<!DOCTYPE html><html lang="en"><head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="shortcut icon" href="/img/favicon.ico">
<meta name="keywords" content="Finite Element Library,Finite Element Code,Finite Element Method Library,Finite Element Method Code,Finite Element C++,Finite Element Method C++,Finite Element Library C++,Finite Element Code C++,FEM Code,FEM Library,FEM C++">
<meta name="google-site-verification" content="45K56TawSdzbgy-uDtfJdYwp7zaVdL3fIAGYt869wwU">
<!-- <title></title> -->
<title>MFEM - Finite Element Discretization Library</title>
<link href="../../doc/web/bootstrap-mfem.min.css" rel="stylesheet">
<link href="../../doc/web/font-awesome-4.0.3.css" rel="stylesheet">
<link href="../../doc/web/highlight.css" rel="stylesheet">
<link href="../../doc/web/base.css" rel="stylesheet">
<!-- HTML5 shim and Respond.js IE8 support of HTML5 elements and media queries -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.3.0/respond.min.js"></script>
<![endif]-->
<style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
.MathJax_Menu_Button .MathJax_Hover_Arrow {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 4px; -webkit-border-radius: 4px; -moz-border-radius: 4px; -khtml-border-radius: 4px; font-family: 'Courier New',Courier; font-size: 9px; color: #F0F0F0}
.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
.MathJax_Hover_Arrow:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
</style><style type="text/css">#MathJax_About {position: fixed; left: 50%; width: auto; text-align: center; border: 3px outset; padding: 1em 2em; background-color: #DDDDDD; color: black; cursor: default; font-family: message-box; font-size: 120%; font-style: normal; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 15px; -webkit-border-radius: 15px; -moz-border-radius: 15px; -khtml-border-radius: 15px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_About.MathJax_MousePost {outline: none}
.MathJax_Menu {position: absolute; background-color: white; color: black; width: auto; padding: 5px 0px; border: 1px solid #CCCCCC; margin: 0; cursor: default; font: menu; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 5px; -webkit-border-radius: 5px; -moz-border-radius: 5px; -khtml-border-radius: 5px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
.MathJax_MenuItem {padding: 1px 2em; background: transparent}
.MathJax_MenuArrow {position: absolute; right: .5em; padding-top: .25em; color: #666666; font-size: .75em}
.MathJax_MenuActive .MathJax_MenuArrow {color: white}
.MathJax_MenuArrow.RTL {left: .5em; right: auto}
.MathJax_MenuCheck {position: absolute; left: .7em}
.MathJax_MenuCheck.RTL {right: .7em; left: auto}
.MathJax_MenuRadioCheck {position: absolute; left: .7em}
.MathJax_MenuRadioCheck.RTL {right: .7em; left: auto}
.MathJax_MenuLabel {padding: 1px 2em 3px 1.33em; font-style: italic}
.MathJax_MenuRule {border-top: 1px solid #DDDDDD; margin: 4px 3px}
.MathJax_MenuDisabled {color: GrayText}
.MathJax_MenuActive {background-color: #606872; color: white}
.MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus {background-color: #E8E8E8}
.MathJax_ContextMenu:focus {outline: none}
.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
#MathJax_AboutClose {top: .2em; right: .2em}
.MathJax_Menu .MathJax_MenuClose {top: -10px; left: -10px}
.MathJax_MenuClose {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; font-family: 'Courier New',Courier; font-size: 24px; color: #F0F0F0}
.MathJax_MenuClose span {display: block; background-color: #AAA; border: 1.5px solid; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; line-height: 0; padding: 8px 0 6px}
.MathJax_MenuClose:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_MenuClose:hover span {background-color: #CCC!important}
.MathJax_MenuClose:hover:focus {outline: none}
</style><style type="text/css">.MathJax_Preview .MJXf-math {color: inherit!important}
