CN109726496A - A kind of implementation method improving incompressible water model efficiency based on IISPH - Google Patents
A kind of implementation method improving incompressible water model efficiency based on IISPH Download PDFInfo
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Abstract
The present invention is a kind of implementation method that incompressible water model efficiency is improved based on IISPH (Implicit Incompressibility Smooth particle hydrodynamics, implicit incompressible smoothed particle method) method.Speed is introduced into IISPH without dispersion models first, corrects the divergence error of velocity field in each time step of IISPH method；Secondly, by improved density constant model, so that density constant model and speed realize the efficient simulation of incompressible water body without the shared calculating factor of dispersion models.By introducing speed without divergence model, the amendment of divergence error is realized, solves the problems, such as that density error is continuously increased at any time；By improving density constant model, realizes speed and shared without the calculating factor of dispersion models and density constant model, solve the problems, such as redundant computation.The present invention can reach 30.10 frames/second real-time analog rate under 17k particle scale, with PCISPH (Predictive-Corrective Incompressible SPH, based on prediction-correction SPH) method compares, and simulation precision improves 24.17%；Compared with IISPH method, simulation precision improves 16.21%.
Description
Technical field
The present invention relates to field of Computer Graphics, and in particular to one kind is based on IISPH (Implicit
Incompressibility Smooth particle hydrodynamics, implicit incompressible smooth particle fluid dynamic
Learn) improve the implementation method of incompressible water model efficiency.
Background technique
It is limited to the calculated performance and computational efficiency of computer, water model technology is chronically at theoretical research stage.Closely
With the promotion of computer hardware performance and computational efficiency over year, water model is achieved, simultaneous computer graphics
Development further promotes the development of the water model technology based on physics, make its be increasingly becoming the research direction that is concerned it
One.Smoothed particle method (Smooth particle hydrodynamics, SPH) method is a kind of by solving matter
Point group kinetics equation calculates the non-mesh method of fluid (including water body) mass motion state, is able to maintain quality because it exists
Conservation is easy to capture many advantages, such as spray and bubble, is widely used in each neck such as film, game, advertisement and military affairs
Domain.
The pressure that traditional SPH method is subject to using the Solving Equation of State water body particle of perfect gas, so that water body has
Apparent compressibility, so as to cause when simulating extensive water body and Large Scale Motion, water surface occurs significantly not
Naturally it shakes, causes apparent vision distortion, therefore realize the Incoercibility of water body for improving water model sense of reality tool
It is significant.
The existing thinking for realizing water body Incoercibility is broadly divided into two kinds: explicit incompressible method and it is implicit not
Compressible method.Explicit incompressible method passes through direct solution equation of state for liquids or pressure Poisson's equation (Pressure
Poisson Equation, PPE) pressure is solved, and the mode that implicit incompressible equation mostly uses prediction to correct solves pressure.
1. explicit incompressible method
Explicit incompressible method mainly passes through direct solution equation of state for liquids or pressure Poisson's equation, obtains water body
The pressure that particle is subject to, to realize the Incoercibility of water body.
2007, Becker et al. proposed micro- compressible SPH (Weakly Compressible SPH, WCSPH) method
(reference paper 1:Monaghan, J.J.Simulating free surface flows with SPH [J] .Journal of
Computational Physics, 1994,110 (2): 399-406), this method replaces perfect gas using equation of state for liquids
State equation, and by using big stiffness coefficient make water body particle density wave momentum be no more than 1%, to realize
The Incoercibility of water body.WCSPH ensure that the Incoercibility of water model, be obviously improved the true of water model
Sense.But the rigidity of water body dominates CFL (Courant-Friedrichs-Levy) condition, big stiffness coefficient only allows to make
It with small time step, calculates cost and increases with the reduction of compressibility, therefore WCSPH method can not efficient simulation complexity
Water sports.
