/usr/include/trilinos/Intrepid_FunctionSpaceTools.hpp is in libtrilinos-intrepid-dev 12.4.2-2.
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 | // @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_FunctionSpaceTools.hpp
\brief Header file for the Intrepid::FunctionSpaceTools class.
\author Created by P. Bochev and D. Ridzal.
*/
#ifndef INTREPID_FUNCTIONSPACETOOLS_HPP
#define INTREPID_FUNCTIONSPACETOOLS_HPP
#include "Intrepid_ConfigDefs.hpp"
#include "Intrepid_ArrayTools.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_CellTools.hpp"
#include <KokkosRank.hpp>
namespace Intrepid {
/** \class Intrepid::FunctionSpaceTools
\brief Defines expert-level interfaces for the evaluation of functions
and operators in physical space (supported for FE, FV, and FD methods)
and FE reference space; in addition, provides several function
transformation utilities.
*/
class FunctionSpaceTools {
public:
/** \brief Transformation of a (scalar) value field in the H-grad space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of \e HGRAD functions \f$\Phi^*(\widehat{u}_f) = \widehat{u}_f\circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the function set \f$\{\widehat{u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p)
= \widehat{u}_f\circ F^{-1}_{c}(x_{c,p})
= \widehat{u}_f(\widehat{x}_p) = inVals(f,p) \qquad 0\le c < C \,,
\f]
i.e., it simply replicates the values in the user-provided container to every cell.
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeIn>
static void HGRADtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeIn & inVals);
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeIn>
static void HGRADtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeIn & inVals);*/
/** \brief Transformation of a gradient field in the H-grad space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of gradients of \e HGRAD functions
\f$\Phi^*(\nabla\widehat{u}_f) = \left((DF_c)^{-{\sf T}}\cdot\nabla\widehat{u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the gradients of the function set \f$\{\widehat{u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \nabla\widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left((DF_c)^{-{\sf T}}\cdot\nabla\widehat{u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= (DF_c)^{-{\sf T}}(\widehat{x}_p)\cdot\nabla\widehat{u}_f(\widehat{x}_p) \qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HGRADtransformGRAD(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HGRADtransformGRADTemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
*/
/** \brief Transformation of a (vector) value field in the H-curl space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of \e HCURL functions
\f$\Phi^*(\widehat{\bf u}_f) = \left((DF_c)^{-{\sf T}}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left((DF_c)^{-{\sf T}}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= (DF_c)^{-{\sf T}}(\widehat{x}_p)\cdot\widehat{\bf u}_f(\widehat{x}_p) \qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of native basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HCURLtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HCURLtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');*/
/** \brief Transformation of a curl field in the H-curl space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of curls of \e HCURL functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c} DF_{c}\cdot\nabla\times\widehat{\bf u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the curls of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \nabla\times\widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c} DF_{c}\cdot\nabla\times\widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) DF_{c}(\widehat{x}_p)\cdot\nabla\times\widehat{\bf u}_f(\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HCURLtransformCURL(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HCURLtransformCURLTemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');*/
/** \brief Transformation of a (vector) value field in the H-div space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of \e HDIV functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c} DF_{c}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c} DF_{c}\cdot \widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) DF_{c}(\widehat{x}_p)\cdot\widehat{\bf u}_f(\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');*/
/** \brief Transformation of a divergence field in the H-div space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of the divergence of \e HDIV functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c}\nabla\cdot\widehat{\bf u}_{f}\right) \circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the divergencies of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \nabla\cdot\widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c}\nabla\cdot\widehat{\bf u}_{f}\right) \circ F^{-1}_{c} (x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) \nabla\cdot\widehat{\bf u}_{f} (\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformDIV(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformDIVTemp(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
*/
/** \brief Transformation of a (scalar) value field in the H-vol space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of \e HVOL functions
\f$\Phi^*(\widehat{u}_f) = \left(J^{-1}_{c}\widehat{u}_{f}\right) \circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the functions in the set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c}\widehat{u}_{f}\right) \circ F^{-1}_{c} (x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) \widehat{u}_{f} (\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HVOLtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
/** \brief Contracts \a <b>leftValues</b> and \a <b>rightValues</b> arrays on the
point and possibly space dimensions and stores the result in \a <b>outputValues</b>;
this is a generic, high-level integration routine that calls either
FunctionSpaceTools::operatorIntegral, or FunctionSpaceTools::functionalIntegral,
or FunctionSpaceTools::dataIntegral methods, depending on the rank of the
\a <b>outputValues</b> array.
\param outputValues [out] - Output array.
\param leftValues [in] - Left input array.
