The fluid-structure interaction and the fluid flow are studied in cases of a fuel tank sloshing and overturning. A
bi-phase liquid-gas material with an ALE formulation is used to define the interaction between water and air in the
fuel tank.
The purpose of this example is to study the energy propagation and the momentum transfer through several bodies, initially
in contact with each other, subjected to multiple impact. The process of collision and the energetic behavior upon
impact are described using a 3-dimensional mode.
The impact and rebound between balls on a small billiard table is studied. This example deals with the problem of
defining interfaces and transmitting momentum between the balls.
After a quasi-static pre-loading using gravity, a dummy cyclist rides along a plane, then jumps down onto a lower
plane. Sensors are used to simulate the scenario in terms of time.
The purpose of this study is to demonstrate the use of quadratic interface contact using two gears in contact with
identical pitch diameter and straight teeth. Two different contact interfaces are compared.
The problem of a dummy positioning on the seat before a crash analysis is the quasi-static loading which can be resolved
by either Radioss explicit or Radioss implicit solvers.
The crashing of a box beam against a rigid wall is a typical and famous example of simulation in dynamic transient
problems. The purpose for this example is to study the mesh influence on simulation results when several kinds of
shell elements are used.
A square plane subjected to in-plane and out-of-plane static loading is a simple element test. It allows you to highlight
element formulation for elastic and elasto-plastic cases. The under-integrated quadrilateral shells are compared with
the fully-integrated BATOZ shells. The triangles are also studied.
The modeling of a camshaft, which takes the engine's rotary motion and translates it into linear motion for operating
the intake and exhaust valves, is studied.
The ditching of an object into a pool of water is studied using ALE and SPH approaches. The simulation results are
compared to the experimental data and to the analytical results.
A rubber ring resting on a flat rigid surface is pushed down by a circular roller to produce self-contact on the inside
surface of the ring. Then the roller is simultaneously rolled and translated so that crushed ring rolls along the
flat surface.
Separate the whole model into main domain and sub-domain and solve each one with its own timestep. The new Multi-Domain
Single Input Format makes the sub-domain part definition with the /SUBDOMAIN keyword.
The aim of this example is to introduce /INIVOL for initial volume fractions of different materials in multi-material ALE elements, /SURF/PLANE for infinite plane, and fluid structure interaction (FSI) with a Lagrange container.
A heat source moved on one plate. Heat exchanged between a heatsource and a plate through contact, also between a
plate and theatmosphere (water) through convective flux.
Impacts of rotating structures usually happen while the structure is rotating at a steady state. When the structure is
rotating at very high speeds, it is necessary to include the centrifugal force field acting on the structure to correctly
account for the initial stresses in the structure due to rotation.
The impact and rebound between balls on a small billiard table is studied. This example deals with the problem of
defining interfaces and transmitting momentum between the balls.
The model files used in this example are available in:
/radioss/example/09_Billiards
Model Description
Study on Trajectories
The balls' behavior is described using the parameters (angles and velocities) shown in Figure 1. The numerical results are compared with the analytical
solution, assuming a perfect elastic rebound (coefficient of restitution is equal to
1).
Initial Values
V_{1}
0.7m.s^{-1}
V_{2}
1m.s^{-1}
$\theta $_{1}
40°
$\theta $_{2}
30
mass_{ball}
44.514g
Model Method
The balls and the table have the same properties, previously defined for a pool game.
The dimensions of the table are 900 mm x 450 mm x 25 mm and the balls' diameter is
50.8 mm. The balls and the table are meshed with 16-node thick shell elements for
using the TYPE16 Lagrangian interface.
The initial translational velocities are applied to the balls in the
/INIV Engine option. Velocities are projected on the X and Y
axes.
Gravity is considered for the balls (0.00981 mm.ms^{-2}).
The ball/ball and balls/table contact is modeled using the TYPE16 interface
(secondary nodes/main 16-node thick shells contact). The interface defining the
ball/ball contact is shown in Figure 4.
Analytical Solution
Take two balls, 1 and 2 from masses $${m}_{1}$$ and $${m}_{2}$$, moving in the same plane and approaching each other
on a collision course using velocities $${V}_{1}$$ and $${V}_{2}$$, as shown in Figure 5.
Velocities are projected onto the local axes $$n$$ and $$t$$. To obtain the velocities and their direction after
impact, the momentum conservation law is recorded for the two balls:(1)$$-{m}_{1}{V}_{1n}+{m}_{2}{V}_{2n}={m}_{1}{V}_{1n}^{\text{'}}-{m}_{2}{V}_{2n}^{\text{'}}$$
The shock is presumed elastic and without friction. Maintaining the translational
kinetic energy is respected as there is no rotational energy:(3)$$\frac{1}{2}{m}_{1}\left({V}_{1n}^{\text{'}2}+{V}_{1t}^{\text{'}2}\right)+\frac{1}{2}{m}_{2}\left({V}_{2n}^{\text{'}2}+{V}_{2t}^{\text{'}2}\right)=\frac{1}{2}{m}_{1}\left({V}_{1n}^{2}+{V}_{1t}^{2}\right)+\frac{1}{2}{m}_{2}\left({V}_{2n}^{2}+{V}_{2t}^{2}\right)$$
Such equality implies that the recovering capacity of the two balls corresponds to
their tendency to deform.
