partial differential equations for computer animation ... › ~sullivan › part-time-2011 ›...

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University of Freiburg - Institute of Computer Science - Computer Graphics Laboratory

Partial Differential Equationsfor Computer Animation


Acknowledgement

• A major part of this slide set and all accompanyingdemos are courtesy of Matthias Müller, ETH Zurich, NovodeX AG.


Motivation

• Most dynamic effects and physical processes can bedescribed by partial differential equations PDEs

• We give a short overview of a large research field in mathematics and physicsapplied to the fields of animation and computer graphics

• Goals• You know what a PDE is• You know PDEs for specific effects• You know how to solve these PDEs numerically• You can use your knowledge to implement

your own effects and animationsUniversity of Freiburg - Institute of Computer Science - Computer Graphics Laboratory

Introductory Example

• Simulate temperature evolution• One space dimension + time

T

x

cT(x,0)

• Given the temperature distribution T(x,0) at time t=0and wind speed c.

• Find PDE for the temperature evolution T(x,t) at time t !

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Advection

• How does the temperature change in one time step?

• PDE for T(x,t) is 1-d advection (transport) equation.

x

TT(x,t)

T(x,t+∆t)c∆t

∆T

x

xT

tcT

∂∂≈

∆∆−

xTc

tT

∂∂−=

∂∂

xt cTT −=or0→∆t


Analytical Solution

• Any T(x,t) of the form

)(),( ctxftxT −=

solves

• Any T(x,t) of the form

)()0,( 0 xTxT =

• The physical solution also needs to satisfy the initial condition

• Thus, the solution is)(),( 0 ctxTtxT −=

xt cTT −=

• T is obtained by shifting T0 through a distance ct withoutany change of shape of T0 .

• Only simple PDEs can be solved analytically.


Numerical Solution – Finite Differences

..)2,1,0(),,..,1(),,(][ ∈∈∆⋅⋅= tnitthiTiT t• Sample T(x,t):

• Discretize the derivatives (e.g. first-order backward scheme)

xt cTT −=h

iTiTct

iTiT tttt ]1[][][][1 −−−=∆

−+

hiTiTctiTiT

tttt ]1[][][][1 −−⋅∆−=+

• Solving for Tt+1[i] yields the explicit update rule

• Before starting the simulation, initialize T0[i]• Possible boundary conditions are periodic boundaries

]1[]1[],[]0[ tttt TnTnTT =+=


Outline

Mathematical BackgroundWhat is a PDE?Types of PDEsBoundary Conditions

Solution TechniquesAnalytical MethodsNumerical Methods

ExamplesAdvectionDiffusionThe Wave Equation

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Mathematical Definition of a PDE

• A Partial Differential Equation (PDE) is an equation thatdetermines the behavior of u in terms of

• partial derivatives of u, e.g. uxy, utt, • u itself and• the independent variables, e.g. x,y,z,t

xuuu xytt +=• Example:

,..),,(,..),(2

2

2

yxuyx

utut

u xytt ∂∂∂=

∂∂=• Notation:

• Given a function u of two or more independent variables, e.g. u(x,y,z,t)


PDEs in Physics

• Independent variables are • Static Problem: x (1D), x,y (2D), x,y,z (3D)• Dynamic Problem: x,t (1D+1) x,y,t (2D+1) x,y,z,t (3D+1)

• Unknown function u can be • u(x,..) a scalar,

e.g. temperature T, density ρ• u(x,..) = [u(x,..), v(x,..), w(x,..)]T a vector,

e.g. displacement u, velocity v


PDE Classification

• Order• Order of PDE = Order of highest partial derivative

0),(),(),(),(

),(),(),(

7654

321

=+++

+++

yxfuyxfuyxfuyxf

uyxfuyxfuyxf

yx

yyxyxx

• 2nd order linear PDE of 2 independent variables:

uuu xyxx =⋅• Non-linear example

• Linear• u and its partial derivatives only occur linearly• coefficients may be functions of independent variables


PDE Classification

• Classification of 2nd order linear PDEs

),,,,(),(),(2),( yxyyxyxx uuuyxFuyxCuyxBuyxA =++

• Hyperbolic B2-AC > 0• Parabolic B2-AC = 0

• Elliptic B2-AC < 0

• Geometric motivation• Different mathematical and physical behavior• Determines type of boundary conditions needed• For more than 2 independent variables use generalized

criterion (see textbook)

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PDE Classification

Hyperbolic• time-dependent processes• not evolving to a steady state, reversible• propagate behavior undiminished, e. g. wave motion

