elastic_two_layer_interface.cc

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//LIC//====================================================================
// Driver for an adaptive two-dimensional two fluid interface problem,
// where the mesh is deformed using a pseudo-solid node-update strategy
 
// Generic oomph-lib header
#include "generic.h"
 
// Navier-Stokes headers
#include "navier_stokes.h"
 
// Interface headers
#include "fluid_interface.h"
 
// Constitutive law headers
#include "constitutive.h"
 
// Solid headers
#include "solid.h"
 
// The mesh
#include "meshes/rectangular_quadmesh.h"
 
using namespace std;
 
using namespace oomph;
 
 
//==start_of_namespace====================================================
/// Namespace for physical parameters
//========================================================================
namespace Global_Physical_Variables
{
 
 /// Reynolds number
 double Re = 5.0;
 
 /// Strouhal number
 double St = 1.0;
 
 /// Womersley number (Reynolds x Strouhal)
 double ReSt = 5.0;
 
 /// Product of Reynolds number and inverse of Froude number
 double ReInvFr = 5.0;
 
 /// Ratio of viscosity in upper fluid to viscosity in lower
 /// fluid. Reynolds number etc. is based on viscosity in lower fluid.
 double Viscosity_Ratio = 0.1;
 
 /// Ratio of density in upper fluid to density in lower
 /// fluid. Reynolds number etc. is based on density in lower fluid.
 double Density_Ratio = 0.5;
 
 /// Capillary number
 double Ca = 0.01;
 
 /// Direction of gravity
 Vector<double> G(2);
 
 /// Pseudo-solid Poisson ratio
 double Nu = 0.1;
 
} // End of namespace
 
 
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
 
 
//==start_of_specific_mesh_class==========================================
/// Two layer mesh which employs a pseudo-solid node-update strategy.
/// This class is essentially a wrapper to an 
/// ElasticRefineableRectangularQuadMesh, with an additional boundary
/// to represent the interface between the two fluid layers.
//========================================================================
template <class ELEMENT>
class ElasticRefineableTwoLayerMesh :
 public virtual ElasticRefineableRectangularQuadMesh<ELEMENT>
{
 
public:
 
 /// Constructor: Pass number of elements in x-direction, number of
 /// elements in y-direction in bottom and top layer, respectively,
 /// axial length and height of top and bottom layers, a boolean
 /// flag to make the mesh periodic in the x-direction, and pointer 
 /// to timestepper (defaults to Steady timestepper)
 ElasticRefineableTwoLayerMesh(const unsigned &nx, 
                               const unsigned &ny1,
                               const unsigned &ny2, 
                               const double &lx,
                               const double &h1,
                               const double &h2,
                               const bool& periodic_in_x,
                               TimeStepper* time_stepper_pt=
                               &Mesh::Default_TimeStepper)
  : RectangularQuadMesh<ELEMENT>(nx,ny1+ny2,lx,h1+h2,
                                 periodic_in_x,time_stepper_pt),
    ElasticRectangularQuadMesh<ELEMENT>(nx,ny1+ny2,lx,h1+h2,
                                        periodic_in_x,time_stepper_pt),
    ElasticRefineableRectangularQuadMesh<ELEMENT>(nx,ny1+ny2,lx,h1+h2,
                                                  periodic_in_x,
                                                  time_stepper_pt)
  {
   // ----------------------------------------------------
   // Convert all nodes on the interface to boundary nodes
   // ----------------------------------------------------
 
   // Set the number of boundaries to 5
   this->set_nboundary(5);
 
   // Loop over horizontal elements
   for(unsigned e=0;e<nx;e++)
    {
     // Get pointer to element in lower fluid adjacent to interface
     FiniteElement* el_pt = this->finite_element_pt(nx*(ny1-1)+e);
 
     // Determine number of nodes in this element
     const unsigned n_node = el_pt->nnode();
 
