Simulação Física - Dinâmica de Fluidos

Simulacao FısicaDinamica de Fluidos

Waldemar [email protected]

Tecgraf, DI/PUC-Rio

27 de Setembro de 2016

W. Celes, DI/PUC-Rio Dinamica de Fluidos 1

Sumario

Bibliografia:

I Physics Based AnimationErleben et al. (2005)

I Fast Fluid Dynamics Simulation on the GPUHarris (2004)

I Stable FluidsStam (1999)

I Particle-based fluid simulation for interactive applicationsMuller et al. (2003)

I Smoothed particle hydrodynamics: a meshfree particle methodLiu and Liu (2003)


Dinamica de fluidosUm fluido e descrito por campos de valores

I velocidade: u

I pressao: p

I temperatura: T

I massa especıfica (density): ρ

I etc.

Simular um fluido significadeterminar a evolucao doscampos de valores no tempo

I Em especial, o campo develocidade


Equacao do movimentoMudanca de temperatura de um volume do fluido em movimento

I Uma partıcula do contınuo deste volumetem coordenadas: [x(t), y(t), z(t)]

I A mudanca no tempo do campotemperatura e dada por: T : <4 → <

A derivada total em relacao ao tempo e dada por:

dT

dt=

∂T

∂t+∂T

∂x

dx

dt+∂T

∂y

dy

dt+∂T

∂z

dz

dt

Com u = [ u v w ]T , tem-se:

dT

dt=

∂T

∂t+∂T

∂xu +

∂T

∂yv +

∂T

∂zw

I que descreve a mudanca de temperatura do fluido, seguindo o fluido(derivada material ou substancial)






dT

dt=

∂T

∂t+∂T

∂x

dx

dt+∂T

∂y

dy

dt+∂T

∂z

dz

dt


dT

dt=

∂T

∂t+∂T

∂xu +

∂T

∂yv +

∂T

∂zw







dT

dt=

∂T

∂t+∂T

∂x

dx

dt+∂T

∂y

dy

dt+∂T

∂z

dz

dt


dT

dt=

∂T

∂t+∂T

∂xu +

∂T

∂yv +

∂T

∂zw



Operador substancial

O operador substancial de um fluxo u:

d

dt=

∂

∂t+ u.∇

I Aplicado a um valor escalar ou a um valor vetorialcomponente a componente

Assim, a segunda Lei de Newton em termos de derivadasubstancial por unidade de volume e:

ρd

dtu = f

ρ

(∂

∂t+ u.∇

)u = f


Operador substancial

O operador substancial de um fluxo u:

d

dt=

∂

∂t+ u.∇

I Aplicado a um valor escalar ou a um valor vetorialcomponente a componente

Assim, a segunda Lei de Newton em termos de derivadasubstancial por unidade de volume e:

ρd

dtu = f

ρ

(∂

∂t+ u.∇

)u = f


Equacao de Euler para fluidosConsiderando o gradiente de pressao como forca interna:

ρ

(∂

∂t+ u.∇

)u = −∇p

Acrescido da equacao para conservacao da massa (fluidosincompressıveis):

∇ .u = 0

diz-se que velocidade e um campo vetorial com divergencia livre

I Operador gradiente (aplicado a um valor escalar)

∇p =[

∂p∂x

∂p∂y

∂p∂z

]T

I Operador divergente (aplicado a um valor vetorial)

∇ .u =∂u

∂x+∂v

∂y+∂w

∂z


Equacao de Euler para fluidosConsiderando o gradiente de pressao como forca interna:

ρ

(∂

∂t+ u.∇

)u = −∇p

Acrescido da equacao para conservacao da massa (fluidosincompressıveis):

∇ .u = 0

diz-se que velocidade e um campo vetorial com divergencia livre

I Operador gradiente (aplicado a um valor escalar)

∇p =[

∂p∂x

∂p∂y

∂p∂z

]T

I Operador divergente (aplicado a um valor vetorial)

∇ .u =∂u

∂x+∂v

∂y+∂w

∂z


Equacao de Navier-StokesConsidera a viscosidade do fluido (e forca externa)

