Integrating Mesh and Meshfree Methods for Physics Based ...vis.pku.edu.cn/research/publication/pbg2006-simulation.pdf · Integrating Mesh and Meshfree Methods for Physics Based Fracture

Eurographics Symposium on Point-Based Graphics (2006), pp. 1–9M. Botsch, B. Chen (Editors)

Integrating Mesh and Meshfree Methods for Physics BasedFracture and Debris Cloud Simulation

AbstractWe present a hybrid framework for physics-based simulation of fracture and debris clouds. Previous methodsmainly consider bulk fractures. However, in many situations, small fractured pieces and debris are visually impor-tant. Our framework takes a hybrid approach that integrates both tetrahedron-based finite element and particle-based meshfree methods. The simulation starts with a tetrahedral mesh. When the damage of elements reachesa damage failure threshold, the associated nodes are converted into mass-based particles. Molecular dynamicsis used to model particle motion and interaction with other particles and the remaining elements. In rendering,we propose an algorithm of dynamically extracting a polygonal boundary surface for the damaged elements andparticles. Our framework is simple, accurate, and efficient. It avoids the remeshing and stability problems of puremesh-based techniques and other meshfree methods and offers high visual realism.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling—Physically based modeling; I.3.7 [Computer Graphics]: Three-Dimensional Graphics andRealism—Animation

1. Introduction

Realistic animation of material fracture and the subsequentfractured debris clouds remains a challenging problem inComputer Graphics community. Prominent existing meth-ods [MBF04, OH99, PKA∗05] for physics-based fracturesimulation mainly consider bulk fractures. For long term to-tal effect of the material damage/fracture/penetration anima-tion, the failed material is one of the crucial parts of the mod-eling, since the interaction between the failed material andthe unfractured bulk material will cause further fracture ofthe bulk material and dictates the post initial fracture behav-ior of the complete animation. Current fracture simulationapproaches are unable to model the fracture debris and thebehavior of the failed material accurately and efficiently yet.

In this paper we propose a hybrid solution to the anima-tion of the fracture behavior of multiple body interactions.Our approach initially models the entire interaction systemwith the finite element method. When the continuum dam-age of the elements reaches a critical state, the nodes of thedamaged elements are converted into particles and subjectedto the associated pressure. The particles are modeled by themolecular dynamics (MD) method to account for the inter-action between the particles, and the interaction with the un-fractured elements. This generalized framework for animat-

ing the fracture behavior of multiple objects and interactionsis a composite of individual ingredients of the state-of-the-art development of the respective research fields. The com-bination of both mesh and meshfree techniques makes thisapproach a hybrid simulation method. Figure 1 shows sucha simulation result computed by our simulation framework.

Figure 1: A vase shattered onto the ground, where particlesare used to model the fractured debris cloud. A zoomed-inview of the boxed region is shown at the top-right corner.

submitted to Eurographics Symposium on Point-Based Graphics (2006)

2 / Integrating Mesh and Meshfree Methods for Physics BasedFracture and Debris Cloud Simulation

The primary reasons for the choice of the ingredients in ourapproach are: FEM has been well established for model-ing contact problems, but has difficulty in handling discretefields and changing boundary. Meshfree methods are goodat modeling debris clouds. In some physical events such asdeep penetration, debris clouds play an important role forthe visual effects. Thus, the combination of the two methodsbecomes crucial. Our contributions in this paper are:

• The integration of both mesh and meshfree methods, inparticular, the particle based MD method with FEM. Weaccurately model the underlying physics for the anima-tion of the fracture behavior of multiple objects and theirinteractions to provide a realistic scenario of the physics.

• A visualization technique to dynamically extract a polyg-onal boundary surface for failed elements and particles,and a reconstruction of the physical representation of dy-namic systems from the discrete simulation outputs.