     // The last three nodes in this element are those on the interface.
     // Loop over these nodes and convert them to boundary nodes.
     for(unsigned n=0;n<3;n++)
      {
       Node* nod_pt = el_pt->node_pt(n_node-3+n);
       this->convert_to_boundary_node(nod_pt);
       this->add_boundary_node(4,nod_pt);
      }
    } // End of loop over horizontal elements
 
   // Set up the boundary element information
   this->setup_boundary_element_info();
  }
 
}; // End of specific mesh class
 
 
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
 
 
//==start_of_problem_class================================================
/// Two fluid interface problem in a rectangular domain which is
/// periodic in the x direction
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
class InterfaceProblem : public Problem
{
 
public:
 
 /// Constructor
 InterfaceProblem();
 
 /// Destructor (empty)
 ~InterfaceProblem() {}
 
 /// Set initial conditions
 void set_initial_condition();
 
 /// Set boundary conditions
 void set_boundary_conditions();
 
 /// Document the solution
 void doc_solution(DocInfo &doc_info);
 
 /// Do unsteady run up to maximum time t_max with given timestep dt
 void unsteady_run(const double &t_max, const double &dt); 
 
private:
 
 /// No actions required before solve step
 void actions_before_newton_solve() {}
 
 /// No actions required after solve step
 void actions_after_newton_solve() {}
 
 /// Actions before the timestep: For maximum stability, reset
 /// the current nodal positions to be the "stress-free" ones.
 void actions_before_implicit_timestep()
  {
   Bulk_mesh_pt->set_lagrangian_nodal_coordinates();
  }
 
 /// Strip off the interface elements before adapting the bulk mesh
 void actions_before_adapt();
 
 /// Rebuild the mesh of interface elements after adapting the bulk mesh
 void actions_after_adapt();
 
 /// Create the 1d interface elements
 void create_interface_elements();
 
 /// Delete the 1d interface elements
 void delete_interface_elements();
 
 /// Deform the mesh/free surface to a prescribed function
 void deform_free_surface(const double &epsilon, const unsigned &n_periods);
 
 /// Fix pressure in element e at pressure dof pdof and set to pvalue
 void fix_pressure(const unsigned &e,
                   const unsigned &pdof, 
                   const double &pvalue)
  {
   // Fix the pressure at that element
   dynamic_cast<ELEMENT*>(mesh_pt()->element_pt(e))->
                          fix_pressure(pdof,pvalue);
  }
 
 /// Pointer to the (specific) "bulk" mesh
 ElasticRefineableTwoLayerMesh<ELEMENT>* Bulk_mesh_pt;
 
 /// Pointer to the "surface" mesh
 Mesh* Surface_mesh_pt;
 
 // Pointer to the constitutive law used to determine the mesh deformation
 ConstitutiveLaw* Constitutive_law_pt;
 
 /// Width of domain
 double Lx;
 
 /// Trace file
 ofstream Trace_file;
 
}; // End of problem class
 
 
 
//==start_of_constructor==================================================
/// Constructor for two fluid interface problem
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
InterfaceProblem<ELEMENT,TIMESTEPPER>::
InterfaceProblem()
{
 // Allocate the timestepper (this constructs the time object as well)
 add_time_stepper_pt(new TIMESTEPPER);
 
 // Define number of elements in x direction
 const unsigned n_x = 3;
   
 // Define number of elements in y direction in lower fluid (fluid 1)
 const unsigned n_y1 = 3;
 
 // Define number of elements in y direction in upper fluid (fluid 2)
 const unsigned n_y2 = 3;
 
 // Define width of domain and store as class member data
 const double l_x = 1.0;
 this->Lx = l_x;
 
 // Define height of lower fluid layer
 const double h1 = 1.0;
 
 // Define height of upper fluid layer
 const double h2 = 1.0;
 
