CN104809319A - Cloth simulation algorithm based on simplified implicit Euler numerical integration - Google Patents

Abstract

The invention relates to a cloth simulation algorithm based on simplified implicit Euler numerical integration. According to the algorithm, each mass point in the cloth is subjected to force analysis, the resultant force applied to each mass point is calculated, and the speed and position of a target mass point are calculated by utilizing a simplified implicit Euler numerical integration algorithm and are updated in real time, and thus the simulation of the cloth is realized. The cloth simulation algorithm disclosed by the invention has the advantages of solving the problem of stability of a traditional algorithm and a small time slice in explicit Euler integration and also preventing complex calculation in implicit integration.

Description

A kind of Cloth simulation algorithm based on simplifying Implicit Euler numerical integration

Technical field

The invention belongs to Cloth simulation field, being specifically related to a kind of Cloth simulation algorithm based on simplifying Implicit Euler numerical integration.

Background technology

Real-life a lot of behavior can be described with infinitesimal analysis, and textile simulating is also like this.This process is generally first set up the differential equation, is then solved and then realized the emulation of cloth by integral method.

In Cloth simulation, conventional integration method mainly contains: explicit Euler numerical integration method and Implicit Euler numerical integration method.What explicit integration was focused on is the authenticity of simulation result, and its time needed for emulation is not long, and workload is less, but also because this increasing the number of times of iteration.Implicit integration is contrary with dominant integration, and its iterations is fewer than dominant integration, ensure that stability, can produce good simulated effect.But equally also there is shortcoming in implicit integration: because the complexity calculated is higher, cause the calculating scale of this algorithm and amount of calculation to be promoted, and the performance requirement of implicit integration to hardware is higher.Mixed method is after the advantage combining above two kinds of methods, and its performance is improved, and but, its scope that can apply is very limited, is only limitted to the system that the differential equation can be divided into linear processes.

Explicit integral algorithm has a variety of, such as Euler method, mid-point method, fourth-order Runge-Kutta method.Wherein, explicit Euler numerical integration method is the simplest integration method, and it has lower solving precision, and the method has good concurrency and speed faster, and the method calculated amount is little and be easy to realize.But the method can produce very large error, this is also one of major reason of not being widely adopted of explicit Euler algorithm.

Implicit integration algorithm is when differential equation discretize, replaces a kind of integration method of derivative with Backward divided difference.Implicit integration method has unconditional stability, and the most frequently used implicit method is exactly Implicit Euler numerical integration method.Implicit integration, while keeping system stability, can converge on equilibrium state fast, but such implicit method is when particle number is larger, and namely when fabric model is large time, its complexity calculated is quite large.Want to reach real-time effect by the method for implicit integration, need higher computer hardware demand.

Explicit Euler's integral method is one simply and fast integration method.But in the method, if want to obtain good stability, then selected time step must very little.Implicit integration is generally used for the Rigidity solving equation.It avoids the situation of time step larger in simulation calculation, meanwhile, which overcome the problem of stability, but the method also exists the problems such as inefficiency.

Pure explicit Euler numerical integration method or Implicit Euler numerical integration method can not meet the demand of reality.Reason is: explicit Euler numerical integration method can only guarantee calculates fast under the prerequisite of little step-length stability.Although and Implicit Euler integration method avoids the calculating of large step-length, overcoming stability problem, still there is the low problem of counting yield in it.

Summary of the invention

For traditional cloths emulation mode Problems existing, the present invention proposes a kind of Cloth simulation algorithm based on simplifying Implicit Euler numerical integration, by solving the speed of the next time step of each particle and corresponding position, finally the position of particle is dynamically updated, thus realize the emulation of cloth, solve the problem of minor time slice in algorithm stability and explicit Euler's integral, also avoid calculating complicated in implicit integration simultaneously.

