bretherton.cc

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//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
//LIC// 
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//LIC// License as published by the Free Software Foundation; either
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//LIC//====================================================================
// The 2D Bretherton problem
#include<algorithm>
 
// The oomphlib headers   
#include "generic.h"
#include "navier_stokes.h"
#include "fluid_interface.h"
 
// The mesh
#include "meshes/bretherton_spine_mesh.h"
 
using namespace std;
 
using namespace oomph;
 
//======================================================================
/// Namepspace for global parameters
//======================================================================
namespace Global_Physical_Variables
{
 
 /// Reynolds number
 double Re;
 
 /// Womersley = Reynolds times Strouhal
 double ReSt; 
 
 /// Product of Reynolds and Froude number
 double ReInvFr;
 
 /// Capillary number
 double Ca;  
 
 /// Direction of gravity
 Vector<double> G(2);
 
 /// Pointer to film thickness at outflow on the lower wall
 double* H_lo_pt;
 
 /// Pointer to film thickness at outflow on the upper wall
 double* H_up_pt;
 
 /// Pointer to y-position at inflow on the lower wall
 double* Y_lo_pt;
 
 /// Pointer to y-position at inflow on the upper wall
 double* Y_up_pt;
 
 /// Set inflow velocity, based on spine heights at outflow
 /// and channel width at inflow
 void inflow(const Vector<double>& x, Vector<double>& veloc)
 {
#ifdef PARANOID
  std::ostringstream error_stream;
  bool throw_error=false;
  if (H_lo_pt==0) 
   {
    error_stream << "You must set H_lo_pt\n";
    throw_error = true;
   }
  if (H_up_pt==0) 
   {
    error_stream << "You must set H_up_pt\n";
    throw_error = true;
   }
  if (Y_lo_pt==0) 
   {
    error_stream << "You must set Y_lo_pt\n";
    throw_error = true;
   }
  if (Y_up_pt==0) 
   {
    error_stream << "You must set Y_up_pt\n";
    throw_error = true;
   }
 
  //If one of the errors has occured
  if(throw_error)
   {
    throw OomphLibError(error_stream.str(),
                        OOMPH_CURRENT_FUNCTION,
                        OOMPH_EXCEPTION_LOCATION);
   }
#endif
 
 
  // Get average film thickness far ahead
  double h_av=0.5*(*H_lo_pt+*H_up_pt);
 
  // Get position of upper and lower wall at inflow
  double y_up=*Y_up_pt;
  double y_lo=*Y_lo_pt;
 
  // Constant in velocity profile
  double C =6.0*(2.0*h_av+y_lo-y_up)/
   (y_up*y_up*y_up-y_lo*y_lo*y_lo-h_av*y_up*
    y_up*y_up+h_av*y_lo*y_lo*y_lo-3.0*y_lo*y_up*y_up+
    3.0*y_lo*y_lo*y_up+3.0*y_lo*y_up*y_up*h_av-3.0*y_lo*y_lo*y_up*h_av);
 
  // y coordinate
  double y=x[1];
 
  // Parallel, parabolic inflow
  veloc[0]=-1.0+C*(1.0-h_av)*((y_lo-y)*(y_up-y));
  veloc[1]=0.0;
 }
 
}
 
 
 
 
//======================================================================
/// "Bretherton element" is a fluid element (of type ELEMENT) for which
/// the (inflow) velocity  at those nodes that are located on a 
/// specified Mesh boundary is prescribed by Dirichlet boundary 
/// conditions. The key is that prescribed velocity profile can be
/// a function of some external Data -- this dependency must
/// be taken into account when computing the element's Jacobian
/// matrix.
/// 
/// This element type is useful, for instance, in the Bretherton
/// problem, where the parabolic "inflow" profile is a function 
/// of the film thickness (represented by Spine heights) at the
/// "outflow".
//======================================================================
template<class ELEMENT>
class BrethertonElement : public ELEMENT
{
 
public:
 
 /// Typedef for pointer (global) function that specifies the
 /// the inflow
 typedef void (*InflowFctPt)(const Vector<double>& x, Vector<double>& veloc);
 