\param rightValues [in] - Right input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar>
static void integrate(Intrepid::FieldContainer<Scalar> & outputValues,
const Intrepid::FieldContainer<Scalar> & leftValues,
const Intrepid::FieldContainer<Scalar> & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight>
static void integrate(ArrayOut & outputValues,
const ArrayInLeft & leftValues,
const ArrayInRight & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight>
static void integrateTemp(ArrayOut & outputValues,
const ArrayInLeft & leftValues,
const ArrayInRight & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
*/
template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight,int leftrank,int outrank>
struct integrateTempSpec;
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of
two rank-3, 4, or 5 containers with dimensions (C,L,P) and (C,R,P),
or (C,L,P,D1) and (C,R,P,D1), or (C,L,P,D1,D2) and (C,R,P,D1,D2),
and returns the result in a rank-3 container with dimensions (C,L,R).
For a fixed index "C", (C,L,R) represents a rectangular L X R matrix
where L and R may be different.
\code
C - num. integration domains dim0 in both input containers
L - num. "left" fields dim1 in "left" container
R - num. "right" fields dim1 in "right" container
P - num. integration points dim2 in both input containers
D1- vector (1st tensor) dimension dim3 in both input containers
D2- 2nd tensor dimension dim4 in both input containers
\endcode
\param outputFields [out] - Output array.
\param leftFields [in] - Left input array.
\param rightFields [in] - Right input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInFieldsLeft, class ArrayInFieldsRight>
static void operatorIntegral(ArrayOutFields & outputFields,
const ArrayInFieldsLeft & leftFields,
const ArrayInFieldsRight & rightFields,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutFields, class ArrayInFieldsLeft, class ArrayInFieldsRight>
static void operatorIntegralTemp(ArrayOutFields & outputFields,
const ArrayInFieldsLeft & leftFields,
const ArrayInFieldsRight & rightFields,
const ECompEngine compEngine,
const bool sumInto = false);*/
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of a
rank-3, 4, or 5 container and a rank-2, 3, or 4 container, respectively,
with dimensions (C,F,P) and (C,P), or (C,F,P,D1) and (C,P,D1),
or (C,F,P,D1,D2) and (C,P,D1,D2), respectively, and returns the result
in a rank-2 container with dimensions (C,F).
For a fixed index "C", (C,F) represents a (column) vector of length F.
\code
C - num. integration domains dim0 in both input containers
F - num. fields dim1 in fields input container
P - num. integration points dim2 in fields input container and dim1 in tensor data container
D1 - first spatial (tensor) dimension index dim3 in fields input container and dim2 in tensor data container
D2 - second spatial (tensor) dimension index dim4 in fields input container and dim3 in tensor data container
\endcode
\param outputFields [out] - Output fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void functionalIntegral(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void functionalIntegralTemp(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const ECompEngine compEngine,
const bool sumInto = false);
*/
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of two
rank-2, 3, or 4 containers with dimensions (C,P), or (C,P,D1), or
(C,P,D1,D2), respectively, and returns the result in a rank-1 container
with dimensions (C).
\code
C - num. integration domains dim0 in both input containers
P - num. integration points dim1 in both input containers
D1 - first spatial (tensor) dimension index dim2 in both input containers
D2 - second spatial (tensor) dimension index dim3 in both input containers
\endcode
\param outputData [out] - Output data array.
\param inputDataLeft [in] - Left data input array.
\param inputDataRight [in] - Right data input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dataIntegral(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dataIntegralTemp(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const ECompEngine compEngine,
const bool sumInto = false);
*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of cell integrals, by multiplying absolute values
of the user-provided cell Jacobian determinants \a <b>inDet</b> with dimensions (C,P)
with the user-provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) = |\mbox{det}(DF_{c}(\widehat{x}_p))|\omega_{p} \,,
\f]
where \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on a reference cell
(a set of integration points and their associated weights; see
Intrepid::Cubature::getCubature for getting cubature rules on reference cells).
\warning
The user is responsible for providing input arrays with consistent data: the determinants
in \a <b>inDet</b> should be evaluated at integration points on the <b>reference cell</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all containers
P - num. integration points dim1 in all containers
\endcode
\param outVals [out] - Output array with weighted cell measures.
\param inDet [in] - Input array containing determinants of cell Jacobians.
\param inWeights [in] - Input integration weights.
*/
template<class Scalar, class ArrayOut, class ArrayDet, class ArrayWeights>
static void computeCellMeasure(ArrayOut & outVals,
const ArrayDet & inDet,
const ArrayWeights & inWeights);
/*template<class Scalar, class ArrayOut, class ArrayDet, class ArrayWeights>
static void computeCellMeasureTemp(ArrayOut & outVals,
const ArrayDet & inDet,
const ArrayWeights & inWeights);*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of face integrals, based on the provided
cell Jacobian array \a <b>inJac</b> with dimensions (C,P,D,D) and the
provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) =
\left\|\frac{\partial\Phi_c(\widehat{x}_p)}{\partial u}\times
\frac{\partial\Phi_c(\widehat{x}_p)}{\partial v}\right\|\omega_{p} \,,
\f]
where:
\li \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on \b reference
\b face \f$\widehat{\mathcal{F}}\f$, with ordinal \e whichFace relative to the specified parent reference cell;
\li \f$ \Phi_c : R \mapsto \mathcal{F} \f$ is parameterization of the physical face
corresponding to \f$\widehat{\mathcal{F}}\f$; see Section \ref sec_cell_topology_subcell_map.
