Fracture Modeling in Computer Graphics

Lien Muguercia, Carles Bosch1, Gustavo Patow

ViRVIG-UdG, Universitat de Girona, Spain

Abstract

While object deformation has received a lot of attention in Computer Graphics in recent years, with several good surveys thatsummarize the state-of-the-art in the field, a comparable comprehensive literature review is still needed for the related problem ofcrack and fracture modeling. In this paper we present such a review, with a special focus on the latest advances in this area, and acareful analysis of the open issues along with the avenues for further research. With this survey, we hope to provide the communitynot only a fresh view of the topic, but also an incentive to delve into and explore these unsolved problems further.

Keywords: crack modeling, fracture modeling, simulation, survey

1. Introduction1

Physically plausible object deformation and fracture have2

been of central importance in many fields, and particularly in3

Computer Graphics since more than 25 years ago [1, 2]. Differ-4

ent areas, such as architecture and fabrication, usually require5

very precise simulations, for which numerical models have been6

devised using a combination of continuum mechanics, dynam-7

ics, differential geometry, calculus and Computer Graphics, among8

others. As a body can undergo many physical phenomena, frac-9

tures are essential to the movie and video game industries be-10

cause of the explosions or shattering bodies required. In gen-11

eral, the phenomena we study in this survey can be considered12

as ubiquitous, as can be observed in almost every structure,13

from crystals to entire buildings.14

In spite of its importance, the study of fractures is still a15

non-closed problem, with several open issues to be dealt with,16

most of which result from the many approximations and sim-17

plifications introduced to simulate the intrinsic complexities of18

this phenomenon. Advances in this field would open new fron-19

tiers for applications such as simulation and prototyping of frag-20

ile objects, resistance assessment and model resilience studies.21

There are a number of good reviews on deformable models22

in Computer Graphics [1, 3] as well as aging/weathering tech-23

niques [4] that touch on the topic of fracture processes. How-24

ever, we feel that a deep review of the current state-of-the-art25

in crack and fracture modeling techniques is missing. Thus, in26

this paper we aim to fill this gap with a comprehensive review27

of the work done thus far, and, to improve understanding and28

strengthen the relationships among the different works carefully29

classifying them according to several criteria.30

2. Overview31

After an introduction to the mathematical background needed32

to understand the basic principles of object deformation and the33

phenomenon of fracture (Section 3), we present our principal34

classification of the different methods involved in the fracture35

process:36

• Physically-based methods (Section 4), are those that fol-37

low a simulation-based approach to compute the fracture38

opening, propagation and appearance. Among these, we39

sub-classify the state-of-the-art in the field into:40

– Mass-spring models (Section 4.1), where the object41

is approximated by a finite set of masses, pairwise42

joined by springs, each with its own defining pa-43

rameters.44

– Finite element methods (Section 4.2), that partition45

the object into a set of disjoint elements (e.g., tetra-46

hedrons) joining at discrete points. When the prob-47

lem is formulated in terms of these points, then it is48

converted into a set of simpler algebraic equations,49

which are then solved to establish the behavior of50

the system.51

– Meshless methods (Section 4.3), where the model is52

approximated with a set of unconnected calculation53

points that are simulated. The value for any other54

point in the model is obtained by interpolation.55

– Other approaches (Section 4.4) cover those, which56

do not fall into the previous categories, but rather57

follow physical principles for their simulations of58

the fracture process.59

• Geometry-based methods (Section 5), also known as pro-60

cedural methods, seek plausible patterns but are not inter-61

ested in a physically accurate phenomena description.62

• Example-based methods (Section 6) try to mimic real-63

world fractures by copying the behavior observed in real64

phenomena. These methods, which build on both Com-65

puter Vision and Computer Graphics techniques, usually66

extract parameters from images and then apply these to67

generate a new fracture.68

Preprint submitted to Computers & Graphics October 16, 2014

Finally, in our conclusions (Section 7), we present compar-69

isons and further classification schemes to ensure the reader has70

a comprehensive view of the most recent developments in this71

area. This includes Table 2, which provides further details of72

the main techniques reviewed in this survey. Some avenues for73

future research are also outlined at the end.74

3. Background75

In this section we will briefly introduce the main physical76

concepts behind generating and propagating fractures.77

3.1. Stress and Strain78

In continuum mechanics, we define the physical quantities79

of an object as a continuous function in space (and time). In80

general, we define the rest shape of an undeformed object as81

the connected subspace M ⊂ R3 [1, 3, 5]. Each point x ∈ M82

has its own properties defined at its coordinates x inside the ob-83

ject, called material coordinates. When we deform the object,84

we apply forces that move its x points to their new positions x’.85

With the old and new positions we can define the displacement86

vector field on M as u(x) = x’ − x, which represents the posi-87

tional differences between the current point and rest positions.88

Refer to Figure 1 for a graphical representation.89

Figure 1: The displacement vector field u(x).

We usually measure deformation in terms of the so-calledstrain, which we often define as a normalized measure of thebody deformation. This measure represents the displacementbetween particles in the body relative to a reference length. Ba-sically, the strain measures the local deviation of a given de-formation from a rigid-body deformation. As the deformationin different directions might be different, the strain is generallyexpressed as a tensor. In three dimensions, this tensor is of or-der 2. Given the field u(x), we can compute the elastic strainε of a point at a given time, simply by relating it to the gradi-ent ∇u. Observe that ∇u is a 3 × 3 matrix of the derivatives

(∇u)i j = ∂ jui. In Computer Graphics the strain is usually de-fined for small deformations as one of

εG =12

(∇u + [∇u]T + [∇u]T∇u)

εC =12

(∇u + [∇u]T )

The first one (i.e., εG) is called the Green-Lagrange’s strain ten-90

sor and εC the Cauchy strain tensor, is its linearized counterpart.91

Based on the strain tensor, we can compute the stress tensorσ ∈ R3×3, which provides information about the forces actingon a point when the body is deformed. Of course, this relation-ship strongly depends on the properties of the material, and canbe quite complex. In Computer Graphics it is customary to useHook’s law:

σ = E · ε

where E is a rank 4 tensor which relates both tensors σ andε in a linear way, which is useful for small deformations. Onthe other hand, other definitions of stress are required for largedeformations, such as the Piola-Kirchhoff stress tensor, whichexpresses the stress relative to the reference configuration (incontrast to the Cauchy stress tensor that expresses the stress rel-ative to the present configuration); the Biot stress tensor, whichexpresses the forces due to stretch only applied in the unde-formed body per unit undeformed area; or the Kirchhoff stresstensor, which is widely used when there is no change in volumeduring plastic deformation [6]. Another possibility is the SaintVenant-Kirchhoff model:

σ = λTr(εG)I3 + 2µεG

where λ and µ are constants specific to each material and de-92

fine the way it is deformed, and I3 is the 3 × 3 identity ma-93

trix. For more information on these tensors, we refer the inter-94

ested reader to the works authored by Bonet and Wood [6] and95

Chakrabarty [7].96

In general, the stress tensor σ is a symmetric 3 × 3 matrix,97

so it has 3 real eigenvalues. These eigenvalues correspond to98

the stresses in the principal directions, represented by its re-99

spective eigenvectors to the principal stress directions. Positive100

eigenvalues indicate tension, while negative eigenvalues repre-101

sent compression.102

It is possible to compute the body force f for each pointfrom σ as:

f (x, t) = ∇ · σ(x, t)

whose elements are fi =∑

j ∂ jσi j. With these forces f (x, t),we can model deformation using the equations of motion. Ingeneral, these equations are posed in terms of the density ρ ofthe material, which again is a function of position x and time t:

ρ(x, t)∂2

∂t2 x = fi(x, t)

However, for general cases it is almost impossible to find an103

analytical solution, and we must resort to numerical methods,104

which are the subject of the following sections.105

2

Figure 2: Brittle (a) vs. ductile (b) fracture, from O’Brien et al. [8]. (imagecopyright ACM 2002)