 /// Constructor: Call the constructor of the underlying element
 BrethertonElement() : ELEMENT() {}
 
 
 /// Activate the dependency of the "inflow" on the external
 /// data. Pass the vector of pointers to the external Data that affects
 /// the inflow, the id of the boundary on which the inflow
 /// condition is to be applied and the function pointer to 
 /// to the global function that defines the inflow
 void activate_inflow_dependency_on_external_data(
  const Vector<Data*>& inflow_ext_data,
  const unsigned& inflow_boundary,
  InflowFctPt inflow_fct_pt)
  {
   // Copy data that affects the inflow
   unsigned n_ext=inflow_ext_data.size();
   Inflow_ext_data.resize(n_ext);
   for (unsigned i=0;i<n_ext;i++)
    {
     Inflow_ext_data[i]=inflow_ext_data[i];
    }
   // Set inflow boundary
   Inflow_boundary=inflow_boundary;
   // Set fct pointer to inflow condition
   Inflow_fct_pt=inflow_fct_pt;
  }
 
 
 /// Overload assign local equation numbers: Add the dependency 
 /// on the external Data that affects the inflow profile
 void assign_local_eqn_numbers(const bool  &store_local_dof_pt)
  {
   // Call the element's local equation numbering procedure first
   ELEMENT::assign_local_eqn_numbers(store_local_dof_pt);
   
   // Now add the equation numbers for the Data that affects the inflow
   // profile
   
   // Number of local equations so far
   unsigned local_eqn_count = this->ndof();
   
   // Find out max. number of values stored at all Data values 
   // that affect the inflow
   unsigned max_nvalue=0;
   unsigned n_ext=Inflow_ext_data.size();
   for (unsigned i=0;i<n_ext;i++)
    {
     // The external Data:
     Data* data_pt=Inflow_ext_data[i];
     // Number of values
     unsigned n_val=data_pt->nvalue();
     if (n_val>max_nvalue) max_nvalue=n_val;
    }
 
   // Allocate sufficient storage
   Inflow_ext_data_eqn.resize(n_ext,max_nvalue);
   
   //A local queue to store the global equation numbers
   std::deque<unsigned long> global_eqn_number_queue;
 
   // Loop over external Data that affect the "inflow"
   for (unsigned i=0;i<n_ext;i++)
    {
     // The external Data:
     Data* data_pt=Inflow_ext_data[i];
     
     // Loop over number of values:
     unsigned n_val=data_pt->nvalue();
     for (unsigned ival=0;ival<n_val;ival++)
      {
 
       // Is it free or pinned?
       long eqn_number=data_pt->eqn_number(ival);
       if (eqn_number>=0)
        {
         // Add global equation number to local storage
         global_eqn_number_queue.push_back(eqn_number);
         // Store local equation number associated with this external dof
         Inflow_ext_data_eqn(i,ival)=local_eqn_count;
         // Increment counter for local dofs
         local_eqn_count++;
        }
       else
        {
         Inflow_ext_data_eqn(i,ival)=-1;
        }
      }
    }
   
   //Now add our global equations numbers to the internal element storage
   this->add_global_eqn_numbers(global_eqn_number_queue, 
                                GeneralisedElement::Dof_pt_deque);
  }
 
 
 /// Overloaded Jacobian computation: Computes the Jacobian
 /// of the underlying element and then adds the FD 
 /// operations to evaluate the derivatives w.r.t. the Data values
 /// that affect the inflow.
 void get_jacobian(Vector<double>& residuals,
                   DenseMatrix<double>& jacobian)
  {
   // Loop over Data values that affect the inflow
   unsigned n_ext=Inflow_ext_data.size();
 
   // Call "normal" jacobian first.
   ELEMENT::get_jacobian(residuals,jacobian);
   
   if (n_ext==0) return;
   
   // Get ready for FD operations
   Vector<double> residuals_plus(residuals.size());
   double fd_step=1.0e-8;
 
   // Loop over Data values that affect the inflow
   for (unsigned i=0;i<n_ext;i++)
    {
     // Get Data item
     Data* data_pt=Inflow_ext_data[i];
     
