Study on improving three-dimensional dynamic simulation model and equation of flexible fabric

Abstract

The use of the traditional fabric simulation model evidently shows that it cannot accurately reflect the material properties of the real fabric. This is against the background that the simulation result is artificial or an imitation, which leads to a low simulation equation. In order to solve such problems from occurring, there is need for a novel model that is designed to enhance the essential properties required for a flexible fabric, the simulation effect of the fabric, and the efficiency of simulation equation solving. Therefore, the improvement study results will offer a meaningful and practical understanding within the field of garment automation design, three-dimensional animation, virtual fitting to mention but a few.

Keywords

Flexible fabric three-dimensional dynamic simulation model motion equation improvement study

Introduction

Given the present-day rapid development of Internet technology and application, there is an increased number of customers shopping online instead of physically going to the stores. However, there is a serious problem that needs to be looked at critically regarding buying clothes online and having the customers fitting the clothes just like in the normal walk-in store. Therefore, this would mean virtual fitting becomes an essential need in order for the problem to be solved. With virtual fitting, customers get to try on their preferred online clothing item using a virtual fitting room and check whether or not it is suitable before deciding on buying. Nevertheless, in order to create dynamic sports movement and morphing effect of the clothing items, achieving wearer effect on virtual human model in the virtual fitting room, there are steps that need to be followed. The first step is to solve the dynamic simulation problems of clothing. As for the main element of clothing fabric, it is critical to solve its dynamic simulation problem. Currently, the real-time computer simulation of fabric has comprehensive application prospects on clothing item manufacturing and automation,¹ three-dimensional (3D) animation, e-commerce,² virtual reality,^3,4 online game, and other areas.

To date, a number of researchers have done much in-depth research work on simulation of flexible fabric and have also proposed effective methods of doing so. However, given that flexible fabric has complex physical and mechanical properties, its appearance is not fixed as a rigid body appearance changes with different external conditions. As such, it makes the simulation very difficult. Therefore, the assortment of fiber materials composing the fabric, structural complexity, and the irregularities of shape and properties also bring about difficulties to the dynamic simulation. This is against the understanding that the core problem lies on how to construct the simulation model, and the equations that show the intrinsic property of the flexible fabric accurately, and makes the simulation more realistic and more efficient.^5–7

The existing fabric simulation technology is mainly divided into three classes that are based on geometry, physics, and the combination of both geometry and physics. Terzopoulos et al.⁸ described the deformation of flexible object as elastic force, external force, and damping force, which are caused by the inner resistance to deformation. Breen et al.⁹ suggest a particle system model named the spring-mass model, which is based on the analysis of fabric’s physical properties. The spring-mass physical model fully takes into account both the external force of the fabric and the interaction force between the particles inside the fabric and then shows the appearance changes of the fabric through calculating Newton’s force equation.

This is done so that the fabric can make a corresponding change according to the different external forces, and it therefore makes computer simulation of fabric movement possible. This model is widely used by most of the fabric simulations.¹⁰

Traditional fabric spring-mass models analysis

The traditional spring-mass models treats a piece of fabric as a lattice structure with many longitudinal and latitudinal weaves. However, the intersection of the warp and the weft of the lattice are represented by a particle. Particles are connected by three kinds of springs, namely, structural spring, shear spring, and bending spring,^11,12 which restrain the tensile deformation, shear deformation, and bending deformation of fabric, respectively (Figure 1). As for every particle, its motion is controlled by the sum of external force and internal force. While combined with the movement of each particle, it shows the motion of fabric represented by the whole system.

Figure 1.

Conventional fabric spring-mass model and spring: (a) spring mass, (b) structural spring, (c) shear spring, and (d) bending spring.

