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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 Displacement and Rotation</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 = "shortdesc">This section describes the theory of displacements and rotation.
<ul><li><a href = "#tso-c-usr-terms-displRotOvw__tso-c-usr-terms-displRotOvw-anaType" id = "toc_rg" title = "">Analysis Types: Static Linear or Nonlinear Analysis</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 = "For getting the scalar value of a design response or a variable, it is necessary to combine values derived from model properties or analysis results. This could be done getting the maximum or minimum value within a number of load cases or a number of group entities or creating response combinations like relative values, differences or mean values." href = "tso-m-usr-terms-combTerms-sb.htm#tso-m-usr-terms-combTerms-sb">Combined Terms</a></td></tr></table></td></tr></table>






<div class = "body conbody">
<table class = "table" id = "tso-c-usr-terms-displRotOvw__xx926195"><caption/><colgroup><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926195__entry__1"><p>Parameter Name</p></th>
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926195__entry__2"><p>Formula</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__1"><p>DISP_X, DISP_Y, DISP_Z</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__2"><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></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__1"><p>ROT_X, ROT_Y, ROT_Z</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__2"><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></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__1"><p>DISP_X_ABS, DISP_Y_ABS, DISP_Z_ABS</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__2"><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></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__1"><p>DISP_ABS</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926195__entry__2"><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 ">x</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">+</mo><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 ">y</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">+</mo><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 ">z</mi></mrow></msub></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></msqrt></mrow></math></span></td>
</tr>
</tbody></table>

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

<table class = "table" id = "tso-c-usr-terms-displRotOvw__table_E11BB6BA8A504392B5DF84F2C0BA9397"><caption/><colgroup><col style = "width:100%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">K</mi><mi class = "- topic/foreign ">u</mi><mo class = "- topic/foreign ">=</mo><mi class = "- topic/foreign ">F</mi></mrow></math></span></td>
</tr>
</tbody></table>
<p> where K  might be linear or nonlinear.</p>
<p>For displacements and rotations, 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. <span class = "ph">Tosca</span> sensitivities support nonlinear contact, but material 
and large displacement theory are only supported by <span class = "ph">Tosca</span> sensitivities for 
Topology optimization. Sizing, Shape and Bead optimization only allow linear 
material and small displacement theory.
</p>
<table class = "table" id = "tso-c-usr-terms-displRotOvw__xx926216"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926216__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926216__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926216__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926216__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-displRotOvw__xx926216__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__5"><p>S</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-displRotOvw__xx926216__entry__5"><p>S</p></td>
</tr>
</tbody></table>



<p>Displacements and rotations are the primary variables in the FEM solution.
They are also very often the main interest of the FEM-analyst, for example, the
maximal displacement. Displacements and rotations should be defined
using a nodal id, although node groups might also be referenced. Large
node groups can lead to major performance issues, see 
<a class = "xref" href = "tso-m-usr-terms-combTerms-groupOper-sb.htm" title = "For extracting one scalar value from a field of analysis results for a number of group entities into one design response, it is necessary to combine these results using an operator (GROUP_OPER).">Group Operations for Design Responses</a>.</p>
<p>Displacements and rotations can also be referenced in a local coordinate
system. </p>

<p><table class = "Remark" id = "table132"><tr><td class = "Remark"><span class = "run-in.important">Important:
				</span><span class = "notecontent"><ol class = "ol" id = "tso-c-usr-terms-displRotOvw__ol_B6B9B6A399D044FF9BB08871C8BB3C17">
<li class = "li">It is always strongly recommended that the user defines design elements
attached to nodes used in displacement definitions or reaction definitions
(<code class = "ph codeph">DRESP</code>) as frozen elements. This stabilizes the optimization
iterations and often leads to a significant lower number of optimization
iterations.</li>
<li class = "li">When using solver sensitivities, all types of nonlinear analysis are supported. See 
<a class = "xref" href = "tso-c-usr-solver-abaqus-remsensbasedshape.htm#tso-c-usr-solver-abaqus-remsensbasedshape" title = "Additional remarks concerning sensitivity-based optimizations with Abaqus are summarized in this section.">Remarks for Sensitivity-Based Optimizations</a></li>
</ol></span></td></tr></table>
</p>
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