CN102324108A - Non-stretch cloth simulation oriented mass-spring model constraint method - Google Patents

Abstract

The invention provides a non-stretch cloth simulation oriented mass-spring model constraint method. By adopting the method, deformation repair of a structural spring in a mass-spring model can be realized by correcting the mass point position of the mass-spring model. The mass-spring model constraint method specifically comprises the following steps: in order to accelerate the convergence speed of constraint implementation, giving an optimal relaxation factor based on a topological structure and deformation of the structural spring and the like in the mass-spring model; according to deformation of a single structural spring, calculating components of position correction quantity of mass points at two endpoints along the spring direction; superposing the components of the position correction quantity of the same mass point; multiplying the superposition sum by the optimal relaxation factor, and taking the result as the position correction quantity of the mass point; and finally updating the mass point position so as to realize deformation repair of the structural spring. The mass-spring model constraint method has the beneficial effects that conservation between linear momentum and angular momentum of cloth can be ensured and the non-stretch cloth can be efficiently and stably simulated during the constraint implementation process, thus the method can be widely applied to non-stretch cloth simulation in virtual reality systems for film and video production, digital entertainment and other purposes.

Description

A kind of particle-spring model constrained procedure towards no stretch material emulation

Technical field

The invention belongs to Computer Simulation and virtual reality field, specifically a kind of particle-spring model constrained procedure towards no stretch material emulation.

Background technology

Utilize the method for computer graphics and virtual reality technology simulation of natural real human body motion process to be called human motion emulation; Specifically comprise the computation model of setting up human body and accessory thereof; The physical motion process of simulation virtual people natural reality under specifying constraint, and in the virtual environment that computing machine generates with three-dimensional picture mode this motion process that appears true to nature.Yet the natural reality human body surface has 80% area to be covered by cloth, so the emulation true to nature of cloth plays key effect in human motion emulation true to nature.

Cloth is natural or the netted braid of artificial fiber.On mechanical characteristic, cloth has some obvious characteristics such as anisotropy, incompressible, not bending resistance of tension.Cloth is a flexible article, is easy to generate various folds.Different with rigid objects, cloth can produce very big deformation under the effect of external force, and also has nothing in common with each other with the deformation of different materials, different manufacturing approach manufacturing gained cloth.Because the complicacy of cloth deformation can't be constructed a model unification, that can be suitable for various situation.Main difficulty is that cloth raw material and weaving method thereof have nothing in common with each other, and the discreteness of experimental data is often bigger, is not easy to construct a more unified mechanical model.

Since people such as Terzopoulos proposed first cloth realistic model in 1987; The researchist has done a large amount of experiments and research; The various physical parameters of cloth deformation have under external force been obtained; And developed some physical models and described these deformation, mainly contain the particle-spring model etc. of the artificial representatives such as parameterized model, French state-run computing machine and robotization research institute Provot and the Choi of Seoul, South Korea university of artificial representative such as the Volino of University of Geneva.For the application of cloth model in animation, only require to produce relatively more real cloth animation effect, and requirement is not very high to the accuracy on the mechanics; And,, require model enough simple in order to produce the cloth animation of interactivity.Therefore, particle-spring model is extensively adopted because of it is simple.

Tension not transverse property is the key characteristic of cloth.According to this characteristic, whether people are easy to distinguish deformable bodies is cloth.Yet particle-spring model can be stretched to a certain extent, is difficult to show realistically the stretch-proof characteristic of cloth based on this model, thereby presses for the emulation mode of the no stretch material of research.The cloth that emulation does not have tensile properties mainly contains two kinds of schemes: a kind of is the alternative particle-spring model of model that research can be satisfied this specific character, and another kind is based on the spring deformation constrained procedure of particle-spring model.Existing research shows adopts other model to substitute particle-spring model, shows the no tensile properties of cloth, can reduce the stability and the efficient of cloth analogue system greatly.Therefore, many choose show no tensile properties based on the spring deformation constrained procedure of particle-spring model.

