计算机图形学流体模拟：第二版

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"Fluid Simulation for Computer Graphics, Second Edition" by Robert Bridson 本书是计算机图形学领域关于流体模拟的权威著作，由罗伯特·布里德森（Robert Bridson）撰写，第二版对原有的内容进行了更新和扩展。流体模拟在游戏开发、电影特效、科学研究等多个领域都有着广泛的应用，它允许我们以视觉上逼真的方式呈现水、烟雾、火焰等动态效果。在计算机图形学中，流体模拟是一个复杂而深奥的主题，涉及到计算流体动力学（CFD）、数值方法、物理建模等多个方面。罗伯特·布里德森的这本书详细介绍了如何在计算机中实现这些复杂的物理现象。书中涵盖了以下关键知识点： 1. **基础理论**：介绍流体力学的基本原理，如纳维-斯托克斯方程（Navier-Stokes equations），以及用于简化这些问题的近似方法，如拉格朗日粒子追踪和欧拉方法。 2. **数值方法**：讨论了求解流体方程的数值技术，如有限差分、有限体积和粒子系统。这些方法在保持计算效率的同时，力求准确地模拟流体行为。 3. **GPU加速**：随着GPU计算能力的增强，书中可能涉及如何利用GPU并行计算来加速流体模拟，提高实时性能。 4. **稳定性和误差控制**：探讨如何处理数值稳定性问题，避免模拟过程中的振荡和不稳定现象，以及如何减少计算误差。 5. **应用案例**：通过实例展示如何将流体模拟应用于实际项目，如游戏、动画和视觉特效，以及如何结合其他图形技术，如光照和渲染，提升整体视觉效果。 6. **编程实践**：提供代码示例或伪代码，帮助读者理解和实现流体模拟算法，可能包括开源库如OpenFOAM或FLEX的相关使用。 7. **版权信息**：书中引用了多个来源的信息，并尊重所有原始作品的版权，如果在出版过程中未得到某些材料的使用许可，作者和出版社表示歉意。 "Fluid Simulation for Computer Graphics, Second Edition" 是一个全面且深入的学习资源，适合计算机图形学研究人员、游戏开发者、动画师以及对流体模拟感兴趣的任何技术爱好者。通过学习本书，读者可以掌握创建逼真流体效果所需的技术和理论知识。

The Equations of Fluids

Fluids surround us, from the a ir we breathe to the oceans covering two

thirds of the Earth, and are at the heart of some of the most beautiful and

impressive phenomena we know. From water splashing, to ﬁre a nd smoke

swirling, ﬂuids have beco me an important part o f computer graphics. This

book aims to cover the basics of simulating these eﬀects for a nimation. So

let’s jump rig ht in with the fundamental equa tions governing their motion.

Most ﬂuid ﬂow of interest in animation is governed by the famous in-

compressible Navier-Stokes equations, a set of partia l diﬀerential equations

that a re supposed to ho ld throug hout the ﬂuid. The eq uations are usually

written as

∂~u

∂t

+ ~u · ∇~u +

∇p = ~g + ν∇ · ∇~u, (1.1)

∇ · ~u = 0. (1.2)

These may appear pretty complicated at ﬁrs t glance! We’ll soon break them

down into easy-to-understand par ts (and in Appendix B provide a more

rigorous explanation), but ﬁrst let’s begin by deﬁning what each symbol

means.

1.1 Symbols

The letter ~u is traditionally used in ﬂuid mechanics for the velocity of the

ﬂuid. Why no t ~v? It’s hard to say, but it ﬁts another useful convention

to call the three components of 3D velocity (u, v, w), just as the three

components of position ~x are often taken to be (x, y, z).

The Greek letter ρ stands for the density of the ﬂuid. For water, this is

roughly 1000 kg/m

, and for air in usual sea-level conditions this is roughly

1.3 kg/m

, a ratio of about 700 : 1.

It’s worth emphasizing right away my insistence on using real units

(meters, kilograms, etc.): long experience has shown me that it is well

worth keeping all quantities in a solver implicitly in SI units, rather than

just set to arbitrary values. It is tempting when starting to program a

4 1. The Equations of Fluid s

new solver to just ﬁll in unit-less values like 1 for physical quantities such

as density or to drop them altogether from expre ssions, whether you’re

operating fro m a quick-and-dirty just-make-it-work point of view o r a more

mathematically founded non-dimensionalization rationale

. However, this

often comes back to haunt you when simulations don’t quite look right, or

need to be resized, or adjusted in other ways w he re it’s not clear which of a

plethora of nonphysical parameters need to be tweaked. We’ll discuss the

ramiﬁcations of this in algorithm design as well throughout the book.

