Stability

The stability of the numerical algorithm depends on the size of the time step used for time integration . For brick elements, Radioss uses the following equation to calculate the size of the time step:(1)

h \leq k \frac{l}{c (α + \sqrt{α^{2} + 1})}

This is the same form as the Courant condition for damped materials. The characteristic length of a particular element is computed using:(2)

l = \frac{E l e m e n t V o l u m e}{L a r g e s t S i d e S u r f a c e}

For a 6-sided brick, this length is equal to the smallest distance between two opposite faces.

The terms inside the parentheses in the denominator are specific values for the damping of the material:

$α = \frac{2 v}{ρ c l}$
$ν$ effective kinematic viscosity
$c = \sqrt{\frac{1 \partial p}{ρ \partial ρ}}$ for fluid materials
$c = \sqrt{\frac{K}{ρ} + \frac{4}{3} \frac{μ}{ρ}} = \sqrt{\frac{λ + 2 μ}{ρ}}$ for a solid elastic material
$K$ is the bulk modulus
$λ$ , $μ$ are Lame moduli

The scaling factor $k = 0.90$ , is used to prevent strange results that may occur when the time step is equal to the Courant condition. This value can be altered by the user.