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.

Shell elements behave in two ways, either membrane or bending behavior. The Mindlin plate
elements that are used by Radioss account for bending and
transverse shear deformation. Hence, they can be used to model thick and thin plates.

Membrane Behavior

The membrane strain rates for Mindlin plate elements are defined as:(1)

The bending behavior in plate elements is described using the amount of curvature. The
curvature rates of the Mindlin plate elements are defined as:(6)

As with the membrane strain rate, the bending strain rate is computed from the
velocity field vector. However, the velocity field vector for the bending strain rate
contains rotational velocities, as well as translations:(14)

With the exact formula for inertia (Equation 19 to Equation 22), the solution tends to
diverge for large rotation rates. Belytschko proposed a way to stabilize the solution by
setting ${I}_{xx}$
=
${I}_{yy}$, that is, to consider the rectangle as a square with respect
to the inertia calculation only. This introduces an error into the formulation. However, if
the aspect ratio is small the error will be minimal. In Radioss
a better stabilization is obtained by:(23)