Example problem: A rigid cylinder rotating in a viscous fluid beneath a free surface.

Detailed documentation to be written. Here's the driver code...

(This problem is solved using spatially adaptive elements with a pseudo-elastic remesh strategy)

//LIC// ====================================================================
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//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC// Copyright (C) 2006-2025 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
//LIC// This library is distributed in the hope that it will be useful,
//LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
//LIC// 
//LIC// You should have received a copy of the GNU Lesser General Public
//LIC// License along with this library; if not, write to the Free Software
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//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//A driver program to solve the problem of a cylinder rotating near a free
//surface
 
#include "generic.h"
#include "navier_stokes.h"
#include "solid.h"
#include "fluid_interface.h"
 
using namespace oomph;
 
 
//=start_of_namespace================================================
/// Namespace for physical parameters
//===================================================================
namespace Global_Physical_Variables
{
 
 /// Pseudo-solid Poisson ratio
 double Nu=0.1;
 
 /// Direction of the wall normal vector
 Vector<double> Wall_normal;
 
 /// Function that specifies the wall unit normal
 void wall_unit_normal_fct(const Vector<double> &x, 
                      Vector<double> &normal)
 {
  normal=Wall_normal;
 }
 
} // end_of_namespace
 
 
//My own Ellipse class
class GeneralEllipse : public GeomObject
{
private:
 //Internal data to store the centre and semi-axes
 double *centre_x_pt, *centre_y_pt, *a_pt, *b_pt;
 
public:
 
 //Constructor
 GeneralEllipse(const double &centre_x, const double &centre_y,
                const double &a, const double &b)
  : GeomObject(1,2), centre_x_pt(0), centre_y_pt(0), a_pt(0), b_pt(0)
  {
   centre_x_pt = new double(centre_x);
   centre_y_pt = new double(centre_y);
   a_pt = new double(a);
   b_pt = new double(b);
  }
 
 //Destructor
 ~GeneralEllipse()
  {
   delete centre_x_pt;
   delete centre_y_pt;
   delete a_pt;
   delete b_pt;
  }
 
 //Return the position
 void position(const Vector<double> &xi, Vector<double> &r) const
  {
   r[0] = *centre_x_pt + *a_pt*cos(xi[0]);
   r[1] = *centre_y_pt + *b_pt*sin(xi[0]);
  }
};
 
 
//A Domain
class CylinderAndInterfaceDomain : public Domain
{
public:
 double centre_x, centre_y;
 
private:
 
 double Lower_left[2], Lower_right[2], Lower_mid_left[2], Lower_mid_right[2];
 double Upper_left[2], Upper_right[2], Upper_mid_left[2], Upper_mid_right[2];
 double Lower_centre_left[2], Lower_centre_right[2];
 double Upper_centre_left[2], Upper_centre_right[2];
 
 
 /// Geometric object that represents the rotating cylinder
 GeomObject* Cylinder_pt;
 
public:
 
 //Constructor, pass the length and height of the domain
 CylinderAndInterfaceDomain(const double &Length, const double &Height)
  {   
 
   centre_x = Length/2.0;
   centre_y = Height/2.0; //3.0*Height/4.0;
  //Create a new ellipse object to represent the rotating cylinder
   Cylinder_pt = new GeneralEllipse(centre_x,centre_y,0.2*Height,0.2*Height);
 
   //Set some basic coordinates
 
   Lower_left[0] = 0.0;
   Lower_left[1] = 0.0;
   
   Upper_left[0] = 0.0;
   Upper_left[1] = Height;
 
   Lower_right[0] = Length;
   Lower_right[1] = 0.0;
   
   Upper_right[0] = Length;
   Upper_right[1] = Height;
 
 
   //Let's just do some mid coordinates
   Lower_mid_left[0] = Length/10.0;
   Lower_mid_left[1] = 0.0;
 
   Upper_mid_left[0] = Length/10.0;
   Upper_mid_left[1] = Height;
 
   Vector<double> xi(1), f(2);
   xi[0] = -3.0*atan(1.0);
   Cylinder_pt->position(xi,f);
 
   Lower_centre_left[0] = f[0];
   Lower_centre_left[1] = f[1];
   
   xi[0] = 3.0*atan(1.0);
   Cylinder_pt->position(xi,f);
 
   Upper_centre_left[0] = f[0];
   Upper_centre_left[1] = f[1];
 
 
   Lower_mid_right[0] = 9.0*Length/10.0;
   Lower_mid_right[1] = 0.0;
 
   Upper_mid_right[0] = 9.0*Length/10.0;
   Upper_mid_right[1] = Height;
 
 
   xi[0] = -1.0*atan(1.0);
   Cylinder_pt->position(xi,f);
 
   Lower_centre_right[0] = f[0];
   Lower_centre_right[1] = f[1];
   
   xi[0] = 1.0*atan(1.0);
   Cylinder_pt->position(xi,f);
 
   Upper_centre_right[0] = f[0];
   Upper_centre_right[1] = f[1];
   
   //There are six macro elements
   Macro_element_pt.resize(6); 
 
