Hi everyone, I hope you enjoy the video! This is a topic I’ve been wanting to tackle for ages, but have always found a bit intimidating to get started with. So I’m happy to have finally dived in! Let me know if you have any suggestions for improving it, or ideas for future projects. In other news, I’ve recently created a Ko-Fi page where it’s possible to support my work with a one-off contribution (as opposed to the monthly pledges on Patreon). No pressure of course, but if that’s something you’d like and are able to do, you can find the page here ko-fi.com/sebastianlague

The near-density trick was something I came up with for my master's thesis 18 years ago, and nobody mentioned that they were using it before today. It took a long time, but it was worth it in the end because I got my name in one of best videos on youtube. Thank you Sebastian.

The worst part about Sebastian Lague videos is that they end.

I love how one of the bugs you try to fix is just emergent surface tension.

He is the Bob Ross of Coding. He is like: „Lets put a happy little rekursive non-linear differencial equation of sixt order here.“ I love it!

Sebastian Lague never disappoints. I am astonished by how much a single man can accomplish, and at the same time, how much he can teach others. The 48 minutes that this video lasted felt more like 10 because of how enjoyable and informative it was. I can't wait to see how the finished product will look like. Amazing work!

The real insane part of these videos isn’t the coding is the visualisation of the coding at the same time to make it so that non coders can understand clearly, I love these videos

Good throwback to my old times as a molecular dynamics scientist :) Heating up of your simulation: This is normal and happens for different reasons, the biggest contribution is certainly that your forces are cut off and discretized. There is a difference for incoming and outgoing particles (whenever particles bounce, they heat up a little bit, because not everything is perfectly symmetrical over time). The only way to solve this with 100% correct physics is to go via action integral solutions, which is absolutely not feasible for a simulation of that size, that is why even physicists don't use that for molecular dynamics (MD). The easy and cheap way here to go is to introduce a thermostat, which for example every 100 or 10k frames readjusts the temperature, by shifting the speed of each particle a bit more in the direction of the desired speed. In your case you don't need to care about real temperature, just pick an average speed that you are aiming for, which means you still set the temperature but don't know the value in terms of K or °C. Incompressibility: Be happy you don't have it. You don't want it. It is not a good thing for discrete particle simulations. This is something for totally different methods like finite volume/element simulations, density functional and such things. If you have discrete particles exchanging forces, don't try incompressibility. It is unphysical and it only works in more abstract and less close to real physics. Even in these fields it ends up with singularities and problems everywhere, ask people who try to model turbulence ;-) Behavior at the boundaries: This is no surprise. You have this behavior because the particles at the edges do not have any force pulling on them from the other side. They only have forces pulling on them from the inside and along the boundary but not from outside. This increases their local density there. In MD simulations there are several solutions for this. Often simulations have periodic boundary conditions, then this problem is mostly gone. In your case this is not a solution. You can have a background field that is outside of the box, which is calculated from what is happening inside the box, fast fourier transform is used to calculate that cheaply, this basically means that you copy your box abstractly into each direction, so the particles feel long distance forces from all sides. This is really important for MD simulation to get proper bulk behavior of materials, but in your case this is absolutely not necessary as you do not care if your fluid has a realistic volume viscosity, density, temperature / pressure relation etc. pp. You don't care about physical correctness, so you don't need to go that costly route (especially coding it from scratch). Another solution is to not have "neutral" walls. Your walls can have forces too, which results in the walls being sticky or repulsive, changing the behavior. The easiest solution would be to put ghost particles on the walls, which just emit the same forces as your other particles, but do not receive anything and do not move ever. This is often used in simulation of pores and confined materials. The ghost particles will certainly imprint a structure on the wall (and pull real particles into that structure), which you might not desire. In that case a homogenous wall force might be more what you are looking for, that is basically the same thing, but smooth, so that particles do not stick to certain points on the wall, but just still glide, without higher local density.

This is probably the only channel where I can watch an almost hour long video and not get bored for a tiny second

Super cool video! We need part 2 👁👄👁

Sebastian comes back every once in a while like a Santa to bless us with heavenly vibe

I personally would love to see a "deep dive walkthrough" on you porting this to the GPU. It seems like a great vehicle for introducing a lot of people to GPU programming. You mostly did that part "off screen" here - those details would be hugely interesting, I think.

It's so fun to watch these as a physicist! On the one hand, if feels like you get just enough of the physics to follow along with your project, and I, on the other hand, get just enough of the programming to follow along with your video. It's also so cool to see you implementing these ideas. One of my favorite youtube series for sure.

