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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 Constraint for Difference between Two Displacements</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">Aim of this optimization task is to obtain the structure with
maximal stiffness subject to the constraint on the difference of the
displacements of two nodes.</span>

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<ul><li><a href = "#tso-c-user-TopOpt-StaAna-DiffDisp__tso-c-user-TopOpt-StaAna-DiffDisp-OptPro" id = "toc_rg" title = "">Formulation of the Optimization Problem</a></li></ul>
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<p>This relative displacement constraint is used in the stiffness optimization
task of the shift fork model, see the following figure.</p>
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<p> The two arms of the shift fork have unequal displacements. This is
caused by the different lengths of the arms. In order to get a working
component, the displacements of the two arms must be nearly identical.
If the displacements are different, then the gear wheel is unable to
be moved with the shift-fork.</p>
<div class = "section" id = "tso-c-user-TopOpt-StaAna-DiffDisp__tso-c-user-TopOpt-StaAna-DiffDisp-OptPro"><h2 class = "title sectiontitle">Formulation of the Optimization Problem</h2>

<p>The optimization problem will be solved with the sensitivity-based
topology optimization. The optimization task is formulated as:</p>
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<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">min</mi><mo class = "- topic/foreign ">⁡</mo><mo class = "- topic/foreign ">(</mo><mi class = "- topic/foreign ">U</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 ">u</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</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 "><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 ">max</mi><mo class = "- topic/foreign ">⁡</mo><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">D</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">f</mi><mi class = "- topic/foreign ">f</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">c</mi><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span></p><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">l</mi><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">v</mi><mi class = "- topic/foreign ">o</mi><mi class = "- topic/foreign ">l</mi><mi class = "- topic/foreign ">_</mi><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">s</mi><mi class = "- topic/foreign ">t</mi><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">c</mi><mi class = "- topic/foreign ">t</mi></mrow></math></span></p></td>
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<p>with U being the sum of the strain energy over all elements, <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 "><mn class = "- topic/foreign ">1</mn></mrow></msub></mrow></math></span> 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 "><mn class = "- topic/foreign ">2</mn></mrow></msub></mrow></math></span> the displacements of
the nodes in a given direction (x, y, or z), Vol the relative material
volume, vol_restrict the value of the volume constraint 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 ">max</mi><mo class = "- topic/foreign ">⁡</mo><mtext class = "- topic/foreign "> </mtext><mi class = "- topic/foreign ">D</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">f</mi><mi class = "- topic/foreign ">f</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">n</mi><mi class = "- topic/foreign ">c</mi><mi class = "- topic/foreign ">e</mi></mrow></msub></mrow></math></span> the constraint value.
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