Speeding up Cloth Simulation by Linearizing the Bending Function of the Physical

Speeding up Cloth Simulation by Linearizing the Bending Function of the Physical

Mass-Spring Model

Asmaa A. ElBadrawy and Elsayed E. Hemayed

Computer Engineering Department

Faculty of Engineering, Cairo University

Giza, Egypt

[email protected], [email protected]

Abstract—We introduce a new enhancement to speed upthe physical mass-spring model for efficient cloth simulation.The mass-spring model is known for its accuracy but slowness.The proposed technique tackles the computation burden of themodel by developing a linearization solution for the bendingfunction. The proposed technique is about six times faster thanthe traditional technique of directly evaluating the complexphysical models. The detailed of the proposed technique andits experimental results are discussed in this paper.

Keywords-Cloth simulation; bending function; physical mass-spring model;

I. INTRODUCTION

The problem of cloth simulation has more than one aspect.

This includes modeling the intrinsic properties of cloth,

handling external forces that influence textile behavior, and

handling self-collisions among textile and collisions with

other objects in the scene. Fast cloth simulation requires

handling all these aspects in an efficient way, and at the

same time maintaining plausible cloth appearance. Fast cloth

simulation is a vital requirement for a lot of interactive

applications. For example, apparel simulation for online

fitting room applications, and the computer game industry.

Physical models for cloth simulation [5], [6], [16] man-

aged to model and simulate the intrinsic properties of tex-

tiles. This enables the simulation of differnet types of textiles

by changing the model parameters. Physical models usually

produce plausible simulation results. However, they are not

efficeint in terms of performance. This is because they

use sophisticated non-linear physical equations for modeling

cloth behavior, and they use integration schemes with large

numbers of iterations. Therefore, they are not suitable for

interactive applications.

The goal of our work is to develop a technique for

fast cloth simulation while preserving -to an acceptable

extent- the ability to simulate different types of textiles. To

simulate different textiles, we use a physical model to model

intrinsic cloth properties. However, we do not implement

the underlying complex equations as they are. We first

linearize the sophisticates models, then use the linearized

versions to compute the internal forces acting on textiles

via interpolation. We also empower our technique by using a

velocity-less Verlet integration scheme which is more stable

and requires less computations per integration step compared

to other techniques. We use the same physical model that

was proposed by Choi and Ko [5]. Our technique proved to

be six times faster than the straight forward implementation

of the model. Our technique also preserved the ability to

simulate different kinds of textiles by altering the values of

the coefficients of the physical model.

This paper is organized as follows: section 2 discusses

the related research work. Section 3 presents the proposed

approach. Section 4 presents and discusses the experimental

results. Section 5 concludes the paper.

II. RELATED WORK

The problem of cloth simulation has been mainly handled

by two different models: the geometric model [11], [21] and

the physical model [1], [5], [6], [9], [10], [13]–[16], [18].

Geometric models (sometimes named Mathematical Mod-

els) focus on the appearance of cloth. They try to mimic

the external shape of cloth like folds and wrinkles, and

represent these apparent properties by geometrical equations.

They do not consider the intrinsic properties of cloth. They

consider cloth as three dimensional curves and try to find

mathematical models for these curves. They are not able

to simulate the physical properties of cloth. Accordingly,

simulating different kinds of textiles using geometric models

is not a direct task, and there are no guarantees for physical

correctness.

In physical models, various structural studies were done

on different textiles attempting to measure and model their

intrinsic behavior. The physical model preserves the internal

properties of the textile and thus enabling different textile

simulation with different motion using the same model.

This is a great advantage over geometric model. However,

Physical models are far less efficient compared to geometric

models. One simulation step using a physical model takes

more CPU cycles compared to a geometric model, and they

are accordingly not suitable for interactive applications. This

is because of the complexity of the physical equations under-

lying the physical models. Another problem with physical

2011 International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission

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model is using integration schemes with large numbers of

iterations.

