Particle Simulation

Repo

This program is a particle simulation achieved by employing Newton's law of universal gravitation.  The simulation is implemented in C++, using SFML to create the visualization.

The simulation uses a numerical integration method to update particle positions and velocities over time.  To ensure accuracy and stability, a variable time step is employed. This time step is calculated based on the interval between subsequent render loop iterations. By adapting to the frame rate, the simulation remains robust against fluctuating render times and avoids potential numerical issues that can arise from fixed time steps, particularly when dealing with fast-moving particles.  By adjusting parameters like particle mass, initial velocity, and the collision model, a wide range of physical phenomena can be simulated.  

Each render loop iteration is updated as follows: 

1. For each particle, the euclidean distance to every other particle is calculated. 

   • If the euclidean distance between the particles is less than the sum of their radii, a collision is occurring. Depending on the selected collision model, the result of collisions will vary. 

     • No Collision:  Particles pass through each other without interaction.  The force of attraction between particles that are intersecting is not calculated to avoid a divide-by-zero error if the particles were to land on the same point.  Additionally, the force of attraction calculated using Newtonian gravity approaches infinity as the distance between particles approaches zero.  This can cause extreme acceleration values that are not desirable for this simulation. 

     • Elastic Collision:  Particles bounce off each other.  The elastic collision model assumes perfect elastic collisions.  The final velocity in x and y is calculated for each particle with an equation derived using the conservation of momentum and the conservation of kinetic energy. 

     • Inelastic Collision:  One particle absorbs the other to create a new, larger particle with mass equal to the sum of the two colliding particles.  The x and y components of the final velocity of the new particle are calculated using an equation derived from the conservation of momentum and the conservation of kinetic energy. 

   • If a particle is at the bounds of the render window and has a velocity vector pointing towards the edge, it will collide with the wall. The x or y component of velocity is negated accordingly to cause the particle to reflect off of the walls of the render window.

   • The collision resolution step modifies particle velocities instantaneously, allowing the rest of the update function to proceed without interruption.

2. For each particle, the force of gravity from every other particle is calculated using Newton's law of gravitation.  These values are summed to generate a total force in the x and y directions for the current particle.  

3. If friction is enabled, the force due to friction is calculated using Stoke's law.  Each particle is approximated as a sphere, and the force due to drag is calculated using the radius of the particle and a viscosity value that is configurable to produce different results. 

4. The total x and y components of force on the particle (the force from gravity minus the force from drag, if enabled) are used to calculate the instantaneous acceleration of the particle according to Newton's second law. 

5. The velocity of each particle is updated by multiplying the instantaneous acceleration by the elapsed time between iterations of the render loop. 

6. The change in position in x and y of each particle is calculated by multiplying the instantaneous acceleration by the elapsed time between render window updates and adding the newly updated velocity. 

7. The x and y components of the change in position are added to the current position of the particle, determining its new position for the current render loop iteration.