Implementation of Structural Assemblies by Simultaneous Projection and Density-Based Topology Optimization

<div>Optimized topologies and compliance for various constraints values. (a) <span class="nowrap"><svg height="11.6217pt" id="M225" style="vertical-align:-3.272881pt" version="1.1" viewbox="-0.0498162 -8.34882 104.19 11.6217" width="104.19pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.202,3.132)"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 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transform="matrix(.013,0,0,-0.013,82.145,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,88.385,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,91.349,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g><g transform="matrix(.013,0,0,-0.013,97.59,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg>.</span> (A) <i>f</i> = 194, <i>r</i><sub>min</sub> = 10, <i>r</i><sub>ɸ</sub> = 10, <i>r</i><sub>max</sub> = 10, <span class="nowrap"><svg height="11.7782pt" id="M226" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 40.7613 11.7782" width="40.7613pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.668,3.132)"><path d="M498 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(B) Shape of the optimized projection area. (b) <span class="nowrap"><svg height="11.6217pt" id="M227" style="vertical-align:-3.272881pt" version="1.1" viewbox="-0.0498162 -8.34882 104.19 11.6217" width="104.19pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.202,3.132)"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 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transform="matrix(.013,0,0,-0.013,38.713,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.954,0)"><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,-0.013,50.097,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,57.299,3.132)"><path d="M518 50L491 51C452 54 444 60 444 110V444C429 441 405 437 377 434C348 430 314 427 287 426V403L323 397C356 392 365 386 365 333V99C331 66 293 51 257 51C212 51 169 75 169 164V299C169 366 169 413 172 444C156 441 128 437 101 433C75 430 50 427 29 426V403L57 397C82 391 90 386 90 333V137C90 29 147 -12 214 -12C241 -12 262 -4 291 13S342 48 365 65V-6L371 -12C390 -7 415 1 441 8C468 15 496 21 518 24V50Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,62.094,3.132)"><path d="M152 404V712C115 698 54 683 7 677V654C71 648 73 642 73 579V24C128 -2 179 -12 220 -12C353 -12 471 92 471 238C471 357 381 449 274 449C262 449 249 446 233 439L152 404ZM152 374C170 384 202 393 230 393C313 393 382 326 382 213C382 97 330 26 246 26C194 26 165 62 158 81C154 91 152 101 152 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23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g></svg>.</span> (A) <i>f</i> = 229, <i>r</i><sub>min</sub> = 3, <i>r</i><sub>ɸ</sub> = 3, <i>r</i><sub>max</sub> = 5, <span class="nowrap"><svg height="11.7782pt" id="M228" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 40.7613 11.7782" width="40.7613pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.668,3.132)"><path d="M498 658H277V626C335 621 342 616 342 579C220 574 46 510 46 330C46 148 226 88 342 86C342 41 334 37 271 32V0H498V32C435 37 427 41 427 86C546 88 723 153 723 337C723 518 550 575 427 579C427 616 434 621 498 626V658ZM342 123C255 129 141 183 141 338C141 489 257 538 342 542V123ZM427 542C512 538 629 489 629 333C629 178 512 129 427 123V542Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.822,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.085,0)"><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 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 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(B) Shape of the optimized projection area.</div>

Journal of Mathematics

fig6

Figure 6

Figure 6: Implementation of Structural Assemblies by Simultaneous Projection and Density-Based Topology Optimization 