Parabolic• time-dependent processes• evolving to a steady state, irreversible• dissipative, e. g. heat diffusion

Elliptic• time-independent• already in steady state


Boundary Conditions

• Generally, there are many u which solve the PDE

• In physical applications only one solution is expected

x

t

D

x0

x1

• Typically u is only defined in a region D

• The physical solution is required to satisfy certainconditions on the boundary δD of D

Initial condition)()0,( xuxu o=

)(),( 11 tftxu =

Value prescribedDirichlet condition

)(),( 00 tftxux =Derivative prescribedNeumann condition


Outline





Analytical Solution Techniques

• Analytical solutions only exist for small and simple problems

• linear equations, simple geometry, initial and boundary conditions

• Examples for analytical solution techniques (see textbooks):• Method of separation of variables• Theory of characteristics• Green function method• Laplace transform

• Real-world problems are typically solved numerically

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Numerical Solution - Overview

GoverningEquationsIC / BC

DiscretizationSystem ofAlgebraicEquations

Equation(Matrix)Solver

Approx.Solution

ContinuousSolution

Finite DifferenceFinite VolumeFinite ElementSpectralBoundary Element

DiscreteNodal Values

ExplicitsolutionCG


Discretization

Spatial derivatives• Finite-difference method FDM• Finite-element method FEM• Finite-volume method FVM• Spectral methods• Boundary element methods• …

Time derivatives• Finite-difference methods


The Finite Difference Method

• Simplest method to understand and implement

),,(],,[ tthjhiutjiu ∆⋅⋅⋅=

• Sample function u on a regular grid, e.g.

,...)2,1,0(),,..,1(),,..,1( ∈∈∈ tmjni

• with grid spacing h and time step ∆t

u[i,j,t]h

h

x

y



• Approximation of the temporal derivative

• Approximation of spatial derivatives (three possibilities)

)(],,[]1,,[

],,[ tOt

tjiutjiutjiut ∆+

∆−+

=

)(],,[],,1[

],,[ hOh

tjiutjiutjiux +

−+= forward scheme

)(],,1[],,[

],,[ hOh

tjiutjiutjiux +

−−= backward scheme

)(2

],,1[],,1[],,[ 2hO

htjiutjiu

tjiux +−−+

= central scheme

• If information moves from left to right (as in ut= -cux)backward is upwind (upwind is best choice)

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• Approximation of higher derivatives

)(],,1[],,[2],,1[

],,1[],,[],,[

22 hO

htjiutjiutjiu

htjiutjiu

tjiu xxxx

+−+−+

=

−−=

• Higher-order approximations

)(12

],,2[],,1[8],,1[8],,2[],,[ 4hO

htjiutjiutjiutjiu

tjiux ++−++−−−

=

)(12

],,2[],,1[16],,[30],,1[16],,2[],,[ 4

2hO

htjiutjiutjiutjiutjiu

tjiuxx ++−++−−+−−

=



• PDE becomes system of algebraic equationsfor the grid values u[i,j,t]

• Linear PDE becomes system of linear equationsfor the grid values u[i,j,t]

• Simple stability rule: Information must not travel morethan one grid cell in one time step

cht /<∆→<∆ htc


Explicit Schemes – Linear Scalar Advection

• upwind (c>0)

0]1,[],[],[],1[

=∆

−−+

∆−+

xituitu

ct

ituitu ( )]1,[],[],[],1[ −−∆∆

−=+ ituituxt

cituitu

• downwind (c>0)

0],[]1,[],[],1[

=∆

−++

∆−+

xituitu

ct

ituitu ( )],[]1,[],[],1[ ituituxt

cituitu −+∆∆

−=+

• centered (Forward time centered space FTCS)

02

]1,[]1,[],[],1[=

∆−−+

+∆

−+x

ituituc

tituitu ( )]1,[]1,[

2],[],1[ −−+

∆∆

−=+ ituituxt

cituitu

• Leap-frog

02

]1,[]1,[2

],1[],1[=

∆−−+

+∆

−−+x

ituituc

tituitu ( )]1,[]1,[],1[],1[ −−+

∆∆

−−=+ ituituxt

cituitu


Explicit Schemes – Linear Scalar Advection• Lax-Wendroff

( ) ( )]1,[],[2]1,[21

]1,[]1,[2

],[],1[2

−+−+

∆∆

+−−+∆∆

−=+ ituituituxt

cituituxt

cituitu

• Beam-Warming (c>0)