However, with the current existing spring-mass model, the macro behavior of fabrics is fully taken into account, but the deformation behavior of different types, structures, and different styles of fabrics are not accurately described. This is against the background that when designing the spring, the coefficients of three kinds of springs are fixed, which results in the true characteristics of the fabric not being accurately simulated and leads to the simulation losing the sense of reality. As for simulation calculation, it needs to solve complex differential equation with many iterations to obtain the result. However, it is computationally expensive and inefficient. Therefore, it is necessary to improve the traditional spring-mass model for fabric simulation, so that the model can reflect the simulation effect of fabric with different types and structures.

Study on improving spring-mass model of fabric

An effective method of fabric simulation should consider all the possible factors that might affect the true shape of the fabric and then reconstruct the appearance of the fabric deformation in the computer. This can be done starting with the stress analysis in the fabric movement process so as to improve the fabric spring-mass simulation model. According to previous research findings, in the fabric movement process, some forces are external. These external forces include the gravity and wind force, and the others are internal, such as tensile, shear, and bending forces. The elastic deformation force of the spring in the spring-mass model can be calculated using Hooke’s law. Assuming the particle P_i and the set of adjacent particles is R, then P_i subjected to the elastic deformation force F_s is

F_{s} = \sum_{i \in R} K_{s} (| \bar{p_{i} p_{j}} | - {| \bar{p_{i} p_{j}} |}_{0}) N_{\bar{P_{i} P_{j}}}

(1)

where $| \bar{p_{i} p_{j}} |$ denotes the distance between the points P_i and P_j at time t, ${| \bar{p_{i} p_{j}} |}_{0}$ denotes the initial distance between the points P_i and P_j at time zero, $N_{\bar{P_{i} P_{j}}}$ denotes the unit vector from P_i to P_j, and $K_{s}$ denotes the elastic deformation coefficient of the spring. As it can be seen from formula (1), in our simulations of the spring-mass model, three different springs were simulated with the fabric tensile properties, bending performance, and shear performance, then each spring can be set by their own different elastic coefficient $K_{s}$ . In addition, according to previous research results, it is evident that the relationship between the tensile force and the elongation of each fibrous material constituting the fabric is non-linear (Figure 2). Elasticity coefficient $K_{s}$ of the same spring is suitably changed in accordance with the difference of tensile elongation during the simulation. According to previous research findings and the test experiments¹³ of fabric performance using different fabric that influences simulation effect, it shows that performance parametric curves of various fabrics can be calculated and then the slope of various curves is taken as the elastic coefficient of the fabric. This follows adopting the way of “using measurement data to estimate the optimal parameter”¹⁴ and obtaining an optimal value using the least squares method. This is regarding the different slopes measured on the same curve, so as to figure out the elastic coefficient of the spring $K_{s}$ . Therefore, the intrinsic properties of the fabric can be presented much more accurately, which makes the fabric to be more real.

Figure 2.

The tensile curve of different types of fabrics.

Through the research findings from predecessors and the project team that test the performance parameters of various types of fabrics in different types of fabrics, it is evident that the performance parameter curves of various types of fabrics are obtained and then the slope of various curves is taken as the elastic coefficient of the fabric. However, measured data to estimate the optimal parameters method, uses the different slopes measured on the same curve, and the least squares method to estimate an optimal value, so as to determine the spring’s elastic coefficient.

In order to simplify the model, we can prove through experiments that two shear springs and a shear spring have little effect on the simulation performance of the fabric. Therefore, in the improved spring-mass model, a shear spring is discarded, and finally, the improved fabric spring-mass model is shown in Figure 3.

Figure 3.

An improved fabric spring-mass model.

Study on motion equation of fabric simulation

After the former 3D dynamic force analysis of flexible fabric and the analysis of constructing improved simulation model, there is still need to solve the motion equation of model. Therefore, the common solution to solve motion equation is based on the model of force, which calculates accelerated speed a_i of mass P_i according to Newton’s second law a = F/m, and list partial differential equation, while using each of numerical methods to figure out the location and the speed of each time.

