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.

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.

There are four types of rigid walls available in Radioss:

Infinite Plane

Infinite Cylinder with Diameter D

Sphere with Diameter D

Parallelogram

Each wall can be fixed or moving.

A kinematic condition is applied on each impacted secondary node. Therefore, a secondary node
cannot have another kinematic condition; unless these conditions are applied in orthogonal
directions.

Fixed Rigid Wall

A fixed wall is a pure kinematic option on all impacted secondary nodes. It is defined using two
points, M and M1. These define the normal, as shown in Figure 1.

Moving Rigid Wall

A moving rigid wall is defined by a node number, N, and a point, M1. This allows a normal to be
calculated, as shown in Figure 2.

The motion of node N can be specified with fixed velocity, or with an initial velocity. For
simplification, an initial velocity and a mass may be given at the wall definition level.

A moving wall is a main secondary option. Main node defines the wall position at each time step
and imposes velocity on impacted secondary nodes. Impacted secondary node forces are applied to
the main node. The secondary node forces are computed with momentum conservation. The mass of
the secondary nodes is not transmitted to the main node, assuming a large rigid wall mass
compared to the impacted secondary node mass.

Secondary Node Penetration

Secondary node penetration must be checked. Figure 3 shows how penetration is checked.

If penetration occurs, a new velocity must be computed. This new velocity is computed using one
of three possible situations.

Sliding

Sliding with Friction

Tied

For a node which is allowed to slide along the face of the rigid wall, the new velocity ${\stackrel{\rightharpoonup}{V}}^{\prime}$ is given by:(1)

A friction coefficient can be applied between a sliding node and the rigid wall. The friction
models are developed in Interface Friction.

For a node that is defined as tied, once the secondary node contacts the rigid wall, its velocity
is the same as that of the wall. The node and the wall are tied. Therefore:(2)

$${\stackrel{\rightharpoonup}{V}}^{\prime}=0$$

Rigid Wall Impact Force

The force exerted by nodes impacting onto a rigid wall is found by calculating the impulse
by:(3)