Several methods were developed to speed up the cloth

simulation using physical models or to enhance its compu-

tations stability. For example, Barbic and Popovic [1] and Oh

et al. [15] addressed the problem of numerical damping in

cloth simulation. Goldenthal et al. [9] addressed the problem

of textile overstretching. Volino and Thalmann [20] proposed

a linear approach suited for the implicit integration schemes

used in particle systems such as cloth. Bergou et al. [2]

proposed an isometric bending model that reduces cloth

simulation times up to three-fold. Garg et al. [8] proposed

a hinge-based model to speed up the cloth simulation.

Bridson et al. [3] enhanced the realism of cloth simulation by

producing a cloth simulation with many folds and wrinkles.

Maintaining high simulation accuracy would inevitably

lead to an increase in computational time and vice versa.

We are developing an approach that runs in real time and

yet preserves –to some extent- the appealing look of different

textiles that result from high simulation accuracy of different

textiles achieving the performance requirements for cloth

simulation, as described by Volino and Thalmann [19];

low computational time, acceptable accuracy, and numerical

stability.

III. PROPOSED APPROACH

The cloth is represented using the mass-spring physical

model [5], [15]. It is represented as a matrix of particles

connected by a group of springs. The springs connecting

the particles are: stretch, sheer, and bending springs. The

structure, sheer, and bending connections for particle Pij

are illustrated in figure 1.

Figure 1. Connection for Pij : Stretch and sheer connections are respe-sented by straight lines, and bending connections are represented by curvedlines.

Newton’s law of motion is used to define the system

behavior. Newton’s law of motion stated that the total force

acting on a mass particle is equal to the mass of this particle

multiplied by its acceleration as shown in the equation:

F = m× a (1)

The forces acting on a particle are divided into internal

and external forces. The internal forces are the stretching,

sheering, and bending forces that exist in the springs con-

necting the particles. The external forces can be any external

factors acting on the cloth like gravity and wind. We just

account for acceleration due to gravity and air friction as

the external forces acting on cloth.

For modeling the internal forces, we use the mechanical

equations that were suggested by Choi and Ko [5]. That is,

we use linear spring equations for modeling the stretching

and sheer behavior and use a non-linear complex spring

equation for modeling the bending behavior. We use this

bending model as a case to apply our general technique to.

According to Choi and Ko [5], the force acting on a mass

point i due to a structural or sheer spring connecting it with

particle j is defined by the equation:

fi =

{

ks(|xij | − L)xij

|xij |, |xij | ≥ L

0 |xij | ≺ L(2)

Where ks represents the stiffness coefficient for stretch

and sheer, xij is the vector connecting the two mass points,

—xij— represents the current length of the spring connect-

ing the two mass points (i.e. the distance between the two

mass points), and L is the actual length of the spring under

no tension or compression forces. The force acting on a mass

point i due to a bending spring connecting it with particle

j is defined by the equation:

fi = kbk2(cos(

kL

2)− sinc(

kL

2))−1

xij

|xij |= fb(|xij |)

xij

|xij |(3)

Where kb is the flexural rigidity and k represents the

curvature of the spring connecting the mass points i and

j due to the exerted bending force:

k =2

Lsinc−1(

|xij |

L),where sinc(x) =

sin(x)

x(4)

The modified version of the bending magnitude f∗b in

equation 5 is used to mimic the imperfection in the structure

of the fabric equation 3 as pointed out by Choi and Ko

[5]. The magnitute curve for the modified bending force is

plotted in figure 2.

f∗b =

{

cb(|xij | − L) , fb < cb(|xij | − L)fb , otherwise

(5)

where cb is a constant that is usually assigned a value

comparable to ks.

