Public Member Functions |
Static Public Member Functions |
Public Attributes |
Protected Types |
Protected Member Functions |
Protected Attributes |
Static Protected Attributes |
Private Member Functions |
Friends
libMesh::QGauss Class Reference
#include <quadrature_gauss.h>
Inheritance diagram for libMesh::QGauss:
Public Member Functions | |
| QGauss (const unsigned int _dim, const Order _order=INVALID_ORDER) | |
| ~QGauss () | |
| 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 |
| virtual void | init (const ElemType type=INVALID_ELEM, unsigned int p_level=0) |
| virtual void | init (const Elem &elem, const std::vector< Real > &vertex_distance_func, unsigned int p_level=0) |
| Order | get_order () const |
| void | print_info (std::ostream &os=libMesh::out) const |
| void | scale (std::pair< Real, Real > old_range, std::pair< Real, Real > new_range) |
| virtual bool | shapes_need_reinit () |
Static Public Member Functions | |
| static UniquePtr< QBase > | build (const std::string &name, const unsigned int _dim, const Order _order=INVALID_ORDER) |
| static UniquePtr< QBase > | build (const QuadratureType _qt, const unsigned int _dim, const Order _order=INVALID_ORDER) |
| static void | print_info (std::ostream &out=libMesh::out) |
| static std::string | get_info () |
| static unsigned int | n_objects () |
| static void | enable_print_counter_info () |
| static void | disable_print_counter_info () |
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) |
| libmesh_error_msg ("ERROR: Seems as if this quadrature rule \nis not implemented for 2D.") | |
| void | tensor_product_hex (const QBase &q1D) |
| void | tensor_product_prism (const QBase &q1D, const QBase &q2D) |
| void | increment_constructor_count (const std::string &name) |
| void | increment_destructor_count (const std::string &name) |
Protected Attributes | |
| 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 |
| static bool | _enable_print_counter = true |
Private Member Functions | |
| void | init_1D (const ElemType _type=INVALID_ELEM, unsigned int p_level=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 | dunavant_rule (const Real rule_data[][4], const unsigned int n_pts) |
| void | dunavant_rule2 (const Real *wts, const Real *a, const Real *b, const unsigned int *permutation_ids, const unsigned int n_wts) |
| void | keast_rule (const Real rule_data[][4], const unsigned int n_pts) |
Friends | |
| std::ostream & | operator<< (std::ostream &os, const QBase &q) |
Detailed Description
This class implemenets specific orders of Gauss quadrature. Gauss quadrature rules of Order p have the property of integrating polynomials of degree p exactly.
Definition at line 43 of file quadrature_gauss.h.
Member Typedef Documentation
typedef std::map<std::string, std::pair<unsigned int, unsigned int> > libMesh::ReferenceCounter::Counts [protected, inherited] |
Data structure to log the information. The log is identified by the class name.
Definition at line 113 of file reference_counter.h.
Constructor & Destructor Documentation
| libMesh::QGauss::QGauss | ( | const unsigned int | _dim, |
| const Order | _order = INVALID_ORDER |
||
| ) | [inline] |
Constructor. Declares the order of the quadrature rule.
Definition at line 105 of file quadrature_gauss.h.
References libMesh::QBase::_dim, libMesh::EDGE2, and libMesh::QBase::init().
: QBase(d,o) { // explicitly call the init function in 1D since the // other tensor-product rules require this one. // note that EDGE will not be used internally, however // if we called the function with INVALID_ELEM it would try to // be smart and return, thinking it had already done the work. if (_dim == 1) init(EDGE2); }
| libMesh::QGauss::~QGauss | ( | ) | [inline] |
Member Function Documentation
| UniquePtr< QBase > libMesh::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 UniquePtr<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 42 of file quadrature_build.C.
Referenced by libMesh::InfFE< Dim, T_radial, T_map >::attach_quadrature_rule().
{
return QBase::build (Utility::string_to_enum<QuadratureType> (type),
_dim,
_order);
}
| UniquePtr< QBase > libMesh::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 UniquePtr<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 53 of file quadrature_build.C.
References libMesh::FIRST, libMesh::FORTYTHIRD, libMesh::QBase::libmesh_error_msg(), libMesh::out, libMesh::QCLOUGH, libMesh::QCONICAL, libMesh::QGAUSS, libMesh::QGAUSS_LOBATTO, libMesh::QGRID, libMesh::QGRUNDMANN_MOLLER, libMesh::QJACOBI_1_0, libMesh::QJACOBI_2_0, libMesh::QMONOMIAL, libMesh::QSIMPSON, libMesh::QTRAP, libMesh::THIRD, and libMesh::TWENTYTHIRD.
{
switch (_qt)
{
case QCLOUGH:
{
#ifdef DEBUG
if (_order > TWENTYTHIRD)
{
libMesh::out << "WARNING: Clough quadrature implemented" << std::endl
<< " up to TWENTYTHIRD order." << std::endl;
}
#endif
return UniquePtr<QBase>(new QClough(_dim, _order));
}
case QGAUSS:
{
#ifdef DEBUG
if (_order > FORTYTHIRD)
{
libMesh::out << "WARNING: Gauss quadrature implemented" << std::endl
<< " up to FORTYTHIRD order." << std::endl;
}
#endif
return UniquePtr<QBase>(new QGauss(_dim, _order));
}
case QJACOBI_1_0:
{
#ifdef DEBUG
if (_order > FORTYTHIRD)
{
libMesh::out << "WARNING: Jacobi(1,0) quadrature implemented" << std::endl
<< " up to FORTYTHIRD order." << std::endl;
}
if (_dim > 1)
{
libMesh::out << "WARNING: Jacobi(1,0) quadrature implemented" << std::endl
<< " in 1D only." << std::endl;
}
#endif
return UniquePtr<QBase>(new QJacobi(_dim, _order, 1, 0));
}
case QJACOBI_2_0:
{
#ifdef DEBUG
if (_order > FORTYTHIRD)
{
libMesh::out << "WARNING: Jacobi(2,0) quadrature implemented" << std::endl
<< " up to FORTYTHIRD order." << std::endl;
}
if (_dim > 1)
{
libMesh::out << "WARNING: Jacobi(2,0) quadrature implemented" << std::endl
<< " in 1D only." << std::endl;
}
#endif
return UniquePtr<QBase>(new QJacobi(_dim, _order, 2, 0));
}
case QSIMPSON:
{
#ifdef DEBUG
if (_order > THIRD)
{
libMesh::out << "WARNING: Simpson rule provides only" << std::endl
<< " THIRD order!" << std::endl;
}
#endif
return UniquePtr<QBase>(new QSimpson(_dim));
}
case QTRAP:
{
#ifdef DEBUG
if (_order > FIRST)
{
libMesh::out << "WARNING: Trapezoidal rule provides only" << std::endl
<< " FIRST order!" << std::endl;
}
#endif
return UniquePtr<QBase>(new QTrap(_dim));
}
case QGRID:
return UniquePtr<QBase>(new QGrid(_dim, _order));
case QGRUNDMANN_MOLLER:
return UniquePtr<QBase>(new QGrundmann_Moller(_dim, _order));
case QMONOMIAL:
return UniquePtr<QBase>(new QMonomial(_dim, _order));
case QGAUSS_LOBATTO:
return UniquePtr<QBase>(new QGaussLobatto(_dim, _order));
case QCONICAL:
return UniquePtr<QBase>(new QConical(_dim, _order));
default:
libmesh_error_msg("ERROR: Bad qt=" << _qt);
}
libmesh_error_msg("We'll never get here!");
return UniquePtr<QBase>();
}
| void libMesh::ReferenceCounter::disable_print_counter_info | ( | ) | [static, inherited] |
Definition at line 106 of file reference_counter.C.
References libMesh::ReferenceCounter::_enable_print_counter.
Referenced by libMesh::LibMeshInit::LibMeshInit().
{
_enable_print_counter = false;
return;
}
| void libMesh::QGauss::dunavant_rule | ( | const Real | rule_data[][4], |
| const unsigned int | n_pts | ||
| ) | [private] |
The Dunavant rule is for triangles. It takes permutation points and weights in a specific format as input and fills the pre-sized _points and _weights vectors. This function is only used internally by the TRI geometric elements.
Definition at line 260 of file quadrature_gauss.C.
References libMesh::QBase::_points, libMesh::QBase::_weights, and libMesh::QBase::libmesh_error_msg().
Referenced by init_2D().
{
// The input data array has 4 columns. The first 3 are the permutation points.
// The last column is the weights for a given set of permutation points. A zero
// in two of the first 3 columns implies the point is a 1-permutation (centroid).
// A zero in one of the first 3 columns implies the point is a 3-permutation.
// Otherwise each point is assumed to be a 6-permutation.
// Always insert into the points & weights vector relative to the offset
unsigned int offset=0;
for (unsigned int p=0; p<n_pts; ++p)
{
// There must always be a non-zero entry to start the row
libmesh_assert_not_equal_to ( rule_data[p][0], static_cast<Real>(0.0) );
// A zero weight may imply you did not set up the raw data correctly
libmesh_assert_not_equal_to ( rule_data[p][3], static_cast<Real>(0.0) );
// What kind of point is this?
// One non-zero entry in first 3 cols ? 1-perm (centroid) point = 1
// Two non-zero entries in first 3 cols ? 3-perm point = 3
// Three non-zero entries ? 6-perm point = 6
unsigned int pointtype=1;
if (rule_data[p][1] != static_cast<Real>(0.0))
{
if (rule_data[p][2] != static_cast<Real>(0.0))
pointtype = 6;
else
pointtype = 3;
}
switch (pointtype)
{
case 1:
{
// Be sure we have enough space to insert this point
libmesh_assert_less (offset + 0, _points.size());
// The point has only a single permutation (the centroid!)
_points[offset + 0] = Point(rule_data[p][0], rule_data[p][0]);
// The weight is always the last entry in the row.
_weights[offset + 0] = rule_data[p][3];
offset += 1;
break;
}
case 3:
{
// Be sure we have enough space to insert these points
libmesh_assert_less (offset + 2, _points.size());
// Here it's understood the second entry is to be used twice, and
// thus there are three possible permutations.
_points[offset + 0] = Point(rule_data[p][0], rule_data[p][1]);
_points[offset + 1] = Point(rule_data[p][1], rule_data[p][0]);
_points[offset + 2] = Point(rule_data[p][1], rule_data[p][1]);
for (unsigned int j=0; j<3; ++j)
_weights[offset + j] = rule_data[p][3];
offset += 3;
break;
}
case 6:
{
// Be sure we have enough space to insert these points
libmesh_assert_less (offset + 5, _points.size());
// Three individual entries with six permutations.
_points[offset + 0] = Point(rule_data[p][0], rule_data[p][1]);
_points[offset + 1] = Point(rule_data[p][0], rule_data[p][2]);
_points[offset + 2] = Point(rule_data[p][1], rule_data[p][0]);
_points[offset + 3] = Point(rule_data[p][1], rule_data[p][2]);
_points[offset + 4] = Point(rule_data[p][2], rule_data[p][0]);
_points[offset + 5] = Point(rule_data[p][2], rule_data[p][1]);
for (unsigned int j=0; j<6; ++j)
_weights[offset + j] = rule_data[p][3];
offset += 6;
break;
}
default:
libmesh_error_msg("Don't know what to do with this many permutation points!");
}
}
}
| void libMesh::QGauss::dunavant_rule2 | ( | const Real * | wts, |
| const Real * | a, | ||
| const Real * | b, | ||
| const unsigned int * | permutation_ids, | ||
| const unsigned int | n_wts | ||
| ) | [private] |
Definition at line 185 of file quadrature_gauss.C.
References libMesh::QBase::_points, libMesh::QBase::_weights, and libMesh::QBase::libmesh_error_msg().
Referenced by init_2D().
{
// Figure out how many total points by summing up the entries
// in the permutation_ids array, and resize the _points and _weights
// vectors appropriately.
unsigned int total_pts = 0;
for (unsigned int p=0; p<n_wts; ++p)
total_pts += permutation_ids[p];
// Resize point and weight vectors appropriately.
_points.resize(total_pts);
_weights.resize(total_pts);
// Always insert into the points & weights vector relative to the offset
unsigned int offset=0;
for (unsigned int p=0; p<n_wts; ++p)
{
switch (permutation_ids[p])
{
case 1:
{
// The point has only a single permutation (the centroid!)
// So we don't even need to look in the a or b arrays.
_points [offset + 0] = Point(1.0L/3.0L, 1.0L/3.0L);
_weights[offset + 0] = wts[p];
offset += 1;
break;
}
case 3:
{
// For this type of rule, don't need to look in the b array.
_points[offset + 0] = Point(a[p], a[p]); // (a,a)
_points[offset + 1] = Point(a[p], 1.L-2.L*a[p]); // (a,1-2a)
_points[offset + 2] = Point(1.L-2.L*a[p], a[p]); // (1-2a,a)
for (unsigned int j=0; j<3; ++j)
_weights[offset + j] = wts[p];
offset += 3;
break;
}
case 6:
{
// This type of point uses all 3 arrays...
_points[offset + 0] = Point(a[p], b[p]);
_points[offset + 1] = Point(b[p], a[p]);
_points[offset + 2] = Point(a[p], 1.L-a[p]-b[p]);
_points[offset + 3] = Point(1.L-a[p]-b[p], a[p]);
_points[offset + 4] = Point(b[p], 1.L-a[p]-b[p]);
_points[offset + 5] = Point(1.L-a[p]-b[p], b[p]);
for (unsigned int j=0; j<6; ++j)
_weights[offset + j] = wts[p];
offset += 6;
break;
}
default:
libmesh_error_msg("Unknown permutation id: " << permutation_ids[p] << "!");
}
}
}
| void libMesh::ReferenceCounter::enable_print_counter_info | ( | ) | [static, inherited] |
Methods to enable/disable the reference counter output from print_info()
Definition at line 100 of file reference_counter.C.
References libMesh::ReferenceCounter::_enable_print_counter.
{
_enable_print_counter = true;
return;
}
| unsigned int libMesh::QBase::get_dim | ( | ) | const [inline, inherited] |
- Returns:
- the dimension of the quadrature rule.
Definition at line 125 of file quadrature.h.
References libMesh::QBase::_dim.
Referenced by libMesh::InfFE< Dim, T_radial, T_map >::attach_quadrature_rule(), libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), and libMesh::QConical::conical_product_tri().
{ return _dim; }
| ElemType libMesh::QBase::get_elem_type | ( | ) | const [inline, inherited] |
- Returns:
- the current element type we're set up for
Definition at line 106 of file quadrature.h.
References libMesh::QBase::_type.
{ return _type; }
| std::string libMesh::ReferenceCounter::get_info | ( | ) | [static, inherited] |
Gets a string containing the reference information.
Definition at line 47 of file reference_counter.C.
References libMesh::ReferenceCounter::_counts, and libMesh::Quality::name().
Referenced by libMesh::ReferenceCounter::print_info().
{
#if defined(LIBMESH_ENABLE_REFERENCE_COUNTING) && defined(DEBUG)
std::ostringstream oss;
oss << '\n'
<< " ---------------------------------------------------------------------------- \n"
<< "| Reference count information |\n"
<< " ---------------------------------------------------------------------------- \n";
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;
oss << "| " << name << " reference count information:\n"
<< "| Creations: " << creations << '\n'
<< "| Destructions: " << destructions << '\n';
}
oss << " ---------------------------------------------------------------------------- \n";
return oss.str();
#else
return "";
#endif
}
| Order libMesh::QBase::get_order | ( | ) | const [inline, inherited] |
- Returns:
- the order of the quadrature rule.
Definition at line 183 of file quadrature.h.
References libMesh::QBase::_order, and libMesh::QBase::_p_level.
Referenced by libMesh::InfFE< Dim, T_radial, T_map >::attach_quadrature_rule().
| unsigned int libMesh::QBase::get_p_level | ( | ) | const [inline, inherited] |
- Returns:
- the current p refinement level we're initialized with
Definition at line 112 of file quadrature.h.
References libMesh::QBase::_p_level.
{ return _p_level; }
| const std::vector<Point>& libMesh::QBase::get_points | ( | ) | const [inline, inherited] |
- Returns:
- a
std::vectorcontaining the quadrature point locations on a reference object.
Definition at line 131 of file quadrature.h.
References libMesh::QBase::_points.
Referenced by libMesh::QClough::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QMonomial::init_2D(), init_3D(), and libMesh::QMonomial::init_3D().
{ return _points; }
| std::vector<Point>& libMesh::QBase::get_points | ( | ) | [inline, inherited] |
- Returns:
- a
std::vectorcontaining the quadrature point locations on a reference object as a writeable reference.
Definition at line 137 of file quadrature.h.
References libMesh::QBase::_points.
{ return _points; }
| const std::vector<Real>& libMesh::QBase::get_weights | ( | ) | const [inline, inherited] |
- Returns:
- a
std::vectorcontaining the quadrature weights.
Definition at line 142 of file quadrature.h.
References libMesh::QBase::_weights.