This paper is organized as follows. We discuss the relatedwork in Section 2. In Section 3 we first give an outline of thesimulation framework, and then present the physical lawsgoverning the simulation. In Section 4 we propose a visual-ization technique to extract the polygonal boundary surfacesfor the particles. Experimental results are presented in Sec-tion 5. Finally we conclude the paper in Section 6.

2. Related Work

Previous work in fracture simulation can be classified asnon-physics, FEM, and meshfree methods. Non-physicsbased methods are popular in the early stage of fracturemodeling. Neff and Fiume [NF99] model the destructed wallstructure caused by a spherical blast wave using an artificialpattern generator. Hirota et al. [HTK98] use a mass-springmodel to create static crack patterns. Smith et al. [SWB01]and Norton et al. [NTB∗91] model dynamic cracks, such asa teapot shattered onto the ground, using a constraint basedmethod and a mass-spring method, respectively. Althoughthese methods allow for easy control of fracture patterns andare simple and fast, they do not provide very realistic results.

Terzopoulos and Fleischer [TF88] have pioneered physicsbased simulation by using finite differences to handle plas-tic deformation and cloth tearing. Later on, based on re-search from computational mechanics, researchers started towork on finite element methods that directly approximatethe equations of continuum mechanics. O’Brien et al. arethe first to apply this technique for simulating brittle frac-tures [OH99] and ductile fractures [OBH02]. To conformwith the fracture lines that are derived from the principalstresses, elements are dynamically cut and remeshed. How-ever, remeshing can be expensive and reduces the time stepfor simulation. To avoid these issues, Molino et al. [MBF04]propose a virtual node algorithm, where elements are du-plicated and fracture surfaces are embedded in the copiedtetrahedra. Built upon the recent advances in meshfree meth-

ods for computational mechanics, Pauly et al. [PKA∗05] ex-tend the meshfree surface deformation framework of Mülleret al. [MKN∗04] to deform solid objects that fracture. Intheir system, both sparse simulation nodes and dense bound-ary points are maintained. To adapt the simulation nodes tothe fractures, new simulation nodes are inserted. Similarly,new points are generated at the crack interface. The explicitmodeling of advancing crack fronts and associated fracturesurfaces makes it easy to generate highly detailed, complexfracture patterns, but requires handling of topological opera-tions on crack fronts. In our application domain, topologymaintaining operations are very difficult to achieve. Notethat it is also possible to model fracture surfaces implic-itly in a meshfree fracturing method, such as the level setmethod [VXB02]. However, it is expensive to model a largeamount of tiny fractures using such techniques.

Particle-based methods have been widely used in computergraphics for simulating a wide range of amorphous phe-nomena, such as fluid [FF01], smoke [FSJ01], and explo-sion [FOA03]. In these methods, particle motion is the pri-mary concern and is computed by physical equations suchas the Navier-Stokes. Particles have also been used for gran-ular material simulation. Bell et al. [BYM05] use anisotrop-ical particles and molecular dynamics equations to simulatesplashing and avalanche of sands. They also compute inter-acting forces between particles and rigid objects by sam-pling the boundary surfaces of rigid objects into particles,but no fractures are involved. For deformable solid model-ing, Desbrun and Cani [DC95] model soft inelastic materialsusing particle systems coated with a smooth iso-surface. Inall these methods, particles are generated in the beginning,which is much different from our simulation method thatgenerates particles from damaged elements. Another distinctdifference is that in these systems particles are used to modelamorphous objects and can be rendered using texture splatsor isosurfaces. In our method, particles are given a clear,sharp, polygonal boundary to provide meaningful simulationof the actual fragmented pieces.

The meshfree/particle method was initially developed by thecomputational mechanics community. A detailed summaryof this technique is beyond the scope of this paper. For someexcellent reviews, please refer to [LIU02]. In hybrid par-ticle/FEM interaction, Johnson et al. [JBS00, JBS01] pro-pose a generalized particle algorithm (GPA) for high veloc-ity impact and other dynamics problems, where a variationof the smooth particle hydrodynamics (SPH) method is usedto model particle motion, which is different from the molec-ular dynamics method used in our framework.

