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Example 16 - Dummy Positioning

Summary

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. If deformation remains small, a linear analysis may be used as a simple approach to determine the position after applying gravity force. However, this method is not valid if the contact surface between the dummy and the seat is not correctly estimated before analysis. When comparing the implicit and explicit solvers, it's shown that the implicit computation enables saving time in the computation. However, the rigid body modes of the dummy must be controlled. This is not the case if the explicit solver is used.

The following studies are depicted in this tutorial

EXPLICIT Solver

IMPLICIT Solver

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EXPLICIT Solver

Title

EXPLICIT solver

Number

16.1

Brief Description

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

Keywords

• Shell, brick, beam, dummy

• Quasi-static analysis by explicit, kinetic and dynamic relaxation, Rayleigh damping

• Type 7 interface (symmetric)

• Kelvin-Voigt visco-elastic model (/MAT/LAW35), linear elastic law (/MAT/LAW1)

RADIOSS Options

• Added mass (/ADMAS)

• Boundary conditions (/BCS)

• Dynamic relaxation (/DYREL)

• Gravity (/GRAV)

• Kinetic relaxation (/KEREL)

• Material definition (/MAT)

• Rayleigh damping (/DAMP)

• Rigid body (/RBODY)

Input File

Rayleigh_damping: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/EXPLICIT_solv

er/RAYLEIGH/.../SEAT_RAYLEIGH*

Dynamic_relaxation: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/EXPLICIT_solv

er/DYREL/SEAT_DYREL*

Kinetic_relaxation: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/EXPLICIT_solv

er/KEREL/SEAT_KEREL*

Without_damping: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/EXPLICIT_solv

er/Without_damping/SEAT*

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RADIOSS Version

51f

Technical / Theoretical Level

Advanced

Overview

Aim of the Problem

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.

Physical Problem 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.

Units: mm, s, ton, N, MPa

Fig 1: Problem studied.

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 for seat brace - both the columns and the floor are made of steel with the following properties (/MAT/LAW1):

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• Young’s modulus: 210000 MPa

• Poisson’s ratio: 0.3

• Density: 7.8 x 10-9 Gkg/l

The seat columns have the following characteristics:

• Area: 2580 mm2

• Inertia: IXX = 554975 mm4 ; IYY = 554975 mm4 ; IZZ = 937908 mm4

The thickness for the seat back and the floor:

• Brace thickness = 2 mm

• Floor thickness = 1 mm

The seat cushion is made of foam which can be described using the generalized Kelvin-Voigt model. The material properties of the foam are:

• Young’s modulus: 0.2 MPa

• Poisson’s ratio: 0

• Density: 4.3 x 10-11 Gkg/l

• E1 and E2: 0 MPa

• Tangent modulus: 0.25 MPa

• Viscosity in pure shear: 10000 MPa/s

• C1 = C2 = C3 = 1 (visco-elastic bulk viscosity)

RADIOSS material law 35 is used. The open cell foam option is not active (IFlag = 0) and the pressure is read using the following input curve:

Table 1: Pressure versus compression curve.

Compression -100000 -10 0 3000 209000 210000

Pressure -1000 -1000 0 7.633 7.633 18.5

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

Based on the Navier equation, law 35 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.

The schematic model in Fig 2 describes the generalized Kelvin-Voigt model where a time-dependent spring working in parallel with a Navier dashpot is put in a series with a nonlinear rate-dependent spring.

Fig 2: Generalized Kelvin-Voigt model – Law 35.

Two pressure computations are available in RADIOSS for foam having no open cells. The expression used by default is:

See the RADIOSS Theory Manual for explanation of coefficients.

Pressure may also be computed using the pressure versus compression curve defined by you. The compression, is defined as:

where, is the density at a time t, and 0 is the initial density.

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Analysis, Assumptions and Modeling Description

Modeling Methodology

The model consists of two subsets:

• 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).

Fig 3: Model mesh. Fig 4: Model mesh. (Perspective view – Shaded display) (Profile view – Edges display)

The seat cushion is meshed with 70 brick elements defined by general type 14 solid property.