 // Build and assign the "bulk" mesh (the "true" boolean flag tells
 // the mesh constructor that the domain is periodic in x)
 Bulk_mesh_pt = new ElasticRefineableTwoLayerMesh<ELEMENT>
  (n_x,n_y1,n_y2,l_x,h1,h2,true,time_stepper_pt());
 
 // Create and set the error estimator for spatial adaptivity
 Bulk_mesh_pt->spatial_error_estimator_pt() = new Z2ErrorEstimator;
 
 // Set the maximum refinement level for the mesh to 4
 Bulk_mesh_pt->max_refinement_level() = 4;
 
 // Create the "surface" mesh that will contain only the interface
 // elements. The constructor just creates the mesh without giving
 // it any elements, nodes, etc.
 Surface_mesh_pt = new Mesh;
 
 // Create interface elements at the boundary between the two fluids,
 // and add them to the surface mesh
 create_interface_elements();
 
 // Add the two sub meshes to the problem
 add_sub_mesh(Bulk_mesh_pt);
 add_sub_mesh(Surface_mesh_pt);
 
 // Combine all sub-meshes into a single mesh
 build_global_mesh();
 
 // --------------------------------------------
 // Set the boundary conditions for this problem
 // --------------------------------------------
 
 // All values are free by default -- just pin the ones that have
 // Dirichlet conditions here
 
 // Determine number of mesh boundaries
 const unsigned n_boundary = Bulk_mesh_pt->nboundary();
 
 // Loop over mesh boundaries
 for(unsigned b=0;b<n_boundary;b++)
  {
   // Determine number of nodes on boundary b
   const unsigned n_node = Bulk_mesh_pt->nboundary_node(b);
 
   // Loop over nodes on boundary b
   for(unsigned n=0;n<n_node;n++)
    {
     // Fluid boundary conditions:
     // --------------------------
 
     // Pin x-component of velocity (no slip/penetration)
     // on all boundaries other than the interface (b=4)
     if(b!=4) { Bulk_mesh_pt->boundary_node_pt(b,n)->pin(0); }
 
     // Pin y-component of velocity on both solid boundaries (no penetration)
     if(b==0 || b==2) { Bulk_mesh_pt->boundary_node_pt(b,n)->pin(1); }
 
     // Solid boundary conditions:
     // --------------------------
 
     // Pin vertical mesh position on both solid boundaries (no penetration)
     if(b==0 || b==2) { Bulk_mesh_pt->boundary_node_pt(b,n)->pin_position(1); }
 
    } // End of loop over nodes on boundary b
  } // End of loop over mesh boundaries
 
 // Pin horizontal position of all nodes
 const unsigned n_node = Bulk_mesh_pt->nnode();
 for(unsigned n=0;n<n_node;n++) { Bulk_mesh_pt->node_pt(n)->pin_position(0); }
 
 // Define a constitutive law for the solid equations: generalised Hookean
 Constitutive_law_pt = new GeneralisedHookean(&Global_Physical_Variables::Nu);
 
 // ----------------------------------------------------------------
 // Complete the problem setup to make the elements fully functional
 // ----------------------------------------------------------------
 
 // Compute number of bulk elements in lower/upper fluids
 const unsigned n_lower = n_x*n_y1;
 const unsigned n_upper = n_x*n_y2;
 
 // Loop over bulk elements in lower fluid
 for(unsigned e=0;e<n_lower;e++)
  {
   // Upcast from GeneralisedElement to the present element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(Bulk_mesh_pt->element_pt(e));
 
   // Set the Reynolds number
   el_pt->re_pt() = &Global_Physical_Variables::Re;
 
   // Set the Womersley number
   el_pt->re_st_pt() = &Global_Physical_Variables::ReSt;
 
   // Set the product of the Reynolds number and the inverse of the
   // Froude number
   el_pt->re_invfr_pt() = &Global_Physical_Variables::ReInvFr;
 
   // Set the direction of gravity
   el_pt->g_pt() = &Global_Physical_Variables::G;
 
