Animating Exploding Objects

Oleg Mazarak Claude Martins John Amanatides

Department of Computer ScienceYork University

AbstractThe paper explores the physically-based modeling of ablast wave impact on surrounding objects. We proposea connected voxel representation of objects to modelexplosions that result in realistic solid debris, ratherthan flat polygons. The paper also presents improvedfracture algorithms capable of accounting for the dam-age of multiple explosions. The important implementa-tion issues and the results of the simulation are dis-cussed.

Key words: Explosions, blast waves, connected voxels,spring-mass particle model, rigid bodies, solid model-ing, physically-based modeling

1 IntroductionDespite the growing interest in explosions, especially invisual effects, recent publications on the topic canhardly be found. Ever since the pioneering work ofReeves [Reeves83], who proposed using particle sys-tems to model explosions, the area of explosion anima-tion seems to have been abandoned by researchers.Particles are very effective at modeling certain after-effects of an explosion—fire and smoke—and creatorsof visual effects for commercial software have concen-trated mainly on perfecting these two phenomena.

An important visual aspect of an explosion is theblast wave it generates. Although the blast wave itself isalmost invisible, its influence on surrounding objects iseasily discernible and results in multiple deformations,breakage and debris. Realistic simulation of the blastwave impact and debris formation can enhance the vis-ual perception of the explosion by a viewer. Unfortu-nately, there is currently no tool on the market for mod-eling realistic debris. Existing commercial solutionseither create flat polygonal debris, which is no longersatisfactory for the growing demands of graphic artists,or force users to design debris templates explicitly,based on their best understanding of the explosion phe-nomenon.

This paper is intended to fill the gap and generateinterest inside the graphics community in the physi-cally-based modeling of explosions, by presenting amodel for the physically accurate simulation of debrisformation and motion. In this work, however, we dis-

cuss only the elementary blast wave equations, whichare used in our simulation to provide a well-balancedtrade-off between physical accuracy and simulationspeed. The full scientific simulation of the real physicalprocesses associated with explosions would be unnec-essarily complex and too computationally expensive forthe field of computer graphics, which is focused mainlyon the visualization of the blast wave impact on sur-rounding objects.

We introduce a connected voxel representation ofobjects, which allows the creation of realistic volumet-ric debris and eliminates certain disadvantages of thespring-mass particle model that arise when attemptingto model rigid bodies. We also show how to incorporateour blast wave model with a rigid body motion simula-tor to produce realistic animation of flying debris.

2 Theory of Blast WavesAny rapid release of energy in air, such as an explosion,generates a blast wave. The front of this blast waveexhibits a nearly discontinuous increase in pressure,density, and temperature, and is called a shock front.

The transmission of real blast waves in air is an in-herently nonlinear process involving nonlinear equa-tions of motion. In this sense, a blast wave differs quitesignificantly from an acoustic wave, which involvesonly infinitesimal pressure changes and moves at sonicvelocity.

An ideal blast wave, which can be used as a suffi-cient approximation, is generated under the assump-tions that the medium—air—is still and homogenous,and that the source is spherically symmetric. The shockfront of an ideal blast wave is perfectly spherical, andtherefore the characteristics of the blast wave are func-tions only of distance R from the center of the source,and the simulation time t [Baker73].

2.1 Blast Wave Pressure ProfileSince the blast wave’s impact on surrounding objects isinfluenced primarily by pressure, we are mainly inter-ested in simulating the pressure changes across theshock front.

The pressure profile generated by an ideal blastwave at some location removed from the center of ex-plosion is shown in Figure 2-1. Before the shock front

reaches the given point, the pressure is equal to the am-bient pressure p0. At arrival time ta, the pressure risesdiscontinuously to the peak value of p0 + Ps

+. Thequantity Ps

+ is called the peak overpressure. The pres-sure then decays to ambient in total time ta + T+ (posi-tive phase), drops to a partial vacuum of value p0 - Ps

-

(negative phase), and eventually returns to the ambientpressure p0 in total time ta + T+ + T -.

