airy_cantilever.cc

Go to the documentation of this file.

//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented, 
//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
//LIC// This library is distributed in the hope that it will be useful,
//LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
//LIC// 
//LIC// You should have received a copy of the GNU Lesser General Public
//LIC// License along with this library; if not, write to the Free Software
//LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
//LIC// 02110-1301  USA.
//LIC// 
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//Driver for Airy cantilever beam problem
 
//Oomph-lib includes
#include "generic.h"
#include "solid.h"
#include "constitutive.h"
 
//The mesh
#include "meshes/rectangular_quadmesh.h"
 
using namespace std;
 
using namespace oomph;
 
namespace oomph
{
 
//=================start_wrapper==================================
/// Wrapper class for solid elements to modify their output 
/// functions.
//================================================================
template <class ELEMENT>
class MySolidElement : public virtual ELEMENT
{
 
public:
 
 /// Constructor: Call constructor of underlying element
 MySolidElement() : ELEMENT() {};
 
 /// Overload output function:
 void output(std::ostream &outfile, const unsigned &n_plot)
  {
 
   // Element dimension
   unsigned el_dim = this->dim();
 
   Vector<double> s(el_dim);
   Vector<double> x(el_dim);
   DenseMatrix<double> sigma(el_dim,el_dim);
   
   switch(el_dim)
    {
     
    case 2:
 
     //Tecplot header info 
     outfile << "ZONE I=" << n_plot << ", J=" << n_plot << std::endl;
     
     //Loop over element nodes
     for(unsigned l2=0;l2<n_plot;l2++)
      {
       s[1] = -1.0 + l2*2.0/(n_plot-1);
       for(unsigned l1=0;l1<n_plot;l1++)
        {
         s[0] = -1.0 + l1*2.0/(n_plot-1);
         
         // Get Eulerian coordinates and stress
         this->interpolated_x(s,x);
         this->get_stress(s,sigma);
 
         //Output the x,y,..
         for(unsigned i=0;i<el_dim;i++) 
          {outfile << x[i] << " ";}
 
         // Output stress
         outfile << sigma(0,0) << " "
                 << sigma(1,0) << " "
                 << sigma(1,1) << " "
                 << std::endl;
        }
      }
 
     break;
     
    default:
 
     std::ostringstream error_message;
     error_message << "Output for dim !=2 not implemented" << std::endl;
     throw OomphLibError(error_message.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
  
  }
 
};
 
 
 
//===========start_face_geometry==============================================
/// FaceGeometry of wrapped element is the same as the underlying element
//============================================================================
template<class ELEMENT>
class FaceGeometry<MySolidElement<ELEMENT> > :
 public virtual FaceGeometry<ELEMENT>
{
public:
 
 /// Constructor [this was only required explicitly
 /// from gcc 4.5.2 onwards...]
 FaceGeometry() : FaceGeometry<ELEMENT>() {}
 
};
 
 
}
 
 
 
 
 
 
/// ////////////////////////////////////////////////////////////////////
/// ////////////////////////////////////////////////////////////////////
/// ////////////////////////////////////////////////////////////////////
 
 
 
//=======start_namespace==========================================
/// Global variables
//================================================================
namespace Global_Physical_Variables
{
 
 /// Half height of beam
 double H=0.5;
 
 /// Length of beam
 double L=10.0;
 
 /// Pointer to constitutive law
 ConstitutiveLaw* Constitutive_law_pt;
 
 /// Elastic modulus
 double E=1.0;
 
 /// Poisson's ratio
 double Nu=0.3;
 
 /// Uniform pressure
 double P = 0.0;
 
 /// Constant pressure load. The arguments to this function are imposed
 /// on us by the SolidTractionElements which allow the traction to 
 /// depend on the Lagrangian and Eulerian coordinates x and xi, and on the 
 /// outer unit normal to the surface. Here we only need the outer unit
 /// normal.
 void constant_pressure(const Vector<double> &xi, const Vector<double> &x,
                        const Vector<double> &n, Vector<double> &traction)
 {
  unsigned dim = traction.size();
  for(unsigned i=0;i<dim;i++)
   {
    traction[i] = -P*n[i];
   }
 } // end traction
 
 
 /// Non-dim gravity
 double Gravity=0.0;
 
 /// Non-dimensional gravity as body force
 void gravity(const double& time, 
              const Vector<double> &xi, 
              Vector<double> &b)
 {
  b[0]=0.0;
  b[1]=-Gravity;
 }
 
} //end namespace
 
 
 
