Simulation-Based Engineering LabDepartment of Mechanical Engineering

University of Wisconsin–Madison

Technical Report TR-2017-02

Using Chrono::FSI and Chrono::Vehicle for the Analysis of Sloshing

Phenomena in Vehicle Dynamics Maneuvers

Hammad Mazhar, Radu Serban and Dan Negrut

Version: July 10, 2017

Abstract

We present several simulations carried out using Chrono::FSI and Chrono::Vehiclethat highlight the two way coupling between the dynamics of a vehicle and the dynamicsof a fluid that is free to move in a tank rigidly attached to the vehicle; i.e., a vehiclesloshing problem. The simulations are carried out in Chrono [1] using a monolithicsolution; i.e., there is no co-simulation, the entire set of equations of motion are solvedat the same time. The dynamics of the fluid phase is formulated using the massand momentum balance equations, the latter known as the Navier-Stokes equations.The fluid herein is considered inviscid. The vehicle dynamics is formulated using aset of differential complementarity equations, which represent the combination of anindex 3 set of differential algebraic equations with complementarity conditions thatcapture the Coulomb friction model [2]. The problem is discretized in space usinga meshless approach – the Smoothed Particle Hydrodynamics (SPH) method [3, 4].Incompressibility is enforced via kinematic constraint equations, which are stated interms of the velocity of the SPH particles and enforce a constant fluid density constraint[5]. The set of equations is discretized in time using a symplectic half-implicit Eulerscheme [6]. The overall discretized problem leads to a Cone Complementarity Problemthat is solved using a Barzilai-Borwein type method [7].

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Contents

1 Introduction 3

2 Simulation Model 4

3 Constant radius turn 7

4 Double Lane Change 9

5 Driving on Gravel 11

6 Conclusions and future work 13

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1 Introduction

There are three approaches that have been tested in Chrono [1] to handle the fluid-solidinteraction (FSI) problem. What all of them share is a reliance on the Smoothed ParticleHydrodynamics (SPH) method [3,4] for the spatial discretization of partial differential equa-tions that govern the dynamics of the fluid phase. As expected, a spatial discretization ifphysics-agnostic, which means that there is no specified mechanism for handling the incom-pressibility attribute associated with the dynamics of the fluid phase. We have experimentedwith three ways of handling incompressibility in Chrono. The first one, used in [8–10] re-lies on a penalty approach, which uses an equation of state to evaluate the pressure p as afunction of the deviation of the density ρ from a nominal value ρ0:

p =c2sρ0γ

[(ρ

ρ0

)γ

− 1

], (1)

where cs is the numerical speed of sound and γ adjusts the stiffness of the pressure-densityrelationship. We identify this approach as WCSPH, from weakly compressible SPH. The term“weakly” emphasizes the following attribute of this method: owing to the penalty methodused to enforce incompressibility, the value of the density will always fluctuate around thenominal value ρ0. Additionally, depending on the value of the stiffness parameter γ, theoverall spatial discretization of the problem can lead to a stiff ordinary differential problem.This in turn requires very small time integration step to maintain numerical stability.

A second approach enforces the incompressibility condition explicitly; i.e., via an equa-tions of the form:

ρ̇ = 0 , (2)

which is a consequence of the mass balance equation and solenoidal attribute of the flowfor incompressible fluids. Using the SPH approximation is used in Eq. (2) yields a set ofconstraints imposed on the particle velocities. These kinematic constraint equations, whichare of a algebraic nature, are appended to the equations of motion much in the way kinematicconstraint equations are added whenever one encounters a mechanical joint in a multi-bodysystem [11]. We identify this approach as CFSPH, from constrained fluid SPH.

A third approach goes back to the idea of a split integration time step, which follows anapproach proposed for computational fluid dynamics in [12]. SPH variations on this themeare discussed in [13,14], or more recently in [15]. This approach uses an implicit integrationformula, which calls for the solution of a nonlinear algebraic problem at each time step. Weidentify this approach as ISPH, from implicit SPH.

Herein, we are not concerned with numerical solution aspects or a comparison of WCSPH,CFSPH, ISPH. Instead, we use the constraint fluid approach as implemented in Chrono totest its robustness in conjunction with a multi-physics and multi-scale problem: vehicledynamics with fluid sloshing. To this end, we will present three simulations: constant radiusturn, double lane change, and driving on gravel. In all these scenarios, a tank partially filledwith water is attached to the vehicle. These setups lead to a two way coupling between thedynamics of the vehicle and of the fluid in the tank.

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Axle

Upright

Spindle

Lower Control

Arm

Chassis

Upper Control

Arm

Shock

Shaft-Body Connector

3D Rigid Body

Revolute Joint

Spherical Joint

1D Shaft Element

Figure 1: Double wishbone suspension diagram.

