# Lagrangian and Corotational Formulations

- Total Lagrangian description (TL)
- The FEM equations are formulated with respect to a fixed reference configuration which is not changed throughout the analysis. The initial configuration is often used; but in special cases the reference could be an artificial base configuration.
- Updated Lagrangian description (UL)
- The reference is the last known (accepted) solution. It is kept fixed over a step and updated at the end of each step.
- Corotational description (CR)
- The FEM equations of each element are referred to two systems. A fixed or base configuration is used as in TL to compute the rigid body motion of the element. Then the deformed current state is referred to the corotated configuration obtained by the rigid body motion of the initial reference.

By default, Radioss uses a large strain, large displacement formulation with explicit time integration. The large displacement formulation is obtained by computing the derivative of the shape functions at each cycle. The large strain formulation is derived from the incremental strain computation. Hence, stress and strains are true stresses and true strains.

In the updated Lagrangian formulation, the Lagrangian coordinates are considered instantaneously coincident with the Eulerian spatial $x$ coordinates. This leads to the following simplifications:

The derivatives are with respect to the spatial (Eulerian) coordinates. The weak form involves integrals over the deformed or current configuration. In the total Lagrangian formulation, the weak form involves integrals over the initial (reference) configuration and derivatives are taken with respect to the material coordinates.

The corotational kinematic description is the most recent of the formulations in geometrically
nonlinear structural analysis. It decouples small strain material nonlinearities from geometric
nonlinearities and handles naturally the question of frame indifference of anisotropic behavior
due to fabrication or material nonlinearities. Several important works outline the various
versions of CR formulation. ^{1}
^{2}
^{3}
^{4}
^{5}

^{1}Belytschko T. and Hsieh B.H.,

Nonlinear Transient Finite Element Analysis with convected coordinates, Int. Journal Num. Methods in Engineering, 7 255-271, 1973.

^{2}Argyris J.H.,

An excursion into the large rotations, Computer Methods in Applied Mechanics and Engineering, 32, 85-155, 1982.

^{3}Crisfield M.A.,

A consistent corotational formulation for nonlinear three-dimensional beam element, Computer Methods in Applied Mechanics and Engineering, 81, 131-150, 1990.

^{4}Simo J.C.,

A finite strain beam formulation, Part I: The three-dimensional dynamic problem, Computer Methods in Applied Mechanics and Engineering, 49, 55-70, 1985.

^{5}Wempner G.A.,

F.E., finite rotations and small strains of flexible shells, IJSS, 5, 117-153, 1969.