Volume 28 (2009), Number 7

Pacific Graphics 2009
S. Lee, D. Lischinski, and Y. Yu(Guest Editors)

Simulation of Tearing Cloth with Frayed Edges
Napaporn Metaaphanon†† The University of Tokyo

Yosuke Bando†,‡Bing-Yu Chen§‡ TOSHIBA Corporation

Tomoyuki Nishita†§ National Taiwan University

Abstract
Woven cloth can commonly be seen in daily life and also in animation. Unless prevented in some way, woven clothusually frays at the edges. However, in computer graphics, woven cloth is typically modeled as a continuum sheet,
which is not suitable for representing frays. This paper proposes a model that allows yarn movement and slippage
during cloth tearing. Drawing upon techniques from textile and mechanical engineering fields, we model cloth as
woven yarn crossings where each yarn can be independently torn when the strain limit is reached. To make themodel practical for graphics applications, we simulate only tearing part of cloth with a yarn-level model using asimple constrained mass-spring system for computational efficiency. We designed conditions for switching from astandard continuum sheet model to our yarn-level model, so that frays can be initiated and propagated along thetorn lines. Results show that our method can achieve plausible tearing cloth animation with frayed edges.
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism—Animation; I.6.8 [Simulation and Modeling]: Types of Simulation—Animation

1. Introduction
In the field of computer graphics, researches about cloth aremainly focused on the deformation of cloth as a continuum.
Typical models represent cloth as a sheet of elastic material, which means that only macroscopic geometric featuresof cloth could be modeled. Since these kinds of models allow compressing, stretching, out-of-plane and also in-planebending, they are suitable for representing woven cloth ingeneral cases. However, in some cases, treating cloth as awhole is not sufficient; the case of tearing cloth is one ofthem.
Torn edges of woven cloth usually fray as shown in Figure 1(a). The weave becomes ruined, and yarns can easily
move out. This behavior is caused by weaving, which is a
special characteristic of woven cloth. Much work in computer graphics assumes that slippage does not occur between
yarns at their crossings including the edges. Hence, frayededges of woven cloth are normally ignored, and only smoothtorn edges are produced, resulting in rubber-like appearanceof tear.
† e-mail: {noimeta, ybando, nis}@nis-lab.is.s.u-tokyo.ac.jp‡ e-mail: yosuke1.bando@toshiba.co.jp§ e-mail: robin@ntu.edu.tw
c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.
Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and
350 Main Street, Malden, MA 02148, USA.

(a) Frays at torn edges(b) Plain weave

Figure 1: (a) A real example of woven cloth and how it looks
when torn. (b) Schematics of the plain weave.

In this paper, we propose a cloth model that can capture
the dynamic behavior of fraying at edges when woven cloth
is being torn. In order to allow yarn movement and slippageat frayed edges, we adapt techniques from textile and mechanical engineering fields, and we model cloth as woven
yarn crossings so that each yarn can be independently tornwhen the strain limit is reached. The primary contribution ofthis paper is that, as opposed to the typical yarn-level models in the textile industry with complicated formulation for
pursuing mechanical accuracy, we use a simple constrained
mass-spring system, and simulate only tearing part of cloth