The pressure profile curve can be accurately de-scribed by the modified Friedlander equation[Baker73]:

p(t) = p0 + Ps+ (1 – t / T+ ) e bt T− +/ (2.1)

where time t is measured from time of arrival ta. Theblast wave parameters Ps

+, ta, T+, and b allow freedom

to customize the pressure profile curve and the initialdecay rate for different explosions, at different dis-tances from the source.

Given an ideal point source explosion, the peakoverpressure Ps

+ initially decays as the inverse of thecube of the radius 1/R3, and as the shock approaches thestrength of a sound pulse, the decay approximates theinverse of the radius 1/R.

2.2 Dynamic Fracture of ObjectsThe mechanism of body fracture and fragmentationunder sharp blast wave loads is different from that un-der ordinary static loads. Blast wave propagation cre-ates strong tensile stresses inside a solid object. Sincemost materials are substantially weaker under tensionthan under compression, the process of dynamic frac-turing, called spalling, occurs. Spalling explains theeffect of multiple, nearly uniformly distributed fractur-ing of solid bodies hit by a blast wave. Further propa-gation of the wave causes the fractures to grow, and thebody is eventually fragmented into many parts. Due tothe initial presence of multiple micro-fractures, calledflaws, existing in most materials, spalling usually oc-

curs for stresses lower than theoretical predictions[DGS96].

Davison and Stevens [Meyers94] proposed the con-cept of a continuum measure of spalling, based on re-viewing and systematizing existing spall criteria. Davi-son and Stevens’ theory is phenomenological in thesense that no detailed mechanisms for the initiation andpropagation of fracture is incorporated. According totheir compound-damage accumulation hypothesis, thenumber of cracks (damage D) inside a flawed micro-structure grows exponentially:

]1)[exp(* −Ε= tCBDD (2.2)

|)|(2

100 σσσσ −+−=Ε (2.3)

where D* represents the damage at total separation, ormaximum possible damage, coefficients B and C arematerial-dependent constants, σ is the actual stresscaused by the blast wave load, and 0σ is a critical stress

below which there is no damage.

3 Modeling

3.1 Blast Wave ModelThe blast wave model employs the Friedlander equation(Equation 2.1) to compute physically accurate pressurechanges at any distance R from the source of the explo-sion. The blast wave parameters required by (2.1) areobtained from experimental data for a reference explo-sion of one ton of TNT (trinitrotoluene) in a standardatmosphere [Kinney62]. The experimental data containsvalues for peak overpressure Ps

+, expected arrival timeta, positive phase duration T+, and the pressure decaycoefficient b, measured at certain distances R1, R2, …,Rn, from the source. The corresponding parameters forany arbitrary distance R are obtained by interpolation.

Time and distance related parameters for explosionswith different yields than one ton of TNT are derivedfrom the reference explosion data by using an appropri-ate scaling factor as determined by the scaling laws[Kinney62]. This scaling factor is normally equal to theinverse of the cube root of the energy of the derivedexplosion.

In order to use our blast wave model in a dynamicssimulator, the forces exerted on objects have to be de-rived from the pressure values. This process, however,is better understood after discussing object representa-tion. The force computation is presented in detail insection 4.1.

Figure 2-1. Pressure-time curve of an ideal blastwave

p0 + P+SP

RESSURE

TIME

0

Positive phase

p(t) Negative phase

p0

p0 – P-S

ta ta + T+ ta + T+ + T- 0

3.2 Object RepresentationModeling exploding objects requires creating a struc-ture that incorporates some connectivity information, inorder to simulate the object’s fracture and fragmenta-tion. A good example of a connectivity-based repre-sentation is the spring-mass particle model, which wasinitially proposed by Terzopoulos [TPBF87]. A well-known problem with the spring-mass particle model isits poor ability to model rigid, inflexible objects. In-creasing the spring stiffness coefficient to make theobject more rigid leads to the “stiffness” of the gov-erning ordinary differential equations (ODEs), whichbecome difficult to solve. In such cases, the slightestimbalance in the system eventually causes the system tocrash.