This sophisticated bending model yields realistic bending

behavior and gives plausible results. The main problem with

this bending model is that it is a non-linear; it includes

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Figure 2. The magnitude curve for the bending force given by equation5.

cosine, sinc, and sinc inverse functions that require a lot

of computations. Our technique works by linearizing the

bending function fb, figure 2. The negative sign of the

bending magnitude fb means that the distance xij is less

than the spring length due to bending. For the bending

behavior, xij can only be less than or equal to L. So the

bending magnitude fb is defined for the values ofxij

Lwhich

lies on the interval [0, 1]. To linearize this curve, we divide

the interval [0, 1] into a group of subintervals. The actual

value of fb is computed at the points defining the beginning

and ending of each interval. The two points representing

an interval is used to specify the slope and intercept of the

straight line connecting them. Then for any value ofxij

L,

we compute its corresponding fb value approximately by

applying the equation of the corresponding straight line. For

more details in how to approximate trigonometric functions

using precomputed values see Sewell [17]. Our technique is

divided into two steps:

1) Step I is an initialization step. It is performed once

before the simulation starts. This step is responsible

for dividing the interval [0, 1] –which is the interval

of the values ofxij

L- into subintervals.

2) Step II is executed once with each simulation time

step. It is responsible for computing the bending force

at each time step. It is the main part that saves com-

putations compared to calculating the bending force

according to the original equation 5. The pseudocode

is listed in figure 3.

A. Computational Savings

To compute the savings obtained by Step II, we compute

the difference in the number of performed arithmetic and

logic operations in Step II and compare it to the number

of performed operations using the original equation 5. It

should be clear that the cosine, sinc, and sinc inverse

xij ←current distance between mass particles i and jv ←

xij

L

interval← [start, end]i inPts that v lies within

line← line of intervalbaij ←approximate bending force between particles i and

jbaij ←substitute v in the equation of line defined by its

slope and intercept

Figure 3. Pseudocode for step II: Computing Approximate Bending Force

functions in equations 3 and 4 are evaluated using Taylor

series expansion. The sinc inverse function was calculated

according to the series specified by Cantrell [4]:

f(x) =

2x+3

10x3 +

321

2800x5 +

3197

56000x7 +

445617

13798400x9+

1766784699

89689600000x11 +

317184685563

25113088000000x13+

14328608561991

1707689984000000x15 +

6670995251837391

1165411287040x17

(6)

The series listed in equation 6 requires 101 multiplica-

tions, 9 divisions, 9 additions, 1 subtraction and 1 square

root operation. Table I summarizes the operations performed

in both techniques: equation 5, and the proposed approach.

Table IA COMPARISON FOR THE OPERATIONS PERFORMED BY EQUATION 5

AND STEP II IN OUR PROPOSED TECHNIQUE

Operation Equation 5from [5]

Step II in our technique

+ 9 1

− 2 0

× 105 1

÷ 13 0√sqrt 1 0

cos 1 0

sinc 1 0

comparisons 3 Avg =number intervals

2,

Max = number intervals

From table I, we can conclude that the computations done

by equation 5 are almost eleven times the computations done

by Step II in the proposed algorithm. It is clear that Step

II in the proposed technique saves a lot of computations.

Notice that the cost of Step I in the proposed approach is

considered equal to zero. This is because this step is only

performed once at the initialization. We are only concerned

with the operations performed with each time step.

The intrinsic damping force of cloth is also taken into

account as specified by Choi and Ko [5]. The internal

damping force exerted on a particle i due to its interaction

with particle j is given by the equation:

fi = −kd(vi − vj) (7)

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Where kd is the damping stiffness coefficient, vi is the

velocity of particle i, and vj is the velocity of particle j.

Back to Newton’s law of motion: Fi = mi ∗ ai, where

Fi represents the summation of all the vector forces acting

on the particle i, mi is a scalar that represents the mass

of particle i, and ai represents the acceleration vector of

particle i. Now given Fi and mi, we can compute ai. Then

we can integrate ai using a numerical integration method to

get the velocity vector vi then integrate once more to get

the position vector xi of particle i.In our technique, we use a velocity-less Verlet integration

scheme for time integration [10]. Velocity-less means that

the velocity is approximated by the old and new positions

of the mass particles. This way we only do one integration

time step to get the positions instead of doing two integration

steps. Verlet integration is mainly used to integrate Newton’s

law of motion. It offers great stability compared to other

integration techniques. This integration scheme was used by

Gaber [7] and Jakobsen [10]:

x(t+△t) = 2x(t)− x(t−△t) + a(t)△t2 +O(△t4) (8)