3.2. Brittle and ductile fractures106

We can define an elastic material as one that will return to107

its original shape when the external forces on it cease to exist.108

To the contrary, a plastic material, will not go back to its orig-109

inal configuration. Real materials usually have a limited elas-110

tic behavior, and if deformed beyond a certain threshold (called111

elastic limit or yield point), they will undergo a plastic deforma-112

tion. If the material is deformed further, there is another limit,113

called the failure threshold σmax, which is the point at which the114

material fractures. This failure threshold is a material-specific115

parameter. If the elastic limit and the failure threshold are close116

to each other, then the material will undergo a small (almost117

negligible) elastic deformation before fracturing, and the ma-118

terial is termed brittle. A brittle fracture releases most of its119

elastic energy thus allowing the crack to proceed further into120

the material and reducing the energy required to break it. As a121

consequence, brittle objects generally shatter.122

On the other hand, if the thresholds are adequately sepa-123

rated, such that the failure threshold is much larger than the124

elastic limit, then the object undergoes an elastic deformation,125

followed by a plastic one, before being fractured. In this case126

the material is termed ductile (Refer to Figure 2).127

Examples of ductile materials include structural steel, as128

well as many alloys of other metals. They usually exhibit a129

very linear stress-strain relationship up to the well defined yield130

point, as shown in Figure 3. The stress value plotted in curve131

(A) was obtained by dividing the load P by the cross-sectional132

area A0 of the specimen measured before any deformation had133

taken place (compare to its tensorial version defined above).134

The linear portion of the curve shown in the left is the elastic135

region, and the slope is the modulus of elasticity or Young’s136

Modulus. The curve typically decreases slightly after passing137

the yield point (Point 2 in the figure). As deformation contin-138

ues, the stress increases on account of strain hardening until139

it reaches the ultimate strength (Point 1 in the figure). Until140

this point, the cross-sectional area decreases uniformly and ran-141

domly. However, beyond this point a neck forms where the lo-142

cal cross-sectional area decreases more quickly than the rest of143

the sample, which results in an increased true stress. Since the144

cross-sectional area of the specimen decreases as P increases,145

the stress plotted in curve (A) does not represent the true stress146

in the specimen, shown as curve (B). The difference between147

this engineering stress (P/A0) and the true stress (P/A) becomes148

apparent in ductile materials after yield has started. While the149

engineering stress, which is directly proportional to the load150

P, decreases with P during the necking phase, the true stress,151

which is proportional to P but also inversely proportional to A,152

is observed to keep increasing until fracture of the specimen oc-153

curs (at point 3 in the figure). The actual fracture point is in the154

same vertical line as the visual fracture point. It is easily appre-155

ciated that ductile fractures require greater amounts of energy156

to break as the result of the considerable energy used for plas-157

tic deformation, so in fact these objects usually tear [9]. For-158

mally, we can say that fracture will happen, in both cases, when159

the maximum stress is greater than the failure threshold σmax.160

Other factors, such as material density and damping factors, are161

usually taken into account to simulate fracture processes.162

Figure 3: Approximate stress vs. strain. Left: typical curve for steel. Here,(A) is the apparent curve (computed with σ = P/A0, where P is the load andA0 the cross section of the sample before any deformation) and (B) is the realcurve (computed with σ = P/A, with A the real cross section of the samblebeing deformed). Right: brittle material [9]. In the figures, numbered regionsrepresent: 1. Ultimate Strength; 2. Yield Strength; 3. Fracture point (at σmax);4. Strain hardening region; 5. Necking region. Adapted from [10]

To simplify computations, it is customary to split the totalstrain ε into its elastic and plastic parts, as ε = εe + εp. As aconsequence, one option commonly used for the yield point isthe von Mises yield criterion [11], defined as

γ1 <∣∣∣∣∣∣εe −

Tr(εe)3

I3∣∣∣∣∣∣ (1)

3

where γ1 is the yield threshold, Tr(.) is the trace of a matrix,and ||.|| is the Frobenius norm. Basically, this last expressionmeasures the deviation of the elastic strain from an initial ref-erence configuration. The second term, to the right, essentiallyeliminates any dilation effect, concentrating only on shape dis-tortions. For strains beyond this point, plastic deformation oc-curs. To handle the plastic part of the deformation, O’Brien [8]proposes measuring the change in the plastic strain as

∆εp =||ε′|| − γ1

||ε′||ε′

where ||ε′|| is the right hand side of Equation 1. Then, the limiton the maximum plastic deformation can be enforced by updat-ing the plastic strain with the threshold γ2 as

εp := (εp + ∆εp)min(1,

γ2

||εp + ∆εp||

)Thus, the difference between the total strain and the plastic163

strain is the current elastic strain. By introducing γ2 we are164

limiting the plastic strain, but not the total strain, which can be165

larger and produce fracture.166

Both brittle and ductile fracture have been widely addressed167

in Computer Graphics, although their different properties often168

led to the use of different methods to simulate their behavior.169

Refer to Section 4 for further details.170

3.3. Visual aspects of fracture simulation171

The presence of cracks and fractures on objects may provide172

a great deal of information on their properties, interactions with173

other objects, usage or history. Objects being impacted by other174

objects are expected to show deformation and/or fracture, and175

crack patterns are also very common on aged environments sub-176

ject to fatigue conditions. Properly simulating this phenomenon177

is thus important to improving the perceived realism of virtual178

environments.179

Physically-based simulation is especially effective for mod-180

eling and animating fractured objects, which tend to have many181

degrees of freedom. Animating these phenomena with keyfram-182

ing and motion capture would be much more difficult [12]. Most183

simulation techniques reviewed in this survey draw heavily from184

the field of fracture mechanics and its literature, which abstracts185

the micro-scale nature of fracture to a macroscopic level based186

on a continuum model. The requirements of engineering ap-187

plications, however, are different from the ones in graphics ap-188

plications. In Computer Graphics, simulation techniques rely189

on simplifications that would be unacceptable in an engineer-190

ing context. In the continuum approach, the scale of the ef-191

fects being modeled is significantly greater than the scale of the192

materials composition. Even if macroscopic fractures can be193

significantly influenced by effects occurring at the microscale,194

ignoring the microscopic effects in favour of a macroscopic de-195

scription is often reasonable, as our interest is on the graphical196

appearance rather than its physical correctness. Similarly, nu-197

merical accuracy is less important compared to issues such as198

visual appearance, ease of use, or computational efficiency.199

In order to achieve good visual quality, we must rely on200

representations able to capture realistic fracture patterns. A201

common limitation present in earlier approaches is to limit the202

propagation of fractures to the boundaries of the initial mesh,203

which tends to create visible artifacts. Irregular shaped shards204

can only be achieved by allowing fractures to propagate in ar-205

bitrary directions. Rendering realistic fractures from the results206

of a simulation may also pose its own problems. In particular,207

when the fracture is not represented by the boundaries of the208

original mesh elements, then we have a dissociation that results209

in two different meshes being used: a simulation mesh and a210

rendering mesh. In such situations, special care is needed to211

maintain consistency between both representations.212

3.4. Related reading material213

For a more in-depth review of the related background, the214

interested reader is referred to some previous surveys. In partic-215

ular, the works by Gibson and Mirtich [1] and Nealen et al. [3]216

present the most significant contributions for physically-based217

deformable models in recent decades. Akin to our classifica-218

tion, they also focus on finite element methods, mass-spring-219

based methods, meshfree methods, coupled particle systems220

and reduced deformable models based on modal analysis. They221

also make a particular connection with the fundamental con-222

cept of time discretization, and present a number of application223

areas such as cloth, hair, virtual surgery, deformation and frac-224

ture. The background in the recent thesis by Glondu [5] should225

also be taken into account, as model deformation is presented226

and previous work is carefully classified. Finally, the survey by227

Merillou and Ghazanfarpour [4] presents a very detailed sur-228

vey on aging and weathering techniques where methods dealing229

with crack formation are also reviewed.230

4. Physically-based methods231

Effective simulation of fracture processes is one of the most232

accurate ways of approaching the problem. The main tech-233

niques usually used to solve this complex problem are mass-234

spring models, finite element methods and meshless methods.235

In general, an analytical solution is not possible, so we must236

resort to numerical approximations to solve the equations for237

the material. One option is explicit integration schemes, which238

provide explicitly the values for a given time step in function of239

the previous step. These are usually not unconditionally stable:240

they converge only for small values of the integration step. Two241

examples are the explicit Euler integration [13] and the Runge-242

Kutta integration [14]. To the contrary, implicit schemes pose243

a system of equations that must be solved, but provide stability244

for larger time steps, some even arbitrarily large. An example245

of these is the implicit Euler method [13]. The choice made be-246

tween the two has implications not only on the stability of the247

system but also on its computational time.248

4.1. Mass-spring models249

Mass-spring models are one of the simplest ways to modela deformable body. Mass-spring models are characterized bydiscretizing M into a finite set of particles {pi; 1 ≤ i ≤ n}. Eachparticle pi has its own mass mi and position xi, and are pairwise