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

with a yarn-level model in order to make the model computationally efficient and practical for computer graphics applications. We designed conditions for switching from a standard continuum sheet model to our yarn-level model, so thatyarns around the torn area become able to slip over eachother, which in turn induces slippage of other neighboringyarns as tearing proceeds. That is, fraying can be initiatedand propagated through cloth in our model. We demonstratethat our method can produce visually pleasing tearing clothanimation by showing simulation results of cloth under various types of load which leads to different dynamic behaviors
of fraying.
2. Related Work
Since cloth tearing simulation is related with both of cloth
simulation and fracture simulation, they are introduced separately in this section.
2.1. Cloth Simulation
Traditionally, cloth models in computer graphics treat woven cloth as a sheet of material. Early work by Terzopoulos et al. [TPBF87, TF88a, TF88b] simulated deformableobjects including cloth based on finite difference schemes.
Since then, a number of researchers have addressed different aspects of cloth simulation, including computational
efficiency [VCMT95, VMT00], numerical stability [BW98,
CK02], collision handling [VMT00, BFA02, BWK03], clothbending and buckling [GHDS03, BMF03, CK02, ZJW08],
cloth inextensibility [Pro95, BFA02, MHHR07, GHF∗ 07],and dressing synthetic actors [CYMTT92, VMTJT96].
While all of the above continuum approaches are efficient
for representing most characteristics of cloth, they are notsuitable for modeling frays or raveling edges of torn cloth.
Although cloth tearing simulation using a continuum modelis possible, only smooth torn edges can be generated.
In textile and mechanical engineering fields, however,yarn-level models are commonplace. The pioneering workof Peirce [Pei37] modeled geometric relationships among
circular yarns at yarn crossings. A number of modified formsof Peirce’s model have been developed: a non-circular crosssection model by Kemp [Kem58], a three-dimensional yarncrossing structure by Kawabata et al. [KNK73], and a curvemodel of yarn crossings by Warren [War90], for instance.
These models are designed to investigate the elastic properties of fabrics at equilibrium, while we use a yarn-crossing
model similar to these to simulate dynamic fraying behaviorof tearing cloth.
Some researchers simulated dynamic cloth tearing in thecontext of analyzing ballistic properties of bulletproof woven fabric armors. The work of Zeng et al. [ZTS06] bears
many similarities to ours, which models each yarn as a chainof linear elements whose nodal points are placed at yarn
crossings. Frays can be seen in the simulated results aroundthe bullet hole. Zhang et al. [ZBZ08] used a similar but
more complicated model in which each yarn is discretized
into hexahedral finite elements. The primary differences of
our work are the simplicity and practicality of the proposedmodel for graphics applications: we simulate only tearing
part of cloth with a yarn-level model using a simple constrained mass-spring system for computational efficiency,with tailored conditions for switching from a standard continuum sheet model to the yarn-level model.
Some work from the computer graphics community conducted yarn- or strand-level simulation.
Breen et al. [BHW94] used a particle-based model that
represents yarn crossings to simulate static shapes of draped
woven cloth. Kaldor et al. [KJM08] presented yarn-level
simulation of knitted cloth that produced detailed andvisually intriguing deformation of knitwear. Unfortunately,
as mentioned in their paper, the mechanical structures andbehaviors of knits are totally different from woven cloth,and therefore we need to devise a separate model for woven
cloth. Selle et al. [SLF08] proposed a hair model thatcan efficiently and stably simulate every one of a hundred
thousand strands on a human head. None of the above
methods treat tearing and fraying of yarns.

2.2. Fracture Simulation
In computer graphics literature, fracture simulation has
been researched on several kinds of materials. Terzopoulos and Fleischer [TF88a, TF88b] demonstrated atechnique to tear sheets of paper and cloth-like material. O’Brien and Hodgins [OH99] simulated fracture of brittle objects based on finite element methods,
which was extended to include ductile fracture [OBH02].
Smith et al. [SWB01] also simulated shattering of brittle objects efficiently by constraint forces. Norton et al. [NTB∗ 91]
used a mass-spring system to model solid objects that broke
under large strains, whereas Hirota et al. [HTK98] modeled
the drying process of mud with cracks. Unfortunately, noneof the above methods modeled strand-like material, and they
are not directly applicable to simulating dynamic fraying behavior of tearing cloth.

3. Woven Cloth Modeling
As described in Section 1, a two-level model is used for simulating tearing cloth: the base cloth model, which is a continuum sheet model for simulating untorn part of cloth; andthe yarn-level model for simulating raveling edges. In thissection, we start by reviewing the basic structure of woven
cloth, and then explain about each level of our cloth modeland how and when to switch from the base cloth model tothe yarn-level model.
c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

3.1. Structure of Woven Cloth

Table 1: Parameters used in our model.

