501 lines
82 KiB
HTML
501 lines
82 KiB
HTML
<!DOCTYPE html><html lang="en"><head>
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<meta charset="utf-8">
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<meta http-equiv="X-UA-Compatible" content="IE=edge">
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<meta name="viewport" content="width=device-width, initial-scale=1.0">
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<link rel="shortcut icon" href="/img/favicon.ico">
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<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++">
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<meta name="google-site-verification" content="45K56TawSdzbgy-uDtfJdYwp7zaVdL3fIAGYt869wwU">
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<!-- <title></title> -->
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<title>MFEM - Finite Element Discretization Library</title>
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<link href="../../doc/web/bootstrap-mfem.min.css" rel="stylesheet">
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<link href="../../doc/web/font-awesome-4.0.3.css" rel="stylesheet">
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<link href="../../doc/web/highlight.css" rel="stylesheet">
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<link href="../../doc/web/base.css" rel="stylesheet">
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<!-- HTML5 shim and Respond.js IE8 support of HTML5 elements and media queries -->
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<!--[if lt IE 9]>
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<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
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<script src="https://oss.maxcdn.com/libs/respond.js/1.3.0/respond.min.js"></script>
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<![endif]-->
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<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}
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.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}
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.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
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.MathJax_Hover_Arrow:hover {color: white!important; border: 2px solid #CCC!important}
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.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
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</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')}
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#MathJax_About.MathJax_MousePost {outline: none}
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.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')}
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.MathJax_MenuItem {padding: 1px 2em; background: transparent}
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.MathJax_MenuArrow {position: absolute; right: .5em; padding-top: .25em; color: #666666; font-size: .75em}
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.MathJax_MenuActive .MathJax_MenuArrow {color: white}
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.MathJax_MenuArrow.RTL {left: .5em; right: auto}
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.MathJax_MenuCheck {position: absolute; left: .7em}
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.MathJax_MenuCheck.RTL {right: .7em; left: auto}
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.MathJax_MenuRadioCheck {position: absolute; left: .7em}
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.MathJax_MenuRadioCheck.RTL {right: .7em; left: auto}
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.MathJax_MenuLabel {padding: 1px 2em 3px 1.33em; font-style: italic}
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.MathJax_MenuRule {border-top: 1px solid #DDDDDD; margin: 4px 3px}
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.MathJax_MenuDisabled {color: GrayText}
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.MathJax_MenuActive {background-color: #606872; color: white}
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.MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus {background-color: #E8E8E8}
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.MathJax_ContextMenu:focus {outline: none}
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.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
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#MathJax_AboutClose {top: .2em; right: .2em}
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.MathJax_Menu .MathJax_MenuClose {top: -10px; left: -10px}
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.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}
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.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}
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.MathJax_MenuClose:hover {color: white!important; border: 2px solid #CCC!important}
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.MathJax_MenuClose:hover span {background-color: #CCC!important}
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.MathJax_MenuClose:hover:focus {outline: none}
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</style><style type="text/css">.MathJax_Preview .MJXf-math {color: inherit!important}
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</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}
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.MJX_Assistive_MathML.MJX_Assistive_MathML_Block {width: 100%!important}
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</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')}
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#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)}
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#MathJax_ZoomFrame {position: relative; display: inline-block; height: 0; width: 0}
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#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)}
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</style><style type="text/css">.MathJax_Preview {color: #888}