3. Simulation

3.1. Overview

Consider an interactive system of at least two individual con-tinuum objects with initial velocities and subjected to a grav-


/ Integrating Mesh and Meshfree Methods for Physics BasedFracture and Debris Cloud Simulation 3

ity field. The objects are discretized in space by finite ele-ment meshes with all the surface elements treated as mas-ter contact segments and all the surface nodes treated asslave contact nodes. Contact search is performed in everytime step to find the corresponding contactable master seg-ment and slave node pairs. When impact occurs, the objectssubjected to the contact forces undergo finite strain visco-elasto-plastic deformation. If the accumulated effective plas-tic strain of any given Gauss point of the element reachesthe threshold value, the continuum damage effect needs tobe accounted for the deviatoric stress. At the same time,the equation of state (EOS) is considered for the hydrostaticpressure of the material to account for the shock physics.When the damage of a Gauss point reaches a critical value,the material of the corresponding finite element is consid-ered failed. The material of the finite element can no longerhandle any shear stresses, but still sustain the hydrostaticpressure. At this time, since the finite element cannot with-stand any shear stresses, it mostly likely undergoes severedistortion and causes the simulation to stop. Therefore, it isremoved from the continuum body. The mass and the hydro-static pressure of the failed finite element are distributed tothe corresponding nodes of the element. The boundary sur-faces are regenerated as new master contact segments. Thefinite element nodes of the failed elements are then treated asmass based particles using the rule described in Section 3.2.Particles are modeled by the molecular dynamics method toaccount for the interactions amongst the mass particles andbetween the mass particles and the existing master contactsegments. They are not divided any further.

One may suggest to convert failed elements into rigid bodiesand simulate them using a rigid body engine. Although it issimpler, it does not account for all possible physical equa-tions representing the model and the underlying mechanics.The conversion of the failed elements is to continually modelthe pressure field resulting from the equation of state. Con-verting to rigid bodies or using a simple approach will notcorrectly model the underlying physics.

3.2. Element/Particle Conversion

In [JBS00], a straightforward strategy is given to convert afailed element into a particle. A particle inherits the massof the element and is positioned in the element center. Inour system, we remove a failed element from the volumet-ric mesh, too. Whether or not to generate a particle dependson its surrounding. The particle conversion rule is: when allthe elements connected with a node fail, we then convert thisnode into a particle. The particle carries a quarter mass fromall its connecting elements and inherits the node’s currentposition. In Figure 2, we compare the two different convert-ing strategies. Although particles should have different ge-ometric shapes, accurately evaluating their actual geometricshapes is very expensive and does not contribute much tothe correctness of the simulation, since particles are small.

By treating the particles as isotropic spheres in the simula-tion stage, the computation cost is tremendously reduced.

Figure 2: Two element/particle converting strategies. (Left)One element to one particle. (Right) Our strategy, where allconnecting elements must fail. Because of the strict limita-tion, only one particle is generated.

The advantages of our converting strategy are: (1) Becauseof the restrictions we have enforced, much fewer particlesare generated than in the method of [JBS00], while the ele-ment count remains the same. Therefore, the computationalcost is reduced. (2) There are no new vertices generated,which eases the dynamic memory management. Many ma-trices do not need to be recomputed. (3) This kind of particleconversion makes the visualization technique discussed inSection 4 possible. Instead of rendering isotropic spheres,we extract a polygonal boundary for each particle to achievemore realistic visualization for the fragmented pieces.

3.3. Governing Equation

The governing equation describes the force balance relationamongst the space-discretized mathematical material pointsand is essentially Newton’s second law: F = ma, where F isthe force acting on the discrete material point, m is the asso-ciated mass of the discrete material point, and a is the accel-eration of the discrete material point. Therefore, the govern-ing equation of the problem is given as the follows.

