title
Coding Challenge #132: Fluid Simulation

description
In this video, I follow Mike Ash’s guide to Fluid Simulation and port Jos Stam’s "Real-Time Fluid Dynamics for Games" algorithm and code in Processing (Java). Code: https://thecodingtrain.com/challenges/132-fluid-simulation 🕹️ p5.js Web Editor Sketch: https://editor.p5js.org/codingtrain/sketches/9kVfB4BF2 🎥 Previous video: https://youtu.be/0j86zuqqTlQ?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH 🎥 Next video: https://youtu.be/bl3nc_a1nvs?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH 🎥 All videos: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH References: 💾 GitHub Issue: https://github.com/CodingTrain/Rainbow-Topics/issues/178 💻 Real-Time Fluid Dynamics for Games: http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf 💻 MSAFluid: https://www.memo.tv/msafluid/ 💻 Lily Pad: https://github.com/weymouth/lily-pad 📓 Fluid Simulation for Dummies: https://mikeash.com/pyblog/fluid-simulation-for-dummies.html Videos: 🎥 Why Laminar Flow is AWESOME: https://youtu.be/y7Hyc3MRKno 🎥 What DO we know about turbulence?: https://youtu.be/_UoTTq651dE 🚂 Perlin Noise: https://youtu.be/Qf4dIN99e2w 🔴 Livestream Archive: https://youtu.be/_LyC2l8g0R8 Related Coding Challenges: 🚂 #24 Perlin Noise Flow Field: https://youtu.be/BjoM9oKOAKY 🚂 #102 2D Water Ripple: https://youtu.be/BZUdGqeOD0w 🚂 #103 Fire Effect: https://youtu.be/X0kjv0MozuY Timestamps: 0:00 Introduction 0:59 Topic suggestion from deardanielxd 3:30 Mike Ash's "Fluid For Dummies" thesis 6:42 Incompressible fluid 8:45 Velocity field 9:12 Density of dye 11:14 Port the code to Processing 15:47 addDensity() function 19:34 Diffuse 23:35 Project 25:52 Advect 30:17 Set bounds 33:05 Mirror velocity in edge layers 36:19 Time set function 38:58 Render the density 44:10 Add fade 46:40 Add perlin noise 48:00 Add Pvector 50:35 Recap and next steps 52:22 Adjustments to code Editing by Mathieu Blanchette Animations by Jason Heglund Music from Epidemic Sound 🚂 Website: http://thecodingtrain.com/ 👾 Share Your Creation! https://thecodingtrain.com/guides/passenger-showcase-guide 🚩 Suggest Topics: https://github.com/CodingTrain/Suggestion-Box 💡 GitHub: https://github.com/CodingTrain 💬 Discord: https://discord.gg/hPuGy2g 💖 Membership: http://youtube.com/thecodingtrain/join 🛒 Store: https://standard.tv/codingtrain 🖋️ Twitter: https://twitter.com/thecodingtrain 📸 Instagram: https://www.instagram.com/the.coding.train/ 🎥 Coding Challenges: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH 🎥 Intro to Programming: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6Zy51Q-x9tMWIv9cueOFTFA 🔗 p5.js: https://p5js.org 🔗 p5.js Web Editor: https://editor.p5js.org/ 🔗 Processing: https://processing.org 📄 Code of Conduct: https://github.com/CodingTrain/Code-of-Conduct This description was auto-generated. If you see a problem, please open an issue: https://github.com/CodingTrain/thecodingtrain.com/issues/new #fluidsimulation #turbulentfluidflow #processing

detail
{'title': 'Coding Challenge #132: Fluid Simulation', 'heatmap': [{'end': 2649.87, 'start': 2582.621, 'weight': 0.74}], 'summary': 'A coding challenge explores fluid simulation, covering fundamentals, dynamics, solving navier-stokes equation, refactoring, and visualization, aiming to implement incompressible fluid dynamics with arrays and classes for potential visual opportunities.', 'chapters': [{'end': 389.765, 'segs': [{'end': 62.571, 'src': 'embed', 'start': 1.17, 'weight': 1, 'content': [{'end': 11.183, 'text': 'Hello, and welcome to a coding challenge, a calm, soothing, although somewhat turbulent coding challenge called fluid simulation.', 'start': 1.17, 'duration': 10.013}, {'end': 12.905, 'text': 'Now, I have something to admit to you.', 'start': 11.263, 'duration': 1.642}, {'end': 15.949, 'text': "I don't really understand how any of this stuff works.", 'start': 12.925, 'duration': 3.024}, {'end': 24.759, 'text': "I did make this at one point, and so I'm hoping in this coding challenge to recreate exactly this, which will be a basis on which, hopefully,", 'start': 16.81, 'duration': 7.949}, {'end': 27.602, 'text': 'a lot of other interesting ideas will come about.', 'start': 24.759, 'duration': 2.843}, {'end': 33.649, 'text': "This idea came into mind when I recently saw Smarter Every Day's video on laminar flow.", 'start': 28.163, 'duration': 5.486}, {'end': 38.073, 'text': 'You know, I love laminar flow and all, but Team Turbulence for life.', 'start': 33.709, 'duration': 4.364}, {'end': 41.576, 'text': "All right, so there's a wonderful 3Blue1Brown.", 'start': 38.994, 'duration': 2.582}, {'end': 44.838, 'text': 'Do I just reference the same other YouTube channels every single time? Yes, I do.', 'start': 41.596, 'duration': 3.242}, {'end': 50.142, 'text': "But there's also an excellent 3Blue1Brown video on Turbulence, which I would also recommend.", 'start': 45.098, 'duration': 5.044}, {'end': 53.124, 'text': 'So let me give you some background here.', 'start': 50.602, 'duration': 2.522}, {'end': 57.547, 'text': 'So first of all, there is a GitHub issue thread, which started by DearDanielXD.', 'start': 53.384, 'duration': 4.163}, {'end': 62.571, 'text': 'from 2016, Latzman-Boltzmann Methods for Fluid Simulation.', 'start': 58.868, 'duration': 3.703}], 'summary': 'Coding challenge to recreate fluid simulation, inspired by smarter every day and 3blue1brown videos.', 'duration': 61.401, 'max_score': 1.17, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1170.jpg'}, {'end': 107.331, 'src': 'embed', 'start': 83.883, 'weight': 0, 'content': [{'end': 90.767, 'text': 'Real-time fluid dynamics for games, I believe this is a SIGGRAPH paper, 2003, somebody fact check me on that.', 'start': 83.883, 'duration': 6.884}, {'end': 101.33, 'text': "And it's built on top of this idea, of these Navier-Stokes equations, which are these partial differential equations that describe fluid dynamics,", 'start': 91.907, 'duration': 9.423}, {'end': 107.331, 'text': "and there's actually a one million dollar challenge for proving that this can or cannot be solved in three dimensions.", 'start': 101.33, 'duration': 6.001}], 'summary': 'Real-time fluid dynamics for games using navier-stokes equations, linked to a one million dollar challenge.', 'duration': 23.448, 'max_score': 83.883, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ83883.jpg'}, {'end': 228.216, 'src': 'embed', 'start': 197.438, 'weight': 4, 'content': [{'end': 202.539, 'text': 'And I have spent some time kind of like poking around in this rabbit hole in the last week.', 'start': 197.438, 'duration': 5.101}, {'end': 210.604, 'text': "The article that I found that I kind of enjoyed the most in terms of style was Mike Ash's article called Fluid Simulation for Dummies,", 'start': 202.779, 'duration': 7.825}, {'end': 217.208, 'text': "which is actually a port, not a port, but a version of Joss Stam's paper,", 'start': 210.604, 'duration': 6.604}, {'end': 228.216, 'text': "but actually turning it into 3D and how to render that 3D with paralyzing computing power is all here in his master's thesis.", 'start': 217.208, 'duration': 11.008}], 'summary': "Explored rabbit hole and found mike ash's 3d fluid simulation article based on joss stam's paper.", 'duration': 30.778, 'max_score': 197.438, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ197438.jpg'}, {'end': 264.911, 'src': 'embed', 'start': 240.709, 'weight': 5, 'content': [{'end': 247.696, 'text': 'What I try to do, what I want to do I asked Mike Ash for permission on Twitter.', 'start': 240.709, 'duration': 6.987}, {'end': 252.28, 'text': "I think it's at Mike Ash, but I'll include a link in this video's description, if I could go through this in a video.", 'start': 247.997, 'duration': 4.283}, {'end': 259.507, 'text': "And so what I'm going to do is I'm going to go through this in the video and mostly just kind of like copy-paste this code, which is written,", 'start': 252.561, 'duration': 6.946}, {'end': 264.911, 'text': 'I believe, C++ or C, something like that, some object-oriented C flavor language.', 'start': 259.507, 'duration': 5.404}], 'summary': 'Request made to mike ash for permission to use code in video', 'duration': 24.202, 'max_score': 240.709, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ240709.jpg'}, {'end': 353.302, 'src': 'embed', 'start': 295.136, 'weight': 6, 'content': [{'end': 299.7, 'text': 'I just want to copy-paste the code, change the syntax around, get it rendering, and play with it.', 'start': 295.136, 'duration': 4.564}, {'end': 302.203, 'text': 'Number two is I want to kind of refactor the code.', 'start': 300.021, 'duration': 2.182}, {'end': 303.164, 'text': 'That will be in another video.', 'start': 302.223, 'duration': 0.941}, {'end': 306.888, 'text': "I'm actually going to refactor it with a kind of modern approach.", 'start': 303.364, 'duration': 3.524}, {'end': 308.25, 'text': 'Maybe modern is the wrong word.', 'start': 307.129, 'duration': 1.121}, {'end': 308.991, 'text': 'I mean, this was done.', 'start': 308.27, 'duration': 0.721}, {'end': 316.115, 'text': 'over 10 years ago, but I want to use object-oriented