Sebastian didn’t end his video with “cheers”. My conclusion is that he’s been kidnapped and had to send a secret message that only we’d understand something was weird about. The chessbots must have finally got him.

Developing these prototypes requires a lot of work. The fact you go the extra mile to create visual graphs on what is currently taking place using the Editor itself and no editing tricks is ridiculous, you take a great deal of pride in your work! You are a very smart man!

I love how the water particles simulate adhesion and cohesion naturally… so cool… (33:05)

The way you timed up the music with the waves at 45:58 was so clean 🤌 love the production quality you put into these videos

This is extremely cool! There's something about the combination of particle-based systems and fluid simulations that is fascinating. I ran into a lot of the same problems while trying to implement a particle-based fluid visualization where the underlying simulation is not actually particle based. A lot of the techniques you showed would definitely be applicable to this problem and I'll look into them a bit more. My solution was just giving up on trying to match the density with the particles and instead make the particles act more like dust that is pulled and pushed by the fluid's motion. It works alright but suffers from its own set of issues... Anyway, great video as always!

An absolutely classic Lague video

I did a compsci degree at a decent university many years back... this video is astounding quality and right up there in terms of lecturer quality... and it's free!

As someone who works with SPH I am almost bursting of joy. Thank you so much @SebastianLague Consider using Artificial Viscosity (Monaghan et al.), diffusive density terms (Antuono et al.) and particle shifting technique (Sun et al.) to improve your simulation. Make sure your CFL conditions are correct. You can use kernel corrections to improve accuracy at the fluid boundary (Sheppard or CSPM (expensive)). For surface tension, use the CSF model (Morris is a simple enough model).

Have been waiting for a fluid simulation video for 10 years, thank you Sebastian.

You're like the Bob Ross of coding! Always a great day when you upload 😁

Very impressive. I also like the fact that you don't hide your mistakes.

Something i think would be cool to see is a simulation with different liquids, different colors, viscosity, weight, ect, and have them collide together to see how they diffuse and mix together 😮

This is truly inspiring and the production quality is amazing as well, thank you! I also picked up a few notes for spatial search optimisations I'm excited to try out in my little evolution sandbox simulator

It would be cool to see how different particles dispersed in the fluid. Like, start with 50% red particles and 50% blue and just watch them mix and diffuse.

Here's a rogue idea for how the fluid sticks to the bounding box: Make the walls of the bounding box a little bit "elusive" by making it strongly repulse the particles instead of preventing the particles to go past it. This will prevent the particles from sticking to it and instead should bounce off of it. Hopefully.

Absolutely wonderful video! I really like that you kept many of the fumbles and mistakes in the video and showed the thought process exactly the way an experienced programmer might or should, instead of just showing us the complex end result. Really looking forward to seeing how you turn the particles into a rendered result. Great video!

Not only did he learn these concepts and teach them, he built an interactive tool to use as a presentation aid.

0:38 Sebastian: "Let's keep track of the particle's position and velocity." Heisenberg: "Am I a Joke to you?"

Just came across your channel and as a fluid dynamics student who's starting his thesis using SPH and I'm super excited to see this and I love your approach! It's an awesome video and I'm definitely gonna stick around for more coding adventures! Thank you and have a lovely day

I barely understand anything about how coding works, but your videos are not only relaxing, but somehow make it so easy to understand whats actually happening. Always are great day when a new Coding Adventure releases! Also, cute cat.

Sebastian Lague: new video \0/ All programmers while watching: “This part of my life, this little part, is called happiness” Thank you.

Immediately in the beginning I thought you were going to ask me to focus on the circle, relax my muscles and feel my feet touching the ground, to breathe in and out slowly and to let my thoughts pass by like little clouds in a blue blue sky. I'm relaxed now.

You have genuinely changed my life, your videos have been a landmark for my development as a programmer and also reaffirming my interest in game development. I started game development at first just through loving games, but you have a way of presenting game development that made me love it as much as a love games themselves. Keep doing it❤️

That 3blue1brown plush was so fitting with the math refresh, glad to see him also enjoy the obscure math problems and their explanation

I love how humble and determined he is.

This is absolutely awesome. Even i wrote some fluid simulations many years before, i learned a lot from that again and i think this is the best introduction into SPH fluid simulations at all. All that visualizations of the particles, kernels, the smoothing radius, the density, the cell spatial structure makes it much more easy to understand. I did visualizations too, but used it not enough to understand everything that is going on. Also it was so nice to see, that tuning the parameters can result in a lot of weird and funny results and interestingly enough, i had similar ones - like gliding hundreds of particles on the boundary in one line.

