claudioperez/geometrictransformations

Geometrically exact finite element analysis.

GeometricTransformations

A framework for geometrically exact finite elment 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.

The procedure is demonstrated with the Cosserat rod model, furnishing a novel finite element formulation that rectifies the lack of objectivity exhibited by existing finite elements without violating the director constraints or compromising the symmetry of the tangent stiffness at equilibrium. The framework is element-independent, allowing its implementation as a wrapper to existing element libraries without modification of the element state determination procedures.

Elements/

This directory contains the finite elements:

• DisplShear3dFrm_wCS 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.

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_Column.

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