Yap abt project
I chose a Smoothed-particle hydrodynamic model for my first implementation because handling the fluid as a set of particles was more familiar to me. This is known as a Lagrangian approach due to the "computational mesh" or fluid being a collection of individual points with their own data that is manipulated according the equations that govern the system. This means its a easier to handle free surfaces like breaking waves, b we can kinda ignore mass conservation, c interactions with other moving bodies are easy, and fluid properties are easy.
a) It's particle based so boundaries are defined inherently - so basically it solves itself
b) Each particle has the same mass
No mass enters/exits the system
c) Same logic as a. These interactions are calculated based on nearby particles
Yap abt how I suck at math so other people derived the equations for me and it was still hard to understand
This is done by looping through all the particles and for each particle i, we loop through all the particles again j and sum their density contributions, apply some weight to it and multiply by the mass m of particle j.
r is the vector pointing from particle i to particle j so ||r|| is the distance between the two particles.
W is the smoothing kernel, which is a type of weighted moving average. This spreads the influence of one particle over some small radius h. Particles outside that radius will be ignored. k is the kernel constant for the poly6 kernel, which was designed by Müller et al for particle density approximations.
Since the density at each point is known, the pressure can then be solved for using an approximation of the Tait Equation of State. k is the gas constant and ρ0 is the density at equilibrium. This can be linearized as:
We want to find the total force for every particle. We do so by summing the forces due to pressure, viscosity, and gravity.
Once again we loop through all particles i, and for each i, we loop through all other particles j and sum their pressure, viscosity force contributions.
In our simulation, pressure is responsible for the force that pushes particles from regions of high concentration to regions of low concentration. Mathematically, this is represented as a gradient, whose value at a point gives the direction and magnitude of the maximum rate of increase. So the pressure can be represented by:
But we are dealing with a bunch of particles, not a continuous field, so we must adapt it for a discrete field.
But this force is not symmetric, which will lead to an unstable fluid. The symmetrical form of the force on particle i due to particle j is:
W is now a gradient of the Spiky smoothing kernel, where h is the same as earlier.
Viscosity is a fluid's resistance to flow. Imagine identical two streams of water inside a box clashing against each other. We know momentum is always conserved in a collision, so the fluid must eventually come to rest because the net momentum is zero before the collision (assuming identical water streams). In a continuous fluid this is represented as the laplacian of the velocity field, which is basically how much the gradient changes around a point (kinda like local extrema). The force also depends on η, the dynamic viscosity coefficient. This interaction is known as momentum diffusion.
Of course this must be discretized and stablized
Note the influence of the neighbors mass and density must be accounted for, as well as the velocity difference. Another smoothing kernel W is used, but now the laplacian is applied to it instead of the velocity due to the discretization. This will dictate how the velocity is to "smooth out" over space.
Remember to divide the force of gravity mg by the density of the object.
We will use the simplest numerical integration approach, the Euler Method, and apply it to Newton's Second Law and kinematics equations for acceleration, velocity, and position.
Müller, M., Charypar, D., & Gross, M. (2003). Particle‑based fluid simulation for interactive applications.
https://matthias-research.github.io/pages/publications/sca03.pdf
Schuermann, Lucas V. (May 2016). Particle-Based Fluid Simulation with SPH. Writing. https://lucasschuermann.com/writing/particle-based-fluid-simulation