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SAE TECHNICALPAPER SERIES 1999-01-0104

The DyMesh Method for Three-DimensionalMulti-Vehicle Collision Simulation

Allen R. YorkA.R. York Engineering, Inc

Terry D. DayEngineering Dynamics Corp.

Reprinted From: Accident Reconstruction: Technology and Animation IX(SP-1407)

International Congress and ExpositionDetroit, Michigan

March 1-4, 1999

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1999-01-0104

The DyMesh Method for Three-DimensionalMulti-Vehicle Collision Simulation

Allen R. YorkA.R. York Engineering, Inc

Terry D. DayEngineering Dynamics Corp.

Copyright © 1999 Society of Automotive Engineers, Inc.

ABSTRACT

Two-dimensional collision simulation has been used suc-cessfully for two decades. Two- and three-dimensionalmomentum methods are also well known. Three-dimen-sional collision simulation can be accomplished usingfinite element methods, but this is not practical for inter-active collision simulation due to long mesh generationtimes and run times which may take several days. Thispaper presents an approach to collision simulation usinga new algorithm to track interacting vehicle surfacemeshes. Three-dimensional forces due to vehicle crushare taken into account during the solution and the dam-age profile is visualized at run time. The new collisionalgorithm is portable in that it takes as input vehicle mate-rial properties and surface geometries and calculatesfrom their interaction three-dimensional forces andmoments at the vehicle center of gravity. Intervehiclemesh forces may be calculated from a user-definedforce-deflection relationship. The derivation is discussed.The paper includes examples using arbitrarily shapedthree-dimensional bodies as a proof of concept. Samplesare also included using three-dimensional vehiclemeshes. The simulations are shown to agree favorablywith theory, test, and finite element results.

INTRODUCTION

MOTOR VEHICLE CRASHES are the leading cause ofdeath for persons below the age of 40. In the U.S.,370,000 people have died on the nations highways since1990. Another 30 million have been injured.[1]1 The cost,in both real dollars and human suffering, is enormous.Clearly, because of its impact on our society, it is impor-tant that we focus on improving our understanding of thecause of highway crashes.

Engineers and physicists have understood the mechan-ics of vehicle collisions for several decades. Until the mid-seventies, collisions were analyzed by impulse-momen-tum methods using a slide rule or hand-held calculator.The 2- and 3-dimensional momentum equations can bearranged to calculate impact velocities for two or morevehicles given a set of exit velocities. Research by Emori[2] suggested energy-based methods that were ultimatelyturned into an algorithm by Campbell, first in his Ph.D.thesis [3] and later in his important SAE paper [4] pub-lished while he was at General Motors. McHenry devel-oped the CRASH computer program [5] around these two(momentum and energy) methods. The CRASH modelhas been extended, first by NHTSA [6] and then by oth-ers [7,8,9]; still it is based on the momentum and energymethods in use for several decades. Additional research-ers [10,11, 12] have programmed the momentum equa-tions to calculate exit velocities from assumed impactvelocities.

In the mid-seventies, McHenry also developed the firstcollision simulation model, SMAC [13]. Collision simula-tion was a fundamentally new approach to collision anal-ysis made possible by the digital computer. Collisionsimulation methods calculate the collision forces andmoments acting on the vehicle, then solve the equationsof motion at small, user-defined time intervals. TheSMAC algorithm has been extended by others [14,15,16],and is still the most popular simulation method in usetoday.

Finite element technology has existed for decades. It hasbeen during the last decade or so that large-scaledynamic simulations with the finite element method havebecome possible. The processor speeds and memorywith modern workstations and desktop personal comput-ers allow dynamic finite element simulations with reason-able fidelity. Nonlinear material properties, largedeformations, and complex contact conditions are typicalof these analyses. These types of simulations are doneby vehicle designers and manufacturers.1. Numbers in brackets designate references found at the

end of the paper.

2

Safety engineers have not used the finite elementmethod outside of the realm of manufacturing anddesign. A single collision simulation typically takes sev-eral hours or even days to process on high-speed supercomputers.[29] In addition, developing a structural modelfor a single vehicle may take several weeks or more.Thus, the finite element method is not practical for use bythose researchers who reconstruct collisions that occuron public roads.

Therefore, reconstructions are still performed using sim-ple momentum and energy methods, or extended ver-sions of the SMAC algorithm.

There exists room to improve the current methods forreconstructing collisions. However, the quantam leapfrom current methods to using a complex finite elementcode is not practical. A compromise approach has beendeveloped employing techniques from finite elementtechnology, and from the current methods, while still mak-ing a large advance in the state-of-the-art. This approachis called DyMesh (Dynamic Mechanical shell).

The purpose of this paper is to provide a detaileddescription of the DyMesh model. The process of inte-grating DyMesh into a typical simulation model is alsoprovided. Finally, examples of results using DyMesh arepresented, first on simple 3-dimensional shapes, andthen on vehicle vs barrier and vehicle vs vehicle crashes.

GENERAL DESCRIPTION

This section gives an overview of the two main featuresof the DyMesh method - collision detection and collisionforces.

COLLISION DETECTION – The DyMesh method isbased on a collision algorithm which uses a triangularmesh defined by discretizing the exterior surfaces of avehicle(s). The algorithm detects the interaction of vehi-cles by monitoring the positions of nodes, or vertices, andsurfaces representing the exterior of the vehicles. Theconcept of a master surface and slave node is used. Thismirrors proven technology used with contact algorithmsfor many years.

Contact algorithms are necessary in finite element analy-sis to simulate the contact and impact between bodies.These algorithms are widely used in explicit Lagrangianfinite element codes such as PRONTO3D [17,18], EPIC-3 [19], and DYNA3D [20] and have been used success-fully to simulate problems such as crush, impact, andpenetration [21,22,23]. Contact algorithms are also usedin static or quasi-static codes like JAC [24] and NIKE [25].

