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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>Objective Function for Sensitivity-Based Shape Optimization</h1></td></tr><tr><td class = "DocHeader4" colspan = "2"/></tr><tr><td class = "DocHeader3"><table class = "DocHeaderIntro" id = "table12"><tr><td class = "Intro1Only"><p class = "header"><p class = "abstract">
<span class = "shortdesc">
The objective function describes the optimization target.
In general, one scalar value (sometimes combined from other scalars)
is to be maximized or minimized. 
</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-shape-objFunc-sens__tso-c-usr-shape-objFun-ovw" id = "toc_rg" title = "">Overview</a></li><li><a href = "#tso-c-usr-shape-objFunc-sens__tso-c-usr-shape-objFun-terms" id = "toc_rg" title = "">Objective Function Terms</a></li><li><a href = "#tso-c-usr-shape-objFunc-sens__tso-c-user-ShapeSensOpt-ObjFun-MinMax" id = "toc_rg" title = "">Differences in Objective Target Formulation</a></li></ul>
</p></td></tr></table></td><td class = "DocHeader2"><table class = "DocTopicsSeeAlso" id = "table13"><tr><td class = "TopicsTitle">See Also</td></tr><tr><td><a title = "The objective function defines the values to be maximized or minimized during optimization. This function might depend on analysis results, geometric values or combinations of those." href = "tso-m-usr-terms-objfunc-sb.htm#tso-m-usr-terms-objfunc-sb">Objective Function</a></td></tr><tr><td><a title = "Most design responses are only available for certain analysis types. Others are independent of the analysis type because they are directly linked to the geometry of the FE-model. Each design response represents one scalar value which can be extracted from the model information (like the volume) or from the FE-results." href = "tso-m-usr-terms-dresps-sb.htm#tso-m-usr-terms-dresps-sb">Design Responses</a></td></tr><tr><td><a title = "Additional remarks concerning sensitivity-based optimizations with Abaqus are summarized in this section." href = "tso-c-usr-solver-abaqus-remsensbasedshape.htm#tso-c-usr-solver-abaqus-remsensbasedshape">Remarks for Sensitivity-Based Optimizations</a></td></tr></table><table class = "DocTopicsInOtherGuides" id = "table14"><tr><td height = "8px"/></tr><tr><td class = "TopicsTitleNormal">In Other Guides</td></tr><tr><td><a title = "Defines the objective function of the optimization." href = "../TsoCmdMap/tso-r-cmd-objFunc.htm#tso-r-cmd-objFunc">OBJ_FUNC</a></td></tr><tr><td><a title = "The table lists the properties of each variable type and which optimization strategy could be applied to which variable." href = "../TsoCmdMap/tso-r-cmd-dresp-optStrat.htm#tso-r-cmd-dresp-optStrat">Allowed Optimization Strategy for Design Responses</a></td></tr></table></td></tr></table>




<div class = "body conbody">

<div class = "section" id = "tso-c-usr-shape-objFunc-sens__tso-c-usr-shape-objFun-ovw"><h2 class = "title sectiontitle">Overview</h2>

<p>The <code class = "ph codeph">OBJECTIVE FUNCTION</code> is the function, which value can be
maximized or minimized during the optimization. This function depends on the
results of the FE analysis. Therefore, the values of interest must be derived
from the FE results in order to define an objective function and functional
constraints. The results of the FE analysis (total displacement, energy, etc.)
for determining the objective function and functional constraints are called
’design responses’. A single value from the design response must be specified
for the definition of the objective function using the command <code class = "ph codeph">DRESP</code>(Design Response).</p>
</div>


<div class = "section" id = "tso-c-usr-shape-objFunc-sens__tso-c-usr-shape-objFun-terms"><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">Objective Function Terms</h2>

<p><span class = "ph">Tosca Structure.shape</span> 
                          controller allows optimization on different stress hypotheses, strain formulations, and damage results. The most used equivalent stress is von Mises
                          (<code class = "ph codeph">SIG_MISES</code>).</p>
<p>The supported types of design responses usable in an objective function for sensitivity-based shape optimization are listed in the table below.</p>

