fe_lagrange_shape_1d.c

一个用来实现偏微分方程中网格的计算库

// $Id: fe_lagrange_shape_1D.C 2789 2008-04-13 02:24:40Z roystgnr $

// The libMesh Finite Element Library.
// Copyright (C) 2002-2007  Benjamin S. Kirk, John W. Peterson
  
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
  
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
  
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA


// C++ inlcludes

// Local includes
#include "fe.h"
#include "elem.h"




template <>
Real FE<1,LAGRANGE>::shape(const ElemType,
			   const Order order,
			   const unsigned int i,
			   const Point& p)
{
  const Real xi = p(0);

	
  switch (order)
    {
      // Lagrange linears
    case FIRST:
      {
	libmesh_assert (i<2);
	
	switch (i)
	  {
	  case 0:
	    return .5*(1. - xi);
	  
	  case 1:
	    return .5*(1. + xi);
	   
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }

      
	      
      // Lagrange quadratics
    case SECOND:
      {
	libmesh_assert (i<3);
	
	switch (i)
	  {
	  case 0:
	    return .5*xi*(xi - 1.);
	    
	  case 1:
	    return .5*xi*(xi + 1);
	   
	  case 2:
	    return (1. - xi*xi);
	    
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }

      

      // Lagrange cubics
    case THIRD:
      {
	libmesh_assert (i<4);
	
	switch (i)
	  {
	  case 0:
	    return 9./16.*(1./9.-xi*xi)*(xi-1.);
	   
	  case 1:
	    return -9./16.*(1./9.-xi*xi)*(xi+1.);
	   
	  case 2:
	    return 27./16.*(1.-xi*xi)*(1./3.-xi);
	   
	  case 3:
	    return 27./16.*(1.-xi*xi)*(1./3.+xi);
	   
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }
      
    default:
      {
	std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
	libmesh_error();
      }
    }

  
  libmesh_error();
  return 0.;
}



template <>
Real FE<1,LAGRANGE>::shape(const Elem* elem,
			   const Order order,
			   const unsigned int i,
			   const Point& p)
{
  libmesh_assert (elem != NULL);
  
  return FE<1,LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}



template <>
Real FE<1,LAGRANGE>::shape_deriv(const ElemType,
				 const Order order,
				 const unsigned int i,
				 const unsigned int j,
				 const Point& p)
{
  // only d()/dxi in 1D!
  
  libmesh_assert (j == 0);
	
  const Real xi = p(0);

	
  switch (order)
    {      
      // Lagrange linear shape function derivatives
    case FIRST:
      {
	libmesh_assert (i<2);
	
	switch (i)
	  {
	  case 0:
	    return -.5;
	   
	  case 1:
	    return .5;
	    
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }
	
	      
      // Lagrange quadratic shape function derivatives
    case SECOND:
      {
	libmesh_assert (i<3);
	
	switch (i)
	  {
	  case 0:
	    return xi-.5;
	    
	  case 1:
	    return xi+.5;
	    
	  case 2:
	    return -2.*xi;
	   
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }


      // Lagrange cubic shape function derivatives
    case THIRD:
      {
	libmesh_assert (i<4);
	
	switch (i)
	  {
	  case 0:
	    return -9./16.*(3.*xi*xi-2.*xi-1./9.);
	    
	  case 1:
	    return -9./16.*(-3.*xi*xi-2.*xi+1./9.);
	   
	  case 2:
	    return 27./16.*(3.*xi*xi-2./3.*xi-1.);
	    
	  case 3:
	    return 27./16.*(-3.*xi*xi-2./3.*xi+1.);
	    
	  default:
	    std::cerr << "Invalid shape function index!" << std::endl;
	    libmesh_error();
	  }
      }

      
    default:
      {
	std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
	libmesh_error();
      }
    }

  libmesh_error();
  return 0.;
}



template <>
Real FE<1,LAGRANGE>::shape_deriv(const Elem* elem,
				 const Order order,
				 const unsigned int i,
				 const unsigned int j,
				 const Point& p)
{
  libmesh_assert (elem != NULL);
  
  return FE<1,LAGRANGE>::shape_deriv(elem->type(),
				     static_cast<Order>(order + elem->p_level()), i, j, p);
}




template <>
Real FE<1,LAGRANGE>::shape_second_deriv(const ElemType,
					const Order order,
					const unsigned int i,
					const unsigned int j,
					const Point& p)
{
  // Don't need to switch on j.  1D shape functions
  // depend on xi only!

  const Real xi = p(0);
  libmesh_assert (j == 0);
  
  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
	// All second derivatives of linears are zero....
	return 0.;
      }

      // quadratic Lagrange shape functions
    case SECOND:
      {
	switch (i)
	  {
	  case 0:
	    return 1.;
	    
	  case 1:
	    return 1.;
	    
	  case 2:
	    return -2.;
	    
	  default:
	    {
	      std::cerr << "Invalid shape function index requested!"
			<< std::endl;
	      libmesh_error();
	    }
	  }
      } // end case SECOND

    case THIRD:
     {
      switch (i)
        {
        case 0:
            return -9./16.*(6.*xi-2);
            
        case 1:
            return -9./16.*(-6*xi-2.);

        case 2:
            return 27./16.*(6*xi-2./3.);
            
        case 3:
            return 27./16.*(-6*xi-2./3.);
            
        default:
          {
            std::cerr << "Invalid shape function index requested!"
                      << std::endl;
            libmesh_error();
          }
        }
     } // end case THIRD


    default:
      {
	std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
	libmesh_error();
      }
    } // end switch (order)
  
  libmesh_error();
  return 0.;
}



template <>
Real FE<1,LAGRANGE>::shape_second_deriv(const Elem* elem,
				        const Order order,
				        const unsigned int i,
				        const unsigned int j,
				        const Point& p)
{
  libmesh_assert (elem != NULL);
  
  return FE<1,LAGRANGE>::shape_second_deriv(elem->type(),
				            static_cast<Order>(order + elem->p_level()), i, j, p);
}