   // Build the macro elements
   for (unsigned i=0;i<6;i++)
    {Macro_element_pt[i]= new QMacroElement<2>(this,i);}
  }
 
 // Destructor: Empty; cleanup done in base class
 ~CylinderAndInterfaceDomain() {}
 
 //Private little interpolation problem
 void linear_interpolate(double Left[2], double Right[2],
                         const double &s, Vector<double> &f)
  {
   for(unsigned i=0;i<2;i++)
    {
     f[i] = Left[i] + (Right[i] - Left[i])*0.5*(s+1.0);
    }
  }
 
   
 
 // Sort out the vector representation of the i-th macro element
 void macro_element_boundary(const unsigned &time,
                             const unsigned &m,
                             const unsigned &direction,
                             const Vector<double> &s,
                             Vector<double>& f)
 {
 
  using namespace QuadTreeNames;
 
#ifdef WARN_ABOUT_SUBTLY_CHANGED_OOMPH_INTERFACES
   // Warn about time argument being moved to the front
   OomphLibWarning(
    "Order of function arguments has changed between versions 0.8 and 0.85",
    "CylinderAndInterfaceDomain::macro_element_boundary(...)",
    OOMPH_EXCEPTION_LOCATION);
#endif
 
  Vector<double> xi(1);  
 
  //Switch on the macro element
  switch(m)
   {
    //Macro element 0, is the left-hand film
   case 0:
    
    switch(direction)
     {
     case N:
      linear_interpolate(Upper_left,Upper_mid_left,s[0],f);
      break;
 
     case S:  
       linear_interpolate(Lower_left,Lower_mid_left,s[0],f);
      break;
 
     case W:
       linear_interpolate(Lower_left,Upper_left,s[0],f);
      break;
 
     case E:
       linear_interpolate(Lower_mid_left,Upper_mid_left,s[0],f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
 
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
    
    break;
    
    //Macro element 1, is immediately left of the cylinder
   case 1:
    
    switch(direction)
     {
     case N:
       linear_interpolate(Upper_mid_left,Upper_centre_left,s[0],f);
      break;
 
     case S:  
       linear_interpolate(Lower_mid_left,Lower_centre_left,s[0],f);
      break;
 
     case W:
       linear_interpolate(Lower_mid_left,Upper_mid_left,s[0],f);
      break;
 
     case E:
      xi[0] = 5.0*atan(1.0) - 2.0*atan(1.0)*0.5*(1.0+s[0]);
      Cylinder_pt->position(xi,f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
      
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
    
    break;
 
   //Macro element 2, is immediately above the cylinder
   case 2:
    
    switch(direction)
     {
     case N:
       linear_interpolate(Upper_mid_left,Upper_mid_right,s[0],f);
      break;
      
     case S:  
      xi[0] = 3.0*atan(1.0) - 2.0*atan(1.0)*0.5*(1.0+s[0]);
      Cylinder_pt->position(xi,f);
      break;
 
     case W:
       linear_interpolate(Upper_centre_left,Upper_mid_left,s[0],f);
      break;
 
     case E:
       linear_interpolate(Upper_centre_right,Upper_mid_right,s[0],f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
      
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
    
    break;
 
    //Macro element 3, is immediately right of the cylinder
   case 3:
 
    switch(direction)
     {
     case N:
       linear_interpolate(Upper_centre_right,Upper_mid_right,s[0],f);
      break;
      
     case S:  
       linear_interpolate(Lower_centre_right,Lower_mid_right,s[0],f);
      break;
 
     case W:
      xi[0] = -atan(1.0) + 2.0*atan(1.0)*0.5*(1.0+s[0]);
      Cylinder_pt->position(xi,f);
      break;
 
     case E:
       linear_interpolate(Lower_mid_right,Upper_mid_right,s[0],f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
 
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
 
    break;
    
    //Macro element 4, is immediately below cylinder
   case 4:
    
    switch(direction)
     {
     case N:
      //linear_interpolate(Lower_centre_left,Lower_centre_right,s[0],f);
      xi[0] = -3.0*atan(1.0) + 2.0*atan(1.0)*0.5*(1.0+s[0]);
      Cylinder_pt->position(xi,f);
      break;
      
     case S:  
       linear_interpolate(Lower_mid_left,Lower_mid_right,s[0],f);
      break;
 
     case W:
       linear_interpolate(Lower_mid_left,Lower_centre_left,s[0],f);
      break;
 
     case E:
       linear_interpolate(Lower_mid_right,Lower_centre_right,s[0],f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
 
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }  
 
    break;
 
    //Macro element 5, is right film
   case 5:
    
    switch(direction)
     {
     case N:
       linear_interpolate(Upper_mid_right,Upper_right,s[0],f);
      break;
      
     case S:  
       linear_interpolate(Lower_mid_right,Lower_right,s[0],f);
      break;
 
     case W:
       linear_interpolate(Lower_mid_right,Upper_mid_right,s[0],f);
      break;
 
     case E:
       linear_interpolate(Lower_right,Upper_right,s[0],f);
      break;
 
     default:
      std::ostringstream error_stream;
      error_stream << "Direction is incorrect: " << direction << std::endl;
 
