# RD-E: 1601 EXPLICIT Solver

A dummy is sat down via gravity using the quasi-static load treatment.

The topic of this study concerns quasi-static load treatment using kinetic relaxation, dynamic relaxation and Rayleigh damping. The explicit solutions provided by the three different approaches will be compared and analyzed.

## Options and Keywords Used

- Shell, brick, beam, and dummy
- Quasi-static analysis by explicit, kinetic and dynamic relaxation (/KEREL and /DYREL), and Rayleigh damping (/DAMP)
- Symmetric interface (/INTER/TYPE7)
- Kelvin-Voigt visco-elastic model (/MAT/LAW35 (FOAM_VISC)) and linear elastic law (/MAT/LAW1 (ELAST))
- Added mass (/ADMAS)
- Boundary conditions (/BCS)
- Gravity (/GRAV)
- Material definition (Materials)
- Rigid body (/RBODY)

The goal is to set the body on the seat using a quasi-static approach in order to obtain static equilibrium. The positioning phase is not included in this study. Thus, all nodes of the dummy are placed in a global rigid body in order to maintain the dummy's initial configuration.

When the `ICoG`
flag is set to 1 for the rigid body of the seat, the center of gravity is computed
using the main and secondary node coordinates, and the main node is moved to the
center of gravity, where mass and inertia are placed.

When the `ICoG`
flag is set to 3 for the rigid body of the dummy, the center of gravity is set at
the main node coordinates defined by you. The added masses and added inertia are
transmitted to the main node coordinates.

The main node coordinates and skew are extracted from the pelvis part of the original rigid body.

Static analysis: quasi-static treatment of gravity loading up to static equilibrium.

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

- Kinetic relaxation (/KEREL)
- Dynamic relaxation (/DYREL)
- Rayleigh damping (/DAMP)

## Kinetic Relaxation Method

## Dynamic Relaxation Method

- $\beta $
- Relaxation value (recommended default value 1)
- $$T$$
- The period to be damped (less than or equal to the highest period of the system)

This option is activated in the Engine file (*_0001.rad) using /DYREL (inputs: $\beta $ and $$T$$).

## Rayleigh Damping Method

Where, $$a$$ and $\beta $ are the pre-defined constants.

- $${\xi}_{i}$$
- The $$i$$
^{th}being the damping ratio of the system - $${\omega}_{i}$$
- The $$i$$
^{th}being the natural frequency of the system.

If several frequencies are available, an average of computed values, $$a$$ and $\beta $ may be used.

This model of proportional damping is not recommended for complex structures and does not enable good experimental retiming.

This option is activated in the Engine file (*_0001.rad) using /DAMP (inputs data: $$a$$ and $\beta $).

## Parameters Used

- First case: $$a$$ = 10 and $\beta $ = 10
- Second case: $$a$$ = 0 and $\beta $ = 10
- Third case: $$a$$ = 10 and $\beta $ = 0
- Fourth case: $$a$$ = 20 and $\beta $ = 0

- First case
- [C] = 10[M] + 10[K]
- Second case
- [C] = 10[K]
- Third case
- [C] = 10[M]
- Fourth case
- [C] = 20[M]

## Input Files

## Model Description

The purpose is to position a dummy on a foam seat under the gravity field using a quasi-static approach prior to a possible dynamic crash simulation.

The dummy weighs 80 kg (173.4 lbs.). The material introduced does not represent the physical case; however, the global weight of the dummy is respected. As the dummy deformation is neglected in this loading phase, simplifying the material characterizations has no incidence on the simulation.

**Material Properties**- Young's modulus
- 210000 $\left[\mathrm{MPa}\right]$
- Poisson's ratio
- 0.3
- Density
- 7.8 x 10
^{-9}Gkg/l

- Area
- 2580 mm
^{2} - Inertia
- I
_{XX}= 554975 mm^{4}

- Brace thickness
- 2 mm
- Floor thickness
- 1 mm

**Material Properties**- Young's modulus
- 0.2 $\left[\mathrm{MPa}\right]$
- Poisson's ratio
- 0
- Density
- 4.3 x 10
^{-11}Gkg/l - E
_{1}and E_{2} - 0 $\left[\mathrm{MPa}\right]$
- Tangent modulus
- 0.25 $\left[\mathrm{MPa}\right]$
- Viscosity in pure shear
- 10000 MPa/s
- C
_{1}= C_{2}= C_{3}= 1 - Visco-elastic bulk viscosity

