QMonomial

Section: C Library Functions (3) Updated: Thu Apr 7 2011 Local index Up

NAME

QMonomial -

SYNOPSIS

Inherits QBase.

Public Member Functions

QMonomial (const unsigned int _dim, const Order _order=INVALID_ORDER)

~QMonomial ()

QuadratureType type () const

ElemType get_elem_type () const

unsigned int get_p_level () const

unsigned int n_points () const

unsigned int get_dim () const

const std::vector< Point > & get_points () const

std::vector< Point > & get_points ()

const std::vector< Real > & get_weights () const

std::vector< Real > & get_weights ()

Point qp (const unsigned int i) const

Real w (const unsigned int i) const

void init (const ElemType _type=INVALID_ELEM, unsigned int p_level=0)

Order get_order () const

void print_info (std::ostream &os=std::cout) const

void scale (std::pair< Real, Real > old_range, std::pair< Real, Real > new_range)

Static Public Member Functions

static AutoPtr< QBase > build (const std::string &name, const unsigned int _dim, const Order _order=INVALID_ORDER)

static AutoPtr< QBase > build (const QuadratureType _qt, const unsigned int _dim, const Order _order=INVALID_ORDER)

static void print_info ()

static std::string get_info ()

static unsigned int n_objects ()

Public Attributes

bool allow_rules_with_negative_weights

Protected Types

typedef std::map< std::string, std::pair< unsigned int, unsigned int > > Counts

Protected Member Functions

virtual void init_0D (const ElemType _type=INVALID_ELEM, unsigned int p_level=0)

void increment_constructor_count (const std::string &name)

void increment_destructor_count (const std::string &name)

Protected Attributes

std::cerr<< 'ERROR: Seems as if this quadrature rule'<< std::endl<< ' is not implemented for 2D.'<< std::endl;libmesh_error();}#endif virtual void init_3D(const ElemType, unsigned int=0)#ifndef DEBUG{}#else{std::cerr<< 'ERROR: Seems as if this quadrature rule'<< std::endl<< ' is not implemented for 3D.'<< std::endl;libmesh_error();}#endif void tensor_product_quad(const QBase &q1D);void tensor_product_hex(const QBase &q1D);void tensor_product_prism(const QBase &q1D, const QBase &q2D);const unsigned int _dim;const Order _order;ElemType _type;unsigned int _p_level;std::vector< Point > _points

std::vector< Real > _weights

Static Protected Attributes

static Counts _counts

static Threads::atomic< unsigned int > _n_objects

static Threads::spin_mutex _mutex

Private Member Functions

void init_1D (const ElemType, unsigned int=0)

void init_2D (const ElemType _type=INVALID_ELEM, unsigned int p_level=0)

void init_3D (const ElemType _type=INVALID_ELEM, unsigned int p_level=0)

void wissmann_rule (const Real rule_data[][3], const unsigned int n_pts)

void stroud_rule (const Real rule_data[][3], const unsigned int *rule_symmetry, const unsigned int n_pts)

void kim_rule (const Real rule_data[][4], const unsigned int *rule_id, const unsigned int n_pts)

Friends

std::ostream & operator<< (std::ostream &os, const QBase &q)

Detailed Description

This class defines alternate quadrature rules on 'tensor-product' elements (QUADs and HEXes) which can be useful when integrating monomial finite element bases.

While tensor product rules are ideal for integrating bi/tri-linear, bi/tri-quadratic, etc. (i.e. tensor product) bases (which consist of incomplete polynomials up to degree= dim*p) they are not optimal for the MONOMIAL or FEXYZ bases, which consist of complete polynomials of degree=p.

This class is implemented to provide quadrature rules which are more efficient than tensor product rules when they are available, and fall back on Gaussian quadrature rules when necessary.

A number of these rules have been helpfully collected in electronic form by:

Prof. Ronald Cools Katholieke Universiteit Leuven, Dept. Computerwetenschappen http://www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html

(A username and password to access the tables is available by request.)

We also provide the original reference for each rule, as available, in the source code file.

Author:

: John W. Peterson, 2008

Definition at line 62 of file quadrature_monomial.h.

Member Typedef Documentation

typedef std::map<std::string, std::pair<unsigned int, unsigned int> > ReferenceCounter::Counts [protected, inherited]Data structure to log the information. The log is identified by the class name.

Definition at line 105 of file reference_counter.h.

Constructor & Destructor Documentation

QMonomial::QMonomial (const unsigned int_dim, const Order_order = INVALID_ORDER)Constructor. Declares the order of the quadrature rule.

Definition at line 32 of file quadrature_monomial.C.

                                    : QBase(d,o)
{
}

QMonomial::~QMonomial ()Destructor.

Definition at line 39 of file quadrature_monomial.C.

{
}

Member Function Documentation

AutoPtr< QBase > QBase::build (const std::string &name, const unsigned int_dim, const Order_order = INVALID_ORDER) [static, inherited]Builds a specific quadrature rule, identified through the name string. An AutoPtr<QBase> is returned to prevent a memory leak. This way the user need not remember to delete the object. Enables run-time decision of the quadrature rule. The input parameter name must be mappable through the Utility::string_to_enum<>() function.

Definition at line 36 of file quadrature_build.C.

References Utility::string_to_enum< QuadratureType >().

Referenced by InfFE< Dim, T_radial, T_map >::attach_quadrature_rule().

{
  return QBase::build (Utility::string_to_enum<QuadratureType> (type),
                       _dim,
                       _order);
}

AutoPtr< QBase > QBase::build (const QuadratureType_qt, const unsigned int_dim, const Order_order = INVALID_ORDER) [static, inherited]Builds a specific quadrature rule, identified through the QuadratureType. An AutoPtr<QBase> is returned to prevent a memory leak. This way the user need not remember to delete the object. Enables run-time decision of the quadrature rule.

Definition at line 47 of file quadrature_build.C.

References libMeshEnums::FIRST, libMeshEnums::FORTYTHIRD, libMeshEnums::QCLOUGH, libMeshEnums::QGAUSS, libMeshEnums::QJACOBI_1_0, libMeshEnums::QJACOBI_2_0, libMeshEnums::QSIMPSON, libMeshEnums::QTRAP, libMeshEnums::THIRD, and libMeshEnums::TWENTYTHIRD.

{
  switch (_qt)
    {
      
    case QCLOUGH:
      {
#ifdef DEBUG
        if (_order > TWENTYTHIRD)
          {
            std::cout << 'WARNING: Clough quadrature implemented' << std::endl
                      << ' up to TWENTYTHIRD order.' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QClough(_dim, _order));
        return ap;
      }

    case QGAUSS:
      {

#ifdef DEBUG
        if (_order > FORTYTHIRD)
          {
            std::cout << 'WARNING: Gauss quadrature implemented' << std::endl
                      << ' up to FORTYTHIRD order.' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QGauss(_dim, _order));
        return ap;
      }

    case QJACOBI_1_0:
      {

#ifdef DEBUG
        if (_order > TWENTYTHIRD)
          {
            std::cout << 'WARNING: Jacobi(1,0) quadrature implemented' << std::endl
                      << ' up to TWENTYTHIRD order.' << std::endl;
          }

        if (_dim > 1)
          {
            std::cout << 'WARNING: Jacobi(1,0) quadrature implemented' << std::endl
                      << ' in 1D only.' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QJacobi(_dim, _order, 1, 0));
        return ap;
      }

    case QJACOBI_2_0:
      {

#ifdef DEBUG
        if (_order > TWENTYTHIRD)
          {
            std::cout << 'WARNING: Jacobi(2,0) quadrature implemented' << std::endl
                      << ' up to TWENTYTHIRD order.' << std::endl;
          }

        if (_dim > 1)
          {
            std::cout << 'WARNING: Jacobi(2,0) quadrature implemented' << std::endl
                      << ' in 1D only.' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QJacobi(_dim, _order, 2, 0));
        return ap;
      }

    case QSIMPSON:
      {

#ifdef DEBUG
        if (_order > THIRD)
          {
            std::cout << 'WARNING: Simpson rule provides only' << std::endl
                      << ' THIRD order!' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QSimpson(_dim));
        return ap;
      }

    case QTRAP:
      {

#ifdef DEBUG
        if (_order > FIRST)
          {
            std::cout << 'WARNING: Trapezoidal rule provides only' << std::endl
                      << ' FIRST order!' << std::endl;
          }
#endif

        AutoPtr<QBase> ap(new QTrap(_dim));
        return ap;
      }


    default:
      { 
        std::cerr << 'ERROR: Bad qt=' << _qt << std::endl;
        libmesh_error();
      }
    }


  libmesh_error();
  AutoPtr<QBase> ap(NULL);
  return ap;
}

unsigned int QBase::get_dim () const [inline, inherited]Returns:

: the dimension of the quadrature rule.

Definition at line 121 of file quadrature.h.

Referenced by InfFE< Dim, T_radial, T_map >::attach_quadrature_rule(), QConical::conical_product_pyramid(), QConical::conical_product_tet(), and QConical::conical_product_tri().

{ return _dim;  }

ElemType QBase::get_elem_type () const [inline, inherited]Returns:

: the current element type we're set up for

Definition at line 103 of file quadrature.h.

    { return _type; }

std::string ReferenceCounter::get_info () [static, inherited]Gets a string containing the reference information.

Definition at line 45 of file reference_counter.C.