Referenced by libMesh::QClough::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QMonomial::init_2D(), init_3D(), and libMesh::QMonomial::init_3D().
{ return _weights; }
| std::vector<Real>& libMesh::QBase::get_weights | ( | ) | [inline, inherited] |
- Returns:
- a
std::vectorcontaining the quadrature weights.
Definition at line 147 of file quadrature.h.
References libMesh::QBase::_weights.
{ return _weights; }
| void libMesh::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 163 of file reference_counter.h.
References libMesh::ReferenceCounter::_counts, libMesh::Quality::name(), and libMesh::Threads::spin_mtx.
Referenced by libMesh::ReferenceCountedObject< RBParametrized >::ReferenceCountedObject().
{
Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
std::pair<unsigned int, unsigned int>& p = _counts[name];
p.first++;
}
| void libMesh::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 176 of file reference_counter.h.
References libMesh::ReferenceCounter::_counts, libMesh::Quality::name(), and libMesh::Threads::spin_mtx.
Referenced by libMesh::ReferenceCountedObject< RBParametrized >::~ReferenceCountedObject().
{
Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
std::pair<unsigned int, unsigned int>& p = _counts[name];
p.second++;
}
| void libMesh::QBase::init | ( | const ElemType | type = INVALID_ELEM, |
| unsigned int | p_level = 0 |
||
| ) | [virtual, inherited] |
Initializes the data structures to contain a quadrature rule for an object of type type.
Definition at line 28 of file quadrature.C.
References libMesh::QBase::_dim, libMesh::QBase::_p_level, libMesh::QBase::_type, libMesh::QBase::init_0D(), libMesh::QBase::init_1D(), libMesh::QBase::init_2D(), libMesh::QBase::init_3D(), and libMesh::QBase::libmesh_error_msg().
Referenced by libMesh::QBase::init(), libMesh::QClough::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QGaussLobatto::init_2D(), libMesh::QTrap::init_2D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QSimpson::init_2D(), libMesh::QGrid::init_2D(), libMesh::QMonomial::init_2D(), libMesh::QGaussLobatto::init_3D(), libMesh::QTrap::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), libMesh::QGrid::init_3D(), libMesh::QMonomial::init_3D(), QGauss(), libMesh::QGaussLobatto::QGaussLobatto(), libMesh::QJacobi::QJacobi(), libMesh::QSimpson::QSimpson(), and libMesh::QTrap::QTrap().
{
// 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_msg("Invalid dimension _dim = " << _dim);
}
}
| void libMesh::QBase::init | ( | const Elem & | elem, |
| const std::vector< Real > & | vertex_distance_func, | ||
| unsigned int | p_level = 0 |
||
| ) | [virtual, inherited] |
Initializes the data structures for a specific, potentially cut element. The array vertex_distance_func contains vertex values of a signed distance function that cuts the element. This interface is indended to be extended by derived classes that can cut the element into subelements, for example, and constuct a composite quadrature rule for the cut element.
Definition at line 72 of file quadrature.C.
References libMesh::QBase::init(), and libMesh::Elem::type().
{
// dispatch generic implementation
this->init(elem.type(), p_level);
}
| void libMesh::QBase::init_0D | ( | const ElemType | type = INVALID_ELEM, |
| unsigned int | p_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 82 of file quadrature.C.
References libMesh::QBase::_points, and libMesh::QBase::_weights.
Referenced by libMesh::QBase::init().
| void libMesh::QGauss::init_1D | ( | const ElemType | type = INVALID_ELEM, |
| unsigned int | p_level = 0 |
||
| ) | [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 libMesh::QBase.
Definition at line 30 of file quadrature_gauss_1D.C.
References libMesh::QBase::_order, libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::CONSTANT, libMesh::EIGHTH, libMesh::EIGHTTEENTH, libMesh::ELEVENTH, libMesh::FIFTEENTH, libMesh::FIFTH, libMesh::FIRST, libMesh::FORTIETH, libMesh::FORTYFIRST, libMesh::FORTYSECOND, libMesh::FORTYTHIRD, libMesh::FOURTEENTH, libMesh::FOURTH, libMesh::QBase::libmesh_error_msg(), libMesh::NINETEENTH, libMesh::NINTH, libMesh::SECOND, libMesh::SEVENTEENTH, libMesh::SEVENTH, libMesh::SIXTEENTH, libMesh::SIXTH, libMesh::TENTH, libMesh::THIRD, libMesh::THIRTEENTH, libMesh::THIRTIETH, libMesh::THIRTYEIGHTH, libMesh::THIRTYFIFTH, libMesh::THIRTYFIRST, libMesh::THIRTYFOURTH, libMesh::THIRTYNINTH, libMesh::THIRTYSECOND, libMesh::THIRTYSEVENTH, libMesh::THIRTYSIXTH, libMesh::THIRTYTHIRD, libMesh::TWELFTH, libMesh::TWENTIETH, libMesh::TWENTYEIGHTH, libMesh::TWENTYFIFTH, libMesh::TWENTYFIRST, libMesh::TWENTYFOURTH, libMesh::TWENTYNINTH, libMesh::TWENTYSECOND, libMesh::TWENTYSEVENTH, libMesh::TWENTYSIXTH, and libMesh::TWENTYTHIRD.
{
//----------------------------------------------------------------------
// 1D quadrature rules
switch(_order + 2*p)
{
case CONSTANT:
case FIRST:
{
_points.resize (1);
_weights.resize(1);
_points[0](0) = 0.;
_weights[0] = 2.;
return;
}
case SECOND:
case THIRD:
{
_points.resize (2);
_weights.resize(2);
_points[0](0) = -5.7735026918962576450914878050196e-01L; // -sqrt(3)/3
_points[1] = -_points[0];
_weights[0] = 1.;
_weights[1] = _weights[0];
return;
}
case FOURTH:
case FIFTH:
{
_points.resize (3);
_weights.resize(3);
_points[ 0](0) = -7.7459666924148337703585307995648e-01L;
_points[ 1](0) = 0.;
_points[ 2] = -_points[0];
_weights[ 0] = 5.5555555555555555555555555555556e-01L;
_weights[ 1] = 8.8888888888888888888888888888889e-01L;
_weights[ 2] = _weights[0];
return;
}
case SIXTH:
case SEVENTH:
{
_points.resize (4);
_weights.resize(4);
_points[ 0](0) = -8.6113631159405257522394648889281e-01L;
_points[ 1](0) = -3.3998104358485626480266575910324e-01L;
_points[ 2] = -_points[1];
_points[ 3] = -_points[0];
_weights[ 0] = 3.4785484513745385737306394922200e-01L;
_weights[ 1] = 6.5214515486254614262693605077800e-01L;
_weights[ 2] = _weights[1];
_weights[ 3] = _weights[0];
return;
}
case EIGHTH:
case NINTH:
{
_points.resize (5);
_weights.resize(5);
_points[ 0](0) = -9.0617984593866399279762687829939e-01L;
_points[ 1](0) = -5.3846931010568309103631442070021e-01L;
_points[ 2](0) = 0.;
_points[ 3] = -_points[1];
_points[ 4] = -_points[0];
_weights[ 0] = 2.3692688505618908751426404071992e-01L;
_weights[ 1] = 4.7862867049936646804129151483564e-01L;
_weights[ 2] = 5.6888888888888888888888888888889e-01L;
_weights[ 3] = _weights[1];
_weights[ 4] = _weights[0];
return;
}
case TENTH:
case ELEVENTH:
{
_points.resize (6);
_weights.resize(6);
_points[ 0](0) = -9.3246951420315202781230155449399e-01L;
_points[ 1](0) = -6.6120938646626451366139959501991e-01L;
_points[ 2](0) = -2.3861918608319690863050172168071e-01L;
_points[ 3] = -_points[2];
_points[ 4] = -_points[1];
_points[ 5] = -_points[0];
_weights[ 0] = 1.7132449237917034504029614217273e-01L;
_weights[ 1] = 3.6076157304813860756983351383772e-01L;
_weights[ 2] = 4.6791393457269104738987034398955e-01L;
_weights[ 3] = _weights[2];
_weights[ 4] = _weights[1];
_weights[ 5] = _weights[0];
return;
}
case TWELFTH:
case THIRTEENTH:
{
_points.resize (7);
_weights.resize(7);
_points[ 0](0) = -9.4910791234275852452618968404785e-01L;
_points[ 1](0) = -7.4153118559939443986386477328079e-01L;
_points[ 2](0) = -4.0584515137739716690660641207696e-01L;
_points[ 3](0) = 0.;
_points[ 4] = -_points[2];
_points[ 5] = -_points[1];
_points[ 6] = -_points[0];
_weights[ 0] = 1.2948496616886969327061143267908e-01L;
_weights[ 1] = 2.7970539148927666790146777142378e-01L;
_weights[ 2] = 3.8183005050511894495036977548898e-01L;
_weights[ 3] = 4.1795918367346938775510204081633e-01L;
_weights[ 4] = _weights[2];
_weights[ 5] = _weights[1];
_weights[ 6] = _weights[0];
return;
}
case FOURTEENTH:
case FIFTEENTH:
{
_points.resize (8);
_weights.resize(8);
_points[ 0](0) = -9.6028985649753623168356086856947e-01L;
_points[ 1](0) = -7.9666647741362673959155393647583e-01L;
_points[ 2](0) = -5.2553240991632898581773904918925e-01L;
_points[ 3](0) = -1.8343464249564980493947614236018e-01L;
_points[ 4] = -_points[3];
_points[ 5] = -_points[2];
_points[ 6] = -_points[1];
_points[ 7] = -_points[0];
_weights[ 0] = 1.0122853629037625915253135430996e-01L;
_weights[ 1] = 2.2238103445337447054435599442624e-01L;
_weights[ 2] = 3.1370664587788728733796220198660e-01L;
_weights[ 3] = 3.6268378337836198296515044927720e-01L;
_weights[ 4] = _weights[3];
_weights[ 5] = _weights[2];
_weights[ 6] = _weights[1];
_weights[ 7] = _weights[0];
return;
}
case SIXTEENTH:
case SEVENTEENTH:
{
_points.resize (9);
_weights.resize(9);
_points[ 0](0) = -9.6816023950762608983557620290367e-01L;
_points[ 1](0) = -8.3603110732663579429942978806973e-01L;
_points[ 2](0) = -6.1337143270059039730870203934147e-01L;
_points[ 3](0) = -3.2425342340380892903853801464334e-01L;
_points[ 4](0) = 0.;
_points[ 5] = -_points[3];
_points[ 6] = -_points[2];
_points[ 7] = -_points[1];
_points[ 8] = -_points[0];
_weights[ 0] = 8.1274388361574411971892158110524e-02L;
_weights[ 1] = 1.8064816069485740405847203124291e-01L;
_weights[ 2] = 2.6061069640293546231874286941863e-01L;
_weights[ 3] = 3.1234707704000284006863040658444e-01L;
_weights[ 4] = 3.3023935500125976316452506928697e-01L;
_weights[ 5] = _weights[3];
_weights[ 6] = _weights[2];
_weights[ 7] = _weights[1];
_weights[ 8] = _weights[0];
return;
}
case EIGHTTEENTH:
case NINETEENTH:
{
_points.resize (10);
_weights.resize(10);
_points[ 0](0) = -9.7390652851717172007796401208445e-01L;
_points[ 1](0) = -8.6506336668898451073209668842349e-01L;
_points[ 2](0) = -6.7940956829902440623432736511487e-01L;
_points[ 3](0) = -4.3339539412924719079926594316578e-01L;
_points[ 4](0) = -1.4887433898163121088482600112972e-01L;
_points[ 5] = -_points[4];
_points[ 6] = -_points[3];
_points[ 7] = -_points[2];
_points[ 8] = -_points[1];
_points[ 9] = -_points[0];
_weights[ 0] = 6.6671344308688137593568809893332e-02L;
_weights[ 1] = 1.4945134915058059314577633965770e-01L;
_weights[ 2] = 2.1908636251598204399553493422816e-01L;
_weights[ 3] = 2.6926671930999635509122692156947e-01L;
_weights[ 4] = 2.9552422471475287017389299465134e-01L;
_weights[ 5] = _weights[4];
_weights[ 6] = _weights[3];
_weights[ 7] = _weights[2];
_weights[ 8] = _weights[1];
_weights[ 9] = _weights[0];
return;
}
case TWENTIETH:
case TWENTYFIRST:
{
_points.resize (11);
_weights.resize(11);
_points[ 0](0) = -9.7822865814605699280393800112286e-01L;
_points[ 1](0) = -8.8706259976809529907515776930393e-01L;
_points[ 2](0) = -7.3015200557404932409341625203115e-01L;
_points[ 3](0) = -5.1909612920681181592572566945861e-01L;
_points[ 4](0) = -2.6954315595234497233153198540086e-01L;
_points[ 5](0) = 0.;
_points[ 6] = -_points[4];
_points[ 7] = -_points[3];
_points[ 8] = -_points[2];
_points[ 9] = -_points[1];
_points[10] = -_points[0];
_weights[ 0] = 5.5668567116173666482753720442549e-02L;
_weights[ 1] = 1.2558036946490462463469429922394e-01L;
_weights[ 2] = 1.8629021092773425142609764143166e-01L;
_weights[ 3] = 2.3319376459199047991852370484318e-01L;
_weights[ 4] = 2.6280454451024666218068886989051e-01L;
_weights[ 5] = 2.7292508677790063071448352833634e-01L;
_weights[ 6] = _weights[4];
_weights[ 7] = _weights[3];
_weights[ 8] = _weights[2];
_weights[ 9] = _weights[1];
_weights[10] = _weights[0];
return;
}
case TWENTYSECOND:
case TWENTYTHIRD:
{
_points.resize (12);
_weights.resize(12);
_points[ 0](0) = -9.8156063424671925069054909014928e-01L;
_points[ 1](0) = -9.0411725637047485667846586611910e-01L;
_points[ 2](0) = -7.6990267419430468703689383321282e-01L;
_points[ 3](0) = -5.8731795428661744729670241894053e-01L;
_points[ 4](0) = -3.6783149899818019375269153664372e-01L;
_points[ 5](0) = -1.2523340851146891547244136946385e-01L;
_points[ 6] = -_points[5];
_points[ 7] = -_points[4];
_points[ 8] = -_points[3];
_points[ 9] = -_points[2];
_points[10] = -_points[1];
_points[11] = -_points[0];
_weights[ 0] = 4.7175336386511827194615961485017e-02L;
_weights[ 1] = 1.0693932599531843096025471819400e-01L;
_weights[ 2] = 1.6007832854334622633465252954336e-01L;
_weights[ 3] = 2.0316742672306592174906445580980e-01L;
_weights[ 4] = 2.3349253653835480876084989892483e-01L;
_weights[ 5] = 2.4914704581340278500056243604295e-01L;
_weights[ 6] = _weights[5];
_weights[ 7] = _weights[4];
_weights[ 8] = _weights[3];
_weights[ 9] = _weights[2];
_weights[10] = _weights[1];
_weights[11] = _weights[0];
return;
}
case TWENTYFOURTH:
case TWENTYFIFTH:
{
_points.resize (13);
_weights.resize(13);
_points[ 0](0) = -9.8418305471858814947282944880711e-01L;
_points[ 1](0) = -9.1759839922297796520654783650072e-01L;
_points[ 2](0) = -8.0157809073330991279420648958286e-01L;
_points[ 3](0) = -6.4234933944034022064398460699552e-01L;
_points[ 4](0) = -4.4849275103644685287791285212764e-01L;
_points[ 5](0) = -2.3045831595513479406552812109799e-01L;
_points[ 6](0) = 0.;
_points[ 7] = -_points[5];
_points[ 8] = -_points[4];
_points[ 9] = -_points[3];
_points[10] = -_points[2];
_points[11] = -_points[1];
_points[12] = -_points[0];
_weights[ 0] = 4.0484004765315879520021592200986e-02L;
_weights[ 1] = 9.2121499837728447914421775953797e-02L;
_weights[ 2] = 1.3887351021978723846360177686887e-01L;
_weights[ 3] = 1.7814598076194573828004669199610e-01L;
_weights[ 4] = 2.0781604753688850231252321930605e-01L;
_weights[ 5] = 2.2628318026289723841209018603978e-01L;
_weights[ 6] = 2.3255155323087391019458951526884e-01L;
_weights[ 7] = _weights[5];
_weights[ 8] = _weights[4];
_weights[ 9] = _weights[3];
_weights[10] = _weights[2];
_weights[11] = _weights[1];
_weights[12] = _weights[0];
return;
}
case TWENTYSIXTH:
case TWENTYSEVENTH:
{
_points.resize (14);
_weights.resize(14);
_points[ 0](0) = -9.8628380869681233884159726670405e-01L;
_points[ 1](0) = -9.2843488366357351733639113937787e-01L;
_points[ 2](0) = -8.2720131506976499318979474265039e-01L;
_points[ 3](0) = -6.8729290481168547014801980301933e-01L;
_points[ 4](0) = -5.1524863635815409196529071855119e-01L;
_points[ 5](0) = -3.1911236892788976043567182416848e-01L;
_points[ 6](0) = -1.0805494870734366206624465021983e-01L;
_points[ 7] = -_points[6];
_points[ 8] = -_points[5];
_points[ 9] = -_points[4];
_points[10] = -_points[3];
_points[11] = -_points[2];
_points[12] = -_points[1];
_points[13] = -_points[0];