The basic operations performed in a contact algorithmare location and restoration. Location involves searchingfor surfaces in contact with one another and defining thecontact geometry such as depth of penetration. Restora-tion involves finding the penetration depth and movingthe slave node (vertex) back to the slave surface. In thiscontext, we are using the classic definition of restoration

as it is used by finite element contact algorithms. It is notat all related to restitution. Restoration is what causes thesurfaces to remain in contact. The terminology used inthe literature usually addresses one of the surfaces asthe master and the other the slave. In three-dimensions,surfaces may be defined by four-noded quadrilateral ele-ments or three-noded triangular elements. In the finiteelement method, quadrilateral surface elements are oneside of a solid brick or hex element, and triangular ele-ments are one side of a solid tetrahedral element.

Various algorithms treat the surfaces and forces usingdifferent methods [23,25]. The general idea is to move, orcause to move, one or both of the interpenetrating sur-faces represented by discrete nodes so that the surfacesare in line-to-line contact. In Taylor’s method [17], forcesproportional to penetration depth are applied to slavenodes, which accelerate the node out of the penetratedsurface. The slave node force is also apportioned to thenodes of the master surface. The enforcement of this no-penetration condition may occur over one or severaltimesteps. Belytschko [19] moves slave nodes onto themaster surface and modifies the velocity of the node by

where . The displacement of the node

defines the displacement vector and the current time

step is the value for . Then the momentum associatedwith modifying the slave node velocity is apportioned tothe master nodes.

Contact enforcement in finite element methods isachieved on a local level. That is, depending on the par-ticular method, forces, velocities, and/or momentum areprescribed to nodes and elements on the surface of thediscretized body.

There have been many algorithms proposed to search forcontact as it can be the most expensive component of thealgorithm. An automatic global contact algorithm isdesired to minimize the user definition of pairs of contactsurfaces. Efficiency is important as the potential for muchunnecessary and time-consuming searching exists in aglobal approach to contact.

There are many cases of contact, such as with cornersand self-contact, that may cause problems for contactalgorithms. An algorithm that monitors the velocity vec-tors of nodes penetrating a surface reduces the uncer-tainty involved in determining contact for many cases[26].

Figure 1 shows two bodies discretized on their exteriorwith nodes and elements. Consider the left body to be

the master and the right to be the slave. At time tk thebodies are moving towards one another with velocities

vm and vs, but there is no contact. At severalslave nodes have penetrated the master surfaces. Slave

node s has penetrated a distance p. The tilde on indicates that this is not the configuration at the end ofthe time step. The penetrating slave nodes have yet to bemoved or restored to the master surface.

∆v ∆v ∆x ∆t⁄=

∆x

∆t

t tk 1+

=

tk 1+

3

Figure 1. Master Surface-Slave Node Concept

Once penetration [Figure 2(a)] is detected, the pushbackdirection is determined, and slave node s is restored tobe consistent with the kinematic constraints between thetwo surfaces. In most explicit, finite element, dynamicscodes a force is applied to the slave node in a directionnormal to the master surface nm [Figure 2(b)], which isusually used as the pushback direction. This force accel-erates the node to the outside of the master surface. Thevector nm passes through the point on the master surfaceto which the slave node projects. Another method used infinite element codes applies the slave node force in adirection of its normal ns. In this work since the impactforces are accelerating the CG and not individual nodesseparately, nodes are restored by adjusting their three-dimensional coordinates along the pushback direction,and then calculating a force. It has been found that forthis application an effective pushback direction is in theopposite direction of the node’s velocity. In this case, theintersection of the line defined by the velocity vector withthe master surface is calculated, and the slave node ismoved to this point of intersection. This point of intersec-tion is called the contact point.

COLLISION FORCES – After the slave nodes are

restored [Figure 2(c)], the vector is calculated. Theforce on the slave node is a function of the vehicle stiff-ness parameters, the volume of material associated withthe crush of slave node s, and where

.

A difference between explicit, dynamic, finite elementcodes and this vehicle simulation algorithm is that therestoring force magnitude depends on the cumulative as opposed to depending only on the instantaneous localsurface penetration p (Figure 1).

During loading, the collision force at each slave node isbased on a general purpose, 3rd-order polynomial force-deflection relationship,

(1)

Figure 2. Restoration Phase

master surface slave nodes

p

t tk 1+

=t tk

=

vm

vs s

δk 1+

∆δδ

∆δδ δo δk 1+–=

∆δδ

Fcoll k0 k1∆δ k2∆δ2k3∆δ3 κ ∆δ( )·

+ + + +=

t tk 1+

= t tk 1+=

s s nm

ns

(a) (b) (c)

fsfm

δk 1+

δo

4

where k0, k1, k2 and k3 are the polynomial coefficients(note that if k2=k3=0, the model reduces to that describedby Campbell [4]), and is a damping coefficient. Theforce-deflection relationship may also include a satura-tion force, Fmax, a saturation deflection, dmax, and veloc-ity- dependent node damping. A null band, d0, is used forsmall node deflections (typically 0.5 inches or less). Aforce-deflection model for loading and unloading isshown in Figure 3.

Figure 3. Force-deflection Relationship Using a 3rd Order Polynomial with Saturation Force and Deflection and Linear Unloading.

During unloading, the collision force is determined by anunloading slope, Ku. Specification of the unloading slopeallows the model to account for restitution (a largeunloading slope causes minimal restitution, while asmaller unloading slope would cause higher restitution).Note that the unloading slope must cause the residualnode deformation to be zero or positive. The presence ofunloading is determined by monitoring the deformationrate.

Tangential forces may also exist between interactingmeshes. The tangential force is calculated as the productof the normal component of the collision force, Fcoll(equation 1), and the friction coefficients for interactingnodes. The direction of the tangential force is determinedfrom the direction of the relative node velocities.

Polynomial coefficients can be estimated from traditionalcrash test data and by estimating a crush height. For bar-rier crashes, this height is normally set to 30 inches (seeFigure 4). Thus, by dividing the traditional A and B coeffi-cients by 30 inches and setting k2 and k3 equal to zero,reasonable estimates for ko and k1 are obtained. Notethat because the collision forces are impulsive (i.e., occurover a short time interval), a reasonable range of esti-mates does not significantly affect the total computedspeed change, but does affect the peak acceleration andduration of the collision (see comparisons with barrierand pole collisions later in this paper).

Figure 4. Height of Crush Region

NUMERICAL IMPLEMENTATION

This section discusses the numerical procedures used toimplement the previously outlined methodology. The top-ics include initialization, collision detection, and the calcu-lation of collision and friction forces.