programming, vector, like the P vector class in processing.', 'start': 310.452, 'duration': 5.663}, {'end': 325.54, 'text': "I think there's some ways that I can redo the code to make it a bit more readable than this particular style that uses a lot of esoteric variable naming.", 'start': 316.135, 'duration': 9.405}, {'end': 328.542, 'text': "So that's number two.", 'start': 327.621, 'duration': 0.921}, {'end': 340.131, 'text': 'And then number three, I want to then apply this logic to my flow field example from the Nature of Code book.', 'start': 328.622, 'duration': 11.509}, {'end': 346.957, 'text': 'So if I could take the fluid simulation, turn it into a vector field where I could just toss particles in that fly around,', 'start': 340.391, 'duration': 6.566}, {'end': 349.038, 'text': 'I think some visual opportunities will come of that.', 'start': 346.957, 'duration': 2.081}, {'end': 349.939, 'text': "So that's three parts.", 'start': 349.099, 'duration': 0.84}, {'end': 351.32, 'text': 'Just get the thing working.', 'start': 350.34, 'duration': 0.98}, {'end': 353.302, 'text': "That's what I'm about to attempt to do right now.", 'start': 351.561, 'duration': 1.741}], 'summary': 'Refactor code using object-oriented programming and apply logic to flow field example for visual opportunities.', 'duration': 58.166, 'max_score': 295.136, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ295136.jpg'}, {'end': 400.27, 'src': 'embed', 'start': 374.716, 'weight': 10, 'content': [{'end': 382.24, 'text': "But you'll see a lot of the implementations of this use shaders or WebGL and all sorts of fancy tricks that I'm not going to get into.", 'start': 374.716, 'duration': 7.524}, {'end': 386.523, 'text': "But if you know about that stuff and can build on top of whatever I'm doing, then fantastic.", 'start': 382.28, 'duration': 4.243}, {'end': 389.084, 'text': "All right, so I'm about to get started coding.", 'start': 387.023, 'duration': 2.061}, {'end': 389.765, 'text': 'But before I do that,', 'start': 389.124, 'duration': 0.641}, {'end': 396.508, 'text': "I've written in advance a bunch of the concepts that are involved in this implementation that I want to make sure that I don't forget to mention.", 'start': 389.765, 'duration': 6.743}, {'end': 400.27, 'text': 'The first one that I think is really important is this idea of an incompressible fluid.', 'start': 396.808, 'duration': 3.462}], 'summary': 'The implementation may involve shaders, webgl, and incompressible fluid concepts.', 'duration': 25.554, 'max_score': 374.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ374716.jpg'}], 'start': 1.17, 'title': 'Fluid simulation and code refactoring', 'summary': 'Introduces a coding challenge for fluid simulation, explores related resources, and involves refactoring c++ code in processing for potential visual opportunities.', 'chapters': [{'end': 174.506, 'start': 1.17, 'title': 'Fluid simulation: coding challenge and turbulence', 'summary': "Introduces a coding challenge to recreate a fluid simulation as a basis for various ideas, referencing influential resources such as 3blue1brown and joss stan's article on real-time fluid dynamics for games, based on navier-stokes equations.", 'duration': 173.336, 'highlights': ["Joss Stan's article on real-time fluid dynamics for games, serves as the basis for the fluid simulation, referencing Navier-Stokes equations. The article by Joss Stan on real-time fluid dynamics for games is a foundational resource for the fluid simulation, built on the concepts of Navier-Stokes equations.", 'Introduction of coding challenge for recreating a fluid simulation as a basis for various ideas. The coding challenge is introduced to recreate a fluid simulation, aiming to serve as a foundation for various ideas and concepts.', 'Referencing influential resources such as 3Blue1Brown and Smarter Every Day for insights on laminar flow and turbulence. The chapter references influential resources such as 3Blue1Brown and Smarter Every Day for insights on laminar flow and turbulence, providing a diverse range of perspectives.', 'Mention of GitHub issue thread started by DearDanielXD on Latzman-Boltzmann Methods for Fluid Simulation. A GitHub issue thread started by DearDanielXD is mentioned, focusing on Latzman-Boltzmann Methods for Fluid Simulation, indicating collaborative discussions in the field.']}, {'end': 259.507, 'start': 174.646, 'title': 'Exploring fluid simulation and rendering', 'summary': "Explores various resources related to fluid simulation, including gabriel wymouth's lilypad project, mike ash's article on fluid simulation for dummies, and the process of seeking permission to use the article's content in a video.", 'duration': 84.861, 'highlights': ["Gabriel Wymouth's Lilypad project, used in the three blue one brown video on turbulence, is mentioned as a significant resource for fluid simulation.", "Mike Ash's article on Fluid Simulation for Dummies is highlighted as a valuable resource, offering a 3D version of Joss Stam's paper and insights into rendering with parallel computing power.", "The process of seeking permission from Mike Ash to use his article's content in a video is discussed, emphasizing the importance of respecting intellectual property rights.", 'The chapter delves into the rabbit holes of exploring various resources related to fluid simulation, showcasing a curiosity-driven approach to research and learning.']}, {'end': 389.765, 'start': 259.507, 'title': 'Refactoring c++ code in processing', 'summary': 'Involves refactoring c++ code in processing environment to make it more modern and readable, and then applying the logic to a flow field example for potential visual opportunities.', 'duration': 130.258, 'highlights': ['The chapter involves refactoring C++ code in Processing environment to make it more modern and readable, and then applying the logic to a flow field example for potential visual opportunities.', 'The speaker plans to copy-paste the code, change the syntax around, get it rendering, and play with it as the first step.', 'The speaker also aims to refactor the code with a modern approach, using object-oriented programming and vector class in processing, to make it more readable than the existing style.', 'The third part involves applying the refactored logic to a flow field example from the Nature of Code book to explore visual opportunities.', 'The speaker mentions potential performance issues and intends to keep things low resolution, while acknowledging that other implementations may use shaders or WebGL for performance enhancements.']}], 'duration': 388.595, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1170.jpg', 'highlights': ["Joss Stan's article on real-time fluid dynamics for games, serves as the basis for the fluid simulation, referencing Navier-Stokes equations.", 'Introduction of coding challenge for recreating a fluid simulation as a basis for various ideas.', 'Referencing influential resources such as 3Blue1Brown and Smarter Every Day for insights on laminar flow and turbulence.', 'Mention of GitHub issue thread started by DearDanielXD on Latzman-Boltzmann Methods for Fluid Simulation.', "Mike Ash's article on Fluid Simulation for Dummies is highlighted as a valuable resource, offering a 3D version of Joss Stam's paper and insights into rendering with parallel computing power.", "The process of seeking permission from Mike Ash to use his article's content in a video is discussed, emphasizing the importance of respecting intellectual property rights.", 'The chapter involves refactoring C++ code in Processing environment to make it more modern and readable, and then applying the logic to a flow field example for potential visual opportunities.', 'The speaker plans to copy-paste the code, change the syntax around, get it rendering, and play with it as the first step.', 'The speaker also aims to refactor the code with a modern approach, using object-oriented programming and vector class in processing, to make it more readable than the existing style.', 'The third part involves applying the refactored logic to a flow field example from the Nature of Code book to explore visual opportunities.', 'The speaker mentions potential performance issues and intends to keep things low resolution, while acknowledging that other implementations may use shaders or WebGL for performance enhancements.']