It achieved real time simulation and was used in computer

games. But the authors of [7], [10] fixed the stiffness

coefficients of the springs and assumed that they are equal

to infinity. This increased the numerical stability of the

integration scheme, even with large time steps. On the

other hand, this limitation (fixing the stiffness coefficients)

prevents the simulation of different kinds of textiles. To

overcome this limitation, we remove this assumption and

deal with non-constant stiffness coefficients. To do this,

we assume different values for the stiffness coefficients

of different forces, then compute the force vectors, apply

Newton’s law of motion, and finally integrate to get the

velocity and position vectors of all cloth particles. But

removing this assumption would compromise the numerical

stability of the integration technique. This can be overcome

by slightly decreasing the time step to the extent that

preserves numerical stability and at the same time still be

large compared to the time steps used with other numerical

integration techniques.

To show the computational savings acquired by utilizing

the Verlet Integration scheme, we compare it with other

integration techniques. Given the total force vector acting

on a particle and the particle mass, we want to compute the

position vector of that particle by integration. The number

of arithmetic operations is counted for each technique for a

single integration operation. The comparison is carried out

between (1) Verlet Integration Scheme, (2) Euler Integration

Scheme, (3) Mid Point Integration Scheme, and (4) Fourth

Order Runge-Kutta Integration Scheme. The first scheme is

implemented by Gaber [7] and Jakobsen [10], and in this

work. The other three schemes were implemented by Lander

[12] who implemented a simple cloth simulation system

based on the spring-mass model and a simple collision

detection scheme for detecting collisions with a sphere. He

used three different integration schemes for time integration:

Euler, Mid Point, and Fourth Order Runge-Kutta. Table II

shows the number of performed operations by each scheme.

It is clear from table II that the Verlet scheme requires the

least number of computations per single integration time step

resulting in smaller simulation time compared to the other

techniques, even the Euler method.

Table IISUMMARY FOR THE NUMBER OF OPERATIONS REQUIRED BY EACH

INTEGRATION SCHEME TO COMPUTE THE NEW POSITION OF A MASS

PARTICLE.

Operation Verlet Euler Mid-Point

4thorderRunge-Kutta

IntegrationSteps

1 2 2 2

× 4 7 14 48

÷ 0 0 1 2

+ 6 6 12 42

− 3 0 0 0

ForceCalculations

1 2 2 3

Total 14 15 29 95

IV. EXPERIMENTAL RESULTS

In this section, we present the experimental results for

simulating different textile. The proposed approach was

tested on a machine with an Intel Pentium M735A Processor,

Intel Graphics Media Accelerator 900 with Shared Memory

of up to 128MB, 1GB of RAM, Windows XP operating

system, and OpenGL/C++ for graphics rendering.

We show four experiments; cloth hanged from 2 points,

cloth falling on a box, cloth falling on a sphere and an

experiment to analyze the trade off between speed and

quality. In the four experiments we used the following

parameters:

• Cloth matrix consists of 31× 31 particles.

• Cloth particles are connected by stretching, sheering

and bending springs.

• The friction with air is computed according to Gaber

[7].

• Verlet velocity-less integration scheme is used to inte-

grate particle positions with each time step.

• The integration time step is equal to 0.1 second.

• The particle mass is equal to 1 gram.

• The acceleration due to gravity is equal to 9.8 m/sec2

• The internal cloth damping is implemented as proposed

by Choi and Ko [5] with the internal damping coeffi-

cient kd equal to 0.0002.

A. Experiment 1: Cloth Hanged from 2 Points

This experiment shows how the proposed approach saved

enormous computations in handling the internal forces of

104

cloth . The cloth sheet is hanged from 2 points. The

simulation starts from a horizontal position and the cloth

sheet moves under the effect of gravity till it reaches an

equilibrium vertical position, see figure 4.

The time taken to compute the bending forces of all

the particles in a time step was measured for the original

equation 5, and for the proposed approach. The average

computation time of 64 time steps was 33.625 msec for

the implementation of the original equation, and was 5.875

msec for our “linearization and interpolation” technique.

