Line data    Source code

       1             : // The libMesh Finite Element Library.
       2             : // Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
       3             : 
       4             : // This library is free software; you can redistribute it and/or
       5             : // modify it under the terms of the GNU Lesser General Public
       6             : // License as published by the Free Software Foundation; either
       7             : // version 2.1 of the License, or (at your option) any later version.
       8             : 
       9             : // This library is distributed in the hope that it will be useful,
      10             : // but WITHOUT ANY WARRANTY; without even the implied warranty of
      11             : // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      12             : // Lesser General Public License for more details.
      13             : 
      14             : // You should have received a copy of the GNU Lesser General Public
      15             : // License along with this library; if not, write to the Free Software
      16             : // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
      17             : 
      18             : 
      19             : 
      20             : // C++ includes
      21             : 
      22             : // Local includes
      23             : #include "libmesh/quadrature_gauss_lobatto.h"
      24             : 
      25             : namespace libMesh
      26             : {
      27             : 
      28       21444 : void QGaussLobatto::init_1D()
      29             : {
      30             :   //----------------------------------------------------------------------
      31             :   // 1D quadrature rules
      32       27437 :   switch(get_order())
      33             :     {
      34             :       // Since Gauss-Lobatto rules must include the endpoints of the
      35             :       // domain, there is no 1-point rule.  The two-point
      36             :       // Gauss-Lobatto rule is equivalent to the trapezoidal rule.
      37        5119 :     case CONSTANT:
      38             :     case FIRST:
      39             :       {
      40        5119 :         _points.resize (2);
      41        5119 :         _weights.resize(2);
      42             : 
      43        5119 :         _points[0](0) = -1;
      44        5119 :         _points[1]    = -_points[0];
      45             : 
      46        5119 :         _weights[0]   = 1.;
      47        5119 :         _weights[1]   = _weights[0];
      48             : 
      49        5119 :         return;
      50             :       }
      51             : 
      52             :       // The three-point Gauss-Lobatto rule is equivalent to Simpsons' rule.
      53             :       // It can integrate cubic polynomials exactly.
      54        5240 :     case SECOND:
      55             :     case THIRD:
      56             :       {
      57        5240 :         _points.resize (3);
      58        5240 :         _weights.resize(3);
      59             : 
      60        5240 :         _points[0](0) = -1;
      61        5240 :         _points[1]    = 0;
      62        5240 :         _points[2]    = -_points[0];
      63             : 
      64        5240 :         _weights[0]   = Real(1)/3;
      65        5240 :         _weights[1]   = Real(4)/3;
      66        5240 :         _weights[2]   = _weights[0];
      67        5240 :         return;
      68             :       }
      69             : 
      70             :       // The four-point Gauss-Lobatto rule can integrate 2*4-3 =
      71             :       // 5th-order polynomials exactly.
      72        5105 :     case FOURTH:
      73             :     case FIFTH:
      74             :       {
      75        5105 :         _points.resize (4);
      76        5105 :         _weights.resize(4);
      77             : 
      78        5105 :         _points[ 0](0) = -1;
      79        5105 :         _points[ 1](0) = -std::sqrt(Real(1)/5); // ~ -0.4472135955
      80        5105 :         _points[ 2]    = -_points[1];
      81        5105 :         _points[ 3]    = -_points[0];
      82             : 
      83        5105 :         _weights[ 0]   = Real(1)/6;
      84        5105 :         _weights[ 1]   = Real(5)/6;
      85        5105 :         _weights[ 2]   = _weights[1];
      86        5105 :         _weights[ 3]   = _weights[0];
      87             : 
      88        5105 :         return;
      89             :       }
      90             : 
      91             :       // The five-point Gauss-Lobatto rule can integrate 2*5-3 =
      92             :       // 7th-order polynomials exactly.
      93        5105 :     case SIXTH:
      94             :     case SEVENTH:
      95             :       {
      96        5105 :         _points.resize (5);
      97        5105 :         _weights.resize(5);
      98             : 
      99        5105 :         _points[ 0](0) = -1;
     100        5105 :         _points[ 1](0) = -std::sqrt(Real(3)/7); // ~ -0.6546536707
     101        5105 :         _points[ 2](0) = 0.;
     102        5105 :         _points[ 3]    = -_points[1];
     103        5105 :         _points[ 4]    = -_points[0];
     104             : 
     105        5105 :         _weights[ 0]   = 1/Real(10);
     106        5105 :         _weights[ 1]   = Real(49)/90;
     107        5105 :         _weights[ 2]   = Real(32)/45;
     108        5105 :         _weights[ 3]   = _weights[1];
     109        5105 :         _weights[ 4]   = _weights[0];
     110             : 
     111        5105 :         return;
     112             :       }
     113             : 
     114             :       // 2*6-3 = 9
     115          98 :     case EIGHTH:
     116             :     case NINTH:
     117             :       {
     118          98 :         _points.resize (6);
     119          98 :         _weights.resize(6);
     120             : 
     121          98 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     122          98 :         _points[ 1](0) = -7.6505532392946469285100297395934e-01_R;
     123          98 :         _points[ 2](0) = -2.8523151648064509631415099404088e-01_R;
     124          98 :         _points[ 3]    = -_points[2];
     125          98 :         _points[ 4]    = -_points[1];
     126          98 :         _points[ 5]    = -_points[0];
     127             : 
     128          98 :         _weights[ 0]   = 6.6666666666666666666666666666667e-02_R;
