APPLICATION OF PARTICLE SYSTEMS FOR MODELING OBJECTS OF A DYNAMIC NATURE

Abstract

Introduction. The modeling of dynamically behaving objects represents one of the
most challenging problems in computer graphics, physics, and engineering. Dynamic systems
encompass a wide range of phenomena, including fluid and gas flows, combustion processes,
explosions, smoke, rain, and snow. Simulating such objects in real time, while preserving their
physical behavior and visual plausibility, is a key aspect of modern animation, simulation, and game
mechanics. Particle systems have established themselves as a highly effective approach to creating
realistic models of such objects, enabling a level of detail that is difficult to achieve through
conventional modeling techniques. The relevance of this research is reinforced by the growing
demand for realistic visualization of natural phenomena in cinematography, computer games, virtual reality, and engineering, as well as by the critical role of dynamic simulation in fields where physical
experimentation may be prohibitively expensive or technically infeasible.
Methods. The Störmer-Verlet method is a numerical integration technique for solving Newton's
equations of motion. Originally employed in 1791 by Jean Baptiste Delambre and subsequently
refined by Verlet in the 1960s for molecular dynamics, the method provides excellent numerical
stability along with time reversibility and preservation of the symplectic structure of phase space,
without significant additional computational cost compared to the simple Euler method. A notable
constraint is that it operates only with a fixed time step. All computational approaches based on the
Verlet method treat simulated objects as systems of particles connected by flexible links rather than as
rigid bodies. Based on the requirements of the implementation, the C programming language together
with the raylib library were selected as the development tools.
Results. Two programs were developed and implemented. The first models a fountain: particles
are generated with random masses and initial velocities scaled inversely by mass to ensure physical
consistency, with coordinates updated each frame via Verlet integration according to Newton's second
law. The second program models gravitational interaction between particles, computing pairwise
gravitational forces and accumulating accelerations before updating positions to ensure physical
synchronicity. A sub-stepping strategy with time increment Δt = 1/steps and a custom double-precision
vector type Vector2D were introduced to meet the higher accuracy demands of this task. A collision
resolution function was also added, separating overlapping particle pairs by a distance proportional
to their masses to realistically approximate momentum transfer, with iteration count bounded to
prevent numerical instability.
Conclusions. The Störmer–Verlet method demonstrated high performance, sufficient accuracy,
and ease of modification across all presented tasks, confirming its suitability for modeling dynamic
objects. Particle systems, due to their flexibility, scalability, and ease of implementation, find broad
application from cinematography and the gaming industry to science and engineering. Further
research is expected to yield more realistic dynamic models and new opportunities for interactive
real-time simulation with the involvement of artificial intelligence algorithms.