\warning
The user is responsible for providing input arrays with consistent data: the Jacobians
in \a <b>inJac</b> should be evaluated at integration points on the <b>reference face</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
Cubature rules on reference faces are defined by a two-step process:
\li A cubature rule is defined on the parametrization domain \e R of the face
(\e R is the standard 2-simplex {(0,0),(1,0),(0,1)} or the standard 2-cube [-1,1] X [-1,1]).
\li The points are mapped to a reference face using Intrepid::CellTools::mapToReferenceSubcell
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all input containers
P - num. integration points dim1 in all input containers
D - spatial dimension dim2 and dim3 in Jacobian container
\endcode
\param outVals [out] - Output array with weighted face measures.
\param inJac [in] - Input array containing cell Jacobians.
\param inWeights [in] - Input integration weights.
\param whichFace [in] - Index of the face subcell relative to the parent cell; defines the domain of integration.
\param parentCell [in] - Parent cell topology.
*/
template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeFaceMeasure(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichFace,
const shards::CellTopology & parentCell);
/* template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeFaceMeasureTemp(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichFace,
const shards::CellTopology & parentCell);*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of edge integrals, based on the provided
cell Jacobian array \a <b>inJac</b> with dimensions (C,P,D,D) and the
provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) =
\left\|\frac{d \Phi_c(\widehat{x}_p)}{d s}\right\|\omega_{p} \,,
\f]
where:
\li \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on \b reference
\b edge \f$\widehat{\mathcal{E}}\f$, with ordinal \e whichEdge relative to the specified parent reference cell;
\li \f$ \Phi_c : R \mapsto \mathcal{E} \f$ is parameterization of the physical edge
corresponding to \f$\widehat{\mathcal{E}}\f$; see Section \ref sec_cell_topology_subcell_map.
\warning
The user is responsible for providing input arrays with consistent data: the Jacobians
in \a <b>inJac</b> should be evaluated at integration points on the <b>reference edge</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
Cubature rules on reference edges are defined by a two-step process:
\li A cubature rule is defined on the parametrization domain \e R = [-1,1] of the edge.
\li The points are mapped to a reference edge using Intrepid::CellTools::mapToReferenceSubcell
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all input containers
P - num. integration points dim1 in all input containers
D - spatial dimension dim2 and dim3 in Jacobian container
\endcode
\param outVals [out] - Output array with weighted edge measures.
\param inJac [in] - Input array containing cell Jacobians.
\param inWeights [in] - Input integration weights.
\param whichEdge [in] - Index of the edge subcell relative to the parent cell; defines the domain of integration.
\param parentCell [in] - Parent cell topology.
*/
template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeEdgeMeasure(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichEdge,
const shards::CellTopology & parentCell);
/* template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeEdgeMeasureTemp(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichEdge,
const shards::CellTopology & parentCell);*/
/** \brief Multiplies fields \a <b>inVals</b> by weighted measures \a <b>inMeasure</b> and
returns the field array \a <b>outVals</b>; this is a simple redirection to the call
FunctionSpaceTools::scalarMultiplyDataField.
\param outVals [out] - Output array with scaled field values.
\param inMeasure [in] - Input array containing weighted measures.
\param inVals [in] - Input fields.
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeMeasure, class ArrayTypeIn>
static void multiplyMeasure(ArrayTypeOut & outVals,
const ArrayTypeMeasure & inMeasure,
const ArrayTypeIn & inVals);
/* template<class Scalar, class ArrayTypeOut, class ArrayTypeMeasure, class ArrayTypeIn>
static void multiplyMeasureTemp(ArrayTypeOut & outVals,
const ArrayTypeMeasure & inMeasure,
const ArrayTypeIn & inVals);*/
/** \brief Scalar multiplication of data and fields; please read the description below.
There are two use cases:
\li
multiplies a rank-3, 4, or 5 container \a <b>inputFields</b> with dimensions (C,F,P),
(C,F,P,D1) or (C,F,P,D1,D2), representing the values of a set of scalar, vector
or tensor fields, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P),
representing the values of scalar data, OR
\li
multiplies a rank-2, 3, or 4 container \a <b>inputFields</b> with dimensions (F,P),
(F,P,D1) or (F,P,D1,D2), representing the values of a scalar, vector or a
tensor field, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P),
representing the values of scalar data;
the output value container \a <b>outputFields</b> is indexed by (C,F,P), (C,F,P,D1)
or (C,F,P,D1,D2), regardless of which of the two use cases is considered.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\note The argument <var><b>inputFields</b></var> can be changed!
This enables in-place multiplication.
\param outputFields [out] - Output (product) fields array.
\param inputData [in] - Data (multiplying) array.
\param inputFields [in] - Input (being multiplied) fields array.
\param reciprocal [in] - If TRUE, <b>divides</b> input fields by the data
(instead of multiplying). Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void scalarMultiplyDataField(ArrayOutFields & outputFields,
ArrayInData & inputData,
ArrayInFields & inputFields,
const bool reciprocal = false);
/** \brief Scalar multiplication of data and data; please read the description below.