ρ

(∂

∂t+ u.∇

)u = −∇p + µ∇2u + f

I ρ: massa especıfica (density)I µ: viscosidade dinamica

I Viscosidade cinematica: ν = µ/ρ

I Operador Laplaciano (aplicado a um valor escalar):

∇2u =∂2u

∂x2+∂2u

∂y2+∂2u

∂z2

I Equacoes de Poisson:

∇2x = b


Formulacao EulerianaGrade regular 2D

Gradiente:

∇p =[

pi+1,j−pi−1,j

2δpi,j+1−pi,j−1

2δ

]

Divergente:

∇ .u =ui+1,j − ui−1,j

2δ+

vi ,j+1 − vi ,j−1

2δ

Laplaciano:

∇2u =ui+1,j − 2ui ,j + ui−1,j

δ2+

ui ,j+1 − 2ui ,j + ui ,j−1

δ2

=ui+1,j + ui−1,j + ui ,j+1 + ui ,j−1 − 4ui ,j

δ2


Solucao da equacao de Navier-StokesEquacao de Navier-Stokes

ρ

(∂

∂t+ u.∇

)u = −∇p + µ∇2u + f

∂u

∂t= −(u.∇) u− 1

ρ∇p + ν∇2u +

1

ρf

Decomposicao de Helmholtz-HodgeI Considerando um domınio D, sendo seu contorno (∂D) suave

e diferenciavel com normal n, um campo vetorial qualquer, w,pode ser decomposto na soma de dois campos vetoriais:

I Um campo vetorial divergente livre, vI Sendo v . n = 0 em ∂D

I O gradiente de um campo escalar, p

w = v +∇p


Solucao da equacao de Navier-StokesEquacao de Navier-Stokes

ρ

(∂

∂t+ u.∇

)u = −∇p + µ∇2u + f

∂u

∂t= −(u.∇) u− 1

ρ∇p + ν∇2u +

1

ρf

Decomposicao de Helmholtz-HodgeI Considerando um domınio D, sendo seu contorno (∂D) suave

e diferenciavel com normal n, um campo vetorial qualquer, w,pode ser decomposto na soma de dois campos vetoriais:

I Um campo vetorial divergente livre, vI Sendo v . n = 0 em ∂D

I O gradiente de um campo escalar, p

w = v +∇p


Solucao da equacao de Navier-Stokes

Procedimento:

I Dado o estado inicial, u e p, calcula velocidade divergente: w

I Calcula campo de pressao

w = u +∇pAplicando o operador divergente nos dois lados:

∇ .w = ∇ . (u +∇p) = ∇ .u +∇2p

∇2p = ∇ .w

I Converte a velocidade resultante numa divergente livreI Para atender a equacao de conservacao de massa

u = w −∇p


Solucao da equacao de Navier-StokesComo calcular w

Operador de projecao PI Projeta vetor divergente w no seu divergente livre v

Pw = Pu + P(∇p)

como Pw = Pu = u, tem-se: P(∇p) = 0Aplicando o operador de projecao na equacao de Navier-Stokes:

P∂u

∂t= P

(−(u.∇) u− 1

ρ∇p + ν∇2u + f

)

∂u

∂t= P

(−(u.∇) u + ν∇2u + f

)

I Adveccao, difusao (dissipacao) e aplicacao da forca

S(u) = P F D A(u)



Termo de adveccao

I Como as partıculas do fluidos sao transportadas pelo campode velocidade do proprio fluido

Transporte de cada partıcula do fluidoI Metodo de Euler explıcito: r(t + h) = r(t) + u(t) h

I Instavel!

I Metodo de Euler implıcito: r(t + h) = r(t) + u(t + h) h

Inversao do problema

I Qual quantidade passara a ocupar uma determinada posicaodo grid?

q(x, t + h) = q(x− u(x, t) h, t)



Termo de adveccao

I Como as partıculas do fluidos sao transportadas pelo campode velocidade do proprio fluido

Transporte de cada partıcula do fluidoI Metodo de Euler explıcito: r(t + h) = r(t) + u(t) h

I Instavel!