   // Set the constitutive law
   el_pt->constitutive_law_pt() = Constitutive_law_pt;
 
  } // End of loop over bulk elements in lower fluid
 
 // Loop over bulk elements in upper fluid
 for(unsigned e=n_lower;e<(n_lower+n_upper);e++)
  {
   // Upcast from GeneralisedElement to the present element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(Bulk_mesh_pt->element_pt(e));
 
   // Set the Reynolds number
   el_pt->re_pt() = &Global_Physical_Variables::Re;
 
   // Set the Womersley number
   el_pt->re_st_pt() = &Global_Physical_Variables::ReSt;
 
   // Set the product of the Reynolds number and the inverse of the
   // Froude number
   el_pt->re_invfr_pt() = &Global_Physical_Variables::ReInvFr;
 
   // Set the viscosity ratio
   el_pt->viscosity_ratio_pt() = &Global_Physical_Variables::Viscosity_Ratio;
 
   // Set the density ratio
   el_pt->density_ratio_pt() = &Global_Physical_Variables::Density_Ratio;
 
   // Set the direction of gravity
   el_pt->g_pt() = &Global_Physical_Variables::G;
 
   // Set the constitutive law
   el_pt->constitutive_law_pt() = Constitutive_law_pt;
 
  } // End of loop over bulk elements in upper fluid
 
 // Set the pressure in the first element at 'node' 0 to 0.0
 fix_pressure(0,0,0.0);
 
 // Apply the boundary conditions
 set_boundary_conditions();
 
 // Set up equation numbering scheme
 cout << "Number of equations: " << assign_eqn_numbers() << std::endl;
 
} // End of constructor
 
 
 
//==start_of_set_initial_condition========================================
/// Set initial conditions: Set all nodal velocities to zero and
/// initialise the previous velocities and nodal positions to correspond
/// to an impulsive start
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::set_initial_condition()
{
 // Determine number of nodes in mesh
 const unsigned n_node = mesh_pt()->nnode();
 
 // Loop over all nodes in mesh
 for(unsigned n=0;n<n_node;n++)
  {
   // Loop over the two velocity components
   for(unsigned i=0;i<2;i++)
    {
     // Set velocity component i of node n to zero
     mesh_pt()->node_pt(n)->set_value(i,0.0);
    }
  }
 
 // Initialise the previous velocity values and nodal positions
 // for timestepping corresponding to an impulsive start
 assign_initial_values_impulsive();
 
} // End of set_initial_condition
 
 
 
//==start_of_set_boundary_conditions======================================
/// Set boundary conditions: Set both velocity components
/// to zero on the top and bottom (solid) walls and the horizontal
/// component only to zero on the side (periodic) boundaries
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::set_boundary_conditions()
{
 // Determine number of mesh boundaries
 const unsigned n_boundary = Bulk_mesh_pt->nboundary();
 
 // Loop over mesh boundaries
 for(unsigned b=0;b<n_boundary;b++)
  {
   // Determine number of nodes on boundary b
   const unsigned n_node = Bulk_mesh_pt->nboundary_node(b);
   
   // Loop over nodes on boundary b
   for(unsigned n=0;n<n_node;n++)
    {
     // Set x-component of the velocity to zero on all boundaries
     // other than the interface (b=4)
     if(b!=4) { Bulk_mesh_pt->boundary_node_pt(b,n)->set_value(0,0.0); }
     
     // Set y-component of the velocity to zero on both solid
     // boundaries (top and bottom)
     if(b==0 || b==2)
      {
       Bulk_mesh_pt->boundary_node_pt(b,n)->set_value(1,0.0);
      }
    } // End of loop over nodes on boundary b
  } // End of loop over mesh boundaries
 
} // End of set_boundary_conditions
 
 
 
//==start_of_actions_before_adapt=========================================
/// Strip off the interface elements before adapting the bulk mesh
//========================================================================
template<class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::actions_before_adapt()
{
 // Delete the interface elements and wipe the surface mesh
 delete_interface_elements();
 