For achieving the above object, the concrete technical scheme of the present invention is as follows: a kind of Cloth simulation algorithm based on simplifying Implicit Euler numerical integration, comprises the steps:

1) force analysis: carry out force analysis to particle each in t cloth, stressedly comprises spring force damping force gravity and wind-force wherein, described spring force computing formula is as follows:

$F_{\mathrm{si}}^t={\mathrm{\Σ}}_{\∀j|(i,j)\∈E}{k}_{i,j}\left(\right|x_j-x_i|-l_{i,j}^0)\frac{x_j-x_i}{|x_j-x_i|}$

from the spring force of particle j suffered by t particle i, k _i,jthe spring constant between particle i and particle j, x _iwith x _jrepresent the position of particle i and particle j respectively, l _i,jfor the initial length of spring between particle i and particle j;

Described damping force computing formula is as follows:

$F_{\mathrm{di}}^t=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_{\mathrm{ij}}(v_i-v_j)$

D _ijthe elasticity coefficient between particle i and particle j, v _iwith v _jthe particle i speed corresponding with particle j;

Described gravity computing formula is as follows:

$F_{\mathrm{gi}}^t=m_{i}g$

M _ifor the quality of particle i, g is acceleration of gravity;

Described wind-force computing formula is as follows:

$F_{\mathrm{wi}}^t=k_w(v_w-v_i)$

K _wfor wind factor, v _wfor wind speed, v _ifor the speed of particle i;

2) particle is calculated with joint efforts suffered: calculate making a concerted effort suffered by particle according to the stressing conditions of particle, make a concerted effort computing formula is as follows:

$F_i^t=F_{\mathrm{si}}^t+F_{\mathrm{di}}^t+F_{\mathrm{gi}}^t+F_{\mathrm{wi}}^t$

3) target particle velocity difference is calculated: utilize and simplify the speed difference that Implicit Euler numerical integration algorithm calculates particle, particle i is at t and the speed difference in t+h moment computing formula as follows:

$\mathrm{\Δ}{v}_i^{t+h}=\frac{F_i^{t}h+h^{2}k{\mathrm{\Σ}}_{(i,j)\∈E}{F}_j^{t}h/(m_j+h^2{\mathrm{kn}}_j)}{m_i+h^2{\mathrm{kn}}_i}$

In formula, h represents the time difference of adjacent states, n _iand n _jrepresent the particle number be connected with particle j with particle i respectively;

4) target particle postition difference is calculated: the position difference calculating particle according to the speed difference of particle, particle i is at t and the position difference DELTA x in t+h moment ^t+hcomputing formula is as follows:

Δx ^t+h＝(v ^t+Δv ^t+h)h

In formula, v ^trepresent the speed of particle in t, Δ v ^t+hfor the difference of t and t+h hourly velocity;

5) speed and the position of each particle is upgraded: according to speed difference and position difference, real-time update is carried out to particle velocities all in cloth and position, complete the motion simulation of cloth.

A kind of Cloth simulation algorithm based on simplifying Implicit Euler numerical integration that the present invention proposes, solves the problem of minor time slice in the stability problem of traditional algorithm and explicit Euler's integral, also avoids calculating complicated in implicit integration simultaneously.

Accompanying drawing explanation

Fig. 1 is based on the Cloth simulation algorithm flow chart simplifying Implicit Euler numerical integration.

Embodiment

Below in conjunction with accompanying drawing, the invention will be further described.

Figure 1 shows that the Cloth simulation algorithm flow chart based on simplifying Implicit Euler numerical integration, comprising the following steps:

1) force analysis: carry out force analysis to particle each in t cloth, stressedly comprises spring force damping force gravity and wind-force wherein, described spring force computing formula is as follows:

$F_{\mathrm{si}}^t={\mathrm{\Σ}}_{\∀j|(i,j)\∈E}{k}_{i,j}\left(\right|x_j-x_i|-l_{i,j}^0)\frac{x_j-x_i}{|x_j-x_i|}$

from the spring force of particle j suffered by t particle i, k _i,jthe spring constant between particle i and particle j, x _iwith x _jrepresent the position of particle i and particle j respectively, l _i,jfor the initial length of spring between particle i and particle j;

Described damping force computing formula is as follows:

$F_{\mathrm{di}}^t=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_{\mathrm{ij}}(v_i-v_j)$

D _ijthe elasticity coefficient between particle i and particle j, v _iwith v _jthe particle i speed corresponding with particle j;

Described gravity computing formula is as follows:

$F_{\mathrm{gi}}^t=m_{i}g$

M _ifor the quality of particle i, g is acceleration of gravity;

Described wind-force computing formula is as follows:

$F_{\mathrm{wi}}^t=k_w(v_w-v_i)$

K _wfor wind factor, v _wfor wind speed, v _ifor the speed of particle i;