Ma = Fexternal −Finternal +Fbody +Fcontact (1)

where M is the mass matrix, a is the acceleration vector,Fexternal is the external force vector due to external load-ing, Finternal is the internal force vector and is a result fromthe constitutive equation due to the deformation, Fbody isthe body force vector due to the gravity field, and Fcontactis the contact force vector due to the contact between thedeformable bodies and the contact between the deformablebodies and the debris particles.

The mathematical modeling of the failed material or de-bris clouds proposed here is adopted from the mathemati-cal modeling of atomic interaction of the molecular dynam-ics method as described in [AW57]. For a particular massbased particle i, the governing equation is Newton’s equa-tion of motion given by,

F i = miai = mid2xi

dt2 (2)

where F i is the force exerted on the particle i, mi is the massof the particle, ai is the acceleration of the particle, and xi



is the center point of the particle. The force F i is the sumof all the interaction forces between particle i and the neigh-boring particles. The interaction force between particle i andparticle j is given by,

F i j =−∇rU(ri j) (3)

where ri j = ‖xi − x j‖ is the distance between particles iand j, and U(ri j) is the potential energy function (see Sec-tion 3.4.3 for further explanation). Each of the particles alsoserves as the slave node for the contact formulation. The con-tact forces also need to be accounted for when the gap vectorof the associated slave node to the master segment is zero.

3.4. Constitutive Models

The simulation described in this exposition is modeledin conjunction with the following three aspects: (i) theCoulomb friction law for modeling the contact-impact in-teraction, (ii) the continuum damage mechanics based hy-drodynamic equation for modeling the continuum materialbehavior, and (iii) the EOS induced soft-sphere potentialfor modeling the failure material behavior. The continuumequation is discretized using linear tetrahedral or linear hex-ahedral elements, since higher order elements cannot modelcontact physics.

3.4.1. Friction Law

The contact between the continuum bodies and the contactbetween the failed material and the continuum bodies aremodeled by the following Coulomb friction law.

‖σ τ‖= µc‖σN‖ (4)

where σN is the normal contact stress, σ τ is the tangentialcontact stress, and µc is the Coulomb friction coefficient.

3.4.2. Hydrodynamic Equation

In the governing equation, Equation (1), the internal forcevector, Finternal is a result from the constitutive equationwhich dictates the behavior of the material. The material be-havior of the continuum bodies is modeled by the continuumdamage mechanics based hydrodynamic equation proposed,which is given as

σ =σ

1−ϕ= σD + σH

σ∇

D = 2GDeD

σH =−p(ρ,e)I

(5)

where σ is the effective stress tensor, σ is the Cauchy stresstensor, ϕ is the isotropic damage state variable which is theratio between the continuum damaged volume representedby the micro-void volume and the volume of the material(see Figure 3 for visualization), σD is the deviatoric portionof the effective stress tensor, σH is the hydrostatic portionof the effective stress tensor, G is the shear modulus, De

D

is the elastic part for the deviatoric portion of the velocitystrain tensor, p is the hydrostatic pressure, ρ is the materialdensity, e is the internal energy per unit mass, I is the ranktwo identity tensor, and the symbol •∇ denotes the Truesdellstress rate [Tru52]. The combination of the continuum dam-age mechanics with the hydrodynamic equation and the useof the Truesdell stress rate for the hydrodynamic equationare novel and yield meaningful and accurate physics for thisapplication.

Figure 3: Visualizing the scalar damage values of the sim-ulation nodes using 1D texture-mapping.

The deviatoric portion of the hydrodynamic equation incor-porates the Truesdell stress rate hypo-elasto-plastic model,the Lemaitre plastic-damage model, and the Johnson-Cookfracture model together which account for the temperatureeffect. The temperature of the material is governed by thefollowing temperature equation.

ρcT = ηeσ : (DD −DeD) (6)

where ρ is the material density, c is the material heat capac-ity, DD is the velocity strain, De

D is the elastic part of thevelocity strain, and ηe = 0.9 is efficiency ratio [BCHW93].Equation (6) represents that the temperature change of thesystem (left hand side) is equal to a portion of the dissipatedenergy of the system (right hand side).

