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A co-simulation framework for high-performance, high-fidelity simulation of ground vehicle-terrain interaction Radu Serban, Nicholas Olsen, and Dan Negrut University of Wisconsin - Madison Madison, WI 53706, USA [serban,nicholas.olsen,negrut]@wisc.edu Antonio Recuero The Goodyear Tire & Rubber Co. Akron, OH 44316, USA antonio [email protected] Paramsothy Jayakumar U.S. Army TARDEC Warren, MI 48397, USA [email protected]

ABSTRACT Assessing the mobility of off-road vehicles is a complex task that most often falls back on semi-empirical approaches to quantify the tire-terrain interaction. In this paper, we introduce a high-fidelity ground vehicle mobility simulation framework that uses physics-based models of the vehicle, tires, and terrain to factor in both tire flexibility and soil deformation. The tires are modeled using a nonlinear finite element approach that involves layers of orthotropic shell elements. The soil is represented as a large collection of rigid elements that mutually interact through contact, friction, and cohesive forces. The high-fidelity vehicle models incorporate suspension, steering, driveline, and powertrain models, and can be driven through driver inputs to their appropriate subsystems. To alleviate the prohibitive computational costs that a coupled simulation of the overall problem (resulting in a multi-physics, multi-scale dynamical system with millions of degrees of freedom), we proposed a decoupled approach implemented as an explicit, force–displacement co-simulation framework. The resulting software capability is built upon and available in the open-source, multi-physics software package Chrono [1, 2]. We present details on the modeling and ensuing physical models of all involved components; provide an overview of the relevant Chrono capabilities leveraged here, including a discussion on parallel computing aspects; and describe the co-simulation framework which is developed over a distributed, MPI-based layer. We demonstrate the ability of the resulting simulation capability in providing high-fidelity mobility analysis results at a computational cost considerably smaller than that of a fully-coupled simulation, by means of several representative full-vehicle simulations on soft soil, including a straight-line acceleration test, obstaclecrossing scenario, and a constant radius turning maneuver. STO-MP-AVT-265 UNCLASSIFIED: Distribution Statement A. Approved for public release; distribution is unlimited. #28937

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PHYSICS-BASED GROUND VEHICLE MOBILITY ANALYSIS

Vehicle dynamics simulations are often focused on modeling a number of physical phenomena of one single system. Some approaches aim to model the reaction of some vehicle’s degrees of freedom in the presence of prescribed irregularities, or the influence of a variety of terrains on the steady state response of a wheel. These approaches shed light on the specific mechanisms that drive some physical phenomena. Sometimes, however, the range of physical interactions happening is broad and complex, in scale as well as in nature. In these cases, a combination of formulations and computer implementations are needed to model and analyze the coupled dynamics of these systems, e.g., the vehicle and the terrain. In vehicle–terrain interaction, not only the flexibility of the tires, the deformation of the terrain, and the suspension geometry play important roles, but also the interaction among them. In particular, this type of approach where the interface between the vehicle and the ground is modeled in more detail is of importance in the field of terramechanics. Indeed, key to the interest of the mobility engineer is the ability to predict whether a vehicle under certain physical conditions is able to negotiate soft soil. Empirical and semi-empirical approaches are available to carry out such a go/no–go type predictions, but they preclude, or only roughly account for, a multitude of aspects, including vehicle and terrain dynamics, suspension geometry, tire stiffness and footprint, and many others. Recently, computational methods which rely on semi-empirical approaches have been developed for the modeling of tire mobility. Senatore and Sandu [3] use such methods to predict multipass effects on vehicle mobility. Other tests on tire–deformable ground dynamics [4] have shown the effects of parameter variation on mobility metrics for a single tire. Some trends can be recognized: The drawbar pull coefficient increases with soil compaction, but decreases for larger inflation pressure and larger loads. Other studies have demonstrated that the drawbar pull peaks at a larger longitudinal slip on granular terrain compared to tires traveling on rigid ground [5]. These complex terrain effects require the consideration of the granular behavior of soil, which can be modeled via discrete element method (DEM) and be tuned via physical parameters, such as cohesive and friction parameters. Soil modeling for terramechanics application has also been the subject of validation efforts, either using discrete elements or Lagrangian finite elements [6, 7]. Along these lines, soft soil contact models were proposed to alleviate the numerical burden of soil models in multibody simulations [8]. Tire-wise, detailed finite element models of single tires were also used for the analysis of mobility [9]. Small deformation finite element formulations that take advantage of the symmetry of tire geometry have also been presented [10]. Other multibody system approaches to tire flexibility involve finite strain shell elements, such as those based on the absolute nodal coordinate formulation (ANCF) [11] and on geometrically exact formulations [12]. As evidenced by the literature, there are several levels of fidelity in the available methods. Nonetheless, there seems to be a lack in the state of the art of higher-fidelity vehicle–terrain simulations. Here, we propose an approach where the dynamics of a vehicle model, modeled as a three-dimensional detailed multibody system model, the tires, represented by finite strain multi-layered, orthotropic shell elements, and the soil, modeled as millions of cohesive elements, interact mutually giving rise to a framework that minimizes empirical assumptions and parameters. This framework allows for obtaining numerical mobility measures of practical use, such as sinkage, drawbar pull, terrain resistance, and required driver’s inputs. These mobility measures can be naturally retrieved as reaction forces at mechanical joints, as contact forces, and via controllers. This detailed modeling of vehicles running over deformable terrain can be key to the assessment of whether a vehicle is able to successfully negotiate certain terrain and its corresponding maximum speed. This framework thus follows the proposed directions for the development of an updated NATO Reference Mobility Model (NRMM) [13]; i.e., the Next Generation NRMM (NG-NRMM), which will draw on more powerful and varied computer architectures [14]. The entire numerical framework is developed in Chrono [1, 2], a multi-physics open-source software package. The high-fidelity deformable-terrain vehicle mobility analysis in Chrono leads to a multi-physics, multiscale problem in several millions degrees of freedom. To solve this and similar problems arising in multibody dynamics, finite element procedures, and granular dynamics, Chrono leverages parallel computing at several 12 - 2