     // Loop over values
     unsigned n_val=data_pt->nvalue();
     for (unsigned ival=0;ival<n_val;ival++)
      {
       // Local equation number
       int local_eqn=Inflow_ext_data_eqn(i,ival);
       
       // Dof or pinned?
       if (local_eqn>=0)
        {
         //get pointer to the data value
         double *value_pt = data_pt->value_pt(ival);
 
         //Backup Data value
         double backup = *value_pt;
         
         // Do FD step
         *value_pt += fd_step;
         
         // Re-assign the inflow velocities for nodes in this element
         reassign_inflow();
         
                  
         // Fill in the relevant column in the Jacobian matrix
         unsigned n_dof = this->ndof();
         //Zero the residuals
         for(unsigned idof=0;idof<n_dof;idof++) {residuals_plus[idof] = 0.0;}
         // Re-compute the element residual
         this->get_residuals(residuals_plus);
 
         for(unsigned idof=0;idof<n_dof;idof++)
          {
           jacobian(idof,local_eqn)=
            (residuals_plus[idof]-residuals[idof])/fd_step;
          }
         
         //Reset spine height
         *value_pt = backup;
         
        }
       // Note: Re-assignment of inflow is done on the fly during next loop
      }
    }
   
   // Final re-assign for the inflow velocities for nodes in this element
   reassign_inflow();
   
  }
 
 
private:
 
 
 
 /// For all nodes that are located on specified boundary
 /// re-assign the inflow velocity, using the function pointed to
 /// by the function pointer
 void reassign_inflow()
  { 
   // Loop over all nodes in element -- if they are
   // on inflow boundary, re-assign their velocities
   Vector<double> x(2);
   Vector<double> veloc(2);
   unsigned n_nod = this->nnode();
   for (unsigned j=0;j<n_nod;j++)
    {
     Node* nod_pt = this->node_pt(j);
 
     if(nod_pt->is_on_boundary(Inflow_boundary))
      {
#ifdef PARANOID
           for (unsigned i=0;i<2;i++)
            {
             if (nod_pt->eqn_number(0)>=0)
              {
               std::ostringstream error_stream;
               error_stream 
                << "We're assigning a Dirichlet condition for the " 
                << i << "-th "
                << "velocity, even though it is not pinned!\n" 
                << "This can't be right! I'm bailing out..." 
                << std::endl;
 
               throw OomphLibError(error_stream.str(),
                                   OOMPH_CURRENT_FUNCTION,
                                   OOMPH_EXCEPTION_LOCATION);
              }
            }
#endif           
           // Get inflow profile 
           x[0]=nod_pt->x(0);
           x[1]=nod_pt->x(1);
           Inflow_fct_pt(x,veloc);
           nod_pt->set_value(0,veloc[0]);
           nod_pt->set_value(1,veloc[1]);
      }
    }
  }
 
 
 /// Storage for the external Data that affects the inflow
 Vector<Data*> Inflow_ext_data;
 
 /// Storage for the local equation numbers associated the Data
 /// values that affect the inflow
 DenseMatrix<int> Inflow_ext_data_eqn;
 
 /// Number of the inflow boundary in the global mesh
 unsigned Inflow_boundary;
 
 /// Function pointer to the global function that specifies the
 /// inflow velocity profile on the global mesh boundary Inflow_boundary
 InflowFctPt Inflow_fct_pt;
 
};
 
 
 
// Note: Specialisation must go into namespace!
namespace oomph
{
 
//=======================================================================
/// Face geometry of the Bretherton 2D Crouzeix_Raviart spine elements
//=======================================================================
template<>
class FaceGeometry<BrethertonElement<SpineElement<QCrouzeixRaviartElement<2> > > >: public virtual QElement<1,3>
{
  public:
 FaceGeometry() : QElement<1,3>() {}
};
 
 
 
//=======================================================================
/// Face geometry of the Bretherton 2D Taylor Hood spine elements
//=======================================================================
template<>
class FaceGeometry<BrethertonElement<SpineElement<QTaylorHoodElement<2> > > >: public virtual QElement<1,3>
{
  public:
 FaceGeometry() : QElement<1,3>() {}
};
 
