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.
The definition of numerically stability is similar to the stability of mechanical systems. A numerical procedure is stable
if small perturbations of initial data result in small changes in the numerical solution.
Radioss uses elements with a lumped mass approach. This reduces computational time considerably as no matrix inversion is necessary
to compute accelerations.
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 question "how far can a body be dropped without incurring damage?" is frequently asked
in the packaging manufacturing for transportation of particles. The problem is similar in
landing of aircrafts. It can be studied by an analytical approach where the dropping body is
modeled by a simple mass-spring system (Figure 1). If is the dropping height, and the mass of the body and the stiffness representing the
contact between the body and the ground, the equation of the motion can be represented by a
simple one DOF differential equation as long as the spring remains in contact with
floor:
(1)
In this equation the damping effects are neglected to simplify the solution. The general
solution of the differential equation is written as:(2)
Where, the constants A, B and C are determined by the initial conditions:(3)
At t=0 ≥
,
,
Where, is the natural frequency of the system: (4)
Introducing these initial solutions into Equation 3, the following result are
obtained:(5)
The same problem can be resolved by the numerical procedure explained in this section.
Considering at first the following numerical values for the mass, the stiffness, the
dropping height and the gravity:(6)
From Equation 1, the dynamic
equilibrium equation or equation of motion is obtained as:(7)
For the first time step the initial conditions are defined by Equation 3. Using a constant time
step the mass motion can be computed. It is compared to the
analytical solution given by Equation 5 in Figure 2. The difference between the two results shows the time
discretization error.