Physics-Based Animation Prof. Rahul Narain About me Rahul Narain - - PowerPoint PPT Presentation

• Prof. Rahul Narain

COL781 guest lecture:

Physics-Based Animation

About me

• Rahul Narain


http://rahul.narain.name/
 narain@cse.iitd.ac.in

• Assistant professor in CS&E


Bharti IIA-517

• Research interests:


computer graphics, physics-based animation,
 numerical methods

Turbulent fluids and
 granular materials Cloth
 modeling Parallel algorithms for
 interactive simulation Crowd
 simulation

Basic types of animation

• Manual
• e.g. keyframing

• Recorded
• e.g. motion capture

• Algorithmic
• e.g. physics-based

Physics-based animation (a.k.a. simulation)

Basic types of animation

• Manual
• e.g. keyframing

• Recorded
• e.g. motion capture

• Algorithmic
• e.g. physics-based

more automation less artistic control

Physics-based animation (a.k.a. simulation)

Today’s agenda

• Basic principles of physics-based animation
• Mass-spring systems
• Preview of other techniques

Discretization

How to represent the continuous motion of an object?

x(0) x(t) Motion of a particle: x : ℝ → ℝ3 Flow field of a fluid: v : ℝ3 × ℝ → ℝ3 v(x, t)

Discretization

Discretization of space Discretization of time

• Time stepping

particles grids meshes

Discretization

Objects represented as collections of particles

• ith particle has mass mi, position xi, velocity vi

Time divided into discrete time steps

• Usually t0, t1 = t0 + ∆t, t2 = t0 + 2∆t, …

Given positions, velocities at tn,
 compute positions, velocities at tn+1

Equations of motion

Newton’s second law: f = ma Or, for each particle: fi = total force on particle i
 (including forces from environment and from other particles) Two questions:

• How to define fi?
• How to solve d2xi/dt2 = …?

d2xi dt2 = fi mi

Recap: Single-particle dynamics

Consider a point mass attached to a fixed point via a spring

• Total force = mg − ks(‖x‖ − ℓ0)e − kd(v · e)e + fuser
• Call this f(x(t), v(t))

dv/dt = f/m dx/dt = v . Simplest time stepping scheme (forward Euler): vnew = v + f(x, v)/m ∆t xnew = x + v ∆t .

fspring mg

Recap: Single-particle dynamics

What you covered in an earlier class is slightly difgerent: vnew = v + f(x, v)/m ∆t xnew = x + vnew ∆t . This is sometimes called semi-implicit Euler or symplectic Euler, actually much better than forward Euler.

Multi-particle dynamics

For each particle, Let's define and similarly v and f. Then we can just write where M = diag(m1 I, m2 I, …)

dvi dt = fi mi dxi dt = vi

dv dt = M−1f dx dt = v

x =    x1 x2 . . .   

Time stepping

Can still apply the same scheme: vnew = v + M−1 f(x, v) ∆t xnew = x + vnew ∆t . Algorithm:

• Compute f = total force acting on each particle
• For each particle, update v, then update x

Mass-spring systems

Collection of particles with springs
 connecting them

xi xj fij fji

Mass-spring cloth

Simulate as particles but render as smooth surface

Mass-spring cloth

Create grid of particles Add springs to model internal forces structural springs

Mass-spring cloth

Create grid of particles Add springs to model internal forces shear springs

Mass-spring cloth

Create grid of particles Add springs to model internal forces bending springs

Mass-spring cloth

Create grid of particles Add springs to model internal forces bending springs

Other applications

Other applications

Advanced topics

• Collision handling
• Time integration schemes
• Realistic materials using the finite element method
• Fluids

Collision handling

Collision detection: check if objects
 intersect, if so return collision point
 and normal

• Discrete vs. continuous

Collision response: apply normal & frictional forces to fix it

• Penalty forces / collision impulses / more complex algorithms

Time integration schemes

Semi-implicit Euler has a “speed limit”:
 If ∆t or spring const too large, solution blows up! Unconditionally stable method: backward Euler: vnew = v + f(xnew, vnew)/m ∆t xnew = x + vnew ∆t . Requires solving a system of equations! (Newton’s method) Analysis of time stepping schemes:

• Accuracy vs. stability vs. conservation vs. compute cost

Finite elements

Instead of springs, divide object into collection of volumetric “elements” (triangles in 2D, tetrahedra in 3D)

• Spring rest length → element rest shape
• Spring constant → stress-strain relationship

X1 x1 X2 X3 x2 x3

Fluids

Particles

• Mass, velocity, pressure, etc.

stored on particles

• Particles move with fluid

Grids

• Velocity, pressure, etc. stored
• n grid vertices / cells
• Grid doesn’t move

Forces: pressure, viscosity, surface tension, …

Online notes for further reading

Foundations:

• Bargteil and Shinar, “An Introduction to Physics-based Animation”


https://cal.cs.umbc.edu/Courses/PhysicsBasedAnimation/ Finite elements for animation:

• Sifakis and Barbič, “FEM Simulation of 3D Deformable Solids”


http://www.femdefo.org/ Fluid simulation:

• Bridson and Müller-Fischer, “Fluid Simulation for Computer

Animation”
 https://www.cs.ubc.ca/~rbridson/fluidsimulation/