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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>Measure for Intermediate Densities</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">
The <code class = "ph codeph">DRESP</code> named <code class = "ph codeph">DENSITY_MEASURE</code>
can be applied for measuring the amount of intermediate densities.
</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-terms-densMeasure__tso-c-usr-terms-densMeasure-anaType" id = "toc_rg" title = "">Analysis-Independent Design Response </a></li><li><a href = "#tso-c-usr-terms-densMeasure__tso-c-usr-terms-densMeasure-parFile" id = "toc_rg" title = ""> Command in Parameter File</a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">


<p>
The <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> is defined
in the following way:
</p>
<p>
<table class = "table"><caption/><colgroup><col style = "width:100%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry">
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn></mrow><mrow class = "- topic/foreign "><mtext class = "- topic/foreign ">Volume</mtext></mrow></mfrac><mstyle displaystyle = "false" class = "- topic/foreign "><mrow class = "- topic/foreign "><munder class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∫</mo></mrow><mrow class = "- topic/foreign "><mtext class = "- topic/foreign ">Volume</mtext></mrow></munder><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">ρ</mi><mo class = "- topic/foreign ">)</mo></mrow></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">v</mi></mrow></mrow></mstyle><mo class = "- topic/foreign ">≈</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn></mrow><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">l</mi></mrow></msub></mrow></mfrac><mrow class = "- topic/foreign "><munderover class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">l</mi></mrow></msub></mrow></munderover><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow></msub><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow></mrow></mrow></mrow></math></span>
</td>
</tr>
</tbody></table>

</p>

<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 ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span>
 is the relative density
in each element of the given user-defined element group and 
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">l</mi></mrow></msub></mrow></math></span>
 is the number of elements in the defined element group, respectively.
</p>
<div class = "section" id = "tso-c-usr-terms-densMeasure__tso-c-usr-terms-densMeasure-anaType"><h2 class = "title sectiontitle">Analysis-Independent Design Response </h2>

<p>
For the density measure 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-densMeasure__xx945843"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx945843__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx945843__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx945843__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx945843__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx945843__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__3"/>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__4"/>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__5"/>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__2"><p> S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__3"/>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__4"/>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx945843__entry__5"/>
</tr>
</tbody></table>

<p>The <code class = "ph codeph">DRESP</code> named <code class = "ph codeph">DENSITY_MEASURE</code>
can be applied for measuring the amount of intermediate densities. The
<code class = "ph codeph">DENSITY_MEASURE</code> can be used in the <code class = "ph codeph">OBJ_FUNC</code>
or <code class = "ph codeph">CONSTRAINT</code> for minimizing and surpassing the number
of elements containing intermediate densities. This design response can
be used for additional penalization of intermediate elements and thereby,
obtain more clearly defined solid/void conceptual designs.
</p>
<p><code class = "ph codeph">DENSITY_MEASURE</code> is aimed at advanced dynamic applications
like modal eigenfrequency analysis and frequency response analysis.
</p>
<table class = "table" id = "tso-c-usr-terms-densMeasure__xx756326"><caption/><colgroup><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx756326__entry__1"><p>intermediate density of all elements</p></th>
<th class = "entry" id = "tso-c-usr-terms-densMeasure__xx756326__entry__2"><p>value of <code class = "ph codeph">DENSITY_MEASURE</code></p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__1"><p>0.5</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__2"><p>1.0</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__1"><p>1.0</p></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__2"><p>0.0</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__1"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub></mrow></math></span></td>
<td class = "entry" headers = "tso-c-usr-terms-densMeasure__xx756326__entry__2"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mo class = "- topic/foreign ">≈</mo><mn class = "- topic/foreign ">4</mn><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub></mrow></math></span></td>
</tr>
</tbody></table>