Existing constrained procedure mainly comprises two types, and one type is based on the deformation constrained procedure of tri patch, the another kind of partial spring constrained procedure that is based on the limit.Because based on the cloth that particle-spring model is represented, finally all be to draw with the form of tri patch.So people have proposed the deformation constrained procedure based on tri patch, the deformation on each limit of constraint tri patch.Because the quantity on the limit of the tri patch of connection particle is mostly more than or equal to 6, if use the strict deformation constraint based on tri patch, the degree of freedom of particle can reach capacity, and will cause stiff emulation cloth.Therefore, can only emulation allow the cloth of certain tensile deformation degree based on the deformation constrained procedure of tri patch, like woollen goods etc.The researchist finds that cloth has anisotropy, can in to a certain degree, be stretched in some direction, and be difficult to be stretched in some direction, like bast fiber fabrics.Therefore, people have proposed the partial spring constrained procedure based on the limit, have avoided too much causing the degree of freedom of particle to reach capacity because of restricted spring quantity.Based on the spring deformation of direction of warp and weft in the partial spring constrained procedure constraint particle-spring model on limit, this warp thread and weft yarn with fabric is corresponding, can show the no tensile properties of cloth more realistically.

Since the state-run computing machine of nineteen ninety-five France and robotization research institute Provot propose the partial spring constrained procedure based on your iteration of Gauss's Saden first, obtained certain development based on the research of the partial spring constrained procedure on limit.The spring deformation fast projection method that proposes based on the partial spring constrained procedure of your iteration of Gauss's Saden and researchists such as the Rony Goldenthal of Columbia Univ USA in 2007 that the representative achievement of this respect has that the Thomas Jakobsen of the constrained procedure based on the implicit expression integration that the David Baraff of institute of robot of Carnegie Mellon University in 1998 proposes, calendar year 2001 Copenhagen, Denmark IO Interactive company proposes based on Lagrangian mechanics constraint.Constrained procedure based on the implicit expression integration can bring extra damping force, makes cloth look like rubber; Constrained procedure based on your iteration of Gauss's Saden can be introduced extra linear momentum and angular momentum in the constraint implementation process, the cloth of emulation can jitter phenomenon occur because of the momentum nonconservation.

Summary of the invention

In order to overcome the deficiency of prior art, the invention provides a kind of particle-spring model constrained procedure towards no stretch material emulation.The objective of the invention is to deformation quantity according to the topological sum structure spring of particle-spring model; Revise the particle postition; Implement the spring deformation constraint apace, guarantee the linear momentum and the conservation of angular momentum of cloth in the constraint implementation process, do not have stretch material for emulation efficiently and stably and lay the foundation.

The technical scheme that the present invention adopts is:

A kind of particle towards no stretch material emulation-spring model constrained procedure may further comprise the steps based on the no stretch material simulation process of this method:

(1) cloth and virtual environment initialization;

(2) particle force analysis: according to the residing state of particle-spring, the bending force of the pulling force of the suffered structure spring of analysis particle, the shearing force of shear spring, flexural spring, air damping power, the internally-damped power of cloth, gravity etc.;

(3) particle postition prediction: the integration method prediction particle postition of adopting position-based and acceleration;

(4) not having the constraint of stretching implements: utilize particle-spring model constrained procedure, revise the particle postition, realize the spring deformation constraint;

(5) cloth self collision and collision are handled: adopt the cloth self collision and the collision processing method of the robust of Robert's proposition to realize;

(6) cloth and virtual environment are played up: adopt OPENGL real-time rendering and POV-RAY off-line to play up dual mode and play up cloth and environment thereof;

(7) judge whether next time step arrives, if return (2) step; If not, then finish;

Wherein, the concrete steps of the particle described in the step (4)-spring model constrained procedure are following:

(A1) according to the deformation of single structure spring, calculate the component of two-end-point particle along the position correction amount of spring direction, the component of position correction amount abbreviates component as;

(A2) based on the syntople of spring and particle, the component that will belong to same particle is superimposed;

(A3) based on topological structure, structure spring deformation, the particle component stack sum of structure spring in particle-spring model, estimate optimum relaxation factor;

(A4) superpose sum and optimum relaxation factor of component multiplied each other, the result is as the position correction amount of particle;

(A5) upgrade the particle postition according to the particle postition correction that obtains, realize the spring deformation reparation.

