generalised_newtonian_refineable_navier_stokes_elements.cc

Go to the documentation of this file.

// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
// LIC// This library is distributed in the hope that it will be useful,
// LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
#include "generalised_newtonian_refineable_navier_stokes_elements.h"
 
 
namespace oomph
{
  //===================================================================
  /// 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)
  //===================================================================
  template<unsigned DIM>
  void RefineableGeneralisedNewtonianNavierStokesEquations<DIM>::
    get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one)
  {
    // Resize and initialise
    unsigned n_dof = ndof();
 
    if ((which_one == 0) || (which_one == 1))
      press_mass_diag.assign(n_dof, 0.0);
    if ((which_one == 0) || (which_one == 2))
      veloc_mass_diag.assign(n_dof, 0.0);
 
    // Pointers to hang info object
    HangInfo* hang_info_pt = 0;
 
    // Number of master nodes and weight for shape fcts
    unsigned n_master = 1;
    double hang_weight = 1.0;
 
    // find out how many nodes there are
    unsigned n_node = nnode();
 
    // Set up memory for veloc shape functions
    Shape psi(n_node);
 
    // Find number of pressure dofs
    unsigned n_pres = this->npres_nst();
 
    // Pressure shape function
    Shape psi_p(n_pres);
 
    // Local coordinates
    Vector<double> s(DIM);
 
    // find the indices at which the local velocities are stored
    Vector<unsigned> u_nodal_index(DIM);
    for (unsigned i = 0; i < DIM; i++)
    {
      u_nodal_index[i] = this->u_index_nst(i);
    }
 
    // Which nodal value represents the pressure? (Negative if pressure
    // is not based on nodal interpolation).
    int p_index = this->p_nodal_index_nst();
 
    // Local array of booleans that are true if the l-th pressure value is
    // hanging (avoid repeated virtual function calls)
    bool pressure_dof_is_hanging[n_pres];
 
    // If the pressure is stored at a node
    if (p_index >= 0)
    {
      // Read out whether the pressure is hanging
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = pressure_node_pt(l)->is_hanging(p_index);
      }
    }
    // Otherwise the pressure is not stored at a node and so cannot hang
    else
    {
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = false;
      }
    }
 
 
    // Number of integration points
    unsigned n_intpt = integral_pt()->nweight();
 
    // Integer to store the local equations no
    int local_eqn = 0;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Get determinant of Jacobian of the mapping
      double J = J_eulerian_at_knot(ipt);
 
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Premultiply weights and Jacobian
      double W = w * J;
 
 
      // Do we want the velocity one?
      if ((which_one == 0) || (which_one == 2))
      {
        // Get the velocity shape functions
        shape_at_knot(ipt, psi);
 
 
        // Number of master nodes and storage for the weight of the shape
        // function
        unsigned n_master = 1;
        double hang_weight = 1.0;
 
        // Loop over the nodes for the test functions/equations
        //----------------------------------------------------
        for (unsigned l = 0; l < n_node; l++)
        {
          // Local boolean to indicate whether the node is hanging
          bool is_node_hanging = node_pt(l)->is_hanging();
 
          // If the node is hanging
          if (is_node_hanging)
          {
            hang_info_pt = node_pt(l)->hanging_pt();
 
            // Read out number of master nodes from hanging data
            n_master = hang_info_pt->nmaster();
          }
          // Otherwise the node is its own master
          else
          {
            n_master = 1;
          }
 
          // Loop over the master nodes
          for (unsigned m = 0; m < n_master; m++)
          {
            // Loop over velocity components for equations
            for (unsigned i = 0; i < DIM; i++)
            {
              // Get the equation number
              // If the node is hanging
              if (is_node_hanging)
              {
                // Get the equation number from the master node
                local_eqn = this->local_hang_eqn(
                  hang_info_pt->master_node_pt(m), u_nodal_index[i]);
                // Get the hang weight from the master node
                hang_weight = hang_info_pt->master_weight(m);
              }
              // If the node is not hanging
              else
              {
                // Local equation number
                local_eqn = this->nodal_local_eqn(l, u_nodal_index[i]);
 
                // Node contributes with full weight
                hang_weight = 1.0;
              }
 
              // If it's not a boundary condition...
              if (local_eqn >= 0)
              {
                //       //Loop over the veclocity shape functions
                //       for(unsigned l=0; l<n_node; l++)
                //        {
                //         //Loop over the velocity components
                //         for(unsigned i=0; i<DIM; i++)
                //          {
                //           local_eqn = nodal_local_eqn(l,u_nodal_index[i]);
 
                //           //If not a boundary condition
                //           if(local_eqn >= 0)
                //            {
 
 
                // Add the contribution
                veloc_mass_diag[local_eqn] += pow(psi[l] * hang_weight, 2) * W;
              }
            }
          }
        }
      }
 
      // Do we want the pressure one?
      if ((which_one == 0) || (which_one == 1))
      {
        // Get the pressure shape functions
        this->pshape_nst(s, psi_p);
 
        // Loop over the pressure shape functions
        for (unsigned l = 0; l < n_pres; l++)
        {
          // If the pressure dof is hanging
          if (pressure_dof_is_hanging[l])
          {
            // Pressure dof is hanging so it must be nodal-based
            // Get the hang info object
            hang_info_pt = this->pressure_node_pt(l)->hanging_pt(p_index);
 
            // Get the number of master nodes from the pressure node
            n_master = hang_info_pt->nmaster();
          }
          // Otherwise the node is its own master
          else
          {
            n_master = 1;
          }
 
          // Loop over the master nodes
          for (unsigned m = 0; m < n_master; m++)
          {
            // Get the number of the unknown
            // If the pressure dof is hanging
            if (pressure_dof_is_hanging[l])
            {
              // Get the local equation from the master node
              local_eqn =
                this->local_hang_eqn(hang_info_pt->master_node_pt(m), p_index);
              // Get the weight for the node
              hang_weight = hang_info_pt->master_weight(m);
            }
            else
            {
              local_eqn = this->p_local_eqn(l);
              hang_weight = 1.0;
            }
 
            // If the equation is not pinned
            if (local_eqn >= 0)
            {
              //       //Loop over the veclocity shape functions
              //       for(unsigned l=0; l<n_pres; l++)
              //        {
              //         // Get equation number
              //         local_eqn = p_local_eqn(l);
 
              //         //If not a boundary condition
              //         if(local_eqn >= 0)
              //          {
 
 
              // Add the contribution
              press_mass_diag[local_eqn] += pow(psi_p[l] * hang_weight, 2) * W;
            }
          }
        }
      }
    }
  }
 
