\partial t^D} \right) \psi_{lm}(\xi_n) \ dv \mbox{ \ \ \ \ for \ \ \ $l=1,...,N, \ \ m=1,...,K$} \f]
    /// where \f$ N \f$ is the number of nodes in the mesh and \f$ K \f$
    /// the number of generalised nodal coordinates. The initial shape
    /// of the solid body, \f$ {\bf R}^{(IC)},\f$ and its time-derivatives
    /// are specified via the \c GeomObject that is stored in the
    /// \c SolidFiniteElement::SolidInitialCondition object. The latter also
    /// stores the order of the time-derivative \f$ D \f$ to be assigned.
    void fill_in_generic_jacobian_for_solid_ic(Vector<double>& residuals,
                                               DenseMatrix<double>& jacobian,
                                               const unsigned& flag);
 
    /// Set the number of types required to interpolate the
    /// Lagrangian coordinates
    void set_nnodal_lagrangian_type(const unsigned& nlagrangian_type)
    {
      Nnodal_lagrangian_type = nlagrangian_type;
    }
 
    /// Pointer to the element's "undeformed" macro element (NULL by default)
    MacroElement* Undeformed_macro_elem_pt;
 
    /// Calculate the mapping from local to lagrangian coordinates,
    /// given the derivatives of the shape functions w.r.t. local coorindates.
    /// Return the determinant of the jacobian, the jacobian  and inverse
    /// jacobian
    virtual double local_to_lagrangian_mapping(
      const DShape& dpsids,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& inverse_jacobian) const
    {
      // Assemble the jacobian
      assemble_local_to_lagrangian_jacobian(dpsids, jacobian);
      // Invert the jacobian
      return invert_jacobian_mapping(jacobian, inverse_jacobian);
    }
 
    /// Calculate the mapping from local to lagrangian coordinates,
    /// given the derivatives of the shape functions w.r.t. local coordinates,
    /// Return only the determinant of the jacobian and the inverse of the
    /// mapping (ds/dx)
    double local_to_lagrangian_mapping(
      const DShape& dpsids, DenseMatrix<double>& inverse_jacobian) const
    {
      // Find the dimension of the element
      unsigned el_dim = dim();
      // Assign memory for the jacobian
      DenseMatrix<double> jacobian(el_dim);
      // Calculate the jacobian and inverse
      return local_to_lagrangian_mapping(dpsids, jacobian, inverse_jacobian);
    }
 
    /// Calculate the mapping from local to Lagrangian coordinates given
    /// the derivatives of the shape functions w.r.t the local coorindates.
    /// 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.
    /// This function returns the determinant of the jacobian, the jacobian
    /// and the inverse jacobian.
    virtual double local_to_lagrangian_mapping_diagonal(
      const DShape& dpsids,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& inverse_jacobian) const;
 
 
    /// Assigns local equation numbers for the generic solid
    /// local equation numbering schemes.
    /// If the boolean flag is true the the degrees of freedom are stored in
    /// Dof_pt
    virtual void assign_solid_local_eqn_numbers(const bool& store_local_dof);
 
    /// Classifies dofs locally for solid specific aspects.
    void describe_solid_local_dofs(std::ostream& out,
                                   const std::string& current_string) const;
 
    /// Pointer to object that specifies the initial condition
    SolidInitialCondition* Solid_ic_pt;
 
  public:
    /// Access function that returns the local equation number that
    /// corresponds to the j-th coordinate of the k-th position-type at the
    /// n-th local node.
    inline int position_local_eqn(const unsigned& n,
                                  const unsigned& k,
                                  const unsigned& j) const
    {
#ifdef RANGE_CHECKING
      std::ostringstream error_message;
      bool error = false;
      if (n >= nnode())
      {
        error = true;
        error_message << "Range Error: Nodal number " << n
                      << " is not in the range (0," << nnode() << ")";
      }
 
      if (k >= nnodal_position_type())
      {
        error = true;
        error_message << "Range Error: Position type " << k
                      << " is not in the range (0," << nnodal_position_type()
                      << ")";
      }
 
      if (j >= nodal_dimension())
      {
        error = true;
        error_message << "Range Error: Nodal coordinate " << j
                      << " is not in the range (0," << nodal_dimension() << ")";
      }
 
      if (error)
      {
        // Throw the error
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Return the value
      return Position_local_eqn[(n * nnodal_position_type() + k) *
                                  nodal_dimension() +
                                j];
    }
 
  protected:
    /// Overload the fill_in_contribution_to_jacobian() function to
    /// use finite
    /// differences to calculate the solid residuals by default
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Add the contribution to the residuals
      fill_in_contribution_to_residuals(residuals);
 
      // Solve for the consistent acceleration in Newmark scheme?
      if (Solve_for_consistent_newmark_accel_flag)
      {
        fill_in_jacobian_for_newmark_accel(jacobian);
        return;
      }
 