_weights[ 0] = 3.5119460331751863031832876138192e-02L;
_weights[ 1] = 8.0158087159760209805633277062854e-02L;
_weights[ 2] = 1.2151857068790318468941480907248e-01L;
_weights[ 3] = 1.5720316715819353456960193862384e-01L;
_weights[ 4] = 1.8553839747793781374171659012516e-01L;
_weights[ 5] = 2.0519846372129560396592406566122e-01L;
_weights[ 6] = 2.1526385346315779019587644331626e-01L;
_weights[ 7] = _weights[6];
_weights[ 8] = _weights[5];
_weights[ 9] = _weights[4];
_weights[10] = _weights[3];
_weights[11] = _weights[2];
_weights[12] = _weights[1];
_weights[13] = _weights[0];
return;
}
case TWENTYEIGHTH:
case TWENTYNINTH:
{
_points.resize (15);
_weights.resize(15);
_points[ 0](0) = -9.8799251802048542848956571858661e-01L;
_points[ 1](0) = -9.3727339240070590430775894771021e-01L;
_points[ 2](0) = -8.4820658341042721620064832077422e-01L;
_points[ 3](0) = -7.2441773136017004741618605461394e-01L;
_points[ 4](0) = -5.7097217260853884753722673725391e-01L;
_points[ 5](0) = -3.9415134707756336989720737098105e-01L;
_points[ 6](0) = -2.0119409399743452230062830339460e-01L;
_points[ 7](0) = 0.;
_points[ 8] = -_points[6];
_points[ 9] = -_points[5];
_points[10] = -_points[4];
_points[11] = -_points[3];
_points[12] = -_points[2];
_points[13] = -_points[1];
_points[14] = -_points[0];
_weights[ 0] = 3.0753241996117268354628393577204e-02L;
_weights[ 1] = 7.0366047488108124709267416450667e-02L;
_weights[ 2] = 1.0715922046717193501186954668587e-01L;
_weights[ 3] = 1.3957067792615431444780479451103e-01L;
_weights[ 4] = 1.6626920581699393355320086048121e-01L;
_weights[ 5] = 1.8616100001556221102680056186642e-01L;
_weights[ 6] = 1.9843148532711157645611832644384e-01L;
_weights[ 7] = 2.0257824192556127288062019996752e-01L;
_weights[ 8] = _weights[6];
_weights[ 9] = _weights[5];
_weights[10] = _weights[4];
_weights[11] = _weights[3];
_weights[12] = _weights[2];
_weights[13] = _weights[1];
_weights[14] = _weights[0];
return;
}
case THIRTIETH:
case THIRTYFIRST:
{
_points.resize (16);
_weights.resize(16);
_points[ 0](0) = -9.8940093499164993259615417345033e-01L;
_points[ 1](0) = -9.4457502307323257607798841553461e-01L;
_points[ 2](0) = -8.6563120238783174388046789771239e-01L;
_points[ 3](0) = -7.5540440835500303389510119484744e-01L;
_points[ 4](0) = -6.1787624440264374844667176404879e-01L;
_points[ 5](0) = -4.5801677765722738634241944298358e-01L;
_points[ 6](0) = -2.8160355077925891323046050146050e-01L;
_points[ 7](0) = -9.5012509837637440185319335424958e-02L;
_points[ 8] = -_points[7];
_points[ 9] = -_points[6];
_points[10] = -_points[5];
_points[11] = -_points[4];
_points[12] = -_points[3];
_points[13] = -_points[2];
_points[14] = -_points[1];
_points[15] = -_points[0];
_weights[ 0] = 2.7152459411754094851780572456018e-02L;
_weights[ 1] = 6.2253523938647892862843836994378e-02L;
_weights[ 2] = 9.5158511682492784809925107602246e-02L;
_weights[ 3] = 1.2462897125553387205247628219202e-01L;
_weights[ 4] = 1.4959598881657673208150173054748e-01L;
_weights[ 5] = 1.6915651939500253818931207903033e-01L;
_weights[ 6] = 1.8260341504492358886676366796922e-01L;
_weights[ 7] = 1.8945061045506849628539672320828e-01L;
_weights[ 8] = _weights[7];
_weights[ 9] = _weights[6];
_weights[10] = _weights[5];
_weights[11] = _weights[4];
_weights[12] = _weights[3];
_weights[13] = _weights[2];
_weights[14] = _weights[1];
_weights[15] = _weights[0];
return;
}
case THIRTYSECOND:
case THIRTYTHIRD:
{
_points.resize (17);
_weights.resize(17);
_points[ 0](0) = -9.9057547531441733567543401994067e-01L;
_points[ 1](0) = -9.5067552176876776122271695789580e-01L;
_points[ 2](0) = -8.8023915372698590212295569448816e-01L;
_points[ 3](0) = -7.8151400389680140692523005552048e-01L;
_points[ 4](0) = -6.5767115921669076585030221664300e-01L;
_points[ 5](0) = -5.1269053708647696788624656862955e-01L;
_points[ 6](0) = -3.5123176345387631529718551709535e-01L;
_points[ 7](0) = -1.7848418149584785585067749365407e-01L;
_points[ 8](0) = 0.;
_points[ 9] = -_points[7];
_points[10] = -_points[6];
_points[11] = -_points[5];
_points[12] = -_points[4];
_points[13] = -_points[3];
_points[14] = -_points[2];
_points[15] = -_points[1];
_points[16] = -_points[0];
_weights[ 0] = 2.4148302868547931960110026287565e-02L;
_weights[ 1] = 5.5459529373987201129440165358245e-02L;
_weights[ 2] = 8.5036148317179180883535370191062e-02L;
_weights[ 3] = 1.1188384719340397109478838562636e-01L;
_weights[ 4] = 1.3513636846852547328631998170235e-01L;
_weights[ 5] = 1.5404576107681028808143159480196e-01L;
_weights[ 6] = 1.6800410215645004450997066378832e-01L;
_weights[ 7] = 1.7656270536699264632527099011320e-01L;
_weights[ 8] = 1.7944647035620652545826564426189e-01L;
_weights[ 9] = _weights[7];
_weights[10] = _weights[6];
_weights[11] = _weights[5];
_weights[12] = _weights[4];
_weights[13] = _weights[3];
_weights[14] = _weights[2];
_weights[15] = _weights[1];
_weights[16] = _weights[0];
return;
}
case THIRTYFOURTH:
case THIRTYFIFTH:
{
_points.resize (18);
_weights.resize(18);
_points[ 0](0) = -9.9156516842093094673001600470615e-01L;
_points[ 1](0) = -9.5582394957139775518119589292978e-01L;
_points[ 2](0) = -8.9260246649755573920606059112715e-01L;
_points[ 3](0) = -8.0370495897252311568241745501459e-01L;
_points[ 4](0) = -6.9168704306035320787489108128885e-01L;
_points[ 5](0) = -5.5977083107394753460787154852533e-01L;
_points[ 6](0) = -4.1175116146284264603593179383305e-01L;
_points[ 7](0) = -2.5188622569150550958897285487791e-01L;
_points[ 8](0) = -8.4775013041735301242261852935784e-02L;
_points[ 9] = -_points[8];
_points[10] = -_points[7];
_points[11] = -_points[6];
_points[12] = -_points[5];
_points[13] = -_points[4];
_points[14] = -_points[3];
_points[15] = -_points[2];
_points[16] = -_points[1];
_points[17] = -_points[0];
_weights[ 0] = 2.1616013526483310313342710266452e-02L;
_weights[ 1] = 4.9714548894969796453334946202639e-02L;
_weights[ 2] = 7.6425730254889056529129677616637e-02L;
_weights[ 3] = 1.0094204410628716556281398492483e-01L;
_weights[ 4] = 1.2255520671147846018451912680020e-01L;
_weights[ 5] = 1.4064291467065065120473130375195e-01L;
_weights[ 6] = 1.5468467512626524492541800383637e-01L;
_weights[ 7] = 1.6427648374583272298605377646593e-01L;
_weights[ 8] = 1.6914238296314359184065647013499e-01L;
_weights[ 9] = _weights[8];
_weights[10] = _weights[7];
_weights[11] = _weights[6];
_weights[12] = _weights[5];
_weights[13] = _weights[4];
_weights[14] = _weights[3];
_weights[15] = _weights[2];
_weights[16] = _weights[1];
_weights[17] = _weights[0];
return;
}
case THIRTYSIXTH:
case THIRTYSEVENTH:
{
_points.resize (19);
_weights.resize(19);
_points[ 0](0) = -9.9240684384358440318901767025326e-01L;
_points[ 1](0) = -9.6020815213483003085277884068765e-01L;
_points[ 2](0) = -9.0315590361481790164266092853231e-01L;
_points[ 3](0) = -8.2271465653714282497892248671271e-01L;
_points[ 4](0) = -7.2096617733522937861709586082378e-01L;
_points[ 5](0) = -6.0054530466168102346963816494624e-01L;
_points[ 6](0) = -4.6457074137596094571726714810410e-01L;
_points[ 7](0) = -3.1656409996362983199011732884984e-01L;
_points[ 8](0) = -1.6035864564022537586809611574074e-01L;
_points[ 9](0) = 0.;
_points[10] = -_points[8];
_points[11] = -_points[7];
_points[12] = -_points[6];
_points[13] = -_points[5];
_points[14] = -_points[4];
_points[15] = -_points[3];
_points[16] = -_points[2];
_points[17] = -_points[1];
_points[18] = -_points[0];
_weights[ 0] = 1.9461788229726477036312041464438e-02L;
_weights[ 1] = 4.4814226765699600332838157401994e-02L;
_weights[ 2] = 6.9044542737641226580708258006013e-02L;
_weights[ 3] = 9.1490021622449999464462094123840e-02L;
_weights[ 4] = 1.1156664554733399471602390168177e-01L;
_weights[ 5] = 1.2875396253933622767551578485688e-01L;
_weights[ 6] = 1.4260670217360661177574610944190e-01L;
_weights[ 7] = 1.5276604206585966677885540089766e-01L;
_weights[ 8] = 1.5896884339395434764995643946505e-01L;
_weights[ 9] = 1.6105444984878369597916362532092e-01L;
_weights[10] = _weights[8];
_weights[11] = _weights[7];
_weights[12] = _weights[6];
_weights[13] = _weights[5];
_weights[14] = _weights[4];
_weights[15] = _weights[3];
_weights[16] = _weights[2];
_weights[17] = _weights[1];
_weights[18] = _weights[0];
return;
}
case THIRTYEIGHTH:
case THIRTYNINTH:
{
_points.resize (20);
_weights.resize(20);
_points[ 0](0) = -9.9312859918509492478612238847132e-01L;
_points[ 1](0) = -9.6397192727791379126766613119728e-01L;
_points[ 2](0) = -9.1223442825132590586775244120330e-01L;
_points[ 3](0) = -8.3911697182221882339452906170152e-01L;
_points[ 4](0) = -7.4633190646015079261430507035564e-01L;
_points[ 5](0) = -6.3605368072651502545283669622629e-01L;
_points[ 6](0) = -5.1086700195082709800436405095525e-01L;
_points[ 7](0) = -3.7370608871541956067254817702493e-01L;
_points[ 8](0) = -2.2778585114164507808049619536857e-01L;
_points[ 9](0) = -7.6526521133497333754640409398838e-02L;
_points[10] = -_points[9];
_points[11] = -_points[8];
_points[12] = -_points[7];
_points[13] = -_points[6];
_points[14] = -_points[5];
_points[15] = -_points[4];
_points[16] = -_points[3];
_points[17] = -_points[2];
_points[18] = -_points[1];
_points[19] = -_points[0];
_weights[ 0] = 1.7614007139152118311861962351853e-02L;
_weights[ 1] = 4.0601429800386941331039952274932e-02L;
_weights[ 2] = 6.2672048334109063569506535187042e-02L;
_weights[ 3] = 8.3276741576704748724758143222046e-02L;
_weights[ 4] = 1.0193011981724043503675013548035e-01L;
_weights[ 5] = 1.1819453196151841731237737771138e-01L;
_weights[ 6] = 1.3168863844917662689849449974816e-01L;
_weights[ 7] = 1.4209610931838205132929832506716e-01L;
_weights[ 8] = 1.4917298647260374678782873700197e-01L;
_weights[ 9] = 1.5275338713072585069808433195510e-01L;
_weights[10] = _weights[9];
_weights[11] = _weights[8];
_weights[12] = _weights[7];
_weights[13] = _weights[6];
_weights[14] = _weights[5];
_weights[15] = _weights[4];
_weights[16] = _weights[3];
_weights[17] = _weights[2];
_weights[18] = _weights[1];
_weights[19] = _weights[0];
return;
}
case FORTIETH:
case FORTYFIRST:
{
_points.resize (21);
_weights.resize(21);
_points[ 0](0) = -9.9375217062038950026024203593794e-01L;
_points[ 1](0) = -9.6722683856630629431662221490770e-01L;
_points[ 2](0) = -9.2009933415040082879018713371497e-01L;
_points[ 3](0) = -8.5336336458331728364725063858757e-01L;
_points[ 4](0) = -7.6843996347567790861587785130623e-01L;
_points[ 5](0) = -6.6713880419741231930596666999034e-01L;
_points[ 6](0) = -5.5161883588721980705901879672431e-01L;
_points[ 7](0) = -4.2434212020743878357366888854379e-01L;
_points[ 8](0) = -2.8802131680240109660079251606460e-01L;
_points[ 9](0) = -1.4556185416089509093703098233869e-01L;
_points[10](0) = 0.;
_points[11] = -_points[9];
_points[12] = -_points[8];
_points[13] = -_points[7];
_points[14] = -_points[6];
_points[15] = -_points[5];
_points[16] = -_points[4];
_points[17] = -_points[3];
_points[18] = -_points[2];
_points[19] = -_points[1];
_points[20] = -_points[0];
_weights[ 0] = 1.6017228257774333324224616858471e-02L;
_weights[ 1] = 3.6953789770852493799950668299330e-02L;
_weights[ 2] = 5.7134425426857208283635826472448e-02L;
_weights[ 3] = 7.6100113628379302017051653300183e-02L;
_weights[ 4] = 9.3444423456033861553289741113932e-02L;
_weights[ 5] = 1.0879729916714837766347457807011e-01L;
_weights[ 6] = 1.2183141605372853419536717712572e-01L;
_weights[ 7] = 1.3226893863333746178105257449678e-01L;
_weights[ 8] = 1.3988739479107315472213342386758e-01L;
_weights[ 9] = 1.4452440398997005906382716655375e-01L;
_weights[10] = 1.4608113364969042719198514768337e-01L;
_weights[11] = _weights[9];
_weights[12] = _weights[8];
_weights[13] = _weights[7];
_weights[14] = _weights[6];
_weights[15] = _weights[5];
_weights[16] = _weights[4];
_weights[17] = _weights[3];
_weights[18] = _weights[2];
_weights[19] = _weights[1];
_weights[20] = _weights[0];
return;
}
case FORTYSECOND:
case FORTYTHIRD:
{
_points.resize (22);
_weights.resize(22);
_points[ 0](0) = -9.9429458548239929207303142116130e-01L;
_points[ 1](0) = -9.7006049783542872712395098676527e-01L;
_points[ 2](0) = -9.2695677218717400052069293925905e-01L;
_points[ 3](0) = -8.6581257772030013653642563701938e-01L;
_points[ 4](0) = -7.8781680597920816200427795540835e-01L;
_points[ 5](0) = -6.9448726318668278005068983576226e-01L;
_points[ 6](0) = -5.8764040350691159295887692763865e-01L;
_points[ 7](0) = -4.6935583798675702640633071096641e-01L;
_points[ 8](0) = -3.4193582089208422515814742042738e-01L;
_points[ 9](0) = -2.0786042668822128547884653391955e-01L;
_points[10](0) = -6.9739273319722221213841796118628e-02L;
_points[11] = -_points[10];
_points[12] = -_points[9];
_points[13] = -_points[8];
_points[14] = -_points[7];
_points[15] = -_points[6];
_points[16] = -_points[5];
_points[17] = -_points[4];
_points[18] = -_points[3];
_points[19] = -_points[2];
_points[20] = -_points[1];
_points[21] = -_points[0];
_weights[ 0] = 1.4627995298272200684991098047185e-02L;
_weights[ 1] = 3.3774901584814154793302246865913e-02L;
_weights[ 2] = 5.2293335152683285940312051273211e-02L;
_weights[ 3] = 6.9796468424520488094961418930218e-02L;
_weights[ 4] = 8.5941606217067727414443681372703e-02L;
_weights[ 5] = 1.0041414444288096493207883783054e-01L;
_weights[ 6] = 1.1293229608053921839340060742175e-01L;
_weights[ 7] = 1.2325237681051242428556098615481e-01L;
_weights[ 8] = 1.3117350478706237073296499253031e-01L;
_weights[ 9] = 1.3654149834601517135257383123152e-01L;
_weights[10] = 1.3925187285563199337541024834181e-01L;
_weights[11] = _weights[10];
_weights[12] = _weights[9];
_weights[13] = _weights[8];
_weights[14] = _weights[7];
_weights[15] = _weights[6];
_weights[16] = _weights[5];
_weights[17] = _weights[4];
_weights[18] = _weights[3];
_weights[19] = _weights[2];
_weights[20] = _weights[1];
_weights[21] = _weights[0];
return;
}
default:
libmesh_error_msg("Quadrature rule " << _order << " not supported!");
}
}
| void libMesh::QGauss::init_2D | ( | const ElemType | = INVALID_ELEM, |
| unsigned int | = 0 |
||
| ) | [private, virtual] |
Initializes the 2D 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. Should not be pure virtual since a derived quadrature rule may only be defined in 1D. If not redefined, gives an error (when DEBUG defined) when called.
