Real-time Physics Round-up Revised Schedule of Demos TUESDAY 06/03 - - PowerPoint PPT Presentation

Real-time Physics Round-up

Revised Schedule of Demos TUESDAY 06/03

Session #1 (11-1pm)

Name Time

Shane 11:30 Carlos 11:40 Dermot 11:50 Andy 12:00 Kevin 12:10 Colin 12:20 Owen 12:30 Brendan 12:40 Gavin 12:50

Session #2 (2-3pm)

Name Time Name Time Jan 13:00 Stephen 13:10 Darragh 13:20 Siddiqur 13:30 Dariaus 13:40 Gianluca 13:50 Jeff W 14:00 Rick 14:10 Luis 14:20 Geoff K 14:30 Marco 14:40

Please note Small changes (in red) due to a Postgraduate Committee meeting scheduled before this

Finite Element Simulation

Mass spring

Pros:

Easy to implement Reasonable fast

Some problems:

Lack of correct preservation of volume Behaviour is heavily dependent on mesh topology and choice of springs

Finite Element Simulation

A more physically accurate approach –popular technique in physics and engineering Employs finite element method to solve the partial differential equations which govern the dynamics of an elastic material Objects are modelled as a three-dimensional elastic continuum by breaking it into a large number of solid elements which fit together, and solving for the stresses and strains in each element using a model of the material. Pixelux use a finite-element-based approach for DMM, using a tetrahedral mesh

Pixelux Demo, Parker & O’Brien 2009

Some impressive FEM based effects used in Force Unleashed in their 2009 paper at SCA. This is a great example of cutting edge research deployed in a game.

Video and paper: http://graphics.eecs.berkeley.edu/site_root/papers/Parker-RTD-2009-08/ Eric G. Parker and James F. O'Brien. "Real-Time Deformation and Fracture in a Game Environment". In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pages 156–166, August 2009.

Continuum mechanics

Branch of physics that studies the behavior of materials modeled as a continuum

Solid: There exists some equilibrium state /rest position for the body Characterised by Elasticity / Plasticity Fluids: Characterised by Viscosity

Deformable properties

Stress within a solid object is the Magnitude of force applied on an object divided by the surface area over which it acts

Has units of pressure

Strain is the fractional deformation caused by stress Elasticity: Tendency of a body to return to its original shape after the forces causing deformation are removed

Opposite of plasticity: tendency to undergo permanent deformation due to forces

Hooke’s Law in a bit more detail

Hooke’s Law: Extension of spring is directly propotional to the load placed on it f = -kx Force per area applied is proportional to its relative elongation

Can be rewritten Where is the applied stress and the resulting strain E: elasticisty tensor N/m2 defined by young’s modulus

Measuring Deformations

An object in its equilibrium configuration is defined as a connected subset M of 3 .

Coordinates x є M of a point on the

• bject are called material co-
• rdinates of the point

When object deforms, points move to p(x), the spatial world co-ordinates We can represent displacement vectors u(x) = p(x) – x

u defines the displacement field

(x,y,z) p(x,y,z) Deform geometry from its

• riginal configuration

u

Rotation and translation are solid movements and not deformations – spatially constant displacement field Relative movement of point with respect to its neighbours

Strain

The relative elongation (or compression) of the material in 3d A magnitude to measure deformation

A function of the displacement field u(x) Not constant inside a body Derived from the spatial variation or spatial derivatives of the displacement field

stretched compressed

Strain in 3D

In 3D, the displacement function has 3 components

N.b. A point getting compressed in one direction may be depressed in another

Strain cannot be expressed as a single scalar but by a symmetric 3x3 matrix:

Diagonal describes stretch Off-diagonals: shear

Strain in 3D

Strain tensor

Non linear tensor Green tensor:

Invariant to rigid body motions

Linear tensor

Cauchy strain tensor

Invariant to translations Not invariant to rotations!!! Linearization of Green Strain

Where Gradient

• f displacement

field:

Strain in 3D

Green strain tensor vs. Cauchy strain tensor

Stress

Describes forces acting inside object Stress is defined as a force per unit area: As with strain, represented by a 3x3 stress matrix: To get force per unit area with a specific plane defined by normal n:

Axioms of Continuum Mechanics

• 1. A material continuum remains continuum under the

action of forces.