Using the above simplified particle-spring model to describe the fabric model that is combined by many fictitious triangular net’s vertex, also named particle. Each of the particle connects with its neighboring particle by three kinds of springs, each particle’s motion state is the result of all forces exerted, the internal force—such as the elastic force and damping force—and the external forces—such as gravity and wind force. However, the comprehensive effect of all particles’ movement reflects the deformation of the fabric form on the entire system.¹⁵ According to the improved simulation model built above, a variety of forces exerted to the fabric are analyzed and studied and we get the motion equation

m \frac{\partial^{2} X}{\partial t^{2}} = F_{in} (X, t) + F_{out} (X, t)

(2)

where X presents particle’s vector; $X \in R^{3}$ ; m is the quality of particle; $F_{in} (X, t)$ is the internal force that particle has encountered, primarily referring to spring force and damping force. $F_{out} (X, t)$ is the external force and wind force that particle has encountered. All of them change with particle’s location and time.

Internal force

Spring force

In order to make simulation fabric produce deformation, we consider any two particles on the fabric’s surface that are connected by the spring. One particle can produce a same direction force and a reverse direction force to its neighboring particle.

Suppose particle P_i and neighboring particle P_j are connected by a spring, according to Hooke’s law, its spring force is in direct proportion to the length of spring

F_{s} = (k_{s (i, j)} (| X_{j} - X_{i} |) - ℓ_{i, j}^{0}) \frac{(x_{j} - x_{i})}{| x_{j} - x_{i} |}

(3)

Between these, $k_{s (i, j)}$ is the elastic coefficient between particle P_i and neighboring particle P_j. X_i and X_j represent the locations of particle P_i and neighboring particle P_j, and $ℓ_{i, j}^{0}$ is the original length of the spring.

Damping force

Damping force can reduce excess volatility of particle that was caused by every force of the fabric. It also can maintain the stability of system and the functions of particle speed. The damping force F_d between fabric particle P_i and the neighboring fabric particle P_j of fabric is

F_{d} = \sum_{j = 0}^{n} k_{s (i, j)} (v_{i} - v_{j}) = - k_{d} \frac{\partial X}{\partial t}

(4)

where $k_{s (i, j)}$ is the elastic coefficient between particle P_i and neighboring particle P_j, v_i is the speed of particle P_i, and v_j is the speed of particle P_j.

External force

In the process of fabric simulation motion, it encounters a lot of effects of external force, such as gravity and wind force.

Gravity

All objects have mass and are influenced by the gravity. The fabric mass in the spring-mass simulation model is formed by the above particle. Each particle suffered gravity F_g

F_{g} = m_{i} g

(5)

where m_i is the mass of particle p_i and g is the acceleration due to gravity.

Wind force

According to the aerodynamics and the hydromechanics,¹⁶ the calculation of wind force would be very complex in the wind field. Therefore, in order to simplify the calculation, without influencing the authenticity of simulation, there is need to design a wind field in parallel direction where every deformation of fabric can be ignored when fabric suffered wind force. In this wind field, the wind force F_w that fabric suffered can be represented like this

F_{w} = k_{w} (v_{w} - v_{i})

(6)

where k_w is the wind factor, v_w is wind speed, and v_i is the speed of particle p_i. According to the above force condition analysis and research of fabric particle, the resultant force F_i of fabric particle P_i at time t

F_{i} = F_{out} (X, t) + F_{in} (X, t) = F_{s} + F_{d} + F_{g} + F_{w}

(7)

Equation (2) can be extended as follows

m \frac{\partial^{2} X}{\partial t^{2}} + k_{d} \frac{\partial X}{\partial t} = F_{S} + F_{g} + F_{w}

(8)

Equation (8) is the whole fabric simulation model, and it needs numerical method to figure out the answer.

Analysis of simulation results

We built an improved spring-mass model simulation system based on VC++ development platform and OpenGL 3D graphics library. In this simulation system, we choose fabrics which were created from silk, wool, cotton, and synthetic fabric to be used in the experiment.

