Vega FEM Library |

**NEW: Vega FEM 2.1 released on Aug 15, 2014. New features described below.**

Vega is a computationally efficient and stable C/C++ physics library for three-dimensional deformable object simulation. It is designed to model large deformations, including geometric and material nonlinearities, and can also efficiently simulate linear systems. Vega contains about 100,000 lines of code, and is open-source and free. It is released under the 3-clause BSD license, which means that it can be used freely both in academic research and in commercial applications.

Vega implements several widely used methods for simulation of **large deformations** of 3D **solid** deformable objects:

Method | Description | Citations |

co-rotational linear FEM (optional: exact tangent stiffness matrix) | large deformations, uses polar decomposition of the deformation gradient to extract large rotations, linear strain-stress relationship | [MG04]; exact matrix: [Bar12] (similar to [CPSS10]) |

co-rotational orthotropic (anisotropic) FEM elasticity | same as above, but the strain-stress relationship is orthotropic, i.e., models anisotropic materials with different stiffnesses in three orthogonal directions | [LB14] |

invertible isotropic nonlinear FEM | large deformations, nonlinear isotropic strain-stress relationship, e.g., Saint-Venant Kirchhoff, neo-Hookean, Mooney-Rivlin models; includes a mechanism to handle material inversions | [ITF04], [TSIF05] |

Saint-Venant Kirchhoff FEM deformable model | large deformations, Green-Lagrange strain, linear strain-stress relationship, informally referred to as isotropic geometrically nonlinear FEM | [Bar07] |

linear FEM elasticity | standard small-deformation FEM, linear Cauchy strain, linear strain-stress relationship | [Sha90] |

mass-spring system | standard mass-spring code; optionally, the network can be built as edges of a tet mesh | many |

For any 3D tetrahedral or cubic mesh, Vega can compute both internal elastic forces and their gradients (tangent stiffness matrix), in any deformed configuration. Different parts of the mesh can be assigned arbitrary material properties. Vega can also timestep these models in time under any user-specified forces, using several provided integrators: implicit backward Euler [BW98], implicit Newmark [Wri02], explicit central differences [Wri02] and symplectic Euler. All models include support for multi-core computing. In addition to linear materials, Vega provides neo-Hookean and Mooney-Rivlin nonlinear material models; arbitrary nonlinear material models can be added to Vega. For isotropic hyperelastic materials, this is as easy as defining an energy function, and its first and second derivatives.

Since version 2.0, Vega FEM also supports **model reduction** [BJ05], **cloth** simulation (Baraff-Witkin cloth solver, [BW98]), and **rigid body dynamics**.
Isotropic materials now also include (optional) compression resistance ([KTY09] and several prior references), which makes it possible to simulate very soft
materials undergoing large compression in a stable way.

Since version 2.1, Vega FEM also supports **orthotropic materials** [LB14], i.e., anisotropic materials that exhibit different stiffnesses in three orthogonal directions.
Also included is a binary .vegb format, faster loading of volumetric meshes with many regions, and support for the clang compiler (Mac).
We also fixed known bugs and made numerous code improvements.

Vega is a **middleware** physics library. It is aimed at researchers and engineers with some preexisting
knowledge in numerical simulation, computer graphics and/or solid mechanics, who can integrate
Vega into their projects. The strength of Vega lies in its many C/C++ libraries which depend minimally on
each other, and are in most cases independently reusable. Vega contains about 50,000 lines of C/C++ code.
It compiles under Windows, Linux and Mac OS X, and has no external dependencies.

Most of Vega was written by Jernej Barbič. Other code contributors include Fun Shing Sin, Daniel Schroeder, Andy Pierce, Yuyu Xu, Yijing Li, Hongyi Xu, Yili Zhao, Christopher Twigg and Somya Sharma.