Finite Elements in Fluids
General
Notices
The oral exams are scheduled for Februrary 6, 20 and March 13, all Wednesdays.
Lectures
- 2013.01.30 Lecture 13: Steady and unsteady Navier-Stokes problem (6.6, 6.7)
Supplemental reading: Stability Analysis of Scalar Advection-Diffusion Equation (PDF)
- 2013.01.23 Lecture 12: Stationary Stokes problem (6.5.1, 6.5.2), compatibility conditions, pressure stabilization (6.5.3–8)
- 2013.01.16 Lecture 11: Viscous incompressible flow (6.1, 6.2, 6.3, 6.4).
- 2013.01.09 Lecture 10: Modified equation method (3.5.3), unsteady advection-diffusion-reaction problems (5.1, 5.2), error analysis of Galerkin formulation of θ-family methods (5.4.2), stabilization of the semi-discrete scheme (5.4.5)
Note that the equation (5.16) in the book has a sign mistake; it should read:
G_\theta = \frac{\mathcal{M}(\xi) + (1-\theta) \left( \mathcal{K}(\xi,d) - \mathcal{A}(\xi,C) - r \mathcal{M}(\xi) \right)}{\mathcal{M}(\xi) - \theta \left( \mathcal{K}(\xi,d) - \mathcal{A}(\xi,C) - r \mathcal{M}(\xi) \right)}
which leads to an amplification factor:
G_\theta = \frac{1 - \frac{2}{3} \sin^2 \frac{\xi}{2} + (1-\theta) \left( - 4 d \sin^2 \frac{\xi}{2} - i C \sin \xi - r ( 1 - \frac{2}{3} \sin^2 \frac{\xi}{2}) \right)}{1 - \frac{2}{3} \sin^2 \frac{\xi}{2} - \theta \left( - 4 d \sin^2 \frac{\xi}{2} - i C \sin \xi - r ( 1 - \frac{2}{3} \sin^2 \frac{\xi}{2}) \right)}
- 2012.12.19 Lecture 9: Fourier's error analysis for hyperbolic problems (3.5.1, 3.5.2)
- 2012.12.12 Lecture 8: unsteady advection problems (3.1, 3.2, 3.4), space-time formulations (3.10), evaluations
- 2012.12.05 No lecture; extended exercise session instead
- 2012.11.28 Lecture 7: stabilized methods for transport problems (2.4), origins of stabilization: finite increment calculus (2.5.1), variational multiscale (2.5.3)
Reading: Onate2000a.pdf, Hughes95a.pdf (RWTH only)
- 2012.11.21 Lecture 6: Galerkin versus exact stencils and upwinding (2.3.1, 2.3.2), balancing diffusion (2.3.3)
- 2012.11.14 Lecture 5: strong, weak and Galerkin form of advection-diffusion equation (2.1, 2.2)
- 2012.11.07 No lecture; optional Q&A session at 12:15.
- 2012.10.31 Lecture 4: weak and Galerkin forms of Poisson equation (1.5.2, 1.5.3, 1.5.4), implementation (1.5.5)
Reading: FE integrals slides
- 2012.10.24 Lecture 3: conservation laws (1.4), strong form of Poisson equation (1.5.1)
- 2012.10.17 Lecture 2: reference frames (1.3.1, 1.3.2, 1.3.3), Reynolds transport theorem (1.3.4)
Reading: Huerta88a.pdf (RWTH only)
- 2012.10.10 Lecture 1: introduction; FE history
Reading: Clough2004a.pdf, Zienkiewicz2004a.pdf, Krylov41a.pdf, Williamson80a.pdf (RWTH only)
Exercises
- 2012.10.24 No exercise due to lecture by Tom Hughes.
- 2012.10.17 Introduction to exercises.
Tools
- XNS simulation code: download link for Linux on Intel (64-bit), Linux on Intel (32-bit), Mac OS X on PowerPC, or Mac OS X on Intel (RWTH only; at CATS, use
/usr/local/bin/xns). - Pager visualization code: download link for Linux on Intel (64-bit), Linux on Intel (32-bit), Mac OS X on PowerPC, or Mac OS X on Intel (at CATS, use
/usr/local/bin/pager). - A simple XNS Emacs mode based on the XNS input page .
Additional Reading
- Howard Elman et al., Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2004
- Alfio Quarteroni and Alberto Valli, Numerical Approximation of Partial Differential Equations, Springer, 1997
- Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987
- Olivier Pirronneau, Finite Element Methods for Fluids, Wiley, 1989