There are two use cases:
\li
multiplies a rank-2, 3, or 4 container \a <b>inputDataRight</b> with dimensions (C,P),
(C,P,D1) or (C,P,D1,D2), representing the values of a set of scalar, vector
or tensor data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P),
representing the values of scalar data, OR
\li
multiplies a rank-1, 2, or 3 container \a <b>inputDataRight</b> with dimensions (P),
(P,D1) or (P,D1,D2), representing the values of scalar, vector or
tensor data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P),
representing the values of scalar data;
the output value container \a <b>outputData</b> is indexed by (C,P), (C,P,D1) or (C,P,D1,D2),
regardless of which of the two use cases is considered.
\code
C - num. integration domains
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\note The arguments <var><b>inputDataLeft</b></var>, <var><b>inputDataRight</b></var> can be changed!
This enables in-place multiplication.
\param outputData [out] - Output data array.
\param inputDataLeft [in] - Left (multiplying) data array.
\param inputDataRight [in] - Right (being multiplied) data array.
\param reciprocal [in] - If TRUE, <b>divides</b> input fields by the data
(instead of multiplying). Default: FALSE.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void scalarMultiplyDataData(ArrayOutData & outputData,
ArrayInDataLeft & inputDataLeft,
ArrayInDataRight & inputDataRight,
const bool reciprocal = false);
/** \brief Dot product of data and fields; please read the description below.
There are two use cases:
\li
dot product of a rank-3, 4 or 5 container \a <b>inputFields</b> with dimensions (C,F,P)
(C,F,P,D1) or (C,F,P,D1,D2), representing the values of a set of scalar, vector
or tensor fields, by the values in a rank-2, 3 or 4 container \a <b>inputData</b> indexed by
(C,P), (C,P,D1), or (C,P,D1,D2) representing the values of scalar, vector or
tensor data, OR
\li
dot product of a rank-2, 3 or 4 container \a <b>inputFields</b> with dimensions (F,P),
(F,P,D1) or (F,P,D1,D2), representing the values of a scalar, vector or tensor
field, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P), (C,P,D1) or
(C,P,D1,D2), representing the values of scalar, vector or tensor data;
the output value container \a <b>outputFields</b> is indexed by (C,F,P),
regardless of which of the two use cases is considered.
For input fields containers without a dimension index, this operation reduces to
scalar multiplication.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\param outputFields [out] - Output (dot product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void dotMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields);
/** \brief Dot product of data and data; please read the description below.
There are two use cases:
\li
dot product of a rank-2, 3 or 4 container \a <b>inputDataRight</b> with dimensions (C,P)
(C,P,D1) or (C,P,D1,D2), representing the values of a scalar, vector or a
tensor set of data, by the values in a rank-2, 3 or 4 container \a <b>inputDataLeft</b> indexed by
(C,P), (C,P,D1), or (C,P,D1,D2) representing the values of scalar, vector or
tensor data, OR
\li
dot product of a rank-2, 3 or 4 container \a <b>inputDataRight</b> with dimensions (P),
(P,D1) or (P,D1,D2), representing the values of scalar, vector or tensor
data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D1) or
(C,P,D1,D2), representing the values of scalar, vector, or tensor data;
the output value container \a <b>outputData</b> is indexed by (C,P),
regardless of which of the two use cases is considered.
For input fields containers without a dimension index, this operation reduces to
scalar multiplication.
\code
C - num. integration domains
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\param outputData [out] - Output (dot product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dotMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight);
/** \brief Cross or outer product of data and fields; please read the description below.
There are four use cases:
\li
cross product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-3
container \a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
cross product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-3 container
\a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-3
container \a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-3 container
\a <b>inputData</b> indexed by (C,P,D), representing the values of vector data;
for cross products, the output value container \a <b>outputFields</b> is indexed by
(C,F,P,D) in 3D (vector output) and by (C,F,P) in 2D (scalar output);
for outer products, the output value container \a <b>outputFields</b> is indexed by (C,F,P,D,D).
\code
C - num. integration domains
F - num. fields
P - num. integration points
D - spatial dimension, must be 2 or 3
\endcode
\param outputFields [out] - Output (cross or outer product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void vectorMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields);
/** \brief Cross or outer product of data and data; please read the description below.
There are four use cases:
\li
cross product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-3
container \a <b>inputDataLeft</b> indexed by (C,P,D) representing the values of vector data, OR
\li
cross product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-3 container
\a <b>inputDataLeft</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-3
container \a <b>inputDataLeft</b> indexed by (C,P,D) representing the values of vector data, OR
\li
outer product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-3 container
\a <b>inputDataLeft</b> indexed by (C,P,D), representing the values of vector data;
for cross products, the output value container \a <b>outputData</b> is indexed by
(C,P,D) in 3D (vector output) and by (C,P) in 2D (scalar output);
for outer products, the output value container \a <b>outputData</b> is indexed by (C,P,D,D).