 
  //========================================================================
  /// Add element's contribution to the elemental
  /// residual vector and/or Jacobian matrix.
  /// flag=1: compute both
  /// flag=0: compute only residual vector
  //========================================================================
  template<unsigned DIM>
  void RefineableGeneralisedNewtonianNavierStokesEquations<DIM>::
    fill_in_generic_residual_contribution_nst(Vector<double>& residuals,
                                              DenseMatrix<double>& jacobian,
                                              DenseMatrix<double>& mass_matrix,
                                              unsigned flag)
  {
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Get continuous time from timestepper of first node
    double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
    // Find out how many pressure dofs there are
    unsigned n_pres = this->npres_nst();
 
    // Get the indices at which the velocity components are stored
    unsigned u_nodal_index[DIM];
    for (unsigned i = 0; i < DIM; i++)
    {
      u_nodal_index[i] = this->u_index_nst(i);
    }
 
    // Which nodal value represents the pressure? (Negative if pressure
    // is not based on nodal interpolation).
    int p_index = this->p_nodal_index_nst();
 
    // Local array of booleans that are true if the l-th pressure value is
    // hanging (avoid repeated virtual function calls)
    bool pressure_dof_is_hanging[n_pres];
    // If the pressure is stored at a node
    if (p_index >= 0)
    {
      // Read out whether the pressure is hanging
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = pressure_node_pt(l)->is_hanging(p_index);
      }
    }
    // Otherwise the pressure is not stored at a node and so cannot hang
    else
    {
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = false;
      }
    }
 
    // Set up memory for the shape and test functions
    Shape psif(n_node), testf(n_node);
    DShape dpsifdx(n_node, DIM), dtestfdx(n_node, DIM);
 
 
    // Set up memory for pressure shape and test functions
    Shape psip(n_pres), testp(n_pres);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Set the Vector to hold local coordinates
    Vector<double> s(DIM);
 
    // Get Physical Variables from Element
    // Reynolds number must be multiplied by the density ratio
    double scaled_re = this->re() * this->density_ratio();
    double scaled_re_st = this->re_st() * this->density_ratio();
    double scaled_re_inv_fr = this->re_invfr() * this->density_ratio();
    double visc_ratio = this->viscosity_ratio();
    Vector<double> G = this->g();
 
    // Integers that store the local equations and unknowns
    int local_eqn = 0, local_unknown = 0;
 
    // Pointers to hang info objects
    HangInfo *hang_info_pt = 0, *hang_info2_pt = 0;
 
    // Local boolean for ALE (or not)
    bool ALE_is_disabled_flag = this->ALE_is_disabled;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Call the derivatives of the shape and test functions
      double J = this->dshape_and_dtest_eulerian_at_knot_nst(
        ipt, psif, dpsifdx, testf, dtestfdx);
 
      // Call the pressure shape and test functions
      this->pshape_nst(s, psip, testp);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Calculate local values of the pressure and velocity components
      //--------------------------------------------------------------
      double interpolated_p = 0.0;
      Vector<double> interpolated_u(DIM, 0.0);
      Vector<double> interpolated_x(DIM, 0.0);
      Vector<double> mesh_veloc(DIM, 0.0);
      Vector<double> dudt(DIM, 0.0);
      DenseMatrix<double> interpolated_dudx(DIM, DIM, 0.0);
 
      // Calculate pressure
      for (unsigned l = 0; l < n_pres; l++)
      {
        interpolated_p += this->p_nst(l) * psip[l];
      }
 
 
      // Calculate velocities and derivatives
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over directions
        for (unsigned i = 0; i < DIM; i++)
        {
          // Get the nodal value
          double u_value = this->nodal_value(l, u_nodal_index[i]);
          interpolated_u[i] += u_value * psif[l];
          interpolated_x[i] += this->nodal_position(l, i) * psif[l];
          dudt[i] += this->du_dt_nst(l, i) * psif[l];
 
          // Loop over derivative directions for velocity gradients
          for (unsigned j = 0; j < DIM; j++)
          {
            interpolated_dudx(i, j) += u_value * dpsifdx(l, j);
          }
        }
      }
 
      if (!ALE_is_disabled_flag)
      {
        // Loop over nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned i = 0; i < DIM; i++)
          {
            mesh_veloc[i] += this->dnodal_position_dt(l, i) * psif[l];
          }
        }
      }
 
      // The strainrate used to compute the second invariant
      DenseMatrix<double> strainrate_to_compute_second_invariant(DIM, DIM, 0.0);
 
      // the strainrate used to calculate the second invariant
      // can be either the current one or the one extrapolated from
      // previous velocity values
      if (!this->Use_extrapolated_strainrate_to_compute_second_invariant)
      {
        this->strain_rate(s, strainrate_to_compute_second_invariant);
      }
      else
      {
        this->extrapolated_strain_rate(ipt,
                                       strainrate_to_compute_second_invariant);
      }
 
      // Calculate the second invariant
      double second_invariant = SecondInvariantHelper::second_invariant(
        strainrate_to_compute_second_invariant);
 
      // Get the viscosity according to the constitutive equation
      double viscosity = this->Constitutive_eqn_pt->viscosity(second_invariant);
 
      // Get the user-defined body force terms
      Vector<double> body_force(DIM);
      this->get_body_force_nst(time, ipt, s, interpolated_x, body_force);
 
      // Get the user-defined source function
      double source = this->get_source_nst(time, ipt, interpolated_x);
 
      // The generalised Newtonian viscosity differentiated w.r.t.
      // the unknown velocities
      DenseMatrix<double> dviscosity_dunknown(DIM, n_node, 0.0);
 
      if (!this->Use_extrapolated_strainrate_to_compute_second_invariant)
      {
        // Calculate the derivate of the viscosity w.r.t. the second invariant
        double dviscosity_dsecond_invariant =
          this->Constitutive_eqn_pt->dviscosity_dinvariant(second_invariant);
 
        // calculate the strainrate
        DenseMatrix<double> strainrate(DIM, DIM, 0.0);
        this->strain_rate(s, strainrate);
 
        // pre-compute the derivative of the second invariant w.r.t. the
        // entries in the rate of strain tensor
        DenseMatrix<double> dinvariant_dstrainrate(DIM, DIM, 0.0);
 
        // setting up a Kronecker Delta matrix, which has ones at the diagonal
        // and zeros off-diagonally
        DenseMatrix<double> kroneckerdelta(DIM, DIM, 0.0);
 
        for (unsigned i = 0; i < DIM; i++)
        {
          for (unsigned j = 0; j < DIM; j++)
          {
            if (i == j)
            {
              // Set the diagonal entries of the Kronecker delta
              kroneckerdelta(i, i) = 1.0;
 
              double tmp = 0.0;
              for (unsigned k = 0; k < DIM; k++)
              {
                if (k != i)
                {
                  tmp += strainrate(k, k);
                }
              }
              dinvariant_dstrainrate(i, i) = tmp;
            }
            else
            {
              dinvariant_dstrainrate(i, j) = -strainrate(j, i);
            }
          }
        }
 
        // a rank four tensor storing the derivative of the strainrate
        // w.r.t. the unknowns
        RankFourTensor<double> dstrainrate_dunknown(DIM, DIM, DIM, n_node);
 