     129          98 :         _weights[ 1]   = 3.7847495629784698031661280821202e-01_R;
     130          98 :         _weights[ 2]   = 5.5485837703548635301672052512131e-01_R;
     131          98 :         _weights[ 3]   = _weights[2];
     132          98 :         _weights[ 4]   = _weights[1];
     133          98 :         _weights[ 5]   = _weights[0];
     134             : 
     135          98 :         return;
     136             :       }
     137             : 
     138             :       // 2*7-3 = 11
     139          63 :     case TENTH:
     140             :     case ELEVENTH:
     141             :       {
     142          63 :         _points.resize (7);
     143          63 :         _weights.resize(7);
     144             : 
     145          63 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     146          63 :         _points[ 1](0) = -8.3022389627856692987203221396747e-01_R;
     147          63 :         _points[ 2](0) = -4.6884879347071421380377188190877e-01_R;
     148          63 :         _points[ 3](0) = 0.;
     149          63 :         _points[ 4]    = -_points[2];
     150          63 :         _points[ 5]    = -_points[1];
     151          63 :         _points[ 6]    = -_points[0];
     152             : 
     153          63 :         _weights[ 0]   = 4.7619047619047619047619047619048e-02_R;
     154          63 :         _weights[ 1]   = 2.7682604736156594801070040629007e-01_R;
     155          63 :         _weights[ 2]   = 4.3174538120986262341787102228136e-01_R;
     156          63 :         _weights[ 3]   = 4.8761904761904761904761904761905e-01_R;
     157          63 :         _weights[ 4]   = _weights[2];
     158          63 :         _weights[ 5]   = _weights[1];
     159          63 :         _weights[ 6]   = _weights[0];
     160             : 
     161          63 :         return;
     162             :       }
     163             : 
     164             :       // 2*8-3 = 13
     165          42 :     case TWELFTH:
     166             :     case THIRTEENTH:
     167             :       {
     168          42 :         _points.resize (8);
     169          42 :         _weights.resize(8);
     170             : 
     171          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     172          42 :         _points[ 1](0) = -8.7174014850960661533744576122066e-01_R;
     173          42 :         _points[ 2](0) = -5.9170018143314230214451073139795e-01_R;
     174          42 :         _points[ 3](0) = -2.0929921790247886876865726034535e-01_R;
     175          42 :         _points[ 4]    = -_points[3];
     176          42 :         _points[ 5]    = -_points[2];
     177          42 :         _points[ 6]    = -_points[1];
     178          42 :         _points[ 7]    = -_points[0];
     179             : 
     180          42 :         _weights[ 0]   = 3.5714285714285714285714285714286e-02_R;
     181          42 :         _weights[ 1]   = 2.1070422714350603938299206577576e-01_R;
     182          42 :         _weights[ 2]   = 3.4112269248350436476424067710775e-01_R;
     183          42 :         _weights[ 3]   = 4.1245879465870388156705297140221e-01_R;
     184          42 :         _weights[ 4]   = _weights[3];
     185          42 :         _weights[ 5]   = _weights[2];
     186          42 :         _weights[ 6]   = _weights[1];
     187          42 :         _weights[ 7]   = _weights[0];
     188             : 
     189          42 :         return;
     190             :       }
     191             : 
     192             :       // 2*9-3 = 15
     193          84 :     case FOURTEENTH:
     194             :     case FIFTEENTH:
     195             :       {
     196          84 :         _points.resize (9);
     197          84 :         _weights.resize(9);
     198             : 
     199          84 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     200          84 :         _points[ 1](0) = -8.9975799541146015731234524441834e-01_R;
     201          84 :         _points[ 2](0) = -6.7718627951073775344588542709134e-01_R;
     202          84 :         _points[ 3](0) = -3.6311746382617815871075206870866e-01_R;
     203          84 :         _points[ 4](0) = 0.;
     204          84 :         _points[ 5]    = -_points[3];
     205          84 :         _points[ 6]    = -_points[2];
     206          84 :         _points[ 7]    = -_points[1];
     207          84 :         _points[ 8]    = -_points[0];
     208             : 
     209          84 :         _weights[ 0]   = 2.7777777777777777777777777777778e-02_R;
     210          84 :         _weights[ 1]   = 1.6549536156080552504633972002921e-01_R;
     211          84 :         _weights[ 2]   = 2.7453871250016173528070561857937e-01_R;
     212          84 :         _weights[ 3]   = 3.4642851097304634511513153213972e-01_R;
     213          84 :         _weights[ 4]   = 3.7151927437641723356009070294785e-01_R;
     214          84 :         _weights[ 5]   = _weights[3];
     215          84 :         _weights[ 6]   = _weights[2];
     216          84 :         _weights[ 7]   = _weights[1];
     217          84 :         _weights[ 8]   = _weights[0];
     218             : 
     219          84 :         return;
     220             :       }
     221             : 
     222             :       // 2*10-3 = 17
     223          42 :     case SIXTEENTH:
     224             :     case SEVENTEENTH:
     225             :       {
     226          42 :         _points.resize (10);
     227          42 :         _weights.resize(10);
     228             : 
     229          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     230          42 :         _points[ 1](0) = -9.1953390816645881382893266082234e-01_R;
     231          42 :         _points[ 2](0) = -7.3877386510550507500310617485983e-01_R;
     232          42 :         _points[ 3](0) = -4.7792494981044449566117509273126e-01_R;
     233          42 :         _points[ 4](0) = -1.6527895766638702462621976595817e-01_R;
     234          42 :         _points[ 5]    = -_points[4];
     235          42 :         _points[ 6]    = -_points[3];
     236          42 :         _points[ 7]    = -_points[2];
     237          42 :         _points[ 8]    = -_points[1];
     238          42 :         _points[ 9]    = -_points[0];
     239             : 
     240          42 :         _weights[ 0]   = 2.2222222222222222222222222222222e-02_R;
     241          42 :         _weights[ 1]   = 1.3330599085107011112622717075539e-01_R;
     242          42 :         _weights[ 2]   = 2.2488934206312645211945782173105e-01_R;