I Metodo de Euler implıcito: r(t + h) = r(t) + u(t + h) h

Inversao do problema

I Qual quantidade passara a ocupar uma determinada posicaodo grid?

q(x, t + h) = q(x− u(x, t) h, t)



Termo de difusao

∂u

∂t= ν∇2u

I Metodo explıcito

u(x, t + h) = u(x, t) + ν h∇2u(x, t)

I Metodo implıcito

(I− ν h∇2) u(x, t + h) = u(x, t)


Solucao das equacoes de Poisson

Equacoes de Poisson:

∇2p = ∇ .w(I− ν h∇2) u(x, t + h) = u(x, t)

Metodo de relaxacao

I Iteracao de Jacobi

A x = b, onde A representa o Laplaciano

x(k+1)i ,j =

xki−1,j+xki+1,j+xki,j−1+xki,j+1+α bi,jβ

I Para pressao: x → p, b → ∇ .w, α = −δ2 e β = 4

I Para difusao: x , b → u, α = δ2/νh e β = 4 + α



Termo de forca externa

I Pode-se usar Euler simples:

v(t + h) = v + h f(x, t)



Condicoes de contorno

I Velocidade: deve satisfazer u .n = 0Ao longo de uma borda vertical:

u0,j + u1,j

2= 0 ∀ j ∈ [0,N]

I Para pressao: variacao deve ser nula

p1,j − p0,j

δ= 0



Exemplos (Foster and Metaxas, 1997)

Tipo de objeto un ut T Descricao

Concreto 0 0 T0 Pequena turbulencia

Parede rugosa 0 −ut T0 Grande turbulencia

Plastico liso 0 ut T0 Escoamento laminar

Radiador 0 0 Tuser Conveccao de calor

Janela aberta vuser 0 Tuser Entrada de “vento” quente


Aplicacoes

Injecao de tinta

I Valor escalar passivo: apenas carregado pelo fluido

I Concentracao de tinta (d) sofre adveccao

∂d

∂t= −(u .∇) d

I Adveccao com difusao

∂d

∂t= −(u .∇) d + γ∇2d + S

onde: γ indica coeficiente de difusao e S representa uma fonte


Aplicacoes

Empuxo termico

I Forca devido a variacao da temperatura:

fbuoy = σ (T − T0) j

onde:

I σ: valor escalar constante

I T : campo escalar de temperatura

I T0: temperatura ambiente

I j: direcao vertical


Formulacao LagrangianaSmoothed Particle Hydrodynamics (SPH)

Representacao integral de uma funcaoSeja f (x) uma funcao de campo:

f (x) =

∫

Ωf (x) δ(x− x) dx

onde Ω e o volume da integral e δ e a funcao Delta de Dirac:

δ(x− x) =

1, se x = x

0, se x 6= x


Funcao suavizadora

Substituindo δ por uma “funcao suavizadora” W (x− x, h), tem-se:

< f (x) >=

∫

Ωf (x)W (x− x, h) dx

onde h representa o comprimento caracterıstico

I Define a area de influencia da funcao W

Propriedades de W :

I Par: W (x) = W (−x)

I Normalizada:∫

Ω W (x− x, h) dx = 1

I Convergente: limh→0 W (x− x, h) = δ(x− x)I Compacta: W (x− x, h) = 0, ∀ |x− x| > k h

I A area efetiva da funcao suavizadora e dita domınio de suporte


Representacao integral da derivada de uma funcao

A aproximacao de ∇ . f (x) e dada por:

< ∇ f (x) >=

∫

Ω[∇ f (x)] W (x− x, h) dx

Sabendo-se que: [∇ f ]W = ∇ [f W ]− f ∇W , tem-se:

< ∇ f (x) >=

∫

Ω∇ [f (x)W (x− x, h)] dx−

∫

Ωf (x) ∇W (x− x, h) dx



Teorema da Divergencia

∫

Ω∇ f dΩ =

∫

Sf .n dS Ω

S

n

Logo:

< ∇ f (x) >=

∫

S[f (x)W (x− x, h)] .n dS −

∫

Ωf (x) ∇W (x− x, h) dx



Como W tem suporte compacto, se este nao sobrepor a fronteirado domınio, a integral na superfıcie se anula, ficando com:

< ∇ f (x) >= −∫

Ωf (x)∇W (x− x, h) dx

I A derivada da funcao e “transferida” para a da funcao W

Representacao integral do Laplaciano de uma funcao

I Analogamente:

< ∇2 f (x) >=

∫

Ωf (x) ∇2 W (x− x, h) dx


Aproximacao por partıculasSeja mj a massa da partıcula j :

mj = ∆Vj ρj

A representacao integral pode ser aproximada por:

< f (x) > =

∫

Ωf (x)W (x− x, h) dx

≈N∑

j=1

f (xj)W (x− xj , h) ∆Vj

=N∑

j=1

f (xj)W (x− xj , h)1

ρj(ρj ∆Vj)

=N∑

j=1

mj

ρjf (xj)W (x− xj , h)


SPH: metodo de interpolacao

Valor do campo contınuo e aproximado por media ponderada deamostras na vizinhanca


Aproximacao por partıculasAvaliacao da funcao na posicao da partıcula i :

< f (xi ) >=N∑

j=1

mj

ρjf (xj)W (xi − xj , h)

Gradiente

I Tomado em relacao a partıcula i

< ∇ f (xi ) >=N∑

j=1

mj

ρjf (xj)∇W (xi − xj , h)

Laplaciano

< ∇2 f (xi ) >=N∑

j=1

mj

ρjf (xj)∇2 W (xi − xj , h)


Simulacao de fluido com SPHMuller et al. (2003)

Conservacao de massa

I Considerando o numero de partıcula e a massa de cadapartıculas valores constantes, a equacao de conservacao demassa e naturalmente satisfeita

Conservacao de momento

I Como as partıculas se movem com o fluido, nao e necessario otermo de adveccao da Equacao de Navier-Stokes

ρ∂u

∂t= −∇p + µ∇2u + f


Metodo numerico

Equacao diferencial:

a =du

dt=−∇p + µ∇2u + f

ρ

Integracao Leap-Frog:

vt+h/2i = v

t−h/2i + h ai

xt+hi = xti + h v

t+h/2i


Massa especıfica

Aplicando SPH:

ρi =∑

j

mjρjρj

W (xi − xj , h)

=∑

j

mj W (xi − xj , h)


Forca de pressao

fpressurei = −∇p

Aplicando SPH:

fpressurei = −∑

j

mjpjρj∇W (xi − xj , h)

Para tornar simetrica, pode-se somar as pressoes:

fpressurei = −∑

j

mjpi + pj

2ρj∇W (xi − xj , h)


Forca de pressao

Pressao em cada partıcula

I Lei ideal do gas (usada em Muller et al.)

p = k(ρ− ρ0)

I k: constante do gas (varia com temperatura)I ρ0: massa especıfica em repouso

I Equacao de TaitI Tende a ser mais estavel numericamente

p = B

(ρ

ρ0

γ− 1

), γ = 7

I Monaghan et al.: B = 200gHργ

I Becker et al.: B =ρ0 c2

sγ


Forca de viscosidade

fviscosityi = µ∇2u

Aplicando SPH:

fviscosityi = µ∑

j

mjvjρj∇2W (xi − xj , h)

Para tornar simetrica, pode-se usar a diferenca de velocidade:

fviscosityi = µ∑

j

mjvj − viρj

∇2W (xi − xj , h)


Forca externa

Gravidade:

fexternali = ρ g

I Forcas externas (gravidade, impulso de colisao, etc.) podemser aplicadas diretamente nas partıculas, sem o uso de SPH


Tensao superficialModeladas explicitamente

I Representam as forcas intermoleculares

I Se anulam no interior do domınioI Existem na superfıcie livre

I Na direcao da normal, voltada para o interior do fluido

I Tendem a minimizar a curvatura da superfıcie livre

Proposta de Muller et al.:Campo escalar adicional: “campo de cor”

I Vale 1 nas partıculas e 0 no resto do domınio

cs =∑

j

mjcjρj

W (xi − xj , h)

=∑

j

mj1

ρjW (xi − xj , h)


Tensao superficialA normal da superfıcie e dada pelo gradiente do campo:

n = ∇cs , na direcao do fluido

Curvatura:

