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.

No material data card is required for spring elements. However, the stiffness $k$ and equivalent viscous damping coefficient $c$ are required. The mass $m$ is required if there is any spring translation.

There are three other options defining the type of spring stiffness with the hardening flag:

Linear Stiffness

Nonlinear Stiffness

Nonlinear Elasto-Plastic Stiffness

Likewise, the damping can be either:

Linear

Nonlinear

A spring may also have zero length. However, a one DOF spring must have 2 nodes.

The forces applied on the nodes of a one DOF spring are always colinear with direction through
both nodes; refer to Figure 4.

Time Step

The time of a spring element depends on the values of stiffness, damping and mass.

The critical time step ensures that the stability of the explicit time integration is
maintained, but it does not ensure high accuracy of spring vibration behavior. Only two time
steps are required during one vibration period of a free spring to keep stability. However, if
true sinusoidal reproduction is desired, the time step should be reduced by a factor of at least
5.

If the spring is used to connect the two parts, the spring vibration period increases and the
default spring time step ensures stability and accuracy.

Linear Spring

Function number defining $f\text{}\left(\delta \right)$.

N1=0

The general linear spring is defined by constant mass, stiffness and damping. These are all
required in the property type definition. The relationship between force and spring displacement
is given by:(4)

$$F=k\left(l-{l}_{0}\right)+c\frac{dl}{dt}$$

The stability condition is given by Equation 3:(5)

The hardening flag must be set to 0 for a nonlinear elastic spring. The only difference
between linear and nonlinear elastic spring elements is the stiffness definition. The mass and
damping are defined as constant. However, a function must be defined that relates the force,
$F$, to the displacement of the spring, ($l-{l}_{0}$). It is defined as:(6)

$$F=f\left(l-{l}_{0}\right)+c\frac{dl}{dt}$$

The stability criterion is the same as for the linear spring, but rather than being constant,
the stiffness is displacement dependent:(7)

The hardening flag must be set to 1 in this case and $f\left(l-{l}_{0}\right)$ is defined by a function. Hardening is isotropic if compression
behavior is identical to tensile behavior:(9)

The hardening flag is set to 2 in this case and f$f\left(l-{l}_{0}\right)$ is defined by a function. The hardening is decoupled for
compression and tensile behavior:(10)

The hardening flag is set to 4 in this case and ${f}_{1}\left(l-{l}_{0}\right)$ and ${f}_{2}\left(l-{l}_{0}\right)$ (respectively maximum and minimum yield force) are defined by a
function. The hardening is kinematic if maximum and minimum yield curves are
identical:(11)

The hardening flag is set to 5 in this case and $f\text{}\left(\delta \right)$ and $f2\left({\delta}_{\mathrm{max}}\right)$ (maximum yield force and residual deformation, respectively) are
defined by a function. Uncoupled hardening in compression and tensile behavior with nonlinear
unloading:(12)

$$F=f\left(l-{l}_{0}\right)+C\frac{dl}{dt}$$

With $\delta =l-{l}_{0}$.

Nonlinear Dashpot

The input properties for a nonlinear dashpot are very close to that of a spring. The required
values are:

Mass, $M$.

A function defining the change in force with respect to the spring displacement. This must
be equal to unity:

$f\left(l-{l}_{0}\right)=1$

A function defining the change in force with spring displacement rate,

$g\left(dl/dt\right)$

The hardening flag in the input must be set to zero.

The relationship between force and spring displacement and displacement rate
is:(13)