`I`

_{flag}= 0) and the pressure is read using the following input curve:

Compression $\mu $ | -100000 | -10 | 0 | 3000 | 209000 | 210000 |

Pressure | -1000 | -1000 | 0 | 7.633 | 7.633 | 18.5 |

### Visco-elastic Foam Material Law (/MAT/LAW35)

Based on the Navier equation, LAW35 describes materials using visco-elastic behavior. The effect of the air enclosed is taken into account via a separate pressure versus compression function. Relaxation and creep can be modeled.

Refer to the Radioss Theory Manual for explanation of coefficients.

- $\rho $
- Density at a time $$t$$
- ${\rho}_{0}$
- Initial density

### Model Method

- a dummy composed of 38 parts (limbs and joints).
- a seat comprised of six parts (foam seat back, foam seat cushion, seat back brace, seat bottom brace, seat columns and the floor).

- Quadratic bulk viscosity
- 1.1
- Linear bulk viscosity
- 0.05
- Hourglass viscosity coefficient
- 0.1

The dummy and seat brace are modeled with shell elements, divided into 4871 4-node shells and 203 3-node shells (Dummy: 5004 shells and seat: 70 shells).

**Shell Properties**- Belytschko hourglass formulation
- Hourglass TYPE4,
`I`_{shell}= 4 - Membrane hourglass coefficients
- 0.01 (default value)
- Out-of-plane hourglass
- 0.01 (default value)
- Rotation hourglass coefficient
- 0.01 (default value)

**First interface**- Dummy parts
- Secondary nodes
- Seat
- Main surface

**Second interface**- Dummy parts
- Main surface
- Seat
- Secondary nodes

The gap between the symmetrical interfaces is equal to 5 mm, while a gap of 0.5 mm is set for the other interface.

**Symmetric Interface Properties**- Coulomb friction (
`Fric`flag) - 0.3
- Critical damping coefficient (
`Visc`flag) - 0.05
- Scale factor for stiffness (
`Stfac`flag) - 1
- Sorting factor (
`Bumult`flag) - 0.20

Refer to the Radioss Theory Manual and Starter Input for further information about the definition of the TYPE7 interface.

## Results

### Curve and Animation Results Obtained using Kinetic Relaxation: /KEREL

### Results Obtained using Dynamic Relaxation: /DYREL

The period T to be damped is estimated from the velocity curves (highest period).

### Results Obtained using Rayleigh Damping: /DAMP

### Comparison of the Different Approaches

### Conclusion

It is undeniable that the damping methods used to converge towards static equilibrium provide accurate results, especially in the case of this problem where the low rigidity of the seat caused very little quenched oscillations.

The kinetic relaxation introduced in /KEREL, was relatively effective having a swift convergence of the solution towards a static solution, in addition to being easy to use since no input is required. Stability was obtained at 0.137 s.

The /DYREL and /DAMP options are based on viscous damping conducted for the same response, with convergence in three oscillations. Stability was obtained at 0.75 s. Furthermore, dynamic relaxation and the Rayleigh damping methods are basically equivalent in this problem, due to the low stiffness of the seat cushion (Young's modulus is equal to 0.2 $\left[\mathrm{MPa}\right]$), which breaks the balance between the mass and the weight stiffness in the Rayleigh assumption. Moreover, the boundary conditions and the loading applied on the model lead to a problem described using a predominant natural frequency. Thus, only one parameter, $$a$$ is needed to describe this physical behavior, which reverts back to the dynamic relaxation assumption.

- Dynamic relaxation:
$$\left[C\right]=\frac{2\beta}{T}\left[M\right]=11.1\left[M\right]$$

- Rayleigh damping:
$$\left[C\right]=\alpha \left[M\right]+\beta \left[K\right]\approx \alpha \left[M\right]$$

10[M]

The approaches available in Radioss provided after convergence a single solution, namely displacement of the dummy by -12.66 mm along the Z-axis and an identical deformation of the seat cushion.