        // loop over the nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // loop over the velocity components
          for (unsigned k = 0; k < DIM; k++)
          {
            // loop over the entries of the rate of strain tensor
            for (unsigned i = 0; i < DIM; i++)
            {
              for (unsigned j = 0; j < DIM; j++)
              {
                // initialise the entry to zero
                dstrainrate_dunknown(i, j, k, l) = 0.0;
 
                // loop over velocities and directions
                for (unsigned m = 0; m < DIM; m++)
                {
                  for (unsigned n = 0; n < DIM; n++)
                  {
                    dstrainrate_dunknown(i, j, k, l) +=
                      0.5 *
                      (kroneckerdelta(i, m) * kroneckerdelta(j, n) +
                       kroneckerdelta(i, n) * kroneckerdelta(j, m)) *
                      kroneckerdelta(m, k) * dpsifdx(l, n);
                  }
                }
              }
            }
          }
        }
 
        // Now calculate the derivative of the viscosity w.r.t. the unknowns
        // loop over the nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // loop over the velocities
          for (unsigned k = 0; k < DIM; k++)
          {
            // loop over the entries in the rate of strain tensor
            for (unsigned i = 0; i < DIM; i++)
            {
              for (unsigned j = 0; j < DIM; j++)
              {
                dviscosity_dunknown(k, l) += dviscosity_dsecond_invariant *
                                             dinvariant_dstrainrate(i, j) *
                                             dstrainrate_dunknown(i, j, k, l);
              }
            }
          }
        }
      }
 
 
      // MOMENTUM EQUATIONS
      //==================
 
      // Number of master nodes and storage for the weight of the shape function
      unsigned n_master = 1;
      double hang_weight = 1.0;
 
      // Loop over the nodes for the test functions/equations
      //----------------------------------------------------
      for (unsigned l = 0; l < n_node; l++)
      {
        // Local boolean to indicate whether the node is hanging
        bool is_node_hanging = node_pt(l)->is_hanging();
 
        // If the node is hanging
        if (is_node_hanging)
        {
          hang_info_pt = node_pt(l)->hanging_pt();
 
          // Read out number of master nodes from hanging data
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Loop over velocity components for equations
          for (unsigned i = 0; i < DIM; i++)
          {
            // Get the equation number
            // If the node is hanging
            if (is_node_hanging)
            {
              // Get the equation number from the master node
              local_eqn = this->local_hang_eqn(hang_info_pt->master_node_pt(m),
                                               u_nodal_index[i]);
              // Get the hang weight from the master node
              hang_weight = hang_info_pt->master_weight(m);
            }
            // If the node is not hanging
            else
            {
              // Local equation number
              local_eqn = this->nodal_local_eqn(l, u_nodal_index[i]);
 
              // Node contributes with full weight
              hang_weight = 1.0;
            }
 
            // If it's not a boundary condition...
            if (local_eqn >= 0)
            {
              // Temporary variable to hold the residuals
              double sum = 0.0;
 
              // Add the user-defined body force terms
              sum += body_force[i];
 
              // Add the gravitational body force term
              sum += scaled_re_inv_fr * G[i];
 
              // Add in the inertial term
              sum -= scaled_re_st * dudt[i];
 
              // Convective terms, including mesh velocity
              for (unsigned k = 0; k < DIM; k++)
              {
                double tmp = scaled_re * interpolated_u[k];
                if (!ALE_is_disabled_flag)
                {
                  tmp -= scaled_re_st * mesh_veloc[k];
                }
                sum -= tmp * interpolated_dudx(i, k);
              }
 
              // Add the pressure gradient term
              // Potentially pre-multiply by viscosity ratio
              if (this->Pre_multiply_by_viscosity_ratio)
              {
                sum = visc_ratio * viscosity *
                      (sum * testf[l] + interpolated_p * dtestfdx(l, i)) * W *
                      hang_weight;
              }
              else
              {
                sum = (sum * testf[l] + interpolated_p * dtestfdx(l, i)) * W *
                      hang_weight;
              }
 
              // Add in the stress tensor terms
              // The viscosity ratio needs to go in here to ensure
              // continuity of normal stress is satisfied even in flows
              // with zero pressure gradient!
              for (unsigned k = 0; k < DIM; k++)
              {
                sum -= visc_ratio * viscosity *
                       (interpolated_dudx(i, k) +
                        this->Gamma[i] * interpolated_dudx(k, i)) *
                       dtestfdx(l, k) * W * hang_weight;
              }
 
              // Add contribution
              residuals[local_eqn] += sum;
 
              // CALCULATE THE JACOBIAN
              if (flag)
              {
                // Number of master nodes and weights
                unsigned n_master2 = 1;
                double hang_weight2 = 1.0;
                // Loop over the velocity nodes for columns
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  // Local boolean to indicate whether the node is hanging
                  bool is_node2_hanging = node_pt(l2)->is_hanging();
 
                  // If the node is hanging
                  if (is_node2_hanging)
                  {
                    hang_info2_pt = node_pt(l2)->hanging_pt();
                    // Read out number of master nodes from hanging data
                    n_master2 = hang_info2_pt->nmaster();
                  }
                  // Otherwise the node is its own master
                  else
                  {
                    n_master2 = 1;
                  }
 
                  // Loop over the master nodes
                  for (unsigned m2 = 0; m2 < n_master2; m2++)
                  {
                    // Loop over the velocity components
                    for (unsigned i2 = 0; i2 < DIM; i2++)
                    {
                      // Get the number of the unknown
                      // If the node is hanging
                      if (is_node2_hanging)
                      {
                        // Get the equation number from the master node
                        local_unknown = this->local_hang_eqn(
                          hang_info2_pt->master_node_pt(m2), u_nodal_index[i2]);
                        // Get the hang weights
                        hang_weight2 = hang_info2_pt->master_weight(m2);
                      }
                      else
                      {
                        local_unknown =
                          this->nodal_local_eqn(l2, u_nodal_index[i2]);
                        hang_weight2 = 1.0;
                      }
 
                      // If the unknown is non-zero
                      if (local_unknown >= 0)
                      {
                        // Add contribution to Elemental Matrix
                        jacobian(local_eqn, local_unknown) -=
                          visc_ratio * viscosity * this->Gamma[i] *
                          dpsifdx(l2, i) * dtestfdx(l, i2) * W * hang_weight *
                          hang_weight2;
 
                        // Now add in the inertial terms
                        jacobian(local_eqn, local_unknown) -=
                          scaled_re * psif[l2] * interpolated_dudx(i, i2) *
                          testf[l] * W * hang_weight * hang_weight2;
 
                        if (
                          !this
                             ->Use_extrapolated_strainrate_to_compute_second_invariant)
                        {
                          for (unsigned k = 0; k < DIM; k++)
                          {
                            jacobian(local_eqn, local_unknown) -=
                              visc_ratio * dviscosity_dunknown(i2, l2) *
                              (interpolated_dudx(i, k) +
                               this->Gamma[i] * interpolated_dudx(k, i)) *
                              dtestfdx(l, k) * W * hang_weight * hang_weight2;
                          }
                        }
 