\code
C - num. integration domains
P - num. integration points
D - spatial dimension, must be 2 or 3
\endcode
\param outputData [out] - Output (cross or outer product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void vectorMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight);
/** \brief Matrix-vector or matrix-matrix product of data and fields; please read the description below.
There are four use cases:
\li
matrix-vector product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-vector product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-5 container \a <b>inputFields</b> with dimensions (C,F,P,D,D),
representing the values of a set of tensor fields, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-4 container \a <b>inputFields</b> with dimensions (F,P,D,D),
representing the values of a tensor field, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data;
for matrix-vector products, the output value container \a <b>outputFields</b> is
indexed by (C,F,P,D);
for matrix-matrix products the output value container \a <b>outputFields</b> is
indexed by (C,F,P,D,D).
\remarks
The rank of \a <b>inputData</b> implicitly defines the type of tensor data:
\li rank = 2 corresponds to a constant diagonal tensor \f$ diag(a,\ldots,a) \f$
\li rank = 3 corresponds to a nonconstant diagonal tensor \f$ diag(a_1,\ldots,a_d) \f$
\li rank = 4 corresponds to a full tensor \f$ \{a_{ij}\}\f$
\note It is assumed that all tensors are square!
\note The method is defined for spatial dimensions D = 1, 2, 3
\code
C - num. integration domains
F - num. fields
P - num. integration points
D - spatial dimension
\endcode
\param outputFields [out] - Output (matrix-vector or matrix-matrix product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
\param transpose [in] - If 'T', use transposed left data tensor; if 'N', no transpose. Default: 'N'.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void tensorMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const char transpose = 'N');
/** \brief Matrix-vector or matrix-matrix product of data and data; please read the description below.
There are four use cases:
\li
matrix-vector product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-vector product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-4 container \a <b>inputDataRight</b> with dimensions (C,P,D,D),
representing the values of a set of tensor data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-3 container \a <b>inputDataRight</b> with dimensions (P,D,D),
representing the values of tensor data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data;
for matrix-vector products, the output value container \a <b>outputData</b>
is indexed by (C,P,D);
for matrix-matrix products, the output value container \a <b>outputData</b>
is indexed by (C,P,D1,D2).
\remarks
The rank of <b>inputDataLeft</b> implicitly defines the type of tensor data:
\li rank = 2 corresponds to a constant diagonal tensor \f$ diag(a,\ldots,a) \f$
\li rank = 3 corresponds to a nonconstant diagonal tensor \f$ diag(a_1,\ldots,a_d) \f$
\li rank = 4 corresponds to a full tensor \f$ \{a_{ij}\}\f$
\note It is assumed that all tensors are square!
\note The method is defined for spatial dimensions D = 1, 2, 3
\code
C - num. integration domains
P - num. integration points
D - spatial dimension
\endcode
\param outputData [out] - Output (matrix-vector product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
\param transpose [in] - If 'T', use transposed tensor; if 'N', no transpose. Default: 'N'.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void tensorMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const char transpose = 'N');
/* template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void tensorMultiplyDataDataTemp(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const char transpose = 'N');
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight,int outvalRank>
struct tensorMultiplyDataDataTempSpec;
/** \brief Applies left (row) signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,L), to the operator
<var><b>inoutOperator</b></var> indexed by (C,L,R).
Mathematically, this method computes the matrix-matrix product
\f[
\mathbf{K}^{c} = \mbox{diag}(\sigma^c_0,\ldots,\sigma^c_{L-1}) \mathbf{K}^c
\f]
where \f$\mathbf{K}^{c} \in \mathbf{R}^{L\times R}\f$ is array of matrices indexed by
cell number \e c and stored in the rank-3 array \e inoutOperator, and
\f$\{\sigma^c_l\}_{l=0}^{L-1}\f$ is array of left field signs indexed by cell number \e c
and stored in the rank-2 container \e fieldSigns;
see Section \ref sec_pullbacks for discussion of field signs. This operation is
required for operators generated by \e HCURL and \e HDIV-conforming vector-valued
finite element basis functions; see Sections \ref sec_pullbacks and Section
\ref sec_ops for applications of this method.
\code
C - num. integration domains
L - num. left fields
R - num. right fields
\endcode
\param inoutOperator [in/out] - Input / output operator array.
\param fieldSigns [in] - Left field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyLeftFieldSigns(ArrayTypeInOut & inoutOperator,
const ArrayTypeSign & fieldSigns);
/** \brief Applies right (column) signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,R), to the operator
<var><b>inoutOperator</b></var> indexed by (C,L,R).