• Lax-Friedrich

• CFL number (Courant-Friedrichs-Lewy), important in stability analysis

( ) ( )]2,[]1,[2],[21

]2,[]1,[4],[32

],[],1[2

−+−−

∆∆

+−+−−∆∆

−=+ ituituituxt

cituituituxt

cituitu

( ) ( )]1,[]1,[2

]1,[]1,[21

],1[ −−+∆∆

−−−+=+ ituituxt

cituituitu

xt

c∆∆

=σ

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Outline





Advection in 2D and 3D

• In 2D and 3D, speed c is a vector with direction ncand length c

xt cTT −=• For 1D advection we had

TTt ∇⋅−= c

• Since c = cnc the general advection PDE reads

• We have the spatial derivative of T in direction nc

)( TcT ct ∇⋅−= n• Thus, we have

,TTc

c

∇⋅=∂∂ nn

∂∂∂∂

=∇

=∇

yx

TT

Ty

x

//

,:2Din

Nabla operator


Finite Differences Solution

−−+−−∆−=+

hjiTjiTc

hjiTjiTctjiTjiT yx

tt ]1,[],[],1[],[],[],[1

• The update rule in 2D

TyxTt ∇⋅−= ),(c

• Make the wind velocity a function of the location

• T is advected by a general wind field• Use velocity array, replace cx by cx[i,j] and cy by cy[i,j]• PDE still first order and linear


Diffusion in 1D

• Conduction law: The mass flow through an area A (flux) is proportional to the gradient of the density normal to A:

• The density ρ(x) describes the concentration of a substance in a tube with cross section A.

xx x+dx

Am=ρ(x)Adx

A

xxdxxx kkdtA

AdxxddtAdm ρρρ −==

+

)(

xxxxdxxx

t kdx

k ρρρ

ρ =−

= +xxt kρρ =

Conductivity

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Diffusion in 3D

• The diffusion PDE in 3D

• Intuition for second spatial derivatives• Constant concentration gradient ρ(x)=a+bx has no effect

Flux but no change Flux and positive change

ρρρρρρ ∆=∇=++= kkk zzyyxxt2)(

Laplace operator



21 ],[4]1,[]1,[],1[],1[],[],[

hktjiji jijijijijitt ρρρρρρρ −−+++−++∆+=+


• Intuition:If the average concentration of the neighbor cells is larger than the concentration of the cell, its concentration increases and vice versa.


The 1D Wave Equation

• Function u(x) is displacement of the string normal to x-axis

u

• Assuming small displacements and constant stress σ

xu Auf σ≈• Component in u-direction

xxdxxxtt uAuAuAdx σσρ −=+

)( xxtt uu σρ =

• Newton’s 2nd law for an infinitesimal segment

• Force acting normal to cross section A is Af σ=

xx x+dx

A

u

Vibrating string

fuf


Analytical Solution

xxtt uu σρ =

)()( ctxfbctxfa −⋅++⋅• with the analytical solution

• For the string we have

xxtt ucu 2=?s

c =• The standard form is where

• so c is the speed with which waves travel

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2D Wave Equation

• The wave equation in 2D

ucucuucu yyxxtt ∆=∇=+= 2222 )(

• Waves propagate with velocity c• For the vibrating string we have c2=σ/ρ



221 ],[4]1,[]1,[],1[],1[],[],[

hctjivjiv jiujiujiujiujiutt −−+++−++∆+=+


• Very simple to implement• Yields cool water surface animation

],[],[],[ 11 jivtjiujiu ttt ++ ∆+=

• Boundary conditions assuming• Periodic:• Mirror:

• Analog for j

],1[],1[],,[],0[ jujnujnuju =+=],[],1[],,1[],0[ jnujnujuju =+=

),..,1( ni ∈


Demo

tw

tt utcvv 221 ∇∆+=+

• Combination of wave equation, diffusion, advection (1D)

xxwtt ucu 2−=

xxdt ucu =

xat ucu −=

( )ta

td

ttt ucucvtuu ∇−∇+∆+= ++ 211

• Explicit Leap-frog integration

wave equation

advection

diffusion


Literature

• Alan Jeffrey, „Applied Partial Differential Equations – An Introduction,“ Academic Press, Amsterdam, ISBN 0-12-382252-1

• John D. Anderson, „Computational Fluid Dynamics – TheBasics with Applications,“ McGraw-Hill Inc., New York, ISBN 0-07-001685-2

partial differential equations for computer animation ... › ~sullivan › part-time-2011 ›...

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