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<a name = "hj-top"> </a><table class = "table1" id = "table11"><tr><td><table class = "DocHeader"><tr><td class = "DocHeader1" colspan = "2"><h1>Volume Overview</h1></td></tr><tr><td class = "DocHeader4" colspan = "2"/></tr><tr><td class = "DocHeader3" colspan = "2"><table class = "DocThemeIntro" id = "table12"><tr><td class = "Intro1Only"><p class = "header"><p class = "abstract">
<span class = "shortdesc">For standard optimization problems, the volume constraint is
necessary, for example, if the topology optimization is used in order to minimize
the compliance (thus to maximize the stiffness) and no volume constraint
is used, the optimizer simply fills up the whole design area with material.</span>

</p>
<ul><li><a href = "#tso-c-usr-terms-volume__tso-c-usr-terms-volume-anaType" id = "toc_rg" title = "">Analysis-Independent Design Response </a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">
<table class = "table" id = "tso-c-usr-terms-volume__xx944603"><caption/><colgroup><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-volume__xx944603__entry__1"><p>Parameter Name</p></th>
<th class = "entry" id = "tso-c-usr-terms-volume__xx944603__entry__2"><p>Formula</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944603__entry__1"><p>VOLUME</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944603__entry__2"><span class = "ph inlineequation"><math class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow></msub></math></span></td>
</tr>
</tbody></table>

<p>where <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span> is the element volume.
For topology optimization, design elements are scaled with the current
relative density. </p>
<div class = "section" id = "tso-c-usr-terms-volume__tso-c-usr-terms-volume-anaType"><h2 class = "title sectiontitle">Analysis-Independent Design Response </h2>

<p>For the volume, the following table shows the allowed combinations
between the strategy and the items <code class = "ph codeph">OBJ_FUNC</code> and <code class = "ph codeph">CONSTRAINT</code>
with C for controller and S for sensitivity-based optimization. </p>
<table class = "table" id = "tso-c-usr-terms-volume__xx944634"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-volume__xx944634__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-volume__xx944634__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-volume__xx944634__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-volume__xx944634__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-volume__xx944634__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__2"><p>C, S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__3"><p>C, S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__4"/>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__5"><p>S</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__2"><p>C, S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__3"><p>C, S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__4"/>
<td class = "entry" headers = "tso-c-usr-terms-volume__xx944634__entry__5"><p>S</p></td>
</tr>
</tbody></table>


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