The same year, Adams et al. propose incompressible SPH (Incompressible SPH, ISPH) method (reference paper
2:X.Y.Hu, N.A.Adams.An incompressible multi-phase SPH method [J] .Journal of
Computational Physics,2007,227(1):264–278.).This method is moored by conjugate gradient direct solution pressure
Loose measure journey, to obtain the pressure that particle is subject to.ISPH method calculates other power other than pressure as resultant force first, secondly leads to
The midrange speed that the resultant force calculates particle is crossed, it is then discrete to Poisson's equation progress, calculate the pressure that particle is subject to.It is different from
WCSPH, ISPH allow using biggish time step, but this method need in each time step to complicated formulas and
Equation group is solved, and computing cost is huge, cannot achieve the efficient simulation of incompressible water body.
2015, Kang et al. will introduce ISPH method (reference paper 3:Kang N, Sagong without divergence condition
D.Incompressible SPH using the Divergence-Free Condition[J].Computer Graphics
Forum, 2015,33 (7): 219-228), this method is implemented when solving the equation of momentum without divergence condition.Guarantee that volume can not
A speed without divergence field is kept while compression, is performed simultaneously Incoercibility on location class and velocity stage.Compared to
ISPH, this method can greatly accelerate the convergence rate of Pressure solution, reduce the number of iterations of solver, to improve water body
Simulation precision.But this method is easy to appear particle density at water body Free Surface and is underestimated, to generate in simulation process
Phenomena such as particle buildup.
2. implicit incompressible method
Implicit incompressible method mainly passes through solution pressure Poisson's equation indirectly, obtains the pressure that water body particle is subject to,
To realize the Incoercibility of water body.
2009, Solenthaler et al. was proposed based on prediction-correction SPH (Predictive-Corrective
Incompressible SPH, PCISPH) method (reference paper 4:Solenthaler B, Pajarola R.Predictive-
corrective incompressible SPH[J].Acm Transactions on Graphics,2009,28(3):1-
6), this method is divided into two stages: forecast period and calibration phase.In forecast period, this method uses other power outside pressure
The attributes, i.e. the prediction physics of particle such as speed, the position of future time step-length particle are calculated as resultant force, and according to the resultant force
Attribute.Then iteration correction circulation is executed, when the error of particle density is greater than given error amount, is pressed by applying to particle
Power adjusts the speed of particle, and correction circulation is exited if averag density error is in given error range.Finally utilize solution
Pressure and other power out recalculates the particle property of future time step-length.This method avoid direct solution pressure Poissons
The computing cost of equation, but solve pressure indirectly and need to undergo successive ignition, computing cost is still increased to a certain extent.
2012, He et al. proposed a kind of local Poisson SPH (Local Poisson for solving Incoercibility water body
SPH, LPSPH) method (reference paper 5:X.He, N.Liu, S.Li, et al.Local poisson SPH for viscous
incompressible fluids[J].Computer Graphics Forum,2012,31(6):1948–1958).This method
Integrated form is converted by Poisson's equation first；Then using discretization method by continuous integral it is equations turned be discrete summation,
Avoid the computing cost of direct solution overall situation pressure Poisson's equation.
2014, Ihmsen et al. proposed a kind of pressure projection formula, i.e., implicit incompressible SPH (Implicit
Incompressibility SPH, IISPH) method (reference paper 6:Ihmsen M, Cornelis J, Solenthaler B,
et al.Implicit Incompressible SPH[J].IEEE Transactions on Visualization and
Computer Graphics, 2014,20 (3): 426-435.), this method by the symmetrical SPH pressure of continuity equation and SPH from
Dephasing combines, and obtains the discrete form of pressure Poisson's equation, and solve pressure using relaxation Jacobi algorithm.With throwing before
Shadow scheme is compared, and the method increase the convergence rates of solver, in addition, this method is based on speed rather than position is close to correct
Deviation is spent, this improves the stability of time integral to a certain extent.But the velocity field that this method is calculated is in the presence of scattered
Spending error reduces simulation precision to increase the mean iterative number of time of density correction.
2014, Ihmsen et al. proposed IISPH-LFIP (Implicit Incompressibility SPH-Fluid
Implicit Particle, IISPH-FLIP) method (reference paper 7:Cornelis J, Ihmsen M, Peer A, et
al.IISPH-FLIP for incompressible fluids[J].Computer Graphics Forum,2014,33
(2):255-262.).This method carries out pressure projection using implicit incompressible method, and uses the implicit particle (Fluid of fluid
Implicit Particle, FLIP) solver progress BORDER PROCESSING, realize the simulation of incompressible water body.This new combination
Solve the problems, such as two of existing SPH and FLIP solver, the i.e. efficiency in the conservation of mass and SPH in FLIP.The party
The IISPH-FLIP solver that method proposes can simulate incompressible water body, can quantify, non density variation is less than
0.1%.
