# Courant Condition Stability

Radioss uses elements with a lumped mass approach. This reduces computational time considerably as no matrix inversion is necessary to compute accelerations.

The integration scheme used by Radioss is based on the central difference integration scheme which is conditionally stable, that is, the time step must be small enough to assure that the solution does not grow boundlessly.

Where, $K$ and $M$ are respectively the stiffness and the mass matrices of the system.

The time step restriction given by Equation 1 was derived considering a linear system (Explicit Scheme Stability), but the result is also applicable to nonlinear analysis since on a given step the resolution is linear. However, in nonlinear analysis the stiffness properties change during the response calculation. These changes in the material and the geometry influence the value of ${\omega}_{\mathrm{max}}$ and in this way the critical value of the time step.

The above point can be easily pointed out by using a nonlinear spring with increasing stiffness in Body Drop Example. It can be shown that the critical time step decreases when the spring becomes stiffer. Therefore, if a constant time step close to the initial critical value is considered, a significant solution error is accumulated over steps when the explicit central difference method is used.

- ${l}_{c}$
- Characteristic element length, representing the shortest road for a wave arriving on a node to cross the element
- $c$
- Speed of sound in the material
- $\text{\Delta}t$
- Time step

This relation is that of Equation 3 and shows that the
critical time step value in the explicit time integration of dynamic equation of motion can
be carried out by the interpretation of shock wave propagation in the material. This is
shown for the first time by Courant. ^{1} In spite of their works are limited to simple
cases, the same procedure can be used for different kinds of finite elements. The
characteristic lengths of the elements are found and Equation 3 is written for all
elements to find the most critical time step over a mesh. Regarding to the type (shape) of
element, the expression of characteristic length is different. Figure 2 shows some typical cases for elements with one integration
point.

^{1}Courant R., Friedrichs K.O., and Levy H., “About the partial Differenzensleichungen Bogdanova of Physics”, Math. A nn., Vol. 100, 32, 1928.