quadrature_monomial_3D.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
 
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
 
 
// Local includes
#include "libmesh/quadrature_monomial.h"
#include "libmesh/quadrature_gauss.h"
 
namespace libMesh
{
 
 
void QMonomial::init_3D()
{
 
  switch (_type)
    {
      //---------------------------------------------
      // Hex quadrature rules
    case HEX8:
    case HEX20:
    case HEX27:
      {
        switch(get_order())
          {
 
            // The CONSTANT/FIRST rule is the 1-point Gauss "product" rule...we fall
            // through to the default case for this rule.
 
          case SECOND:
          case THIRD:
            {
              // A degree 3, 6-point, "rotationally-symmetric" rule by
              // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //
              // Warning: this rule contains points on the boundary of the reference
              // element, and therefore may be unsuitable for some problems.  The alternative
              // would be a 2x2x2 Gauss product rule.
              const Real data[1][4] =
                {
                  {1, 0, 0, Real(4)/3}
                };
 
              const unsigned int rule_id[1] = {
                1 // (x,0,0) -> 6 permutations
              };
 
              _points.resize(6);
              _weights.resize(6);
 
              kim_rule(data, rule_id, 1);
              return;
            } // end case SECOND,THIRD
 
          case FOURTH:
          case FIFTH:
            {
              // A degree 5, 13-point rule by Stroud,
              // AH Stroud, "Some Fifth Degree Integration Formulas for Symmetric Regions II.",
              // Numerische Mathematik 9, pp. 460-468 (1967).
              //
              // This rule is provably minimal in the number of points.  The equations given for
              // the n-cube on pg. 466 of the paper for mu/gamma and gamma are wrong, at least for
              // the n=3 case.  The analytical values given here were computed by me [JWP] in Maple.
 
              // Convenient intermediate values.
              const Real sqrt19 = std::sqrt(Real(19));            // ~4.35889894354
              const Real tp     = std::sqrt(71440 + 6802*sqrt19); // ~317.945326454
 
              // Point data for permutations.
              const Real eta    =  0;
 
              const Real lambda =  std::sqrt(Real(1919)/3285 - 148*sqrt19/3285 + 4*tp/3285);
              // 8.8030440669930978047737818209860e-01_R;
 
              const Real xi     = -std::sqrt(Real(1121)/3285 +  74*sqrt19/3285 - 2*tp/3285);
              // -4.9584817142571115281421242364290e-01_R;
 
              const Real mu     =  std::sqrt(Real(1121)/3285 +  74*sqrt19/3285 + 2*tp/3285);
              // 7.9562142216409541542982482567580e-01_R;
 
              const Real gamma  =  std::sqrt(Real(1919)/3285 - 148*sqrt19/3285 - 4*tp/3285);
              // 2.5293711744842581347389255929324e-02_R;
 
              // Weights: the centroid weight is given analytically.  Weight B (resp C) goes
              // with the {lambda,xi} (resp {gamma,mu}) permutation.  The single-precision
              // results reported by Stroud are given for reference.
 
              const Real A      = Real(32)/19; // ~1.684210560
              // Stroud: 0.21052632  * 8.0;
 
              const Real B      = Real(1) / (Real(260072)/133225  - 1520*sqrt19/133225 + (133 - 37*sqrt19)*tp/133225);
              // 5.4498735127757671684690782180890e-01_R; // Stroud: 0.068123420 * 8.0 = 0.544987360;
 
              const Real C      = Real(1) / (Real(260072)/133225  - 1520*sqrt19/133225 - (133 - 37*sqrt19)*tp/133225);
              // 5.0764422766979170420572375713840e-01_R; // Stroud: 0.063455527 * 8.0 = 0.507644216;
 
              _points.resize(13);
              _weights.resize(13);
 
              unsigned int c=0;
 
              // Point with weight A (origin)
              _points[c] = Point(eta, eta, eta);
              _weights[c++] = A;
 
              // Points with weight B
              _points[c] = Point(lambda, xi, xi);
              _weights[c++] = B;
              _points[c] = -_points[c-1];
              _weights[c++] = B;
 
