/usr/include/trilinos/Intrepid_CellTools.hpp is in libtrilinos-intrepid-dev 12.12.1-5.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 | #ifndef INTREPID_CELTOOLS_HPP
#define INTREPID_CELTOOLS_HPP
// @HEADER
// ************************************************************************
//
// Intrepid Package
// Copyright (2007) Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
// license for use of this work by or on behalf of the U.S. Government.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// 3. Neither the name of the Corporation nor the names of the
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
// PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
// LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Questions? Contact Pavel Bochev (pbboche@sandia.gov)
// Denis Ridzal (dridzal@sandia.gov), or
// Kara Peterson (kjpeter@sandia.gov)
//
// ************************************************************************
// @HEADER
/** \file Intrepid_CellTools.hpp
\brief Header file for the Intrepid::CellTools class.
\author Created by P. Bochev and D. Ridzal.
*/
#ifndef INTREPID_CELLTOOLS_HPP
#define INTREPID_CELLTOOLS_HPP
#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_ConfigDefs.hpp"
#include "Intrepid_Types.hpp"
#include "Intrepid_Utils.hpp"
#include "Intrepid_Basis.hpp"
#include "Intrepid_HGRAD_TRI_C1_FEM.hpp"
#include "Intrepid_HGRAD_QUAD_C1_FEM.hpp"
#include "Intrepid_HGRAD_TET_C1_FEM.hpp"
#include "Intrepid_HGRAD_WEDGE_C1_FEM.hpp"
#include "Intrepid_HGRAD_PYR_C1_FEM.hpp"
#include "Intrepid_HGRAD_HEX_C1_FEM.hpp"
#include "Intrepid_HGRAD_LINE_C1_FEM.hpp"
#include "Intrepid_HGRAD_TRI_C2_FEM.hpp"
#include "Intrepid_HGRAD_QUAD_C2_FEM.hpp"
#include "Intrepid_HGRAD_TET_C2_FEM.hpp"
#include "Intrepid_HGRAD_TET_COMP12_FEM.hpp"
#include "Intrepid_HGRAD_WEDGE_C2_FEM.hpp"
#include "Intrepid_HGRAD_WEDGE_I2_FEM.hpp"
#include "Intrepid_HGRAD_HEX_C2_FEM.hpp"
#include "Intrepid_HGRAD_HEX_I2_FEM.hpp"
#include "Intrepid_HGRAD_PYR_I2_FEM.hpp"
#include "Shards_CellTopology.hpp"
#include "Shards_BasicTopologies.hpp"
#include "Teuchos_Assert.hpp"
#include "Teuchos_RCP.hpp"
#include <Intrepid_KokkosRank.hpp>
#ifdef HAVE_INTREPID_KOKKOSCORE
#include "Kokkos_Core.hpp"
#endif
namespace Intrepid {
//nn
//============================================================================================//
// //
// CellTools //
// //
//============================================================================================//
/** \class Intrepid::CellTools
\brief A stateless class for operations on cell data. Provides methods for:
\li computing Jacobians of reference-to-physical frame mappings, their inverses and determinants
\li application of the reference-to-physical frame mapping and its inverse
\li parametrizations of edges and faces of reference cells needed for edge and face integrals,
\li computation of edge and face tangents and face normals on both reference and physical frames
\li inclusion tests for point sets in reference and physical cells.
*/
template<class Scalar>
class CellTools {
private:
//============================================================================================//
// //
// Parametrization coefficients of edges and faces of reference cells //
// //
//============================================================================================//
/** \brief Returns array with the coefficients of the parametrization maps for the edges or faces
of a reference cell topology.
See CellTools<Scalar>::setSubcellParametrization and Section \ref sec_cell_topology_subcell_map
more information about parametrization maps.
\param subcellDim [in] - dimension of subcells whose parametrization map is returned
\param parentCell [in] - topology of the reference cell owning the subcells
\return FieldContainer<double> with the coefficients of the parametrization map for all subcells
of the specified dimension.
*/
static const FieldContainer<double>& getSubcellParametrization(const int subcellDim,
const shards::CellTopology& parentCell);
/** \brief Defines orientation-preserving parametrizations of reference edges and faces of cell
topologies with reference cells.
Given an edge {V0, V1} of some reference cell, its parametrization is a mapping from
[-1,1] onto the edge. Parametrization of a triangular face {V0,V1,V2} is mapping from
the standard 2-simplex {(0,0,0), (1,0,0), (0,1,0)}, embedded in 3D onto that face.
Parametrization of a quadrilateral face {V0,V1,V2,V3} is mapping from the standard
2-cube {(-1,-1,0),(1,-1,0),(1,1,0),(-1,1,0)}, embedded in 3D, onto that face.
This method computes the coefficients of edge and face parametrization maps and stores
them in static arrays owned by CellTools<Scalar>::getSubcellParametrization method.
All mappings are affine and orientation-preserving, i.e., they preserve the tangent
and normal directions implied by the vertex order of the edge or the face relative to
the reference cell:
\li the tangent on [-1,1] from -1 in the direction of 1 is mapped to a tangent on edge {V0,V1}
from V0 in the direction of V1 (the forward direction of the edge determined by its
start and end vertices)
\li the normal in the direction of (0,0,1) to the standard 2-simplex {(0,0,0),(1,0,0),(0,1,0)}
and the standard 2-cube {(-1,-1,0),(1,-1,0),(1,1,0),(-1,1,0)} is mapped to a normal
on {V0,V1,V2} and {V0,V1,V2,V3}, determined according to the right-hand rule
(see http://mathworld.wolfram.com/Right-HandRule.html for definition of right-hand rule
and Section \ref Section sec_cell_topology_subcell_map for further details).
Because faces of all reference cells supported in Intrepid are affine images of either
the standard 2-simplex or the standard 2-cube, the coordinate functions of the respective
parmetrization maps are linear polynomials in the parameter variables (u,v), i.e., they
are of the form \c F_i(u,v)=C_0(i)+C_1(i)u+C_2(i)v; \c 0<=i<3 (face parametrizations
are supported only for 3D cells, thus parametrization maps have 3 coordinate functions).
As a result, application of these maps is independent of the face type which is convenient
for cells such as Wedge or Pyramid that have both types of faces. Also, coefficients of
coordinate functions for all faces can be stored together in the same array.
\param subcellParametrization [out] - array with the coefficients of the parametrization map
\param subcellDim [in] - dimension of the subcells being parametrized (1 or 2)
\param parentCell [in] - topology of the parent cell owning the subcells.
*/
static void setSubcellParametrization(FieldContainer<double>& subcellParametrization,
const int subcellDim,
const shards::CellTopology& parentCell);
//============================================================================================//
// //
// Validation of input/output arguments for CellTools methods //
// //
//============================================================================================//
/** \brief Validates arguments to Intrepid::CellTools::setJacobian
\param jacobian [in] - rank-4 (C,P,D,D) array or rank-3 (P,D,D) array required
\param points [in] - rank-2 (P,D) or rank-3 (C,P,D) array required
\param cellWorkset [in] - rank-3 (C,N,D) array required
\param whichCell [in] - default = -1 or 0 <= whichCell < C required
\param cellTopo [in] - cell topology with a reference cell required
*/
template<class ArrayJac, class ArrayPoint, class ArrayCell>
static void validateArguments_setJacobian(const ArrayJac & jacobian,
const ArrayPoint & points,
const ArrayCell & cellWorkset,
const int & whichCell,
const shards::CellTopology & cellTopo);
/** \brief Validates arguments to Intrepid::CellTools::setJacobianInv
\param jacobianInv [in] - rank and dimensions must match jacobian array
\param jacobian [in] - rank-4 (C,P,D,D) array or rank-3 (P,D,D) array required
*/
template<class ArrayJacInv, class ArrayJac>
static void validateArguments_setJacobianInv(const ArrayJacInv & jacobianInv,
const ArrayJac & jacobian);
/** \brief Validates arguments to Intrepid::CellTools::setJacobianDet
\param jacobianDet [in] - rank = (jacobian rank - 2) required
\param jacobian [in] - rank-4 (C,P,D,D) array or rank-3 (P,D,D) array required
*/
template<class ArrayJacDet, class ArrayJac>
static void validateArguments_setJacobianDetArgs(const ArrayJacDet & jacobianDet,
const ArrayJac & jacobian);
/** \brief Validates arguments to Intrepid::CellTools::mapToPhysicalFrame
\param physPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param refPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param cellWorkset [in] - rank-3 (C,N,D) array required
\param whichCell [in] - default = -1 or 0 <= whichCell < C required
\param cellTopo [in] - cell topology with a reference cell required
*/
template<class ArrayPhysPoint, class ArrayRefPoint, class ArrayCell>
static void validateArguments_mapToPhysicalFrame(const ArrayPhysPoint & physPoints,
const ArrayRefPoint & refPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int& whichCell);
/** \brief Validates arguments to Intrepid::CellTools::mapToReferenceFrame with default initial guess.
\param physPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param refPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param cellWorkset [in] - rank-3 (C,N,D) array required
\param whichCell [in] - default = -1 or 0 <= whichCell < C required
\param cellTopo [in] - cell topology with a reference cell required
*/
template<class ArrayRefPoint, class ArrayPhysPoint, class ArrayCell>
static void validateArguments_mapToReferenceFrame(const ArrayRefPoint & refPoints,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int& whichCell);
/** \brief Validates arguments to Intrepid::CellTools::mapToReferenceFrame with user-defined initial guess.
\param physPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param initGuess [in] - rank and dimensions must match those of physPoints
\param refPoints [in] - rank-3 (C,P,D) array or rank-2 (P,D) array required
\param cellWorkset [in] - rank-3 (C,N,D) array required
\param whichCell [in] - default = -1 or 0 <= whichCell < C required
\param cellTopo [in] - cell topology with a reference cell required
*/
template<class ArrayRefPoint, class ArrayInitGuess, class ArrayPhysPoint, class ArrayCell>
static void validateArguments_mapToReferenceFrame(const ArrayRefPoint & refPoints,
const ArrayInitGuess & initGuess,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int& whichCell);
/** \brief Validates arguments to Intrepid::CellTools::checkPointwiseInclusion
\param inCell [out] - rank-1 (P) array required
\param physPoints [in] - rank-2 (P,D) array required
\param cellWorkset [in] - rank-3 (C,N,D) array required
\param whichCell [in] - 0 <= whichCell < C required
\param cellTopo [in] - cell topology with a reference cell required
*/
template<class ArrayIncl, class ArrayPoint, class ArrayCell>
static void validateArguments_checkPointwiseInclusion(ArrayIncl & inCell,
const ArrayPoint & physPoints,
const ArrayCell & cellWorkset,
const int & whichCell,
const shards::CellTopology & cell);
public:
/** \brief Default constructor.