}, {'end': 771.735, 'segs': [{'end': 413.338, 'src': 'embed', 'start': 389.765, 'weight': 0, 'content': [{'end': 396.508, 'text': "I've written in advance a bunch of the concepts that are involved in this implementation that I want to make sure that I don't forget to mention.", 'start': 389.765, 'duration': 6.743}, {'end': 400.27, 'text': 'The first one that I think is really important is this idea of an incompressible fluid.', 'start': 396.808, 'duration': 3.462}, {'end': 407.034, 'text': "An incompressible fluid is a fluid that's Density must remain constant throughout, like water.", 'start': 400.67, 'duration': 6.364}, {'end': 412.198, 'text': "so, for example, if you have water in a balloon And you squeeze that balloon, the water's kind of like come out.", 'start': 407.034, 'duration': 5.164}, {'end': 413.338, 'text': "It's not compressible.", 'start': 412.198, 'duration': 1.14}], 'summary': "The importance of incompressible fluid in fluid dynamics, e.g., water's constant density.", 'duration': 23.573, 'max_score': 389.765, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ389765.jpg'}, {'end': 453.373, 'src': 'embed', 'start': 429.629, 'weight': 3, 'content': [{'end': 440.139, 'text': 'I should also mention that the goal here is not necessarily to, with scientific accuracy, simulate true fluid mechanics,', 'start': 429.629, 'duration': 10.51}, {'end': 447.907, 'text': 'but rather to create the illusion and feeling of that through some remote connection to that actual scientific accuracy.', 'start': 440.139, 'duration': 7.768}, {'end': 453.373, 'text': "But I'm sure whatever I do will be less accurate than what people have done before me based on how I know the way I make things.", 'start': 447.967, 'duration': 5.406}], 'summary': 'Goal: create illusion of fluid mechanics, less accurate than previous work.', 'duration': 23.744, 'max_score': 429.629, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ429629.jpg'}, {'end': 632.496, 'src': 'embed', 'start': 595.003, 'weight': 1, 'content': [{'end': 598.126, 'text': "We're also able to visualize it just as a vector field, which I will do at some point.", 'start': 595.003, 'duration': 3.123}, {'end': 606.315, 'text': "But the dye is what's going to give it more of this smoky-like quality by visualizing the amount of dye as it moves throughout the fluid.", 'start': 598.347, 'duration': 7.968}, {'end': 607.857, 'text': 'So this is the basic idea.', 'start': 606.676, 'duration': 1.181}, {'end': 614.624, 'text': "So the first thing I need to do is get an array to store all of the x's and y's.", 'start': 607.877, 'duration': 6.747}, {'end': 620.45, 'text': 'of the vector field and the amount of dye for every single one of these spots.', 'start': 615.788, 'duration': 4.662}, {'end': 626.933, 'text': "And in the example, it's done with three separate arrays, an array of x's, an array of y's, and an array of densities.", 'start': 620.89, 'duration': 6.043}, {'end': 632.496, 'text': 'And it kind of works like a cellular automata simulation, where I need the previous state and the next state,', 'start': 627.193, 'duration': 5.303}], 'summary': 'Visualizing fluid flow with dye using arrays and vector field.', 'duration': 37.493, 'max_score': 595.003, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ595003.jpg'}, {'end': 707.547, 'src': 'embed', 'start': 672.299, 'weight': 2, 'content': [{'end': 673.24, 'text': 'This is what I was talking about.', 'start': 672.299, 'duration': 0.941}, {'end': 677.144, 'text': "So this C++ structure, I'm going to take it into processing.", 'start': 673.48, 'duration': 3.664}, {'end': 678.746, 'text': "I'm going to add setup.", 'start': 677.745, 'duration': 1.001}, {'end': 680.749, 'text': "I'm going to add draw.", 'start': 679.908, 'duration': 0.841}, {'end': 685.024, 'text': "I'm going to say size 250.", 'start': 681.923, 'duration': 3.101}, {'end': 688.684, 'text': "Actually, you know what I'm going to do? I'm going to create a variable called n.", 'start': 685.024, 'duration': 3.66}, {'end': 695.005, 'text': 'This will appear in the code, which is kind of like the width and height of the square.', 'start': 688.684, 'duration': 6.321}, {'end': 695.685, 'text': 'So it will be five.', 'start': 695.025, 'duration': 0.66}, {'end': 699.206, 'text': 'So n is going to be, in this case, what did I say, 256.', 'start': 695.725, 'duration': 3.481}, {'end': 707.547, 'text': 'And then actually in processing, if you use a variable for the dimensions that you want to put in size,', 'start': 699.206, 'duration': 8.341}], 'summary': "Using c++ structure in processing with dimensions set to 250 and a variable 'n' set to 256.", 'duration': 35.248, 'max_score': 672.299, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ672299.jpg'}], 'start': 389.765, 'title': 'Fluid simulation fundamentals', 'summary': 'Covers incompressible fluid simulation and its relevance, addressing potential limitations. it also introduces basics like grid representation, velocity vectors, and dye concept, aiming to visualize fluid dynamics and implement these concepts using arrays and classes in a coding environment.', 'chapters': [{'end': 453.373, 'start': 389.765, 'title': 'Incompressible fluid simulation', 'summary': 'Discusses the concept of incompressible fluid and its relevance in fluid simulation, aiming to create the illusion of true fluid mechanics while acknowledging potential limitations.', 'duration': 63.608, 'highlights': ['The concept of incompressible fluid is crucial in fluid simulation, where the density must remain constant throughout, as exemplified by the behavior of water in comparison to air, which is compressible.', "The goal of the simulation is to create the illusion and feeling of true fluid mechanics, rather than achieving scientific accuracy, while acknowledging potential limitations based on the creator's approach and understanding."]}, {'end': 771.735, 'start': 453.813, 'title': 'Fluid simulation basics', 'summary': 'Introduces the basics of fluid simulation, including grid representation, velocity vectors, and the concept of dye, and discusses the implementation of these concepts using arrays and classes in a coding environment, aiming to visualize fluid dynamics.', 'duration': 317.922, 'highlights': ['The fluid is represented in a grid of pixels, with velocity vectors defining the flow direction, and a dye vector field used to visualize the fluid dynamics. The fluid is represented in a grid of pixels, with velocity vectors defining the flow direction. The dye vector field is used to visualize the fluid dynamics, with the density of the dye representing an additional visualization element.', 'Implementation involves storing velocity and dye data in separate arrays, akin to a cellular automata simulation, with the need for previous and next state arrays, highlighting the complexity of the code and potential for future refactoring. Implementation involves storing velocity and dye data in separate arrays, akin to a cellular automata simulation, with the need for previous and next state arrays, highlighting the complexity of the code and potential for future refactoring.', 'Discussion on transitioning C++ code to Processing, including defining the grid dimensions, creating a new class, and addressing language differences such as pointers. Discussion on transitioning C++ code to Processing, including defining the grid dimensions, creating a new class, and addressing language differences such as pointers.']}], 'duration': 381.97, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ389765.jpg', 'highlights': ['The concept of incompressible fluid is crucial in fluid simulation, where the density must remain constant throughout, as exemplified by the behavior of water in comparison to air, which is compressible.', 'The fluid is represented in a grid of pixels, with velocity vectors defining the flow direction, and a dye vector field used to visualize the fluid dynamics.', 'Discussion on transitioning C++ code to Processing, including defining the grid dimensions, creating a new class, and addressing language differences such as pointers.', "The goal of the simulation is to create the illusion and feeling of true fluid mechanics, rather than achieving scientific accuracy, while acknowledging potential limitations based on the creator's approach and understanding.", 'Implementation involves storing velocity and dye data in separate arrays, akin to a cellular automata simulation, with the need for previous and next state arrays, highlighting the complexity of the code and potential for future refactoring.']