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
 
    break;
    
   default:
    std::ostringstream error_stream;
      error_stream << "Wrong domain number: " << m<< std::endl;
 
      throw OomphLibError(error_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
   }
 
 }
 
};
 
//Now I need to actually create a Mesh
template<class ELEMENT>
class CylinderAndInterfaceMesh : public virtual SolidMesh
{
 double Height;
 
protected:
 //Pointer to the domain
 CylinderAndInterfaceDomain* Domain_pt;
 
public:
 
 //Access function to the domain
 CylinderAndInterfaceDomain* domain_pt() {return Domain_pt;}
 
 //Constructor, 
 CylinderAndInterfaceMesh(const double &length, const double &height,
                            TimeStepper* time_stepper_pt) : Height(height)
  {
   //Create the domain
   Domain_pt = new CylinderAndInterfaceDomain(length,height);
 
   //Initialise the node counter
   unsigned node_count=0;
   //Vectors Used to get data from domains
   Vector<double> s(2), r(2);
   
   //Setup temporary storage for the Node
   Vector<Node *> Tmp_node_pt;
 
   //Now blindly loop over the macro elements and associate and finite
   //element with each
   unsigned Nmacro_element = Domain_pt->nmacro_element();
   for(unsigned e=0;e<Nmacro_element;e++)
    {
     //Create the FiniteElement and add to the Element_pt Vector
     Element_pt.push_back(new ELEMENT);
 
     //Read out the number of linear points in the element
     unsigned Np = 
      dynamic_cast<ELEMENT*>(finite_element_pt(e))->nnode_1d();
     
     //Loop over nodes in the column
     for(unsigned l1=0;l1<Np;l1++)
      {
       //Loop over the nodes in the row
       for(unsigned l2=0;l2<Np;l2++)
        {
         //Allocate the memory for the node
         Tmp_node_pt.push_back(finite_element_pt(e)->
                           construct_node(l1*Np+l2,time_stepper_pt));
         
         //Read out the position of the node from the macro element
         s[0] = -1.0 + 2.0*(double)l2/(double)(Np-1);
         s[1] = -1.0 + 2.0*(double)l1/(double)(Np-1);
         Domain_pt->macro_element_pt(e)->macro_map(s,r);
         
         //Set the position of the node
         Tmp_node_pt[node_count]->x(0) = r[0];
         Tmp_node_pt[node_count]->x(1) = r[1];
         
         //Increment the node number
         node_count++;
        }
      }
    } //End of loop over macro elements
 
   //Now the elements have been created, but there will be nodes in 
   //common, need to loop over the common edges and sort it, by reassigning
   //pointers and the deleting excess nodes
   
   //Read out the number of linear points in the element
   unsigned Np = 
    dynamic_cast<ELEMENT*>(finite_element_pt(0))->nnode_1d();
 
   //DelaunayEdge between Elements 0 and 1
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 1 to be the same as in element 0
     finite_element_pt(1)->node_pt(Np*n)
      = finite_element_pt(0)->node_pt(n*Np+Np-1);
 
     //Remove the nodes in element 1 from the temporaray node list
     delete Tmp_node_pt[Np*Np + Np*n];
     Tmp_node_pt[Np*Np + Np*n] = 0;
    }
 
   //DelaunayEdge between Elements 1 and 2
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 2 to be the same as in element 1
     finite_element_pt(2)->node_pt(n*Np)
      = finite_element_pt(1)->node_pt((Np-1)*Np+Np-1-n);
 
     //Remove the nodes in element 2 from the temporaray node list
     delete Tmp_node_pt[2*Np*Np + n*Np];
     Tmp_node_pt[2*Np*Np + n*Np] = 0;
    }
   
   //DelaunayEdge between Elements 1 and 4
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 4 to be the same as in element 1
     finite_element_pt(4)->node_pt(n*Np)
      = finite_element_pt(1)->node_pt(n);
 
     //Remove the nodes in element 2 from the temporaray node list
     delete Tmp_node_pt[4*Np*Np + n*Np];
     Tmp_node_pt[4*Np*Np + n*Np] = 0;
    }
 
   //DelaunayEdge between Element 2 and 3
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 3 to be the same as in element 2
     finite_element_pt(3)->node_pt(Np*(Np-1)+n)
      = finite_element_pt(2)->node_pt(Np*n+Np-1);
 
     //Remove the nodes in element 3 from the temporaray node list
     delete Tmp_node_pt[3*Np*Np + Np*(Np-1)+n];
     Tmp_node_pt[3*Np*Np + Np*(Np-1)+n] = 0;
    }
 
 
   //DelaunayEdge between Element 4 and 3
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 3 to be the same as in element 4
     finite_element_pt(3)->node_pt(n)
      = finite_element_pt(4)->node_pt(Np*(Np-n-1)+Np-1);
 
     //Remove the nodes in element 3 from the temporaray node list
     delete Tmp_node_pt[3*Np*Np + n];
     Tmp_node_pt[3*Np*Np + n] = 0;
    }
 