The letter p stands for pressure, the force per unit area that the ﬂuid

exerts on anything.

The letter ~g is the familiar acceleration due to gravity, usually

(0, −9.81, 0) m/s

. Now is a good time to mention that in this b ook we’ll

take a s a convention that the y-axis is pointing vertically upwards, and the

x- a nd z-axes are horizontal. We should add that in anima tion, a dditio nal

control acc elerations (to make the ﬂuid behave in some desired way) might

be added on top of gravity — we’ll lump all of these into the one symbol ~g.

More generally, people call these body forces, because they ar e applied

throughout the whole body of ﬂuid, not just on the surfaces.

The Greek letter ν is technically called the kinematic viscosity. It

measures how viscous the ﬂuid is. Fluids like molas ses have high viscosity,

and ﬂuids like mercury have low visco sity: it measures how much the ﬂuid

resists deforming while it ﬂows (or more intuitively, how diﬃcult it is to

stir).

1.2 The Momentum Equation

The ﬁrst diﬀerential equation (1.1), which is actually three in one wrapped

up as a vector e quation, is called the momentum equation. This really

is good old Newton’s equation

F = m~a in disguise. It tells us how the ﬂuid

accelera tes due to the fo rces acting on it. We’ll try to break this down

before moving onto the second diﬀerential equation (1.2), which is called

the incompressi bility condition.

Let’s ﬁrst imagine we are simulating a ﬂuid using a particle system

(later in the book we will actually use this as a practical metho d, but

for now let’s just use it as a thought experiment). Each particle might

represent a little blob of ﬂuid. It would have a mass m, a volume V , and

a velocity ~u. To integrate the system forward in time all we need is to

Non-dimensionalization is a strategy for mathematically simplifying physical equa-

tions by rewriting all quantities as ratios to characteristic values of the problem at the

hand, like the usual density of the ﬂuid and the width of the container, rather than using

SI units. This can reduce the number of constants that appear in the equations to the

minimal set that matter, making some analysis much easier.

1.2. The Momentum Equation 5

ﬁgure out what the forces acting on each particle are:

F = m~a then tells

us how the particle accelerates, from which we get its motion. We’ll write

the acceleration of the particle in slightly odd notation (which we’ll later

relate to the momentum equation above):

~a ≡

D~u

The big D de rivative notation is called the material de rivative (more on

this later). Newton’s law is now

D~u

F .

So what are the forces acting on the par ticle ? The simplest is of course

gravity: m~g. However, it gets interesting when we consider how the rest of

the ﬂuid a lso exerts force: how the particle interacts with other particles

nearby.

The ﬁrst of the ﬂuid forces is pressure. High-pressure regions push on

lower-pressure reg ions. Note that what we r eally care about is the net

force on the particle: for example, if the pressure is equal in every di-

rection there’s going to be a net force of zero and no acceleration due to

pressure. We only see an eﬀect on the ﬂuid particle when there is an im-

balance, i.e. higher pressure on one side of the particle than on the other

side, resulting in a fo rce po inting away from the high pressure and toward

the low pressure. In the appendices we show how to rigorously derive

this, but for now let’s just point out that the simplest way to measure

the imbalance in pressure at the position of the particle is s imply to take

the negative gradient of pressure: −∇p. (Recall from calculus that the

gradient is in the direction of “steepe st ascent,” thus the negative gradi-

ent points away from high-pressure regions toward low-pressure regions.)

We’ll need to integrate this over the volume of our blob of ﬂuid to get

the pressure force. As a simple approximation, we’ll just multiply by the

volume V . You might be asking yourself, but what is the pr essure? We’ll

skip over this until later, whe n we talk about incompressibility, but for now

you can think of it being whatever it takes to keep the ﬂuid at constant

volume.

The other ﬂuid force is due to viscosity. A viscous ﬂuid tries to resist

deforming. Later we will derive this in more depth, but for now let’s

intuitively develop this as a force that tries to make our particle move at

the average veloc ity of the nearby particles, i.e., that tries to minimize

diﬀerences in velocity be tween nearby bits of ﬂuid. You may remember

from image proce ssing, digital geometry processing, the physics of diﬀusion

or heat dissipation, or many other domains, that the diﬀerential operator

that measures how far a quantity is from the average around it is the

6 1. The Equations of Fluid s

Laplacian ∇ · ∇. (Now is a good time to mention that there is a quick

review of vector calculus in the appendices, including diﬀerential opera tors

like the Laplacian.) This will provide our viscous force then, once we’ve

integrated it over the volume o f the blob. We’ll use the dynamic viscosity

coeﬃcient, which is denoted with the Greek letter µ (dynamic means we’re

getting a force out of it; the kinematic viscosity from befor e is used to get

an acceleration instead). I’ll note here that near the surface of a liquid

(where there isn’t a complete neighbo rhood around the blob) and for ﬂuids

with variable viscosity, this term ends up being a little more complicated;

see Chapter 10 for more details.