INITIALIZATION – Initialization calculations are onlydone once. To initialize the collision algorithm the vectorsfrom the vehicle CG to each node are calculated. These

vectors are denoted by , where i is the node number,

and j is the vehicle index. The superscript indicates the

time level is zero. The vectors are used to calculate thecrush forces, and will be described in more detail later.

Next, the connected surface neighbors to each node aredetermined for each vehicle. Figure 5 illustrates the con-cept for node i. There are six Level 1 neighbors (in thisexample), denoted where 1 indicates a Level 1 or

connected neighbor, n ranges from one to the number ofneighbors, and j is the vehicle index. Level 2 neighborsare also stored in memory. These are surfaces that sharea node with a connected neighbor. The vehicle meshconnectivity that defines which nodes form a triangularsurface does not change during a simulation. Thus,neighbors remain constant for a given mesh. Theseneighbor data are used to associate an area with a pene-trating node and sometimes are used in the algorithmthat searches to determine through which surface a nodehas penetrated. In addition, at the beginning of eachtimestep the area and outward normal for each surface iscalculated.

COLLISION DETECTION – A gross collision detection isused during the simulation to initiate the detailed collisionalgorithm. A bounding box is placed around each vehicle.When these boxes overlap, the detailed collision algo-rithm is used.

When the collision algorithm is invoked to search for colli-sions, a neighborhood around each node is searched forpotential surfaces through which it may have penetrated.This neighborhood is a rectangular bounding box whose

κ

deflection

For

ce

dmax

Fmax

Ku

H

δ i j,o

δ

N1n j,

5

dimensions can be varied depending on the problem. Foreach slave node the potential contact surfaces are testedto determine which one, if any, it has penetrated. Thesequence of tests is described below.

Figure 5. Neighbor Surfaces of Node i

First, the intersection of the line formed by the velocityvector of a slave node with the surface in question is cal-culated. This intersection or contact point, , isdefined as

(2)

where is the vector from the origin to the slave node,

v is the velocity vector, and is a parameter given by

(3)

where is a vector originating at any one of the

three triangular surface nodes (i=1,2, or 3), and n is theoutward normal vector to the surface. The possibilityexists for the denominator in equation 3 to be zero. In thiscase the velocity is parallel to the surface and there is nointersection between the two.

Several tests are made to determine if this contact point,, is valid. First, the vector from the contact point to

the slave node must be in the same direction to within180 degrees of the velocity vector. Otherwise, the slavenode could not have possibly penetrated the surface inquestion. This check amounts to calculating the dot prod-uct of the two vectors and ignoring those surfaces forwhich the dot product is negative. Another simple checkrequires that the slave node be inside the potential sur-face. This is done by taking the dot product of the vectornormal to the surface with the vector from the contactpoint to the slave node. In this case the dot product mustbe less than zero.

Figure 6. Contact Point Inside Surface

Next, the area formed by the three triangles and the con-tact point is calculated and compared to the area of thetriangular surface in question (Figure 6). If

the surface is still considered a

candidate for contacting the slave node.

Another check determines if the distance from the slavenode to the contact point is too far to be realistic This dis-tance is denoted by the vector h, shown as a dashed linein Figure 6. The average velocity of the master surfaceand the velocity of the slave node are used to determinea maximum distance that could possibly exist betweenthe contact point and the slave node. If this distance istoo great, the surface is no longer considered for contactwith the slave node. If multiple surfaces meet all theabove criteria, the one that minimizes the distancebetween the contact point and slave node is chosen.

COLLISION FORCES – There are no collision forcesuntil there is deformation. Deformation occurs when aslave node is restored to its non-penetrating positionwhich is defined to be on the contact or master surface.Figure 7(a) illustrates slave node s penetrating a surfacein two-dimensions. In Figure 7(b) the penetrating nodes

are restored to the contact points and the vector is

shown for node s. The change in the vector at eachnode is used to calculate collision forces.

In this work an omni-directional stiffness is used althoughthe use of orthotropic or other spatially varying stiff-nesses is not precluded. The slave node collision force,

, due to the deformation follows a tri-linear

curve that includes unloading. A vector component is

in a state of loading when the deformation rate, , is

greater than a critical deformation rate . For this

loading regime the collision force is given by

(4)

N16,j

N12,j

N 13,j

N14,j

N15,j

N11,j

i

N21,j

N22,j

N25,j

N24,j

N23,j

N26,j connected orLevel 1neighbor

Level 2neighbor

xcp

xcp xs vα+=

xs

α

αx i xs–( ) n⋅

v n⋅------------------------------=

x i xs–

xcp

12

3

A1A2

A3

contact point

slave nodev

triangular (contact)master surface

z

y

x

h

A1 A2 A3+ + A123=

∆δδδ

Fs coll, ∆δ

δ

∆δ· i

∆δ· c

Fs coll, A B ∆δδ+( )Assign ∆δδ( )1=

6

Figure 7. (a) Penetrating Slave Node and (b) Restored Slave Node in Two-Dimensions

where 1 is a unit vector in the direction of , is the

area associated with the penetrating slave node, is

the absolute value of , and are

the modified stiffness coefficients, and applies

the sign of each component to each force compo-nent. The area associated with each penetrating slavenode is approximated by

(5)

where Nn is the number of connected neighbors, and Aiis the area of the ith neighbor surface. In the loadingregime individual vector components contribute toforces in their respective directions (omni-directional stiff-ness).

To unload in exactly the same direction as the slave nodewas loaded would require knowledge of the entire defor-mation history of the node. An efficient compromiseapproach used by DyMesh is to define the unloadingdirection to be the vector originating at the current slavenode location and ending at the original slave node loca-tion, in the local vehicle coordinate system. When theslave node deformation rate falls below the critical defor-

mation rate, , the vectors grow a distance

defined by the unloading slope Ku in Figure 3. The growthof the vector is the maximum magnitude of the collisionforce divided by the unloading slope,

. When the vector unloads

or grows, in most cases, it will cause a contact conditionto be enforced with the surface of the other vehicle. This

contact condition reduces the growth of the vector. The

corresponding force is calculated based on the new total

length of the vector and the unloading slope Ku. Each

time step in the unloading regime the vector will grow

until the total unloading distance is .