 
   //DelaunayEdge between Element 3 and 5
   for(unsigned n=0;n<Np;n++)
    {
     //Set the nodes in element 5 to be the same as in element 3
     finite_element_pt(5)->node_pt(n*Np)
      = finite_element_pt(3)->node_pt(Np*n+Np-1);
 
     //Remove the nodes in element 5 from the temporaray node list
     delete Tmp_node_pt[5*Np*Np + n*Np];
     Tmp_node_pt[5*Np*Np + n*Np] = 0;
    }
 
   //Now set the actual true nodes
   for(unsigned n=0;n<node_count;n++)
    {
     if(Tmp_node_pt[n]!=0) {Node_pt.push_back(Tmp_node_pt[n]);}
    }
 
 
 
   //Finally set the nodes on the boundaries
   set_nboundary(5);
   
   for(unsigned n=0;n<Np;n++)
    {
     //Left hand side
     Node* temp_node_pt = finite_element_pt(0)->node_pt(n*Np);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(3,temp_node_pt);
     
 
     //Right hand side
     temp_node_pt = finite_element_pt(5)->node_pt(n*Np+Np-1);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(1,temp_node_pt);
     
     //LH part of lower boundary
     temp_node_pt = finite_element_pt(0)->node_pt(n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(0,temp_node_pt);
 
     //First part of upper boundary
     temp_node_pt = finite_element_pt(0)->node_pt(Np*(Np-1)+n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(2,temp_node_pt);
     
     //First part of hole boundary
     temp_node_pt = finite_element_pt(4)->node_pt(Np*(Np-1)+n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(4,temp_node_pt);
    }
 
   for(unsigned n=1;n<Np;n++)
    {
     //Middle of lower boundary
     Node* temp_node_pt = finite_element_pt(4)->node_pt(n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(0,temp_node_pt);
     
     //Middle of upper boundary                                
     temp_node_pt = finite_element_pt(2)->node_pt(Np*(Np-1)+n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(2,temp_node_pt);
 
     //Next part of hole
     temp_node_pt = finite_element_pt(3)->node_pt(n*Np);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(4,temp_node_pt);
    }
 
   for(unsigned n=1;n<Np;n++)
    {
     //Final part of lower boundary
     Node* temp_node_pt = finite_element_pt(5)->node_pt(n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(0,temp_node_pt);
     
     //Middle of upper boundary                                
     temp_node_pt = finite_element_pt(5)->node_pt(Np*(Np-1)+n);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(2,temp_node_pt);
     
     //Next part of hole
     temp_node_pt = finite_element_pt(2)->node_pt(Np-n-1);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(4,temp_node_pt);
    }
 
   for(unsigned n=1;n<Np-1;n++)
    {
     //Final part of hole
     Node* temp_node_pt = finite_element_pt(1)->node_pt(Np*(Np-n-1)+Np-1);
     this->convert_to_boundary_node(temp_node_pt);
     add_boundary_node(4,temp_node_pt);
    }
   
   //Now loop over all the nodes and set their Lagrangian coordinates
   unsigned Nnode = nnode();
   for(unsigned n=0;n<Nnode;n++)
   {
    //Cast node to an elastic node
    SolidNode* temp_pt = static_cast<SolidNode*>(Node_pt[n]);
    for(unsigned i=0;i<2;i++)
     {temp_pt->xi(i) = temp_pt->x(i);}
   }
 }
 
 
};
 
//Now let's do the adaptive mesh
template<class ELEMENT>
class RefineableCylinderAndInterfaceMesh :
 public CylinderAndInterfaceMesh<ELEMENT>, public RefineableQuadMesh<ELEMENT>
{
public: 
 
 // Constructor
 RefineableCylinderAndInterfaceMesh(const double &length, const double &height,
                                    TimeStepper* time_stepper_pt) :
  CylinderAndInterfaceMesh<ELEMENT>(length,height,time_stepper_pt) 
  {
 
   // Nodal positions etc. were created in constructor for
   // Cylinder...<...>. Need to setup adaptive information.
 
   // Loop over all elements and set macro element pointer
   for (unsigned e=0;e<6;e++)
    {
     dynamic_cast<ELEMENT*>(this->element_pt(e))->
      set_macro_elem_pt(this->Domain_pt->macro_element_pt(e));
    }
 
   // Setup quadtree forest for mesh refinement
   this->setup_quadtree_forest();
   
   // Setup the boundary element info
   this->setup_boundary_element_info();
 
  }
 
 
 /// Destructor: Empty
 virtual ~RefineableCylinderAndInterfaceMesh() {}
 
 
};
 
template<class ELEMENT>
class RefineableRotatingCylinderProblem : public Problem
{
private:
 double Length, Height;
 
 //Constitutive law used to determine the mesh deformation
 ConstitutiveLaw *Constitutive_law_pt;
 
 Data* Traded_pressure_data_pt;
 
public:
 
 double Re, Ca, ReInvFr, Bo;
 
 double Omega;
 
 double Volume;
 
 double Angle;
 