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levels: (i) x86-AVX acceleration; (ii) multi-core, shared memory OpenMP [15]; (iii) GPU-based parallel computing using CUDA [16]; and, (iv) distributed parallel computing via MPI [17]. Specifically, the nonlinear FEA component, which is computationally taxing when evaluating the tire internal force and Jacobian, is accelerated with OpenMP directives. Granular dynamics, including both the collision detection phase and the numerical solution of the discretized equations of motion, can be performed in parallel either on the GPU, or in a multi-core shared-memory OpenMP fashion, or else in a distributed MPI framework. Despite the attempts to exploit the available computer power, this higher-fidelity approach faces the burden of long wall-clock run times. To address these computational challenges posed by coupled, all-at-once simulations, we adopt here a splitting of the problem, in different sub-systems (from both physical and computational perspectives), into an explicit force–displacement co-simulation framework. This paper is organized as follows. Vehicle, tire, and soil formulations are described in Section 2. The design and implementation of the co-simulation frameworks for the simulation of single tires and off-road vehicles are presented in Section 3. Representative full-vehicle simulations on soft soil are discussed in Section 4, where a straight-line acceleration scenario, a constant radius turn maneuver, and an obstacle negotiation are described and analyzed. Section 5 outlines the conclusions of this investigation and the necessary future work.

2

VEHICLE AND TERRAIN MODELING

The physics modeling and simulation capabilities are provided by the multiphysics open-source package Chrono [1]. The core functionality of Chrono provides support for the modeling, simulation, and visualization of rigid multibody systems, with additional capabilities offered through optional modules. These modules provide support for additional classes of problems (e.g., finite element analysis and fluid-solid interaction), for modeling and simulation of specialized systems (such as ground vehicles and granular dynamics problems), or provide specialized parallel computing algorithms (multi-core, GPU, and distributed) for large-scale simulations.

2.1

Vehicle multi-body modeling

Built as a Chrono extension module, Chrono::Vehicle [18] is a C++ middleware library focused on the modeling, simulation, and visualization of ground vehicles. Chrono::Vehicle provides a collection of templates for various topologies of both wheeled and tracked vehicle subsystems, as well as support for modeling of rigid, flexible, and granular terrain, support for closed-loop and interactive driver models, and run-time and off-line visualization of simulation results. Modeling of vehicle systems is done in a modular fashion, with a vehicle defined as an assembly of instances of various subsystems (suspension, steering, driveline, etc.). Flexibility in modeling is provided by adopting a template-based design. In Chrono::Vehicle, templates are parameterized models that define a particular implementation of a vehicle subsystem. As such, a template defines the basic modeling elements (bodies, joints, force elements), imposes the subsystem topology, prescribes the design parameters, and implements the common functionality for a given type of subsystem (e.g. suspension) particularized to a specific template (e.g. double wishbone). For wheeled vehicle systems, templates are provided for the following subsystems: suspension (double wishbone, reduced double wishbone using distance constraints, multi-link, solid-axle, McPhearson strut, semi-trailing arm); steering (Pitman arm, rack-and-pinion); driveline (2WD and 4WD shaft-based using specialized Chrono modeling elements, simplified kinematic driveline); wheel (simply a carrier for additional mass and inertia appended to the suspension’s spindle body and, optionally, visualization information); brake (simple model using a constant torque modulated by the driver braking input). Chrono::Vehicle offers a variety of tire models and associated templates, ranging from rigid tires (with either a cylindrical contact shape, or a mesh defined in a Wavefront input file), to empirical and semiSTO-MP-AVT-265

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empirical models (such as Pacejka [19] and Fiala [20]), to fully deformable tires modeled with finite elements (using either an ANCF formulation [21], or else a co-rotational formulation). The latter FEA tire models allow for detailed specification of tire geometry and material properties, as well as specification of flexible 3D tire tread patterns. For modeling tracked vehicles, templates are provided for suspension, road wheels, sprockets, track shoe assemblies, and idler (with tensioner). Different suspension configurations are available, including torsion spring with linear or rotational dampers and a hydropneumatic suspension template, and both single- and double-pin track shoe models are supported. Sprocket profiles are defined as 2D curves (parameterized differently for sprockets engaging single- and double-pin track shoes) and the sprocket-track shoe contact is processed with a custom collision detection algorithm. Chrono::Vehicle provides functionality for automatic assembly of the track over the vehicle’s wheels to eliminate the burden of consistent initialization of all track shoe bodies. For additional flexibility and to facilitate inclusion in larger simulation frameworks, Chrono::Vehicle allows formally separating various systems, such as the vehicle itself, powertrain, tires, terrain, driver, and provides the inter-system communication API for a co-simulation framework based on force-displacement couplings. For consistency, these systems are themselves templatized: vehicle (a collection of references to instantiations of templates for its constitutive subsystems, with specific versions provided for wheeled and tracked vehicles); powertrain (shaft-based template using an engine model based on speed-torque curves, torque converter based on capacity factor and torque ratio curves, and transmission parameterized by an arbitrary number of forward gear ratios and a single reverse gear ratio); driver (interactive driver model using inputs from keyboard for real-time simulation, file-based driver model using interpolated driver inputs as functions of time, closed-loop driver models using PID controllers for path following). Various approaches for terrain modeling are supported in Chrono::Vehicle. The parameterized template for rigid terrain allows specification of a flat profiles, arbitrary geometry specified as a Wavefront format mesh object, or else profiles constructed from height-field information provided as gray images. We provide support for deformable terrain at various degrees of accuracy and computational efficiency. An expeditious, but lower accuracy option is given by the Chrono::Vehicle extension of the SCM (Soil Contact Model) technique [22] in which contact forces between the soil and the vehicle components (tires or track) are computed based on Bekker’s empirical terramechanics formulae [23]. At the other extreme, Chrono::Vehicle can be easily interfaced to the granular dynamics support in Chrono to allow simulations of ground vehicles over granular terrain using either a compliant- or a rigid-body approach to the frictional contact problem [1]. An alternative high-fidelity approach to modeling deformable terrain is provided by templates for specifying FEA terrain patches, leveraging the support provided in Chrono::FEA. Chrono::Vehicle provides visualization support both for run-time, interactive simulations (an extension of the Chrono Irrlicht optional module) and for high-quality post-processing rendering (using for example the POV-Ray ray-tracing package, see Fig. 1).