Further, the component of all particles is found the solution under same state in the step (A1), guarantees can keep in the spring deformation repair process linear momentum and the conservation of angular momentum of cloth, makes that the cloth of emulation is highly stable.

Further, belonging to the component of same particle in the step (A2), is to calculate according to the structure spring deformation that connects this particle; Therefore, component stack is local but not the overall situation, and most of details of cloth can keep down in constraint is implemented, and the cloth of emulation has a large amount of folds.

Further; The described topological structure of step (A3), structure spring deformation, particle component stack sum based on structure spring in particle-spring model; Estimate optimum relaxation factor; Specifically, optimum relaxation factor is by two factors: the structure spring constitutes in the mould of particle component stack sum and the ratio of structure spring deformation amount summation, the particle-spring neighbours territory topological structure, and the estimation of the simple combination of these two factors obtains.

Further; The component stack sum and the optimum relaxation factor of the described position correction amount of step (A4) multiply each other; Obtain the position correction amount of particle; The speed of convergence of spring deformation constraint is accelerated, and estimated that the optimum relaxation factor that obtains can repeated application implement in same time step inner spring deformation constraint iteration, thereby improve the efficient of spring deformation constraint enforcement.

Principle of the present invention is: based on the constrained procedure of particle-spring model; The structure spring that deformation takes place is returned to its car space; Its process comprises according to single structure spring deformation amount calculates the component of particle along the position correction amount of spring direction; Be superimposed according to the syntople of structure spring and particle, the component that will belong to the position correction amount of same particle; Utilize the topological structure of structure spring in mould, structure spring deformation and the particle-spring model of component stack sum of particle postition correction to estimate optimum relaxation factor; The component stack sum of the position correction amount that utilize to obtain utilizes the particle postition correction that obtains to upgrade particle postition, the reparation of realization spring deformation with optimum relaxation factor multiplies each other, its result is as the position correction amount of particle.

Particle-spring model comprises three types of springs: flexural spring, shear spring and structure spring.Flexural spring is in order to describe the counter-bending attribute of cloth, and it receives force direction outside the normal plane of particle place curved surface, and elasticity coefficient is generally smaller.Shear spring and structure spring receive force direction in the normal plane of particle place curved surface.The structure spring is in order to describe the tensile characteristics of cloth along the filling yarn direction, and cloth generally is difficult for being stretched in this direction, so its elasticity coefficient is general bigger.Shear spring is in order to describe the anti-shearing characteristic of cloth, and elasticity coefficient is moderate.In most of clothes, do not have obvious stretching phenomenon along warp yarn direction and weft direction.On the other hand, if shear spring is retrained by spring deformation also, because particle degree of freedom supersaturation problem, the result of particle-spring model emulation is a duricrust, can't realize soft cloth emulation.Therefore, the present invention only retrains the structure spring of particle-spring model.

The car space of spring is meant the length that spring keeps under the situation that does not receive external force and internal force.Flexural spring is just strong effect in its length less than its car space only, and shear spring and structure spring are only in the strong effect of ability during greater than its car space of its length.Therefore, the car space among the present invention is meant that three types of springs do not receive under the situation of external force and internal force the length that the structure spring can keep in particle-spring model.

The particle postition correction is that component sum and the optimum relaxation factor of the particle postition correction that calculates of all the structure springs according to the particle adjacency multiplies each other and obtain.The direction of each component is along the spring direction: when spring is compressed, the component direction of particle postition correction is by pointing to both ends of the spring in the middle of the spring; In the middle of spring is stretched, and the direction of particle postition correction component is pointed to by the two ends of spring.

Optimum relaxation factor mainly receives two factor affecting: the neighbours territory topological structure that restricted spring (being the structure spring) is formed, the mould of particle postition correction component stack sum and the ratio of spring deformation amount summation.Therefore, estimate optimum relaxation factor according to these two factors among the present invention.Because optimum relaxation factor is found the solution simply, reusability is good, in constraint is implemented, can accelerate convergent speed greatly, thereby improve the efficient of no stretch material emulation.