//=============begin_problem============================================ 
/// Problem class for the cantilever "beam" structure.
//====================================================================== 
template<class ELEMENT>
class CantileverProblem : public Problem
{
 
public:
 
 /// Constructor:
 CantileverProblem();
 
 /// Update function (empty)
 void actions_after_newton_solve() {}
 
 /// Update function (empty)
 void actions_before_newton_solve() {}
 
 /// Access function for the solid mesh
 ElasticRefineableRectangularQuadMesh<ELEMENT>*& solid_mesh_pt() 
  {return Solid_mesh_pt;} 
 
 /// Access function to the mesh of surface traction elements
 SolidMesh*& traction_mesh_pt(){return Traction_mesh_pt;} 
 
 /// Actions before adapt: Wipe the mesh of traction elements
 void actions_before_adapt();
 
 /// Actions after adapt: Rebuild the mesh of traction elements
 void actions_after_adapt();
 
 /// Doc the solution
 void doc_solution();
 
private:
 
 /// Pass pointer to traction function to the
 /// elements in the traction mesh
 void set_traction_pt();
 
 /// Create traction elements
 void create_traction_elements();
 
 /// Delete traction elements
 void delete_traction_elements();
 
 /// Trace file
 ofstream Trace_file;
 
 /// Pointers to node whose position we're tracing
 Node* Trace_node_pt;
 
 /// Pointer to solid mesh
 ElasticRefineableRectangularQuadMesh<ELEMENT>* Solid_mesh_pt;
 
 /// Pointer to mesh of traction elements
 SolidMesh* Traction_mesh_pt;
 
 /// DocInfo object for output
 DocInfo Doc_info;
 
};
 
 
//===========start_of_constructor======================================= 
/// Constructor: 
//====================================================================== 
template<class ELEMENT>
CantileverProblem<ELEMENT>::CantileverProblem() 
{
 
 // Create the mesh
 
 // # of elements in x-direction
 unsigned n_x=20;
 
 // # of elements in y-direction
 unsigned n_y=2;
 
 // Domain length in x-direction
 double l_x= Global_Physical_Variables::L;
 
 // Domain length in y-direction
 double l_y=2.0*Global_Physical_Variables::H;
 
 // Shift mesh downwards so that centreline is at y=0:
 Vector<double> origin(2);
 origin[0]=0.0;
 origin[1]=-0.5*l_y;
 
 //Now create the mesh 
 solid_mesh_pt() = new ElasticRefineableRectangularQuadMesh<ELEMENT>(
  n_x,n_y,l_x,l_y,origin);
 
 // Set error estimator
 solid_mesh_pt()->spatial_error_estimator_pt()=new Z2ErrorEstimator;
 
 //Assign the physical properties to the elements before any refinement
 //Loop over the elements in the main mesh
 unsigned n_element =solid_mesh_pt()->nelement();
 for(unsigned i=0;i<n_element;i++)
  {
   //Cast to a solid element
   ELEMENT *el_pt = dynamic_cast<ELEMENT*>(solid_mesh_pt()->element_pt(i));
   
   // Set the constitutive law
   el_pt->constitutive_law_pt() =
    Global_Physical_Variables::Constitutive_law_pt;
 
   //Set the body force
   el_pt->body_force_fct_pt() = Global_Physical_Variables::gravity;
  }
 
 
 // Choose a control node: The last node in the solid mesh
 unsigned nnod=solid_mesh_pt()->nnode();
 Trace_node_pt=solid_mesh_pt()->node_pt(nnod-1);
 
 // Refine the mesh uniformly
 solid_mesh_pt()->refine_uniformly();
 
 // Construct the traction element mesh
 Traction_mesh_pt=new SolidMesh;
 create_traction_elements();
 
 // Pass pointer to traction function to the elements
 // in the traction mesh
 set_traction_pt();
 
 // Solid mesh is first sub-mesh
 add_sub_mesh(solid_mesh_pt());
 
 // Add traction sub-mesh
 add_sub_mesh(traction_mesh_pt());
 
 // Build combined "global" mesh
 build_global_mesh();
 
 // Pin the left boundary (boundary 3) in both directions
 unsigned n_side = mesh_pt()->nboundary_node(3);
 
 // Loop over the nodes
 for(unsigned i=0;i<n_side;i++)
  {
   solid_mesh_pt()->boundary_node_pt(3,i)->pin_position(0);
   solid_mesh_pt()->boundary_node_pt(3,i)->pin_position(1);
  }
 