Once the collision force is determined for a slave node,the force is distributed to the three nodes of the corre-sponding master surface. This distribution is accom-plished using standard finite element shape functionswhich apportion the force to the nodes. The force magni-tude on master nodes is

(6)

where is the magnitude of the slave node colli-

sion force and Li is the shape function associated withmaster node i. The shape functions are given by

(7)

where Ai is the sub-area as shown in Figure 6. The forcesFi are applied in a direction normal to the master surface.

An elegant feature of the loading and unloading is that itis handled naturally through the contact algorithm. Colli-sion forces are only present when there is contactbetween the vehicles. The unloading methodology isrealistic in that it results in a body pushing itself awayfrom the other body in a reasonable and controllablemanner.

FRICTION FORCES – Friction forces are only presentwhen there is contact between a slave node and mastersurface. The relative tangential velocity between the mas-ter surface and slave node at the contact point is deter-mined. If the relative velocity is zero there is no friction. Ifthere are nonzero friction forces, ,

(8)

are applied in equal and opposite directions along the rel-ative tangential velocity vector where n is the master sur-face normal, and is the friction coefficient.

DYMESH FLOW CHART – Figure 8 shows the main ele-ments of the DyMesh algorithm which are enclosed in thedashed box. DyMesh is only invoked if the gross collisioncheck indicates a collision is occurring or just about tooccur. If there is no collision or when DyMesh has com-pleted calculating the collision and friction forces the sim-ulation program continues by calculating other externalforces such as aerodynamic drag and tire forces (seesection entitled Typical System Model).

δo

δ1∆δδ

(a) (b)

slavemaster

sformerposition

∆δδ As

∆δδ∆δ A A H⁄= B B H⁄=

sign ∆δδ( )∆δδ

As13--- Ai

i 1=

Nn

∑=

∆δ

∆δδ· ∆δc·< δ

δu max Fs coll,( ) Ku⁄= δ

δ

δδ

δu

Fi Fs coll, Li=

Fs coll,

Li Ai Ajj 1=

3

∑⁄=

Ff

Ff Fs coll, n⋅( )µ=

µ

7

Figure 8. DyMesh Flow Chart

The DyMesh algorithm can be called in a variety of ways.For example, DyMesh can be called twice each timestep(swapping the master and slave objects), or it can becalled once each timestep, alternating the master andslave objects. The latter approach reduces computationtime, and was used in this work.

DYMESH OUTPUT – The output from DyMesh is a 3-dimensional force vector acting at each of the damagednodes or vertices. Since the current vertex coordinatesare known, it is a simple manner to calculate the forcesand moments acting at the vehicle CG:

(9)

(10)

Here, i is a damaged vertex, Fi is the force on the dam-aged vertex, Nd is the number of damaged vertices, andxi, yi, and zi are the vertex vehicle-fixed coordinates.

TYPICAL SYSTEM MODEL

This section discusses how the output from DyMesh canbe easily coupled into a vehicle simulation program. Sim-ulation models normally include four common compo-nents:

• main control routine,

• numerical integration routine,

• free-body analysis of the objects,

• acceleration calculation.

A flow chart for a typical simulation model is shown inFigure 9. The DyMesh algorithm contributes to the forcecalculations. It should be noted that these componentsare required by all simulations, whether they are used tomodel humans, vehicles, even the weather. The primarydifference is in the free-body analysis of the objects. Inour case, the objects are vehicles and our analysisincludes six degrees of freedom for the collision forcesand moments acting on each sprung mass.

Figure 9. Simulation Flow Chart

collisiongross

check define potentialslave nodes &

master surfaces

calc surfacenormals andsurface area

find surfacesthru which nodeshave penetrated

find surfacesin neighborhoodof slave nodes

collisionno

col

lisio

n

restore nodes in contact &calc forces move node

calc ∆δδcalc Fcollcalc Ff

initialization

return;calcotherforces

DyMesh

calc xcp

Ai∑ Aijk?=

v n⋅ 0?=

v h⋅ 0?>n h⋅ 0?<

Fx cg, Fix coll,

i 1=

Nd

∑=

Fy cg, Fiy coll,

i 1=

Nd

∑=

Fz cg, Fiz coll,

i 1=

Nd

∑=

Mx cg, F– iy coll,zi Fiz coll,

yi+( )i 1=

Nd

∑=

My cg, F– iz coll,xi Fix coll,

zi+( )i 1=

Nd

∑=

Mz cg, F– ix coll,yi Fiy coll,

xi+( )i 1=

Nd

∑=

From INPUT

CONTROLLINGPROGRAM NUMERICAL

FORCECALCS

DERIVATIVECALCS

INTEGRATION

OTHER SPECIALROUTINES

8

These four basic system components are describedbelow.

MAIN CONTROL ROUTINE – The main control routinedoes just what its name suggests: it controls thesequence of calculations. The following tasks are per-formed by the typical control routine:

• initializes all variables

• calls the numerical integration routine

• calls the output routine

• performs logical checks that affect execution.

DyMesh requires that several parameters be initialized,including the initial (undeformed) mesh coordinates, ver-tex velocities, and forces and deflections material param-eters (force vs deflection, friction, damping) for eachvertex. Calling the numerical integration starts the pro-cess that updates the velocities and positions for eachtimestep (see below). The output routine updates the cur-rent simulation results for each timestep. This stepincludes updating the current vertex coordinates so theymay be visualized as vehicle damage. Logical checksinclude termination conditions and setting various flags tocontrol execution.

NUMERICAL INTEGRATION – The numerical integra-tion routine causes the free-body analysis to be executedfor each timestep. Typically a fourth order routine is used,meaning that the free body analysis is performed fourtimes per timestep. After the accelerations have beencomputed (see below), they are returned to the numericalintegration routine to be integrated. This integration pro-cess updates the velocity and position for the nexttimestep. Any valid numerical integration method may beused. Common methods include Runge-Kutta, Adams-Moulton and Hamming's Modified Predictor-Corrector.

FREE BODY ANALYSIS – DyMesh calculates the colli-sion forces and moments acting on each vehicle. Thisprocess was described above in detail.