     243          42 :         _weights[ 3]   = 2.9204268367968375787558225737444e-01_R;
     244          42 :         _weights[ 4]   = 3.2753976118389745665651052791689e-01_R;
     245          42 :         _weights[ 5]   = _weights[4];
     246          42 :         _weights[ 6]   = _weights[3];
     247          42 :         _weights[ 7]   = _weights[2];
     248          42 :         _weights[ 8]   = _weights[1];
     249          42 :         _weights[ 9]   = _weights[0];
     250             : 
     251          42 :         return;
     252             :       }
     253             : 
     254             :       // 2*11-3 = 19
     255          42 :     case EIGHTTEENTH:
     256             :     case NINETEENTH:
     257             :       {
     258          42 :         _points.resize (11);
     259          42 :         _weights.resize(11);
     260             : 
     261          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     262          42 :         _points[ 1](0) = -9.3400143040805913433227413609938e-01_R;
     263          42 :         _points[ 2](0) = -7.8448347366314441862241781610846e-01_R;
     264          42 :         _points[ 3](0) = -5.6523532699620500647096396947775e-01_R;
     265          42 :         _points[ 4](0) = -2.9575813558693939143191151555906e-01_R;
     266          42 :         _points[ 5](0) = 0.;
     267          42 :         _points[ 6]    = -_points[4];
     268          42 :         _points[ 7]    = -_points[3];
     269          42 :         _points[ 8]    = -_points[2];
     270          42 :         _points[ 9]    = -_points[1];
     271          42 :         _points[10]    = -_points[0];
     272             : 
     273          42 :         _weights[ 0]   = 1.8181818181818181818181818181818e-02_R;
     274          42 :         _weights[ 1]   = 1.0961227326699486446140344958035e-01_R;
     275          42 :         _weights[ 2]   = 1.8716988178030520410814152189943e-01_R;
     276          42 :         _weights[ 3]   = 2.4804810426402831404008486642187e-01_R;
     277          42 :         _weights[ 4]   = 2.8687912477900808867922240333154e-01_R;
     278          42 :         _weights[ 5]   = 3.0021759545569069378593188116998e-01_R;
     279          42 :         _weights[ 6]   = _weights[4];
     280          42 :         _weights[ 7]   = _weights[3];
     281          42 :         _weights[ 8]   = _weights[2];
     282          42 :         _weights[ 9]   = _weights[1];
     283          42 :         _weights[10]   = _weights[0];
     284             : 
     285          42 :         return;
     286             :       }
     287             : 
     288             :       // 2*12-3 = 21
     289          42 :     case TWENTIETH:
     290             :     case TWENTYFIRST:
     291             :       {
     292          42 :         _points.resize (12);
     293          42 :         _weights.resize(12);
     294             : 
     295          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     296          42 :         _points[ 1](0) = -9.4489927222288222340758013830322e-01_R;
     297          42 :         _points[ 2](0) = -8.1927932164400667834864158171690e-01_R;
     298          42 :         _points[ 3](0) = -6.3287615303186067766240485444366e-01_R;
     299          42 :         _points[ 4](0) = -3.9953094096534893226434979156697e-01_R;
     300          42 :         _points[ 5](0) = -1.3655293285492755486406185573969e-01_R;
     301          42 :         _points[ 6]    = -_points[5];
     302          42 :         _points[ 7]    = -_points[4];
     303          42 :         _points[ 8]    = -_points[3];
     304          42 :         _points[ 9]    = -_points[2];
     305          42 :         _points[10]    = -_points[1];
     306          42 :         _points[11]    = -_points[0];
     307             : 
     308          42 :         _weights[ 0]   = 1.5151515151515151515151515151515e-02_R;
     309          42 :         _weights[ 1]   = 9.1684517413196130668342594134079e-02_R;
     310          42 :         _weights[ 2]   = 1.5797470556437011516467106270034e-01_R;
     311          42 :         _weights[ 3]   = 2.1250841776102114535830207736687e-01_R;
     312          42 :         _weights[ 4]   = 2.5127560319920128029324441214760e-01_R;
     313          42 :         _weights[ 5]   = 2.7140524091069617700028833849960e-01_R;
     314          42 :         _weights[ 6]   = _weights[5];
     315          42 :         _weights[ 7]   = _weights[4];
     316          42 :         _weights[ 8]   = _weights[3];
     317          42 :         _weights[ 9]   = _weights[2];
     318          42 :         _weights[10]   = _weights[1];
     319          42 :         _weights[11]   = _weights[0];
     320             : 
     321          42 :         return;
     322             :       }
     323             : 
     324             :       // 2*13-3 = 23
     325          42 :     case TWENTYSECOND:
     326             :     case TWENTYTHIRD:
     327             :       {
     328          42 :         _points.resize (13);
     329          42 :         _weights.resize(13);
     330             : 
     331          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     332          42 :         _points[ 1](0) = -9.5330984664216391189690546475545e-01_R;
     333          42 :         _points[ 2](0) = -8.4634756465187231686592560709875e-01_R;
     334          42 :         _points[ 3](0) = -6.8618846908175742607275903956636e-01_R;
     335          42 :         _points[ 4](0) = -4.8290982109133620174693723363693e-01_R;
     336          42 :         _points[ 5](0) = -2.4928693010623999256867370037423e-01_R;
     337          42 :         _points[ 6](0) = 0.;
     338          42 :         _points[ 7]    = -_points[5];
     339          42 :         _points[ 8]    = -_points[4];
     340          42 :         _points[ 9]    = -_points[3];
     341          42 :         _points[10]    = -_points[2];
     342          42 :         _points[11]    = -_points[1];
     343          42 :         _points[12]    = -_points[0];
     344             : 
     345          42 :         _weights[ 0]   = 1.2820512820512820512820512820513e-02_R;
     346          42 :         _weights[ 1]   = 7.7801686746818927793588988333134e-02_R;
     347          42 :         _weights[ 2]   = 1.3498192668960834911991476258937e-01_R;
     348          42 :         _weights[ 3]   = 1.8364686520355009200749425874681e-01_R;
     349          42 :         _weights[ 4]   = 2.2076779356611008608553400837940e-01_R;