                        // Extra diagonal components if i2=i
                        if (i2 == i)
                        {
                          // Mass matrix entries
                          // Again note the positive sign because the mass
                          // matrix is taken on the other side of the equation
                          if (flag == 2)
                          {
                            mass_matrix(local_eqn, local_unknown) +=
                              scaled_re_st * psif[l2] * testf[l] * W *
                              hang_weight * hang_weight2;
                          }
 
                          // du/dt term
                          jacobian(local_eqn, local_unknown) -=
                            scaled_re_st *
                            node_pt(l2)->time_stepper_pt()->weight(1, 0) *
                            psif[l2] * testf[l] * W * hang_weight *
                            hang_weight2;
 
                          // Extra advective terms
                          for (unsigned k = 0; k < DIM; k++)
                          {
                            double tmp = scaled_re * interpolated_u[k];
                            if (!ALE_is_disabled_flag)
                            {
                              tmp -= scaled_re_st * mesh_veloc[k];
                            }
 
                            jacobian(local_eqn, local_unknown) -=
                              tmp * dpsifdx(l2, k) * testf[l] * W *
                              hang_weight * hang_weight2;
                          }
 
                          // Extra viscous terms
                          for (unsigned k = 0; k < DIM; k++)
                          {
                            jacobian(local_eqn, local_unknown) -=
                              visc_ratio * viscosity * dpsifdx(l2, k) *
                              dtestfdx(l, k) * W * hang_weight * hang_weight2;
                          }
                        }
                      }
                    }
                  }
                }
 
                // Loop over the pressure shape functions
                for (unsigned l2 = 0; l2 < n_pres; l2++)
                {
                  // If the pressure dof is hanging
                  if (pressure_dof_is_hanging[l2])
                  {
                    hang_info2_pt =
                      this->pressure_node_pt(l2)->hanging_pt(p_index);
                    // Pressure dof is hanging so it must be nodal-based
                    // Get the number of master nodes from the pressure node
                    n_master2 = hang_info2_pt->nmaster();
                  }
                  // Otherwise the node is its own master
                  else
                  {
                    n_master2 = 1;
                  }
 
                  // Loop over the master nodes
                  for (unsigned m2 = 0; m2 < n_master2; m2++)
                  {
                    // Get the number of the unknown
                    // If the pressure dof is hanging
                    if (pressure_dof_is_hanging[l2])
                    {
                      // Get the unknown from the master node
                      local_unknown = this->local_hang_eqn(
                        hang_info2_pt->master_node_pt(m2), p_index);
                      // Get the weight from the hanging object
                      hang_weight2 = hang_info2_pt->master_weight(m2);
                    }
                    else
                    {
                      local_unknown = this->p_local_eqn(l2);
                      hang_weight2 = 1.0;
                    }
 
                    // If the unknown is not pinned
                    if (local_unknown >= 0)
                    {
                      if (this->Pre_multiply_by_viscosity_ratio)
                      {
                        jacobian(local_eqn, local_unknown) +=
                          visc_ratio * viscosity * psip[l2] * dtestfdx(l, i) *
                          W * hang_weight * hang_weight2;
                      }
                      else
                      {
                        jacobian(local_eqn, local_unknown) +=
                          psip[l2] * dtestfdx(l, i) * W * hang_weight *
                          hang_weight2;
                      }
                    }
                  }
                }
 
              } // End of Jacobian calculation
 
            } // End of if not boundary condition statement
 
          } // End of loop over components of non-hanging node
 
        } // End of loop over master nodes
 
      } // End of loop over nodes for equations
 
 
      // CONTINUITY EQUATION
      //===================
 
      // Loop over the pressure shape functions
      for (unsigned l = 0; l < n_pres; l++)
      {
        // If the pressure dof is hanging
        if (pressure_dof_is_hanging[l])
        {
          // Pressure dof is hanging so it must be nodal-based
          // Get the hang info object
          hang_info_pt = this->pressure_node_pt(l)->hanging_pt(p_index);
 
          // Get the number of master nodes from the pressure node
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Get the number of the unknown
          // If the pressure dof is hanging
          if (pressure_dof_is_hanging[l])
          {
            // Get the local equation from the master node
            local_eqn =
              this->local_hang_eqn(hang_info_pt->master_node_pt(m), p_index);
            // Get the weight for the node
            hang_weight = hang_info_pt->master_weight(m);
          }
          else
          {
            local_eqn = this->p_local_eqn(l);
            hang_weight = 1.0;
          }
 
          // If the equation is not pinned
          if (local_eqn >= 0)
          {
            // Source term
            residuals[local_eqn] -= source * testp[l] * W * hang_weight;
 
            // Loop over velocity components
            for (unsigned k = 0; k < DIM; k++)
            {
              // Potentially pre-multiply by viscosity ratio
              if (this->Pre_multiply_by_viscosity_ratio)
              {
                residuals[local_eqn] += visc_ratio * viscosity *
                                        interpolated_dudx(k, k) * testp[l] * W *
                                        hang_weight;
              }
              else
              {
                residuals[local_eqn] +=
                  interpolated_dudx(k, k) * testp[l] * W * hang_weight;
              }
            }
 
            // CALCULATE THE JACOBIAN
            if (flag)
            {
              unsigned n_master2 = 1;
              double hang_weight2 = 1.0;
              // Loop over the velocity nodes for columns
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Local boolean to indicate whether the node is hanging
                bool is_node2_hanging = node_pt(l2)->is_hanging();
 
                // If the node is hanging
                if (is_node2_hanging)
                {
                  hang_info2_pt = node_pt(l2)->hanging_pt();
                  // Read out number of master nodes from hanging data
                  n_master2 = hang_info2_pt->nmaster();
                }
                // Otherwise the node is its own master
                else
                {
                  n_master2 = 1;
                }
 
                // Loop over the master nodes
                for (unsigned m2 = 0; m2 < n_master2; m2++)
                {
                  // Loop over the velocity components
                  for (unsigned i2 = 0; i2 < DIM; i2++)
                  {
                    // Get the number of the unknown
                    // If the node is hanging
                    if (is_node2_hanging)
                    {
                      // Get the equation number from the master node
                      local_unknown = this->local_hang_eqn(
                        hang_info2_pt->master_node_pt(m2), u_nodal_index[i2]);
                      hang_weight2 = hang_info2_pt->master_weight(m2);
                    }
                    else
                    {
                      local_unknown =
                        this->nodal_local_eqn(l2, u_nodal_index[i2]);
                      hang_weight2 = 1.0;
                    }
 