Explicit incompressible water model method direct solution equation of state for liquids or pressure Poisson's equation calculate complicated
Degree is high；Implicit incompressible water model method by the way of being corrected such as the methods of PCISPH, IISPH using prediction, but is to solve for
The number of iterations is excessive in the process, and iteration all needs to execute time-consuming huge proximate particle finding step every time, cause calculation amount with
The increase of population sharply increase.To sum up, although existing incompressible water model method realizes the incompressible of water body
Property, but when applying Incoercibility, expense is too big, reduces simulation precision.
Summary of the invention
The present invention proposes to improve the implementation method of incompressible water model efficiency based on IISPH, first by by speed
No dispersion models introduce IISPH method, correct the divergence error of speed in IISPH method, avoid density error with the time
Increase is continuously increased, to reduce the mean iterative number of time of effective density error；Then the density constant model of IISPH is carried out
It improves, variable density amount is solved by rate of change of the density, so that improved density constant model is able to use speed without scattered
The calculating factor that model solution goes out is spent, and then avoids redundant computation, improves incompressible water model efficiency.It specifically includes
Following steps:
Step 1: speed is introduced without dispersion models, to correct the divergence error of IISPH method
Na Wei-Stokes (Navier-Stokes, N-S) equation is kept for describing viscous incompressible fluid momentum
The permanent equation of motion.The equation is mainly used to describe the movement of liquids and gases, can also be with substantially according to Newton's second law
Statement are as follows: the momentum change of infinitely small volume fluid is the effect summation of its gravity, viscous force, pressure and other active forces.N-S
Equation describes the rule of the fluid motion in nature, and the essence based on physics makes it be highly suitable for description nature
The natural fluids such as phenomenon, such as water body, smog, air-flow.
Shown in the three-dimensional representations of N-S equation such as formula (1) and formula (2).
Wherein u represents the speed of particle, and ρ represents the density of particle, and p indicates the pressure of particle, that is, is applied to unit area
On pressure, f indicates the external force (such as gravity) that is subject to of fluid.Operator is the dot product of space vector,Representation space vector
To the local derviation of Spatial Dimension, i.e.,Also known as gradient, operatorFor Laplace operator.
Formula (1) describes conservation of momentum when fluid motion, and right formula indicates all field of force amounts of fluid.On the right of equation
First item represent pressure difference between particle, Section 2 indicates the viscous force between particle, and Section 3 is external force (such as gravity).Its physics
Meaning are as follows: the movement of water body particle is codetermined by pressure, viscous force and external force.
Formula (2) is known as incompressibility, which indicates that the speed divergence of water body during the motion is 0, i.e., at some
In time step, inflow and discharge in water body unit volume do not change, to guarantee water body during the motion
Incoercibility.
The modified particle rapidity of IISPH is unsatisfactory for there are divergence errorThe present invention is by introducing speed without scattered
Model is spent, and calculates pressure term using the model, the divergence error of particle rapidity is modified to realize, suffered by particle i
The size of pressure is determined by formula (3).
WhereinIndicate pressure suffered by particle i,Indicate the volume of water body particle,Indicate barometric gradient.It introducesThe stiffness coefficient being modified to the speed v of particle i is indicated, shown in the calculating formula such as formula (4) that barometric gradient can be obtained.
Wherein mjIndicate the quality of proximate particle j,Indicate the gradient of kernel function,As meet speed without divergence item
The stiffness coefficient of part.
Particle meets the conservation of momentum in the process of movement.Force analysis is carried out it is found that pressure suffered by particle i to particle i
Power is the sum of the pressure of proximate particle j, and interparticle pressure is a pair of of interaction force.Due to interaction force size phase
Deng contrary, therefore, pressure suffered by particle i and particle i meet the pressure of proximate particle j
WhereinIndicate particle i to the pressure of particle j, it means that pressure as a kind of interparticle active force, within
The form of power exists, and the sum of pressure of all particles is 0.First and last from water body, which does not generate water body itself
The change of kinetic energy meets kinetic energy conservation law.Size and interparticle distance dependent of the particle i to proximate particle pressure, are asked
It solves shown in formula such as formula (5).
