RD-E: 0903 Interfaces Study

Comparison of results obtained using different interfaces.

Options and Keywords Used

To reduce the dynamic effect, dynamic relaxation can be used (/DYREL in the Engine file). A diagonal damping matrix proportional to the mass matrix is introduced into the dynamic equation:(1)

[M] {\ddot{u}} + [C] {\dot{u}} + [K] {u} = F

(2)

[C] = \frac{2 β}{T} [M]

With,

$β$: The relaxation value by default, equal to 1
$T$: The period to be damped (less than or equal to the largest period of the system)

Thus, a viscous stress tensor is added to the stress tensor:(3)

σ_{i j}^{v i s c o u s} = (λ + 2 μ) {\dot{ε}}_{k k} δ_{i j} + 2 μ {\dot{ε}}_{i j}

In an explicit code, the application of the dashpot force modifies the velocity equation:

$V_{t + Δ t / 2} = V_{t - Δ t / 2} + γ_{t} Δ t$: without relaxation
$V_{t + Δ t / 2} = (1 - 2 ω) V_{t - Δ t / 2} + (1 - ω) γ_{t} Δ t$: with relaxation

with $ω = β \frac{Δ t}{t}$

This option is activated in the Engine file (*_0001.rad) using /DYREL.

Inputs
$β$: = 1
$T$: = 0.2

The dynamic problem (impact between balls) is considered in a second run managed by the second Engine file (*_0002.rad) with a time running from 30 ms to 130 ms.

Input Files

The input files used in this example include:

Inter_7_Penalty: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_7_Penalty/TEST7P*
Inter_7_Lagrangian: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_7_Lagrangian/TEST7L*
Inter_16_tied: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_16_tied/TEST16T*
Inter_16_sliding: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_16_sliding/TEST16S*
Inter_17_tied: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_17_tied/TEST17ST*
Inter_17_sliding: <install_directory>/hwsolvers/demos/radioss/examples/09_Billiards/Contact_modelling/Inter_17_sliding/TEST17S*

Model Description

The balls and the table have the same properties as previously defined. The dimensions of the table are 900 mm x 450 mm x 25 mm and the balls' diameter is 50.8 mm.

Six interfaces are used to model the contacts (ball/ball and balls/table):

Interfaces Used in the Problems
TYPE16 (Lagrange Multipliers) tied or sliding: Slave nodes/master solids contact
TYPE17 (Lagrange Multipliers) tied or sliding: Slave 16-node shells/master 16-node shells contact
TYPE7 (Lagrange Multipliers): Slave nodes/master surface contact
TYPE7 (Penalty) sliding: Slave nodes/master surface contact

Model Method

The TYPE16 interface defines contact between a group of nodes (slaves) and a curved surface of quadratic elements (master part). The TYPE17 interface is used for modeling a surface-to-surface contact. For both interfaces, the Lagrange Multipliers method is used to apply the contact conditions; gaps are not required. Contact between the balls and the table is set as tied or sliding. Contact between the balls themselves is always considered as sliding. The TYPE7 interface enables the simulation of the most general contact types occurring between a master surface and a set of slave nodes. The Coulomb friction between surfaces is not modeled here (sliding contact) and the gap is fixed at 0.1 mm. The other parameters are set to default values.

The TYPE7 interface with the Penalty method is not available with 16-node thick shell elements. Thus, brick elements replace the 16-nodes shells in this case (check in the input file).

Contact modeling between balls (always sliding).

The symmetrical interface definition is not recommended when using the Lagrange Multipliers method (TYPE16, TYPE17 and TYPE7-Lag). The problem using the interface with the Penalty method uses two interfaces to model the symmetrical impact.

Interface: Slave (red) and Master (blue) Objects
TYPE16 - tied: Slave: nodes; Master: solids (16-node shell)
TYPE16 - sliding: Slave: nodes; Master: solids (16-node shell)
TYPE17 - tied: Slave: 16-node shell; Master: 16-node shell
TYPE17 - sliding: Slave: 16-node shell; Master: 16-node shell
TYPE7 - Lagrange Multipliers: Slave: nodes; Master: surface (segments)
TYPE7 - Penalty method: Slave: nodes; Master: surface (segments)

Contact between the balls and the table (sliding or tied depending on the problem):

Interface: Slave (red) and Master (blue) Objects
TYPE16 - tied: Slave: nodes; Master: solids (16-node shell)
TYPE16 - sliding: Slave: nodes; Master: solids (16-node shell)
TYPE17 - tied: Slave: 16-node shell; Master: 16-node shell
TYPE17 - sliding: Slave: 16-node shell; Master: 16-node shell
TYPE7 - Lagrange Multipliers: Slave: nodes; Master: surface (segments)
TYPE7 - Penalty method: Slave: nodes; Master: surface (segments)

Pre-loading: quasi-static gravity loading to reach static equilibrium.

The explicit time integration scheme starts with nodal acceleration computation. It is efficient for the simulation of dynamic loadings. However, a quasi-static simulation via a dynamic resolution method needs to minimize the dynamic effects for converging towards static equilibrium and describes the pre-loading case before the dynamic analysis. Thus, the quasi-static solution of gravity loading on the model shows a steady state in the transient response.

Results

Kinetic Energy Transmission between Balls during Collision

Conclusion

	Interface 16 Tied	Interface 16 Sliding	Interface 17 Tied	Interface 17 Sliding	Interface 7 Lagrange Multipliers	Interface 7 Penalty
Cycles	241392	241385	241387	241385	241385	773099
Error on Energy	-30.8%	-1.4%	-55.5%	-10.8%	-1.2%	-46.1%
Rolling	yes	no	yes	no	no	no
Momentum Transmission	partial	quasi-perfect	partial	good	good	partial
Quadratic surface	master side	master side	master and slave sides	master and slave sides	no	no

A non-elastic collision appears using the TYPE7 interface Penalty method. After impact, each ball has about half of the initial velocity. The momentum transmission is partial and can be improved by increasing the stiffness of the interface despite the hourglass energy and degradation of the energy assessment.

Error on energy is more noticeable for interfaces using the Tied option, due to taking into account the rolling simulation.

This study shows the high sensitivity of the numerical algorithms for the modeling impact on elastic balls. Regarding the interface type, the kinematics of the problem and the transmission of momentum are more or less satisfactory. TYPE16 interface allows good results to be obtained.