}, {'end': 1164.258, 'segs': [{'end': 814.066, 'src': 'embed', 'start': 771.756, 'weight': 0, 'content': [{'end': 773.979, 'text': 'This is now a pointer to an array.', 'start': 771.756, 'duration': 2.223}, {'end': 776.684, 'text': "You'll see in a second.", 'start': 775.642, 'duration': 1.042}, {'end': 780.431, 'text': "And then, and by the way, I'm going to take out all the z's.", 'start': 778.708, 'duration': 1.723}, {'end': 784.711, 'text': 'I should mention so these variables.', 'start': 782.25, 'duration': 2.461}, {'end': 794.953, 'text': 'this is a diffusion amount, which is a variable to control how the velocity and the vectors and the dye diffuses throughout the fluid.', 'start': 784.711, 'duration': 10.242}, {'end': 796.534, 'text': 'This is viscosity.', 'start': 795.374, 'duration': 1.16}, {'end': 800.715, 'text': 'Viscosity is the thickness of the fluid, so playing with that can change the behavior as well.', 'start': 796.834, 'duration': 3.881}, {'end': 802.816, 'text': 'Dt is the time step.', 'start': 801.155, 'duration': 1.661}, {'end': 806.82, 'text': "in all of my physics simulations I've always done it just have the time step of one.", 'start': 802.816, 'duration': 4.004}, {'end': 814.066, 'text': "But I think you need a smaller time step to be able to get the the simulation to behave somewhat accurately, so that that'll come up later.", 'start': 806.82, 'duration': 7.246}], 'summary': 'Variables control fluid behavior, like diffusion amount & viscosity. smaller time steps for accurate simulation.', 'duration': 42.31, 'max_score': 771.756, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ771756.jpg'}, {'end': 909.169, 'src': 'embed', 'start': 877.241, 'weight': 1, 'content': [{'end': 882.905, 'text': "So this is going to be this, and it's n times n times n because his is in three dimensions.", 'start': 877.241, 'duration': 5.664}, {'end': 892.954, 'text': 'So I know I could do like a find and replace, but this is like so crazy that I want to do this This is like very meditative for me.', 'start': 884.366, 'duration': 8.588}, {'end': 903.684, 'text': 'Okay, and this receives three arguments when you create the fluid.', 'start': 893.815, 'duration': 9.869}, {'end': 909.169, 'text': 'You create it with a time step, a diffusion, and a viscosity.', 'start': 904.425, 'duration': 4.744}], 'summary': 'Fluid creation involves 3 arguments: time step, diffusion, and viscosity.', 'duration': 31.928, 'max_score': 877.241, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ877241.jpg'}, {'end': 1094.996, 'src': 'embed', 'start': 1062.164, 'weight': 2, 'content': [{'end': 1068.469, 'text': "and then we're basically going to say hey, this dot, what is it called? Vx?", 'start': 1062.164, 'duration': 6.305}, {'end': 1069.71, 'text': "oh wait, no, that's velocity, sorry,", 'start': 1068.469, 'duration': 1.241}, {'end': 1077.632, 'text': 'This dot density at index, add some amount.', 'start': 1069.75, 'duration': 7.882}, {'end': 1079.013, 'text': 'This is really simple.', 'start': 1078.133, 'duration': 0.88}, {'end': 1080.454, 'text': 'This is like a really simple function.', 'start': 1079.033, 'duration': 1.421}, {'end': 1084.035, 'text': 'Just add some density to this spot, this amount of density to this spot.', 'start': 1080.494, 'duration': 3.541}, {'end': 1094.996, 'text': 'Then we can also do this add velocity, which is basically the same thing, but just with an amount x and an amount y.', 'start': 1085.831, 'duration': 9.165}], 'summary': 'A simple function to add density and velocity to a spot, with specific amounts.', 'duration': 32.832, 'max_score': 1062.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1062164.jpg'}, {'end': 1150.487, 'src': 'embed', 'start': 1122.983, 'weight': 4, 'content': [{'end': 1126.187, 'text': 'some of this code is going to get a lot worse soon enough.', 'start': 1122.983, 'duration': 3.204}, {'end': 1130.151, 'text': 'Okay, now, all right.', 'start': 1128.249, 'duration': 1.902}, {'end': 1132.954, 'text': 'Ah, look at this, so here, now we can take a moment.', 'start': 1130.912, 'duration': 2.042}, {'end': 1136.899, 'text': 'Here are the three main operations.', 'start': 1133.475, 'duration': 3.424}, {'end': 1139.741, 'text': 'Diffuse, project, advect.', 'start': 1138.18, 'duration': 1.561}, {'end': 1141.282, 'text': "Let's do them one at a time.", 'start': 1139.841, 'duration': 1.441}, {'end': 1142.843, 'text': 'So we can read this.', 'start': 1141.822, 'duration': 1.021}, {'end': 1145.004, 'text': "I mean, it's useful to read Mike Ash's description.", 'start': 1142.943, 'duration': 2.061}, {'end': 1146.105, 'text': 'Put a drop of soy sauce.', 'start': 1145.064, 'duration': 1.041}, {'end': 1148.046, 'text': 'Soy sauce gives a way of thinking about dye.', 'start': 1146.545, 'duration': 1.501}, {'end': 1150.487, 'text': "And you'll notice that doesn't stay still, but it spreads out.", 'start': 1148.366, 'duration': 2.121}], 'summary': 'Code will deteriorate soon, discussing three main operations: diffuse, project, advect.', 'duration': 27.504, 'max_score': 1122.983, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1122983.jpg'}], 'start': 771.756, 'title': 'Fluid dynamics and operations', 'summary': 'Introduces pointers to arrays and discusses variables like diffusion amount, viscosity, and time step in fluid dynamics simulations. it also covers the creation of a fluid object, including memory allocation and function setup, as well as the addition of density and velocity to specific locations. additionally, it introduces the main operations of diffusion, projection, and advection in fluid simulation.', 'chapters': [{'end': 814.066, 'start': 771.756, 'title': 'Pointer to array and fluid dynamics', 'summary': 'Introduces a pointer to an array and discusses the variables including diffusion amount, viscosity, and time step in fluid dynamics simulations.', 'duration': 42.31, 'highlights': ['Viscosity affects fluid thickness and behavior, offering control over simulations.', 'Diffusion amount controls velocity and dye diffusion in the fluid.', 'Time step adjustment can improve simulation accuracy.']}, {'end': 1164.258, 'start': 814.066, 'title': 'Fluid constructor and operations', 'summary': 'Discusses the creation of a fluid object, including memory allocation and function setup, as well as the addition of density and velocity to specific locations. it also introduces the main operations of diffusion, projection, and advection in fluid simulation.', 'duration': 350.192, 'highlights': ['The creation of the fluid object involves memory allocation for fluid values in three dimensions, with the constructor receiving arguments for time step, diffusion, and viscosity.', 'The addition of density and velocity to specific locations in the fluid is achieved through simple functions, with the use of a two-dimensional to one-dimensional index conversion method.', 'The main operations of diffusion, projection, and advection are introduced, with a focus on the concept of diffusion affecting both dye and velocity in the fluid simulation.']}], 'duration': 392.502, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ771756.jpg', 'highlights': ['Viscosity affects fluid thickness and behavior, offering control over simulations.', 'The creation of the fluid object involves memory allocation for fluid values in three dimensions, with the constructor receiving arguments for time step, diffusion, and viscosity.', 'The addition of density and velocity to specific locations in the fluid is achieved through simple functions, with the use of a two-dimensional to one-dimensional index conversion method.', 'Diffusion amount controls velocity and dye diffusion in the fluid.', 'The main operations of diffusion, projection, and advection are introduced, with a focus on the concept of diffusion affecting both dye and velocity in the fluid simulation.', 'Time step adjustment can improve simulation accuracy.']}, {'end': 1603.293, 'segs': [{'end': 1238.568, 'src': 'embed', 'start': 1209.375, 'weight': 0, 'content': [{'end': 1214.657, 'text': "Final iteration, int, iterations, like we can just put that as one, but let's leave that as like 10 right now.", 'start': 1209.375, 'duration': 5.282}, {'end': 1229.443, 'text': 'So the idea here is that this function knows how to diffuse any arbitrary array of numbers x based on its previous values, x,', 'start': 1216.277, 'duration': 13.166}, {'end': 1233.105, 'text': 'zero based on a diffusion amount and a time step.', 'start': 1229.443, 'duration': 3.662}, {'end': 1238.568, 'text': "But you'll notice that what it does is immediately calls another function called linear solve.", 'start': 1233.565, 'duration': 5.003}], 'summary': 'Function diffuses array of numbers based on previous values, implementing linear solve.', 'duration': 29.193, 'max_score': 1209.375, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1209375.jpg'}, {'end': 1299.184, 'src': 'embed', 'start': 1268.093, 'weight': 1, 'content': [{'end': 1270.033, 'text': "I was reading the comments in Mike Ash's paper.", 'start': 1268.093, 'duration': 1.94}, {'end': 1272.514, 'text': "There's like this Gauss-Seidel technique.", 'start': 1270.053, 'duration': 2.461}, {'end': 1274.834, 'text': 'Some people did his code using this.', 'start': 1272.534, 'duration': 2.3}, {'end': 1286.418, 'text': "I forget which technique his code is particularly using, but it's a way of basically solving those linear equations for fluid dynamics.", 'start': 1274.935, 'duration': 11.483}, {'end': 1290.499, 'text': 'in the space of this grid.', 'start': 1288.638, 'duration': 1.861}, {'end': 1299.184, 'text': "So that's kind of all I want to say about it, but it's needed for every single, for this diffusion algorithm.", 'start': 1290.599, 'duration': 8.585}], 'summary': 'Gauss-seidel technique used to solve linear equations for fluid dynamics in a grid space, essential for diffusion algorithm.', 'duration': 31.091, 'max_score': 1268.093, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1268093.jpg'}, {'end': 1338.066, 'src': 'embed', 'start': 1314.093, 'weight': 2, 'content': [{'end': 1320.718, 'text': "So I'm now going to go and just grab the linear solve code from here.", 'start': 1314.093, 'duration': 6.625}, {'end': 1321.518, 'text': "I'm going to grab this.", 'start': 1320.818, 'duration': 0.7}, {'end': 1322.999, 'text': "And you can sort of see what it's doing.", 'start': 1321.538, 'duration': 1.461}, {'end': 