quadrature_conical.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2019 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
 
// 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
 
 
// libMesh includes
#include "libmesh/quadrature_conical.h"
#include "libmesh/quadrature_gauss.h"
#include "libmesh/quadrature_jacobi.h"
#include "libmesh/enum_quadrature_type.h"
 
namespace libMesh
{
 
// See also the files:
// quadrature_conical_2D.C
// quadrature_conical_3D.C
// for additional implementation.
 
QuadratureType QConical::type() const
{
  return QCONICAL;
}
 
void QConical::init_1D(const ElemType, unsigned int)
{
  QGauss gauss1D(1, get_order());
 
  // Swap points and weights with the about-to-be destroyed rule.
  _points.swap(gauss1D.get_points());
  _weights.swap(gauss1D.get_weights());
}
 
 
 
// Builds and scales a Gauss rule and a Jacobi rule.
// Then combines them to compute points and weights
// of a 2D conical product rule.
void QConical::conical_product_tri()
{
  // Be sure the underlying rule object was built with the same dimension as the
  // rule we are about to construct.
  libmesh_assert_equal_to (this->get_dim(), 2);
 
  QGauss  gauss1D(1, get_order());
  QJacobi jac1D(1, get_order(), 1, 0);
 
  // The Gauss rule needs to be scaled to [0,1]
  std::pair<Real, Real> old_range(-1.0L, 1.0L);
  std::pair<Real, Real> new_range( 0.0L, 1.0L);
  gauss1D.scale(old_range,
                new_range);
 
  // Now construct the points and weights for the conical product rule.
 
  // Both rules should have the same number of points.
  libmesh_assert_equal_to (gauss1D.n_points(), jac1D.n_points());
 
  // Save the number of points as a convenient variable
  const unsigned int np = gauss1D.n_points();
 
  // Both rules should be between x=0 and x=1
  libmesh_assert_greater_equal (gauss1D.qp(0)(0), 0.0);
  libmesh_assert_less_equal (gauss1D.qp(np-1)(0), 1.0);
  libmesh_assert_greater_equal (jac1D.qp(0)(0), 0.0);
  libmesh_assert_less_equal (jac1D.qp(np-1)(0), 1.0);
 
  // Resize the points and weights vectors
  _points.resize(np * np);
  _weights.resize(np * np);
 
  // Compute the conical product
  unsigned int gp = 0;
  for (unsigned int i=0; i<np; i++)
    for (unsigned int j=0; j<np; j++)
      {
        _points[gp](0) = jac1D.qp(j)(0);                          //s[j];
        _points[gp](1) = gauss1D.qp(i)(0) * (1.-jac1D.qp(j)(0)); //r[i]*(1.-s[j]);
        _weights[gp]   = gauss1D.w(i) * jac1D.w(j);              //A[i]*B[j];
        gp++;
      }
}
 
 
 
 
// Builds and scales a Gauss rule and a Jacobi rule.
// Then combines them to compute points and weights
// of a 3D conical product rule for the Tet.
void QConical::conical_product_tet()
{
  // Be sure the underlying rule object was built with the same dimension as the
  // rule we are about to construct.
  libmesh_assert_equal_to (this->get_dim(), 3);
 
  QGauss  gauss1D(1, get_order());
  QJacobi jacA1D(1, get_order(), /*alpha=*/1, /*beta=*/0);
  QJacobi jacB1D(1, get_order(), /*alpha=*/2, /*beta=*/0);
 
  // The Gauss rule needs to be scaled to [0,1]
  std::pair<Real, Real> old_range(-1.0L, 1.0L);
  std::pair<Real, Real> new_range( 0.0L, 1.0L);
  gauss1D.scale(old_range,
                new_range);
 
  // Now construct the points and weights for the conical product rule.
 