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#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}
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#MathJax_MSIE_Frame {position: absolute; top: 0; left: 0; width: 0px; z-index: 101; border: 0px; margin: 0px; padding: 0px}
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.MathJax_Error {color: #CC0000; font-style: italic}
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</style><style type="text/css">.MJXp-script {font-size: .8em}
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.MJXp-right {-webkit-transform-origin: right; -moz-transform-origin: right; -ms-transform-origin: right; -o-transform-origin: right; transform-origin: right}
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.MJXp-bold {font-weight: bold}
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.MJXp-italic {font-style: italic}
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.MJXp-scr {font-family: MathJax_Script,'Times New Roman',Times,STIXGeneral,serif}
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.MJXp-frak {font-family: MathJax_Fraktur,'Times New Roman',Times,STIXGeneral,serif}
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.MJXp-sf {font-family: MathJax_SansSerif,'Times New Roman',Times,STIXGeneral,serif}
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.MJXp-cal {font-family: MathJax_Caligraphic,'Times New Roman',Times,STIXGeneral,serif}
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.MJXp-mono {font-family: MathJax_Typewriter,'Times New Roman',Times,STIXGeneral,serif}
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.MJXp-largeop {font-size: 150%}
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.MJXp-largeop.MJXp-int {vertical-align: -.2em}
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.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}
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.MJXp-display {display: block; text-align: center; margin: 1em 0}
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.MJXp-math span {display: inline-block}
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.MJXp-box {display: block!important; text-align: center}
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.MJXp-box:after {content: " "}
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.MJXp-rule {display: block!important; margin-top: .1em}
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.MJXp-char {display: block!important}
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.MJXp-mo {margin: 0 .15em}
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.MJXp-mfrac {margin: 0 .125em; vertical-align: .25em}
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.MJXp-denom {display: inline-table!important; width: 100%}
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.MJXp-denom > * {display: table-row!important}
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.MJXp-surd {vertical-align: top}
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.MJXp-surd > * {display: block!important}
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.MJXp-script-box > * {display: table!important; height: 50%}
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.MJXp-script-box > * > * {display: table-cell!important; vertical-align: top}
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.MJXp-script-box > *:last-child > * {vertical-align: bottom}
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.MJXp-script-box > * > * > * {display: block!important}
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.MJXp-mphantom {visibility: hidden}
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.MJXp-munderover {display: inline-table!important}
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.MJXp-over {display: inline-block!important; text-align: center}
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.MJXp-over > * {display: block!important}
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.MJXp-munderover > * {display: table-row!important}
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.MJXp-mtable {vertical-align: .25em; margin: 0 .125em}
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.MJXp-mtable > * {display: inline-table!important; vertical-align: middle}
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.MJXp-mtr {display: table-row!important}
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.MJXp-mtd {display: table-cell!important; text-align: center; padding: .5em 0 0 .5em}
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.MJXp-mtr > .MJXp-mtd:first-child {padding-left: 0}
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.MJXp-mtr:first-child > .MJXp-mtd {padding-top: 0}
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.MJXp-mlabeledtr {display: table-row!important}
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.MJXp-mlabeledtr > .MJXp-mtd:first-child {padding-left: 0}
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.MJXp-mlabeledtr:first-child > .MJXp-mtd {padding-top: 0}
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.MJXp-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
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.MJXp-scale0 {-webkit-transform: scaleX(.0); -moz-transform: scaleX(.0); -ms-transform: scaleX(.0); -o-transform: scaleX(.0); transform: scaleX(.0)}
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.MJXp-scale1 {-webkit-transform: scaleX(.1); -moz-transform: scaleX(.1); -ms-transform: scaleX(.1); -o-transform: scaleX(.1); transform: scaleX(.1)}
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.MJXp-scale2 {-webkit-transform: scaleX(.2); -moz-transform: scaleX(.2); -ms-transform: scaleX(.2); -o-transform: scaleX(.2); transform: scaleX(.2)}
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.MJXp-scale3 {-webkit-transform: scaleX(.3); -moz-transform: scaleX(.3); -ms-transform: scaleX(.3); -o-transform: scaleX(.3); transform: scaleX(.3)}
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.MJXp-scale4 {-webkit-transform: scaleX(.4); -moz-transform: scaleX(.4); -ms-transform: scaleX(.4); -o-transform: scaleX(.4); transform: scaleX(.4)}
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.MJXp-scale5 {-webkit-transform: scaleX(.5); -moz-transform: scaleX(.5); -ms-transform: scaleX(.5); -o-transform: scaleX(.5); transform: scaleX(.5)}