The hydrostatic pressure is determined from the EOS alongwith the material internal energy equation [WC55]. The rateof internal energy per current unit mass is given as

e =1ρ

(σD : D− p tr(D)) (7)

where p is the hydrostatic pressure, and tr(D) is the trace ofthe velocity strain. And, the Mie-Grüneisen EOS consideredhere is given by

p(ρ,e) = (1− 12

γµ)pH + γρe (8)

where µ = η − 1, η = ρ/ρ0, ρ0 is the material densityat the initial reference state, γ is the Grüneisen parameter,and pH is the Hugoniot pressure. Additionally, for numer-ical shock wave computation, the artificial viscosity is also



needed to stabilize the numerical oscillations [vR50] en-countered when the continuum body is discretized in spaceby numerical methods such as FEM.

3.4.3. Interaction Potential Energy Function

The classical molecular dynamics method was developed forthe interaction between discrete particles. It has the follow-ing advantages: (1) Accurate physical modeling; (2) Easyimplementation; (3) Minimal computational complexity; (4)Symmetry and robustness; (5) Readily interfaces with clas-sical finite elements. The failed material resulting from thecontinuum damage model is in the form of the debris parti-cles and subjected to the resulting hydrostatic pressure field.The interaction of the failed materials is readily suitable tobe modeled by the molecular dynamics method. The inter-action potential energy function is appropriately modeled bythe soft-sphere pair potential energy function given by thefollowing form,

U(ri j) = α(ri

ri j)γ (9)

where α is the strength of interaction, ri is the radius of themass particle ball, and γ is the repulsion parameter. Sincethe failed material debris particles are subjected to the hy-drostatic pressure field, a novel soft-sphere pair potential en-ergy function is proposed as the follows.

U(ri j) =−12

p jr jVi

(ri j

r2i j

)(10)

where p j ≤ 0 is the hydrostatic pressure of particle j, r j isthe radius of particle j, Vi is the volume of particle i, ri jis the direction vector between particle i and j, and ri j isthe distance between particle i and j. The negative pressureis by mathematical definition. Therefore, the force acted onparticle i under the pressure field for particle j is given bythe following expression.

Fi j =−p jr jVi

(ri j

r3i j

)(11)

Most of the existing meshfree methods result in un-symmetric system equations, therefore the convergence ofthe mathematical modeling becomes problem-dependent.Here our MD approach results in symmetric system equa-tions thus guarantees convergence.

4. Particle Geometry Reconstruction

Although the above simulation method generates physically-correct results, directly rendering particles as spheres doesnot convey realistic appearance. Figure 1 shows such an ex-ample. There are two major reasons. First, modeling parti-cles as spheres is an acceptable simplification assumption inthe simulation process. However, rendering spherical parti-cles makes the phenomena unrealistic. Second, due to the

vertex/particle conversion constraints, there will be holes onthe boundary surface where the voided elements are not freeenough to be released as particles. If the tetrahedral mesh istessellated fine enough, this may be less a visual problem.However, an overly-refined mesh is too costly for computa-tion.

To provide better visualization, we devise a technique to re-construct geometric shapes for particles. For each flying-outparticle, we find out all the voided tetrahedra it is attachedto in the original mesh. Tetrahedral splitting is performed onthese elements. We first extract one piece from each tetrahe-dron, then combine these pieces as the geometric shape ofthat particle. At the same time, pieces are also given backto the element faces connecting the voided elements. Forvoided elements without particles generated, if the verticesare broken into different pieces, we apply a similar recon-struction strategy. Using this technique, a polygonal bound-ary surface for the fractured pieces is constructed.

Our reconstruction algorithm is extended from the multi-material interface surface computing algorithm [NF97],where a splitting surface is extracted when the vertices ofeach element of a tetrahedral mesh are classified as differentmaterials. We take the same spirit, but apply it to a differentscenario, where each flying-out or broken vertex of an el-ement is treated as a distinct material. Furthermore, our ex-tension includes timing-varying and topology-changing han-dling of the data set.