*/
CellTools(){ };
/** \brief Destructor
*/
~CellTools(){ };
//============================================================================================//
// //
// Jacobian, inverse Jacobian and Jacobian determinant //
// //
//============================================================================================//
/** \brief Computes the Jacobian matrix \e DF of the reference-to-physical frame map \e F.
There are three use cases:
\li Computes Jacobians \f$DF_{c}\f$ of the reference-to-physical map for a \b specified physical
cell \f${\mathcal C}\f$ from a cell workset on a \b single set of reference points stored
in a rank-2 (P,D) array;
\li Computes Jacobians \f$DF_{c}\f$ of the reference-to-physical map for \b all physical cells
in a cell workset on a \b single set of reference points stored in a rank-2 (P,D) array;
\li Computes Jacobians \f$DF_{c}\f$ of the reference-to-physical map for \b all physical cells
in a cell workset on \b multiple reference point sets having the same number of points,
indexed by cell ordinal, and stored in a rank-3 (C,P,D) array;
For a single point set in a rank-2 array (P,D) and \c whichCell set to a valid cell ordinal
relative to \c cellWorkset returns a rank-3 (P,D,D) array such that
\f[
\mbox{jacobian}(p,i,j) = [DF_{c}(\mbox{points}(p))]_{ij} \quad \mbox{for $0\le c < C$ - fixed}
\f]
For a single point set in a rank-2 array (P,D) and \c whichCell=-1 (default value) returns
a rank-4 (C,P,D,D) array such that
\f[
\mbox{jacobian}(c,p,i,j) = [DF_{c}(\mbox{points}(p))]_{ij} \quad c=0,\ldots, C
\f]
For multiple sets of reference points in a rank-3 (C,P,D) array returns
rank-4 (C,P,D,D) array such that
\f[
\mbox{jacobian}(c,p,i,j) = [DF_{c}(\mbox{points}(c,p))]_{ij} \quad c=0,\ldots, C
\f]
This setting requires the default value \c whichCell=-1.
Requires cell topology with a reference cell. See Section \ref sec_cell_topology_ref_map_DF
for definition of the Jacobian.
The default \c whichCell = -1 forces computation of all cell Jacobians and
requiers rank-4 output array. Computation of single cell Jacobians is forced by
selecting a valid cell ordinal \c whichCell and requires rank-3 output array.
\warning
The points are not required to be in the reference cell associated with the specified
cell topology. CellTools provides several inclusion tests methods to check whether
or not the points are inside a reference cell.
\param jacobian [out] - rank-4/3 array with dimensions (C,P,D,D)/(P,D,D) with the Jacobians
\param points [in] - rank-2/3 array with dimensions (P,D)/(C,P,D) with the evaluation points
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param whichCell [in] - cell ordinal (for single cell Jacobian computation); default is -1
*/
/*
#ifdef HAVE_INTREPID_KOKKOSCORE
template<class ArrayJac, class ArrayPoint, class ArrayCell, bool typecheck>
struct setJacobianTempSpecKokkos;
template<class Scalar1,class Scalar2,class Scalar3,class Layout,class MemorySpace>
static void setJacobianTemp(Kokkos::View<Scalar1,Layout,MemorySpace> & jacobian,
const Kokkos::View<Scalar2,Layout,MemorySpace> & points,
const Kokkos::View<Scalar3,Layout,MemorySpace> & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
#endif */
template<class ArrayJac, class ArrayPoint, class ArrayCell, bool typecheck>
struct setJacobianTempSpec;
/*template<class ArrayJac, class ArrayPoint, class ArrayCell>
static void setJacobianTemp(ArrayJac & jacobian,
const ArrayPoint & points,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);*/
template<class ArrayJac, class ArrayPoint, class ArrayCell>
static void setJacobian(ArrayJac & jacobian,
const ArrayPoint & points,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
template<class ArrayJac, class ArrayPoint, class ArrayCell>
static void setJacobian(ArrayJac & jacobian,
const ArrayPoint & points,
const ArrayCell & cellWorkset,
const Teuchos::RCP< Basis< Scalar, FieldContainer<Scalar> > > HGRAD_Basis,
const int & whichCell = -1);
/** \brief Computes the inverse of the Jacobian matrix \e DF of the reference-to-physical frame map \e F.
Returns rank-4 or rank-3 array with dimensions (C,P,D,D) or (P,D,D) such that
\f[
\mbox{jacobianInv}(c,p,*,*) = \mbox{jacobian}^{-1}(c,p,*,*) \quad c = 0,\ldots, C
\quad\mbox{or}\quad
\mbox{jacobianInv}(p,*,*) = \mbox{jacobian}^{-1}(p,*,*)
\f]
\param jacobianInv [out] - rank-4/3 array with dimensions (C,P,D,D)/(P,D,D) with the inverse Jacobians
\param jacobian [in] - rank-4/3 array with dimensions (C,P,D,D)/(P,D,D) with the Jacobians
*/
template<class ArrayJacInv, class ArrayJac>
static void setJacobianInv(ArrayJacInv & jacobianInv,
const ArrayJac & jacobian);
/* template<class ArrayJacInv, class ArrayJac>
static void setJacobianInvTemp(ArrayJacInv & jacobianInv,
const ArrayJac & jacobian);
*/
/** \brief Computes the determinant of the Jacobian matrix \e DF of the reference-to-physical frame map \e F.
Returns rank-2 or rank-1 array with dimensions (C,P)/(P) such that
\f[
\mbox{jacobianDet}(c,p) = \mbox{det}(\mbox{jacobian}(c,p,*,*)) \quad c=0,\ldots, C
\quad\mbox{or}\quad
\mbox{jacobianDet}(p) = \mbox{det}(\mbox{jacobian}(p,*,*))
\f]
\param jacobianDet [out] - rank-2/1 array with dimensions (C,P)/(P) with Jacobian determinants
\param jacobian [in] - rank-4/3 array with dimensions (C,P,D,D)/(P,D,D) with the Jacobians
*/
template<class ArrayJacDet, class ArrayJac>
static void setJacobianDet(ArrayJacDet & jacobianDet,
const ArrayJac & jacobian);
/* template<class ArrayJacDet, class ArrayJac>
static void setJacobianDetTemp(ArrayJacDet & jacobianDet,
const ArrayJac & jacobian);*/
//============================================================================================//
// //
// Reference-to-physical frame mapping and its inverse //
// //
//============================================================================================//
/** \brief Computes \e F, the reference-to-physical frame map.
There are 3 use cases:
\li Applies \f$ F_{c} \f$ for a \b specified physical cell \f${\mathcal C}\f$ from a cell
workset to a \b single point set stored in a rank-2 (P,D) array;
\li Applies \f$ F_{c} \f$ for \b all cells in a cell workset to a \b single point set stored
in a rank-2 (P,D) array;
\li Applies \f$ F_{c} \f$ for \b all cells in a cell workset to \b multiple point sets having
the same number of points, indexed by cell ordinal, and stored in a rank-3 (C,P,D) array;
For a single point set in a rank-2 array (P,D) and \c whichCell set to a valid cell ordinal
relative to \c cellWorkset returns a rank-2 (P,D) array such that
\f[
\mbox{physPoints}(p,d) = \Big(F_c(\mbox{refPoint}(p,*)) \Big)_d \quad \mbox{for $0\le c < C$ - fixed}
\f]
For a single point set in a rank-2 array (P,D) and \c whichCell=-1 (default value) returns
a rank-3 (C,P,D) array such that
\f[
\mbox{physPoints}(c,p,d) = \Big(F_c(\mbox{refPoint}(p,*)) \Big)_d \quad c=0,\ldots, C
\f]
For multiple point sets in a rank-3 (C,P,D) array returns a rank-3 (C,P,D) array such that
\f[
\mbox{physPoints}(c,p,d) = \Big(F_c(\mbox{refPoint}(c,p,*)) \Big)_d \quad c=0,\ldots, C
\f]
This corresponds to mapping multiple sets of reference points to a matching number of
physical cells and requires the default value \c whichCell=-1.
Requires cell topology with a reference cell. See Section \ref sec_cell_topology_ref_map
for definition of the mapping function. Presently supported cell topologies are
\li 1D: \c Line<2>
\li 2D: \c Triangle<3>, \c Triangle<6>, \c Quadrilateral<4>, \c Quadrilateral<9>
\li 3D: \c Tetrahedron<4>, \c Tetrahedron<10>, \c Hexahedron<8>, \c Hexahedron<27>
The default \c whichCell = -1 requires rank-3 output array and
forces application of all reference-to-physical frame mappings corresponding to the
cells stored in \c cellWorkset. Application of a single mapping is forced by selecting
a valid cell ordinal \c whichCell and requires rank-2 input/output point arrays.
\warning
The array \c refPoints represents an arbitrary set of points in the reference
frame that are not required to be in the reference cell corresponding to the
specified cell topology. As a result, the images of these points under a given
reference-to-physical map are not necessarily contained in the physical cell that
is the image of the reference cell under that map. CellTools provides several
inclusion tests methods to check whether or not the points are inside a reference cell.
\param physPoints [out] - rank-3/2 array with dimensions (C,P,D)/(P,D) with the images of the ref. points
\param refPoints [in] - rank-3/2 array with dimensions (C,P,D)/(P,D) with points in reference frame
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param whichCell [in] - ordinal of the cell that defines the reference-to-physical map; default is -1
\todo Implement method for non-standard (shell, beam, etc) topologies.
*/
template<class ArrayPhysPoint, class ArrayRefPoint, class ArrayCell>
static void mapToPhysicalFrame(ArrayPhysPoint & physPoints,
const ArrayRefPoint & refPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
template<class ArrayPhysPoint, class ArrayRefPoint, class ArrayCell>
static void mapToPhysicalFrame(ArrayPhysPoint & physPoints,
const ArrayRefPoint & refPoints,
const ArrayCell & cellWorkset,
const Teuchos::RCP< Basis< Scalar, FieldContainer<Scalar> > > HGRAD_Basis,
const int & whichCell = -1);
/** \brief Computes \e F, the reference-to-physical frame map.