</style><style type="text/css">.MJX_Assistive_MathML {position: absolute!important; top: 0; left: 0; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0 0 0!important; border: 0!important; height: 1px!important; width: 1px!important; overflow: hidden!important; display: block!important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none}
.MJX_Assistive_MathML.MJX_Assistive_MathML_Block {width: 100%!important}
</style><style type="text/css">#MathJax_Zoom {position: absolute; background-color: #F0F0F0; overflow: auto; display: block; z-index: 301; padding: .5em; border: 1px solid black; margin: 0; font-weight: normal; font-style: normal; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; box-sizing: content-box; box-shadow: 5px 5px 15px #AAAAAA; -webkit-box-shadow: 5px 5px 15px #AAAAAA; -moz-box-shadow: 5px 5px 15px #AAAAAA; -khtml-box-shadow: 5px 5px 15px #AAAAAA; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_ZoomOverlay {position: absolute; left: 0; top: 0; z-index: 300; display: inline-block; width: 100%; height: 100%; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
#MathJax_ZoomFrame {position: relative; display: inline-block; height: 0; width: 0}
#MathJax_ZoomEventTrap {position: absolute; left: 0; top: 0; z-index: 302; display: inline-block; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
</style><style type="text/css">.MathJax_Preview {color: #888}
#MathJax_Message {position: fixed; left: 1px; bottom: 2px; background-color: #E6E6E6; border: 1px solid #959595; margin: 0px; padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto; white-space: nowrap}
#MathJax_MSIE_Frame {position: absolute; top: 0; left: 0; width: 0px; z-index: 101; border: 0px; margin: 0px; padding: 0px}
.MathJax_Error {color: #CC0000; font-style: italic}
</style><style type="text/css">.MJXp-script {font-size: .8em}
.MJXp-right {-webkit-transform-origin: right; -moz-transform-origin: right; -ms-transform-origin: right; -o-transform-origin: right; transform-origin: right}
.MJXp-bold {font-weight: bold}
.MJXp-italic {font-style: italic}
.MJXp-scr {font-family: MathJax_Script,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-frak {font-family: MathJax_Fraktur,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-sf {font-family: MathJax_SansSerif,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-cal {font-family: MathJax_Caligraphic,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-mono {font-family: MathJax_Typewriter,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-largeop {font-size: 150%}
.MJXp-largeop.MJXp-int {vertical-align: -.2em}
.MJXp-math {display: inline-block; line-height: 1.2; text-indent: 0; font-family: 'Times New Roman',Times,STIXGeneral,serif; white-space: nowrap; border-collapse: collapse}
.MJXp-display {display: block; text-align: center; margin: 1em 0}
.MJXp-math span {display: inline-block}
.MJXp-box {display: block!important; text-align: center}
.MJXp-box:after {content: " "}
.MJXp-rule {display: block!important; margin-top: .1em}
.MJXp-char {display: block!important}
.MJXp-mo {margin: 0 .15em}
.MJXp-mfrac {margin: 0 .125em; vertical-align: .25em}
.MJXp-denom {display: inline-table!important; width: 100%}
.MJXp-denom > * {display: table-row!important}
.MJXp-surd {vertical-align: top}
.MJXp-surd > * {display: block!important}
.MJXp-script-box > * {display: table!important; height: 50%}
.MJXp-script-box > * > * {display: table-cell!important; vertical-align: top}
.MJXp-script-box > *:last-child > * {vertical-align: bottom}
.MJXp-script-box > * > * > * {display: block!important}
.MJXp-mphantom {visibility: hidden}
.MJXp-munderover {display: inline-table!important}
.MJXp-over {display: inline-block!important; text-align: center}
.MJXp-over > * {display: block!important}
.MJXp-munderover > * {display: table-row!important}