2.2

Deformable tire modeling

The nonlinear finite element tire is modeled with volumetric shell elements using an ANCF approach that can describe translations, rotations, and finite deformation using Lagrangian coordinates referred to an inertial frame [24]. The four-node, bilinear, and continuum mechanics-based shell element used here was recently developed in [25] and validated for tire dynamics applications in [26, 11]. The element geometry is described by the interpolation of a position vector and a position vector gradient defined to be oriented along the fiber direction at each of the four nodes. An arbitrary point in a shell element i is defined as a ∂ri i i function of the global position of the nodes, ri , and its transverse gradient vector, riz = ∂z i (x , y ). The position and cross section orientation of the element are expressed in terms of the position of its ∂ri i i i i mid plane, rim , and a gradient vector defining shear deformation, ∂z i (x , y ). Vectors ep and eg gather the element’s position and gradient coordinates, which yield the position and orientation of an arbitrary point 12 - 4

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Figure 1: Chrono::Vehicle visualization of an off-road vehicle with POV-Ray. (xi , y i ) as follows: rim (xi , y i ) ∂ri (xi , y i ) ∂z i

= Sim (xi , y i )eip = Sim (xi , y i )eig ,

(1)

where xi and y i refer to element i ’s local coordinates in the parametric space, Sim = [S1i I S2i I S3i I S4i I] is a shape function matrix, S1i ... S4i are bilinear shape functions, and I, is the 3×3 identity matrix [25]. The position of an arbitrary point in the shell may then be described as ri (xi , y i , z i ) = Si (xi , y i , z i )ei , where the resulting shape function matrix is given by Si = [Sim z i Sim ] and the final coordinate vector of element i is defined as ei = [(eip )T (eig )T ]T .    T Using these element kinematics, the Green-Lagrange strain tensor becomes Ei = 12 Fi Fi − I , where Fi is the deformation gradient tensor defined as the partial derivation of the current configuration, r, with respect to a reference configuration, X. The deformation gradient tensor can be defined as Fi =  −1 ∂ri ∂ri ∂Xi = , where the initial, possibly distorted, configuration of the tire profile, Xi , is considered i i i ∂X ∂x ∂x as undeformed (reference). Numerical techniques are used to alleviate shear and thickness lockings in the element. Namely, assumed natural strain and enhanced strain formulations, which find justification in the mixed variational principle by Hu–Washizu [27], are added to guarantee accuracy. The computer implementation of this element allows defining an arbitrary number of layers of orthotropic materials. For the numerical results presented in this investigation, simplified tire material properties are used [28]. Lumping of orthotropic layers is sometimes used [29, 30] to reduce the computational cost of the tire material forces, but it has limitations as to the description of transverse shear properties [31]. Details on this element’s software implementation in Chrono and its verification are provided in Refs. [32, 33].

2.3

Granular dynamics and terrain modeling

In this work, the focus is on modeling and simulating the terrain and the tire-terrain interaction using highfidelity, fully-resolved granular dynamics simulations, employing the so-called Discrete element Method (DEM). Meaningful mobility simulations require large enough terrain patches and small enough particle dimensions that result in DEM problems involving frictional contact with millions of degrees of freedom. Broadly speaking, computational methods for DEM at this scale can be categorized into two classes: penalty-based (also known as a compliant-body approach; denoted here by DEM-P) and complementaritybased (also known as a rigid-body approach; denoted here by DEM-C). While differing in the underlying formulation employed for modeling and generating the normal and tangential forces at the contact interface, STO-MP-AVT-265

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and thus leading to different mathematical models and different problem sizes, both methods rely crucially on efficient methods for proximity calculation. This common algorithmic step provides a complete geometric characterization of the interaction between neighboring bodies, taking into account the current system state and specification of the contact shapes associated with all interacting bodies. An overview of these two formulations is provided in the remainder of this section. As described below, the DEM-P and DEM-C approaches are very unlike each other: a local vs. global view of the frictional contact interaction, compliant vs. rigid treatment of bodies for contact purposes, force-acceleration vs. impulse-momentum formulation of the resulting equations of motion. They each have their advantages, as well as shortcomings. Backed by a large body of literature and widely used in commercial DEM software packages, DEM-P has the attractive features of not requiring additional system states and allowing straightforward implementations that are easily scalable, as the numerical solution remains decoupled. However, they typically require small integration step-sizes, due to the resulting stiff differential equations and pose difficulties in identifying appropriate model parameters, especially for large heterogeneous systems, as well as in treating shapes with non-spherical geometry. On the other hand, DEM-C solutions permit integration with larger step-sizes, as they do not suffer from limitations due to numerical stability, rely on a small number of model parameters (effectively only the friction coefficient and cohesion properties), and have no underlying assumptions on shape geometry. But DEM-C requires a much more involved and expensive solution of an optimization problem at each step, increases the problem size by a significant factor, and, especially when friction is present, lack a unique solution (a direct consequence of the rigid-body assumption, see for example [34]). Penalty-based frictional contact. Penalty methods begin with a relaxation of the rigid body assumption [35, 36]. A regularization approach, DEM-P assumes local body deformation at the contact point. Employing the finite element method to characterize this deformation would incur a stiff computational cost. Instead, an approximation is employed, using information generated during the collision detection stage of the solution, and the local body deformation is related to the depth of inter-penetration between two otherwise rigid contact shapes. In order to apply results from Hertz contact theory, valid for spheresphere interactions, the contact shapes are further approximated by their local radius of curvature at the contact point [37]. This approach yields a general methodology for computing the normal and tangential forces at the contact point. As an example, a viscoelastic model based on Hertzian contact theory takes the form p ¯ n (Kn δn − Cn mv Fn = pRδ ¯ n) (2) ¯ Ft = Rδn (−Kt δ t − Ct mv ¯ t) , for the normal (n) and tangential (t) directions, respectively. Here, δ is the overlap of two interacting bodies; ¯ and m R ¯ represent the effective radius of curvature and mass, respectively; and v is the relative velocity at the contact point [38]. For the materials in contact, the normal and tangential stiffness and damping coefficients Kn , Kt , Cn , and Ct are obtained, through various constitutive laws, from physically-measurable quantities, such as Young’s modulus, Poisson ratio, and the coefficient of restitution [37, 39]. For granular dynamics via DEM-P, the equations of motion need not be changed. Indeed, Fn and Ft are treated as any external forces and directly factored in the momentum balance. For details on the specific DEM-P implementation in Chrono, see [40]. Complementarity-based frictional contact. The DEM-C class of methods for granular dynamics fully embraces the rigid body assumption and formulates the frictional contact problem by imposing nonpenetration conditions which result in unilateral constraints between each pair of interacting shapes. The derivation begins by defining a so-called gap function Φi which describes the separation between the two shapes involved in a potential contact i in the current active contact set. Then, a complementarity condition is imposed for contact i stating that either the distance between the two shapes is greater than zero, in which case the normal force is zero, or vice-versa; i.e., if the distance is zero, the contact force is nonzero. 12 - 6

