fe_hermite_shape_1d.c

一个用来实现偏微分方程中网格的计算库

// $Id: fe_hermite_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"
#include "utility.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;

// Compute the static coefficients for an element
void hermite_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(-1));
  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)
    {
      dxdxi[p] = 0;
      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] += elem->point(i)(0) * ddxi;
        }
    }

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


} // end anonymous namespace



template<>
Real FEHermite<1>::hermite_raw_shape_second_deriv
 (const unsigned int i, const Real xi)
{
  using Utility::pow;

  switch (i)
    {
      case 0:
        return 1.5 * xi;
      case 1:
        return -1.5 * xi;
      case 2:
        return 0.5 * (-1. + 3.*xi);
      case 3:
        return 0.5 * (1. + 3.*xi);
      case 4:
        return (8.*xi*xi + 4.*(xi*xi-1.))/24.;
      case 5:
        return (8.*xi*xi*xi + 12.*xi*(xi*xi-1.))/120.;
//      case 6:
//        return (8.*pow<4>(xi) + 20.*xi*xi*(xi*xi-1.) +
//          2.*(xi*xi-1)*(xi*xi-1))/720.;
      default:
        Real denominator = 720., xipower = 1.;
        for (unsigned n=6; n != i; ++n)
          {
            xipower *= xi;
            denominator *= (n+1);
          }
        return (8.*pow<4>(xi)*xipower +
                (8.*(i-4)+4.)*xi*xi*xipower*(xi*xi-1.) +
                (i-4)*(i-5)*xipower*(xi*xi-1.)*(xi*xi-1.))/denominator;
    }

  libmesh_error();
  return 0.;
}



template<>
Real FEHermite<1>::hermite_raw_shape_deriv
 (const unsigned int i, const Real xi)
{
  switch (i)
    {
      case 0:
        return 0.75 * (-1. + xi*xi);
      case 1:
        return 0.75 * (1. - xi*xi);
      case 2:
        return 0.25 * (-1. - 2.*xi + 3.*xi*xi);
      case 3:
        return 0.25 * (-1. + 2.*xi + 3.*xi*xi);
      case 4:
        return 4.*xi * (xi*xi-1.)/24.;
      case 5:
        return (4*xi*xi*(xi*xi-1.) + (xi*xi-1.)*(xi*xi-1.))/120.;
//      case 6:
//        return (4*xi*xi*xi*(xi*xi-1.) + 2*xi*(xi*xi-1.)*(xi*xi-1.))/720.;
      default:
        Real denominator = 720., xipower = 1.;
        for (unsigned n=6; n != i; ++n)
          {
            xipower *= xi;
            denominator *= (n+1);
          }
        return (4*xi*xi*xi*xipower*(xi*xi-1.) +
                (i-4)*xi*xipower*(xi*xi-1.)*(xi*xi-1.))/denominator;
    }

  libmesh_error();
  return 0.;
}

template<>
Real FEHermite<1>::hermite_raw_shape
 (const unsigned int i, const Real xi)
{
  switch (i)
    {
      case 0:
        return 0.25 * (2. - 3.*xi + xi*xi*xi);
      case 1:
        return 0.25 * (2. + 3.*xi - xi*xi*xi);
      case 2:
        return 0.25 * (1. - xi - xi*xi + xi*xi*xi);
      case 3:
        return 0.25 * (-1. - xi + xi*xi + xi*xi*xi);
      // All high order terms have the form x^(p-4)(x^2-1)^2/p!
      case 4:
        return (xi*xi-1.) * (xi*xi-1.)/24.;
      case 5:
        return xi * (xi*xi-1.) * (xi*xi-1.)/120.;
//      case 6:
//        return xi*xi * (xi*xi-1.) * (xi*xi-1.)/720.;
      default:
        Real denominator = 720., xipower = 1.;
        for (unsigned n=6; n != i; ++n)
          {
            xipower *= xi;
            denominator *= (n+1);
          }
        return (xi*xi*xipower*(xi*xi-1.)*(xi*xi-1.))/denominator;
                
    }

  libmesh_error();
  return 0.;
}


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



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

  hermite_compute_coefs(elem);

  const ElemType type = elem->type();

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

	      switch (i)
		{
		case 0:
		  return FEHermite<1>::hermite_raw_shape(0, p(0));
		case 1:
		  return d1xd1x * FEHermite<1>::hermite_raw_shape(2, p(0));
		case 2:
		  return FEHermite<1>::hermite_raw_shape(1, p(0));
		case 3:
                  return d2xd2x * FEHermite<1>::hermite_raw_shape(3, p(0));
		default:
		  return FEHermite<1>::hermite_raw_shape(i, p(0));
		}
	    }
	  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,HERMITE>::shape_deriv(const ElemType,
				const Order,			    
				const unsigned int,
				const unsigned int,
				const Point&)
{
  std::cerr << "Hermite elements require the real element\n"
	    << "to construct gradient-based degrees of freedom."
	    << std::endl;

  libmesh_error();
  return 0.;
}



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

  hermite_compute_coefs(elem);

  const ElemType type = elem->type();
  
  const Order totalorder = static_cast<Order>(order + elem->p_level());
  
  switch (totalorder)
    {      
      // Hermite cubic shape functions
    case THIRD:
      {
	switch (type)
	  {
	    // C1 functions on the C1 cubic edge
	  case EDGE2:
	  case EDGE3:
	    {
	      switch (i)
		{
		case 0:
		  return FEHermite<1>::hermite_raw_shape_deriv(0, p(0));
		case 1:
		  return d1xd1x * FEHermite<1>::hermite_raw_shape_deriv(2, p(0));
		case 2:
		  return FEHermite<1>::hermite_raw_shape_deriv(1, p(0));
		case 3:
                  return d2xd2x * FEHermite<1>::hermite_raw_shape_deriv(3, p(0));
		default:
		  return FEHermite<1>::hermite_raw_shape_deriv(i, p(0));
		}
	    }
	  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,HERMITE>::shape_second_deriv(const Elem* elem,
                                       const Order order,
                                       const unsigned int i,
                                       const unsigned int,
                                       const Point& p)
{
  libmesh_assert (elem != NULL);

  hermite_compute_coefs(elem);

  const ElemType type = elem->type();
  
  const Order totalorder = static_cast<Order>(order + elem->p_level());
  
  switch (totalorder)
    {      
      // Hermite cubic shape functions
    case THIRD:
      {
	switch (type)
	  {
	    // C1 functions on the C1 cubic edge
	  case EDGE2:
	  case EDGE3:
	    {
	      switch (i)
		{
		case 0:
		  return FEHermite<1>::hermite_raw_shape_second_deriv(0, p(0));
		case 1:
		  return d1xd1x * FEHermite<1>::hermite_raw_shape_second_deriv(2, p(0));
		case 2:
		  return FEHermite<1>::hermite_raw_shape_second_deriv(1, p(0));
		case 3:
                  return d2xd2x * FEHermite<1>::hermite_raw_shape_second_deriv(3, p(0));
		default:
		  return FEHermite<1>::hermite_raw_shape_second_deriv(i, p(0));
		}
	    }
	  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.;
}