Reimplemented from libMesh::QBase.
Definition at line 28 of file quadrature_gauss_2D.C.
References libMesh::QBase::_order, libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::CONSTANT, dunavant_rule(), dunavant_rule2(), libMesh::EDGE2, libMesh::EIGHTH, libMesh::EIGHTTEENTH, libMesh::ELEVENTH, libMesh::FIFTEENTH, libMesh::FIFTH, libMesh::FIRST, libMesh::FOURTEENTH, libMesh::FOURTH, libMesh::QBase::get_points(), libMesh::QBase::get_weights(), libMesh::QBase::init(), libMesh::QBase::libmesh_error_msg(), libMesh::NINETEENTH, libMesh::NINTH, libMesh::QUAD4, libMesh::QUAD8, libMesh::QUAD9, libMesh::Real, libMesh::SECOND, libMesh::SEVENTEENTH, libMesh::SEVENTH, libMesh::SIXTEENTH, libMesh::SIXTH, libMesh::TENTH, libMesh::THIRD, libMesh::THIRTEENTH, libMesh::THIRTIETH, libMesh::TRI3, libMesh::TRI3SUBDIVISION, libMesh::TRI6, libMesh::TWELFTH, libMesh::TWENTIETH, libMesh::TWENTYEIGHTH, libMesh::TWENTYFIFTH, libMesh::TWENTYFOURTH, libMesh::TWENTYNINTH, libMesh::TWENTYSECOND, libMesh::TWENTYSEVENTH, libMesh::TWENTYSIXTH, and libMesh::TWENTYTHIRD.
{
#if LIBMESH_DIM > 1
//-----------------------------------------------------------------------
// 2D quadrature rules
switch (type_in)
{
//---------------------------------------------
// Quadrilateral quadrature rules
case QUAD4:
case QUAD8:
case QUAD9:
{
// We compute the 2D quadrature rule as a tensor
// product of the 1D quadrature rule.
//
// For QUADs, a quadrature rule of order 'p' must be able to integrate
// bilinear (p=1), biquadratic (p=2), bicubic (p=3), etc. polynomials of the form
//
// (x^p + x^{p-1} + ... + 1) * (y^p + y^{p-1} + ... + 1)
//
// These polynomials have terms *up to* degree 2p but they are *not* complete
// polynomials of degree 2p. For example, when p=2 we have
// 1
// x y
// x^2 xy y^2
// yx^2 xy^2
// x^2y^2
QGauss q1D(1,_order);
q1D.init(EDGE2,p);
tensor_product_quad( q1D );
return;
}
//---------------------------------------------
// Triangle quadrature rules
case TRI3:
case TRI3SUBDIVISION:
case TRI6:
{
switch(_order + 2*p)
{
case CONSTANT:
case FIRST:
{
// Exact for linears
_points.resize(1);
_weights.resize(1);
_points[0](0) = 1.0L/3.0L;
_points[0](1) = 1.0L/3.0L;
_weights[0] = 0.5;
return;
}
case SECOND:
{
// Exact for quadratics
_points.resize(3);
_weights.resize(3);
// Alternate rule with points on ref. elt. boundaries.
// Not ideal for problems with material coefficient discontinuities
// aligned along element boundaries.
// _points[0](0) = .5;
// _points[0](1) = .5;
// _points[1](0) = 0.;
// _points[1](1) = .5;
// _points[2](0) = .5;
// _points[2](1) = .0;
_points[0](0) = 2.0L/3.0L;
_points[0](1) = 1.0L/6.0L;
_points[1](0) = 1.0L/6.0L;
_points[1](1) = 2.0L/3.0L;
_points[2](0) = 1.0L/6.0L;
_points[2](1) = 1.0L/6.0L;
_weights[0] = 1.0L/6.0L;
_weights[1] = 1.0L/6.0L;
_weights[2] = 1.0L/6.0L;
return;
}
case THIRD:
{
// Exact for cubics
_points.resize(4);
_weights.resize(4);
// This rule is formed from a tensor product of
// appropriately-scaled Gauss and Jacobi rules. (See
// also the QConical quadrature class, this is a
// hard-coded version of one of those rules.) For high
// orders these rules generally have too many points,
// but at extremely low order they are competitive and
// have the additional benefit of having all positive
// weights.
_points[0](0) = 1.5505102572168219018027159252941e-01L;
_points[0](1) = 1.7855872826361642311703513337422e-01L;
_points[1](0) = 6.4494897427831780981972840747059e-01L;
_points[1](1) = 7.5031110222608118177475598324603e-02L;
_points[2](0) = 1.5505102572168219018027159252941e-01L;
_points[2](1) = 6.6639024601470138670269327409637e-01L;
_points[3](0) = 6.4494897427831780981972840747059e-01L;
_points[3](1) = 2.8001991549907407200279599420481e-01L;
_weights[0] = 1.5902069087198858469718450103758e-01L;
_weights[1] = 9.0979309128011415302815498962418e-02L;
_weights[2] = 1.5902069087198858469718450103758e-01L;
_weights[3] = 9.0979309128011415302815498962418e-02L;
return;
// The following third-order rule is quite commonly cited
// in the literature and most likely works fine. However,
// we generally prefer a rule with all positive weights
// and an equal number of points, when available.
//
// (allow_rules_with_negative_weights)
// {
// // Exact for cubics
// _points.resize(4);
// _weights.resize(4);
//
// _points[0](0) = .33333333333333333333333333333333;
// _points[0](1) = .33333333333333333333333333333333;
//
// _points[1](0) = .2;
// _points[1](1) = .6;
//
// _points[2](0) = .2;
// _points[2](1) = .2;
//
// _points[3](0) = .6;
// _points[3](1) = .2;
//
//
// _weights[0] = -27./96.;
// _weights[1] = 25./96.;
// _weights[2] = 25./96.;
// _weights[3] = 25./96.;
//
// return;
// } // end if (allow_rules_with_negative_weights)
// Note: if !allow_rules_with_negative_weights, fall through to next case.
}
// A degree 4 rule with six points. This rule can be found in many places
// including:
//
// J.N. Lyness and D. Jespersen, Moderate degree symmetric
// quadrature rules for the triangle, J. Inst. Math. Appl. 15 (1975),
// 19--32.
//
// We used the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
// to generate additional precision.
case FOURTH:
{
const unsigned int n_wts = 2;
const Real wts[n_wts] =
{
1.1169079483900573284750350421656140e-01L,
5.4975871827660933819163162450105264e-02L
};
const Real a[n_wts] =
{
4.4594849091596488631832925388305199e-01L,
9.1576213509770743459571463402201508e-02L
};
const Real b[n_wts] = {0., 0.}; // not used
const unsigned int permutation_ids[n_wts] = {3, 3};
dunavant_rule2(wts, a, b, permutation_ids, n_wts); // 6 total points
return;
}
// Exact for quintics
// Can be found in "Quadrature on Simplices of Arbitrary
// Dimension" by Walkington.
case FIFTH:
{
const unsigned int n_wts = 3;
const Real wts[n_wts] =
{
static_cast<Real>(9.0L/80.0L),
static_cast<Real>(31.0L/480.0L + std::sqrt(15.0L)/2400.0L),
static_cast<Real>(31.0L/480.0L - std::sqrt(15.0L)/2400.0L)
};
const Real a[n_wts] =
{
0., // 'a' parameter not used for origin permutation
static_cast<Real>(2.0L/7.0L + std::sqrt(15.0L)/21.0L),
static_cast<Real>(2.0L/7.0L - std::sqrt(15.0L)/21.0L)
};
const Real b[n_wts] = {0., 0., 0.}; // not used
const unsigned int permutation_ids[n_wts] = {1, 3, 3};
dunavant_rule2(wts, a, b, permutation_ids, n_wts); // 7 total points
return;
}
// A degree 6 rule with 12 points. This rule can be found in many places
// including:
//
// J.N. Lyness and D. Jespersen, Moderate degree symmetric
// quadrature rules for the triangle, J. Inst. Math. Appl. 15 (1975),
// 19--32.
//
// We used the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
// to generate additional precision.
//
// Note that the following 7th-order Ro3-invariant rule also has only 12 points,
// which technically makes it the superior rule. This one is here for completeness.
case SIXTH:
{
const unsigned int n_wts = 3;
const Real wts[n_wts] =
{
5.8393137863189683012644805692789721e-02L,
2.5422453185103408460468404553434492e-02L,
4.1425537809186787596776728210221227e-02L
};
const Real a[n_wts] =
{
2.4928674517091042129163855310701908e-01L,
6.3089014491502228340331602870819157e-02L,
3.1035245103378440541660773395655215e-01L
};
const Real b[n_wts] =
{
0.,
0.,
6.3650249912139864723014259441204970e-01L
};
const unsigned int permutation_ids[n_wts] = {3, 3, 6}; // 12 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// A degree 7 rule with 12 points. This rule can be found in:
//
// K. Gatermann, The construction of symmetric cubature
// formulas for the square and the triangle, Computing 40
// (1988), 229--240.
//
// This rule, which is provably minimal in the number of
// integration points, is said to be 'Ro3 invariant' which
// means that a given set of barycentric coordinates
// (z1,z2,z3) implies the quadrature points (z1,z2),
// (z3,z1), (z2,z3) which are formed by taking the first
// two entries in cyclic permutations of the barycentric
// point. Barycentric coordinates are related in the
// sense that: z3 = 1 - z1 - z2.
//
// The 12-point sixth-order rule for triangles given in
// Flaherty's (http://www.cs.rpi.edu/~flaherje/FEM/fem6.ps)
// lecture notes has been removed in favor of this rule
// which is higher-order (for the same number of
// quadrature points) and has a few more digits of
// precision in the points and weights. Some 10-point
// degree 6 rules exist for the triangle but they have
// quadrature points outside the region of integration.
case SEVENTH:
{
_points.resize (12);
_weights.resize(12);
const unsigned int nrows=4;
// In each of the rows below, the first two entries are (z1, z2) which imply
// z3. The third entry is the weight for each of the points in the cyclic permutation.
const Real rule_data[nrows][3] = {
{6.2382265094402118e-02, 6.7517867073916085e-02, 2.6517028157436251e-02}, // group A
{5.5225456656926611e-02, 3.2150249385198182e-01, 4.3881408714446055e-02}, // group B
{3.4324302945097146e-02, 6.6094919618673565e-01, 2.8775042784981585e-02}, // group C
{5.1584233435359177e-01, 2.7771616697639178e-01, 6.7493187009802774e-02} // group D
};
for (unsigned int i=0, offset=0; i<nrows; ++i)
{
_points[offset + 0] = Point(rule_data[i][0], rule_data[i][1]); // (z1,z2)
_points[offset + 1] = Point(1.-rule_data[i][0]-rule_data[i][1], rule_data[i][0]); // (z3,z1)
_points[offset + 2] = Point(rule_data[i][1], 1.-rule_data[i][0]-rule_data[i][1]); // (z2,z3)
// All these points get the same weight
_weights[offset + 0] = rule_data[i][2];
_weights[offset + 1] = rule_data[i][2];
_weights[offset + 2] = rule_data[i][2];
// Increment offset
offset += 3;
}
return;
// // The following is an inferior 7th-order Lyness-style rule with 15 points.
// // It's here only for completeness and the Ro3-invariant rule above should
// // be used instead!
// const unsigned int n_wts = 3;
// const Real wts[n_wts] =
// {
// 2.6538900895116205835977487499847719e-02L,
// 3.5426541846066783659206291623201826e-02L,
// 3.4637341039708446756138297960207647e-02L
// };
//
// const Real a[n_wts] =
// {
// 6.4930513159164863078379776030396538e-02L,
// 2.8457558424917033519741605734978046e-01L,
// 3.1355918438493150795585190219862865e-01L
// };
//
// const Real b[n_wts] =
// {
// 0.,
// 1.9838447668150671917987659863332941e-01L,
// 4.3863471792372471511798695971295936e-02L
// };
//
// const unsigned int permutation_ids[n_wts] = {3, 6, 6}; // 15 total points
//
// dunavant_rule2(wts, a, b, permutation_ids, n_wts);
//
// return;
}
// Another Dunavant rule. This one has all positive weights. This rule has
// 16 points while a comparable conical product rule would have 5*5=25.
//
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case EIGHTH:
{
const unsigned int n_wts = 5;
const Real wts[n_wts] =
{
7.2157803838893584125545555244532310e-02L,
4.7545817133642312396948052194292159e-02L,
5.1608685267359125140895775146064515e-02L,
1.6229248811599040155462964170890299e-02L,
1.3615157087217497132422345036954462e-02L
};
const Real a[n_wts] =
{
0.0, // 'a' parameter not used for origin permutation
4.5929258829272315602881551449416932e-01L,
1.7056930775176020662229350149146450e-01L,
5.0547228317030975458423550596598947e-02L,
2.6311282963463811342178578628464359e-01L,
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
7.2849239295540428124100037917606196e-01L
};
const unsigned int permutation_ids[n_wts] = {1, 3, 3, 3, 6}; // 16 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Another Dunavant rule. This one has all positive weights. This rule has 19
// points. The comparable conical product rule would have 25.
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case NINTH:
{
const unsigned int n_wts = 6;
const Real wts[n_wts] =
{
4.8567898141399416909620991253644315e-02L,
1.5667350113569535268427415643604658e-02L,
1.2788837829349015630839399279499912e-02L,
3.8913770502387139658369678149701978e-02L,
3.9823869463605126516445887132022637e-02L,
2.1641769688644688644688644688644689e-02L
};
const Real a[n_wts] =
{
0.0, // 'a' parameter not used for origin permutation
4.8968251919873762778370692483619280e-01L,
4.4729513394452709865106589966276365e-02L,
4.3708959149293663726993036443535497e-01L,
1.8820353561903273024096128046733557e-01L,
2.2196298916076569567510252769319107e-01L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
0.,
7.4119859878449802069007987352342383e-01L
};
const unsigned int permutation_ids[n_wts] = {1, 3, 3, 3, 3, 6}; // 19 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Another Dunavant rule with all positive weights. This rule has 25
// points. The comparable conical product rule would have 36.
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case TENTH:
{
const unsigned int n_wts = 6;
const Real wts[n_wts] =
{
4.5408995191376790047643297550014267e-02L,
1.8362978878233352358503035945683300e-02L,
2.2660529717763967391302822369298659e-02L,
3.6378958422710054302157588309680344e-02L,
1.4163621265528742418368530791049552e-02L,
4.7108334818664117299637354834434138e-03L
};
const Real a[n_wts] =
{
0.0, // 'a' parameter not used for origin permutation
4.8557763338365737736750753220812615e-01L,
1.0948157548503705479545863134052284e-01L,
3.0793983876412095016515502293063162e-01L,
2.4667256063990269391727646541117681e-01L,
6.6803251012200265773540212762024737e-02L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
5.5035294182099909507816172659300821e-01L,
7.2832390459741092000873505358107866e-01L,
9.2365593358750027664630697761508843e-01L
};
const unsigned int permutation_ids[n_wts] = {1, 3, 3, 6, 6, 6}; // 25 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Dunavant's 11th-order rule contains points outside the region of
// integration, and is thus unacceptable for our FEM calculations.