      // Allocate storage for the full residuals (residuals of entire element)
      unsigned n_dof = ndof();
      Vector<double> full_residuals(n_dof);
      // Get the residuals for the entire element
      get_residuals(full_residuals);
      // Get the solid entries in the jacobian using finite differences
      fill_in_jacobian_from_solid_position_by_fd(full_residuals, jacobian);
      // There could be internal data
      //(finite-difference the lot by default)
      fill_in_jacobian_from_internal_by_fd(full_residuals, jacobian, true);
      // There could also be external data
      //(finite-difference the lot by default)
      fill_in_jacobian_from_external_by_fd(full_residuals, jacobian, true);
      // There could also be nodal data
      fill_in_jacobian_from_nodal_by_fd(full_residuals, jacobian);
    }
 
 
    /// Use finite differences to calculate the Jacobian entries
    /// corresponding to the solid positions. This version assumes
    /// that the residuals vector has already been computed
    virtual void fill_in_jacobian_from_solid_position_by_fd(
      Vector<double>& residuals, DenseMatrix<double>& jacobian);
 
 
    /// Use finite differences to calculate the Jacobian entries
    /// corresponding to the solid positions.
    void fill_in_jacobian_from_solid_position_by_fd(
      DenseMatrix<double>& jacobian)
    {
      // Allocate storage for a residuals vector and initialise to zero
      unsigned n_dof = ndof();
      Vector<double> residuals(n_dof, 0.0);
      // Get the residuals for the entire element
      get_residuals(residuals);
      // Call the jacobian calculation
      fill_in_jacobian_from_solid_position_by_fd(residuals, jacobian);
    }
 
    /// Function that is called before the finite differencing of
    /// any solid position data. This may be overloaded to update any dependent
    /// data before finite differencing takes place.
    virtual inline void update_before_solid_position_fd() {}
 
    /// Function that is call after the finite differencing of
    /// the solid position data. This may be overloaded to reset any dependent
    /// variables that may have changed during the finite differencing.
    virtual inline void reset_after_solid_position_fd() {}
 
    /// Function called within the finite difference loop for
    /// the solid position dat after a change in any values in the n-th
    ///  node.
    virtual inline void update_in_solid_position_fd(const unsigned& i) {}
 
    /// Function called within the finite difference loop for
    /// solid position data after the values in the i-th node
    /// are reset. The default behaviour is to call the update function.
    virtual inline void reset_in_solid_position_fd(const unsigned& i)
    {
      update_in_solid_position_fd(i);
    }
 
 
  private:
    /// Assemble the jacobian matrix for the mapping from local
    /// to lagrangian coordinates, given the derivatives of the shape function
    virtual void assemble_local_to_lagrangian_jacobian(
      const DShape& dpsids, DenseMatrix<double>& jacobian) const;
 
 
    /// Assemble the the "jacobian" matrix of second derivatives, given
    /// the second derivatives of the shape functions w.r.t. local coordinates
    virtual void assemble_local_to_lagrangian_jacobian2(
      const DShape& d2psids, DenseMatrix<double>& jacobian2) const;
 
    /// Pointer to function that computes the "multiplier" for the
    /// inertia terms in the consistent determination of the initial
    /// conditions for Newmark timestepping.
    MultiplierFctPt Multiplier_fct_pt;
 
    /// Array to hold the local equation number information for the
    /// solid equations, whatever they may be.
    // ALH: (This is here so that the numbering can be done automatically)
    int* Position_local_eqn;
 
    /// The Lagrangian dimension of the nodes stored in the element,
    /// / i.e. the number of Lagrangian coordinates
    unsigned Lagrangian_dimension;
 
    /// The number of coordinate types requried to intepolate the
    /// Lagrangian coordinates in the element.  For Lagrange elements
    /// it is 1 (the default). It must be over-ridden by using
    /// the set_nlagrangian_type() function in the constructors of elements
    /// that use generalised coordinate, e.g. for 1D Hermite elements
    /// Nnodal_position_types =2.
    unsigned Nnodal_lagrangian_type;
 
  protected:
    /// Flag to indicate which system of equations to solve
    /// when assigning initial conditions for time-dependent problems.
    /// If true, solve for the history value that
    /// corresponds to the acceleration in the Newmark scheme by demanding
    /// that the PDE is satisifed at the initial time. In this case
    /// the Jacobian is replaced by the mass matrix.
    bool Solve_for_consistent_newmark_accel_flag;
 
  private:
    /// Access to the "multiplier" for the
    /// inertia terms in the consistent determination of the initial
    /// conditions for Newmark timestepping.
    inline double multiplier(const Vector<double>& xi)
    {
      // If no function has been set, return 1
      if (Multiplier_fct_pt == 0)
      {
        return 1.0;
      }
      else
      {
        // Evaluate function pointer
        return (*Multiplier_fct_pt)(xi);
      }
    }
  };
 