Woven cloth is made up of two sets of yarns called warps
and wefts. Warps are lengthwise yarns that run verticallyin woven fabrics, while wefts are crosswise yarns that run
horizontally under and over parallel warps. Various mechanical properties of cloth are determined by the characteristics of yarns themselves and how they are woven or interlaced [HB00]. The simplest fundamental weave structure isplain weave, in which each weft simply goes over one warp
and under the next repetitively as shown in Figure 1(b).
For simplicity of explanation, we will assume plain weave
in what follows; applying our method to other types of
weave is straightforward, though. We will show an exampleof twill weave in Section 6.

Cloth thickness T
0.01 cm
Cloth areal density ρ
0.25 g/cm2
Number of yarns per unit length σ
10-15 cm−1
Structural spring constant kstruct
6000 g/s2
Bend spring constant kbend
5.0 g/s2
Shear spring constant kshear
2.0 g/s2
Strain threshold for tearing θtear
5-10%
Strain threshold for model transition θtrans4.5-9%
Distance threshold for mass couples θdist
0.02 cm
Number of constraint projections N pro j
30-50
Time step ∆t
1/240 s3.3. Yarn-Level Model
3.2. Base Cloth Model
As continuum sheet models often suffice for simulating
undamaged cloth, we model untorn part of cloth using
a standard mass-spring system as shown in Figure 2(a).
Three types of springs are attached to each mass: structural
springs, bend springs and shear springs. Structural springs
connect immediate neighbors, while bend springs connect
every other particles. Shear springs connect diagonal particles, and are only active under compression [ZJW08]. Table 1 summarizes the spring constants we used, along withother parameters which will be used in this paper. For numerical stability, the magnitude of the structural spring constant kstruct is kept moderate, and the tensile stiffness of clothis enforced by strain limiting, as will be described in Section 4.1. For consistency with the yarn-level model, we use
quadrilateral meshes of yarn-level resolution. Using generaltriangular meshes and coarser resolution for the base clothmodel is left as future work.
For tearing simulation, we assign strain threshold θtear
randomly between 5% and 10% to each structural spring,
and we cut springs whose strain exceeds their thresholds.

At the yarn level, individual yarns are modeled explicitly as
warps or wefts crossing over the others (see Figure 2(b)).
Each yarn is composed of a series of mass particles andsprings. At the rest state, one particle from a warp and another particle from a weft are located at their crossover point(we collectively call them a mass couple). Initially, the twoparticles in a mass couple are separated by the cloth thickness T in the direction of the cloth normal vector n̂c at thecrossing. The normal vector n̂c is calculated as the average ofthe normal vectors of the four triangles around the crossover
point as shown in Figure 3(a). In our algorithm, the normal
vector n̂c is calculated only at most once per crossover point
when a particle in the base cloth model is split into a mass
couple in the yarn-level model.
As we use the yarn-level model to represent frays, we donot strictly constrain a mass couple to stay close to eachother, and we allow two states for each mass couple, referred
to as loosely connected and disconnected. In the loosely connected state, we apply constraints so that the warp and weft
springs attached to a mass couple can slide over each other,which will be explained below in detail. When a mass couple draws apart beyond the distance threshold θdist , the stateof that mass couple is changed to disconnected. In the disconnected state, no constraints are applied.

nˆ c(a) Base cloth modelpb
q a tb(b) Yarn-level model

Figure 2: The two models used in our method. (a) The basecloth model with mass particles (blue), structural springs(orange), shear springs (green) and bend springs (red). (b)The yarn-level model with mass couples. The green balls andsprings represent warps, while the red balls and springs represent wefts.

c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

ta
nˆ s

(a) Cloth normalpaqb(b) Spring pair’s normal

Figure 3: Two types of normal vectors used in our method.
(a) The normal vector n̂c of cloth. (b) The normal vector n̂s
for the closest pair of a warp spring and a weft spring.

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

(a)

(b)

(c)

(d)

Figure 4: Transition from the base cloth model to the yarn-level model. (a) The base cloth model is stretched and one of thestructural springs is cut. (b) The two particles at both ends of the cut spring, plus their adjacent particles, are split. (c) One of
the mass couples becomes disconnected as highlighted in purple. (d) The neighboring particles are also split.