//
// This 30-point, 11th-order rule was obtained by me [JWP] using the code in
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
//
// Note: the 28-point 11th-order rule obtained by Zhang in the paper above
// does not appear to be unique. It is a solution in the sense that it
// minimizes the error in the least-squares minimization problem, but
// it involves too many unknowns and the Jacobian is therefore singular
// when attempting to improve the solution via Newton's method.
case ELEVENTH:
{
const unsigned int n_wts = 6;
const Real wts[n_wts] =
{
3.6089021198604635216985338480426484e-02L,
2.1607717807680420303346736867931050e-02L,
3.1144524293927978774861144478241807e-03L,
2.9086855161081509446654185084988077e-02L,
8.4879241614917017182977532679947624e-03L,
1.3795732078224796530729242858347546e-02L
};
const Real a[n_wts] =
{
3.9355079629947969884346551941969960e-01L,
4.7979065808897448654107733982929214e-01L,
5.1003445645828061436081405648347852e-03L,
2.6597620190330158952732822450744488e-01L,
2.8536418538696461608233522814483715e-01L,
1.3723536747817085036455583801851025e-01L
};
const Real b[n_wts] =
{
0.,
0.,
5.6817155788572446538150614865768991e-02L,
1.2539956353662088473247489775203396e-01L,
1.2409970153698532116262152247041742e-02L,
5.2792057988217708934207928630851643e-02L
};
const unsigned int permutation_ids[n_wts] = {3, 3, 6, 6, 6, 6}; // 30 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Another Dunavant rule with all positive weights. This rule has 33
// points. The comparable conical product rule would have 36 (ELEVENTH) or 49 (TWELFTH).
//
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case TWELFTH:
{
const unsigned int n_wts = 8;
const Real wts[n_wts] =
{
3.0831305257795086169332418926151771e-03L,
3.1429112108942550177135256546441273e-02L,
1.7398056465354471494664198647499687e-02L,
2.1846272269019201067728631278737487e-02L,
1.2865533220227667708895461535782215e-02L,
1.1178386601151722855919538351159995e-02L,
8.6581155543294461858210504055170332e-03L,
2.0185778883190464758914349626118386e-02L
};
const Real a[n_wts] =
{
2.1317350453210370246856975515728246e-02L,
2.7121038501211592234595134039689474e-01L,
1.2757614554158592467389632515428357e-01L,
4.3972439229446027297973662348436108e-01L,
4.8821738977380488256466206525881104e-01L,
2.8132558098993954824813069297455275e-01L,
1.1625191590759714124135414784260182e-01L,
2.7571326968551419397479634607976398e-01L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
0.,
6.9583608678780342214163552323607254e-01L,
8.5801403354407263059053661662617818e-01L,
6.0894323577978780685619243776371007e-01L
};
const unsigned int permutation_ids[n_wts] = {3, 3, 3, 3, 3, 6, 6, 6}; // 33 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Another Dunavant rule with all positive weights. This rule has 37
// points. The comparable conical product rule would have 49 points.
//
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// A second rule with additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case THIRTEENTH:
{
const unsigned int n_wts = 9;
const Real wts[n_wts] =
{
3.3980018293415822140887212340442440e-02L,
2.7800983765226664353628733005230734e-02L,
2.9139242559599990702383541756669905e-02L,
3.0261685517695859208964000161454122e-03L,
1.1997200964447365386855399725479827e-02L,
1.7320638070424185232993414255459110e-02L,
7.4827005525828336316229285664517190e-03L,
1.2089519905796909568722872786530380e-02L,
4.7953405017716313612975450830554457e-03L
};
const Real a[n_wts] =
{
0., // 'a' parameter not used for origin permutation
4.2694141425980040602081253503137421e-01L,
2.2137228629183290065481255470507908e-01L,
2.1509681108843183869291313534052083e-02L,
4.8907694645253934990068971909020439e-01L,
3.0844176089211777465847185254124531e-01L,
1.1092204280346339541286954522167452e-01L,
1.6359740106785048023388790171095725e-01L,
2.7251581777342966618005046435408685e-01L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
0.,
6.2354599555367557081585435318623659e-01L,
8.6470777029544277530254595089569318e-01L,
7.4850711589995219517301859578870965e-01L,
7.2235779312418796526062013230478405e-01L
};
const unsigned int permutation_ids[n_wts] = {1, 3, 3, 3, 3, 6, 6, 6, 6}; // 37 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Another Dunavant rule. This rule has 42 points, while
// a comparable conical product rule would have 64.
//
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
case FOURTEENTH:
{
const unsigned int n_wts = 10;
const Real wts[n_wts] =
{
1.0941790684714445320422472981662986e-02L,
1.6394176772062675320655489369312672e-02L,
2.5887052253645793157392455083198201e-02L,
2.1081294368496508769115218662093065e-02L,
7.2168498348883338008549607403266583e-03L,
2.4617018012000408409130117545210774e-03L,
1.2332876606281836981437622591818114e-02L,
1.9285755393530341614244513905205430e-02L,
7.2181540567669202480443459995079017e-03L,
2.5051144192503358849300465412445582e-03L
};
const Real a[n_wts] =
{
4.8896391036217863867737602045239024e-01L,
4.1764471934045392250944082218564344e-01L,
2.7347752830883865975494428326269856e-01L,
1.7720553241254343695661069046505908e-01L,
6.1799883090872601267478828436935788e-02L,
1.9390961248701048178250095054529511e-02L,
1.7226668782135557837528960161365733e-01L,
3.3686145979634500174405519708892539e-01L,
2.9837288213625775297083151805961273e-01L,
1.1897449769695684539818196192990548e-01L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
0.,
0.,
7.7060855477499648258903327416742796e-01L,
5.7022229084668317349769621336235426e-01L,
6.8698016780808783735862715402031306e-01L,
8.7975717137017112951457163697460183e-01L
};
const unsigned int permutation_ids[n_wts]
= {3, 3, 3, 3, 3, 3, 6, 6, 6, 6}; // 42 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// This 49-point rule was found by me [JWP] using the code in:
//
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
//
// A 54-point, 15th-order rule is reported by
//
// Stephen Wandzura, Hong Xiao,
// Symmetric Quadrature Rules on a Triangle,
// Computers and Mathematics with Applications,
// Volume 45, Number 12, June 2003, pages 1829-1840.
//
// can be found here:
// http://people.scs.fsu.edu/~burkardt/f_src/wandzura/wandzura.f90
//
// but this 49-point rule is superior.
case FIFTEENTH:
{
const unsigned int n_wts = 11;
const Real wts[n_wts] =
{
2.4777380743035579804788826970198951e-02L,
9.2433943023307730591540642828347660e-03L,
2.2485768962175402793245929133296627e-03L,
6.7052581900064143760518398833360903e-03L,
1.9011381726930579256700190357527956e-02L,
1.4605445387471889398286155981802858e-02L,
1.5087322572773133722829435011138258e-02L,
1.5630213780078803020711746273129099e-02L,
6.1808086085778203192616856133701233e-03L,
3.2209366452594664857296985751120513e-03L,
5.8747373242569702667677969985668817e-03L
};
const Real a[n_wts] =
{
0.0, // 'a' parameter not used for origin
7.9031013655541635005816956762252155e-02L,
1.8789501810770077611247984432284226e-02L,
4.9250168823249670532514526605352905e-01L,
4.0886316907744105975059040108092775e-01L,
5.3877851064220142445952549348423733e-01L,
2.0250549804829997692885033941362673e-01L,
5.5349674918711643207148086558288110e-01L,
7.8345022567320812359258882143250181e-01L,
8.9514624528794883409864566727625002e-01L,
3.2515745241110782862789881780746490e-01L
};
const Real b[n_wts] =
{
0.,
0.,
0.,
0.,
0.,
1.9412620368774630292701241080996842e-01L,
9.8765911355712115933807754318089099e-02L,
7.7663767064308164090246588765178087e-02L,
2.1594628433980258573654682690950798e-02L,
1.2563596287784997705599005477153617e-02L,
1.5082654870922784345283124845552190e-02L
};
const unsigned int permutation_ids[n_wts]
= {1, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6}; // 49 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Dunavant's 16th-order rule contains points outside the region of
// integration, and is thus unacceptable for our FEM calculations.
//
// This 55-point, 16th-order rule was obtained by me [JWP] using the code in
//
// Additional precision obtained from the code in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
//
// Note: the 55-point 16th-order rule obtained by Zhang in the paper above
// does not appear to be unique. It is a solution in the sense that it
// minimizes the error in the least-squares minimization problem, but
// it involves too many unknowns and the Jacobian is therefore singular
// when attempting to improve the solution via Newton's method.
case SIXTEENTH:
{
const unsigned int n_wts = 12;
const Real wts[n_wts] =
{
2.2668082505910087151996321171534230e-02L,
8.4043060714818596159798961899306135e-03L,
1.0850949634049747713966288634484161e-03L,
7.2252773375423638869298219383808751e-03L,
1.2997715227338366024036316182572871e-02L,
2.0054466616677715883228810959112227e-02L,
9.7299841600417010281624372720122710e-03L,
1.1651974438298104227427176444311766e-02L,
9.1291185550484450744725847363097389e-03L,
3.5568614040947150231712567900113671e-03L,
5.8355861686234326181790822005304303e-03L,
4.7411314396804228041879331486234396e-03L
};
const Real a[n_wts] =
{
0.0, // 'a' parameter not used for centroid weight
8.5402539407933203673769900926355911e-02L,
1.2425572001444092841183633409631260e-02L,
4.9174838341891594024701017768490960e-01L,
4.5669426695387464162068900231444462e-01L,
4.8506759880447437974189793537259677e-01L,
2.0622099278664205707909858461264083e-01L,
3.2374950270039093446805340265853956e-01L,
7.3834330556606586255186213302750029e-01L,
9.1210673061680792565673823935174611e-01L,
6.6129919222598721544966837350891531e-01L,
1.7807138906021476039088828811346122e-01L
};
const Real b[n_wts] =
{
0.0,
0.0,
0.0,
0.0,
0.0,
3.2315912848634384647700266402091638e-01L,
1.5341553679414688425981898952416987e-01L,
7.4295478991330687632977899141707872e-02L,
7.1278762832147862035977841733532020e-02L,
1.6623223223705792825395256602140459e-02L,
1.4160772533794791868984026749196156e-02L,
1.4539694958941854654807449467759690e-02L
};
const unsigned int permutation_ids[n_wts]
= {1, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6}; // 55 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
}
// Dunavant's 17th-order rule has 61 points, while a
// comparable conical product rule would have 81 (16th and 17th orders).
//
// It can be found here:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
//
// Zhang reports an identical rule in:
// L. Zhang, T. Cui, and H. Liu. "A set of symmetric quadrature rules
// on triangles and tetrahedra" Journal of Computational Mathematics,
// v. 27, no. 1, 2009, pp. 89-96.
//
// Note: the 61-point 17th-order rule obtained by Dunavant and Zhang
// does not appear to be unique. It is a solution in the sense that it
// minimizes the error in the least-squares minimization problem, but
// it involves too many unknowns and the Jacobian is therefore singular
// when attempting to improve the solution via Newton's method.
//
// Therefore, we prefer the following 63-point rule which
// I [JWP] found. It appears to be more accurate than the
// rule reported by Dunavant and Zhang, even though it has
// a few more points.
case SEVENTEENTH:
{
const unsigned int n_wts = 12;
const Real wts[n_wts] =
{
1.7464603792572004485690588092246146e-02L,
5.9429003555801725246549713984660076e-03L,
1.2490753345169579649319736639588729e-02L,
1.5386987188875607593083456905596468e-02L,
1.1185807311917706362674684312990270e-02L,
1.0301845740670206831327304917180007e-02L,
1.1767783072977049696840016810370464e-02L,
3.8045312849431209558329128678945240e-03L,
4.5139302178876351271037137230354382e-03L,
2.2178812517580586419412547665472893e-03L,
5.2216271537483672304731416553063103e-03L,
9.8381136389470256422419930926212114e-04L
};
const Real a[n_wts] =
{
2.8796825754667362165337965123570514e-01L,
4.9216175986208465345536805750663939e-01L,
4.6252866763171173685916780827044612e-01L,
1.6730292951631792248498303276090273e-01L,
1.5816335500814652972296428532213019e-01L,
1.6352252138387564873002458959679529e-01L,
6.2447680488959768233910286168417367e-01L,
8.7317249935244454285263604347964179e-01L,
3.4428164322282694677972239461699271e-01L,
9.1584484467813674010523309855340209e-02L,
2.0172088013378989086826623852040632e-01L,
9.6538762758254643474731509845084691e-01L
};
const Real b[n_wts] =
{
0.0,
0.0,
0.0,
3.4429160695501713926320695771253348e-01L,
2.2541623431550639817203145525444726e-01L,
8.0670083153531811694942222940484991e-02L,
6.5967451375050925655738829747288190e-02L,
4.5677879890996762665044366994439565e-02L,
1.1528411723154215812386518751976084e-02L,
9.3057714323900610398389176844165892e-03L,
1.5916814107619812717966560404970160e-02L,
1.0734733163764032541125434215228937e-02L
};
const unsigned int permutation_ids[n_wts]
= {3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6}; // 63 total points
dunavant_rule2(wts, a, b, permutation_ids, n_wts);
return;
// _points.resize (61);
// _weights.resize(61);
// // The raw data for the quadrature rule.
// const Real p[15][4] = {
// { 1./3., 0., 0., 0.033437199290803e+00 / 2.0}, // 1-perm
// {0.005658918886452e+00, 0.497170540556774e+00, 0., 0.005093415440507e+00 / 2.0}, // 3-perm
// {0.035647354750751e+00, 0.482176322624625e+00, 0., 0.014670864527638e+00 / 2.0}, // 3-perm
// {0.099520061958437e+00, 0.450239969020782e+00, 0., 0.024350878353672e+00 / 2.0}, // 3-perm
// {0.199467521245206e+00, 0.400266239377397e+00, 0., 0.031107550868969e+00 / 2.0}, // 3-perm
// {0.495717464058095e+00, 0.252141267970953e+00, 0., 0.031257111218620e+00 / 2.0}, // 3-perm
// {0.675905990683077e+00, 0.162047004658461e+00, 0., 0.024815654339665e+00 / 2.0}, // 3-perm
// {0.848248235478508e+00, 0.075875882260746e+00, 0., 0.014056073070557e+00 / 2.0}, // 3-perm
// {0.968690546064356e+00, 0.015654726967822e+00, 0., 0.003194676173779e+00 / 2.0}, // 3-perm
// {0.010186928826919e+00, 0.334319867363658e+00, 0.655493203809423e+00, 0.008119655318993e+00 / 2.0}, // 6-perm
// {0.135440871671036e+00, 0.292221537796944e+00, 0.572337590532020e+00, 0.026805742283163e+00 / 2.0}, // 6-perm
// {0.054423924290583e+00, 0.319574885423190e+00, 0.626001190286228e+00, 0.018459993210822e+00 / 2.0}, // 6-perm
// {0.012868560833637e+00, 0.190704224192292e+00, 0.796427214974071e+00, 0.008476868534328e+00 / 2.0}, // 6-perm
// {0.067165782413524e+00, 0.180483211648746e+00, 0.752351005937729e+00, 0.018292796770025e+00 / 2.0}, // 6-perm
// {0.014663182224828e+00, 0.080711313679564e+00, 0.904625504095608e+00, 0.006665632004165e+00 / 2.0} // 6-perm
// };
// // Now call the dunavant routine to generate _points and _weights
// dunavant_rule(p, 15);
// return;
}
// Dunavant's 18th-order rule contains points outside the region and is therefore unsuitable
// for our FEM calculations. His 19th-order rule has 73 points, compared with 100 points for
// a comparable-order conical product rule.
//
// It was copied 23rd June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/dunavant/dunavant.f90
case EIGHTTEENTH:
case NINETEENTH:
{
_points.resize (73);
_weights.resize(73);
// The raw data for the quadrature rule.
const Real rule_data[17][4] = {
{ 1./3., 0., 0., 0.032906331388919e+00 / 2.0}, // 1-perm
{0.020780025853987e+00, 0.489609987073006e+00, 0., 0.010330731891272e+00 / 2.0}, // 3-perm
{0.090926214604215e+00, 0.454536892697893e+00, 0., 0.022387247263016e+00 / 2.0}, // 3-perm
{0.197166638701138e+00, 0.401416680649431e+00, 0., 0.030266125869468e+00 / 2.0}, // 3-perm
{0.488896691193805e+00, 0.255551654403098e+00, 0., 0.030490967802198e+00 / 2.0}, // 3-perm
{0.645844115695741e+00, 0.177077942152130e+00, 0., 0.024159212741641e+00 / 2.0}, // 3-perm
{0.779877893544096e+00, 0.110061053227952e+00, 0., 0.016050803586801e+00 / 2.0}, // 3-perm
{0.888942751496321e+00, 0.055528624251840e+00, 0., 0.008084580261784e+00 / 2.0}, // 3-perm
{0.974756272445543e+00, 0.012621863777229e+00, 0., 0.002079362027485e+00 / 2.0}, // 3-perm
{0.003611417848412e+00, 0.395754787356943e+00, 0.600633794794645e+00, 0.003884876904981e+00 / 2.0}, // 6-perm
{0.134466754530780e+00, 0.307929983880436e+00, 0.557603261588784e+00, 0.025574160612022e+00 / 2.0}, // 6-perm
{0.014446025776115e+00, 0.264566948406520e+00, 0.720987025817365e+00, 0.008880903573338e+00 / 2.0}, // 6-perm
{0.046933578838178e+00, 0.358539352205951e+00, 0.594527068955871e+00, 0.016124546761731e+00 / 2.0}, // 6-perm
{0.002861120350567e+00, 0.157807405968595e+00, 0.839331473680839e+00, 0.002491941817491e+00 / 2.0}, // 6-perm
{0.223861424097916e+00, 0.075050596975911e+00, 0.701087978926173e+00, 0.018242840118951e+00 / 2.0}, // 6-perm
{0.034647074816760e+00, 0.142421601113383e+00, 0.822931324069857e+00, 0.010258563736199e+00 / 2.0}, // 6-perm
{0.010161119296278e+00, 0.065494628082938e+00, 0.924344252620784e+00, 0.003799928855302e+00 / 2.0} // 6-perm
};
// Now call the dunavant routine to generate _points and _weights
dunavant_rule(rule_data, 17);
return;
}
// 20th-order rule by Wandzura.
