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.

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.

Truss elements are simple two node linear members that only take axial extension or compression.
Figure 1 shows a truss element.

Property Input

The only property required by a truss element is the cross-sectional area. This value will change
as the element is deformed. The cross sectional area is computed using:(1)

Determining the stability of truss elements is very simple. The characteristic length is defined
as the length of the element, that is, the distance between N1 and N2 nodes.(2)

$$\text{\Delta}t\le \frac{L\left(t\right)}{C}$$

Where,

$L\left(t\right)$

Current truss length

$C=\sqrt{\frac{E}{\rho}}$

Sound speed

Rigid Body Motion

The rigid body motion of a truss element as shown in Figure 2 shows the different velocities of
nodes 1 and 2. It is the relative velocity difference between the two nodes that produces a
strain in the element, namely e_{x}.

Strain

The strain rate, as shown in Figure 2, is defined as:(3)

A generalized force-strain graph can be seen in Figure 3. The force rate under elastic
deformation is given by:(5)

$${\dot{F}}_{x}=EA{\dot{\epsilon}}_{x}$$

Where,

$E$

Elastic modulus

$A$

Cross-sectional area

In the plastic region, the force rate is given by:(6)

$${\dot{F}}_{x}={E}_{t}A{\dot{\epsilon}}_{x}$$

Where,

${E}_{t}$

Gradient of the material curve at the deformation point

In a general case, it is possible to introduce a gap distance in the truss definition. If gap is
not null, the truss is activated when the length of the element is equal to the initial
length minus the gap value. This results a force-strain curve shown in Figure 3(b).