• 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).

Using a dummy in the model, the /DEL/SHELL/1 option should be activated in the Engine file to avoid a small time step, due to the low density of material defining the dummy envelope.

The shell properties are:

• Belytschko hourglass formulation (Hourglass type 4, Ishell = 4).

• Membrane hourglass coefficients: 0.01 (default value)

• Out-of-plane hourglass: 0.01 (default value)

• Rotation hourglass coefficient: 0.01 (default value)

Contacts between the dummy and the seat cushion, as well as between the foot and the floor, use type 7 interface models with the penalty method. Additionally, symmetrical contact between the body and seat is achieved by creating two complementary interfaces, as shown below:

First interface: Dummy parts: slave nodes / seat: master surface

Second interface: Dummy parts: master surface / seat: slave nodes

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Fig 5: Contacts modeling with type 7 symmetrical interface

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

The type 7 interface allows sliding to occur between surfaces. A Coulomb friction can be introduced; in addition, a critical viscous damping coefficient can be defined to damp sliding.

The symmetric interfaces properties are:

• 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

• Maximum impacted segment / node (Multimp flag) = 4

See the RADIOSS Theory Manual and Starter Input for further information about the definition of the type 7 interface.

RADIOSS Options Used

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.

In order to save the CPU, a second global rigid body includes parts of the seat and the floor; except for the seat cushion parts, which will only have active elements during simulation.

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Fig 6: Set up of both rigid bodies.

When the ICoG flag is set to 1 for the rigid body of the seat, the center of gravity is computed using the master and slave node coordinates, and the master 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 master node coordinates defined by you. The added masses and added inertia are transmitted to the master node coordinates.

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

Gravity is applied to all nodes of the model. A function defines gravity acceleration in the z direction versus

time. Gravity is activated by /GRAV in the D00 file.

Fig 7: Input gravity function (-9810 mm.s

-2 ) and nodes selection (yellow).

The six rigid body modes of the seat are removed by completely fixing the rigid body master node attached to the seat. In order to limit the out-of-plane vibrations, the master node of the dummy's rigid body is fixed in translation along the Y axis.

Fig 8: Boundary conditions on the rigid bodies’ master nodes.

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

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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.

To reduce the dynamic effect, three options are available in the Engine file:

• Kinetic relaxation (/KEREL)

• Dynamic relaxation (/DYREL)

• Rayleigh damping (/DAMP)

Kinetic Relaxation Method

All velocities are set to zero each time the kinetic energy reaches a maximum value. This option is activated in the Engine file using /KEREL (input is not required).

Fig 9: Kinetic relaxation method with /KEREL (also named energy discrete relaxation).

Dynamic Relaxation Method

Dynamic loading is damped by introducing a diagonal damping matrix, proportional to mass matrix, in the dynamic equation:

with:

is the relaxation value (recommended default value 1).

T is the period to be damped (less than or equal to the highest period of the system).

Thus, a viscous stress tensor is added to the stress tensor:

Using an explicit code, application of the dashpot force reduces the velocity equation modification:

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This option is activated in the D01 file using /DYREL (inputs: and T).

Rayleigh Damping Method

Dynamic loading is damped by introducing a damping matrix, proportional to the mass and stiffness matrix, in the dynamic equation. This simplified approach will allow you to reduce the global equilibrium equation to n-uncoupled equations by using an orthogonal transformation. This damping is said to be proportionally uncoupled.

where, and are the pre-defined constants.

The orthogonal transformation using this proportional damping assumption leads to:

with: i is the ith being the damping ratio of the system

i is the ith being the natural frequency of the system

Fig 10: Rayleigh type damping.

If you have some experimental results, the proportionality factors, and are found by evaluating the damping for a pair of the most significant frequencies used. Thus, two equations with two unknown variables are obtained:

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If several frequencies are available, an average of computed values, and 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 D01 Engine file using /DAMP (inputs data: and ).