 
//=======================================================================
/// Face geometry of the Face geometry of the 
///  the Bretherton 2D Crouzeix_Raviart spine elements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<BrethertonElement<
 SpineElement<QCrouzeixRaviartElement<2> > > > >: public virtual PointElement
{
  public:
 FaceGeometry() : PointElement() {}
};
 
 
 
//=======================================================================
/// Face geometry of face geometry of 
/// the Bretherton 2D Taylor Hood spine elements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<BrethertonElement<SpineElement<QTaylorHoodElement<2> > > > >: public virtual PointElement
{
  public:
 FaceGeometry() : PointElement() {}
};
 
 
}
 
/// //////////////////////////////////////////////////////////////////
/// //////////////////////////////////////////////////////////////////
/// //////////////////////////////////////////////////////////////////
 
 
//======================================================================
/// Bretherton problem
//======================================================================
template<class ELEMENT>
class BrethertonProblem : public Problem
{
 
 
public:
 
 /// Constructor: 
 BrethertonProblem();
 
 /// Spine heights/lengths are unknowns in the problem so their
 /// values get corrected during each Newton step. However,
 /// changing their value does not automatically change the
 /// nodal positions, so we need to update all of them
 void actions_before_newton_convergence_check()
  {
   // Update
   Bulk_mesh_pt->node_update();
 
   // Apply inflow on boundary 1
   unsigned ibound=1;
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);
   
   // Coordinate and velocity
   Vector<double> x(2);
   Vector<double> veloc(2);
 
   // Loop over all nodes
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Get nodal position
     x[0]=mesh_pt()->boundary_node_pt(ibound,inod)->x(0);
     x[1]=mesh_pt()->boundary_node_pt(ibound,inod)->x(1);
 
     // Get inflow profile
     Global_Physical_Variables::inflow(x,veloc);
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(0,veloc[0]);
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(1,veloc[1]);
    }
  }
 
 /// Update before solve: empty
 void actions_before_newton_solve() {}
 
 /// Update after solve can remain empty, because the update 
 /// is performed automatically after every Newton step.
 void actions_after_newton_solve() {}
 
 /// Fix pressure value l in element e to value p_value
 void fix_pressure(const unsigned &e, const unsigned &l, 
                   const double &pvalue)
  {
   //Fix the pressure at that element
   dynamic_cast<ELEMENT *>(Bulk_mesh_pt->element_pt(e))->
    fix_pressure(l,pvalue);
  }
 
 
 /// Activate the dependency of the inflow velocity on the 
 /// spine heights at the outflow
 void activate_inflow_dependency();
 
 /// Run a parameter study; perform specified number of steps
 void parameter_study(const unsigned& nsteps); 
 
 /// Doc the solution
 void doc_solution(DocInfo& doc_info);
 
 
private:
 
 /// Pointer to control element
 ELEMENT* Control_element_pt;
 
 /// Trace file
 ofstream Trace_file;
 
 /// Pointer to bulk mesh
 BrethertonSpineMesh<ELEMENT,SpineLineFluidInterfaceElement<ELEMENT> >* 
 Bulk_mesh_pt;
 
};
 
 
//====================================================================
/// Problem constructor
//====================================================================
template<class ELEMENT>
BrethertonProblem<ELEMENT>::BrethertonProblem()
{
 
 // Number of elements in the deposited film region
 unsigned nx1=24;
 
 // Number of elements in the bottom part of the transition region
 unsigned nx2=6;
 
 // Number of elements in channel region
 unsigned nx3=12;
 
 // Number of elements in the vertical part of the transition region
 // (=half the number of elements in the liquid filled region ahead
 // of the finger tip)
 unsigned nhalf=4;
 
 // Number of elements through thickness of deposited film
 unsigned nh=3;
 
 // Thickness of deposited film
 double h=0.1;
 
 // Create wall geom objects
 GeomObject* lower_wall_pt=new StraightLine(-1.0);
 GeomObject* upper_wall_pt=new StraightLine( 1.0);
 
 // Start coordinate on wall
 double xi0=-4.0;
 
 // End of transition region on wall
 double xi1=1.5;
 