4

connected with springs, each with its own stiffness, dampingfactor and rest length. Each particle is set to a classic equilib-rium equation relating the internal forces (the springs) and theexternal forces (gravity, collisions) with its mass:

ρi∂2xi

∂t2 = fi ,

where fi is the sum of the forces acting on the particle pi, andρi is the mass associated with the i-th particle. In general, fiis described as the sum of two main terms: the external forces(e.g., gravity or collisions) and the internal forces, which comefrom the springs attached to the particle pi. In general, springsfollow Hooke’s law [15], which can be stated as:

fi = k(|∆xi j| − li j)∆xi j

||∆xi j||,

with k being the spring constant that characterizes its stiffness,250

li j its original or rest length, and ∆xi j = xi−x j its current length,251

measured as the difference between the particle positions. As252

all the particles are connected through the springs, the equa-253

tions lead to a system of coupled ordinary differential equations,254

which is solved through numerical integration methods. When255

the limit of a spring exceeds a given threshold, the breakage is256

simply performed by removing the corresponding connection257

between the particles.258

In Computer Graphics, some of the first approaches for sim-259

ulating fracture relied on mass-spring representations [16, 17,260

18, 19, 20]. Norton et al. [16], for instance, place lattices of261

cubic cells around the surface of objects and connect them with262

several spring configurations. They simulate brittle fracture due263

to impacts and stretching effects by removing an entire cube of264

springs at once, which may produce visible artifacts. For the265

final surface, they classify the faces of the model as belonging266

to the interior or the fracture surface, generating and rendering267

a polygon in the latter.268

Hirota et al. [17, 18] focus on the simulation of drying ma-269

terials and their temporal cracking. A set of shrinking and270

contraction effects modify the physical properties of a material271

based on a set of measurements, which results in the opening272

and propagation of cracks (Refer to Figure 4 for an example).273

For this, both a bi-layered representation [17] and a 3D mass-274

spring model [18] have been examined. For visualization, Hi-275

rota et al. rely on a marching tetrahedra-like algorithm that uses276

the broken points on the springs to compute the crack shape.277

Aoki et al. [21] later incorporate a moisture diffusion model to278

guide the drying process. Federl and Prusinkiewicz [19, 20]279

also use a similar bi-layered approach for modeling tree bark,280

but based on wedge elements and element removal.281

Mazarak et al. [22] simulate fractures and debris produced282

by explosions based on a similar strategy, where objects are283

modeled with voxels connected through rigid links, represent-284

ing infinitely stiff springs (Refer to Figure 5). Based on the285

strength of a spherical blast wave and a set of heuristics, the286

yield limit of each link is subsequently decreased until fracture287

appears. The spring-based model is later improved on [23] by288

using adaptive voxel shapes and including rotational forces.289

Figure 4: Comparison between the cracks obtained on a real drying clay (top)and the corresponding simulation (bottom), from [18]. (image copyrightSpringel-Verlag 2000)

Figure 5: Graph representation of the connected voxel model, from Mazarak etal. [22]. (image copyright Mazarak 1999)

Smith et al. [24] propose a system that connects point-masses290

with distance-preserving constraints rather than a grid of stiff291

springs. Instead of computing displacements, they evaluate the292

forces that these constraints exert in response to the impulses,293

and use strength thresholds to indicate when and where objects294

will break. In contrast to explicit methods and the use of elas-295

tic meshes, these rigid constraints coupled with a quasi-static296

approach allow them to quickly compute a solution. Figure 6297

shows a table broken using this approach.298

Figure 6: Breaking table from Smith et al. [24]. (image copyright CGF 2001)

Mass-spring models and similar approaches are sometimes299

preferred because of their simplicity and faster results, which300

is especially true for real-time fracture. However, these ap-301

5

proaches suffer from several limitations in terms of mechanical302

behavior and visual quality. They do not provide a direct way303

of resisting shear or bending, and it is difficult to express im-304

portant material properties such as the stress-strain relationship.305

Furthermore, the exact location and orientation of the fracture306

are unknown, and fracture surfaces are often restricted to the307

boundaries in the initial mesh structure, exhibiting directional308

artifacts (Refer to Figure 4). As a result, these techniques can309

only realistically model effects that occur on a scale much larger310

than the inter-node spacing. Continuous models such as the311

Finite Element Method, as described in the following section,312

directly account for most of these issues.313

4.2. Finite element methods314

The Finite Element Methods (FEM) partition the model M315

into a set of disjoint elements (usually tetrahedrons). These el-316

ements join at discrete points, usually called node points. The317

FEM method proceeds by defining the problem on the finite set318

of node points, instead of on the original mesh, resulting in a319

set of algebraic equations that are solved numerically. Rather320

than solving a continuous problem, these methods look to solve321

the discrete positions of the node points. First, the unknown x322

is approximated by x =∑

i xibi, where xi are the positions of323

the node points and bi are fixed nodal basis functions that are324

1 at node i and 0 everywhere else. While replacing x in the325

above equations for motion results in a set of simpler algebraic326

equations, no sets of basis functions will yield the solution in327

the general case, so all methods aim at reducing the error in-328

troduced by the approximations employed. This approximation329

error is measured by replacing the resulting approximation into330

the original equations. One example is the Galerkin method331

[25] that treats finding xi as an optimization problem where the332

error is minimized. Refer to Swenson and Ingrafiea [26] for333

further theoretical details about Galerkin methods.334

In their pioneering work, Terzopoulos and Fleisher [2, 27]335

model viscoelastic, plastic and fracture behaviors based on a336

continuous formulation, where both FEM and finite differences337

are proposed for their discretization. One of their formulations338

is based on a hybrid decomposition of the model into rigid and339

deformable components which, in turn, improves the numeri-340

cal conditioning of rigid objects. This enables a wide range of341

deformable behaviors from highly elastic to nearly rigid to be342

covered, while fracture is shown on torn paper and cloth sheets.343

Later, O’Brien and Hodgins [28] propose a model that uses344

FEM with the theory, which has found widespread use in the345

graphics literature, developed by Griffith [29] and Irwin [30].346

Their approach is based on linear elastic fracture mechanics and347

non-linear finite element analysis, where 3D objects are repre-348

sented as tetrahedral elements, refer to Figure 7. A separation349

tensor is used to decide where and how fractures should appear350

or propagate. The separation tensor is built from tensile and351

compressive forces obtained from the stress tensor. Remeshing352

is finally applied in order to enable unconstrained crack paths,353

thus achieving high visual quality.354

The same approach has been applied to simulate shattering355

due to explosions [31] and later extended to incorporate plastic356

deformation [12, 8]. By means of an additive model, the total357

Figure 7: FEM discretization into tetrahedra. Surface (a) and object interior(b); O’Brien [28]. (image copyright ACM 1999)

strain is separated into elastic and plastic components, easily358

allowing the simulation of ductile fracture. Muller et al. [32]359

include a further co-rotational formulation that avoids visible360

artifacts during large deformations. As a result, the rotation on361

a per-element basis is factored out, allowing for linear deforma-362

tion analysis independent of rotation.363

For real-time purposes, common solutions rely on instanta-364

neous fracture models, which combine rigid bodies with quasi-365

static stress analysis [33, 34, 35]. Such a hybrid approach treats366

objects as rigid bodies between collisions, while using static367

FEM during collision events. Instantaneous fracture models368

can be appropriate for stiff materials, which tend to exhibit369

small deformations. However, the elastic stiffness matrix can370

have null spaces (rigid translations and rotations that do not de-371

form the model) that may affect computations. Muller et al. [33]372

address this issue by anchoring elements far from the collision373

event, while Bao et al. [34] directly identify and eliminate them374

during the solution; typically based on the conjugate gradient375

method. Refer to Figure 8 for an example of FEM-based frac-376

ture from [34]. In the context of sound simulation, Zheng and377

James [35] propose a sparse least-squares solver to produce378

fracture sounds at near audio rates, which they combine with379

Voronoi-based fracture patterns. Iben and O’Brien [36] use a380

quasi-static system for simulating elastic relaxation and shrink-381

age after cracking, while the stress field is defined heuristically,382

thus providing substantial control for the user in terms of the383

position and patterns of the cracks.384

Figure 8: An example of a FEM-based fracture with copious small shards; Baoet al. [34]. (image copyright IEEE 2007)