1325.22, 'text': "So I'm going to grab this function.", 'start': 1323.019, 'duration': 2.201}, {'end': 1326.661, 'text': "I'm going to bring it in here.", 'start': 1325.78, 'duration': 0.881}, {'end': 1328.121, 'text': 'By the way, this is kind of how I work.', 'start': 1326.681, 'duration': 1.44}, {'end': 1329.762, 'text': 'I kind of want to understand this more.', 'start': 1328.322, 'duration': 1.44}, {'end': 1333.664, 'text': 'But I feel like I need to just port the code and play around with it.', 'start': 1330.322, 'duration': 3.342}, {'end': 1338.066, 'text': "And then maybe I could do some more research about what the equations actually are and how it's working.", 'start': 1333.824, 'duration': 4.242}], 'summary': 'The speaker plans to port and understand the linear solve code and may conduct further research on the equations and its functionality.', 'duration': 23.973, 'max_score': 1314.093, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1314093.jpg'}, {'end': 1488.478, 'src': 'embed', 'start': 1457.502, 'weight': 3, 'content': [{'end': 1460.804, 'text': "And then now I also don't need k.", 'start': 1457.502, 'duration': 3.302}, {'end': 1463.225, 'text': "I only need i and j because I'm just doing two dimensions.", 'start': 1460.804, 'duration': 2.421}, {'end': 1465.327, 'text': "I don't need k.", 'start': 1464.866, 'duration': 0.461}, {'end': 1468.603, 'text': "I don't need these two.", 'start': 1467.742, 'duration': 0.861}, {'end': 1475.248, 'text': "And then, I don't have set boundaries yet, I'll add that.", 'start': 1468.623, 'duration': 6.625}, {'end': 1477.85, 'text': 'Linear solve is the same thing, but without this.', 'start': 1475.548, 'duration': 2.302}, {'end': 1484.094, 'text': "And I think I can, when I do set bounds, I'll probably take out the n.", 'start': 1477.87, 'duration': 6.224}, {'end': 1488.478, 'text': "Then I don't need k, I don't need k, I don't need k.", 'start': 1484.094, 'duration': 4.384}], 'summary': 'Using i and j for two dimensions, removing k and setting boundaries for linear solve.', 'duration': 30.976, 'max_score': 1457.502, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1457502.jpg'}, {'end': 1609.856, 'src': 'embed', 'start': 1578.88, 'weight': 4, 'content': [{'end': 1583.163, 'text': 'As with diffusion, advection applies both to the dye and to the velocities itself.', 'start': 1578.88, 'duration': 4.283}, {'end': 1591.147, 'text': "So what's really the difference between diffusion and advection? Well, diffusion is just this idea of spreading out.", 'start': 1583.583, 'duration': 7.564}, {'end': 1594.989, 'text': 'But advection is actually the motion associated with the velocities.', 'start': 1591.507, 'duration': 3.482}, {'end': 1599.252, 'text': "They're obviously related and they both happen together, but those are separate things.", 'start': 1595.249, 'duration': 4.003}, {'end': 1603.293, 'text': "So let's go grab the advection code.", 'start': 1600.412, 'duration': 2.881}, {'end': 1609.856, 'text': 'Oh, what? Seriously? Okay.', 'start': 1604.134, 'duration': 5.722}], 'summary': 'Advection and diffusion are related but separate processes in the context of motion and spreading out.', 'duration': 30.976, 'max_score': 1578.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1578880.jpg'}], 'start': 1164.658, 'title': 'Solving navier-stokes equation and porting linear solve functions', 'summary': 'Discusses solving the navier-stokes equation, focusing on diffusion and linear solving techniques such as gauss-seidel and porting linear solve and project functions for 2d application, with mention of advection process and its differences from diffusion.', 'chapters': [{'end': 1313.773, 'start': 1164.658, 'title': 'Navier-stokes equation solver', 'summary': 'Discusses the function for solving the navier-stokes equation, focusing on diffusion and linear solving techniques, with mention of specific techniques like gauss-seidel, to spread out values in a cellular automata-like manner.', 'duration': 149.115, 'highlights': ['The function is designed to diffuse any arbitrary array of numbers based on its previous values, diffusion amount, and time step. The function is intended to diffuse arbitrary arrays of numbers based on previous values, diffusion amount, and time step.', 'The function immediately calls another function called linear solve to solve linear equations for fluid dynamics. The function calls another function called linear solve to solve linear equations for fluid dynamics.', 'Mention of techniques like Gauss-Seidel for solving linear equations in fluid dynamics. The mention of techniques like Gauss-Seidel for solving linear equations in fluid dynamics.']}, {'end': 1603.293, 'start': 1314.093, 'title': 'Porting linear solve and project functions', 'summary': 'Involves porting the linear solve and project functions, making modifications to the code for 2d application, and discussing the advection process and its differences from diffusion.', 'duration': 289.2, 'highlights': ['The process of porting the linear solve and project functions for 2D application involves removing unnecessary variables such as m and k, and making adjustments to the brackets and components of the code.', 'The discussion on advection highlights its role in moving cells by following velocities back in time, and differentiates advection from diffusion by emphasizing its association with motion and velocities.']}], 'duration': 438.635, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1164658.jpg', 'highlights': ['The function is designed to diffuse any arbitrary array of numbers based on its previous values, diffusion amount, and time step.', 'Mention of techniques like Gauss-Seidel for solving linear equations in fluid dynamics.', 'The function immediately calls another function called linear solve to solve linear equations for fluid dynamics.', 'The process of porting the linear solve and project functions for 2D application involves removing unnecessary variables such as m and k, and making adjustments to the brackets and components of the code.', 'The discussion on advection highlights its role in moving cells by following velocities back in time, and differentiates advection from diffusion by emphasizing its association with motion and velocities.']}, {'end': 2144.065, 'segs': [{'end': 1688.999, 'src': 'embed', 'start': 1660.237, 'weight': 1, 'content': [{'end': 1666.762, 'text': "When we look through the code, but I'm going to take out the u, I don't need the three, and I don't need the z.", 'start': 1660.237, 'duration': 6.525}, {'end': 1670.225, 'text': "Oh my god, I don't need the k, I don't need the k.", 'start': 1666.762, 'duration': 3.463}, {'end': 1676.49, 'text': "So, alright, so I don't need the k part of the loop, and I don't need this last thing, and I don't need this z.", 'start': 1670.225, 'duration': 6.265}, {'end': 1685.476, 'text': "and I need the x and the i's, the y's and the j's, and I don't need the z's and the k's.", 'start': 1679.309, 'duration': 6.167}, {'end': 1688.999, 'text': "and I need the S's and the T's.", 'start': 1687.418, 'duration': 1.581}], 'summary': 'Removing unnecessary variables and focusing on x, y, i, j, s, and t.', 'duration': 28.762, 'max_score': 1660.237, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1660237.jpg'}, {'end': 1854.837, 'src': 'embed', 'start': 1831.528, 'weight': 4, 'content': [{'end': 1839.799, 'text': 'And so we need some sort of mechanism for what we do with the edge cells and how we deal with those velocity vectors or things moving.', 'start': 1831.528, 'duration': 8.271}, {'end': 1842.943, 'text': 'And basically, we want to add some kind of bouncing.', 'start': 1840.08, 'duration': 2.863}, {'end': 1847.409, 'text': 'And then we also need to deal with the corner cells even differently.', 'start': 1843.003, 'duration': 4.406}, {'end': 1848.631, 'text': 'in a different way.', 'start': 1848.03, 'duration': 0.601}, {'end': 1854.837, 'text': "So we can read, this is short for set bounds, and it's a way to keep your fluid from leaking out of your box.", 'start': 1848.891, 'duration': 5.946}], 'summary': 'Developing mechanism to handle edge and corner cells, including adding bouncing and setting bounds to prevent fluid leakage.', 'duration': 23.309, 'max_score': 1831.528, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1831528.jpg'}, {'end': 2059.654, 'src': 'embed', 'start': 2031.739, 'weight': 0, 'content': [{'end': 2034.141, 'text': 'So this set boundary function is actually much simpler.', 'start': 2031.739, 'duration': 2.402}, {'end': 2043.005, 'text': 'It is just reversing the velocities in the last in all the edge layers according to edge columns.', 'start': 2034.681, 'duration': 8.324}, {'end': 2044.306, 'text': 'edge columns are rows.', 'start': 2043.005, 'duration': 1.301}, {'end': 2047.447, 'text': 'according to the next two, the edge column or row.', 'start': 2044.306, 'duration': 3.141}, {'end': 2054.63, 'text': 'And then for the corners, it actually just does an average of the two neighbors, and that keeps it this like walled box.', 'start': 2048.348, 'duration': 6.282}, {'end': 2058.033, 