There are 3 use cases:
\li Applies \f$ F_{c} \f$ for a \b specified physical cell \f${\mathcal C}\f$ from a cell
workset to a \b single point set stored in a rank-2 (P,D) array;
\li Applies \f$ F_{c} \f$ for \b all cells in a cell workset to a \b single point set stored
in a rank-2 (P,D) array;
\li Applies \f$ F_{c} \f$ for \b all cells in a cell workset to \b multiple point sets having
the same number of points, indexed by cell ordinal, and stored in a rank-3 (C,P,D) array;
For a single point set in a rank-2 array (P,D) and \c whichCell set to a valid cell ordinal
relative to \c cellWorkset returns a rank-2 (P,D) array such that
\f[
\mbox{physPoints}(p,d) = \Big(F_c(\mbox{refPoint}(p,*)) \Big)_d \quad \mbox{for $0\le c < C$ - fixed}
\f]
For a single point set in a rank-2 array (P,D) and \c whichCell=-1 (default value) returns
a rank-3 (C,P,D) array such that
\f[
\mbox{physPoints}(c,p,d) = \Big(F_c(\mbox{refPoint}(p,*)) \Big)_d \quad c=0,\ldots, C
\f]
For multiple point sets in a rank-3 (C,P,D) array returns a rank-3 (C,P,D) array such that
\f[
\mbox{physPoints}(c,p,d) = \Big(F_c(\mbox{refPoint}(c,p,*)) \Big)_d \quad c=0,\ldots, C
\f]
This corresponds to mapping multiple sets of reference points to a matching number of
physical cells and requires the default value \c whichCell=-1.
Requires cell topology with a reference cell. See Section \ref sec_cell_topology_ref_map
for definition of the mapping function. Presently supported cell topologies are
\li 1D: \c Line<2>
\li 2D: \c Triangle<3>, \c Triangle<6>, \c Quadrilateral<4>, \c Quadrilateral<9>
\li 3D: \c Tetrahedron<4>, \c Tetrahedron<10>, \c Hexahedron<8>, \c Hexahedron<27>
The default \c whichCell = -1 requires rank-3 output array and
forces application of all reference-to-physical frame mappings corresponding to the
cells stored in \c cellWorkset. Application of a single mapping is forced by selecting
a valid cell ordinal \c whichCell and requires rank-2 input/output point arrays.
\warning
The array \c refPoints represents an arbitrary set of points in the reference
frame that are not required to be in the reference cell corresponding to the
specified cell topology. As a result, the images of these points under a given
reference-to-physical map are not necessarily contained in the physical cell that
is the image of the reference cell under that map. CellTools provides several
inclusion tests methods to check whether or not the points are inside a reference cell.
\param physPoints [out] - rank-3/2 array with dimensions (C,P,D)/(P,D) with the images of the ref. points
\param refPoints [in] - rank-3/2 array with dimensions (C,P,D)/(P,D) with points in reference frame
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param whichCell [in] - ordinal of the cell that defines the reference-to-physical map; default is -1
\todo Implement method for non-standard (shell, beam, etc) topologies.
*/
/* template<class ArrayPhysPoint, class ArrayRefPoint, class ArrayCell>
static void mapToPhysicalFrameTemp(ArrayPhysPoint & physPoints,
const ArrayRefPoint & refPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
*/
template<class ArrayPhysPoint, class ArrayRefPoint, class ArrayCell, int refRank,int phyptsrank>
struct mapToPhysicalFrameTempSpec;
/** \brief Computes \f$ F^{-1}_{c} \f$, the inverse of the reference-to-physical frame map
using a default initial guess.
There are two use cases:
\li Applies \f$ F^{-1}_{c} \f$ for a \b specified physical cell \f${\mathcal C}\f$ from a
cell workset to a \b single set of points stored in a rank-2 (P,D) array;
\li Applies \f$ F^{-1}_{c} \f$ for \b all cells in a cell workset to \b multiple point sets
having the same number of points, indexed by cell ordinal, and stored in a rank-3
(C,P,D) array (default mode).
For a single point set in a rank-2 array (P,D) returns a rank-2 (P,D) array such that
\f[
\mbox{refPoints}(p,d) = \Big(F^{-1}_c(physPoint(p,*)) \Big)_d
\f]
The \c whichCell argument selects the physical cell and is required to be a valid cell
ordinal for \c cellWorkset array.
For multiple point sets in a rank-3 (C,P,D) array returns a rank-3 (C,P,D) array such that
\f[
\mbox{refPoints}(c,p,d) = \Big(F^{-1}_c(physPoint(c,p,*)) \Big)_d
\f]
The default value \e whichCell=-1 selects this mode.
Requires cell topology with a reference cell. See Section \ref sec_cell_topology_ref_map
for definition of the mapping function. Presently supported cell topologies are
\li 1D: \c Line<2>
\li 2D: \c Triangle<3>, \c Triangle<6>, \c Quadrilateral<4>, \c Quadrilateral<9>
\li 3D: \c Tetrahedron<4>, \c Tetrahedron<10>, \c Hexahedron<8>, \c Hexahedron<27>
\warning
Computation of the inverse map in this method uses default selection of the initial guess
based on cell topology:
\li \c Line topologies: line center (0)
\li \c Triangle topologies: the point (1/3, 1/3)
\li \c Quadrilateral topologies: the point (0, 0)
\li \c Tetrahedron topologies: the point (1/6, 1/6, 1/6)
\li \c Hexahedron topologies: the point (0, 0, 0)
\li \c Wedge topologies: the point (1/2, 1/2, 0)
For some cells with extended topologies, these initial guesses may not be good enough
for Newton's method to converge in the allotted number of iterations. A version of this
method with user-supplied initial guesses is also available.
\warning
The array \c physPoints represents an arbitrary set (or sets) of points in the physical
frame that are not required to belong in the physical cell (cells) that define(s) the reference
to physical mapping. As a result, the images of these points in the reference frame
are not necessarily contained in the reference cell corresponding to the specified
cell topology.
\param refPoints [out] - rank-3/2 array with dimensions (C,P,D)/(P,D) with the images of the physical points
\param physPoints [in] - rank-3/2 array with dimensions (C,P,D)/(P,D) with points in physical frame
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param whichCell [in] - ordinal of the cell that defines the reference-to-physical map; default is -1
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\todo Implement method for non-standard (shell, beam, etc) topologies.
*/
template<class ArrayRefPoint, class ArrayPhysPoint, class ArrayCell>
static void mapToReferenceFrame(ArrayRefPoint & refPoints,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
template<class ArrayRefPoint, class ArrayPhysPoint, class ArrayCell>
static void mapToReferenceFrame(ArrayRefPoint & refPoints,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const Teuchos::RCP< Basis< Scalar, FieldContainer<Scalar> > > HGRAD_Basis,
const int & whichCell = -1);
/** \brief Computation of \f$ F^{-1}_{c} \f$, the inverse of the reference-to-physical frame map
using user-supplied initial guess.
There are two use cases:
\li Applies \f$ F^{-1}_{c} \f$ for a \b specified physical cell \f${\mathcal C}\f$ from a
cell workset to a \b single set of points stored in a rank-2 (P,D) array;
\li Applies \f$ F^{-1}_{c} \f$ for \b all cells in a cell workset to \b multiple point sets
having the same number of points, indexed by cell ordinal, and stored in a rank-3
(C,P,D) array (default mode).
For a single point set in a rank-2 array (P,D) returns a rank-2 (P,D) array such that
\f[
\mbox{refPoints}(p,d) = \Big(F^{-1}_c(physPoint(p,*)) \Big)_d
\f]
The \c whichCell argument selects the physical cell and is required to be a valid cell
ordinal for \c cellWorkset array.
For multiple point sets in a rank-3 (C,P,D) array returns a rank-3 (C,P,D) array such that
\f[
\mbox{refPoints}(c,p,d) = \Big(F^{-1}_c(physPoint(c,p,*)) \Big)_d
\f]
The default value \c whichCell=-1 selects this mode.
Requires cell topology with a reference cell. See Section \ref sec_cell_topology_ref_map
for definition of the mapping function. Presently supported cell topologies are
\li 1D: \c Line<2>
\li 2D: \c Triangle<3>, \c Triangle<6>, \c Quadrilateral<4>, \c Quadrilateral<9>
\li 3D: \c Tetrahedron<4>, \c Tetrahedron<10>, \c Hexahedron<8>, \c Hexahedron<27>
\warning
The array \c physPoints represents an arbitrary set (or sets) of points in the physical
frame that are not required to belong in the physical cell (cells) that define(s) the reference
to physical mapping. As a result, the images of these points in the reference frame
are not necessarily contained in the reference cell corresponding to the specified
cell topology.
\param refPoints [out] - rank-3/2 array with dimensions (C,P,D)/(P,D) with the images of the physical points
\param initGuess [in] - rank-3/2 array with dimensions (C,P,D)/(P,D) with the initial guesses for each point
\param physPoints [in] - rank-3/2 array with dimensions (C,P,D)/(P,D) with points in physical frame
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param whichCell [in] - ordinal of the cell that defines the reference-to-physical map; default is -1
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
*/
template<class ArrayRefPoint, class ArrayInitGuess, class ArrayPhysPoint, class ArrayCell>
static void mapToReferenceFrameInitGuess(ArrayRefPoint & refPoints,
const ArrayInitGuess & initGuess,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const shards::CellTopology & cellTopo,
const int & whichCell = -1);
template<class ArrayRefPoint, class ArrayInitGuess, class ArrayPhysPoint, class ArrayCell>
static void mapToReferenceFrameInitGuess(ArrayRefPoint & refPoints,
const ArrayInitGuess & initGuess,
const ArrayPhysPoint & physPoints,
const ArrayCell & cellWorkset,
const Teuchos::RCP< Basis< Scalar, FieldContainer<Scalar> > > HGRAD_Basis,
const int & whichCell = -1);
/** \brief Computes parameterization maps of 1- and 2-subcells of reference cells.
Applies \f$\hat{\Phi}_i\f$, the parametrization map of a subcell \f$\hat{\mathcal{S}}_i\f$
from a given reference cell, to a set of points in the parametrization domain
\e R of \f$\hat{\mathcal{S}}_i\f$. Returns a rank-2 array with dimensions
(P,D) where for 1-subcells:
\f[
{subcellPoints}(p,*) = \hat{\Phi}_i(t_p) \quad\mbox{and}\quad
\hat{\Phi}_i(t_p) = \left\{
\begin{array}{ll}
(\hat{x}(t_p),\hat{y}(t_p),\hat{z}(t_p)) & \mbox{for 3D parent cells}\\[1.5ex]
(\hat{x}(t_p),\hat{y}(t_p)) & \mbox{for 2D parent cells}
\end{array} \right.
