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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>Typical Tasks for Static Analysis</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 deals with the typical tasks for linear static analysis types. Only some very common tasks are described here.</span>

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<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-optProb" id = "toc_rg" title = "">Minimizing Mass with Stiffness Restrictions</a></li><li><a href = "#tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-def" id = "toc_rg" title = "">Necessary Definitions</a></li><li><a href = "#tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-ResConv" id = "toc_rg" title = "">Result and Convergence</a></li></ul>
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<div class = "section" id = "tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-optProb"><h2 class = "title sectiontitle">Minimizing Mass with Stiffness Restrictions</h2>

<p>
If the structure with the minimal volume (mass) subject to displacement
constraints (corresponding to a restriction on the mechanical stiffness)
is sought, the optimization task is formulated as follows:
</p><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 ">min</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mstyle displaystyle = "false" class = "- topic/foreign "><mrow class = "- topic/foreign "><mstyle displaystyle = "false" class = "- topic/foreign "><mrow class = "- topic/foreign "><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 "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">,</mo><mi class = "- topic/foreign ">n</mi></mrow></munder><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">Vol</mo></mrow></mrow></mstyle></mrow></mstyle></mrow></mstyle><mo class = "- topic/foreign ">)</mo></mrow></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 ">j</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 "><mtext class = "- topic/foreign ">restrict,j</mtext></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><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><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><mtext class = "- topic/foreign "> </mtext><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">j</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">…</mo><mi class = "- topic/foreign ">m</mi><mo class = "- topic/foreign ">⁢</mo></mrow></math></span></p></td>
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<p>where Vol is the relative material volume of an element in 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 ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></msub></mrow></math></span> is 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 "><mtext class = "- topic/foreign ">restrict,j</mtext></mrow></msub></mrow></math></span> is the restriction for
the nodal displacement of the node j.</p>
<p>A model for the minimization of the relative material volume under
the displacement of the loaded nodes is presented in the example below.</p>
<p>The following figures show different <span class = "ph">load case</span>
with its constraints and the corresponding formula. The amplitude of
the distortion is defined by two variables: u1 and u2 give the displacement
at the two pints where the load is applied. The indices x and z are directions
in local or global coordinate systems. </p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p>Bending stiffness </p><br/><img class = "image" id = "tso-c-usr-sizing-tasksStatic-minMass__image_945C4077FEA14FC083B364616298E4BA" src = "../TsoUserImages/size_typtasks_bendingstiff2.png" width = "334" height = "174"/><br/><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac 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 ">z</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></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 ">z</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">1</mn></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mspace indentalign = "left" linebreak = "newline" class = "- topic/foreign "/></mrow></math></span> target bending</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>Torsional stiffness </p><br/><img class = "image" id = "tso-c-usr-sizing-tasksStatic-minMass__image_C3D92B8F45E9426EA0DB0C17E329E790" src = "../TsoUserImages/size_typtasks_torsstiff2.png" width = "334" height = "244"/><br/><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac 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 ">z</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></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 ">z</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">1</mn></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mspace linebreak = "newline" indentalign = "left" class = "- topic/foreign "/></mrow></math></span> target torsional</p></td>
</tr>
<tr class = "row">
<td class = "entry"><p>Axial stiffness</p><br/><img class = "image" id = "tso-c-usr-sizing-tasksStatic-minMass__image_6FB4E8351C844BD4B37686B41F23E88C" src = "../TsoUserImages/size_typtasks_axialstiff2.png" width = "334" height = "246"/><br/><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac 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 ">x</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></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 ">x</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">1</mn></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></mfrac><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">⁢</mo><mtext class = "- topic/foreign "> </mtext><mspace indentalign = "left" linebreak = "newline" class = "- topic/foreign "/></mrow></math></span> target axial</p></td>
</tr>
</tbody></table>

</div>

<div class = "section" id = "tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-def"><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>Two design responses are needed: </p>
<ul class = "ul" id = "tso-c-usr-sizing-tasksStatic-minMass__ul_FC8D74C51B3F4EDEA2ED0FCF5184C8C4">
<li class = "li"> The first design response is the design response representing the relative
material volume of the design area. This design response is the objective
function that is to be minimized.</li>
<li class = "li">The second design response is the displacement of the loaded node.
The absolute displacement is used in this case. Alternatively, the displacement
in the direction of the load (in the example it is z- or x-direction
of a local coordinate system) can be used. The restriction to one direction
reduces the number of <span class = "ph">load case</span>
generated by the optimization system because using absolute displacement
leads to the need of 3 additional <span class = "ph">load case</span>
in order to calculate the sensitivities while the displacement in z-direction
only requires one additional pseudo-<span class = "ph">load case</span>
for the calculation of the sensitivities. The constraint is then defined
as an inequality constraint with an absolute value for the displacement
value.</li>
</ul>
</div>

<div class = "section" id = "tso-c-usr-sizing-tasksStatic-minMass__tso-c-usr-sizing-tasksStatic-minMass-ResConv"><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">Result and Convergence</h2>

<p>The objective function (minimize mass - mass normalized) is 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-usr-sizing-tasksStatic-minMass__image_CA1B7CF433734809AB092A5B427A65C0" src = "../TsoUserImages/aa0bec12.jpg" width = "585" height = "201"/><br/></td>
</tr>
</tbody></table>
<p>The constraints (stiffness restrictions) are shown in the next 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-usr-sizing-tasksStatic-minMass__image_78B1FCF4A88A4334828B20CC5F7C8A2F" src = "../TsoUserImages/aa0bec1f.jpg" width = "585" height = "242"/><br/></td>
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</tbody></table>


<p> In the figure above, you can see the gradient of the normalized mass about
15 Iterations. In this example, the reduction amounts about 15%. Also,
the gradients of the normalized three stiffness constraints for the
three <span class = "ph">load case</span>
(bending, axial and torsional) are illustrated. What you can see is,
that they are fulfilled.</p>
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