CN108536940A - A kind of method for building up of indoor smog diffusion model - Google Patents
A kind of method for building up of indoor smog diffusion model Download PDFInfo
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Abstract
The present invention discloses a kind of method for building up of indoor smog diffusion model, the structure of smog diffusion model based on boundary constraint, make up the external force factor that previous smoke model is ignored, in view of wind-force, influence of the air drag to smog movement, and extension is artificially added, in distortionless, the real-time of raising smog diffusion simulations.Simultaneously, it is contemplated that the limitation of indoor boundary condition handles the physical quantity of boundary, keeps the process that the smog simulated is spread indoors more true and natural.
Description
Technical field
The invention belongs to field of Computer Graphics more particularly to a kind of method for building up of indoor smog diffusion model.
Background technology
In recent years, with the quickening of the increase of population and pace of construction, the density of building is increasing, fire hazard
It greatly increases, fire has become a kind of City Disasters frequently occurred.Since fire has great destructive power and certain
The features such as dangerous, people are difficult to be analyzed and predicted the diffusion process of the intensity of a fire and smog indoors.Therefore, it makes full use of
Computer Simulation fire hazard aerosol fog simulates smog flow direction in certain circumstances and concentration variation, contributes to researcher clear
The variation tendency for grasping to Chu smog in the scene of a fire, to instructing fire safety evaluating, fire attack work that all there is very important reality
Meaning.
The simulation of smog is the important component of fire scenario simulation.Smog is a kind of common natural phenomena, is had
The features such as geometry is not fixed, and motion feature is irregular, and movement is affected by the surrounding environment.Therefore, to the simulation of smog one
It is directly the difficult point and hot spot of field of Computer Graphics.
The method that domestic and foreign scholars constantly probe into smoke simulation, it is proposed that based on the method for dividing shape, Kernel-based methods texture
Method, the method based on particIe system, the method based on cellular automaton and the method based on physical model, wherein by answering extensively
With being method based on particIe system and based on the method for physical model with most important method.
Method based on particIe system is put forward for the first time by Reeves, for simulating irregular fuzzy objective, basic thought
It is that smog is regarded as particle cluster, particle cluster is molecular by numerous grains, is simulated by drawing the movement locus of each particle
The motion process of smog.This method can fully demonstrate the dynamic and randomness of irregularly shaped object, and thought is fairly simple,
It is easy to implement, but the geometric properties of particle are too simple, the object range that can be constructed is also very limited, and the result simulated is not
It is enough true.
Method based on physical model is the physical characteristic according to smog, in conjunction with the correlation theory in Fluid Mechanics Computation,
With mechanical model Na Wei-Stokes (Navier-Stokes) equation group of fluid motion, abbreviation N-S equations, to describe cigarette
The diffusion process of mist.This method more can accurately simulate true effect, make smog movement it is true to nature, naturally, still
N-S equations are solved on grid, need prodigious calculation amount, than relatively time-consuming, so that cannot meet smoke simulation reality well
The requirement of when property.Further research with the rapid development and brainstrust of computer hardware to GPU, people start with GPU
Concurrency and programmability improve the solution efficiency of N-S equations, preferably meet the requirement of real-time.
Currently, most of methods based on physical model are confined to simulation smoke itself, seldom consider smog in specific field
Spread condition in scape；When establishing smoke model, numerical dissipation caused by N-S equations is solved in order to make up, people are usual
Whirlpool restraining force is added in flow field, carrys out generated cyclonic effect in simulation smoke motion process, but to smog diffusion
Detailed simulation is not deep enough.
In conclusion the method use scope based on particIe system is limited, the smog simulated is untrue；Based on physics mould
The method of type is confined to simulation smoke itself, does not account for the diffusion process of smog under specific boundary conditions, meanwhile, it is building
When mould, it is contemplated that external force factor it is not comprehensive.Therefore, it currently needs to establish a kind of new smog diffusion model, it is above-mentioned to solve
Problem.