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.MJXp-scale6 {-webkit-transform: scaleX(.6); -moz-transform: scaleX(.6); -ms-transform: scaleX(.6); -o-transform: scaleX(.6); transform: scaleX(.6)}
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.MJXp-scale7 {-webkit-transform: scaleX(.7); -moz-transform: scaleX(.7); -ms-transform: scaleX(.7); -o-transform: scaleX(.7); transform: scaleX(.7)}
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.MJXp-scale8 {-webkit-transform: scaleX(.8); -moz-transform: scaleX(.8); -ms-transform: scaleX(.8); -o-transform: scaleX(.8); transform: scaleX(.8)}
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.MJXp-scale9 {-webkit-transform: scaleX(.9); -moz-transform: scaleX(.9); -ms-transform: scaleX(.9); -o-transform: scaleX(.9); transform: scaleX(.9)}
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.MathJax_PHTML .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
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</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%}
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.MathJax_SVG .MJX-monospace {font-family: monospace}
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.MathJax_SVG .MJX-sans-serif {font-family: sans-serif}
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#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}
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.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}
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.MathJax_SVG * {transition: none; -webkit-transition: none; -moz-transition: none; -ms-transition: none; -o-transition: none}
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.mjx-svg-href {fill: blue; stroke: blue}
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.MathJax_SVG_Processing {visibility: hidden; position: absolute; top: 0; left: 0; width: 0; height: 0; overflow: hidden; display: block!important}
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.MathJax_SVG_Processed {display: none!important}
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.MathJax_SVG_ExBox {display: block!important; overflow: hidden; width: 1px; height: 60ex; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0}
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.MathJax_SVG_LineBox {display: table!important}
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.MathJax_SVG_LineBox span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MathJax_SVG .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
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</style></head>
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<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="MJMATHI-42" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 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218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path stroke-width="1" id="MJMATHI-47" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 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<li class="main active"><a href="#tensor-based-operator-assembly-and-evaluation">Tensor-Based Operator Assembly and Evaluation</a></li>
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<h1 id="tensor-based-operator-assembly-and-evaluation">Tensor-Based Operator Assembly and Evaluation</h1>
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<h2 id="overview">Overview</h2>
|
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<p>The high performance (HPC) versions of the example codes in the <code>miniapps/performance</code> directory
|
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use a set of templated classes to efficiently implement the inner-most portion
|
||
(<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"><msup><mi>B</mi><mi>T</mi></msup><mi>D</mi><mi>B</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.842ex" height="2.378ex" viewBox="0 -908.4 2945.7 1023.7" role="img" focusable="false" style="vertical-align: -0.268ex;" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="513"></use><use xlink:href="#MJMATHI-44" x="1357" y="0"></use><use xlink:href="#MJMATHI-42" x="2186" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mi>T</mi></msup><mi>D</mi><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">B^T D B</script>) of the fundamental finite element (FE) operator decomposition:</p>
|
||
<p><img alt="" src="../../doc/web/FEDecomposition.png"></p>
|
||
<p>We take advantage of the tensor-product structure of both the finite element
|
||
basis and the quadrature rule to efficiently apply the action of <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>B</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.115ex" viewBox="0 -795.1 759.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">B</script> without
|
||
necessarily computing its entries. This is generally know as <em><a href="http://www.sciencedirect.com/science/article/pii/0021999180900054">sum
|
||
factorization</a></em>.
|
||
In the case where we pre-compute and store the <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>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">D</script> matrix, we call the algorithm
|
||
<em>partial assembly</em>.</p>
|
||
<h2 id="template-implementation">Template implementation</h2>
|
||
<p>Below is a short description of the header files that are part of the initial
|
||
reference implementation of the tensor-based finite element assembly and
|
||
evaluation algorithms.</p>
|
||
<h3 id="configtconfighpp">config/tconfig.hpp</h3>
|
||
<p>Configuration macros including the specification of the template blocking
|
||
size (currently used in the matrix-matrix multiply).</p>
|
||
<h3 id="generaltassignhpp">general/tassign.hpp</h3>
|
||
<p>Helper templated function to allow single implementation of multiple
|
||
assignment operations on tensor entries.</p>
|
||
<h3 id="linalgtlayouthpp">linalg/tlayout.hpp</h3>
|
||
<p>Classes describing fixed size tensor layouts. Implemented are standard
|
||
strided layouts for 1D/2D/3D/4D tensors. Layouts support reshape and
|
||
sub-tensor operations independent of the actual data storage.</p>
|
||
<p>This header also contains another set of "vector layout" classes for
|
||
converting scalar data indices into multi-component (vector) data indices.
