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.

The need of simple and efficient element in nonlinear analysis of shells undergoing large
rotations is evident in crash and sheet metal forming simulations. The constant-moment plate
elements fit this need. One of the famous concepts in this field is that of Batoz et al.
^{1} known under DKT elements where DKT stands for Discrete
Kirchhoff Triangle. The DKT12 element ^{1}, ^{2} has a total of 12 DOFs. The discrete Kirchhoff plate
conditions are imposed at three mid-point of each side. The element makes use of rotational
DOF. at each edge to take into account the bending effects. A simplified three-node element
without rotational DOF is presented in ^{3}. The rotational DOF is computed with the help of
out-of-plane translational DOF in the neighbor elements. This attractive approach is used in
Radioss in the development of element SH3N6 which based on
DKT12.

Strain Computation

Consider two adjacent coplanar elements with a common edge i-j as shown in Figure 2. Due to out-of-plane displacements of nodes $m$ and $k$, the elements rotate around the side i-j. The angles between
final and initial positions of the elements are respectively ${\alpha}_{m}$ and ${\alpha}_{k}$ for corresponding opposite nodes $m$ and $k$. Assuming, a constant curvature for both of elements, the
rotation angles ${\theta}_{m}$ and ${\theta}_{k}$ related to the bending of each element around the common side
are obtained by:(1)

Consider the triangle element in Figure 2. The outward normal vectors at the three sides are defined and denoted $n1$, $n2$ and $n3$. The normal component strain due to the bending around the
element side is obtained using plate assumption:(4)

The six mid-side rotations ${\alpha}_{i}$ are related to the out-of-plane displacements of the six apex nodes as
shown in Figure 3 by the following relation:(5)

Where, $\left(\begin{array}{ccc}{h}_{1}& {h}_{2}& {h}_{3}\end{array}\right)$, $\left(\begin{array}{ccc}{q}_{1}& {h}_{4}& {q}_{2}\end{array}\right)$,
$\left(\begin{array}{ccc}{r}_{2}& {h}_{5}& {r}_{3}\end{array}\right)$
and $\left(\begin{array}{ccc}{s}_{3}& {h}_{6}& {s}_{1}\end{array}\right)$ are respectively the heights of the triangles (1,2,3), (1,4,2), (2,5,3)
and (3,6,1).

The non-null components of strain tensor in the local element reference are related to the
normal components of strain by the following relation: ^{1}^{3}(6)

As the side rotation of the element is computed using the out-of-plane displacement of the
neighbor elements, the application of clamped or free boundary conditions needs a particular
attention. It is natural to consider the boundary conditions on the edges by introducing a
virtual and symmetric element outside of the edge as described in Figure 4. In the case of free rotation at the edge, the
normal strain ${\epsilon}_{nk}$ is vanished. From Equation 4, this leads
to:(7)

$${\alpha}_{k}=-{\alpha}_{m}$$

In Equation 5 the fourth row of the
matrix is then changed to:(8)

The clamped condition is introduced by the symmetry in out-of-plane displacement, that is, ${w}_{m}={w}_{k}$. This implies ${\alpha}_{k}={\alpha}_{m}$. The fourth row of the matrix in Equation 5 is then changed
to:(9)

Sabourin F. and Brunet M., “Analysis of plates and shells with a
simplified three-node triangle element”, Thin-walled Structures, Vol. 21, pp.
209-223, Elsevier, 1995.