tensioned_string.cc

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//LIC// ====================================================================
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//LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
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//LIC//====================================================================
//Driver function for a simple beam proble
 
//OOMPH-LIB includes
#include "generic.h"
#include "beam.h"
#include "meshes/one_d_lagrangian_mesh.h"
 
using namespace std;
 
using namespace oomph;
 
//========start_of_namespace========================
/// Namespace for physical parameters
//==================================================
namespace Global_Physical_Variables
{
 /// Non-dimensional thickness
 double H;
 
 /// 2nd Piola Kirchhoff pre-stress
 double Sigma0;
 
 /// Pressure load
 double P_ext;
 
 /// Load function: Apply a constant external pressure to the beam
 void load(const Vector<double>& xi, const Vector<double> &x,
           const Vector<double>& N, Vector<double>& load)
 {
  for(unsigned i=0;i<2;i++) {load[i] = -P_ext*N[i];}
 }
 
} // end of namespace
 
//======start_of_problem_class==========================================
/// Beam problem object
//======================================================================
class ElasticBeamProblem : public Problem
{
public:
 
 /// Constructor: The arguments are the number of elements, 
 /// the length of domain
 ElasticBeamProblem(const unsigned &n_elem, const double &length);
 
 /// Conduct a parameter study
 void parameter_study();
 
 /// Return pointer to the mesh
 OneDLagrangianMesh<HermiteBeamElement>* mesh_pt() 
  {return dynamic_cast<OneDLagrangianMesh<HermiteBeamElement>*>
    (Problem::mesh_pt());}
 
 /// No actions need to be performed after a solve
 void actions_after_newton_solve() {}
 
 /// No actions need to be performed before a solve
 void actions_before_newton_solve() {}
 
private:
 
 /// Pointer to the node whose displacement is documented
 Node* Doc_node_pt;
 
 /// Length of domain (in terms of the Lagrangian coordinates)
 double Length;
 
 /// Pointer to geometric object that represents the beam's undeformed shape
 GeomObject* Undef_beam_pt;
 
}; // end of problem class
 
 
//=============start_of_constructor=====================================
/// Constructor for elastic beam problem
//======================================================================
ElasticBeamProblem::ElasticBeamProblem(const unsigned &n_elem,
                                       const double &length) : Length(length)
{
 // Set the undeformed beam to be a straight line at y=0
 Undef_beam_pt=new StraightLine(0.0); 
 
 // Create the (Lagrangian!) mesh, using the geometric object
 // Undef_beam_pt to specify the initial (Eulerian) position of the
 // nodes.
 Problem::mesh_pt() = 
  new OneDLagrangianMesh<HermiteBeamElement>(n_elem,length,Undef_beam_pt);
 
 // Set the boundary conditions: Each end of the beam is fixed in space
 // Loop over the boundaries (ends of the beam)
 for(unsigned b=0;b<2;b++)
  {
   // Pin displacements in both x and y directions
   // [Note: The mesh_pt() function has been overloaded
   //  to return a pointer to the actual mesh, rather than
   //  a pointer to the Mesh base class. The current mesh is derived
   //  from the SolidMesh class. In such meshes, all access functions
   //  to the nodes, such as boundary_node_pt(...), are overloaded
   //  to return pointers to SolidNodes (whose position can be
   //  pinned) rather than "normal" Nodes.]
   mesh_pt()->boundary_node_pt(b,0)->pin_position(0); 
   mesh_pt()->boundary_node_pt(b,0)->pin_position(1); 
  }
 
 //Find number of elements in the mesh
 unsigned n_element = mesh_pt()->nelement();
 
 //Loop over the elements to set physical parameters etc.
 for(unsigned e=0;e<n_element;e++)
  {
   // Upcast to the specific element type
   HermiteBeamElement *elem_pt = 
    dynamic_cast<HermiteBeamElement*>(mesh_pt()->element_pt(e));
   
   // Set physical parameters for each element:
   elem_pt->sigma0_pt() = &Global_Physical_Variables::Sigma0;
   elem_pt->h_pt() = &Global_Physical_Variables::H;
 
   // Set the load Vector for each element
   elem_pt->load_vector_fct_pt() = &Global_Physical_Variables::load;
 
   // Set the undeformed shape for each element
   elem_pt->undeformed_beam_pt() = Undef_beam_pt;
  } // end of loop over elements
 