Constraint applying process for a loosely connected mass
couple starts from determining the closest pair of a warp
spring and a weft spring attached to the mass couple asshown in Figure 3(b). Since there are at most two warp andtwo weft springs per mass couple, a maximum of four warpweft spring pairs need to be checked so as to identify theclosest one. Afterwards, the normal vector n̂s is obtained bynormalizing the vector between the closest points. The constraints are applied in the direction of normal vector n̂s to one
spring and in the opposite direction −n̂s to the other, so thatthe closest distance will be kept constant (i.e., cloth thickness T ). However, instead of applying constraints directly tothe closest points, they are applied to the spring endpoints,
weighted by the barycentric coordinates [BFA02]. Let a andb be the closest warp-weft spring pair, and pa , qa , pb , andqb be their endpoints. If the closest points are at the fraction ta and tb along the edges pa qa and pb qb as shown in
Figure 3(b), the new positions of the endpoints can be calculated as:pnew= pa + (1 − ta )(d − T )n̂s ,aqnew= qa + (ta )(d − T )n̂s ,apnew= pb − (1 − tb )(d − T )n̂s ,bqnew= qb − (tb )(d − T )n̂s ,b(1)where d is the closest distance between the two springs.

When a mass couple becomes disconnected as describedin Section 3.3, the mass splitting propagates to its neighbors
as shown in Figures 4(c)(d). The propagation stops at particles with a different weaving pattern. In other words, if thewarp is over the weft at a disconnected mass couple, we split
masses until reaching points where a weft is over a warp.
Let mbase , xbase , and vbase denote the mass, position, andvelocity of a mass particle in the base model. We also use
subscripts warp and we f t to indicate those of a mass couple
in the yarn-level model. As described in Section 3.3, we split
a mass particle in the cloth normal direction n̂c as:
T
n̂c ,
2
T
xwe f t = xbase − n̂c .
(2)2Note that the signs (addition or subtraction) in Eq. (2) may
be opposite depending on the weave pattern of a fabric. We
set mwarp = mwe f t = mbase /2 for conservation of mass, andvwarp = vwe f t = vbase for conservation of linear and angularmomenta, as shown in Figure 5(a). In fact, letting P and L
denote linear and angular momenta, we have:xwarp = xbase +Pwarp + Pwe f t = mwarp vwarp + mwe f t vwe f tmm= base vbase + base vbase2
2= mbase vbase = Pbase ,(3)

and also:
3.4. Model Transition
The transition from the base cloth model to the yarn-level
model occurs in over-strained regions, where weave becomes loose. The mass particles at both ends of a structural
spring whose strain exceeds the threshold θtrans , as well as
their immediate neighbor particles, are split into mass couples as shown in Figures 4(a)(b). Although Figures 4(a)(b)show for simplicity the case where the model transition occurs at the same time when a spring is cut, we actually set thethreshold θtrans slightly less than the spring cutting thresholdθtear as listed in Table 1, so that weave loosening can occur
before cloth begins to tear.

Lwarp + Lwe f t = xwarp × mwarp vwarp + xwe f t × mwe f t vwe f t
Tm= (xbase + n̂c ) × base vbase
22Tm+ (xbase − n̂c ) × base vbase22= xbase × mbase vbase = Lbase .
(4)
When splitting a mass particle, we reattach the springs toa mass couple as follows. As shown in Figure 5(a), structural
springs are separated into warp and weft springs, and attached to the corresponding particles. Reattachment of bend
springs goes in a similar manner to that of structural springs.
c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

If a bend spring is on warp, we attach it to the warp particle; otherwise we attach it to the weft particle as shown in
Figure 5(b). Shear springs need duplication before reattachment. Two duplicated shear springs connect particles in thesame plane (upper or lower one), or connect to the same unsplit particle as shown in Figure 5(c). Because the number ofsprings is doubled, the shear spring constant becomes a halfof the original one (i.e., kshear ) to preserve the same stiffness.

