A simple fluid-structure interaction problem: Finite Reynolds number flow, driven by an oscillating ring.

This is a very simple fluid-structure interaction problem: We study the finite-Reynolds number internal flow generated by an oscillating ring. The wall motion only has a single degree of freedom: The ring's average radius, which needs to be adjusted to conserve mass. [This is a warm-up problem for the full fluid structure interaction problem discussed in the next example]. We compare the predictions for the flow field against asymptotic results.

The nodal positions in the fluid domain is updated by MacroElements.

//LIC// ====================================================================
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//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
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//LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
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//LIC//====================================================================
// Driver for 2D Navier Stokes flow, driven by oscillating ring
// with pseudo-elasticity: The mean radius of ring is adjusted to
// allow conservation of volume (area).
 
// Oomph-lib includes
#include "generic.h"
#include "navier_stokes.h"
 
//Need to instantiate templated mesh
#include "meshes/quarter_circle_sector_mesh.h"
 
//Include namespace containing Sarah's asymptotics 
#include "osc_ring_sarah_asymptotics.h"
 
 
using namespace std;
 
using namespace oomph;
 
using namespace MathematicalConstants;
 
  
 
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
  
 
//==================================================
/// Namespace for physical parameters
//==================================================
namespace Global_Physical_Variables
{
 
 /// Reynolds number
 double Re=100.0;  // ADJUST_PRIORITY 
 
 /// Reynolds x Strouhal number
 double ReSt=100.0; // ADJUST_PRIORITY 
 
}
 
 
 
 
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
 
 
//====================================================================
/// Driver for oscillating ring problem: Wall performs oscillations
/// that resemble eigenmodes of freely oscillating ring and drives
/// viscous fluid flow. Mean radius of wall is adjustable and
/// responds to a pressure value in the fluid to allow for
/// mass conservation. 
//====================================================================
template<class ELEMENT>
class OscRingNStProblem : public Problem
{
 
public:
 
 /// Constructor: Pass timestep and function pointer to the 
 /// solution that provides the initial conditions for the fluid
 OscRingNStProblem(const double& dt,
                   FiniteElement::UnsteadyExactSolutionFctPt IC_fct_pt);
 
 /// Destructor (empty)
 ~OscRingNStProblem(){}
 
 /// Get pointer to wall as geometric object
 GeomObject* wall_pt()
  {
   return Wall_pt;
  }
 
 /// Update after solve (empty)
 void actions_after_newton_solve(){}
 
 /// Update the problem specs before solve (empty)
 void actions_before_newton_solve(){}
 
 /// Update the problem specs before checking Newton
 /// convergence: Update the fluid mesh and re-set velocity
 ///  boundary conditions -- no slip velocity on the wall means
 /// that the velocity on the wall is dependent.
 void actions_before_newton_convergence_check() 
 {
  // Update the fluid mesh -- auxiliary update function for algebraic
  // nodes automatically updates no slip condition.
  fluid_mesh_pt()->node_update(); 
 }
 
 
 /// Update the problem specs after adaptation:
 /// Set auxiliary update function that applies no slip on all 
 /// boundary nodes and choose fluid pressure dof that drives
 /// the wall deformation
 void actions_after_adapt() 
  {
   // Ring boundary: No slip; this also implies that the velocity needs
   // to be updated in response to wall motion. This needs to be reset
   // every time the mesh is changed -- there's no mechanism by which
   // auxiliary update functions are copied to newly created nodes.
   // (that's because unlike boundary conditions, they don't 
   // occur exclusively at boundaries)
   unsigned ibound=1;
   {
    unsigned num_nod= fluid_mesh_pt()->nboundary_node(ibound);
    for (unsigned inod=0;inod<num_nod;inod++)
     {
      fluid_mesh_pt()->boundary_node_pt(ibound,inod)->
       set_auxiliary_node_update_fct_pt(
        FSI_functions::apply_no_slip_on_moving_wall); 
     }
   }
   
   // Set the reference pressure as the constant pressure in element 0
   dynamic_cast<PseudoBucklingRingElement*>(Wall_pt)
    ->set_reference_pressure_pt(fluid_mesh_pt()->element_pt(0)
                                ->internal_data_pt(0));
  }
   
