Simulation in Computer Graphics Partial Differential Equations · Simulation in Computer Graphics Partial Differential Equations. ... the physical solution is required to satisfy

Matthias Teschner

Computer Science DepartmentUniversity of Freiburg

Simulation in Computer Graphics

Partial Differential Equations

University of Freiburg – Computer Science Department – Computer Graphics - 2

Motivation

various dynamic effects and physical processesare described by partial differential equations PDEs e.g., wave propagation, advection, diffusion

we consider PDEs that describe the time rate of change of a quantity at fixed positions


Motivation

this is a short overview of a large research field in mathematics and physics applied 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 theses PDFs with finite differences


Outline

introduction 1D advection

mathematical background definition

types

boundary conditions

numerical solution methods

examples advection

diffusion

wave equation


Advection

simulation of temperate evolution

consider temperature at position and time

start with known temperatures at time 0:

consider some wind speed

how to compute at arbitrary position and time


Advection Equation in 1D

consider temperature change within a small time step at a fixed position

temperature is only advected no additional effects

T depends on two variables(space and time) ∂/ ∂t denotes the time derivative∂/ ∂x denotes the space derivative


Analytical Solution

define an initial condition temperature distribution at all positions at time 0

at a position and time is obtained byshifting the temperature distribution at time 0 by the distance


Numerical Solution with Finite DifferencesDiscretization

consider / sample the function at discrete positions with distance and discretetimes with time step

approximate the partial derivatives with finite differences


Numerical Solution with Finite DifferencesApproximate Computation of T

can be solved for

from the initial condition, temperatureis known at all sample positions at some time

i.e., for all positions , we can compute

if we have computed for all positions ,we can compute

and so on …


Numerical Solution with Finite DifferencesBoundary Conditions

at time , the temperature is known at sample positions:

however, the computation of the temperature at requires the temperature at

setting boundary conditions / define missing values e.g., periodic boundaries

temperature leaving/entering the right-hand side of the simulation domain, enters/leaves the left-hand side of the domain


Outline



types

boundary conditions


examples advection

diffusion

wave equation


Definition

a partial differential equation PDE describes the behavior of an unknown multivariable function using partial derivatives of , e.g.

the function and

the independent variables, e.g.

example:

notation

ordinary differential equations are a special casefor functions that depend on one variable


PDEs in Physics

independent variables are static problems: (1D), (2D), (3D)

dynamic problems: (1D+1), (2D+1), (3D+1)

unknown functions are scalars, e.g. temperature , density

vectors, e.g. displacement , velocity


PDE Classification

order given by the highest order of a partial derivative

linearity the function and its partial derivatives only occur linearly

coefficients may be functions of independent variables

e.g. second-order linear PDE of functionwith two independent variables general form

non-linear example


PDE Classification

second-order PDEs

can be classified into hyperbolic

parabolic

elliptic

classification is geometrically motivated

is characterized by different mathematical and physical behavior

determines type of required boundary conditions


PDE Classification

hyperbolic time-dependent processes

reversible, not evolving to a steady state

undiminished propagation

e.g., wave motion

parabolic time-dependent processes

irreversible, evolving to a steady state

dissipative

e.g., heat diffusion

elliptic time-independent

already in a steady state


Boundary Conditions

generally, many functions solve a PDE

physical applications expect one solution

is typically defined in a simulation domain

the physical solution is required to satisfy certainconditions on the boundary of

initial condition

Dirichlet condition

Neumann condition


Numerical Solution - Overview

GoverningEquationsIC / BC

DiscretizationSystem ofAlgebraicEquations

Equation(Matrix)Solver

Approx.Solution

Finite Difference

Finite Volume

Finite Element

Spectral

Boundary Element

DiscreteNodal Values

e.g., CGcontinuousform


Discretization

time derivatives finite differences

spatial derivatives finite differences FDM, finite elements FEM, finite volumes FVM

example 1D advection

continuous form

discretizing the time derivative with FD

discretizing the spatial derivative with FD

algebraic equation


2D Sampling

function is reconstructed at fixed sample positions at fixed time points

simplest case: sampling of on a regular grid e.g., 2D+1 dimensions

grid spacing and time step


2D Finite Difference Approximations

time derivative

spatial derivatives

if information moves from left to right, backward is upwind, otherwise forward is upwind

upwind is typically preferred

time marching

forward scheme

backward scheme

central scheme


2D Finite Difference Approximations

higher-order derivatives, e.g.

higher-order approximations, e.g.


Explicit Solution Schemes - 1D Advection

advection equation

upwind

simple stability rule information must not travel more than one grid cell per time step

CFL number

time step should be sufficiently small to result in

discretization

solver (solution scheme)



downwind

centered, FTCS (forward time centered space)

Leap-frog



Lax-Wendroff

Beam-Warming

Lax-Friedrich


Outline



types

boundary conditions


examples advection

diffusion

wave equation


Advection in 2D and 3D

1D advection

generally, the velocity is a vector with direction and length

spatial derivative of in direction

therefore,

2D 3D


2D Finite Difference Solution Scheme

advection equation

discretization

solution scheme

variant velocity can vary with the location

PDE still first order and linear


Diffusion in 1D

density describes the concentration of asubstance in a tube with cross section

conduction law: the mass flow through an area (flux)is prop. to the neg. gradient of the density normal to

k - conductivity


Diffusion in 2D and 3D

equation

intuition for second spatial derivative

constant gradient, i.e. , has no effect

Laplaceoperator

Flux, no change Flux, positive change

2D 3D


2D Finite Difference Solution Scheme

diffusion equation

discretization

solution scheme

intuition if the average value of the neighboring cells is larger than

the cell value, the value increases and vice versa


Wave Equation in 1D

function is the displacement of the string normal to

assuming small displacement and constant stress

force acting normal to cross section is

component in -direction

Newton’s Second Law for an infinitesimal segment

Vibrating string


Wave Equation in 2D and 3D

1D wave equation

2D:

3D:

2D finite difference solution scheme

c – speed of wave propagation


Demo

combination of wave equation, advection, and diffusion

update rule (spatial derivatives not discretized)

wave equation

advection equation

diffusion equation


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

Documents

Simulation in Computer Graphics Partial Differential Equations · Simulation in Computer Graphics Partial Differential Equations. ... the physical solution is required to satisfy