Wherein xjIndicate that the three-dimensional coordinate of particle j, solver must correct the speed of particle by changing pressure value, thus
So that the speed of particle meets without divergence.Shown in change rate such as formula (6) of the density relative to the time.
Wherein WijIndicate smoothing kernel function, it is ensured that the Incoercibility of water body, i.e. density do not change, then formula (6)
The density acquired should be 0 relative to the change rate of time.V in formula (6)i-vjIndicate interparticle relative speed difference, the speed difference
Caused by pressure difference, there are difference for the pressure that different particles are subject to, therefore cause interparticle speed different.Caused by pressure
Velocity variable such as formula (7) shown in.
Wherein Δ viIt indicates the velocity variable as caused by pressure, formula (7) is substituted into formula (6), obtain formula (8).
The size of particle pressure can be calculated by formula (3), convolution (6) rate of change of the density is 0, can be solved
Stiffness coefficient；Using the stiffness coefficient correct particle rapidity, enable to speed divergence be 0 so that density relative to
The change rate of time is 0.Formula (4) are substituted into formula (3), are obtainedFormula is substituted into again
(8), formula (9) are obtained.
Stiffness coefficient is solved according to formula (9)As shown in formula (10).
WhereinAnd αiSize only it is related with particle position.
Formula (10) is speed without dispersion models, which goes out so that speed meets the stiffness coefficient without divergence
The stiffness coefficient guarantees that rate of change of the density does not fluctuate water body in the process of movement, to realize water for calculating pressure
The Incoercibility of body.
The resultant force that particle is subject to finally is calculated according to formula (11), and recalculates the speed of particle with joint efforts using this.
WhereinIndicate the suffered resultant force of particle i,Indicate the resultant force of other power in addition to pressure,Indicate particle i
The pressure for the proximate particle j being subject to.
Step 2 improves density constant model, makes it with speed without the shared calculating factor of dispersion models, to correct IISPH
Density error.
In speed without in dispersion models, particle rapidity is corrected without the stiffness coefficient of divergence using speed is met, to protect
It demonstrate,proves rate of change of the density and is less than given divergence error amount ηdiv.Need to calculate the α factor, this hair when carrying out speed without divergence amendment
The density constant Model Reuse α of bright proposition calculates the factor, is further corrected to the density of water body particle.
The present invention calculates variable density amount by rate of change of the density and time step, then the density value ρ of t+ time Δt
′i, as shown in formula (12).
Speed is substituted into formula (12) without the rate of change of the density solution formula (8) of dispersion models, and combines density constant condition
ρi=ρ0, obtain the density error ρ ' in Δ t time stepi-ρ0, as shown in formula (13).
Wherein ρ0It indicates particle initial density, formula (3), (4) is substituted into formula (13).
Stiffness coefficient is obtained by formula (14)Solution formula such as formula (15) shown in.
WhereinFor the stiffness coefficient for meeting water body density constant, α is speed without the calculating factor in dispersion models, same
In one time step, which is only calculated once, and reduction computes repeatedly caused time overhead.
Formula (15) is density constant model, which goes out the stiffness coefficient so that density constantThe stiffness coefficient
The pressure being subject to for calculating particle guarantees that variable density amount does not fluctuate water body in the process of movement, to realize water
The Incoercibility of body.
As formula (11) it is found that shown in such as formula of resultant force suffered by particle after density constant Modifying model (16).
It uses resultant force suffered by the particle after correction to be modified the speed of particle, can be obtained in conjunction with Newton's second law
T+ time Δt particle rapidity v ' after amendmentiSolution formula, such as shown in (17).
It is calculated according to above-mentionedAnd formula (17) is to the speed v of particleiIt is modified, obtains revised particle
Speed v 'i。
The present invention combines speed without dispersion models formula (11) with improved density constant modular form (15), realizes pair
The amendment of speed divergence error and density error, to reduce redundancy meter while reducing density revision the number of iterations
It calculates, the simulation precision of incompressible water body can be effectively improved.Test result shows that method proposed by the present invention can not in simulation
When compressing water body, simulation precision is better than IISPH and PCISPH method.