Invention content
The present invention proposes a kind of method for building up of the smog diffusion model based on boundary constraint, makes up previous smoke model
The external force factor ignored, it is contemplated that wind-force, influence of the air drag to smog movement, and extension is artificially added, not
In the case of distortion, the real-time of smog diffusion simulations is improved.Simultaneously, it is contemplated that the limitation of indoor boundary condition, to boundary
Physical quantity handled, keep the process that the smog simulated is spread indoors more true and natural.Below from based on boundary
Two aspects of solution of the structure of the smog diffusion model of constraint and the smog diffusion model based on boundary constraint carry out specifically
It is bright.
1) structure of the smog diffusion model based on boundary constraint
In the case of divulging information indoors, during smog is spread indoors, can inevitably it be influenced by wind field.
In order to realize the interactive process of wind and smog, wind field space is divided into several uniform grids by the present invention, in each grid
The size and Orientation of wind speed is constant, then corresponding wind-force calculation formula is：
Wherein ρairFor atmospheric density, v is wind speed.
When smog moves in wind field, frictional force is generated with air, i.e., is influenced by air drag, although the power is to cigarette
The influence of mist diffusion is smaller, but in order to enhance the sense of reality of smog diffusion, and air resistance is added in the present invention in smog diffusion model
Power.According to Ni Gulazi trial curves, fluid resistance is directly proportional to speed how many power to depend on flow velocity.The speed of smog diffusion
It is relatively small, therefore the first power for obtaining air drag and smog speed u is directly proportional, i.e.,
fairdrag=-ku
Wherein k is resistance coefficient, and u is the speed of smog.
Smog moves in space indoors, in addition to by gravity, the influence of heat buoyancy there is also diffusion phenomena, macroscopic view
On show as smog and be diffused into the small place of density from the big place of density, i.e., spread to the opposite direction of density gradient, then
Automatically the entire interior space is filled.Therefore, it is necessary to a power to describe the diffusion process of smog, accelerate the diffusion of smog indoors
Process, improves the real-time of smoke simulation, and corresponding formula is：
Wherein s is diffusion coefficient,For the gradient of density contrast, negative sign indicates opposite with density gradient direction.
Since interior is a restricted clearance, boundary condition is complicated, and smog inevitably connects during diffusion
Wall is contacted, therefore, it is necessary to fully consider the motion conditions after smog touches wall, to the density of boundary, temperature, pressure
Etc. physical quantitys handled accordingly, corresponding formula is：
Wherein φ indicates speed, and pressure, the physical quantitys such as density and temperature, a, b are constant, and c is given boundary value.
2) solution of the smog diffusion model based on boundary constraint
Since interior is a restricted clearance, boundary condition is complicated, and smog inevitably connects during diffusion
It contacts wall, therefore, needs the corresponding borderline physical quantity of update after each update solves the physical quantity in domain.For boundary
For the speed of upper smog, after smog encounters wall, smog should not have the velocity component across wall, perpendicular to the speed of wall
It is 0 to spend component.The density of smog on boundary, temperature, pressure are physical quantitys corresponding to the grid cell adjacent with the boundary element
Average value.
It is three-dimensional, therefore, the present invention expands to two-dimentional smoke simulation algorithm since domain is indoor space
Three-dimensional, it is using finite difference method, the interior space is discrete as a series of small cubical set, and according to N-S equations
To the speed of each cube grid, density, temperature is calculated, and each time step iteration is primary, until reaching user
The maximum time of setting.
The present invention simplifies the physical model of smog using Helmholtz-Hodge decomposition theorems, is broken down into advection, outside
Power, diffusion project four, select suitable method to solve this four parts respectively, reduce the complexity of equation solution, reach
The effect that smog is simulated in real time.When solving advective term, using Semi Lagrangian scheme, by Euler method and Lagrangian method
Advantage combines, and introduces the thought of particIe system, regards the flow field grid in Euler method as particle in Lagrangian method
Particle, by the method reversely tracing particle of back-flow velocity field, overcome Euler method time step it is excessive when unstability, together
When ensure that the sense of reality of simulation.External force term is solved using forward difference method, diffusion term is solved using the method for implicit iterative
With projection item.