2 Simulation Model

For all of the applications presented in this section the a nineteen-body Chrono::VehicleHMMWV model is used. This model consists of a 2086 kg chassis connected to four wheelsthrough a double wishbone suspension, see Fig. 1. This suspension is modeled using fourrigid bodies representing the upper and lower control arms, the suspension upright, andthe spindle. The upper and lower control arms are connected to the chassis using revolutejoints, with a spring damper connected to the lower control arm. The upright is connectedto the upper and lower control arms using spherical joints. An additional revolute jointconnects the upright to the spindle which is then connected to a one dimensional elementrepresenting the axle using a rigid body to shaft joint. Two more bodies and three joints(revolute, universal, and a revolute-spherical composite) are used in the Pitman arm steeringmechanism. The tireods are modeled using distance constraints.

The vehicle is driven by a four wheel drive powertrain model that accurately models thetorque and inertial properties of the engine and the associated shafts in the model. Thedifferentials and gears are modeled using Chrono specialized constraints. Figure 2 shows aschematic of the model and how it connects to the chassis and axles.

The vehicle uses a driver model that locks the steering and attempts to maintain, via aPID controller, a constant speed of 2 m/s; the controlled inputs are the throttle and braking.The PID gains were 4.0 , 1.0 , 0.0 , for the proportional, integral and derivative components,respectively. The four wheel drive powertrain provides better traction when the vehicleenters the fluid and allows the PID controller to maintain the speed of the vehicle in a stablemanner.

The collision geometry used for this vehicle consists of a convex decomposition of thechassis and the lugged wheel. For the chassis the original visual mesh was simplified by

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Front 2 Front 1Central

Diff

Gear Box

CrankEngineChassis

Rear Diff

Rear 2Rear 1FrontDiff

Front Right Spindle

Front Right Spindle

Front Right Spindle

Front Right Spindle

Torque Converter

Driveline

Powertrain

Shaft-Body Connector

3D Rigid Body

Conical Gear

1D Shaft Element

Output

Input

Rear Right Axle

Rear Left Axle

Front Right Axle

Front Left Axle

Figure 2: Four wheel drive powertrain and driveline diagram.

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Figure 3: Top Right: Original mesh used for visualization. Bottom Right: Simplified trianglemesh used for convexification. Bottom Left: Convexified HMMWV mesh.

hand to remove extra geometry and was sliced into several sections for processing by v-hacd,a robust and controllable convex hull generation algorithm [16]. This algorithm generateda set of convex hulls which were used for collision detection against other rigid objects andthe fluid. Figure 3 shows the original mesh, followed by the simplified mesh, and the finalresult of the convex decomposition.

The rigid tank of fluid is attached to the back of the vehicle chassis. Each of the248 889 SPH markers has a mass of 2.39 × 10−3 kg. The fluid rest density was 1000 kg/m3

and the SPH kernel radius was set at 0.016 m. The material drops in the container is sim-ulated for 7 s with a time step of 0.001 s. The fluid is initialized as a cylindrical volume of0.55 m3 and a mass of 550 kg. As the HMMWV is not designed to hold this extra weight thespring stiffnesses for the suspension were increased from 167 062 N/m to 211 099 N/m for thefront, and from 369 149 N/m to 466 455 N/m for the back. These stiffnesses ensure that thesame amount of suspension displacement takes place with the added weight.

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3 Constant radius turn

In this simulation the vehicle drives at a target speed of 35 mph or 15.65 m/s performing aconstant radius turn of 100 ft or 30.48 m. The friction coefficient between the wheels and theground is µ = 1.0 so that the vehicle has enough traction to perform the maneuver.

Figure 4: Vehicle performing a constant radius turn with a tank of water at a velocity of35 mph.

Results for the position and velocity of the vehicle are shown in Figs. 5 - 8. Figure 5shows the position of the vehicle on the xy plane as it drives in a circle.

−40 −35 −30 −25 −20 −15 −10 −5 0 5 10 15 20 25 30 35 40−40

−20

0

20

Pos X [m]

Pos

Y[m

]

Figure 5: The position of the vehicle on the XY plane as it performs the turning maneuver.

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0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

time [s]

Pos

X[m

]

Figure 6: The position of the vehicle in the X direction over time as it performs the turningmaneuver.

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

time [s]

Pos

Y[m

]

Figure 7: The position of the vehicle in the Y direction over time as it performs the turningmaneuver.

0 2 4 6 8 10 12 14 16 18 200

5

10

Time [s]

Velocity

[m/s]

Figure 8: The speed of the vehicle as it performs the turning maneuver over time. Thevehicle it attempting to reach a target speed of 35 mph

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0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

Time [s]

Throttle/B

rake

Figure 9: Normalized throttle of the vehicle as it performs the turning maneuver over time.