4.1. Tetrahedral Splitting Configuration

We first discuss how to compute the splitting surface in astatic volumetric mesh. In many finite-element simulationapplications, such as computational fluid dynamics and hy-drodynamics, researchers are concerned with reconstructinga boundary surface between multiple materials from simu-lation results. To represent the material information in thegrid cells, each mesh vertex can have an associated tuple(α1,α2, ...,αm) as the material classification, where m is thenumber of materials. In general, each αi is a fractional num-ber and it is assumed that α1 +α2 + · · ·+αm = 1, as shownin [BDS∗03]. In the simplest case, a binary classification isused so that each vertex belongs to exactly one material.The Nielson-Frank algorithm calculates a splitting surfacefor unstructured grids in such a scenario, which is a gener-alization of the well-known Marching Cubes (or MarchingTetrahedra) algorithm [LC87]. The edges of a tetrahedronwith differently classified endpoints are intersected by thesplitting surface. Similarly, the faces of a tetrahedron withthree vertices classified differently, are assumed to be inter-sected by the surface in the middle of the triangle. If each ofthe four vertices has a different class, the boundary surfacesintersect in the interior of the tetrahedron. The resulting mid-edge, mid-face, and mid-tetrahedron intersections are trian-gulated to form a piecewise approximation surface.



(a) (b)

(c) (d)

Figure 4: The 4 splitting cases of a tetrahedron: (a) 3-1, (b)2-2, (c) 2-1-1, and (d) 1-1-1-1.

Since there are only 4 vertices per element, four cases areidentified in [NF97]: (1) 3-1, three vertices are of one classand one other vertex is of another class; (2) 2-2, two verticesare of one class and two vertices are of another class; (3) 2-1-1, two vertices of one class and the other two vertices areof second and third classes; and (4) 1-1-1-1, each vertex is ofa different class. As the data set is static, once the case num-ber of an element is determined, it will not change. Figure 4illustrates the four cases.

When we are dealing with time varying data sets, the classi-fication of one element may start from case 1 to case 2, andfinally to case 4. If we follow the above splitting configura-tion, the split objects may exhibit changes in shape, whichwill cause visual inconsistency in animation. We need co-herent splitting so that the split objects can smoothly transit.Therefore, we always use the splitting configuration of case4 to construct the splitting surface, even though more tri-angles are used. Figure 5 shows our configuration for 3-1,2-1-1, and 1-1-1-1 cases from left to right.

Figure 5: The three cases for consecutive splitting of a tetra-hedron.

4.2. Dynamic Tetrahedral Splitting

In each time step, we scan the tetrahedral mesh for failedelements. In these elements, we first determine the split caseof each element, then determine the split points, finally we

compute the split pieces according to the local coordinate ofeach component.

Determine the splitting case: The case number of a voidedtetrahedron is determined by the sum of the number of par-ticle vertices and the broken vertex sets. Each particle is adisconnected piece. For the remaining vertices that are stillin the volume mesh, we further check whether they are di-rectly connected or not. The checking is easily achievedsince these vertices are on the boundary surface of an object.By searching the triangle edges of the boundary surface, wecan group these vertices into disconnected clusters. For in-stance, if only one vertex is converted into a particle, it isclassified as case 1. If there is one particle vertex and twodisconnected clusters of the remaining vertices, it is case 3.Because of the dynamic property of the volumetric mesh,this operation is executed once in each time step.

Determine the split points: Our system keeps a copy of thevolumetric mesh in its initial, un-deformed position. Sincethe element is already voided, in this step we use the orig-inal tetrahedron. In a straight-forward way, we can use themid-edge, mid-face and mid-tetrahedron as the split points,as used in [NF97]. To generate more accurate split positionsand let the cutting better conform to the simulation result,we use the scalar damage value computed at each vertex orparticle (from Equation (5)). Then, based on a user-specifieddamage threshold value, we compute these bisecting posi-tions and store them in the barycentric coordinates of thetetrahedron. The barycentric coordinate values are then usedin the next step. We note that for each split point this com-putation is executed once during the whole process.