References ReferenceCounter::_counts, and Quality::name().

Referenced by ReferenceCounter::print_info().

{
#if defined(LIBMESH_ENABLE_REFERENCE_COUNTING) && defined(DEBUG)

  std::ostringstream out;
  
  out << '
      << ' ---------------------------------------------------------------------------- 
      << '| Reference count information                                                |
      << ' ---------------------------------------------------------------------------- ;
  
  for (Counts::iterator it = _counts.begin();
       it != _counts.end(); ++it)
    {
      const std::string name(it->first);
      const unsigned int creations    = it->second.first;
      const unsigned int destructions = it->second.second;

      out << '| ' << name << ' reference count information:
          << '|  Creations:    ' << creations    << '
          << '|  Destructions: ' << destructions << ';
    }
  
  out << ' ---------------------------------------------------------------------------- ;

  return out.str();

#else

  return '';
  
#endif
}

Order QBase::get_order () const [inline, inherited]Returns:

: the order of the quadrature rule.

Definition at line 167 of file quadrature.h.

Referenced by InfFE< Dim, T_radial, T_map >::attach_quadrature_rule().

{ return static_cast<Order>(_order + _p_level); }

unsigned int QBase::get_p_level () const [inline, inherited]Returns:

: the current p refinement level we're initialized with

Definition at line 109 of file quadrature.h.

    { return _p_level; }

const std::vector<Point>& QBase::get_points () const [inline, inherited]Returns:

: a std::vector containing the quadrature point locations on a reference object.

Definition at line 127 of file quadrature.h.

References QBase::_points.

Referenced by FE< Dim, T >::edge_reinit(), QClough::init_1D(), init_2D(), QGauss::init_2D(), QClough::init_2D(), init_3D(), QGauss::init_3D(), InfFE< Dim, T_radial, T_map >::init_face_shape_functions(), InfFE< Dim, T_radial, T_map >::reinit(), and REINIT_ERROR().

{ return _points;  }

std::vector<Point>& QBase::get_points () [inline, inherited]Returns:

: a std::vector containing the quadrature point locations on a reference object as a writeable reference.

Definition at line 133 of file quadrature.h.

References QBase::_points.

{ return _points;  }

const std::vector<Real>& QBase::get_weights () const [inline, inherited]Returns:

: a std::vector containing the quadrature weights.

Definition at line 138 of file quadrature.h.

References QBase::_weights.

{ return _weights; }

std::vector<Real>& QBase::get_weights () [inline, inherited]Returns:

: a std::vector containing the quadrature weights.

Definition at line 143 of file quadrature.h.

References QBase::_weights.

{ return _weights; }

void ReferenceCounter::increment_constructor_count (const std::string &name) [inline, protected, inherited]Increments the construction counter. Should be called in the constructor of any derived class that will be reference counted.

Definition at line 149 of file reference_counter.h.

References ReferenceCounter::_counts, Quality::name(), and Threads::spin_mtx.

Referenced by ReferenceCountedObject< Value >::ReferenceCountedObject().

{
  Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
  std::pair<unsigned int, unsigned int>& p = _counts[name];

  p.first++;
}

void ReferenceCounter::increment_destructor_count (const std::string &name) [inline, protected, inherited]Increments the destruction counter. Should be called in the destructor of any derived class that will be reference counted.

Definition at line 167 of file reference_counter.h.

References ReferenceCounter::_counts, Quality::name(), and Threads::spin_mtx.

Referenced by ReferenceCountedObject< Value >::~ReferenceCountedObject().

{
  Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
  std::pair<unsigned int, unsigned int>& p = _counts[name];

  p.second++;
}

void QBase::init (const ElemType_type = INVALID_ELEM, unsigned intp_level = 0) [inherited]Initializes the data structures to contain a quadrature rule for an object of type type.

Definition at line 26 of file quadrature.C.

References QBase::init_0D(), QBase::init_1D(), and QBase::init_2D().

Referenced by FE< Dim, T >::edge_reinit(), QClough::init_1D(), QTrap::init_2D(), QSimpson::init_2D(), init_2D(), QGrid::init_2D(), QGauss::init_2D(), QClough::init_2D(), QTrap::init_3D(), QSimpson::init_3D(), init_3D(), QGrid::init_3D(), QGauss::init_3D(), InfFE< Dim, T_radial, T_map >::init_face_shape_functions(), QGauss::QGauss(), QJacobi::QJacobi(), QSimpson::QSimpson(), QTrap::QTrap(), InfFE< Dim, T_radial, T_map >::reinit(), and REINIT_ERROR().

{
  // check to see if we have already
  // done the work for this quadrature rule
  if (t == _type && p == _p_level)
    return;
  else
    {
      _type = t;
      _p_level = p;
    }
    
  
  
  switch(_dim)
    {
    case 0:
      this->init_0D(_type,_p_level);

      return;
      
    case 1:
      this->init_1D(_type,_p_level);

      return;
      
    case 2:
      this->init_2D(_type,_p_level);

      return;

    case 3:
      this->init_3D(_type,_p_level);

      return;

    default:
      libmesh_error();
    }
}

void QBase::init_0D (const ElemType_type = INVALID_ELEM, unsigned intp_level = 0) [protected, virtual, inherited]Initializes the 0D quadrature rule by filling the points and weights vectors with the appropriate values. Generally this is just one point with weight 1.

Definition at line 70 of file quadrature.C.

References QBase::_points, and QBase::_weights.

Referenced by QBase::init().

{
  _points.resize(1);
  _weights.resize(1);
  _points[0] = Point(0.);
  _weights[0] = 1.0;
}

void QMonomial::init_1D (const ElemType_type, unsignedp_level = 0) [inline, private, virtual]Initializes the 1D quadrature rule by filling the points and weights vectors with the appropriate values. The order of the rule will be defined by the implementing class. It is assumed that derived quadrature rules will at least define the init_1D function, therefore it is pure virtual.

Implements QBase.

Definition at line 85 of file quadrature_monomial.h.

  {
    // See about making this non-pure virtual in the base class?
    libmesh_error();
  }

void QMonomial::init_2D (const ElemType_type = INVALID_ELEM, unsigned intp_level = 0) [private, virtual]More efficient rules for QUADs

Reimplemented from QBase.

Definition at line 27 of file quadrature_monomial_2D.C.

References QBase::_points, QBase::_weights, libMeshEnums::EIGHTH, libMeshEnums::ELEVENTH, libMeshEnums::FIFTEENTH, libMeshEnums::FIFTH, libMeshEnums::FOURTEENTH, libMeshEnums::FOURTH, QBase::get_points(), QBase::get_weights(), QBase::init(), libMeshEnums::NINTH, libMeshEnums::QUAD4, libMeshEnums::QUAD8, libMeshEnums::QUAD9, libMeshEnums::SECOND, libMeshEnums::SEVENTEENTH, libMeshEnums::SEVENTH, libMeshEnums::SIXTEENTH, libMeshEnums::SIXTH, stroud_rule(), libMeshEnums::TENTH, libMeshEnums::THIRTEENTH, libMeshEnums::TWELFTH, and wissmann_rule().

{

  switch (_type)
    {
      //---------------------------------------------
      // Quadrilateral quadrature rules
    case QUAD4:
    case QUAD8:
    case QUAD9:
      {
        switch(_order + 2*p)
          {
          case SECOND:
            {
              // A degree=2 rule for the QUAD with 3 points.
              // A tensor product degree-2 Gauss would have 4 points.
              // This rule (or a variation on it) is probably available in
              //
              // A.H. Stroud, Approximate calculation of multiple integrals,
              // Prentice-Hall, Englewood Cliffs, N.J., 1971.
              //
              // though I have never actually seen a reference for it.
              // Luckily it's fairly easy to derive, which is what I've done
              // here [JWP].
              const Real
                s=std::sqrt(1./3.),
                t=std::sqrt(2./3.);
              
              const Real data[2][3] =
                {
                  {0.0,  s,  2.0},
                  {  t, -s,  1.0}
                };

              _points.resize(3);
              _weights.resize(3);

              wissmann_rule(data, 2);
              
              return;
            } // end case SECOND


            
          // For third-order, fall through to default case, use 2x2 Gauss product rule.
          // case THIRD:
          //   {
          //   }  // end case THIRD
            
          case FOURTH:
            {
              // A pair of degree=4 rules for the QUAD 'C2' due to
              // Wissmann and Becker. These rules both have six points.
              // A tensor product degree-4 Gauss would have 9 points.
              //
              // J. W. Wissmann and T. Becker, Partially symmetric cubature 
              // formulas for even degrees of exactness, SIAM J. Numer. Anal.  23
              // (1986), 676--685.                                           
              const Real data[4][3] =
                {
                  // First of 2 degree-4 rules given by Wissmann
                  {0.0000000000000000e+00,  0.0000000000000000e+00,  1.1428571428571428e+00},
                  {0.0000000000000000e+00,  9.6609178307929590e-01,  4.3956043956043956e-01},
                  {8.5191465330460049e-01,  4.5560372783619284e-01,  5.6607220700753210e-01},
                  {6.3091278897675402e-01, -7.3162995157313452e-01,  6.4271900178367668e-01}
                  //
                  // Second of 2 degree-4 rules given by Wissmann.  These both
                  // yield 4th-order accurate rules, I just chose the one that
                  // happened to contain the origin.
                  // {0.000000000000000, -0.356822089773090,  1.286412084888852}, 
                  // {0.000000000000000,  0.934172358962716,  0.491365692888926},
                  // {0.774596669241483,  0.390885162530071,  0.761883709085613},
                  // {0.774596669241483, -0.852765377881771,  0.349227402025498} 
                };