     350          42 :         _weights[ 5]   = 2.4401579030667635645857814836016e-01_R;
     351          42 :         _weights[ 6]   = 2.5193084933344673604413864154124e-01_R;
     352          42 :         _weights[ 7]   = _weights[5];
     353          42 :         _weights[ 8]   = _weights[4];
     354          42 :         _weights[ 9]   = _weights[3];
     355          42 :         _weights[10]   = _weights[2];
     356          42 :         _weights[11]   = _weights[1];
     357          42 :         _weights[12]   = _weights[0];
     358             : 
     359          42 :         return;
     360             :       }
     361             : 
     362             :       // 2*14-3 = 25
     363          42 :     case TWENTYFOURTH:
     364             :     case TWENTYFIFTH:
     365             :       {
     366          42 :         _points.resize (14);
     367          42 :         _weights.resize(14);
     368             : 
     369          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     370          42 :         _points[ 1](0) = -9.5993504526726090135510016201542e-01_R;
     371          42 :         _points[ 2](0) = -8.6780105383034725100022020290826e-01_R;
     372          42 :         _points[ 3](0) = -7.2886859909132614058467240052088e-01_R;
     373          42 :         _points[ 4](0) = -5.5063940292864705531662270585908e-01_R;
     374          42 :         _points[ 5](0) = -3.4272401334271284504390340364167e-01_R;
     375          42 :         _points[ 6](0) = -1.1633186888370386765877670973616e-01_R;
     376          42 :         _points[ 7]    = -_points[6];
     377          42 :         _points[ 8]    = -_points[5];
     378          42 :         _points[ 9]    = -_points[4];
     379          42 :         _points[10]    = -_points[3];
     380          42 :         _points[11]    = -_points[2];
     381          42 :         _points[12]    = -_points[1];
     382          42 :         _points[13]    = -_points[0];
     383             : 
     384          42 :         _weights[ 0]   = 1.0989010989010989010989010989011e-02_R;
     385          42 :         _weights[ 1]   = 6.6837284497681284634070660746053e-02_R;
     386          42 :         _weights[ 2]   = 1.1658665589871165154099667065465e-01_R;
     387          42 :         _weights[ 3]   = 1.6002185176295214241282099798759e-01_R;
     388          42 :         _weights[ 4]   = 1.9482614937341611864033177837588e-01_R;
     389          42 :         _weights[ 5]   = 2.1912625300977075487116252395417e-01_R;
     390          42 :         _weights[ 6]   = 2.3161279446845705888962835729264e-01_R;
     391          42 :         _weights[ 7]   = _weights[6];
     392          42 :         _weights[ 8]   = _weights[5];
     393          42 :         _weights[ 9]   = _weights[4];
     394          42 :         _weights[10]   = _weights[3];
     395          42 :         _weights[11]   = _weights[2];
     396          42 :         _weights[12]   = _weights[1];
     397          42 :         _weights[13]   = _weights[0];
     398             : 
     399          42 :         return;
     400             :       }
     401             : 
     402             :       // 2*15-3 = 27
     403          42 :     case TWENTYSIXTH:
     404             :     case TWENTYSEVENTH:
     405             :       {
     406          42 :         _points.resize (15);
     407          42 :         _weights.resize(15);
     408             : 
     409          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     410          42 :         _points[ 1](0) = -9.6524592650383857279585139206960e-01_R;
     411          42 :         _points[ 2](0) = -8.8508204422297629882540163148223e-01_R;
     412          42 :         _points[ 3](0) = -7.6351968995181520070411847597629e-01_R;
     413          42 :         _points[ 4](0) = -6.0625320546984571112352993863673e-01_R;
     414          42 :         _points[ 5](0) = -4.2063805471367248092189693873858e-01_R;
     415          42 :         _points[ 6](0) = -2.1535395536379423822567944627292e-01_R;
     416          42 :         _points[ 7](0) = 0.;
     417          42 :         _points[ 8]    = -_points[6];
     418          42 :         _points[ 9]    = -_points[5];
     419          42 :         _points[10]    = -_points[4];
     420          42 :         _points[11]    = -_points[3];
     421          42 :         _points[12]    = -_points[2];
     422          42 :         _points[13]    = -_points[1];
     423          42 :         _points[14]    = -_points[0];
     424             : 
     425          42 :         _weights[ 0]   = 9.5238095238095238095238095238095e-03_R;
     426          42 :         _weights[ 1]   = 5.8029893028601249096880584025282e-02_R;
     427          42 :         _weights[ 2]   = 1.0166007032571806760366617078880e-01_R;
     428          42 :         _weights[ 3]   = 1.4051169980242810946044680564367e-01_R;
     429          42 :         _weights[ 4]   = 1.7278964725360094905207709940835e-01_R;
     430          42 :         _weights[ 5]   = 1.9698723596461335609250034650741e-01_R;
     431          42 :         _weights[ 6]   = 2.1197358592682092012743007697722e-01_R;
     432          42 :         _weights[ 7]   = 2.1704811634881564951495021425091e-01_R;
     433          42 :         _weights[ 8]   = _weights[6];
     434          42 :         _weights[ 9]   = _weights[5];
     435          42 :         _weights[10]   = _weights[4];
     436          42 :         _weights[11]   = _weights[3];
     437          42 :         _weights[12]   = _weights[2];
     438          42 :         _weights[13]   = _weights[1];
     439          42 :         _weights[14]   = _weights[0];
     440             : 
     441          42 :         return;
     442             :       }
     443             : 
     444             :       // 2*16-3 = 29
     445          42 :     case TWENTYEIGHTH:
     446             :     case TWENTYNINTH:
     447             :       {
     448          42 :         _points.resize (16);
     449          42 :         _weights.resize(16);
     450             : 
     451          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     452          42 :         _points[ 1](0) = -9.6956804627021793295224273836746e-01_R;
     453          42 :         _points[ 2](0) = -8.9920053309347209299462826151985e-01_R;
     454          42 :         _points[ 3](0) = -7.9200829186181506393108827096315e-01_R;
     455          42 :         _points[ 4](0) = -6.5238870288249308946788321964058e-01_R;