Methods based on quasi-static analysis are much cheaper385

than a fully dynamic simulation, but ignoring the elastic en-386

ergy released during fracture, which might produce unrealistic387

fracture patterns. Parker and O’Brien [37] propose a simplified388

version of previous FEM techniques [8, 32] for use in video-389

games and real-time simulations. Their approach relies on a390

6

linearized semi-implicit solver and a parallelized implementa-391

tion of the conjugate gradient method, where objects are parti-392

tioned into islands and processed independently. This leads to393

a fast and robust simulation of fracture on stiff and soft mate-394

rials. Another common approach is to use a simplified model,395

which can be embedded into the original mesh for the simula-396

tion [32, 38, 37], which allows the use of lower resolutions and397

simpler representations in order to obtain faster results without398

compromising visual quality. Refer to Figure 9 for a represen-399

tation of an embedded mesh.400

Figure 9: An embedded object mesh representation; Parker and O’Brien [37].The simulated mesh is based on tetrahedrons, while a triangle-based represen-tation models the final fracture patterns. (image copyright ACM 2009)

While continuous remeshing offers high visual quality it has401

many limitations in terms of performance and simulation sta-402

bility. Stability can be guaranteed by limiting the directions of403

cracks [28], but the model can easily grow in complexity and404

become computationally expensive. Wicke et al. [39] propose405

a conservative local re-meshing algorithm that tries to replace406

as few tetrahedra as possible. This maintains high tetrahedron407

quality and limits the accumulation of numerical error. For vi-408

sualization (and collision detection), they keep track of a sec-409

ond mesh which shares the topology with the simulation mesh,410

but introduce different nodal positions. The rendering mesh411

also needs to be remeshed to maintain an appropriate geometric412

quality. Refer to Figure 10 to see an example of re-meshing.413

Other approaches simply split the mesh along element bound-414

aries [33] or even remove whole elements [20, 40], which, while415

they can be very fast, compromise visual accuracy. Embed-416

ded methods may help in mitigating these effects, as the qual-417

ity of the simulation does not need to match the visual quality418

[37, 32, 38]. Molino et al. [41] propose a virtual node algorithm419

that relies on a lattice for modeling the extra geometry, which420

serves to avoid the generation of ill-conditioned elements. The421

material within an element is fragmented by creating several422

Figure 10: Adaptive mesh refinement helps a ball to crash through differentductile plates; Wicke et al. [39]. (image copyright ACM 2010)

replicas of the element and assigning a portion of material to423

each one. These elements can then be directly used for vi-424

sualization purposes. However, the geometry increases with425

both the subdivision and the added hidden nodes, thus affect-426

ing the performance. The approach by Molino et al. can be427

seen as a generalization of extended FEM (XFEM) [42], and428

has been used in other works [34] (see Figure 8). Level sets429

can be also very effective for handling topological changes, as430

demonstrated by Hegemann et al. [43]. Here, the level set is de-431

fined in the undeformed configuration of the object, and evolves432

in material space to represent the transition from undamaged to433

failed material. The tetrahedralization is done in a regular lat-434

tice and the elements are duplicated to handle both damaged435

and healthy regions. Refer to Figure 11 for an embedded mesh436

representation.437

Figure 11: Embedded Lagrangian mesh stages for their material; Hegemann etal. [43]. (image copyright SCM 2013)

4.3. Meshless methods438

Meshless methods appear as an alternative to enhance FEMs,where M is discretized in a set of points without connectivityinformation between them. In these methods, the value at anypoint in the interior of the body is retrieved through the inter-polation of these calculation points. Basically, if we have com-puted certain values ξi at the calculation points xi, we can obtainthis value at any other point x by computing

φ(x) =∑

i

ωri (x)ξi ,

where the ωri are a set of kernel functions that allow us to com-439

pute the interpolation at point x from the values ξi at points xi.440

Usually, these kernel functions depend on a parameter r that is441

1 at the evaluation point and goes to 0 at range r.442

7

There are a number of features of these methods that make443

them favorable for fracture simulation, such as avoiding the444

complex re-meshing operations and the problems associated to445

element cutting and mesh alignment common to FEM, as de-446

scribed in the work by Belytschko et al. [44]. In the field of447

mechanics, Sukumar et al. [45] proposed a particle-based ap-448

proach to model the physical behavior around a crack; while449

Belytschko and Tabbara [46] resort entirely to meshless meth-450

ods. The methods later developed in the field of computer451

graphics were based on these seminal works.452

Muller et al. [47] introduce a meshless framework for the453

animation of elastic and plastic deformation, which supports454

a wide range of materials. The spatial derivatives of the dis-455

placement field are computed using a Moving Least Squares456

(MLS) [48] procedure based on a linear basis. These deriva-457

tives are subsequently used to obtain strains, stresses and elastic458

forces at each simulated point. Topological changes are handled459

with resampling, in order to deal with undersampled and over-460

sampled regions. Pauly et al. [49] build from the method by461

Muller et al. to include brittle and ductile fractures, which is462

achieved by continuously adding surface samples during crack463

propagation and dynamically adapting the shape functions around464

them. Resampling is also conducted around the fracture sur-465

faces, as shown in Figure 12.466

Figure 12: Volume sampling: Octree decomposition (a), initial adaptive oc-tree sampling (b), sampling after local repulsion (c), and dynamic re-samplingduring fracture process (d); Pauly et al. [49]. (image copyright ACM 2005)

Liu et al. [50] propose a quasi-static solution for stiff ma-467

terials by treating brittle objects as fully rigid bodies. The lo-468

cal Petrov-Galerkin method (MLPG) [25] is used in order to469

avoid dealing with large neighborhoods of point masses, and a470

simple damage-based fracture model is proposed. This model471

clusters particles previously classified as damaged or undam-472

aged to generate the new fragments. Chen et al. [51] focus on473

a similar problem but using Smoothed Particle Hydrodynamics474

(SPH) [52]. Their formulation is used to analyze local stress475

tensors induced by collisions. Based on a tetrahedralization of476

the material, particles are linked with up to four neighborhood477

particles and fracture is simulated by breaking these links ap-478

propriately. A clustering approach is also conducted to group479

damaged points, in order to resample the point set before gener-480

ating the fracture surface. Even with the resampling, directional481

artifacts are easily spotted.482

Efficiently dealing with proximity information is paramount483

for meshless approaches. Preliminary approaches proposed the484

use of spatial hashing to retrieve neighboring points within a485

specific distance, or a local caching scheme for exploiting tem-486

poral coherence [47, 49]. Steinemann et al. [53] use a visibility487

graph instead, which is applied both for cutting and fracturing488

deformable objects. Figure 13 shows an object splitted using489

this approach.490

Figure 13: Pumpkins splitted into pieces, from Steinemann et al. [53]. (imagecopyright ACM 2006)

In terms of rendering, Pauly et al. [49] adapt a CSG ren-491

dering technique for point-sampled surfaces [54], while Liu et492

al. [50] embed a detailed triangle mesh into the particles, where493

the triangles move with the particles using the same rigid body494

parameters.495

Despite all the advantages of Meshless methods, the fact496

that essential boundary conditions cannot be as straightforwardly497

implemented as mesh-based methods can be a problem given498

that its shape functions are not interpolating. Eventually, the499

computational cost would be higher than in FEM. One thing500

these methods have in common is that again we find the disasso-501

ciation between the simulation representation and the geometry502

used for rendering (e.g., the rendering mesh). In general, these503

methods have to generate that mesh from the set of sampling504

points, and the number of points can grow in an unpredictable505

manner, which can pose serious problems to the process.506

4.4. Other approaches507

This section covers fracture methods that rely on physically-508

based approaches but do not belong to any of the above classi-509

fications.510

Glondu et al. [55, 56], for instance, use modal analysis to511

simulate real-time brittle fracture. They base the fracture initi-512

ation method on this analysis and use a fast energy-based frac-513

ture propagation algorithm. To generate the fragments and their514

geometric surfaces, they rely on an implicit representation that515

efficiently models the surfaces obtained. Refer to Figure 14 for516

a representacion of a crack propagation, from [55]. The gen-517

erated mesh is constructed by evaluating the type of fractured518

surface in a way very similar to a marching-tetrahedra algo-519

rithm; mean while the deviations of the cracks are handled with520

a noise function. As modal analysis uses an analytical solution521

for computing deformation, no integration method is necessary.522

As a result, modal analysis leads to very fast deformable simu-523

lations, although the range of deformation tends to be limited.524

Ning et al. [57] handle heterogeneous materials using a mov-525

able cellular automata (MCA) approach [58]. The object is dis-526

cretized into spherical particles, called the MCA, which react527

to the deformation. In order to obtain accurate simulations, a528

large number of automata are often needed, which results in a529

significant computational cost. Ning and colleagues use CUDA530

to achieve real-time performance.531

8

Figure 14: Propagation of one crack in a coarse mesh. The colored section rep-resents a fracture that cut the body straight through the physical mesh; Glonduet al. [55]. (image copyright IEEE 2012)

