Results are calculated for each time step or cycle in a simulation. Therefore, the
smaller the time step, the longer a simulation will take to solve because more
cycles and calculations are done. As discussed in

Dynamic Analysis of the

Radioss Theory Manual, a direction integration
method is used to solve the equations of motion. The direct integration method used
in

Radioss is derived from Newmark time integration
scheme. This method solves the equations of motion using a step-by-step procedure
using a numerically stable time step,

$\text{\Delta}t$
.

Numerical Stability of Undamped Systems of the

Radioss Theory
Manual shows that a system without damping will remain stable if

$\text{\Delta}t\le \frac{2}{{\omega}_{\mathrm{max}}}$
. Where,

${\omega}_{\mathrm{max}}$
is the highest angular frequency in the system. For a
discrete finite element simulation, the solution remains stable if the shock wave
traveling through the mesh does not travel through more than one element during one
time step. In this way, the shock wave does not miss any nodes when traveling
through the mesh and thus excites all the frequencies in the finite element mesh.
Using the speed of sound in a material

$c$
and the characteristic element length

${l}_{c}$
of a finite element, the time for the wave to travel across one element length
is:

(1)
$$\text{\Delta}t=\frac{{l}_{c}}{c}$$

For the discrete solution to remain stable, the time step should be less than or
equal to the time needed for the wave to travel across one element:

(2)
$$\text{\Delta}t\le \frac{{l}_{c}}{c}$$