 
  //========================================================================
  /// FaceElements are elements that coincide with the faces of
  /// higher-dimensional "bulk" elements. They are used on
  /// boundaries where additional non-trivial boundary conditions need to be
  /// applied. Examples include free surfaces, and applied traction conditions.
  /// In many cases, FaceElements need to evaluate to quantities
  /// in the associated bulk elements. For instance, the evaluation
  /// of a shear stresses on 2D FaceElement requires the evaluation
  /// of velocity derivatives in the associated 3D volume element etc.
  /// Therefore we store a pointer to the associated bulk element,
  /// and information about the relation between the local
  /// coordinates in the face and bulk elements.
  //========================================================================
  class FaceElement : public virtual FiniteElement
  {
    /// Typedef for the function that translates the face coordinate
    /// to the coordinate in the bulk element
    typedef void (*CoordinateMappingFctPt)(const Vector<double>& s,
                                           Vector<double>& s_bulk);
 
    /// Typedef for the function that returns the partial derivative
    /// of the local coordinates in the bulk element
    /// with respect to the coordinates along the face.
    /// In addition this function returns an index of one of the
    /// bulk local coordinates that varies away from the edge
    typedef void (*BulkCoordinateDerivativesFctPt)(
      const Vector<double>& s,
      DenseMatrix<double>& ds_bulk_dsface,
      unsigned& interior_direction);
 
  private:
    /// Pointer to a function that translates the face coordinate
    /// to the coordinate in the bulk element
    CoordinateMappingFctPt Face_to_bulk_coordinate_fct_pt;
 
    /// Pointer to a function that returns the derivatives of the local
    /// "bulk" coordinates with respect to the local face coordinates.
    BulkCoordinateDerivativesFctPt Bulk_coordinate_derivatives_fct_pt;
 
    /// Vector holding integers to translate additional position types
    /// to those of the "bulk" element; e.g. Bulk_position_type(1) is
    /// the position type of the "bulk" element that corresponds to face
    /// position type 1. This is required in QHermiteElements, where the slope
    /// in the direction of the 1D element is face position type 1, but may be
    /// position type 1 or 2 in the "bulk" element, depending upon which face,
    /// we are located.
    Vector<unsigned> Bulk_position_type;
 
    /// Sign of outer unit normal (relative to cross-products of tangent
    /// vectors in the corresponding "bulk" element.
    int Normal_sign;
 
    /// Index of the face
    int Face_index;
 
    /// Ignores the warning when the tangent vectors from
    /// continuous_tangent_and_outer_unit_normal may not be continuous
    /// as a result of the unit normal being aligned with the z axis.
    /// This can be avoided by supplying a general direction vector for
    /// the tangent vector via set_tangent_direction(...).
    static bool Ignore_discontinuous_tangent_warning;
 
  protected:
    /// The boundary number in the bulk mesh to which this element is attached
    unsigned Boundary_number_in_bulk_mesh;
 
#ifdef PARANOID
 
    /// Has the Boundary_number_in_bulk_mesh been set? Only included if
    /// compiled with PARANOID switched on.
    bool Boundary_number_in_bulk_mesh_has_been_set;
 
#endif
 
    /// Pointer to the associated higher-dimensional "bulk" element
    FiniteElement* Bulk_element_pt;
 
    /// List of indices of the local node numbers in the "bulk" element
    /// that correspond to the local node numbers in the FaceElement
    Vector<unsigned> Bulk_node_number;
 
    /// A vector that will hold the number of data values at the nodes
    /// that are associated with the "bulk" element. i.e. not including any
    /// additional degrees of freedom that might be required for extra equations
    /// that are being solved by
    /// the FaceElement.
    // NOTE: This breaks if the nodes have already been resized
    // before calling the face element, i.e. two separate sets of equations on
    // the face!
    Vector<unsigned> Nbulk_value;
 
    /// A general direction pointer for the tangent vectors.
    /// This is used in the function continuous_tangent_and_outer_unit_normal()
    /// for creating continuous tangent vectors in spatial dimensions.
    /// The general direction is projected on to the surface. This technique is
    /// not required in two spatial dimensions.
    Vector<double>* Tangent_direction_pt;
 
    /// Helper function adding additional values for the unknowns
    /// associated with the FaceElement. This function also sets the map
    /// containing the position of the first entry of this face element's
    /// additional values.The inputs are the number of additional values
    /// and the face element's ID. Note the same number of additonal values
    /// are allocated at ALL nodes.
    void add_additional_values(const Vector<unsigned>& nadditional_values,
                               const unsigned& id)
    {
      // How many nodes?
      const unsigned n_node = nnode();
 