 /// Run the time integration for ntsteps steps
 void unsteady_run(const unsigned &ntsteps, const bool& restarted,
                   DocInfo& doc_info);
 
 /// Set initial condition (incl previous timesteps) according
 /// to specified function. 
 void set_initial_condition();
 
 /// Doc the solution
 void doc_solution(DocInfo& doc_info);
 
 /// Access function for the fluid mesh
 MacroElementNodeUpdateRefineableQuarterCircleSectorMesh<ELEMENT>* fluid_mesh_pt()
  {
   return Fluid_mesh_pt; 
  }
 
 /// Dump problem data.
 void dump_it(ofstream& dump_file, DocInfo doc_info);
 
 /// Read problem data.
 void restart(ifstream& restart_file);
 
private:
 
 /// Write header for trace file
 void write_trace_file_header();
 
 /// Function pointer to set the intial condition
 FiniteElement::UnsteadyExactSolutionFctPt IC_Fct_pt;
 
 /// Pointer to wall
 GeomObject* Wall_pt;
 
 /// Pointer to fluid mesh
 MacroElementNodeUpdateRefineableQuarterCircleSectorMesh<ELEMENT>* Fluid_mesh_pt;
 
 /// Pointer to wall mesh (contains only a single GeneralisedElement)
 Mesh* Wall_mesh_pt;
 
 /// Trace file
 ofstream Trace_file;
 
 /// Pointer to node on coarsest mesh on which velocity is traced
 Node* Veloc_trace_node_pt;
 
 /// Pointer to node in symmetry plane on coarsest mesh at 
 /// which velocity is traced
 Node* Sarah_veloc_trace_node_pt;
 
};
 
 
//========================================================================
/// Constructor: Pass (constant) timestep and function pointer to the solution 
/// that provides the initial conditions for the fluid.
//========================================================================
template<class ELEMENT>
OscRingNStProblem<ELEMENT>::OscRingNStProblem(const double& dt, 
FiniteElement::UnsteadyExactSolutionFctPt IC_fct_pt) : IC_Fct_pt(IC_fct_pt)
{ 
 
 // Period of oscillation
 double T=1.0;
 
 //Allocate the timestepper
 add_time_stepper_pt(new BDF<4>);
 
 // Initialise timestep -- also sets the weights for all timesteppers
 // in the problem.
 initialise_dt(dt);
 
 // Parameters for pseudo-buckling ring 
 double eps_buckl=0.1;  // ADJUST_PRIORITY 
 double ampl_ratio=-0.5;  // ADJUST_PRIORITY 
 unsigned n_buckl=2; // ADJUST_PRIORITY 
 double r_0=1.0;
 
 // Build wall geometric element
 Wall_pt=new PseudoBucklingRingElement(eps_buckl,ampl_ratio,n_buckl,r_0,T,
                                       time_stepper_pt());
 
 // Fluid mesh is suspended from wall between these two Lagrangian
 // coordinates:
 double xi_lo=0.0;
 double xi_hi=2.0*atan(1.0);
 
 // Fractional position of dividing line for two outer blocks in fluid mesh
 double fract_mid=0.5;
 
 // Build fluid mesh
 Fluid_mesh_pt=new MacroElementNodeUpdateRefineableQuarterCircleSectorMesh<ELEMENT>(
  Wall_pt,xi_lo,fract_mid,xi_hi,time_stepper_pt());
 
 // Set error estimator
 Z2ErrorEstimator* error_estimator_pt=new Z2ErrorEstimator;
 Fluid_mesh_pt->spatial_error_estimator_pt()=error_estimator_pt;
  
 // Fluid mesh is first sub-mesh
 add_sub_mesh(Fluid_mesh_pt);
 
 // Build wall mesh 
 Wall_mesh_pt=new Mesh;
 
 // Wall mesh is completely empty: Add Wall element in its GeneralisedElement
 // incarnation
 Wall_mesh_pt->add_element_pt(dynamic_cast<GeneralisedElement*>(Wall_pt));
 
 // Wall mesh is second sub-mesh
 add_sub_mesh(Wall_mesh_pt);
 
 // Combine all submeshes into a single Mesh
 build_global_mesh();
 