     456          42 :         _points[ 5](0) = -4.8605942188713761178189078584687e-01_R;
     457          42 :         _points[ 6](0) = -2.9983046890076320809835345472230e-01_R;
     458          42 :         _points[ 7](0) = -1.0132627352194944784303300504592e-01_R;
     459          42 :         _points[ 8]    = -_points[7];
     460          42 :         _points[ 9]    = -_points[6];
     461          42 :         _points[10]    = -_points[5];
     462          42 :         _points[11]    = -_points[4];
     463          42 :         _points[12]    = -_points[3];
     464          42 :         _points[13]    = -_points[2];
     465          42 :         _points[14]    = -_points[1];
     466          42 :         _points[15]    = -_points[0];
     467             : 
     468          42 :         _weights[ 0]   = 8.3333333333333333333333333333333e-03_R;
     469          42 :         _weights[ 1]   = 5.0850361005919905403244919565455e-02_R;
     470          42 :         _weights[ 2]   = 8.9393697325930800991052080166084e-02_R;
     471          42 :         _weights[ 3]   = 1.2425538213251409834953633265731e-01_R;
     472          42 :         _weights[ 4]   = 1.5402698080716428081564494048499e-01_R;
     473          42 :         _weights[ 5]   = 1.7749191339170412530107566952836e-01_R;
     474          42 :         _weights[ 6]   = 1.9369002382520358431691359885352e-01_R;
     475          42 :         _weights[ 7]   = 2.0195830817822987148919912541094e-01_R;
     476          42 :         _weights[ 8]   = _weights[7];
     477          42 :         _weights[ 9]   = _weights[6];
     478          42 :         _weights[10]   = _weights[5];
     479          42 :         _weights[11]   = _weights[4];
     480          42 :         _weights[12]   = _weights[3];
     481          42 :         _weights[13]   = _weights[2];
     482          42 :         _weights[14]   = _weights[1];
     483          42 :         _weights[15]   = _weights[0];
     484             : 
     485          42 :         return;
     486             :       }
     487             : 
     488             :       // 2*17-3 = 31
     489          42 :     case THIRTIETH:
     490             :     case THIRTYFIRST:
     491             :       {
     492          42 :         _points.resize (17);
     493          42 :         _weights.resize(17);
     494             : 
     495          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     496          42 :         _points[ 1](0) = -9.7313217663141831415697950187372e-01_R;
     497          42 :         _points[ 2](0) = -9.1087999591557359562380250639773e-01_R;
     498          42 :         _points[ 3](0) = -8.1569625122177030710675055323753e-01_R;
     499          42 :         _points[ 4](0) = -6.9102898062768470539491935737245e-01_R;
     500          42 :         _points[ 5](0) = -5.4138539933010153912373340750406e-01_R;
     501          42 :         _points[ 6](0) = -3.7217443356547704190723468073526e-01_R;
     502          42 :         _points[ 7](0) = -1.8951197351831738830426301475311e-01_R;
     503          42 :         _points[ 8](0) = 0.;
     504          42 :         _points[ 9]    = -_points[7];
     505          42 :         _points[10]    = -_points[6];
     506          42 :         _points[11]    = -_points[5];
     507          42 :         _points[12]    = -_points[4];
     508          42 :         _points[13]    = -_points[3];
     509          42 :         _points[14]    = -_points[2];
     510          42 :         _points[15]    = -_points[1];
     511          42 :         _points[16]    = -_points[0];
     512             : 
     513          42 :         _weights[ 0]   = 7.3529411764705882352941176470588e-03_R;
     514          42 :         _weights[ 1]   = 4.4921940543254209647400954623212e-02_R;
     515          42 :         _weights[ 2]   = 7.9198270503687119190264429952835e-02_R;
     516          42 :         _weights[ 3]   = 1.1059290900702816137577270522008e-01_R;
     517          42 :         _weights[ 4]   = 1.3798774620192655905620157495403e-01_R;
     518          42 :         _weights[ 5]   = 1.6039466199762153951632836586475e-01_R;
     519          42 :         _weights[ 6]   = 1.7700425351565787043694574536329e-01_R;
     520          42 :         _weights[ 7]   = 1.8721633967761923589208848286062e-01_R;
     521          42 :         _weights[ 8]   = 1.9066187475346943329940724702825e-01_R;
     522          42 :         _weights[ 9]   = _weights[7];
     523          42 :         _weights[10]   = _weights[6];
     524          42 :         _weights[11]   = _weights[5];
     525          42 :         _weights[12]   = _weights[4];
     526          42 :         _weights[13]   = _weights[3];
     527          42 :         _weights[14]   = _weights[2];
     528          42 :         _weights[15]   = _weights[1];
     529          42 :         _weights[16]   = _weights[0];
     530             : 
     531          42 :         return;
     532             :       }
     533             : 
     534             :       // 2*18-3 = 33
     535          42 :     case THIRTYSECOND:
     536             :     case THIRTYTHIRD:
     537             :       {
     538          42 :         _points.resize (18);
     539          42 :         _weights.resize(18);
     540             : 
     541          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     542          42 :         _points[ 1](0) = -9.7610555741219854286451892434170e-01_R;
     543          42 :         _points[ 2](0) = -9.2064918534753387383785462543128e-01_R;
     544          42 :         _points[ 3](0) = -8.3559353521809021371364636232794e-01_R;
     545          42 :         _points[ 4](0) = -7.2367932928324268130621036530207e-01_R;
     546          42 :         _points[ 5](0) = -5.8850483431866176117353589319356e-01_R;
     547          42 :         _points[ 6](0) = -4.3441503691212397534228713674067e-01_R;
     548          42 :         _points[ 7](0) = -2.6636265287828098416766533202560e-01_R;
     549          42 :         _points[ 8](0) = -8.9749093484652111022645010088562e-02_R;
     550          42 :         _points[ 9]    = -_points[8];
     551          42 :         _points[10]    = -_points[7];
     552          42 :         _points[11]    = -_points[6];
     553          42 :         _points[12]    = -_points[5];
     554          42 :         _points[13]    = -_points[4];
     555          42 :         _points[14]    = -_points[3];
     556          42 :         _points[15]    = -_points[2];
