Martin Schreiber, Kristof Unterweger...Martin Schreiber, Kristof Unterweger 16.04.2012 M. Schreiber,...

Technische Universitat Munchen

PSE Game PhysicsSession (1)

Martin Schreiber, Kristof Unterweger

16.04.2012

M. Schreiber, K. Unterweger: PSE Game Physics

Session (1), 16.04.2012 1


Outline

Mathematics and PhysicsPhysics and Mathematical ModelingMathematical Description: Ordinary Differential EquationsTime Discretisation: Finite Difference SchemesProperties of Finite Difference Schemes: Consistency, Stabilityand Convergence

C++Live demonstration in eclipseTemplates


Session (1), 16.04.2012 2


Physical systems - Our modelPhysics engine - What do we talk about here?


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Physical systems - Our model

Physics engine - What do we talk about here?

• Starting point: Physical system with certain properties• Determine laws which hold for these properties

Our physical system:• Set of rigid bodies n, n ∈ {1, ...,N}, N ∈ N• Each body is described by

• its position ~x ∈ RD

• its velocity ~v ∈ RD

• its mass m ∈ R• its geometrical form (we first only consider spheres!)


Session (1), 16.04.2012 4


Property relations (1)

What do we know about the relations of these properties?

• Position ~x(t) changes according to the body’s velocity ~v(t) overtime t

• Velocity of a rigid body changes due to• collisions with other rigid bodies (more information in the

next lab)• external forces acting on the bodies, e.g. gravity


Session (1), 16.04.2012 5


Property relations (2)How do we put our knowledge about the properties into amathematical description?

taken from http://tiramisu.mirellacastellano.com/wp-content/uploads/2010/10/cartoon-car.giftaken from http://www.allmystery.de/dateien/uh60967,1292866473,obelix latein plakat klein 1.jpg

Velocity ⇔ change of position over time ⇔ d~xdt

Forces ⇔ change in velocity over time ⇔ d~vdt


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Conservation laws - The equations of motion

d~xdt = ~v

d~vdt = 1

m

(~Fcoll + ~Fext

) (1)

with external forces ~Fext , collisional forces ~Fcoll andinitial conditions ~x(t = 0) := ~x0, ~v(t = 0) := v0.

Remark: Within the lab, we use the equations of motion asmathematical model that we try to solve.The difference between the “real-world solution” andthe mathematical solution of our model is called model error.


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Mathematical description - BasicsHow do we solve Eq.(1)?

→ A little bit of formalism...• Definition:

dfdt := lim

dt→0

f (t+dt)−f (t)dt is called (time) derivative of a function f : R→ R.

Analogously, the derivative for a vector-type function can be defined asthe derivatives of each vector component.

• Definition:An equation of the type F (t , df

dt ,d2fdt2 , ...,

d l fdt l ) = 0 is called ordinary

differential equation of order l .• Definition:

A system of equations of the form

Fk (t ,df1dt

, ...,dfNdt

, ....,d l f1dt l , ...,

d l fNdt l ) = 0, k = 1, ...,K

with functions f1(t), ..., fN(t) is called a system of ordinary differentialequations.


Session (1), 16.04.2012 8


Let’s consider our mathematical model...

d~xdt = ~v

d~vdt = 1

m

(~Fcoll + ~Fext

) (1)

How can we describe our Eq.(1) in terms of our definitions?


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Let’s consider our mathematical model...

Answer: We need to deal with a system of ordinarydifferential equations of order 1, consistingof 2 · D · N equations.

Which general approaches are there to solve the system?• Analytically→ Not always possible (many rigid bodies, complex

geometries, complex forces); available for simple problems• Numerically→ Can handle many rigid bodies, complex

geometries, complex forces; in most cases: cannot solve theproblem exactly, but up to a certain accuracy


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Towards the numerical solution:The three steps of discretisation

• Step 1:Instead of solving the problem on a certain time interval [0, tend ],we try to solve it only at discrete points in timet0 := 0 < t1 < t2 < ... < tT := tend .In our case, ti := t0 + i ·∆t , i = 0, ...,T , ∆t := tT−t0

T timestep• Step 2:

We try to find discrete approximate solutions ~xh(t), ~vh(t) that are“close” to the exact solutions ~x(t), ~v(t).

• Step 3:We try to find a discrete form for our differential operators fromEq.(1).

Remark: The error between the exact solution of our mathematicalmodel and the solution of our discretised problem is calleddiscretisation error.


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Towards the numerical solution:The three steps of discretisation

How can we approximate the differential expressions from Eq.(1)?