 Vector<double> G;
 
 /// Constructor: Pass flag to indicate if you want
 /// a constant source function or the tanh profile.
 RefineableRotatingCylinderProblem(const double &length, const double &height);
 
 /// Update the problem specs after solve (empty)
 void actions_after_newton_solve() {}
 
 /// Update the problem specs before solve: 
 void actions_before_newton_solve() {set_boundary_conditions();}
 
 /// Strip off the interface before adaptation
 void actions_before_adapt() 
  {
   this->delete_volume_constraint_elements();
   this->delete_free_surface_elements();
  }
 
 void actions_after_adapt() {finish_problem_setup(); this->rebuild_global_mesh();}
 
 /// Complete problem setup: Setup element-specific things 
 /// (source fct pointers etc.)
 void finish_problem_setup();
 
 //Access function for the mesh
 RefineableCylinderAndInterfaceMesh<ELEMENT>* Bulk_mesh_pt;
 
 //Access function for surface mesh
 Mesh* Surface_mesh_pt;
 
 //Access function for point mesh
 Mesh* Point_mesh_pt;
 
 /// The volume constraint mesh 
 Mesh* Volume_constraint_mesh_pt;
 
 void set_boundary_conditions();
 
 void solve();
 
 /// Create the volume constraint elements
 void create_volume_constraint_elements()
  {
   //The single volume constraint element
   VolumeConstraintElement* vol_constraint_element = 
    new VolumeConstraintElement(&Volume,Traded_pressure_data_pt,0);
   Volume_constraint_mesh_pt->add_element_pt(vol_constraint_element);
   
   //Loop over all boundaries (or just 1 and 2 why?)
   for(unsigned b=0;b<4;b++)
    {
     // How many bulk fluid elements are adjacent to boundary b?
     unsigned n_element = Bulk_mesh_pt->nboundary_element(b);
     
     // Loop over the bulk fluid elements adjacent to boundary b?
     for(unsigned e=0;e<n_element;e++)
      {
       // Get pointer to the bulk fluid element that is 
       // adjacent to boundary b
       ELEMENT* bulk_elem_pt = dynamic_cast<ELEMENT*>(
        Bulk_mesh_pt->boundary_element_pt(b,e));
       
       //Find the index of the face of element e along boundary b
       int face_index = Bulk_mesh_pt->face_index_at_boundary(b,e);
       
       // Create new element
       ElasticLineVolumeConstraintBoundingElement<ELEMENT>* el_pt =
        new ElasticLineVolumeConstraintBoundingElement<ELEMENT>(
         bulk_elem_pt,face_index);   
       
       //Set the "master" volume control element
       el_pt->set_volume_constraint_element(vol_constraint_element);
       
       // Add it to the mesh
       Volume_constraint_mesh_pt->add_element_pt(el_pt);     
      }
    }
  }
 
 void delete_volume_constraint_elements()
  {
   unsigned n_element = Volume_constraint_mesh_pt->nelement();
   for(unsigned e=0;e<n_element;e++)
    {
     delete Volume_constraint_mesh_pt->element_pt(e);
    }
   Volume_constraint_mesh_pt->flush_element_and_node_storage();
  }
 
 void create_free_surface_elements()
  {
   //Find number of elements adjacent to upper boundary
   unsigned n_boundary_element = Bulk_mesh_pt->nboundary_element(2);
   //The boundary elements do no necessarily come in order, so we will
   //need to detect the element adjacent to boundary 1.
   //The index of that element in our array will be stored in this variable
   //(initialised to a negative and therefore invalid number)
   int final_element_index=-1;
   //Loop over the elements adjacent to the boundary
   for(unsigned e=0;e<n_boundary_element;e++)
    {
     //Create the free surface element (on face 2)
     FiniteElement *free_surface_element_pt 
      = new ElasticLineFluidInterfaceElement<ELEMENT>
      (Bulk_mesh_pt->boundary_element_pt(2,e),
       Bulk_mesh_pt->face_index_at_boundary(2,e));
     //Push it back onto the stack
     Surface_mesh_pt->add_element_pt(free_surface_element_pt);       
 
      //Check whether the element is on the boundary 1
      unsigned n_node = free_surface_element_pt->nnode();
      //Only need to check the end nodes
      if((free_surface_element_pt->node_pt(0)->is_on_boundary(1)) ||
         (free_surface_element_pt->node_pt(n_node-1)->is_on_boundary(1)))
       {
          final_element_index=e;
       }
    }
 
   unsigned Nfree = Surface_mesh_pt->nelement();
   oomph_info << Nfree << " free surface elements assigned" << std::endl;
 
    if(final_element_index == -1)
     {
      throw OomphLibError("No Surface Element adjacent to boundary 1\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
   
    //Make the edge point
    FiniteElement* point_element_pt= 
     dynamic_cast<ElasticLineFluidInterfaceElement<ELEMENT>*>
     (Surface_mesh_pt->element_pt(final_element_index))
     ->make_bounding_element(1);
    