Detailed description of the invention
Fig. 1 is the implementation method flow chart that incompressible water model efficiency is improved based on IISPH.
Fig. 2 is improved density constant model algorithm figure.
Fig. 3 is speed without dispersion models algorithm pattern.
Fig. 4 is incompressible water body system module design figure.
Fig. 5 is PCISPH, IISPH and the method for the present invention simulation precision comparison diagram.
Fig. 6 is the method for the present invention multithreading efficiency comparative figure.
Fig. 7 is that three-dimensional scenic water droplet falls simulation and effect picture.
Specific embodiment
Technical solution of the present invention and result are described in further detail below in conjunction with drawings and examples.
The present invention proposes a kind of incompressible water model method of the speed without divergence, first by by speed without divergence mould
Type introduces IISPH method, corrects the divergence error of speed in IISPH method, avoids density error continuous as time increases
Increase, to reduce the mean iterative number of time of effective density error；Then the density constant model of IISPH is improved, is led to
It crosses rate of change of the density and solves variable density amount, so that improved density constant model is able to use speed and asks without dispersion models
The calculating factor solved, therefore redundant computation is avoided, incompressible water model efficiency can be significantly improved.
Step 1: speed is introduced into IISPH without dispersion models, IISPH density constant model is improved, constructs incompressible water
Body phantom frame, as shown in Figure 1.
Step 1.1, proximate particle search is executed, proximate particle collection of each water body particle in current time step is obtained
Close Ni(t)。
Step 1.2, according to the proximate particle set N of each particlei(t), the resultant force other than the pressure that particle is subject to is calculated,
And predetermined speed and predicted density of particle are calculated according to the resultant force.
Step 1.3, in conjunction with density constant Modifying model particle rapidity, the shared calculating factor is calculated first, and according to the meter
It calculates the factor and calculates stiffness coefficient, which enables to density error to be less than given density error value；Then using should
Stiffness coefficient is corrected the speed of particle, so that the variable quantity of density | ρ 'i(t+Δt)-ρ0| less than given density error
Value ηρ。
Step 1.4, the particle rapidity obtained in conjunction with speed without dispersion models amendment step (1.3) is counted using step (1.3)
The shared calculating factor calculated calculates stiffness coefficient, which enables to rate of change of the density error to be less than given divergence
Error amount ηdiv, then using the speed of stiffness coefficient correction particle, so that rate of change of the density is less than given divergence error amount
ηdiv。
Step 1.5, all particles are traversed, particle rapidity is calculated according to above-mentioned, which is updated to lower a period of time
Between step-length particle rapidity, and other attributes of more new particle.
Step 2: in conjunction with improved density constant model, effective density error, as shown in Figure 2.
Step 2.1, all particles are traversed, in conjunction with its proximate particle set Ni(t), averag density variable quantity is calculated.
Step 2.2, if averag density variable quantity is less than given density error value (ρavg-ρ0<ηρ) or the number of iterations
Iter is not less than 2, then exits density constant model, no to then follow the steps 2.3.
Step 2.3, for each particle, its proximate particle set N is traversedi(t), corresponding stiffness coefficient is calculatedAnd
According to the stiffness coefficient calculate the particle and proximate particle suffered by pressure.
Step 2.4, it is followed according to the pressure correction particle rapidity that step 2.3 is calculated using revised particle rapidity
Ring executes step 2.1-2.4, until variable density amount is less than given density error (ρavg-ρ0<ηρ)。
Step 3: in conjunction with speed without dispersion models, divergence error is corrected, as shown in Figure 3.
Step 3.1, all particles are traversed, in conjunction with its proximate particle set Ni(t), the rate of change of the density of particle is calculated.
Step 3.2, if rate of change of the density is less than or equal to given divergence error amountOr the number of iterations is not small
In 1, then rate of withdraw is no to then follow the steps 3.3 without dispersion models.
Step 3.3, for each particle, its proximate particle set N is traversedi(t), corresponding stiffness coefficient is calculatedAnd
According to the stiffness coefficient calculate the particle and proximate particle suffered by pressure.