Perform local split: As the tetrahedron is already voided,it is incorrect to split the tetrahedron using the current po-sitions of the four vertices. For each splitting component,we need to setup a local, virtual tetrahedron for it. The vir-tual tetrahedron is built based on the current positions of thevertices in this component. Then, a real split is performed onthe virtual tetrahedron. Since we assume a particle is a mass-point and does not have any rotational momentum, the localcoordinate is fixed at the time when it is converted from amesh vertex into a particle. After that, there is no change toits local coordinate. Therefore, we simply record this coor-dinate for each particle at the time of its creation.

For the mesh vertices, they are translating and rotating dur-ing the simulation. To setup the virtual tetrahedron for meshvertices, we need to find a base triangle in each time stept. In split case 1, the base triangle is formed by the 3 meshvertices since they are in one cluster. In case 2, for each clus-ter we need to find a triangle on the boundary surface whichshares the two vertices as one of its edges. For each of thesingle-vertex clusters in case 3 and 4, we need to find a trian-gle on the boundary surface which shares that vertex. Afterlocating the base triangle on the boundary surface, we com-pute a differential representation for each vertex in the ele-ment that is not used to form the base triangle. In this com-



putation, we use the initial, un-deformed vertex positions attime step 0. The vector M d0 is given by

M d0 = d0 − (a0 +b0 + c0)/3 (12)

where a0, b0 and c0 are vertices of the base triangle, d0 isthe vertex to be evaluated, M d0 is the differential vector ofd0. See Figure 6 for illustration. Then, the differential vectorM d0 is used to compute d′

t, a virtual, deformed positions ofd0 at time step t:

M dt = k[M d0 ·Nx,M d0 ·Ny,M d0 ·Nz] (13)

d′t = (at +bt + ct)/3+M d

′t (14)

where at, bt, and ct are the deformed positions, k is the ra-tio of the perimeter of the deformed triangle 4atbtct to theoriginal triangle 4a0b0c0, and Nx, Ny, and Nz are the threeaxis of the triangle atbtct’s local frame in world coordinates.Once all the vertices of the virtual tetrahedron are ready,we split the virtual tetrahedron using the barycentric coordi-nate values computed in advance. The split results are storedas vectors relative to the real mesh vertices. Since there areseveral tetrahedra sharing one edge, a timestamp is used oneach tetrahedron edge to avoid redundant computation andto keep consistency of the split positions.

a0 d0

c0b0

△d0

a0 d0

c0b0

△d0at

dt’

ct

bt

△dt

dt

Figure 6: Computing the virtual position of a particle vertexd0 in time step t. (left) Vertex d0 and its base triangle. (right)The deformed position dt and the virtual position d′

t.

5. Experimental Results

We have implemented the simulation and visualization al-gorithms in C++. The experiments are conducted on a 2.8GIntel Xeon PC with 1G RAM. The models used for the simu-lation are created using NETGEN, a publicly available meshgeneration package [Sch97]. We use a linear Complemen-tary Conjugate-Gradient method for collision detection of el-ements. Our explicit integration is stable with non-shrinkingstep length. In visualization, since the geometry reconstruc-tion algorithm generates many tiny triangles, we perform anonline vertex clustering simplification on the output data toreduce the triangle count. In addition, a feature-preservingsurface smoothing algorithm is performed on the simplifiedmesh. The accompanying video contains animations corre-sponding to the examples. The images are rendered using anopen source ray-tracer, POV-Ray (http://www.povray.org).

t=3ms t=6ms

t=9ms t=12ms

t=15ms

Figure 7: Simulation of a vase shattered onto the ground,where geometric shapes of the particles are reconstructedand rendered. Images are rendered at different time steps (insimulation time).