Mathematically, this method computes the matrix-matrix product
\f[
\mathbf{K}^{c} = \mathbf{K}^c \mbox{diag}(\sigma^c_0,\ldots,\sigma^c_{R-1})
\f]
where \f$\mathbf{K}^{c} \in \mathbf{R}^{L\times R}\f$ is array of matrices indexed by
cell number \e c and stored in the rank-3 container \e inoutOperator, and
\f$\{\sigma^c_r\}_{r=0}^{R-1}\f$ is array of right field signs indexed by cell number \e c
and stored in the rank-2 container \e fieldSigns;
see Section \ref sec_pullbacks for discussion of field signs. This operation is
required for operators generated by \e HCURL and \e HDIV-conforming vector-valued
finite element basis functions; see Sections \ref sec_pullbacks and Section
\ref sec_ops for applications of this method.
\code
C - num. integration domains
L - num. left fields
R - num. right fields
\endcode
\param inoutOperator [in/out] - Input / output operator array.
\param fieldSigns [in] - Right field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyRightFieldSigns(ArrayTypeInOut & inoutOperator,
const ArrayTypeSign & fieldSigns);
/** \brief Applies field signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,F), to the function
<var><b>inoutFunction</b></var> indexed by (C,F), (C,F,P),
(C,F,P,D1) or (C,F,P,D1,D2).
Returns
\f[
\mbox{inoutFunction}(c,f,*) = \mbox{fieldSigns}(c,f)*\mbox{inoutFunction}(c,f,*)
\f]
See Section \ref sec_pullbacks for discussion of field signs.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - spatial dimension
D2 - spatial dimension
\endcode
\param inoutFunction [in/out] - Input / output function array.
\param fieldSigns [in] - Right field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyFieldSigns(ArrayTypeInOut & inoutFunction,
const ArrayTypeSign & fieldSigns);
/* template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyFieldSignsTemp(ArrayTypeInOut & inoutFunction,
const ArrayTypeSign & fieldSigns);
*/
/** \brief Computes point values \a <b>outPointVals</b> of a discrete function
specified by the basis \a <b>inFields</b> and coefficients
\a <b>inCoeffs</b>.
The array \a <b>inFields</b> with dimensions (C,F,P), (C,F,P,D1),
or (C,F,P,D1,D2) represents the signed, transformed field (basis) values at
points in REFERENCE frame; the \a <b>outPointVals</b> array with
dimensions (C,P), (C,P,D1), or (C,P,D1,D2), respectively, represents
values of a discrete function at points in PHYSICAL frame.
The array \a <b>inCoeffs</b> dimensioned (C,F) supplies the coefficients
for the field (basis) array.
Returns rank-2,3 or 4 array such that
\f[
outPointValues(c,p,*) = \sum_{f=0}^{F-1} \sigma_{c,f} u_{c,f}(x_p)
\f]
where \f$\{u_{c,f}\}_{f=0}^{F-1} \f$ is scalar, vector or tensor valued finite element
basis defined on physical cell \f$\mathcal{C}\f$ and \f$\{\sigma_{c,f}\}_{f=0}^{F-1} \f$
are the field signs of the basis functions; see Section \ref sec_pullbacks.
This method implements the last step in a four step process; please see Section
\ref sec_evaluate for details about the first three steps that prepare the
necessary data for this method.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - spatial dimension
D2 - spatial dimension
\endcode
\param outPointVals [out] - Output point values of a discrete function.
\param inCoeffs [in] - Coefficients associated with the fields (basis) array.
\param inFields [in] - Field (basis) values.
*/
template<class Scalar, class ArrayOutPointVals, class ArrayInCoeffs, class ArrayInFields>
static void evaluate(ArrayOutPointVals & outPointVals,
const ArrayInCoeffs & inCoeffs,
const ArrayInFields & inFields);
}; // end FunctionSpaceTools
} // end namespace Intrepid
// include templated definitions
#include <Intrepid_FunctionSpaceToolsDef.hpp>
#endif
/***************************************************************************************************
** **
** D O C U M E N T A T I O N P A G E S **
** **
**************************************************************************************************/
/**
\page function_space_tools_page Function space tools
<b>Table of contents </b>
\li \ref sec_fst_overview
\li \ref sec_pullbacks
\li \ref sec_measure
\li \ref sec_evaluate
\section sec_fst_overview Overview
Intrepid::FunctionSpaceTools is a stateless class of \e expert \e methods for operations on finite
element subspaces of \f$H(grad,\Omega)\f$, \f$H(curl,\Omega)\f$, \f$H(div,\Omega)\f$ and \f$L^2(\Omega)\f$.
In Intrepid these spaces are referred to as \e HGRAD, \e HCURL, \e HDIV and \e HVOL. There are four
basic groups of methods:
- Transformation methods provide implementation of pullbacks for \e HGRAD, \e HCURL, \e HDIV and \e HVOL
finite element functions. Thease are essentialy the "change of variables rules" needed to transform
values of basis functions and their derivatives defined on a reference element \f$\widehat{{\mathcal C}}\f$
to a physical element \f${\mathcal C}\f$. See Section \ref sec_pullbacks for details
- Measure computation methods implement the volume, surface and line measures required for computation
of integrals in the physical frame by changing variables to reference frame. See Section \ref sec_measure
for details.
- Integration methods implement the algebraic operations to compute ubiquitous integrals of finite element
functions: integrals arising in bilinear forms and linear functionals.
