pressure corrected sph for fluid - korea...

Pressure corrected SPH for fluidanimation

Analyzed by Po-Ram Kim

2 March 2010

Korea UniversityComputer Graphics Lab.

Kai Bao, Hui Zhang,Lili Zhengand Enhua Wu

Korea UniversityComputer Graphics Lab. Po-Ram Kim | 2 March 2010 | # 2KUCG |

Abstract

• We present pressure scheme for the SPH for fluid animation

• In conventional SPH method, EOS are used§ For volume conservation, high speeds of sound are

required à numerical instability

• In the paper, a new extra pressure correction scheme is proposed


Abstract

• Smoother pressure distribution and more efficient simulation are achieved

• The proposed method has been used to simulate free surface problems

• Surface tension and fluid fragmentation can be well handled


Introduction

• SPH is a method to capture characteristics and thus becomes more and more widely used

• To simulate free surface flow, one crucial problem is to ensure the incompressibility of the flow§ There are two main ways to achieve this

• The divergence of the velocity field à zero‗ Poisson equation : time consuming solution

• Keeping the fluid density constant‗ Poisson equation : time consuming solution‗ Moving Particle Semi-implicit (MPS) method


Introduction

• In the conventional SPH method§ EOS are used to directly associate the pressure with

particle density• The time-consuming solution of Poisson equation is avoided :

widely used‗ But, it proved to be hard to guarantee the incompressibility of free

surface flow

§ Tait’s equation• The volume of fluid is generally well conserved

‗ But , a high sound speed has to be used àthe time step is too small‗ little difference in density will lead to large variation in pressure


Introduction

• In the paper§ An iterative pressure correction scheme is proposed

to be used along with the EOS§ The local pressure disturbance is made to propagate

to the neighboring areas • Smoother pressure distribution• Incompressible fluid• More accurate and efficient simulation

§ Smaller sound speeds and larger time steps are made possible

§ Enhancement in the surface tension model


Previous Work

• In Eulerain methods§ The incompressibility is easy to enforce§ While the mass conservation for small features is not

well guaranteed§ Widely used in the physically based animation and

many fluid phenomena


Previous Work

• In Largrangian particles(SPH)§ First introduced for highly deformable bodies§ Grids are required during the computation§ Mass conservation is naturally guaranteed § Surface tracking techniques are required§ Large deformation and violent fragmentation can be

handled§ Interactive simulation of free surface flow was

achieved using SPH by Muller et al


Previous Work

• Free-surface flow

• Highly viscous fluids


Previous Work

• Solid-fluid coupling

• An adaptive sampling technique


Methodology

Basic SPH Formations

• Lagrangian form of the Navier – Stokes equation§ Conservation of mass

§ Conservation of momentum

v : velocity vector , p : pressure , ρ : fluid density , g : gravitational acceleration vector,: kinetic viscosity


Methodology


• To evaluate the value f at an arbitrary position x§ an interpolation is applied with the neighboring

particles: Particle approximation

fj : the value of f at the position of particle j , W : smoothing kernel functionm : mass , ρ : density

i

j


Methodology


• By applying the SPH particle approximation to the momentum equation(equation (2))

i

j

pj : the pressure of particle i , : direction gradient to particle i


Methodology

Density Computation

• Two main approaches to determine the density of particles in the traditional SPH§ The density summation method

§ Tracking the evolution of the density through the continuity equation


Methodology

Density Computation

• The density summation method (ßwidely used)§ Advantage

• It conserves the mass exactly

§ Disadvantage • It suffers from particle deficiency near the boundary

• In the paper, the continuity density approach (equation(5)) is used


Methodology

Equations of State

• Incompressible fluid§ liquids

• Compressible fluid§ Gases

• A theoretically incompressible flow is practically compressible§ Artificial compressibility is introduced

• Weakly compressible• The pressure is determined with EOS• This approach for free surface flowàThe volume of the flow is hard to be well conserved


Methodology

Equations of State

• Equation of State

• Tait’s equation

àThe variations of density remain smallàThe volume of the fluid is generally well conserved

kp = c2 , c : the sound of speed , ρ0 : reference density


Methodology

Equations of State

• Tait’s equation§ Small deviation in density field will result in large

fluctuation in pressure§ Noisy pressure distribution will be obtained

• Numerical instability


Methodology

Equations of State

• Time step

• With the Tait’s equation, § A high speed of sound is required

• To keep density fluctuation lowà Small time step has to be used

§ To keep the density variation under the order of 1%• Sound speed = 10 * (maximum possible velocity)• In ref[8]

‗ Time step = 4.52*10-4

CFL conditionViscous force conditionExternal force condition


Methodology

Pressure Correction Equation

• For a truly incompressible flow§ ρ = constant à§ equation(1) àà divergence-free field

§ To obtain a divergence free field, the classical prejection method is used

• However, solving poisson equation proves to be very time consuming

0=dtdr

0v =×Ñ

*2 vdt

p ×Ñ=Ñr

v* : intermediate velocity field without applying the pressure in momentum equation


Methodology


• To resolve § The noisy pressure disturbance§ Instability arising from the EOS

§ To avoid the expensive solution of global Poisson equation

• A flexible pressure correction equation is presented


Methodology


• By substituting the equation(6) into continuity equation(equation(1)), the following equation could be obtained

0

01)1(

1

)(

2

0

=×Ñ+

=×Ñ+

=

=

=

-=

v

v

r

r

r

r

rr

cdtdpdtdp

k

equationdtdp

kdtd

dtdk

dtdp

kp

p

p

p

p

02 =×Ñ+ vcdtdp r (9)

tdtdII dffd )()( =

Variational method


Methodology


• Equation(9) can be written in SPH form as

• If the computation is convergent, RHS of equation(10) should be zero

• A pressure correction value could be obtained by

Since the pressure correction scheme is iterative, a counting number n is introduced