              _points[c] = Point(xi, lambda, xi);
              _weights[c++] = B;
              _points[c] = -_points[c-1];
              _weights[c++] = B;
 
              _points[c] = Point(xi, xi, lambda);
              _weights[c++] = B;
              _points[c] = -_points[c-1];
              _weights[c++] = B;
 
              // Points with weight C
              _points[c] = Point(mu, mu, gamma);
              _weights[c++] = C;
              _points[c] = -_points[c-1];
              _weights[c++] = C;
 
              _points[c] = Point(mu, gamma, mu);
              _weights[c++] = C;
              _points[c] = -_points[c-1];
              _weights[c++] = C;
 
              _points[c] = Point(gamma, mu, mu);
              _weights[c++] = C;
              _points[c] = -_points[c-1];
              _weights[c++] = C;
 
              return;
 
 
              //       // A degree 5, 14-point, "rotationally-symmetric" rule by
              //       // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //       // Was also reported in Stroud's 1971 book.
              //       const Real data[2][4] =
              // {
              //   {7.95822425754221463264548820476135e-01_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 8.86426592797783933518005540166204e-01_R},
              //   {7.58786910639328146269034278112267e-01_R, 7.58786910639328146269034278112267e-01_R, 7.58786910639328146269034278112267e-01_R, 3.35180055401662049861495844875346e-01_R}
              // };
 
              //       const unsigned int rule_id[2] = {
              // 1, // (x,0,0) -> 6 permutations
              // 4  // (x,x,x) -> 8 permutations
              //       };
 
              //       _points.resize(14);
              //       _weights.resize(14);
 
              //       kim_rule(data, rule_id, 2);
              //       return;
            } // end case FOURTH,FIFTH
 
 
          case SIXTH:
          case SEVENTH:
            {
              if (allow_rules_with_negative_weights)
                {
                  // A degree 7, 31-point, "rotationally-symmetric" rule by
                  // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
                  // This rule contains a negative weight, so only use it if such type of
                  // rules are allowed.
                  const Real data[3][4] =
                    {
                      {0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, -1.27536231884057971014492753623188e+00_R},
                      {5.85540043769119907612630781744060e-01_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R,  8.71111111111111111111111111111111e-01_R},
                      {6.94470135991704766602025803883310e-01_R, 9.37161638568208038511047377665396e-01_R, 4.15659267604065126239606672567031e-01_R,  1.68695652173913043478260869565217e-01_R}
                    };
 
                  const unsigned int rule_id[3] = {
                    0, // (0,0,0) -> 1 permutation
                    1, // (x,0,0) -> 6 permutations
                    6  // (x,y,z) -> 24 permutations
                  };
 
                  _points.resize(31);
                  _weights.resize(31);
 
                  kim_rule(data, rule_id, 3);
                  return;
                } // end if (allow_rules_with_negative_weights)
 
 
              // A degree 7, 34-point, "fully-symmetric" rule, first published in
              // P.C. Hammer and A.H. Stroud, "Numerical Evaluation of Multiple Integrals II",
              // Mathematical Tables and Other Aids to Computation, vol 12., no 64, 1958, pp. 272-280
              //
              // This rule happens to fall under the same general
              // construction as the Kim rules, so we've re-used
              // that code here.  Stroud gives 16 digits for his rule,
              // and this is the most accurate version I've found.
              //
              // For comparison, a SEVENTH-order Gauss product rule
              // (which integrates tri-7th order polynomials) would
              // have 4^3=64 points.
              const Real
                r  = std::sqrt(Real(6)/7),                                     // ~0.92582009977
                s  = std::sqrt((Real(960) - 3*std::sqrt(Real(28798))) / 2726), // ~0.40670318642
                t  = std::sqrt((Real(960) + 3*std::sqrt(Real(28798))) / 2726), // ~0.73411252875
                B1 = Real(8624)/29160,                                         // ~0.29574759945
                B2 = Real(2744)/29160,                                         // ~0.09410150891
                B3 = 8*(774*t*t - 230)/(9720*(t*t-s*s)),                       // ~0.41233386227
                B4 = 8*(230 - 774*s*s)/(9720*(t*t-s*s));                       // ~0.22470317477
 
              const Real data[4][4] =
                {
                  {r,  0,  0, B1},
                  {r,  r,  0, B2},
                  {s,  s,  s, B3},
                  {t,  t,  t, B4}
                };
 
              const unsigned int rule_id[4] = {
                1, // (x,0,0) -> 6 permutations
                2, // (x,x,0) -> 12 permutations
                4, // (x,x,x) -> 8 permutations
                4  // (x,x,x) -> 8 permutations
              };
 