5. Geometry-based methods532

Geometry-based methods, also known as procedural tech-533

niques, are able to produce patterns of cracks and fractures that534

are visually plausible but without relying on the actual physical535

process or its accurate simulation. Physical simulation is of-536

ten considered to be computationally demanding and does not537

provide sufficient control over fracture propagation. Procedural538

methods, on the other hand, rely on tools that offer considerable539

control of the patterns of cracks obtained as well as the size and540

shape of the fragments. This often translates in the generation541

of customized patterns by means of a simple set of parameters.542

Figure 15: Real (left) and synthetic (right) bark; Lefebvre and Neyret [67].(image copyright The Eurographics Association 2002)

In the context of fragmentation induced by explosions, Neff543

and Fiume [59] propose a simple recursive pattern generator544

that divides a planar region into polygonal shards, where cracks545

fork according to a user-specified angle. Gobron and Chiba546

[60] propose a semi-physical approach to simulate cracks on547

layered surfaces. These layers are divided into cells and a 2D548

directional stress distribution is assigned to each one based on a549

set of geometric properties. Cracks then open based on a clas-550

sical stress threshold and propagate using a relaxation process551

and a set of heuristics. For rendering, the crack segments are552

projected onto the cells. Then, each cell crossed by a segment553

is divided into micro-cells, and these are used for anti-aliased554

rendering while taking into account shading and refraction ef-555

fects. A similar strategy has been used for propagating cracks556

on layered surfaces in the context of paint cracking and peeling557

[71]. Lefebvre and Neyret [67] also rely on a semi-physical ap-558

proach for bark generation, refer to Figure 15. They model the559

bark with a set of strips parallel to the growing direction, where560

the epidermal elements are treated as semi-rigid. The fracture561

criterion is based on the relative lengthening of elements and562

which corresponds to the Griffith energetic approach [29]. Once563

generated, the bark surface can be rendered directly from any564

viewpoint.565

Some authors focus on non-photorealistic (NPR) techniques566

for simulating cracking on images. Wyvill [70], for instance,567

simulates crack patterns on images of wax paintings, where568

their placement is based on a Distance Transformation along569

with a set of heuristics. Mould [72] simulates cracks by means570

of Voronoi regions and weighted edges. For rendering cracks,571

two approaches are proposed: texture transfer, which involves572

extracting the texture from one image for its application into573

another; and the modulation of an existing texture based on the574

distance from the crack locations. Figure 16 shows two crack575

patterns generated by this method.576

Figure 16: Different crack patterns by manipulating the site distributions, asshown in [72]. (image copyright ACM 2005)

Martinet et al. [61] propose an interactive approach for mod-577

eling both cracks and fractures on 3D objects. Their patterns are578

modeled using a set of skeletons that are applied onto the object579

by means of a set of Boolean operations. Cracks also include580

an associated profile curve that is swept along the pattern before581

its subtraction from the object. Once generated, fragments are582

converted into polygons for fast rendering. Refer to Figure 17583

for a comparison between a real object and a synthetic model.584

A similar representation is also used in [64], but storing the pat-585

terns into an atlas for their later application and animation.586

Figure 17: A real clay vase (left) and a synthetic model (right). Martinet et al.[61]. (image copyright IEEE 2004)

For real-time fracture, especially in computer games, a com-587

mon approach is to rely on pre-fracturing, also known as pre-588

scoring. Objects are fractured during the modeling stage, and589

9

Behavior 2D 3D Pre-fractured Not pre-fractured Voronoi Semi-physicalBrittle [59] [60] [61] [62] [63] [64] [65] [63] [66] [59] [60] [62] [63] [60] [67]

[67] [68] [64] [65] [66] [62] [69] [68] [67] [61] [66] [65] [68]Ductile [70] - - [70] - -

Table 1: Summary of geometry-based methods exposed in Section 5.

the pre-fractured versions replace the original objects based on590

impacts and similar events happening at run-time. This ap-591

proach is very fast and provides a lot of control for the artists,592

but the obtained patterns tend to be fixed. Most of these pat-593

terns habitually rely on a Voronoi partitioning of the object594

[65, 63, 66, 62], and are readily available in software such as595

Maya [73] or Houdini [74].596

In order to handle non-centered events, pre-fractured pat-597

terns can be aligned with specific locations or even adapted598

to a specific object or bounding-box [69, 62]. These patterns599

ease the modeling stage, as only a small set of (user-provided)600

generic fracture patterns is necessary, but it does require a dy-601

namic cutting of the mesh. In order to cut the mesh, Muller et602

al. [62] propose fast boolean operations while Oh et al. resort to603

recursive spatial decompositions [66]. The approach from Oh604

et al. further interconnects fragments as in [24], where connec-605

tions can be dynamically broken based on relative velocities.606

With respect to the visualization aspects, as with other meth-607

ods, Muller et al. [62] further relie on a second mesh for render-608

ing, represented by an approximate convex hull decomposition.609

Su et al. [69] incorporate a level set into the fractured object610

that is combined with the target object for collision purposes.611

The level set is triangulated by either fast dual-contouring or612

by marching cubes. Figure 18 shows an overview of the pre-613

fractured patterns aligniation proposed by [62].614

Figure 18: Overview of Muller et al. [62] fracture algorithm. (image copyrightACM 2013)

Along the same lines, Valette et al. [68] use pre-scoring for615

modeling cracks on soils. Cracks propagate inside a voxelized616

representation of the terrain according to a precomputed pattern617

of paths, where shrinkage is incorporated to guide the progres-618

sion. Rendering is performed by transforming the terrain into619

a mesh, and then incorporating the cracks as surface polygons.620

An interesting aspect of this work is its validation against real621

patterns. The validation is based on a set of statistics taken over622

area densities and the length and connectivity of these patterns,623

which could be used for validating other methods.624

In order to avoid expensive physical computations authors625

tend to bank on a procedural technique. However, the freedom626

to make some assumptions and provide the user with tools to627

control and guide the processes, can bring physical limitations628

with non-real behaviors and visual limitations. To avoid some629

of these assumptions, an example-based method could be ap-630

plied, where the idea is to extract information from examples in631

order to reproduce or guide the fracture. Refer to Section 6 for632

further details.633

Given the large variety of procedural solutions, Table 1 sum-634

marizes the various papers that were presented in this section635

and classified according to different criteria. This includes sim-636

ulated behavior (brittle vs. ductile), whether they are 2D or 3D,637

pre-fractured or not, based on a Voronoi diagram, and whether638

they rely on a semi-physical approach. Notice how ductile frac-639

ture has barely been addressed in this context.640

6. Example-based methods641

Example-based techniques build from both Computer Vi-642

sion and Computer Graphics fields. In the case of cracks and643

fractures, the purpose is to use real examples, often in the form644

of photographs, to obtain similar patterns or to guide the fractur-645

ing process itself. For a comprehensive review of image-based646

techniques we refer the reader to Shum and Kang [75].647

The direct mapping of crack patterns onto 3D models has648

been addressed by Hsien-Hsi and Wen-Kai [76]. Given an im-649

age of a crack pattern, they vectorize the image and extract the650

corresponding pattern in the form of a graph. The graph can651

then be interactively projected onto the surface from a parabolic652

bounding volume, rendered as two projected quads for each653

crack edge, and enhanced with bump mapping. Note that other654

procedural techniques could also be useful for mapping crack655

patterns [61, 64]. Wang et al. [77] focus on reconstructing656

tree bark from an input image, where bark features are seg-657

mented using texton analysis. An interactive system is pro-658

posed to transform these features into a height-field represen-659

tation, which is then used for rendering.660

Figure 19: Paint cracking, with context of paint thickness on a frog; [78]. (im-age copyright ACM 2007)

