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Octree Simulation with Barnes-Hut Algorithm

Overview

This project implements the Barnes-Hut algorithm for simulating gravitational interactions in a system of stars using an octree data structure. The algorithm efficiently calculates the gravitational forces between particles by approximating distant interactions, reducing the computational complexity from O(N^2) to O(N log N).

This simulation calculates accelerations and gravitational forces and visualizes the system in 3D using Matplotlib. The Barnes-Hut algorithm is widely used in astrophysics and particle simulations for its scalability and efficiency when dealing with large systems.

3D Simulation Demo

Simulation Demo

Octree Structure

Gravitational Forces

Features

  • Octree data structure to partition space and optimize calculations.
  • Barnes-Hut algorithm to calculate gravitational forces and accelerations efficiently.
  • 3D visualization of the simulation using Matplotlib.

Mathematical Formulas

The simulation uses fundamental principles of gravitational forces and accelerations to model the motion of stars.

1. Gravitational Force

The gravitational force between two stars is given by Newton's law of universal gravitation:

F = G * (m_1 * m_2) / r^2

Where:

  • F is the gravitational force between two stars.
  • G is the gravitational constant.
  • m_1 and m_2 are the masses of the two stars.
  • r is the distance between the two stars.

2. Acceleration Due to Gravity

The acceleration a experienced by a star due to gravitational force is calculated using Newton's second law:

a = F / m

Where:

  • F is the gravitational force calculated above.
  • m is the mass of the star.

3. Barnes-Hut Approximation

The Barnes-Hut algorithm approximates the gravitational interaction between distant stars using a single "center of mass" for groups of stars. The force calculation is simplified based on a theta parameter, which controls the threshold for approximating distant interactions:

theta = s / d

Where:

  • s is the size of the octant (or group of stars).
  • d is the distance between the center of mass of the octant and the star.

If theta is smaller than a given threshold, the octant is approximated as a single mass, and the force is calculated directly. Otherwise, the interaction is recursively computed with the children of the octree.

Usage

To run the simulation, simply execute the Python script:

Requirements

  • Python 3
  • OPP Concepts
  • Matplotlib
  • NumPy

Installation

  1. Clone the repository:

    git clone https://github.com/yChaaby/Octrees_Similation.git

  2. Install required dependencies:

    pip install matplotlib numpy

Usage

  1. To run the simulation:

    python simulation.py

  2. The simulation will display a 3D visualization of the stars and their interactions.

License

None ... it's all yours