//
// Stephen Wandzura, Hong Xiao,
// Symmetric Quadrature Rules on a Triangle,
// Computers and Mathematics with Applications,
// Volume 45, Number 12, June 2003, pages 1829-1840.
//
// Wandzura's work extends the work of Dunavant by providing degree
// 5,10,15,20,25, and 30 rules with positive weights for the triangle.
//
// Copied on 3rd July 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/wandzura/wandzura.f90
case TWENTIETH:
{
// The equivalent concial product rule would have 121 points
_points.resize (85);
_weights.resize(85);
// The raw data for the quadrature rule.
const Real rule_data[19][4] = {
{0.33333333333333e+00, 0.0, 0.0, 0.2761042699769952e-01 / 2.0}, // 1-perm
{0.00150064932443e+00, 0.49924967533779e+00, 0.0, 0.1779029547326740e-02 / 2.0}, // 3-perm
{0.09413975193895e+00, 0.45293012403052e+00, 0.0, 0.2011239811396117e-01 / 2.0}, // 3-perm
{0.20447212408953e+00, 0.39776393795524e+00, 0.0, 0.2681784725933157e-01 / 2.0}, // 3-perm
{0.47099959493443e+00, 0.26450020253279e+00, 0.0, 0.2452313380150201e-01 / 2.0}, // 3-perm
{0.57796207181585e+00, 0.21101896409208e+00, 0.0, 0.1639457841069539e-01 / 2.0}, // 3-perm
{0.78452878565746e+00, 0.10773560717127e+00, 0.0, 0.1479590739864960e-01 / 2.0}, // 3-perm
{0.92186182432439e+00, 0.03906908783780e+00, 0.0, 0.4579282277704251e-02 / 2.0}, // 3-perm
{0.97765124054134e+00, 0.01117437972933e+00, 0.0, 0.1651826515576217e-02 / 2.0}, // 3-perm
{0.00534961818734e+00, 0.06354966590835e+00, 0.93110071590431e+00, 0.2349170908575584e-02 / 2.0}, // 6-perm
{0.00795481706620e+00, 0.15710691894071e+00, 0.83493826399309e+00, 0.4465925754181793e-02 / 2.0}, // 6-perm
{0.01042239828126e+00, 0.39564211436437e+00, 0.59393548735436e+00, 0.6099566807907972e-02 / 2.0}, // 6-perm
{0.01096441479612e+00, 0.27316757071291e+00, 0.71586801449097e+00, 0.6891081327188203e-02 / 2.0}, // 6-perm
{0.03856671208546e+00, 0.10178538248502e+00, 0.85964790542952e+00, 0.7997475072478163e-02 / 2.0}, // 6-perm
{0.03558050781722e+00, 0.44665854917641e+00, 0.51776094300637e+00, 0.7386134285336024e-02 / 2.0}, // 6-perm
{0.04967081636276e+00, 0.19901079414950e+00, 0.75131838948773e+00, 0.1279933187864826e-01 / 2.0}, // 6-perm
{0.05851972508433e+00, 0.32426118369228e+00, 0.61721909122339e+00, 0.1725807117569655e-01 / 2.0}, // 6-perm
{0.12149778700439e+00, 0.20853136321013e+00, 0.66997084978547e+00, 0.1867294590293547e-01 / 2.0}, // 6-perm
{0.14071084494394e+00, 0.32317056653626e+00, 0.53611858851980e+00, 0.2281822405839526e-01 / 2.0} // 6-perm
};
// Now call the dunavant routine to generate _points and _weights
dunavant_rule(rule_data, 19);
return;
}
// 25th-order rule by Wandzura.
//
// Stephen Wandzura, Hong Xiao,
// Symmetric Quadrature Rules on a Triangle,
// Computers and Mathematics with Applications,
// Volume 45, Number 12, June 2003, pages 1829-1840.
//
// Wandzura's work extends the work of Dunavant by providing degree
// 5,10,15,20,25, and 30 rules with positive weights for the triangle.
//
// Copied on 3rd July 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/wandzura/wandzura.f90
// case TWENTYFIRST: // fall through to 121 point conical product rule below
case TWENTYSECOND:
case TWENTYTHIRD:
case TWENTYFOURTH:
case TWENTYFIFTH:
{
// The equivalent concial product rule would have 169 points
_points.resize (126);
_weights.resize(126);
// The raw data for the quadrature rule.
const Real rule_data[26][4] = {
{0.02794648307317e+00, 0.48602675846341e+00, 0.0, 0.8005581880020417e-02 / 2.0}, // 3-perm
{0.13117860132765e+00, 0.43441069933617e+00, 0.0, 0.1594707683239050e-01 / 2.0}, // 3-perm
{0.22022172951207e+00, 0.38988913524396e+00, 0.0, 0.1310914123079553e-01 / 2.0}, // 3-perm
{0.40311353196039e+00, 0.29844323401980e+00, 0.0, 0.1958300096563562e-01 / 2.0}, // 3-perm
{0.53191165532526e+00, 0.23404417233737e+00, 0.0, 0.1647088544153727e-01 / 2.0}, // 3-perm
{0.69706333078196e+00, 0.15146833460902e+00, 0.0, 0.8547279074092100e-02 / 2.0}, // 3-perm
{0.77453221290801e+00, 0.11273389354599e+00, 0.0, 0.8161885857226492e-02 / 2.0}, // 3-perm
{0.84456861581695e+00, 0.07771569209153e+00, 0.0, 0.6121146539983779e-02 / 2.0}, // 3-perm
{0.93021381277141e+00, 0.03489309361430e+00, 0.0, 0.2908498264936665e-02 / 2.0}, // 3-perm
{0.98548363075813e+00, 0.00725818462093e+00, 0.0, 0.6922752456619963e-03 / 2.0}, // 3-perm
{0.00129235270444e+00, 0.22721445215336e+00, 0.77149319514219e+00, 0.1248289199277397e-02 / 2.0}, // 6-perm
{0.00539970127212e+00, 0.43501055485357e+00, 0.55958974387431e+00, 0.3404752908803022e-02 / 2.0}, // 6-perm
{0.00638400303398e+00, 0.32030959927220e+00, 0.67330639769382e+00, 0.3359654326064051e-02 / 2.0}, // 6-perm
{0.00502821150199e+00, 0.09175032228001e+00, 0.90322146621800e+00, 0.1716156539496754e-02 / 2.0}, // 6-perm
{0.00682675862178e+00, 0.03801083585872e+00, 0.95516240551949e+00, 0.1480856316715606e-02 / 2.0}, // 6-perm
{0.01001619963993e+00, 0.15742521848531e+00, 0.83255858187476e+00, 0.3511312610728685e-02 / 2.0}, // 6-perm
{0.02575781317339e+00, 0.23988965977853e+00, 0.73435252704808e+00, 0.7393550149706484e-02 / 2.0}, // 6-perm
{0.03022789811992e+00, 0.36194311812606e+00, 0.60782898375402e+00, 0.7983087477376558e-02 / 2.0}, // 6-perm
{0.03050499010716e+00, 0.08355196095483e+00, 0.88594304893801e+00, 0.4355962613158041e-02 / 2.0}, // 6-perm
{0.04595654736257e+00, 0.14844322073242e+00, 0.80560023190501e+00, 0.7365056701417832e-02 / 2.0}, // 6-perm
{0.06744280054028e+00, 0.28373970872753e+00, 0.64881749073219e+00, 0.1096357284641955e-01 / 2.0}, // 6-perm
{0.07004509141591e+00, 0.40689937511879e+00, 0.52305553346530e+00, 0.1174996174354112e-01 / 2.0}, // 6-perm
{0.08391152464012e+00, 0.19411398702489e+00, 0.72197448833499e+00, 0.1001560071379857e-01 / 2.0}, // 6-perm
{0.12037553567715e+00, 0.32413434700070e+00, 0.55549011732214e+00, 0.1330964078762868e-01 / 2.0}, // 6-perm
{0.14806689915737e+00, 0.22927748355598e+00, 0.62265561728665e+00, 0.1415444650522614e-01 / 2.0}, // 6-perm
{0.19177186586733e+00, 0.32561812259598e+00, 0.48261001153669e+00, 0.1488137956116801e-01 / 2.0} // 6-perm
};
// Now call the dunavant routine to generate _points and _weights
dunavant_rule(rule_data, 26);
return;
}
// 30th-order rule by Wandzura.
//
// Stephen Wandzura, Hong Xiao,
// Symmetric Quadrature Rules on a Triangle,
// Computers and Mathematics with Applications,
// Volume 45, Number 12, June 2003, pages 1829-1840.
//
// Wandzura's work extends the work of Dunavant by providing degree
// 5,10,15,20,25, and 30 rules with positive weights for the triangle.
//
// Copied on 3rd July 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/wandzura/wandzura.f90
case TWENTYSIXTH:
case TWENTYSEVENTH:
case TWENTYEIGHTH:
case TWENTYNINTH:
case THIRTIETH:
{
// The equivalent concial product rule would have 256 points
_points.resize (175);
_weights.resize(175);
// The raw data for the quadrature rule.
const Real rule_data[36][4] = {
{0.33333333333333e+00, 0.0, 0.0, 0.1557996020289920e-01 / 2.0}, // 1-perm
{0.00733011643277e+00, 0.49633494178362e+00, 0.0, 0.3177233700534134e-02 / 2.0}, // 3-perm
{0.08299567580296e+00, 0.45850216209852e+00, 0.0, 0.1048342663573077e-01 / 2.0}, // 3-perm
{0.15098095612541e+00, 0.42450952193729e+00, 0.0, 0.1320945957774363e-01 / 2.0}, // 3-perm
{0.23590585989217e+00, 0.38204707005392e+00, 0.0, 0.1497500696627150e-01 / 2.0}, // 3-perm
{0.43802430840785e+00, 0.28098784579608e+00, 0.0, 0.1498790444338419e-01 / 2.0}, // 3-perm
{0.54530204829193e+00, 0.22734897585403e+00, 0.0, 0.1333886474102166e-01 / 2.0}, // 3-perm
{0.65088177698254e+00, 0.17455911150873e+00, 0.0, 0.1088917111390201e-01 / 2.0}, // 3-perm
{0.75348314559713e+00, 0.12325842720144e+00, 0.0, 0.8189440660893461e-02 / 2.0}, // 3-perm
{0.83983154221561e+00, 0.08008422889220e+00, 0.0, 0.5575387588607785e-02 / 2.0}, // 3-perm
{0.90445106518420e+00, 0.04777446740790e+00, 0.0, 0.3191216473411976e-02 / 2.0}, // 3-perm
{0.95655897063972e+00, 0.02172051468014e+00, 0.0, 0.1296715144327045e-02 / 2.0}, // 3-perm
{0.99047064476913e+00, 0.00476467761544e+00, 0.0, 0.2982628261349172e-03 / 2.0}, // 3-perm
{0.00092537119335e+00, 0.41529527091331e+00, 0.58377935789334e+00, 0.9989056850788964e-03 / 2.0}, // 6-perm
{0.00138592585556e+00, 0.06118990978535e+00, 0.93742416435909e+00, 0.4628508491732533e-03 / 2.0}, // 6-perm
{0.00368241545591e+00, 0.16490869013691e+00, 0.83140889440718e+00, 0.1234451336382413e-02 / 2.0}, // 6-perm
{0.00390322342416e+00, 0.02503506223200e+00, 0.97106171434384e+00, 0.5707198522432062e-03 / 2.0}, // 6-perm
{0.00323324815501e+00, 0.30606446515110e+00, 0.69070228669389e+00, 0.1126946125877624e-02 / 2.0}, // 6-perm
{0.00646743211224e+00, 0.10707328373022e+00, 0.88645928415754e+00, 0.1747866949407337e-02 / 2.0}, // 6-perm
{0.00324747549133e+00, 0.22995754934558e+00, 0.76679497516308e+00, 0.1182818815031657e-02 / 2.0}, // 6-perm
{0.00867509080675e+00, 0.33703663330578e+00, 0.65428827588746e+00, 0.1990839294675034e-02 / 2.0}, // 6-perm
{0.01559702646731e+00, 0.05625657618206e+00, 0.92814639735063e+00, 0.1900412795035980e-02 / 2.0}, // 6-perm
{0.01797672125369e+00, 0.40245137521240e+00, 0.57957190353391e+00, 0.4498365808817451e-02 / 2.0}, // 6-perm
{0.01712424535389e+00, 0.24365470201083e+00, 0.73922105263528e+00, 0.3478719460274719e-02 / 2.0}, // 6-perm
{0.02288340534658e+00, 0.16538958561453e+00, 0.81172700903888e+00, 0.4102399036723953e-02 / 2.0}, // 6-perm
{0.03273759728777e+00, 0.09930187449585e+00, 0.86796052821639e+00, 0.4021761549744162e-02 / 2.0}, // 6-perm
{0.03382101234234e+00, 0.30847833306905e+00, 0.65770065458860e+00, 0.6033164660795066e-02 / 2.0}, // 6-perm
{0.03554761446002e+00, 0.46066831859211e+00, 0.50378406694787e+00, 0.3946290302129598e-02 / 2.0}, // 6-perm
{0.05053979030687e+00, 0.21881529945393e+00, 0.73064491023920e+00, 0.6644044537680268e-02 / 2.0}, // 6-perm
{0.05701471491573e+00, 0.37920955156027e+00, 0.56377573352399e+00, 0.8254305856078458e-02 / 2.0}, // 6-perm
{0.06415280642120e+00, 0.14296081941819e+00, 0.79288637416061e+00, 0.6496056633406411e-02 / 2.0}, // 6-perm
{0.08050114828763e+00, 0.28373128210592e+00, 0.63576756960645e+00, 0.9252778144146602e-02 / 2.0}, // 6-perm
{0.10436706813453e+00, 0.19673744100444e+00, 0.69889549086103e+00, 0.9164920726294280e-02 / 2.0}, // 6-perm
{0.11384489442875e+00, 0.35588914121166e+00, 0.53026596435959e+00, 0.1156952462809767e-01 / 2.0}, // 6-perm
{0.14536348771552e+00, 0.25981868535191e+00, 0.59481782693256e+00, 0.1176111646760917e-01 / 2.0}, // 6-perm
{0.18994565282198e+00, 0.32192318123130e+00, 0.48813116594672e+00, 0.1382470218216540e-01 / 2.0} // 6-perm
};
// Now call the dunavant routine to generate _points and _weights
dunavant_rule(rule_data, 36);
return;
}
// By default, we fall back on the conical product rules. If the user
// requests an order higher than what is currently available in the 1D
// rules, an error will be thrown from the respective 1D code.
default:
{
// The following quadrature rules are generated as
// conical products. These tend to be non-optimal
// (use too many points, cluster points in certain
// regions of the domain) but they are quite easy to
// automatically generate using a 1D Gauss rule on
// [0,1] and two 1D Jacobi-Gauss rules on [0,1].
QConical conical_rule(2, _order);
conical_rule.init(type_in, p);
// Swap points and weights with the about-to-be destroyed rule.
_points.swap (conical_rule.get_points() );
_weights.swap(conical_rule.get_weights());
return;
}
}
}
//---------------------------------------------
// Unsupported type
default:
libmesh_error_msg("Element type not supported!:" << type_in);
}
#endif
}
| void libMesh::QGauss::init_3D | ( | const ElemType | = INVALID_ELEM, |
| unsigned int | = 0 |
||
| ) | [private, virtual] |
Initializes the 3D 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. Should not be pure virtual since a derived quadrature rule may only be defined in 1D. If not redefined, gives an error (when DEBUG defined) when called.
Reimplemented from libMesh::QBase.
Definition at line 28 of file quadrature_gauss_3D.C.
References libMesh::QBase::_order, libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::QBase::allow_rules_with_negative_weights, libMesh::CONSTANT, libMesh::EDGE2, libMesh::EIGHTH, libMesh::FIFTH, libMesh::FIRST, libMesh::FOURTH, libMesh::QBase::get_points(), libMesh::QBase::get_weights(), libMesh::HEX20, libMesh::HEX27, libMesh::HEX8, libMesh::QBase::init(), keast_rule(), libMesh::QBase::libmesh_error_msg(), libMesh::PRISM15, libMesh::PRISM18, libMesh::PRISM6, libMesh::PYRAMID13, libMesh::PYRAMID14, libMesh::PYRAMID5, libMesh::Real, libMesh::SECOND, libMesh::SEVENTH, libMesh::SIXTH, libMesh::QBase::tensor_product_hex(), libMesh::QBase::tensor_product_prism(), libMesh::TET10, libMesh::TET4, libMesh::THIRD, and libMesh::TRI3.
