Conservation of Momentum

Conservation of momentum, expressed in terms of a finite element formulation, is given by:(1)

$${\displaystyle \underset{V}{\int }{\Phi }_I\left(\rho \frac{\partial {\nu }_i}{\partial t}-\frac{\partial {\sigma }_{ij}}{\partial x_j}-\rho b{}_i\right)dV=0}$$

Where,

${\Phi }_I$

Weight functions

$\rho$

Material density

$\nu$

Velocity

${\sigma }_{ij}$

Stress matrix

$b_i$

Body acceleration vector

$V$

Volume

This can be rewritten in a form similar to the explicit Lagrangian formulation with the addition of a new nodal force $F^{trm}$ , accounting for transport of momentum:(2)

$$M\frac{\partial }{\partial t}\nu =\left\{F^{ext}\right\}-\left\{F^{\mathrm{int}}\right\}+\left\{F^{bod}\right\}+\left\{F^{hgr}\right\}+\left\{F^{trm}\right\}$$

Where,

$\left\{F^{trm}\right\}={\displaystyle \sum f^{trm}}$

Transport of momentum forces

By default, since version 2018.0 a streamline upwind technique is used to computed transportation forces (SUPG). This formulation enables to get rid of false diffusion (numerical issue with classical upwind technique). Nevertheless, SUPG method can be disabled in order to retrieve numerical solution obtained with classical upwind technique from versions prior to 2017. For this purpose, Engine keyword must be define: /ALE/SUPG/OFF. Using this keyword, momentum transport forces are computed using the classical upwind technique which was the default method up to version 2017.(3)

$$F_{iI}^{trm}=\left(1+{\eta }_I\right)\rho {\Phi }_I\left(w_j-v_j\right)\frac{\partial v_i}{\partial x_j}V$$

The classical upwinding technique is introduced to add numerical diffusion to the scheme, which otherwise is generally under diffuse and thus unstable.(4)

$${\eta }_I=\eta sign\left[\frac{\partial {\Phi }_I}{\partial x_j}\left(v_j-w_j\right)\right]$$

Full upwind $\eta =1$ (default value) is generally used and can be tuned with ${\eta }_1$ parameter from /UPWIND keyword (now obsolete with SUPG formulation).

Figure 1.