Step 3.4, it is followed according to the pressure correction particle rapidity that step 3.3 is calculated using revised particle rapidity
Ring executes step 3.1-3.3, until speed divergence error is less than given divergence error amount
Step 4: utilizing Step 2: three physical models solved, construct incompressible water sports scene, output knot
Fruit simultaneously shows.
Step 2: the incompressible water model solved in three, be only based on mathematical physics solve it is obtaining as a result, from
It visually describes, is made of a series of particles, the water body scene in reality can not realistically be presented.Therefore it also needs to upper
It states model and carries out corresponding Rendering operations, to obtain realistic water body, skill is rendered using screen space in the present invention
Art realizes water body rendering, to show fluid effect true to nature.
All tests of the present invention are carried out on Dell's desktop computer, and CPU model is Intel Core i7-4790
3.6GHz, video card model AMD RadeonTMR5 240。
Fig. 4 is incompressible water body system module design figure, and whole system is divided into 4 modules: initialization module, prediction school
Positive module, margin processing module and rendering module.It is initial to generate particle for initialization module primary recipient parameter set by user
The attributes such as information, including particle position, pressure, density, speed.
Prediction correction module combines speed without dispersion models, density constant model, and the module is in each time step
It executes once, generates the particle rapidity of future time step-length, be broadly divided into two parts: preprocessing part and speed calculation part
Point.Wherein preprocessing part mainly calculates and stores only data relevant to particle position, this partial data is in the time step
It is interior to change, it can be to avoid computing repeatedly by way of precomputation.Speed calculating section operating speed is without divergence mould
Predetermined speed of type and density constant Modifying model particle, so that the speed after correction meets speed without divergence and density
It is constant.
Margin processing module is used to correct the speed after predicted correction module correction, to avoid particle in movement
Solid boundaries are penetrated in the process.Whether detection particle first can collide with boundary, by based on repulsion if colliding
The collision response Modifying model particle rapidity of power, to guarantee the not penetrability of Vessel Boundary；It need not if not colliding
Execute collision response model.
Rendering module mainly realizes water body rendering, including the building of anisotropy core surface, depth according to particle location information
Smoothly, apply illumination and draw day sylphon, improve the smoothness of water surface and the sense of reality of water body rendering.
Speed is introduced IISPH without dispersion models by the present invention, and is improved the density constant model of IISPH, this
Kind improves the implementation method of incompressible water model efficiency based on IISPH, can the more efficient incompressible water body of simulation
Moving scene.
It is realized using PCISPH algorithm, IISPH algorithm and the method for the present invention and is fallen the simulation of model to water droplet respectively, it should
Model is made of 17074 particles, and test includes 12 independent experiments, and the average frame per second that 12 results are obtained by calculation accelerates
Than.Table 1 is that water droplet falls modeling frame per second contrast table, and Fig. 5 is corresponding line chart.It can be calculated by data in table 1,
The average frame per second of PCISPH algorithm is 24.24fps, and the average frame per second of IISPH algorithm is 25.91fps, and incompressible water body accelerates
The average frame per second of analogy method is 30.10fps.Compared to PCISPH algorithm, the computational efficiency of this paper method is improved
24.17%, compared to IISPH algorithm, the computational efficiency of this paper method improves 16.21%.
1 water droplet of table falls modeling frame per second contrast table
The present invention improves proximate particle search, kernel function and its gradient solution efficiency using multithreading.In order to verify this hair
The validity and high efficiency that bright program multithreading is realized, are respectively started 1,4,8 thread and simulate to 7104 particles, obtain
Simulation frame per second figure shown in fig. 6.It will be appreciated from fig. 6 that being unable to reach simulation in real time when 1 thread of starting and requiring；Start 4 threads
When, simulation precision significantly improves, and meets requirement of real-time；When starting 8 threads, simulation precision is further increased, to test
The validity and high efficiency of multi-threading of the present invention are demonstrate,proved.
Fig. 7, which depicts three-dimensional scenic water droplet and falls, simulates effect picture, and Fig. 7 (a) indicates water droplet original state, Fig. 7 (b), Fig. 7
(c) fall for water droplet and with the effect that repeatedly rebounds after collision on the ground, Fig. 7 (d) is that water body is gradually recovered stable effect, by scheming
It is found that water body do not occur in the process of movement particle penetration boundary, particle adhesion boundary, particle boundary aggregation etc. it is disorderly
Random phenomenon.