\quad t_p \in R = [-1,1] \,;
\f]
for 2-subcells:
\f[
{subcellPoints}(p,*) = \hat{\Phi}_i(u_p,v_p)\quad\mbox{and}\quad
\hat{\Phi}_i(u_p,v_p) = (\hat{x}(u_p,v_p), \hat{y}(u_p,v_p), \hat{z}(u_p, v_p))
\quad (u_p,v_p)\in R
\f]
and
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if face is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if face is Quadrilateral}
\end{array}\right.
\f]
\remarks
\li Parametrization of 1-subcells is defined for all 2D and 3D cell topologies with reference
cells, including special 2D and 3D topologies such as shell and beams.
\li Parametrization of 2-subcells is defined for all 3D cell topologies with reference cells,
including special 3D topologies such as shells.
To map a set of points from a parametrization domain to a physical subcell workset, apply
CellTools<Scalar>::mapToPhysicalFrame to the output of this method. This will effectively
apply the parametrization map \f$ \Phi_{c,i} = F_{c}\circ\hat{\Phi}_i \f$
of each subcell in the workset to \c paramPoints. Here <var>c</var> is
ordinal of a parent cell, relative to subcell workset, and <var>i</var> is subcell
ordinal, relative to a reference cell, of the subcell workset. See Section
\ref sec_cell_topology_subcell_wset for definition of subcell workset and Section
\ref sec_cell_topology_subcell_map for definition of parametrization maps.
\param refSubcellPoints [out] - rank-2 (P,D1) array with images of parameter space points
\param paramPoints [in] - rank-2 (P,D2) array with points in 1D or 2D parameter domain
\param subcellDim [in] - dimension of the subcell where points are mapped to
\param subcellOrd [in] - subcell ordinal
\param parentCell [in] - cell topology of the parent cell.
*/
template<class ArraySubcellPoint, class ArrayParamPoint>
static void mapToReferenceSubcell(ArraySubcellPoint & refSubcellPoints,
const ArrayParamPoint & paramPoints,
const int subcellDim,
const int subcellOrd,
const shards::CellTopology & parentCell);
/** \brief Computes constant tangent vectors to edges of 2D or 3D reference cells.
Returns rank-1 array with dimension (D), D=2 or D=3; such that
\f[
{refEdgeTangent}(*) = \hat{\bf t}_i = {\partial\hat{\Phi}_i(t)\over\partial t}\,,
\f]
where \f$\hat{\Phi}_i : R =[-1,1]\mapsto \hat{\mathcal E}_i\f$ is the parametrization map
of the specified reference edge \f$\hat{\mathcal E}_i\f$, given by
\f[
\hat{\Phi}_i(t) = \left\{\begin{array}{ll}
(\hat{x}(t),\hat{y}(t),\hat{z}(t)) & \mbox{for 3D parent cells} \\[1ex]
(\hat{x}(t),\hat{y}(t)) & \mbox{for 2D parent cells} \\[1ex]
\end{array}\right.
\f]
The length of computed edge tangents is one-half the length of their associated edges:
\f[
|\hat{\bf t}_i | = {1\over 2} |\hat{\mathcal E}_i |\,.
\f]
Because the edges of all reference cells are always affine images of [-1,1],
the edge tangent is constant vector field.
\param refEdgeTangent [out] - rank-1 array (D) with the edge tangent; D = cell dimension
\param edgeOrd [in] - ordinal of the edge whose tangent is computed
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArrayEdgeTangent>
static void getReferenceEdgeTangent(ArrayEdgeTangent & refEdgeTangent,
const int & edgeOrd,
const shards::CellTopology & parentCell);
/** \brief Computes pairs of constant tangent vectors to faces of a 3D reference cells.
Returns 2 rank-1 arrays with dimension (D), D=3, such that
\f[
{refFaceTanU}(*) = \hat{\bf t}_{i,u} = {\partial\hat{\Phi}_i(u,v)\over\partial u} =
\left({\partial\hat{x}(u,v)\over\partial u},
{\partial\hat{y}(u,v)\over\partial u},
{\partial\hat{z}(u,v)\over\partial u} \right) ;
\f]
\f[
{refFaceTanV}(*) = \hat{\bf t}_{i,v} = {\partial\hat{\Phi}_i(u,v)\over \partial v} =
\left({\partial\hat{x}(u,v)\over\partial v},
{\partial\hat{y}(u,v)\over\partial v},
{\partial\hat{z}(u,v)\over\partial v} \right)\,;
\f]
where \f$\hat{\Phi}_i: R \mapsto \hat{\mathcal F}_i\f$
is the parametrization map of the specified reference face \f$\hat{\mathcal F}_i\f$ given by
\f[
\hat{\Phi}_i(u,v) =(\hat{x}(u,v),\hat{y}(u,v),\hat{z}(u,v))
\f]
and
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if $\hat{\mathcal F}_i$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if $\hat{\mathcal F}_i$ is Quadrilateral} \,.
\end{array}\right.
\f]
Because the faces of all reference cells are always affine images of \e R ,
the coordinate functions \f$\hat{x},\hat{y},\hat{z}\f$ of the parametrization map
are linear and the face tangents are constant vectors.
\param refFaceTanU [out] - rank-1 array (D) with (constant) tangent in u-direction
\param refFaceTanV [out] - rank-1 array (D) with (constant) tangent in v-direction
\param faceOrd [in] - ordinal of the face whose tangents are computed
\param parentCell [in] - cell topology of the parent 3D reference cell
*/
template<class ArrayFaceTangentU, class ArrayFaceTangentV>
static void getReferenceFaceTangents(ArrayFaceTangentU & refFaceTanU,
ArrayFaceTangentV & refFaceTanV,
const int & faceOrd,
const shards::CellTopology & parentCell);
/** \brief Computes constant normal vectors to sides of 2D or 3D reference cells.
A side is defined as a subcell of dimension one less than that of its parent cell.
Therefore, sides of 2D cells are 1-subcells (edges) and sides of 3D cells
are 2-subcells (faces).
Returns rank-1 array with dimension (D), D = 2 or 3 such that
\f[
{refSideNormal}(*) = \hat{\bf n}_i =
\left\{\begin{array}{rl}
\displaystyle
\left({\partial\hat{\Phi}_i(t)\over\partial t}\right)^{\perp}
& \mbox{for 2D parent cells} \\[2ex]
\displaystyle
{\partial\hat{\Phi}_{i}\over\partial u} \times
{\partial\hat{\Phi}_{i}\over\partial v} & \mbox{for 3D parent cells}
\end{array}\right.
\f]
where \f$ (u_1,u_2)^\perp = (u_2, -u_1)\f$, and \f$\hat{\Phi}_i: R \mapsto \hat{\mathcal S}_i\f$
is the parametrization map of the specified reference side \f$\hat{\mathcal S}_i\f$ given by
\f[
\hat{\Phi}_i(u,v) =
\left\{\begin{array}{rl}
(\hat{x}(t),\hat{y}(t)) & \mbox{for 2D parent cells} \\[1ex]
(\hat{x}(u,v),\hat{y}(u,v),\hat{z}(u,v)) & \mbox{for 3D parent cells}
\end{array}\right.
\f]
For sides of 2D cells \e R=[-1,1] and for sides of 3D cells
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if $\hat{\mathcal S}_i$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if $\hat{\mathcal S}_i$ is Quadrilateral} \,.
\end{array}\right.
\f]
For 3D cells the length of computed side normals is proportional to side area:
\f[
|\hat{\bf n}_i | = \left\{\begin{array}{rl}
2 \mbox{Area}(\hat{\mathcal F}_i) & \mbox{if $\hat{\mathcal F}_i$ is Triangle} \\[1ex]
\mbox{Area}(\hat{\mathcal F}_i) & \mbox{if $\hat{\mathcal F}_i$ is Quadrilateral} \,.
\end{array}\right.
\f]
For 2D cells the length of computed side normals is proportional to side length:
\f[
|\hat{\bf n}_i | = {1\over 2} |\hat{\mathcal F}_i |\,.
\f]
Because the sides of all reference cells are always affine images of \e R ,
the coordinate functions \f$\hat{x},\hat{y},\hat{z}\f$ of the parametrization maps
are linear and the side normal is a constant vector.
\remark
- For 3D cells the reference side normal coincides with the face normal computed by
CellTools<Scalar>::getReferenceFaceNormal and these two methods are completely equivalent.
- For 2D cells the reference side normal is defined by \f$\hat{{\bf n}}= \hat{\bf t}^\perp = (t_2,-t_1)\f$
where \f$\hat{{\bf t}}=(t_1,t_2)\f$ is the tangent vector computed by
CellTools<Scalar>::getReferenceEdgeTangent. Therefore, the pair
\f$(\hat{{\bf n}},\hat{{\bf t}})\f$ is positively oriented.
\param refSideNormal [out] - rank-1 array (D) with (constant) side normal
\param sideOrd [in] - ordinal of the side whose normal is computed
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArraySideNormal>
static void getReferenceSideNormal(ArraySideNormal & refSideNormal,
const int & sideOrd,
const shards::CellTopology & parentCell);
/** \brief Computes constant normal vectors to faces of 3D reference cell.
Returns rank-1 array with dimension (D), D=3 such that
\f[
{refFaceNormal}(*) = \hat{\bf n}_i = {\partial\hat{\Phi}_{i}\over\partial u} \times
{\partial\hat{\Phi}_{i}\over\partial v}
\f]
where \f$\hat{\Phi}_i: R \mapsto \hat{\mathcal F}_i\f$
is the parametrization map of the specified reference face \f$\hat{\mathcal F}_i\f$ given by
\f[
\hat{\Phi}_i(u,v) =(\hat{x}(u,v),\hat{y}(u,v),\hat{z}(u,v))
\f]
and
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if ${\mathcal F}$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if ${\mathcal F}$ is Quadrilateral} \,.
\end{array}\right.
\f]
The length of computed face normals is proportional to face area:
\f[
|\hat{\bf n}_i | = \left\{\begin{array}{rl}
2 \mbox{Area}(\hat{\mathcal F}_i) & \mbox{if $\hat{\mathcal F}_i$ is Triangle} \\[1ex]
\mbox{Area}(\hat{\mathcal F}_i) & \mbox{if $\hat{\mathcal F}_i$ is Quadrilateral} \,.