              _points.resize(34);
              _weights.resize(34);
 
              kim_rule(data, rule_id, 4);
              return;
 
 
              //      // A degree 7, 38-point, "rotationally-symmetric" rule by
              //      // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //      //
              //      // This rule is obviously inferior to the 34-point rule above...
              //      const Real data[3][4] =
              //{
              //  {9.01687807821291289082811566285950e-01_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 2.95189738262622903181631100062774e-01_R},
              //  {4.08372221499474674069588900002128e-01_R, 4.08372221499474674069588900002128e-01_R, 4.08372221499474674069588900002128e-01_R, 4.04055417266200582425904380777126e-01_R},
              //  {8.59523090201054193116477875786220e-01_R, 8.59523090201054193116477875786220e-01_R, 4.14735913727987720499709244748633e-01_R, 1.24850759678944080062624098058597e-01_R}
              //};
              //
              //      const unsigned int rule_id[3] = {
              //1, // (x,0,0) -> 6 permutations
              //4, // (x,x,x) -> 8 permutations
              //5  // (x,x,z) -> 24 permutations
              //      };
              //
              //      _points.resize(38);
              //      _weights.resize(38);
              //
              //      kim_rule(data, rule_id, 3);
              //      return;
            } // end case SIXTH,SEVENTH
 
          case EIGHTH:
            {
              // A degree 8, 47-point, "rotationally-symmetric" rule by
              // Kim and Song, Comm. Korean Math. Soc vol. 13, no. 4, 1998, pp. 913-931.
              //
              // A EIGHTH-order Gauss product rule (which integrates tri-8th order polynomials)
              // would have 5^3=125 points.
              const Real data[5][4] =
                {
                  {0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 4.51903714875199690490763818699555e-01_R},
                  {7.82460796435951590652813975429717e-01_R, 0.00000000000000000000000000000000e+00_R, 0.00000000000000000000000000000000e+00_R, 2.99379177352338919703385618576171e-01_R},
                  {4.88094669706366480526729301468686e-01_R, 4.88094669706366480526729301468686e-01_R, 4.88094669706366480526729301468686e-01_R, 3.00876159371240019939698689791164e-01_R},
                  {8.62218927661481188856422891110042e-01_R, 8.62218927661481188856422891110042e-01_R, 8.62218927661481188856422891110042e-01_R, 4.94843255877038125738173175714853e-02_R},
                  {2.81113909408341856058098281846420e-01_R, 9.44196578292008195318687494773744e-01_R, 6.97574833707236996779391729948984e-01_R, 1.22872389222467338799199767122592e-01_R}
                };
 
              const unsigned int rule_id[5] = {
                0, // (0,0,0) -> 1 permutation
                1, // (x,0,0) -> 6 permutations
                4, // (x,x,x) -> 8 permutations
                4, // (x,x,x) -> 8 permutations
                6  // (x,y,z) -> 24 permutations
              };
 
              _points.resize(47);
              _weights.resize(47);
 
              kim_rule(data, rule_id, 5);
              return;
            } // end case EIGHTH
 
 
            // By default: construct and use a Gauss quadrature rule
          default:
            {
              // Break out and fall down into the default: case for the
              // outer switch statement.
              break;
            }
 
          } // end switch(_order + 2*p)
      } // end case HEX8/20/27
 
      libmesh_fallthrough();
 
      // By default: construct and use a Gauss quadrature rule
    default:
      {
        QGauss gauss_rule(3, _order);
        gauss_rule.init(*this);
 
        // Swap points and weights with the about-to-be destroyed rule.
        _points.swap (gauss_rule.get_points() );
        _weights.swap(gauss_rule.get_weights());
 
        return;
      }
    } // end switch (_type)
}
 
} // namespace libMesh