    //Add it to the stack
    Point_mesh_pt->add_element_pt(point_element_pt);
  }
 
 //Function to delete the free surface elements
 void delete_free_surface_elements()
  {
   //Find the number of traction elements
   unsigned Nfree_surface = Surface_mesh_pt->nelement();
   
   //The traction elements are ALWAYS? stored at the end
   //So delete and remove them, add one to get rid of the constraint
   for(unsigned e=0;e<Nfree_surface;e++)
    {
     delete Surface_mesh_pt->element_pt(e);
    }
   Surface_mesh_pt->flush_element_and_node_storage();
 
   delete Point_mesh_pt->element_pt(0);
   Point_mesh_pt->flush_element_and_node_storage();
  }
 
 
};
 
 
//========================================================================
/// Constructor for adaptive Poisson problem in deformable fish-shaped
/// domain. Pass bool to indicate if we want a constant source
/// function or the one that produces a tanh step.
//========================================================================
template<class ELEMENT>
RefineableRotatingCylinderProblem<ELEMENT>::RefineableRotatingCylinderProblem(
 const double &length, const double &height) : Length(length), Height(height),
                                               Re(0.0), Ca(0.001), 
                                               ReInvFr(0.0),
                                               Bo(0.0), Omega(1.0), 
                                               Volume(12.0),
                                               Angle(1.57)
{ 
 Global_Physical_Variables::Wall_normal.resize(2);
 Global_Physical_Variables::Wall_normal[0] = 1.0; 
 Global_Physical_Variables::Wall_normal[1] = 0.0;
 
 G.resize(2);
 G[0] = 0.0; G[1] = -1.0;
 
 /// Set the initial value of the ReInvFr = Bo/Ca
 ReInvFr = Bo/Ca;
 
 /// Build a linear solver: Use HSL's MA42 frontal solver
 //linear_solver_pt() = new HSL_MA42;
 
 //Set the constituive law
 Constitutive_law_pt = new GeneralisedHookean(&Global_Physical_Variables::Nu);
 
 
 /// Switch off full doc for frontal solver
 //static_cast<HSL_MA42*>(linear_solver_pt())->disable_doc_stats();
 
 //Allocate the timestepper (no timedependence)
 add_time_stepper_pt(new Steady<0>);
   
 // Build mesh
 Bulk_mesh_pt= 
  new RefineableCylinderAndInterfaceMesh<ELEMENT>(length,height,
                                                  Problem::time_stepper_pt());
 
 // Set error estimator
 Z2ErrorEstimator* error_estimator_pt=new Z2ErrorEstimator;
 Bulk_mesh_pt->spatial_error_estimator_pt()=error_estimator_pt;
  
 
 //Refine the problem a couple of times
 bool update_all_solid_nodes=true;
 Bulk_mesh_pt->refine_uniformly();
 Bulk_mesh_pt->node_update(update_all_solid_nodes);
 Bulk_mesh_pt->refine_uniformly();
 Bulk_mesh_pt->node_update(update_all_solid_nodes);
 //Bulk_mesh_pt->refine_uniformly();
 //refine_uniformly();
 //refine_uniformly();
  
 // Loop over all elements and unset macro element pointer
 unsigned Nelement = Bulk_mesh_pt->nelement();
 for(unsigned e=0;e<Nelement;e++)
  {
   dynamic_cast<ELEMENT*>(Bulk_mesh_pt->element_pt(e))->
    set_macro_elem_pt(0);
  }
 
 
 //The external pressure is a piece of global data
 Traded_pressure_data_pt = new Data(1);
 this->add_global_data(Traded_pressure_data_pt);
 
 // Complete the build of all elements so they are fully functional
 
 Surface_mesh_pt = new Mesh;
 Point_mesh_pt = new Mesh;
 Volume_constraint_mesh_pt = new Mesh;
 
 finish_problem_setup();
  
 this->add_sub_mesh(Bulk_mesh_pt);
 this->add_sub_mesh(Surface_mesh_pt);
 this->add_sub_mesh(Point_mesh_pt);
 this->add_sub_mesh(Volume_constraint_mesh_pt);
 
 this->build_global_mesh();
 
 //Attach the boundary conditions to the mesh
 oomph_info <<"Number of equations: " << assign_eqn_numbers() << std::endl; 
 
}
 
 
//========================================================================
/// Complete build of Poisson problem:
/// Loop over elements and setup pointers to source function
///
//========================================================================
template<class ELEMENT>
void RefineableRotatingCylinderProblem<ELEMENT>::finish_problem_setup()
{ 
 //Now sort out the free surface
 this->create_free_surface_elements();
 //Create the volume constraint elements
 this->create_volume_constraint_elements();
 
 // Set the boundary conditions for this problem: All nodes are
 // free by default -- just pin the ones that have Dirichlet conditions
 // here. 
 