|
||
They are used to describe the layout of vector <code>GridFunctions</code> on global
|
||
degrees of freedom (similar to the <code>Ordering</code> class). In the FE operator
|
||
decomposition, these classes are used by the templated <code>*_FiniteElementSpace</code>
|
||
classes (see <code>fem/tfespace.hpp</code> below) to implement the actions of <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>G</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.827ex" height="2.115ex" viewBox="0 -795.1 786.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">G</script> and
|
||
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.216ex" height="2.378ex" viewBox="0 -908.4 1384.7 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1112" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-5">G^T</script> in the case of vector (multi-component) input and/or output fields.</p>
|
||
<h3 id="linalgtmatrixhpp">linalg/tmatrix.hpp</h3>
|
||
<p>Small matrix operations, defined by specializations: determinant, adjugate,
|
||
etc. Matrix-matrix multiply, <code>C = A.B</code>, simple and blocked version.</p>
|
||
<h3 id="linalgttensorhpp">linalg/ttensor.hpp</h3>
|
||
<p>Fixed-size container classes <code>TVector</code>, <code>TMatrix</code>, <code>TTensor3</code>, <code>TTensor4</code> for
|
||
1D/2D/3D/4D tensors stored in column-major layout.
|
||
Element-wise tensor operations: <code>A {=,+=,*=} scalar</code>; <code>A {=,+=,*=} B</code>. The
|
||
latter allows different input and output layouts. With suitable layouts this
|
||
can be used to permute (transpose) tensors, extract sub-tensors, etc.
|
||
The tensor contraction/product operations:</p>
|
||
<ul>
|
||
<li><code>Mult_1_2</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>&#x2211;</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.84ex" height="2.904ex" viewBox="0 -851.8 8972.8 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2391" y="0"></use><g transform="translate(3448,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5103,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(6774,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1093" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1372" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-6">C_{i,j,k} = \sum_s A_{s,j} B_{i,s,k}</script></li>
|
||
<li><code>Mult_2_1</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>&#x2211;</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.84ex" height="2.904ex" viewBox="0 -851.8 8972.8 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2391" y="0"></use><g transform="translate(3448,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5103,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use></g></g><g transform="translate(6727,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1160" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1439" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-7">C_{i,j,k} = \sum_s A_{i,s} B_{s,j,k}</script></li>
|
||
<li><code>TensorAssemble</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>&#x2211;</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.643ex" height="2.904ex" viewBox="0 -851.8 11471.3 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1145" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="1424" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2799" y="0"></use><g transform="translate(3856,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5511,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="748" y="0"></use></g></g><g transform="translate(7135,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(8806,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="521" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="800" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1269" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="1547" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub><mtext> </mtext><mtext> </mtext></math></span></span><script type="math/tex" id="MathJax-Element-8">C_{i,k,j,l} = \sum_s A_{s,i} A_{s,j} B_{k,s,l}~~</script> and
|
||
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>&#xA0;</mtext><mtext>&#xA0;</mtext><msub><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>&#x2211;</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.824ex" height="2.904ex" viewBox="0 -851.8 11549.3 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(500,0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><g transform="translate(828,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1145" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="1424" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g></g><use xlink:href="#MJMAIN-3D" x="3412" y="0"></use><g transform="translate(4469,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(6124,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use></g></g><g transform="translate(7748,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(9428,0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="521" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="800" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1269" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="1547" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext> </mtext><mtext> </mtext><msub><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-9">~~D_{i,k,j,l} = \sum_s A_{i,s} B_{s,j} C_{k,s,l}</script></li>