• 2. Stress and strain can be defined everywhere in the

body.

• 3. Stress at a point is related to the strain and the rate of

change of strain with respect to time at the same point.

Stress at any point in the body depends only on the deformation in the immediate neighborhood of that point. The stress-strain relationship may be considered separately, though it may be influenced by temperature, electric charge, ion transport, etc.

Continuum Mechanics

The full continuum model of a deformable object considers the equilibrium of a general body acted on by external forces. The

• bject reaches equilibrium when its potential energy is at a

minimum. The total potential energy P of a deformable system is P =  - W where  is the total strain energy of the deformable object, and W is the work done by external loads on the deformable object. In order to determine the shape of the object at equilibrium, both are expressed in terms of the object deformation, which is represented by a function of the material displacement over the

• bject. The system potential reaches a minimum when dP w.r.t.

displacement function is zero.

Continuum Mechanics

What we need to solve is the momentum equation for a finite element within a continuum

Other Deformable Techniques

Shape Matching

Match rest state to deformed state of a point cloud Replace energies by geometric constraints and forces by distances of current positions to goal positions Versatile and efficient (speed and memory)

Constraint-based

Not to be forgotten, character animation is one of the most common instances of “deformable” object in games Not discussed here but essentially the physics boils down to a hierarchy of rigid bodies, held together with constraints Geometric operations are used to associate the rigid body with the actual deforming rendered geometry E.g. Matrix palette skinning

For more on constraints: see Baraff et al Siggraph Course Notes on PBM

Image (c) Francois Faure

Recommended Reading

Matthias Muller ran a course on real-time Physics at Siggraph 2007 (included a segment by Jos Stam) The course notes provide an excellent review of Real-time Deformable Objects Supplemental Reading: http://www.matthiasmueller.info/realtimephysics/index.html In particular : Chapters 3, 4, and 5 Link to video of Matthias talk (slightly cut down):

https://www.scss.tcd.ie/John.Dingliana/cs7057/muellertalk.mov

De Defor

• rma

mable le ob

• bject

ct Col

• llisi

ision

• ns

Deformable Object Collision Detection

Almost entirely separate problem from Rigid Bodies

Different Data Structures Rigid body data structures usually cannot be computed on the fly strictly depend on the shape of the object Different techniques Different optimizations Different Granularity Self Intersections possible

A typical feature of deformable objects:

Dynamic change of topology and/or (from the point of view

• f a mesh) change of surface tesselation

Strategies

Bounding volume hierarchies (AABB trees, OBB trees, sphere trees) Grids, either uniform (using hashing for memory efficiency) or hierarchical (e.g. Octree, kd-tree) Coherence-exploiting schemes, such as sweep and prune with insertion sort, or tree-tree collisions with front tracking. Hybrid methods involving a combination of various of these schemes, e.g. a coarse AABB tree plus sweep-and- prune with coherence between colliding leaves.

Hierarchies

Axis Aligned Bounding Boxes are the most used, because they are:

Easiest to compute Easiest to check for interference

They do not provide the best fitting volume, but this concept becomes a bit fuzzy for deformable objects The manner in which the bounding volumes is computed is the most notable difference among the various approaches

Best fit : sphere Best fit : box

AABB Trees

Problem statement: the surface of an object is tessellated with polygons The hierarchy of boxes can be quickly updated using the following property:

Let Sm(R) be the smallest AABB of a region R and r1, r1 two regions.

