Partial Differential Equations for Computer Animation

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

Partial Differential Equations

for Computer Animation

Acknowledgement

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

Motivation

• Most dynamic effects and physical processes can be described by partial differential equations PDEs • We give a short overviewof

a large research fieldin mathematics and physics

applied to the fields of animationand 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 animations

Introductory Example

• Simulate temperature evolution • One space dimension + time

T

x c T(x,0)

• Given the temperature distributionT(x,0)at time t=0 and wind speedc.

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Advection

• How does the temperature change in one time step?

• PDE forT(x,t)is 1-d advection(transport) equation. x T T(x,t) T(x,t+∆t) c∆t ∆T x x T t c T ∂ ∂ ≈ ∆ ∆ − x T c t T ∂ ∂ − = ∂ ∂ x t cT T=− or 0 → ∆t

Analytical Solution

• AnyT(x,t)of the form ) ( ) , (xt f x ct T = − solves

• AnyT(x,t)of the form

) ( ) 0 , (x T0 x T =

• The physical solution also needs to satisfy theinitial condition

• Thus, the solution is ) ( ) , (xt T0x ct T = − x t cT T=−

•Tis obtained by shifting T0through a distance ctwithout

any change of shape of T0 .

• Only simple PDEs can be solved analytically.

Numerical Solution – Finite Differences

..) 2 , 1 , 0 ( ), ,.., 1 ( ), , ( ] [i =Ti⋅ht⋅∆t i∈ n t∈ Tt • Sample T(x,t):

• Discretize the derivatives (e.g. first-order backward scheme) x t cT T=− h i T i T c t i T i Tt t t t ] 1 [ ] [ ] [ ] [ 1 ₋ ₋ − = ∆ − + h i T i T c t i T i T t t t t [] [ 1] ] [ ] [ 1 ₌ ₋_∆ _⋅ − − +

• Solving forTt+1_[i]_{yields the explicit update rule}

• Before starting the simulation, initializeT0_[i]

• Possible boundary conditions areperiodic boundaries ] 1 [ ] 1 [ ], [ ] 0 [ t t t t T n T n T T = + =

Outline

Mathematical Background What is a PDE? Types of PDEs Boundary Conditions Solution Techniques Analytical Methods Numerical Methods Examples Advection Diffusion

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Mathematical Definition of a PDE

• A Partial Differential Equation (PDE) is an equation that

determines the behavior of u in terms of

• partial derivatives of u, e.g. uxy, utt, • uitself and

• the independent variables, e.g. x,y,z,t x uu utt= xy+ • Example: ,..), , ( ,..), ( 2 2 2 y x u y x u t u t utt xy ∂ ∂ ∂ = ∂ ∂ = • Notation:

• Given a functionuof twoor 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 functionucan be

• u(x,..)a scalar,

e.g. temperatureT, densityρ

• u(x,..)= [u(x,..), v(x,..), w(x,..)]T_{a vector,} e.g. displacementu, velocityv

PDE Classification

• Order

• Order of PDE = Order of highest partial derivative

0 ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 7 6 5 4 3 2 1 = + + + + + + y x f u y x f u y x f u y x f u y x f u y x f u y x f y x yy xy xx

• 2nd_{order linear PDE of 2 independent variables:}

u u uxx⋅ xy= • 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 ) , (x yuxx B x yuxy Cx yuyy F xyuux uy A + + = • 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

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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 aremanyuwhich solve the PDE • In physical applications onlyone solution is expected

x

t D

x0 x1

• Typically u is only defined in a regionD

• The physical solution is required to satisfy certain

conditions on the boundaryδDof D

Initialcondition ) ( ) 0 , (x u x u = o ) ( ) , (x1t f1t u = Value prescribed Dirichletcondition ) ( ) , (x0t f0t ux = Derivative prescribed Neumanncondition

Outline

The Wave Equation

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

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Numerical Solution - Overview

Governing Equations IC / BC Discretization System of Algebraic Equations Equation (Matrix) Solver Approx. Solution Continuous Solution Finite Difference Finite Volume Finite Element Spectral Boundary Element Discrete Nodal Values Explicit solution CG

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

) , , ( ] , , [i jt ui h j ht t u = ⋅ ⋅ ⋅∆

• Sample functionuon a regular grid, e.g.