 // Extract pointer to node at center of mesh (this node exists
 // at all refinement levels and can be used to doc continuous timetrace
 // of velocities)
 unsigned nnode=fluid_mesh_pt()->finite_element_pt(0)->nnode();
 Veloc_trace_node_pt=fluid_mesh_pt()->finite_element_pt(0)->node_pt(nnode-1);
 
 //  Extract pointer to node in symmetry plane (this node exists
 // at all refinement levels and can be used to doc continuous timetrace
 // of velocities)
 unsigned nnode_1d=dynamic_cast<ELEMENT*>(
  fluid_mesh_pt()->finite_element_pt(0))->nnode_1d();
 Sarah_veloc_trace_node_pt=fluid_mesh_pt()->
  finite_element_pt(0)->node_pt(nnode_1d-1);
 
 // The "pseudo-elastic" wall element is "loaded" by a pressure.
 // Use the "constant" pressure component in Crouzeix Raviart
 // fluid element as that pressure. 
 dynamic_cast<PseudoBucklingRingElement*>(Wall_pt)
  ->set_reference_pressure_pt(fluid_mesh_pt()->element_pt(0)
                              ->internal_data_pt(0));
 
 
 // Set the boundary conditions for this problem:
 //----------------------------------------------
 
 // All nodes are free by default -- just pin the ones that have 
 // Dirichlet conditions here. 
 
 // Bottom boundary: 
 unsigned ibound=0;
  {
   unsigned num_nod= fluid_mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Pin vertical velocity
     {
       fluid_mesh_pt()->boundary_node_pt(ibound,inod)->pin(1); 
     }
    }
  }
  
 // Ring boundary: No slip; this also implies that the velocity needs
 // to be updated in response to wall motion
 ibound=1;
  {
   unsigned num_nod=fluid_mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Which node are we dealing with?
     Node* node_pt=fluid_mesh_pt()->boundary_node_pt(ibound,inod);
 
     // Set auxiliary update function pointer to apply no-slip condition
     // to velocities whenever nodal position is updated
     node_pt->set_auxiliary_node_update_fct_pt(
      FSI_functions::apply_no_slip_on_moving_wall);
     
     // Pin both velocities
     for (unsigned i=0;i<2;i++)
      {
       node_pt->pin(i); 
      }
    }
  }
                        
 // Left boundary:
 ibound=2;
  {
   unsigned num_nod=fluid_mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Pin horizontal velocity
      {
       fluid_mesh_pt()->boundary_node_pt(ibound,inod)->pin(0);
      }
    }
  }
 
 
 // Complete the build of all elements so they are fully functional
 //----------------------------------------------------------------
 
 //Find number of elements in mesh
 unsigned n_element = fluid_mesh_pt()->nelement();
 
 // Loop over the elements to set up element-specific 
 // things that cannot be handled by constructor
 for(unsigned i=0;i<n_element;i++)
  {
   // Upcast from FiniteElement to the present element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(fluid_mesh_pt()->element_pt(i));
 
   //Set the Reynolds number, etc
   el_pt->re_pt() = &Global_Physical_Variables::Re;
   el_pt->re_st_pt() = &Global_Physical_Variables::ReSt;
 
  }
 
 //Attach the boundary conditions to the mesh
 cout <<"Number of equations: " << assign_eqn_numbers() << std::endl; 
 
 
 // Set parameters for Sarah's asymptotic solution
 //-----------------------------------------------
 
 // Amplitude of the oscillation
 SarahBL::epsilon=static_cast<PseudoBucklingRingElement*>(Wall_pt)->
  eps_buckl();
 
 // Womersley number is the same as square root of Reynolds number
 SarahBL::alpha=sqrt(Global_Physical_Variables::Re);
 
 // Amplitude ratio
 SarahBL::A=static_cast<PseudoBucklingRingElement*>(Wall_pt)->ampl_ratio();
 
 // Buckling wavenumber
 SarahBL::N=static_cast<PseudoBucklingRingElement*>(Wall_pt)->n_buckl_float();
 
 // Frequency of oscillation (period is one)
 SarahBL::Omega=2.0*MathematicalConstants::Pi;
 
}
 
 
 
 
 
//========================================================================
/// Set initial condition: Assign previous and current values
/// of the velocity from the velocity field specified via
/// the function pointer.
///
/// Values are assigned so that the velocities and accelerations
/// are correct for current time.
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::set_initial_condition()
{ 
 
