CS-184: Computer Graphics Lecture #18: Simulation Basics Prof. - - PowerPoint PPT Presentation

CS-184: Computer Graphics

Lecture #18: Simulation Basics

• Prof. James O’Brien

University of California, Berkeley

V2016-F-18-1.0

(With some slides from Prof. Ren Ng who is really an awesome guy.)

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Today

• Introduction to Simulation
• Basic particle systems
• Time integration (simple version)

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• Generate motion of objects using numerical simulation

methods

Physically Based Animation

xt+∆t = xt +∆t vt + 1 2∆t2at

g v

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Physically Based Animation

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Example: Cloth Simulation

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Example: Fluids

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Macklin and Müller, Position Based Fluids

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Example: Fracture

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Example: Fracture

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Particle Systems

• Single particles are very simple
• Large groups can produce interesting effects
• Supplement basic ballistic rules
• Collisions
• Interactions
• Force fields
• Springs
• Others...

Karl Sims, SIGGRAPH 1990

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• Single particles are very simple
• Large groups can produce interesting effects
• Supplement basic ballistic rules
• Collisions
• Interactions
• Force fields
• Springs
• Others...

Particle Systems

Feldman, Klingner, O’Brien, SIGGRAPH 2005

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Basic Particles

• Basic governing equation
• is a sum of a number of things
• Gravity: constant downward force proportional to mass
• Simple drag: force proportional to negative velocity
• Particle interactions: particles mutually attract and/or repell
• Beware complexity!
• Force fields
• Wind forces
• User interaction

¨ x = 1 mf

f

O(n2)

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Basic Particles

• Properties other than position
• Color
• Temp
• Age
• Differential equations also needed to govern these

properties

• Collisions and other constrains directly modify position

and/or velocity

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Particle Rules

14 Bryan E. Feldman, James F . O'Brien, and Okan Arikan. "Animating Suspended Particle Explosions". In Proceedings of ACM SIGGRAPH 2003, pages 708–715, August 2003.

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Gravitational Attraction

• Newton’s universal law of gravitation
• Gravitational pull between particles

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Fg = Gm1m2 d2 G = 6.67428 × 10−11 Nm2kg−2

d

m1 m2

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Example: Galaxy Simulation

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Disk galaxy simulation, NASA Goddard

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Integration

• Euler’s Method
• Simple
• Commonly used
• Very inaccurate
• Most often goes unstable

xt+∆t = xt + ∆t ˙ xt ˙ xt+∆t = ˙ xt + ∆t ¨ xt

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Integration

• For now let’s pretend
• Velocity (rather than acceleration) is a function of force

f = mv ˙ x = f(x, t)

Witkin and Baraff

Note: Second order ODEs can be turned into first order ODEs using extra variables.

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Integration

• For now let’s pretend
• Velocity (rather than acceleration) is a function of force

f = mv ˙ x = f(x, t)

Witkin and Baraff

Start

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Integration

• With numerical integration, errors accumulate
• Euler integration is particularly bad

x := x + ∆t f(x, t)

Witkin and Baraff

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Integration

• Stability issues can also arise
• Occurs when errors lead to larger errors
• Often more serious than error issues

˙ x = [ − sin(ωt) , − cos(ωt) ]

Witkin and Baraff

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• Modified Euler

Integration

˙ xt+∆t = ˙ xt + ∆t ¨ xt xt+∆t = xt + ∆t 2 ( ˙ xt + ˙ xt+∆t) xt+∆t = xt + ∆t ˙ xt + (∆t)2 2 ¨ xt 22

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Integration

• Midpoint method

a. Compute half Euler step b.

• Eval. derivative at halfway

c. Retake step

• Other methods
• Verlet
• Runge-Kutta
• And many others...

a b c

Witkin and Baraff

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Integration

• Implicit methods
• Informally (incorrectly) called backward methods
• Use derivatives in the future for the current step

xt+∆t = xt + ∆t ˙ xt+∆t ˙ xt+∆t = ˙ xt + ∆t ¨ xt+∆t

¨ xt+∆t = A(xt+∆t, ˙ xt+∆t, t + ∆t) ˙ xt+∆t = V(xt+∆t, ˙ xt+∆t, t + ∆t)

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Integration

• Implicit methods
• Informally (incorrectly) called backward methods
• Use derivatives in the future for the current step
• Solve nonlinear problem for and
• This is fully implicit backward Euler
• Many other implicit methods exist...
• Modified Euler is partially implicit as is Verlet

˙ xt+∆t = ˙ xt + ∆t A(xt+∆t, ˙ xt+∆t, t + ∆t)

xt+∆t

˙ xt+∆t

˙ xt+∆t = ˙ xt + ∆t V(xt+∆t, ˙ xt+∆t, t + ∆t)

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Temp Slide

Need to draw reverse diagrams....