<p><code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> yields the
value 1.0 for the intermediate density 0.5 of all elements.</p>
<p><code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> yields the
value 0.0 for the intermediate density 1.0 of all elements.</p>
<p><code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> yields the
value 
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mo class = "- topic/foreign ">≈</mo><mn class = "- topic/foreign ">4</mn><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub></mrow></math></span> 
  for the intermediate minimum density 
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo></mrow></msub></mrow></math></span> of all elements.
</p>
<p>The following figure shows the <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code>
depending on element density with all elements having the same intermediate
density.
</p>
<table class = "table" id = "tso-c-usr-terms-densMeasure__xx756358"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry">
<br/><img class = "image" id = "tso-c-usr-terms-densMeasure__image_012D8F7A3EE446FE86ACF79145480C06" src = "../TsoUserImages/terms_density_measure.png" width = "185" height = "120"/><br/>
</td>
</tr>
</tbody></table>

</div>

<div class = "section" id = "tso-c-usr-terms-densMeasure__tso-c-usr-terms-densMeasure-parFile"><p><map name = "FPMap1"><area href = "#hj-top" title = "Back to Top" shape = "rect" coords = "416, 0, 435, 10"/></map><span class = "itemsprite"/></p><h2 class = "title sectiontitle"> Command in Parameter File</h2>

<p>The <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> is defined
in the following way:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME         = DRESP_DENSITY_MEASURE
 DEF_TYPE        = SYSTEM
 TYPE            = DENSITY_MEASURE
 EL_GROUP        = ………
 GROUP_OPER      = SUM
END_
</code>
</pre>

<p>The <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> can then
be applied in the objective function or in the constraint in the following
way:</p>
<pre class = "codeblock">
<code class = "ph codeph">
OBJ_FUNC&gt;
 ID_NAME     = DENSITY_MEASURE_ONJ
 DRESP       = ....
 DRESP       = DRESP_DENSITY_MEASURE, &lt;weighting_factor&gt;
 TARGET      = MIN
END_
</code>
</pre>
<p>Or</p>
<pre class = "codeblock">
<code class = "ph codeph">
CONSTRAINT
 ID_NAME       = DENSITY_MEASURE_constraint
 DRESP         = DRESP_DENSITY_MEASURE
 MAGNITUDE     = ABS
 LE_VALUE      = 0.05
END_
</code>
</pre>

<p>The design response <code class = "ph codeph">DENSITY_MEASURE</code> drives the optimization towards a local minimum and therefore needs to be used with care.</p>
  
<table class = "Remark" id = "table132"><tr><td class = "Remark"><span class = "run-in.important">Important:
				</span><span class = "notecontent">
<ol class = "ol" id = "tso-c-usr-terms-densMeasure__ol_AD66AE94DD5B4DD191E19C6B491D5FBD">
<li class = "li">
The <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> is only
for advanced applications. In particular, it is aimed at advanced dynamic
applications like modal eigenfrequency analysis and frequency response
analysis.
</li>
<li class = "li">
The weighting factor must be carefully chosen when applying the
<code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> in the objective
function. Otherwise, only one of the objective terms would be optimized.
</li>
<li class = "li">
The constraint value must be carefully chosen when applying the
<code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code> as constraint.
Otherwise, there might be no feasible solution.
</li>
<li class = "li">
The constraint value for <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code>
cannot be lower than <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo><mtext class = "- topic/foreign "> </mtext><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">s</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">t</mi><mi class = "- topic/foreign ">y</mi></mrow></math></span> (<code class = "ph codeph">DENSITY_LOWER</code>)
which is default 
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">4</mn><mo class = "- topic/foreign ">⋅</mo><mn class = "- topic/foreign ">0.001</mn><mo class = "- topic/foreign ">⋅</mo><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo><mn class = "- topic/foreign ">0.001</mn><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">≈</mo><mn class = "- topic/foreign ">0.004</mn></mrow></math></span>.
</li>
<li class = "li">
Normally, the <code class = "ph codeph">DRESP</code> <code class = "ph codeph">DENSITY_MEASURE</code>
yields a significant higher number of optimization iterations.
</li>
</ol>
</span></td></tr></table>

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