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.

Element degeneration is the collapsing of an element by one or more edges. For example: making an eight node element into
a seven node element by giving nodes 7 and 8 the same node number.

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 Jaumann objective stress tensor derivative ${\dot{\sigma}}_{ij}^{v}$ is the corrected true stress rate tensor without rotational
effects. The constitutive law is directly applied to the Jaumann stress rate tensor.

Deviatoric stresses and pressure (Stresses in Solids) are computed separately and
related by:(4)

$${\sigma}_{ij}={s}_{ij}-p{\delta}_{ij}$$

Where,

${s}_{ij}$

Deviatoric stress tensor

$p$

Pressure or mean stress - defined as positive in compression

${\delta}_{ij}$

Substitution tensor or unit matrix

Co-rotational Formulation

A co-rotational formulation for bricks is a formulation where rigid body rotations are
directly computed from the element's node positions. Objective stress and strain tensors are
computed in the local (co-rotational) frame. Internal forces are computed in the local frame
and then rotated to the global system.

So, when co-rotational formulation is used, Deviatoric Stress Calculation, Equation 2${\dot{\sigma}}_{ij}={\dot{\sigma}}_{ij}^{v}+{\dot{\sigma}}_{ij}^{r}$ reduces to:(5)

$${\dot{\sigma}}_{ij}={\dot{\sigma}}_{ij}^{v}$$

Where,

${\dot{\sigma}}_{ij}^{v}$

Jaumann objective stress tensor derivative expressed in the co-rotational frame

Figure 1 orthogonalization, when one of the
r, s, t directions is orthogonal to the two other directions.

When large rotations occur, this formulation is more accurate than the global formulation,
for which the stress rotation due to rigid body rotational velocity is computed in an
incremental way.

Co-rotational formulation avoids this kind of problem. Consider this test:

The increment of the rigid body rotation vector during time step $\text{\Delta}t$ is:(6)

So, shear stress is sinusoidal and is not strictly increasing.

So, it is recommended to use co-rotational formulation, especially for visco-elastic
materials such as foams, even if this formulation is more time consuming than the global
one.

Co-rotational Formulation
and Orthotropic Material

When orthotropic material and global formulation are
used, the fiber is attached to the first direction of the isoparametric frame and the fiber
rotates a different way depending on the element numbering.

On the other hand, when the co-rotational formulation is used, the orthotropic
frame keeps the same orientation with respect to the local (co-rotating) frame, and is
therefore also co-rotating.

^{1}Wilkins M., “Calculation of elastic plastic flow” LLNL, University
of California UCRL-7322, 1981.