2) particle is calculated with joint efforts suffered: calculate making a concerted effort suffered by particle according to the stressing conditions of particle, make a concerted effort computing formula is as follows:

$F_i^t=F_{\mathrm{si}}^t+F_{\mathrm{di}}^t+F_{\mathrm{gi}}^t+F_{\mathrm{wi}}^t$

3) target particle velocity difference is calculated: utilize and simplify the speed difference that Implicit Euler numerical integration algorithm calculates particle, particle i is at t and the speed difference in t+h moment computing formula as follows:

$\mathrm{\Δ}{v}_i^{t+h}=\frac{F_i^{t}h+h^{2}k{\mathrm{\Σ}}_{(i,j)\∈E}{F}_j^{t}h/(m_j+h^2{\mathrm{kn}}_j)}{m_i+h^2{\mathrm{kn}}_i}$

In formula, h represents the time difference of adjacent states, n _iand n _jrepresent the particle number be connected with particle j with particle i respectively;

4) target particle postition difference is calculated: the position difference calculating particle according to the speed difference of particle, particle i is at t and the position difference DELTA x in t+h moment ^t+hcomputing formula is as follows:

Δx ^t+h＝(v ^t+Δv ^t+h)h

In formula, v ^trepresent the speed of particle in t, Δ v ^t+hfor the difference of t and t+h hourly velocity;

5) speed and the position of each particle is upgraded: according to speed difference and position difference, real-time update is carried out to particle velocities all in cloth and position, complete the motion simulation of cloth.

Claims (1)

1., based on the Cloth simulation algorithm simplifying Implicit Euler numerical integration, it is characterized in that comprising the steps:

1) force analysis: carry out force analysis to particle each in t cloth, stressedly comprises spring force damping force gravity and wind-force wherein, described spring force computing formula is as follows:

$F_{\mathrm{si}}^t={\mathrm{\Σ}}_{\∀j|(i,j)\∈E}{k}_{i,j}\left(\right|x_j-x_i|-l_{i,j}^0)\frac{x_j-x_i}{|x_j-x_i|}$

from the spring force of particle j suffered by t particle i, k _i,jthe spring constant between particle i and particle j, x _iwith x _jrepresent the position of particle i and particle j respectively, l _i,jfor the initial length of spring between particle i and particle j;

Described damping force computing formula is as follows:

$F_{\mathrm{di}}^t=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_{\mathrm{ij}}(v_i-v_j)$

D _ijthe elasticity coefficient between particle i and particle j, v _iwith v _jthe particle i speed corresponding with particle j;

Described gravity computing formula is as follows:

$F_{\mathrm{gi}}^t=m_{i}g$

M _ifor the quality of particle i, g is acceleration of gravity;

Described wind-force computing formula is as follows:

$F_{\mathrm{wi}}^t=k_w(v_w-v_i)$

K _wfor wind factor, v _wfor wind speed, v _ifor the speed of particle i;

2) particle is calculated with joint efforts suffered: calculate making a concerted effort suffered by particle according to the stressing conditions of particle, make a concerted effort computing formula is as follows:

$F_i^t=F_{\mathrm{si}}^t+F_{\mathrm{di}}^t+F_{\mathrm{gi}}^t+F_{\mathrm{wi}}^t;$

3) target particle velocity difference is calculated: utilize and simplify the speed difference that Implicit Euler numerical integration algorithm calculates particle, particle i is at t and the speed difference in t+h moment computing formula as follows:

${{\mathrm{\Δv}}_i}^{t+h}=\frac{F_i^{t}h+h^{2}k{\mathrm{\Σ}}_{(i,j)\∈E}{F}_j^{t}h/(m_j+h^2{\mathrm{kn}}_j)}{m_i+h^2{\mathrm{kn}}_i}$

H represents the time difference of adjacent states, n _iand n _jrepresent the particle number be connected with particle j with particle i respectively;

4) target particle postition difference is calculated: the position difference calculating particle according to the speed difference of particle, particle i is at t and the position difference DELTA x in t+h moment ^t+hcomputing formula is as follows:

Δx ^t+h＝(v ^t+Δv ^t+h)h

V ^trepresent the speed of particle in t, Δ v ^t+hfor the difference of t and t+h hourly velocity;

5) speed and the position of each particle is upgraded: according to speed difference and position difference, real-time update is carried out to particle velocities all in cloth and position, complete the motion simulation of cloth.