              _points.resize(6);
              _weights.resize(6);

              wissmann_rule(data, 4);
              
              return;
            } // end case FOURTH



            
          case FIFTH:
            {
              // A degree 5, 7-point rule due to Stroud.  
              //
              // A.H. Stroud, Approximate calculation of multiple integrals,
              // Prentice-Hall, Englewood Cliffs, N.J., 1971.
              //
              // This rule is provably minimal in the number of points.
              // A tensor-product rule accurate for 'bi-quintic' polynomials would have 9 points.
              const Real data[3][3] =
                {
                  {               0.L,                  0.L, 8.L  /  7.L}, // 1
                  {               0.L, std::sqrt(14.L/15.L), 20.L / 63.L}, // 2
                  {std::sqrt(3.L/5.L),   std::sqrt(1.L/3.L), 20.L / 36.L}  // 4
                };

              const unsigned int symmetry[3] = {
                0, // Origin
                7, // Central Symmetry
                6  // Rectangular
              };
              
              _points.resize (7);
              _weights.resize(7);

              stroud_rule(data, symmetry, 3);
              
              return;
            } // end case FIFTH



            
          case SIXTH:
            {
              // A pair of degree=6 rules for the QUAD 'C2' due to
              // Wissmann and Becker. These rules both have 10 points.
              // A tensor product degree-6 Gauss would have 16 points.
              //
              // J. W. Wissmann and T. Becker, Partially symmetric cubature 
              // formulas for even degrees of exactness, SIAM J. Numer. Anal.  23
              // (1986), 676--685.                                           
              const Real data[6][3] =
                {
                  // First of 2 degree-6, 10 point rules given by Wissmann
                  // {0.000000000000000,  0.836405633697626,  0.455343245714174},
                  // {0.000000000000000, -0.357460165391307,  0.827395973202966},
                  // {0.888764014654765,  0.872101531193131,  0.144000884599645},
                  // {0.604857639464685,  0.305985162155427,  0.668259104262665},
                  // {0.955447506641064, -0.410270899466658,  0.225474004890679},
                  // {0.565459993438754, -0.872869311156879,  0.320896396788441} 
                  //
                  // Second of 2 degree-6, 10 point rules given by Wissmann.
                  // Either of these will work, I just chose the one with points
                  // slightly further into the element interior.
                  {0.0000000000000000e+00,  8.6983337525005900e-01,  3.9275059096434794e-01},
                  {0.0000000000000000e+00, -4.7940635161211124e-01,  7.5476288124261053e-01},
                  {8.6374282634615388e-01,  8.0283751620765670e-01,  2.0616605058827902e-01},
                  {5.1869052139258234e-01,  2.6214366550805818e-01,  6.8999213848986375e-01},
                  {9.3397254497284950e-01, -3.6309658314806653e-01,  2.6051748873231697e-01},
                  {6.0897753601635630e-01, -8.9660863276245265e-01,  2.6956758608606100e-01}
                };

              _points.resize(10);
              _weights.resize(10);

              wissmann_rule(data, 6);

              return; 
            } // end case SIXTH



            
          case SEVENTH:
            {
              // A degree 7, 12-point rule due to Tyler, can be found in Stroud's book  
              //
              // A.H. Stroud, Approximate calculation of multiple integrals,
              // Prentice-Hall, Englewood Cliffs, N.J., 1971.
              //
              // This rule is fully-symmetric and provably minimal in the number of points.
              // A tensor-product rule accurate for 'bi-septic' polynomials would have 16 points.
              const Real
                r  = std::sqrt(6.L/7.L),
                s  = std::sqrt( (114.L - 3.L*std::sqrt(583.L)) / 287.L ),
                t  = std::sqrt( (114.L + 3.L*std::sqrt(583.L)) / 287.L ),
                B1 = 196.L / 810.L,
                B2 = 4.L * (178981.L + 2769.L*std::sqrt(583.L)) / 1888920.L,
                B3 = 4.L * (178981.L - 2769.L*std::sqrt(583.L)) / 1888920.L;
              
              const Real data[3][3] =
                {
                  {r, 0.0, B1}, // 4
                  {s, 0.0, B2}, // 4
                  {t, 0.0, B3}  // 4
                };

              const unsigned int symmetry[3] = {
                3, // Full Symmetry, (x,0)
                2, // Full Symmetry, (x,x)
                2  // Full Symmetry, (x,x)
              };
              
              _points.resize (12);
              _weights.resize(12);

              stroud_rule(data, symmetry, 3);

              return;
            } // end case SEVENTH



            
          case EIGHTH:
            {
              // A pair of degree=8 rules for the QUAD 'C2' due to
              // Wissmann and Becker. These rules both have 16 points.
              // A tensor product degree-6 Gauss would have 25 points.
              //
              // J. W. Wissmann and T. Becker, Partially symmetric cubature 
              // formulas for even degrees of exactness, SIAM J. Numer. Anal.  23
              // (1986), 676--685.                                           
              const Real data[10][3] =
                {
                  // First of 2 degree-8, 16 point rules given by Wissmann
                  // {0.000000000000000,  0.000000000000000,  0.055364705621440},
                  // {0.000000000000000,  0.757629177660505,  0.404389368726076},
                  // {0.000000000000000, -0.236871842255702,  0.533546604952635},
                  // {0.000000000000000, -0.989717929044527,  0.117054188786739},
                  // {0.639091304900370,  0.950520955645667,  0.125614417613747},
                  // {0.937069076924990,  0.663882736885633,  0.136544584733588},
                  // {0.537083530541494,  0.304210681724104,  0.483408479211257},
                  // {0.887188506449625, -0.236496718536120,  0.252528506429544},
                  // {0.494698820670197, -0.698953476086564,  0.361262323882172},
                  // {0.897495818279768, -0.900390774211580,  0.085464254086247}
                  //
                  // Second of 2 degree-8, 16 point rules given by Wissmann.
                  // Either of these will work, I just chose the one with points
                  // further into the element interior.
                  {0.0000000000000000e+00,  6.5956013196034176e-01,  4.5027677630559029e-01},
                  {0.0000000000000000e+00, -9.4914292304312538e-01,  1.6657042677781274e-01},
                  {9.5250946607156228e-01,  7.6505181955768362e-01,  9.8869459933431422e-02},
                  {5.3232745407420624e-01,  9.3697598108841598e-01,  1.5369674714081197e-01},
                  {6.8473629795173504e-01,  3.3365671773574759e-01,  3.9668697607290278e-01},
                  {2.3314324080140552e-01, -7.9583272377396852e-02,  3.5201436794569501e-01},
                  {9.2768331930611748e-01, -2.7224008061253425e-01,  1.8958905457779799e-01},
                  {4.5312068740374942e-01, -6.1373535339802760e-01,  3.7510100114758727e-01},
                  {8.3750364042281223e-01, -8.8847765053597136e-01,  1.2561879164007201e-01}
                };

              _points.resize(16);
              _weights.resize(16);

              wissmann_rule(data, /*10*/ 9);

              return; 
            } // end case EIGHTH




          case NINTH:
            {
              // A degree 9, 17-point rule due to Moller.  
              //
              // H.M. Moller,  Kubaturformeln mit minimaler Knotenzahl,
              // Numer. Math.  25 (1976), 185--200.
              //
              // This rule is provably minimal in the number of points.
              // A tensor-product rule accurate for 'bi-ninth' degree polynomials would have 25 points.
              const Real data[5][3] =
                {
                  {0.0000000000000000e+00, 0.0000000000000000e+00, 5.2674897119341563e-01}, // 1
                  {6.3068011973166885e-01, 9.6884996636197772e-01, 8.8879378170198706e-02}, // 4
                  {9.2796164595956966e-01, 7.5027709997890053e-01, 1.1209960212959648e-01}, // 4
                  {4.5333982113564719e-01, 5.2373582021442933e-01, 3.9828243926207009e-01}, // 4
                  {8.5261572933366230e-01, 7.6208328192617173e-02, 2.6905133763978080e-01}  // 4
                };

              const unsigned int symmetry[5] = {
                0, // Single point
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4  // Rotational Invariant
              };
              
              _points.resize (17);
              _weights.resize(17);

              stroud_rule(data, symmetry, 5);

              return;
            } // end case NINTH




          case TENTH:
          case ELEVENTH:
            {
              // A degree 11, 24-point rule due to Cools and Haegemans.  
              //
              // R. Cools and A. Haegemans, Another step forward in searching for
              // cubature formulae with a minimal number of knots for the square,
              // Computing 40 (1988), 139--146.
              // 
              // P. Verlinden and R. Cools, The algebraic construction of a minimal
              // cubature formula of degree 11 for the square, Cubature Formulas
              // and their Applications (Russian) (Krasnoyarsk) (M.V. Noskov, ed.),
              // 1994, pp. 13--23.
              //
              // This rule is provably minimal in the number of points.
              // A tensor-product rule accurate for 'bi-tenth' or 'bi-eleventh' degree polynomials would have 36 points.
              const Real data[6][3] =
                {
                  {6.9807610454956756e-01, 9.8263922354085547e-01, 4.8020763350723814e-02}, // 4
                  {9.3948638281673690e-01, 8.2577583590296393e-01, 6.6071329164550595e-02}, // 4
                  {9.5353952820153201e-01, 1.8858613871864195e-01, 9.7386777358668164e-02}, // 4
                  {3.1562343291525419e-01, 8.1252054830481310e-01, 2.1173634999894860e-01}, // 4
                  {7.1200191307533630e-01, 5.2532025036454776e-01, 2.2562606172886338e-01}, // 4
                  {4.2484724884866925e-01, 4.1658071912022368e-02, 3.5115871839824543e-01}  // 4
                };

              const unsigned int symmetry[6] = {
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4  // Rotational Invariant
              };
              