Description of the drawings
Fig. 1 is the flow chart that the present invention solves indoor smog diffusion model；
Fig. 2 is the present invention schematic diagram discrete to the interior space；
Fig. 3 is the schematic diagram that the present invention solves advective term；
Fig. 4 is the schematic diagram that the present invention solves diffusion term；
Fig. 5 is the schematic diagram of borderline region of the present invention；
Fig. 6 is the present invention design sketch that smog is spread in resolution ratio is 124*124*84 grid cells.
Specific implementation mode
To make the objectives, technical solutions, and advantages of the present invention clearer, with reference to specific step and attached drawing,
The present invention is described in more detail.
The present invention proposes a kind of method for building up of the smog diffusion model based on boundary constraint, and the model is in original smog
On the basis of model, wind-force, air drag and extension are added in external force f, simultaneously, it is contemplated that the limitation of indoor boundary condition,
The physical quantity of boundary is handled, indoor smog diffusion simulations system is designed and Implemented, structure is specifically included and is based on boundary
The smog diffusion model of constraint, solves smog diffusion model and three parts are drawn in indoor smog diffusion.
1) the smog diffusion model based on boundary constraint is built
By analyzing the movement characteristic of smog, diffusion process and the principal element for influencing smog movement, use is incompressible
N-S equation groups establish indoor smog diffusion physical model.Since N-S equations are partial differential equation, the equation is being solved
In the process, it will produce a large amount of numerical dissipation.Therefore, in order to ensure the sense of reality of smoke simulation, the present invention is in original N-S equations
On the basis of group, by wind-force, the gravity of smog, heat buoyancy, whirlpool restraining force, air drag and extension are as external force, supplement
The details characteristic lost in smog diffusion process, the cyclonic effect of enhancing smog diffusion.The gravity of smog is related with density, and heat is floating
Power is related with temperature, thus the present invention takes into account density, temperature with the variation of velocity field when establishing model.Meanwhile it examining
The limitation for considering indoor boundary condition handles the physical quantity of boundary, the mistake for making the smog simulated spread indoors
Cheng Gengwei is true and natural.
fwindFor wind-force, ρairFor atmospheric density, v is wind speed.
fairdrag=-ku (1.2)
fairdragFor air drag, k is resistance coefficient, and u is smog speed.
fdiffFor extension, s is diffusion coefficient,For the gradient of density contrast.
fbouy=-α ρ y+ β (T-Tamb)y (1.4)
fbouyFor heat buoyancy, equation the right first item indicates that the gravity of smog, Section 2 indicate temperature and ambient enviroment temperature
Buoyancy caused by the temperature difference between degree.Y=(0,1,0) indicates that direction straight up, α, β are respectively intended to control density and temperature
To the influence degree of heat buoyancy, ρ is the density of smog, and T is the temperature of smog, TambFor environment temperature.
fconf=ε h (N × ω) (1.5)
fconfFor whirlpool restraining force, ε indicates that the intensity of whirlpool restraining force, h are grid spacing, curlCurl
Normal line vectorThe unit normal vector of curl
Above five power is added in external force f and is obtained：
F=fbouy+fconf+fwind+fairdrag+fdiff (1.6)
Formula (1.1), (1.2), (1.3), (1.4), (1.5), (1.6), (1.7), (1.8), (1.9), (1.10),
(1.11) indoor smog diffusion model is collectively formed, wherein u is the speed of smog, and p is pressure, and v is viscosity coefficient, and f is external force,
ρ is the density of smog, and T is the temperature of smog, ρ0,T0The respectively extraneous density supplemented to mesh space, temperature.Y=(0,1,
0) indicate that direction straight up, α, β are respectively intended to control density and temperature to the influence degree of heat buoyancy, TambFor environment temperature
Degree, φ indicate speed, and pressure, the physical quantitys such as density and temperature, a, b are constant, and c is given boundary value, and ε indicates whirlpool constraint
The intensity of power, h are grid spacing, curlThe normal line vector of curlThe unit normal vector of curl
Formula (1.1), (1.2), (1.3), (1.4), (1.5) describe the circular of external force suffered by smog.
Formula (1.6) describes external force suffered by smog, respectively heat buoyancy, whirlpool restraining force, wind-force, air drag and expansion
Dissipate power.