|
||
<li><code>TensorProduct</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.569ex" height="2.772ex" viewBox="0 -795.1 7564.4 1193.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2799" y="0"></use><g transform="translate(3856,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g></g><g transform="translate(6005,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="691" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-10">C_{i,j,k,l} = A_{i,j,k} B_{j,l}</script></li>
|
||
</ul>
|
||
<h3 id="meshtmeshhpp">mesh/tmesh.hpp</h3>
|
||
<p>The Mesh object templated by the finite element space and layout of the
|
||
nodes. Provides <code>MatchesGeometry()</code> and <code>MatchesNodes()</code> functions to verify
|
||
if compiled and runtime mesh agree.</p>
|
||
<h3 id="femtintruleshpp">fem/tintrules.hpp</h3>
|
||
<p>Quadrature rules templated by geometry (triangles, quads, etc.) and
|
||
integration order.</p>
|
||
<h3 id="femtfehpp">fem/tfe.hpp</h3>
|
||
<p>H1 and L2 finite elements templated by geometry and polynomial order.</p>
|
||
<h3 id="femtfespacehpp">fem/tfespace.hpp</h3>
|
||
<p>Template <code>*_FiniteElementSpace</code> classes providing the mappings between global
|
||
and local (element) degrees of freedom for H1 continuous and L2 discontinuous
|
||
spaces. In the FE operator decomposition, these classes provide the element
|
||
local action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.827ex" height="2.115ex" viewBox="0 -795.1 786.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">G</script> (<code>Extract</code> methods) and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.216ex" height="2.378ex" viewBox="0 -908.4 1384.7 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1112" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-12">G^T</script> (<code>Assemble</code> methods).</p>
|
||
<h3 id="femtcoefficienthpp">fem/tcoefficient.hpp</h3>
|
||
<p>Templated versions of classes derived from the abstract class <code>Coefficient</code>.
|
||
It encapsulates physical quantities like material properties, sources,
|
||
boundary/initial conditions, etc. Its main functionality is to evaluate the
|
||
coefficient at all quadrature points in an element, which is then used in the
|
||
evaluation of the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-13">D</script> matrix.</p>
|
||
<h3 id="femteltranshpp">fem/teltrans.hpp</h3>
|
||
<p>Element transformation class, templated on a mesh type and an integration
|
||
rule. It is constructed from a mesh (e.g. class <code>TMesh</code>) and shape evaluator
|
||
(e.g. class <code>ShapeEvaluator</code>) objects. Allows computation of physical
|
||
coordinates and Jacobian matrices corresponding to the reference integration
|
||
points. The desired result (a combination of coordinates and/or Jacobian
|
||
matrices at quadrature points, element attribute and/or element index) is
|
||
specified through the template sub-class <code>Result</code> and stored in an object of
|
||
the same type. The idea of this approach is to eliminate unnecessary
|
||
evaluations if they are not needed. The need is determined based on what the
|
||
particular "users" need. The "users" are the templated <code>Coefficient</code> and
|
||
<code>Kernel</code> (see <code>fem/tbilininteg.hpp</code> below) classes which specify what they
|
||
need through static constant boolean variables, e.g. <code>uses_coordinates</code>,
|
||
<code>uses_Jacobians</code>, etc.</p>
|
||
<h3 id="femtevaluatorhpp">fem/tevaluator.hpp</h3>
|
||
<p>Classes for evaluating FE basis, <code>ShapeEvaluator</code>, and finite element
|
||
functions, <code>FieldEvaluator</code>, and their derivatives at quadrature points,
|
||
templated by a finite element class and an integration rule class. These
|
||
correspond to the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.115ex" viewBox="0 -795.1 759.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-14">B</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>G</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.591ex" height="2.115ex" viewBox="0 -795.1 1546 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMATHI-47" x="759" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-15">BG</script> matrices above. Quads and hexes use the
|
||
tensor-product structure for fast evaluation.</p>
|
||
<h3 id="femtbilininteghpp">fem/tbilininteg.hpp</h3>
|
||
<p><code>Kernel</code> classes (e.g. mass, diffusion) that represent the matrix <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-16">D</script> from
|
||
the above FE operator decomposition. These classes also specify the <em>type</em> of
|
||
the local operator that needs to be applied before and after the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-17">D</script> matrix -
|
||