 //Pin bottom and cylinder
  unsigned num_bound = Bulk_mesh_pt->nboundary();
  for(unsigned ibound=0;ibound<num_bound;ibound+=4)
   {
    unsigned num_nod= Bulk_mesh_pt->nboundary_node(ibound);
    for (unsigned inod=0;inod<num_nod;inod++)
     {
      Bulk_mesh_pt->boundary_node_pt(ibound,inod)->pin(0);
      Bulk_mesh_pt->boundary_node_pt(ibound,inod)->pin(1);
     }
   }
  
  //Pin u and v on LHS
  {
   unsigned num_nod= Bulk_mesh_pt->nboundary_node(3);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     Bulk_mesh_pt->boundary_node_pt(3,inod)->pin(0);
     //Bulk_mesh_pt->boundary_node_pt(3,inod)->pin(1);
    }
  }
  
  //Pin u and v on RHS
  {
   unsigned num_nod= Bulk_mesh_pt->nboundary_node(1);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     Bulk_mesh_pt->boundary_node_pt(1,inod)->pin(0);
     Bulk_mesh_pt->boundary_node_pt(1,inod)->pin(1);
    }
  }
 
  
  dynamic_cast<FluidInterfaceBoundingElement*>
   (Point_mesh_pt->element_pt(0))->set_contact_angle(&Angle);
 
  dynamic_cast<FluidInterfaceBoundingElement*>
   (Point_mesh_pt->element_pt(0))->ca_pt() = &Ca;
 
  
  dynamic_cast<FluidInterfaceBoundingElement*>
   (Point_mesh_pt->element_pt(0))->
   wall_unit_normal_fct_pt() = &Global_Physical_Variables::wall_unit_normal_fct;
 
  //Pin one pressure
  dynamic_cast<ELEMENT*>(Bulk_mesh_pt->element_pt(0))->fix_pressure(0,0.0);
 
  //Loop over the lower boundary and pin nodal positions in both directions
  unsigned num_nod= Bulk_mesh_pt->nboundary_node(0);
  for (unsigned inod=0;inod<num_nod;inod++)
   {
    Bulk_mesh_pt->boundary_node_pt(0,inod)->pin_position(0);
    Bulk_mesh_pt->boundary_node_pt(0,inod)->pin_position(1);
   }
  
  //Loop over the RHS side and pin in x and y
  num_nod= Bulk_mesh_pt->nboundary_node(1);
  for (unsigned inod=0;inod<num_nod;inod++)
   {
    Bulk_mesh_pt->boundary_node_pt(1,inod)->pin_position(0);
    //Bulk_mesh_pt->boundary_node_pt(1,inod)->pin_position(1);
  }
  
  
 //Loop over the LHS side and pin in x
  num_nod= Bulk_mesh_pt->nboundary_node(3);
 for (unsigned inod=0;inod<num_nod;inod++)
  {
   Bulk_mesh_pt->boundary_node_pt(3,inod)->pin_position(0);
   //Bulk_mesh_pt->boundary_node_pt(3,inod)->pin_position(1);
  }
 
 //Loop over the cylinder and pin nodal positions in both directions
 num_nod= Bulk_mesh_pt->nboundary_node(4);
 for (unsigned inod=0;inod<num_nod;inod++)
  {
   Bulk_mesh_pt->boundary_node_pt(4,inod)->pin_position(0);
   Bulk_mesh_pt->boundary_node_pt(4,inod)->pin_position(1);
  }
 
 
 //Find number of elements in mesh
 unsigned Nfluid = Bulk_mesh_pt->nelement();
 //Find the number of free surface elements
 unsigned Nfree = Surface_mesh_pt->nelement();
 
    // Loop over the elements to set up element-specific 
    // things that cannot be handled by constructor
    for(unsigned i=0;i<Nfluid;i++)
     {
      // Upcast from FiniteElement to the present element
      ELEMENT *temp_pt = dynamic_cast<ELEMENT*>(Bulk_mesh_pt->element_pt(i));
      
      //Set the source function pointer
      temp_pt->re_pt() = &Re;
      temp_pt->re_invfr_pt() = &ReInvFr;
      temp_pt->g_pt() = &G;
      
      //Assign the mesh deformation constitutive law
      temp_pt->constitutive_law_pt() = Constitutive_law_pt;
      
     }
    
    
    // Pin the redundant solid pressures (if any)
    PVDEquationsBase<2>::pin_redundant_nodal_solid_pressures(
     Bulk_mesh_pt->element_pt());
    
    //Loop over the free surface elements
    for(unsigned i=0;i<Nfree;i++)
     {
      // Upcast from FiniteElement to the present element
      ElasticLineFluidInterfaceElement<ELEMENT> *temp_pt = 
       dynamic_cast<ElasticLineFluidInterfaceElement<ELEMENT>*>
       (Surface_mesh_pt->element_pt(i));
      //Set the Capillary number
      temp_pt->ca_pt() = &Ca;
      
      //Pass the Data item that contains the external pressure
      temp_pt->set_external_pressure_data(this->global_data_pt(0));
     }
   
}
 
template<class ELEMENT>
void RefineableRotatingCylinderProblem<ELEMENT>::set_boundary_conditions()
{ 
 //Only bother to set non-zero velocity on the cylinder
 unsigned Nnode = Bulk_mesh_pt->nboundary_node(4);
 for(unsigned n=0;n<Nnode;n++)
  {
   //Get x and y
   double x = Bulk_mesh_pt->boundary_node_pt(4,n)->x(0);
   double y = Bulk_mesh_pt->boundary_node_pt(4,n)->x(1);
 