The hierarchy is updated in O(n) time Note: this is not the same as rebuilding the hierarchy

[Van Den Bergen 1998] Public domain library: http://www.win.tue.nl/~gino/solid/index.html

) ( )) ( ) ( (

2 1 2 1

r r Sm r Sm r Sm Sm   

refitting rebuilding

BD-Trees

Bounded Deformation Tree, or BD- Tree, can perform collision detection with reduced deformable models at costs comparable to collision detection with rigid objects

BD-Tree Update

Preprocess:

Compute Bounding Box of leaves Compute parent BB in bottom-up order

At Run Time – Refit Tree

Refitting AABB takes constant time - Ten times faster than rebuilding it Refitted tree has more overlap

BD-Tree Summary

Doug L. James and Dinesh K. Pai. BD-Tree: Output-Sensitive Collision Detection for Reduced Deformable Models. ACM Transactions on Graphics (SIGGRAPH 2004), 23(3), August 2004.

Distance Fields

Specify (signed) distance to a closed surface for all points in the field Fast evaluation of distances and normals Independent of complexity of object

Distance Fields

Many techniques for storing

Uniform 3d grid Octree Bsp tree

Distance values calculated at each grid and then tri- linearly interpolated Some Resolution problems

Solution Adaptive Sampling (ADF)

Fuhrman, Sobottka, Gross. Distance Fields for Rapid Collision Detection in Physically Based Modeling International Conference on Computer Graphics and Vision. Graphicon 2003.

• S. Fisher, M. Lin.

Deformed Distance Fields for Simulation of Non-Penetrating Flexible BodiesEG Symposium on Comptuer Animation 2001.

Distance Fields

DiFi: Fast 3D Distance Field Computation Using Graphics Hardware by Avneesh Sud, Miguel A. Otaduy, and Dinesh Manocha.

References

Collision Detection for Deformable Objects

• M. Teschner, S. Kimmerle, B. Heidelberger, G. Zachmann,

Laks Raghupathi, A. Fuhrmann, Marie-Paule Cani, François Faure, N. Magnetat-Thalmann, W. Strasser, P. Volino Computer Graphics Forum, Volume 24, Number 1, page 61--81 - March 2005 http://www- evasion.inrialpes.fr/Publications/2005/TKHZRFCFMSV05/ STARmain.pdf

Fluid Simulation – Basic Concepts

Some slides are based on Notes by : (1) Marcos Garcia (2) Richard Lee (3) Matthias Mueller (4) Michael Su

Granular Flow e.g. sand [Zhu, Bridson] Animating Sand as a Fluid, SIGGRAPH 05 Smoke [Fedkiw, Stam, Jensen] Visual Simulation of Smoke, SIGGRAPH 01 Newtonian Fluids [Tessendorf] Simulating Ocean Water, SIGGRAPH 01 Non-newtonian Fluid [Goktekin, Bargteil, O'Brien] A Method for Animating Viscoelastic Fluids, SIGGRAPH 04 Microscopic Effects [Wang, Mucha, Turk] Water Drops on Surfaces, SIGGRAPH 05

Different types of fluids and representative papers at SIGGRAPH

Fluid Characteristics (1)

Basic properties

Pressure Density Viscosity (subject to shear stress) Surface tension

Different types of fluids:

Incompressible (divergence-free) fluids: Fluids doesn’t change volume (very much). Compressible fluids: Fluids change their volume significantly. Viscous fluids: Fluids tend to resist a certain degrees of deformation

Fluid Characteristics (2)

Inviscid (Ideal) fluids: Fluids don’t have resistance to the shear stress Turbulent flow: Flow that appears to have chaotic and random changes Laminar (streamline) flow: Flow that has smooth behavior Newtonian fluids: Fluids continue

to flow, regardless of the force acting on it

laminar turbulent

Fluid Characteristics (3)

Non-Newtonian fluids: Fluids that have non-constant viscosity Phase Transition: Fluids may change

physical behavior under different environmental conditions.

