This scheme is only used with the ALE formulation (Arbitrary Lagrangian Eulerian) and in the CFD
version of
Radioss. The force is calculated using the
relation:
(1)
$${}^{trm}F{}_{i}^{I}=\left(1+{\eta}_{I}\right)\rho {\Phi}_{I}\left({w}_{j}{v}_{j}\right)\text{\hspace{0.17em}}\frac{\partial {v}_{i}}{\partial {X}_{j}}\text{\hspace{0.17em}}V$$
Where,
 $w$
 Grid velocity

$\nu $
 Material velocity

$V$
 Element volume

$\eta $
 Upwind coefficient (userdefined, default = 1 for full upwind)
When a Lagrangian formulation is used, the values of
${w}_{j}$
and ${\nu}_{j}$
are
equal. Thus, Equation 1 is equal
to zero.
Upwinding Technique
An upwinding technique is introduced to add numerical diffusion to the scheme; otherwise it is
generally under diffusive and thus unstable. The upwind coefficient used in
Equation 1 is
calculated by:
(2)
$${\eta}_{I}=\eta sign\left(\frac{\partial {\Phi}_{I}}{\partial {X}_{j}}\left({v}_{j}{w}_{j}\right)\right)$$
Development of a less diffusive flux calculation is currently under
investigation.
(3)
$${F}_{i}^{I}={\sigma}_{ij}{\displaystyle \underset{V}{\int}\frac{\partial {\Phi}_{I}}{\partial {X}_{j}}dV}$$
This option is activated with the flag INTEG (only in the CFD version).