\end{array}\right.
\f]
Because the faces of all reference cells are always affine images of \e R ,
the coordinate functions \f$\hat{x},\hat{y},\hat{z}\f$ of the parametrization map
are linear and the face normal is a constant vector.
\remark
The method CellTools::getReferenceFaceTangents computes the reference face tangents
\f${\partial\hat{\Phi}_{i}/\partial u}\f$ and \f${\partial\hat{\Phi}_{i}/\partial v}\f$.
\param refFaceNormal [out] - rank-1 array (D) with (constant) face normal
\param faceOrd [in] - ordinal of the face whose normal is computed
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArrayFaceNormal>
static void getReferenceFaceNormal(ArrayFaceNormal & refFaceNormal,
const int & faceOrd,
const shards::CellTopology & parentCell);
/** \brief Computes non-normalized tangent vectors to physical edges in an edge workset
\f$\{\mathcal{E}_{c,i}\}_{c=0}^{N}\f$; (see \ref sec_cell_topology_subcell_wset for definition of edge worksets).
For every edge in the workset the tangents are computed at the points
\f${\bf x}_p = F_c(\hat{\Phi}_i(t_p))\in\mathcal{E}_{c,i}\f$ that are images of points
from <var>R=[-1,1]</var> on edge \f$\mathcal{E}_{c,i}\f$. Returns rank-3 array with
dimensions (C,P,D1), D1=2 or D1=3 such that
\f[
{edgeTangents}(c,p,d) =
DF_c(\hat{\Phi}_i(t_p))\cdot {\partial{\hat{\Phi}}_{i}(t_p)\over\partial t}\,; \qquad t_p \in R
\f]
In this formula:
\li \f$ DF_c \f$ is Jacobian of parent cell \f${\mathcal C}\f$ that owns physical edge \f${\mathcal E}_{c,i}\f$;
\li \f$ {\partial{\hat{\Phi}}_{i}/\partial t}\f$ is the (constant) tangent to reference edge
\f$\hat{\mathcal E}_i\f$; see CellTools<Scalar>::getReferenceEdgeTangent that has the
same local ordinal as the edges in the workset;
\li \f$ \hat{\Phi}_i R\mapsto\hat{\mathcal E}_i \f$ is parametrization of reference edge \f$\hat{\mathcal E}_i\f$;
\warning
\c worksetJacobians must contain the values of \f$DF_c(\hat{\Phi}_i(t_p))\f$,
where \f$ t_p \in R=[-1,1] \f$, i.e., Jacobians of the parent cells evaluated at points
that are located on reference edge \f$\hat{\mathcal E}_i\f$ having the same local ordinal as
the edges in the workset.
\param edgeTangents [out] - rank-3 array (C,P,D1) with tangents on workset edges
\param worksetJacobians [in] - rank-4 array (C,P,D1,D1) with Jacobians evaluated at ref. edge points
\param worksetEdgeOrd [in] - edge ordinal, relative to ref. cell, of the edge workset
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArrayEdgeTangent, class ArrayJac>
static void getPhysicalEdgeTangents(ArrayEdgeTangent & edgeTangents,
const ArrayJac & worksetJacobians,
const int & worksetEdgeOrd,
const shards::CellTopology & parentCell);
/* template<class ArrayEdgeTangent, class ArrayJac>
static void getPhysicalEdgeTangentsTemp(ArrayEdgeTangent & edgeTangents,
const ArrayJac & worksetJacobians,
const int & worksetEdgeOrd,
const shards::CellTopology & parentCell); */
/** \brief Computes non-normalized tangent vector pairs to physical faces in a face workset
\f$\{\mathcal{F}_{c,i}\}_{c=0}^{N}\f$; (see \ref sec_cell_topology_subcell_wset for definition of face worksets).
For every face in the workset the tangents are computed at the points
\f${\bf x}_p = F_c(\hat{\Phi}_i(u_p,v_p))\in\mathcal{F}_{c,i}\f$ that are images of points
from the parametrization domain \e R on face \f$\mathcal{F}_{c,i}\f$.
Returns 2 rank-3 arrays with dimensions (C,P,D), D=3 such that
\f[
{faceTanU}(c,p,d) = DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_i\over\partial u};\qquad
{faceTanV}(c,p,d) = DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_{i}\over\partial v}\,;
\qquad (u_p, v_p) \in R \,.
\f]
In this formula:
\li \f$ DF_c \f$ is Jacobian of parent cell \f${\mathcal C}\f$ that owns physical face \f${\mathcal F}_{c,i}\f$;
\li \f$ {\partial\hat{\Phi}_i/\partial u}, {\partial\hat{\Phi}_i/\partial v}\f$ are the (constant)
tangents on reference face \f$\hat{\mathcal F}_i\f$; see CellTools<Scalar>::getReferenceFaceTangents;
that has the same local ordinal as the faces in the workset;
\li \f$ \hat{\Phi}_i : R\mapsto \hat{\mathcal F}_i\f$ is parametrization of reference face \f$\hat{\mathcal F}_i\f$;
\li \e R is the parametrization domain for reference face \f$\hat{\mathcal F}_i\f$:
\f[
R =
\left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if $\hat{\mathcal F}_i$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if $\hat{\mathcal F}_i$ is Quadrilateral}
\end{array}\right.
\f]
\warning
\c worksetJacobians must contain the values of \f$DF_c(\hat{\Phi}_i(u_p,v_p))\f$,
where \f$(u_p,v_p)\in R\f$, i.e., Jacobians of the parent cells evaluated at points
that are located on reference face \f$\hat{\mathcal F}_i\f$ having the same local ordinal as
the faces in the workset.
\param faceTanU [out] - rank-3 array (C,P,D), image of ref. face u-tangent at workset faces
\param faceTanV [out] - rank-3 array (C,P,D), image of ref. face u-tangent at workset faces
\param worksetJacobians [in] - rank-4 array (C,P,D,D) with Jacobians at ref. face points
\param worksetFaceOrd [in] - face ordinal, relative to ref. cell, of the face workset
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArrayFaceTangentU, class ArrayFaceTangentV, class ArrayJac>
static void getPhysicalFaceTangents(ArrayFaceTangentU & faceTanU,
ArrayFaceTangentV & faceTanV,
const ArrayJac & worksetJacobians,
const int & worksetFaceOrd,
const shards::CellTopology & parentCell);
/* template<class ArrayFaceTangentU, class ArrayFaceTangentV, class ArrayJac>
static void getPhysicalFaceTangentsTemp(ArrayFaceTangentU & faceTanU,
ArrayFaceTangentV & faceTanV,
const ArrayJac & worksetJacobians,
const int & worksetFaceOrd,
const shards::CellTopology & parentCell);
*/
/** \brief Computes non-normalized normal vectors to physical sides in a side workset
\f$\{\mathcal{S}_{c,i}\}_{c=0}^{N}\f$.
For every side in the workset the normals are computed at the points
\f${\bf x}_p = F_c(\hat{\Phi}_i(P_p))\in\mathcal{S}_{c,i}\f$ that are images of points
from the parametrization domain \e R on side \f$\mathcal{S}_{c,i}\f$.
A side is defined as a subcell of dimension one less than that of its parent cell.
Therefore, sides of 2D cells are 1-subcells (edges) and sides of 3D cells are 2-subcells (faces).
Returns rank-3 array with dimensions (C,P,D), D = 2 or 3, such that
\f[
{sideNormals}(c,p,d) =
\left\{\begin{array}{crl}
\displaystyle
\left(DF_c(\hat{\Phi}_i(t_p))\cdot
{\partial{\hat{\Phi}}_{i}(t_p)\over\partial t}\right)^{\perp} & t_p\in R
& \mbox{for 2D parent cells} \\[2ex]
\displaystyle
\left( DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_i\over\partial u}\right) \times
\left( DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_i\over\partial v}\right) \,;
& (u_p, v_p) \in R & \mbox{for 3D parent cells}
\end{array}\right.
\f]
In this formula:
\li \f$ DF_c \f$ is Jacobian of parent cell \f${\mathcal C}\f$ that owns physical side \f${\mathcal S}_{c,i}\f$;
\li For 2D cells: \f$ {\partial{\hat{\Phi}}_{i}/\partial t}\f$ is the (constant) tangent to reference side (edge)
\f$\hat{\mathcal S}_i\f$; see CellTools<Scalar>::getReferenceEdgeTangent, that has the
same local ordinal as the sides in the workset;
\li For 3D cells: \f$ {\partial\hat{\Phi}_i/\partial u}, {\partial\hat{\Phi}_i/\partial v}\f$ are the (constant)
tangents on reference side (face) \f$\hat{\mathcal S}_i\f$; see CellTools<Scalar>::getReferenceFaceTangents,
that has the same local ordinal as the sides in the workset;
\li \f$ \hat{\Phi}_i : R\mapsto \hat{\mathcal S}_i\f$ is parametrization of reference side \f$\hat{\mathcal S}_i\f$;
\li \e R is the parametrization domain for reference side \f$\hat{\mathcal S}_i\f$. For
2D parent cells \e R=[-1,1] and for 3D parent cells
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if $\hat{\mathcal S}_i$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if $\hat{\mathcal S}_i$ is Quadrilateral}
\end{array}\right.
\f]
\remark
- For 3D cells the physical side normals coincides with the face normals computed by
CellTools<Scalar>::getPhysicalFaceNormals and these two methods are completely equivalent.
- For 2D cells the physical side normals are defined by \f${\bf n}=(t_2,-t_1)\f$
where \f${\bf t}=(t_1,t_2)\f$ are the physical edge tangents computed by
CellTools<Scalar>::getPhysicalEdgeTangents. Therefore, the pairs \f$({\bf n},{\bf t})\f$ are positively oriented.
\warning
\c worksetJacobians must contain the values of \f$DF_c(\hat{\Phi}_i(P_p))\f$,
where \f$P_p\in R\f$, i.e., Jacobians of the parent cells evaluated at points
that are located on reference side \f$\hat{\mathcal S}_i\f$ having the same local ordinal as
the sides in the workset.