.MJXp-mtable {vertical-align: .25em; margin: 0 .125em}
.MJXp-mtable > * {display: inline-table!important; vertical-align: middle}
.MJXp-mtr {display: table-row!important}
.MJXp-mtd {display: table-cell!important; text-align: center; padding: .5em 0 0 .5em}
.MJXp-mtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-mlabeledtr {display: table-row!important}
.MJXp-mlabeledtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mlabeledtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
.MJXp-scale0 {-webkit-transform: scaleX(.0); -moz-transform: scaleX(.0); -ms-transform: scaleX(.0); -o-transform: scaleX(.0); transform: scaleX(.0)}
.MJXp-scale1 {-webkit-transform: scaleX(.1); -moz-transform: scaleX(.1); -ms-transform: scaleX(.1); -o-transform: scaleX(.1); transform: scaleX(.1)}
.MJXp-scale2 {-webkit-transform: scaleX(.2); -moz-transform: scaleX(.2); -ms-transform: scaleX(.2); -o-transform: scaleX(.2); transform: scaleX(.2)}
.MJXp-scale3 {-webkit-transform: scaleX(.3); -moz-transform: scaleX(.3); -ms-transform: scaleX(.3); -o-transform: scaleX(.3); transform: scaleX(.3)}
.MJXp-scale4 {-webkit-transform: scaleX(.4); -moz-transform: scaleX(.4); -ms-transform: scaleX(.4); -o-transform: scaleX(.4); transform: scaleX(.4)}
.MJXp-scale5 {-webkit-transform: scaleX(.5); -moz-transform: scaleX(.5); -ms-transform: scaleX(.5); -o-transform: scaleX(.5); transform: scaleX(.5)}
.MJXp-scale6 {-webkit-transform: scaleX(.6); -moz-transform: scaleX(.6); -ms-transform: scaleX(.6); -o-transform: scaleX(.6); transform: scaleX(.6)}
.MJXp-scale7 {-webkit-transform: scaleX(.7); -moz-transform: scaleX(.7); -ms-transform: scaleX(.7); -o-transform: scaleX(.7); transform: scaleX(.7)}
.MJXp-scale8 {-webkit-transform: scaleX(.8); -moz-transform: scaleX(.8); -ms-transform: scaleX(.8); -o-transform: scaleX(.8); transform: scaleX(.8)}
.MJXp-scale9 {-webkit-transform: scaleX(.9); -moz-transform: scaleX(.9); -ms-transform: scaleX(.9); -o-transform: scaleX(.9); transform: scaleX(.9)}
.MathJax_PHTML .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
</style><style type="text/css">.MathJax_SVG_Display {text-align: center; margin: 1em 0em; position: relative; display: block!important; text-indent: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; width: 100%}
.MathJax_SVG .MJX-monospace {font-family: monospace}
.MathJax_SVG .MJX-sans-serif {font-family: sans-serif}
#MathJax_SVG_Tooltip {background-color: InfoBackground; color: InfoText; border: 1px solid black; box-shadow: 2px 2px 5px #AAAAAA; -webkit-box-shadow: 2px 2px 5px #AAAAAA; -moz-box-shadow: 2px 2px 5px #AAAAAA; -khtml-box-shadow: 2px 2px 5px #AAAAAA; padding: 3px 4px; z-index: 401; position: absolute; left: 0; top: 0; width: auto; height: auto; display: none}
.MathJax_SVG {display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; padding: 0; margin: 0}
.MathJax_SVG * {transition: none; -webkit-transition: none; -moz-transition: none; -ms-transition: none; -o-transition: none}
.mjx-svg-href {fill: blue; stroke: blue}
.MathJax_SVG_Processing {visibility: hidden; position: absolute; top: 0; left: 0; width: 0; height: 0; overflow: hidden; display: block!important}
.MathJax_SVG_Processed {display: none!important}
.MathJax_SVG_ExBox {display: block!important; overflow: hidden; width: 1px; height: 60ex; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0}
.MathJax_SVG_LineBox {display: table!important}
.MathJax_SVG_LineBox span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MathJax_SVG .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
</style></head>