     557          42 :         _points[16]    = -_points[1];
     558          42 :         _points[17]    = -_points[0];
     559             : 
     560          42 :         _weights[ 0]   = 6.5359477124183006535947712418301e-03_R;
     561          42 :         _weights[ 1]   = 3.9970628810914066137599176410101e-02_R;
     562          42 :         _weights[ 2]   = 7.0637166885633664999222960167786e-02_R;
     563          42 :         _weights[ 3]   = 9.9016271717502802394423605318672e-02_R;
     564          42 :         _weights[ 4]   = 1.2421053313296710026339635889675e-01_R;
     565          42 :         _weights[ 5]   = 1.4541196157380226798300321049443e-01_R;
     566          42 :         _weights[ 6]   = 1.6193951723760248926432670670023e-01_R;
     567          42 :         _weights[ 7]   = 1.7326210948945622601061440382668e-01_R;
     568          42 :         _weights[ 8]   = 1.7901586343970308229381880694353e-01_R;
     569          42 :         _weights[ 9]   = _weights[8];
     570          42 :         _weights[10]   = _weights[7];
     571          42 :         _weights[11]   = _weights[6];
     572          42 :         _weights[12]   = _weights[5];
     573          42 :         _weights[13]   = _weights[4];
     574          42 :         _weights[14]   = _weights[3];
     575          42 :         _weights[15]   = _weights[2];
     576          42 :         _weights[16]   = _weights[1];
     577          42 :         _weights[17]   = _weights[0];
     578             : 
     579          42 :         return;
     580             :       }
     581             : 
     582             :       // 2*19-3 = 35
     583          42 :     case THIRTYFOURTH:
     584             :     case THIRTYFIFTH:
     585             :       {
     586          42 :         _points.resize (19);
     587          42 :         _weights.resize(19);
     588             : 
     589          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     590          42 :         _points[ 1](0) = -9.7861176622208009515263406311022e-01_R;
     591          42 :         _points[ 2](0) = -9.2890152815258624371794025879655e-01_R;
     592          42 :         _points[ 3](0) = -8.5246057779664609308595597004106e-01_R;
     593          42 :         _points[ 4](0) = -7.5149420255261301416363748963394e-01_R;
     594          42 :         _points[ 5](0) = -6.2890813726522049776683230622873e-01_R;
     595          42 :         _points[ 6](0) = -4.8822928568071350277790963762492e-01_R;
     596          42 :         _points[ 7](0) = -3.3350484782449861029850010384493e-01_R;
     597          42 :         _points[ 8](0) = -1.6918602340928157137515415344488e-01_R;
     598          42 :         _points[ 9](0) = 0.;
     599          42 :         _points[10]    = -_points[8];
     600          42 :         _points[11]    = -_points[7];
     601          42 :         _points[12]    = -_points[6];
     602          42 :         _points[13]    = -_points[5];
     603          42 :         _points[14]    = -_points[4];
     604          42 :         _points[15]    = -_points[3];
     605          42 :         _points[16]    = -_points[2];
     606          42 :         _points[17]    = -_points[1];
     607          42 :         _points[18]    = -_points[0];
     608             : 
     609          42 :         _weights[ 0]   = 5.8479532163742690058479532163743e-03_R;
     610          42 :         _weights[ 1]   = 3.5793365186176477115425569035122e-02_R;
     611          42 :         _weights[ 2]   = 6.3381891762629736851695690418317e-02_R;
     612          42 :         _weights[ 3]   = 8.9131757099207084448008790556153e-02_R;
     613          42 :         _weights[ 4]   = 1.1231534147730504407091001546378e-01_R;
     614          42 :         _weights[ 5]   = 1.3226728044875077692604673390973e-01_R;
     615          42 :         _weights[ 6]   = 1.4841394259593888500968064366841e-01_R;
     616          42 :         _weights[ 7]   = 1.6029092404406124197991096818359e-01_R;
     617          42 :         _weights[ 8]   = 1.6755658452714286727013727774026e-01_R;
     618          42 :         _weights[ 9]   = 1.7000191928482723464467271561652e-01_R;
     619          42 :         _weights[10]   = _weights[8];
     620          42 :         _weights[11]   = _weights[7];
     621          42 :         _weights[12]   = _weights[6];
     622          42 :         _weights[13]   = _weights[5];
     623          42 :         _weights[14]   = _weights[4];
     624          42 :         _weights[15]   = _weights[3];
     625          42 :         _weights[16]   = _weights[2];
     626          42 :         _weights[17]   = _weights[1];
     627          42 :         _weights[18]   = _weights[0];
     628             : 
     629          42 :         return;
     630             :       }
     631             : 
     632             :       // 2*20-3 = 37
     633          42 :     case THIRTYSIXTH:
     634             :     case THIRTYSEVENTH:
     635             :       {
     636          42 :         _points.resize (20);
     637          42 :         _weights.resize(20);
     638             : 
     639          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     640          42 :         _points[ 1](0) = -9.8074370489391417192544643858423e-01_R;
     641          42 :         _points[ 2](0) = -9.3593449881266543571618158493063e-01_R;
     642          42 :         _points[ 3](0) = -8.6687797808995014130984721461629e-01_R;
     643          42 :         _points[ 4](0) = -7.7536826095205587041431752759469e-01_R;
     644          42 :         _points[ 5](0) = -6.6377640229031128984640332297116e-01_R;
     645          42 :         _points[ 6](0) = -5.3499286403188626164813596182898e-01_R;
     646          42 :         _points[ 7](0) = -3.9235318371390929938647470381582e-01_R;
     647          42 :         _points[ 8](0) = -2.3955170592298649518240135692709e-01_R;
     648          42 :         _points[ 9](0) = -8.0545937238821837975944518159554e-02_R;
     649          42 :         _points[10]    = -_points[9];
     650          42 :         _points[11]    = -_points[8];
     651          42 :         _points[12]    = -_points[7];
     652          42 :         _points[13]    = -_points[6];
     653          42 :         _points[14]    = -_points[5];
     654          42 :         _points[15]    = -_points[4];
     655          42 :         _points[16]    = -_points[3];
     656          42 :         _points[17]    = -_points[2];
     657          42 :         _points[18]    = -_points[1];