Texture synthesis is a standard practice for mapping gen-661

eral example-based textures [79]. Synthesis techniques may662

rely on many types of input data and can be performed in both663

space and time. Enrique et al. [80], for instance, rely on time-664

lapse images to synthesize temporal phenomena including paint665

10

Figure 20: Overview of the optimization process from [81]. Parameter vectorp is optimized iteratively. After each simulation, the statistics of the simulationare compared to the statistics of the reference image. (image copyright TheAuthors 2012)

cracking. Lu et al. [78] resort to similar data but to analyze666

how these phenomena might be influenced by a set of context667

parameters. These parameters are measured during the acqui-668

sition process and include local geometrical properties of the669

object. Given a target 3D object, these kinds of properties serve670

to guide the placement of features like cracks. Refer to Figure671

19 for an example.672

More recently, Glondu et al. [81] proposed using real ex-673

amples to guide a physical simulation of fracture. Rather than674

matching the input pattern, they focus on matching a set of675

statistics captured from the example. By means of a user study,676

they evaluate which statistics are more relevant for visual simi-677

larity, using measures similar to [68]. Using an extension from678

[55], these statistics are then used in an inverse procedure to679

estimate the simulation parameters that generate such patterns.680

Refer to Figure 20 for an overview of this process.681

7. Conclusions682

It is well known that creating good and realistic motion,683

whatever the field of application, is a challenging task due to684

its complexity. Our survey addresses the particular problem of685

simulating and animating cracks and fractures, and provides a686

survey of the most relevant techniques proposed in the litera-687

ture.688

There is no single generic solution or representation that689

can cover all cases well because the behavior of each mate-690

rial can vary in various ways, depending on multiple factors.691

Thus, our classification is based on the way the simulation of692

these phenomena is performed. One section was dedicated to693

physically-based works, which can be classified according to694

the simulation method. A further section was dedicated to those695

approaches that renounce physical accuracy preferring to select696

a procedural technique in order to gain a more interactive ap-697

plication. The final classifications focus on approaches based698

on examples (i.e., data, images, etc). Theses approaches are699

looking to replicate observed patterns, consequently avoiding700

tedious physical experiments and computations.701

We can see that FEM methods are some of the most pop-702

ular techniques, while meshless approaches are gaining more703

and more relevance. This is surely because of their ability to704

correctly represent cracks and fracture patterns. Models derived705

from the equations of continuum mechanics are important as it706

is possible to quantify the precision of the simulation, since the707

parameters of the model can be obtained from experiments con-708

ducted on real world objects, even if it is not trivial to obtain all709

the necessary quantities. Some authors, on the other hand, still710

prefer to work on representations that limit their accuracy but711

provide the user with more freedom and interactivity. Geomet-712

ric approaches are one such example.713

In Table 2, we provide a comparison of the main techniques714

reviewed in this paper; which can help the reader to easily eval-715

uate the differences between them. For each method, we report716

the dimensions the method operates on, the underlying repre-717

sentation, the simulated behavior, the method or solver used718

by the technique, and the estimated performance. The latter719

represents the speed of a method on simulating a fracture pat-720

tern, where a higher number of dots represents a better perfor-721

mance/speed. This classification is roughly based on timings722

provided by the authors as well as other considerations such723

as model size or hardware specifications. Methods with higher724

performance (5 dots) are able to simulate fractures in the order725

of a few milliseconds, while slower methods (1 dot) may take726

several seconds or minutes. Please note that this performance727

can only be considered as a very rough indication of the real728

performance. The timings reported by these methods depend729

on many aspects, including the parameters chosen for the simu-730

lations, the numerical approach, the number of iterations, or any731

code optimizations. The main purpose of this score is thus to732

provide an intuitive way to compare their behavior rather than733

a precise estimation.734

Finally, we devise some current challenges and open prob-735

lems regarding the simulation of fracture in Computer Graph-736

ics.737

Scalability. Despite recent advances in the field, physically-738

based methods are still too time consuming to deal with large739

scenes. Real-time approaches either rely on fast geometric ap-740

proaches or on reducing the physical accuracy to obtain greater741

performance. Even with these approximations, the complexity742

of the simulated objects is still limited, so there is a need to deal743

with very complex scenes such as outdoor scenarios.744

Parameter tweaking. Obtaining a specific pattern often745

implies trial and error of the parameters offered by each method,746

which can be tedious for artists. Image-based methods are able747

to directly map real patterns onto the objects, although they748

often lack physical validity. There is a need to provide more749

intuitive ways to edit these patterns and to capture them from750

measured data. A recent approach [81] represents a first step751

towards combining both approaches and capturing such param-752

eters from images, although there is still a great deal of room753

for improvement.754

Fracture Surface. Generating fracture surface is an impor-755

tant challenge that needs specific solutions. Little of the pre-756

11

Type/Method Dim. Representation Behavior Method/Solver PerformanceMass-spring[16] 3D deformed lattices brittle Euler n/a[17] 2D two-layer brittle n/a ••

[18] 3D lattice (cubic cells) brittle n/a ••

[21] 3D tetrahedra brittle finite diff (moisture) + quantum mechanics ••

[22] 3D interconnected voxels brittle Euler ••••

[23] 3D arbitrarily shaped voxels brittle Euler •••••

[24] 3D lattice w/ rigid constraints brittle quasi-static •••

FEM[2] 2D mesh ductile semi-implicit n/a[28, 8] 3D tetrahedra brit./duct. Euler or 2nd-order Taylor •

[33] 3D tetrahedra brittle rigid body (Euler) + quasi-static ••••

[20] 2.5D wedge elements brittle static •

[41] 2D/3D thin sells/tetrahedra ductile Newmark ••

[32] 3D tetrahedra ductile implicit Euler ••••

[38] 3D surface + cube mesh brittle implicit Euler •••

[34] 3D mesh/tetrahedra + level set ductile rigid body + quasi-static •••

[36] 2D triangle mesh brittle Euler ••

[37] 3D tetrahedra ductile backward Euler •••••

[35] 3D tetrahedra brittle quasi-static •

[43] 3D tetrahedra + level set ductile Euler •••

Meshless[47] 2D/3D point-based ductile Leap-Frog / implicit •••

[49] 3D point-based + surfels ductile Leap-Frog •••

[50] 3D point-based + mesh brittle quasi-static (Local Petrov-Galerkin) ••

[51] 3D point-based + tetrahedra brittle SPH ••••

Others[57] 3D spherical particles brittle movable cellular automata ••••

[55] 3D mesh + implicit brittle modal analysis •••••

Geometry-based[59] 2D flat mesh + grid brittle iterative propagation •••

[60] 2.5D mesh + multi-layer cells brittle cellular automata ••

[67] 2.5D bark slices brittle procedural + quasi-static ••

[72] 2D image - Voronoi ••••

[61] 2D/3D mesh + implicit brittle manual + boolean operations ••••

[64] 2.5D mesh + implicit brittle manual + boolean operations ••

[70] 2D image ductile distance transform •••••

[68] 2.5D height map + implicit brittle quasi-static + precomputed paths •••

[69] 3D mesh + level set brittle pre-scoring •••••

[66] 3D mesh brittle recursive decomposition + SPH •••

[62] 3D mesh brittle pre-scoring + convex decomposition ••••

Example-based[77] 2.5D image - texton analysis + user assistance ••

[76] 2D image - vectorization + projection n/a[78] 2D mesh + texture - texture synthesis •

[81] 3D mesh + implicit brittle optimization + simulation •••

Table 2: Comparison between different fracture methods. Refer to the text for a detailed description.

