Time Step

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, $\text{Δ}t$ . Numerical Stability of Undamped Systems of the Radioss Theory Manual shows that a system without damping will remain stable if $\text{Δ}t\le \frac{2}{{\omega }_{\max }}$ . Where, ${\omega }_{\max }$ 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:

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:

This stability criterion is often called the Courant condition after the research first done by Courant et al. in 1928. ¹