An Efficient Approach for Solving Mesh Optimization Problems Using Newton’s Method

<table class="algorithm-group"><tr><td><table class="algorithm" id="alg2"><tr><td colspan="2">Find a spectral decomposition of <svg height="11.8532pt" id="M86" style="vertical-align:-0.2130003pt" version="1.1" viewbox="-0.0498162 -11.6402 53.859 11.8532" width="53.859pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z" id="g113-73"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="840" vert-adv-y="840"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,15.169,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" 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horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg>, where <svg height="9.04777pt" id="M87" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.99795 10.5164 9.04777" width="10.5164pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="766" vert-adv-y="766"></glyph.data></g></svg> is a diagonal matrix. </td></tr><tr><td colspan="2"><svg height="13.8794pt" id="M88" style="vertical-align:-2.239201pt" version="1.1" viewbox="-0.0498162 -11.6402 66.2748 13.8794" width="66.2748pt" 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310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z" id="g113-84"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="472" vert-adv-y="472"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,38.274,0)"><path d="M162 -163V703H101V-163H162Z" id="g113-9"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="263" vert-adv-y="263"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,41.837,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path><glyph.data 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-12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg></td></tr></table></td></tr></table>

<div>HM2: Changing the signs of negative eigenvalues.</div>

Mathematical Problems in Engineering

alg2

Algorithm 2

Algorithm 2: An Efficient Approach for Solving Mesh Optimization Problems Using Newton’s Method 