- Methods for algebraic and vector-algebraic operations on multi-dimensional arrays with finite element
function values. These methods are used to prepare multidimensional arrays with data and finite
element function values for the integration routines. They also include evaluation methods to compute
finite element function values at some given points in physical frame; see Section \ref sec_evaluate.
\section sec_pullbacks Pullbacks
Notation in this section follows the standard definition of a finite element space by Ciarlet; see
<var> The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, SIAM, 2002. </var>
Given a reference cell \f$\{\widehat{{\mathcal C}},\widehat{P},\widehat{\Lambda}\}\f$ with a basis
\f$\{\widehat{u}_i\}_{i=0}^n\f$, the basis \f$\{{u}_i\}_{i=0}^n\f$ of \f$\{{\mathcal C},P,\Lambda\}\f$ is defined
as follows:
\f[
u_i = \sigma_i \Phi^*(\widehat{u}_i), \qquad i=1,\ldots,n \,.
\f]
In this formula \f$\{\sigma_i\}_{i=0}^n\f$, where \f$\sigma_i = \pm 1\f$, are the \e field \e signs,
and \f$\Phi^*\f$ is the \e pullback ("change of variables") transformation. For scalar spaces
such as \e HGRAD and \e HVOL the field signs are always equal to 1 and can be disregarded. For vector
field spaces such as \e HCURL or \e HDIV, the field sign of a basis function can be +1 or -1,
depending on the orientation of the physical edge or face, associated with the basis function.
The actual form of the pullback depends on which one of the four function spaces \e HGRAD, \e HCURL,
\e HDIV and \e HVOL is being approximated and is computed as follows. Let \f$F_{\mathcal C}\f$
denote the reference-to-physical map (see Section \ref sec_cell_topology_ref_map);
\f$DF_{\mathcal C}\f$ is its Jacobian (see Section \ref sec_cell_topology_ref_map_DF) and
\f$J_{\mathcal C} = \det(DF_{\mathcal C})\f$. Then,
\f[
\begin{array}{ll}
\Phi^*_G : HGRAD(\widehat{{\mathcal C}}) \mapsto HGRAD({\mathcal C})&
\qquad \Phi^*_G(\widehat{u}) = \widehat{u}\circ F^{-1}_{\mathcal C} \\[2ex]
\Phi^*_C : HCURL(\widehat{{\mathcal C}}) \mapsto HCURL({\mathcal C})&
\qquad \Phi^*_C(\widehat{\bf u}) = \left((DF_{\mathcal C})^{-{\sf T}}\cdot\widehat{\bf u}\right)\circ F^{-1}_{\mathcal C} \\[2ex]
\Phi^*_D : HDIV(\widehat{{\mathcal C}}) \mapsto HDIV({\mathcal C})&
\qquad \Phi^*_D(\widehat{\bf u}) = \left(J^{-1}_{\mathcal C} DF_{\mathcal C}\cdot\widehat{\bf u}\right)\circ F^{-1}_{\mathcal C}
\\[2ex]
\Phi^*_S : HVOL(\widehat{{\mathcal C}}) \mapsto HVOL({\mathcal C})&
\qquad \Phi^*_S(\widehat{u}) = \left(J^{-1}_{\mathcal C} \widehat{u}\right) \circ F^{-1}_{\mathcal C} \,.
\end{array}
\f]
Intrepid supports pullbacks only for cell topologies that have reference cells; see
\ref cell_topology_ref_cells.
\section sec_measure Measure
In Intrepid integrals of finite element functions over cells, 2-subcells (faces) and 1-subcells (edges)
are computed by change of variables to reference frame and require three different kinds of measures.
-# The integral of a scalar function over a cell \f${\mathcal C}\f$
\f[
\int_{{\mathcal C}} f(x) dx = \int_{\widehat{{\mathcal C}}} f(F(\widehat{x})) |J | d\widehat{x}
\f]
requires the volume measure defined by the determinant of the Jacobian. This measure is computed
by Intrepid::FunctionSpaceTools::computeCellMeasure
-# The integral of a scalar function over 2-subcell \f$\mathcal{F}\f$
\f[
\int_{\mathcal{F}} f(x) dx = \int_{R} f(\Phi(u,v))
\left\|\frac{\partial\Phi}{\partial u}\times \frac{\partial\Phi}{\partial v}\right\| du\,dv
\f]
requires the surface measure defined by the norm of the vector product of the surface tangents. This
measure is computed by Intrepid::FunctionSpaceTools::computeFaceMeasure. In this formula \e R is the parametrization
domain for the 2-subcell; see Section \ref sec_cell_topology_subcell_map for details.
-# The integral of a scalar function over a 1-subcell \f$\mathcal{E}\f$
\f[
\int_{\mathcal{E}} f(x) dx = \int_{R} f(\Phi(s)) \|\Phi'\| ds
\f]
requires the arc measure defined by the norm of the arc tangent vector. This measure is computed
by Intrepid::FunctionSpaceTools::computeEdgeMeasure. In this formula \e R is the parametrization
domain for the 1-subcell; see Section \ref sec_cell_topology_subcell_map for details.