 // Choose node at which displacement is documented (halfway along -- provided
 // we have an odd number of nodes; complain if this is not the
 // case because the comparison with the exact solution will be wrong 
 // otherwise!)
 unsigned n_nod=mesh_pt()->nnode();
 if (n_nod%2!=1)
  {
   cout << "Warning: Even number of nodes " << n_nod << std::endl;
   cout << "Comparison with exact solution will be misleading..." << std::endl;
  }
 Doc_node_pt=mesh_pt()->node_pt((n_nod+1)/2-1);
 
 // Assign the global and local equation numbers
 cout << "# of dofs " << assign_eqn_numbers() << std::endl;
 
} // end of constructor
 
 
//=======start_of_parameter_study==========================================
/// Solver loop to perform parameter study
//=========================================================================
void ElasticBeamProblem::parameter_study()
{
 // Over-ride the default maximum value for the residuals
 Problem::Max_residuals = 1.0e10;
 
 // Set the increments in control parameters
 double pext_increment = 0.001;
 
 // Set initial values for control parameters 
 Global_Physical_Variables::P_ext = 0.0 - pext_increment;
 
 // Create label for output
 DocInfo doc_info;
 
 // Set output directory -- this function checks if the output
 // directory exists and issues a warning if it doesn't.
 doc_info.set_directory("RESLT");
 
 // Open a trace file
 ofstream trace("RESLT/trace_beam.dat");
 
 // Write a header for the trace file
 trace << 
  "VARIABLES=\"p_e_x_t\",\"d\"" << 
  ", \"p_e_x_t_(_e_x_a_c_t_)\"" << std::endl;
 
 // Output file stream used for writing results
 ofstream file;
 // String used for the filename
 char filename[100]; 
 
 // Loop over parameter increments
 unsigned nstep=10;
 for(unsigned i=1;i<=nstep;i++)
  {
   // Increment pressure
   Global_Physical_Variables::P_ext += pext_increment;
   
   // Solve the system
   newton_solve();
    
   // Calculate exact solution for `string under tension' (applicable for
   // small wall thickness and pinned ends)
 
   // The tangent of the angle beta
   double tanbeta =-2.0*Doc_node_pt->x(1)/Length;
 
   double exact_pressure = 0.0;
   //If the beam has deformed, calculate the pressure required
   if(tanbeta!=0)
    {
      
      //Calculate the opening angle alpha
      double alpha = 2.0*atan(2.0*tanbeta/(1.0-tanbeta*tanbeta));
 
      // Jump back onto the main branch if alpha>180 degrees
      if (alpha<0) alpha+=2.0*MathematicalConstants::Pi;
 
     // Green strain:
     double gamma=0.5*(0.25*alpha*alpha/(sin(0.5*alpha)*sin(0.5*alpha))-1.0);
 
     //Calculate the exact pressure
     exact_pressure=Global_Physical_Variables::H*
      (Global_Physical_Variables::Sigma0+gamma)*alpha/Length;
    } 
   
   // Document the solution
   sprintf(filename,"RESLT/beam%i.dat",i);
   file.open(filename);
   mesh_pt()->output(file,5);
   file.close();
   
   // Write trace file: Pressure, displacement and exact solution
   // (for string under tension)
   trace << Global_Physical_Variables::P_ext  << " " 
         << abs(Doc_node_pt->x(1))
         << " " << exact_pressure 
         << std::endl;
  }
 
} // end of parameter study
 
//========start_of_main================================================
/// Driver for beam (string under tension) test problem 
//=====================================================================
int main()
{
 
 // Set the non-dimensional thickness 
 Global_Physical_Variables::H=0.01; 
 
 // Set the 2nd Piola Kirchhoff prestress
 Global_Physical_Variables::Sigma0=0.1; 
 
 // Set the length of domain
 double L = 10.0;
 
 // Number of elements (choose an even number if you want the control point 
 // to be located at the centre of the beam)
 unsigned n_element = 10;
 
 // Construst the problem
 ElasticBeamProblem problem(n_element,L);
 
 // Check that we're ready to go:
 cout << "\n\n\nProblem self-test ";
 if (problem.self_test()==0) 
  {
   cout << "passed: Problem can be solved." << std::endl;
  }
 else 
  {
   throw OomphLibError("Self test failed",
                       OOMPH_CURRENT_FUNCTION,
                       OOMPH_EXCEPTION_LOCATION);
  }
 
 // Conduct parameter study
 problem.parameter_study();
 
} // end of main