vious work has resolved this problem in an elegant way. For757

instance, in FEM methods, element cutting and local remesh-758

ing are time consuming and may lead to sliver elements. Even759

the use of virtual nodes cannot be regarded as a perfect defini-760

tive solution. In general, they all have artifacts that appear and761

that are related to the size of elements. Hence, there is still a762

pending issue crucial to this point.763

Validation. Fracture methods proposed in Computer Graph-764

ics are rarely validated against real physical data or even from a765

perceptual point of view. This is common for most simulation766

approaches as well as aging/weathering techniques [4]. Some767

of the very few works that deal with validation are Valette et768

12

al. [82], who compare their patterns obtained in lab conditions769

against real patterns, or Ramanarayanan et al. [83], who pro-770

vided a framework to link physically accurate to visually accu-771

rate models using a Visual Equivalence Predictor. Having a set772

of standard tests based on real-life examples could clearly help773

here.774

Implementation. Developing a fracture technique often775

implies starting from scratch and the techniques available are776

rarely made public. This also makes the comparison with pre-777

vious approaches much more difficult. Given the complexity778

of some of these methods, promoting the availability of such779

techniques and the reusability of libraries would help to further780

advance the field.781

Fracture simulation is still a very active research area, and782

we believe this will continue to be so in the future. Given its783

applications in many areas, there is a lot of interest in develop-784

ing new techniques that offer more realism and greater interac-785

tivity. In this survey, we have revised the state-of-the-art in the786

field and we have provided insights on its present circumstances787

as well as on some of the problems that still have to be faced.788

We hope this field will further evolve and provide us with more789

exciting results.790

Acknowledgements791

We would like to thank the anonymous reviewers for their792

valuable comments. This work was partially funded by the793

TIN2013-47137-C2-2-P project from Ministerio de Economıa794

y Competitividad, Spain, and a Beatriu de Pinos grant from the795

Catalan Government, Spain.796

Bibliography797

[1] Gibson, S.F., Mirtich, B.. A survey of deformable modeling in computer798

graphics. Tech. Rep. TR-97-19; MERL, Cambridge, MA; 1997.799

[2] Terzopoulos, D., Fleischer, K.. Modeling inelastic deforma-800

tion: Viscolelasticity, plasticity, fracture. SIGGRAPH Comput Graph801

1988;22(4):269–278.802

[3] Nealen, A., Muller, M., Keiser, R., Boxerman, E., Carlson, M.. Physi-803

cally Based Deformable Models in Computer Graphics. Computer Graph-804

ics Forum 2006;25(4):809–836.805

[4] Merillou, S., Ghazanfarpour, D.. A survey of aging and weathering phe-806

nomena in computer graphics. Computers & Graphics 2008;32(2):159–807

174.808

[5] Glondu, L.. Physically-based and real-time simulation of brittle fracture809

for interactive applications. Ph.D. thesis; INRIA Rennes - Ens Cachan,810

France; 2012.811

[6] Bonet, J., Wood, R.. Nonlinear Continuum Mechanics for Finite Element812

Analysis. Cambridge University Press; 2008. ISBN 9781139467544.813

[7] Chakrabarty, J.. Theory of Plasticity. Elsevier Science; 2006. ISBN814

9780080481364.815

[8] O’Brien, J.F., Bargteil, A.W., Hodgins, J.K.. Graphical model-816

ing and animation of ductile fracture. ACM Transactions Graphics817

2002;21(3):291–294.818

[9] Beer, F., Jr. Johnston, E., DeWolf, J., Mazurek, D.. Mechanics of819

Materials. McGraw-Hill Education; 2011. ISBN 9780073380285.820

[10] Wikipedia, . Stressstrain curve. 2014. [accessed 14-August-2014]; URL821

http://en.wikipedia.org/wiki/Stress-strain curve.822

[11] von Mises, R.. Mechanik der festen Korper im plastisch- deformablen823

Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu824

Gottingen, Mathematisch-Physikalische Klasse 1913;1913:582–592.825

[12] O’Brien, J.F., Hodgins, J.K.. Animating fracture. Commun ACM826

2000;43(7):68–75.827

[13] Hairer, E., Nørsett, S., Wanner, G.. Solving Ordinary Differential Equa-828

tions I Nonstiff problems. Springer; 2000.829

[14] Atkinson, K.E.. An Introduction to Numerical Analysis. New York:830

Wiley; 1978.831

[15] Petroski, H.. Invention by design : How Engineers Get from Thought to832

Thing. Harvard University Press,; 1996.833

[16] Norton, A., Turk, G., Bacon, R., Gerth, J., Sweeney, P.. Animation of834

fracture by physical modeling. The Visual Computer 1991;7:210–219.835

[17] Hirota, K., Tanoue, Y., Kaneko, T.. Generation of crack patterns with a836

physical model. The Visual Computer 1998;14(3):126–137.837

[18] Hirota, K., Tanoue, Y., Kaneko, T.. Simulation of three-dimensional838

cracks. The Visual Computer 2000;16:371–378.839

[19] Federl, P., Prusinkiewicz, P.. A texture model for cracked surfaces,840

with an application to tree bark. In: In Proceedings of Western Computer841

Graphics Symposium. 1996, p. 23–29.842

[20] Federl, P., Prusinkiewicz, P.. Modelling fracture formation in bi-layered843

materials, with applications to tree bark and drying mud. In: In Proceed-844

ings of Western Computer Graphics Symposium. 2002, p. 29–35.845

[21] Aoki, K., Dong, N.H., Kaneko, T., Kuriyama, S.. Physically based sim-846

ulation of cracks on drying 3d solids. Computer Graphics International847

Conference 2004;:357–364.848

[22] Mazarak, O., Martins, C., Amanatides, J.. Animating exploding objects.849

In: In Proceedings of Graphics Interface. Morgan Kaufmann Publishers850

Inc; 1999, p. 211–218.851

[23] Martins, C., Buchanan, J., Amanatides, J.. Visually believable explo-852

sions in real time. Computer Animation, 2001 The Fourteenth Conference853

on Computer Animation Proceedings 2001;.854

[24] Smith, J., Witkin, A.P., Baraff, D.. Fast and controllable simulation of855

the shattering of brittle objects. Comput Graph Forum 2001;20(2):81–90.856

[25] Mikhlin, S.. Variational methods in mathematical physics. International857

series of monographs in pure and applied mathematics; Pergamon Press;858

[distributed by Macmillan, New York]; 1964.859

[26] Swenson, D.V., Ingrafiea, A.R.. Modeling mixed-mode dynamic crack860

propogation using finite elements- theory and applications. vol. 3. 1988,861

p. 381–397.862

[27] Terzopoulos, D., Fleischer, K.. Deformable models. The Visual Com-863

puter 1988;4(6):306–331.864

[28] O’Brien, J., Hodgins, J.. Graphical model and animation of brittle frac-865

ture. In: In Proceedings of SIGGRAPH’99. 1999, p. 137–146.866

[29] Griffith, A.. The Phenomena of Rupture and Flow in Solids. Philosophi-867

cal Transactions of the Royal Society, London 1921;A221:163–198.868

[30] Irwin, G.R.. Analysis of Stresses and Strains Near the End of a Crack869

Traversing a Plate. J Appl Mech 1957;.870

[31] Yngve, G.D., O’Brien, J.F., Hodgins, J.K.. Animating explosions. In:871

Proceedings of ACM SIGGRAPH 2000. 2000, p. 29–36.872

[32] Muller, M., Gross, M.. Interactive virtual materials. In: GI ’04- Pro-873

ceedings of Graphics Interface 2004. 2004, p. 239–246.874

[33] Muller, M., McMillan, L., Dorsey, J., Jagnow, R.. Real-time simu-875

lation of deformation and fracture of stiff materials. In: Proceedings of876

the Eurographic workshop on Computer animation and simulation. New877

York, NY, USA: Springer-Verlag New York, Inc. ISBN 3-211-83711-6;878

2001, p. 113–124.879

[34] Bao, Z., Hong, J.M., Teran, J., Fedkiw, R.. Fracturing rigid materials.880

IEEE Trans Vis Comput Graph 2007;13(2):370–378.881

[35] Zheng, C., James, D.L.. Rigid-body fracture sound with precom-882

puted soundbanks. ACM Transactions on Graphics (Proceedings of SIG-883

GRAPH 2010) 2010;29(3).884

[36] Iben, H.N., O’Brien, J.F.. Generating surface crack patterns. Graph885

Models 2009;71:198–208.886

[37] Parker, E.G., O’Brien, J.F.. Real-time deformation and fracture in a game887

environment. In: Symposium on Computer Animation. ACM; 2009, p.888

165–175.889

[38] Muller, M., Teschner, M., Gross, M.. Physically-based simulation890

of objects represented by surface meshes. In: Proc. Computer Graphics891

International. 2004, p. 156–165.892

[39] Wicke, M., Ritchie, D., Klingner, B.M., Burke, S., Shewchuk, J.R.,893

OBrien, J.F.. Dynamic local remeshing for elastoplastic simulation.894

ACM Trans Graph 2010;29:49–1–49–11.895

[40] Federl, P., Prusinkiewicz, P.. Finite element model of fracture formation896