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This models the disjunctive nature of unilateral contact constraints: if contact supports load, then contact is maintained; if contact breaks, then the contact force vanishes. The Coulomb friction model is posed via the principle of maximum dissipation (PMD) [41], which for contact i involves the tangential friction force components and the relative motion of the two bodies in contact. Also including bilateral constraints, modeling the usual kinematic joints in multibody dynamics, expressed here through algebraic conditions of the form Ψi = 0, this leads to a differential variational inequality (DVI) problem [42]: q˙ = L(q)v P P Mv˙ = f (t, q, v) + γ bi,b ∇Ψi + (b γi,n Di,n + γ bi,u Di,u + γ bi,w Di,w ) i∈B

i∈A

i ∈ B : Ψi (t, q) = 0 i∈A : 0≤γ bi,n ⊥ Φi (q) ≥ 0 (b γi,u , γ bi,w ) = √ argmin vT (x Di,u + y Di,w )

(3)

x2 +y 2 ≤µi γ bi,n

The first condition in Eq. (3) relates, through a linear transformation L, the generalized positions q and velocities v, while the force balance in the second equation ties the inertial forces, involving the generalized mass matrix M to the applied external forces f , the joint reaction forces due to the bilateral constraints in the set B (involving the Lagrange multipliers γ bi,b ), and the contact and friction forces for all contacts in the set A, expressed here in terms of the projectors Di,n , Di,u , and Di,w . The joint constraints, represented as a set of algebraic equations, and the non-penetration conditions, expressed as complementarity conditions, comprise the third and fourth equations, respectively. Finally, the optimization problem in the last equation of (3) expresses the PMD for isotropic Coulomb friction, where µi is the coefficient of friction for contact i. See [43] for additional details on the DEM-C formulation in Chrono. The numerical solution of the resulting problem is challenging and continues to be an area of active research. The numerical method adopted in Chrono is introduced in [34]. Upon time discretization followed by a relaxation of the complementarity constraints, the numerical problem is posed as a conically constrained quadratic optimization problem, which is subsequently solved using projected Barzilai-Borwein or Nesterov algorithms [44]. This step represents the computational bottleneck of the DEM-C approach, as it involves the solution of a large optimization problem at every simulation time step.

2.4

Parallel computing aspects in large-scale granular dynamics

Shared memory parallel granular dynamics. Support for shared-memory parallel computing for large-scale granular dynamics application is provided through the optional module Chrono::Parallel, using OpenMP directives and the Thrust library with the OpenMP back-end. The underlying design philosophy of Chrono::Parallel is that it relies on the core Chrono library and API for all its modeling capabilities, but uses different data structures and algorithms to facilitate and leverage OpenMP parallelization. More precisely, Chrono::Parallel replaces the array-of-structures (AoS) data representation in Chrono with a structure-ofarrays (SoA) data layout which is more suitable for efficient loop parallelization, as well as allowing the use of packed SIMD instructions on modern CPUs. Chrono::Parallel relies on several external libraries, including OpenMP [15] (for parallel for loops), Thrust [45] (for parallel sort, scan, and reduction algorithms), and Blaze [46] (for high-performance sparse matrix representation). Chrono::Parallel supports a large set of the modeling elements available in Chrono; in particular, these include rigid bodies with frictional contact (both penalty- and complementarity-based), arbitrary contact shapes, kinematic joints, force elements (springs, actuators, etc.), as well as 1-D shaft elements required for simulation of complete Chrono::Vehicle models of both wheeled and tracked vehicle systems. Most DVI solvers in Chrono have counterpart parallel implementations in Chrono::Parallel, including Barzilai-Borwein STO-MP-AVT-265

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and Nesterov-type, but the only time stepping currently available is a semi-implicit Euler scheme. Because of this latter restriction, Chrono::FEA finite-element models cannot be currently tackled with Chrono::Parallel. To facilitate the accurate modeling of particulate or granular materials used in Chrono simulations, a module called Chrono::Granular is available to provides pre-processing tools for specification of granular materials with either preset or user-defined distributions for particle size and shape, physical properties, and contact material characteristics. Chrono::Granular also includes methods for efficient sampling of spatial regions (prismatic, cylindrical, spherical) using different point arrangements, including regular grids, hexagonal close-packed, as well as Poisson disk sampling to produce tightly-packed points with guaranteed minimum separation.

