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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>Overview of Energy Stiffness Measure</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">This section describes the theory of <code class = "ph codeph">ENERGY_STIFF_MEASURE</code>.
</span>

</p>
<ul><li><a href = "#tso-c-usr-terms-dresps-enStiffMsr__tso-c-usr-terms-dresps-enStiffMsrOvw-anaType" id = "toc_rg" title = "">Analysis Types: Static Linear or Nonlinear Analysis</a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">
<table class = "table" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924795"><caption/><colgroup><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924795__entry__1"><p>Parameter Name</p></th>
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924795__entry__2"><p>Formula</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924795__entry__1"><p>ENERGY_STIFF_MEASURE</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924795__entry__2"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">c</mi><mo class = "- topic/foreign ">=</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">1</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mi class = "- topic/foreign ">P</mi><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">−</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">1</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mi class = "- topic/foreign ">R</mi><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">*</mo></mrow></math></span></td>
</tr>
</tbody></table>

<div class = "section" id = "tso-c-usr-terms-dresps-enStiffMsr__tso-c-usr-terms-dresps-enStiffMsrOvw-anaType"><h2 class = "title sectiontitle">Analysis Types: Static Linear or Nonlinear Analysis</h2>

<p>For energy stiffness 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>
<p>For Bead and Shape, this design response is available only with 
  <span class = "ph">Abaqus</span> sensitivities.</p>
<table class = "table" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__3"><p>S*</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__4"><p>S*</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__5"><p>S</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-dresps-enStiffMsr__xx924938__entry__5"><p>S</p></td>
</tr>
</tbody></table>
<p><code class = "ph codeph">ENERGY_STIFF_MEASURE</code> describes a new stiffness measure
without physical meaning for handling of prescribed displacement in stiffness optimization. </p>
<p>For stiffness optimization of structures with only external loading,
the strain energy should be minimized 
(see <a class = "xref" href = "tso-m-usr-terms-dresps-compliance-sb.htm" title = "This section describes the theory of compliance optimization.">Compliance (Stiffness Optimization)</a>):

<span class = "ph inlineequation"><math class = "- topic/foreign "><mtext class = "- topic/foreign ">min </mtext><mfenced open = "(" close = ")" separators = "" class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac></mfenced></math></span>  

where P is the external loading and u is the corresponding nodal deflections
of the loaded nodes.</p>
<p>If a <span class = "ph">load case</span> 
is driven by prescribed displacements, the elastic energy/compliance will only decrease
if the structure is made softer. If only prescribed displacements are present without
external loading, the strain energy should be maximized to obtain optimal results:
<span class = "ph inlineequation"><math class = "- topic/foreign "><mtext class = "- topic/foreign ">max </mtext><mfenced open = "(" close = ")" separators = "" class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi><mo class = "- topic/foreign ">⋅</mo><msup class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">*</mo></msup></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac></mfenced></math></span>
where u* are the nodal prescribed displacements being different from zero
and R are the corresponding nodal reaction forces. </p>

<p>The physical strain energy with both external load and prescribed
displacements is described as follows but might not lead to optimal stiffness
results:   

<span class = "ph inlineequation"><math class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mo class = "- topic/foreign ">+</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi><mo class = "- topic/foreign ">⋅</mo><msup class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">*</mo></msup></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac></math></span>
</p>

<p>The new stiffness measure (<code class = "ph codeph">ENERGY_STIFF_MEASURE</code>)
combines the first two approaches in the following optimization formulation: 
<span class = "ph inlineequation"><math class = "- topic/foreign "><mtext class = "- topic/foreign ">min </mtext><mfenced open = "(" close = ")" separators = "" class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi><mo class = "- topic/foreign ">⋅</mo><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mo class = "- topic/foreign ">-</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi><mo class = "- topic/foreign ">⋅</mo><msup class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">*</mo></msup></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac></mfenced></math></span>
</p>

<table class = "Remark" id = "table132"><tr><td class = "Remark"><span class = "run-in.important">Important:
				</span><span class = "notecontent"><p><ul class = "ul" id = "tso-c-usr-terms-dresps-enStiffMsr__ol_0017829A65D54E869E01ED86E7BFB0DF">
<li class = "li">Always minimize energy stiffness measure
                (<code class = "ph codeph">TARGET = MIN</code>) in the objective function independent on external
              loading, prescribed displacement, and thermal loading. This requires less user effort
              compared to total strain energy.</li>
<li class = "li">Always apply energy stiffness measure to <code class = "ph codeph">ALL_ELEMENTS</code>.</li>
<li class = "li">Energy stiffness measure can also be applied in a min-max optimization
formulation (<code class = "ph codeph">TARGET = MINMAX</code>).</li>
<li class = "li">The energy stiffness measure is also allowed in constraints.</li>
 <li class = "li">Thermal loading is allowed as long the thermal loading
              is independent on material distribution.</li>

<li class = "li">If only external loading is present in a given <span class = "ph">load case</span>, the energy stiffness measure (<code class = "ph codeph">TYPE = ENERGY_STIFF_MEASURE</code>) is
              equal to the total strain energy (<code class = "ph codeph">TYPE = STRAIN_ENERGY</code>) also called
              the compliance.</li>
<li class = "li">If only prescribed displacements are present in a given <span class = "ph">load case</span>
then the energy stiffness measure (<code class = "ph codeph">TYPE = ENERGY_STIFF_MEASURE</code>)
is equal to the minus total strain energy (<code class = "ph codeph">TYPE = STRAIN_ENERGY</code>)
also called the compliance.</li>
<li class = "li">If both external loading and prescribed displacements
              are present in a given <span class = "ph">load case</span>, the energy stiffness measure (<code class = "ph codeph">TYPE = ENERGY_STIFF_MEASURE</code>) is not
              equal to the total strain energy (<code class = "ph codeph">TYPE = STRAIN_ENERGY</code>).</li>
<li class = "li">The energy stiffness measure is not a physical energy
              measure.</li>
<li class = "li">The energy stiffness measure is not available for the controller-based
optimization strategy.</li>
</ul></p></span></td></tr></table>

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