 // Elastic wall: We're starting from a given initial state in which
 // the wall is undeformed. If set_initial_condition() is called again
 // after mesh refinement for first timestep, this needs to be reset.
 dynamic_cast<PseudoBucklingRingElement*>(Wall_pt)->set_R_0(1.0); 
 
 // Backup time in global timestepper
 double backed_up_time=time_pt()->time();
         
 // Past history for velocities needs to be established for t=time0-deltat, ...
 // Then provide current values (at t=time0) which will also form
 // the initial guess for first solve at t=time0+deltat
 
 // Vector of exact solution values (includes pressure)
 Vector<double> soln(3);
 Vector<double> x(2);
 
 //Find number of nodes in mesh
 unsigned num_nod = fluid_mesh_pt()->nnode();
 
 // Get continuous times at previous timesteps
 int nprev_steps=time_stepper_pt()->nprev_values();
 Vector<double> prev_time(nprev_steps+1);
 for (int itime=nprev_steps;itime>=0;itime--)
  {
   prev_time[itime]=time_pt()->time(unsigned(itime));
  }
 
 // Loop over current & previous timesteps (in outer loop because
 // the mesh might also move)
 for (int itime=nprev_steps;itime>=0;itime--)
  {
   double time=prev_time[itime];
   
   // Set global time (because this is how the geometric object refers 
   // to continous time 
   time_pt()->time()=time;
   
   cout << "setting IC at time =" << time << std::endl;
   
   // Update the fluid mesh for this value of the continuous time
   // (The wall object reads the continous time from the same
   // global time object)
   fluid_mesh_pt()->node_update(); 
   
   // Loop over the nodes to set initial guess everywhere
   for (unsigned jnod=0;jnod<num_nod;jnod++)
    {
     
     // Get nodal coordinates
     x[0]=fluid_mesh_pt()->node_pt(jnod)->x(0);
     x[1]=fluid_mesh_pt()->node_pt(jnod)->x(1);
 
     // Get intial solution
     (*IC_Fct_pt)(time,x,soln);
     
     // Loop over velocity directions (velocities are in soln[0] and soln[1]).
     for (unsigned i=0;i<2;i++)
      {
       fluid_mesh_pt()->node_pt(jnod)->set_value(itime,i,soln[i]);
      }
     
     // Loop over coordinate directions
     for (unsigned i=0;i<2;i++)
      {
       fluid_mesh_pt()->node_pt(jnod)->x(itime,i)=x[i];
      }
    } 
  }
 
 // Reset backed up time for global timestepper
 time_pt()->time()=backed_up_time;
 
}
 
 
 
 
 
//========================================================================
/// Doc the solution
///
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::doc_solution(DocInfo& doc_info)
{ 
 
 cout << "Doc-ing step " <<  doc_info.number()
      << " for time " << time_stepper_pt()->time_pt()->time() << std::endl;
 
 ofstream some_file;
 char filename[100];
 
 // Number of plot points
 unsigned npts;
 npts=5; 
 
 
 // Output solution on fluid mesh
 //-------------------------------
 sprintf(filename,"%s/soln%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 //some_file.precision(20);
 some_file.open(filename);
 unsigned nelem=fluid_mesh_pt()->nelement();
 for (unsigned ielem=0;ielem<nelem;ielem++)
  {
   dynamic_cast<ELEMENT*>(fluid_mesh_pt()->element_pt(ielem))->
    full_output(some_file,npts);
  }
 some_file.close();
  
 
 // Plot wall posn
 //---------------
 sprintf(filename,"%s/Wall%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 
 unsigned nplot=100;
 Vector<double > xi_wall(1);
 Vector<double > r_wall(2);
  for (unsigned iplot=0;iplot<nplot;iplot++)
   {
    xi_wall[0]=0.5*Pi*double(iplot)/double(nplot-1);
    Wall_pt->position(xi_wall,r_wall);
    some_file << r_wall[0] << " " << r_wall[1] << std::endl;
   }
  some_file.close();
 