4 Double Lane Change

Figure 10: Vehicle performing a double lane change with a tank of water at a velocity of15 mph

In this simulation the vehicle drives at a target speed of 15 mph or 6.7 m/s performinga double lane change maneuver – first turning to the left and then the right traveling adistance of 295.276 ft or 90 m. At 6 s the vehicle turns left and travels a distance of 5 m,straightens out and then immediately turns back to the original lane it was traveling in. Thefriction between the wheels and the ground was µ = 1.0

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0 2 4 6 8 10 12 14 16 18 20

−40

−20

0

20

40

time [s]

Pos

X[m

]

Figure 11: X position of vehicle during double lane change maneuver.

0 2 4 6 8 10 12 14 16 18 20

−126

−124

−122

−120

time [s]

Pos

Y[m

]

Figure 12: Y position of vehicle during double lane change maneuver.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

Time [s]

Velocity

[m/s]

Figure 13: Velocity of vehicle during double lane change maneuver.

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0 2 4 6 8 10 12 14 16 18 20

−1

−0.5

0

0.5

1

Time [s]

Throttle/B

rake

Figure 14: Normalized throttle and brake vehicle inputs during double lane change maneuver.

5 Driving on Gravel

In this simulation the vehicle drives over a bed of gravel at a speed of 10 mph or 4.47 m/s.The size of the gravel bed is 8 m × 3 m × 0.2 m and it contains elements with a density of2500 kg/m3; the element geometry is two connected spheres with radii of 0.02 m. The joinedspheres allow the elements to interlock. This interlocking is important since it preventsthe vehicle from “sinking” into the soil when the wheels press into the gravel. We haveobserved this sinking in situations where the elements were simple spheres. Note that thefriction coefficient between the gravel particles is µ = 1.0 to approximate the rough contactinteraction at the gravel-gravel interface.

Figure 15: Vehicle driving on gravel with a tank of water at a velocity of 10 mph.

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0 2 4 6 8 10 12 14 16 18 20

0

5

10

time [s]

Pos

X[m

]

Figure 16: X position of the vehicle as it drives over the gravel pit.

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

Time [s]

Velocity

[m/s]

Figure 17: Velocity of vehicle as it drives over the gravel pit.

0 2 4 6 8 10 12 14 16 18 20

−1

−0.5

0

0.5

1

Time [s]

Throttle/B

rake

Figure 18: Normalized throttle and brake inputs for vehicle driving over gravel.

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6 Conclusions and future work

We showcased the use of Chrono in a collection of simulations that pertain to the fluid-solidinteraction problem. Reflecting our lab’s interest in vehicle dynamics, we highlight howChrono can draw on its Chrono::FSI and Chrono::Vehicle modules to enable the simulation ofsloshing phenomena in vehicle dynamics studies. The most challenging test case is the onewhere the vehicle operates on granular terrain. This represents a multi-physics, multi-scaleproblem. Indeed, we are handling in one monolithic solution multiple phases – fluid andsolid – whose dynamics are coupled. Moreover, friction and contact come into play as thevehicle moves on granular terrain. Secondly, this application has a multi-scale attribute – inthe same set of equations we handle the dynamics of a 2500 kg vehicle and the dynamics ofnumerous gravel elements. The inertia properties of the problem components span four tofive orders of magnitude.

In terms of future work, it remains to understand to what extent the results obtainedusing Chrono match field measured data. At this time, we do not have any experimental datato validate the simulations discussed herein. In lieu of validation data for vehicle mobilitystudies, we have validated the numerical solution via a sequence of basic test cases suchas incompressibility, dam break, and sloshing experiments. The results of this validationeffort, which are reported in an upcoming paper [5], provide an encouraging first assessmentof the accuracy of the constrained fluid approach used herein. Finally, in the context offluid-structure interaction, an ongoing but parallel effort is under way to understand how anSPH approach, like the one embraced here, compares against the immersed boundary (IM)method [17] and the arbitrary lagrangian-eulerian (ALE) method [18].

References

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//sbel.wisc.edu/documents/TR-2016-12.pdf, Simulation-Based Engineering Labo-ratory, University of Wisconsin-Madison, 2016.

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[4] L. B. Lucy, “A numerical approach to the testing of the fission hypothesis,” The Astro-nomical Journal, vol. 82, pp. 1013–1024, 1977.

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[5] H. Mazhar, A. Pazouki, M. Rakhsha, P. Jayakumar, and D. Negrut, “A differentialvariational approach for handling fluid-solid interaction problems via Smoothed ParticleHydrodynamics,” Journal of Computational Physics (under review), vol. 0, p. 0, 2017.

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[18] J. Donea, S. Giuliani, and J.-P. Halleux, “An arbitrary lagrangian-eulerian finite ele-ment method for transient dynamic fluid-structure interactions,” Computer methods inapplied mechanics and engineering, vol. 33, no. 1-3, pp. 689–723, 1982.

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