Thank you for the journey. It was great and satisfying to watch how you handle pretty complex problems of this project.

I've done some fluid mechincs lately, and was utterly shocked (positively, of course) when I saw how fluid like you got it to behave sheerly through trying to enforce incompressibility. Very impressive!

this guy can make the most complicated concepts digestible and he has a calming voice and amazing editing

My handsome fellow youtuber, I subscribed in your channel because of some cool Blender rigging tutorials and... here we are, 8 years later and you still amazes me with your videos. You make programing a lot more fun, thank you

Hi, Huge respect for your work. I am doing right now my thesis in fluid dynamics. I have written already about all math basics. In your video I have not only visualisation but also you share with everyone how to creat code for it. During my studies I was writing the code for movement of the buoy on the waves. I spend lots of time to do it with supervision of my porfesor. I felt like a child id the fog. After your video I have comprehend all knowlege from my Naval Architect studies. Huge respect and thank you thousand times.

The production value, education, and accessibility of Sebastian Lague’s videos is astounding. Keep up the great work! ❤

I feel like "on the brink of pandemonium" describes many of your projects 😂. This was both technically interesting and beautiful, like so many things you make.

I recently have come across to this channel after the fluid simulation being recommended multiple times but I'm simply amazed by how skilled you are. You go on to the depth to show every piece of code and your mathematical explanations are remarkable. I'm pretty sure you can create a black hole simulation as well. That'd be fantastic😂

amazing man; way to go; I can't wait to see the future video; more videos on fluids are always welcome!

It's always a good day when you post a new video. As always the effort you put in and the amazing simple explanation of a complex topic is much appreciated so thank you a lot for everything. What a legend

Fluid simulation is probably one of the holy grails of indie gamedev. Great work!

I keep recommending this video to people on reddit - and then usually end up watching it again myself, because it's just so damn good. Thanks for the videos Mr Lague.

I am in complete awe at this. This might just be the most informative video ever created on fluid simulation. Thank you immensely for sharing this with us.

47 minute Sebastian Lague video? Heck yeah! Love seeing these very charming and interesting programming projects you do. It's a lot of fun!

Awesome video! I've been reading a lot on fluid simulation, mainly focusing on gases. The reason using the predicted position works is because either way, you're incurring some O(dt^2) error, but using the predicted position has an error which tends to slow down the particles, while the initial position tends to speed them up. Consider a particle sliding in a bowl. If you use the position at the start of a timestep, it'll be higher up the bowl and see a larger speeding-up force when it's sliding downhill, and when it's sliding uphill, it'll be lower down and see a weaker slowing-down force. So you end up consistently speeding up the particle. The opposite happens when you use the predicted position. And that's also why the simulations at 30:00 diverge - the smaller the timestep, the more similar the predicted/initial positions become, and the closer you are to the exact value. The most physically realistic solution is to use a higher-order integrator (e.g. RK4) so your error becomes O(dt^5) and small enough to neglect. Then apply some viscosity to achieve the damping you were previously getting from your O(dt^2) error. (There are also other "symplectic" integrators which are designed around conserving energy, which you can also get higher-order versions of.)

I’d love a video on how these animations are made. Beautiful.

God I love these videos! Love seeing you go through the steps and explain your thinking. This was a blast watching, can't wait to see what you do next

I work doing Computational Fluid Dynamics and this is just awesome. I just wish you'd talked a bit more about conservation of momentum, cause I feel it's one of the most important things for fluid simulations, given that conservation of mass is already accomplished by using a SPH. Otherwise, great work!!! I pretty much look forward to the next episode on fluid dynamics

For the boundaries, you could try getting WolframAlpha to calculate int_{-1}^{x} W(x,y,z) dx, which would give you the density if there was a uniform sea of particles on the other side of the wall

So cool. Love the project and seeing your thought process throughout! Great job!

Absolutely amazing. One of the fastest three quarters of an hour I’ve experienced. I’m loving the iteration: from idea, to concept, to good enough, a bit of polish, and on to the next idea. I can’t wait till the next one!

First I must say this was an awesome video, thanks for making it. Now that I've released my new game on steam I have some more free time to do experiments and your video motivated me greatly. Happy to be a patreon, your content is always amazing.