Thus the proposed technique is almost six times faster than

the direct implementation method. The speed up illustrated

in this experiment applies to all the following experiments.

B. Experiment 2: Cloth Falling on a Box

This experiment shows the effect of changing the bending

stiffness coefficient on the behavior –and hence- the appear-

ance of cloth. In this experiment, the cloth sheet falls under

the effect of gravity and collides with a box. The value of

the bending stiffness coefficient kb is tested for 2 different

values: 0.0 and 0.1. The resulting effects are shown in figure

5. It is clear from the circled parts that the bending behavior

of the fabric changes when the bending stiffness coefficient

is changed. The fabric with higher bending stiffness shows

higher resistance to bending at the corners of the table.

C. Experiment 3: Cloth Falling on a Sphere

This experiment shows the effect of changing the bending

stiffness coefficient on the behavior –and hence- the appear-

ance of cloth. In this experiment, the cloth sheet falls under

the effect of gravity and collides with a sphere. The value of

the bending stiffness coefficient kb is tested for 3 different

values: 0.0, 0.1, and 0.3. The resulting effects are shown in

figure 6. Notice the difference in the behavior at the edges

of the cloth sheet.

In this experiment, it was observed that the stability of the

simulation degraded as the value of the bending stiffness

coefficient increases. For that reason, the time step was

adjusted (to a smaller value) for larger bending stiffness

coefficient. This guarantees the stability of the simulation.

In this experiment, a time step of 0.1 second was used with

the bending stiffness coefficient values 0.1 and 0.2. While a

time step of 0.05 second was used with the bending stiffness

coefficient value of 0.3.

D. Experiment 4: Speed Vs Quality Tradeoff

The bending force curve is linearized by dividing it into

a group of intervals, where each interval is represented by a

line segment. This experiment investigates the effect of using

different numbers of line segments to approximate the curve.

It is expected that as the number of segments approximating

the curve increases, the accuracy of the simulation increases

(i.e. which yields a better bending behavior). In case the

line segments are being searched linearly, then using a

larger number of segments will increase the time required

to compute the bending force. In this regard, the segments

data can be stored in a hash table to speed up the segments’

search.

Figure 7 shows different numbers of line segments used

to approximate the bending curve influence the bending

behavior. Each row shows the cloth draping behaviour on

a sphere at different time steps. As the number of segments

approximating the curve increases, the cloth’s bending be-

havior gets better. A number of segments as small as 2 makes

cloth sheet behaves similar to a paper sheet. As the number

increases, the bending behavior gets better and closer to the

actual, non-approximated behavior developed by Choi and

Ko [5].

V. CONCLUSION AND FUTURE WORK

In this paper we proposed a fast technique for cloth

simulation. The proposed work extended the physical mass-

spring model technique to speed up its computation by

linearizing the bending function. Results showed that the

proposed technique was almost six times faster than the

traditional technique of directly evaluating the complex

physical models. Results showed the effect of changing

the bending stiffness coefficient on the behavior of cloth.

We conclude from these results that allowing for a large

range for the bending coefficients require decreasing the

time step even with a numerical technique that shows high

stability like Verlet integration. This means that we have to

compromise between the range of bending stiffness (which

enables simulating a larger range of behaviors) and the speed

of the simulation. Textiles with higher stiffness coefficients

should be simulated at smaller time steps. However, the

velocity-less Verlet integration scheme we utilized is still

faster than other integration schemes. The overall proposed

approach can be embedded into an apparel design and

modeling tool or a virtual fitting room module to quickly

and stably simulate clothes.

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Figure 4. Simulation results for cloth hanged from 2 points

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Figure 5. Different behavior of cloth draping on a box for different bending stiffness.

Figure 6. Cloth draping behavior on a sphere for different bending stiffnessKb (0.0, 0.1, 0.3). Each column shows the draping behaviour at differenttime steps

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Figure 7. The effect of using different numbers of segments to approximate

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