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.

Multistrand elements are n-node springs where matter is assumed to slide through the
nodes.

It could be used for belt modelization by taking nodes upon the dummy. Friction may be defined at
all or some nodes. When nodes are taken upon a dummy in order to modelize a belt, this allows
friction to be modelized between the belt and the dummy.

Internal Forces Computation

Nodes are numbered from 1 to $n$, and strands are numbered from 1 to n-1 (strand $k$ goes from node N_{k} to node N_{k+1}).

Averaged Force

The averaged force in the multistrand is computed as:

Linear spring $F=\frac{K}{{L}^{0}}\delta +\frac{C}{{L}^{0}}\stackrel{\dot{}}{\delta}$

Nonlinear spring $F=f\left(\epsilon \right)\cdot g\left(\dot{\epsilon}\right)+\frac{C}{{L}^{0}}\dot{\delta}$

with ${l}_{k}^{0}$ the length of the unconstrained strand $k$, $\delta \epsilon =\epsilon \left(t\right)-\epsilon \left(t-1\right)$ and $\delta {\epsilon}_{k}=\delta t{u}_{k}\cdot \left({v}_{k+1}-{v}_{k}\right)$.

Where, ${u}_{k}$ is the unitary vector from node N_{k} to node
N_{k+1}.

Assuming:(4)

$$\frac{{l}_{k}}{{l}_{k}^{0}}=\frac{L}{{L}^{0}}$$

Where, ${l}_{k}$ is the actual length of strand $k$.

When equation Equation 6 is not satisfied, $\left|\text{\Delta}{F}_{k-1}-\text{\Delta}{F}_{k}\right|$ is reset to $\left(2F+\text{\Delta}{F}_{k-1}+\text{\Delta}{F}_{k}\right)\mathrm{tanh}\left(\frac{\beta \mu}{2}\right)$.

All the $\text{\Delta}{F}_{k}$ (k=1, n-1) are modified in order to satisfy all conditions upon $\text{\Delta}{F}_{k-1}-\text{\Delta}{F}_{k}$ (k=2, n-1), plus the following condition on the force integral
along the multistrand element:(7)

This process could fail to satisfy Equation 6 after the $\text{\Delta}{F}_{k}(k=1,n-1)$ modification, since no iteration is made. However, in such a case
one would expect the friction condition to be satisfied after a few time steps.

Note: Friction expressed upon strands (giving a friction coefficient $\mu $ along strand $k$) is related to pulley friction by adding a friction coefficient $\mu /2$ upon each nodes N_{k} and N_{k+1}.

Time Step

Stability of a multistrand element is expressed as:(8)