fe_clough_shape_1d.c

一个用来实现偏微分方程中网格的计算库

// $Id: fe_clough_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"


// Anonymous namespace for persistant variables.
// This allows us to cache the global-to-local mapping transformation
// This should also screw up multithreading royally
namespace
{
  static unsigned int old_elem_id = libMesh::invalid_uint;
  // Coefficient naming: d(1)d(2n) is the coefficient of the
  // global shape function corresponding to value 1 in terms of the
  // local shape function corresponding to normal derivative 2
  static Real d1xd1x, d2xd2x;

Real clough_raw_shape_second_deriv(const unsigned int basis_num,
                                   const unsigned int deriv_type,
                                   const Point& p);
Real clough_raw_shape_deriv(const unsigned int basis_num,
                            const unsigned int deriv_type,
                            const Point& p);
Real clough_raw_shape(const unsigned int basis_num,
                      const Point& p);


// Compute the static coefficients for an element
void clough_compute_coefs(const Elem* elem)
{
  // Coefficients are cached from old elements
  if (elem->id() == old_elem_id)
    return;

  old_elem_id = elem->id();

  const Order mapping_order        (elem->default_order());
  const ElemType mapping_elem_type (elem->type());
  const int n_mapping_shape_functions =
    FE<1,LAGRANGE>::n_shape_functions(mapping_elem_type,
				      mapping_order);

  // Degrees of freedom are at vertices and edge midpoints
  std::vector<Point> dofpt;
  dofpt.push_back(Point(0));
  dofpt.push_back(Point(1));

  // Mapping functions - first derivatives at each dofpt
  std::vector<Real> dxdxi(2);
  std::vector<Real> dxidx(2);

  for (int p = 0; p != 2; ++p)
    {
      for (int i = 0; i != n_mapping_shape_functions; ++i)
        {
          const Real ddxi = FE<1,LAGRANGE>::shape_deriv 
            (mapping_elem_type, mapping_order, i, 0, dofpt[p]);
          dxdxi[p] += dofpt[p](0) * ddxi;
        }
    }

  // Calculate derivative scaling factors
  
  d1xd1x = dxdxi[0];
  d2xd2x = dxdxi[1];
}


  // Return shape function second derivatives on the unit interval
Real clough_raw_shape_second_deriv(const unsigned int basis_num,
                            const unsigned int deriv_type,
                            const Point& p)
{
  Real xi = p(0);

  switch (deriv_type)
  {

  // second derivative in xi-xi direction
  case 0:
  switch (basis_num)
    {
      case 0:
        return -6 + 12*xi;
      case 1:
        return 6 - 12*xi;
      case 2:
        return -4 + 6*xi;
      case 3:
        return -2 + 6*xi;
    }
  }

  libmesh_error();
  return 0.;
}



Real clough_raw_shape_deriv(const unsigned int basis_num,
                            const unsigned int deriv_type,
                            const Point& p)
{
  Real xi = p(0);

  switch (deriv_type)
  {
  case 0:
  switch (basis_num)
    {
      case 0:
        return -6*xi + 6*xi*xi;
      case 1:
        return 6*xi - 6*xi*xi;
      case 2:
        return 1 - 4*xi + 3*xi*xi;
      case 3:
        return -2*xi + 3*xi*xi;
    }
  }

  libmesh_error();
  return 0.;
}

Real clough_raw_shape(const unsigned int basis_num,
                      const Point& p)
{
  Real xi = p(0);

  switch (basis_num)
    {
      case 0:
        return 1 - 3*xi*xi + 2*xi*xi*xi;
      case 1:
        return 3*xi*xi - 2*xi*xi*xi;
      case 2:
        return xi - 2*xi*xi + xi*xi*xi;
      case 3:
        return -xi*xi + xi*xi*xi;
    }

  libmesh_error();
  return 0.;
}

  
} // end anonymous namespace



template <>
Real FE<1,CLOUGH>::shape(const ElemType,
			     const Order,
			     const unsigned int,
			     const Point&)
{
  std::cerr << "Clough-Tocher elements require the real element\n"
	    << "to construct gradient-based degrees of freedom."
	    << std::endl;
  
