Pergamon Comput. & Graphics, Vol. 21. No. 6, pp. 769-775. 1997

if+ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: s0097-8493(97)ooo56-3

0097~8493197 $17.00 + 0.00

Technical Section

ANIMATION OF STOCHASTIC MOTION OF 3-D CLOTH OBJECTS

LI LING

School of Applied Science, Nanyang Technological University, Nanyang Avenue. Singapore 639798

e-mail: aslli@ntuvax.ntu.edu.:;g

Abstract---This paper presents a physically-based aerodynamic force model for the realistic animation of a three-dimensional cloth object, such as a skirt. The model is based on the theory of low speed aerodynamics with some modifications to model the unsteady movement of a 3-D cloth object in air flow. The consideration of the solid object in the vicinity of the cloth, the porosity of the cloth and the collision detection of the cloth with other objects and with itself are aIs- included. To achieve more realistic animation, a stochastic wind model has been proposed to mod.el the free stream air flow. Realistic animation results of a skirt moving in air flow have been obtained by applying these models. (c 1997 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

Computer animation of cloth movement has at- tracted much interest in recent years since it is a typical example of the animation of deformable surfaces. One of the earliest examples was presented by Weil [I] who defined the shape of the cloth surface by applying a relaxation process to the control points. More recently, researchers have found that physically-based models could represent closely the natural movement of cloth objects. Terzopolous et a/. [2] employed mechanical laws to construct differential equations that model the behavior of non-rigid curves, surfaces and solids as a function of time. Aono [3] simulated the wrinkle propagation on a cloth surface based on the equilibrium equation in elasticity theory. Kunii and Gotoba [4] modelled the formulation process of wrinkles on cloth by integrat- ing both geometric and physical modelling techni- ques. Another important approach is the particle based model used by Breen et al. [5] to realistically simulate the draping behaviour of cloth surfaces on solid objects. Thalmann and Thalmann [6] and Carignan et al. [7] simulated the interaction of clothes with a synthetic moving human body. Techniques were also reported to account for collisions between cloth and the moving human body [8]. Volino et al. [9] presented a model to deal with difficult situations with respect to cloth defor- mation and collisions, like a wrinkled fabric falling on the ground.

It can be noticed that in all these efforts, one important aspect has been overlooked, namely the determination of forces acting on the cloth as a result of the relative motion of air flow over the cloth surface. In viewing this, we proposed an aerody- namic force model to describe the movement of cloth objects such as a curtain or a flag immersed in air

flow [lo, Ill. In that model, the cloth objects are topologically 2-dimensional in the sense that their initial shapes are assumed to be planar and they remain open surfaces in the whole motion process. Another limitation of the model is that the free stream velocity of air flow was simulated by artificial mathematical functions which make the animation process not completely physically-based and restrict the application of the aerodynamic force model.

An extension and improvement of the previous work is presented in this paper. The aerodynamic force model is extended to handle 3-D cloth objects such as a skirt. The effect of solid objects in the vicinity of the cloth object is also carefully studied and discussed. Realistic wind models such as those used in wind engineering are adapted to simulate the air flow velocity in a free field. By combining the wind model, the 3-D aerodynamic force model and the cloth deformation model, the motion of a 3-D cloth object such as a skirt can be animated realistically based on physical laws.

The paper has been structured as follows. In the next section, the 3-D aerodynamic force model is discussed. In Section 3, the effect of the solid object in the vicinity of the cloth to the air flow field is considered. Section 4 briefly describes the wind model. Section 5 presents an example of the proposed models applied to a skirt hung around a cylinder. Finally, Section 6 concludes the paper and suggests future research.

2. THREE-DIMENSIONAL AERODYNAMIC FORCE MODEL

When a piece of cloth is immersed in air flow, the pressure field around it affects its movement. The distributed forces of air flow on the cloth surface cannot be predefined since the force distribution of the air flow and the changing shape of the cloth

769

770 L. Li

surface actively interact with each other. Hence, to animate the cloth motion realistically, an aerody- namic force model based on theoretical aerody- namics should be constructed. The aerodynamics of air flow around a moving object is unsteady, non- linear, complicated and highly turbulent, especially when the shape of the object changes arbitrarily. Nevertheless, some common fundamental physical principles are deeply entrenched and lead to basic equations for describing air flow motion. These principles are:

(4

(b)

Cc)

Mass is conserved. It can be neither created nor destroyed. Newton’s second law: force = mass x accelera- tion. Energy is conserved. It can only change from one form to another.