This condition equals one of the elastic impacts, with no energy loss. Maintaining
the system's energy gives:(4)$$\left({V}_{2n}^{\text{'}}-{V}_{1n}^{\text{'}}\right)=-\left({V}_{2n}-{V}_{1n}\right)$$
This relation means that the normal component of the relative velocity changes into
its opposite during the elastic shock (coefficient of restitution value e is equal
to the unit).
The following equations must be checked for normal components:(5)$$\begin{array}{l}{V}_{2n}^{\text{'}}={V}_{1n}^{\text{'}}=-\left({V}_{2n}-{V}_{1n}\right)\\ {m}_{2}{V}_{2n}^{\text{'}}+{m}_{1}{V}_{1n}^{\text{'}}={m}_{2}{V}_{2n}+{m}_{1}{V}_{1n}\end{array}$$
The equations system using V'_{1} and V'_{2} as unknowns is easily
solved:(6)$$\begin{array}{l}{V}_{2n}^{\text{'}}=\left(\frac{{m}_{2}-{m}_{1}}{{m}_{2}+{m}_{1}}\right){V}_{2n}+\left(\frac{2{m}_{1}}{{m}_{1}+{m}_{2}}\right){V}_{1n}\\ {V}_{1n}^{\text{'}}=\left(\frac{{m}_{1}-{m}_{2}}{{m}_{1}+{m}_{2}}\right){V}_{1n}+\left(\frac{2{m}_{2}}{{m}_{1}+{m}_{2}}\right){V}_{2n}\end{array}$$
It should be noted that these relations depend upon the masses ratio.
As the balls do not suffer from velocity change in the t-direction, maintaining the
tangential component of each sphere's velocity provides:(7)$$\begin{array}{l}{V}_{1t}^{\text{'}}={V}_{1t}={V}_{1}\mathrm{cos}{\theta}_{1}\\ {V}_{2t}^{\text{'}}={V}_{2t}={V}_{2}\mathrm{cos}{\theta}_{2}\end{array}$$
The norms of velocities after shock result from the following
relations.(8)$${V}_{1}^{\text{'}}={\left({\left({V}_{1n}^{\text{'}}\right)}^{2}+{\left({V}_{1t}^{\text{'}}\right)}^{2}\right)}^{1/2}$$(9)$${V}_{2}^{\text{'}}={\left({\left({V}_{2n}^{\text{'}}\right)}^{2}+{\left({V}_{2t}^{\text{'}}\right)}^{2}\right)}^{1/2}$$
In this example, balls have the same mass: m_{1} = m_{2}.
Therefore, $${V}_{2}^{\text{'}}={V}_{1n}$$ and $${V}_{1n}^{\text{'}}={V}_{2n}$$.
The norms of the velocities are given using the following relations, depending on the
initial velocities and angles. Used to determine the analytical solutions (angles
and velocities after collision):(10)$${V}_{1}^{\text{'}}={\left({\left({V}_{2}\right)}^{2}{\mathrm{sin}}^{2}{\theta}_{2}+\left({V}_{1}\right){\mathrm{cos}}^{2}{\theta}_{1}\right)}^{1/2}$$(11)$${V}_{2}^{\text{'}}={\left({\left({V}_{1}\right)}^{2}{\mathrm{sin}}^{2}{\theta}_{1}+\left({V}_{2}\right){\mathrm{cos}}^{2}{\theta}_{2}\right)}^{1/2}$$
By recording the projection of the velocities, directions after shock can be
evaluated using relation. Used to determine the analytical solutions (angles and
velocities after collision):(12)$${\theta}_{1}^{\text{'}}=\mathrm{arcsin}\left(\frac{{V}_{2}}{{V}_{1}}\mathrm{sin}{\theta}_{2}\right)$$(13)$${\theta}_{2}^{\text{'}}=\mathrm{arcsin}\left(\frac{{V}_{1}}{{V}_{2}}\mathrm{sin}{\theta}_{1}\right)$$
Results
Numerical Results Comparison with the Analytical Solution
Figure 6 shows the trajectories of the balls' center point obtained using
numerical simulation before and after collision.
For given initial values of $${V}_{1}$$, $${V}_{2}$$, $\theta $_{1} and $\theta $_{2}, simulation results are reported in Table 1.
Table 1. Comparison of Results for After Collision
Numerical Results
Analytical Solution
$\theta $_{ 1'}
42.27°
44.72°
$\theta $_{ 2'}
26.75°
26.48°
V _{1'}
0.731 m/s
.731 m/s
V _{2'}
0.969 m/s
0.977 m/s
Conclusion
The simulation corroborates with the analytical solution. The 16-node thick shells are
fully-integrated elements without hourglass energy. This modeling provides a good
transmission of momentum. However, the TYPE16 interface does not take into account
the quadratic surface on the secondary side (ball 2), due to the node to thick shell
contact. Accurate results are obtained for a collision without penetrating the
quadratic surface of the secondary side in order to confirm impact between the
spherical bodies.