\param sideNormals [out] - rank-3 array (C,P,D), normals at workset sides
\param worksetJacobians [in] - rank-4 array (C,P,D,D) with Jacobians at ref. side points
\param worksetSideOrd [in] - side ordinal, relative to ref. cell, of the side workset
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArraySideNormal, class ArrayJac>
static void getPhysicalSideNormals(ArraySideNormal & sideNormals,
const ArrayJac & worksetJacobians,
const int & worksetSideOrd,
const shards::CellTopology & parentCell);
/** \brief Computes non-normalized normal vectors to physical faces in a face workset
\f$\{\mathcal{F}_{c,i}\}_{c=0}^{N}\f$; (see \ref sec_cell_topology_subcell_wset for definition of face worksets).
For every face in the workset the normals are computed at the points
\f${\bf x}_p = F_c(\hat{\Phi}_i(u_p,v_p))\in\mathcal{F}_{c,i}\f$ that are images of points
from the parametrization domain \e R on face \f$\mathcal{F}_{c,i}\f$.
Returns rank-3 array with dimensions (C,P,D), D=3, such that
\f[
{faceNormals}(c,p,d) =
\left( DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_i\over\partial u}\right) \times
\left( DF_c(\hat{\Phi}_i(u_p, v_p))\cdot {\partial\hat{\Phi}_i\over\partial v}\right) \,;
\qquad (u_p, v_p) \in R \,.
\f]
In this formula:
\li \f$ DF_c \f$ is Jacobian of parent cell \f${\mathcal C}\f$ that owns physical face \f${\mathcal F}_{c,i}\f$;
\li \f$ {\partial\hat{\Phi}_i/\partial u}, {\partial\hat{\Phi}_i/\partial v}\f$ are the (constant)
tangents on reference face \f$\hat{\mathcal F}_i\f$; see CellTools<Scalar>::getReferenceFaceTangents;
that has the same local ordinal as the faces in the workset;
\li \f$ \hat{\Phi}_i : R\mapsto \hat{\mathcal F}_i\f$ is parametrization of reference face \f$\hat{\mathcal F}_i\f$;
\li \e R is the parametrization domain for reference face \f$\hat{\mathcal F}_i\f$:
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if $\hat{\mathcal F}_i$ is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if $\hat{\mathcal F}_i$ is Quadrilateral}
\end{array}\right.
\f]
\warning
\c worksetJacobians must contain the values of \f$DF_c(\hat{\Phi}_i(u_p,v_p))\f$,
where \f$(u_p,v_p)\in R\f$, i.e., Jacobians of the parent cells evaluated at points
that are located on reference face \f$\hat{\mathcal F}_i\f$ having the same local ordinal as
the faces in the workset.
\param faceNormals [out] - rank-3 array (C,P,D), normals at workset faces
\param worksetJacobians [in] - rank-4 array (C,P,D,D) with Jacobians at ref. face points
\param worksetFaceOrd [in] - face ordinal, relative to ref. cell, of the face workset
\param parentCell [in] - cell topology of the parent reference cell
*/
template<class ArrayFaceNormal, class ArrayJac>
static void getPhysicalFaceNormals(ArrayFaceNormal & faceNormals,
const ArrayJac & worksetJacobians,
const int & worksetFaceOrd,
const shards::CellTopology & parentCell);
/* template<class ArrayFaceNormal, class ArrayJac>
static void getPhysicalFaceNormalsTemp(ArrayFaceNormal & faceNormals,
const ArrayJac & worksetJacobians,
const int & worksetFaceOrd,
const shards::CellTopology & parentCell);
*/
//============================================================================================//
// //
// Inclusion tests //
// //
//============================================================================================//
/** \brief Checks if a point belongs to a reference cell.
Requires cell topology with a reference cell.
\param point [in] - spatial coordinates of the point tested for inclusion
\param pointDim [in] - spatial dimension of that point
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param threshold [in] - "tightness" of the inclusion test
\return 1 if the point is in the closure of the specified reference cell and 0 otherwise.
*/
static int checkPointInclusion(const Scalar* point,
const int pointDim,
const shards::CellTopology & cellTopo,
const double & threshold = INTREPID_THRESHOLD);
/** \brief Checks if a set of points belongs to a reference cell.
Requires cell topology with a reference cell. See Intrepid::CellTools::checkPointwiseInclusion
for admissible ranks and dimensions of the input point array.
\param points [in] - rank-1, 2 or 3 array (point, vector of points, matrix of points)
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param threshold [in] - "tightness" of the inclusion test
\return 1 if all points are in the closure of the specified reference cell
0 if at least one point is outside the closure of the reference cell
*/
template<class ArrayPoint>
static int checkPointsetInclusion(const ArrayPoint & points,
const shards::CellTopology & cellTopo,
const double & threshold = INTREPID_THRESHOLD);
/** \brief Checks every point in a set for inclusion in a reference cell.
Requires cell topology with a reference cell. Admissible ranks and dimensions of the
input point array and the corresponding rank and dimension of the output array are as follows:
\verbatim
|-------------------|-------------|-------------|-------------|
| rank: (in)/(out) | 1/1 | 2/1 | 3/2 |
|-------------------|-------------|-------------|-------------|
| points (in) | (D) | (I, D) | (I, J, D) |
|-------------------|-------------|-------------|-------------|
| inRefCell (out) | (1) | (I) | (I, J) |
|------------------ |-------------|-------------|-------------|
\endverbatim
Example: if \c points is rank-3 array with dimensions (I, J, D), then
\f[
\mbox{inRefCell}(i,j) =
\left\{\begin{array}{rl}
1 & \mbox{if $points(i,j,*)\in\hat{\mathcal{C}}$} \\[2ex]
0 & \mbox{if $points(i,j,*)\notin\hat{\mathcal{C}}$}
\end{array}\right.
\f]
\param inRefCell [out] - rank-1 or 2 array with results from the pointwise inclusion test
\param refPoints [in] - rank-1,2 or 3 array (point, vector of points, matrix of points)
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param threshold [in] - "tightness" of the inclusion test
*/
template<class ArrayIncl, class ArrayPoint>
static void checkPointwiseInclusion(ArrayIncl & inRefCell,
const ArrayPoint & points,
const shards::CellTopology & cellTopo,
const double & threshold = INTREPID_THRESHOLD);
/** \brief Checks every point in a set or multiple sets for inclusion in physical cells from a cell workset.
There are two use cases:
\li Checks every point from a \b single point set, stored in a rank-2 array (P,D) for inclusion
in a \b specified physical cell \f${\mathcal C}\f$ from a cell workset.
\li Checks every point from \b multiple point sets indexed by a cell ordinal, and stored in a rank-3
(C,P,D) array, for inclusion in the physical cell having the same cell ordinal, for \b all
cells in a cell workset.
For a single point set in a rank-2 array (P,D) and \c whichCell set to a valid cell ordinal
relative to \c cellWorkset returns a rank-1 (P) array such that
\f[
\mbox{inCell}(p) =
\left\{\begin{array}{rl}
1 & \mbox{if $points(p,*)\in {\mathcal{C}}$} \\ [2ex]
0 & \mbox{if $points(p,*)\notin {\mathcal{C}}$}
\end{array}\right.
\f]
For multiple point sets in a rank-3 array (C,P,D) and \c whichCell=-1 (default value)
returns a rank-2 (C,P) array such that
\f[
\mbox{inCell}(c,p) =
\left\{\begin{array}{rl}
1 & \mbox{if $points(c,p,*)\in {\mathcal{C}}$} \\ [2ex]
0 & \mbox{if $points(c,p,*)\notin {\mathcal{C}}$}
\end{array}\right.
\f]
\param inCell [out] - rank-1 array with results from the pointwise inclusion test
\param points [in] - rank-2 array with dimensions (P,D) with the physical points
\param cellWorkset [in] - rank-3 array with dimensions (C,N,D) with the nodes of the cell workset
\param cellTopo [in] - cell topology of the cells stored in \c cellWorkset
\param whichCell [in] - ordinal of the cell used in the inclusion test
\param threshold [in] - tolerance for inclusion tests on the input points
*/
template<class ArrayIncl, class ArrayPoint, class ArrayCell>
static void checkPointwiseInclusion(ArrayIncl & inCell,
const ArrayPoint & points,
const ArrayCell & cellWorkset,
const shards::CellTopology & cell,
const int & whichCell = -1,
const double & threshold = INTREPID_THRESHOLD);
/** \brief Retrieves the Cartesian coordinates of a reference cell vertex.
Requires cell topology with a reference cell. Vertex coordinates are always returned
as an (x,y,z)-triple regardlesss of the actual topological cell dimension. The unused
coordinates are set to zero, e.g., vertex 0 of Line<2> is returned as {-1,0,0}.
\param cell [in] - cell topology of the cell
\param vertexOrd [in] - vertex ordinal
\return pointer to array with the (x,y,z) coordinates of the specified reference vertex
*/
static const double* getReferenceVertex(const shards::CellTopology& cell,
const int vertexOrd);
/** \brief Retrieves the Cartesian coordinates of all vertices of a reference subcell.
Returns rank-2 array with the Cartesian coordinates of the vertices of the
specified reference cell subcell. Requires cell topology with a reference cell.
\param subcellVertices [out] - rank-2 (V,D) array with the Cartesian coordinates of the reference subcell vertices
\param subcellDim [in] - dimension of the subcell; 0 <= subcellDim <= parentCell dimension
\param subcellOrd [in] - ordinal of the subcell
\param parentCell [in] - topology of the cell that owns the subcell
\remark When \c subcellDim = dimension of the \c parentCell this method returns the Cartesian
coordinates of the vertices of the reference cell itself.
Note that this requires \e subcellOrd=0.
*/
template<class ArraySubcellVert>
static void getReferenceSubcellVertices(ArraySubcellVert & subcellVertices,
const int subcellDim,
const int subcellOrd,
const shards::CellTopology& parentCell);
/** \brief Retrieves the Cartesian coordinates of a reference cell node.
Returns Cartesian coordinates of a reference cell node. Requires cell topology
with a reference cell. Node coordinates are always returned as an (x,y,z)-triple
regardlesss of the actual topological cell dimension. The unused coordinates are
set to zero, e.g., node 0 of Line<2> is returned as {-1,0,0}.
\remark
Because the nodes of a cell with a base topology coincide with its vertices, for cells
with base topology this method is equivalent to CellTools<Scalar>::getReferenceVertex.
\param cell [in] - cell topology of the cell
\param vertexOrd [in] - node ordinal
\return pointer to array with the (x,y,z) coordinates of the specified reference vertex
*/
static const double* getReferenceNode(const shards::CellTopology& cell,
const int nodeOrd);
/** \brief Retrieves the Cartesian coordinates of all nodes of a reference subcell.
