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bachelor-degree

My Bachelor Degree thesis project.

Data structures for bodies interactions

Summary

The $n$-body problem is a class of problems still continuously studied in the field of mathematical physics, but which has multiple possible applications ranging from celestial mechanics to applied engineering. These problems concern the possible interaction that occurs between a number of bodies greater than two, which makes the modeling and formulation of the laws that govern these processes rather complex. Analytically, solving such problems means constructing systems of various differential equations, to which initial conditions must be assigned. Computationally, this methodology is very expensive to address numerically. For this reason, numerically simplified models are used, i.e. models that significantly reduce the number of operations to be carried out, so as to reduce computational complexity and, possibly, calculation times.

This thesis work concerned the development of an algorithm for the efficient and effective solution of an n-body problem based on the Barnes-Hut approach. The problem considered consisted in the calculation of the electric force field established by a non-uniform network of charges all of the same sign.

Wanting to solve an $n$-body problem numerically exactly would have a computational cost equal to $O(n^2)$. The Barnes-Hut algorithm has the merit of reducing the computational complexity to a value proportional to $O(n \log{n})$ at the cost of a light and controllable light presented with precision thanks to the parameter $\theta$.

The discussed approach was implemented in a code written in MATLAB language and the performance of the Barnes-Hut algorithm was compared with an exact calculation of particle interactions in terms of accuracy and computational cost.

The code

The code is divided in two parts:

  1. quadtree and Barnes-Hut algorithm part;
  2. operations count part.

Quadtree and Barnes-Hut algorithm

This part shows how quadtree and Barnes-Hut algorithm work. To run this part, open the quadtree_main.m file on MATLAB and run it.

Operations count

This part compares the complexity of the Barnes-Hut algorithm with the exact method. To run this part, open the opCount_main.m file on MATLAB and run it.