GeometricTransformations

A framework for geometrically exact finite element analysis

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This is a partial repository; an upload of the full project is in progress

This project implements a framework for simulating differential equations on nonlinear manifolds.

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 by 2025. Development is ongoing at https://github.com/claudioperez/OpenSeesRT.

Elements/

This directory contains the finite elements:

• ExactShear3dFrm Displacement-interpolated shear 3d frame element with Cosserat strains; this element implements the three geometrically exact beam formulations presented in Appendix B.
• GeomWrap3dFrm Wrapper for 3d frame elements. This element implements the element wrapper from Section 6 using Algorithm 1 from Section 4.
• GeomTran3dFrm Wrapper for 3d frame elements that are formulated in a basic coordinate system. This element implements the element wrapper from Section 6 using Algorithm 1 from Section 4.

Geometry/

• GeomTran3dFrm_Pull This function implements the $g$ operation of the geometric transformations by calling the following functions:
  • Transform_RotPull
  • Transform_IsoPull
  • Tran3dFrm_IsoPull
• GeomTran3dFrm_Push This function implements the $\mathbf{a}_g$ and $\mathbf{k}_g$ operations of the geometric transformations.
  • Tran3dFrm_IsoPush

Rotations/

This directory contains the rotation functions described in Appendix A of Perez and Filippou (2024).

Examples/

This directory contains scripts to reproduce the examples of the paper.

1. Invariance
2. 1. Plane Cantilever - Point moment
   2. Plane Cantilever - Transverse Force
3. Space Cantilever - Point moment and force
4. Curved Cantilever - Point force (Bathe's problem)
5. Hockling of a flexible rod.

For an example using "basic" elements, see E00_Cantilever.

• Perez CM, Filippou FC. On nonlinear geometric transformations of finite elements. Int J Numer Methods Eng. 2024;e7506. doi: 10.1002/nme.7506