A GPU-Accelerated 3D Mesh Deformation Method Based on Radial Basis Function Interpolation

<table class="algorithm-group"><tr><td><table class="algorithm" id="alg2"><tr><td> </td><td>function LinearSolver <svg height="11.5564pt" id="M71" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 30.4641 11.5564" width="30.4641pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.71,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.854,0)"><path d="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.822,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></td></tr><tr><td> </td><td>/∗ Solve the linear equation system with the form of <svg height="9.44253pt" id="M72" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.23613 38.6151 9.44253" width="38.6151pt" 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="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.212,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.23,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.492,0)"><path d="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g></svg> ∗/</td></tr><tr><td> </td><td>Input: Matrix <svg height="8.69224pt" id="M73" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.64243 9.31206 8.69224" width="9.31206pt" 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="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g></svg> of dimension <span class="nowrap"><svg height="12.5794pt" id="M74" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 93.7965 12.5794" width="93.7965pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.959,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.729,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.265,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.505,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,42.909,0)"><path d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"></path></g><g transform="matrix(.013,0,0,-0.013,53.445,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,57.943,0)"><path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,64.404,3.132)"><path d="M460 334C460 396 434 451 378 451C330 451 241 408 148 300H146L237 679C241 697 241 710 232 710C213 710 153 684 67 675L66 646H95C141 646 145 642 134 595L39 170C23 97 31 54 46 33C64 8 100 -12 137 -12C178 -12 234 3 298 43C391 101 460 222 460 334ZM371 320C371 204 316 89 248 51C230 41 208 37 192 37C143 37 102 72 119 166C124 194 129 215 135 235C202 323 298 392 335 392C353 392 371 372 371 320Z"></path></g><g transform="matrix(.013,0,0,-0.013,72.174,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,82.711,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,88.951,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>,</span> vector <svg height="9.42947pt" id="M75" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.23613 7.06767 9.42947" width="7.06767pt" 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="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g></svg></td></tr><tr><td> </td><td>Output: vector <svg height="6.1673pt" id="M76" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt" 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="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g></svg></td></tr><tr><td> </td><td>cusolverDnDgetrf <span class="nowrap"><svg height="11.5564pt" id="M77" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 18.3418 11.5564" width="18.3418pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.71,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>;</span> //LU decomposition of matrix <svg height="8.69224pt" id="M78" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.64243 9.31206 8.69224" width="9.31206pt" 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="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g></svg></td></tr><tr><td> </td><td>cusolverDnDgetrs <span class="nowrap"><svg height="11.5564pt" id="M79" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 43.1083 11.5564" width="43.1083pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.688,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.831,0)"><path d="M723 650H473V615C541 611 564 591 570 556C575 526 580 488 580 417V310C580 143 528 53 409 53C301 53 247 140 247 290V508C247 602 257 608 330 615V650H29V615C103 608 113 602 113 508V272C113 155 140 81 196 36C244 -1 307 -15 373 -15C444 -15 512 10 558 58C614 118 630 202 630 308V417C630 488 634 525 639 556C644 591 669 612 723 615V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,26.344,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.487,0)"><path d="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.455,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>;</span> //solving linear equation system: <svg height="9.47515pt" id="M80" style="vertical-align:-0.2390203pt" version="1.1" viewbox="-0.0498162 -9.23613 46.1051 9.47515" width="46.1051pt" 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="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g><g transform="matrix(.013,0,0,-0.013,7.19,0)"><path d="M723 650H473V615C541 611 564 591 570 556C575 526 580 488 580 417V310C580 143 528 53 409 53C301 53 247 140 247 290V508C247 602 257 608 330 615V650H29V615C103 608 113 602 113 508V272C113 155 140 81 196 36C244 -1 307 -15 373 -15C444 -15 512 10 558 58C614 118 630 202 630 308V417C630 488 634 525 639 556C644 591 669 612 723 615V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.702,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.72,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.982,0)"><path d="M196 417V703C139 687 58 673 7 667V636C63 632 68 623 68 566V24C137 -2 188 -12 244 -12C390 -12 504 97 504 243C504 355 418 454 297 454C277 454 261 447 233 434L196 417ZM196 382C211 387 221 389 238 389C301 389 365 339 365 212C365 87 316 27 262 28C230 29 207 59 201 75C198 83 196 94 196 108V382Z"></path></g></svg></td></tr></table></td></tr></table>

<div> Kernel function for solving the linear equation system of equation (<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/6018008/figures/#EEq4">4</a>).</div>

Mathematical Problems in Engineering

alg2

Algorithm 2

Algorithm 2: A GPU-Accelerated 3D Mesh Deformation Method Based on Radial Basis Function Interpolation 