The basic governing equations are obtained from these principles. To analyze the fluid mechanics of air flow around a moving piece of cloth, the appropriate mathematical model is Reynold’s averaged Navier- Stokes equations [12]. The how field variables at any point in the flow can be obtained by solving these equations. The equations are very complicated and their solution is further complicated by the fact that the cloth is in arbitrary motion.

In reality, the air flow over a cloth object is normally at low speed, i.e. below 72 km h-t. Therefore, to analyze the flow field around the cloth object in order to animate the cloth motion, the air flow can be reasonably simplified to be incompres- sible, irrotational and inviscid as justified by low-

speed aerodynamics theory. Being incompressible means the density of the air flow is the same everywhere; being irrotational means the fluid elements have no angular velocity; and being inviscid means the flow involves no friction, thermal conduc- tion or diffusion. Practically there is no flow which is incompressible, irrotational and inviscid. But in low speed aerodynamics, it is found that these assump- tions yield approximate flow field analysis without detrimental loss of accuracy [12].

Under these assumptions, the aerodynamic model describing the air flow field around an object with arbitrary shape can be represented mathematically [IZ, 131 as

V24 = 0 (potential equation) 0.1)

u = g, 1’ = C9, w = 2 (velocity components) 4

Cl.21

p + ipV2 + p $ = const. (Bernoulli’s equation)

0.3)

where 4 is a scalar function called the velocity

potent:.al such that the flow velocity V is given by the gradient of 4, i.e. V4 = V; U,V,W are the X,Y,Z components of the flow velocity V; p is the pressure of flow at any point; p is the density of air flow; and t is time.

Obviously. the irrotational, incompressible air flow fields around different cloth shapes, such as a curtain or a skirt, should be distinctly different. However, the flows around these different shapes are all governed by the same aerodynamic model such as represented in Equations (1.1 t( 1.3). The different flows over different shapes are represented by the boundary conditions. Although the governing equa- tions for the different flows are the same, the boundary conditions conforming to different geo- metric shapes yield different flow field solutions. The flow is bounded by the free stream flow far away from the cloth object and the shape of the cloth surface. Therefore, two set of boundary conditions apply. The first boundary condition means that far away from the object, the disturbance due to the existence of the cloth to the air flow is negligible. Mathematically this is written as

lim, V4 = V, (2.1)

where V, is the free stream velocity of the air flow. The second set of boundary conditions applies on the cloth surface. They represents the fact that the air flow going through the cloth surface is proportional to the surface’s porosity ratio. Mathematically this second boundary condition is written as

(V4 - V,t).n = pV,.n (2.2)

where V,, is the kinematic velocity, p (0 <p < 1) is the porosity ratio, and n is the normal of the cloth surface. The force distributions of the air flow on the cloth surface of any shape can be obtained by solving the aerodynamic force model in the following manner. First, the potential equation (1.1) is solved subject to the boundary conditions. Next, the velocity field of the air flow can be obtained from Equation (1.2). Finally, the unsteady Bernoulli equation (1.3) is used to calculate the pressure field

of the flow on the surface. The force distribution of the air flow on the cloth surface can then be found from the information of the pressure field on the surface.

For an object whose shape changes arbitrarily, there is no general analytical solution for the potential equation of flow around it without major geometric simplifications. Hence, the numerical solution is used based on the fact that the potential equation is a linear differential equation. This means that the principle of superposition permits a combi- nation of any of the equation’s particular solutions to be its solution. Therefore, a complicated irrota- tional, incompressible flow can be synthesized by combining a number of elementary vortex flows [ 121. The air flow field around a cloth subject can then be numerically calculated by discretizing the space

Animation of stochastic motion of 3-D cloth objects 771

surrounding the cloth and the cloth itself into quadrilateral panels and distributing vortex flows on them. The solution is reduced to finding the strengths of the vortices so that the boundary condition on the cloth surface is satisfied. This solution method is called the panel method [12, 131.

Because we are dealing with a three-dimensional cloth object, the distribution of the vortex rings on the cloth surface and the wake vortex rings shed along the trailing edge of the object are very different from those for 2-dimensional objects. The panel system for a 3-D skirt-like cloth object is shown in Fig. 1.

