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<html xmlns:mml = "http://www.w3.org/1998/Math/MathML" lang = "en"><head><meta charset = "UTF-8"/><meta name = "copyright" content = "(C) Copyright 2020"/><meta name = "DC.rights.owner" content = "(C) Copyright 2020"/><meta name = "DC.type" content = "concept"/><meta name = "abstract" content = "To check if the sensitivity-based sizing optimization has converged, two stop criteria are implemented. The user can specify if both stop criteria should be fulfilled or just one of the stop criteria must be fulfilled for the optimization algorithm to stop. Default is that both criteria should be fulfilled before the optimization algorithm stops. To specify that just one of the criteria must be fulfilled set STOP_CRITERION_LEVEL equal to SINGLE. The default value for STOP_CRITERION_LEVEL is BOTH."/><meta name = "description" content = "To check if the sensitivity-based sizing optimization has converged, two stop criteria are implemented. The user can specify if both stop criteria should be fulfilled or just one of the stop criteria must be fulfilled for the optimization algorithm to stop. Default is that both criteria should be fulfilled before the optimization algorithm stops. To specify that just one of the criteria must be fulfilled set STOP_CRITERION_LEVEL equal to SINGLE. The default value for STOP_CRITERION_LEVEL is BOTH."/><meta name = "DC.format" content = "HTML5"/><meta name = "DC.identifier" content = "tso-c-usr-sizing-StopCond-ConvCrit"/><meta name = "DC.language" content = "en"/><link rel = "stylesheet" type = "text/css" href = "../DSDocUI_XML34.css"/><title>Convergence Criteria</title>
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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>Convergence Criteria </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">To check if the sensitivity-based sizing optimization has
converged, two stop criteria are implemented. The user can specify if both stop
criteria should be fulfilled or just one of the stop criteria must be
fulfilled for the optimization algorithm to stop. Default is that both criteria
should be fulfilled before the optimization algorithm stops. To specify that
just one of the criteria must be fulfilled set
<code class = "ph codeph">STOP_CRITERION_LEVEL</code> equal to
<code class = "ph codeph">SINGLE</code>. The default value for
<code class = "ph codeph">STOP_CRITERION_LEVEL</code> is
<code class = "ph codeph">BOTH</code>.
</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-stop-Default" id = "toc_rg" title = "">Default Convergence Criteria</a></li><li><a href = "#tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvergenceCriteria" id = "toc_rg" title = "">Convergence Criteria</a></li><li><a href = "#tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ObjFunc" id = "toc_rg" title = "">Stop Criterion: Change in Objective Function</a></li><li><a href = "#tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ChangeDV" id = "toc_rg" title = "">Stop Criterion: Change in Element Thickness </a></li><li><a href = "#tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ItConvCheck" id = "toc_rg" title = "">Start Iteration for Convergence Check</a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">

<div class = "section" id = "tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-stop-Default"><h2 class = "title sectiontitle">Default Convergence Criteria</h2>

<p>
For sensitivity-based optimization, the optimization stops automatically when the following convergence criteria are fulfilled.
More detailed explanations of the criteria can be found in the sections beneath.
<pre class = "codeblock">
<code class = "ph codeph">
OPT_PARAM
 ID_NAME                  = NO_OPT_PARAM
 OPTIMIZE                 = MY_OPTIMIZE
 STOP_CRITERION_LEVEL     = BOTH
 STOP_CRITERION_OBJ       = 0.001
 STOP_CRITERION_THICKNESS = 0.005
 STOP_CRITERION_ITER      = 4
END_
</code>
</pre>
</p>
</div>

  
<div class = "section" id = "tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvergenceCriteria"><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">Convergence Criteria</h2>


<p>By default, the stop criterion measuring the change in the objective function
and the criterion measuring the change in the design variables should both be fulfilled before the optimization
algorithm stops. To specify that just one of the criteria must be
fulfilled, set <code class = "ph codeph">STOP_CRITERION_LEVEL</code> equal to <code class = "ph codeph">SINGLE</code>.
The default value for <code class = "ph codeph">STOP_CRITERION_LEVEL is BOTH</code>.</p>
<p>In the example below, the optimization algorithm should stop when just one of the criteria
is fulfilled</p>
<pre class = "codeblock">
<code class = "ph codeph">
OPT_PARAM
 ID_NAME              = opt_params
 OPTIMIZE             = id_of_optimize
 STOP_CRITERION_LEVEL = SINGLE
END_
</code>
</pre>
</div>

  
<div class = "section" id = "tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ObjFunc"><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">Stop Criterion: Change in Objective Function</h2>

<p> One of the available stop criteria is a measurement based on the change in the objective
function from one optimization design cycle to the next design cycle. This stop criterion
is defined by </p>
<table class = "table" id = "tso-c-usr-sizing-StopCond-ConvCrit__d197e25"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry">
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">|</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow></msub><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi><mo class = "- topic/foreign ">−</mo><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">|</mo></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">|</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow></msub><mo class = "- topic/foreign ">|</mo></mrow></mfrac><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">STOP_CRITERION_OBJ</mi></mrow></math></span></p>
</td>
</tr>
</tbody></table>