Distributed memory parallel granular dynamics. For large granular problems that would otherwise require prohibitively long simulation times, Chrono provides an optional module for MPI [17] distributedmemory simulation. Chrono::Distributed provides an additional level of parallelization on top of the framework of Chrono::Parallel in the form of a hybrid distributed/shared memory programming model, in particular an MPI/OpenMP approach. The module is designed to utilize high-performance computing clusters with powerful nodes to efficiently simulate systems with billions of degrees of freedom. Chrono::Distributed splits the domain of a large-scale granular system and assigns each node of the cluster to perform computations only on bodies in its sub-domain. Within each MPI rank, Chrono::Distributed closely mirrors the shared-memory algorithms from Chrono::Parallel for collision detection and numerical integration using OpenMP. Minimal data is sent between nodes with MPI to update proxies for bodies on the border of two sub-domains. This module exclusively uses DEM-P because the method’s decoupled nature allows for a clean division of work between nodes. The primary intention of this module is to provide support for high-resolution terrain simulation for vehicle-ground interaction testing.

Parallel collision detection. Collision detection (proximity calculation) is a critical algorithmic step in both the DEM-P and DEM-C approaches to large-scale granular dynamics simulations. The role of collision detection is to identify all (potential) collisions at a given time and then geometrically characterize the interaction of the identified colliding shape pairs by calculating closest points, normals and radii of curvature at contact points, etc. A naive approach to collision detection, falling back on testing all pairs of contact shapes in the system, has O(n2 ) complexity and is therefore unsuitable for anything but very small systems. To address the issue of computational efficiency, most collision detection systems use (at least) a two-phase process. In a first phase, so-called broad-phase, pairs of shapes that cannot be interacting are quickly discarded; this phase relies on bounding shapes (spheres, axis-aligned bounding boxes, or objectoriented bounding boxes) and specialized data structures (such as dynamic trees or hierarchical grids). In a second narrow-phase, the candidate pairs identified by the broad-phase are then fully characterized from a geometrical point of view, using either analytical methods for shapes with simple geometry, or else specialized algorithms such as Gilbert-Johnson-Keerthi (GJK) or Minkovski-Portal Refinement (MPR) for general convex shapes. Chrono::Parallel implements a binning algorithm for the broad-phase (see [47] for implementation details), suitable for both multi-core and GPU shared-memory parallel environments, and analytical algorithms with a fall-back on MPR for the narrow-phase collision detection. Faithful to its design principles, Chrono::Distributed does not require an additional, specialized distributed collision detection algorithm. Instead, with appropriate replication of shared bodies on neighboring MPI ranks, the Chrono::Parallel shared memory parallel collision detection described above can be used, independently and in parallel, on all cluster nodes. 12 - 8

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3

CO-SIMULATION FRAMEWORK FOR MOBILITY ANALYSIS

To address the inherent multi-scale nature of the off-road mobility problem, Chrono has an additional cosimulation layer which decouples the vehicle, tire, and terrain subsystems. This approach allows further acceleration of the simulation by (i) allowing each subsystem to advance its state using a suitable integration time step; (ii) using different integration schemes (e.g., an implicit, adaptive HHT scheme for FEA tires and a semi-implicit Euler scheme for the granular terrain), as called for by the particular dynamics problem; and (iii) leveraging different and independent parallelization techniques, as dictated by the structure of each subsystem. Owing to its modular design, the co-simulation framework facilitates the swapping between a shared memory, OpenMP-based simulation of the granular terrain (using the Chrono::Parallel module), and an MPI-based distributed terrain simulation (provided by the Chrono::Distributed module). To allow different and complementary types of mobility analyses, we provide two incarnations of this co-simulation framework. The first one is a model of a single-tire test rig and uses two subsystems, while the second one targets full-vehicle mobility analysis and separates the vehicle, the terrain, and each tire into a separate co-simulation subsystem.

3.1

Single-tire test rig

A first co-simulation framework, tailored for a tire test rig, is a two-way co-simulation scheme in which two MPI nodes control the simultaneous and independent simulation of the rig and tire on the one hand, and the granular terrain, on the other hand. The two nodes communicate through MPI messages the required synchronization data, namely positions and velocities of the tire FEA mesh in one direction, and tire contact forces at mesh vertices in the opposite direction. The rig mechanism, which is composed of a sequence of rigid bodies, joints, and actuators, enables one to control the toe angle, tire angular velocity, and rig linear velocity. The separation of systems, communication pattern, and independent simulation loops for the tire test rig are illustrated in Fig. 2. For a complete description of the co-simulation single-tire test rig software framework, see [28].

RIG node Creation rig mechanism + tire

MPI Tire contact material properties

TERRAIN node Creation container + (granular material)

Settling phase (granular material)

Simulation loop

Terrain initial (settled) height Initialization Initialize tire above terrain

Initialization Create proxy bodies Tire contact surface specification Mesh vertex positions and velocities

Simulation loop

Synchronize

Synchronize Vertex indices and contact forces

Advance

Simulation loop

Advance

Figure 2: Co-simulation framework for tire rig–terrain system. The dynamics of the two component subsystem are advanced in time independently and in parallel, in an explicit force–displacement co-simulation setup.

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3.2

Full vehicle

The second co-simulation setup supports full-vehicle mobility simulations. As illustrated in Fig. 3, it implements a three-way communication mechanism. The MPI process controlling the vehicle subsystem provides wheel states and receives cumulative tire forces at the wheel center to/from all tire nodes. The tire MPI processes communicate with the terrain node to exchange FEA tire mesh vertex states, in one direction; and tire-terrain normal and tangential contact forces, in the opposite direction. The simulation of the granular terrain and the tire-terrain interaction can be conducted either on one multi-core shared-memory node using OpenMP threads, or else can be dispatched to a distributed memory HPC cluster, organized in a separate MPI communicator.

Figure 3: Co-simulation framework for full vehicle–terrain system. The simulation is separated in three component classes (vehicle, tires, terrain) and a 3-way co-simulation scheme is employed for synchronization.