     658          42 :         _points[19]    = -_points[0];
     659             : 
     660          42 :         _weights[ 0]   = 5.2631578947368421052631578947368e-03_R;
     661          42 :         _weights[ 1]   = 3.2237123188488941491605028117294e-02_R;
     662          42 :         _weights[ 2]   = 5.7181802127566826004753627173243e-02_R;
     663          42 :         _weights[ 3]   = 8.0631763996119603144776846113721e-02_R;
     664          42 :         _weights[ 4]   = 1.0199149969945081568378120573289e-01_R;
     665          42 :         _weights[ 5]   = 1.2070922762867472509942970500239e-01_R;
     666          42 :         _weights[ 6]   = 1.3630048235872418448978079298903e-01_R;
     667          42 :         _weights[ 7]   = 1.4836155407091682581471301373397e-01_R;
     668          42 :         _weights[ 8]   = 1.5658010264747548715816989679364e-01_R;
     669          42 :         _weights[ 9]   = 1.6074328638784574900772672644908e-01_R;
     670          42 :         _weights[10]   = _weights[9];
     671          42 :         _weights[11]   = _weights[8];
     672          42 :         _weights[12]   = _weights[7];
     673          42 :         _weights[13]   = _weights[6];
     674          42 :         _weights[14]   = _weights[5];
     675          42 :         _weights[15]   = _weights[4];
     676          42 :         _weights[16]   = _weights[3];
     677          42 :         _weights[17]   = _weights[2];
     678          42 :         _weights[18]   = _weights[1];
     679          42 :         _weights[19]   = _weights[0];
     680             : 
     681          42 :         return;
     682             :       }
     683             : 
     684             :       // 2*21-3 = 39
     685          42 :     case THIRTYEIGHTH:
     686             :     case THIRTYNINTH:
     687             :       {
     688          42 :         _points.resize (21);
     689          42 :         _weights.resize(21);
     690             : 
     691          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     692          42 :         _points[ 1](0) = -9.8257229660454802823448127655541e-01_R;
     693          42 :         _points[ 2](0) = -9.4197629695974553429610265066144e-01_R;
     694          42 :         _points[ 3](0) = -8.7929475532359046445115359630494e-01_R;
     695          42 :         _points[ 4](0) = -7.9600192607771240474431258966036e-01_R;
     696          42 :         _points[ 5](0) = -6.9405102606222323262731639319467e-01_R;
     697          42 :         _points[ 6](0) = -5.7583196026183068692702187033809e-01_R;
     698          42 :         _points[ 7](0) = -4.4411578327900210119451634960735e-01_R;
     699          42 :         _points[ 8](0) = -3.0198985650876488727535186785875e-01_R;
     700          42 :         _points[ 9](0) = -1.5278551580218546600635832848567e-01_R;
     701          42 :         _points[10](0) = 0.;
     702          42 :         _points[11]    = -_points[9];
     703          42 :         _points[12]    = -_points[8];
     704          42 :         _points[13]    = -_points[7];
     705          42 :         _points[14]    = -_points[6];
     706          42 :         _points[15]    = -_points[5];
     707          42 :         _points[16]    = -_points[4];
     708          42 :         _points[17]    = -_points[3];
     709          42 :         _points[18]    = -_points[2];
     710          42 :         _points[19]    = -_points[1];
     711          42 :         _points[20]    = -_points[0];
     712             : 
     713          42 :         _weights[ 0]   = 4.7619047619047619047619047619048e-03_R;
     714          42 :         _weights[ 1]   = 2.9184840098505458609458543613171e-02_R;
     715          42 :         _weights[ 2]   = 5.1843169000849625072722971852830e-02_R;
     716          42 :         _weights[ 3]   = 7.3273918185074144252547861041894e-02_R;
     717          42 :         _weights[ 4]   = 9.2985467957886065301137664149214e-02_R;
     718          42 :         _weights[ 5]   = 1.1051708321912333526700048678439e-01_R;
     719          42 :         _weights[ 6]   = 1.2545812119086894801515753570800e-01_R;
     720          42 :         _weights[ 7]   = 1.3745846286004134358089961741515e-01_R;
     721          42 :         _weights[ 8]   = 1.4623686244797745926727053063439e-01_R;
     722          42 :         _weights[ 9]   = 1.5158757511168138445325068150529e-01_R;
     723          42 :         _weights[10]   = 1.5338519033217494855158440506754e-01_R;
     724          42 :         _weights[11]   = _weights[9];
     725          42 :         _weights[12]   = _weights[8];
     726          42 :         _weights[13]   = _weights[7];
     727          42 :         _weights[14]   = _weights[6];
     728          42 :         _weights[15]   = _weights[5];
     729          42 :         _weights[16]   = _weights[4];
     730          42 :         _weights[17]   = _weights[3];
     731          42 :         _weights[18]   = _weights[2];
     732          42 :         _weights[19]   = _weights[1];
     733          42 :         _weights[20]   = _weights[0];
     734             : 
     735          42 :         return;
     736             :       }
     737             : 
     738             :       // 2*22-3 = 41
     739          42 :     case FORTIETH:
     740             :     case FORTYFIRST:
     741             :       {
     742          42 :         _points.resize (22);
     743          42 :         _weights.resize(22);
     744             : 
     745          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     746          42 :         _points[ 1](0) = -9.8415243845764617655228962221207e-01_R;
     747          42 :         _points[ 2](0) = -9.4720428399922868052421376661573e-01_R;
     748          42 :         _points[ 3](0) = -8.9006229019090447052965782577909e-01_R;
     749          42 :         _points[ 4](0) = -8.1394892761192113604544184805614e-01_R;
     750          42 :         _points[ 5](0) = -7.2048723996120215811988189639847e-01_R;
     751          42 :         _points[ 6](0) = -6.1166943828425897122621160586993e-01_R;
     752          42 :         _points[ 7](0) = -4.8981487518990234980875123568327e-01_R;
     753          42 :         _points[ 8](0) = -3.5752071013891953806095728024018e-01_R;
     754          42 :         _points[ 9](0) = -2.1760658515928504178795509346539e-01_R;
     755          42 :         _points[10](0) = -7.3054540010898334761088790464107e-02_R;