Other forces acting on the vehicle may include tire forces,aerodynamic forces and suspension forces. After eachforce producer is executed, the results are summed toproduce the total vehicle-fixed forces and moments act-ing on the vehicle.

ACCELERATION VECTOR – Each force vector has anassociated acceleration vector. According to Newton's2nd law, the acceleration in each direction is equal to theproduct of the force and mass (linear acceleration) or theproduct of the moment and rotational inertia (rotation).Mathematically these equations of motion are expressedas:

(11)

(12)

where m is the sprung mass, u, v and w are forward, lat-eral, and vertical velocity, p is roll (about x), q is pitch(about y), r is yaw (about z), and Ijj is inertia about axis j,all in vehicle-fixed coordinates.

To solve the equations, the mass matrix is inverted andthe accelerations are computed using the SimSol (simul-taneous solution) function. [28]

OUTPUT – The primary output from DyMesh is the cur-rent vertex coordinates and level of force acting at eachvertex on the mesh in the vehicle-fixed coordinate sys-tem. The vertex coordinates allow the damaged mesh tobe visualized at each timestep. Intermediate values avail-able from DyMesh include vertex deflection and deflec-tion rates.

EXAMPLES

The first two examples use simple geometries to demon-strate the algorithm. The first is a cube impacting a rigidwall. Next, two colliding spheres are simulated. Thenthree examples are presented with vehicle meshes. Thefirst of these is a simulation of a 35 mph barrier impacttest on a Ford Escort. Then a collision between a Festivaand a telephone pole is simulated. Finally, a two-vehiclesimulation is presented.

In these examples piecewise-linear force-deflection rela-tionships are used. For vehicles, the standard two-dimen-sional force-per-width versus deformation relationship[Figure 10(a)] is extended to three-dimensions usingequation 1 and assuming . The three-

dimensional model takes the form shown in Figure 10(b).The non-permanent deformation represented by G

( ) is accounted for in the unloading, and theforces are per unit area instead of width. Conversion tothe three-dimensional form is achieved by dividing A andB by the height H (Figure 4) of the vehicle crushed whenthe stiffness parameters were generated. The portion ofthe curve with a slope Ku is the unloading path.

Fx∑ m u· vr wq+–( )=

Fy∑ m v· ur wp–+( )=

Fz∑ m w· uq v– p+( )=

Mx∑ Ixxp· qr Izz Iyy–( )+=

My∑ Iyyq· pr Ixx Izz–( )+=

Mz∑ Izzr· pq Iyy Ixx–( )+=

k2 k3 κ 0= = =

G A2

2B⁄=

9

(a)

(b)

Figure 10. (a) Standard 2D and (b) New 3D Force Deflection Curve

The examples shown use an omni-directional stiffness.That is, if a node is deformed equally in all three direc-tions the force in each direction will be equal. The capa-bility is for each node to have unique stiffnesses, and thestiffnesses for each node may be a function of the coordi-nate direction. Also, as stated earlier, the force model isnot limited to piecewise linear segments. More complexquadratic or cubic functions may be defined by using anonzero k2 or k3 in equation 1.

CUBE-WALL COLLISION – This simulation is of a unitcube impacting a rigid wall at a velocity of 2 (consistentunits). A cross-section view of the problem set-up isshown in Figure 11. The cube is made up of 8 verticesand 12 elements. This simple problem provides anopportunity to compare the DyMesh algorithm results totheoretical results.

Figure 11. Cube Collision with a Rigid Wall

All stiffness parameters are zero except for . Thecube has a mass of 0.3. The initial kinetic energy is

(13)

The unloading slope is set to twice the loading slope.Therefore, the cube should rebound from the wall withone-half of the original kinetic energy. Also, the maximumdeformation can be calculated based on conservation ofenergy. Each of the four vertices will absorb one-quarterof the energy. Equating the energy used in deformingeach vertex to kinetic energy gives

(14)

where the factor of one-half comes from knowing one-half of the energy is used to induce permanent deforma-

tion. Since the loading slope, , is 25, the maximum the-oretical deformation can be solved for in equation 14 by

substituting which gives

(15)

Since only one-half of the damage will be left followingrestitution, the final deformation is given as

(16)

The energy and deformation history for a vertex in theDyMesh simulation are shown in Figures 12 and 13. Itcan be seen that the final kinetic energy is 0.3, or one-half of the original value of 0.6, which agrees with theory.In the lower plot, the maximum deformation is 0.1095which agrees with theory. The final deformation, ,

is 0.0574 which differs by 4.8% from the theoretical valueof 0.05475.

0

500

1000

1500

2000

2500

-5 0 5 10 15 20

Permanent Crush (in.)

Fo

rce

pe

r W

idth

(lb

/in.)

0

10

20

30

40

50

0 5 10 15 20 25

Permanent Crush (in.)

For

ce p

er A

rea

(lb/s

q in

.)

wall

unitcube

vx=2

B 25=

KE 12--- 0.3( )4 0.6 .= =

12---Fcoll∆δ 0.6

4--------=

B

Fcoll 25∆δ=

12---25 ∆δmax( )2 0.6

4--------=

or

∆δmax 0.1095 .=

∆δfinal

∆δmax

2--------------- 0.05475 .= =

∆δfinal

10

Figure 12. Cube-Wall Results - Kinetic Energy

Figure 13. Cube-Wall Results - Deformation

SPHERE-TO-SPHERE COLLISION – This example illus-trates the DyMesh algorithm on two spheres discretizedby 1,452 triangular surfaces. The problem set-up isshown in Figure 14(a). Both left and right spheres have a

mass of 0.1, a radius of 0.5, a stiffness , and avelocity of 3 (relative velocity of 6) in the x direction. Atime step of two milliseconds is used.

Two simulations are presented - without and with restitu-tion. Ideally, the interface between the colliding spheresshould be flat. Figure 14 shows the two spheres at t=0and t=0.7, and Figure 15 shows two views of the leftsphere at t=0.7, without restitution. The surface areawhich contacted the right sphere is nearly flat.

