/ALE/GRID/STANDARD

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1. Fictitious springs are introduced on solid elements to control grid velocities.
   These springs are nonlinear elastic viscous. To ensure stability, their stiffness is computed from time step. The two types of springs are edge and anti-shear springs.

   1. Edge springs
      The forces for an edge spring are a function of its length variation during time.(1)

      $$\text{Δ}{F}_{edge}=\mathrm{k}\left(h\right)\cdot \left(w_2-w_1\right)dt$$

      Where,

      $w_1,w_2$ are grid velocities on nodes N₁ and N₂, respectively.

      $h$ is the N₁ distance from opposite face

      $dt$ is the time step

      and $k\left(h\right)$ is the spring stiffness $\mathrm{k}\left(h\right)=k_{critical}$

      If $h$ is inferior to the characteristic length $l_c$ and N₁ is moving toward the opposite face then,(2)

      $$\mathrm{k}\left(h\right)=\frac{1}{{\lambda }^2}\left[\gamma +\left(\gamma -1\right){\left(\frac{h-l_c}{l_c}\right)}^3\right]k_{critical}$$
      $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\lambda }^2$}\right.$ is the stability factor taking into account the damping factor $\beta$ , the scale factor $\alpha$ , and time step $dt$ 4

      
      
      Figure 2.

      otherwise, $\mathrm{k}\left(h\right)=\frac{\gamma }{{\lambda }^2}{k}_{critical}$

      
      
      Figure 3.

   2. Anti-shear springs

      The anti-shear forces $F_{shear}$ are computed from node penetration. gap is $\raisebox{1ex}{$l_c$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ from opposite face.

      The value of $F_{shear}$ is:

      (3)
      $$F_{shear}=gap\cdot \mathrm{k}\left(h\right)$$
      and (4)

      $$\mathrm{k}\left(h\right)=\frac{\gamma }{{\lambda }^2}\left[\gamma +\left(\gamma -1\right){\left(\frac{h-l_c}{l_c}\right)}^3\right]k_{critical}$$

      
      
      Figure 4.

   3. Viscous Damping
      Viscous forces are computed from a critical damping corresponding to the upper bound for stiffness: $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\lambda }^2$}\right.$ (5)

      $$F_{viscous}=\beta \alpha \left(\sqrt{1+{\beta }^2}-\beta \right)\left(w_2-w_1\right)$$
   4. Grid Velocity
      The grid velocity is then updated according to:(6)

      $$w_{n+1}=w_n+\frac{\left(\text{Δ}{F}_{edge}+F_{shear}+F_{viscous}\right)dt}{m}$$

      Where, $m$ is fictitious mass on node from springs (automatically computed during Starter).

2. Increasing $\alpha =1$ , the maximum stiffness will be increased. The scale factor $\alpha$ determines the maximum stiffness for a given spring at zero length. The scale factor ensures that the critical stability value is not exceeded (to avoid time step decrease).
3. This flag is acting on stiffness shape. Stiffness is linear with $\gamma$ = 0. Moreover, increasing $\gamma$ , the lower bound stiffness for edge spring will be increased. Springs have a critical stiffness at zero length (this corresponds to a unitary factor). For a length greater than or equal to the characteristic length, the spring stiffness is the critical stiffness multiplied by $\gamma$ .
4. It is recommended to use small values for $\beta$ , otherwise damping may become over critical. The stability factor is:(7)
   $$\lambda =\frac{dt}{{\alpha }_0\left(\sqrt{1+{\beta }^2}-\beta \right)}$$
5. $l_c$ defines the length below which:
   • Edge spring stiffness is increased $h<l_c$
   • Anti-shear spring is activated: $h<\raisebox{1ex}{$l_c$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$
6. All these parameters can be modified during an Engine restart (/ALE/GRID/STANDARD).
7. Mesh auto correction. It is possible to give more weight to anti-shear forces by either:
   • Setting $l_c$ parameter close to the mesh size
   • Setting a negative value for $\alpha$ parameter (elastic forces on edges are set to 0 at the first cycle of current run)
8. This method assumes a homogeneous spring repartition around each node. This is not the case when connecting two meshes, where topologies are different.