v baseof the simulation without tearing forces, and determine thenumber of sweeps N pro j needed to limit the extension of all
springs in the system below the threshold θtear . We use thisnumber N pro j for simulating tearing cloth, and we cut overstrained springs beyond θtear even after N pro j sweeps of theconstraint projection are performed. Similar to the idea ofstrain limiting ordering presented by Selle et al. for hair simulation [SLF08], we apply strain limiting in increasing orderof distance to fixed mass particles (e.g., those attached to theballs and bars in Figures 6 and 7), as springs around fixed
particles are most likely to be elongated.

mbase
4.2. Collision Handling
Since this paper is mainly concerned with cloth tearing andfraying, we assume objects other than cloth consist of rigid
bodies, and we only handle collisions between cloth particles and rigid body faces. Although this allows penetrationof springs into rigid bodies, no visual artifacts were seen in
our experiments.

v basembase v base2 mbase

2
(a)

(b)

(c)

Figure 5: Splitting mass particles (from top to bottom). (a)Structural spring reattachment. (b) Bend spring reattachment. (c) Shear spring duplication and reattachment.

4. Cloth Simulation

The self-collision type in our model is edge-edge collision. To detect a collision between two moving edges x1 x2
and x3 x4 , we check if there is time t between two consecutive frames (at t = 0 and 1) at which the both edges aretouching. When two edges are in contact, the four endpoints
are coplanar. Therefore, assuming the endpoints move withconstant velocities v1 , v2 , v3 and v4 between the frames, thecollision time t can be obtained by solving the following
equation:

In each simulation loop, our algorithm performs as follows:(x31 + tv31 ) · (x21 + tv21 ) × (x43 + tv43 ) = 0,

1. Apply forces to cloth particles.
2. Update velocities.
3. Damp velocities.
4. Update positions.
5. Apply constraints (Eq. (1) and Section 4.1).
6. Process collisions (Section 4.2).
7. Apply model transition and cut springs according to thestrain values of the springs (Section 3.4).
8. Post-manipulate velocities.

where xi j and vi j are short for x j − xi and v j − vi . To compute the roots t, instead of floating-point arithmetic, we useinterval arithmetic for robust calculation [Eri04]. All roots
outside of [0, 1] are discarded. If multiple roots are left, theleast one is chosen as the colliding time. If there is no root,no collision has occurred.

For time integration of cloth dynamics, we use 8 explicit
Euler steps with time step ∆t = 1/240 s to produce 30 fps
animation. Stability is achieved by the constraint projection
method for strain limiting and the post velocity manipulationof Müller et al. [MHHR07] (Steps 3, 5, and 8).

To capture the details of weaving and to prevent possible visual seams at the border between the two-level models, thewhole cloth is rendered as a number of interleaved yarns.
Warp and weft positions for base-model masses are calculated using Eq. (2). Each yarn is drawn as a Catmull-Rom
spline that interpolates warp/weft particles. Although interpolated splines are not guaranteed to be free from mutual
penetration, no noticeable artifacts were seen in our rendering results.

4.1. Strain Limiting
We impose constraints on structural springs to prevent themfrom stretching excessively. Each constraint is solved independently one after the other according to [MHHR07]. Sincecorrecting the length of one spring can cause the length ofother springs to change, multiple “sweeps” of the constraint
projection are necessary, and results can depend on the order in which springs are processed. We first perform test-run
c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

(5)

5. Yarn Rendering

For yarn shading, we use the method presented in[HDK∗ 06] that blends between Kajiya-Kay [KK89] andLambertian shading. This shader requires the tangent vector of yarns, which we can calculate as the derivative of the
spline equation. We applied alpha blending to the tips of torn
yarns in order to create the stretched appearance.

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

6. Results
Figures 6-11 show our results. Simulation parameters arelisted in Table 1, and some performance statistics are summarized in Table 2. All simulations were performed on adesktop PC with an Intel Core 2 Duo 2GHz CPU.

