GeometricTransformations
A framework for geometrically exact finite element analysis
This project implements a framework for simulating differential equations on
nonlinear manifolds.
<!--
Theoretical developments give rise to a modular computational
framework for composing coordinate transformations and manifold
parameterizations.
-->
This repository contains the source code for the investigation presented by Perez and Filippou (2024).
The MATLAB implementation requires the FEDEASLab toolbox for nonlinear structural analysis
which may be obtained from the author upon reasonable request.
A completely open-source C++ implementation of the framework is under development
for OpenSees and expected to be completed in 2025. Development is ongoing
at https://github.com/peer-open-source/xara.
All renderings were created with the veux library.
<dl>
<dt><a href="./Elements">Elements/</a></dt>
<dd>This directory contains the finite elements:
<ul>
<li><a href="./Elements/ExactShear3dFrm.m"><code>ExactShear3dFrm</code></a>
Displacement-interpolated shear 3d frame element with Cosserat strains;
this element implements the three geometrically exact beam formulations
presented in Appendix B. The implementation of this element in OpenSeesRT is documented <a href="https://opensees.stairlab.io/user/manual/model/elements/frame/ExactFrame.html">here</a>.
</li>
<li><a href="./Elements/GeomWrap3dFrm.m"><code>GeomWrap3dFrm</code></a> Wrapper for 3d frame elements. This element
implements the <em>element wrapper</em> from Section 6 using Algorithm 1
from Section 4.
</li>
<li><a href="./Elements/GeomTran3dFrm.m"><code>GeomTran3dFrm</code></a> Wrapper for 3d frame elements
that are formulated in a <em>basic</em> coordinate system. This element
implements the <em>element wrapper</em> from Section 6 using Algorithm 1
from Section 4.
</li>
</ul>
</dd>
<dt><a href="./Geometry">Geometry/</a></dt>
<dd>
<ul>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/GeomTran3dFrm_Pull.m"><code>GeomTran3dFrm_Pull</code></a>
This function implements the $g$ operation of the geometric transformations by calling the following functions:
<ul>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/Transform_RotPull.m"><code>Transform_RotPull</code></a></li>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/Transform_IsoPull.m"><code>Transform_IsoPull</code></a></li>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/Tran3dFrm_IsoPull.m"><code>Tran3dFrm_IsoPull</code></a></li>
</ul>
</li>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/GeomTran3dFrm_Push.m"><code>GeomTran3dFrm_Push</code></a>
This function implements the $\mathbf{a}_g$ and $\mathbf{k}_g$ operations of the geometric transformations.
<ul>
<li><a href="/claudioperez/FiniteRotationLab/blob/master/Geometry/Tran3dFrm_IsoPush.m"><code>Tran3dFrm_IsoPush</code></a></li>
</ul>
</li>
</ul>
</dd>
<dt><a href="./Rotations">Rotations/</a></dt>
<dd>This directory contains the rotation functions described in Appendix A of Perez and Filippou (2024).
</dd>
<dt><a href="./Examples">Examples/</a></dt>
<dd>
This directory contains scripts to reproduce the examples of the paper.
<ol>
<li><a href="./Examples/E10_Invariance.m">Invariance</a></li>
<li><ol>
<li><a href="./Examples/E21_PlaneMoment.m">Plane Cantilever - Point moment</a></li>
<li><a href="./Examples/E22_PlaneTransverse.m">Plane Cantilever - Transverse Force</a></li>
</ol></li>
<li><a href="./Examples/E30_HelicalForms.m">Space Cantilever - Point moment and force</a></li>
<li><a href="./Examples/E40_BatheCantilever.m">Curved Cantilever - Point force (Bathe's problem)</a></li>
<li><a href="./Examples/E50_Hockling.m">Hockling of a flexible rod.</a></li>
</ol>
</dd>
</dl>
For an example using "basic" elements, see <a href="./Examples/E00_Cantilever.m"><code>E00_Cantilever</code></a>.
<table align="center" style="border: 0;">
<tr style="background-color:rgba(0, 0, 0, 0);">
<td style="background-color:rgba(0, 0, 0, 0);" colspan="3">
<a>
<img src="./Figures/Figure_4a.png"
width="600" alt="OpenSeesRT Logo">
</a>
</td>
</tr>
</table>
- Perez CM, Filippou FC. On nonlinear geometric transformations of finite elements. Int
J Numer Methods Eng. 2024;e7506. doi:
10.1002/nme.7506