 // Pin the redundant solid pressures (if any)
 PVDEquationsBase<2>::pin_redundant_nodal_solid_pressures(
  solid_mesh_pt()->element_pt());
 
 //Attach the boundary conditions to the mesh
 cout << assign_eqn_numbers() << std::endl; 
 
 // Set output directory
 Doc_info.set_directory("RESLT");
 
 // Open trace file
 char filename[100];   
 sprintf(filename,"%s/trace.dat",Doc_info.directory().c_str());
 Trace_file.open(filename);
 
 
} //end of constructor
 
 
//=====================start_of_actions_before_adapt======================
/// Actions before adapt: Wipe the mesh of traction elements
//========================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::actions_before_adapt()
{
 // Kill the traction elements and wipe surface mesh
 delete_traction_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=======================
///  Actions after adapt: Rebuild the mesh of traction elements
//========================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::actions_after_adapt()
{
 // Create traction elements from all elements that are 
 // adjacent to boundary 2 and add them to surface meshes
 create_traction_elements();
 
 // Rebuild the Problem's global mesh from its various sub-meshes
 rebuild_global_mesh();
 
 // Pin the redundant solid pressures (if any)
 PVDEquationsBase<2>::pin_redundant_nodal_solid_pressures(
  solid_mesh_pt()->element_pt());
 
 // Set pointer to prescribed traction function for traction elements
 set_traction_pt();
 
}// end of actions_after_adapt
 
 
 
//==================start_of_set_traction_pt==============================
/// Set pointer to traction function for the relevant
/// elements in the traction mesh
//========================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::set_traction_pt()
{
 // Loop over the elements in the traction element mesh
 // for elements on the top boundary (boundary 2)
 unsigned n_element=traction_mesh_pt()->nelement();
 for(unsigned i=0;i<n_element;i++)
  {
   //Cast to a solid traction element
   SolidTractionElement<ELEMENT> *el_pt = 
    dynamic_cast<SolidTractionElement<ELEMENT>*>
    (traction_mesh_pt()->element_pt(i));
 
   //Set the traction function
   el_pt->traction_fct_pt() = Global_Physical_Variables::constant_pressure;
  }
 
}// end of set traction pt
 
 
 
//============start_of_create_traction_elements==============================
/// Create traction elements 
//=======================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::create_traction_elements()
{
 // Traction elements are located on boundary 2:
 unsigned b=2;
 
 // How many bulk elements are adjacent to boundary b?
 unsigned n_element = solid_mesh_pt()->nboundary_element(b);
 
 // Loop over the bulk elements adjacent to boundary b
 for(unsigned e=0;e<n_element;e++)
  {
   // Get pointer to the bulk element that is adjacent to boundary b
   ELEMENT* bulk_elem_pt = dynamic_cast<ELEMENT*>(
    solid_mesh_pt()->boundary_element_pt(b,e));
   
   //Find the index of the face of element e along boundary b
   int face_index = solid_mesh_pt()->face_index_at_boundary(b,e);
      
   // Create new element and add to mesh
   Traction_mesh_pt->add_element_pt(new SolidTractionElement<ELEMENT>
                                    (bulk_elem_pt,face_index));   
  }  
 
 // Pass the pointer to the traction function to the traction elements
 set_traction_pt();
 
} // end of create_traction_elements
 
 
 
 
//============start_of_delete_traction_elements==============================
/// Delete traction elements and wipe the  traction meshes
//=======================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::delete_traction_elements()
{
 // How many surface elements are in the surface mesh
 unsigned n_element = Traction_mesh_pt->nelement();
 
 // Loop over the surface elements
 for(unsigned e=0;e<n_element;e++)
  {
   // Kill surface element
   delete Traction_mesh_pt->element_pt(e);
  }
 
 // Wipe the mesh
 Traction_mesh_pt->flush_element_and_node_storage();
 
} // end of delete_traction_elements
 
 
 
//==============start_doc===========================================
/// Doc the solution
//==================================================================
template<class ELEMENT>
void CantileverProblem<ELEMENT>::doc_solution()
{
 
 ofstream some_file;
 char filename[100];
 
 // Number of plot points
 unsigned n_plot = 5; 
 
 // Output shape of and stress in deformed body
 //--------------------------------------------
 sprintf(filename,"%s/soln%i.dat",Doc_info.directory().c_str(),
         Doc_info.number());
 some_file.open(filename);
 solid_mesh_pt()->output(some_file,n_plot);
 some_file.close();
 