 // End of liquid filled region (inflow) on wall
 double xi2=5.0;
 
 //Now create the mesh
 Bulk_mesh_pt = new  BrethertonSpineMesh<ELEMENT,
  SpineLineFluidInterfaceElement<ELEMENT> > 
  (nx1,nx2,nx3,nh,nhalf,h,lower_wall_pt,upper_wall_pt,xi0,0.0,xi1,xi2);
 
 // Make bulk mesh the global mesh
 mesh_pt()=Bulk_mesh_pt;
 
 // Store the control element
 Control_element_pt=Bulk_mesh_pt->control_element_pt();
 
 
 // Set pointers to quantities that determine the inflow profile
 
 // Film thickness at outflow on lower wall:
 Global_Physical_Variables::H_lo_pt=
  Bulk_mesh_pt->spine_pt(0)->spine_height_pt()->value_pt(0);
 
 // Film thickness at outflow on upper wall:
 unsigned last_spine=Bulk_mesh_pt->nfree_surface_spines()-1;
 Global_Physical_Variables::H_up_pt=
  Bulk_mesh_pt->spine_pt(last_spine)->spine_height_pt()->value_pt(0);
 
 // Current y-position on lower wall at inflow
 unsigned ibound=1;
 Global_Physical_Variables::Y_lo_pt=
  Bulk_mesh_pt->boundary_node_pt(ibound,0)->x_pt(0,1);
 
 // Current y-position on upper wall at inflow
 unsigned nnod=Bulk_mesh_pt->nboundary_node(ibound);
 Global_Physical_Variables::Y_up_pt=
  Bulk_mesh_pt->boundary_node_pt(ibound,nnod-1)->x_pt(0,1);
 
 // Activate dependency of inflow on spine heights at outflow
 activate_inflow_dependency();
 
 // Set the boundary conditions for this problem: All nodes are
 // free by default -- just pin the ones that have Dirichlet conditions
 // here
 
 // No slip on boundaries 0 1 and 2
 for(unsigned ibound=0;ibound<=2;ibound++)
  {
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     mesh_pt()->boundary_node_pt(ibound,inod)->pin(0);
     mesh_pt()->boundary_node_pt(ibound,inod)->pin(1);
    }
  }
 
 // Uniform, parallel outflow on boundaries 3 and 5
 for(unsigned ibound=3;ibound<=5;ibound+=2)
  {
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     mesh_pt()->boundary_node_pt(ibound,inod)->pin(0);
     mesh_pt()->boundary_node_pt(ibound,inod)->pin(1);
    }
  }
 
 // Pin central spine
 unsigned central_spine=(Bulk_mesh_pt->nfree_surface_spines()-1)/2;
 Bulk_mesh_pt->spine_pt(central_spine)->spine_height_pt()->pin(0);
 
 
 // No slip in moving frame on boundaries 0 and 2
 for (unsigned ibound=0;ibound<=2;ibound+=2)
  {
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Parallel flow
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(0,-1.0);
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(1, 0.0);
    }
  }
   
 
 // Parallel, uniform outflow on boundaries 3 and 5
 for (unsigned ibound=3;ibound<=5;ibound+=2)
  {
   unsigned num_nod=mesh_pt()->nboundary_node(ibound);
   for (unsigned inod=0;inod<num_nod;inod++)
    {
     // Parallel inflow
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(0,-1.0);
     mesh_pt()->boundary_node_pt(ibound,inod)->set_value(1, 0.0);
    }
  }
   
 
 
 //Complete the problem setup to make the elements fully functional
 
 //Loop over the elements in the layer
 unsigned n_bulk=Bulk_mesh_pt->nbulk();
 for(unsigned i=0;i<n_bulk;i++)
  {
   //Cast to a fluid element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(Bulk_mesh_pt->
                                           bulk_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;
   el_pt->re_invfr_pt() = &Global_Physical_Variables::ReInvFr;
   el_pt->g_pt() = &Global_Physical_Variables::G;
  }
 