The full-vehicle co-simulation framework implements an explicit force–displacement co-simulation scheme between the following MPI processes: [vehicle] A single MPI process is responsible for a sequential integration of the dynamics of the vehicle model which includes the chassis, powertrain, driveline, steering and suspension subsystems, up to the wheel spindles. At initialization, the vehicle node receives the terrain height at the initial vehicle location and sends to each tire node the initial state of the corresponding wheel spindle body. At synchronization, the vehicle process communicates only with the tire MPI processes, as described below. [tire] For simulating the ANCF deformable tires, the system launches a number of MPI processes equal to the number of vehicle wheels. At system initialization, each tire node sends to the terrain MPI process information on the tire FEA mesh topology (number of FEA nodes and element connectivity), as well as tire contact material properties. The tire nodes are at the core of the explicit force–displacement co-simulation scheme adopted here. Indeed, at each synchronization time, each tire node sends to the vehicle node a generalized tire force (force and moment as applied to the center of the associated wheel) and receives the current wheel body status (linear and angular position and velocity). Concurrently, each tire node sends to the terrain node current mesh vertex locations and velocities and receives from the terrain node a (variable-length) array of resultant contact forces to be applied to the tire FEA mesh vertices. 12 - 10

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[terrain] A single MPI process is responsible for orchestrating the simulation of the granular terrain. This single process may in turn spawn a new MPI sub-communicator for a distributed-memory granular dynamics simulation with Chrono::Distributed. During initialization, an optional dynamic settling phase may be performed, before the vehicle and tires are instantiated. At synchronization, the terrain process communicates only with the tire MPI processes, as described above. Since the relative time spent in data exchange through MPI communication is small (see Section 4), the co-simulation approach significantly speeds up the overall simulation by reducing the effective computer time to, roughly, that of the slowest MPI process. Depending on the size of the terrain patch and particle dimensions, this can be either the terrain node (performing a granular dynamics simulation) or one of the tire nodes (performing an FEA dynamic analysis).

3.3

Software implementation considerations

The subsystems in the co-simulation framework are implemented as MPI processes, each of them responsible for simulating one of the above components. Communication at synchronization times is performed through exchange of MPI messages of relatively small size; as such, no effort has been made to use asynchronous communication, instead relying on plain-vanilla MPI Send and MPI Recv functions. Indeed, even in the extreme case where inter-process communication occurs at the high-frequency rate corresponding to the small integration step sizes required by the FEA solution on the tire nodes (∼ 10−5 s), the cost of all non-computational operations is approximately 5% of the total wall clock time and this includes both MPI communication as well as all disk I/O operations. The architecture and design of the MPI co-simulation solution presented here were also modulated by the assumption that relatively powerful hardware nodes will be available. In other words, it is assumed that each MPI process can run on a powerful enough processor (e.g., multi-socket, with 20 hardware threads) to enable a hybrid MPI/OpenMP parallel computation model. With such a setup, a tire FEA computation can be locally accelerated by using parallel OpenMP threads in the internal force and Jacobian evaluations, while the granular terrain process can leverage the OpenMP parallel implementation in Chrono::Parallel. Additional built-in capabilities include the ability for check-pointing to disk files the results of the granular terrain simulation (important for multi-pass effect investigations) and support for data collection from each MPI process separately (for post-processing and visualization). Run-time visualization support is provided through the Chrono::OpenGL optional module in Chrono, while post-processing rendering can be generated with either POV-Ray [48] or ParaView [49].

4

FULL VEHICLE MOBILITY EXPERIMENTS

The single-tire co-simulation framework described in Section 3.1 was used for several numerical experiments to obtain mobility metrics such as draw-bar pull and terrain resistance, investigate deformable tire behavior (strain and stress distributions), and quantify multi-pass effects on terrain compaction and vehicle performance. A comprehensive description and interpretation of these tests is provided in [28]. We reproduce here a representative set of results from [28], related to the effect of soil cohesion on tire mobility performance. Figure 4 shows snapshots from a a simulation involving three successive runs over the same terrain segment. The evolutions of the terrain sinkage and of the available drawbar pull with the number of passes are summarized in Fig. 5. The soil shows consolidation, with incremental sinkage values higher for initial passes; it is also observed that, due to the process of consolidation, the available drawbar pull is larger for the later passes. The remainder of this section is dedicated to results obtained with the full-vehicle co-simulation framework. The off-road vehicle model used in these numerical experiments has an overall mass 2,550 kg and includes front and rear double-wishbone suspensions and a Pitman arm steering mechanism. Each of the four tires is slick and is modeled with a mesh of 90 × 24 multi-layered, orthotropic ANCF shell elements [11], STO-MP-AVT-265

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Figure 4: Snapshots from successive passes over the same terrain segment.

Figure 5: Evolution of steady-state values of sinkage and drawbar pull for multiple passes. Sinkage values presented are referred to the initial terrain height.

with the section geometry created by cubic splines. The default tire internal pressure is assumed to be 200 kPa. The tire input data is specified through a user data file in JSON format [50] and consists of complete tire specification, including tire profile, inflation pressure, contact stiffness and damping parameters, number and thickness of orthotropic material layers and their elastic material properties. At the time publication, development of Chrono::Distributed is still being completed and we are therefore unable to provide results using a distributed MPI simulation of the terrain co-simulation subsystem. All simulations and numerical experiments presented in the remainder of this section were conducted using a multi-core DEM-P terrain simulation (see also Section 4.4 for a discussion of DEM-P vs. DEM-C in the context of the present co-simulation framework).

4.1

Straight line acceleration

In this numerical scenario, the vehicle throttle input (scaled in [0, 1]) is gradually increased to 80%, starting at t = 0.5 s. The vehicle moves forward on a 15 m x 3 m patch of granular terrain consisting of approximately 500,000 spherical particles of 12 mm radius, initially organized in 15 layers. Soil cohesion is set at 100 kPa. The simulation of each individual subsystem (vehicle, terrain, and tires) is advanced with an integration step size ∆t = 4 · 10−5 s, which is also used as the inter-process synchronization interval. A snapshot of the entire vehicle/terrain system in the middle of this acceleration maneuver is shown in Fig. 6. During the course of this simulation scenario, three main dynamic events need to be taken into account: (i) during the first second of simulation the entire vehicle drops onto the deformable terrain, causing abrupt dynamic forces; (ii) at around 3.4 s, the rear tires fall onto the ditch carved by the front tires, changing the pitch angle of the vehicle; and (iii) at 4.70 s, the vehicle switches gears for the first time. These dynamic 12 - 12

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Figure 6: Snapshot of the entire system at t = 3 s. In this simulation, the multilayered material has been lumped into one single orthotropic layer. The tire FEA meshes are color-coded with the current radial strain distributions. events can be identified in the plots of Fig. 7 which provide the longitudinal and vertical position of the chassis over the simulation duration.