     756          42 :         _points[11]    = -_points[10];
     757          42 :         _points[12]    = -_points[9];
     758          42 :         _points[13]    = -_points[8];
     759          42 :         _points[14]    = -_points[7];
     760          42 :         _points[15]    = -_points[6];
     761          42 :         _points[16]    = -_points[5];
     762          42 :         _points[17]    = -_points[4];
     763          42 :         _points[18]    = -_points[3];
     764          42 :         _points[19]    = -_points[2];
     765          42 :         _points[20]    = -_points[1];
     766          42 :         _points[21]    = -_points[0];
     767             : 
     768          42 :         _weights[ 0]   = 4.3290043290043290043290043290043e-03_R;
     769          42 :         _weights[ 1]   = 2.6545747682501757911627904520543e-02_R;
     770          42 :         _weights[ 2]   = 4.7214465293740752123775734864792e-02_R;
     771          42 :         _weights[ 3]   = 6.6865605864553076012404194157097e-02_R;
     772          42 :         _weights[ 4]   = 8.5090060391838447815711236095748e-02_R;
     773          42 :         _weights[ 5]   = 1.0150057480164767437243730374960e-01_R;
     774          42 :         _weights[ 6]   = 1.1574764465393906659003636772146e-01_R;
     775          42 :         _weights[ 7]   = 1.2752769665343027553084445930883e-01_R;
     776          42 :         _weights[ 8]   = 1.3658968861374142668617736220617e-01_R;
     777          42 :         _weights[ 9]   = 1.4274049227136140033623599356679e-01_R;
     778          42 :         _weights[10]   = 1.4584901944424179361642043947997e-01_R;
     779          42 :         _weights[11]   = _weights[10];
     780          42 :         _weights[12]   = _weights[9];
     781          42 :         _weights[13]   = _weights[8];
     782          42 :         _weights[14]   = _weights[7];
     783          42 :         _weights[15]   = _weights[6];
     784          42 :         _weights[16]   = _weights[5];
     785          42 :         _weights[17]   = _weights[4];
     786          42 :         _weights[18]   = _weights[3];
     787          42 :         _weights[19]   = _weights[2];
     788          42 :         _weights[20]   = _weights[1];
     789          42 :         _weights[21]   = _weights[0];
     790             : 
     791          42 :         return;
     792             :       }
     793             : 
     794             :       // 2*23-3 = 43
     795          42 :     case FORTYSECOND:
     796             :     case FORTYTHIRD:
     797             :       {
     798          42 :         _points.resize (23);
     799          42 :         _weights.resize(23);
     800             : 
     801          42 :         _points[ 0](0) = -1.0000000000000000000000000000000e+00_R;
     802          42 :         _points[ 1](0) = -9.8552715587873257808146276673810e-01_R;
     803          42 :         _points[ 2](0) = -9.5175795571071020413563967985143e-01_R;
     804          42 :         _points[ 3](0) = -8.9945855804034501095016032034737e-01_R;
     805          42 :         _points[ 4](0) = -8.2965109665128588622320061929000e-01_R;
     806          42 :         _points[ 5](0) = -7.4369504117206068394516354306700e-01_R;
     807          42 :         _points[ 6](0) = -6.4326364446013620847614553360277e-01_R;
     808          42 :         _points[ 7](0) = -5.3031177113684416813011532015230e-01_R;
     809          42 :         _points[ 8](0) = -4.0703793791447482919595048821510e-01_R;
     810          42 :         _points[ 9](0) = -2.7584154894579306710687763267914e-01_R;
     811          42 :         _points[10](0) = -1.3927620404066839859186261298277e-01_R;
     812          42 :         _points[11](0) = 0.;
     813          42 :         _points[12]    = -_points[10];
     814          42 :         _points[13]    = -_points[9];
     815          42 :         _points[14]    = -_points[8];
     816          42 :         _points[15]    = -_points[7];
     817          42 :         _points[16]    = -_points[6];
     818          42 :         _points[17]    = -_points[5];
     819          42 :         _points[18]    = -_points[4];
     820          42 :         _points[19]    = -_points[3];
     821          42 :         _points[20]    = -_points[2];
     822          42 :         _points[21]    = -_points[1];
     823          42 :         _points[22]    = -_points[0];
     824             : 
     825          42 :         _weights[ 0]   = 3.9525691699604743083003952569170e-03_R;
     826          42 :         _weights[ 1]   = 2.4248600771531736517399658937097e-02_R;
     827          42 :         _weights[ 2]   = 4.3175871170241834748876465612042e-02_R;
     828          42 :         _weights[ 3]   = 6.1252477129554206381382847440355e-02_R;
     829          42 :         _weights[ 4]   = 7.8135449475569989741934255347965e-02_R;
     830          42 :         _weights[ 5]   = 9.3497246163512341833500706906697e-02_R;
     831          42 :         _weights[ 6]   = 1.0703910172433651153518362791547e-01_R;
     832          42 :         _weights[ 7]   = 1.1849751066274913130212600472426e-01_R;
     833          42 :         _weights[ 8]   = 1.2764947470175887663614855305567e-01_R;
     834          42 :         _weights[ 9]   = 1.3431687263860381990156489770071e-01_R;
     835          42 :         _weights[10]   = 1.3836993638580739452350273386294e-01_R;
     836          42 :         _weights[11]   = 1.3972978001274736514015970647975e-01_R;
     837          42 :         _weights[12]   = _weights[10];
     838          42 :         _weights[13]   = _weights[9];
     839          42 :         _weights[14]   = _weights[8];
     840          42 :         _weights[15]   = _weights[7];
     841          42 :         _weights[16]   = _weights[6];
     842          42 :         _weights[17]   = _weights[5];
     843          42 :         _weights[18]   = _weights[4];
     844          42 :         _weights[19]   = _weights[3];
     845          42 :         _weights[20]   = _weights[2];
     846          42 :         _weights[21]   = _weights[1];
     847          42 :         _weights[22]   = _weights[0];
     848             : 
     849          42 :         return;
     850             :       }
     851             : 
     852           0 :     default:
     853           0 :       libmesh_error_msg("Quadrature rule " << _order << " not supported!");
     854             :     }
     855             : }
     856             : 
     857             : } // namespace libMesh