 
 // Doc Sarah's asymptotic solution 
 //--------------------------------
 sprintf(filename,"%s/exact_soln%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 fluid_mesh_pt()->output_fct(some_file,npts,
                       time_stepper_pt()->time_pt()->time(), 
                       SarahBL::full_exact_soln);
 some_file.close();
 
 
 
 
 // Get position of control point
 //------------------------------
 Vector<double> r(2);
 Vector<double> xi(1);
 xi[0]=MathematicalConstants::Pi/2.0;
 wall_pt()->position(xi,r);
 
 
 
 // Get total volume (area) of computational domain, energies and average
 //----------------------------------------------------------------------
 // pressure
 //---------
 double area=0.0;
 double press_int=0.0;
 double diss=0.0;
 double kin_en=0.0;
 nelem=fluid_mesh_pt()->nelement();
 
 for (unsigned ielem=0;ielem<nelem;ielem++)
  {
   area+=fluid_mesh_pt()->finite_element_pt(ielem)->size();
   press_int+=dynamic_cast<ELEMENT*>(fluid_mesh_pt()->element_pt(ielem))
    ->pressure_integral();
   diss+=dynamic_cast<ELEMENT*>(fluid_mesh_pt()->element_pt(ielem))->
    dissipation();
   kin_en+=dynamic_cast<ELEMENT*>(fluid_mesh_pt()->element_pt(ielem))->
    kin_energy();
  }
 
 // Total kinetic energy in entire domain
 double global_kin=4.0*kin_en;
 
 // Max/min refinement level
 unsigned min_level;
 unsigned max_level;
 fluid_mesh_pt()->get_refinement_levels(min_level,max_level);
 
 
 // Total dissipation for quarter domain
 double time=time_pt()->time();
 double diss_asympt=SarahBL::Total_Diss_sarah(time)/4.0;
 
 // Asymptotic predictions for velocities at control point
 Vector<double> x_sarah(2);
 Vector<double> soln_sarah(3);
 x_sarah[0]=Sarah_veloc_trace_node_pt->x(0);
 x_sarah[1]=Sarah_veloc_trace_node_pt->x(1);
 SarahBL::exact_soln(time,x_sarah,soln_sarah);
 
 // Doc
 Trace_file << time_pt()->time() 
            << " " << r[1]
            << " " << global_kin 
            << " " << SarahBL::Kin_energy_sarah(time_pt()->time()) 
            << " " << static_cast<PseudoBucklingRingElement*>(Wall_pt)->r_0()
            << " " << area  
            << " " << press_int/area 
            << " " << diss  
            << " " << diss_asympt
            << " " << Veloc_trace_node_pt->x(0)
            << " " << Veloc_trace_node_pt->x(1) 
            << " " << Veloc_trace_node_pt->value(0)
            << " " << Veloc_trace_node_pt->value(1) 
            << " " << fluid_mesh_pt()->nelement() 
            << " " << ndof()
            << " " << min_level 
            << " " << max_level
            << " " << fluid_mesh_pt()->nrefinement_overruled() 
            << " " << fluid_mesh_pt()->max_error()  
            << " " << fluid_mesh_pt()->min_error()  
            << " " << fluid_mesh_pt()->max_permitted_error() 
            << " " << fluid_mesh_pt()->min_permitted_error() 
            << " " << fluid_mesh_pt()->max_keep_unrefined()
            << " " << doc_info.number()
            << " " << Sarah_veloc_trace_node_pt->x(0) 
            << " " << Sarah_veloc_trace_node_pt->x(1) 
            << " " << Sarah_veloc_trace_node_pt->value(0) 
            << " " << Sarah_veloc_trace_node_pt->value(1)
            << " " << x_sarah[0]
            << " " << x_sarah[1]
            << " " << soln_sarah[0]
            << " " << soln_sarah[1]
            << " " 
            << static_cast<PseudoBucklingRingElement*>(Wall_pt)->r_0()-1.0
            << std::endl;
 
 
 // Output fluid solution on coarse-ish mesh 
 //-----------------------------------------
 
 // Extract all elements from quadtree representation
 Vector<Tree*> all_element_pt;
 fluid_mesh_pt()->forest_pt()->
  stick_all_tree_nodes_into_vector(all_element_pt);
 
 // Build a coarse mesh
 Mesh* coarse_mesh_pt = new Mesh();
 
