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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 Constraints</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">Optimization tasks normally have restrictions that limit the values of the design
            responses or linear combinations of the design responses. If the constraint is not
            fulfilled, the optimization result is not feasible. </span>

</p>
<ul><li><a href = "#tso-c-usr-terms-constraint__tso-c-user-TermConstraint" id = "toc_rg" title = "">Constraint Formulations</a></li></ul>
</p></td></tr></table></td><td class = "DocHeader2"><table class = "DocTopicsSeeAlso" id = "table13"><tr><td class = "TopicsTitle">See Also</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 = "Definition of an equality or inequality (less or equal/ greater or equal) constraint. The constraint is defined with respect to a design response." href = "../TsoCmdMap/tso-r-cmd-constraint.htm#tso-r-cmd-constraint">CONSTRAINT</a></td></tr></table></td></tr></table>




<div class = "body conbody">
<p><span class = "ph">Tosca Structure</span>
allows inequality constraints in all sensitivity-based algorithms. Equality constraints are allowed only in the controller-based approaches.</p>
<p>This means that the item <code class = "ph codeph">EQ_VALUE</code> defining the equality
value might only be used for controller-based optimization. <code class = "ph codeph">LE_VALUE</code>
and <code class = "ph codeph">GE_VALUE</code> are to be used for the upper and lower
values of constraints in sensitivity-based optimization algorithms.
</p>
<p>An equality constraint is given as:</p>
<table class = "table" id = "tso-c-usr-terms-constraint__xx639419"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ψ</mi><mo class = "- topic/foreign ">=</mo><mi class = "- topic/foreign ">φ</mi></mrow></math></span></td>
</tr>
</tbody></table>

<p> where <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">φ</mi></mrow></math></span> is the value of the
design response. Inequality constraints are given as:                                                   
</p>
<table class = "table" id = "tso-c-usr-terms-constraint__xx639494"><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 "><mi mathvariant = "normal" class = "- topic/foreign ">Ψ</mi><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">φ</mi></mrow></math></span></p><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ψ</mi><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">φ</mi></mrow></math></span></p></td>
</tr>
</tbody></table>

<p>The <code class = "ph codeph">MAGNITUDE</code> can be defined as <code class = "ph codeph">ABS</code>
or <code class = "ph codeph">REL</code>, short for absolute or relative value of the
design response in the constraint. When using the relative value the
design response is normalized with respect to the initial value of the
design response (design response value of optimization iteration 0).
</p>

<div class = "section" id = "tso-c-usr-terms-constraint__tso-c-user-TermConstraint"><h2 class = "title sectiontitle">Constraint Formulations</h2>

<p> The following tables describe how a constraint for the respective design response is formulated.
    The design responses can be constrained using equality constraint, lower equal
    constraint, or greater equal constraint. <table class = "Remark" id = "table132"><tr><td class = "Remark"><span class = "run-in.important">Important:
				</span><span class = "notecontent"> The equality
    constraint is allowed only for the controller-based topology optimization. The
    lower equal or greater equal constraints are allowed for the sensitivity-based
    approach. The constraint values can be defined as absolute values or relative
    with respect to the corresponding values of the optimization start model.
    </span></td></tr></table>

</p>
<p>Moreover, for the sensitivity-based approach new combinations using <code class = "ph codeph">VAR_OPER</code>
or <code class = "ph codeph">GROUP_OPER</code> for the
<code class = "ph codeph">DRESP</code> can also be applied in the constraints.</p>
<table class = "table" id = "tso-c-usr-terms-constraint__ag571345"><caption/><colgroup><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571345__entry__1"><p> Compliance terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571345__entry__2"><p>Material volume terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571345__entry__3"><p>Eigenfrequency terms for constraints</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571345__entry__1"><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 ">C</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></msub><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">C</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">C</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 ">C</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571345__entry__2">
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">V</mi><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><mi class = "- topic/foreign ">V</mi><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">V</mi><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571345__entry__3">
<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 ">k</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 ">k</mi></mrow></msub><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 "><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 ">f</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><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 ">k</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 "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">≥</mo><mo class = "- topic/foreign ">△</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">*</mo></mrow></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><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 ">k</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 "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">≤</mo><mo class = "- topic/foreign ">△</mo><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">)</mo></mrow></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">p</mi></mrow></mfrac><mi class = "- topic/foreign ">ln</mi><mo class = "- topic/foreign ">⁡</mo><mo class = "- topic/foreign ">(</mo><mstyle displaystyle = "true" class = "- topic/foreign "><mrow class = "- topic/foreign "><munderover class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow></munderover><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">p</mi><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></msub></mrow></msup></mrow></mstyle><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
</td>
</tr>
</tbody></table>