κ =−∇2cs|n|

Tensao superficial:

tsurface = σ κn

|n|onde: σ= coeficiente de tensao da interface

Forca de superfıcieI Multiplica tensao por um campo escalar normalizado δs = |n|:

fsurface = σ κn = −σ∇2csn

|n|


Tensao superficial

Proposta de Becker:Modelada como forca de coesao entre as partıculas

dvidt

= − κ

mi

∑

j

mj(xi − xj) W (xi − xj)


Funcoes suavizadoresFuncao de uso geral

I Tem a vantagem de so usar r 2

Wpoly6(r, h) =315

64πh9

(h2 − r2)3, 0 ≤ r ≤ h

0

Funcao para avaliacao da pressaoI Evita agrupamento de partıculas com derivada positiva

para r pequenos

Wspiky (r, h) =15

πh6

(h − r)3, 0 ≤ r ≤ h

0

Funcao para avaliacao da viscosidadeI Evita Laplaciano com valor negativo

Wvisco(r, h) =15

2πh3

− r3

2h3 + r2

h2 + h2r − 1, 0 ≤ r ≤ h

0

Müller et al / Particle-Based Fluid Simulation for Interactive Applications

everywhere else. This field is called color field in the litera-

ture. For the smoothed color field we get:

cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)

The gradient field of the smoothed color field

n= ∇cs (16)

yields the surface normal field pointing into the fluid and the

divergence of n measures the curvature of the surface

κ =−∇2cs

|n| . (17)

The minus is necessary to get positive curvature for con-

vex fluid volumes. Putting it all together, we get for the sur-

face traction:

tsurface = σκ n|n| (18)

To distribute the surface traction among particles near the

surface and to get a force density we multiply by a normal-

ized scalar field δs = |n| which is non-zero only near thesurface. For the force density acting near the surface we get

fsurface = σκn= −σ∇2

cSn

|n| (19)

Evaluating n/|n| at locations where |n| is small causes nu-merical problems. We only evaluate the force if |n| exceedsa certain threshold.

3.4. External Forces

Our simulator supports external forces such as gravity, col-

lision forces and forces caused by user interaction. These

forces are applied directly to the particles without the use of

SPH. When particles collide with solid objects such as the

glass in our examples, we simply push them out of the ob-

ject and reflect the velocity component that is perpendicular

to the object’s surface.

3.5. Smoothing Kernels

Stability, accuracy and speed of the SPH method highly de-

pend on the choice of the smoothing kernels. The kernels

we use have second order interpolation errors because they

are all even and normalized (see Fig. 2). In addition, kernels

that are zero with vanishing derivatives at the boundary are

conducive to stability. Apart from those constraints, one is

free to design kernels for special purposes. We designed the

following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)

and use it in all but two cases. An important feature of this

simple kernel is that r only appears squared which means

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

Figure 2: The three smoothing kernels Wpoly6, Wspiky and

Wviscosity (from left to right) we use in our simulations. The

thick lines show the kernels, the thin lines their gradients

in the direction towards the center and the dashed lines the

Laplacian. Note that the diagrams are differently scaled. The

curves show 3-d kernels along one axis through the center

for smoothing length h= 1.

that is can be evaluated without computing square roots in

distance computations. However, if this kernel is used for the

computation of the pressure forces, particles tend to build

clusters under high pressure. As particles get very close to

each other, the repulsion force vanishes because the gradient

of the kernel approaches zero at the center. Desbrun2 solves

this problem by using a spiky kernel with a non vanishing

gradient near the center. For pressure computations we use

Debrun’s spiky kernel

Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)

that generates the necessary repulsion forces. At the bound-

ary where it vanishes it also has zero first and second deriva-

tives.

