Real-time animation of low-Reynolds-number flow using Smoothed Particle Hydrodynamics presentation by 薛德明 Dominik Seifert B97902122.

Computational Fluid Dynamics

Real-time animation of low-Reynolds-number flowusing Smoothed Particle Hydrodynamics

presentation by薛德明

Dominik SeifertB97902122

Visualization of my resultshttp://csie.ntu.edu.tw/~b97122/archive/fluid_dynamics/capture.mp4

My codehttp://csie.ntu.edu.tw/~b97122/archive/fluid_dynamics/OpenTissue_backup.rar

My motivation: From Dusthttp://www.youtube.com/watch?v=CfKQCAxizrA

The frameworkOpenTissueOpenTissue is an open source simulation

framework with a simple, working SPH implementation

I added some features to their implementation

Their original implementation is described in this 88 page document:“Lagrangian Fluid Dynamics Using Smoothed

Particle Hydrodynamics”

Smoothed Particle HydrodynamicsQuick reviewNote: SPH particles are not actual particles!

They are really fluid samples of constant massParticles are placed inside a container to

represent a fluidEvery particle is then assigned a set of initial

propertiesAfter every time step (a few milliseconds),

update the properties of all particlesUse interpolation methods to solve the Navier-

Stokes equation to find force contributions, then integrate to find new velocity and position

SPHParticle PropertiesSupport Radius (constant)

Minimum interaction distance between particlesMass (constant; ensures Conservation of Mass

explicitly)Position (varying)Surface normal (varying)Velocity (varying)Acceleration (varying)Sum of external forces (varying)Density (varying due to constant mass and varying

volume)Pressure (varying)Viscosity (constant)

Smoothed Particle HydrodynamicsSolver pseudocode

Smoothed Particle HydrodynamicsThe pressure problemFluids cannot be accurately incompressiblePressure value approximated by Ideal Gas Law:

k called “gas-stiffness”Entails assumptions of gas in steady stateDoes not consider weight pressureCauses “pulsing” because of lagging interplay between

gravity and pressure forceLarge gas-stiffness can reduce/eliminate the lag and the

pulsingAlternatively, take density to the power of heat capacity

ratioBut high pressure requires a smaller time-step and thus

makes the simulation more expensive

pV = nRT ⇒ p=( k ' )V = k ρ

My contributions IFluid-fluid interactionsIn OpenTissue, a system can be comprised of only a

single fluidI changed the code to support more than one fluid at a

timeThe math and physics are mostly the same, except for:

ViscosityKernel support radius

Still missing:Surface tension interfaces between fluids of different polarity

But I still spent three days on changing the framework due to heavily templated C++ code

My contributions IIFluid-solid interactionsIn OpenTissue only supports static (immovable)

objectsI wanted to add the ability to add objects that

interact with the fluid, objects that can float, sink etc.

Solid dynamics are very complicated! What are…Tensile Strength? Compressive Strength? Young’s modulus?

I came up with an intuitive but not quite correct approach

My contributions IIRestoring forcePlace an invisible spring between every

two particles that are close to each other initially

Store the initial distance between every two “neighboring” particles

Add a new spring force component that contributes:

k * abs(current_distance – original_distance)

Works for very few particles, but not for many

My contributions IIIControl Volumes - OverviewControl volumes are used in the analysis of

fluid flow phenomenaThey are used to represent the conservation

laws in integral form Conservation of mass over a given (control)

volume c.v. with surface area c.s.:

= density; u = velocity; n = surface normal

0= ddt ∫c.v.

ρ dV + ∮c.s.

ρ(u⋅n̂ )dA

My contributions IIIControl Volumes in SPHVolume integrators are easy: Simply

accumulate all contributions of all particles in volume

Area integrators are trickierTime derivative can be obtained via difference

quotients: for any property Fluid properties at some point in the field

can be obtained by interpolation

d ηd t

≈Δ ηΔ t

=ηT0+ 1−ηT0

Δ tη

My contributions IIIField evaluator function

ValueType evaluate(vector pos, real radius, ParticleProperty A): ParticleContainer particles; search(pos, radius, particles);

ValueType res = ValueType(0); foreach particle p in particles: real W = DefaultKernel.evaluate(pos - p.position); res += (p.*A)() * p.mass/p.density * W;

return res;

This required me to change the spatial partioning grid to support queries at arbitrary locations

After that, I only had to implement the famous SPH field evaluation template for some property A:

Translates to:

A( x)=∑j∈ΩA jV jW (x− x j , h)=∑

j∈ΩA jm j

ρ j

W (x−x j , h)

My contributions IIIArea Integrator

Goal: Find average of property at discretely sampled points

I went for an evenly distributing samplerAliasing is not an issue, don’t need random

sampling# of samples ns should be proportional to # of

particles that can fit into the surface:

So we get:

D f̄ =∫D

f (A)dA

∮c.s.

