Topology Optimization Using Parabolic Aggregation Function with Independent-Continuous-Mapping Method

<table>(a) The elemental strain energy <svg height="7.9499998" id="M360" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 7.1624999 7.9499998" width="7.1624999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350
q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" id="x1D452"></path></g>
</svg> distribution for the MBB-beam problem. (b) The independent continuous topological design variables <svg height="10.3875" id="M361" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 5.9250002 10.3875" width="5.9250002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.125)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g>
</svg>-distribution for the MBB-beam problem. (c) Stress <svg height="8.3000002" id="M362" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 9.7250004 8.3000002" width="9.7250004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.038)"><path d="M548 455q-32 -77 -74 -77q-37 0 -103 10l-3 -2q36 -30 48.5 -62t12.5 -80q0 -103 -75 -179.5t-175 -76.5q-70 0 -113 45t-43 128q0 115 83.5 200.5t209.5 85.5q31 0 77.5 -4t67.5 -4q36 0 62 30zM350 274q0 54 -17 86q-19 35 -57 35q-50 0 -90 -37.5t-59 -91.5t-19 -109
q0 -61 24.5 -96.5t65.5 -35.5q51 0 87.5 46.5t50.5 100.5t14 102z" id="x1D70E"></path></g>
</svg> distribution for the MBB-beam problem.</table>

Mathematical Problems in Engineering

fig13

Figure 13

Figure 13: Topology Optimization Using Parabolic Aggregation Function with Independent-Continuous-Mapping Method 