Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of
nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box.
The purpose of this manual is to describe the numerical methods included in Radioss.

The historical shell element in Radioss is a simple bilinear Mindlin plate element coupled with a reduced integration scheme using one integration point.
It is applicable in a reliable manner to both thin and moderately thick shells.

Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.

The stability of solution concerns the evolution of a process subjected to small perturbations. A process is considered
to be stable if small perturbations of initial data result in small changes in the solution. The theory of stability
can be applied to a variety of computational problems.

A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.

Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.

The performance criterion in the computation was always an essential point in the architectural conception of Radioss. At first, the program has been largely optimized for the vectored super-calculators like CRAY. Then, a first parallel
version SMP made possible the exploration of shared memory on processors.

Hourglass modes are element distortions that have zero strain energy. The 4 node shell
element has 12 translational modes, 3 rigid body modes (1, 2, 9), 6 deformation modes (3, 4,
5, 6, 10, 11) and 3 hourglass modes (7, 8, 12).

Along with the translational modes, the 4 node shell has 12 rotational modes: 4 out of
plane rotation modes (1, 2, 3, 4), 2 deformation modes (5, 6), 2 rigid body or deformation
modes (7, 8) and 4 hourglass modes (9, 10, 11, 12).

Hourglass Viscous Forces

Hourglass resistance forces are usually either viscous or stiffness related. The viscous
forces relate to the rate of displacement or velocity of the elemental nodes, as if the
material was a highly viscous fluid. The viscous formulation used by Radioss is the same as that outlined by Kosloff and Frasier ^{1}. Refer to Hourglass Modes. An hourglass normalized vector is defined
as:(1)

Radioss can apply a stiffness force to resist hourglass modes.
This acts in a similar fashion to the viscous resistance, but uses the elastic material
stiffness and node displacement to determine the size of the force. The formulation is the
same as that outlined by Flanagan et al. ^{2} Refer to Flanagan-Belytschko Formulation. The hourglass resultant forces are defined as:(5)

This formulation is analogous to the hourglass viscous force scheme. The
hourglass angular velocity rate is defined for the main hourglass modes as:(8)

Where, ${h}_{r}$ is the shell rotation hourglass coefficient.

^{1}Kosloff D. and Frazier G., “Treatment of hourglass pattern in low
order finite element code”, International Journal for Numerical and
Analytical Methods in Geomechanics, 1978.

^{2}Flanagan D. and Belytschko T., “A Uniform Strain Hexahedron and
Quadrilateral with Orthogonal Hourglass Control”, Int. Journal Num.
Methods in Engineering, 17 679-706, 1981.