<table class = "table" id = "tso-c-usr-terms-constraint__ag571373"><caption/><colgroup><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571373__entry__1"><p> Displacement terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571373__entry__2"><p>Reaction force terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag571373__entry__3"><p>Internal force terms for constraints</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571373__entry__1">
<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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><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 "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><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 ">θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">θ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup 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 ">i</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup 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 ">i</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><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 "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">*</mo><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 "><mi class = "- topic/foreign ">i</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 ">i</mi></mrow></msub><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 "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 "><mi class = "- topic/foreign ">i</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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><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 ">α</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">1</mn></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">α</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">+</mo><mn class = "- topic/foreign ">...</mn><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 ">i</mi></mrow></msub><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 ">α</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">1</mn></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">α</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">+</mo><mn class = "- topic/foreign ">...</mn><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571373__entry__2">
<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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">R</mi><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">R</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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">R</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag571373__entry__3">
<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 ">i</mi></mrow>
</msub><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><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 ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup 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 ">i</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><mo class = "- topic/foreign ">⁢</mo><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "/></msubsup></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><mo class = "- topic/foreign ">≤</mo><mi class = "- topic/foreign ">F</mi><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup><mo class = "- topic/foreign ">+</mo><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msubsup></mrow></msqrt><mo class = "- topic/foreign ">≥</mo><mi class = "- topic/foreign ">F</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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><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 ">i</mi><mo class = "- topic/foreign ">,</mo><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 ">F</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt><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 ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></mrow></math></span></p>
</td>
</tr>
</tbody></table>

<table class = "table" id = "tso-c-usr-terms-constraint__ag580591"><caption/><colgroup><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-constraint__ag580591__entry__1"><p>Von Mises stress terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag580591__entry__2"><p>Center of gravity terms for constraints</p></th>
<th class = "entry" id = "tso-c-usr-terms-constraint__ag580591__entry__3"><p>Moment of inertia terms for constraints</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag580591__entry__1">
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi><mi class = "- topic/foreign ">a</mi><mi class = "- topic/foreign ">x</mi><mo class = "- topic/foreign ">|</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mi class = "- topic/foreign ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">v</mi><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">s</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">s</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">)</mo><mi class = "- topic/foreign ">²</mi></mrow></msup></mrow></msub></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mi class = "- topic/foreign ">f</mi><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">)</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">f</mi></mrow></msub><mo class = "- topic/foreign ">)</mo><mi class = "- topic/foreign ">²</mi></mrow></mfrac><mo class = "- topic/foreign ">⋅</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">f</mi></mrow></msub><mo class = "- topic/foreign ">|</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">r</mi><mi class = "- topic/foreign ">e</mi><mi class = "- topic/foreign ">f</mi></mrow></msub></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag580591__entry__2">
<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 ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><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 ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
</td>
<td class = "entry" headers = "tso-c-usr-terms-constraint__ag580591__entry__3">
<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 ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≤</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub><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 ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub><mo class = "- topic/foreign ">⁢</mo><mo class = "- topic/foreign ">≥</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
</td>
</tr>
</tbody></table>
  
<table class = "table"><caption/><colgroup><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-constraint__tso-c-user-TermConstraint__entry__19"><p>Plastic strain terms for constraints</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-constraint__tso-c-user-TermConstraint__entry__19">
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">3</mn></mrow></mfrac><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup><mo class = "- topic/foreign ">:</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup></mrow></mrow></msqrt><mo class = "- topic/foreign ">≤</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup><mo class = "- topic/foreign ">*</mo></mrow></math></span></p>
<p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">[</mo><msqrt class = "- topic/foreign "><mrow class = "- topic/foreign "><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">3</mn></mrow></mfrac><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup><mo class = "- topic/foreign ">:</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup></mrow></mrow></msqrt><mo class = "- topic/foreign ">≥</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ε</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">pl</mi></mrow></msup><mo class = "- topic/foreign ">*</mo><mo class = "- topic/foreign ">]</mo></mrow></math></span></p>
</td>

</tr>
</tbody></table>





<p><table class = "Remark" id = "table132"><tr><td class = "Remark"><span class = "run-in.important">Important:
				</span><span class = "notecontent"><ul class = "ul">
<li class = "li">Constraints defined using relative values always refer to the design response
of the start model for the optimization.</li>
<li class = "li">The element densities in the optimization start model might be modified compared to your original
                            model (for example, when no volume constraint is present they are set to 50% of
                            the original density). Take this into account when defining, for
                            example, relative displacement or frequency constraints. This behavior
                            can be controlled by the user with the parameter
                                <code class = "ph codeph">DENSITY_INITIAL</code> in the <code class = "ph codeph">OPT_PARAM</code>
                            command. For more information, see <a class = "xref" href = "../TsoCmdMap/tso-r-cmd-optParam.htm#tso-r-cmd-optParam" title = "Defines parameters to control optimization.">OPT_PARAM</a>. In general, it is
                            not recommended that you use relative constraints in a topology
                            optimization, with an exception for volume and mass constraints.</li>
<li class = "li">The values marked with <code class = "ph codeph">*</code> are the constraint values defined by the user.</li>
<li class = "li">Design responses marked with <code class = "ph codeph">**</code> are only allowed using <span class = "ph">Abaqus</span> sensitivities.</li>
<li class = "li">Constraint expressions in <code class = "ph codeph">[]</code> are supported as constraint but do not apply to
                            the standard cases.</li></ul></span></td></tr></table>
</p>
</div>


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