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FEniCS Tutorial: Navier-Stokes Equation for Lid-Driven Cavity

The incompressible Navier-Stokes equations describe the evolution of fluid motion. In this video, we will simulate them using the Finite Element Method and Chorin's Projection using the FEniCS package in Python. Here is the resulting code: https://github.com/Ceyron/machine-lea... Computational Fluid Dynamics (=CFD) is concerned with the simulation (=quantitative prediction) of the Partial Differential Equations (=PDEs) that underpin fluid mechanics, the Navier-Stokes Equations. Those set of PDEs are particularly challenging to solve due to multiple reasons, they are non-linear, form a saddle-point structure (and in the case of turbulence: are multi-scale). The Finite Element Method presents a methodology to discretize the spatial derivatives in it. We will combine this with a simple Euler time step, for which we treat the convection fully explicit and the diffusion fully implicit. We will enforce incompressibility as required by the continuity equation by the usage of Chorin's Projection method. This method gives rise to pressure Poisson problem. ------- 📝 : Check out the GitHub Repository of the channel, where I upload all the handwritten notes and source-code files (contributions are very welcome): https://github.com/Ceyron/machine-lea... 📢 : Follow me on LinkedIn or Twitter for updates on the channel and other cool Machine Learning & Simulation stuff:   / felix-koehler   and   / felix_m_koehler   💸 : If you want to support my work on the channel, you can become a Patreon here:   / mlsim   ------- Timestamps: 00:00 Intro 00:30 Navier-Stokes Equations 01:11 About Lid-Driven Cavity & BC 02:27 Solution Strategy with Weak Forms 07:12 Taylor-Hood Elements & Saddle Point Problems 08:29 Choose Time Step size carefully 08:55 Imports 09:25 Simulation Parameters 10:13 Some Boilerplate 10:26 Define Mesh 11:01 Set up Function Spaces (with Taylor-Hood Elements) 12:26 Define Trial & Test Functions 13:32 Boundary Conditions (Stationary & Moving Wall) 18:11 Solution Fields 19:47 Weak Form of Momentum Equation 22:55 Weak Form of Pressure Poisson Problem 24:57 Weak Form of Velocity Projection/Correction 27:09 Time Loop Setup 27:25 (1) Solve for tentative velocity 28:18 (2) Solve for pressure 28:50 (3) Correct velocities for incompressibility 29:12 (4) Advance in time 29:31 Interactive visualization 30:07 First Run + Discussion 31:22 Pre-Computing assembly of system matrices 36:14 Second Run + Small Bug Fix 36:53 Adjusting Linear Solver and Preconditioner 38:06 Third Run + Admiring Speedup 38:35 Outro