{
#if LIBMESH_DIM == 3
//-----------------------------------------------------------------------
// 3D quadrature rules
switch (type_in)
{
//---------------------------------------------
// Hex quadrature rules
case HEX8:
case HEX20:
case HEX27:
{
// We compute the 3D quadrature rule as a tensor
// product of the 1D quadrature rule.
QGauss q1D(1,_order);
q1D.init(EDGE2,p);
tensor_product_hex( q1D );
return;
}
//---------------------------------------------
// Tetrahedral quadrature rules
case TET4:
case TET10:
{
switch(_order + 2*p)
{
// Taken from pg. 222 of "The finite element method," vol. 1
// ed. 5 by Zienkiewicz & Taylor
case CONSTANT:
case FIRST:
{
// Exact for linears
_points.resize(1);
_weights.resize(1);
_points[0](0) = .25;
_points[0](1) = .25;
_points[0](2) = .25;
_weights[0] = .1666666666666666666666666666666666666666666667L;
return;
}
case SECOND:
{
// Exact for quadratics
_points.resize(4);
_weights.resize(4);
const Real a = .585410196624969;
const Real b = .138196601125011;
_points[0](0) = a;
_points[0](1) = b;
_points[0](2) = b;
_points[1](0) = b;
_points[1](1) = a;
_points[1](2) = b;
_points[2](0) = b;
_points[2](1) = b;
_points[2](2) = a;
_points[3](0) = b;
_points[3](1) = b;
_points[3](2) = b;
_weights[0] = .0416666666666666666666666666666666666666666667;
_weights[1] = _weights[0];
_weights[2] = _weights[0];
_weights[3] = _weights[0];
return;
}
// Can be found in the class notes
// http://www.cs.rpi.edu/~flaherje/FEM/fem6.ps
// by Flaherty.
//
// Caution: this rule has a negative weight and may be
// unsuitable for some problems.
// Exact for cubics.
//
// Note: Keast (see ref. elsewhere in this file) also gives
// a third-order rule with positive weights, but it contains points
// on the ref. elt. boundary, making it less suitable for FEM calculations.
case THIRD:
{
if (allow_rules_with_negative_weights)
{
_points.resize(5);
_weights.resize(5);
_points[0](0) = .25;
_points[0](1) = .25;
_points[0](2) = .25;
_points[1](0) = .5;
_points[1](1) = .16666666666666666666666666666666666666666667;
_points[1](2) = .16666666666666666666666666666666666666666667;
_points[2](0) = .16666666666666666666666666666666666666666667;
_points[2](1) = .5;
_points[2](2) = .16666666666666666666666666666666666666666667;
_points[3](0) = .16666666666666666666666666666666666666666667;
_points[3](1) = .16666666666666666666666666666666666666666667;
_points[3](2) = .5;
_points[4](0) = .16666666666666666666666666666666666666666667;
_points[4](1) = .16666666666666666666666666666666666666666667;
_points[4](2) = .16666666666666666666666666666666666666666667;
_weights[0] = -.133333333333333333333333333333333333333333333;
_weights[1] = .075;
_weights[2] = _weights[1];
_weights[3] = _weights[1];
_weights[4] = _weights[1];
return;
} // end if (allow_rules_with_negative_weights)
else
{
// If a rule with postive weights is required, the 2x2x2 conical
// product rule is third-order accurate and has less points than
// the next-available positive-weight rule at FIFTH order.
QConical conical_rule(3, _order);
conical_rule.init(type_in, p);
// Swap points and weights with the about-to-be destroyed rule.
_points.swap (conical_rule.get_points() );
_weights.swap(conical_rule.get_weights());
return;
}
// Note: if !allow_rules_with_negative_weights, fall through to next case.
}
// Originally a Keast rule,
// Patrick Keast,
// Moderate Degree Tetrahedral Quadrature Formulas,
// Computer Methods in Applied Mechanics and Engineering,
// Volume 55, Number 3, May 1986, pages 339-348.
//
// Can also be found the class notes
// http://www.cs.rpi.edu/~flaherje/FEM/fem6.ps
// by Flaherty.
//
// Caution: this rule has a negative weight and may be
// unsuitable for some problems.
case FOURTH:
{
if (allow_rules_with_negative_weights)
{
_points.resize(11);
_weights.resize(11);
// The raw data for the quadrature rule.
const Real rule_data[3][4] = {
{0.250000000000000000e+00, 0., 0., -0.131555555555555556e-01}, // 1
{0.785714285714285714e+00, 0.714285714285714285e-01, 0., 0.762222222222222222e-02}, // 4
{0.399403576166799219e+00, 0., 0.100596423833200785e+00, 0.248888888888888889e-01} // 6
};
// Now call the keast routine to generate _points and _weights
keast_rule(rule_data, 3);
return;
} // end if (allow_rules_with_negative_weights)
// Note: if !allow_rules_with_negative_weights, fall through to next case.
}
// Walkington's fifth-order 14-point rule from
// "Quadrature on Simplices of Arbitrary Dimension"
//
// We originally had a Keast rule here, but this rule had
// more points than an equivalent rule by Walkington and
// also contained points on the boundary of the ref. elt,
// making it less suitable for FEM calculations.
case FIFTH:
{
_points.resize(14);
_weights.resize(14);
// permutations of these points and suitably-modified versions of
// these points are the quadrature point locations
const Real a[3] = {0.31088591926330060980, // a1 from the paper
0.092735250310891226402, // a2 from the paper
0.045503704125649649492}; // a3 from the paper
// weights. a[] and wt[] are the only floating-point inputs required
// for this rule.
const Real wt[3] = {0.018781320953002641800, // w1 from the paper
0.012248840519393658257, // w2 from the paper
0.0070910034628469110730}; // w3 from the paper
// The first two sets of 4 points are formed in a similar manner
for (unsigned int i=0; i<2; ++i)
{
// Where we will insert values into _points and _weights
const unsigned int offset=4*i;
// Stuff points and weights values into their arrays
const Real b = 1. - 3.*a[i];
// Here are the permutations. Order of these is not important,
// all have the same weight
_points[offset + 0] = Point(a[i], a[i], a[i]);
_points[offset + 1] = Point(a[i], b, a[i]);
_points[offset + 2] = Point( b, a[i], a[i]);
_points[offset + 3] = Point(a[i], a[i], b);
// These 4 points all have the same weights
for (unsigned int j=0; j<4; ++j)
_weights[offset + j] = wt[i];
} // end for
{
// The third set contains 6 points and is formed a little differently
const unsigned int offset = 8;
const Real b = 0.5*(1. - 2.*a[2]);
// Here are the permutations. Order of these is not important,
// all have the same weight
_points[offset + 0] = Point(b , b, a[2]);
_points[offset + 1] = Point(b , a[2], a[2]);
_points[offset + 2] = Point(a[2], a[2], b);
_points[offset + 3] = Point(a[2], b, a[2]);
_points[offset + 4] = Point( b, a[2], b);
_points[offset + 5] = Point(a[2], b, b);
// These 6 points all have the same weights
for (unsigned int j=0; j<6; ++j)
_weights[offset + j] = wt[2];
}
// Original rule by Keast, unsuitable because it has points on the
// reference element boundary!
// _points.resize(15);
// _weights.resize(15);
// _points[0](0) = 0.25;
// _points[0](1) = 0.25;
// _points[0](2) = 0.25;
// {
// const Real a = 0.;
// const Real b = 0.333333333333333333333333333333333333333;
// _points[1](0) = a;
// _points[1](1) = b;
// _points[1](2) = b;
// _points[2](0) = b;
// _points[2](1) = a;
// _points[2](2) = b;
// _points[3](0) = b;
// _points[3](1) = b;
// _points[3](2) = a;
// _points[4](0) = b;
// _points[4](1) = b;
// _points[4](2) = b;
// }
// {
// const Real a = 0.7272727272727272727272727272727272727272727272727272727;
// const Real b = 0.0909090909090909090909090909090909090909090909090909091;
// _points[5](0) = a;
// _points[5](1) = b;
// _points[5](2) = b;
// _points[6](0) = b;
// _points[6](1) = a;
// _points[6](2) = b;
// _points[7](0) = b;
// _points[7](1) = b;
// _points[7](2) = a;
// _points[8](0) = b;
// _points[8](1) = b;
// _points[8](2) = b;
// }
// {
// const Real a = 0.066550153573664;
// const Real b = 0.433449846426336;
// _points[9](0) = b;
// _points[9](1) = a;
// _points[9](2) = a;
// _points[10](0) = a;
// _points[10](1) = a;
// _points[10](2) = b;
// _points[11](0) = a;
// _points[11](1) = b;
// _points[11](2) = b;
// _points[12](0) = b;
// _points[12](1) = a;
// _points[12](2) = b;
// _points[13](0) = b;
// _points[13](1) = b;
// _points[13](2) = a;
// _points[14](0) = a;
// _points[14](1) = b;
// _points[14](2) = a;
// }
// _weights[0] = 0.030283678097089;
// _weights[1] = 0.006026785714286;
// _weights[2] = _weights[1];
// _weights[3] = _weights[1];
// _weights[4] = _weights[1];
// _weights[5] = 0.011645249086029;
// _weights[6] = _weights[5];
// _weights[7] = _weights[5];
// _weights[8] = _weights[5];
// _weights[9] = 0.010949141561386;
// _weights[10] = _weights[9];
// _weights[11] = _weights[9];
// _weights[12] = _weights[9];
// _weights[13] = _weights[9];
// _weights[14] = _weights[9];
return;
}
// This rule is originally from Keast:
// Patrick Keast,
// Moderate Degree Tetrahedral Quadrature Formulas,
// Computer Methods in Applied Mechanics and Engineering,
// Volume 55, Number 3, May 1986, pages 339-348.
//
// It is accurate on 6th-degree polynomials and has 24 points
// vs. 64 for the comparable conical product rule.
//
// Values copied 24th June 2008 from:
// http://people.scs.fsu.edu/~burkardt/f_src/keast/keast.f90
case SIXTH:
{
_points.resize (24);
_weights.resize(24);
// The raw data for the quadrature rule.
const Real rule_data[4][4] = {
{0.356191386222544953e+00 , 0.214602871259151684e+00 , 0., 0.00665379170969464506e+00}, // 4
{0.877978124396165982e+00 , 0.0406739585346113397e+00, 0., 0.00167953517588677620e+00}, // 4
{0.0329863295731730594e+00, 0.322337890142275646e+00 , 0., 0.00922619692394239843e+00}, // 4
{0.0636610018750175299e+00, 0.269672331458315867e+00 , 0.603005664791649076e+00, 0.00803571428571428248e+00} // 12
};
// Now call the keast routine to generate _points and _weights
keast_rule(rule_data, 4);
return;
}
// Keast's 31 point, 7th-order rule contains points on the reference
// element boundary, so we've decided not to include it here.
//
// Keast's 8th-order rule has 45 points. and a negative
// weight, so if you've explicitly disallowed such rules
// you will fall through to the conical product rules
// below.
case SEVENTH:
case EIGHTH:
{
if (allow_rules_with_negative_weights)
{
_points.resize (45);
_weights.resize(45);
// The raw data for the quadrature rule.
const Real rule_data[7][4] = {
{0.250000000000000000e+00, 0., 0., -0.393270066412926145e-01}, // 1
{0.617587190300082967e+00, 0.127470936566639015e+00, 0., 0.408131605934270525e-02}, // 4
{0.903763508822103123e+00, 0.320788303926322960e-01, 0., 0.658086773304341943e-03}, // 4
{0.497770956432810185e-01, 0., 0.450222904356718978e+00, 0.438425882512284693e-02}, // 6
{0.183730447398549945e+00, 0., 0.316269552601450060e+00, 0.138300638425098166e-01}, // 6
{0.231901089397150906e+00, 0.229177878448171174e-01, 0.513280033360881072e+00, 0.424043742468372453e-02}, // 12
{0.379700484718286102e-01, 0.730313427807538396e+00, 0.193746475248804382e+00, 0.223873973961420164e-02} // 12
};
// Now call the keast routine to generate _points and _weights
keast_rule(rule_data, 7);
return;
} // end if (allow_rules_with_negative_weights)
// Note: if !allow_rules_with_negative_weights, fall through to next case.
}
// Fall back on Grundmann-Moller or Conical Product rules at high orders.
default:
{
if ((allow_rules_with_negative_weights) && (_order + 2*p < 34))
{
// The Grundmann-Moller rules are defined to arbitrary order and
// can have significantly fewer evaluation points than concial product
// rules. If you allow rules with negative weights, the GM rules
// will be more efficient for degree up to 33 (for degree 34 and
// higher, CP is more efficient!) but may be more susceptible
// to round-off error. Safest is to disallow rules with negative
// weights, but this decision should be made on a case-by-case basis.
QGrundmann_Moller gm_rule(3, _order);
gm_rule.init(type_in, p);
// Swap points and weights with the about-to-be destroyed rule.
_points.swap (gm_rule.get_points() );
_weights.swap(gm_rule.get_weights());
return;
}
else
{
// The following quadrature rules are generated as
// conical products. These tend to be non-optimal
// (use too many points, cluster points in certain
// regions of the domain) but they are quite easy to
// automatically generate using a 1D Gauss rule on
// [0,1] and two 1D Jacobi-Gauss rules on [0,1].
QConical conical_rule(3, _order);
conical_rule.init(type_in, p);
// Swap points and weights with the about-to-be destroyed rule.
_points.swap (conical_rule.get_points() );
_weights.swap(conical_rule.get_weights());
return;
}
}
}
} // end case TET4,TET10
//---------------------------------------------
// Prism quadrature rules
case PRISM6:
case PRISM15:
case PRISM18:
{
// We compute the 3D quadrature rule as a tensor
// product of the 1D quadrature rule and a 2D
// triangle quadrature rule
QGauss q1D(1,_order);
QGauss q2D(2,_order);
// Initialize
q1D.init(EDGE2,p);
q2D.init(TRI3,p);
tensor_product_prism(q1D, q2D);
return;
}
//---------------------------------------------
// Pyramid
case PYRAMID5:
case PYRAMID13:
case PYRAMID14:
{
// We compute the Pyramid rule as a conical product of a
// Jacobi rule with alpha==2 on the interval [0,1] two 1D
// Gauss rules with suitably modified points. The idea comes
// from: Stroud, A.H. "Approximate Calculation of Multiple
// Integrals."
QConical conical_rule(3, _order);
conical_rule.init(type_in, p);
// Swap points and weights with the about-to-be destroyed rule.
_points.swap (conical_rule.get_points() );
_weights.swap(conical_rule.get_weights());
return;
}
//---------------------------------------------
// Unsupported type
default:
libmesh_error_msg("ERROR: Unsupported type: " << type_in);
}
#endif
}
| void libMesh::QGauss::keast_rule | ( | const Real | rule_data[][4], |
| const unsigned int | n_pts | ||
| ) | [private] |
The Keast rule is for tets. It takes permutation points and weights in a specific format as input and fills the pre-sized _points and _weights vectors. This function is only used internally by the TET geometric elements.
Definition at line 33 of file quadrature_gauss.C.
References libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::QBase::libmesh_error_msg(), and libMesh::Real.
Referenced by init_3D().
{
// Like the Dunavant rule, the input data should have 4 columns. These columns
// have the following format and implied permutations (w=weight).
// {a, 0, 0, w} = 1-permutation (a,a,a)
// {a, b, 0, w} = 4-permutation (a,b,b), (b,a,b), (b,b,a), (b,b,b)
// {a, 0, b, w} = 6-permutation (a,a,b), (a,b,b), (b,b,a), (b,a,b), (b,a,a), (a,b,a)
// {a, b, c, w} = 12-permutation (a,a,b), (a,a,c), (b,a,a), (c,a,a), (a,b,a), (a,c,a)
// (a,b,c), (a,c,b), (b,a,c), (b,c,a), (c,a,b), (c,b,a)
// Always insert into the points & weights vector relative to the offset
unsigned int offset=0;
for (unsigned int p=0; p<n_pts; ++p)
{
// There must always be a non-zero entry to start the row
libmesh_assert_not_equal_to (rule_data[p][0], static_cast<Real>(0.0));
// A zero weight may imply you did not set up the raw data correctly
libmesh_assert_not_equal_to (rule_data[p][3], static_cast<Real>(0.0));
// What kind of point is this?