,...) 2 , 1 , 0 ( ), ,.., 1 ( ), ,.., 1 ( ∈ ∈ ∈ n j m t i

• withgrid spacinghand time step∆t u[i,j,t]

h h

x y

The Finite Difference Method

• Approximation of the temporal derivative

• Approximation of spatial derivatives (three possibilities)

) ( ] , , [ ] 1 , , [ ] , , [ O t t t j i u t j i u t j i ut _∆ + ∆ − + = ) ( ] , , [ ] , , 1 [ ] , , [ Oh h t j i u t j i u t j i ux + − + = forwardscheme ) ( ] , , 1 [ ] , , [ ] , , [ Oh h t j i u t j i u t j i ux + − − = backwardscheme ) ( 2 ] , , 1 [ ] , , 1 [ ] , , [ 2 h O h t j i u t j i u t j i ux + − − + = centralscheme

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

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The Finite Difference Method

• Approximation of higher derivatives

) ( ] , , 1 [ ] , , [ 2 ] , , 1 [ ] , , 1 [ ] , , [ ] , , [ 2 2 Oh h t j i u t j i u t j i u h t j i u t j i u t j i u x x xx + − + − + = − − = • Higher-order approximations ) ( 12 ] , , 2 [ ] , , 1 [ 8 ] , , 1 [ 8 ] , , 2 [ ] , , [ 4 h O h t j i u t j i u t j i u t j i u t j i ux + + − + + − − − = ) ( 12 ] , , 2 [ ] , , 1 [ 16 ] , , [ 30 ] , , 1 [ 16 ] , , 2 [ ] , , [ 4 2 Oh h t j i u t j i u t j i u t j i u t j i u t j i uxx + + − + + − − + − − =

The Finite Difference Method

• PDE becomessystem of algebraic equations for the grid valuesu[i,j,t]

• LinearPDE becomessystem of linear equations

for the grid valuesu[i,j,t]

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

c h t< / ∆ → < ∆t h c

Explicit Schemes – Linear Scalar

Advection

• upwind (c>0) 0 ] 1 , [ ] , [ ] , [ ] , 1 [ ₌ ∆ − − + ∆ − + x i t u i t u c t i t u i t u ₍ ₎ ] 1 , [ ] , [ ] , [ ] , 1 [ − − ∆ ∆ − = + uti uti x t c i t u i t u • downwind (c>0) 0 ] , [ ] 1 , [ ] , [ ] , 1 [ ₌ ∆ − + + ∆ − + x i t u i t u c t i t u i t u ₍ ₎ ] , [ ] 1 , [ ] , [ ] , 1 [ uti uti x t c i t u i t u + − ∆ ∆ − = +

• centered (Forward time centered space FTCS)

0 2 ] 1 , [ ] 1 , [ ] , [ ] , 1 [ ₌ ∆ − − + + ∆ − + x i t u i t u c t i t u i t u ₍ ₎ ] 1 , [ ] 1 , [ 2 ] , [ ] , 1 [ + − − ∆ ∆ − = + uti uti x t c i t u i t u • Leap-frog 0 2 ] 1 , [ ] 1 , [ 2 ] , 1 [ ] , 1 [ ₌ ∆ − − + + ∆ − − + x i t u i t u c t i t u i t u ₍ ₎ ] 1 , [ ] 1 , [ ] , 1 [ ] , 1 [ + − − ∆ ∆ − − = + uti uti x t c i t u i t u

Explicit Schemes – Linear Scalar

Advection

• Lax-Wendroff ( ) ([, 1] 2[,] [, 1]) 2 1 ] 1 , [ ] 1 , [ 2 ] , [ ] , 1 [ 2 − + − +       ∆ ∆ + − − + ∆ ∆ − = + uti uti uti x t c i t u i t u x t c i t u i t u • Beam-Warming (c>0) • Lax-Friedrich

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

( ) ([,] 2[, 1] [, 2]) 2 1 ] 2 , [ ] 1 , [ 4 ] , [ 3 2 ] , [ ] , 1 [ 2 − + − −       ∆ ∆ + − + − − ∆ ∆ − = + uti uti uti x t c i t u i t u i t u x t c i t u i t u ( ) ([, 1] [, 1]) 2 ] 1 , [ ] 1 , [ 2 1 ] , 1 [ + − − ∆ ∆ − − − + = + uti uti x t c i t u i t u i t u x t c ∆ ∆ = σ

