A Novel Virtual Node Hexahedral Element with Exact Integration and Octree Meshing

<table class="table-group" id="tab2"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Number of elements in the patch</td><td class="align_center">Error norm in displacement</td><td class="align_center">Error norm in temperature</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">6</td><td class="align_center"><span style="width: 64.653ptpx;"><svg height="11.9413pt" id="M231" style="vertical-align:-0.3499002pt" version="1.1" viewbox="-0.0498162 -11.5914 64.653 11.9413" width="64.653pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 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Patch test results for VPHE elements (<svg height="11.6412pt" id="M230" style="vertical-align:-0.04979992pt" version="1.1" viewbox="-0.0498162 -11.5914 12.262 11.6412" width="12.262pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" vert-adv-y="553"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg> error norm in displacement and temperature).

Mathematical Problems in Engineering

tab2

Table 2

Table 2: A Novel Virtual Node Hexahedral Element with Exact Integration and Octree Meshing 