 //Loop over all elements and check if their refinement level is
 //equal to the mesh's minimum refinement level
 nelem=all_element_pt.size();
 for (unsigned ielem=0;ielem<nelem;ielem++)
  {
   Tree* el_pt=all_element_pt[ielem];
   if (el_pt->level()==min_level)
    {
     coarse_mesh_pt->add_element_pt(el_pt->object_pt());
    }
  }
 
 // Output fluid solution on coarse mesh
 sprintf(filename,"%s/coarse_soln%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 nelem=coarse_mesh_pt->nelement();
 for (unsigned ielem=0;ielem<nelem;ielem++)
  {
   dynamic_cast<ELEMENT*>(coarse_mesh_pt->element_pt(ielem))->
    full_output(some_file,npts);
  }
 some_file.close();
 
 // Write restart file
 sprintf(filename,"%s/restart%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 dump_it(some_file,doc_info);
 some_file.close();
 
}
 
 
 
 
 
 
//========================================================================
/// Dump the solution
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::dump_it(ofstream& dump_file,DocInfo doc_info)
{
 
 // Dump refinement status of mesh
 //fluid_mesh_pt()->dump_refinement(dump_file);
 
 // Call generic dump()
 Problem::dump(dump_file);
  
}
 
 
 
//========================================================================
/// Read solution from disk
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::restart(ifstream& restart_file)
{
 // Refine fluid mesh according to the instructions in restart file
 //fluid_mesh_pt()->refine(restart_file);
 
 // Rebuild the global mesh
 //rebuild_global_mesh();
 
 // Read generic problem data
 Problem::read(restart_file);
 
//   // Complete build of all elements so they are fully functional
//  finish_problem_setup();
 
 //Assign equation numbers
 //cout <<"\nNumber of equations: " << assign_eqn_numbers() 
 //     << std::endl<< std::endl; 
}
 
 
 
//========================================================================
/// Driver for timestepping the problem: Fixed timestep but 
/// guaranteed spatial accuracy. Beautiful, innit?
///
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::unsteady_run(const unsigned& ntsteps,
                                              const bool& restarted,
                                              DocInfo& doc_info)
{ 
 
 // Open trace file
 char filename[100];   
 sprintf(filename,"%s/trace.dat",doc_info.directory().c_str());
 Trace_file.open(filename);
 
 // Max. number of adaptations per solve
 unsigned max_adapt;
 
 // Max. number of adaptations per solve
 if (restarted)
  {
   max_adapt=0;
  }
 else
  {
   max_adapt=1; 
  }
 
 // Max. and min. error for adaptive refinement/unrefinement
 fluid_mesh_pt()->max_permitted_error()=  0.5e-2;   
 fluid_mesh_pt()->min_permitted_error()=  0.5e-3;  
 
 // Don't allow refinement to drop under given level
 fluid_mesh_pt()->min_refinement_level()=2;
 
 // Don't allow refinement beyond given level 
 fluid_mesh_pt()->max_refinement_level()=6; 
 
 // Don't bother adapting the mesh if no refinement is required
 // and if less than ... elements are to be merged.
 fluid_mesh_pt()->max_keep_unrefined()=20;
 
 
 // Get max/min refinement levels in mesh
 unsigned min_refinement_level;
 unsigned max_refinement_level;
 fluid_mesh_pt()->get_refinement_levels(min_refinement_level,
                                        max_refinement_level);
 
 cout << "\n Initial mesh: min/max refinement levels: " 
      << min_refinement_level << " " << max_refinement_level << std::endl << std::endl;
 
 // Doc refinement targets
 fluid_mesh_pt()->doc_adaptivity_targets(cout);
 
 // Write header for trace file
 write_trace_file_header();
 
 // Doc initial condition
 doc_solution(doc_info);
 doc_info.number()++;
 
 // Switch off doc during solve
 doc_info.disable_doc();
 
 // If we set first to true, then initial guess will be re-assigned
 // after mesh adaptation. Don't want this if we've done a restart.
 bool first;
 bool shift;
 if (restarted)
  {
   first=false;
   shift=false;
   // Move time back by dt to make sure we're re-solving the read-in solution
   time_pt()->time()-=time_pt()->dt();
  }
 else
  {
   first=true;
   shift=true;
  }
 