Parameters Used

In this example, and are set to the following values:

• First case: = 10 and = 10

• Second case: = 0 and = 10

• Third case: = 10 and = 0

• Fourth case: = 20 and = 0

The resulting assumptions are:

• First case: [C] = 10[M] + 10[K]

• Second case: [C] = 10[K]

• Third case: [C] = 10[M]

• Fourth case: [C] = 20[M]

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Simulation Results and Conclusions

Curves and Animations

Results Obtained using Kinetic Relaxation: /KEREL

Fig 11: Z-displacement of the rigid body’s master node on dummy (node 14199).

Fig 12: Kinetic energy of global model.

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Results Obtained using Dynamic Relaxation: /DYREL

Fig 13: Z-displacement of the rigid body’s master node on dummy (node 14199).

Fig 14: Z-velocity of the rigid body’s master node on dummy (node 14199).

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

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Results Obtained using Rayleigh Damping: /DAMP

Fig 15: Z-displacement of the rigid body’s master node on dummy (node 14199)

Fig 16: Z-velocity of rigid body’s master node on dummy (node 14199)

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Comparison of the Different Approaches

Fig 17: Comparison of the nodal displacements’ display on the seat at time t = 1.48 s

Fig 18: Comparison of damping on displacement obtained using the three static approaches (Z-displacement of the rigid body’s master node on dummy: node 14199)

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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 MPa), 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, is needed to describe this physical behavior, which reverts back to the dynamic relaxation assumption.

Using =1 and T =0.18s for dynamic relaxation and =10 for Rayleigh damping, you achieve:

• Dynamic relaxation:

• Rayleigh damping:

[C] = [M] + [K] ≈ [M]

≈ 10[M]

In conclusion, 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.

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IMPLICIT Solver

Title

IMPLICIT solver

Number

16.2

Brief Description

A dummy is sat down via gravity using the implicit approach (static).

Keywords

• Shell, brick, beam, spring, dummy

• Linear and nonlinear static solution by implicit solver

• Type 7 interface (symmetric), tied interface (type 2)

• Kelvin-Voigt visco-elastic model (/MAT/LAW35), linear elastic law (/MAT/LAW1)

RADIOSS Options

• Concentrated load (/CLOAD)

• Imposed displacement (/IMPDISP)

• Time step control method for implicit (/IMPL/DT)

• Initial time step for implicit (/IMPL/DTINI)

• Static linear implicit solution (/IMPL/LINEAR)

• Static nonlinear implicit solution (/IMPL/NONLIN)

• Print frequency for implicit (/IMPL/PRINT)

• Implicit solver method (/IMPL/SOLVER)

• Gravity (/GRAV)

Input File

Linear_implicit_model: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/IMPLICIT_solv

er/Linear/SEAT_IMPL_LIN*

Nonlinear_implicit_model: <install_directory>/demos/hwsolvers/radioss/16_Dummy_Positioning/IMPLICIT_solv

er/Nonlinear/

• Imposed_displacement: //.../Imposed_displacement/SEAT_IMPL_DISP*

• Concentrated_load: //.../Concentrated_load/SEAT_IMPL_CLOAD*

• Gravity_loading: //.../Gravity/SEAT_IMPL_GRAV*

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RADIOSS Version

51f

Technical / Theoretical Level

Advanced

Linear and Nonlinear Analysis by Implicit Solver

The main advantages of implicit resolution are:

• Unconditional stable scheme

• Large time step

• Treatment of the static problem

However, the implicit algorithm uses a global resolution which requires convergence for each time step and has low robustness in comparison to the explicit (null pivots, divergence for high nonlinearities, etc.).

The implicit methods result in solving a linear system for each time step, which is relatively expensive but enables a large time step: few expensive calculations. The explicit method treats linear or nonlinear systems depending on the problem. It is less expensive and faster for each step, but requires short time steps to ensure stability of the scheme that has many inexpensive cycles.

• Implicit integration scheme: Newmark

This scheme is unconditionally stable, the stability condition being independent of the time step choice. See the RADIOSS Theory Manual for further information about the Newmark scheme.