Claims (4)
1. one kind based on IISPH (Implicit Incompressibility Smooth particle hydrodynamics,
Implicit incompressible smoothed particle method) improve the implementation method of incompressible water model efficiency, it is characterised in that
Realize that steps are as follows:
Step 1: speed is introduced into IISPH without dispersion models, constructs incompressible water model frame
Firstly, building incompressible water model frame based on IISPH.IISPH is divided to for prediction and two stages of correction, prediction rank
Section calculates particle density and speed using other power in addition to pressure, and calibration phase can not by apply that pressure realizes water body
Compressibility.
Secondly, improving the density constant model of IISPH, it is made, without the shared calculating factor of dispersion models, to reduce redundancy meter with speed
It calculates, improves water model efficiency.
Finally, speed is introduced into incompressible water model frame without dispersion models.Exist for the speed that IISPH is solved
Speed is introduced into IISPH method without dispersion models, corrects the divergence error of particle by the problem of divergence error.
Step 2: speed is introduced without dispersion models, to correct the divergence error of IISPH method
The modified particle rapidity of IISPH is unsatisfactory for there are divergence errorThe present invention is by introducing speed without divergence mould
Type, and pressure term is calculated using the model, the divergence error of particle rapidity is modified to realize, particle i pressure
Size determined by formula (1).
WhereinIndicate pressure suffered by particle i,Indicate the volume of water body particle,Indicate barometric gradient.It introducesIt indicates
To the stiffness coefficient that the speed v of particle i is modified, shown in the calculating formula such as formula (2) that barometric gradient can be obtained.
Wherein mjIndicate the quality of proximate particle j,Indicate the gradient of kernel function,As meet speed without divergence condition
Stiffness coefficient.
Particle meets the conservation of momentum in the process of movement.Force analysis is carried out it is found that pressure suffered by particle i is to particle i
The sum of the pressure of proximate particle j, and interparticle pressure is a pair of of interaction force.Since interaction force is equal in magnitude,
Contrary, therefore, pressure suffered by particle i and particle i meet the pressure of proximate particle j
WhereinIndicate particle i to the pressure of particle j, it means that pressure is as a kind of interparticle active force, with internal force
Form exists, and the sum of pressure of all particles is 0.First and last from water body, which does not generate water body kinetic energy itself
Change, meet kinetic energy conservation law.Size and interparticle distance dependent of the particle i to proximate particle pressure, solve public
Shown in formula such as formula (3).
Wherein xjIndicate that the three-dimensional coordinate of particle j, solver must correct the speed of particle by changing pressure value, so that
The speed of particle meets without divergence.Shown in change rate such as formula (4) of the density relative to the time.
Wherein WijIndicate smoothing kernel function, it is ensured that the Incoercibility of water body, i.e. density do not change, then formula (4) acquires
Density should be 0 relative to the change rate of time.V in formula (4)i-vjIndicate interparticle relative speed difference, the speed difference is by pressing
Poor by force to cause, there are difference for the pressure that different particles are subject to, therefore cause interparticle speed different.The speed as caused by pressure
It spends shown in variable quantity such as formula (5).
Wherein Δ viIt indicates the velocity variable as caused by pressure, formula (5) is substituted into formula (4), obtain formula (6).
The size of particle pressure can be calculated by formula (1), convolution (4) rate of change of the density is 0, can solve rigidity
Coefficient；Particle rapidity is corrected using the stiffness coefficient, enabling to the divergence of speed is 0, so that density is relative to the time
Change rate be 0.Formula (2) are substituted into formula (1), are obtainedFormula (6) are substituted into again, are obtained
To formula (7).
Stiffness coefficient is solved according to formula (7)As shown in formula (8).
WhereinAnd αiSize only it is related with particle position.
Formula (8) is speed without dispersion models, and the model is by solving stiffness coefficientAnd pressure is calculated using the stiffness coefficient,
To guarantee that rate of change of the density does not fluctuate water body in the process of movement, that is, guarantee the Incoercibility of water body.
