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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>About Minimizing a Reaction or Internal Force</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 aim of this optimization is to minimize the internal force
of spot-weld elements. Both the relative material volume constraint and
a displacement constraint are used. </span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-user-TopOpt-StaAna-MinRF__tso-c-user-TopOpt-StaAna-MinRF-OptPro" id = "toc_rg" title = "">Formulation of the Optimization Problem</a></li><li><a href = "#tso-c-user-TopOpt-StaAna-MinRF__tso-c-user-TopOpt-StaAna-MinRF-NecDef" id = "toc_rg" title = "">Necessary Definitions</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 = "This section describes the theory of reaction forces." href = "tso-c-usr-terms-reactForceOvw.htm#tso-c-usr-terms-reactForceOvw">Overview of Reaction Force</a></td></tr><tr><td><a title = "This section describes the theory of internal forces." href = "tso-c-usr-terms-intForceOvw.htm#tso-c-usr-terms-intForceOvw">Overview of Internal Force</a></td></tr></table></td></tr></table>




<div class = "body conbody">
<p>A simple box model with 16 spot-weld elements is optimized.</p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><br/><img class = "image" id = "tso-c-user-TopOpt-StaAna-MinRF__image_59B1AA0991AE4BFB85F1414367C235CB" src = "../TsoUserImages/ag09742f.jpg" width = "500"/><br/></td>
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</tbody></table>
<p>Displacement plots for <span class = "ph">load cases</span> 1 and 2 are pictured in the following figure.</p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><br/><img class = "image" id = "tso-c-user-TopOpt-StaAna-MinRF__image_93903365FC864976B70FE172BFA172F6" src = "../TsoUserImages/ag097468.jpg" width = "500"/><br/></td>
</tr>
</tbody></table>
<p> The box is loaded with 2 forces on the two sides. The edges of the
bottom rectangle are fixed.</p>
<div class = "section" id = "tso-c-user-TopOpt-StaAna-MinRF__tso-c-user-TopOpt-StaAna-MinRF-OptPro"><h2 class = "title sectiontitle">Formulation of the Optimization Problem</h2>

<p>The optimization problem can be formulated in different ways: </p>
<ul class = "ul"><li class = "li">The first possible formulation is to minimize the maximum spot-weld
force under a volume constraint. The second formulation is to minimize
the volume while restricting the forces in the spot-weld elements. Additionally,
a displacement constraint is applied.</li><li class = "li">The second formulation is presented in the following:<table class = "table"><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 "><mo class = "- topic/foreign ">(</mo><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">V</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">l</mi><mo class = "- topic/foreign ">)</mo></mrow></math></span></p><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi></mrow></msub><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">L</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1...</mn><mi class = "- topic/foreign ">m</mi></mrow></math></span></p><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi></mrow></msub><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">L</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1...</mn><mi class = "- topic/foreign ">k</mi></mrow></math></span></p></td>
</tr>
</tbody></table>

<p>with Vol being the relative material volume of the design area, <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span> the nodal force in an
element in a given direction, <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">L</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">t</mi></mrow></msub></mrow></math></span> the constraint force,
<span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">d</mi><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span> the nodal displacement,
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 ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">L</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">m</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">t</mi></mrow></msub></mrow></math></span> the restriction for
the nodal displacement.</p></li></ul>


</div>

<div class = "section" id = "tso-c-user-TopOpt-StaAna-MinRF__tso-c-user-TopOpt-StaAna-MinRF-NecDef"><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">Necessary Definitions</h2>

<p>Four design responses are needed in order to set up the optimization
task.</p>
<ul class = "ul" id = "tso-c-user-TopOpt-StaAna-MinRF__ul_71EDA2687A014D1B947AF4B691588844">
<li class = "li"> The first design response is the relative material volume of the design
area. This relative material volume design response is then used in the
objective function that is to be minimized.</li>
<li class = "li">The second design response is the maximum internal force over the
nodes of a given node group belonging to the elements mentioned in the
given element group.</li>
<li class = "li">The third and fourth design responses are the displacements of the
loaded nodes that are then used in the constraints. These design responses
are defined as the total displacements for each <span class = "ph">load case</span>
and the appropriate loaded node.</li>
</ul>
</div>






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