(a) Forward vehicle position.

(b) Vertical vehicle position.

Figure 7: Position of the vehicle center of mass during the straight-line acceleration test. The plots in Fig. 8a show the resultant of the normal vertical forces on the two left tires. The highest jumps on the vertical forces correspond to the fall of the rear tires on the initial front wheel sinkage and the gear shift. The resulting terrain resistance forces and net forces on the left tires are depicted in Fig. 8b which shows an increase in the mobility performance of the rear tires after 3.45 s; this effect is due to the consolidation of the terrain caused by the rolling of the front tires (see the multi-pass effects of Fig. 5).

4.2

Obstacle crossing

In this new simulation scenario, the test vehicle overcomes a heavy cylindrical obstacle placed on its way. The time step for this simulation is ∆t = 3.5 · 10−5 s. The terrain cohesion is assumed to be 100 kPa, the STO-MP-AVT-265

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(a) Resultant of normal vertical forces on left tires.

(b) Terrain resistance and net forces on left tires.

Figure 8: Forces acting on the left vehicle tires during the straight-line acceleration test. dimensions of the terrain patch is 20 m × 3.6 m, the radii of the spherical bodies are 12 mm, and the friction coefficient is µ = 0.9. The cylindrical obstacle, with a radius of 0.3 m and a mass of 1,200 kg is placed in front of the vehicle, offset towards the path of the left wheels. A snapshot of the entire vehicle negotiating the obstacle is shown in Fig. 9.

Figure 9: Snapshot depicting the vehicle negotiating the cylindrical obstacle. Longitudinal and lateral vehicle positions are plotted in Figs. 10. The throttle input is ramped, starting at t = 0.5 s, to 80% level and consequently, the vehicle starts accelerating until it is faced with the obstacle. The vehicle dynamic behavior can be observed in Fig. 11a: two sharp reductions in forward velocity can be identified at x = 0 m and x = 2.5 m, where the front-left and rear-left tires, respectively, negotiate the obstacle. Similarly, the same effects of the obstacle-crossing on the vertical vehicle position is depicted in Fig. 11b. The resultant forces of the granular terrain and obstacle acting on the left tires are shown in Fig. 12. Figure 12a shows the influence of the obstacle on the net terrain longitudinal force exerted on the tires: a positive value refers to a propulsion forces, whereas a negative one implies a resistance action. The normal 12 - 14

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Figure 10: Position of the vehicle center of mass during obstacle crossing.

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(a) Forward vehicle velocity as function of position.

(b) Vertical vehicle position as function of position.

Figure 11: Vehicle states as functions of forward vehicle position during obstacle crossing. vertical force is plotted in Fig. 12b and shows that, at some instants of time, the vehicle’s left tires lose contact with the soil.

4.3

Constant radius turn maneuver

Steady turning of an off-road vehicle over a circular path is one of the standard maneuvers used to assess basic cornering features, such as understeering and load transfer. Of interest is also the driver input needed to keep the vehicle on a circular trajectory of a specific radius. All these responses vary with several system parameters, including tire materials and geometry, vehicle suspension, and soil behavior. This subsection demonstrates the leverage of this framework for this case scenario. In this numerical experiment, the off-road vehicle is initially supported by an 8-meter long rigid platform and starts accelerating until it negotiates the cohesive, granular material. A proportional integral derivative (PID) controller, acting as a closed-loop driver model, produces the steering input required to maintain the STO-MP-AVT-265

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(b) Vertical normal forces acting on left tires.

Figure 12: Forces acting on the left vehicle tires during obstacle crossing.

vehicle on a circular trajectory of 12 m of radius and the throttle/braking inputs required to maintain a constant vehicle speed. The granular terrain, of dimensions 28 m × 28 m, is initialized in 8 layers of spherical bodies of 3.6 cm of radius. A constant cohesive pressure of 100 kPa is assumed. Snapshots of the vehicle during the constant radius cornering are shown in Fig. 13 which depict the terrain sinkage caused by the tire footprints. The driver steering and throttle inputs generated by the PID controller, normalized from -1 to 1 (from maximum steering to the right to maximum steering to the left) and from 0 to 1, respectively, are shown in Fig. 14a. At the beginning, the vehicle accelerates at full throttle until it reaches the target speed of 7.5 m/s. The steering input increases sharply at about 2.5 s into the simulation, where the vehicle comes across the prescribed circular path. After some transient effects, both driver’s inputs stabilize. The trajectory of the center of gravity of the vehicle on a horizontal plane is plotted in Fig. 14b. The evolution of the forward speed of the vehicle with time as well as its speed with respect to the global X axis are depicted in Figs. 15a and Figs. 15b, respectively. Initially, the forward speed drops as the initial velocity of the vehicle chassis and tire are transmitted to other initially still mechanical components. After that, the forward speed increases until it stabilizes on the deformable terrain.

Figure 13: Snapshot depicting the vehicle during the constant velocity turning maneuver.

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(a) PID controller-induced steering and throttle inputs.

(b) Vehicle trajectory.

Figure 14: Constant velocity turning maneuver.

(a) Vehicle speed as function of time.

(b) Vehicle speed as function of X global coordinate.

Figure 15: Evolution of the vehicle velocity during the turning maneuver.