Another difficulty with the spring-mass particlemodel is the fact that it represents only the object’sskeleton and requires the association of a surface to theparticles before the modeled object can be rendered.This problem can be avoided when the simulation speedis not important, or when the number of objects in thescene does not change during the simulation and thesurfaces can be computed during the initialization step.Simulating an explosion, however, entails the continualcreation of new objects—debris—during the simula-tion. Computing surfaces to these multiple bodies dy-namically can get unacceptably expensive for thesimulation because the debris usually has a complexshape, and the accurate computation of the exact sur-faces is non-trivial.

To overcome the difficulties described above, wepropose modeling exploding bodies with connectedvoxels. The connected voxel model is a simple volu-metric representation of objects, based on spatial occu-pancy enumeration [Foley90]. Connectivity informationis preserved by the introduction of links between adja-cent voxels. Links can be thought of as springs that aremade infinitely stiff (in physical terms), and which donot allow any flexibility in the body, keeping adjacentvoxels attached firmly together.

Objects represented with connected voxels are ani-mated in a different way than objects represented withthe spring-mass particle model. In particle dynamics,each particle is considered an independent moving ob-ject, which does not make sense for connected voxels.Connected voxels are immovable with respect to eachother and can be grouped into more complex structures,namely bodies. The world position of a particular voxel

is computed from the position and orientation of thebody to which that voxel belongs.

Voxel-represented objects can be displayed by asso-ciating a desired shape to voxels, for example, a cube.Rendering time is linear on the number of voxels; how-ever, only boundary voxels are visible and require ren-dering. Since our representation already contains theadjacency information, the determination of visiblesurfaces is simply based on the observation that bound-ary voxels lack at least one neighbor out of six. Thecorresponding face is visible and has to be rendered.

Voxels, in principle, can be made as small as re-quired in order to reduce the “staircased edges” prob-lem. If simulation speed is not of primary concern, ei-ther the final image can be anti-aliased, or more sophis-ticated volume-rendering algorithms can be used[Kaufman91].

3.3 Simulating Fracture with Connected VoxelsThe fracture of an object is simulated by “breaking”links in the connected voxels structure, which imitatesthe formation of cracks inside a solid. When the numberof broken links grows, the object may be split intofragments. In order to compute the split fragments, theentire scene is represented with a connectivity graph. Itsnodes are associated with voxels and its arcs with thelinks that connect voxels. The connected components ofsuch a graph correspond to the independent bodies inthe scene.

Although simulating fracture with connectivity-based models, in particular with spring-mass particlemodels, is not new to computer graphics ([TF88],[Norton91]), simulating fracture with connected voxelsis different in the sense that using links requires findingnew criteria for making a decision on when to break alink. We utilize the capabilities of our blast wave modelto compute the pressure at the midpoint of each link. Alink is broken whenever the pressure exceeds the link’syield limit, which is the maximum pressure that the linkcan withstand before breaking.

Figure 3-1. Connected voxels

Initial graph

Connectedcomponents

Figure 3-2. Graph representation of the connected voxel model

In our work, we have experimented with severalfracture algorithms. These fracture algorithms producedifferent kinds of debris, and therefore the most appro-priate one may be chosen for any particular case.

The simplest one relies on the variable link yieldlimit, which is computed by perturbing some randomvalue, generated within a certain range, around somemean value [TF88]:

yield_limit1 = mean_yield ± rand(var_yield) (3.1)

Assigning a variable yield limit creates a portion ofweak links, which naturally simulates the existence offlaws in the material structure.

In the improved algorithm, links that encounter thefirst explosion are weakened explicitly by reducingtheir yield limit, which makes them an easy target forsubsequent explosions. This method simulates partialdamage to the object, which does not necessarily resultin link breakage.

A further improvement of the fracture algorithm ac-counts for the orientation of the links relative to thedirection of the blast wave. In real explosions, the ten-sile forces created by the blast wave inside an object aremuch stronger in the direction parallel to the directionof the wave than normal to it. Based on this observa-tion, links that are parallel to the direction of the waveare weakened by the orientation factor, computed as adot product of the wave’s radius vector and the linkvector.