Formula (1.7) and formula (1.8) are respectively mass-conservation equation and momentum conservation equation.
Formula (1.9) and formula (1.10) respectively describe smog density p and temperature T with velocity field motion process.
Formula (1.11) describes the speed of borderline smog, temperature, the computational methods of density and pressure.
According to Helmholtz-Hodge decomposition theorems, it is 0 that any one vector field w, which can be broken into a divergence,
The sum of the gradient of vector field u and a scalar field p.
I.e.Wherein
Both sides are same to take divergence to obtain,
Define a projection operatorOne vector field w is projected on its component u without divergence.I.e.
So full scale equation can be reduced to：
Algorithm flow：Three (advective term, external force term, diffusion terms) in the bracket of the right are calculated successively, obtain velocity field w,
Then it projects, that is, solves equationAsk p, last basisObtain the vector field u that divergence is 0.
2) smog diffusion model is solved
Solve thought：As shown in Figure 1, center first indoors initializes mist source, the initial velocity of smog is set, it is close
Degree and temperature, secondly when calculating the speed of smog movement, for a new time point, respectively to the speed at a upper time point
Apply advective term, external force term successively, the influence of diffusion term and projection item finally solves the close of smog with the method for solving advective term
Spend field and temperature field.Whole process often calculates once, and time step adds 1, until program exits.Below to main solution
Step is described in detail.
2.1) mesh generation
The present invention is indoor smog diffusion simulations, and domain is three-dimensional, as shown in Fig. 2, using collocated grid pair
After the discrete region, a series of small cube set will be obtained.Regard each cube grid cell as a particle, speed
Degree u, pressure p, temperature T, the physical quantitys such as density p are all defined on the center of grid cell.
2.2) advective term solves
Advective term refers to speed, density, temperature on the direction of fluid velocity field by fluid etc. and fluid relevant one
A little physical quantitys are from a localized transmissions to another place.
The thought for introducing particIe system calculates advective term using Semi -Lagrangian method：Particle need not be calculated when current
Between section which position be moved to, but regard each cube grid cell as a particle, chased after from each grid cell inverse time
The position of its last moment is found by velocity field, and the amount of the position is copied into current location in track Particles Moving track：
Q (x, t+ Δ t)=q (x-u (x, t) Δ t, t)
Q indicates that speed, density, temperature or any physical quantity carried by fluid, x indicate the current space of particle in above formula
Position, u indicate the speed of fluid.
As shown in figure 3, circles mark is the particle for needing to calculate advection, X is to track the velocity field of the particle inverse time to arrive
The position of last moment.Because cannot ensure that each inverse time tracking can reach grid element center, interpolation calculation is needed.For
Two-dimensional grid needs the value progress bilinear interpolation to 4 mesh points adjacent around the position to obtain approximation.For three-dimensional
Grid selects to carry out Tri linear interpolation as sampled point with eight vertex of the nearest grid cell in the position and obtain approximation.