                    // If the unknown is not pinned
                    if (local_unknown >= 0)
                    {
                      if (this->Pre_multiply_by_viscosity_ratio)
                      {
                        jacobian(local_eqn, local_unknown) +=
                          visc_ratio * viscosity * dpsifdx(l2, i2) * testp[l] *
                          W * hang_weight * hang_weight2;
                      }
                      else
                      {
                        jacobian(local_eqn, local_unknown) +=
                          dpsifdx(l2, i2) * testp[l] * W * hang_weight *
                          hang_weight2;
                      }
                    }
                  }
                }
              }
 
              // NO PRESSURE CONTRIBUTION TO THE JACOBIAN
 
            } // End of jacobian calculation
          }
        }
      } // End of loop over pressure variables
 
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Compute derivatives of elemental residual vector with respect
  /// to nodal coordinates.
  /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
  /// Overloads the FD-based version in the FE base class.
  //======================================================================
  template<unsigned DIM>
  void RefineableGeneralisedNewtonianNavierStokesEquations<
    DIM>::get_dresidual_dnodal_coordinates(RankThreeTensor<double>&
                                             dresidual_dnodal_coordinates)
  {
    // Return immediately if there are no dofs
    if (ndof() == 0)
    {
      return;
    }
 
    // Determine number of nodes in element
    const unsigned n_node = nnode();
 
    // Get continuous time from timestepper of first node
    double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
    // Determine number of pressure dofs in element
    const unsigned n_pres = this->npres_nst();
 
    // Find the indices at which the local velocities are stored
    unsigned u_nodal_index[DIM];
    for (unsigned i = 0; i < DIM; i++)
    {
      u_nodal_index[i] = this->u_index_nst(i);
    }
 
    // Which nodal value represents the pressure? (Negative if pressure
    // is not based on nodal interpolation).
    const int p_index = this->p_nodal_index_nst();
 
    // Local array of booleans that are true if the l-th pressure value is
    // hanging (avoid repeated virtual function calls)
    bool pressure_dof_is_hanging[n_pres];
 
    // If the pressure is stored at a node
    if (p_index >= 0)
    {
      // Read out whether the pressure is hanging
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = pressure_node_pt(l)->is_hanging(p_index);
      }
    }
    // Otherwise the pressure is not stored at a node and so cannot hang
    else
    {
      for (unsigned l = 0; l < n_pres; ++l)
      {
        pressure_dof_is_hanging[l] = false;
      }
    }
 
    // Set up memory for the shape and test functions
    Shape psif(n_node), testf(n_node);
    DShape dpsifdx(n_node, DIM), dtestfdx(n_node, DIM);
 
    // Set up memory for pressure shape and test functions
    Shape psip(n_pres), testp(n_pres);
 
    // Determine number of shape controlling nodes
    const unsigned n_shape_controlling_node = nshape_controlling_nodes();
 
    // Deriatives of shape fct derivatives w.r.t. nodal coords
    RankFourTensor<double> d_dpsifdx_dX(
      DIM, n_shape_controlling_node, n_node, DIM);
    RankFourTensor<double> d_dtestfdx_dX(
      DIM, n_shape_controlling_node, n_node, DIM);
 
    // Derivative of Jacobian of mapping w.r.t. to nodal coords
    DenseMatrix<double> dJ_dX(DIM, n_shape_controlling_node);
 
    // Derivatives of derivative of u w.r.t. nodal coords
    RankFourTensor<double> d_dudx_dX(DIM, n_shape_controlling_node, DIM, DIM);
 
    // Derivatives of nodal velocities w.r.t. nodal coords:
    // Assumption: Interaction only local via no-slip so
    // X_ij only affects U_ij.
    DenseMatrix<double> d_U_dX(DIM, n_shape_controlling_node, 0.0);
 
    // Determine the number of integration points
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Vector to hold local coordinates
    Vector<double> s(DIM);
 
    // Get physical variables from element
    // (Reynolds number must be multiplied by the density ratio)
    double scaled_re = this->re() * this->density_ratio();
    double scaled_re_st = this->re_st() * this->density_ratio();
    double scaled_re_inv_fr = this->re_invfr() * this->density_ratio();
    double visc_ratio = this->viscosity_ratio();
    Vector<double> G = this->g();
 
    // FD step
    double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
 
    // Pre-compute derivatives of nodal velocities w.r.t. nodal coords:
    // Assumption: Interaction only local via no-slip so
    // X_ij only affects U_ij.
    bool element_has_node_with_aux_node_update_fct = false;
 
    std::map<Node*, unsigned> local_shape_controlling_node_lookup =
      shape_controlling_node_lookup();
 
    // FD loop over shape-controlling nodes
    for (std::map<Node*, unsigned>::iterator it =
           local_shape_controlling_node_lookup.begin();
         it != local_shape_controlling_node_lookup.end();
         it++)
    {
      // Get node
      Node* nod_pt = it->first;
 
      // Get its number
      unsigned q = it->second;
 
      // Only compute if there's a node-update fct involved
      if (nod_pt->has_auxiliary_node_update_fct_pt())
      {
        element_has_node_with_aux_node_update_fct = true;
 
        // Current nodal velocity
        Vector<double> u_ref(DIM);
        for (unsigned i = 0; i < DIM; i++)
        {
          u_ref[i] = *(nod_pt->value_pt(u_nodal_index[i]));
        }
 
        // FD
        for (unsigned p = 0; p < DIM; p++)
        {
          // Make backup
          double backup = nod_pt->x(p);
 
          // Do FD step. No node update required as we're
          // attacking the coordinate directly...
          nod_pt->x(p) += eps_fd;
 
          // Do auxiliary node update (to apply no slip)
          nod_pt->perform_auxiliary_node_update_fct();
 
          // Evaluate
          d_U_dX(p, q) =
            (*(nod_pt->value_pt(u_nodal_index[p])) - u_ref[p]) / eps_fd;
 
          // Reset
          nod_pt->x(p) = backup;
 
          // Do auxiliary node update (to apply no slip)
          nod_pt->perform_auxiliary_node_update_fct();
        }
      }
    }
 
    // Integer to store the local equation number
    int local_eqn = 0;
 
    // Pointers to hang info object
    HangInfo* hang_info_pt = 0;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      const double w = integral_pt()->weight(ipt);
 
      // Call the derivatives of the shape and test functions
      const double J =
        this->dshape_and_dtest_eulerian_at_knot_nst(ipt,
                                                    psif,
                                                    dpsifdx,
                                                    d_dpsifdx_dX,
                                                    testf,
                                                    dtestfdx,
                                                    d_dtestfdx_dX,
                                                    dJ_dX);
 
      // Call the pressure shape and test functions
      this->pshape_nst(s, psip, testp);
 
      // Calculate local values of the pressure and velocity components
      // Allocate
      double interpolated_p = 0.0;
      Vector<double> interpolated_u(DIM, 0.0);
      Vector<double> interpolated_x(DIM, 0.0);
      Vector<double> mesh_velocity(DIM, 0.0);
      Vector<double> dudt(DIM, 0.0);
      DenseMatrix<double> interpolated_dudx(DIM, DIM, 0.0);
 