 // Rebuild the Problem's global mesh from its various sub-meshes
 rebuild_global_mesh();
 
} // End of actions_before_adapt
 
 
 
//==start_of_actions_after_adapt==========================================
/// Rebuild the mesh of interface elements after adapting the bulk mesh
//========================================================================
template<class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::actions_after_adapt()
{
 // Create the interface elements
 this->create_interface_elements();
 
 // Rebuild the Problem's global mesh from its various sub-meshes
 rebuild_global_mesh();
 
 // Pin horizontal displacement of all nodes
 const unsigned n_node = Bulk_mesh_pt->nnode();
 for(unsigned n=0;n<n_node;n++) { Bulk_mesh_pt->node_pt(n)->pin_position(0); }
 
 // Unpin all fluid pressure dofs
 RefineableNavierStokesEquations<2>::
  unpin_all_pressure_dofs(Bulk_mesh_pt->element_pt());
 
 // Pin redudant fluid pressure dofs
 RefineableNavierStokesEquations<2>::
  pin_redundant_nodal_pressures(Bulk_mesh_pt->element_pt());
 
 // Now set the pressure in the first element at 'node' 0 to 0.0
 fix_pressure(0,0,0.0);
 
 // Pin the redundant solid pressures (if any)
 PVDEquationsBase<2>::pin_redundant_nodal_solid_pressures(
  Bulk_mesh_pt->element_pt());
 
 // Reset the boundary conditions
 set_boundary_conditions();
 
} // End of actions_after_adapt
 
 
 
//==start_of_create_interface_elements====================================
/// Create interface elements between the two fluids in the mesh
/// pointed to by Bulk_mesh_pt and add the elements to the Mesh object
/// pointed to by Surface_mesh_pt.
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::create_interface_elements()
{
 // Determine number of bulk elements adjacent to interface (boundary 4)
 const unsigned n_element = this->Bulk_mesh_pt->nboundary_element(4);
 
 // Loop over those elements adjacent to the interface
 for(unsigned e=0;e<n_element;e++)
  {
   // Get pointer to the bulk element that is adjacent to the interface
   ELEMENT* bulk_elem_pt = dynamic_cast<ELEMENT*>(
    this->Bulk_mesh_pt->boundary_element_pt(4,e));
 
   // We only want to attach interface elements to the bulk elements
   // which are BELOW the interface, and so we filter out those above by
   // referring to the viscosity_ratio_pt
   if(bulk_elem_pt->viscosity_ratio_pt()
      !=&Global_Physical_Variables::Viscosity_Ratio)
    {
     // Find index of the face of element e that corresponds to the interface
     const int face_index = this->Bulk_mesh_pt->face_index_at_boundary(4,e);
     
     // Create the interface element
     FiniteElement* interface_element_element_pt =
      new ElasticLineFluidInterfaceElement<ELEMENT>(bulk_elem_pt,face_index);
 
     // Add the interface element to the surface mesh
     this->Surface_mesh_pt->add_element_pt(interface_element_element_pt);
    }
  }
 
 // --------------------------------------------------------
 // Complete the setup to make the elements fully functional
 // --------------------------------------------------------
 
 // Determine number of 1D interface elements in mesh
 const unsigned n_interface_element = this->Surface_mesh_pt->nelement();
 
 // Loop over the interface elements
 for(unsigned e=0;e<n_interface_element;e++)
  {
   // Upcast from GeneralisedElement to the present element
   ElasticLineFluidInterfaceElement<ELEMENT>* el_pt = 
    dynamic_cast<ElasticLineFluidInterfaceElement<ELEMENT>*>
    (Surface_mesh_pt->element_pt(e));
 
   // Set the Strouhal number
   el_pt->st_pt() = &Global_Physical_Variables::St;
 