RADIOSS has a linear and a nonlinear solver. Only static computations are available and loading should be defined as a monotonous increasing time function for nonlinear analysis.

The main computational methods available in RADIOSS:

• Cholesky (direct method, linear solver)

• Preconditioned Conjugate Gradient (linear solver)

• Modified Newton-Raphson method (nonlinear solver)

The precondition methods for linear solver available in RADIOSS:

• No preconditioned

• Diagonal Jacobi

• Incomplete Cholesky

• Stabilized incomplete Cholesky

• Factored Approximate Inverse (by default)

You should define the tolerance and stop criterion for the linear and nonlinear solver (residual).

Strategies of resolution for nonlinear static computation / time step control:

• Iterations number limit for updating stiffness matrix

• Convergence iterations number for increasing time step

• Convergence iterations number for decreasing time step

• Increase time step factor

• Decrease time step factor

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• Minimum time step

• Maximum time step

• Initial time step

The nonlinear solver uses the modified Newton-Raphson method and the resolution is based on sparse iterative techniques.

Fig 19: Newton-Raphson resolution in the case of load control technique.

The modified Newton-Raphson method is based on maintaining the tangent matrix for all iterations and can be combined with the line search acceleration technique for accelerating convergence.

Piloting techniques available in RADIOSS:

• Displacement norm control

• Arc-length control

An automatic time step control is used.

Static Analysis and Implicit Options

This example deals with two implicit analyses:

• A static linear computation (loading by gravity),

• A static nonlinear computation (three computations are performed: dummy positioning using an

imposed displacement, followed by a concentrated load and a gravity loading).

An adapted modeling methodology is set up for each analysis. Contact with the different interfaces depends on the computations taken into account and then the material can be updated.

The goal for this analysis is to propose a modeling method for different loading cases, with specific input data used in the implicit strategies. The studies by linear implicit and nonlinear implicit using imposed

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displacement are no longer comparable with results obtained by explicit due to the different physical approaches. Comparisons are only valid for the positioning by gravity loading.

Linear Static Analysis

Type 7 interface uses nonlinear algorithms to check contact. Thus, in order to be used in a linear solver, it is replaced by a type 2 tied interface which creates kinematic conditions between slave nodes and master surfaces. Gravity loading is studied.

Fig 20: Type 2 tied interface linear contact for dummy / seat cushion modeling.

The visco-elastic law 35 (generalized Kelvin-Voigt model) describing the foam of a seat is converted into a linear plastic law 1 (properties are maintained):

• Young’s modulus: 0.2 MPa

• Poisson’s ratio: 0

• Density: 4.3 x 10-11 k g/l

You can select BATOZ formulations for the shell elements and HA8 formulations using 2x2x2 integration points for the brick elements.

The linear implicit methods used are:

Implicit type: Static linear Linear solver: Direct Cholesky Precondition method: Factored Approximate Inverse Stop criteria: Relative residual of preconditioned matrix Tolerance: 10-6

The implicit options used in the Engine file are:

Results

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Only one animation corresponds to the static solution.

Fig 21: Linear static implicit solution of gravity loading (type 2 interface is used).

It should be noted that this modeling contact slightly modifies the problem which is no longer comparable with the previous explicit models.

Table 1: Indication of time computation.

Explicit Solver - /DYREL

Implicit Solver – Linear

Normalized CPU

170 1

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Nonlinear Static Analysis

Positioning Using an Imposed Displacement

The modeling methodology defined in the explicit studies is maintained (visco-elastic material law, type 7 interface, etc.). Brick elements are modeled by default element formulation.

Fig 22: Nonlinear contact modeling with auto-impacting type 7 interface.

In addition to the constant gravity load, an imposed displacement along the Z-axis is applied on the master node of the global rigid body covering the dummy. This approach allows computation to converge and the rigid body modes to be removed (no null pivot). An input curve for the imposed displacement is required. The boundary conditions on master node 14199 are: 110 111.