              _points.resize (24);
              _weights.resize(24);

              stroud_rule(data, symmetry, 6);

              return;
            } // end case TENTH,ELEVENTH




          case TWELFTH:
          case THIRTEENTH:
            {
              // A degree 13, 33-point rule due to Cools and Haegemans.  
              //
              // R. Cools and A. Haegemans, Another step forward in searching for
              // cubature formulae with a minimal number of knots for the square,
              // Computing 40 (1988), 139--146.
              // 
              // A tensor-product rule accurate for 'bi-12' or 'bi-13' degree polynomials would have 49 points.
              const Real data[9][3] =
                {
                  {0.0000000000000000e+00, 0.0000000000000000e+00, 3.0038211543122536e-01}, // 1
                  {9.8348668243987226e-01, 7.7880971155441942e-01, 2.9991838864499131e-02}, // 4
                  {8.5955600564163892e-01, 9.5729769978630736e-01, 3.8174421317083669e-02}, // 4
                  {9.5892517028753485e-01, 1.3818345986246535e-01, 6.0424923817749980e-02}, // 4
                  {3.9073621612946100e-01, 9.4132722587292523e-01, 7.7492738533105339e-02}, // 4
                  {8.5007667369974857e-01, 4.7580862521827590e-01, 1.1884466730059560e-01}, // 4
                  {6.4782163718701073e-01, 7.5580535657208143e-01, 1.2976355037000271e-01}, // 4
                  {7.0741508996444936e-02, 6.9625007849174941e-01, 2.1334158145718938e-01}, // 4
                  {4.0930456169403884e-01, 3.4271655604040678e-01, 2.5687074948196783e-01}  // 4
                };

              const unsigned int symmetry[9] = {
                0, // Single point
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4, // Rotational Invariant
                4  // Rotational Invariant
              };
              
              _points.resize (33);
              _weights.resize(33);

              stroud_rule(data, symmetry, 9);

              return;
            } // end case TWELFTH,THIRTEENTH




          case FOURTEENTH:
          case FIFTEENTH:
            {
              // A degree-15, 48 point rule originally due to Rabinowitz and Richter,
              // can be found in Cools' 1971 book.
              //
              // A.H. Stroud, Approximate calculation of multiple integrals,
              // Prentice-Hall, Englewood Cliffs, N.J., 1971.
              //
              // The product Gauss rule for this order has 8^2=64 points.
              const Real data[9][3] =
                {
                  {9.915377816777667e-01L, 0.0000000000000000e+00,  3.01245207981210e-02L}, // 4
                  {8.020163879230440e-01L, 0.0000000000000000e+00,  8.71146840209092e-02L}, // 4
                  {5.648674875232742e-01L, 0.0000000000000000e+00, 1.250080294351494e-01L}, // 4
                  {9.354392392539896e-01L, 0.0000000000000000e+00,  2.67651407861666e-02L}, // 4
                  {7.624563338825799e-01L, 0.0000000000000000e+00,  9.59651863624437e-02L}, // 4
                  {2.156164241427213e-01L, 0.0000000000000000e+00, 1.750832998343375e-01L}, // 4
                  {9.769662659711761e-01L, 6.684480048977932e-01L,  2.83136372033274e-02L}, // 4
                  {8.937128379503403e-01L, 3.735205277617582e-01L,  8.66414716025093e-02L}, // 4
                  {6.122485619312083e-01L, 4.078983303613935e-01L, 1.150144605755996e-01L}  // 4
                };

              const unsigned int symmetry[9] = {
                3, // Full Symmetry, (x,0)
                3, // Full Symmetry, (x,0)
                3, // Full Symmetry, (x,0)
                2, // Full Symmetry, (x,x)
                2, // Full Symmetry, (x,x)
                2, // Full Symmetry, (x,x)
                1, // Full Symmetry, (x,y)
                1, // Full Symmetry, (x,y)
                1, // Full Symmetry, (x,y)
              };
              
              _points.resize (48);
              _weights.resize(48);

              stroud_rule(data, symmetry, 9);

              return;
            } //          case FOURTEENTH, FIFTEENTH:



            
          case SIXTEENTH:
          case SEVENTEENTH:
            {
              // A degree 17, 60-point rule due to Cools and Haegemans.  
              //
              // R. Cools and A. Haegemans, Another step forward in searching for
              // cubature formulae with a minimal number of knots for the square,
              // Computing 40 (1988), 139--146.
              // 
              // A tensor-product rule accurate for 'bi-14' or 'bi-15' degree polynomials would have 64 points.
              // A tensor-product rule accurate for 'bi-16' or 'bi-17' degree polynomials would have 81 points.
              const Real data[10][3] =
                {
                  {9.8935307451260049e-01, 0.0000000000000000e+00, 2.0614915919990959e-02}, // 4
                  {3.7628520715797329e-01, 0.0000000000000000e+00, 1.2802571617990983e-01}, // 4
                  {9.7884827926223311e-01, 0.0000000000000000e+00, 5.5117395340318905e-03}, // 4
                  {8.8579472916411612e-01, 0.0000000000000000e+00, 3.9207712457141880e-02}, // 4
                  {1.7175612383834817e-01, 0.0000000000000000e+00, 7.6396945079863302e-02}, // 4
                  {5.9049927380600241e-01, 3.1950503663457394e-01, 1.4151372994997245e-01}, // 8
                  {7.9907913191686325e-01, 5.9797245192945738e-01, 8.3903279363797602e-02}, // 8
                  {8.0374396295874471e-01, 5.8344481776550529e-02, 6.0394163649684546e-02}, // 8
                  {9.3650627612749478e-01, 3.4738631616620267e-01, 5.7387752969212695e-02}, // 8
                  {9.8132117980545229e-01, 7.0600028779864611e-01, 2.1922559481863763e-02}, // 8
                };

              const unsigned int symmetry[10] = {
                3, // Fully symmetric (x,0)
                3, // Fully symmetric (x,0)
                2, // Fully symmetric (x,x)
                2, // Fully symmetric (x,x)
                2, // Fully symmetric (x,x)
                1, // Fully symmetric (x,y) 
                1, // Fully symmetric (x,y)
                1, // Fully symmetric (x,y)
                1, // Fully symmetric (x,y)
                1  // Fully symmetric (x,y)
              };
              
              _points.resize (60);
              _weights.resize(60);

              stroud_rule(data, symmetry, 10);

              return;
            } // end case FOURTEENTH through SEVENTEENTH
            
            
            
            // By default: construct and use a Gauss quadrature rule
          default:
            {
              // Break out and fall down into the default: case for the
              // outer switch statement.
              break;
            }
            
          } // end switch(_order + 2*p)
      } // end case QUAD4/8/9

      
      // By default: construct and use a Gauss quadrature rule
    default:
      {
        QGauss gauss_rule(2, _order);
        gauss_rule.init(_type, p);

        // Swap points and weights with the about-to-be destroyed rule.
        _points.swap (gauss_rule.get_points() );
        _weights.swap(gauss_rule.get_weights());

        return;
      }
    } // end switch (_type)
}

void QMonomial::init_3D (const ElemType_type = INVALID_ELEM, unsigned intp_level = 0) [private]More efficient rules for HEXes

Definition at line 27 of file quadrature_monomial_3D.C.

References QBase::_points, QBase::_weights, QBase::allow_rules_with_negative_weights, libMeshEnums::EIGHTH, libMeshEnums::FIFTH, libMeshEnums::FOURTH, QBase::get_points(), QBase::get_weights(), libMeshEnums::HEX20, libMeshEnums::HEX27, libMeshEnums::HEX8, QBase::init(), kim_rule(), libMeshEnums::SECOND, libMeshEnums::SEVENTH, libMeshEnums::SIXTH, and libMeshEnums::THIRD.

{

  switch (_type)
    {
      //---------------------------------------------
      // Hex quadrature rules
    case HEX8:
    case HEX20:
    case HEX27:
      {
        switch(_order + 2*p)
          {

            // The CONSTANT/FIRST rule is the 1-point Gauss 'product' rule...we fall
            // through to the default case for this rule.
            
          case SECOND:
          case THIRD:
            {
              // A degree 3, 6-point, 'rotationally-symmetric' rule by
              // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //
              // Warning: this rule contains points on the boundary of the reference
              // element, and therefore may be unsuitable for some problems.  The alternative
              // would be a 2x2x2 Gauss product rule.
              const Real data[1][4] =
                {
                  {1.0L, 0.0L, 0.0L, 4.0L/3.0L}  
                };

              const unsigned int rule_id[1] = {
                1 // (x,0,0) -> 6 permutations
              };

              _points.resize(6);
              _weights.resize(6);

              kim_rule(data, rule_id, 1);
              return;
            } // end case SECOND,THIRD

          case FOURTH:
          case FIFTH:
            {
              // A degree 5, 13-point rule by Stroud,
              // AH Stroud, 'Some Fifth Degree Integration Formulas for Symmetric Regions II.',
              // Numerische Mathematik 9, pp. 460-468 (1967).
              //
              // This rule is provably minimal in the number of points.  The equations given for
              // the n-cube on pg. 466 of the paper for mu/gamma and gamma are wrong, at least for
              // the n=3 case.  The analytical values given here were computed by me [JWP] in Maple.