'text': "So in the end, it's kind of going to perform the same way.", 'start': 2055.071, 'duration': 2.962}, {'end': 2059.654, 'text': 'I just had this extra loops in there.', 'start': 2058.053, 'duration': 1.601}], 'summary': 'Set boundary function simplifies by reversing velocities and averaging neighbors for consistent performance.', 'duration': 27.915, 'max_score': 2031.739, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2031739.jpg'}], 'start': 1604.134, 'title': 'Refactoring and setting boundaries for simulation', 'summary': 'Discusses refactoring code for simulation and setting boundaries for fluid simulation. it covers removing unnecessary variables, simplifying the process, and ensuring fluid containment within a box, with a focus on improving efficiency and reducing complexity.', 'chapters': [{'end': 1810.795, 'start': 1604.134, 'title': 'Refactoring code for simulation', 'summary': 'Discusses the process of refactoring code for a simulation, removing unnecessary variables and steps to simplify the process, with a focus on improving efficiency and reducing complexity.', 'duration': 206.661, 'highlights': ['The process involves removing unnecessary variables like z, k, and u to simplify the code and reduce complexity.', 'Identifying and eliminating unnecessary steps, such as unnecessary flooring of float numbers, to streamline the code and improve efficiency.', 'The focus is on simplifying the code by removing unnecessary variables and steps, with the aim of improving the efficiency of the simulation process.']}, {'end': 2144.065, 'start': 1811.638, 'title': 'Set boundaries for fluid simulation', 'summary': 'Discusses the implementation of the set bounds function in a fluid simulation, addressing concerns around containing fluid within a box, handling edge cells, and ensuring the fluid does not leak out of the box, with a focus on refactoring the code for improved simplicity and efficiency.', 'duration': 332.427, 'highlights': ['The set bounds function is written to keep the fluid from leaking out of the box by reversing the velocities in the edge layers and averaging the velocities of the corner cells, ensuring a closed box. The set bounds function ensures a closed box by reversing velocities in edge layers and averaging velocities of corner cells, preventing fluid leakage.', 'The chapter emphasizes the need for refactoring the set bounds function to achieve improved simplicity and efficiency in the fluid simulation. The chapter highlights the importance of refactoring to improve simplicity and efficiency in the fluid simulation code.', 'Discussion on the necessity of handling edge cells and corner cells differently for the set bounds function in the fluid simulation. The chapter discusses the need to handle edge cells and corner cells differently within the set bounds function for the fluid simulation.']}], 'duration': 539.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ1604134.jpg', 'highlights': ['The set bounds function ensures a closed box by reversing velocities in edge layers and averaging velocities of corner cells, preventing fluid leakage.', 'The process involves removing unnecessary variables like z, k, and u to simplify the code and reduce complexity.', 'The focus is on simplifying the code by removing unnecessary variables and steps, with the aim of improving the efficiency of the simulation process.', 'Identifying and eliminating unnecessary steps, such as unnecessary flooring of float numbers, to streamline the code and improve efficiency.', 'The chapter emphasizes the need for refactoring the set bounds function to achieve improved simplicity and efficiency in the fluid simulation.', 'Discussion on the necessity of handling edge cells and corner cells differently for the set bounds function in the fluid simulation.']}, {'end': 3258.679, 'segs': [{'end': 2211.746, 'src': 'embed', 'start': 2144.845, 'weight': 1, 'content': [{'end': 2149.649, 'text': 'I want to add those two things together times S zero, and those two things together times S one.', 'start': 2144.845, 'duration': 4.804}, {'end': 2151.831, 'text': 'All right, no errors.', 'start': 2150.11, 'duration': 1.721}, {'end': 2155.434, 'text': 'No errors, no errors.', 'start': 2154.013, 'duration': 1.421}, {'end': 2158.676, 'text': 'This is, by the way, what I really like to avoid doing.', 'start': 2156.294, 'duration': 2.382}, {'end': 2161.438, 'text': "I don't like to code forever and ever and ever without testing.", 'start': 2158.776, 'duration': 2.662}, {'end': 2169.865, 'text': 'so if this actually works when I start to render it, I will be completely and totally gobsmacked.', 'start': 2161.438, 'duration': 8.427}, {'end': 2175.59, 'text': 'Okay, so now I need a time step function.', 'start': 2171.508, 'duration': 4.082}, {'end': 2183.354, 'text': 'So I need a function time step to step through every moment of time with the fluid.', 'start': 2176.29, 'duration': 7.064}, {'end': 2191.157, 'text': "Going back to Mike Ash's paper, there is a step function which I'm going to grab.", 'start': 2184.894, 'duration': 6.263}, {'end': 2206.464, 'text': "Of course I just wrote it out here, but what I want is void step And I have all these as variables, but let's change that to this dot.", 'start': 2191.177, 'duration': 15.287}, {'end': 2209.285, 'text': 'Replace float space star.', 'start': 2207.344, 'duration': 1.941}, {'end': 2211.746, 'text': "I know you can't see this.", 'start': 2210.926, 'duration': 0.82}], 'summary': 'Developing a time step function for fluid simulation, aiming to avoid coding errors and ensure proper testing.', 'duration': 66.901, 'max_score': 2144.845, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2144845.jpg'}, {'end': 2277.128, 'src': 'embed', 'start': 2247.47, 'weight': 0, 'content': [{'end': 2250.073, 'text': 'The one and the two is controlling the set bounds function.', 'start': 2247.47, 'duration': 2.603}, {'end': 2255.739, 'text': "Then project, which is clean everything up to make sure it's the same amount of fluid everywhere, the velocities.", 'start': 2250.694, 'duration': 5.045}, {'end': 2257.3, 'text': 'Then run advection.', 'start': 2256.119, 'duration': 1.181}, {'end': 2266.064, 'text': 'Advect on the velocities x and y, then clean all that up, then Diffuse the density and advect the density.', 'start': 2258.181, 'duration': 7.883}, {'end': 2271.166, 'text': "the density doesn't need to project step because the density Doesn't remain consistent around.", 'start': 2266.064, 'duration': 5.102}, {'end': 2274.607, 'text': 'so the dye that actually is moving around it is inconsistent.', 'start': 2271.166, 'duration': 3.441}, {'end': 2277.128, 'text': 'So we should really be in.', 'start': 2274.607, 'duration': 2.521}], 'summary': 'Controlling set bounds, cleaning, and advecting velocities and density in a fluid simulation.', 'duration': 29.658, 'max_score': 2247.47, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2247470.jpg'}, {'end': 2391.228, 'src': 'embed', 'start': 2360.981, 'weight': 3, 'content': [{'end': 2366.625, 'text': "So I'm actually going to like lower it and let's actually, let's make the scale 10.", 'start': 2360.981, 'duration': 5.644}, {'end': 2372.928, 'text': 'So now we should have, well if I comment this out, I should have 640 by 640 window, okay.', 'start': 2366.625, 'duration': 6.303}, {'end': 2378.251, 'text': 'So now fluid render d.', 'start': 2373.729, 'duration': 4.522}, {'end': 2381.073, 'text': "I'm going to go here and I'm going to add a function in the fluid class.", 'start': 2378.251, 'duration': 2.822}, {'end': 2385.606, 'text': 'I kind of want to take all these functions out and put them in a separate tab.', 'start': 2382.065, 'duration': 3.541}, {'end': 2391.228, 'text': "I'm going to call this more fluid.", 'start': 2388.307, 'duration': 2.921}], 'summary': 'Adjust window to 640 by 640 and reorganize functions in a separate tab.', 'duration': 30.247, 'max_score': 2360.981, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2360981.jpg'}, {'end': 2455.931, 'src': 'embed', 'start': 2428.16, 'weight': 5, 'content': [{'end': 2435.022, 'text': "I'm going to draw a rectangle at x, y, scale, which is really a square, scale, scale.", 'start': 2428.16, 'duration': 6.862}, {'end': 2441.745, 'text': "By the way, did you know that processing now has the function square in it? I'm going to draw a square.", 'start': 2436.062, 'duration': 5.683}, {'end': 2443.066, 'text': 'Get rid of that.', 'start': 2442.506, 'duration': 0.56}, {'end': 2444.486, 'text': "That's going to work.", 'start': 2443.606, 'duration': 0.88}, {'end': 2445.467, 'text': "I've never used this before.", 'start': 2444.546, 'duration': 0.921}, {'end': 2448.008, 'text': 'This is a moment of occasion four hours into this video.', 'start': 2445.487, 'duration': 2.521}, {'end': 2454.99, 'text': "Then I'm going to say fill 255, 100.", 'start': 2448.448, 'duration': 6.542}, {'end': 2455.931, 'text': 'I just want to see.', 'start': 2454.991, 'duration': 0.94}], 'summary': 'Using processing to draw a square, fill 255, 100.', 'duration': 27.771, 'max_score': 