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Outline

The Wave Equation

Advection in 2D and 3D

• In 2D and 3D, speedcis a vector with directionnc and lengthc

x

t cT

T =− • For 1D advection we had

T Tt=−c⋅∇

• Sincec= cncthe general advection PDE reads • We have the spatial derivative of Tin directionnc

) ( T c Tt=− nc⋅∇ • Thus, we have , T T c c ∇ ⋅ = ∂ ∂ n n      ∂ ∂ ∂ ∂ = ∇       = ∇ y x T T T y x / / , : 2D in Nabla operator

Finite Differences Solution

      − − ₊ − − ∆ − = + h j i T j i T c h j i T j i T c t j i T j i Tt1_[_, _] t_[_, _] _x [, ] [ 1, ] _y [, ] [, 1] • The update rule in 2D

T y x Tt=−c( , )⋅∇

• Make the wind velocity a function of the location

•Tis advected by a generalwind field

• Use velocity array, replacecxbycx[i,j]and cybycy[i,j] • PDE still first order and linear

Diffusion in 1D

• Conduction law: Themass flowthrough an areaA(flux) isproportionalto thegradient of the densitynormal to A:

• The densityρ(x)describes the concentration of a substance in a tube with cross sectionA.

x x x+dx A m=ρ(x)Adx A x x dx x x k k dtA Adx x d dtA dm₌ ρ ₌ _ρ ₋ _ρ + ) ( xx x x dx x x t k dx kρ ρ ρ ρ = + − = xx t kρ ρ = Conductivity

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Diffusion in 3D

• The diffusion PDE in 3D

• Intuition forsecondspatial derivatives

• Constant concentration gradientρ(x)=a+bxhas no effect

Flux but no change Flux and positive change

ρ ρ ρ ρ ρ ρt=k xx+ yy+ zz =k∇ =k∆ 2 ) ( Laplace operator

Finite Differences Solution

2 1 [ 1,] [ 1,] [, 1] [, 1] 4[,] ] , [ ] , [ h k t j i j i t i j i j ij ij ij t _ρ ρ ρ ρ ρ ρ ρ+ = +∆ + + − + + + − −

• The update rule in 2D

• 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 stringnormal to x-axis u

• Assuming small displacementsand constant stress σ

x u Au f ≈σ • Component in u-direction x x dx x x tt Au Au u Adx σ σ ρ ) = ₊ − ( ρutt=σuxx

• Newton’s 2nd_{law for an infinitesimal segment}

• Force acting normal to cross section Ais f=σA x x x+dx A u Vibrating string f_u f

Analytical Solution

xx tt u u σ ρ = ) ( ) (x ct b f x ct f a⋅ + + ⋅ −

• with the analytical solution • For the string we have

xx tt c u u ₌ 2 ? s c= • The standard form is where

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2D Wave Equation

• The wave equation in 2D u c u c u u c utt= xx+ yy = ∇ = ∆ 2 2 2 2 ) (

• Waves propagate with velocityc • For the vibrating string we havec2=σ/ρ

Finite Differences Solution

2 2 1 [ 1,] [ 1,] [, 1] [, 1] 4[,] ] , [ ] , [ h c t j i v j i vt+ ₌ t ₊_∆ ui+ j+ui− j+uij+ +uij− −uij • The update rule in 2D

• Very simple to implement

• Yields cool water surface animation ] , [ ] , [ ] , [ 1 1 j i v t j i u j i ut+ ₌ t ₊_∆ t+

• Boundary conditions assuming • Periodic: • Mirror: • Analog forj ] , 1 [ ] , 1 [ ], , [ ] , 0 [ j un j un j u j u = + = ] , [ ] , 1 [ ], , 1 [ ] , 0 [ j u j un j un j u = + = ) ,.., 1 ( n i∈

Demo

t w t t u tc v v+1₌ ₊_∆ 2_∇2

• Combination of wave equation, diffusion, advection (1D) xx w tt cu u =− 2 xx d t cu u = x a t cu u =−

(

t

)

a t d t t t u c u c v t u u+1= +∆ +1+ ∇2 − ∇

• 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 – The Basics with Applications,“ McGraw-Hill Inc., New York, ISBN 0-07-001685-2