   // Set the Capillary number
   el_pt->ca_pt() = &Global_Physical_Variables::Ca;
 
  } // End of loop over interface elements
 
} // End of create_interface_elements
 
 
 
//==start_of_delete_interface_elements====================================
/// Delete the interface elements and wipe the surface mesh
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::delete_interface_elements()
{
 // Determine number of interface elements
 const unsigned n_interface_element = Surface_mesh_pt->nelement();
 
 // Loop over interface elements and delete
 for(unsigned e=0;e<n_interface_element;e++)
  {
   delete Surface_mesh_pt->element_pt(e);
  }
 
 // Wipe the mesh
 Surface_mesh_pt->flush_element_and_node_storage();
 
} // End of delete_interface_elements
 
 
 
//==start_of_deform_free_surface==========================================
/// Deform the mesh/free surface to a prescribed function
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::
deform_free_surface(const double &epsilon,const unsigned &n_periods)
{
 // Determine number of nodes in the "bulk" mesh
 const unsigned n_node = Bulk_mesh_pt->nnode();
 
 // Loop over all nodes in mesh
 for(unsigned n=0;n<n_node;n++)
  {
   // Determine eulerian position of node
   const double current_x_pos = Bulk_mesh_pt->node_pt(n)->x(0);
   const double current_y_pos = Bulk_mesh_pt->node_pt(n)->x(1);
   
   // Determine new vertical position of node
   const double new_y_pos = current_y_pos
    + (1.0-fabs(1.0-current_y_pos))*epsilon
    *(cos(2.0*n_periods*MathematicalConstants::Pi*current_x_pos/Lx));
   
   // Set new position
   Bulk_mesh_pt->node_pt(n)->x(1) = new_y_pos;
  }
} // End of deform_free_surface
 
 
 
//==start_of_doc_solution=================================================
/// Doc the solution
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::doc_solution(DocInfo &doc_info)
{ 
 
 // Output the time
 cout << "Time is now " << time_pt()->time() << std::endl;
 
 // Upcast from GeneralisedElement to the present element
 ElasticLineFluidInterfaceElement<ELEMENT>* el_pt = 
  dynamic_cast<ElasticLineFluidInterfaceElement<ELEMENT>*>
  (Surface_mesh_pt->element_pt(0));
 
 // Document time and vertical position of left hand side of interface
 // in trace file
 Trace_file << time_pt()->time() << " "
            << el_pt->node_pt(0)->x(1) << std::endl;
 
 ofstream some_file;
 char filename[100];
 
 // Set number of plot points (in each coordinate direction)
 const unsigned npts = 5;
 
 // Open solution output file
 sprintf(filename,"%s/soln%i.dat",
         doc_info.directory().c_str(),doc_info.number());
 some_file.open(filename);
 
 // Output solution to file
 Bulk_mesh_pt->output(some_file,npts);
 
 // Close solution output file
 some_file.close();
 
 // Open interface solution output file
 sprintf(filename,"%s/interface_soln%i.dat",
         doc_info.directory().c_str(),doc_info.number());
 some_file.open(filename);
 
 // Output solution to file
 Surface_mesh_pt->output(some_file,npts);
 
 // Close solution output file
 some_file.close();
 
} // End of doc_solution
 
 
 
//==start_of_unsteady_run=================================================
/// Perform run up to specified time t_max with given timestep dt
//========================================================================
template <class ELEMENT, class TIMESTEPPER>
void InterfaceProblem<ELEMENT,TIMESTEPPER>::
unsteady_run(const double &t_max, const double &dt)
{
 
 // Set value of epsilon
 const double epsilon = 0.1;
 
 // Set number of periods for cosine term
 const unsigned n_periods = 1;
 
 // Deform the mesh/free surface
 deform_free_surface(epsilon,n_periods);
 
 // Initialise DocInfo object
 DocInfo doc_info;
 
 // Set output directory
 doc_info.set_directory("RESLT");
 