2.3) external force term solves
External force term is solved, first respectively to heat buoyancy fbouy, whirlpool restraining force fconf, wind-force fwind, air drag fairdrag
With extension fdiffExternal force f is solved and merged into, equation is secondly solvedIt is solved using forward difference method.Speed
U is in x-axis, and y-axis, the speed in z-axis direction is respectively u, v, w then to be had for (i, j, k) a cube grid cell
Abbreviation obtains：
2.4) diffusion term solves
Diffusion term indicates the exchange process of matter energy between adjacent mesh：It flows into and flows out.It can be expressed as with equation：
As shown in figure 4, in two-dimensional grid, the quality of grid cell (i, j) be flowed into itself due to adjacent cells and
Increase, also due to flow out to its adjacent cells and reduce, with the thought, can in analogy to three-dimensional grid,
Speed u is in x-axis, and the speed of y-axis, z-axis direction is respectively u, v, w, for (i, j, k) a cube grid cell
For, x-axis direction speed u explicit solutions obtain：
Abbreviation obtains：
When time step Δ t is excessive or viscosity v is excessive, this method is unstable, so being asked using implicit method
It solves：
Abbreviation obtains：
Assuming that grid separation delta x=Δ y=Δ z, using Jucobi iterative method to above-mentioned equation solution, when progress (n+1)th
When secondary iteration：
2.5) projection item solves
Projection item is solved, pressure Poisson's equation is actually solvedPressure Poisson's equation is shaped like Ax=b
Large sparse matrix equation group, wherein x are physical quantity to be asked, and the pressure p being equivalent in pressure Poisson's equation, A is a matrix,
The Laplace operator being equivalent in pressure Poisson's equationB is a constant vector, the speed being equivalent in pressure Poisson's equation
Spend the divergence of fieldTo pressure Poisson's equationIt is obtained using centered difference formal expansion：
Assuming that grid separation delta x=Δ y=Δ z, solve above-mentioned equation using Jucobi iterative method, work as progress
When (n+1)th iteration：
According to pressure Poisson's equationThe pressure p solved substitutes into formulaTo acquire divergence
For 0 velocity field u.By formulaInIt is obtained using centered difference formal expansion：
2.6) boundary condition
Since interior is a limited space, so, in solving model, opposite side is required for after each time step
The physical quantitys such as the speed at boundary, density, temperature and pressure are updated.As shown in figure 5, the interior space is divided into (M+2) * (N
+ 2) * (O+2) a grid cell.Boundary condition is as follows：
1. the boundary condition on 8 vertex
| X (0,0,0)=(X (1,0,0)+X (0,1,0)+X (0,0,1))/3 |
| X (M+1,0,0)=(X (M, 0,0)+X (M+1,1,0)+X (M+1,0,1))/3 |
| X (0,0, O+1)=(X (0,0, O)+X (0,1, O+1)+X (1,0, O+1))/3 |
| X (M+1,0, O+1)=(X (M, 0, O+1)+X (M+1,1, O+1)+X (M+1,0, O))/3 |
| X (0, N+1,0)=(X (0, N, 0)+X (1, N+1,0)+X (0, N+1,1))/3 |
| X (0, N+1, O+1)=(X (0, N+1, O)+X (1, N+1, O+1)+X (0, N, O+1))/3 |
| X (M+1, N+1,0)=(X (M, N+1,0)+X (M+1, N, 0)+X (M+1, N+1,1))/3 |
| X (M+1, N+1, O+1)=(X (M, N+1, O+1)+X (M+1, N, O+1)+X (M+1, N+1, O))/3 |
2. removing the boundary condition of the rib on 8 vertex
| X (i, 0,0)=(X (i, 1,0)+X (i, 0,1))/2 |
| X (i, 0, O+1)=(X (i, 1, O+1))+X (i, 0, O))/2 |
| X (i, N+1,0)=(X (i, N, 0)+X (i, N+1,1))/2 |
| X (i, N+1, O+1)=(X (i, N+1, O)+X (i, N, O+1))/2 |
| X (0, j, 0)=(X (1, j, 0)+X (0, j, 1))/2 |
| X (0, j, O+1)=(X (0, j, O)+X (1, j, O+1))/2 |
| X (M+1, j, 0)=(X (M, j, 0)+X (M+1, j, 1))/2 |
| X (M+1, j, O+1)=(X (M+1, j, O)+X (M, j, O+1))/2 |
| X (0,0, k)=(X (1,0, k)+X (0,1, k))/2 |
| X (0, N+1, k)=(X (0, N, k)+X (1, N+1, k))/2 |
| X (M+1,0, k)=(X (M, 0, k)+X (M+1,1, k))/2 |
| X (M+1, N+1, k)=(X (M, N+1, k)+X (M+1, N, k))/2 |
3. removing the boundary condition of rib
| U (0, i, j)=- u (1, i, j) | Y (0, i, j)=Y (1, i, j) |
| U (M+1, i, j)=- u (M, i, j) | Y (M+1, i, j)=Y (M, i, j) |
| V (i, 0, j)=- v (i, 1, j) | Y (i, 0, j)=Y (i, 1, j) |
| V (i, N+1, j)=- v (i, N, j) | Y (i, N+1, j)=Y (i, N, j) |
| W (i, j, 0)=- w (i, j, 1) | Y (i, j, 0)=Y (i, j, 1) |
| W (i, j, O+1)=- w (i, j, O) | Y (i, j, O+1)=Y (i, j, O) |
Wherein X indicates speed, pressure, the physical quantitys such as temperature and density, other physical quantitys of Y expressions in addition to speed, u tables
Showing the speed in x-axis direction, v indicates the speed on y-axis direction, and w indicates the speed on z-axis direction, 1≤i≤M, 1≤j≤N, 1
≤ k≤O, (i, j, k) indicate the label of grid cell.
