Improved Dynamical Particle Swarm Optimization Method for Structural Dynamics

<table class="fixed-width table-group" id="tab3"><tr><td><table class="table"><colgroup><col style="width:6.00em"/><col style="width:11.71em"/><col style="width:10.21em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Design case</td><td class="align_center">Objective(s)</td><td class="align_center">Constraints</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">1</td><td class="align_center">Min. weight: (<span class="nowrap"><svg height="13.8758pt" id="M124" style="vertical-align:-5.48782pt" version="1.1" viewbox="-0.0498162 -8.38798 26.9561 13.8758" width="26.9561pt" 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="M493 503C489 551 484 614 483 650H43V622C120 616 128 611 128 525V126C128 40 120 34 40 28V0H312V28C221 34 213 40 213 126V307H316C407 307 412 296 424 227H453V420H424C412 355 407 346 316 346H213V584C213 613 216 616 246 616H322C398 616 419 607 436 579C449 559 455 539 464 499L493 503Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.877,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.554,3.132)"><path d="M156 406V710C117 695 54 682 9 675V647C73 642 75 635 75 573V25C131 -2 180 -12 223 -12C359 -12 478 92 478 240C478 357 387 451 278 451C265 451 251 447 236 440L156 406ZM156 373C175 384 208 392 235 392C315 392 387 328 387 214C387 97 332 30 251 31C197 31 170 66 162 84C158 93 156 104 156 118V373Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.158,3.132)"><path d="M139 535C170 535 192 560 192 589C192 621 169 644 140 644C109 644 86 621 86 589C86 560 110 535 139 535ZM35 387C99 379 105 374 105 307V114C105 22 103 -35 96 -78C89 -113 82 -131 70 -146C55 -162 30 -174 -1 -181C-16 -184 -23 -192 -23 -207C-23 -232 0 -253 15 -253C24 -253 33 -249 41 -242C63 -223 127 -148 162 -48C181 6 186 47 186 117V453C143 435 93 422 36 414L35 387Z"></path></g><rect height="0.3985" width="2.74024" x="19.0997" y="2.73319"></rect><g transform="matrix(.0091,0,0,-0.0091,21.84,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg>)</span></td><td class="align_center">Structural and seismic</td></tr><tr><td class="align_left" colspan="3"><hr/></td></tr><tr><td class="align_left" rowspan="2">2</td><td class="align_center">Min. weight: (<span class="nowrap"><svg height="13.8758pt" id="M125" style="vertical-align:-5.48782pt" version="1.1" viewbox="-0.0498162 -8.38798 26.9561 13.8758" width="26.9561pt" 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="M493 503C489 551 484 614 483 650H43V622C120 616 128 611 128 525V126C128 40 120 34 40 28V0H312V28C221 34 213 40 213 126V307H316C407 307 412 296 424 227H453V420H424C412 355 407 346 316 346H213V584C213 613 216 616 246 616H322C398 616 419 607 436 579C449 559 455 539 464 499L493 503Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.877,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.554,3.132)"><path d="M156 406V710C117 695 54 682 9 675V647C73 642 75 635 75 573V25C131 -2 180 -12 223 -12C359 -12 478 92 478 240C478 357 387 451 278 451C265 451 251 447 236 440L156 406ZM156 373C175 384 208 392 235 392C315 392 387 328 387 214C387 97 332 30 251 31C197 31 170 66 162 84C158 93 156 104 156 118V373Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.158,3.132)"><path d="M139 535C170 535 192 560 192 589C192 621 169 644 140 644C109 644 86 621 86 589C86 560 110 535 139 535ZM35 387C99 379 105 374 105 307V114C105 22 103 -35 96 -78C89 -113 82 -131 70 -146C55 -162 30 -174 -1 -181C-16 -184 -23 -192 -23 -207C-23 -232 0 -253 15 -253C24 -253 33 -249 41 -242C63 -223 127 -148 162 -48C181 6 186 47 186 117V453C143 435 93 422 36 414L35 387Z"></path></g><rect height="0.3985" width="2.74024" x="19.0997" y="2.73319"></rect><g transform="matrix(.0091,0,0,-0.0091,21.84,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg>)</span></td><td class="align_center" rowspan="2">Structural and seismic</td></tr><tr><td class="align_center">Min. input energy: (<span class="nowrap"><svg height="13.8758pt" id="M126" style="vertical-align:-5.48782pt" version="1.1" viewbox="-0.0498162 -8.38798 26.9561 13.8758" width="26.9561pt" 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="M493 503C489 551 484 614 483 650H43V622C120 616 128 611 128 525V126C128 40 120 34 40 28V0H312V28C221 34 213 40 213 126V307H316C407 307 412 296 424 227H453V420H424C412 355 407 346 316 346H213V584C213 613 216 616 246 616H322C398 616 419 607 436 579C449 559 455 539 464 499L493 503Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.877,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.554,3.132)"><path d="M156 406V710C117 695 54 682 9 675V647C73 642 75 635 75 573V25C131 -2 180 -12 223 -12C359 -12 478 92 478 240C478 357 387 451 278 451C265 451 251 447 236 440L156 406ZM156 373C175 384 208 392 235 392C315 392 387 328 387 214C387 97 332 30 251 31C197 31 170 66 162 84C158 93 156 104 156 118V373Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.158,3.132)"><path d="M139 535C170 535 192 560 192 589C192 621 169 644 140 644C109 644 86 621 86 589C86 560 110 535 139 535ZM35 387C99 379 105 374 105 307V114C105 22 103 -35 96 -78C89 -113 82 -131 70 -146C55 -162 30 -174 -1 -181C-16 -184 -23 -192 -23 -207C-23 -232 0 -253 15 -253C24 -253 33 -249 41 -242C63 -223 127 -148 162 -48C181 6 186 47 186 117V453C143 435 93 422 36 414L35 387Z"></path></g><rect height="0.3985" width="2.74024" x="19.0997" y="2.73319"></rect><g transform="matrix(.0091,0,0,-0.0091,21.84,3.132)"><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"></path></g></svg>)</span></td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Test cases for the single and multiobjective design of the 3-story shear building.</div>

Mathematical Problems in Engineering

tab3

Table 3

Table 3: Improved Dynamical Particle Swarm Optimization Method for Structural Dynamics 