A different approach to simulating fracture is an al-gorithm that does not rely on the link yield limit, butrather on observed fracture patterns. The algorithm em-ploys the compound-damage accumulation hypothesisof Davison and Stevens, discussed in section 2.2(Equations 2.2 and 2.3). The incremental damage D iscomputed for each voxel to determine the number oflinks that have to be broken at the current simulationstep in order to produce a realistic simulation of dy-namic fracture. Here, D* represents the maximum num-ber of links that can be broken (six per voxel), and theparameter Ε is calculated as follows:

)_)(_)((2

1yieldmeantpyieldmeantp −+−=Ε (3.2)

where p(t) is a function that returns the pressure value(see section 3.1) at the center of a given voxel at time t.

The fractures inside a solid object are computed dy-namically. At each time step, the fracture simulator isrun in order to determine whether some links should bebroken. If at least one link is broken at this stage, werun a breadth-first search to compute the fragments.

4 Implementation

4.1 AnimationA blast wave exerts forces on an object similar to thosecreated by a very strong wind. Our model evaluatesthese forces at the centers of the voxels, as a product ofthe pressure generated by the wave and the area of thevoxels projected onto the surface of the wave. Assum-ing a spherical blast wave, the force vector is congruentto the blast wave radius vector:

R

RAtpF

�

�

�

⋅⋅= )( (4.1)

where p(t) is a function that returns the pressure value(see section 3.1) at the center of a given voxel at timet, R

�

is a radius vector from the source of the explosionto the voxel, and A is the projected surface area of thevoxel. Simultaneously, we compute the torque on thebody as a result of the force acting on a specific voxelof the body:

FrT�

�

�

×= (4.2)

where r�

is the relative position of the voxel to thecenter of mass of the body. The force and torque vec-tors only need to be computed for boundary voxels,which approximate the surface of the body. The dy-namic simulator then uses the summed up force andtorque vectors to update the body position and orienta-tion respectively.

In our model, we used the generic rigid body motionsimulator described in [BW97]. We do not present thebasic mechanics equations here. The reader can refer to[BW97], [Haug92], and [Shabana89] for a completedescription of the rigid body dynamics. In order tosolve the governing ODEs, a desirable integrationmethod can be chosen. We opted for Euler’s method,the simplest one, because we wanted to see how themodel behaves in the basic case.

Using quaternions to represent rotation instead ofrotation matrices is more advantageous, because thequaternion normalization step can be used to eliminatethe rotation accumulation error, which may occur dueto the integration process. More information on quater-nions can be found in [Shoemake85] and [BW97].

The linear and angular momentum of the newly cre-ated body fragments are computed as weighted portionsof the initial momentum of the body before it has beenfragmented, which agrees with the law of conservationof momentum and ensures smooth motion of the splitfragments.

4.2 Collision DetectionA discrete representation of the bodies using voxels isemployed to simplify collision detection. In this work,we chose not to concentrate on efficient collision de-tection, and instead used a simple pair-wise test be-tween body voxels. Collision between two bodies isconsidered to occur when the distance between theirvoxels becomes less than the expected limit. The dis-tance between two voxels is defined as the length of thedistance vector connecting the centers of the voxels.

To prevent inter-penetration of the bodies, theproper collision response is computed. The collisionresponse algorithm changes the linear and angular mo-mentum of the bodies discontinuously ([BW97],[MC95]), simulating the collision with the little to nodeformation of the colliding objects specific to rigidbodies. The computation of collision response requiresthe determination of the collision region, which in-cludes computing the point of collision and the vectornormal to the body surfaces at this point. To avoid thetime-consuming procedure of surface reconstruction,we assume that the surfaces of the bodies are smooth.We approximate the collision normal with the distancevector between the colliding voxel, and the collisionpoint with the midpoint of the distance vector (Figure4-1).

Due to the extremely high velocities of the bodieshit by the blast wave, some bodies may interpenetratenoticeably in a single time step. This problem is solvedby reducing the time step adaptively until the inter-penetration is kept within a certain tolerance.