  libmesh_error();
  return 0.;
}



template <>
Real FE<1,CLOUGH>::shape(const Elem* elem,
			     const Order order,
			     const unsigned int i,
			     const Point& p)
{
  libmesh_assert (elem != NULL);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());
  
  switch (totalorder)
    {      
      // 3rd-order C1 cubic element
    case THIRD:
      {
	switch (type)
	  {
	    // C1 functions on the C1 cubic edge
	  case EDGE2:
	  case EDGE3:
	    {
	      libmesh_assert (i<4);

	      switch (i)
		{
		case 0:
		  return clough_raw_shape(0, p);
		case 1:
		  return clough_raw_shape(1, p);
		case 2:
		  return d1xd1x * clough_raw_shape(2, p);
		case 3:
                  return d2xd2x * clough_raw_shape(3, p);
		default:
		  libmesh_error();
		}
	    }
	  default:
            std::cerr << "ERROR: Unsupported element type!" << std::endl;
	    libmesh_error();
	  }
      }
      // by default throw an error
    default:
      std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
      libmesh_error();
    }
  
  libmesh_error();
  return 0.;
}



template <>
Real FE<1,CLOUGH>::shape_deriv(const ElemType,
				   const Order,			    
				   const unsigned int,
				   const unsigned int,
				   const Point&)
{
  std::cerr << "Clough-Tocher elements require the real element\n"
	    << "to construct gradient-based degrees of freedom."
	    << std::endl;

  libmesh_error();
  return 0.;
}



template <>
Real FE<1,CLOUGH>::shape_deriv(const Elem* elem,
				   const Order order,
				   const unsigned int i,
				   const unsigned int j,
				   const Point& p)
{
  libmesh_assert (elem != NULL);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());
  
  switch (totalorder)
    {      
      // 3rd-order C1 cubic element
    case THIRD:
      {
	switch (type)
	  {
	    // C1 functions on the C1 cubic edge
	  case EDGE2:
	  case EDGE3:
	    {
	      switch (i)
		{
		case 0:
		  return clough_raw_shape_deriv(0, j, p);
		case 1:
		  return clough_raw_shape_deriv(1, j, p);
		case 2:
		  return d1xd1x * clough_raw_shape_deriv(2, j, p);
		case 3:
                  return d2xd2x * clough_raw_shape_deriv(3, j, p);
		default:
		  libmesh_error();
		}
	    }
	  default:
            std::cerr << "ERROR: Unsupported element type!" << std::endl;
	    libmesh_error();
	  }
      }
      // by default throw an error
    default:
      std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
      libmesh_error();
    }
  
  libmesh_error();
  return 0.;
}



template <>
Real FE<1,CLOUGH>::shape_second_deriv(const Elem* elem,
                                      const Order order,
                                      const unsigned int i,
                                      const unsigned int j,
                                      const Point& p)
{
  libmesh_assert (elem != NULL);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());
  
  switch (totalorder)
    {      
      // 3rd-order C1 cubic element
    case THIRD:
      {
	switch (type)
	  {
	    // C1 functions on the C1 cubic edge
	  case EDGE2:
	  case EDGE3:
	    {
	      switch (i)
		{
		case 0:
		  return clough_raw_shape_second_deriv(0, j, p);
		case 1:
		  return clough_raw_shape_second_deriv(1, j, p);
		case 2:
		  return d1xd1x * clough_raw_shape_second_deriv(2, j, p);
		case 3:
                  return d2xd2x * clough_raw_shape_second_deriv(3, j, p);
		default:
		  libmesh_error();
		}
	    }
	  default:
            std::cerr << "ERROR: Unsupported element type!" << std::endl;
	    libmesh_error();
	  }
      }
      // by default throw an error
    default:
      std::cerr << "ERROR: Unsupported polynomial order!" << std::endl;
      libmesh_error();
    }
  
  libmesh_error();
  return 0.;
}