      // loop over the nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // Assign the required number of additional nodes to the node
        dynamic_cast<BoundaryNodeBase*>(this->node_pt(n))
          ->assign_additional_values_with_face_id(nadditional_values[n], id);
      }
    }
 
 
  public:
    /// Constructor: Initialise all appropriate member data
    FaceElement()
      : Face_to_bulk_coordinate_fct_pt(0),
        Bulk_coordinate_derivatives_fct_pt(0),
        Normal_sign(0),
        Face_index(0),
        Boundary_number_in_bulk_mesh(0),
        Bulk_element_pt(0),
        Tangent_direction_pt(0)
    {
      // Check whether things have been set
#ifdef PARANOID
      Boundary_number_in_bulk_mesh_has_been_set = false;
#endif
 
      // Bulk_position_type[0] is always 0 (the position)
      Bulk_position_type.push_back(0);
    }
 
    /// Empty virtual destructor
    virtual ~FaceElement() {}
 
 
    /// Broken copy constructor
    FaceElement(const FaceElement&) = delete;
 
    /// Broken assignment operator
    /*void operator=(const FaceElement&) = delete;*/
 
    /// Access function for the boundary number in bulk mesh
    inline const unsigned& boundary_number_in_bulk_mesh() const
    {
      return Boundary_number_in_bulk_mesh;
    }
 
 
    /// Set function for the boundary number in bulk mesh
    inline void set_boundary_number_in_bulk_mesh(const unsigned& b)
    {
      Boundary_number_in_bulk_mesh = b;
#ifdef PARANOID
      Boundary_number_in_bulk_mesh_has_been_set = true;
#endif
    }
 
    /// In a FaceElement, the "global" intrinsic coordinate
    /// of the element along the boundary, when viewed as part of
    /// a compound geometric object is specified using the
    /// boundary coordinate defined by the mesh.
    /// Note: Boundary coordinates will have been set up when
    /// creating the underlying mesh, and their values will have
    /// been stored at the nodes.
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      // Vector in which to hold the intrinsic coordinate
      Vector<double> zeta(this->dim());
 
      // Get the k-th generalised boundary coordinate at node n
      this->node_pt(n)->get_coordinates_on_boundary(
        Boundary_number_in_bulk_mesh, k, zeta);
 
      // Return the individual coordinate
      return zeta[i];
    }
 
    /// Return the Jacobian of mapping from local to global
    /// coordinates at local position s.
    /// Overloaded from FiniteElement.
    double J_eulerian(const Vector<double>& s) const;
 
    /// Return the Jacobian of the mapping from local to global
    /// coordinates at the ipt-th integration point
    /// Overloaded from FiniteElement.
    double J_eulerian_at_knot(const unsigned& ipt) const;
 
    /// Check that Jacobian of mapping between local and Eulerian
    /// coordinates at all integration points is positive.
    void check_J_eulerian_at_knots(bool& passed) const;
 
    /// Return FE interpolated coordinate x[i] at local coordinate s.
    /// Overloaded to get information from bulk.
    double interpolated_x(const Vector<double>& s, const unsigned& i) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Return Eulerian coordinate as computed by bulk
      return bulk_element_pt()->interpolated_x(s_bulk, i);
    }
 
    /// Return FE interpolated coordinate x[i] at local coordinate s
    /// at previous timestep t (t=0: present; t>0: previous timestep).
    /// Overloaded to get information from bulk.
    double interpolated_x(const unsigned& t,
                          const Vector<double>& s,
                          const unsigned& i) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Return Eulerian coordinate as computed by bulk
      return bulk_element_pt()->interpolated_x(t, s_bulk, i);
    }
 
    /// Return FE interpolated position x[] at local coordinate s as
    /// Vector Overloaded to get information from bulk.
    void interpolated_x(const Vector<double>& s, Vector<double>& x) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Get Eulerian position vector
      bulk_element_pt()->interpolated_x(s_bulk, x);
    }
 
    /// Return FE interpolated position x[] at local coordinate s
    /// at previous timestep t as Vector (t=0: present; t>0: previous timestep).
    /// Overloaded to get information from bulk.
    void interpolated_x(const unsigned& t,
                        const Vector<double>& s,
                        Vector<double>& x) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Get Eulerian position vector
      bulk_element_pt()->interpolated_x(t, s_bulk, x);
    }
 
    /// Return t-th time-derivative of the
    /// i-th FE-interpolated Eulerian coordinate at
    /// local coordinate s. Overloaded to get information from bulk.
    double interpolated_dxdt(const Vector<double>& s,
                             const unsigned& i,
                             const unsigned& t)
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Return Eulerian coordinate as computed by bulk
      return bulk_element_pt()->interpolated_dxdt(s_bulk, i, t);
    }
 