<table class = "table"><caption/><colgroup><col/><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph uicontrol">Static analysis</span></td>
<td class = "entry"><span class = "ph uicontrol">Description</span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>CENTER_GRAVITY_X</p>
                      <p>CENTER_GRAVITY_Y</p>
                      <p>CENTER_GRAVITY_Z</p>
</td>
<td class = "entry"><p>Center of gravity design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>DISP_ABS</p>
                      <p>DISP_X</p>
                      <p>DISP_Y</p>
                      <p>DISP_Z</p>
                      <p>DISP_X_ABS</p>
                      <p>DISP_Y_ABS</p>
                      <p>DISP_Z_ABS</p></td>
<td class = "entry"><p>Displacement design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_XX</p>
                      <p>INERTIA_XY</p>
                      <p>INERTIA_XZ</p>
                      <p>INERTIA_YY</p>
                      <p>INERTIA_YZ</p>
                      <p>INERTIA_ZZ</p></td>
<td class = "entry"><p>Moment of inertia design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INTERNAL_FORCE_ABS</p>
                      <p>INTERNAL_FORCE_X</p>
                      <p>INTERNAL_FORCE_Y</p>
                      <p>INTERNAL_FORCE_Z</p>
                      <p>INTERNAL_FORCE_X_ABS</p>
                      <p>INTERNAL_FORCE_Y_ABS</p>
                      <p>INTERNAL_FORCE_Z_ABS</p></td>
<td class = "entry"><p>Internal force design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INTERNAL_MOMENT_X</p>
                      <p>INTERNAL_MOMENT_Y</p>
                      <p>INTERNAL_MOMENT_Z</p>
                      <p>INTERNAL_MOMENT_X_ABS</p>
                      <p>INTERNAL_MOMENT_Y_ABS</p>
                      <p>INTERNAL_MOMENT_Z_ABS</p></td>
<td class = "entry"><p>Internal moment design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>PEMAG</p></td>
<td class = "entry"><p>Plastic strain magnitude design response**</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>REACTION_FORCE_ABS</p>
                      <p>REACTION_FORCE_X</p>
                      <p>REACTION_FORCE_Y</p>
                      <p>REACTION_FORCE_Z</p>
                      <p>REACTION_FORCE_X_ABS</p>
                      <p>REACTION_FORCE_Y_ABS</p>
                      <p>REACTION_FORCE_Z_ABS</p></td>
<td class = "entry"><p>Reaction force design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>REACTION_MOMENT_X</p>
                      <p>REACTION_MOMENT_Y</p>
                      <p>REACTION_MOMENT_Z</p>
                      <p>REACTION_MOMENT_X_ABS</p>
                      <p>REACTION_MOMENT_Y_ABS</p>
                      <p>REACTION_MOMENT_Z_ABS</p></td>
<td class = "entry"><p>Reaction moment design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>ROT_ABS</p>
                      <p>ROT_X</p>
                      <p>ROT_Y</p>
                      <p>ROT_Z</p>
                      <p>ROT_X_ABS</p>
                      <p>ROT_Y_ABS</p>
                      <p>ROT_Z_ABS</p></td>
<td class = "entry"><p>Rotation design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>STRAIN_ENERGY</p></td>
<td class = "entry"/>
</tr>
<tr class = "row">
<td class = "entry"><p>SIG_1</p></td>
<td class = "entry"><p>Maximum principal stress.</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>SIG_3</p></td>
<td class = "entry"><p>Minimum principal stress.</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>SIG_MISES</p>
                      <p>SIG_SENS_MISES</p></td>
<td class = "entry"><p>Von Mises Stress design responses</p>
                      <p>For <code class = "ph codeph">SIG_MISES</code>, <code class = "ph codeph">SIG_SENS_MISES</code> is used for sensitivity calculation.</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>SIG_SIGNED_MISES</p></td>
<td class = "entry"><p>Signed von Mises stress failure criteria.</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>SIG_GLINKA_EEQ</p>
	                  <p>SIG_GLINKA_PEEQ</p>
	                  <p>SIG_GLINKA_SEQ</p>
	                  <p>SIG_NEUBER_EEQ</p>
	                  <p>SIG_NEUBER_PEEQ</p>
	                  <p>SIG_NEUBER_SEQ</p></td>
<td class = "entry"><p>Glinka and Neuber formulations for equivalent strain (_EEQ), stress (_SEQ) and plastic strain (_PEEQ) using the plastic correction factor, respectively**.</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>STRAIN_ENERGY</p></td>
<td class = "entry"><p>Strain energy design responses</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>WEIGHT</p></td>
<td class = "entry"><p>Weight design response</p></td>
</tr>

<tr class = "row">
<td class = "entry"><span class = "ph uicontrol">Modal analysis</span></td>
<td class = "entry"><span class = "ph uicontrol">Description</span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>DYN_FREQ</p></td>
<td class = "entry"><p>Dynamic frequency design response</p></td>
</tr>





<tr class = "row">
<td class = "entry"><span class = "ph uicontrol">Analysis independent</span></td>
<td class = "entry"><span class = "ph uicontrol">Description</span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>VOLUME</p></td>
<td class = "entry"><p>Volume design response</p></td>
</tr>

</tbody></table>

<p>Additional design response types</p>
<table class = "table" id = "tso-c-usr-shape-objFunc-sens__table_shape_sens_add_dresps"><caption/><colgroup><col/><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph uicontrol">Modal analysis</span></td>
<td class = "entry"><span class = "ph uicontrol">Description</span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>DYN_FREQ_KREISSEL</p></td>
<td class = "entry"><p>Dynamic frequency design response</p>
</td>
</tr>
</tbody></table>
<p><span class = "ph uicontrol">Note:</span></p>
<p><ul class = "ul">
<li class = "li">Stress responses (<code class = "ph codeph">*SIG*</code>) are only supported by Tetrahedron 4 and 10 as well as
    Hexahedron 8. This does not hold if solver sensitivities are used. 
    Take a look into the related topics.
    </li>
<li class = "li"><code class = "ph codeph">GBL_SIG_MISES_SENS</code> responses are based on element groups.</li>
<li class = "li"><code class = "ph codeph">SIG_*</code> will generate a pseudo load for EACH element.</li>
<li class = "li">Design responses marked with <code class = "ph codeph">**</code> are only allowed using 
    <span class = "ph">Abaqus</span> sensitivities.</li>


</ul></p>
</div>

<div class = "section" id = "tso-c-usr-shape-objFunc-sens__tso-c-user-ShapeSensOpt-ObjFun-MinMax"><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">Differences in Objective Target Formulation</h2>

The objective formulation for the
optimization varies depending on the objective target such as minimization/ maximization or a
Min-Max/ Max-Min definition. For further details and formulas, see 
<a class = "xref" href = "tso-c-usr-terms-minMaxObjFunc.htm" title = "This section deals with the minimization or maximization of the objective function.">Minimization or Maximization of an Objective Function</a> and 
<a class = "xref" href = "tso-c-usr-terms-multidiscObjFunc.htm" title = "This section deals with multidisciplinary objective functions.">Multidisciplinary Objectives (Minmax and Maxmin Formulations)</a>. </div>

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