2428.16, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2428160.jpg'}, {'end': 2649.87, 'src': 'heatmap', 'start': 2582.621, 'weight': 0.74, 'content': [{'end': 2587.162, 'text': "Oh, y, let's see what happens here.", 'start': 2582.621, 'duration': 4.541}, {'end': 2589.443, 'text': "Oh, okay, I don't mind that so much.", 'start': 2587.402, 'duration': 2.041}, {'end': 2591.784, 'text': 'We got an array out of bounds exception.', 'start': 2590.083, 'duration': 1.701}, {'end': 2607.178, 'text': "So what went out of bounds? One thing I can do here, where's my index function? x equals constrain x to zero and minus one.", 'start': 2592.924, 'duration': 14.254}, {'end': 2608.7, 'text': "Let's just put a constrain in here.", 'start': 2607.238, 'duration': 1.462}, {'end': 2615.168, 'text': "I don't know whether I have a bug in my code or this is actually necessary, but let's do that.", 'start': 2608.72, 'duration': 6.448}, {'end': 2619.614, 'text': 'Ah, there we go! Oh my god.', 'start': 2617.251, 'duration': 2.363}, {'end': 2643.306, 'text': 'Ah! I think what I want to do is add something that always like, yeah, because look at this.', 'start': 2619.914, 'duration': 23.392}, {'end': 2644.787, 'text': "I'm going to always like fade out.", 'start': 2643.446, 'duration': 1.341}, {'end': 2647.489, 'text': "I'm going to add something to like fade.", 'start': 2645.307, 'duration': 2.182}, {'end': 2649.87, 'text': "I'm going to add a function called fadeD.", 'start': 2647.509, 'duration': 2.361}], 'summary': 'Troubleshooting code for array out of bounds exception and adding a function called faded.', 'duration': 67.249, 'max_score': 2582.621, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2582621.jpg'}, {'end': 2886.226, 'src': 'embed', 'start': 2850.019, 'weight': 7, 'content': [{'end': 2852.9, 'text': 'We could actually make this like a random amount of density, maybe.', 'start': 2850.019, 'duration': 2.881}, {'end': 2859.942, 'text': 'And then add the velocity according to Perlin noise.', 'start': 2855.661, 'duration': 4.281}, {'end': 2866.924, 'text': 'I am going to create a variable called t, which is kind of like the offset through the noise space.', 'start': 2862.143, 'duration': 4.781}, {'end': 2872.463, 'text': "And I'm going to have t, oh I should be using Ah, I changed my mind.", 'start': 2867.584, 'duration': 4.879}, {'end': 2886.226, 'text': "I'm going to use a P vector and I'm going to say angle equals noise of T times two pi.", 'start': 2872.903, 'duration': 13.323}], 'summary': 'Creating random density and adding velocity using perlin noise for vector calculation.', 'duration': 36.207, 'max_score': 2850.019, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2850019.jpg'}, {'end': 2972.685, 'src': 'embed', 'start': 2933.436, 'weight': 9, 'content': [{'end': 2934.657, 'text': 'Yeah, this is kind of the idea.', 'start': 2933.436, 'duration': 1.221}, {'end': 2939.56, 'text': "I'm not seeing, I just wanted it to kind of like shoot it out randomly.", 'start': 2934.677, 'duration': 4.883}, {'end': 2942.502, 'text': "Let's take out the fade.", 'start': 2941.161, 'duration': 1.341}, {'end': 2943.963, 'text': 'Let me try.', 'start': 2942.542, 'duration': 1.421}, {'end': 2951.928, 'text': "Someone's telling me I swapped vx and vy at line 96 and 97.", 'start': 2946.845, 'duration': 5.083}, {'end': 2953.109, 'text': 'Ah, okay, thank you.', 'start': 2951.928, 'duration': 1.181}, {'end': 2955.69, 'text': "So that's an important fix.", 'start': 2953.849, 'duration': 1.841}, {'end': 2958.592, 'text': "I mean, visually we're getting some result anyway.", 'start': 2956.551, 'duration': 2.041}, {'end': 2961.734, 'text': 'Let me go back to making it.', 'start': 2960.313, 'duration': 1.421}, {'end': 2967.823, 'text': 'Let me do this.', 'start': 2967.363, 'duration': 0.46}, {'end': 2971.084, 'text': 'Let me add.', 'start': 2970.204, 'duration': 0.88}, {'end': 2972.685, 'text': 'All right.', 'start': 2971.104, 'duration': 1.581}], 'summary': 'Discussion about code adjustments and fixing vx and vy swap at line 96 and 97.', 'duration': 39.249, 'max_score': 2933.436, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2933436.jpg'}, {'end': 3039.335, 'src': 'embed', 'start': 3010.849, 'weight': 6, 'content': [{'end': 3013.51, 'text': 'Okay, that was too, the velocity was too strong.', 'start': 3010.849, 'duration': 2.661}, {'end': 3016.251, 'text': 'Okay, so here we go.', 'start': 3014.771, 'duration': 1.48}, {'end': 3018.052, 'text': 'This is more what I was looking to try to do.', 'start': 3016.271, 'duration': 1.781}, {'end': 3021.574, 'text': "I've got sort of Perlin noise controlling this angle.", 'start': 3018.672, 'duration': 2.902}, {'end': 3023.855, 'text': "I'm just like dropping in dye constantly.", 'start': 3021.634, 'duration': 2.221}, {'end': 3026.036, 'text': "It'd be sort of interesting to like sprinkle dye everywhere.", 'start': 3023.875, 'duration': 2.161}, {'end': 3028.757, 'text': "There's so much you could do with color, to drop particles in.", 'start': 3026.356, 'duration': 2.401}, {'end': 3030.858, 'text': "So let's recap what I've done.", 'start': 3029.157, 'duration': 1.701}, {'end': 3036.301, 'text': 'I wanted to have some type of visualization of turbulent fluid.', 'start': 3030.878, 'duration': 5.423}, {'end': 3039.335, 'text': 'I think I mostly have it.', 'start': 3037.674, 'duration': 1.661}], 'summary': 'Creating a visualization of turbulent fluid with perlin noise control and constant dye dropping.', 'duration': 28.486, 'max_score': 3010.849, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ3010849.jpg'}, {'end': 3090.309, 'src': 'embed', 'start': 3063.101, 'weight': 11, 'content': [{'end': 3069.623, 'text': 'I would like to, and then also I would like to create a particle simulation on top of this,', 'start': 3063.101, 'duration': 6.522}, {'end': 3073.984, 'text': 'where I can drop in particles that sort of fly around based on the vector field.', 'start': 3069.623, 'duration': 4.361}, {'end': 3081.386, 'text': 'That is more like the reference on the GitHub thread.', 'start': 3074.845, 'duration': 6.541}, {'end': 3086.048, 'text': "So I'm kind of imagining something like this, where I'm actually just going to draw particles,", 'start': 3082.127, 'duration': 3.921}, {'end': 3090.309, 'text': 'moving around according to the vector field and see what that looks like.', 'start': 3086.048, 'duration': 4.261}], 'summary': 'Create a particle simulation with particles moving based on a vector field.', 'duration': 27.208, 'max_score': 3063.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ3063101.jpg'}, {'end': 3154.705, 'src': 'embed', 'start': 3114.65, 'weight': 10, 'content': [{'end': 3120.657, 'text': "I think if anything this was useful to sort of see the process of trying to figure out somebody else's code written in another language and port it.", 'start': 3114.65, 'duration': 6.007}, {'end': 3125.602, 'text': 'We could try porting this to JavaScript and see if it performs even with just like raw 2D canvas.', 'start': 3121.478, 'duration': 4.124}, {'end': 3126.643, 'text': "I'd be curious to see that.", 'start': 3125.642, 'duration': 1.001}, {'end': 3128.365, 'text': 'So go ahead and do that.', 'start': 3127.464, 'duration': 0.901}, {'end': 3133.849, 'text': 'Thank you to some comments from many people in the chat, but specifically Kay Weekman,', 'start': 3129.366, 'duration': 4.483}, {'end': 3138.072, 'text': 'who pointed out that I do not have turbulence if the viscosity is zero.', 'start': 3133.849, 'duration': 4.223}, {'end': 3140.274, 'text': 'So I now adjusted some things.', 'start': 3138.492, 'duration': 1.782}, {'end': 3144.517, 'text': 'So I created a viscosity of, let me just zoom into this.', 'start': 3140.594, 'duration': 3.923}, {'end': 3149.06, 'text': "I created a viscosity, a pretty low viscosity, but it's a non-zero viscosity.", 'start': 3144.877, 'duration': 4.183}, {'end': 3154.705, 'text': "I also sort of changed the way I'm adding the density and removed that fade.", 'start': 3149.64, 'duration': 5.065}], 'summary': 'Porting code to javascript, adjusting viscosity and density based on feedback.', 'duration': 40.055, 'max_score': 3114.65, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ3114650.jpg'}], 'start': 2144.845, 'title': 'Fluid simulation and visualization', 'summary': 'Covers implementing fluid simulation functions, rendering a fluid simulation, adding velocity and perlin noise, and visualizing turbulent fluid, with emphasis on testing and programming adjustments, for a 640x640 window and future plans for sophisticated solvers.', 'chapters': [{'end': 2271.166, 'start': 2144.845, 'title': 'Fluid simulation function implementation', 'summary': 'Outlines the process of implementing a fluid simulation function, including diffusion, advection, and the step function, and emphasizes the importance of testing before rendering.', 'duration': 126.321, 'highlights': ['The process involves diffusing the velocities based on the time step and viscosity, controlling the set bounds function, cleaning up to ensure uniform fluid distribution, and advecting the velocities and density, with iterations set at four.', 'Emphasizes the importance of testing before rendering to ensure the effectiveness of the implemented code.', "Describes the need for a time step function to step through every moment of time with the fluid, and references Mike Ash's paper for a step function implementation."]