    /// Compte t-th time-derivative of the
    /// FE-interpolated Eulerian coordinate vector at
    /// local coordinate s.  Overloaded to get information from bulk.
    void interpolated_dxdt(const Vector<double>& s,
                           const unsigned& t,
                           Vector<double>& dxdt)
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Get Eulerian position vector
      bulk_element_pt()->interpolated_dxdt(s_bulk, t, dxdt);
    }
 
    /// Sign of outer unit normal (relative to cross-products of tangent
    /// vectors in the corresponding "bulk" element.
    int& normal_sign()
    {
      return Normal_sign;
    }
 
    /// Return sign of outer unit normal (relative to cross-products
    /// of tangent vectors in the corresponding "bulk" element. (const version)
    int normal_sign() const
    {
      return Normal_sign;
    }
 
    /// Index of the face (a number that uniquely identifies the face
    /// in the element)
    int& face_index()
    {
      return Face_index;
    }
 
    /// Index of the face (a number that uniquely identifies the face
    /// in the element) (const version)
    int face_index() const
    {
      return Face_index;
    }
 
    /// Public access function for the tangent direction pointer.
    const Vector<double>* tangent_direction_pt() const
    {
      return Tangent_direction_pt;
    }
 
    /// Set the tangent direction vector.
    void set_tangent_direction(Vector<double>* tangent_direction_pt)
    {
#ifdef PARANOID
      // Check that tangent_direction_pt is not null.
      if (tangent_direction_pt == 0)
      {
        std::ostringstream error_message;
        error_message << "The pointer tangent_direction_pt is null.\n";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
 
      // Check that the vector is the correct size.
      // The size of the tangent vector.
      unsigned tang_dir_size = tangent_direction_pt->size();
      unsigned spatial_dimension = this->nodal_dimension();
      if (tang_dir_size != spatial_dimension)
      {
        std::ostringstream error_message;
        error_message << "The tangent direction vector has size "
                      << tang_dir_size << "\n"
                      << "but this element has spatial dimension "
                      << spatial_dimension << ".\n";
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
 
      if (tang_dir_size == 2)
      {
        std::ostringstream warning_message;
        warning_message
          << "The spatial dimension is " << spatial_dimension << ".\n"
          << "I do not need a tangent direction vector to create \n"
          << "continuous tangent vectors in two spatial dimensions.";
        OomphLibWarning(warning_message.str(),
                        OOMPH_CURRENT_FUNCTION,
                        OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Set the direction vector for the tangent.
      Tangent_direction_pt = tangent_direction_pt;
    }
 
    /// Turn on warning for when there may be discontinuous tangent
    /// vectors from continuous_tangent_and_outer_unit_normal(...)
    void turn_on_warning_for_discontinuous_tangent()
    {
      FaceElement::Ignore_discontinuous_tangent_warning = true;
    }
 
    /// Turn off warning for when there may be discontinuous tangent
    /// vectors from continuous_tangent_and_outer_unit_normal(...)
    void turn_off_warning_for_discontinuous_tangent()
    {
      FaceElement::Ignore_discontinuous_tangent_warning = false;
    }
 
    /// 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 continuous_tangent_and_outer_unit_normal(
      const Vector<double>& s,
      Vector<Vector<double>>& tang_vec,
      Vector<double>& unit_normal) const;
 
    /// 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 continuous_tangent_and_outer_unit_normal(
      const unsigned& ipt,
      Vector<Vector<double>>& tang_vec,
      Vector<double>& unit_normal) const;
 
    /// Compute outer unit normal at the specified local coordinate
    void outer_unit_normal(const Vector<double>& s,
                           Vector<double>& unit_normal) const;
 
    /// Compute outer unit normal at ipt-th integration point
    void outer_unit_normal(const unsigned& ipt,
                           Vector<double>& unit_normal) const;
 
    /// Pointer to higher-dimensional "bulk" element
    FiniteElement*& bulk_element_pt()
    {
      return Bulk_element_pt;
    }
 
 
    /// Pointer to higher-dimensional "bulk" element (const version)
    FiniteElement* bulk_element_pt() const
    {
      return Bulk_element_pt;
    }
 
    // Clang specific pragma's
#ifdef __clang__
#pragma clang diagnostic push
#pragma clang diagnostic ignored "-Woverloaded-virtual"
#endif
 
    /// Return the pointer to the function that maps the face
    /// coordinate to the bulk coordinate
    CoordinateMappingFctPt& face_to_bulk_coordinate_fct_pt()
    {
      return Face_to_bulk_coordinate_fct_pt;
    }
 
    /// Return the pointer to the function that maps the face
    /// coordinate to the bulk coordinate (const version)
    CoordinateMappingFctPt face_to_bulk_coordinate_fct_pt() const
    {
      return Face_to_bulk_coordinate_fct_pt;
    }
 