The 3-D cloth surface is divided into MxN small panels. The vortex rings are placed on every panel in a similar manner. The four vertices of the vortex ring are not necessarily on the same plane. For a panel ABCD as shown in Fig. l(b), the vortex ring is placed on it such that the leading segment of the vortex ring links points which have distances 4 [ADI and i(BCI respectively from the leading edge of the panel. It has been found in theoretical aerodynamics that the flow boundary condition should apply to a special point on the vortex ring. This special point is called the collocation point. It is located at the centre of the line which links the points having distances i IAD/ and $ IBCl respectively from the leading edge of the panel. At every instant of time, one wake vortex ring is shed from every trailing edge vortex ring. Once the wake vortex rings are created, their strengths are conserved. They go downstream together with the flow, further and further away from the cloth. ‘Their influence on the cloth move- ment becomes less and less; after a certain time, they can be considered as having no contribution to the motion of the cloth.

This vortex ring panel structure remains present during the whole calculation procedure regardless of the manner in which the cloth is moved or deformed since the Your vortices of the vertex ring are not required to be co-planar.

A system of linear algebraic equations can be constructed by applying the boundary condition (2.2) at the collocation point of every panel. The normal velocity component at each collocation point is a combination of the free stream velocity, the velocity induced by all the vortex rings including the wake vortex rings, and the kinematic velocity of this panel. As described in detail in our previous papers [lO][l 11, the velocity induced by a vortex ring on panel (i, j) at the collocation point of panel K can be calculated ,as VTH(K, i, j, Tij), where TV is the vortex ring strength of panel (i, j). The velocity induced by wake vortex L at the collocation point of panel K can be calculated as VTH(K, L. I-,,.,), where r,c, is the vortex ring strength for wake vortex L which is known from the previous time step. Therefore, at the collocation point of panel K, the linear algebraic equation can be written as

V, + >IVTH(i,j, K,rii) IJ

+ >I VTH(L, K, r,,.,) - VreI .nK = pV,.nK .‘.

(3)

The same boundary condition can be applied to all the panels to construct a system of linear algebraic equations in the Mx N unknown values of the vortex ring strengths TV These equations can be solved by any linear algebraic equation solution technique,

+l;Fl , collocation point

leading A/- B

segment /

Fig. 1. Panel system with vortex rings on a skirt surface.

712 L. Li

Fig. 2. Image of vortex in the vicimty of a planar wall

such as Gauss Elimination, to yield the values of the I iis. From this information, the distributed force of the air flow on the cloth surface can be found as described in Li et al. [lo].

3. EFFECT OF SOLID OBJECTS IN THE VICINITY OF THE CLOTH

The flow field analysis discussed above calculates the force distribution of air flow on a cloth surface when the cloth is in open space. If there are some solid objects existing in the vicinity of the cloth, additional conditions must be enforced to ensure that the air flow cannot go through the solid objects, i.e. the normal components of the flow velocities on the additional boundaries must be equal to zero. These additional boundary conditions are modelled by Katz and Plutking’s Method of Images [13]. The idea of this method is briefly described as follows.

As shown in Fig. 2, a vortex of strength I induces velocity Vat an arbitrary point P on the planar wall. Since the air flow cannot pass through the solid wall, the normal component of the flow velocity should be brought to zero. In the Method of Images, the existence of the solid wall is modelled by the image vortex of strength -F, which is the image of the original vortex about the wall. At point P, the normal components of the velocity induced by the original vortex and the image vortex are of the same magnitude but in opposite direction. These compo- nents thus cancel each other, thereby enforcing no through flow at the wall and incorporating the wall into the physics.

A 3-D skirt is often worn on a human body. As the first attempt to analyze the effect of additional solids to the air flow, a cylindrical solid object as shown in Fig. 3 is assumed to be in the vicinity of the cloth. For a cylinclrical solid object in the vicinity of cloth, the additional boundary conditions require that no flow goes through the cylinder.

Similar to the case where a planar wall is in the vicinity of the cloth, the additional boundary conditions on the cylindrical solid can be modelled

by the Method of Images by using different kinds of image vortices [ 131. First consider the cross-section of the cylinder. The centre of the circle is assumed to fall on the origin. For a clockwise vortex of strength I at point PO = [xe, ze] outside a circle as shown in Fig. 4.(a), the image point is at the point

P,,g = gx,. $]

where r is the radius of the circle and a is the distance from the origin to the point PO. The image vortex system then consists of an anticlockwise image vortex of the same strength at the image point Pimp, and a clockwise image vortex of the same strength at the origin. These three vortices line up along the same radial line from the origin.