Putting it all together, here’s how a blob of ﬂuid moves:

D~u

= m~g − V ∇p + V µ∇ · ∇~u.

Obviously we’re making errors when we approximate a ﬂuid w ith a small

ﬁnite number of particles. We will take the limit then as our number of

particles goes to inﬁnity and the size of each blob goes to zero . O f cours e,

this clearly makes a diﬀerent sort o f error, as real ﬂuids are in fact comp osed

of a (very large) ﬁnite number of mo lecules! But this limit, which we call

the continuum model, has the advantages of ma thematical conciseness

and independence from the exact number of blobs, and has been shown

experimentally to be in extraordinarily close agre ement with reality in a

vast range of scenarios. However, taking the continuum limit does pose a

problem in our particle e quation, because the mass m and volume V of the

particle must then go to zero, and we are left with nothing meaningful. We

can ﬁx this by ﬁrst dividing the equation by the volume, and then taking

the limit. Remembering m/V is just the ﬂuid density ρ, we get

D~u

= ρ~g − ∇p + µ∇ · ∇~u.

Looking familiar? We’ll div ide by the de nsity and rearrange the terms a

bit to get

D~u

∇p = ~g +

∇ · ∇~u.

To simplify things even a little mor e, we’ll deﬁne the kinematic viscosity

as ν = µ/ρ to get

D~u

∇p = ~g + ν∇ · ∇~u.

We’ve almost made it back to the momentum equation! In fact this

form, using the material derivative D/Dt, is a ctually more important to us

in computer graphics and will guide us in solving the eq uation numerically.

But we still will want to understand what the mater ial derivative is and

1.3. Lagrangian and Eulerian Viewpoints 7

how it relates back to the traditional form of the momentum equation. For

that, we’ll need to understand the diﬀerence be tween the Lagrangian and

Eulerian viewpoints.

1.3 Lagrangian and Eulerian Viewpoints

When we think about a continuum (like a ﬂuid or a deformable solid)

moving, there are two approaches to tracking this motion: the Lagrangian

viewpoint and the Eulerian viewpoint.

The Lagrangian approach, named after the French mathematician La-

grange, is what you’re proba bly most familiar with. It treats the continuum

just like a particle system. Each point in the ﬂuid or solid is labeled as a

separate particle, with a position ~x and a velocity ~u. You could even think

of each particle as being one molecule of the ﬂuid. Nothing too sp ecial here!

Solids are almost always simulated in a Lagrangian way, with a discrete set

of particles usually connected up in a mesh.

The Eulerian appr oach, named after the Swiss mathematician Euler,

takes a diﬀerent tactic that’s usually used for ﬂuids. Instead of tracking

each particle, we instead look at ﬁxed points in space and see how mea-

surements of ﬂuid quantities, such as density, velocity, tempe rature, etc.,

at those points change in time. The ﬂuid is probably ﬂowing past those

points, contributing one sort of change: for example, as a warm ﬂuid moves

past followed by a cold ﬂuid, the temperature at the ﬁxed point in space

will decrease—even though the temperature of any individual particle in

the ﬂuid is not changing! In addition the ﬂuid variables can be changing

in the ﬂuid, contributing the other sort of change that might be measured

at a ﬁxed point: for example, the temper ature measured at a ﬁxed po int

in space will decrease as the ﬂuid everywhere cools oﬀ.

One way to think of the two viewpoints is in doing a weather report.

In the Lagrang ian viewpoint you’re in a balloon ﬂoating along with the

wind, measuring the pressure and temperature and humidity, etc., of the

air that’s ﬂowing alongside you. In the Eulerian viewpoint you’re stuck on

the ground, measuring the pressure and temperature a nd humidity, etc., of

the air that’s ﬂowing past. Both measurements can create a graph of how

conditions are changing, but the graphs can be c ompletely diﬀerent as they

are measuring the rate of change in fundamentally diﬀerent ways.

Numerically, the Lagrangia n viewpoint corresponds to a particle sys-

tem, with or without a mesh connecting up the par ticle s, and the Eulerian

viewpoint corresp onds to using a ﬁxed grid that does n’t change in space

even as the ﬂuid ﬂows through it.

It might se em the Eulerian approach is unnecessarily complicated: why

not just stick with Lagrangian particle systems? Indeed, there are schemes,

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