<body style=""><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_SVG_Hidden"></div><svg><defs id="MathJax_SVG_glyphs"><path stroke-width="1" id="MJSZ1-2211" d="M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z"></path><path stroke-width="1" id="MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="MJSZ1-222B" d="M113 -244Q113 -246 119 -251T139 -263T167 -269Q186 -269 199 -260Q220 -247 232 -218T251 -133T262 -15T276 155T297 367Q300 390 305 438T314 512T325 580T340 647T361 703T390 751T428 784T479 804Q481 804 488 804T501 805Q552 802 581 769T610 695Q610 669 594 657T561 645Q542 645 527 658T512 694Q512 705 516 714T526 729T538 737T548 742L552 743Q552 745 545 751T525 762T498 768Q475 768 460 756T434 716T418 652T407 559T398 444T387 300T369 133Q349 -38 337 -102T303 -207Q256 -306 169 -306Q119 -306 87 -272T55 -196Q55 -170 71 -158T104 -146Q123 -146 138 -159T153 -195Q153 -206 149 -215T139 -230T127 -238T117 -242L113 -244Z"></path><path stroke-width="1" id="MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="1" id="MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs></svg></div><div id="MathJax_Message" style="display: none;"></div>
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<!-- Collapsed navigation -->
<div class="navbar-header">
<!-- Expander button -->
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<!-- Main title -->
<a class="navbar-brand" href="http://mfem.org">MFEM</a>
</div>
<!-- Expanded navigation -->
<div class="navbar-collapse collapse">
<!-- Main navigation -->
<ul class="nav navbar-nav">
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/features/">Features</a>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/examples/">Examples</a>
</li>
<li class="dropdown active">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Documentation <b class="caret"></b></a>
<ul class="dropdown-menu">
<li>
<a href="http://mfem.org/building/">Building MFEM</a>
</li>
<li>
<a href="http://mfem.org/serial-tutorial/">Serial Tutorial</a>
</li>
<li>
<a href="http://mfem.org/parallel-tutorial/">Parallel Tutorial</a>
</li>
<li>
<a href="http://mfem.org/fem/">Finite Elements</a>
</li>
<li>
<a href="http://mfem.org/electromagnetics/">Electromagnetics</a>
</li>
<li class="active">
<a href="./">Meshing</a>
</li>
<li>
<a href="http://mfem.org/performance/">Performance</a>
</li>
<li>
<a href="http://mfem.org/tools/">Tools</a>
</li>
<li>
<a href="http://mfem.org/toys/">Toys</a>
</li>
<li>
<a href="http://mfem.org/publications/">Publications</a>
</li>
<li>
<a href="http://mfem.org/about/">About</a>
</li>
</ul>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/gallery/">Gallery</a>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/download/">Download</a>
</li>
</ul>
<!-- Search, Navigation and Repo links -->
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://github.com/mfem/mfem/">
<i class="fa fa-github-square"></i>
GitHub
</a>
</li>
</ul>
</div>
</div>
</div>
<div class="container">
<div class="col-md-3 visible-xs visible-sm"><div class="bs-sidebar hidden-print affix well" role="complementary">
<ul class="nav bs-sidenav">
<li class="main active"><a href="#meshing">Meshing</a></li>
<li><a href="#mesh-formats">Mesh formats</a></li>
<li><a href="#meshing-miniapps">Meshing miniapps</a></li>
</ul>
</div></div> <!-- toc on top on mobile -->
<div class="col-md-9" role="main">
<h1 id="meshing">Meshing</h1>
<p>This page provides a brief description of MFEM's mesh formats, meshing mini applications, and related tools.</p>
<h2 id="mesh-formats">Mesh formats</h2>
<p>MFEM supports a number of mesh formats, including:</p>
<ul>
<li>MFEM's <a href="http://mfem.org/mesh-formats/#mfem-mesh-v10">mesh v1.0 format</a> for <a href="http://mfem.org/mesh-formats/#straight-meshes">straight</a> meshes,</li>
<li>MFEM's <a href="http://mfem.org/mesh-formats/#curvilinear-and-more-general-meshes">mesh v1.0 format</a> for <a href="http://mfem.org/mesh-format-v1.x/">arbitrary high-order curvilinear</a> and more general meshes,</li>
<li>MFEM's <a href="http://mfem.org/mesh-formats/#mfem-mesh-v11">mesh v1.1 format</a>, which adds support for non-conforming (AMR) meshes,</li>
<li>MFEM's mesh v1.2 format, which adds support for parallel meshes,</li>