Returns rank-2 array with the Cartesian coordinates of the nodes of the
specified reference cell subcell. Requires cell topology with a reference cell.
\param subcellNodes [out] - rank-2 (N,D) array with the Cartesian coordinates of the reference subcell nodes
\param subcellDim [in] - dimension of the subcell; 0 <= subcellDim <= parentCell dimension
\param subcellOrd [in] - ordinal of the subcell
\param parentCell [in] - topology of the cell that owns the subcell
\remark When \c subcellDim = dimension of the \c parentCell this method returns the Cartesian
coordinates of the nodes of the reference cell itself. Note that this requires \c subcellOrd=0.
*/
template<class ArraySubcellNode>
static void getReferenceSubcellNodes(ArraySubcellNode& subcellNodes,
const int subcellDim,
const int subcellOrd,
const shards::CellTopology& parentCell);
/** \brief Checks if a cell topology has reference cell
\param cell [in] - cell topology
\return 1 if the cell topology has reference cell,
0 oterwise
*/
static int hasReferenceCell(const shards::CellTopology & cellTopo);
//============================================================================================//
// //
// Debug //
// //
//============================================================================================//
/** \brief Prints the reference space coordinates of the vertices of the specified subcell
\param subcellDim [in] - dimension of the subcell where points are mapped to
\param subcellOrd [in] - subcell ordinal
\param parentCell [in] - cell topology of the parent cell.
*/
static void printSubcellVertices(const int subcellDim,
const int subcellOrd,
const shards::CellTopology & parentCell);
/** \brief Prints the nodes of a subcell from a cell workset
*/
template<class ArrayCell>
static void printWorksetSubcell(const ArrayCell & cellWorkset,
const shards::CellTopology & parentCell,
const int& pCellOrd,
const int& subcellDim,
const int& subcellOrd,
const int& fieldWidth = 3);
//============================================================================================//
// //
// Control Volume Coordinates //
// //
//============================================================================================//
/** \brief Computes coordinates of sub-control volumes in each primary cell.
To build the system of equations for the control volume finite element method we
need to compute geometric data for integration over control volumes. A control
volume is polygon or polyhedron that surrounds a primary cell node and has
vertices that include the surrounding primary cells' barycenter, edge midpoints,
and face midpoints if in 3-d.
When using element-based assembly of the discrete equations over the primary mesh,
a single element will contain a piece of each control volume surrounding each of
the primary cell nodes. This piece of control volume (sub-control volume) is
always a quadrilateral in 2-d and a hexahedron in 3-d.
In 2-d the sub-control volumes are defined in the following way:
\verbatim
Quadrilateral primary element:
O________M________O
| | |
| 3 | 2 | B = cell barycenter
| | | O = primary cell nodes
M________B________M M = cell edge midpoints
| | |
| 0 | 1 | sub-control volumes 0, 1, 2, 3
| | |
O________M________O
Triangle primary element:
O
/ \
/ \ B = cell barycenter
/ \ O = primary cell nodes
M 2 M M = cell edge midpoints
/ \ / \
/ \ B / \ sub-control volumes 0, 1, 2
/ | \
/ 0 | 1 \
O________M________O
\endverbatim
In 3-d the sub-control volumes are defined by the primary cell face
centers and edge midpoints. The eight sub-control volumes for a
hexahedron are shown below:
\verbatim
O__________E__________O
/| /| /|
E_|________F_|________E |
/| | /| | /| |
O_|_|______E_|_|______O | | O = primary cell nodes
| | E------|-|-F------|-|-E B = cell barycenter
| |/| | |/| | |/| F = cell face centers
| F-|------|-B-|------|-F | E = cell edge midpoints
|/| | |/| | |/| |
E_|_|______F_|_|______E | |
| | O------|-|-E------|-|-O
| |/ | |/ | |/
| E--------|-F--------|-E
|/ |/ |/
O__________E__________O
\endverbatim
\param subCVCoords [out] - array containing sub-control volume coordinates
\param cellCoords [in] - array containing coordinates of primary cells
\param primaryCell [in] - primary cell topology
*/
template<class ArrayCVCoord, class ArrayCellCoord>
static void getSubCVCoords(ArrayCVCoord & subCVcoords, const ArrayCellCoord & cellCoords,
const shards::CellTopology& primaryCell);
/** \brief Compute cell barycenters.
\param barycenter [out] - array containing cell baycenters
\param cellCoords [in] - array containing cell coordinates
*/
template<class ArrayCent, class ArrayCellCoord>
static void getBarycenter(ArrayCent & barycenter, const ArrayCellCoord & cellCoords);
}; // class CellTools
} // namespace Intrepid
// include templated function definitions
#ifdef HAVE_INTREPID_KOKKOSCORE
#include <Intrepid_CellTools_Kokkos.hpp>
#endif
#include "Intrepid_CellToolsDef.hpp"
#endif
/***************************************************************************************************
** **
** D O C U M E N T A T I O N P A G E S **
** **
**************************************************************************************************/
/**
\page cell_tools_page Cell tools
<b>Table of contents </b>
- \ref cell_topology_sec
- \ref cell_topology_ref_cells
- \ref sec_cell_topology_ref_map
- \ref sec_cell_topology_ref_map_DF
- \ref sec_cell_topology_subcell_map
- \ref sec_cell_topology_subcell_wset
\section cell_topology_sec Cell topologies
The range of admissible cell shapes in Intrepid is restricted to <var>d</var>-dimensional
<strong>polytopes</strong>, <var>d=1,2,3</var>. A polytope is defined by a set of its vertices
\f$ \{ {\bf v}_0,\ldots {\bf v}_V\} \f$ and a <strong>base topology</strong> <var>BT</var> that
defines how these verices are connected into <var>k</var>-dimensional, <var>k < d</var> facets
(k-subcells) of that polytope.
The base topology of any polytope can be extended by augmenting the set of its vertices by
an additional set of points \f$\{ {\bf p}_0,\ldots {\bf p}_P\} \f$. The <strong>extended topology</strong>
<var>ET</var> is defined by specifying the connectivity of the set
\f$ \{ {\bf v}_0,\ldots {\bf v}_V\}\cup \{ {\bf p}_0,\ldots {\bf p}_P\} \f$
relative to the subcells specified by its base topology <var>BT</var>.
The vertices and the extra points are collectively referred to as <strong>nodes</strong>. Thus,
a polytope with <strong>extended topology</strong> <var>ET</var> is defined by a set of nodes
\f$\{{\bf q}_0,\ldots,{\bf q}_N\}\f$, where \f$N = V + P\f$, and a connectivity rule for these nodes.
Intrepid requires any cell to have a valid base topology. The nodes of the cell should always be
ordered by listing its vertices <strong>first</strong>, i.e.,
\f[
\{{\bf q}_0,\ldots,{\bf q}_N\} = \{ {\bf v}_0,\ldots {\bf v}_V,{\bf p}_0,\ldots {\bf p}_P\}
\f]
To manage cell topologies Intrepid uses the Shards package http://trilinos.sandia.gov/packages/shards .
Shards provides definitions for a standard set of base and extended cell topologies plus tools to
construct custom, user defined cell topologies, such as arbitrary polyhedral cells. For further
details see Shards documentation.
\section cell_topology_ref_cells Reference cells
For some cell topologies there exist simple, e.g., polynomial, mappings that allow to obtain any
cell having that topology as an image of a single "standard" cell. We refer to such standard cells
as <strong>reference</strong> cells.
Just like in the general case, a reference cell with a base topology <var>BT</var> is defined by a
set of vertices, and a reference cell with extended topology <var>ET</var> is defined by a set of
nodes that include the original vertices and some additional points.
The actual vertex and node coordinates for the reference cells can be chosen arbitrarily; however,
once selected they should not be changed because in many cases, e.g., in finite element reconstructions,
all calculations are done on a reference cell and then transformed to physical cells by an appropriate
pullback (see Section \ref sec_pullbacks).
In Intrepid base and extended reference cell topologies are defined using the following selections
of vertex and node coordinates:
\verbatim
|=======================|==============================================================================|
| Topology family | reference cell vertices/additional nodes defining extended topology |
|=======================|==============================================================================|
| Line<2> | |
| Beam<2> | {(-1,0,0),(1,0,0)} |
| ShellLine<2> | |
|-----------------------|------------------------------------------------------------------------------|
| Line<3> | |
| Beam<3> | {(0,0,0)} |
| ShellLine<3> | |
|=======================|==============================================================================|
| Triangle<3> | {(0,0,0),(1,0,0),(0,1,0)} |
| ShellTriangle<3> | |
|-----------------------|------------------------------------------------------------------------------|
| Triangle<4> | {(1/3,1/3,0)} |
|.......................|..............................................................................|
| Triangle<6> | {(1/2,0,0),(1/2,1/2,0),(0,1/2,0)} |
| ShellTriangle<6> | |
|=======================|==============================================================================|
| Quadrilateral<4> | {(-1,-1,0),(1,-1,0), (1,1,0),(-1,1,0)} |
| ShellQuadrilateral<4> | |
|-----------------------|------------------------------------------------------------------------------|
| Quadrilateral<8> | {(0,-1,0),(1,0,0),(0,1,0),(-1,0,0)} |
| ShellQuadrilateral<8> | |
|.......................|..............................................................................|
| Quadrilateral<9> | {(0,-1,0),(1,0,0),(0,1,0),(-1,0,0),(0,0,0)} |
| ShellQuadrilateral<9> | |
|=======================|==============================================================================|
| Tetrahedron<4> | {(0,0,0),(1,0,0),(0,1,0),(0,0,1)} |
|-----------------------|------------------------------------------------------------------------------|
| Tetrahedron<8> | {(1/2,0,0),(1/2,1/2,0),(0,1/2,0),(1/3,1/3,1/3)} |
| Tetrahedron<10> | {(1/2,0,0),(1/2,1/2,0),(0,1/2,0),(0,0,1/2),(1/2,0,1/2),(0,1/2,1/2)} |
|=======================|==============================================================================|
| Pyramid<5> | {(-1,-1,0),(1,-1,0),(1,1,0),(-1,1,0),(0,0,1)} |
|-----------------------|------------------------------------------------------------------------------|
| Pyramid<13> | {(0,-1,0),(1,0,0),(0,1,0),(-1,0,0), 1/2((-1,-1,1),(1,-1,1),(1,1,1),(-1,1,1))}|
| Pyramid<14> | all of the above and (0,0,0) |
|=======================|==============================================================================|
| Wedge<6> | {(0,0,-1),(1,0,-1),(0,1,-1),(0,0,1),(1,0,1),(0,1,1)} |
|-----------------------|------------------------------------------------------------------------------|
| Wedge<15> | {(1/2,0,-1),(1/2,1/2,-1),(0,1/2,-1), (0,0,0),(1,0,0),(0,1,0), |
| | (1/2,0, 1),(1/2,1/2, 1),(0,1/2, 1) |
|.......................|..............................................................................|
| Wedge<18> | All of the above plus {(1/2,0,0),(1/2,1/2,0),(0,1/2,0)} |
|=======================|==============================================================================|
| Hexahedron<8> | {(-1,-1,-1),(1,-1,-1),(1,1,-1),(-1,1,-1),(-1,-1,1),(1,-1,1),(1,1,1),(-1,1,1)}|
|-----------------------|------------------------------------------------------------------------------|
| Hexahedron<20> | {(0,-1,-1),(1,0,-1),(0,1,-1),(-1,0,-1), (0,-1,0),(1,0,0),(0,1,0),(-1,0,0), |
| | (0,-1, 1),(1,0, 1),(0,1, 1),(-1,0, 1) } |
|.......................|..............................................................................|
| Hexahedron<27> | All of the above plus center point and face midpoints: |
| | {(0,0,0), (0,0,-1),(0,0,1), (-1,0,0),(1,0,0), (0,-1,0),(0,1,0)} |
|=======================|==============================================================================|
\endverbatim
Finite element reconstruction methods based on pullbacks (see Section \ref sec_pullbacks) are
restricted to the above cell topologies.
