The Application of Shape Gradient for the Incompressible Fluid in Shape Optimization

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="728534.fig.003a" src="https://static.hindawi.com/articles/mpe/volume-2013/728534/figures/728534.fig.003a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b><svg height="12.0375" id="M278" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 21.575001 12.0375" width="21.575001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,5.725,7.9)"><path d="M538 43v-23l-63 -7q-80 -9 -132 -26v65q-39 -37 -71.5 -51.5t-73.5 -14.5q-63 0 -98 38t-35 107v241q0 35 -9 47t-40 18v24h188v-334q0 -76 56 -76q38 0 82 47v276q0 34 -12.5 47t-48.5 16v24h200v-356q0 -34 12.5 -47.5t44.5 -14.5z" id="x1D42E"></path></g>
<g transform="matrix(.012,-0,0,-.012,15.175,11.975)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> for initial shape</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="728534.fig.003b" src="https://static.hindawi.com/articles/mpe/volume-2013/728534/figures/728534.fig.003b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b><svg height="12.0375" id="M279" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 15.9125 12.0375" width="15.9125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.9)"><path d="M538 43v-23l-63 -7q-80 -9 -132 -26v65q-39 -37 -71.5 -51.5t-73.5 -14.5q-63 0 -98 38t-35 107v241q0 35 -9 47t-40 18v24h188v-334q0 -76 56 -76q38 0 82 47v276q0 34 -12.5 47t-48.5 16v24h200v-356q0 -34 12.5 -47.5t44.5 -14.5z" id="x1D42E"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.512,11.975)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> for initial shape</td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="728534.fig.003c" src="https://static.hindawi.com/articles/mpe/volume-2013/728534/figures/728534.fig.003c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b><svg height="12.0375" id="M280" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 15.9125 12.0375" width="15.9125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.9)"><path d="M538 43v-23l-63 -7q-80 -9 -132 -26v65q-39 -37 -71.5 -51.5t-73.5 -14.5q-63 0 -98 38t-35 107v241q0 35 -9 47t-40 18v24h188v-334q0 -76 56 -76q38 0 82 47v276q0 34 -12.5 47t-48.5 16v24h200v-356q0 -34 12.5 -47.5t44.5 -14.5z" id="x1D42E"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.512,11.975)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> for optimal shape</td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="728534.fig.003d" src="https://static.hindawi.com/articles/mpe/volume-2013/728534/figures/728534.fig.003d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b><svg height="12.0375" id="M281" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 15.9125 12.0375" width="15.9125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.9)"><path d="M538 43v-23l-63 -7q-80 -9 -132 -26v65q-39 -37 -71.5 -51.5t-73.5 -14.5q-63 0 -98 38t-35 107v241q0 35 -9 47t-40 18v24h188v-334q0 -76 56 -76q38 0 82 47v276q0 34 -12.5 47t-48.5 16v24h200v-356q0 -34 12.5 -47.5t44.5 -14.5z" id="x1D42E"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.512,11.975)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> for optimal shape</td></tr></table></td></tr></table>

<table>Comparison of the initial shape and optimal shape (<svg height="11.375" id="M277" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 64.637497 11.375" width="64.637497" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M631 18l-4 -26q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -63 14 -77.5t74 -19.5v-28h-262v28q62 5 76 19.5t14 77.5v402q0 62 -13 76t-73 19v28h248q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172q27 -52 89 -144
q49 -71 81 -104q31 -36 72 -47zM212 316h48q73 0 108 29q55 42 55 124q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247z" id="x52"></path></g><g transform="matrix(.017,-0,0,-.017,10.619,11.112)"><path d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3
q14 0 19 4.5t5 15.5q1 41 -23.5 70.5t-65.5 29.5z" id="x65"></path></g><g transform="matrix(.017,-0,0,-.017,25.391,11.112)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,40.095,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,48.254,11.112)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,56.413,11.112)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg>).</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: The Application of Shape Gradient for the Incompressible Fluid in Shape Optimization 