<li>MFEM's format for <a href="http://mfem.org/mesh-formats/#nurbs-meshes">NURBS</a> meshes,</li>
<li>The <a href="http://mfem.org/mesh-formats/#curvilinear-vtk-meshes">VTK</a> unstructured mesh format, for triangular, quadrilateral, tetrahedral and hexahedral meshes,</li>
<li>The <a href="http://gmsh.info/">Gmsh</a> ASCII and binary formats for 2D and 3D meshes.</li>
<li>The <a href="https://cubit.sandia.gov/">CUBIT</a> meshes through the Genesis (NetCDF) binary format.</li>
<li>The <a href="http://sourceforge.net/projects/netgen-mesher/">NETGEN</a> triangular and tetrahedral mesh formats,</li>
<li>The <a href="http://www.truegrid.com/">TrueGrid</a> hexahedral mesh format.</li>
</ul>
<p>Detailed description of these formats can be found on the <a href="http://mfem.org/mesh-formats/">mesh formats</a> page. These formats are also supported by MFEM's native visualization tool, <a href="http://glvis.org/">GLVis</a>.</p>
<h2 id="meshing-miniapps">Meshing miniapps</h2>
<p>The <code>miniapps/meshing</code> directory contains a collection of meshing-related miniapps based on MFEM.</p>
<p>Compared to the <a href="http://mfem.org/examples/">example codes</a>, the miniapps are more complex,
demonstrating more advanced usage of the library. They are intended to be more
representative of MFEM-based application codes. We recommend that new users
start with the example codes before moving to the miniapps.</p>
<p>The current meshing miniapps are described below. Related tools are listed at the <a href="#tools">bottom</a> of the page.</p>
<h3 id="mobius-strip">Mobius Strip</h3>
<p>This miniapp generates various Mobius strip-like surface meshes. It is a good
way to generate complex surface meshes.</p>
<p><img alt="" src="../../doc/web/examples/mobius-strip.png"></p>
<p>Manipulating the mesh topology and
performing mesh transformation are demonstrated. The <code>mobius-strip</code> mesh in the
<code>data</code> directory was generated with this miniapp.</p>
<h3 id="klein-bottle">Klein Bottle</h3>
<p>This miniapp generates three types of Klein bottle surfaces. It is similar to
the mobius-strip miniapp.</p>
<p><img alt="" src="../../doc/web/examples/klein-bottle.png"></p>
<p>The <code>klein-bottle</code> and <code>klein-donut</code> meshes in the
<code>data</code> directory were generated with this miniapp.</p>
<h3 id="toroid">Toroid</h3>
<p>This miniapp generates two types of toroidal volume meshes; one with
triangular cross sections and one with square cross sections.</p>
<p><img alt="" src="../../doc/web/examples/toroid-wedge.png"></p>
<p>A wide variety of toroidal meshes can be generated by varying the
amount of twist as well as the major and minor radii and other
variables. The <code>toroid-wedge</code> and <code>toroid-hex</code> meshes in the <code>data</code> directory
were generated with this miniapp.</p>
<h3 id="extruder">Extruder</h3>
<p>This miniapp creates higher dimensional meshes from lower dimensional meshes
by extrusion.</p>
<p><img alt="" src="../../doc/web/examples/extruded-star.png"></p>
<p>Simple coordinate transformations can also be applied if desired. The initial
mesh can be 1D or 2D. 1D meshes can be extruded in the y-direction first and
then in the z-direction. 2D meshes can be triangular, quadrilateral, or
contain both element types.</p>
<h3 id="shaper">Shaper</h3>
<p>This miniapp performs multiple levels of adaptive mesh refinement to resolve the
interfaces between different "materials" in the mesh, as specified by a given
material() function.</p>
<p><img alt="" src="../../doc/web/examples/shaper.png"></p>
<p>It can be used as a simple initial mesh generator, for example in the case when
the interface is too complex to describe without local refinement. Both
conforming and non-conforming refinements are supported.</p>
<h3 id="mesh-explorer">Mesh Explorer</h3>