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Integration

A(xt+∆t, ˙ xt+∆t) ≈ A(xt, ˙ xt) + C · (∆x) + D · (∆ ˙ x) V(xt+∆t, ˙ xt+∆t) ≈ V(xt, ˙ xt) + A · (∆x) + B · (∆ ˙ x)

  xt+∆t

˙ xt+∆t

   =   xt

˙ xt

   + ∆t       ˙

xt ¨ xt

   +   A B

C D

     ∆x

∆ ˙ x

     

• Semi-Implicit
• Approximate with linearized equations

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Integration

• Explicit methods can be conditionally stable
• Depends on time-step and stiffness of system
• Fully implicit can be unconditionally stable
• May still have large errors
• Semi-implicit can be conditionally stable
• Nonlinearities can cause instability
• Generally more stable than explicit
• Comparable errors as explicit
• Often show up as excessive damping

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Integration

• Integrators can be analyzed in modal domain
• System have different component behaviors
• Integrators impact components differently

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Example: Mass Spring Rope

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Credit: Elizabeth Labelle, https://youtu.be/Co8enp8CH34

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Example: Mass Spring Mesh

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Slide from Ren Ng

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Example: Hair

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Slide from Ren Ng

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Example: Cloth

33 Huamin Wang, Ravi Ramamoorthi, and James F . O'Brien. "Data-Driven Elastic Models for Cloth: Modeling and Measurement". ACM Transactions on Graphics, 30(4):71:1–11, July 2011. Proceedings of ACM SIGGRAPH 2011, Vancouver, BC Canada.

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Example: Clothing on Character

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Strain Limiting

35 Huamin Wang, James F . O'Brien, and Ravi Ramamoorthi. "Multi-Resolution Isotropic Strain Limiting". In Proceedings of ACM SIGGRAPH Asia 2010, pages 160:1–10, December 2010.

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A Simple Spring

• Ideal zero-length spring
• Force pulls points together
• Strength proportional to distance

fa→b = ks(b − a) fb→a = −fa→b

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A Simple Spring

• Energy potential

fa→b = ks(b − a) fb→a = −fa→b

• 2
• 1

1 2

• 2
• 1

1 2 2 4 6 8

• 1

2

fa = raE =  ∂E ∂ax , ∂E ∂ay , ∂E ∂az

• E = 1/2 ks(b − a) · (b − a)

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A Simple Spring

• Energy potential: kinetic vs elastic
• 6
• 4
• 2

2 4 6

• 1
• 0.5

0.5 1

E = 1/2 ks(b − a) · (b − a)

E = 1/2 m(˙ b − ˙ a) · (˙ b − ˙ a)

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Non-Zero Length Springs

fa→b = ks b − a ||b − a|| (||b − a|| − l)

Rest length

E = ks (||b − a|| − l)2

• 2
• 1

1 2

• 2
• 1

1 2 1 2 3

• 1

2

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Comments on Springs

• Springs with zero rest length are linear
• Springs with non-zero rest length are nonliner
• Force magnitude linear w/ discplacement (from rest length)
• Force direction is non-linear
• Singularity at

||b − a|| = 0

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Damping

• “Mass proportional” damping
• Behaves like viscous drag on all motion
• Consider a pair of masses connected by a spring
• How to model rusty vs oiled spring
• Should internal damping slow group motion of the pair?
• Can help stability... up to a point

f = −kd ˙ a

f ˙ a

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Damping

• “Stiffness proportional” damping
• Behaves viscous drag on change in spring length
• Consider a pair of masses connected by a spring
• How to model rusty vs oiled spring
• Should internal damping slow group motion of the pair?

fa = −kd b − a ||b − a||2(b − a) · (˙ b − ˙ a)

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Spring Constants

• Two ways to model a single spring

l ∆l

∆l/2

l/2 l/2 ∆l/2 43

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Spring Constants

• Constant gives inconsistent results with different

discretizations

• Change in length is not what we want to measure
• Strain: change in length as fraction of original length

ks

✏ = ∆l l0 Nice and simple for 1D... 44

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Structures from Springs

• Sheets
• Blocks
• Others

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Structures from Springs

• They behave like what they are (obviously!)

This structure will not resist shearing This structure will not resist out-

• f-plane bending either...

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Structures from Springs

• They behave like what they are (obviously!)

This structure will resist shearing but has anisotopic bias This structure still will not resist

• ut-of-plane bending

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• They behave like what they are (obviously!)

Structures from Springs

This structure will resist shearing Less bias Interference between spring sets This structure still will not resist

• ut-of-plane bending

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• They behave like what they are (obviously!)

Structures from Springs

This structure will resist shearing Less bias Interference between spring sets This structure will resist out-of- plane bending Interference between spring sets Odd behavior How do we set spring constants?

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Edge Springs

1 2 3 4 n e n

1 2 ^ ^ ^

n1

^

n2

^

From Bridson et al., 2003, also see Grinspun et al., 2003 u1 = |E| N1 |N1|2 u2 = |E| N2 |N2|2 u3 = (x1 −x4)·E |E| N1 |N1|2 + (x2 −x4)·E |E| N2 |N2|2 u4 = −(x1 −x3)·E |E| N1 |N1|2 − (x2 −x3)·E |E| N2 |N2|2

Fe

i = ke

|E|2 |N1|+|N2| sin(θ/2)ui,

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Example: Thin Material

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Discrete Shells

SCA 2003 Eitan Grinspun, Anil Hirani, Mathieu Desbrun and Peter Schröder

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Sharp Creases

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Sharp Creases

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Anisotropic remeshing avoids locking.

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Fracture

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Dominated by discretization artifacts Natural appearance

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Variety of Materials

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