(a)

(b)

Figure 14. (a) Initial Positions and (b) Positions at t=0.7

Figure 15. Sphere Geometry After the Collision: (a) Side View and (b) rotated 30 Degrees about the Y Axis

History plots of kinetic energy, velocity, and accelerationare shown in Figure 17. It can be seen that without resti-tution the impact is plastic. Both spheres come to rest fol-lowing the impact, and the acceleration quickly goes tozero.

With restitution both spheres rebound and the accelera-tion more gradually approaches zero as the spheres sep-arate as seen in the plots in Figure 17. The geometry ofthe left sphere during restitution is shown in Figure 16.The discrete time step and alternate swapping of masterand slave surfaces causes slight irregularity in the sur-face as the vectors are lengthened.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 0.05 0.10 0.15

Time

Kin

etic

Ene

rgy

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.00 0.05 0.10 0.15

Time

B 500=

(a) (b)

δ

11

Figure 16. Sphere Geometry During Restitution

VEHICLE-TO-BARRIER COLLISION – In this simulationthe DyMesh method is used to simulate a 1997 FordEscort barrier crash test. A government report fully docu-ments the test.[27] The vehicle weighed 2,963 lb andimpacted the barrier at 35.1 mph (617.8 in/s). The dam-aged region length L was measured to be 53.1 in. Thedamage profile is shown in Figure 18.

Figure 18. Escort Damage Profile

Assuming that the vehicle can withstand a 7 mph colli-sion without damage, the standard stiffness coefficients

are A=501 lb/in and B=115 lb/in2. Assume that the dam-aged vehicle height in the test is 35 in., and the modified

stiffness coefficients are lb/in2 and

lb/in3. These constants will be used in the DyMesh simu-lation.

Figure 19 shows the mesh of the escort and the barrier atthe beginning of the simulation. The Escort mesh has2,075 nodes and 3,827 triangular elements. The barrieris made of eight nodes and six large triangular elements.

To emphasize the fact that only collision forces are beingconsidered, tires are not shown and are not part of thecalculation. The time step used in the simulation is onemillisecond, and the entire simulation takes 40 secondsto run using a 233 MHz Pentium II processor.

Figure 19. Escort and Barrier Mesh

Figure 20 shows the velocity and acceleration history ofthe DyMesh simulation compared with the test. Thechange in velocity in the test was 705 in/s compared with742 in the DyMesh simulation - a difference of 5.2 per-cent. The peak acceleration in the test was -40 G com-pared with -44 G in the DyMesh simulation - a difference

of ten percent. These results suggest the selected and

stiffness coefficients were too stiff. Reducing thesevalues would reduce the peak acceleration and increasethe duration of the collision. Note however, the selectionof the stiffness coefficients have little effect on the totalvelocity change. As a demonstration of this, two addi-tional barrier simulations were run with stiffness coeffi-cients ten percent higher and lower. The peakacceleration differed by as much as 16.8 percent from thenominal case, but the velocity change only differed by 0.5percent at most.

Figure 21 shows the deformed mesh geometry with con-tours of deformation overlaid on the mesh. The maximumdeformation occurs in the center of the bumper, that partof vehicle which protrudes the most. It is interesting tonote the contour changes between 75 ms and 100 ms. At100 ms it is evident from the recession of the darkestcontour that there has been some restitution (i.e., there isless area covered by the darkest contour at 100 ms com-pared with 75 ms).

1 2

3 4

L

C1

C6C6=13.9C5=16.7C4=17.4C3=19.4C2=19.8C1=18.3

A 14.3= B 3.3=

side view

back view

A

B

12

Figure 17. History Plots for Sphere Collision Simulation

Kinetic Energy

0.000.050.100.150.200.250.300.350.400.450.50

0.00 0.05 0.10

ke1

ke2

Velocity

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

0.00 0.05 0.10

vx1

vx2

without restitution with restitution

Kinetic Energy

0.000.050.100.150.200.250.300.350.400.450.50

0.00 0.05 0.10

ke1

ke2

Velocity

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

0.00 0.05 0.10

vx1

vx2

Acceleration (G)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.00 0.05 0.10

AccX1

AccX2

Acceleration (G)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.00 0.05 0.10

AccX1

AccX2

13

Figure 20. Comparison of Velocity and Acceleration History

Figure 21. Escort Mesh Geometry and Deformation Contours

VEHICLE-TO-POLE COLLISION – This simulation is of aFord Festiva collision with a rigid telephone pole. A simi-lar simulation of a Ford Festiva is presented by Grau [29]using the finite element model available from the GeorgeWashington University. The DyMesh Festiva model iscomposed of 5020 nodes and 9403 triangular elements.

-200

-100

0

100

200

300

400

500

600

700

0.00 0.05 0.10 0.15

Time (s)

Vel

ocity

(in

./s)

HyMesh

test

-50

-40

-30

-20

-10

0

10

0.00 0.05 0.10 0.15

Time (s)

Acc

eler

atio

n (G

)

HyMesh

test

DyMesh

FE Model

DyMesh

FE Model

t=0.025

t=0.050

t=0.075

t=0.100

14

The zone of impact has been refined to accuratelyresolve the contact with the simulated telephone pole.The rigid telephone pole is constructed from 1,053 nodesand 2,080 triangular elements and is 8.7 inches in diame-ter. The Festiva is traveling 20 mph when it impacts thetelephone pole as shown in Figure 22.

The mass associated with the Grau finite element simula-tion was suspect; however, for purposes of comparison,the same was used in the DyMesh simulation.[30] The

modified stiffness coefficients used are =9.12 lb/in2

and =1.80 lb/in3. Since the finite element model resultsshowed no rebound velocity from the pole, the unloadingstiffness was set to infinity so that there is no restitution.

Figure 22. Initial Positions for Festiva-Pole Collision

Figure 23 shows the deformed geometry at 20, 80, and100 milliseconds. The rigid pole has been removed fromthis figure for clarity. It can be seen that the vehicle takeson the circular shape of the pole. At 80 ms a few elementedges are seen stretched through the pole. Only nodesor vertices, not edges, have contact prescribed inDyMesh. (Please note the occasional irregularity in thewindshield and window mesh is due to rendering and isnot a result of DyMesh.)