Calculus Review (1)

Gradient ( ): A vector pointing

in the direction of the greatest rate of increment

Divergence ( ): Measure how the

vectors are converging or diverging at a given location (volume density of the outward flux)

                z u y u x u u , ,

u can be a scalar or a vector

z y x            u u u u

u can only be a vector Source, Div(u) > 0 Sink, Div(u) < 0

                z u y u x u u , ,

  

Calculus Review (2)

Laplacian (∆ or ): Divergence of the gradient Finite Difference: Derivative approximation

2



2 2 2 2 2 2 2

z u y u x u u          

u can be a scalar or a vector i i i i

x x u u x u     

  1 1

Useful operators

dy dx

xij xi(j-1) x(i-1)j

Compressible Navier-Stokes

Momentum Equation: Continuity Equation: Equation of State:

Change in velocity Diffusion / Viscosity Advection Pressure Body Forces

 

g u u u u            p t    1 . .

 

.      u   t

) ( f p 

u: the velocity field : kinemaic viscocity : density p: pressure g:external force (e.g. gravity)

Navier-Stokes

Equations of motion taken from Computational Fluid Dynamics (CFD) Computationally expensive to solve

Many state variables required to describe continuum Not quite fast enough for games yet (half of the cost is rendering!)

Various simulation techniques

Grid-based approaches: Eulerian Particle-based approaches: Lagrangian Height-field approaches e.g. shallow water equations

Eulerian Approach

Discretize the domain using finite differences Define scalar & vector fields on the grid

Some Redundancy Grid resolution is critical More complex math

[Stam] Stable Fluids, SIGGRAPH 99

Lagrangian Approach

Treat the fluid as discrete particles Apply interaction forces (i.e. pressure/viscosity) according to certain pre-defined smoothing kernels Evaluations:

Mass / Momentum conservation More intuitive Fast, no linear system solving Connectivity information/Surface reconstruction

[Mueller, Charypar, Gross] Particle-Based Fluid Simulation for Interactive Applications, SCA03

Solver Overview

• 1. Specify initial and boundary conditions
• 2. Solving equations each time-step gives us the fluid

velocity at any point in space (interpolate sample points)

• 3. Use some device to represent fluid, e.g.:
• Scalar field for smoke density
• Marker particles or implicit surface for liquid
• 4. Then use fluid velocity field to move fluid around
• 5. Render

Incompressibility

Real fluids are compressible

Fluid density varies through space Usually not important for animation

Simplify by assuming density field is constant:

No longer need an Equation of State Continuity equation simplifies to: Now the pressure field is whatever it takes to make the velocity field have zero divergence

This constraint-based approach generally gives better results:

Analogous to simulating a rigid body instead of a very stiff deformable body

u=0

 

g u u u u            p t    1 . .

Scalar/Vector fields defined

• n the grid

Advection Body Force Diffusion Pressure Solve

u=0

Inviscid Fluids

Another common simplification for water and smoke is to assume zero viscosity Momentum equation w/o viscosity term: Discrete solution to equations introduces errors which resemble physical viscosity

Called ‘numerical dissipation’ Arises from averaging, interpolation, etc. Much of research in fluids tries to combat this dissipation

g u u u         p t  1 .

The Marker and Cell Method

Originally due to Harlow & Welch, 1965 Introduced to graphics by Foster/Metaxas, 1996 Solution to NSE are computed at fixed grid points Solves for incompressible fluid

MAC Grid Discretization

Discretize fields on Cartesian grid(s)

Speed/accuracy/stability all depend on grid resolution

Offline Fluid Simulations

Typical grid size 2563 cells Linear system with 16 million unknowns With rendering (Photorealistic results)

10 seconds to 50 minutes per frame!

3D MAC simulation is almost impossible in real-time for decent sized grids

For games, consider:

• Computational cost
• Stability
• Memory Consumption

Real-time Fluids

But real-time fluids IS possible

Some Common Real-time Solutions

Procedural Water

Unbounded surfaces, oceans

Heightfield Fluids

Ponds, lakes

Particle Systems

Splashing, spray, puddles, smoke

Procedural Animation

Empirical Models

Simulate effect not the cause

Difficult to do accurate interactions with procedural fluids

Procedural Heightfields

Sum of sinusoidal waves (simple solution to NSE) “Canned” animation “Simulating Ocean Water”