  // All rules should have the same number of points
  libmesh_assert_equal_to (gauss1D.n_points(), jacA1D.n_points());
  libmesh_assert_equal_to (jacA1D.n_points(), jacB1D.n_points());
 
  // Save the number of points as a convenient variable
  const unsigned int np = gauss1D.n_points();
 
  // All rules should be between x=0 and x=1
  libmesh_assert_greater_equal (gauss1D.qp(0)(0), 0.0);
  libmesh_assert_less_equal (gauss1D.qp(np-1)(0), 1.0);
  libmesh_assert_greater_equal (jacA1D.qp(0)(0), 0.0);
  libmesh_assert_less_equal (jacA1D.qp(np-1)(0), 1.0);
  libmesh_assert_greater_equal (jacB1D.qp(0)(0), 0.0);
  libmesh_assert_less_equal (jacB1D.qp(np-1)(0), 1.0);
 
  // Resize the points and weights vectors
  _points.resize(np * np * np);
  _weights.resize(np * np * np);
 
  // Compute the conical product
  unsigned int gp = 0;
  for (unsigned int i=0; i<np; i++)
    for (unsigned int j=0; j<np; j++)
      for (unsigned int k=0; k<np; k++)
        {
          _points[gp](0) = jacB1D.qp(k)(0);                                                  //t[k];
          _points[gp](1) = jacA1D.qp(j)(0)  * (1.-jacB1D.qp(k)(0));                         //s[j]*(1.-t[k]);
          _points[gp](2) = gauss1D.qp(i)(0) * (1.-jacA1D.qp(j)(0)) * (1.-jacB1D.qp(k)(0)); //r[i]*(1.-s[j])*(1.-t[k]);
          _weights[gp]   = gauss1D.w(i)     * jacA1D.w(j)          * jacB1D.w(k);          //A[i]*B[j]*C[k];
          gp++;
        }
}
 
 
 
 
 
// Builds and scales a Gauss rule and a Jacobi rule.
// Then combines them to compute points and weights
// of a 3D conical product rule for the Pyramid.  The integral
// over the reference Pyramid can be written (in LaTeX notation) as:
//
// If := \int_0^1 dz \int_{-(1-z)}^{(1-z)} dy \int_{-(1-z)}^{(1-z)} f(x,y,z) dx (1)
//
// (Imagine a stack of infinitely thin squares which decrease in size as
//  you approach the apex.)  Under the transformation of variables:
//
// z=w
// y=(1-z)v
// x=(1-z)u,
//
// The Jacobian determinant of this transformation is |J|=(1-w)^2, and
// the integral itself is transformed to:
//
// If = \int_0^1 (1-w)^2 dw \int_{-1}^{1} dv \int_{-1}^{1} f((1-w)u, (1-w)v, w) du (2)
//
// The integral can now be approximated by the product of three 1D quadrature rules:
// A Jacobi rule with alpha==2, beta==0 in w, and Gauss rules in v and u.  In this way
// we can obtain 3D rules to any order for which the 1D rules exist.
void QConical::conical_product_pyramid()
{
  // Be sure the underlying rule object was built with the same dimension as the
  // rule we are about to construct.
  libmesh_assert_equal_to (this->get_dim(), 3);
 
  QGauss  gauss1D(1, get_order());
  QJacobi jac1D(1, get_order(), 2, 0);
 
  // These rules should have the same number of points
  libmesh_assert_equal_to (gauss1D.n_points(), jac1D.n_points());
 
  // Save the number of points as a convenient variable
  const unsigned int np = gauss1D.n_points();
 
  // Resize the points and weights vectors
  _points.resize(np * np * np);
  _weights.resize(np * np * np);
 
  // Compute the conical product
  unsigned int q = 0;
  for (unsigned int i=0; i<np; ++i)
    for (unsigned int j=0; j<np; ++j)
      for (unsigned int k=0; k<np; ++k, ++q)
        {
          const Real xi=gauss1D.qp(i)(0);
          const Real yj=gauss1D.qp(j)(0);
          const Real zk=jac1D.qp(k)(0);
 
          _points[q](0) = (1.-zk) * xi;
          _points[q](1) = (1.-zk) * yj;
          _points[q](2) = zk;
          _weights[q]   = gauss1D.w(i) * gauss1D.w(j) * jac1D.w(k);
        }
}
 
} // namespace libMesh