SPH Symmetry Conditions

Multiple SPH Symmetry Conditions

An axi-symmetry condition can be modelized through the use of two conditions with respect to two planes intersecting at the axis of symmetry. A spheric symmetry condition can be modelized through the use of three conditions with respect to three planes intersecting at the center of symmetry.

Nevertheless, these kinds of symmetries are not treated the same way.

For instance, in case of an axi-symmetry condition, not all ghost particles are built around the axis of symmetry. The only symmetric particles of real particles with respect to the two symmetry planes are built.

Therefore, some characteristics of axi-symmetry (respectively, spheric symmetry) conditions can be closed to the axis of symmetry (respectively, the center of symmetry).

Nodes closed to the axis of symmetry (respectively, the center of symmetry) and lying on a symmetry plane (P) can get a normal to (P) velocity which is non-zero, since their neighborhood is not symmetric with respect to plane (P).

Kinematic Boundary Condition

With respect to the previous discussion: adding the kinematic boundary condition an explicit way allows to enforce it.

A kinematic boundary condition will be added to the nodes belonging to the nodes group specified into the /SPHBCS option, so that:

If "Slide" type, the velocity of the node in direction "Dir" is set to zero
If "Tied" type, the velocity of the node in all directions is set to zero

In case of several kinematic boundary conditions applied to the same node through different SPH symmetry conditions, the kinematic boundary conditions are composed automatically, even if the kinematic boundary conditions are applied into non-orthogonal directions.

Figure 3 indicates that if two kinematic boundary conditions are applied to N through two symmetry conditions with respect to planes (P₁) and (P₂), the two boundary conditions are modified so that the velocity in the plane normal to common axis of (P₁) and (P₂) will remain zero.

Note: If one of the two symmetry conditions is a type "Tied" condition, the velocity of N in all directions is set to zero.

It also allows application to the same node, a kinematic boundary condition through a SPH symmetry condition (/SPHBCS) and a standard boundary condition (/BCS) at the same time, as long as the standard boundary condition is not given in a moving skew system, but a fix skew system or the global skew system. The two conditions are then composed the same way.

Part Mass

You must be advised that when a particle lies on a symmetry plane at time t=0, the mass and the initial volume considered for the particles are respectively:(1)

m = \frac{m_{p}}{2}, V = \frac{m_{p}}{2 ρ}

Where, m_p is the mass specified into property set.

When a particle lies on n symmetry planes at time t=0,(2)

m = \frac{m_{p}}{n}, V = \frac{m_{p}}{n_{ρ}}

Ghost particles built from this particle will get the same initial volume and mass.

When $n > 2$ , the previous equation may provide an error on mass and energies output for the part the particles belong to, with respect to the physical model.

Formulation Level

When a symmetry plane is defined, and even if a kinematic condition is set for all particles lying on the symmetry plane, particles lying at time zero inside the domain are theoretically able to cross the symmetry plane.

This is specific to SPH for which stiffness between particles does not increase to an infinite value when particles collapse. So it can occur when the particles, which lie on the symmetry plane let the particles which were inside the domain go through the symmetry plane.

If Ilev=0, particles crossing the symmetry plane will not be (progressively) taken into account anymore in the computation, neither than their symmetric particles which then lie inside the domain.

If Ilev=1, particles which have crossed the symmetry plane rebound an elastic way upon the symmetry plane: their velocity in the normal direction to the plane is set the opposite.

Note: When Ilev=1, it is strongly recommended to associate kinematic condition to all particles lying on symmetry plane at time zero, for computational time reasons.

Maximum Created Number of Ghost Particles

Ghost particles are created at each search for neighbors time within the security distance, and then destroyed when a new search occurs (a new set of ghost particles is then created).

At any search time, all ghost particles which are inside the security distance of any real particle are created.

In practice, some more particles, strictly necessary, are created: a symmetric particle G_i to particle N_i is created, with respect to symmetry plane P, if $\exists j$ neighbor of i:

$d {(N_{i}, (P))}^{2} \leq (1 + α_{s o r t}) \cdot \max {(d_{i} + d_{j})}^{2}$

Where, $d_{i}$ and $d_{j}$ are the smoothing lengths related to particle $i$ and $j$ .

As long as no real particles cross the symmetry plane (all real particles lie on the same side of the symmetry plane), this criteria is sufficient to get all ghost neighbors of all real particles inside the security distance, since:

$d (N_{i}, (P)) \leq d (G_{i}, N_{i})$ for $\forall (i, j)$

And, $d (G_{i}, N_{i}) \leq \sqrt{1 + α_{s o r t}} \cdot \max (d_{i} + d_{j})$

\Rightarrow d (N_{i}, (P)) \leq \sqrt{1 + α_{s o r t}} \cdot \max (d_{i} + d_{j})

Particles, which one can expect to remain far from the symmetry plane all along the simulation, will never be symmetrized. This gives a way to over-estimate the number of particles which will be symmetrized at one time.

When a particle N_i has to be symmetrized with respect to n conditions, the particle N_i gives birth to n ghost particles. The following quantity must remain less than Maxsph (since v14.0.220, "Maxsph" is ignored and the memory is dynamically allocated).(3)

\sum_{n = 1}^{n} n_{p a r t i c l e s}

Where,

$n$: Number of conditions
$n_{p a r t i c l e s}$: Number of particles to be symmetrized with respect to condition $n$

Anyway, the default value which is the number of SPH symmetry conditions multiplied by the number of particles will be enough to treat any problem.

Solid to SPH Options (Sol2SPH)

The solid to SPH option (Sol2SPH) enables you to turn a solid element into particles either in order to increase the time step/robustness of a Lagrangian calculation, while not changing the physics.

Time Step

Two SPH time step methods are available in Radioss:

Particle time step (/DT/SPHCEL)
Nodal time step (/DT/NODA)

In particle time step, stable time step is computed as:(4)

Δ t = Δ T_{s c a} \cdot \min_{i} (\frac{d_{i}}{c_{i} (α_{i} + \sqrt{α_{i}^{2} + 1})})

Where,

$d_{i}$: Smoothing length related to particle $i$
$c_{i}$: Sound speed at location $i$

(5)

α_{i} = q_{a} + \frac{q_{a} \cdot {\bar{μ}}_{i} \cdot d_{i}}{c_{i}}

For time step scale factor $Δ T_{s c a}$ , it is recommended to set it to 0.3.

In nodal time step, stable time step is computed as:(6)

Δ t = Δ T_{s c a} \cdot \sqrt{\frac{2 m}{K^{*}}}

Where,

$m$: Mass for particles
$K^{*}$: Stiffness based on SPH interaction

For time step scale factor $Δ T_{s c a}$ , it is recommended to set it to 0.67.

Thermal Analysis

Heat transfer is now available between SPH particles and finite elements with I_the=1 in /INTER/TYPE7 and /INTER/TYPE21; and with /THERM_STRESS/MAT, thermal expansion in SPH is also possible.