              // Convenient intermediate values.
              const Real sqrt19 = std::sqrt(19.L);
              const Real tp     = std::sqrt(71440.L + 6802.L*sqrt19);
              
              // Point data for permutations.
              const Real eta    =  0.00000000000000000000000000000000e+00L;
              
              const Real lambda =  std::sqrt(1919.L/3285.L - 148.L*sqrt19/3285.L + 4.L*tp/3285.L);
              // 8.8030440669930978047737818209860e-01L; 
              
              const Real xi     = -std::sqrt(1121.L/3285.L +  74.L*sqrt19/3285.L - 2.L*tp/3285.L);
              // -4.9584817142571115281421242364290e-01L; 
              
              const Real mu     =  std::sqrt(1121.L/3285.L +  74.L*sqrt19/3285.L + 2.L*tp/3285.L);
              // 7.9562142216409541542982482567580e-01L;
              
              const Real gamma  =  std::sqrt(1919.L/3285.L - 148.L*sqrt19/3285.L - 4.L*tp/3285.L);
              // 2.5293711744842581347389255929324e-02L; 
              
              // Weights: the centroid weight is given analytically.  Weight B (resp C) goes
              // with the {lambda,xi} (resp {gamma,mu}) permutation.  The single-precision
              // results reported by Stroud are given for reference.

              const Real A      = 32.0L / 19.0L;
              // Stroud: 0.21052632  * 8.0 = 1.684210560;

              const Real B      = 1.L / ( 260072.L/133225.L  - 1520*sqrt19/133225.L + (133.L - 37.L*sqrt19)*tp/133225.L );
              // 5.4498735127757671684690782180890e-01L; // Stroud: 0.068123420 * 8.0 = 0.544987360;
              
              const Real C      = 1.L / ( 260072.L/133225.L  - 1520*sqrt19/133225.L - (133.L - 37.L*sqrt19)*tp/133225.L );
              // 5.0764422766979170420572375713840e-01L; // Stroud: 0.063455527 * 8.0 = 0.507644216;
              
              _points.resize(13);
              _weights.resize(13);

              unsigned int c=0;

              // Point with weight A (origin)
              _points[c] = Point(eta, eta, eta);
              _weights[c++] = A;

              // Points with weight B
              _points[c] = Point(lambda, xi, xi);  
              _weights[c++] = B;
              _points[c] = -_points[c-1];             
              _weights[c++] = B;
              
              _points[c] = Point(xi, lambda, xi);  
              _weights[c++] = B;
              _points[c] = -_points[c-1];             
              _weights[c++] = B;

              _points[c] = Point(xi, xi, lambda);  
              _weights[c++] = B;
              _points[c] = -_points[c-1];             
              _weights[c++] = B;

              // Points with weight C
              _points[c] = Point(mu, mu, gamma);  
              _weights[c++] = C;
              _points[c] = -_points[c-1];             
              _weights[c++] = C;
              
              _points[c] = Point(mu, gamma, mu);  
              _weights[c++] = C;
              _points[c] = -_points[c-1];             
              _weights[c++] = C;

              _points[c] = Point(gamma, mu, mu);  
              _weights[c++] = C;
              _points[c] = -_points[c-1];             
              _weights[c++] = C;
              
              return;

              
//            // A degree 5, 14-point, 'rotationally-symmetric' rule by
//            // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
//            // Was also reported in Stroud's 1971 book.
//            const Real data[2][4] =
//              {
//                {7.95822425754221463264548820476135e-01L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 8.86426592797783933518005540166204e-01L}, 
//                {7.58786910639328146269034278112267e-01L, 7.58786910639328146269034278112267e-01L, 7.58786910639328146269034278112267e-01L, 3.35180055401662049861495844875346e-01L}  
//              };

//            const unsigned int rule_id[2] = {
//              1, // (x,0,0) -> 6 permutations
//              4  // (x,x,x) -> 8 permutations
//            };

//            _points.resize(14);
//            _weights.resize(14);

//            kim_rule(data, rule_id, 2);
//            return;
            } // end case FOURTH,FIFTH

          case SIXTH:
          case SEVENTH:
            {
              if (allow_rules_with_negative_weights)
                {
                  // A degree 7, 31-point, 'rotationally-symmetric' rule by
                  // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
                  // This rule contains a negative weight, so only use it if such type of
                  // rules are allowed.
                  const Real data[3][4] =
                    {
                      {0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, -1.27536231884057971014492753623188e+00L}, 
                      {5.85540043769119907612630781744060e-01L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L,  8.71111111111111111111111111111111e-01L}, 
                      {6.94470135991704766602025803883310e-01L, 9.37161638568208038511047377665396e-01L, 4.15659267604065126239606672567031e-01L,  1.68695652173913043478260869565217e-01L}  
                    };
                  
                  const unsigned int rule_id[3] = {
                    0, // (0,0,0) -> 1 permutation
                    1, // (x,0,0) -> 6 permutations
                    6  // (x,y,z) -> 24 permutations
                  };

                  _points.resize(31);
                  _weights.resize(31);

                  kim_rule(data, rule_id, 3);
                  return;
                } // end if (allow_rules_with_negative_weights)

              
              // A degree 7, 34-point, 'fully-symmetric' rule, first published in
              // P.C. Hammer and A.H. Stroud, 'Numerical Evaluation of Multiple Integrals II',
              // Mathmatical Tables and Other Aids to Computation, vol 12., no 64, 1958, pp. 272-280
              //
              // This rule happens to fall under the same general
              // construction as the Kim rules, so we've re-used
              // that code here.  Stroud gives 16 digits for his rule,
              // and this is the most accurate version I've found.
              //
              // For comparison, a SEVENTH-order Gauss product rule
              // (which integrates tri-7th order polynomials) would
              // have 4^3=64 points.
              const Real
                r  = std::sqrt(6.L/7.L),
                s  = std::sqrt((960.L - 3.L*std::sqrt(28798.L)) / 2726.L),
                t  = std::sqrt((960.L + 3.L*std::sqrt(28798.L)) / 2726.L),
                B1 = 8624.L / 29160.L,
                B2 = 2744.L / 29160.L,
                B3 = 8.L*(774.L*t*t - 230.L)/(9720.L*(t*t-s*s)),
                B4 = 8.L*(230.L - 774.L*s*s)/(9720.L*(t*t-s*s));
                
              const Real data[4][4] =
                {
                  {r, 0.L, 0.L, B1}, 
                  {r,   r, 0.L, B2}, 
                  {s,   s,   s, B3}, 
                  {t,   t,   t, B4}  
                };
                  
              const unsigned int rule_id[4] = {
                1, // (x,0,0) -> 6 permutations
                2, // (x,x,0) -> 12 permutations
                4, // (x,x,x) -> 8 permutations
                4  // (x,x,x) -> 8 permutations
                  };

              _points.resize(34);
              _weights.resize(34);

              kim_rule(data, rule_id, 4);
              return;


//            // A degree 7, 38-point, 'rotationally-symmetric' rule by
//            // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
//            //
//            // This rule is obviously inferior to the 34-point rule above...
//            const Real data[3][4] =
//              {
//                {9.01687807821291289082811566285950e-01L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 2.95189738262622903181631100062774e-01L}, 
//                {4.08372221499474674069588900002128e-01L, 4.08372221499474674069588900002128e-01L, 4.08372221499474674069588900002128e-01L, 4.04055417266200582425904380777126e-01L}, 
//                {8.59523090201054193116477875786220e-01L, 8.59523090201054193116477875786220e-01L, 4.14735913727987720499709244748633e-01L, 1.24850759678944080062624098058597e-01L}  
//              };
//
//            const unsigned int rule_id[3] = {
//              1, // (x,0,0) -> 6 permutations
//              4, // (x,x,x) -> 8 permutations
//              5  // (x,x,z) -> 24 permutations
//            };
//
//            _points.resize(38);
//            _weights.resize(38);
//
//            kim_rule(data, rule_id, 3);
//            return;
            } // end case SIXTH,SEVENTH

          case EIGHTH:
            {
              // A degree 8, 47-point, 'rotationally-symmetric' rule by
              // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //
              // A EIGHTH-order Gauss product rule (which integrates tri-8th order polynomials)
              // would have 5^3=125 points.
              const Real data[5][4] =
                {
                  {0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 4.51903714875199690490763818699555e-01L}, 
                  {7.82460796435951590652813975429717e-01L, 0.00000000000000000000000000000000e+00L, 0.00000000000000000000000000000000e+00L, 2.99379177352338919703385618576171e-01L}, 
                  {4.88094669706366480526729301468686e-01L, 4.88094669706366480526729301468686e-01L, 4.88094669706366480526729301468686e-01L, 3.00876159371240019939698689791164e-01L}, 
                  {8.62218927661481188856422891110042e-01L, 8.62218927661481188856422891110042e-01L, 8.62218927661481188856422891110042e-01L, 4.94843255877038125738173175714853e-02L},  
                  {2.81113909408341856058098281846420e-01L, 9.44196578292008195318687494773744e-01L, 6.97574833707236996779391729948984e-01L, 1.22872389222467338799199767122592e-01L}  
                };

              const unsigned int rule_id[5] = {
                0, // (0,0,0) -> 1 permutation
                1, // (x,0,0) -> 6 permutations
                4, // (x,x,x) -> 8 permutations
                4, // (x,x,x) -> 8 permutations
                6  // (x,y,z) -> 24 permutations
              };