 
    /// Return the pointer to the function that returns the derivatives
    /// of the bulk coordinates wrt the face coordinates
    BulkCoordinateDerivativesFctPt& bulk_coordinate_derivatives_fct_pt()
    {
      return Bulk_coordinate_derivatives_fct_pt;
    }
 
    /// Return the pointer to the function that returns the derivatives
    /// of the bulk coordinates wrt the face coordinates (const version)
    BulkCoordinateDerivativesFctPt bulk_coordinate_derivatives_fct_pt() const
    {
      return Bulk_coordinate_derivatives_fct_pt;
    }
 
#ifdef __clang__
#pragma clang diagnostic pop
#endif
 
    /// Return vector of local coordinates in bulk element,
    /// given the local coordinates in this FaceElement
    Vector<double> local_coordinate_in_bulk(const Vector<double>& s) const;
 
    /// Calculate the vector of local coordinate in the bulk element
    /// given the local coordinates in this FaceElement
    void get_local_coordinate_in_bulk(const Vector<double>& s,
                                      Vector<double>& s_bulk) const;
 
    /// Calculate the derivatives of the local coordinates in the
    /// bulk element with respect to the local coordinates in this FaceElement.
    /// In addition return the index of a bulk local coordinate that varies away
    /// from the face.
    void get_ds_bulk_ds_face(const Vector<double>& s,
                             DenseMatrix<double>& dsbulk_dsface,
                             unsigned& interior_direction) const;
 
    /// Return the position type in the "bulk" element that corresponds
    /// to position type i on the FaceElement.
    unsigned& bulk_position_type(const unsigned& i)
    {
      return Bulk_position_type[i];
    }
 
    /// Return the position type in the "bulk" element that corresponds
    /// to the position type i on the FaceElement. Const version
    const unsigned& bulk_position_type(const unsigned& i) const
    {
      return Bulk_position_type[i];
    }
 
    /// Resize the storage for the bulk node numbers
    void bulk_node_number_resize(const unsigned& i)
    {
      Bulk_node_number.resize(i);
    }
 
    /// Return the bulk node number that corresponds to the n-th
    /// local node number
    unsigned& bulk_node_number(const unsigned& n)
    {
      return Bulk_node_number[n];
    }
 
    /// Return the bulk node number that corresponds to the n-th
    /// local node number (const version)
    const unsigned& bulk_node_number(const unsigned& n) const
    {
      return Bulk_node_number[n];
    }
 
    /// Resize the storage for bulk_position_type to i entries.
    void bulk_position_type_resize(const unsigned& i)
    {
      Bulk_position_type.resize(i);
    }
 
    /// Return the number of values originally stored at local node n
    /// (before the FaceElement added additional values to it (if it did))
    unsigned& nbulk_value(const unsigned& n)
    {
      return Nbulk_value[n];
    }
 
    /// Return the number of values originally stored at local node n
    /// (before the FaceElement added additional values to it (if it did))
    /// (const version)
    unsigned nbulk_value(const unsigned& n) const
    {
      return Nbulk_value[n];
    }
 
    ///  Resize the storage for the number of values originally
    /// stored at the local nodes to i entries.
    void nbulk_value_resize(const unsigned& i)
    {
      Nbulk_value.resize(i);
    }
 
    /// Provide additional storage for a specified number of
    /// values at the nodes of the FaceElement. (This is needed, for
    /// instance, in free-surface elements, if the non-penetration
    /// condition is imposed by Lagrange multipliers whose values
    /// are only stored at the surface nodes but not in the interior of
    /// the bulk element). \c nadditional_data_values[n] specifies the
    /// number of additional values required at node \c n of the
    /// FaceElement. \b Note: Since this function is executed separately
    /// for each FaceElement, nodes that are common to multiple elements
    /// might be resized repeatedly. To avoid this, we only allow
    /// a single resize operation by comparing the number of values stored
    /// at each node to the number of values the node had when it
    /// was simply a member of the associated "bulk"
    /// element. <b>There are cases where this will break!</b> -- e.g. if
    /// a node is common to two FaceElements which require
    /// additional storage for distinct quantities. Such cases
    /// need to be handled by "hand-crafted" face elements.
    void resize_nodes(Vector<unsigned>& nadditional_data_values)
    {
      // Locally cache the number of node
      unsigned n_node = nnode();
      // Resize the storage for values at the nodes:
      for (unsigned l = 0; l < n_node; l++)
      {
        // Find number of values stored at the node
        unsigned Initial_Nvalue = node_pt(l)->nvalue();
        // Read out the number of additional values
        unsigned Nadditional = nadditional_data_values[l];
        // If the node has not already been resized, resize it
        if ((Initial_Nvalue == Nbulk_value[l]) && (Nadditional > 0))
        {
          // Resize the node according to the number of additional values
          node_pt(l)->resize(Nbulk_value[l] + Nadditional);
        }
      } // End of loop over nodes
    }
 