<p> where Θ is objective value, the index n is representing the present optimization iteration and is n-1 is the
previous iteration. When the criterion is lower than <code class = "ph codeph">STOP_CRITERION_OBJ</code>,
the stop criterion is fulfilled. The default for <code class = "ph codeph">STOP_CRITERION_OBJ</code> is
0.001. The value can be changed by the user; for example, </p>
<pre class = "codeblock">
<code class = "ph codeph">
OPT_PARAM
 ID_NAME            = opt_params
 OPTIMIZE           = id_of_optimize
 STOP_CRITERION_OBJ = 0.01
END_
</code>
</pre>
<p>This could be desirable when the number of optimizations should be reduced.</p>
</div>

  
<div class = "section" id = "tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ChangeDV"><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">Stop Criterion: Change in Element Thickness </h2>

<p>Another stop criterion is based on the change in the thicknesses
(design variables) in each element from one optimization iteration to the next
optimization iteration and is defined by:</p>
<table class = "table" id = "tso-c-usr-sizing-StopCond-ConvCrit__aa749688"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mstyle displaystyle = "false" class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi><mi class = "- topic/foreign ">u</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">b</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">r</mi><mo class = "- topic/foreign ">⁢</mo><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">⁢</mo><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">l</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">t</mi><mi class = "- topic/foreign ">s</mi></mrow></msubsup><mrow class = "- topic/foreign "><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 ">n</mi></mrow></msub><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 ">n</mi><mo class = "- topic/foreign ">−</mo><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">|</mo></mrow></mrow></mrow></mstyle></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi><mi class = "- topic/foreign ">u</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">b</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">r</mi><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">⁢</mo><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">l</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">t</mi><mi class = "- topic/foreign ">s</mi></mrow></mfrac><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">S</mi><mi class = "- topic/foreign ">T</mi><mi class = "- topic/foreign ">O</mi><mi class = "- topic/foreign ">P</mi><mi class = "- topic/foreign ">_</mi><mi class = "- topic/foreign ">C</mi><mi class = "- topic/foreign ">R</mi><mi class = "- topic/foreign ">I</mi><mi class = "- topic/foreign ">T</mi><mi class = "- topic/foreign ">E</mi><mi class = "- topic/foreign ">R</mi><mi class = "- topic/foreign ">I</mi><mi class = "- topic/foreign ">O</mi><mi class = "- topic/foreign ">N</mi><mi class = "- topic/foreign ">_</mi><mi class = "- topic/foreign ">T</mi><mi class = "- topic/foreign ">H</mi><mi class = "- topic/foreign ">I</mi><mi class = "- topic/foreign ">C</mi><mi class = "- topic/foreign ">K</mi><mi class = "- topic/foreign ">N</mi><mi class = "- topic/foreign ">E</mi><mi class = "- topic/foreign ">S</mi><mi class = "- topic/foreign ">S</mi></mrow></math></span></p>
</td>
</tr>
</tbody></table>
<p>where the index n is representing the present optimization iteration
and n-1 is the previous iteration. When the criterion is lower than
<code class = "ph codeph">STOP_CRITERION_THICKNESS</code>, the stop criterion is
fulfilled. The default for
<code class = "ph codeph">STOP_CRITERION_THICKNESS</code> is 0.005. The value can be changed by the user; for example,</p>
<pre class = "codeblock">
<code class = "ph codeph">
OPT_PARAM 
 ID_NAME                  = opt_params
 OPTIMIZE                 = id_of_optimize
 STOP_CRITERION_THICKNESS = 0.0001
END_
</code></pre>
</div>

<div class = "section" id = "tso-c-usr-sizing-StopCond-ConvCrit__tso-c-usr-sizing-StopCond-ConvCrit-ItConvCheck"><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">Start Iteration for Convergence Check</h2>

<p>The two convergence criteria above are not checked before a specified number of optimization
iterations have been executed. This means that the optimization will always be executed
until <code class = "ph codeph">STOP_CRITERION_ITER</code> is reached. The number is set by the number of
<code class = "ph codeph">STOP_CRITERION_ITER</code>. To change <code class = "ph codeph">STOP_CRITERION_ITER</code> to,
for example, 10, the following commands could be used: </p>
<pre class = "codeblock">
<code class = "ph codeph">
OPT_PARAM
 ID_NAME             = opt_params
 OPTIMIZE            = id_of_optimize
 STOP_CRITERION_ITER = 10
END_
</code>
</pre>
<p>The default value for
<code class = "ph codeph">STOP_CRITERION_ITER</code> is 4. Note that
<code class = "ph codeph">STOP_CRITERION_ITER</code> must be greater than 2.
</p>
</div>

</div>

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