Viscosity is a phenomenon that is caused by friction and,

thus, decreases the fluid’s kinetic energy by converting it into

heat. Therefore, viscosity should only have a smoothing ef-

fect on the velocity field. However, if a standard kernel is

used for viscosity, the resulting viscosity forces do not al-

ways have this property. For two particles that get close to

each other, the Laplacian of the smoothed velocity field (on

which viscosity forces depend) can get negative resulting in

forces that increase their relative velocity. The artifact ap-

pears in coarsely sampled velocity fields. In real-time appli-

cations where the number of particles is relatively low, this

effect can cause stability problems. For the computation of

viscosity forces we, thus, designed a third kernel:

Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)

whose Laplacian is positive everywhere with the following

c© The Eurographics Association 2003.




cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






Wpoly6(r, h) =315

64πh9

(h2 − r2)3, 0 ≤ r ≤ h

0


para r pequenos

Wspiky (r, h) =15

πh6

(h − r)3, 0 ≤ r ≤ h

0


Wvisco(r, h) =15

2πh3

− r3

2h3 + r2

h2 + h2r − 1, 0 ≤ r ≤ h

0




cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






Wpoly6(r, h) =315

64πh9

(h2 − r2)3, 0 ≤ r ≤ h

0


para r pequenos

Wspiky (r, h) =15

πh6

(h − r)3, 0 ≤ r ≤ h

0


Wvisco(r, h) =15

2πh3

− r3

2h3 + r2

h2 + h2r − 1, 0 ≤ r ≤ h

0




cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)






cS(r) = ∑j

m j1

ρ jW (r− r j,h). (15)


n= ∇cs (16)



κ =−∇2cs

|n| . (17)



face traction:






cSn

|n| (19)


















following kernel

Wpoly6(r,h) =315

64πh9

(h2− r2)3 0≤ r ≤ h

0 otherwise(20)



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

0

2

4

6

8

10

12

14

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

–2

–1

0

1

2

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

















Wspiky(r,h) =15

πh6

(h− r)3 0≤ r ≤ h

0 otherwise,(21)



tives.














Wviscosity(r,h) =15

2πh3

− r3

2h3+ r2

h2+ h2r −1 0≤ r ≤ h

0 otherwise.(22)





Tratamento de colisao

I Muller et al.: Projeta e aplica impulso

I Monaghan et al.: Uso de partıculas de fronteira


Multiplos fluidosMuller et al. (2005)

Forca de viscosidade

pviscosityi =

∑

j

µi + µj2

mjvj − viρj

∇2W (xi − xj , h)

Lıquidos imiscıveisI Adiciona “campo de cor” para caracterizar interfaces

I Exemplo: c iwater = -0.5 , c ioil = 0.5

f interface = −σi ∇2c in

|n|Difusao de calor

∂T

∂t= k ∇2T

∂T

∂t= k

∑

j

mjTj − Ti

ρj∇2W (rij , h)

Ti ← Ti + ∆t∂T

∂t

ρ0 =α

T, sendo α cte


Referencia

Erleben, K., Sporring, J., Henriksen, K., and Dohlmann, H. (2005). Physics BasedAnimation. Charles River Media.

Foster, N. and Metaxas, D. (1997). Modeling the motion of a hot, turbulent gas. InSIGGRAPH ’97: Proceedings of the 24th annual conference on Computer graphicsand interactive techniques, pages 181–188, New York, NY, USA. ACMPress/Addison-Wesley Publishing Co.

Harris, M. (2004). Fast fluid dynamics simulation on the GPU, chapter 38.Addison-Wesley, Boston, USA.

Liu, G. R. and Liu, M. B. (2003). Smoothed particle hydrodynamics: a meshfreeparticle method. World Scientific Publishing Co.

Muller, M., Charypar, D., and Gross, M. (2003). Particle-based fluid simulation forinteractive applications. In SCA ’03: Proceedings of the 2003 ACMSIGGRAPH/Eurographics symposium on Computer animation, pages 154–159,Aire-la-Ville, Switzerland, Switzerland. Eurographics Association.

Muller, M., Solenthaler, B., Keiser, R., and Gross, M. (2005). Particle-based fluid-fluidinteraction. In SCA ’05: Proceedings of the 2005 ACM SIGGRAPH/Eurographicssymposium on Computer animation, pages 237–244, New York, NY, USA. ACM.

Stam, J. (1999). Stable fluids. In SIGGRAPH ’99: Proceedings of the 26th annualconference on Computer graphics and interactive techniques, pages 121–128, NewYork, NY, USA. ACM Press/Addison-Wesley Publishing Co.


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