      // Calculate pressure
      for (unsigned l = 0; l < n_pres; l++)
      {
        interpolated_p += this->p_nst(l) * psip[l];
      }
 
      // Calculate velocities and derivatives:
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over directions
        for (unsigned i = 0; i < DIM; i++)
        {
          // Get the nodal value
          const double u_value = nodal_value(l, u_nodal_index[i]);
          interpolated_u[i] += u_value * psif[l];
          interpolated_x[i] += nodal_position(l, i) * psif[l];
          dudt[i] += this->du_dt_nst(l, i) * psif[l];
 
          // Loop over derivative directions
          for (unsigned j = 0; j < DIM; j++)
          {
            interpolated_dudx(i, j) += u_value * dpsifdx(l, j);
          }
        }
      }
 
      if (!this->ALE_is_disabled)
      {
        // Loop over nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned i = 0; i < DIM; i++)
          {
            mesh_velocity[i] += this->dnodal_position_dt(l, i) * psif[l];
          }
        }
      }
 
      // Calculate derivative of du_i/dx_k w.r.t. nodal positions X_{pq}
 
      // Loop over shape-controlling nodes
      for (unsigned q = 0; q < n_shape_controlling_node; q++)
      {
        // Loop over coordinate directions
        for (unsigned p = 0; p < DIM; p++)
        {
          for (unsigned i = 0; i < DIM; i++)
          {
            for (unsigned k = 0; k < DIM; k++)
            {
              double aux = 0.0;
              for (unsigned j = 0; j < n_node; j++)
              {
                aux +=
                  nodal_value(j, u_nodal_index[i]) * d_dpsifdx_dX(p, q, j, k);
              }
              d_dudx_dX(p, q, i, k) = aux;
            }
          }
        }
      }
 
      // Get weight of actual nodal position/value in computation of mesh
      // velocity from positional/value time stepper
      const double pos_time_weight =
        node_pt(0)->position_time_stepper_pt()->weight(1, 0);
      const double val_time_weight =
        node_pt(0)->time_stepper_pt()->weight(1, 0);
 
      // Get the user-defined body force terms
      Vector<double> body_force(DIM);
      this->get_body_force_nst(time, ipt, s, interpolated_x, body_force);
 
      // Get the user-defined source function
      const double source = this->get_source_nst(time, ipt, interpolated_x);
 
      // Get gradient of body force function
      DenseMatrix<double> d_body_force_dx(DIM, DIM, 0.0);
      this->get_body_force_gradient_nst(
        time, ipt, s, interpolated_x, d_body_force_dx);
 
      // Get gradient of source function
      Vector<double> source_gradient(DIM, 0.0);
      this->get_source_gradient_nst(time, ipt, interpolated_x, source_gradient);
 
 
      // Assemble shape derivatives
      //---------------------------
 
      // MOMENTUM EQUATIONS
      // ------------------
 
      // Number of master nodes and storage for the weight of the shape function
      unsigned n_master = 1;
      double hang_weight = 1.0;
 
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Local boolean to indicate whether the node is hanging
        bool is_node_hanging = node_pt(l)->is_hanging();
 
        // If the node is hanging
        if (is_node_hanging)
        {
          hang_info_pt = node_pt(l)->hanging_pt();
 
          // Read out number of master nodes from hanging data
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Loop over coordinate directions
          for (unsigned i = 0; i < DIM; i++)
          {
            // Get the equation number
            // If the node is hanging
            if (is_node_hanging)
            {
              // Get the equation number from the master node
              local_eqn = this->local_hang_eqn(hang_info_pt->master_node_pt(m),
                                               u_nodal_index[i]);
              // Get the hang weight from the master node
              hang_weight = hang_info_pt->master_weight(m);
            }
            // If the node is not hanging
            else
            {
              // Local equation number
              local_eqn = this->nodal_local_eqn(l, u_nodal_index[i]);
 
              // Node contributes with full weight
              hang_weight = 1.0;
            }
 
            // IF it's not a boundary condition
            if (local_eqn >= 0)
            {
              // Loop over coordinate directions
              for (unsigned p = 0; p < DIM; p++)
              {
                // Loop over shape controlling nodes
                for (unsigned q = 0; q < n_shape_controlling_node; q++)
                {
                  // Residual x deriv of Jacobian
                  // ----------------------------
 
                  // Add the user-defined body force terms
                  double sum = body_force[i] * testf[l];
 
                  // Add the gravitational body force term
                  sum += scaled_re_inv_fr * testf[l] * G[i];
 
                  // Add the pressure gradient term
                  sum += interpolated_p * dtestfdx(l, i);
 
                  // Add in the stress tensor terms
                  // The viscosity ratio needs to go in here to ensure
                  // continuity of normal stress is satisfied even in flows
                  // with zero pressure gradient!
                  for (unsigned k = 0; k < DIM; k++)
                  {
                    sum -= visc_ratio *
                           (interpolated_dudx(i, k) +
                            this->Gamma[i] * interpolated_dudx(k, i)) *
                           dtestfdx(l, k);
                  }
 
                  // Add in the inertial terms
 
                  // du/dt term
                  sum -= scaled_re_st * dudt[i] * testf[l];
 
                  // Convective terms, including mesh velocity
                  for (unsigned k = 0; k < DIM; k++)
                  {
                    double tmp = scaled_re * interpolated_u[k];
                    if (!this->ALE_is_disabled)
                    {
                      tmp -= scaled_re_st * mesh_velocity[k];
                    }
                    sum -= tmp * interpolated_dudx(i, k) * testf[l];
                  }
 
                  // Multiply throsugh by deriv of Jacobian and integration
                  // weight
                  dresidual_dnodal_coordinates(local_eqn, p, q) +=
                    sum * dJ_dX(p, q) * w * hang_weight;
 
                  // Derivative of residual x Jacobian
                  // ---------------------------------
 
                  // Body force
                  sum = d_body_force_dx(i, p) * psif(q) * testf(l);
 
                  // Pressure gradient term
                  sum += interpolated_p * d_dtestfdx_dX(p, q, l, i);
 
                  // Viscous term
                  for (unsigned k = 0; k < DIM; k++)
                  {
                    sum -=
                      visc_ratio * ((interpolated_dudx(i, k) +
                                     this->Gamma[i] * interpolated_dudx(k, i)) *
                                      d_dtestfdx_dX(p, q, l, k) +
                                    (d_dudx_dX(p, q, i, k) +
                                     this->Gamma[i] * d_dudx_dX(p, q, k, i)) *
                                      dtestfdx(l, k));
                  }
 