ρ(u⋅n̂)dA≈ AC1ns

∑j

ns

q j=ACns

∑j

ns

ρ j(u j⋅n̂ j)

ns=ceil(ACA p)

My contributions IIIDisk Integrator

real integrateDiskXZ(real ns, vector2 p_center, real r, field_evalutor f):real q_total = 0real ds = sqrt(PI / ns) * r //int samples_in_diameter = sqrt(ns * 4/PI) //vector2 min = p_center – rfor (int i = 0; i < samples_in_diameter; i++)for (int j = 0; j < samples_in_diameter; j++)vector2 pos = (min.x + i * ds, min.z + j * ds)if (length(pos - p_center) > r) continueq_total += f(pos)

return q_total / ns

In this approach, every kind of surface needs their own integrator

I only have to consider disks in my pipe flow exampleThe disk integrator iterates over the cells of an

imaginary grid that we lay over the disk to find the average of fluid property f

AC=π4

(2 r )2= ns⋅ds2⇒ 2 rds

=√4π ns

My contributions IIIArea Integrator - ConsiderationsNo need to sample over an already sampled set

Can use spatial selection instead:Find all particles in distance d from the surfaceUse scaled smoothing kernel to add up contributions

I was not quite sure how to mathematically scale the kernel, so I went for the sampling approach

I also used the integrator to place the cylindrically-shaped fluid inside the pipe

My contributions IIIConservation of mass

M T0+ 1−M T0

Δ t+ACns

∑j

ns

ρ j(u j⋅n̂ j)

0= ddt ∫c.v.

ρ dV + ∮c.s.

ρ(u⋅n̂ )dAThe integral form:

Becomes:

The first term is the time derivative of Mass inside the c.v.

The second term is the mass flux through the c.v.’s surface area

Particle boundary deficiency,Holes in the fluid andControl Volume - CorrectnessBoundary deficiency:

Since atmosphere and structure are not represented in this model, computations have to cope with a pseudo-vacuum (真空 )

Governing equations are adjusted to cope with the deficiencye.g. Level set function for surface tension considers:

Inside fluid = 1 Outside fluid = 0

C.v.’s must always be completely filled!Fluid volume is never correct which causes “holes” in

the fluid Think: What is the space between the particles/samples? C.v. computations can also never be 100% correct!

Bibliography[1] OpenTissue @ http://www.opentissue.org/

“OpenTissue is a collection of generic algorithms and data structures for rapid development of interactive modeling and simulation.”

[2] Smoothed Particle Hydrodynamics – A Meshfree Particle Method (book)

[3] Lagrangian Fluid Dynamics Using Smoothed Particle Hydrodynamics

[4] Particle-Based Fluid-Fluid Interaction

SummaryGiven high enough gas stiffness, SPH model is OK to

simulate visually appealing real-time flow but is quite inaccurate

OpenTissue SPH implementation is not very mature, lacks a lot of features

I really miss:Accurate pressure valuesCorrect fluid-solid interactionArbitrary geometry

I still cannot create real-time interactive applications involving fluid flow but it was still an insightful endeavour.

What’s next?Surface rendering until next weekThen choose one from the list…

Improve SPH implementation Add generic surfaces (currently only supports implicit primitives) Make it adaptive (choose sample size dynamically) Learn solid dynamics and work on Fluid-Structure interaction

More work on surface rendering Optimized OpenGL/DX/XNA implementation that runs on the GPU

Work on Computational Galactic Dynamics (計算星系動力學 ) with professor 闕志鴻  from the Institute of Astronomy (天文所 ) Simulation of dark matter & fluid during galaxy formation

Work on level sets and the level set method in CFD with professor Yi-Ju 周  from the Institute of Applied Mechanics (應力所 )