    /// Output boundary coordinate zeta
    void output_zeta(std::ostream& outfile, const unsigned& nplot);
  };
 
 
  //========================================================================
  /// SolidFaceElements combine FaceElements and SolidFiniteElements
  /// and overload various functions so they work properly in the
  /// FaceElement context
  //========================================================================
  class SolidFaceElement : public virtual FaceElement,
                           public virtual SolidFiniteElement
  {
  public:
    /// The "global" intrinsic coordinate of the element when
    /// viewed as part of a geometric object should be given by
    /// the FaceElement representation, by default
    /// This final over-ride is required because both SolidFiniteElements
    /// and FaceElements overload zeta_nodal
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
 
    /// Return i-th FE-interpolated Lagrangian coordinate xi[i] at
    /// local coordinate s. Overloaded from SolidFiniteElement. Note that
    /// the Lagrangian coordinates are those defined in the bulk!
    /// For instance, in a 1D FaceElement that is aligned with
    /// the Lagrangian coordinate line xi_0=const, only xi_1 will vary
    /// in the FaceElement. This may confuse you if you (wrongly!) believe that
    /// in a 1D SolidElement there should only a single Lagrangian
    /// coordinate, namely xi_0!
    double interpolated_xi(const Vector<double>& s, const unsigned& i) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Return Lagrangian coordinate as computed by bulk
      return dynamic_cast<SolidFiniteElement*>(bulk_element_pt())
        ->interpolated_xi(s_bulk, i);
    }
 
 
    /// Compute FE interpolated Lagrangian coordinate vector xi[] at
    /// local coordinate s as Vector. Overloaded from SolidFiniteElement. Note
    /// that the Lagrangian coordinates are those defined in the bulk!
    /// For instance, in a 1D FaceElement that is aligned with
    /// the Lagrangian coordinate line xi_0=const, only xi_1 will vary
    /// in the FaceElement. This may confuse you if you (wrongly!) believe that
    /// in a 1D SolidElement there should only a single Lagrangian
    /// coordinate, namely xi_0!
    void interpolated_xi(const Vector<double>& s, Vector<double>& xi) const
    {
      // Local coordinates in bulk element
      Vector<double> s_bulk(dim() + 1);
      s_bulk = local_coordinate_in_bulk(s);
 
      // Get Lagrangian position vector
      dynamic_cast<SolidFiniteElement*>(bulk_element_pt())
        ->interpolated_xi(s_bulk, xi);
    }
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Solid point element
  //=======================================================================
  class SolidPointElement : public virtual SolidFiniteElement,
                            public virtual PointElement
  {
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// FaceGeometry class definition: This policy class is used to allow
  /// construction of face elements that solve arbitrary equations without
  /// having to tamper with the corresponding "bulk" elements.
  /// The geometrical information for the face element must be specified
  /// by each "bulk" element using an explicit specialisation of this class.
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry
  {
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Dummy FaceElement for use with purely geometric operations
  /// such as mesh generation.
  //======================================================================
  template<class ELEMENT>
  class DummyFaceElement : public virtual FaceGeometry<ELEMENT>,
                           public virtual FaceElement
  {
  public:
    /// Constructor, which takes a "bulk" element and the
    /// face index
    DummyFaceElement(FiniteElement* const& element_pt, const int& face_index)
      : FaceGeometry<ELEMENT>(), FaceElement()
    {
      // Attach the geometrical information to the element. N.B. This function
      // also assigns nbulk_value from the required_nvalue of the bulk element
      element_pt->build_face_element(face_index, this);
    }
 
 
    /// Constructor
    DummyFaceElement() : FaceGeometry<ELEMENT>(), FaceElement() {}
 
 
    /// The "global" intrinsic coordinate of the element when
    /// viewed as part of a geometric object should be given by
    /// the FaceElement representation, by default
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
    /// Output nodal coordinates
    void output(std::ostream& outfile)
    {
      outfile << "ZONE" << std::endl;
      unsigned nnod = nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        Node* nod_pt = node_pt(j);
        unsigned dim = nod_pt->ndim();
        for (unsigned i = 0; i < dim; i++)
        {
          outfile << nod_pt->x(i) << " ";
        }
        outfile << std::endl;
      }
    }
 
    /// Output at n_plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FiniteElement::output(outfile, n_plot);
    }
 
    /// C-style output
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C_style output at n_plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
  };
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //=========================================================================
  /// Base class for elements that can specify a drag and torque
  /// (about the origin) -- typically used for immersed particle
  /// computations.
  //=========================================================================
  class ElementWithDragFunction
  {
  public:
    /// Empty constructor
    ElementWithDragFunction() {}
 