The present invention's beneficial effect compared with prior art is: 1, the component of all particle postition corrections is found the solution under same state among the present invention, the stack of the component of particle postition correction is linear; Can guarantee in spring deformation constraint and repair process, can keep the linear momentum and the conservation of angular momentum of cloth; The constrained procedure that possibly cause angular momentum nonconservation and cloth wild effect relatively in the past, particle-spring model constrained procedure that the present invention proposes also can highly stable ground emulation cloth under ultra constraint condition.2, the particle-spring model constrained procedure of the present invention's proposition can be implemented the reparation of deformation spring efficiently, and its high efficiency has benefited from simple optimum relaxation factor evaluation method, estimates that the optimum relaxation factor that obtains can the repeated application neighbours territory fully connected topology that the structure spring is formed in same time step inner spring deformation iteration reparation and particle-spring model.

Description of drawings:

Fig. 1 is the process flow diagram that the present invention does not have stretch material emulation;

Fig. 2 is the process flow diagram that spring deformation constraint of the present invention is implemented;

Fig. 3 is the funtcional relationship of ratio R of mould and the structure spring deformation amount summation of the present invention's component stack sum that relaxation factor ω and particle postition correction are arranged most;

Fig. 4 is spring deformation constraint embodiment figure of the present invention.

Embodiment:

Describe below in conjunction with the accompanying drawing specific embodiments of the invention, so that understand the present invention better.What need point out especially is that in the following description, when perhaps the detailed description that adopts known function and design can desalinate main contents of the present invention, these were described in here and will be left in the basket.

The particle towards no stretch material emulation-spring model constrained procedure that the present invention proposes is in existing cloth analogue system based on particle-spring model, increases the spring deformation constraint, realizes no stretch material emulation.No stretch material simulation process based on particle-spring model constrained procedure comprises that initialization, force analysis, particle postition prediction, nothing stretch and retrain, the cloth collision is handled, the result plays up, and is as shown in Figure 1.Initialization comprises cloth and context initialization thereof; Mainly contain the virtual environment and the object at cloth emulation place, the quality of the position of particle and speed, particle, spring strength coefficient, air damping coefficient, the internally-damped coefficient of cloth, acceleration of gravity, simulation time step-length etc. in particle-spring model.Force analysis comprises the force analysis and the suffered external force analysis of cloth of particle in particle-spring model, specifically comprises bending force, air damping power, the internally-damped power of cloth, gravity of shearing force, the flexural spring of pulling force, the shear spring of structure spring etc.Because the spring deformation that the present invention proposes constraint is moved through the particle postition realize, therefore, adopt the verlet integrator to predict the position of particle, because this integrator is based on particle postition and the prediction of the stressed realization of particle particle postition at next time step.It is to adopt the cloth self collision and the collision processing method of the robust of Robert's proposition to realize that the cloth collision is handled.

Particle-spring model constrained procedure implementation process that the present invention proposes is as shown in Figure 2, comprising: optimum relaxation factor finds the solution, according to single structure spring calculate the particle postition correction component, utilize the component of optimum relaxation factor and particle postition correction to calculate the particle postition correction, the particle postition is upgraded.Because optimum relaxation factor is relevant with the topological structure of forming according to the component sum of spring deformation amount, particle postition correction and restricted spring, below we will introduce how to implement the spring deformation constraint earlier, introduce finding the solution of optimum relaxation factor again.

Consider that at first the constraint of single spring deformation implements, be based upon constraint above the single spring and be in order to make this spring keep its car space, that is:

c _ij＝|p _j-p _i|-l _ij＝0

Wherein, p _i, p _jBe the position that is positioned at spring end points particle i, j, | p _j-p _i| be Euler's distance of end points particle, l _IjIt is the car space of spring.The quality of supposing two particles is respectively m _iAnd m _j, for repairing the deformation spring, particle along the component of the position correction amount of spring direction is:

$\mathrm{\Δ}{p}_{i\left(\mathrm{ij}\right)}=\frac{m_j}{m_i+m_j}\·c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}---\left(1\right)$

${\mathrm{\Δp}}_{j\left(\mathrm{ij}\right)}=-\frac{m_i}{m_i+m_j}\·c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}$