The voxel enumeration algorithm for collision de-tection can be optimized by limiting collision testing toonly boundary voxels (see section 3.2). Further optimi-zation of collision detection with buckets ([Norton91],[MC95]) reduces the total number of required tests sig-nificantly. In this approach, the scene is spatially subdi-vided into smaller regions called buckets, and eachvoxel is assigned the number of the bucket it is in at thecurrent moment in time. Only voxels from the same oradjacent buckets need to be tested for collisions.

Unfortunately, optimization methods that rely onany kind of pre-computation step offer little to no bene-fits for explosion simulation because the number ofbodies in the scene changes continually due to bodyfragmentation. Also, techniques that operate on polyhe-dral objects—such as I-collide [Cohen95] and V-Clip[Mirtich98]—are not directly applicable to our voxel-represented model without the additional step of associ-ating polygon surfaces to the bodies, which can be ex-pensive.

4.3 User ToolsThere is an important trade-off between completephysical accuracy and increased user control over thesimulation. It is frequently the case that some unrealis-tic simulation parameters can be chosen to producemore spectacular visual effects. In our implementation,the user can control the power of the explosion, as wellas the model parameters, such as the mean and variablecomponents of the link yield limit. Other controls in-clude the base simulation time step, voxel mass, dragand gravity coefficients, etc.

5 ResultsWe have generated several video clips showing a blastwave impact on structures modeled with connectedvoxels. The visual results of the simulation are verysimilar to actual footage shot during nuclear bombtesting. Depending on the fracture algorithm used, wewere able to generate various types of explosions. Eventhough we used a simple Euler’s integration method,the simulation was both smooth and robust. Selectedframes from the animations created using the connectedvoxel model are shown in Figures 5-1 and 5-2.

For simple models, the dynamic simulator is capa-ble of generating frames over the real-time threshold.The bottleneck of the simulation is collision detection.Without collision detection, the computational cost ofthe dynamic fracture simulation process has an upperbound of the time required to compute connected com-ponents, which is O(n), where n is the number of voxelsin the scene. Collision detection, however, increases thesimulation time up to O(m2), where m is the number ofboundary voxels of all bodies in the scene. As the num-ber of bodies grows due to fragmentation, so does thenumber of boundary voxels, and the simulation slowsdown. The use of buckets improves the frame rate byapproximately a factor of log m over the non-optimizedcollision detection (see Table 5-1).

Smooth surfaceapproximation

Figure 4-1. Collision approximation

6 Conclusions and Future WorkThe paper has proposed a new model—connected vox-els—for the realistic animation of a blast wave impacton solid objects. Visual realism of the animation isachieved due to both the volumetric representation ofan exploding object, which allows the creation of con-vincing solid debris, and the use of physically-basedmethods to compute object fracture and motion. Im-proved fracture algorithms allow the simulation to takeinto account the damage of multiple explosions. Themodel can be successfully used in games, virtual realityapplications, and visual effects, by choosing the appro-priate level of accuracy in the object representation andthe simulation methods.

Our current blast wave model leaves room for ex-tensions and improvements, which would allow thesimulation of more complex wave interactions with theenvironment. Possible refinements include the simula-tion of non-uniform pressure over the wave front due toshock absorbency of intervening objects, and modelingthe compression of air in front of impacted objects. Theimplementation can be improved with more efficientcollision detection and more sophisticated fracturesimulation algorithms.

7 References[Baker73] Baker, W.E., Explosions in Air, University ofTexas Press, 1973.[Baraff89] Baraff, D., “Analytical Methods for Dy-namic Simulation of Non-penetrating Rigid Bodies” ,SIGGRAPH 89, pp. 223-232, 1989.[BW97] Baraff, D., and Andrew Witkin, “Physically-based Modeling: Principles and Practice” , CourseNotes, SIGGRAPH 97, 1997.[BW98] Baraff, D., and Andrew Witkin, “Large Stepsin Cloth Simulation” , SIGGRAPH 98, pp. 43-54, 1998.[Cohen95] Cohen, J.D., M. C. Lin, D. Manocha, and M.K. Ponamgi, “ I-collide: An interactive and exact colli-sion detection system for large-scaled environments” ,Proceedings of the ACM Symposium on Interactive 3DGraphics, pp. 189-196, 1995.[DGS96] Davison, L., Dennis E. Grady, and MohsenShahinpoor, editors, High Pressure Shock Compression