    /// Empty virtual destructor
    virtual ~ElementWithDragFunction() {}
 
    /// Function that specifies the drag force and the torque about
    /// the origin
    virtual void get_drag_and_torque(Vector<double>& drag_force,
                                     Vector<double>& drag_torque) = 0;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Basic-ified FaceElement, without any of the functionality of
  /// of actual FaceElements -- it's just a  surface element of the
  /// same geometric type as the FaceGeometry associated with
  /// bulk element specified by the template parameter. The element
  /// can be used to represent boundaries without actually being
  /// attached to a bulk element. Used mainly during unstructured
  /// mesh generation.
  //======================================================================
  template<class ELEMENT>
  class FreeStandingFaceElement : public virtual FaceGeometry<ELEMENT>
  {
  public:
    /// Constructor
    FreeStandingFaceElement()
      : FaceGeometry<ELEMENT>(), Boundary_number_in_bulk_mesh(0)
    {
      // Check whether things have been set
#ifdef PARANOID
      Boundary_number_in_bulk_mesh_has_been_set = false;
#endif
    }
 
    /// Access function for the boundary number in bulk mesh
    inline const unsigned& boundary_number_in_bulk_mesh() const
    {
      return Boundary_number_in_bulk_mesh;
    }
 
 
    /// Set function for the boundary number in bulk mesh
    inline void set_boundary_number_in_bulk_mesh(const unsigned& b)
    {
      Boundary_number_in_bulk_mesh = b;
#ifdef PARANOID
      Boundary_number_in_bulk_mesh_has_been_set = true;
#endif
    }
 
    /// In a FaceElement, the "global" intrinsic coordinate
    /// of the element along the boundary, when viewed as part of
    /// a compound geometric object is specified using the
    /// boundary coordinate defined by the mesh.
    /// Note: Boundary coordinates will have been set up when
    /// creating the underlying mesh, and their values will have
    /// been stored at the nodes.
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      // Vector in which to hold the intrinsic coordinate
      Vector<double> zeta(this->dim());
 
      // Get the k-th generalised boundary coordinate at node n
      this->node_pt(n)->get_coordinates_on_boundary(
        Boundary_number_in_bulk_mesh, k, zeta);
 
      // Return the individual coordinate
      return zeta[i];
    }
 
  protected:
    /// The boundary number in the bulk mesh to which this element is attached
    unsigned Boundary_number_in_bulk_mesh;
 
#ifdef PARANOID
 
    /// Has the Boundary_number_in_bulk_mesh been set? Only included if
    /// compiled with PARANOID switched on.
    bool Boundary_number_in_bulk_mesh_has_been_set;
 
#endif
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Pure virtual base class for elements that can be used with
  /// PressureBasedSolidLSCPreconditioner.
  //======================================================================
  class SolidElementWithDiagonalMassMatrix
  {
  public:
    /// Empty constructor
    SolidElementWithDiagonalMassMatrix() {}
 
    /// Virtual destructor
    virtual ~SolidElementWithDiagonalMassMatrix() {}
 
    /// Broken copy constructor
    SolidElementWithDiagonalMassMatrix(
      const SolidElementWithDiagonalMassMatrix&) = delete;
 
    /// Broken assignment operator
    void operator=(const SolidElementWithDiagonalMassMatrix&) = delete;
 
    /// Get the diagonal of whatever represents the mass matrix
    /// in the specific preconditionable element. For Navier-Stokes
    /// elements this is the velocity mass matrix; for incompressible solids
    /// it's the mass matrix; ...
    virtual void get_mass_matrix_diagonal(Vector<double>& mass_diag) = 0;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Pure virtual base class for elements that can be used with
  /// Navier-Stokes Schur complement preconditioner and provide the diagonal
  /// of their velocity and pressure mass matrices -- needs to be defined here
  /// (in generic)  because this applies to a variety of Navier-Stokes
  /// elements (cartesian, cylindrical polar, ...)
  /// that can be preconditioned effectively by the Navier Stokes (!)
  /// preconditioners in the (cartesian) Navier-Stokes directory.
  //======================================================================
  class NavierStokesElementWithDiagonalMassMatrices
  {
  public:
    /// Empty constructor
    NavierStokesElementWithDiagonalMassMatrices() {}
 
    /// Virtual destructor
    virtual ~NavierStokesElementWithDiagonalMassMatrices() {}
 
    /// Broken copy constructor
    NavierStokesElementWithDiagonalMassMatrices(
      const NavierStokesElementWithDiagonalMassMatrices&) = delete;
 
    /// Broken assignment operator
    void operator=(const NavierStokesElementWithDiagonalMassMatrices&) = delete;
 
    /// Compute the diagonal of the velocity/pressure mass matrices.
    /// If which one=0, both are computed, otherwise only the pressure
    /// (which_one=1) or the velocity mass matrix (which_one=2 -- the
    /// LSC version of the preconditioner only needs that one)
    virtual void get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one = 0) = 0;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
} // namespace oomph
 
#endif