\section sec_evaluate Evaluation of finite element fields
To make this example more specific, assume curl-conforming finite element spaces.
Suppose that we have a physical cell \f$\{{\mathcal C},P,\Lambda\}\f$ with a basis
\f$\{{\bf u}_i\}_{i=0}^n\f$. A finite element function on this cell is defined by a set of \e n
coefficients \f$\{c_i\}_{i=0}^n\f$:
\f[
{\bf u}^h(x) = \sum_{i=0}^n c_i {\bf u}_i(x) \,.
\f]
From Section \ref sec_pullbacks it follows that
\f[
{\bf u}^h(x) = \sum_{i=0}^n c_i \sigma_i
\left((DF_{\mathcal C})^{-{\sf T}}\cdot\widehat{\bf u}_i\right)\circ
F^{-1}_{\mathcal C}(x)
= \sum_{i=0}^n c_i \sigma_i
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,,
\f]
where \f$ \widehat{x} = F^{-1}_{\mathcal C}(x) \in \widehat{\mathcal C} \f$ is the pre-image
of \e x in the reference cell.
Consequently, evaluation of finite element functions at a given set of points
\f$\{x_p\}_{p=0}^P \subset {\mathcal C}\f$ comprises of the following four steps:
-# Application of the inverse map \f$F^{-1}_{\mathcal C}\f$ to obtain the pre-images
\f$\{\widehat{x}_p\}_{p=0}^P\f$ of the evaluation points in the reference cell
\f$\widehat{\mathcal{C}}\f$; see Intrepid::CellTools::mapToReferenceFrame
-# Evaluation of the appropriate reference basis set \f$\{\widehat{\bf u}_i\}_{i=1}^n\f$
at the pre-image set \f$\{\widehat{x}_p\}_{p=0}^P\f$; see Intrepid::Basis::getValues
-# Application of the appropriate transformation and field signs. In our example the finite
element space is curl-conforming and the appropriate transformation is implemented in
Intrepid::FunctionSpaceTools::HCURLtransformVALUE. Application of the signs to the
transformed functions is done by Intrepid::FunctionSpaceTools::applyFieldSigns.
-# The final step is to compute the sum of the transformed and signed basis function values
multiplied by the coefficients of the finite element function using
Intrepid::FunctionSpaceTools::evaluate.
Evaluation of adimssible derivatives of finite element functions is completely analogous
and follows the same four steps. Evaluation of scalar finite element functions is simpler
because application of the signes can be skipped for these functions.
\section sec_ops Evaluation of finite element operators and functionals
Assume the same setting as in Section \ref sec_evaluate. A finite element operator defined
by the finite element basis on the physical cell \f$\mathcal{C}\f$ is a matrix
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} = \int_{\mathcal C} {\mathcal L}_L {\bf u}_i(x)\, {\mathcal L}_R {\bf u}_j(x) \, dx \,.
\f]
where \f${\mathcal L}_L\f$ and \f${\mathcal L}_R \f$ are \e left and \e right operators acting on the basis
functions. Typically, when the left and the right basis functions are from the same finite
element basis (as in this example), the left and right operators are the same. If they are set
to \e VALUE we get a mass matrix; if they are set to an admissible differential operator we get
a stiffnesss matrix. Assume again that the basis is curl-conforming and the operators are
set to \e VALUE. Using the basis definition from Section \ref sec_pullbacks we have that
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} = \int_{\widehat{\mathcal C}} \sigma_i \sigma_j
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\cdot
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
It follows that
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} =
\mbox{diag}(\sigma_0,\ldots,\sigma_n)\widehat{\mathbf{K}}^{\mathcal{C}}\mbox{diag}(\sigma_0,\ldots,\sigma_n)
\f]
where
\f[
\widehat{\mathbf{K}}^{\mathcal{C}}_{i,j} = \int_{\widehat{\mathcal C}}
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\cdot
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
is the raw cell operator matrix. The methods Intrepid::FunctionSpaceTools::applyLeftFieldSigns and
Intrepid::FunctionSpaceTools::applyRightFieldSigns apply the left and right diagonal sign matrices to
the raw cell operator.
A finite element operator defined by the finite element basis on the physical cell is a vector
\f[
\mathbf{f}^{\mathcal{C}}_{i} = \int_{\mathcal C} f(x) {\mathcal L}_R u_i(x) \, dx \,.
\f]
Assuming again operator \e VALUE and using the same arguments as above, we see that
\f[
\mathbf{f}^{\mathcal{C}} =
\mbox{diag}(\sigma_0,\ldots,\sigma_n)\widehat{\mathbf{f}}^{\mathcal{C}}\,,
\f]
where
\f[
\widehat{\mathbf{f}}^{\mathcal{C}} = \int_{\widehat{\mathcal C}}
\mathbf{f}\circ F_{\mathcal C}(\widehat{x})
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
is the raw cell functional.
*/
|