\subsection sec_cell_topology_ref_map Reference-to-physical cell mapping
The mapping that takes a given reference cell to a physical cell with the same topology is defined
using a nodal Lagrangian basis corresponding to the nodes of the reference cell. In other words, the
mapping is constructed using basis functions that are dual to the nodes of the reference cell.
Implementation details are as follows.
Assume that \f$ \hat{\mathcal{C}} \f$ is a reference cell with topology <var>T</var> and nodes
\f$\{\hat{{\bf q}}_0,\ldots,\hat{{\bf q}}_{N}\}\f$, and that \f$ \{\hat{\phi}_i\}_{i=0}^{N} \f$ is
the Lagrangian basis dual to these nodes, i.e., \f$ \hat{\phi}_i( \hat{{\bf q}}_j)=\delta_{ij} \f$.
A physical cell \f$\mathcal{C}\f$ with the same topology <var>T</var> as \f$\hat{\mathcal{C}}\f$ is
then defined as the image of \f$ \hat{\mathcal{C}} \f$ under the mapping
\f[
F_\mathcal{C}(\hat{\bf x}) = \sum_{m=0}^{N} {\bf q}_m(\mathcal{C}) \hat{\phi}_m(\hat{\bf x})
\f]
where \f$\{{\bf q}_0(\mathcal{C}),\ldots,{\bf q}_N(\mathcal{C})\}\f$ is the set of <strong>physical nodes</strong>
that defines \f$\mathcal{C}\f$. The number of physical nodes is required to match the number of reference
nodes in the specified cell topology <var>T</var>. The <var>i</var>-th coordinate function of the
reference-to-physical mapping is given by
\f[
\big(F_{\mathcal{C}}(\hat{\bf x})\big)_i =
\sum_{m=0}^{N} \big({\bf q}_m(\mathcal{C})\big)_i\hat{\phi}_m(\hat{\bf x})
\f]
where \f$ \big({\bf q}_m(\mathcal{C})\big)_i \f$ is the <var>i</var>-th spatial coordinate of the <var>m</var>-th node.
For simplicity, unless there's a chance for confusion, the cell symbol will be ommitted from the
designations of physical points and reference-to-physical maps, i.e., we shall simply write
\f$F(\hat{\bf x})\ \mbox{and}\ {\bf q}_m\f$.
\par Summary
\li \f$F(\hat{\bf x})\f$: implemented in Intrepid::CellTools::mapToPhysicalFrame
\li \f$F^{-1}({\bf x})\f$: implemented in Intrepid::CellTools::mapToReferenceFrame
\warning Intrepid::CellTools does not check for non-degeneracy of the physical cell obtained from a
given set of physical nodes. As a result, <var>F</var> is not guaranteed to be a diffeomorphism,
i.e., it may not have a continuously differentiable inverse. In this case some
Intrepid::CellTools methods, such as Intrepid::CellTools::setJacobianInv,
and Intrepid::CellTools::mapToReferenceFrame will fail.
\subsection sec_cell_topology_ref_map_DF Jacobian of the reference-to-physical cell mapping
Intrepid follows the convention that the rows of the Jacobian are the transposed gradients of the
coordinate functions of the mapping, i.e.,
\f[
DF_{ij}(\hat{{\bf x}}) = \frac{\partial F_i(\hat{{\bf x}})}{\partial\hat{{\bf x}}_j}
\f]
In light of the definition of <var>F</var> in Section \ref sec_cell_topology_ref_map, it follows that
\f[
DF_{ij}(\hat{{\bf x}}) = \sum_{m=0}^{N}
({\bf q}_m)_i\frac{\partial\hat{\phi}_m(\hat{{\bf x}})}{\partial\hat{{\bf x}}_j} \,.
\f]
\par Summary
\li \f$DF_{ij}(\hat{{\bf x}}) \f$: implemented in Intrepid::CellTools::setJacobian
\li \f$DF_{ij}^{-1}(\hat{{\bf x}}) \f$: implemented in Intrepid::CellTools::setJacobianInv
\li \f$\mbox{det} DF_{ij}(\hat{{\bf x}}) \f$: implemented in Intrepid::CellTools::setJacobianDet
\subsection sec_cell_topology_subcell_map Parametrization of physical 1- and 2-subcells
Parametrization of a given physical k-subcell \f$\mathcal{S}_i\f$, k=1,2, is a map from a
k-dimensional parametrization domain \e R to that subcell:
\f[
\Phi : R \mapsto \mathcal{S} \,.
\f]
Parametrization domains play role similar to that of reference cells in the sense that they allow
computation of line and surface integrals on 1- and 2-subcells (edges and faces) to be reduced to
computation of integrals on \e R .
Parametrization maps are supported for 1- and 2-subcells (edges and faces) that belong to physical
cells with reference cells. The reason is that these maps are defined by the composition of the
parametrization maps for reference edges and faces with the mapping \e F defined in \ref sec_cell_topology_ref_map.
As a result, parametrization of a given physical k-subcell requires selection of a
<strong>parent cell</strong> that contains the subcell.
\remark
Because a given k-subcell may belong to more than one physical cell, its parent cell is not unique. For a single
k-subcell the choice of a parent cell is not important, however, when dealing with subcell worksets
parent cells must all have the same topology (see \ref sec_cell_topology_subcell_wset for details about
subcell worksets).
Implementation of subcell parametrization is as follows. Assume that \f$\mathcal{S}_i\f$ is a k-subcell
with parent cell \f$\mathcal{C}\f$; \f$\mathcal{\hat{C}}\f$ is the associated reference cell and
\e i is the local ordinal of the subcell relative to the reference cell. To this physical subcell
corresponds a reference subcell \f$\hat{\mathcal{S}}_i\f$ having the same local ordinal. Parametrization of
the reference k-subcell is a map from the k-dimensional parametrization domain \e R to that subcell:
\f[
\hat{\Phi} : R \mapsto \hat{\mathcal{S}}_i \,.
\f]
Parametrization of \f$\mathcal{S}_i\f$ is then defined as
\f[
\Phi = F\circ \hat{\Phi} \,,
\f]
where \e F is the reference-to-physical mapping between the parent cell and its reference cell.
A 1-subcell (edge) always has \c Line<N> topology and so, the parametrization domain for edges is the standard 1-cube:
\f[
R = [-1,1]\,.
\f]
On the other hand, faces of reference cells can have \c Triangle<N> and/or \c Quadrilateral<N> topologies.
Thus, the parametrization domain for a 2-subcell depends on its topology and is either the standard
2-simplex or the standard 2-cube:
\f[
R = \left\{\begin{array}{rl}
\{(0,0),(1,0),(0,1)\} & \mbox{if the face is Triangle} \\[1ex]
[-1,1]\times [-1,1] & \mbox{if the face is Quadrilateral}
\end{array}\right.
\f]
\par Summary
- \f$ \Phi : R \mapsto {\mathcal{S}}_i \f$ requires two steps:
-# Intrepid::CellTools::mapToReferenceSubcell to apply \f$\hat{\Phi}: R \mapsto \hat{\mathcal{S}}_i\f$;
-# Intrepid::CellTools::mapToPhysicalFrame to apply \e F
\subsection sec_cell_topology_subcell_wset Subcell worksets
A subcell workset comprises of 1- or 2-subcells and associated parent cells that satisfy the
following conditions
\li all subcells have the same cell topology;
\li all parent cells have the same cell topology;
\li The parent cell topology has a reference cell;
\li relative to that reference cell, all subcells in the workset have the same local ordinal
Therefore, a subcell workset is defined by
-# collecting a set of 1- or 2-subcells having the same topology
-# selecting a parent cell for every subcell in such a way that
-# all parent cells have the same cell topology
-# all subcells in the workset have the same local ordinal relative to the parent cell topology
Obviously, a subcell can have multiple parent cells. For example, in a mesh consisiting of Triangle<3> cells, every
edge is shared by 2 triangle cells. To define an edge workset we can use either one of the two traingles
sharing the cell.
Suppose now that the mesh comprises of Triangle<3> and Quadrilateral<4> cells and we want to define an
edge workset. Let's say the first few edges in our workset happen to be shared by 2 triangles and so
we select one of them as the parent cell. Now suppose the next edge is shared by a traingle and a
quadrilateral. Because all parent cells in the workset must have the same cell topology we cannot use
the quadrilateral as a parent cell and so we choose the triangle. Finally suppose that one of the candidate
edges for our workset is shared by 2 quadrilaterals. Because of the requirement that all parent cells
have the same topology, we will have to reject this edge because it does not posses a potential parent cell
with the same topology as the rest of the edges in our workset.
A subcell workset is denoted by \f$ \{\mathcal{S}_{c,i}\}_{c=0}^N \f$, where
\li <var>c</var> is parent cell ordinal;
\li <var>i</var> is the local subcell ordinal (relative to the topology of the parent cell) shared
by all subcells in the workset.
*/
#endif
|