4.4

Performance

The obstacle crossing simulation is used here to provide illustrative efficiency measures for a scenario using shared-memory parallelization of the granular terrain calculations. It should be noted, however, that these values are strongly influenced by both the available hardware and the distribution of MPI processes over available nodes; results presented here are provided mostly to assess the effectiveness of the co-simulation strategy in overlaying simultaneous computations. R Xeon R Processor E5The full-vehicle simulation was run on a two-socket node consisting of two Intel 2650 v3 (25M Cache, 2.30 GHz). Two OpenMP threads were assigned to each of the four tires to parallelize the ANCF internal force and Jacobian calculations, as well as the linear system solution for each NewtonRaphson iteration; the MPI process for solving the vehicle dynamics was run single-threaded; finally, 24 OpenMP threads were assigned to the MPI process dedicated for solving the granular material dynamics. For simplicity and to ensure stability of the explicit co-simulation scheme adopted, the communication time steps were selected to be the same as the subsystem integration time steps (namely, 4 · 10−5 s). STO-MP-AVT-265

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For the 7.64 s of simulation, a total wall time of 474,000 s (∼5 days and 12 hours) was recorded. The wall time for the terrain is 96.1% of that of the entire simulation, whereas an average of the wall time for the slowest tire node is 65.2%. Details of computer time for the 3-way co-simulation framework with four ANCF tires and granular terrain are given in the following table: MPI node

Degrees of freedom

Time step (s)

Node wall time (s)

Percentage (%)

Off-road vehicle Granular terrain Tire (slowest node)

145 5.538·106 54,028

3.5·10−5 3.5·10−5 3.5·10−5

321.80 455,354 309,224

0.07 96.06 65.24

Here, only the slowest of the four tire MPI nodes is reported. It may be concluded that the wall time of the tire and terrain MPI nodes is over 95% of that of the entire simulation framework. In this sense, a few aspects need to be considered: (i) while a more sophisticated MPI communication protocol (e.g., using asynchronous send/receives) could be implemented, the relative cost of communication is small and has little impact on overall efficiency, (ii) at each time step, the slowest node may be one of the four tires or the terrain, so it is found challenging to achieve a higher level of overall MPI concurrency; and (iii) for simulations over larger terrain patches, such as the constant radius turn, the granular terrain simulation soon becomes the bottleneck. As mentioned previously, all numerical experiments presented in this section used a penalty-based frictional contact approach for the simulation executed on the terrain node. Within this co-simulation framework, when deformable FEA tires are present, the ability of DEM-C of using larger integration step sizes than those required by a stable DEM-P simulation does not play a relevant role. Indeed, the simulation occurring on the tire nodes (at the small integration step size required by a stable FEA analysis) dictates the overall wall clock time. To illustrate this point, we provide below characteristic timing information for the straight-line acceleration test of Section 4.1, for two different configurations (using the same hardware and node setup as above). In both cases, the integration step size on the tire nodes was kept at 4 · 10−5 s. • DEM-C terrain simulation. In this case, the inter-process communication step size was set at 10−3 s, equal to the integration step size on the terrain node. Advancing the terrain simulation between two synchronization times requires approximately 30 s, while the required 25 internal steps on a tire node require an average of 40 s. • DEM-P terrain simulation. In this case, all processes used an integration step size of 4 · 10−5 s. Extending the required computer times to the above case (i.e., 25 meta-steps required to advance the solution by 10−3 s), we measured simulation times of approximately 28 s for the terrain node and 36 s for a tire node. In other words, no benefit is obtained from the ability of evolving the DEM-C terrain simulation at larger time steps, while the more frequent synchronization with the DEM-P terrain simulation guarantees better stability of the overall explicit co-simulation scheme.

5

CONCLUSIONS AND FUTURE WORK

This contribution describes a co-simulation framework for efficient mobility studies of high-fidelity ground vehicle models, including flexible tires and resolved, granular representations of the deformable soil. The software infrastructure was developed, using a distributed MPI-based layer, as an extension to the opensource Chrono multi-physics package. Based on decoupling the overall multi-scale, multi-physics problem into subsystems whose dynamics are advanced in time independently and in parallel, we implement an explicit force–displacement scheme which effectively alleviates the high computational cost of an equivalent ”all-in-one” coupled simulation. 12 - 18

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The proposed approach enables acceleration of the mobility simulations by (i) allowing each subsystem to advance its state using a suitable integration time step; (ii) allowing the use of different integration schemes for each subsystem, as called for by the particular dynamics problem; and (iii) leveraging different and independent parallelization techniques, as dictated by the structure of each subsystem. Owing to its modular design, the co-simulation framework allows for swapping between a shared memory, OpenMP-based simulation of the granular terrain (using the Chrono::Parallel module), and an MPI-based distributed terrain simulation (provided by the Chrono::Distributed module). Using the latter and assuming reasonable scaling properties of the distributed terrain simulation, the computational time for an arbitrary mobility analysis, is theoretically bounded to the cost of simulating a single deformable tire under the influence of external excitations. Although still significantly more taxing that mobility studies based on empirical or semi-empirical deformable soil models and/or simplified tire models, the high-fidelity simulation capability introduced here enables accurate and predictive vehicle and terrain behavior, as illustrated on several simulation scenarios of a full-vehicle on granular terrain. The problem of ground vehicle mobility using complex terramechanics and high-fidelity vehicle models continues to pose serious challenges, including the need for validation against experimental data, both at the subsystem level (e.g., single-tire rig mechanism) and full-vehicle level, as well as the need to address the still considerable computational requirements. Plans for future research and development in this arena include work on completing the Chrono::Distributed module and incorporating it into this co-simulation framework, as well as future extensions to tackle granular flow problems, which pose additional load-balancing challenges. In addition, additional effort will be dedicated to further improve the efficiency and scalability of the parallel subsystems simulations, by enhanced numerical methods and leveraging new parallel architectures and techniques.

Acknowledgments This research was supported in part by U.S. Army TARDEC Rapid Innovation Fund grant No. W911NF13-R-0011, Topic No. 6a, “Maneuverability Prediction”. Support for the development of Chrono::Vehicle was provided by U.S. Army TARDEC grant W56HZV-08-C-0236.

Disclaimer Reference herein to any specific commercial company, product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the Department of the Army (DoA). The opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or the DoA, and shall not be used for advertising or product endorsement purposes.

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Co-simulation for vehicle-terrain interaction

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