Wherein, p _Ij=p _j-p _i,

Be p _IjVector of unit length.A given relaxation factor ω, the position correction amount of particle is:

Δp _i＝ω·Δp _i(ij)

Δp _j＝ω·Δp _j(ij)

The position renewal of particle is:

$p_i^{\left(1\right)}=p_i+{\mathrm{\Δp}}_i$

$p_j^{\left(1\right)}=p_j+{\mathrm{\Δp}}_j$

An iteration is implemented in constraint through spring deformation, and the constraint of spring becomes:

$c_{\mathrm{ij}}^{\left(1\right)}=|p_j^{\left(1\right)}-p_i^{\left(1\right)}|-l_{\mathrm{ij}}=(1-\mathrm{\ω})\·c_{\mathrm{ij}}$

In like manner, h iteration implemented in constraint through spring deformation, and the constraint of spring becomes:

$c_{\mathrm{ij}}^{\left(h\right)}={(1-\mathrm{\ω})}^h\·c_{\mathrm{ij}}$

Constraint is that convergent means that the constraint of spring levels off to zero, that is:

$\underset{h\→\∞}{\lim }{c}_{\mathrm{ij}}^{\left(h\right)}=c_{\mathrm{ij}}\·\underset{h\→\∞}{\lim }{(1-\mathrm{\ω})}^h=0$

Therefore, relaxation factor ω should be greater than 0, and less than 2.Simultaneously; If ω is greater than 1; The spring of an extended state be through possibly becoming the spring of compressive state after once iterative constrained, the spring of a compressive state causes that through becoming the spring of extended state after once iterative constrained spring jitter phenomenon occurs in the constraint implementation process.Therefore, for fear of jitter phenomenon, obtain stable cloth emulation, relaxation factor ω should be in interval (0,1).

Spring in particle-spring model can be divided three classes: flexural spring, shear spring and structure spring.In order to obtain suitable degree of freedom, the deformation constraint only acts on the structure spring.Structure spring and constitute the topological structure in neighbours territory in abutting connection with particle, every spring connects 2 particles in the topological structure, and each particle connects 4 springs at most.For linear momentum and the angular momentum conservation in the constraint implementation process of realizing cloth, the present invention adopts Jacobi's type iterative particle postition correction.This process comprised for two steps; The first step is independently found the solution the position correction amount component of the corresponding particle of all springs along this spring direction; Second step stacked up these components of each particle respectively; The sum that will superpose again with multiply each other with relaxation factor, take advantage of the result as the position correction amount of particle in this time iteration.

Specifically, hypothesis particle-spring model comprises n particle, m root restricted spring m≤2n, and the position correction amount component of particle is found the solution according to formula (1), and the position correction amount of particle is:

${\mathrm{\Δp}}_i=\mathrm{\ω}\·\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}---\left(2\right)$

Wherein, 1≤i, j≤n upgrades the particle postition then.

Below we introduce directrix momentum I _LinWith accurate angular momentum I _AngProve that cloth linear momentum and angular momentum in the constraint implementation process are conservations.

$I_{\mathrm{lin}}=\underset{i}{\mathrm{\Σ}}{m}_i{\mathrm{\Δp}}_i$

$I_{\mathrm{ang}}=\underset{i}{\mathrm{\Σ}}(p_i-p^{\′})\×m_i{\mathrm{\Δp}}_i$

Wherein p ' is the coordinate of arbitrfary point in the space.According to existing document, if I about momentum conservation _LinIf the linear momentum conservation of=0 cloth is I _AngThe conservation of angular momentum of=0 cloth.With the directrix momentum I in formula (2) and formula (1) the substitution following formula _Lin:

$I_{\mathrm{lin}}=\underset{i}{\mathrm{\Σ}}{m}_i(\mathrm{\ω}\·\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)})=\mathrm{\ω}\·\underset{i}{\mathrm{\Σ}}\underset{j}{\mathrm{\Σ}}{m}_i{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}$

Because the symmetry of constraint and Jacobi iteration is for set { m _iΔ p _{I (ij)}| 1≤i, any element m among j≤n} _iΔ p _{I (ij)}, a unique element m is all arranged _jΔ p _{J (ij)}Corresponding with it.According to formula (1), m is arranged _iΔ p _{I (ij)}+ m _jΔ p _{J (ij)}=0.So I _Lin=0.