3) indoor smog diffusion is drawn
It is that the velocity field of smog and density field are plotted on screen that indoor smog diffusion, which is drawn,.In general, smog
Density is directly proportional to transparency, i.e., density is bigger, and transparency is lower, and the value of transparency is bigger.According to this principle, base of the present invention
Smog is rendered in the density field of smog, the transparent of corresponding points color is replaced with the density approximation on vertex in each discrete grid block
Degree, from x, tri- directions y, z traverse the point in grid, and step-length is grid spacing, draftings mode is quad patch, each
The color on discrete grid block vertex is determined by the density value of respective point, i.e., is led to come the deep or light variation of approximate simulation smog with transparency
It crosses and draws a series of connected quadrangles, simulate the variation of smoke density, to realize the simulation of smog diffusion effect.
Under 64 bit manipulation system of Intel Xeon E5 2.5GHz CPU and Win7, C Plus Plus and open figure are used
Shape interface OpenGL is programmed in VS2010 realizes smog diffusion simulations, and Fig. 6 is the effect of smog diffusion.
Claims (2)
1. a kind of method for building up of indoor smog diffusion model includes the following steps it is characterized in that being：
Step 1, smog diffusion model of the structure based on boundary constraint
On the basis of original N-S equation groups, by wind-force, the gravity of smog, heat buoyancy, whirlpool restraining force, air drag and expansion
Power is dissipated as external force, supplements the details characteristic lost in smog diffusion process；The gravity of smog is related with density simultaneously, heat buoyancy
It is related with temperature, density, temperature are taken into account with the variation of velocity field when establishing model；Simultaneously in view of indoor perimeter strip
The limitation of part handles the physical quantity of boundary, keep the process that the smog simulated is spread indoors more true and from
So,
Wherein, fwindFor wind-force, ρairFor atmospheric density, v is wind speed,
fairdrag=-ku (1.2)
Wherein, fairdragFor air drag, k is resistance coefficient, and u is smog speed,
Wherein, fdiffFor extension, s is diffusion coefficient,For the gradient of density contrast,
fbouy=-α ρ y+ β (T-Tamb)y (1.4)
Wherein, fbouyFor heat buoyancy, equation the right first item indicates that the gravity of smog, Section 2 indicate temperature and ambient enviroment temperature
Buoyancy caused by the temperature difference between degree, y=(0,1,0) indicate that direction straight up, α, β are respectively intended to control density and temperature
To the influence degree of heat buoyancy, ρ is the density of smog, and T is the temperature of smog, TambFor environment temperature.