   //Now find the vector distance to the centre
   double len_x = x - Bulk_mesh_pt->domain_pt()->centre_x;
   double len_y = y - Bulk_mesh_pt->domain_pt()->centre_y;
 
   //Calculate the angle and radius
   double r = sqrt(len_x*len_x + len_y*len_y);
   double theta = atan2(len_y,len_x);
   
   //Now set the velocities
   Bulk_mesh_pt->boundary_node_pt(4,n)->set_value(0,-Omega*r*sin(theta));
   Bulk_mesh_pt->boundary_node_pt(4,n)->set_value(1, Omega*r*cos(theta));
  }
}
 
template<class ELEMENT>
void RefineableRotatingCylinderProblem<ELEMENT>::solve()
{ 
 Newton_solver_tolerance = 1.0e-8;
 //Document the solution
 std::ofstream filenamee("input.dat");
 Bulk_mesh_pt->output(filenamee,5);
 Surface_mesh_pt->output(filenamee,5);
 //Point_mesh_pt->output(filenamee,5);
 filenamee.close();
 
 //Solve the initial value problem
 newton_solve();
 
 std::ofstream filename("first.dat"); 
 Bulk_mesh_pt->output(filename,5); 
 Surface_mesh_pt->output(filename,5); 
 //Point_mesh_pt->output(filename,5);
 filename.close();
 
 //Initialise the value of the arc-length
 double ds=0.001;
 
 std::ofstream trace("trace.dat");
 
 trace << Ca << " " << ReInvFr << " "
       << Bulk_mesh_pt->boundary_node_pt(2,0)->x(1) << std::endl;
 
// bool flag=true, fflag=true;
 
 for(unsigned i=0;i<2;i++)
  {
   if(i<5)
    {
     //Decrease the contact angle
     Angle -= 0.1;
     newton_solve(2);
     //newton_solve();
    }
   else
    {
     //do an arc-length continuation step in Ca
     ds = arc_length_step_solve(&Ca,ds);
    }
 
  //  if(flag)
//     {
//      //Do an arc-length continuation step in ReInvFr
//      ds = arc_length_step_solve(&ReInvFr,ds);
//     }
//    else
//     {
//      //Reset arc-length parameters
//      if(fflag) {reset_arc_length_parameters(); fflag=false;}
//      ds = 0.001;
//      //Now do it in Ca
//      ds = arc_length_step_solve(&Ca,ds);
//     }
 
//    if(Bulk_mesh_pt->boundary_node_pt(2,0)->x(1) < 4.0)
//     {flag=false;}
 
   trace << Ca << " " << ReInvFr << " " << Angle << " "
         << Bulk_mesh_pt->boundary_node_pt(2,0)->x(1) << std::endl;   
   
   char file[100];
   snprintf(file, sizeof(file), "step%i.dat",i);
   filename.open(file);
   Bulk_mesh_pt->output(filename,5);
   Surface_mesh_pt->output(filename,5);
   //Point_mesh_pt->output(filename,5);
   filename.close();
 
   //Now reset the values of the lagrange multipliers and the xi's
   //An updated lagrangian approach
   
   //Now loop over all the nodes and set their Lagrangian coordinates
   unsigned Nnode = Bulk_mesh_pt->nnode();
   for(unsigned n=0;n<Nnode;n++)
    {
     //Cast node to an elastic node
     SolidNode* temp_pt = static_cast<SolidNode*>(Bulk_mesh_pt->node_pt(n));
     for(unsigned j=0;j<2;j++) {temp_pt->xi(j) = temp_pt->x(j);}
    }
 
   //Find the number of free surface elements
   unsigned Nfree = Surface_mesh_pt->nelement();
   //Loop over the free surface elements
   for(unsigned n=0;n<Nfree;n++)
    {
     // Upcast from FiniteElement to the present element
     ElasticLineFluidInterfaceElement<ELEMENT> *temp_pt = 
      dynamic_cast<ElasticLineFluidInterfaceElement<ELEMENT>*>
      (Surface_mesh_pt->element_pt(n));
     unsigned Nnode = temp_pt->nnode();
     //Reset the lagrange multipliers
     for(unsigned j=0;j<Nnode;j++) {temp_pt->lagrange(j) = 0.0;}
    }
  }
 
 //Document the solution
 //filename.open("output.dat");
 //Bulk_mesh_pt->output(filename,5);
 //filename.close();
 trace.close();
}
 
 
///////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////
 
 
 int main()
  {
 
   RefineableRotatingCylinderProblem
    <RefineablePseudoSolidNodeUpdateElement<RefineableQCrouzeixRaviartElement<2>,
    RefineableQPVDElementWithContinuousPressure<2> > > problem(3.0,4.0);
   
   //ofstream filename("mesh.dat");
   //problem.Bulk_mesh_pt->output(filename,5);
 
   problem.solve();
  }

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