              _points.resize(47);
              _weights.resize(47);

              kim_rule(data, rule_id, 5);
              return;
            } // end case EIGHTH

            
            // By default: construct and use a Gauss quadrature rule
          default:
            {
              // Break out and fall down into the default: case for the
              // outer switch statement.
              break;
            }
            
          } // end switch(_order + 2*p)
      } // end case HEX8/20/27

      
      // By default: construct and use a Gauss quadrature rule
    default:
      {
        QGauss gauss_rule(3, _order);
        gauss_rule.init(_type, p);

        // Swap points and weights with the about-to-be destroyed rule.
        _points.swap (gauss_rule.get_points() );
        _weights.swap(gauss_rule.get_weights());

        return;
      }
    } // end switch (_type)
}

void QMonomial::kim_rule (const Realrule_data[][4], const unsigned int *rule_id, const unsigned intn_pts) [private]Rules from Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931. The rules are obtained by considering the group G^{rot} of rotations of the reference hex, and the invariant polynomials of this group.

In Kim and Song's rules, quadrauture points are described by the following points and their unique permutations under the G^{rot} group:

0.) (0,0,0) ( 1 perm ) -> [0, 0, 0] 1.) (x,0,0) ( 6 perms) -> [x, 0, 0], [0, -x, 0], [-x, 0, 0], [0, x, 0], [0, 0, -x], [0, 0, x] 2.) (x,x,0) (12 perms) -> [x, x, 0], [x, -x, 0], [-x, -x, 0], [-x, x, 0], [x, 0, -x], [x, 0, x], [0, x, -x], [0, x, x], [0, -x, -x], [-x, 0, -x], [0, -x, x], [-x, 0, x] 3.) (x,y,0) (24 perms) -> [x, y, 0], [y, -x, 0], [-x, -y, 0], [-y, x, 0], [x, 0, -y], [x, -y, 0], [x, 0, y], [0, y, -x], [-x, y, 0], [0, y, x], [y, 0, -x], [0, -y, -x], [-y, 0, -x], [y, x, 0], [-y, -x, 0], [y, 0, x], [0, -y, x], [-y, 0, x], [-x, 0, y], [0, -x, -y], [0, -x, y], [-x, 0, -y], [0, x, y], [0, x, -y] 4.) (x,x,x) ( 8 perms) -> [x, x, x], [x, -x, x], [-x, -x, x], [-x, x, x], [x, x, -x], [x, -x, -x], [-x, x, -x], [-x, -x, -x] 5.) (x,x,z) (24 perms) -> [x, x, z], [x, -x, z], [-x, -x, z], [-x, x, z], [x, z, -x], [x, -x, -z], [x, -z, x], [z, x, -x], [-x, x, -z], [-z, x, x], [x, -z, -x], [-z, -x, -x], [-x, z, -x], [x, x, -z], [-x, -x, -z], [x, z, x], [z, -x, x], [-x, -z, x], [-x, z, x], [z, -x, -x], [-z, -x, x], [-x, -z, -x], [z, x, x], [-z, x, -x] 6.) (x,y,z) (24 perms) -> [x, y, z], [y, -x, z], [-x, -y, z], [-y, x, z], [x, z, -y], [x, -y, -z], [x, -z, y], [z, y, -x], [-x, y, -z], [-z, y, x], [y, -z, -x], [-z, -y, -x], [-y, z, -x], [y, x, -z], [-y, -x, -z], [y, z, x], [z, -y, x], [-y, -z, x], [-x, z, y], [z, -x, -y], [-z, -x, y], [-x, -z, -y], [z, x, y], [-z, x, -y]

Only two of Kim and Song's rules are particularly useful for FEM calculations: the degree 7, 38-point rule and their degree 8, 47-point rule. The others either contain negative weights or points outside the reference interval. The points and weights, to 32 digits, were obtained from: Ronald Cools' website (http://www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html) and the unique permutations of G^{rot} were computed by me [JWP] using Maple.

Definition at line 214 of file quadrature_monomial.C.

References QBase::_points, QBase::_weights, and QBase::w().

Referenced by init_3D().

{
  for (unsigned int i=0, c=0; i<n_pts; ++i)
    {
      const Real
        x=rule_data[i][0],
        y=rule_data[i][1],
        z=rule_data[i][2],
        w=rule_data[i][3];

      switch(rule_id[i])
        {
        case 0: // (0,0,0) 1 permutation
          {
            _points[c]  = Point( x, y, z);          _weights[c++] = w;
            
            break;
          }
        case 1: //  (x,0,0) 6 permutations
          {
            _points[c] = Point( x, 0., 0.);         _weights[c++] = w;
            _points[c] = Point(0., -x, 0.);         _weights[c++] = w;
            _points[c] = Point(-x, 0., 0.);         _weights[c++] = w;
            _points[c] = Point(0.,  x, 0.);         _weights[c++] = w;
            _points[c] = Point(0., 0., -x);         _weights[c++] = w;
            _points[c] = Point(0., 0.,  x);         _weights[c++] = w;

            break;
          }
        case 2: // (x,x,0) 12 permutations
          {
            _points[c] = Point( x,  x, 0.);         _weights[c++] = w;
            _points[c] = Point( x, -x, 0.);         _weights[c++] = w;
            _points[c] = Point(-x, -x, 0.);         _weights[c++] = w;
            _points[c] = Point(-x,  x, 0.);         _weights[c++] = w;
            _points[c] = Point( x, 0., -x);         _weights[c++] = w;
            _points[c] = Point( x, 0.,  x);         _weights[c++] = w;
            _points[c] = Point(0.,  x, -x);         _weights[c++] = w;
            _points[c] = Point(0.,  x,  x);         _weights[c++] = w;
            _points[c] = Point(0., -x, -x);         _weights[c++] = w;
            _points[c] = Point(-x, 0., -x);         _weights[c++] = w;
            _points[c] = Point(0., -x,  x);         _weights[c++] = w;
            _points[c] = Point(-x, 0.,  x);         _weights[c++] = w;

            break;
          }
        case 3: // (x,y,0) 24 permutations
          {
            _points[c] = Point( x,  y, 0.);         _weights[c++] = w;
            _points[c] = Point( y, -x, 0.);         _weights[c++] = w;
            _points[c] = Point(-x, -y, 0.);         _weights[c++] = w;
            _points[c] = Point(-y,  x, 0.);         _weights[c++] = w;
            _points[c] = Point( x, 0., -y);         _weights[c++] = w;
            _points[c] = Point( x, -y, 0.);         _weights[c++] = w;
            _points[c] = Point( x, 0.,  y);         _weights[c++] = w;
            _points[c] = Point(0.,  y, -x);         _weights[c++] = w;
            _points[c] = Point(-x,  y, 0.);         _weights[c++] = w;
            _points[c] = Point(0.,  y,  x);         _weights[c++] = w;
            _points[c] = Point( y, 0., -x);         _weights[c++] = w;
            _points[c] = Point(0., -y, -x);         _weights[c++] = w;
            _points[c] = Point(-y, 0., -x);         _weights[c++] = w;
            _points[c] = Point( y,  x, 0.);         _weights[c++] = w;
            _points[c] = Point(-y, -x, 0.);         _weights[c++] = w;
            _points[c] = Point( y, 0.,  x);         _weights[c++] = w;
            _points[c] = Point(0., -y,  x);         _weights[c++] = w;
            _points[c] = Point(-y, 0.,  x);         _weights[c++] = w;
            _points[c] = Point(-x, 0.,  y);         _weights[c++] = w;
            _points[c] = Point(0., -x, -y);         _weights[c++] = w;
            _points[c] = Point(0., -x,  y);         _weights[c++] = w;
            _points[c] = Point(-x, 0., -y);         _weights[c++] = w;
            _points[c] = Point(0.,  x,  y);         _weights[c++] = w;
            _points[c] = Point(0.,  x, -y);         _weights[c++] = w;
            
            break;
          }
        case 4: // (x,x,x) 8 permutations
          {
            _points[c] = Point( x,  x,  x);         _weights[c++] = w;
            _points[c] = Point( x, -x,  x);         _weights[c++] = w;
            _points[c] = Point(-x, -x,  x);         _weights[c++] = w;
            _points[c] = Point(-x,  x,  x);         _weights[c++] = w;
            _points[c] = Point( x,  x, -x);         _weights[c++] = w;
            _points[c] = Point( x, -x, -x);         _weights[c++] = w;
            _points[c] = Point(-x,  x, -x);         _weights[c++] = w;
            _points[c] = Point(-x, -x, -x);         _weights[c++] = w;