 // Initialise counter for solutions
 doc_info.number()=0;
 
 // Open trace file
 char filename[100];   
 sprintf(filename,"%s/trace.dat",doc_info.directory().c_str());
 Trace_file.open(filename);
 
 // Initialise trace file
 Trace_file << "time, interface height" << std::endl;
 
 // Initialise timestep
 initialise_dt(dt);
 
 // Set initial condition
 set_initial_condition();
 
 // Maximum number of spatial adaptations per timestep
 unsigned max_adapt = 2;
 
 // Call refine_uniformly twice
 for(unsigned i=0;i<2;i++) { refine_uniformly(); }
 
 // Doc initial solution
 doc_solution(doc_info);
 
 // Increment counter for solutions
 doc_info.number()++;
 
 // Determine number of timesteps
 const unsigned n_timestep = unsigned(t_max/dt);
 
 // Are we on the first timestep? At this point, yes!
 bool first_timestep = true;
 
 // Timestepping loop
 for(unsigned t=1;t<=n_timestep;t++)
  {
   // Output current timestep to screen
   cout << "\nTimestep " << t << " of " << n_timestep << std::endl;
   
   // Take one fixed timestep with spatial adaptivity
   unsteady_newton_solve(dt,max_adapt,first_timestep);
 
   // No longer on first timestep, so set first_timestep flag to false
   first_timestep = false; 
 
   // Reset maximum number of adaptations for all future timesteps
   max_adapt = 1;
 
   // Doc solution
   doc_solution(doc_info);
 
   // Increment counter for solutions 
   doc_info.number()++;
 
  } // End of timestepping loop
 
} // End of unsteady_run
 
 
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////////
 
 
//==start_of_main=========================================================
/// Driver code for two-dimensional two fluid interface problem
//========================================================================
int main(int argc, char* argv[]) 
{
 // Store command line arguments
 CommandLineArgs::setup(argc,argv);
 
 // -----------------------------------------------------------------
 // Define possible command line arguments and parse the ones that
 // were actually specified
 // -----------------------------------------------------------------
 
 // Are we performing a validation run?
 CommandLineArgs::specify_command_line_flag("--validation");
 
 // Parse command line
 CommandLineArgs::parse_and_assign();
 
 // Doc what has actually been specified on the command line
 CommandLineArgs::doc_specified_flags();
 
 // Check that definition of Womersley number is consistent with those
 // of the Reynolds and Strouhal numbers
 if(Global_Physical_Variables::ReSt !=
    Global_Physical_Variables::Re*Global_Physical_Variables::St)
  {
   std::ostringstream error_stream;
   error_stream << "Definition of Global_Physical_Variables::ReSt is "
                << "inconsistant with those\n"
                << "of Global_Physical_Variables::Re and "
                << "Global_Physical_Variables::St." << std::endl;
   throw OomphLibError(error_stream.str(),
                       OOMPH_CURRENT_FUNCTION,OOMPH_EXCEPTION_LOCATION);
  }
 
 /// Maximum time
 double t_max = 0.6;
 
 /// Duration of timestep
 const double dt = 0.0025;
 
 // If we are doing validation run, use smaller number of timesteps
 if(CommandLineArgs::command_line_flag_has_been_set("--validation"))
  {
   t_max = 0.005;
  }
 
 // Set direction of gravity (vertically downwards)
 Global_Physical_Variables::G[0] = 0.0;
 Global_Physical_Variables::G[1] = -1.0;
 
 // Set up the elastic test problem with QCrouzeixRaviartElements,
 // using the BDF<2> timestepper
 InterfaceProblem<RefineablePseudoSolidNodeUpdateElement<
 RefineableQCrouzeixRaviartElement<2>,RefineableQPVDElement<2,3> >, BDF<2> >
  problem;
 
 // Run the unsteady simulation
 problem.unsteady_run(t_max,dt);
 
} // End of main