 //Take the first fixed timestep with specified tolerance for Newton solver
 double dt=time_pt()->dt();
 unsteady_newton_solve(dt,max_adapt,first,shift);
 
 // Switch doc back on
 doc_info.enable_doc();
 
 // Doc initial solution
 doc_solution(doc_info);
 doc_info.number()++;
 
 // Now do normal run; allow for one mesh adaptation per timestep
 max_adapt=1; 
 
 //Loop over the remaining timesteps
 for(unsigned t=2;t<=ntsteps;t++)
  {
 
   // Switch off doc during solve
   doc_info.disable_doc();
 
   //Take fixed timestep
   first=false;
   unsteady_newton_solve(dt,max_adapt,first);
 
   // Switch doc back on
   doc_info.enable_doc();
 
   // Doc solution
   //if (icount%10==0)
    {
     doc_solution(doc_info);
     doc_info.number()++;
    }
  }
 
 // Close trace file
 Trace_file.close();
 
}
 
 
 
 
//========================================================================
/// Write trace file header
//========================================================================
template<class ELEMENT>
void OscRingNStProblem<ELEMENT>::write_trace_file_header()
{
 
 // Doc parameters in trace file
 Trace_file << "# err_max " << fluid_mesh_pt()->max_permitted_error() << std::endl;
 Trace_file << "# err_min " << fluid_mesh_pt()->min_permitted_error() << std::endl;
 Trace_file << "# Re " << Global_Physical_Variables::Re << std::endl;
 Trace_file << "# St " << Global_Physical_Variables::ReSt/
                          Global_Physical_Variables::Re << std::endl;
 Trace_file << "# dt " << time_stepper_pt()->time_pt()->dt() << std::endl;
 Trace_file << "# initial # elements " << mesh_pt()->nelement() << std::endl;
 Trace_file << "# min_refinement_level " 
            << fluid_mesh_pt()->min_refinement_level() << std::endl;
 Trace_file << "# max_refinement_level " 
            << fluid_mesh_pt()->max_refinement_level() << std::endl;
 
 
 
 Trace_file << "VARIABLES=\"time\",\"V_c_t_r_l\",\"e_k_i_n\"";
 Trace_file << ",\"e_k_i_n_(_a_s_y_m_p_t_)\",\"R_0\",\"Area\"" ;
 Trace_file << ",\"Average pressure\",\"Total dissipation (quarter domain)\"";
 Trace_file << ",\"Asymptotic dissipation (quarter domain)\"";
 Trace_file << ",\"x<sub>1</sub><sup>(track)</sup>\"";
 Trace_file << ",\"x<sub>2</sub><sup>(track)</sup>\"";
 Trace_file << ",\"u<sub>1</sub><sup>(track)</sup>\"";
 Trace_file << ",\"u<sub>2</sub><sup>(track)</sup>\"";
 Trace_file << ",\"N<sub>element</sub>\"";
 Trace_file << ",\"N<sub>dof</sub>\"";
 Trace_file << ",\"max. refinement level\"";
 Trace_file << ",\"min. refinement level\"";
 Trace_file << ",\"# of elements whose refinement was over-ruled\"";
 Trace_file << ",\"max. error\"";
 Trace_file << ",\"min. error\"";
 Trace_file << ",\"max. permitted error\"";
 Trace_file << ",\"min. permitted error\"";
 Trace_file << ",\"max. permitted # of unrefined elements\"";
 Trace_file << ",\"doc number\"";
 Trace_file << ",\"x<sub>1</sub><sup>(track2 FE)</sup>\"";
 Trace_file << ",\"x<sub>2</sub><sup>(track2 FE)</sup>\"";
 Trace_file << ",\"u<sub>1</sub><sup>(track2 FE)</sup>\"";
 Trace_file << ",\"u<sub>2</sub><sup>(track2 FE)</sup>\"";
 Trace_file << ",\"x<sub>1</sub><sup>(track2 Sarah)</sup>\"";
 Trace_file << ",\"x<sub>2</sub><sup>(track2 Sarah)</sup>\"";
 Trace_file << ",\"u<sub>1</sub><sup>(track2 Sarah)</sup>\"";
 Trace_file << ",\"u<sub>2</sub><sup>(track2 Sarah)</sup>\"";
 Trace_file << ",\"R<sub>0</sub>-1\"";
 Trace_file << std::endl;
 