Wherein, fconf=ε h (N × ω) (1.5)
Fconf is whirlpool restraining force, and ε indicates that the intensity of whirlpool restraining force, h are grid spacing, curlCurl
Normal line vectorThe unit normal vector of curl
Above five power is added in external force f and is obtained：
F=fbouy+fconf+fwind+fairdrag+fdiff (1.6)
Formula (1.1), (1.2), (1.3), (1.4), (1.5), (1.6), (1.7), (1.8), (1.9), (1.10), (1.11) are total
With indoor smog diffusion model is constituted, wherein u is the speed of smog, and p is pressure, and v is viscosity coefficient, and f is external force, and ρ is smog
Density, T be smog temperature, ρ0, T0The respectively extraneous density supplemented to mesh space, temperature.Y=(0,1,0) indicates perpendicular
Straight upwardly direction, α, β are respectively intended to control density and temperature to the influence degree of heat buoyancy, TambFor environment temperature, φ is indicated
The physical quantitys such as speed, pressure, density and temperature, a, b are constant, and c is given boundary value, and ε indicates the intensity of whirlpool restraining force,
H is grid spacing, curlThe normal line vector of curlThe unit normal vector of curl
Formula (1.1), (1.2), (1.3), (1.4), (1.5) describe the circular of external force suffered by smog,
Formula (1.6) describes external force suffered by smog, respectively heat buoyancy, whirlpool restraining force, wind-force, air drag and diffusion
Power,
Formula (1.7) and formula (1.8) are respectively mass-conservation equation and momentum conservation equation,
Formula (1.9) and formula (1.10) respectively describe smog density p and temperature T with velocity field motion process,
Formula (1.11) describes the speed of borderline smog, temperature, the computational methods of density and pressure,
According to Helmholtz-Hodge decomposition theorems, any one vector field w can be broken into the vector that a divergence is 0
The sum of the gradient of the u and scalar field p in field,
I.e.Wherein
Both sides are same to take divergence to obtain,
Define a projection operatorOne vector field w is projected on its component u without divergence.I.e.
So full scale equation can be reduced to：
Three (advective term, external force term, diffusion terms) in the bracket of the right are calculated successively, are obtained velocity field w, are then projected, that is, solve
EquationAsk p, last basisObtain the vector field u that divergence is 0.
Step 2 solves smog diffusion model
Center first indoors initializes mist source, sets the initial velocity of smog, density and temperature, is secondly calculating smog
When the speed of movement, for a new time point, advective term applied successively to the speed at a upper time point respectively, external force term,
The influence of diffusion term and projection item finally solves density field and the temperature field of smog with the method for solving advective term.Whole process
It often calculates once, time step adds 1, until program exits.
2. the method for building up of indoor smog diffusion model as described in claim 1, it is characterized in that being, step 2 is specially：
2.1) mesh generation
Using collocated grid to the discrete region after, a series of small cube set will be obtained；By each cube grid
Unit regards a particle as, speed u, pressure p, temperature T, and the physical quantitys such as density p are all defined on the center of grid cell；
2.2) advective term solves
Advective term refers to speed, density, temperature on the direction of fluid velocity field by fluid etc. and some relevant objects of fluid
Reason amount is from a localized transmissions to another place；The thought for introducing particIe system calculates advective term using Semi -Lagrangian method：
Regard each cube grid cell as a particle, from tracking Particles Moving of each grid cell inverse time track, passes through speed
The position of its last moment is found in field, and the amount of the position is copied to current location：
Q (x, t+ Δ t)=q (x-u (x, t) Δ t, t)
Wherein, q indicates that speed, density, temperature or any physical quantity carried by fluid, x indicate the current spatial position of particle,
U indicates the speed of fluid；
2.3) external force term solves