 //Loop over 1D interface elements and set capillary number
 unsigned interface_element_pt_range = Bulk_mesh_pt->ninterface_element();
 for(unsigned i=0;i<interface_element_pt_range;i++)
  {
   //Cast to a interface element
   SpineLineFluidInterfaceElement<ELEMENT>* el_pt = 
    dynamic_cast<SpineLineFluidInterfaceElement<ELEMENT>*>
    (Bulk_mesh_pt->interface_element_pt(i));
 
   //Set the Capillary number
   el_pt->ca_pt() = &Global_Physical_Variables::Ca;
  }
 
 //Update the nodal positions, so that the domain is correct
 //(and therefore the repeated node test passes)
 Bulk_mesh_pt->node_update();
 
 //Do equation numbering
 cout << "Number of unknowns: " << assign_eqn_numbers() << std::endl; 
 
}
 
 
 
//========================================================================
/// Activate the dependency of the inflow velocity on the 
/// spine heights at the outflow
//========================================================================
template<class ELEMENT>
void BrethertonProblem<ELEMENT>::activate_inflow_dependency()
{
 
 // Spine heights that affect the inflow
 Vector<Data*> outflow_spines(2);
 
 // First spine
 outflow_spines[0]=Bulk_mesh_pt->spine_pt(0)->spine_height_pt();
 
 // Last proper spine
 unsigned last_spine=Bulk_mesh_pt->nfree_surface_spines()-1;
 outflow_spines[1]=Bulk_mesh_pt->spine_pt(last_spine)->spine_height_pt();;
 
 
 
 /// Loop over elements on inflow boundary (1)
 unsigned ibound=1;
 unsigned nel=Bulk_mesh_pt->nboundary_element(ibound);
 for (unsigned e=0;e<nel;e++)
  {
   // Get pointer to element
   ELEMENT* el_pt=dynamic_cast<ELEMENT*>(Bulk_mesh_pt->
                                         boundary_element_pt(ibound,e));  
   // Activate depency on inflow
   el_pt->activate_inflow_dependency_on_external_data(
    outflow_spines,ibound,&Global_Physical_Variables::inflow);
  }
 
}
 
 
 
//========================================================================
/// Doc the solution
//========================================================================
template<class ELEMENT>
void BrethertonProblem<ELEMENT>::doc_solution(DocInfo& doc_info)
{ 
 
 ofstream some_file;
 char filename[100];
 
 // Number of plot points
 unsigned npts=5; 
 
 // Control coordinate: At bubble tip
 Vector<double> s(2);
 s[0]=1.0;
 s[1]=1.0;
 
 // Last proper spine
 unsigned last_spine=Bulk_mesh_pt->nfree_surface_spines()-1;
 
 // Doc
 Trace_file << " " << Global_Physical_Variables::Ca;
 Trace_file << " " << Bulk_mesh_pt->spine_pt(0)->height();
 Trace_file << " " << Bulk_mesh_pt->spine_pt(last_spine)->height();
 Trace_file << " " << 1.3375*pow(Global_Physical_Variables::Ca,2.0/3.0);
 Trace_file << " " << -Control_element_pt->interpolated_p_nst(s)*
                      Global_Physical_Variables::Ca;
 Trace_file << " " << 1.0+3.8*pow(Global_Physical_Variables::Ca,2.0/3.0);
 Trace_file << " " << Control_element_pt->interpolated_u_nst(s,0);
 Trace_file << " " << Control_element_pt->interpolated_u_nst(s,1);
 Trace_file << std::endl;
 
 
 // Output solution 
 sprintf(filename,"%s/soln%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 Bulk_mesh_pt->output(some_file,npts);
 some_file.close();
 
 
 // Output boundaries
 sprintf(filename,"%s/boundaries%i.dat",doc_info.directory().c_str(),
         doc_info.number());
 some_file.open(filename);
 Bulk_mesh_pt->output_boundaries(some_file);
 some_file.close();
 
}
 
 
 
 
//=============================================================================
/// Parameter study
//=============================================================================
template<class ELEMENT>
void BrethertonProblem<ELEMENT>::parameter_study(const unsigned& nsteps)
{
 
 // Increase maximum residual
 Problem::Max_residuals=100.0;
 
 // Set output directory
 DocInfo doc_info;
 doc_info.set_directory("RESLT");
 doc_info.number()=0;
 