The resultant force that particle is subject to finally is calculated according to formula (9), and recalculates the speed of particle with joint efforts using this.
WhereinIndicate the suffered resultant force of particle i,Indicate the resultant force of other power in addition to pressure,Indicate particle i by
Proximate particle j pressure.
Step 3 improves density constant model, makes it with speed without the shared calculating factor of dispersion models, to correct the close of IISPH
Spend error.
In speed without in dispersion models, particle rapidity is corrected without the stiffness coefficient of divergence using speed is met, to guarantee close
It spends change rate and is less than given divergence error amount ηdiv.Need to calculate the α factor when carrying out speed without divergence amendment, the present invention mentions
Density constant Model Reuse α out calculates the factor, is further corrected to the density of water body particle.
The present invention calculates variable density amount by rate of change of the density and time step, then the density value ρ ' of t+ time ΔtiSuch as formula
(10) shown in.
Speed is substituted into formula (10) without the rate of change of the density solution formula (6) of dispersion models, and combines density constant condition (ρi=
ρ0), obtain the density error ρ ' in Δ t time stepi-ρ0, as shown in formula (11).
Wherein ρ0It indicates particle initial density, formula (1), (2) is substituted into formula (11).
Stiffness coefficient is obtained by formula (12)As shown in formula (13).
WhereinFor the stiffness coefficient for meeting water body density constant, α is speed without the calculating factor in dispersion models, same
In time step, which is only calculated once, therefore can reduce and compute repeatedly caused time overhead.
Formula (15) is density constant model, and the model is by solving stiffness coefficientAnd use the calculated pressure of the stiffness coefficient
Power guarantees the Incoercibility of water body to guarantee that variable density amount does not fluctuate water body in the process of movement.
As formula (11) it is found that shown in such as formula of resultant force suffered by particle after density constant Modifying model (14).
It uses resultant force suffered by the particle after correction to be modified the speed of particle, can must be corrected in conjunction with Newton's second law
T+ time Δt particle rapidity v ' afterwardsiSolution formula, such as shown in (15).
It is calculated according to above-mentionedAnd formula (15) is to the speed v of particleiIt is modified, obtains revised particle rapidity
v′i。
Step 4: utilizing Step 2: three physical models solved, construct incompressible water sports scene, output result is simultaneously
Display.
Step 2: the incompressible water model solved in three, be only based on mathematical physics solve obtain as a result, from vision
Upper description is made of a series of particles, and the water body scene in reality can not realistically be presented.Therefore it also needs to above-mentioned mould
Type carries out corresponding Rendering operations, so that realistic water body is obtained, it is real using screen space Rendering in the present invention
Existing water body rendering, to show fluid effect true to nature.
2. a kind of implementation method for improving incompressible water model efficiency based on IISPH according to claim 1, special
Sign is, incompressible body phantom frame constructed by step 1, specifically: speed being introduced IISPH without dispersion models, and is changed
Into the density constant model of IISPH, make it with speed without the shared calculating factor of dispersion models, to reduce redundant computation.Specifically
It is: calculates other power in addition to pressure first, and calculate the predicted density and predetermined speed of particle；Then density constant mould is used
Type effective density error；Last operating speed corrects divergence error without dispersion models.
3. a kind of implementation method for improving incompressible water model efficiency based on IISPH according to claim 1, special
Sign is, solves the pressure that particle is subject to without dispersion models by speed described in step 2, and then correct particle rapidity, so that
Particle rapidity meets without divergence, specifically: solving the divergence error of particle rapidity first；Secondly by speed without dispersion models
Stiffness coefficient is solved, and further solves the pressure that particle is subject to；The pressure correction particle rapidity is finally used, to obtain nothing
The velocity field of divergence.
4. a kind of implementation method for improving incompressible water model efficiency based on IISPH according to claim 1, special
Sign is that improved density constant model and speed described in step 3 are without the shared calculating factor-alpha of dispersion models, by improved
The pressure that density constant model solution particle is subject to, and then particle rapidity is corrected, so that particle meets density constant, specifically:
By other predicted densities and predetermined speed for making every effort to solve particle in addition to pressure, density constant model solution particle is then used
The pressure being subject to, and according to the pressure correction particle rapidity, realize density constant.
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