}, {'end': 2493.753, 'start': 2271.166, 'title': 'Fluid simulation rendering', 'summary': 'Discusses the process of rendering a fluid simulation, including adding dye, setting up the window size, and creating a function to render the density in a 640x640 window, with various programming adjustments made along the way.', 'duration': 222.587, 'highlights': ['Discussing the process of rendering a fluid simulation, including adding dye, setting up the window size, and creating a function to render the density in a 640x640 window. The chapter covers the steps involved in rendering a fluid simulation, such as adding dye, adjusting the window size to 640x640, and creating a function to render the density.', 'Making programming adjustments, such as changing the scale to 10 and creating a separate tab for the fluid class. The speaker makes programming adjustments by changing the scale to 10 and creating a separate tab for the fluid class to streamline the rendering process.', "Discussing the use of the 'square' function in processing and making adjustments to the visualization, including setting the fill color and stroke. The speaker discusses utilizing the 'square' function in processing and makes adjustments to the visualization by setting the fill color and removing the stroke for better clarity."]}, {'end': 2925.237, 'start': 2495.387, 'title': 'Fluid simulation with velocity and perlin noise', 'summary': 'Covers the implementation of a fluid simulation with added velocity and perlin noise, resulting in a swirling effect and a desire to include randomness in the density addition.', 'duration': 429.85, 'highlights': ['The implementation of fluid simulation with added velocity and Perlin noise results in a swirling effect. The chapter discusses adding velocity to the fluid simulation, resulting in a swirling effect and an attempt to adjust the strength of the vector.', 'Desire to include randomness in the density addition for the fluid simulation. There is a desire to introduce randomness in the density addition for the fluid simulation, considering making the density addition a random amount based on Perlin noise.', 'Adjusting the scale and frame rate for better performance of the fluid simulation. The author adjusts the scale and frame rate to maintain better performance for the fluid simulation, achieving a smoother experience.']}, {'end': 3258.679, 'start': 2933.436, 'title': 'Turbulent fluid visualization', 'summary': 'Discusses the process of visualizing turbulent fluid, including fixing code errors, adjusting viscosity, enhancing density, and future plans for creating a particle simulation and exploring sophisticated solvers.', 'duration': 325.243, 'highlights': ['The code had an important fix regarding swapping vx and vy at line 96 and 97, contributing to the visualization improvement.', 'The viscosity was adjusted to a low, non-zero value to enhance the quality of turbulence in the visualization.', 'Plans for future work include creating a particle simulation based on the vector field and exploring libraries with more sophisticated solvers.', 'The process of trying to understand and port code from another language was discussed, with future considerations for porting to JavaScript and testing performance with raw 2D canvas.']}], 'duration': 1113.834, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/alhpH6ECFvQ/pics/alhpH6ECFvQ2144845.jpg', 'highlights': ['The process involves diffusing velocities based on time step and viscosity, controlling set bounds function, cleaning up for uniform fluid distribution, and advecting velocities and density with four iterations.', 'Emphasizes importance of testing before rendering to ensure effectiveness of implemented code.', "Describes need for a time step function to step through every moment of time with the fluid, and references Mike Ash's paper for step function implementation.", 'Discussing the process of rendering a fluid simulation, including adding dye, setting up the window size, and creating a function to render the density in a 640x640 window.', 'Making programming adjustments, such as changing the scale to 10 and creating a separate tab for the fluid class to streamline the rendering process.', "Discussing the use of the 'square' function in processing and making adjustments to the visualization, including setting the fill color and stroke.", 'The implementation of fluid simulation with added velocity and Perlin noise results in a swirling effect.', 'Desire to include randomness in the density addition for the fluid simulation, considering making the density addition a random amount based on Perlin noise.', 'Adjusting the scale and frame rate for better performance of the fluid simulation.', 'The code had an important fix regarding swapping vx and vy at line 96 and 97, contributing to the visualization improvement.', 'The viscosity was adjusted to a low, non-zero value to enhance the quality of turbulence in the visualization.', 'Plans for future work include creating a particle simulation based on the vector field and exploring libraries with more sophisticated solvers.', 'The process of trying to understand and port code from another language was discussed, with future considerations for porting to JavaScript and testing performance with raw 2D canvas.']}], 'highlights': ["Joss Stan's article on real-time fluid dynamics for games, serves as the basis for the fluid simulation, referencing Navier-Stokes equations.", 'Introduction of coding challenge for recreating a fluid simulation as a basis for various ideas.', 'Referencing influential resources such as 3Blue1Brown and Smarter Every Day for insights on laminar flow and turbulence.', 'Mention of GitHub issue thread started by DearDanielXD on Latzman-Boltzmann Methods for Fluid Simulation.', "Mike Ash's article on Fluid Simulation for Dummies is highlighted as a valuable resource, offering a 3D version of Joss Stam's paper and insights into rendering with parallel computing power.", "The process of seeking permission from Mike Ash to use his article's content in a video is discussed, emphasizing the importance of respecting intellectual property rights.", 'The concept of incompressible fluid is crucial in fluid simulation, where the density must remain constant throughout, as exemplified by the behavior of water in comparison to air, which is compressible.', 'The fluid is represented in a grid of pixels, with velocity vectors defining the flow direction, and a dye vector field used to visualize the fluid dynamics.', 'Discussion on transitioning C++ code to Processing, including defining the grid dimensions, creating a new class, and addressing language differences such as pointers.', 'Viscosity affects fluid thickness and behavior, offering control over simulations.', 'The creation of the fluid object involves memory allocation for fluid values in three dimensions, with the constructor receiving arguments for time step, diffusion, and viscosity.', 'The addition of density and velocity to specific locations in the fluid is achieved through simple functions, with the use of a two-dimensional to one-dimensional index conversion method.', 'The function is designed to diffuse any arbitrary array of numbers based on its previous values, diffusion amount, and time step.', 'Mention of techniques like Gauss-Seidel for solving linear equations in fluid dynamics.', 'The set bounds function ensures a closed box by reversing velocities in edge layers and averaging velocities of corner cells, preventing fluid leakage.', 'The process involves diffusing velocities based on time step and viscosity, controlling set bounds function, cleaning up for uniform fluid distribution, and advecting velocities and density with four iterations.', 'Emphasizes importance of testing before rendering to ensure effectiveness of implemented code.', "Describes need for a time step function to step through every moment of time with the fluid, and references Mike Ash's paper for step function implementation.", 'Discussing the process of rendering a fluid simulation, including adding dye, setting up the window size, and creating a function to render the density in a 640x640 window.', 'The implementation of fluid simulation with added velocity and Perlin noise results in a swirling effect.', 'The code had an important fix regarding swapping vx and vy at line 96 and 97, contributing to the visualization improvement.', 'The viscosity was adjusted to a low, non-zero value to enhance the quality of turbulence in the visualization.', 'Plans for future work include creating a particle simulation based on the vector field and exploring libraries with more sophisticated solvers.', 'The process of trying to understand and port code from another language was discussed, with future considerations for porting to JavaScript and testing performance with raw 2D canvas.']}