According to formula (1), accurate angular momentum formula I _AngWith directrix momentum I _Lin:

$I_{\mathrm{ang}}=\underset{i}{\mathrm{\Σ}}{p}_i\×m_i{\mathrm{\Δp}}_i-\underset{i}{\mathrm{\Σ}}{p}^{\′}\×m_i{\mathrm{\Δp}}_i$

$=\underset{i}{\mathrm{\Σ}}{p}_i\×m_i{\mathrm{\Δp}}_i=\mathrm{\ω}\·\underset{i}{\mathrm{\Σ}}\underset{j}{\mathrm{\Σ}}{p}_i\×m_i{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}$

Be similar to I _Lin=0, since symmetry, I _Ang=0.Therefore, in the implementation process of spring deformation constraint, the linear momentum of cloth and the conservation of angular momentum.

Introduce below and how to find the solution optimum relaxation factor ω.

Remember that acceptable constraint error is e, promptly | c _Ij|≤e.At least want iteration just can make for h time the constraint error in single spring upper spring deformation constraint less than e,

$h=\left\{\begin{array}{cc}0& \left|c_{\mathrm{ij}}\right|\≤e\\ \frac{\ln e-\ln \left|c_{\mathrm{ij}}\right|}{\ln (1-\mathrm{\ω})}& \left|c_{\mathrm{ij}}\right|>e\end{array}\right.$

Therefore, along with relaxation factor ω from 0 to 1 increases, iterations h is reducing always.Simultaneously, when ω=1, in the chain of being made up of two springs, the component sum of the particle postition correction of two-end-point particle possibly cause that particle postition overcorrect even spring deformation constraint do not restrain.Be convergence and the raising speed of convergence of guaranteeing to retrain, must seek the optimal value of relaxation factor, promptly optimum relaxation factor.

Though adopt some method can accurately find the solution optimum relaxation factor, owing on the particle-spring model of description cloth thousands of restricted spring is arranged, the accurate Calculation expense is very big.In order to obtain optimum relaxation factor apace, we adopt estimation approach to obtain the estimated value of optimum relaxation factor rather than exact value.At first the estimator of analysis and formulistic optimum relaxation factor is revised this estimator according to experiment then, finally obtains estimated value.

In particle-spring model of being made up of n particle, m restricted spring, after iteration of spring deformation constraint enforcement, the constraint of spring becomes:

$c_{\mathrm{ij}}^{\left(1\right)}=|(p_j+{\mathrm{\Δp}}_j)-(p_i+{\mathrm{\Δp}}_i)|-l_{\mathrm{ij}}$

$=|(p_j-p_i)+({\mathrm{\Δp}}_j-{\mathrm{\Δp}}_i)|-l_{\mathrm{ij}}$

$=|l_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|-l_{\mathrm{ij}}$

Δ c wherein _Ij=(Δ p _j-Δ p _i).Therefore,

$\left|c_{\mathrm{ij}}^{\left(1\right)}\right|\≤|c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|$

The purpose of spring deformation constraint is to revise the particle postition; Repair the deformation spring; Promptly

so, when

$\underset{i,j}{\mathrm{\Σ}}|c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|=0$

When being met,

can be met.

Because then

$\underset{i,j}{\mathrm{\Σ}}|c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|\≥\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|-\underset{i,j}{\mathrm{\Σ}}\left|{\mathrm{\Δc}}_{\mathrm{ij}}\right|---\left(3\right)$

Again since in neighbours territory net particle count n and spring and count and satisfiedly between the m concern m≤2n, therefore,

$\underset{i,j}{\mathrm{\Σ}}\left|{\mathrm{\Δc}}_{\mathrm{ij}}\right|=\underset{i,j}{\mathrm{\Σ}}|{\mathrm{\Δp}}_j-{\mathrm{\Δp}}_i|\≤4\underset{i}{\mathrm{\Σ}}\left|{\mathrm{\Δp}}_i\right|\≤4\mathrm{\ω}\·\underset{i}{\mathrm{\Σ}}\left|\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|---\left(4\right)$

