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29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Philippos Papaphilippou

Building an N-Body Simulator from scratch

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

The N-Body Problem

• The understanding of the motion of stars, planets and other

space objects has always been desirable

• The N-Body Problem is the problem of predicting the

motion of N objects when the only forces with which they interact are the forces described in Newton's Law of Universal Gravitation

• Currently, the two-body problem has already been solved,

but it remains unsolved for N > 2

• Three-body motions has been described as chaotic

F1 F2

F1 = F2 = G m1⋅m2 d

2

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

The N-Body Problem (2)

• Sir Isaac Newton (1643-1727) has been studying the

movement of planet orbits and concluded that they haven't all been following the motion equations created by the astronomers of the time

• Therefore he concluded that it was the gravitational

forces that were affecting the objects orbits

• The two-body problem has been solved by Johann

Bernoulli (1667-1748)

• The three-body problem has been studied by many,

including Newton (1687), Euler (1767), Lagrange (1772) and Forest Ray Moulton (1917)

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

The N-Body Problem (3)

• The three-body problem remained unsolved but there has

been solutions for a restricted version of the problem that stated that one of the bodies has a negligible mass

• Newton implies in some of his writings that n-body

problem may be unsolvable due to the gravitational forces

• 19th century's King Oscar II of Sweden

announced that a prize would been given to anyone who could solve the n-body problem, mentioning the use of Newton's Law of Universal Gravitation properties

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

N-Body Simulations

• N-Body simulators are tools that astrophysicists and

astronomers use to predict the motions of solar objects

• They include from few-body system simulations such as

for our solar systems to large-scale computations including formations of galaxy structures and effects from dark matter

• In computer science it is described to have a complexity
• f O(N2) which means the time required is proportional

to the square of the number of the bodies in the

• simulation. Though, optimizations exists

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

N-Body Simulations (2)

• The first simulations were done by Sebastian von

Hoerner at the Astronomisches Rechen-Institut in Heidelberg, Germany

• Image: CUDA Code

Samples by NVIDIA include an N-Body simulator

https://developer.nvidia.com/cuda-code-samples

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Parallelization

• Due to the problem's nature, it can

be efficiently parallelized because at each time step there are many computations that can be performed independently for each body in the simulation

• Therefore they are usually

performed in GPUs because of their massive number of processing cores

• They are also often used as

performance benchmark for GPUs

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Implementation

• Each body has 4 parameters

– Mass (kg) – Position in 3D Cartesian space (unit-vector, in meters) – Velocity in each axis (unit-vector, in m/s) – Net Force (unit-vector, in N)

• Random values

– During the program's initialization, the first three get a

random value according to the defined ranges

– Net forces are recalculated according to the first 3

parameters of each object in each time step

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Implementation (2)

• Time steps

– The time in which each objects are recalculated is in every predefined time

step, for example for each 0.1 sec

– Smaller time steps mean more accurate simulation – There is not an optimal time step for the current application but the

selection is based on

• Humanly noticeable inaccuracy
• Calculations overhead
• Interesting motion speeds (in conjunction with the 60 frames per second limit)
• Vectors

– Each calculation is done in unit-vectors because of simplicity in

calculations

– We could also use absolute values and directions but this would led to

additional computations (for example Pythagorean theorem in order to find the motion and forces in each axis)

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Implementation Equations

• Each body has 4 parameters
• Vector form of the Newton's law of universal gravitation

mass = mA position = [ x A y A zA] velocity = [ V x V y V z] Fnet = [ F x F y F z]

⃗

F12 = G m1⋅m2 |r12|

3 (⃗

r1−⃗ r 2)

,where G = 6.67408 ⃗ r1−⃗ r2 = [ x1−x2 y1− y2 z1−z2] |r12| = √(x1−x2)

2+( y1− y2) 2+(z1−z2) 2

Distance Unit vector distance F12 F21

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Implementation Equations (2)

• Vector form of displacement equation
• Vector form of equation for velocity after plastic

collision using momentum

(objects merge and we have conservation of momentum)

⃗

pos A = ⃗ V A⋅t + 1 2⋅⃗ aA⋅t 2 x A = V x⋅t + 1 2⋅ax⋅t2 y A = V y⋅t + 1 2⋅a y⋅t2 z A = V z⋅t + 1 2⋅az⋅t2

⃗

V = m1⋅⃗ V 1 + m2⋅⃗ V 2 m1+m2

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Using JavaFX

• Java is a programming language designed by Sun

Microsystems (now acquired by Oracle Corporation) in 1995

• The main goals of Java was to be general purpose,

platform independent/portable, object-oriented, high- level and widely available

• JavaFX was created in order to replace Swing, the

existing rich content framework creation library

• JavaFX supports 3D graphics

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Using JavaFX (2)

• Advantages of using JavaFX for N-Body simulations

– Excellent simple 3D visuals – Java is an object-oriented language

• Disadvantages

– Not for complex 3D graphics – Not very suitable for parallelization

and performance

• The simulator is based on Oracle's

tutorial “JavaFX: Working with JavaFX Graphics” which explains how to draw simple 3D molecule structures http://docs.oracle.com/javase/8/javafx/graphics-tutorial/sampleapp3d.htm

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Screenshots

• Simulation Controls

Number of Bodies – N Random mass between 0 and … kg Random Velocity between 0 and … m/s (for each dimension) Random Position between 0 and … m (for each axis) Time step in seconds Add more computations for extra accuracy Enable merging (default

• ption),
• therwise

unstable Show motions (memory, CPU

• verhead)

Visualize Velocities Restart Simulation Pause/Resume

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Screenshots (2)

• 2-body simulation

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Screenshots (3)

• 3-body simulation

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Screenshots (4)

• 1000-body simulation

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Screenshots (5)

• Other many-body simulations

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Source Code Statistics

• Lines of code

– 1056 for my simulator (interface, 3D objects, interaction) – 173 for the 3D form (Ready from Oracle's tutorial for JavaFX

3D drawing)

– And some Billion lines for JavaFX itself and the Operating

System

• Estimated development hours

– 30 minutes for JavaFX framework familiarization (as a

computer science student)

– 24 hours of total coding – Two weeks for developing

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

Future Work

• Extra validation that the formulas are correct
• Parallelization using GPUs or/and multi-threading
• Optimize using common optimization techniques
• Save/Export current universe settings
• Prepare presets of realistic data like our solar system
• Give the option to change randomness distribution

curves

• Custom object density
• Other types of collision reactions (such as bouncing etc.)

29/03/2016 Philippos Papaphilippou – FYS 012 – Physics and Applications

END

• Website

– http://www.cs.ucy.ac.cy/~ppapap01/nbody/

• References

– Newton's Law of Universal Gravitation

• Wikipedia aticle https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation
• Learner.org article https://www.learner.org/courses/physics/unit/text.html?unit=3&secNum=3

– Momentum

• Wikipedia article https://en.wikipedia.org/wiki/Momentum

– N-Body problem and simulations

• Wikipedia article https://en.wikipedia.org/wiki/N-body_problem
• Wikipedia article https://en.wikipedia.org/wiki/N-body_simulation

– JavaFX

• JavaFX: Working with JavaFX Graphics - 8 Building a 3D Sample Application

http://docs.oracle.com/javase/8/javafx/graphics-tutorial/sampleapp3d.htm

• Using the JavaFX AnimationTimer

http://blog.netopyr.com/2012/06/14/using-the-javafx-animationtimer/