 
 // Output St. Venant solution
 //---------------------------
 sprintf(filename,"%s/exact_soln%i.dat",Doc_info.directory().c_str(),
         Doc_info.number());
 some_file.open(filename);
 
 // Element dimension
 unsigned el_dim=2;
 
 Vector<double> s(el_dim);
 Vector<double> x(el_dim);
 Vector<double> xi(el_dim);
 DenseMatrix<double> sigma(el_dim,el_dim);
 
 // Constants for exact (St. Venant) solution
 double a=-1.0/4.0*Global_Physical_Variables::P;
 double b=-3.0/8.0*Global_Physical_Variables::P/Global_Physical_Variables::H;
 double c=1.0/8.0*Global_Physical_Variables::P/
  pow(Global_Physical_Variables::H,3);
 double d=1.0/20.0*Global_Physical_Variables::P/Global_Physical_Variables::H;
 
 // Loop over all elements to plot exact solution for stresses
 unsigned nel=solid_mesh_pt()->nelement();
 for (unsigned e=0;e<nel;e++)
  {   
   // Get pointer to element
   SolidFiniteElement* el_pt=dynamic_cast<SolidFiniteElement*>(
    solid_mesh_pt()->element_pt(e));
   
   //Tecplot header info 
   some_file << "ZONE I=" << n_plot << ", J=" << n_plot << std::endl;
   
   //Loop over plot points
   for(unsigned l2=0;l2<n_plot;l2++)
    {
     s[1] = -1.0 + l2*2.0/(n_plot-1);
     for(unsigned l1=0;l1<n_plot;l1++)
      {
       s[0] = -1.0 + l1*2.0/(n_plot-1);
       
       // Get Eulerian and Lagrangian coordinates
       el_pt->interpolated_x(s,x);
       el_pt->interpolated_xi(s,xi);
       
       //Output the x,y,..
       for(unsigned i=0;i<el_dim;i++) 
        {some_file << x[i] << " ";}
       
       // Change orientation of coordinate system relative
       // to solution in lecture notes
       double xx=Global_Physical_Variables::L-xi[0];
       double yy=xi[1];
 
         // Approximate analytical (St. Venant) solution
         sigma(0,0)=c*(6.0*xx*xx*yy-4.0*yy*yy*yy)+
          6.0*d*yy;
         sigma(1,1)=2.0*(a+b*yy+c*yy*yy*yy);
         sigma(1,0)=2.0*(b*xx+3.0*c*xx*yy*yy);
         sigma(0,1)=sigma(1,0);
 
         // Output stress
         some_file << sigma(0,0) << " "
                   << sigma(1,0) << " "
                   << sigma(1,1) << " "
                   << std::endl;
        }
      }
  }
 some_file.close();
 
 // Write trace file: Load/displacement characteristics
 Trace_file << Global_Physical_Variables::P  << " " 
            << Trace_node_pt->x(0) << " " 
            << Trace_node_pt->x(1) << " " 
            << std::endl;
 
 // Increment label for output files
 Doc_info.number()++;
 
} //end doc
 
 
 
//=======start_of_main==================================================
/// Driver for cantilever beam loaded by surface traction and/or
/// gravity
//======================================================================
int main()
{
 
 // Create generalised Hookean constitutive equations
 Global_Physical_Variables::Constitutive_law_pt = 
  new GeneralisedHookean(&Global_Physical_Variables::Nu,
                         &Global_Physical_Variables::E);
 
 //Set up the problem
 CantileverProblem<MySolidElement<RefineableQPVDElement<2,3> > > problem;
 
 
 // Uncomment these as an exercise
 
 // CantileverProblem<MySolidElement<
 //  RefineableQPVDElementWithContinuousPressure<2> > > problem;
 
 // CantileverProblem<MySolidElement<
 //  RefineableQPVDElementWithPressure<2> > > problem;
 
 // Initial values for parameter values
 Global_Physical_Variables::P=0.0; 
 Global_Physical_Variables::Gravity=0.0;
 
 // Max. number of adaptations per solve
 unsigned max_adapt=3;
 
 //Parameter incrementation
 unsigned nstep=5; 
 double p_increment=1.0e-5;   
 for(unsigned i=0;i<nstep;i++)
  {
   // Increment pressure load
   Global_Physical_Variables::P+=p_increment;
 
   // Solve the problem with Newton's method, allowing
   // up to max_adapt mesh adaptations after every solve.
   problem.newton_solve(max_adapt);
 
   // Doc solution
   problem.doc_solution();
 
  }
 
} //end of main