<p>This miniapp is a handy tool to examine, visualize and manipulate a given
mesh.</p>
<p><img alt="" src="../../doc/web/examples/mesh-explorer.png"></p>
<p>Some of its features are:</p>
<ul>
<li>visualizing of mesh materials and individual mesh elements</li>
<li>mesh scaling, randomization, and general transformation</li>
<li>manipulation of the mesh curvature</li>
<li>the ability to simulate parallel partitioning</li>
<li>quantitative and visual reports of mesh quality</li>
</ul>
<h3 id="mesh-optimizer">Mesh Optimizer</h3>
<p>This miniapp performs mesh optimization using the Target-Matrix Optimization
Paradigm (TMOP) by P.Knupp et al., and a global variational minimization
approach.</p>
<p><img alt="" src="../../doc/web/examples/mesh-optimizer.png"></p>
<p>It minimizes the quantity <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo>&#x2211;</mo><mi>T</mi></munder><msub><mo>&#x222B;</mo><mi>T</mi></msub><mi>&#x03BC;</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.925ex" height="3.067ex" viewBox="0 -879.6 6426.1 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1494" y="-405"></use><g transform="translate(1821,0)"><use xlink:href="#MJSZ1-222B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="668" y="-484"></use></g><use xlink:href="#MJMATHI-3BC" x="3058" y="0"></use><use xlink:href="#MJMAIN-28" x="3662" y="0"></use><use xlink:href="#MJMATHI-4A" x="4051" y="0"></use><use xlink:href="#MJMAIN-28" x="4685" y="0"></use><use xlink:href="#MJMATHI-78" x="5074" y="0"></use><use xlink:href="#MJMAIN-29" x="5647" y="0"></use><use xlink:href="#MJMAIN-29" x="6036" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo>∑</mo><mi>T</mi></munder><msub><mo>∫</mo><mi>T</mi></msub><mi>μ</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-1">\sum_T \int_T \mu(J(x))</script>, where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.636ex" height="2.048ex" viewBox="0 -769.9 704.5 881.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-54" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">T</script> are the target
(ideal) elements, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.471ex" height="2.048ex" viewBox="0 -769.9 633.5 881.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">J</script> is the Jacobian of the transformation from the target
to the physical element, and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">\mu</script> is the mesh quality metric.</p>
<p>This metric can measure shape, size or alignment of the region around each
quadrature point. The combination of targets and quality metrics is used to
optimize the physical node positions, i.e., they must be as close as possible to
the shape / size / alignment of their targets.</p>
<script type="text/x-mathjax-config;executed=true">MathJax.Hub.Config({TeX: {equationNumbers: {autoNumber: "all"}}, tex2jax: {inlineMath: [['$','$']]}});</script>
<script type="text/javascript" src=""></script></div>
<div class="col-md-3 hidden-xs hidden-sm"><div class="bs-sidebar hidden-print affix well" role="complementary">
<ul class="nav bs-sidenav">
<li class="main active"><a href="#meshing">Meshing</a></li>
<li><a href="#mesh-formats">Mesh formats</a></li>
<li><a href="#meshing-miniapps">Meshing miniapps</a></li>
</ul>
</div></div>
</div>
<footer class="footer">
<div class="container">
<p class="text-muted alignright">LLNL-WEB-676715</p>
<p class="text-muted alignleft">
Developed by the <a href="/about/">MFEM team</a> at
<a href="http://computation.llnl.gov/casc/">CASC</a>,
<a href="https://www.llnl.gov/">LLNL</a></p>
</div>
</footer>
<script src="../../doc/web/jquery-1.10.2.min.js"></script>
<script src="../../doc/web/bootstrap-3.0.3.min.js"></script>
<script src="../../doc/web/highlight.pack.js"></script>
<script src="../../doc/web/base.js"></script>
<script src="../../doc/web/retina.min.js"></script>
<script src="../search/main.js"></script>
</body></html>