According to traditional spring-mass model, we usually choose three kinds of fixed spring coefficients to simulate the fabric falling from the air coming down below due to the force of gravity and observe the appearance characteristic of the fabric during movement, as well as and the effect of the fabric after falling into the sphere like the ultimate drape and fold. The illustration is shown in Figure 4.

Figure 4.

3D dynamic simulation of fabrics simulated based on traditional spring-mass model.

We use different spring coefficients to simulate silk fabrics and chemical fiber fabrics according to the improved spring-mass model. The above simulation experiments are continuously repeated. The results are shown in Figure 5.

Figure 5.

3D dynamic simulation of fabrics simulated based on improved spring-mass model.

The above experimental results confirm that while using the traditional spring-mass point model, the fabric can only be simulated with a fixed spring coefficient. This is against the understanding that the simulated fabric has a single motion state that cannot reflect the appearance state and characteristics of the fabric composed of different fiber materials. However, the rear spring-mass point model can be used to simulate different fabrics due to the different spring factors in the simulation process, because of the characteristics of different fiber materials, and the different stages of the fabric moving process. In addition to the improved model, the fabric characteristics reduce the operation of a shear spring during the simulation operation. Furthermore, the experimental results also show the improvement of the simulation operation speed as compared with the traditional model.

In order to further study and analyze the simulation effect of the fabric in the actual application using the improved spring-mass model, we simulate a scene with fabric such as skirt on one’s body, fix one end of the fabric and allow gravity and wind force to it. This is done to build a process where the fabric falls down and traces the object using the external force of gravity and wind. This is then followed by setting different spring coefficients into the parameter setting module of the system according to the parameter curves of various materials’ performance obtained by the earlier analysis. The quantities, quality, corresponding damping force, and other datum of particle should also be set. As a result, we can obtain the simulated effect of the fabric during the movement using the wind force produced by the movement process of each particle under the gravitational or wind force. This is done by calculating the cycle changes varying from the internal force, the external force, the accelerated speed, the speed and the location, and the results of immediately dealing with corresponding stress, accelerated speed, speed and other data of each particle when the impact takes place. Figure 6 shows the final 3D simulation effects using different kinds of fabrics.

Figure 6.

3D simulation effect picture of various fabrics.

While using fabrics with different fibers, we can see from the above simulation experiment that we can achieve more real simulated effect by changing the elastic coefficient and other data of each spring in the spring particle model. In addition, the experimental results confirm that when cutting out the shear spring in the spring particle model, there is little effect that it has on the final simulated effect; however, it is able to increase the simulation speed.

Conclusion

In this article, the conventional spring-mass model has been accurately improved, as well as an algorithm that is able to set different elastic coefficients in the simulation equation to express the characteristics of different flexible fabrics is designed. Moreover, the spring-mass point is used to reduce the excess spring. However, the computational complexity of the model has been effectively reduced. Therefore, the experimental results show that this new model can not only better simulate the intrinsic properties of flexible fabrics but also maintain the original simulation results and improve the simulation operation speed.

However, it was also found during the simulation that sometimes the fabric would get penetrated by the collision body due to excessive deformation. As a result, we have attempted to solve the problem of excessive deformation of the fabric by increasing the spring modulus. Although this method eliminates the phenomenon that the fabric is penetrated, it is not realistic enough as compared to using the actual fabric movement. Therefore, we are going to use computer vision and other techniques to further analyze and study the motion state of the fabric, as well as improve the numerical algorithm and correct the simulation results in order to improve the simulation effect of the fabric.

Footnotes

Handling Editor: Shahin Khoddam

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Zhejiang Provincial Natural Science Foundation of China (grant no. LY17F020003) and the National Natural Science Foundation of China (grant no. 61373057).

ORCID iD

Mingjun Wang

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