Hence. if the cloth object is hung around a cylindrical solid object, for every vortex ring includ- ing the vortex rings on the cloth surface and the

Fig. 3. Cylinder in the vicinity of cloth.

Animation of stochastic motion of 3-D :loth objects

69 @I Fig. 4. Two image vortex rings of the vortex ring near a cylinder

wake vortex rings, there are two image vortex rings created on the corresponding image point and the origin, as shown in Fig. 4(b). The positions and the strengths of these image vortex rings are defined from the original vortex rings. When constructing the linear algebraic equations, the velocities induced by the image vortex rings are also included to represent the existence of the cylindrical solid object. Hence, the effect of the cylindrical object in the vicinity of the cloth object is included in the resulting aerodynamic force distributions.

4. WIND MODEL AND SIMULATION

Characteristics of wind velocity have been inten- sively investigated either theoretically or experimen- tally since such characteristics may be of considerable significance to structural design [ 141. In applications, the wind velocities are most com- monly modeled by their power spectral density functions. Many wind velocity power spectral density functions have been obtained based on field or wind tunnel data. These spectral density functions model the characteristics of wind velocities under different environmental conditions, geographical locations and roughness of terrain.

In the present study, a simple wind velocity power spectral density model is employed to model the wind velocities. This model matches the field data well [15] and was successfully applied in computer graphics to synthesize the motion of trees and grass [16]. In this model, the power spectral density functions of wind velocities in the mean wind direction, uo, and in the other two directions perpendicular to the mean wind direction, v. and wo, are given as

S,,(f) z.z co . 2oo” f( 1 + 50”)5’3

(4.1)

S,,(f) = co. 15” f( 1 + 9.5”)5’3

(4.2)

SW?(f) = Co. 3.360

f( 1 + 1ov)s’3 (4.3)

where f is the frequency, u = fz/u(z), in which z is the height of the observation point from the ground, U(Z) is the mean wind velocity at the point, and Co is a normalizatl.on constant. The spatial variation of wind velocities are modeled by the corresponding cross power spectral density function [1.5]. In this study, however, the free stream wind velocities throughout the dimensions of the deformable sur- faces under consideration are assumed to be the same since those dimensions are relatively small as compared to the open space with air flow. Figure 5 shows the corresponding spectral density functions for wind velocities in the three directions.

The wind velocities in the three directions are then stochastically simulated. For a given power spectral density function Sy), the corresponding time history is [14]

X(t) = 2 ~Z?(iiJG COS(tiljt + 4i) (5) i=l

where A.ii, == 2nAf, with Af the frequency increment, and & is a random phase angle uniformly distributed

IO = su - sv - .._.. _ __.

-- SW -----

10 K

1 I---------------

------------A ---.____

--._ -------------_ - ‘>

Fig. 5. Wind velocity fluctuation power spectral density function.

L. Li

> 2-

-151, -4

0 250 500 750 1000 step

II-----i -I?-

5

I I I I 250 500 750 1000 0 250 500 750 1000

step step

Fig. 6. Simulated wind velocity fluctuation in the three directions.

in [0, 2rr]. Figure 6 shows the simulated time histories 5. APPLICATION

of the wind velocities in the three directions. Their The proposed wind model and 3-D aerodynamic power spectral density functions in the frequency force model have been applied to obtain the force domain are very close to the corresponding power distribution of stochastic air flow on a skirt hanging spectral density functions shown in Fig. 5 which are around a cylinder. The deformation of the cloth obtained from field data. under these external forces is computed using the

i k

Fig. 7. Snapshots of skirt mov,ng in air flow.