Figure 24 shows the acceleration history plot of Grau’sfinite element simulation and the DyMesh simulation. Therise time for the DyMesh simulation is slightly longer, butthe agreement in peak acceleration is good with the dif-ference being less than ten percent. In this example theselected stiffness coefficients were too soft, resulting inthe underestimation of peak acceleration and the overes-timation of crush depth. Again, it is clear that the changein velocity agrees well with the finite element model (Fig-ure 25).

Figure 23. Deformed Geometry (pole not shown)

Figure 24. Acceleration Comparison

A

B

t=20ms

t=80ms

t=100ms

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

0.00 0.05 0.10 0.15

Time (s)

Acc

eler

atio

n (G

)

HyMesh

FE Model

DyMesh

FE Model

15

Figure 25. Velocity Comparison

VEHICLE-TO-VEHICLE COLLISION – This offset frontalimpact simulation between two Escorts is a severe test ofthe DyMesh algorithm. The irregular shapes of thebumpers and varying size of elements challenge thealgorithm to maintain a reasonable mesh shape duringdeformation.

Figure 26 shows the initial positions of the two vehicles.Both vehicles are assumed to weigh 2975 lb and have

yaw inertia of 20,000 lb-s2-in. The vehicle’s x-axes areinitially colinear. Both the left and right vehicles have aninitial velocity of 15 mph. Vehicle stiffnesses are

lb/in2 and lb/in3.

A time step of one millisecond is used in the simulation.Twenty milliseconds of simulation time are achieved forevery one minute of elapsed (wall-clock) time when run-ning on a Pentium II, 233 MHz processor. The simulationis run until the X velocity of both vehicles reaches zero.

Figure 27 shows both vehicles at various times during thesimulation. The vehicles begin to rotate due to the offsetforces. At t=88 ms the vehicles are shown contactingeach other as well as with the left vehicle removed toexpose the current damage profile produced by DyMesh.

Correct restoration in the contact algorithm is seen in thedamage profile. Incorrect restoration would result in thevehicle meshes moving through one another withoutdamage. The robustness of the contact algorithm inDyMesh is evidenced by the smooth and continuousappearance of the damage (Figure 28) even though thevehicle surface mesh uses elements of different sizesand aspect ratios.

Figure 26. Initial Positions for the Offset Frontal Impact

Figure 27. Vehicles at Various Times

Figure 29(top) shows the X velocity of each vehicle whichillustrates the symmetry of the collision simulation. Figure

29(bottom) shows the crush area calculated for eachvehicle. It is important to note that the crush areas arecalculated independently for each vehicle. For a symmet-ric collision such as this one, the crush areas would ide-ally be identical. The plot indicates the areas are thesame, except for a small time around 70 milliseconds.

0.0

100.0

200.0

300.0

400.0

0.00 0.05 0.10 0.15

Time (s)

Vel

ocity

(in

./s)

HyMesh

FE Model

DyMesh

FE Model

A 14.0= B 3.3=

t=32ms

t=64ms

t=88ms

t=88ms

16

DISCUSSION

The DyMesh method has the potential to allow more real-istic collision simulations than previously possible in aninteractive simulation environment. Three-dimensionalforces and moments, as well as damage profiles, can bevisualized. Vehicle damage is not as realistic as with thefinite element method (i.e., DyMesh will not simulatehood buckling), but the results for change in velocity aregood. The simulation run times are on the order of sec-onds or minutes, not several hours which is typical offinite element methods.

The overall resolution of the simulation is dependentupon the mesh resolution, as was demonstrated in thetelephone-pole example. Penetration of bodies smallerthan the mesh cannot be resolved by DyMesh.

Figure 28. Close-up Of Damage Profile

The method is not dependent upon the vehicle shapebeing described by a simplified geometric body, such asa parallelepiped. The method holds promise for car-to-barrier, car-to-car, heavy truck and articulated vehiclecollisions. Underride can be simulated in this three-dimensional method.

The DyMesh method requires a vehicle mesh and 3-

dimensional stiffness coefficients, and . Meshes ofreasonable resolution are available from a variety ofsources. [e.g., 31,32]. Initial tests presented in this paper

suggest and calculated from currently available Aand B coefficients are sufficient for speed change calcu-lations required in crash reconstruction. This finding issupported by previous experience with collision simula-tion programs (e.g., [14,16]). Experience has shown

these estimates are also sufficient for use as a startingpoint for rollover simulation [33].

Collision pulses from DyMesh may also be used for otherpurposes, including occupant simulations. For these

uses, the proper selection of and is more important.For these applications, the user must confirm the simu-lated and measured crush depths match. At this time it isnot felt that additional testing is required to use DyMesh.

Figure 29. Velocity and Crush Area Histories

Further work needs to be performed to evaluate the useof DyMesh in collisions wherein the mechanism of con-tact is primarily shear, for example, the case of a passen-ger car sliding under the side of a semi-trailer.

CONCLUSIONS

The DyMesh method is described for calculating thethree-dimensional interactions of colliding vehicles orstructures. This method makes possible a more realisticnumerical calculation and enhances the visual effective-ness of the simulation.

A B

A B

A B

0

200

400

600

800

1000

1200

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

Cru

sh A

rea

(sq

in.)

Veh 2

Veh 1

-300.0

-200.0

-100.0

0.0

100.0

200.0

300.0

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

X V

eloc

ity (

in./s

)

Veh 2

Veh 1

17

A novel method for computing the vehicle deformationsbased on a contact algorithm has been implemented.The force-crush model is derived which transforms theusual two-dimensional relationship to three-dimensions.The equations of motion are presented which depend onthe three-dimensional collision forces output fromDyMesh.

Examples have shown the DyMesh method to be robustand effective. This work lays the foundation for a morethorough validation effort to be undertaken in the nearfuture.

REFERENCES

1. U.S. Dept. of Transportation, National Highway TrafficSafety Administration, Washington, DC, 1998 (1997and 1998 figures are estimated).

2. Emori, R.I., “Analytical Approach to Automobile Colli-sions,” SAE Paper No. 68016, Society of AutomotiveEngineers, Warrendale, 1968.

3. Campbell, K.L., “Energy as a Basis for AccidentSeverity - A Preliminary Study,” Doctoral Thesis, Uni-versity of Wisconsin, Dept. of Mechanical Engineer-ing, Madison, 1972.