• J. Tessendorf .Siggraph

’99 Used in Titanic (also see course notes 2001)

http://www.finelightvisualtechnology.com/docs/coursenotes2004.pdf

Heightfield Fluids

Represent Fluid as a 2D function

Dimensional reduction Impossible to do breaking waves

Try this at home:

=

Heightfield Simulation

Based on simplifications of the full Navier-Stokes equations

Shallow water equations 2D wave equation:

Fast enough for realtime but limited

Particle Fluids

Can define some effects with simple particle system (no-particle- particle interactions):

E.g. Spray / splash

Smoothed Particle Hydrodynamics

Image (c) Richard Lee Particle Simulations that take Navier-Stokes into Account

NSE with Particles

How to formulate Navier-Stokes Eqn. on particles? We need continuous fields, e.g. u(x) We Only have u1, u2, .. un sampled on particles Basic idea:

– Particles induce smooth local fields – Global field is sum of local fields

Particle-particle interaction

For n particles O(n2) potential interactions!

To reduce to linear complexity O(n) define interaction cutoff distance h Also Need some kind of proximity cullling e.g. Spatial hashing

Wave Particles

Cem Yuksel, Donald H. House, and John Keyser, Wave Particles Proceedings of SIGGRAPH 2007 (TOG), 26(3) 2007

GPU heightfield simulation warped to include wave induced flow

Wave forces for interaction with floating particles Scalable: simulation and interaction with 100s

• bject in real-time

http://www.cemyuksel.com/research/waveparticles/

Recommended Reading

Bridson & Mueller: Fluid Simulation for Computer Animation (Sggraph 2007 course):

http://www.cs.ubc.ca/~rbridson/fluidsimulation/

Jos Stam GDC 03 Course notes

http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC0 3.pdf

Tessendorf: Simulating Ocean Water (Siggraph 2004 course notes):

http://www.finelightvisualtechnology.com/docs/coursenotes2004.pdf

Richard Lee (TCD) Introductory Chapters from PhD Thesis on Fluid Simulation and Rendering:

http://www.cs.tcd.ie/John.Dingliana/ET04/Fluids.pdf

Some Current Topics not detailed in the Course

Physically based sound synthesis

Real-time: Bonneel et al Fast Modal Sounds with Scalable Frequency-Domain Synthesis (SIGGRAPH 2008) Van-den Doel et al FOLEYAUTOMATIC: Physically-based Sound Effects for Interactive Simulation and Animation Siggraph 2001 Obrien et al, Synthesizing Sounds from Rigid-Body Simulations – Siggraph 2001

Realistic Fracture Patterns

O'Brien, J. F. and Hodgins, J. K. Animating fracture. Commun. ACM 43, 7 (Jul. 2000), 68-75. O'Brien, J. F. and Hodgins, J. K. Graphical modeling and animation of brittle fracture. Siggraph 1999 Müller, M., McMillan, L., Dorsey, J., and Jagnow, R. Real-time simulation of deformation and fracture of stiff materials. Symposium

• n Comptuer Animation, 2001

In Real-time:

Combining Mocap with dynamics

Demo based on this by Bogdan Bucur (IET 2008-09) http://www.youtube.com/watch?v=wHV3GN8WdFA

ZORDAN, V. B., AND HODGINS, J. K. Motion capture driven simulations that hit and react. Symposium on Computer Animation 2002

Controllable Dynamics

Interactive Manipulation of Rigid Body Simulations Popovic et al Siggraph 2008 Detail-Preserving Fluid Control – Thürey et al – Symposium on compute animation 2006 Many-Worlds Browsing for Control of Multibody Dynamics Christopher D. Twigg Doug L. James (Siggraph 2007)

Physical Interaction

(beyond just the rumblestick)

Novint Falcon apprx. US$ 200 (Gaming Device, 2007) http://www.youtube.com/watch?v=gjAxGVH1JOM

Sensable Phantom OMNI approx USD 5-10k Realistic haptic feedback from virtual simulation Demo available in GV2 of the PHANTOM (just drop by)