During the computation, we output the results on a fixed timeinterval of the simulation time. Information about the inputtetrahedral element size of each example along with the av-erage time required to compute each time step are listed inTable 1. Instead of tracking the crack tips and maintainingthe boundary surface at every step of the simulation, the vi-sualization algorithm operates on the output data only. As-suming that the output time interval is small enough, thisdoes not affect the visualization accuracy and we find thatthe results are satisfactory. By this way we separate the vi-sualization stage from the simulation stage, therefore mak-ing the simulation algorithm simple, clean, and efficient. Infact we have tried most of the meshless methods available inthe literature, such as the moving least-square, element freeGalerkin, smooth particle hydrodynamics, etc. Each methodhas its own deficiency in terms of accuracy and/or stability.



Table 1: Model size and simulation performance. The particle numbers are counted at the end of the simulation.

Tet. Node Particle Frame Time step CPU time perModel # # # # interval (µs) time step (sec.)Vase 40.4K 13.8K 3445 650 40 37.2Plate 23.1K 5.4K 644 220 50 21.5

The proposed framework overcomes these deficiency andprovides accurate and stable results. Besides, in all exam-ples of this paper, the visualization algorithm takes less than1 second per frame. We have also counted the extra trianglesintroduced as the boundary surface for failed elements andparticles. On average a particle will introduce around 100triangles (before simplification), which is determined by thevertex valency in the tetrahedral mesh and the splitting cases.

Figure 7 shows a porcelain vase shattered onto the ground.The vase has a vertical velocity of 25m/s. Since the vaseis brittle, it breaks into several large chunks, together withthousands of small pieces. About 25% of the nodes are con-verted into particles. However, if elements are directly con-verted, the particle count will be much larger. On the top-right corner of Figure 7, a zoomed-in view of the boxed re-gion in this figure is shown to demonstrate the significantappearance difference with Figure 1 for obtaining the actualgeometric shapes of the failed material and debris clouds.

Figure 8 shows a stone plate falling onto the ground with asmall angle, where a side-by-side comparison is given. Theinitial velocity is 20m/s. In the top row, we show the ren-dering results where spheres are used for particle rendering.In the bottom row, the actual geometric shapes are rendered.As demonstrated by this figure and the companion video, thevisual realism is improved significantly.

A general limitation of our current hybrid approach is that adensely tesselated mesh is required as the input. If the inputmesh is too coarse, the simulation results will significantlydepart from the correct solution and will look unnatural. An-other limitation is that the geometric shapes of the particlesare dependent on the initial meshing since the particles areconverted from the damaged elements. If one object is tesse-lated into two different ways, the computed particle shapesare different. This will not be a visual problem as long as thevolume mesh is adequately refined.

6. Concluding Remarks

We have introduced a hybrid framework for fracture simula-tion that generates a large amount of debris particles. Wehave presented the physics-based governing equations forthe hybrid simulation. In visualization, we have further de-veloped a dynamic tetrahedral splitting algorithm to extractgeometric shapes for the particles. Our system is physicallycorrect, computationally efficient, and visually realistic.

We envision many practical applications of our system. It

could be used for simulating, visualizing, and understandingmany phenomena. For example, we could simulate a cometexplorer colliding with the comet kernel, which will causea splendid explosion and generate a large amount of debrisparticles. Another example is to simulate the 911 accidentand gain more understanding of the cause of the catastrophewith progressive damage.

There are a number of avenues for future work, including:

• Incorporation of adaptive meshing for accelerating simu-lation.

• Further improvement of the system efficiency using acombination of explicit and implicit time integration.

• Extension of our system to the simulation of other phe-nomena, such as explosion, fire, smoke, etc.

• Integration of other visualization techniques, such as tex-ture splats, isosurface-based rendering, etc.

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t= 2ms t= 4ms t= 6ms

Figure 8: A stone plate falls onto the ground. The simulation results are rendered using (top row) mesh and spherical particles,and (bottom row) geometric shape reconstruction methods at different time steps (in simulation time).

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