Spatial Modeling

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Spatial Modeling

• Modeling and Simulation of Computer Systems

by Kai Bormann Supervisor Illya Stepanov
22. February 2005
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Contents

• Introduction
• Space-based Approach
• Cellular Automata
• Game of Life
• Margolus Neighborhood
• Lattice Gas Dynamics
• Ising Models
• Entity-based Approach
• L-Systems
• Particle Systems
• Summary

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Introduction
Spatial Models
Entity-based Models
Space-based Models
Space is the object
Entity in the space is the object
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Space-based Approach

• Introduction
• Space-based Approach
• Cellular Automata
• Game of Life
• Margolus Neighborhood
• Lattice Gas Dynamics
• Ising Models
• Entity-based Approach
• L-Systems
• Particle Systems
• Summary

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Cellular Automata (CA)

• Model discrete dynamic systems
• Consists of
• N-dimensional infinite grid
• Regularly ordered cells having m-states
• Rules applied for each cell

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Grid of Cellular Automata
Theory N-dimensional infinite grid Praxis 1 or
2 dimensional grid Problem The grid is not
infintite! Solution Form grid to a ring (1D)
or torus (2D)
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Cells of Automata

• Regularly ordered cells
• Cell structure normally rectangular, but
  triangular or n-angular
• cells are also possible
• Each cell has neighbors
• von Neumann Moore Self-defined
• neighborhood neighborhood neighborhood
•   

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Rules of Cellular Automata

• All rules are applied at each cell at time t
• The next state t1 depends on
• The state of the cell at time t
• The states of the neighboring cells at time t
• The result is in t1

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Example of a CA

• Game of Life developed by the mathematics John
  Horton Conway
• Consists of
• n n square board
• Moore neighborhood
• 2 states
• 2 rules
• birth rule
• death rule

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Game of Life
Generation 0
Generation 1
Generation 25
Generation 50
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Margolus Neighborhood

• A CA with Margolus neighborhood
• Each block has a fix cell number
• The rules are applied for each block
• Blocks are subdivided into phases
• The different phases are overlapping
• Example

Phase 1
Phase 2
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Lattice Gas Dynamics
Using Margolus neighborhood 6 Rules
With collision detecting
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Lattice Gas Dynamics
With this 6 rules it is possible to simulate
diagonal movement
Phase 1, t0
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Lattice Gas Dynamics
With this 6 rules it is possible to simulate
diagonal movement
Phase 2, t1
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Lattice Gas Dynamics
With this 6 rules it is possible to simulate
diagonal movement
Phase 1, t3
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Lattice Gas Dynamics
With this 6 rules it is possible to simulate
diagonal movement
Phase 2, t4
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Ising Model

• Developed by the german physics Ernst Ising
• Describes the behavior of magnet materials
• An increasing temperature leads to a growth of
  the interference factor
• CA can simulate this model
• Consists of
• n n grid
• 2 states
• von Neumann neighborhood

Low High interference factor
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Example of Ising Model

• From http//bartok.ucsc.edu/peter/java/ising/keep/
  ising.html
• States are White and Blue

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Entity-based Approach

• Introduction
• Space-based Approach
• Cellular Automata
• Game of Life
• Margolus Neighborhood
• Lattice Gas Dynamics
• Ising Models
• Entity-based Approach
• L-Systems
• Particle Systems
• Summary

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L-Systems
Developed by the botanist Astrid Lindenmeyer G
(V, ?, P) V Alphabet ? Initial condition P
Production rules Alphabet F Draw a
vertical line with the length d f Does not draw
a line with the lenght d A clockwise rotation
of ? - A counter clockwise rotation of ?
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L-Systems Example 1

• G (F,,-, ?, P) ? 90
• F-F
• p1 F?F-FFF-F

Initial condition
First iteration
Second iteration
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L-Systems Example 2

• Save the current state on the stack.
• Get the saved state from the stack.
• G (F, , -, , , ?, P) ? 26
• F
• p1 F?F F F -F F

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Particle Systems

• Gas Dynamics with Time Slicing
• Gas Dynamics with Event Scheduling
• N-body Methods
• Particle Particle Method
• Particle Mesh Method
• Tree Code Method

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Gas Dynamics with Time Slicing

• Each molecule has
• Initial position
• Velocity
• Straight trajectory
• Collisions and gravity are ignored
• Formula to calculate the new position xnew
  xold vx ?t
• Positions are updates at every ?t

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Gas Dynamics with Event Scheduling

• Event a molecule hits a wall
• Intersection time is calculated
• Molecules are entered into the Event List
• Positions are updated upon each event
• Advantage Simulation is faster
• Problem pay attention at events with the same
  timestamp

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Particle Systems

• Gas Dynamics with Time Slicing
• Gas Dynamics with Event Scheduling
• N-body Methods
• Particle Particle Method
• Particle Mesh Method
• Tree Code Method

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Particle Particle Method

• Calculate the force of each particle
• Calculate for each particle the new position and
  speed
• Update the timestamp
• Advantage accurate and simple
• Disadvantage N(N-1) calculations are needed
• gt complexity O(N2)
• gt bad for a high number of particles

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Optimization with Particle Mesh Method

• Lay a mesh (regular grid) over the particle space
• Calculate the forces of the meshpoints
• Calculate the force of each particle
• Calculate for each particle the new position and
  speed
• Update the timestamp
• Disadvantages
• Improves only if the number of meshpoints (M) is
  smaller than the number of particles (N)
• No collision detecting

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Tree-Code Method

• Demount the rectangles until all particles have
  their own rectangle
• Starting in the leafs
• Calculate the center of
• mass (COM) of each quad-cell
• Calculate the force of each particle
• Calculate for each particle the new position
  and speed
• Update the timestamp
• Complexity O(N log N)

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(COM)
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Summary
Spatial Models
Entity-based Models
Space-based Models
Cellular Automata
L-Systems Particle Systems
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Questions