                  // Convective terms, including mesh velocity
                  for (unsigned k = 0; k < DIM; k++)
                  {
                    double tmp = scaled_re * interpolated_u[k];
                    if (!this->ALE_is_disabled)
                    {
                      tmp -= scaled_re_st * mesh_velocity[k];
                    }
                    sum -= tmp * d_dudx_dX(p, q, i, k) * testf(l);
                  }
                  if (!this->ALE_is_disabled)
                  {
                    sum += scaled_re_st * pos_time_weight * psif(q) *
                           interpolated_dudx(i, p) * testf(l);
                  }
 
                  // Multiply through by Jacobian and integration weight
                  dresidual_dnodal_coordinates(local_eqn, p, q) +=
                    sum * J * w * hang_weight;
 
                } // End of loop over shape controlling nodes q
              } // End of loop over coordinate directions p
 
 
              // Derivs w.r.t. to nodal velocities
              // ---------------------------------
              if (element_has_node_with_aux_node_update_fct)
              {
                // Loop over local nodes
                for (unsigned q_local = 0; q_local < n_node; q_local++)
                {
                  // Number of master nodes and storage for the weight of
                  // the shape function
                  unsigned n_master2 = 1;
                  double hang_weight2 = 1.0;
                  HangInfo* hang_info2_pt = 0;
 
                  // Local boolean to indicate whether the node is hanging
                  bool is_node_hanging2 = node_pt(q_local)->is_hanging();
 
                  Node* actual_shape_controlling_node_pt = node_pt(q_local);
 
                  // If the node is hanging
                  if (is_node_hanging2)
                  {
                    hang_info2_pt = node_pt(q_local)->hanging_pt();
 
                    // Read out number of master nodes from hanging data
                    n_master2 = hang_info2_pt->nmaster();
                  }
                  // Otherwise the node is its own master
                  else
                  {
                    n_master2 = 1;
                  }
 
                  // Loop over the master nodes
                  for (unsigned mm = 0; mm < n_master2; mm++)
                  {
                    if (is_node_hanging2)
                    {
                      actual_shape_controlling_node_pt =
                        hang_info2_pt->master_node_pt(mm);
                      hang_weight2 = hang_info2_pt->master_weight(mm);
                    }
 
                    // Find the corresponding number
                    unsigned q = local_shape_controlling_node_lookup
                      [actual_shape_controlling_node_pt];
 
                    // Loop over coordinate directions
                    for (unsigned p = 0; p < DIM; p++)
                    {
                      double sum = -visc_ratio * this->Gamma[i] *
                                     dpsifdx(q_local, i) * dtestfdx(l, p) -
                                   scaled_re * psif(q_local) *
                                     interpolated_dudx(i, p) * testf(l);
                      if (i == p)
                      {
                        sum -= scaled_re_st * val_time_weight * psif(q_local) *
                               testf(l);
                        for (unsigned k = 0; k < DIM; k++)
                        {
                          sum -=
                            visc_ratio * dpsifdx(q_local, k) * dtestfdx(l, k);
                          double tmp = scaled_re * interpolated_u[k];
                          if (!this->ALE_is_disabled)
                          {
                            tmp -= scaled_re_st * mesh_velocity[k];
                          }
                          sum -= tmp * dpsifdx(q_local, k) * testf(l);
                        }
                      }
 
                      dresidual_dnodal_coordinates(local_eqn, p, q) +=
                        sum * d_U_dX(p, q) * J * w * hang_weight * hang_weight2;
                    }
                  } // End of loop over master nodes
                } // End of loop over local nodes
              } // End of if(element_has_node_with_aux_node_update_fct)
 
 
            } // local_eqn>=0
          }
        }
      } // End of loop over test functions
 
 
      // CONTINUITY EQUATION
      // -------------------
 
      // Loop over the shape functions
      for (unsigned l = 0; l < n_pres; l++)
      {
        // If the pressure dof is hanging
        if (pressure_dof_is_hanging[l])
        {
          // Pressure dof is hanging so it must be nodal-based
          // Get the hang info object
          hang_info_pt = this->pressure_node_pt(l)->hanging_pt(p_index);
 
          // Get the number of master nodes from the pressure node
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Get the number of the unknown
          // If the pressure dof is hanging
          if (pressure_dof_is_hanging[l])
          {
            // Get the local equation from the master node
            local_eqn =
              this->local_hang_eqn(hang_info_pt->master_node_pt(m), p_index);
            // Get the weight for the node
            hang_weight = hang_info_pt->master_weight(m);
          }
          else
          {
            local_eqn = this->p_local_eqn(l);
            hang_weight = 1.0;
          }
 
          // If not a boundary conditions
          if (local_eqn >= 0)
          {
            // Loop over coordinate directions
            for (unsigned p = 0; p < DIM; p++)
            {
              // Loop over nodes
              for (unsigned q = 0; q < n_shape_controlling_node; q++)
              {
                // Residual x deriv of Jacobian
                //-----------------------------
 
                // Source term
                double aux = -source;
 
                // Loop over velocity components
                for (unsigned k = 0; k < DIM; k++)
                {
                  aux += interpolated_dudx(k, k);
                }
 
                // Multiply through by deriv of Jacobian and integration weight
                dresidual_dnodal_coordinates(local_eqn, p, q) +=
                  aux * dJ_dX(p, q) * testp[l] * w * hang_weight;
 
 
                // Derivative of residual x Jacobian
                // ---------------------------------
 
                // Loop over velocity components
                aux = -source_gradient[p] * psif(q);
                for (unsigned k = 0; k < DIM; k++)
                {
                  aux += d_dudx_dX(p, q, k, k);
                }
                // Multiply through by Jacobian and integration weight
                dresidual_dnodal_coordinates(local_eqn, p, q) +=
                  aux * testp[l] * J * w * hang_weight;
              }
            }
 
 
            // Derivs w.r.t. to nodal velocities
            // ---------------------------------
            if (element_has_node_with_aux_node_update_fct)
            {
              // Loop over local nodes
              for (unsigned q_local = 0; q_local < n_node; q_local++)
              {
                // Number of master nodes and storage for the weight of
                // the shape function
                unsigned n_master2 = 1;
                double hang_weight2 = 1.0;
                HangInfo* hang_info2_pt = 0;
 
                // Local boolean to indicate whether the node is hanging
                bool is_node_hanging2 = node_pt(q_local)->is_hanging();
 
                Node* actual_shape_controlling_node_pt = node_pt(q_local);
 
                // If the node is hanging
                if (is_node_hanging2)
                {
                  hang_info2_pt = node_pt(q_local)->hanging_pt();
 
                  // Read out number of master nodes from hanging data
                  n_master2 = hang_info2_pt->nmaster();
                }
                // Otherwise the node is its own master
                else
                {
                  n_master2 = 1;
                }
 
                // Loop over the master nodes
                for (unsigned mm = 0; mm < n_master2; mm++)
                {
                  if (is_node_hanging2)
                  {
                    actual_shape_controlling_node_pt =
                      hang_info2_pt->master_node_pt(mm);
                    hang_weight2 = hang_info2_pt->master_weight(mm);
                  }
 