}
 
 
 
 
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////////
 
 
 
 
 
 
 
//========================================================================
/// Demonstrate how to solve OscRingNSt problem in deformable domain
/// with mesh adaptation
//========================================================================
int main(int argc, char* argv[])
{
 // Store command line arguments
 CommandLineArgs::setup(argc,argv);
 
 //Do a certain number of timesteps per period
 unsigned nstep_per_period=40; // 80;  // ADJUST_PRIORITY
 unsigned nperiod=3;
 
 // Work out total number of steps and timestep
 unsigned nstep=nstep_per_period*nperiod;  
 double dt=1.0/double(nstep_per_period);
 
 // Set up the problem: Pass timestep and Sarah's asymptotic solution for
 // generation of initial condition
 OscRingNStProblem<MacroElementNodeUpdateElement<RefineableQCrouzeixRaviartElement<2> > >
  problem(dt,&SarahBL::full_exact_soln);
 
 
 // Restart?
 //---------
 bool restarted=false;
 
 // Pointer to restart file
 ifstream* restart_file_pt=0;
 
 // No restart
 //-----------
 if (CommandLineArgs::Argc!=2)
  {
   cout << "No restart" << std::endl;
   restarted=false;
 
   // Refine uniformly
   problem.refine_uniformly();
   problem.refine_uniformly();
 
   // Set initial condition on uniformly refined domain (if this isn't done
   // then the mesh contains the interpolation of the initial guess
   // on the original coarse mesh -- if the nodal values were zero in
   // the interior and nonzero on the boundary, then the the waviness of
   // of the interpolated i.g. between the nodes on the coarse mesh
   // gets transferred onto the fine mesh where we can do better
   problem.set_initial_condition();
 
  }
 
 // Restart
 //--------
 else if (CommandLineArgs::Argc==2)
  {
   restarted=true;
 
   // Open restart file
   restart_file_pt=new ifstream(CommandLineArgs::Argv[1],ios_base::in);
   if (restart_file_pt!=0)
    {
     cout << "Have opened " << CommandLineArgs::Argv[1] << 
      " for restart. " << std::endl;
    }
   else
    {
     std::ostringstream error_stream;
     error_stream << "ERROR while trying to open " 
                  << CommandLineArgs::Argv[2] 
                  << " for restart." << std::endl;
 
     throw OomphLibError(error_stream.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
   // Do the actual restart
   problem.restart(*restart_file_pt);
  }
 
 
 // Two command line arguments: do validation run
 if (CommandLineArgs::Argc==3)
  {
    nstep=3;
    cout << "Only doing nstep steps for validation: " << nstep << std::endl;
  }
 
 
 
 
 // Setup labels for output
 DocInfo doc_info;
 
 // Output directory
 doc_info.set_directory("RESLT"); 
 
 
 // Do unsteady run of the problem for nstep steps
 //-----------------------------------------------
 problem.unsteady_run(nstep,restarted,doc_info);
 
 
 // Validate the restart procedure
 //-------------------------------
 if (CommandLineArgs::Argc==3)
  {
   
   // Build problem and restart
   
   // Set up the problem: Pass timestep and Sarah's asymptotic solution for
   // generation of initial condition
   OscRingNStProblem<MacroElementNodeUpdateElement<
    RefineableQCrouzeixRaviartElement<2> > >
    restarted_problem(dt,&SarahBL::full_exact_soln);
   
   // Setup labels for output
   DocInfo restarted_doc_info;
   
   // Output directory
   restarted_doc_info.set_directory("RESLT_restarted"); 
   
   // Step number
   restarted_doc_info.number()=0;
   
   // Copy by performing a restart from old problem
   restart_file_pt=new ifstream("RESLT/restart2.dat");
   
   // Do the actual restart
   restarted_problem.restart(*restart_file_pt);
   
   // Do unsteady run of the problem for one step
   unsigned nstep=2;
   bool restarted=true;
   restarted_problem.unsteady_run(nstep,restarted,restarted_doc_info);
 
  }
 
}
 
 
 
 
 
 
 
 
 
 

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