4. Campbell, K.L., “An Energy Basis for Collision Sever-ity,” SAE Paper No. 74565, Society of AutomotiveEngineers, Warrendale, 1974.

5. McHenry, R.R., “Extensions and Refinements of theCRASH Computer Program Part I, Analytical Recon-struction of Highway Accidents,” PB76-252114,1976.

6. Noga, T., Oppenheim, T., “CRASH3 User's Guideand Technical Manual,” NHTSA, U.S. Dept. of Trans-portation, Washington, DC, 1981.

7. Day, T.D., Hargens, R.L., “An Overview of the WayEDCRASH Computes Delta-V, “SAE Paper No.870045, Society of Automotive Engineers, Warren-dale, 1987.

8. Bigg, G., Moebes, T., WinCrash User's Manual, ARSoftware, Redmond, 1996.

9. Fonda, A.G., “CRASH Extended for Desk and Hand-held Computers,” SAE Paper No. 870044, Society ofAutomotive Engineers, Warrendale, 1987.

10. Woolley, R., “The IMPAC Computer Program for Acci-dent Reconstruction,” SAE Paper No. 850254, Soci-ety of Automotive Engineers, Warrendale, 1985.

11. Smith, G., “Conservation of Momentum Analysis ofTwo-Dimensional Colliding Bodies, With or WithoutTrailers, “SAE Paper No. 940566, Society of Automo-tive Engineers, Warrendale, 1994.

12. Stephan, H., Moser, A., “The Collision and TrajectoryModels of PC-CRASH,” SAE Paper No. 960886,Society of Automotive Engineers, Warrendale, 1996.

13. McHenry, R.R., Jones, I.S., Lynch, J.P., “Mathemati-cal Reconstruction of Highway Accidents - SceneMeasurement and Data Processing System,” Cal-span Report No. ZQ-5341-V-2, DOT HS-801 405,February, 1975.

14. Day, T.D., Hargens, R.L., “An Overview of the WayEDSMAC Computes Delta-V,” SAE Paper No.880069, Society of Automotive Engineers, Warren-dale, 1988.

15. Bigg, G., Moebes, T., WinSmac User's Manual, ARSoftware, Redmond, 1998.

16. Day, T.D., “An Overview of the EDSMAC4 CollisionSimulation Model,” SAE Paper No. 1999-01-0102,Society of Automotive Engineers, Warrendale, 1999.

17. Taylor, L.M., and Flanagan, D.P., “PRONTO 3D AThree-Dimensional Transient Solid Dynamics Pro-gram, SAND87-1912, Sandia National Laboratories,March 1989.

18. S.W. Attaway, “Update of PRONTO 2D and PRONTO3D Transient Solid Dynamics Program,” SandiaNational Laboratories, SAND90-0102, November1994.

19. T. Belytschko and J.I. Lin, “A New Interaction Algo-rithm with Erosion for EPIC-3,” U.S. Army BallisticResearch Laboratory Report BRL-CR-540, February1985.

20. J.O. Hallquist and D.J. Benson, “DYNA3D user’smanual (nonlinear dynamic analysis of structures inthree dimensions,” Lawrence Livermore NationalLaboratory, UCID-19592, Rev. 3, 1987.

21. J.D. Gruda and A.R. York, “Crashworthiness of theAT-400A Shipping Container,” Development, Valida-tion, and Application of Inelastic Methods for Struc-tural Analysis and Design, PVP-Vol 343, 1996International Mechanical Engineering Congress andExposition, pp. 115-121.

22. A.R. York and A.M Slavin, “Design and Analysis of aHigh-Performance Shipping Container for Large Pay-loads,” The 11th International Conference of thePackaging and Transportation of Radioactive Mate-rial (PATRAM ‘95), Dec 3-8, 1995, Las Vegas,Nevada, Proceedings Vol IV, pp. 1736-1743.

23. D.J. Benson and J.O. Hallquist, “A single surfacecontact algorithm for the post-buckling analysis ofshell structures,” Computer Methods in AppliedMechanics and Engineering, 78, pp. 141-163 (1990).

24. J.H. Biffle, “JAC - A Three-Dimensional Finite Ele-ment Computer Program for the Nonlinear Quasi-Static Response of Solids with the Conjugate Gradi-ent Method, Sandia National Laboratories, SAND87-1305.

25. J.O. Hallquist, “NIKE3D: An implicit, finite deforma-tion, finite element code for analyzing the static anddynamic response of three-dimensional solids,”Lawrence Livermore National Laboratory, UCID-18822, Rev. 1., 1984.

26. M.W. Heinstein, S.W. Attaway, J.W. Swegle, and F.J.Mello, “A General-Purpose Contact Detection Algo-rithm for Nonlinear Structural Analysis Codes,” San-dia National Laboratories, SAND92-2141, April 1995.

27. U.S. Department of Transportation, National HighwayTraffic Safety Administration, “New Car AssessmentProgram (NCAP) Frontal Barrier Impact Test,” ReportNo. MGA-96-N018, September 23, 1996.

18

28. HVE Developer’s Toolkit, Version 2, EngineeringDynamics Corporation, Beaverton, OR, October1998.

29. Grau, C.A. and Huston, R.L., “Validation of an Analyt-ical Model of a Right-Angle Collision between a Vehi-cle and a Fixed, Rigid Object,” International Journalof Crashworthiness, V 3 No. 3, pp 249-264.

30. Personal communication with Cesar Grau, December7, 1998.

31. EDVDB Vehicle Database, Version 2, EngineeringDynamics Corporation, Beaverton, OR 1998.

32. Viewpoint Datalabs, Orem UT, 1998.

33. HVE User's Manual, Version 2, Engineering Dynam-ics Corporation, Beaverton, OR, 1998.

ABOUT THE AUTHORS

Allen R. York has worked in the area of computationalmechanics for ten years. Before forming A.R. York Engi-neering, Inc., he worked nine years at Sandia NationalLaboratories. He can be reached by email at: ary-ork@flash.net.

Terry D. Day has worked in the area of motor vehicle han-dling and collision simulation for 18 years. He can bereached at Engineering Dynamics Corporation by email:day@edccorp.com.

TRADEMARKS

DyMesh is a trademark of Engineering Dynamics Corpo-ration.