External force term is solved, first respectively to heat buoyancy fbouy, whirlpool restraining force fconf, wind-force fwind, air drag fairdragAnd expansion
Dissipate power fdiffExternal force f is solved and merged into, equation is secondly solvedIt is solved using forward difference method, speed u is in x
Axis, y-axis, the speed in z-axis direction is respectively u, v, w then to be had for (i, j, k) a cube grid cell
Abbreviation obtains：
2.4) diffusion term solves
Diffusion term indicates the exchange process of matter energy between adjacent mesh：It flows into and flows out, can be expressed as using equation：
In two-dimensional grid, the quality of grid cell (i, j) increases since adjacent cells are flowed into itself, also due to outflow
Reduced to its adjacent cells, with the thought, can in analogy to three-dimensional grid,
For speed u in x-axis, the speed of y-axis, z-axis direction is respectively u, v, w, and (i, j, k) a cube grid cell is come
It says, x-axis direction speed u explicit solutions obtain：
Abbreviation obtains：
When time step Δ t is excessive or viscosity v is excessive, this method is unstable, so being solved using implicit method：
Abbreviation obtains：
Assuming that grid separation delta x=Δ y=Δ z are changed using Jucobi iterative method to above-mentioned equation solution when carrying out (n+1)th time
Dai Shi：
2.5) projection item solves
Projection item is solved, pressure Poisson's equation is actually solvedPressure Poisson's equation is shaped like the big of Ax=b
Type sparse matrix equation group, wherein x is physical quantity to be asked, and the pressure p being equivalent in pressure Poisson's equation, A is a matrix,
The Laplace operator being equivalent in pressure Poisson's equationB is a constant vector, the speed being equivalent in pressure Poisson's equation
Spend the divergence of fieldTo pressure Poisson's equationIt is obtained using centered difference formal expansion：
Assuming that grid separation delta x=Δ y=Δ z, solve above-mentioned equation using Jucobi iterative method, when carry out n-th+
When 1 iteration：
According to pressure Poisson's equationThe pressure p solved substitutes into formulaIt is 0 to acquire divergence
Velocity field u.By formulaInIt is obtained using centered difference formal expansion：
2.6) boundary condition
In solving model, the speed to boundary, density, the physical quantitys such as temperature and pressure are required for after each time step
It is updated, the interior space is divided into (M+2) * (N+2) * (O+2) a grid cell, and boundary condition is as follows：
1. the boundary condition on 8 vertex
X (0,0,0)=(X (1,0,0)+X (0,1,0)+X (0,0,1))/3
X (M+1,0,0)=(X (M, 0,0)+X (M+1,1,0)+X (M+1,0,1))/3
X (0,0, O+1)=(X (0,0, O)+X (0,1, O+1)+X (1,0, O+1))/3
X (M+1,0, O+1)=(X (M, 0, O+1)+X (M+1,1, O+1)+X (M+1,0, O))/3
X (0, N+1,0)=(X (0, N, 0)+X (1, N+1,0)+X (0, N+1,1))/3
X (0, N+1, O+1)=(X (0, N+1, O)+X (1, N+1, O+1)+X (0, N, O+1))/3
X (M+1, N+1,0)=(X (M, N+1,0)+X (M+1, N, 0)+X (M+1, N+1,1))/3
=(X (M, N+1, O+1)+X (M+1, N, O+1)+X (M+1, N+1, O)/3
2. removing the boundary condition of the rib on 8 vertex
X（I, 0,0）=（X (i, 1,0）+X（I, 0,1））/2
X (i, 0, O+1)=(X (i, 1, O+1))+X (i, 0, O))/2
X (i, N+1,0)=(X (i, N, 0)+X (i, N+1,1))/2
X (i, N+1, O+1)=(X (i, N+1, O)+X (i, N, O+1))/2
X (0, j, 0)=(X (1, j, 0)+X (0, j, 1))/2
X (0, j, O+1)=(X (0, j, O)+X (1, j, O+1))/2
X (M+1, j, 0)=(X (M, j, 0)+X (M+1, j, 1))/2
X (M+1, j, O+1)=(X (M+1, j, O)+X (M, j, O+1))/2
X (0,0, k)=(X (1,0, k)+X (0,1, k))/2
X (0, N+1, k)=(X (0, N, k)+X (1, N+1, k))/2
X (M+1,0, k)=(X (M, 0, k)+X (M+1,1, k))/2
X (M+1, N+1, k)=(X (M, N+1, k)+X (M+1, N, k))/2
3. removing the boundary condition of rib
U (0, i, j)=- u (1, i, j) Y (0, i, j)=Y (1, i, j)
U (M+1, i, j)=- u (M, i, j) Y (M+1, i, j)=Y (M, i, j)
V (i, 0, j)=- v (i, 1, j) Y (i, 0, j)=Y (i, 1, j)
V (i, N+1, j)=- v (i, N, j) Y (i, N+1, j)=Y (i, N, j)
W (i, j, 0)=- w (i, j, 1) Y (i, j, 0)=Y (i, j, 1)
W (i, j, O+1)=- w (i, j, O) Y (i, j, O+1)=Y (i, j, O)
Wherein, X indicates speed, pressure, the physical quantitys such as temperature and density, other physical quantitys of Y expressions in addition to speed, u expressions x
Speed in axis direction, v indicate that the speed on y-axis direction, w indicate the speed on z-axis direction, 1≤i≤M, 1≤j≤N, 1≤k
≤ O, (i, j, k) indicate the label of grid cell.
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