these are the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.55ex" height="2.378ex" viewBox="0 -795.1 1528.4 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.531ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6E" x="345" y="0"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-18">B_{in}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.328ex" height="3.036ex" viewBox="0 -908.4 1863.2 1307" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="488"></use><g transform="translate(759,-279)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6F" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-75" x="485" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1058" y="0"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-19">B^T_{out}</script> matrices, respectively. The product
|
||
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup><mi>D</mi><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.802ex" height="3.036ex" viewBox="0 -908.4 4220.2 1307" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="488"></use><g transform="translate(759,-279)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6F" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-75" x="485" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1058" y="0"></use></g><use xlink:href="#MJMATHI-44" x="1863" y="0"></use><g transform="translate(2691,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6E" x="345" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup><mi>D</mi><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-20">B^T_{out} D B_{in}</script> is the local element matrix, which is the result when
|
||
using the <code>BilinearFormIntegrator</code> classes. This specifications of the types
|
||
are given by static constant boolean variables, e.g. <code>in_values</code> and
|
||
<code>out_values</code>. The <code>Kernel</code> classes provide the following methods:</p>
|
||
<ul>
|
||
<li><code>Action</code>: evaluate the action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-21">D</script> <em>without</em> explicitly storing the
|
||
partially assembled data; this is needed for matrix-free action.</li>
|
||
<li><code>Assemble</code>: evaluate the partially assembled data, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-22">D</script>, which is
|
||
kernel-specific: e.g., for mass, the data is one scalar per quadrature
|
||
point; for diffusion, the data is one <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>&#x00D7;</mo><mi>d</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.272ex" height="2.115ex" viewBox="0 -795.1 2269.9 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="745" y="0"></use><use xlink:href="#MJMATHI-64" x="1746" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>×</mo><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-23">d\times d</script> matrix (in <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.216ex" height="2.115ex" viewBox="0 -795.1 523.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">d</script>-dimensions)
|
||
per quadrature point.</li>
|
||
<li><code>MultAssembled</code>: perform the action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-25">D</script> using the pre-computed partially
|
||
assembled data.</li>
|
||
</ul>
|
||
<h3 id="femtbilinearformhpp">fem/tbilinearform.hpp</h3>
|
||
<p>Bilinear form operator, templated on the mesh, finite element space,
|
||
integration rule and bilinear form integrator. Corresponds to the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.115ex" viewBox="0 -795.1 750.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">A</script> matrix
|
||
above. Provides various assembly and evaluation schemes:</p>
|
||
<ul>
|
||
<li><code>MultUnassembled</code>: matrix-free action using the mesh nodes and the input
|
||
vector.</li>
|
||
<li><code>Assemble</code>, <code>MultAssembled</code>: partial assembly and operator action using the
|
||
partially assembled data at quadrature points and the input vector.</li>
|
||
<li><code>AssembleMatrix(DenseTensor &)</code>: assemble the local element matrices and
|
||
store them as <code>DenseTensor</code>.</li>
|
||
<li><code>AssembleMatrix(SparseMatrix &)</code>: assemble the operator in a global (CSR)
|
||
<code>SparseMatrix</code>.</li>
|
||
<li><code>AssembleBilinearForm(BilinearForm &)</code>: assemble element matrices and add
|
||
them to the bilinear form.</li>
|
||
</ul>
|
||
<h3 id="miniappsperformancemakefile">miniapps/performance/makefile</h3>
|
||
<p>By default <code>make</code> builds the example drivers with the compiler used to
|
||
compile MFEM. If <code>g++</code> was used, a pseudo-code dump file with the optimized
|
||
code will be generated (option <code>-fdump-tree-optimized-blocks</code>). The <code>g++</code>
|
||
option <code>--param max-completely-peel-times=3</code> prevents the compiler from
|
||
unrolling innermost loops (of size greater than 3), allowing the compiler to
|
||
vectorize them. Some options for optimization/vectorization with the clang
|
||
compiler are also included.</p>
|
||
<h3 id="miniappsperformanceexcpp">miniapps/performance/ex*.cpp</h3>
|
||
<p>High-performance templated versions of the corresponding <code>examples/ex*.cpp</code>
|
||
example codes.</p>
|
||
<script type="text/x-mathjax-config;executed=true">MathJax.Hub.Config({TeX: {equationNumbers: {autoNumber: "all"}}, tex2jax: {inlineMath: [['$','$']]}});</script>
|
||
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