Table 2: Performance statistics for some of our results. The
averaged times per frame for the whole animation sequence
are shown. The yarn counts are given as “warps × wefts”.
Number of yarns
80 × 40
160 × 80
64 × 64

Fig. 6
Fig. 8Fig. 9In Figure 6, a piece of cloth is grabbed by two balls at thecorners and torn apart. Frays can be seen along the torn line.
Another example is shown as an animation sequence in Figure 9. Figure 7 shows the case where a piece of cloth which isfirmly attached to two bars on the left and right sides is torn
by a thrown ball. Variation of weaving pattern is applied tothe same scene, and the cases of plain weave and twill weave
are shown. Figure 8 shows cloth that is pulled and torn by acube attached to the bottom right corner. Some more examples are shown in Figure 11, along with the corresponding
wire-frame models illustrating how the base model and theyarn-level model are combined.

Simulation time
10.3 sec/frame
67.7 sec/frame
13.1 sec/frame

Figure 8: A piece of cloth is pulled and torn by a cube.

50

50

40

40
Time (sec)Time (sec)Figure 6: A piece of cloth grabbed by two balls at its corners
is torn apart.

Figure 10(a) shows a plot of the average simulation times
per frame against the number of yarn crossings used to simulate tearing cloth as in Figure 6. An approximately linear
relationship is observed, suggesting that our method is scalable. Figure 10(b) shows the time development of the performance of our method when 120 × 60 yarns are used, alongwith the performance obtained by simulating the whole cloth
solely with the yarn-level model. As our yarn-level model
has only the loosely connected and disconnected states formass couples, we added a connected state to this “all-yarn”
model, in which we constrain each mass couple to stay closein order to keep cloth from raveling easily. As shown in theplots, we observed over 2.5 times performance gain in the
beginning of the simulation because the number of particles
is halved, and constraints on mass couples are not necessary.
While the gain gradually decreases as the base model partially switches to the yarn-level model, our two-level model
still runs faster thanks to the remaining base model part.

30
20

20
10

0

0

(a)
Figure 7: Two types of woven cloth are torn apart by athrown ball. Top: plain weave. Bottom: twill weave.

30

10
0
10000
Number of yarn crossings

Two-level modelAll-yarn model

0

10

20
Frame

30

40

(b)

Figure 10: (a) Plot of averaged simulation times against the
number of yarn crossings. (b) Plots of the time developmentof simulation times.

c 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

N. Metaaphanon et al. / Simulation of Tearing Cloth with Frayed Edges

Figure 9: Frames from an animation of tearing cloth grabbed by two balls (from left to right).

(a)

(b)

(c)

(d)

Figure 11: Rendering results (top) and the corresponding wire-frame models (bottom). The base model area is shown inmagenta, while the yarn-level model area is shown in cyan. (a) Tearing cloth in the direction perpendicular to the cloth plane.
(b) Cloth is torn away by being attached to two rigid bars that are moving apart. (c) Cloth is torn open under load at the bottom
right corner. (d) Cloth is torn by a thrown ball.

7. Conclusions and Future Work
We have presented a method for simulating tearing clothwith frayed edges. Our model does not treat cloth as a whole
sheet of material, but as individual warps and wefts that can
cut independently. As a result, frays, the most commonly
seen characteristic of torn woven cloth, are visible. Our primary contribution lies in the combination of two kinds ofmodels: the base cloth model which represents the whole
cloth at the initial state, and the yarn-level model which represents eventually ruined part. Compared to modeling all
yarns directly, the number of nodes at the initial state ishalved, and then the model is gradually refined only aroundtorn part. As a consequence, the resources needed for simulation can be reduced. We have presented a set of modelrefinement criteria with which fraying can be initiated andpropagated through cloth.
While we have demonstrated the benefit of our two-level
model, the run times of our prototype implementation are
still rather long as compared to state-of-the-art cloth simulation methods for graphics applications. We attribute thisc 2009 The Author(s)Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd.

mainly to a large number of constraint projections and also
to our unoptimized code. We would like to further analyze
this issue and to speed up the simulation.
One of the limitations of our method is that the whole
cloth has to be modeled at the yarn-level resolution. A large
number of masses and springs would be required for dense
cloth representation, which can lead to high computational
cost and slow performance. To resolve this limitation, weplan to develop a hierarchical refinement method that can use
a coarser mesh for undamaged part of cloth. We would also
like to adapt our method to more commonly-used, generaltriangular meshes instead of regular quadrilateral meshes.
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