// One non-zero entry in first 3 cols ? 1-perm (centroid) point = 1
// Two non-zero entries in first 3 cols ? 3-perm point = 3
// Three non-zero entries ? 6-perm point = 6
unsigned int pointtype=1;
if (rule_data[p][1] != static_cast<Real>(0.0))
{
if (rule_data[p][2] != static_cast<Real>(0.0))
pointtype = 12;
else
pointtype = 4;
}
else
{
// The second entry is zero. What about the third?
if (rule_data[p][2] != static_cast<Real>(0.0))
pointtype = 6;
}
switch (pointtype)
{
case 1:
{
// Be sure we have enough space to insert this point
libmesh_assert_less (offset + 0, _points.size());
const Real a = rule_data[p][0];
// The point has only a single permutation (the centroid!)
_points[offset + 0] = Point(a,a,a);
// The weight is always the last entry in the row.
_weights[offset + 0] = rule_data[p][3];
offset += pointtype;
break;
}
case 4:
{
// Be sure we have enough space to insert these points
libmesh_assert_less (offset + 3, _points.size());
const Real a = rule_data[p][0];
const Real b = rule_data[p][1];
const Real wt = rule_data[p][3];
// Here it's understood the second entry is to be used twice, and
// thus there are three possible permutations.
_points[offset + 0] = Point(a,b,b);
_points[offset + 1] = Point(b,a,b);
_points[offset + 2] = Point(b,b,a);
_points[offset + 3] = Point(b,b,b);
for (unsigned int j=0; j<pointtype; ++j)
_weights[offset + j] = wt;
offset += pointtype;
break;
}
case 6:
{
// Be sure we have enough space to insert these points
libmesh_assert_less (offset + 5, _points.size());
const Real a = rule_data[p][0];
const Real b = rule_data[p][2];
const Real wt = rule_data[p][3];
// Three individual entries with six permutations.
_points[offset + 0] = Point(a,a,b);
_points[offset + 1] = Point(a,b,b);
_points[offset + 2] = Point(b,b,a);
_points[offset + 3] = Point(b,a,b);
_points[offset + 4] = Point(b,a,a);
_points[offset + 5] = Point(a,b,a);
for (unsigned int j=0; j<pointtype; ++j)
_weights[offset + j] = wt;
offset += pointtype;
break;
}
case 12:
{
// Be sure we have enough space to insert these points
libmesh_assert_less (offset + 11, _points.size());
const Real a = rule_data[p][0];
const Real b = rule_data[p][1];
const Real c = rule_data[p][2];
const Real wt = rule_data[p][3];
// Three individual entries with six permutations.
_points[offset + 0] = Point(a,a,b); _points[offset + 6] = Point(a,b,c);
_points[offset + 1] = Point(a,a,c); _points[offset + 7] = Point(a,c,b);
_points[offset + 2] = Point(b,a,a); _points[offset + 8] = Point(b,a,c);
_points[offset + 3] = Point(c,a,a); _points[offset + 9] = Point(b,c,a);
_points[offset + 4] = Point(a,b,a); _points[offset + 10] = Point(c,a,b);
_points[offset + 5] = Point(a,c,a); _points[offset + 11] = Point(c,b,a);
for (unsigned int j=0; j<pointtype; ++j)
_weights[offset + j] = wt;
offset += pointtype;
break;
}
default:
libmesh_error_msg("Don't know what to do with this many permutation points!");
}
}
}
| libMesh::QBase::libmesh_error_msg | ( | "ERROR: Seems as if this quadrature rule \nis not implemented for 2D." | ) | [protected, inherited] |
Referenced by libMesh::QBase::build(), dunavant_rule(), dunavant_rule2(), libMesh::QBase::init(), libMesh::QGaussLobatto::init_1D(), init_1D(), libMesh::QJacobi::init_1D(), libMesh::QGaussLobatto::init_2D(), libMesh::QTrap::init_2D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QSimpson::init_2D(), libMesh::QConical::init_2D(), libMesh::QGrid::init_2D(), libMesh::QGrundmann_Moller::init_2D(), libMesh::QGaussLobatto::init_3D(), libMesh::QTrap::init_3D(), libMesh::QClough::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), libMesh::QGrid::init_3D(), libMesh::QConical::init_3D(), libMesh::QGrundmann_Moller::init_3D(), keast_rule(), libMesh::QMonomial::kim_rule(), libMesh::QMonomial::stroud_rule(), and libMesh::QJacobi::type().
| static unsigned int libMesh::ReferenceCounter::n_objects | ( | ) | [inline, static, inherited] |
Prints the number of outstanding (created, but not yet destroyed) objects.
Definition at line 79 of file reference_counter.h.
References libMesh::ReferenceCounter::_n_objects.
Referenced by libMesh::LibMeshInit::~LibMeshInit().
{ return _n_objects; }
| unsigned int libMesh::QBase::n_points | ( | ) | const [inline, inherited] |
- Returns:
- the number of points associated with the quadrature rule.
Definition at line 118 of file quadrature.h.
References libMesh::QBase::_points, and libMesh::libmesh_assert().
Referenced by libMesh::ExactSolution::_compute_error(), libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), libMesh::ProjectFEMSolution::operator()(), libMesh::QBase::print_info(), libMesh::QBase::tensor_product_hex(), and libMesh::QBase::tensor_product_prism().
{ libmesh_assert (!_points.empty());
return cast_int<unsigned int>(_points.size()); }
| void libMesh::ReferenceCounter::print_info | ( | std::ostream & | out = libMesh::out | ) | [static, inherited] |
Prints the reference information, by default to libMesh::out.
Definition at line 88 of file reference_counter.C.
References libMesh::ReferenceCounter::_enable_print_counter, and libMesh::ReferenceCounter::get_info().
Referenced by libMesh::LibMeshInit::~LibMeshInit().
{
if( _enable_print_counter ) out_stream << ReferenceCounter::get_info();
}
| void libMesh::QBase::print_info | ( | std::ostream & | os = libMesh::out | ) | const [inline, inherited] |
Prints information relevant to the quadrature rule, by default to libMesh::out.
Definition at line 372 of file quadrature.h.
References libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::libmesh_assert(), libMesh::QBase::n_points(), and libMesh::Real.
Referenced by libMesh::operator<<().
{
libmesh_assert(!_points.empty());
libmesh_assert(!_weights.empty());
Real summed_weights=0;
os << "N_Q_Points=" << this->n_points() << std::endl << std::endl;
for (unsigned int qpoint=0; qpoint<this->n_points(); qpoint++)
{
os << " Point " << qpoint << ":\n"
<< " "
<< _points[qpoint]
<< "\n Weight:\n "
<< " w=" << _weights[qpoint] << "\n" << std::endl;
summed_weights += _weights[qpoint];
}
os << "Summed Weights: " << summed_weights << std::endl;
}
| Point libMesh::QBase::qp | ( | const unsigned int | i | ) | const [inline, inherited] |
- Returns:
- the
quadrature point on the reference object.
Definition at line 152 of file quadrature.h.
References libMesh::QBase::_points.
Referenced by libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), libMesh::QBase::tensor_product_hex(), and libMesh::QBase::tensor_product_prism().
| void libMesh::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 93 of file quadrature.C.
References libMesh::QBase::_dim, libMesh::QBase::_points, libMesh::QBase::_weights, and libMesh::Real.
Referenced by libMesh::QConical::conical_product_tet(), and libMesh::QConical::conical_product_tri().
{
// Make sure we are in 1D
libmesh_assert_equal_to (_dim, 1);
Real
h_new = new_range.second - new_range.first,
h_old = old_range.second - old_range.first;
// Make sure that we have sane ranges
libmesh_assert_greater (h_new, 0.);
libmesh_assert_greater (h_old, 0.);
// Make sure there are some points
libmesh_assert_greater (_points.size(), 0);
// Compute the scale factor
Real scfact = h_new/h_old;
// We're mapping from old_range -> new_range
for (unsigned int i=0; i<_points.size(); i++)
{
_points[i](0) = new_range.first +
(_points[i](0) - old_range.first) * scfact;
// Scale the weights
_weights[i] *= scfact;
}
}
| virtual bool libMesh::QBase::shapes_need_reinit | ( | ) | [inline, virtual, inherited] |
Returns true if the shape functions need to be recalculated.
This can happen if the number of points or their positions change.
By default this will return false.
Definition at line 212 of file quadrature.h.
{ return false; }
| void libMesh::QBase::tensor_product_hex | ( | const QBase & | q1D | ) | [protected, inherited] |
Computes the tensor product quadrature rule [q1D x q1D x q1D] from the 1D rule q1D. Used in the init_3D routines for hexahedral element types.
Definition at line 154 of file quadrature.C.
References libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::QBase::n_points(), libMesh::QBase::qp(), and libMesh::QBase::w().
Referenced by libMesh::QGaussLobatto::init_3D(), libMesh::QTrap::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), and libMesh::QGrid::init_3D().
{
const unsigned int np = q1D.n_points();
_points.resize(np * np * np);
_weights.resize(np * np * np);
unsigned int q=0;
for (unsigned int k=0; k<np; k++)
for (unsigned int j=0; j<np; j++)
for (unsigned int i=0; i<np; i++)
{
_points[q](0) = q1D.qp(i)(0);
_points[q](1) = q1D.qp(j)(0);
_points[q](2) = q1D.qp(k)(0);
_weights[q] = q1D.w(i) * q1D.w(j) * q1D.w(k);
q++;
}
}
| void libMesh::QBase::tensor_product_prism | ( | const QBase & | q1D, |
| const QBase & | q2D | ||
| ) | [protected, inherited] |
Computes the tensor product of a 1D quadrature rule and a 2D quadrature rule. Used in the init_3D routines for prismatic element types.
Definition at line 181 of file quadrature.C.
References libMesh::QBase::_points, libMesh::QBase::_weights, libMesh::QBase::n_points(), libMesh::QBase::qp(), and libMesh::QBase::w().
Referenced by libMesh::QTrap::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), and libMesh::QGrid::init_3D().
{
const unsigned int n_points1D = q1D.n_points();
const unsigned int n_points2D = q2D.n_points();
_points.resize (n_points1D * n_points2D);
_weights.resize (n_points1D * n_points2D);
unsigned int q=0;
for (unsigned int j=0; j<n_points1D; j++)
for (unsigned int i=0; i<n_points2D; i++)
{
_points[q](0) = q2D.qp(i)(0);
_points[q](1) = q2D.qp(i)(1);
_points[q](2) = q1D.qp(j)(0);
_weights[q] = q2D.w(i) * q1D.w(j);
q++;
}
}
| QuadratureType libMesh::QGauss::type | ( | ) | const [inline, virtual] |
- Returns:
QGAUSS
Implements libMesh::QBase.
Definition at line 61 of file quadrature_gauss.h.
References libMesh::QGAUSS.
{ return QGAUSS; }
| Real libMesh::QBase::w | ( | const unsigned int | i | ) | const [inline, inherited] |
- Returns:
- the quadrature weight.
Definition at line 158 of file quadrature.h.
References libMesh::QBase::_weights.
Referenced by libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), libMesh::QBase::tensor_product_hex(), and libMesh::QBase::tensor_product_prism().
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 208 of file quadrature.C.
{
q.print_info(os);
return os;
}
Member Data Documentation
ReferenceCounter::Counts libMesh::ReferenceCounter::_counts [static, protected, inherited] |
Actually holds the data.
Definition at line 118 of file reference_counter.h.
Referenced by libMesh::ReferenceCounter::get_info(), libMesh::ReferenceCounter::increment_constructor_count(), and libMesh::ReferenceCounter::increment_destructor_count().
const unsigned int libMesh::QBase::_dim [protected, inherited] |
The dimension
Definition at line 319 of file quadrature.h.
Referenced by libMesh::QBase::get_dim(), libMesh::QBase::init(), QGauss(), libMesh::QGaussLobatto::QGaussLobatto(), libMesh::QJacobi::QJacobi(), libMesh::QSimpson::QSimpson(), libMesh::QTrap::QTrap(), and libMesh::QBase::scale().
bool libMesh::ReferenceCounter::_enable_print_counter = true [static, protected, inherited] |
Flag to control whether reference count information is printed when print_info is called.
Definition at line 137 of file reference_counter.h.
Referenced by libMesh::ReferenceCounter::disable_print_counter_info(), libMesh::ReferenceCounter::enable_print_counter_info(), and libMesh::ReferenceCounter::print_info().
Threads::spin_mutex libMesh::ReferenceCounter::_mutex [static, protected, inherited] |
Mutual exclusion object to enable thread-safe reference counting.
Definition at line 131 of file reference_counter.h.
Threads::atomic< unsigned int > libMesh::ReferenceCounter::_n_objects [static, protected, inherited] |
The number of objects. Print the reference count information when the number returns to 0.
Definition at line 126 of file reference_counter.h.
Referenced by libMesh::ReferenceCounter::n_objects(), libMesh::ReferenceCounter::ReferenceCounter(), and libMesh::ReferenceCounter::~ReferenceCounter().
const Order libMesh::QBase::_order [protected, inherited] |
The order of the quadrature rule.
Definition at line 324 of file quadrature.h.
Referenced by libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), libMesh::QBase::get_order(), libMesh::QGaussLobatto::init_1D(), libMesh::QClough::init_1D(), init_1D(), libMesh::QGrid::init_1D(), libMesh::QJacobi::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QGaussLobatto::init_2D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QGrid::init_2D(), libMesh::QMonomial::init_2D(), libMesh::QGrundmann_Moller::init_2D(), libMesh::QGaussLobatto::init_3D(), init_3D(), libMesh::QGrid::init_3D(), libMesh::QMonomial::init_3D(), and libMesh::QGrundmann_Moller::init_3D().
unsigned int libMesh::QBase::_p_level [protected, inherited] |
The p level of element for which the current values have been computed.
Definition at line 336 of file quadrature.h.
Referenced by libMesh::QBase::get_order(), libMesh::QBase::get_p_level(), and libMesh::QBase::init().
std::vector<Point> libMesh::QBase::_points [protected, inherited] |
The reference element locations of the quadrature points.
Definition at line 342 of file quadrature.h.
Referenced by libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), dunavant_rule(), dunavant_rule2(), libMesh::QBase::get_points(), libMesh::QGrundmann_Moller::gm_rule(), libMesh::QBase::init_0D(), libMesh::QGaussLobatto::init_1D(), libMesh::QTrap::init_1D(), libMesh::QClough::init_1D(), init_1D(), libMesh::QSimpson::init_1D(), libMesh::QGrid::init_1D(), libMesh::QJacobi::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QTrap::init_2D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QSimpson::init_2D(), libMesh::QGrid::init_2D(), libMesh::QMonomial::init_2D(), libMesh::QTrap::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), libMesh::QGrid::init_3D(), libMesh::QMonomial::init_3D(), libMesh::QGrundmann_Moller::init_3D(), keast_rule(), libMesh::QMonomial::kim_rule(), libMesh::QBase::n_points(), libMesh::QBase::print_info(), libMesh::QBase::qp(), libMesh::QBase::scale(), libMesh::QMonomial::stroud_rule(), libMesh::QBase::tensor_product_hex(), libMesh::QBase::tensor_product_prism(), and libMesh::QMonomial::wissmann_rule().
ElemType libMesh::QBase::_type [protected, inherited] |
The type of element for which the current values have been computed.
Definition at line 330 of file quadrature.h.
Referenced by libMesh::QBase::get_elem_type(), and libMesh::QBase::init().
std::vector<Real> libMesh::QBase::_weights [protected, inherited] |
The value of the quadrature weights.
Definition at line 347 of file quadrature.h.
Referenced by libMesh::QConical::conical_product_pyramid(), libMesh::QConical::conical_product_tet(), libMesh::QConical::conical_product_tri(), dunavant_rule(), dunavant_rule2(), libMesh::QBase::get_weights(), libMesh::QGrundmann_Moller::gm_rule(), libMesh::QBase::init_0D(), libMesh::QGaussLobatto::init_1D(), libMesh::QTrap::init_1D(), libMesh::QClough::init_1D(), init_1D(), libMesh::QSimpson::init_1D(), libMesh::QGrid::init_1D(), libMesh::QJacobi::init_1D(), libMesh::QMonomial::init_1D(), libMesh::QTrap::init_2D(), libMesh::QClough::init_2D(), init_2D(), libMesh::QSimpson::init_2D(), libMesh::QGrid::init_2D(), libMesh::QMonomial::init_2D(), libMesh::QTrap::init_3D(), init_3D(), libMesh::QSimpson::init_3D(), libMesh::QGrid::init_3D(), libMesh::QMonomial::init_3D(), libMesh::QGrundmann_Moller::init_3D(), keast_rule(), libMesh::QMonomial::kim_rule(), libMesh::QBase::print_info(), libMesh::QBase::scale(), libMesh::QMonomial::stroud_rule(), libMesh::QBase::tensor_product_hex(), libMesh::QBase::tensor_product_prism(), libMesh::QBase::w(), and libMesh::QMonomial::wissmann_rule().
bool libMesh::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 229 of file quadrature.h.
Referenced by libMesh::QGrundmann_Moller::init_2D(), init_3D(), libMesh::QMonomial::init_3D(), and libMesh::QGrundmann_Moller::init_3D().
The documentation for this class was generated from the following files:
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