With formula (4) substitution formula (3):

$\mathrm{\ω}\≥\frac{\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|-\underset{i,j}{\mathrm{\Σ}}|c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|}{4\underset{i}{\mathrm{\Σ}}\left|\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|}---\left(5\right)$

In like manner, in order to satisfy inequality

then

$\mathrm{\ω}\≤\frac{\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|+\underset{i,j}{\mathrm{\Σ}}|c_{\mathrm{ij}}\·{\hat{p}}_{\mathrm{ij}}+{\mathrm{\Δc}}_{\mathrm{ij}}|}{4\underset{i}{\mathrm{\Σ}}\left|\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|}.---\left(6\right)$

In conjunction with formula (5), formula (6) and constrained objective:

$\mathrm{\ω}=\frac{\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|}{4\underset{i}{\mathrm{\Σ}}\left|\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|}---\left(7\right)$

With the mould of the component stack sum of R (≠ 0) expression particle postition correction and the ratio of structure spring deformation amount summation:

$R=\frac{\underset{i}{\mathrm{\Σ}}\left|\underset{j}{\mathrm{\Σ}}{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|}{\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|}\≤\frac{\underset{i}{\mathrm{\Σ}}\underset{j}{\mathrm{\Σ}\left|{\mathrm{\Δp}}_{i\left(\mathrm{ij}\right)}\right|}}{\underset{i,j}{\mathrm{\Σ}}\left|c_{\mathrm{ij}}\right|}=1$

Then formula (7) can be by brief note:

$\mathrm{\ω}=\frac{1}{4}\·\frac{1}{R}=\frac{1}{4}+\frac{1}{4}\·\frac{1-R}{R}$

Wherein, 0＜R≤1.If R＜1/4, ω＞1 then, the supposition ω ∈ of this and front (0,1] contradiction.For relaxation factor ω is limited to this zone, must do correction to following formula.

In neighbours' domain structure, each particle possibly connect four restricted springs, and according to basic Jacobi iteration method for solving, optimum relaxation factor should be not less than 0.25.This means if replace second on following formula the right with a function f (R),

$\mathrm{\ω}=\frac{1}{4}\·\frac{1}{R}=\frac{1}{4}+f\left(R\right)$

F (R) >=0 then.On the other hand, if near 1, then the convergence of spring deformation constraint can be very poor like noted earlier relaxation factor ω.So we go up the convergence of test constraint iteration at interval ω ∈ [0.25,0.85] through the experiment of cloth parcel spheroid.Time step is 1.1ms, and spring deformation constrains in each time step inner iteration and implements, and the deformation quantity of online all springs is no more than 1% of its car space up to the neighbours territory.Experiment statistics this process medi-spring deformation retrain average iterations.

The result shows when relaxation factor and is increased near 0.6 from 0.25 that spring deformation retrains average iterations slowly to be reduced; Along with relaxation factor continues to increase, average iterations increases severely then.In other examples, also find similar situation, if relaxation factor greater than 0.75 (ω＞0.75), the non-constant of efficient of spring deformation constraint.Therefore, the upper limit that we limit optimum relaxation factor is 0.75, and provides function f (R),

$f\left(R\right)=\frac{1}{2}\·\sin \left(\frac{\mathrm{\π}}{2}(1-R)\right)$

$\mathrm{\ω}=\frac{1}{4}+\frac{1}{2}\·\sin \left(\frac{\mathrm{\π}}{2}(1-R)\right)$

There is most the funtcional relationship of relaxation factor ω and ratio R as shown in Figure 3.

In no stretch material emulation, the method that the present invention proposes more is competent in short time step.Under the short situation of time step, the variation in the ratio R value goes on foot at one time is less, and the variation in optimum relaxation factor goes on foot at one time is also very little.Therefore, in each time step, optimum relaxation factor is estimated once that this estimated value is repeated to be applied in all iteration in step current time.Each time iteration comprise according to single spring calculate the particle postition correction component, utilize the component of optimum relaxation factor and particle postition correction to calculate the particle postition correction, the particle postition is upgraded.Fig. 4 has showed the spring deformation constraint implementation process of the chain of being made up of 3 particles, 2 springs.