Methodology


• The pressure at the new iteration is written as

• With the pressure correction value, the velocity correction value can be obtained with the momentum equation

ω is the relaxation factor with a value under 1.0

v is the kinetic viscosity


Methodology


• The velocity correction can be obtained by

• The velocity is updated with

Ω is the relaxation factor with a value under 1.0


Methodology


• In each SPH time step§ Equations (11) and (15) are solved iteratively until

convergent§ During the iteration

• Pressure disturbance will propagate to the neighboring particles • Smoother pressure distribution will be obtained

• The pressure correction scheme actually provides a combination of the EOS method and the global pressure Poisson method

• With larger speed of sound, less pressure correction iterations will be required


Methodology

Surface Tension Model

• Surface tension plays a fundamental role in many fluid phenomena§ Fluid breaking§ Droplet dynamics

• The surface tension results from the uneven molecular forces of attraction near the surface

• The surface tension will lead to a net force in the direction of surface normal


Methodology

Surface Tension Model

• In SPH method, widely used form

• Smoother surface tension force

σ : Tension coefficient


Results and Discussions

• All the simulations are performed within a single thread§ Intel Core2 Q6700 CPU § 8GB RAM

• The reference densities in all the simulations

• All the 2D results are rendered with OpenGL• All the 3D results with POVRay



Divergence

• The particles are represented by dots • The velocities of the particles are displayed with line

segments starting from the positions of the particles• Located in the rectangle of 0.2 × 0.2• The initial spacing of the particles is 0.02 • 2520 fluid particles are used in the simulation• A speed of sound of 40 is taken • The time step is 0.001 second



Divergence

Figure1(a-1) Figure1(a-2)

Figure1(a-3) Figure1(a-4)

Initial velocity: (0.5,0.0)



Divergence

Figure1(b-1) Figure1(b-2)

Figure1(b-4)Figure1(b-3)

Initial velocity: (0.5,0.5)



Divergence

§ Usually, several times of the iterations are enough§ c = 5 and dt = 0.008 second

• exactly the same correction results are obtained

We consider the computation has been convergent



Pressure Distribution

• Dam-break flow§ Initial height of water body = 2m§ Initial width of water body = 1m§ Initial particle spacing = 0.02m§ Total 5000 fluid particles are used§ In figure 2

• Figure 2a : the pressure correction scheme is NOT used• Figure 2b : the pressure correction scheme is used• Purple color : the highest pressure• Red color : the lowest pressure




c = 102gH ≈ 62.6m/seconddt

= 8 · 10−5

c = 30 m/seconddt = 2 × 10

−4

figure2




• As shown from the Figure 2,§ Without the pressure correction, the pressure fields

obtained are unphysically noisy§ The pressure noise is significantly reduced § Smoother pressure distribution is achieved



Surface Tension

• The initial side length of cube is 0.005 m• The initial spacing of the particles is 0.0002m • About 12K particles in total are used in the

simulation. • dt = 0.00005 sec• It takes about 0.3 second for one time step of

simulation• As the energy is damped by viscous and numerical

dissipation§ The particles are stable at a spherical shape

• It takes very long time to achieve



Surface Tension

• Evolution of a drop initially in cube shape under effect of surface tension with zero gravity

0.0sec 0.0518sec0.0255 sec

0.0833sec

0.114sec 2.0sec Figure 3



Dam-break on a Wet Bed

• Initial value§ Height = 0.45m§ Length = 0.32m§ Width = 0.4m§ Initial water depth on the bed region = 0.018§ Initial spacing of the particles = 0.006m§ Total # of particle = 267k§ Time step = 0.0005sec§ Simulation time = 7.5 sec



Dam-break on a Wet Bed

Figure 4

• The free surface shape is the main focus of this simulation§ At the initial time

• A mushroom shape in free surface

§ Two breaking waves enclosing voids will be generated

0.0sec

0.19sec

0.34sec

0.72sec



Dam-break Flow on Complicated Topography

• Digital elevation model (DEM) data is used § To generate the terrain surface§ Then a distance field is generated to enforce the solid

boundary condition• Initial velocity of water body is 3m/sec• The initial spacing of particles is 0.04m • Total # of particle is 330k particles• The time step is 0.001 second• When water interacts with the terrain surface

§ Violent breakage and fragmentation occur § Wave propagation and reflection are well produced



Dam-break Flow on Complicated Topography

2.9sec 4.75sec 7.9sec

1.25sec0.5sec0.0sec



CASA 2009

• A simulation of dambreak with an obstacle in “CASA 2009” shape is carried out

• When the water flows over the obstacle, violent breaking is produced

• When the flow settles down§ The shape of the terrain obstacle becomes visible

• The initial spacing of particles is 0.005m • The time step is 0.0005 second• Total # of particle is 310k particles



CASA 2009

Figure 6

0.14sec

1.14sec

0.34sec

4.64sec


Conclusions and Future Work

• In the paper,§ A pressure correction equation is proposed for free

surface flow§ The pressure disturbance incurred by the EOS is

reduced § No solution of pressure Poisson equation is required§ More accurate and efficient simulation is achieved§ The improved SPH method has been used in free

surface and surface tension problem simulation


Conclusions and Future Work

• Our ongoing work § Investigation of numerical properties of the pressure

correction scheme § Its applications to more fluid phenomena, such as

multi-phase flow

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