of Solids II: Dynamic Fracture and Fragmentation,Springer-Verlag, 1996.[Foley90] Foley, J. D., Andries van Dam, Steven K.Feiner, John F. Hughes, Computer Graphics: Principlesand Practice, Addison-Wesley, pp. 549-550, 1990.[Hahn88] Hahn, J. K., “Realistic Animation of RigidBodies” , SIGGRAPH 88, pp. 299-308, 1988.[Haug92] Haug, E. J., Intermediate Dynamics, PrenticeHall, 1992.[Kaufman91] Kaufman, A., Volume Visualization,IEEE Computer Society Press, 1991.[Kinney62] Kinney, G. F., Explosive Shocks in Air, TheMacmillan Company, 1962.[Lawn75] Lawn, B. R., Wilshaw T. R., Fracture ofBrittle Solids, Cambridge University Press, 1975.[MC95] Mirtich, B., Canny, J., “ Impulse-based Simu-lation of Rigid Bodies” , 1995 Symposium on Interactive3D Graphics, pp. 181-188, 1995.[Meyers94] Meyers, M. A., Dynamic Behavior of Ma-terials, John Wiley & Sons, 1994.[Mirtich98] Mirtich, B., “V-Clip: Fast and RobustPolyhedral Collision Detection” , ACM Transactions inGraphics, Vol. 17, No. 3, pp. 177-208, July 1998.[MW88] Moore, M., and Jane Wilhelms, “CollisionDetection and Response for Computer Animation” ,SIGGRAPH 88, pp. 289-298, 1988.[Norton91] Norton, A., Greg Turk, Bob Bacon, JohnGerth, Paula Sweeney, “Animation of Fracture byPhysical Modeling” , Visual Computer, pp. 7:210-219,1991. [PB88] Platt, J.C., and A.H. Barr, “Constraint Methodsfor Flexible Models” , SIGGRAPH 88, 279-288, 1988.[Reeves83] Reeves, W.T., “Particle Systems—A Tech-nique for Modeling a Class of Fuzzy Objects” , SIG-GRAPH 83, pp. 359-376, 1983.[Shabana89] Shabana A. A., Dynamics of MultibodySystems, John Wiley & Sons, 1989.[Shoemake85] Shoemake, K., “Animating Rotationwith Quaternion Curves” , SIGGRAPH 85, pp. 245-254,1985.[TF88] Terzopoulos, D., and K. Fleischer, “ModelingInelastic Deformation: Viscoelasticity, Plasticity, Frac-ture” , SIGGRAPH 88, pp. 269-278, 1988.[TPBF87] Terzopoulos, D., John Platt, Alan Barr, andK. Fleischer, “Elastically Deformable Models” , SIG-GRAPH 87, pp. 205-214, 1987.[WK88] Witkin A., and Michael Kass, “SpacetimeConstraints” , SIGGRAPH 88, pp. 159-168, 1988.

Frames/secNumber ofboundary

voxels

Nocollisiondetection

Collisiondetectionwithoutbuckets

Collisiondetection

with buckets

1 – 10 > 100 > 100 > 10010 – 100 70 20 15

100 - 1,000 15 0.25 51,000 - 10,000 < 1 < 0.017 < 0.33

Table 5-1. Frame rates for connected voxel representation. Theexperiments were performed on an Intel 333 MHz Pentium II,Windows NT 4.0 SP 4, FireGL 1000 Pro graphics accelerator.

(a)

(c)

(e)

(b)

(d)

(f)

Fig. 5-1. Explosion of a rigid body represented by connected voxels

(a)

(c)

(e)

(b)

(d)

(f)

Fig. 5-2. Explosion of multiple rigid objects