Animation of stochastic motion of 3-D cloth objects 775

integrated cloth deformation model described in the modelling of cloth motion in air flow completely Refs. [lo] and [l 11. physically-based. The resulting animation of 3-D

As shown in Fig. 3, a cone-shape skirt is hung cloth objects is realistic and repeatable. around a cylinder in an open space. The cone shape is Future work includes the optimization of the used because of the simplicity to calculate the initial proposed models, giving more attention to the position of the panel vertices. The proposed models collision detection algorithm, and study of a more work in the same way for surfaces with wrinkles and accurate cloth deformation model. creases. The air flows with an angle CI to the central axis of the cylinder. The wind model is used to REFERENCES

simulate the free stream velocities of the stochastic air 1. Weil. J., The synthesis of cloth objects. Computer

flow. The 3-D aerodynamic force model is used to Gruphicp, 1986, 21(4), 49-54.

calculate the force distribution of the air flow on the 2. Terzopoulos, D. and Fleischer. K.. Deformable models.

moving skirt surface. The effect of the existence of the The Visual Computer, 1988, 4, 306332.

cylinder on the air flow field is also included in the air 3. Aono, M., A wrinkle propagation model for cloth. In

Proceedinzs of CGI 1990. 1990. vv. 95-l 15. flow field calculation. The integrated cloth deforma- 4. Kunii, T.-L. “and Gotoba. H., &gularity theoretical

tion model is then used to calculate the displacement modelling and animation of garment wrinkle formation

and deformation of the curtain surface. Collision process. The Visual Computer. 1990, 6, 32&336.

detection and response is included using a simple 5. Breen, D. E., House, D. H. and Getto, P. H., A

method [17] to prevent self-collision of the skirt and particle-based computational model of cloth draping behaviour. In Proceedings @CGI 1991. 1991, pp. 113-

collisions between the skirt and the cylinder. As 133.

shown in Fig. 7, animation of the skirt motion in air 6. Thalmann, N. M. and Thalmann, D., Flashback

flow has been achieved realistically. [videotape]. SIGGRAPH Video Review, ACM SIG-

The implementation of the physical models has GRAPH. New York, 1990.

7. Carignan, M., Yang, Y., Thalmann, N. M. and been done on a SiliconGraphics Crimson work- Thalmann, D., Dressing animated synthetic actors with

station. The skirt was divided into 30x20 panels. The complex deformable clothes. Computer Graphics, 1992,

material properties used in the simulation are 26(2), 919-l 04.

obtained from a textile study institute as follows: 8. Thalmann, N. M. and Yang. Y., Techniques for cloth

animation. In Neti! Trends in Anintation and Visualka- tion. Wiley, New York, 1991.

Mass density of the cloth: 1.54x lo3 kg mp3: 9. Volino, P., Courchesne, M. and Thalmann, N. M.,

Young modulus: 4.2 mPa; Versatile and efficient techniques for simulating cloth

Poisson ratio: 0.25; and other deformable objects. Computer Graphics (Proc. SIGGRAPH), 1995, 1377144.

10. Li, L., Damodaran, M. and Gay, R. K. L.. Aero- and the mass density of the air flow at sea level is a dynamic force models for animating cloth motion in air

standard value. 1.23 kg rne3. flow. The Visual Computer. 1996, 12, 84-104.

To generate a lOOO-step deformation time histories Il. Li, L., Damodaran, M. and Gay. R. K. L., A model for

of the skirt in a gentle wind flow as shown in Fig. 7. animating the motion of cloth. Computers & Graphics, 1996, 20(l). 137-156.

the CPU time is about I min 26.35 s. If the wind 12. Anderson, Jr., J. D.. Fundamentals yf Aero&namies. flow becomes stronger, the computational time McGraw-Hill, New York, 1988.

increases considerably since the computation for 13. Katz, J and Plutking, A., Low-speed Aerodynamics

collision detection and response greatly increases. .from Wing Theory to Panel Method. McGraw-Hill. New York, 1991.

14. Claugh. R. W. and Penzien. J., Dynamics of Structures.

6. CONCLUSION AND DISCUSSION 2nd edn. McGraw-Hill. New York. 1993.

The 3-D aerodynamic force model based on low- 15. Simiu, E. and Scanlan, R. H.. Wind Ej’ects on

speed aerodynamics is able to analyze the interaction Structures: An Introduction to Wind Engineering. Wiley, New York, 1985.

between the deforming 3-D surface and the air flow 16. Shinya, M. and Fournier. A., Stochastic motion-

field around it. The consideration of the existence of motion under the influence of wind. In Eurogruphics ‘9.?

additional solid boundaries enhances the reality of Proceed,kgs. 1992.

the flow field analysis. A realistic stochastic wind 17. Moore, M. and Wilhelm. J.. Collision detection and

response for computer animation. Computer Graphics. model such as those used in wind engineering makes acceptec!.