Fig 23: Imposed displacement along the Z-axis as a monotonous increasing time function.

The nonlinear implicit parameters used are:

Implicit type: Static nonlinear Nonlinear solver: Modified Newton Stop criteria: Relative residual in force Tolerance: 0.01 Update of stiffness matrix: 5 iterations maximum

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Time step control method: Arc-length method and "line-search" Initial time step: 5 s Minimum time step: 0.01 s Maximum time step: no Desired convergence iteration number: 6 Maximum convergence iteration number: 20 Decreasing time step factor: 0.8 Maximum increasing time step scale factor: 1.1 Arc-length: Automatic computation Spring-back option: no

Implicit parameters are set in the Engine file with the options beginning with /IMPL/.

The implicit options used are:

Due to the contact problem, the tolerance value (Tol) is set to 10-2 (default value = 10-3).

Some options are not compatible with the implicit solver. Please refer to RADIOSS Starter Input for more details about implicit options.

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Results

The last animation corresponds to the static solution.

Fig 24: Nonlinear static implicit solution of the imposed displacement.

Note that the Z-displacement of the dummy should not be considered as a result but as an input data (imposed displacement on the master node 14199).

Table 2: Time computation comparison between explicit and implicit computations:

Explicit Solver - /DYREL

Implicit Solver – Nonlinear

Normalized CPU 1.26 1

Number of cycles

(normalized)

56704 (1718) 33 (1)

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Positioning Using a Concentrated Load

The modeling methodology defined in the explicit studies is maintained. The gravity loading is taken into account by applying a constant concentrated load of 813.05N (dummy weight + added masses) on the master node of the rigid body, including the dummy. In order to remove the rigid body modes, the dummy is connected to fixed nodes via type 8 spring elements.

Fig 25: Concentrated load along the Z axis as a monotonous increasing time function.

Fig 26: Springs type 8 defined for removing rigid body modes during implicit computation.

The properties of the general type 8 springs are:

• Linear elastic behavior

• Mass = 1g

• Inertia = 0.001

• Translational stiffness: TX = 1 N/mm

TY = 1 N/mm

TZ = 1 N/mm

• Rotational stiffness: RX = 100 Mg.mm2/(s2.rad)

RY = 100 Mg.mm2/(s2.rad)

RZ = 100 Mg.mm2/(s2.rad)

Implicit options are the same as the previous implicit problem; except for the initial time step is set to: 2s.

Results

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Table 3: Time computation comparison between explicit and implicit computations:

Explicit Solver - /DYREL

Implicit Solver – Nonlinear

Normalized CPU 3.07 1

Number of cycles

(normalized)

56704 (1090) 52 (1)

Z – displacement (master node

dummy)

-12.75 mm -12.49 mm

Positioning Using Gravity Loading

The modeling methodology defined in the implicit model is maintained using a concentrated load. Gravity loading is applied on the slave nodes and the master node of the rigid body, including the dummy. In order to remove the rigid body modes, the dummy is connected to fixed nodes via type 8 spring elements.

Fig 27: Gravity loading as a monotonous increasing time function.

Implicit options are the same as the previous implicit problem (initial time step is set to: 2s).

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Results

Table 4: Time computation comparison between explicit and implicit computations:

Explicit solver - /DYREL

Implicit solver – Nonlinear

Normalized CPU 2.53 1

Number of cycles

(normalized)

56704 (1090) 52 (1)

Z – displacement (master node

dummy)

-12.75 mm -12.42 mm

Fig 28: Convergence results of the X- and Z-displacement of master node 14199 (rigid body dummy) for the implicit models using gravity loading and concentrated load.

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Fig 29: Final dummy position obtained using IMPLICIT (model using gravity loading) and EXPLICIT (model with gravity loading and kinetic relaxation).

Conclusion

This example brings awareness to the use of the RADIOSS implicit solver in resolving quasi-static problems. On the other hand, it illustrates different convergence acceleration techniques when an explicit solver is applied to the quasi-static problems. The advantages and drawbacks of the methods are compared.