A cool project could be building a unity museum where you could walk through and see/interact with all of your different coding adventures projects

This was awesome! It is so inspiring and I want to build something like this myself now. I am really excited to see what you do next with this project! I can't wait to see your updates!

Love your videos, your style of making them is so good. Waiting for part 2 already!

It's a good day when Sebastian uploads.

I find your videos really soothing and engaging. You raise the bar for what's considered high quality educational, yet entertaining content. You explain complicated concepts in simple terms, and turned boring and complicated papers into entertaining work of art, and you're too humble about it. I also love how you explain your thought process, what you tried, what did not work, and how you are equally in love with the journey as you are with the end result. I was watching this with my wife, who has about over a year of programming experience. She commented on how clean, readable and pleasant your code is, even though she did not know the language you're using (she did not know it was C#). When I explained that not only the simulation work is done in C#, but also all the graphs and animations are also done in Unity, she was blown away. I'm a Unity employee btw, and I have shared this video with other colleagues on internal Slack. Good reactions, and a few are already fans and familiar with your work. Thank you for a great video!

DAMN, do you simplify such complex iterations of getting to the point on such an overwhelming subject! I have to date not been ever as happy as now for a UKposts suggestion. And I am embarrassed to have not noticed you sooner. I shall Sub and Bell to hopefully never miss out again. Thank you so much for your contribution!

Fantastic to see the simulation and hearing about how you made it! I can't wait to see where you take this.

Idea for rendering: You could try to implement the marching cubes algorithm to generate a mesh for it. Since you can already calculate the density at any point in space, this might be pretty straightforward. Marching Cubes is essentially used to approximate a boundary of some sort of single value field, like a density field!

When the particles settle down you can see multiple different hexagonal lattices forming, and I think that is really cool, because you see those in real life, if you look at galvanised steel those different shades are each their own lattice arrangement that are reflecting light slightly differently and they come from when the zinc coating was still a liquid, and formed those arrangements on the steel surface.

Always such high quality videos from you. I am looking forward to part 2.

Man I'd love to see how far you can bring this one! It looks amazing! So soothing aswell. Thanks

I made something very like this for my final project at university, using spheres to represent fluid particles with variable density, the posibility of multiple fluids mixed together and sliders for variable gravity. I loved setting the gravity to nothing and watching my fluids float like they were on the space station

You are the best thing about youtube! I love the witty, not-taking-yourself-too-seriously narration and your voice is - as always - heavenly. My wife is not interested in math/programming at all but has commented many times saying you should read audiobooks for a living :) cheers and thanks as always!

great work! rally nice video. you explain it excellently, and make it all really fun and entertaining. i think that a really funky idea would be to connect motion control to it all, so you can shake the entire liquid tank like a water bottle and see it react accordingly on your screen. it would also probably help testing if the liquid motion feels right. thanks for the video!

I do like those kind of video, really nice. Concerning the optimisation i've read about some not a long time ago. Basically, even if we work and leave on a 3D world, our screen will still be in 2D, we only gonna see what is in front of our eyes, and also the first layer. So in case of water, you don't have to render each ball, especially those who are in the middle of the simulation. Of course the computer/the script have to know that they are their, but our eyes will not see the differences. Meaning that you can work with more particules only by rendering the ones we see or we're supposed to see. I'm amazed by your capacity to read & do sketches on paper with those kind of mathematics. I'm also amazed by your smoothness with the code. Amazing wideo.

I really love coding adventures. They have long been my favorite series on youtube. I feel like this community views the world with a healthy sense of curiosity and wonder. In many ways I feel at home watching these.

The editing on this video is on another level!!! Great job Sebastian 👏

Fascinating and fabulous, as ever, Sebastian - thankyou!

I can't explain with words the emotion that runs through me when watching your videos. They are truly fascinating and beautiful

My MSC project back in 1990 was simulating forces in a 3D surface using the Inmos Transputer chip (actually a set of five of them, one controller and four workers). It used a very trivial averaging formula (vertical velocity of cell in next generation is the average of the height differences to its neighbours). It ended up looking quite pretty. After handing the project in, I've barely done anything with computer graphics, sadly. The point of the project was to demonstrate the power of parallelism and message passing rather than accurately model reality. But your adventure is next level and has got me thinking about downloading a games library for Python and reimplementing my MSc project, then moving onto yours. Thanks for the inspiration.

That was absolutely fascinating to watch. The evolved behaviour that comes from such relatively “simple” rules coupled to mass computation is amazing to watch. Excellent explanation of your journey and fantastic visuals made this a riveting watch whilst I waited for my bread to bake. Well done 😊

So inspiring! I'm so happy to found your work and that you share it with us!