 
 // Open trace file
 char filename[100];   
 sprintf(filename,"%s/trace.dat",doc_info.directory().c_str());
 Trace_file.open(filename);
 
 Trace_file << "VARIABLES=\"Ca\",";
 Trace_file << "\"h<sub>bottom</sub>\",\"h<sub>too</sub>\",";
 Trace_file << "\"h<sub>Bretherton</sub>\",\"p<sub>tip</sub>\",";
 Trace_file << "\"p<sub>tip (Bretherton)</sub>\",\"u<sub>stag</sub>\",";
 Trace_file << "\"v<sub>stag</sub>\"";
 Trace_file << "\"<greek>a</greek><sub>bottom</sub>\",";
 Trace_file << "\"<greek>a</greek><sub>top</sub>\"";
 Trace_file << std::endl;
 
 // Initial scaling factor for Ca reduction
 double factor=2.0;
 
 //Doc initial solution
 doc_solution(doc_info);
 
//Loop over the steps
 for (unsigned step=1;step<=nsteps;step++)
  {
   cout << std::endl << "STEP " << step << std::endl;
 
   // Newton method tends to converge is initial stays bounded below
   // a certain tolerance: This is cheaper to check than restarting
   // the Newton method after divergence (we'd also need to back up
   // the previous solution)
   double maxres=100.0;
   while (true)
    {
     cout << "Checking max. res for Ca = " 
          << Global_Physical_Variables::Ca << ".   Max. residual: ";
 
     // Check the maximum residual
     DoubleVector residuals;
     actions_before_newton_solve();
     actions_before_newton_convergence_check();
     get_residuals(residuals);
     double max_res=residuals.max();
     cout << max_res;
 
     // Check what to do 
     if (max_res>maxres)
      {
       cout << ". Too big!" << std::endl;
       // Reset the capillary number
       Global_Physical_Variables::Ca *= factor;       
       // Reduce the factor
       factor=1.0+(factor-1.0)/1.5;
       cout << "New reduction factor: " << factor << std::endl;
       // Reduce the capillary number
       Global_Physical_Variables::Ca /= factor;
       // Try again
       continue;
      }
     // Looks promising: Let's proceed to solve
     else
      {
       cout << ". OK" << std::endl << std::endl;
       // Break out of the Ca adjustment loop
       break;
      }
    }
 
   //Solve
   cout << "Solving for capillary number: " 
        << Global_Physical_Variables::Ca << std::endl;
   newton_solve();
 
   // Doc solution
   doc_info.number()++;
   doc_solution(doc_info);
   
   // Reduce the capillary number
   Global_Physical_Variables::Ca /= factor;
  }
 
}
 
 
 
//======================================================================
/// Driver code for unsteady two-layer fluid problem. If there are
/// any command line arguments, we regard this as a validation run
/// and perform only a single step.
//======================================================================
int main(int argc, char* argv[]) 
{
 
 
 // Store command line arguments
 CommandLineArgs::setup(argc,argv);
 
 // Set physical parameters:
 
 //Set direction of gravity: Vertically downwards
 Global_Physical_Variables::G[0] = 0.0;
 Global_Physical_Variables::G[1] = -1.0;
 
 // Womersley number = Reynolds number (St = 1)
 Global_Physical_Variables::ReSt = 0.0;
 Global_Physical_Variables::Re = Global_Physical_Variables::ReSt;
 
 // The Capillary number
 Global_Physical_Variables::Ca = 0.05;
 
 // Re/Fr -- a measure of gravity...
 Global_Physical_Variables::ReInvFr = 0.0;
 
 //Set up the problem
 BrethertonProblem<BrethertonElement<SpineElement<QCrouzeixRaviartElement<2> > > > problem;
 
 // Self test:
 problem.self_test();
 
 // Number of steps: 
 unsigned nstep;
 if (CommandLineArgs::Argc>1)
  {
   // Validation run: Just one step
   nstep=2;
  }
 else
  {
   // Full run otherwise
   nstep=30;
  }
 
 // Run the parameter study: Perform nstep steps
 problem.parameter_study(nstep);
 
}