            break;
          }
        case 5: // (x,x,z) 24 permutations
          {
            _points[c] = Point( x,  x,  z);         _weights[c++] = w;
            _points[c] = Point( x, -x,  z);         _weights[c++] = w;
            _points[c] = Point(-x, -x,  z);         _weights[c++] = w;
            _points[c] = Point(-x,  x,  z);         _weights[c++] = w;
            _points[c] = Point( x,  z, -x);         _weights[c++] = w;
            _points[c] = Point( x, -x, -z);         _weights[c++] = w;
            _points[c] = Point( x, -z,  x);         _weights[c++] = w;
            _points[c] = Point( z,  x, -x);         _weights[c++] = w;
            _points[c] = Point(-x,  x, -z);         _weights[c++] = w;
            _points[c] = Point(-z,  x,  x);         _weights[c++] = w;
            _points[c] = Point( x, -z, -x);         _weights[c++] = w;
            _points[c] = Point(-z, -x, -x);         _weights[c++] = w;
            _points[c] = Point(-x,  z, -x);         _weights[c++] = w;
            _points[c] = Point( x,  x, -z);         _weights[c++] = w;
            _points[c] = Point(-x, -x, -z);         _weights[c++] = w;
            _points[c] = Point( x,  z,  x);         _weights[c++] = w;
            _points[c] = Point( z, -x,  x);         _weights[c++] = w;
            _points[c] = Point(-x, -z,  x);         _weights[c++] = w;
            _points[c] = Point(-x,  z,  x);         _weights[c++] = w;
            _points[c] = Point( z, -x, -x);         _weights[c++] = w;
            _points[c] = Point(-z, -x,  x);         _weights[c++] = w;
            _points[c] = Point(-x, -z, -x);         _weights[c++] = w;
            _points[c] = Point( z,  x,  x);         _weights[c++] = w;
            _points[c] = Point(-z,  x, -x);         _weights[c++] = w;

            break;
          }
        case 6: // (x,y,z) 24 permutations
          {
            _points[c] = Point( x,  y,  z);         _weights[c++] = w;
            _points[c] = Point( y, -x,  z);         _weights[c++] = w;
            _points[c] = Point(-x, -y,  z);         _weights[c++] = w;
            _points[c] = Point(-y,  x,  z);         _weights[c++] = w;
            _points[c] = Point( x,  z, -y);         _weights[c++] = w;
            _points[c] = Point( x, -y, -z);         _weights[c++] = w;
            _points[c] = Point( x, -z,  y);         _weights[c++] = w;
            _points[c] = Point( z,  y, -x);         _weights[c++] = w;
            _points[c] = Point(-x,  y, -z);         _weights[c++] = w;
            _points[c] = Point(-z,  y,  x);         _weights[c++] = w;
            _points[c] = Point( y, -z, -x);         _weights[c++] = w;
            _points[c] = Point(-z, -y, -x);         _weights[c++] = w;
            _points[c] = Point(-y,  z, -x);         _weights[c++] = w;
            _points[c] = Point( y,  x, -z);         _weights[c++] = w;
            _points[c] = Point(-y, -x, -z);         _weights[c++] = w;
            _points[c] = Point( y,  z,  x);         _weights[c++] = w;
            _points[c] = Point( z, -y,  x);         _weights[c++] = w;
            _points[c] = Point(-y, -z,  x);         _weights[c++] = w;
            _points[c] = Point(-x,  z,  y);         _weights[c++] = w;
            _points[c] = Point( z, -x, -y);         _weights[c++] = w;
            _points[c] = Point(-z, -x,  y);         _weights[c++] = w;
            _points[c] = Point(-x, -z, -y);         _weights[c++] = w;
            _points[c] = Point( z,  x,  y);         _weights[c++] = w;
            _points[c] = Point(-z,  x, -y);         _weights[c++] = w;

            break;
          }
        default:
          {
            std::cerr << 'Unknown rule ID: ' << rule_id[i] << '!' << std::endl;
            libmesh_error();
          }
        } // end switch(rule_id[i])
    }
}

static unsigned int ReferenceCounter::n_objects () [inline, static, inherited]Prints the number of outstanding (created, but not yet destroyed) objects.

Definition at line 76 of file reference_counter.h.

References ReferenceCounter::_n_objects.

Referenced by System::read_serialized_blocked_dof_objects(), and System::write_serialized_blocked_dof_objects().

  { return _n_objects; }

unsigned int QBase::n_points () const [inline, inherited]Returns:

: the number of points associated with the quadrature rule.

Definition at line 115 of file quadrature.h.

References QBase::_points.

Referenced by FEBase::coarsened_dof_values(), QConical::conical_product_pyramid(), QConical::conical_product_tet(), QConical::conical_product_tri(), FEMSystem::eulerian_residual(), InfFE< Dim, T_radial, T_map >::init_face_shape_functions(), FEMSystem::mass_residual(), and QBase::print_info().

    { libmesh_assert (!_points.empty()); return _points.size(); }

void QBase::print_info (std::ostream &os = std::cout) const [inline, inherited]Prints information relevant to the quadrature rule.

Definition at line 350 of file quadrature.h.

References QBase::_points, QBase::_weights, QBase::n_points(), and QBase::qp().

Referenced by operator<<().

{
  libmesh_assert(!_points.empty());
  libmesh_assert(!_weights.empty());

  os << 'N_Q_Points=' << this->n_points() << std::endl << std::endl;
  for (unsigned int qp=0; qp<this->n_points(); qp++)
    {
      os << ' Point ' << qp << ':
         << '  '
         << _points[qp]
         << ' Weight:'
         << '  w=' << _weights[qp] << ' << std::endl;
    }
}

void ReferenceCounter::print_info () [static, inherited]Prints the reference information to std::cout.

Definition at line 83 of file reference_counter.C.

References ReferenceCounter::get_info().

{
#if defined(LIBMESH_ENABLE_REFERENCE_COUNTING) && defined(DEBUG)
  
  std::cout << ReferenceCounter::get_info();
  
#endif
}

Point QBase::qp (const unsigned inti) const [inline, inherited]Returns:

: the $ i^{th} $ quadrature point on the reference object.

Definition at line 148 of file quadrature.h.

References QBase::_points.

Referenced by QConical::conical_product_pyramid(), QConical::conical_product_tet(), QConical::conical_product_tri(), and QBase::print_info().

    { libmesh_assert (i < _points.size()); return _points[i]; }

void QBase::scale (std::pair< Real, Real >old_range, std::pair< Real, Real >new_range) [inherited]Maps the points of a 1D interval quadrature rule (typically [-1,1]) to any other 1D interval (typically [0,1]) and scales the weights accordingly. The quadrature rule will be mapped from the entries of old_range to the entries of new_range.

Definition at line 81 of file quadrature.C.

References QBase::_points, and QBase::_weights.

Referenced by QConical::conical_product_tet(), and QConical::conical_product_tri().

{
  // Make sure we are in 1D
  libmesh_assert(_dim == 1);
  
  // Make sure that we have sane ranges 
  libmesh_assert(new_range.second > new_range.first);
  libmesh_assert(old_range.second > old_range.first);

  // Make sure there are some points
  libmesh_assert(_points.size() > 0);

  // We're mapping from old_range -> new_range 
  for (unsigned int i=0; i<_points.size(); i++)
    {
      _points[i](0) =
        (_points[i](0) - old_range.first) *
        (new_range.second - new_range.first) /
        (old_range.second - old_range.first) +
        new_range.first;
    }

  // Compute the scale factor and scale the weights
  const Real scfact = (new_range.second - new_range.first) /
                      (old_range.second - old_range.first);

  for (unsigned int i=0; i<_points.size(); i++)
    _weights[i] *= scfact;
}

void QMonomial::stroud_rule (const Realrule_data[][3], const unsigned int *rule_symmetry, const unsigned intn_pts) [private]Stroud's rules for QUADs and HEXes can have one of several different types of symmetry. The rule_symmetry array describes how the different lines of the rule_data array are to be applied. The different rule_symmetry possibilities are: 0) Origin or single-point: (x,y) Fully-symmetric, 3 cases: 1) (x,y) -> (x,y), (-x,y), (x,-y), (-x,-y) (y,x), (-y,x), (y,-x), (-y,-x) 2) (x,x) -> (x,x), (-x,x), (x,-x), (-x,-x) 3) (x,0) -> (x,0), (-x,0), (0, x), ( 0,-x) 4) Rotational Invariant, (x,y) -> (x,y), (-x,-y), (-y, x), (y,-x) 5) Partial Symmetry, (x,y) -> (x,y), (-x, y) [x!=0] 6) Rectangular Symmetry, (x,y) -> (x,y), (-x, y), (-x,-y), (x,-y) 7) Central Symmetry, (0,y) -> (0,y), ( 0,-y)

Not all rules with these symmetries are due to Stroud, however, his book is probably the most frequently-cited compendium of quadrature rules and later authors certainly built upon his work.

Definition at line 63 of file quadrature_monomial.C.

References QBase::_points, QBase::_weights, and QBase::w().

Referenced by init_2D().