In a particle-spring model that comprises n particle, the quantity of its restricted spring is no more than 2n; And constraint is each time implemented in the iteration; On each spring, implementing constraint institute's time spent is constant; Therefore the present invention is a linear relationship with particle quantity at particle-iteration institute's time spent of spring model medi-spring deformation constrained procedure of proposing, and the efficient of constraint enforcement is very high.

Although above the illustrative embodiment of the present invention is described; So that the technician in present technique field understands the present invention, but should be clear, the invention is not restricted to the scope of embodiment; To those skilled in the art; As long as various variations appended claim limit and the spirit and scope of the present invention confirmed in, these variations are conspicuous, all utilize innovation and creation that the present invention conceives all at the row of protection.

Claims (5)

1. the particle towards no stretch material emulation-spring model constrained procedure, it is characterized in that: the no stretch material simulation process based on this method may further comprise the steps:

(1) cloth and virtual environment initialization;

(2) particle force analysis: according to the residing state of particle-spring, the bending force of the pulling force of the suffered structure spring of analysis particle, the shearing force of shear spring, flexural spring, air damping power, the internally-damped power of cloth, gravity etc.;

(3) particle postition prediction: the integration method prediction particle postition of adopting position-based and acceleration;

(4) not having the constraint of stretching implements: utilize particle-spring model constrained procedure, revise the particle postition, realize the spring deformation constraint;

(5) cloth self collision and collision are handled: adopt the cloth self collision and the collision processing method of the robust of Robert's proposition to realize;

(6) cloth and virtual environment are played up: adopt OPENGL real-time rendering and POV-RAY off-line to play up dual mode and play up cloth and environment thereof;

(7) judge whether next time step arrives, if return (2) step; If not, then finish;

Wherein, the concrete steps of the particle described in the step (4)-spring model constrained procedure are following:

(A1) according to the deformation of single structure spring, calculate the component of two-end-point particle along the position correction amount of spring direction, the component of position correction amount abbreviates component as;

(A2) based on the syntople of spring and particle, the component that will belong to same particle is superimposed;

(A3) based on topological structure, structure spring deformation, the particle component stack sum of structure spring in particle-spring model, estimate optimum relaxation factor;

(A4) superpose sum and optimum relaxation factor of component multiplied each other, the result is as the position correction amount of particle;

(A5) upgrade the particle postition according to the particle postition correction that obtains, realize the spring deformation reparation.

2. a kind of particle according to claim 1-spring model constrained procedure towards no stretch material emulation; It is characterized in that: the component of all particles is found the solution under same state in the step (A1); Guarantee to keep in the spring deformation repair process linear momentum and the conservation of angular momentum of cloth, make that the cloth of emulation is highly stable.

3. a kind of particle towards no stretch material emulation according to claim 1-spring model constrained procedure is characterized in that: belonging to the component of same particle in the step (A2), is to calculate according to the structure spring deformation that connects this particle; Therefore, component stack is local but not the overall situation, and most of details of cloth can keep down in constraint is implemented, and the cloth of emulation has a large amount of folds.

4. a kind of particle according to claim 1-spring model constrained procedure towards no stretch material emulation; It is characterized in that: the described topological structure of step (A3), structure spring deformation, particle component stack sum based on structure spring in particle-spring model; Estimate optimum relaxation factor; Specifically; Optimum relaxation factor is by two factors: the structure spring constitutes in the mould of particle component stack sum and the ratio of structure spring deformation amount summation, the particle-spring model neighbours territory topological structure, and the estimation of the simple combination of these two factors obtains.

5. a kind of particle according to claim 1-spring model constrained procedure towards no stretch material emulation; It is characterized in that: the component stack sum and the optimum relaxation factor of the described position correction amount of step (A4) multiply each other; Obtain the position correction amount of particle; The speed of convergence of spring deformation constraint is accelerated, and estimated that the optimum relaxation factor that obtains can repeated application implement in same time step inner spring deformation constraint iteration, thereby improve the efficient of spring deformation constraint enforcement.

Images