Nice job! This is amazing, your research, the video editing... beautiful, really

Love the 3Blue1Brown plushie cameo. Been looking forward to this video since you first teased it. Though a couple things to note. Firstly, if it wasn't clear, Sebastian is using Freya's "Shapes" asset to draw the circles, which isn't native to Unity. It's a paid asset. (He mentioned that in his video on how he makes these tutorials.) Secondly, it's not really clear to me how the blurring is done, but I suspect that's also a feature of Shapes as well. Would be nice to have Seb confirm or clarify those points.

That “clumping up against the boundaries” you noticed was actually the surface tension of the fluid as-is! Surface tension is an emergent property of fluids being simulated as individual particles, and is the reason you didn’t have to implement any special math to get those droplets. The tendrils you saw in the high-particle-count simulation were a result of extreme capillary forces dragging the fluid up!

Thank you for this incredible video and explanation. The subject of simulating fluids seems to be beautiful and filled with challenges : how it reacts to different solicitations, external pressure, etc.

It is so beautiful to see how you transform really simple implementation into wonderfully complex models that are really rich in details

you're a literal coding code, every upload is just another mind boggling adventure

even though im currently sick, i still wanna watch sebastian lague, don't wanna miss his greatness, do i?

beyond anything I could produce in a lifetime and visually beautiful. Stunning work. Subscribed.

Amazing and thorough presentation on a complex but very interesting field of computer graphics. I like how you show every step and trying to tackle every problem that arises with different approaches. I would love to see a similar video on other kinds of simulation as well such as Soft Bodies.

It's kind of magical to see it start to look like a real physical thing, when you've seen the math evolve to that point.

Yes!! Was waiting for this one after I had watched Acerola video about ocean simulation and wondered how this fluid simulation project of yours is doing (since you mentioned it back in the AI game challenge). Question answered

Phenomenal new video. Wow. Those little waves emerging in the last part was such a testament to how accurate it's getting!

I've been trying to figure this out for an app I am working on. I love that you released this and explained everything. Thank you!

Hey, physicist with experience in numerics here. First of all, great stuff! A few comments: Pretty sure the "prediction" part of the code is only helpful since it effectively makes the particles repel from their own future versions, which in essence means they experience a global damping force - this both explains why it "calms" the dynamics, by being the only mechanism for the system energy to lower (apart from the more realistic damping you added later), and possibly why it depends on the prediction timestep. I'd recommend just getting rid of this and using only the more realistic damping, unless that doesn't work. Also, did you try using some other spatial partitions, such as KDTrees? They might be better, though honestly I doubt it. Another neat option would be to instead of calculating the actual average quantities per particle, calculate a rough scale approximation for the average quantities per cell, and use that for approximating the effects of any far away particles (say, those further that R/3 or so) - this is similar to the Fast multipole method, though for such local interactions which decay quickly it again might just be wasteful. Oh, and, maybe try some symplectic integration schemes, such as Verlet or even something better (note that even though you have damping forces, which make your system non-Hamiltonian, you could just plug the force directly instead of dV/dq and get nice results) - I didn't mess around with these much, but I've seen claims they greatly improve long term stability for integration (i.e. better conserve energy for the same amount of compute). Oh, and for the boundary, at least against walls you could treat each wall like a really long and flat particle, manually adding a set density from them to the local density estimate. I don't know how to treat the free boundary "collapsing" though, maybe messing around with the near-repulsion term a little?

You can also clearly see convection currents being generated at 46:12 as the boundary volume shrinks, showing that while it probably isn't as realistic as one would like, it is still fairly realistic.

This is absolutely insanely cool. Awesome stuff! It was really cool seeing how issues you were noticing looked quite similar to natural phenomena we see in liquids. You mentioned of course how when you were working with the droplet how you had a crude surface tension, and I was really intrigued by that! At the end of your presentations, in 3D-space, when you tipped the container on its side, it almost looked like the liquid was "working it's way up" the surface, similar to how water will wick it's way up a glass due to it's surface tension. In fact, I wouldn't be surprised if your "little" fluid sim here also unintentionally has capillary action "built-in" due to this. It would be interesting to see how the liquid behaves when put into an area that has something similar in shape to a straw. Very cool stuff, and I look forwards to seeing more on it!

Im a developer myself and this video reminded me of how fun coding can be. Definitely subscribing. Thank you for all the work put into this.