{
  for (unsigned int i=0, c=0; i<n_pts; ++i)
    {
      const Real
        x=rule_data[i][0],
        y=rule_data[i][1],
        w=rule_data[i][2];

      switch(rule_symmetry[i])
        {
        case 0: // Single point (no symmetry)
          {
            _points[c]  = Point( x, y);
            _weights[c++] = w;
            
            break;
          }
        case 1: // Fully-symmetric (x,y)
          {
            _points[c]    = Point( x, y);
            _weights[c++] = w;

            _points[c]    = Point(-x, y);
            _weights[c++] = w;

            _points[c]    = Point( x,-y);
            _weights[c++] = w;

            _points[c]    = Point(-x,-y);
            _weights[c++] = w;
            
            _points[c]    = Point( y, x);
            _weights[c++] = w;

            _points[c]    = Point(-y, x);
            _weights[c++] = w;

            _points[c]    = Point( y,-x);
            _weights[c++] = w;

            _points[c]    = Point(-y,-x);
            _weights[c++] = w;
            
            break;
          }
        case 2: // Fully-symmetric (x,x)
          {
            _points[c]    = Point( x, x);
            _weights[c++] = w;

            _points[c]    = Point(-x, x);
            _weights[c++] = w;

            _points[c]    = Point( x,-x);
            _weights[c++] = w;

            _points[c]    = Point(-x,-x);
            _weights[c++] = w;

            break;
          }
        case 3: // Fully-symmetric (x,0)
          {
            libmesh_assert(y==0.0);

            _points[c]    = Point( x,0.);
            _weights[c++] = w;

            _points[c]    = Point(-x,0.);
            _weights[c++] = w;

            _points[c]    = Point(0., x);
            _weights[c++] = w;

            _points[c]    = Point(0.,-x);
            _weights[c++] = w;

            break;
          }
        case 4: // Rotational invariant
          {
            _points[c]    = Point( x, y);
            _weights[c++] = w;

            _points[c]    = Point(-x,-y);
            _weights[c++] = w;

            _points[c]    = Point(-y, x);
            _weights[c++] = w;

            _points[c]    = Point( y,-x);
            _weights[c++] = w;

            break;
          }
        case 5: // Partial symmetry (Wissman's rules)
          {
            libmesh_assert (x != 0.0);
            
            _points[c]    = Point( x, y);
            _weights[c++] = w;

            _points[c]    = Point(-x, y);
            _weights[c++] = w;

            break;
          }
        case 6: // Rectangular symmetry
          {
            _points[c]    = Point( x, y);
            _weights[c++] = w;

            _points[c]    = Point(-x, y);
            _weights[c++] = w;

            _points[c]    = Point(-x,-y);
            _weights[c++] = w;

            _points[c]    = Point( x,-y);
            _weights[c++] = w;

            break;
          }
        case 7: // Central symmetry
          {
            libmesh_assert (x == 0.0);
            libmesh_assert (y != 0.0);
            
            _points[c]    = Point(0., y);
            _weights[c++] = w;

            _points[c]    = Point(0.,-y);
            _weights[c++] = w;

            break;
          }
        default:
          {
            std::cerr << 'Unknown symmetry!' << std::endl;
            libmesh_error();
          }
        } // end switch(rule_symmetry[i])
    }
}

QuadratureType QMonomial::type () const [inline, virtual]Returns:

: QMONOMIAL

Implements QBase.

Definition at line 80 of file quadrature_monomial.h.

References libMeshEnums::QMONOMIAL.

{ return QMONOMIAL; }

Real QBase::w (const unsigned inti) const [inline, inherited]Returns:

: the $ i^{th} $ quadrature weight.

Definition at line 154 of file quadrature.h.

References QBase::_weights.

Referenced by QConical::conical_product_pyramid(), QConical::conical_product_tet(), QConical::conical_product_tri(), QGauss::init_3D(), QGauss::keast_rule(), kim_rule(), and stroud_rule().

    { libmesh_assert (i < _weights.size()); return _weights[i]; }

void QMonomial::wissmann_rule (const Realrule_data[][3], const unsigned intn_pts) [private]Wissmann published three interesting 'partially symmetric' rules for integrating degree 4, 6, and 8 polynomials exactly on QUADs. These rules have all positive weights, all points inside the reference element, and have fewer points than tensor-product rules of equivalent order, making them superior to those rules for monomial bases.

J. W. Wissman and T. Becker, Partially symmetric cubature formulas for even degrees of exactness, SIAM J. Numer. Anal. 23 (1986), 676--685.

Definition at line 43 of file quadrature_monomial.C.

References QBase::_points, and QBase::_weights.

Referenced by init_2D().

{
  for (unsigned int i=0, c=0; i<n_pts; ++i)
    {
      _points[c]  = Point( rule_data[i][0], rule_data[i][1] );
      _weights[c++] = rule_data[i][2];

      // This may be an (x1,x2) -> (-x1,x2) point, in which case
      // we will also generate the mirror point using the same weight.
      if (rule_data[i][0] != static_cast<Real>(0.0))
        {
          _points[c]  = Point( -rule_data[i][0], rule_data[i][1] );
          _weights[c++] = rule_data[i][2];
        }
    }
}

Friends And Related Function Documentation

std::ostream& operator<< (std::ostream &os, const QBase &q) [friend, inherited]Same as above, but allows you to use the stream syntax.

Definition at line 196 of file quadrature.C.

{
  q.print_info(os);
  return os;
}

Member Data Documentation

ReferenceCounter::Counts ReferenceCounter::_counts [static, protected, inherited]Actually holds the data.

Definition at line 110 of file reference_counter.h.

Referenced by ReferenceCounter::get_info(), ReferenceCounter::increment_constructor_count(), and ReferenceCounter::increment_destructor_count().

Threads::spin_mutex ReferenceCounter::_mutex [static, protected, inherited]Mutual exclusion object to enable thread-safe reference counting.

Definition at line 123 of file reference_counter.h.

Threads::atomic< unsigned int > ReferenceCounter::_n_objects [static, protected, inherited]The number of objects. Print the reference count information when the number returns to 0.

Definition at line 118 of file reference_counter.h.

Referenced by ReferenceCounter::n_objects(), ReferenceCounter::ReferenceCounter(), and ReferenceCounter::~ReferenceCounter().

std::cerr<< 'ERROR: Seems as if this quadrature rule' << std::endl << ' is not implemented for 2D.' << std::endl; libmesh_error(); }#endif virtual void init_3D (const ElemType, unsigned int =0)#ifndef DEBUG {}#else { std::cerr << 'ERROR: Seems as if this quadrature rule' << std::endl << ' is not implemented for 3D.' << std::endl; libmesh_error(); }#endif void tensor_product_quad (const QBase& q1D); void tensor_product_hex (const QBase& q1D); void tensor_product_prism (const QBase& q1D, const QBase& q2D); const unsigned int _dim; const Order _order; ElemType _type; unsigned int _p_level; std::vector<Point> QBase::_points [protected, inherited]

Definition at line 320 of file quadrature.h.

Referenced by QConical::conical_product_pyramid(), QConical::conical_product_tet(), QConical::conical_product_tri(), QGauss::dunavant_rule(), QGauss::dunavant_rule2(), QBase::get_points(), QGrundmann_Moller::gm_rule(), QBase::init_0D(), QTrap::init_1D(), QSimpson::init_1D(), QJacobi::init_1D(), QGrid::init_1D(), QGauss::init_1D(), QClough::init_1D(), QTrap::init_2D(), QSimpson::init_2D(), init_2D(), QGrid::init_2D(), QGauss::init_2D(), QClough::init_2D(), QTrap::init_3D(), QSimpson::init_3D(), init_3D(), QGrid::init_3D(), QGauss::init_3D(), QGauss::keast_rule(), kim_rule(), QBase::n_points(), QBase::print_info(), QBase::qp(), QBase::scale(), stroud_rule(), and wissmann_rule().

std::vector<Real> QBase::_weights [protected, inherited]The value of the quadrature weights.

Definition at line 325 of file quadrature.h.

Referenced by QConical::conical_product_pyramid(), QConical::conical_product_tet(), QConical::conical_product_tri(), QGauss::dunavant_rule(), QGauss::dunavant_rule2(), QBase::get_weights(), QGrundmann_Moller::gm_rule(), QBase::init_0D(), QTrap::init_1D(), QSimpson::init_1D(), QJacobi::init_1D(), QGrid::init_1D(), QGauss::init_1D(), QClough::init_1D(), QTrap::init_2D(), QSimpson::init_2D(), init_2D(), QGrid::init_2D(), QGauss::init_2D(), QClough::init_2D(), QTrap::init_3D(), QSimpson::init_3D(), init_3D(), QGrid::init_3D(), QGauss::init_3D(), QGauss::keast_rule(), kim_rule(), QBase::print_info(), QBase::scale(), stroud_rule(), QBase::w(), and wissmann_rule().

bool QBase::allow_rules_with_negative_weights [inherited]Flag (default true) controlling the use of quadrature rules with negative weights. Set this to false to ONLY use (potentially) safer but more expensive rules with all positive weights.

Negative weights typically appear in Gaussian quadrature rules over three-dimensional elements. Rules with negative weights can be unsuitable for some problems. For example, it is possible for a rule with negative weights to obtain a negative result when integrating a positive function.

A particular example: if rules with negative weights are not allowed, a request for TET,THIRD (5 points) will return the TET,FIFTH (14 points) rule instead, nearly tripling the computational effort required!

Definition at line 203 of file quadrature.h.

Referenced by init_3D(), QGrundmann_Moller::init_3D(), and QGauss::init_3D().