                  // Find the corresponding number
                  unsigned q = local_shape_controlling_node_lookup
                    [actual_shape_controlling_node_pt];
 
                  // Loop over coordinate directions
                  for (unsigned p = 0; p < DIM; p++)
                  {
                    double aux = dpsifdx(q_local, p) * testp(l);
                    dresidual_dnodal_coordinates(local_eqn, p, q) +=
                      aux * d_U_dX(p, q) * J * w * hang_weight * hang_weight2;
                  }
                } // End of loop over (mm) master nodes
              } // End of loop over local nodes q_local
            } // End of if(element_has_node_with_aux_node_update_fct)
          } // End of if it's not a boundary condition
        } // End of loop over (m) master nodes
      } // End of loop over shape functions for continuity eqn
 
    } // End of loop over integration points
  }
 
  //======================================================================
  /// 2D Further build for Crouzeix_Raviart interpolates the internal
  /// pressure dofs from father element: Make sure pressure values and
  /// dp/ds agree between fathers and sons at the midpoints of the son
  /// elements.
  //======================================================================
  template<>
  void PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<
    2>::further_build()
  {
    if (this->tree_pt()->father_pt() != 0)
    {
      // Call the generic further build (if there is a father)
      RefineableGeneralisedNewtonianNavierStokesEquations<2>::further_build();
    }
    // Now do the PRefineableQElement further_build()
    PRefineableQElement<2, 3>::further_build();
 
    // Resize internal pressure storage
    if (this->internal_data_pt(this->P_nst_internal_index)->nvalue() <=
        this->npres_nst())
    {
      this->internal_data_pt(this->P_nst_internal_index)
        ->resize(this->npres_nst());
    }
    else
    {
      Data* new_data_pt = new Data(this->npres_nst());
      delete internal_data_pt(this->P_nst_internal_index);
      internal_data_pt(this->P_nst_internal_index) = new_data_pt;
    }
 
    if (this->tree_pt()->father_pt() != 0)
    {
      // Pointer to my father (in C-R element impersonation)
      PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<2>*
        father_element_pt = dynamic_cast<
          PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<2>*>(
          quadtree_pt()->father_pt()->object_pt());
 
      // If element has same p-order as father then do the projection problem
      // (called after h-refinement)
      if (father_element_pt->p_order() == this->p_order())
      {
        using namespace QuadTreeNames;
 
        // What type of son am I? Ask my quadtree representation...
        int son_type = quadtree_pt()->son_type();
 
        Vector<double> s_father(2);
 
        // Son midpoint is located at the following coordinates in father
        // element:
        switch (son_type)
        {
          case SW:
            // South west son
            s_father[0] = -0.5;
            s_father[1] = -0.5;
            break;
          case SE:
            // South east son
            s_father[0] = 0.5;
            s_father[1] = -0.5;
            break;
          case NE:
            // North east son
            s_father[0] = 0.5;
            s_father[1] = 0.5;
            break;
          case NW:
            // North west son
            s_father[0] = -0.5;
            s_father[1] = 0.5;
            break;
          default:
            throw OomphLibError("Invalid son type in",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
            break;
        }
 
        // Get pressure value in father element
        double press = father_element_pt->interpolated_p_nst(s_father);
 
        // Reset all pressures to zero
        for (unsigned p = 0; p < this->npres_nst(); p++)
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(p, 0.0);
        }
 
        // Set pressure values from father (BENFLAG: projection problem hack)
        if (this->npres_nst() == 1)
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(0, press);
        }
        else if (this->npres_nst() == 3)
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(0, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(1, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(2, press);
        }
        else
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(0, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(1, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(2, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(3, press);
        }
      } // Otherwise this is called after p-refinement
    }
  }
 
  //======================================================================
  /// 2D Further build for Crouzeix_Raviart interpolates the internal
  /// pressure dofs from father element: Make sure pressure values and
  /// dp/ds agree between fathers and sons at the midpoints of the son
  /// elements.
  //======================================================================
  template<>
  void PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<
    3>::further_build()
  {
    if (this->tree_pt()->father_pt() != 0)
    {
      // Call the generic further build (if there is a father)
      RefineableGeneralisedNewtonianNavierStokesEquations<3>::further_build();
    }
    // Now do the PRefineableQElement further_build()
    PRefineableQElement<3, 3>::further_build();
 
    // Resize internal pressure storage
    if (this->internal_data_pt(this->P_nst_internal_index)->nvalue() <=
        this->npres_nst())
    {
      this->internal_data_pt(this->P_nst_internal_index)
        ->resize(this->npres_nst());
    }
    else
    {
      Data* new_data_pt = new Data(this->npres_nst());
      delete internal_data_pt(this->P_nst_internal_index);
      internal_data_pt(this->P_nst_internal_index) = new_data_pt;
    }
 
    if (this->tree_pt()->father_pt() != 0)
    {
      // Pointer to my father (in C-R element impersonation)
      PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<3>*
        father_element_pt = dynamic_cast<
          PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<3>*>(
          octree_pt()->father_pt()->object_pt());
 
      // If element has same p-order as father then do the projection problem
      // (called after h-refinement)
      if (father_element_pt->p_order() == this->p_order())
      {
        using namespace OcTreeNames;
 
        // What type of son am I? Ask my quadtree representation...
        int son_type = octree_pt()->son_type();
 
        Vector<double> s_father(3);
 
 
        // Son midpoint is located at the following coordinates in father
        // element:
        for (unsigned i = 0; i < 3; i++)
        {
          s_father[i] = 0.5 * OcTree::Direction_to_vector[son_type][i];
        }
 
        // Get pressure value in father element
        double press = father_element_pt->interpolated_p_nst(s_father);
 
        // Reset all pressures to zero
        for (unsigned p = 0; p < this->npres_nst(); p++)
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(p, 0.0);
        }
 
        // Set pressure values from father (BENFLAG: projection problem hack)
        if (this->npres_nst() == 1)
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(0, press);
        }
        else
        {
          internal_data_pt(this->P_nst_internal_index)->set_value(0, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(1, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(2, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(3, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(4, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(5, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(6, press);
          internal_data_pt(this->P_nst_internal_index)->set_value(7, press);
        }
      } // Otherwise this is called after p-refinement
    }
  }
 
 
  //====================================================================
  /// / Force build of templates
  //====================================================================
  template class RefineableGeneralisedNewtonianNavierStokesEquations<2>;
  template class RefineableGeneralisedNewtonianNavierStokesEquations<3>;
  template class RefineableGeneralisedNewtonianQTaylorHoodElement<2>;
  template class RefineableGeneralisedNewtonianQTaylorHoodElement<3>;
  template class RefineableGeneralisedNewtonianQCrouzeixRaviartElement<2>;
  template class RefineableGeneralisedNewtonianQCrouzeixRaviartElement<3>;
  template class PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<2>;
  template class PRefineableGeneralisedNewtonianQCrouzeixRaviartElement<3>;
} // namespace oomph