Let’s consider the position vector ~x(t) and define xd (t) := d-thcomponent of vector ~x(t).

Assume that our position evolution is sufficiently “smooth”, i.e. asoften differentiable as we need it to be :-)


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Towards the numerical solution:Finite differences (step 3)

Then, use Taylor expansion for xd (t + ∆t):

xd (t + ∆t) = xd (t) + dxd

dt ∆t + O(∆t2)⇔

dxd

dt = xd (t+∆t)−xd (t)∆t + O(∆t)

(2)

So, we can approximate xd (t + ∆t) from xd (t) as follows:

xd (t + ∆t) = xd (t) + ∆t · vd (t)

Explicit Euler approximation(3)

Remark: Expressions of the form xd (t+∆t)−xd (t)∆t are called

finite differences.


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Towards the numerical solution:Finite difference schemes

Approximation Scheme Error

xd (t+∆t)−xd (t)∆t = vd (t) Explicit Euler O (∆t)

xd (t+∆t)−xd (t)∆t = vd (t + ∆t) Implicit Euler O (∆t)

xd (t+∆t)−xd (t−∆t)2∆t = vd (t) Central difference O

(∆t2

)

Can you think of advantages/ disadvantages of these schemes?


Session (1), 16.04.2012 14


Questions revisited

How “close” are our approximate solutions to the continuous solutionof the mathematical model?

How big should we choose the timestep?

Does our discrete solution approach the continuous solution when wedecrease the timestep?

What happens to numerical errors, that is errors introduced by e.g.the limited computational precision of our computers?


Session (1), 16.04.2012 15


ConsistencyDefinition:A numerical method (for a differential equation) is called consistent,if the solution x∆t (t) according to our solving algorithm approachesthe exact solution x(t) of the mathematical model for ∆t → 0 in asingle timestep, i.e.

lim∆t→0

x∆t (t) = x(t) (4)

taken from http://nl.toonpool.com/user/589/files/the square 556785.jpg


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Consistency: Error estimate

The consistency of our Euler scheme is...

wait for it...

trivial :-), as long as we stick to x(t) ∈ C2:

‖x∆t (t + ∆t)− x(t + ∆t)‖ ≤ C2

maxt

∥∥∥∥d2x(t)dt2

∥∥∥∥dt2


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Stability

Definition:A numerical method is called stable, if numerical errors do notincrease in subsequent timesteps.

The stability of our scheme is...

wait for it...

...here it comes...not that trivial at all :-(


Session (1), 16.04.2012 18


Stability



wait for it...

...here it comes...

not that trivial at all :-(


Session (1), 16.04.2012 18


Stability



wait for it...

...here it comes...not that trivial at all :-(


Session (1), 16.04.2012 18


Stability: Remarks

• Purely implicit discretisation schemestypically are unconditionally stable

• Explicit schemes are only stable in a certaindiscretisation range. For time discretisationschemes, the timestep ∆t often needs to belimited:

∆t < C(Problem) taken from http://www.allmystery.de/dateien/

uh60967,1292866473,obelix latein plakat klein 1.jpg


Session (1), 16.04.2012 19


Convergence

Definition:A numerical method is called convergent, if our numerical methodyields an approximate (global) solution of the continuous problem.

• Consistency, Stability and Convergence are closely related• For linear initial value problems, it holds:

Consistency + Stability⇔ Convergence


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Questions re-revisited

How “close” are our approximate solutions to the continuous solutionof the mathematical model?

How big should we choose the timestep?

Does our discrete solution approach the continuous solution when wedecrease the timestep?

What happens to numerical errors, that is errors introduced by e.g.the limited computational precision of our computers?

Talking about errors: Which errors do you remember?


Session (1), 16.04.2012 21


Outline

Mathematics and PhysicsPhysics and Mathematical ModelingMathematical Description: Ordinary Differential EquationsTime Discretisation: Finite Difference SchemesProperties of Finite Difference Schemes: Consistency, Stabilityand Convergence

C++Live demonstration in eclipseTemplates


Session (1), 16.04.2012 22


C++ introduction

Live demonstration in eclipse


Session (1), 16.04.2012 23


Templates

Generics in Java• Specialized during runtime• ...

Templates in C++• Specialized during compilation time• Advantage: Compiler can run optimizations• Use inlining as much as possible• ...


Session (1), 16.04.2012 24

Martin Schreiber, Kristof Unterweger...Martin Schreiber, Kristof Unterweger 16.04.2012 M. Schreiber,...

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