on growing surfaces. In: In Proceedings of International Conference on897

Computational Science. 2004,.898

13

[41] Molino, N., Bao, Z., Fedkiw, R.. A virtual node algorithm for changing899

mesh topology during simulation. ACM Trans Graph 2004;23:385–392.900

[42] Moes, N., Dolbow, J., Belytschko, T.. A finite element method for crack901

growth without remeshing. International Journal for Numerical Methods902

in Engineering 1999;:131–150.903

[43] Hegemann, J., Jiang, C., Schroeder, C., Teran, J.M.. A level set904

method for ductile fracture. In: Proceedings of the 12th ACM SIG-905

GRAPH/Eurographics Symposium on Computer Animation. SCA ’13;906

New York, NY, USA: ACM; 2013, p. 193–201.907

[44] Belytschko, L.T., Organ, D., Krongauz, Y.. A coupled finite element- el-908

ementfree galerkin method. Computational Mechanics 1995;17(3):186–909

195.910

[45] Sukumar, N., Moran, B., Black, T., Belytschko, L.T.. An element-911

free galerkin method for three-dimensional fracture mechanics. Compu-912

tational Mechanics 1997;:170–175.913

[46] Belytschko, L.T., Tabbara, M.. Element-free galerkin method for wave914

propogation and dynamic fracture. Computer Methods in Applied Me-915

chanics and Engineering 1995;126(1-2):131–153.916

[47] Muller, M., Keiser, R., Nealen, A., Pauly, M., Gross, M., Alexa, M..917

Point based animation of elastic plastic and melting objects. In: Proceed-918

ings of the 2004 ACM SIGGRAPH/Eurographics symposium on Com-919

puter animation. SCA 2004; Eurographics Association. ISBN 3-905673-920

14-2; 2004, p. 141–151.921

[48] Levin, D.. The approximation power of moving least-squares. Math922

Comput 1998;67(224):1517–1531.923

[49] Pauly, M., Keiser, R., Adams, B., Dutre, P., Gross, M., Guibas,924

L.J.. Meshless animation of fracturing solids. ACM Trans Graph925

2005;24(3):957–964.926

[50] Liu, N., He, X., Li, S., Wang, G.. Meshless simulation of brittle927

fracture. Journal of Visualization and Computer Animation 2011;22(2-928

3):115–124.929

[51] Chen, F., Wang, C., Xie, B., Qin, H.. Flexible and rapid anima-930

tion of brittle fracture using the smoothed particle hydrodynamics for-931

mulation. Journal of Visualization and Computer Animation 2013;24(3-932

4):215–224.933

[52] Gingold, R.A., Monaghan, J.J.. Smoothed particle hydrodynamics -934

Theory and application to non-spherical stars. Mon Not Roy Astron Soc935

1977;181:375–389.936

[53] Steinemann, D., Otaduy, M.A., Gross, M.. Fast arbitrary split-937

ting of deforming objects. In: Proceedings of the 2006 ACM SIG-938

GRAPH/Eurographics symposium on Computer animation. SCA ’06; Eu-939

rographics Association; 2006, p. 63–72.940

[54] Wicke, M., Teschner, M., Gross, M.H.. Csg tree rendering for point-941

sampled objects. In: Pacific Conference on Computer Graphics and Ap-942

plications. IEEE Computer Society; 2004, p. 160–168.943

[55] Glondu, L., Marchal, M., Dumont, G.. Real-time simulation of brittle944

fracture using modal analysis. IEEE Transactions on Visualization and945

Computer Graphics 2013;19(2):201–209.946

[56] Glondu, L., Schvartzman, S.C., Marchal, M., Dumont, G., Otaduy,947

M.A.. Fast collision detection for fracturing rigid bodies. IEEE Transac-948

tions on Visualization and Computer Graphics 2013;99(PrePrints):1.949

[57] Ning, J., Xu, H., Wu, B., Zeng, L., Li, S., Xiong, Y.. Modeling950

and animation of fracture of heterogeneous materials based on cuda. The951

Visual Computer 2013;29(4):265–275.952

[58] Psakhie, S., Horie, Y., Korostelev, S., Smolin, A., Dmitriev, A.,953

Shilko, E., et al. Method of movable cellular automata as a tool for954

simulation within the framework of mesomechanics. Russian Physics955

Journal 1995;38(11):1157–1168.956

[59] Neff, M., Fiume, E.. A visual model for blast waves and fracture. In: In957

Proc. of Graph. Interface. 1999, p. 193–202.958

[60] Gobron, S., Chiba, N.. Crack pattern simulation based on 3d surface959

cellular automaton. The Visual Computer 2001;17:287–309.960

[61] Martinet, A., Galin, E., Desbenoit, B., Hakkouche, S.. Procedural mod-961

eling of cracks and fractures. In: Shape Modelling International (Short962

Paper ). Genova, Italy; 2004, p. 346–349.963

[62] Muller, M., Chentanez, N., Kim, T.Y.. Real time dynamic fracture964

with volumetric approximate convex decompositions. ACM Trans Graph965

2013;32(4):115:1–115:10.966

[63] Hellrung, J., Selle, A., Shek, A., Sifakis, E., Teran, J.. Geometric967

fracture modeling in bolt. In: ACM SIGGRAPH Talks (Sketch). 2009,.968

[64] Desbenoit, B., Galin, E., Akkouche, S.. Modeling cracks and fractures.969

The Visual Computer 2005;21(8-10):717–726.970

[65] Raghavachary, S.. Fracture generation on polygonal meshes using971

voronoi polygons. In: ACM SIGGRAPH 2002 Conference Abstracts and972

Applications. New York, NY, USA: ACM; 2002, p. 187–187.973

[66] Oh, S., Shin, S., Jun, H.. Practical simulation of hierarchical brittle974

fracture. Journal of Visualization and Computer Animation 2012;23(3-975

4):291–300.976

[67] Lefebvre, S., Neyret, F.. Synthesizing bark. In: Gibson, S., De-977

bevec, P.E., editors. 13th Eurographics Workshop on Rendering Tech-978

niques (EGSR ’02); vol. 28. Pisa, Italy: Eurographics Association; 2002,979

p. 105 – 116.980

[68] Valette, G., Prevost, S., Lucas, L., Leonard, J.. A dynamic model of981

cracks development based on a 3d discrete shrinkage volume propagation.982

Comput Graph Forum 2007;27(1):47–62.983

[69] Su, J., Schroeder, C., Fedkiw, R.. Energy stability and fracture for984

frame rate rigid body simulations. In: Proceedings of the 2009 ACM SIG-985

GRAPH/Eurographics Symposium on Computer Animation. New York,986

NY, USA: ACM; 2009, p. 155–164.987

[70] Wyvill, B., van Overveld, K., Carpendale, S.. Rendering cracks in batik.988

In: Proceedings of the 3rd international symposium on Non-photorealistic989

animation and rendering. NPAR ’04; New York, NY, USA: ACM. ISBN990

1-58113-887-3; 2004, p. 61–149.991

[71] Paquette, E., Poulin, P., Drettakis, G.. The simulation of paint cracking992

and peeling. In: Graphics Interface 2002. 2002, p. 59–68.993

[72] Mould, D.. Image-guided fracture. In: GI ’05- Proceedings of Graphics994

Interface 2005. 2005, p. 219–226.995

[73] Fracture FX. 2014. URL http://www.fracture-fx.com.996

[74] Houdini. 2014. URL http://www.sidefx.com.997

[75] Shum, H., Kang, S.B.. Review of image-based rendering techniques.998

Proc SPIE 2003;4067:2–13.999

[76] Hsien-Hsi, H., Wen-Kai, T.. A straightforward and intuitive approach on1000

generation and display of crack-like patterns on 3d objects. In: Computer1001

Graphics International. 2006, p. 554–561.1002

[77] Wang, X., Wang, L., Liu, L., Hu, S., Guo, B.. Interactive modeling of1003

tree bark. 11th Pacific Conference on Computer Graphics and Applica-1004

tions 2003;.1005

[78] Lu, J., Georghiades, A.S., Glaser, A., Wu, H., Wei, L.Y., Guo, B.,1006

et al. Context-aware textures. ACM Trans Graph 2007;26(1):3.1007

[79] Wei, L.Y., Lefebvre, S., Kwatra, V., Turk, G., et al. State of the art in1008

example-based texture synthesis. In: Eurographics 2009, State of the Art1009

Report, EG-STAR. 2009, p. 93–117.1010

[80] Enrique, S., Koudelka, M., Belhumeur, P., Dorsey, J., Nayar, S.,1011

Ramamoorthi, R.. Time-varying textures: definition, acquisition, and1012

synthesis. In: ACM SIGGRAPH 2005 Sketches. ACM; 2005, p. 130.1013

[81] Glondu, L., Muguercia, L., Marchal, M., Bosch, C., Rushmeier, H.,1014

Dumont, G., et al. Example-based fractured appearance. Computer1015

Graphics Forum 2012;31(4):1547–1556.1016

[82] Valette, G., Herbin, M., Lucas, L., Leonard, J.. A preliminary approach1017

of 3d simulation of soil surface degradation by rainfall. In: NPH. 2005,1018

p. 41–49.1019

[83] Ramanarayanan, G., Ferwerda, J., Walter, B., Bala, K.. Visual equiv-1020

alence: Towards a new standard for image fidelity. ACM Trans Graph1021

2007;26(3).1022

14