Finite Elements in Fluids

General

• Syllabus • Course ad (PDF) • Campus entry • Last year's FEF

Notices

The oral exams are planned for February 21, March 14 and April 4, 2011, starting at 10:00. Please contact Mike Nicolai to make exact appointment if you don't yet know exact time of your exam. Please bring all exercises which have been graded.

Biweekly Computational Engineering Science seminars take place Mondays, 16:00 in the Rogowski building at Schinkelstr. 2, room 115, starting on October 14, 2010 (the opening seminar is on Thursday).

Lectures

• 2011.02.02 Lecture 13: compatibility conditions, pressure stabilization (6.5.3–8); steady and unsteady Navier-Stokes problem (6.6, 6.7)
• 2011.01.26 Lecture 12: stationary Stokes problem (6.5.1, 6.5.2)
• 2011.01.19 Lecture 11: 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); viscous incompressible flow (6.1, 6.2, 6.3, 6.4)

  The deadline for the second assignment is January 26, 2011! Email the resulting figures with some comments and captions as a PDF report to Mike Nicolai. Questions can be also emailed to Marek.

• 2010.01.12 Lecture 10: Fourier's error analysis for hyperbolic problems (3.5.1, 3.5.2); Modified equation method (3.5.3)
• 2010.12.22 Lecture does not take place this week
• 2010.12.15 Lecture 9: unsteady advection problems (3.1, 3.2, 3.4); Space-time formulations (3.10)

  HiWi ad (RWTH only)

• 2010.12.08 Lecture 8: origins of stabilization: finite increment calculus (2.5.1), variational multiscale (2.5.3), a priori error estimates

  Reading: Onate2000a.pdf, Hughes95a.pdf (RWTH only)

  Stability Analysis of Scalar Advection-Diffusion Equation (PDF)

• 2010.12.01 Lecture 7: balancing diffusion (2.3.3); stabilized methods for transport problems (2.4)
• 2010.11.24 Lecture 6: Galerkin versus exact stencils and upwinding (2.3.1, 2.3.2)
• 2010.11.17 Lecture 5: Galerkin form of Poisson equation; implementation (1.5.5); strong, weak and Galerkin form of advection-diffusion equation (2.1, 2.2)

  Reading: FE integrals slides

• 2010.11.10 Lecture 4: weak and Galerkin forms of Poisson equation, implementation (1.5.2, 1.5.3, 1.5.4)
• 2010.11.03 Lecture 3: conservation laws (1.4), strong form of Poisson equation (1.5.1)
• 2010.10.27 Lecture 2: reference frames (1.3.1, 1.3.2, 1.3.3), Reynolds transport theorem (1.3.4)

  Reading: Huerta88a.pdf (RWTH only)

• 2010.10.20 Lecture 1: introduction; FE history

  Reading: Clough2004a.pdf, Zienkiewicz2004a.pdf, Krylov41a.pdf, Williamson80a.pdf (RWTH only)

Steady advection-diffusion experiments using XNS due 2010.12.22

• Recreate parts of Figure 2.7 and 2.12 by computing and plotting FE solution without and with SUPG, at Peclet numbers 0.25, 0.90 and 5.0, using linear elements:
  1. Download XNS executable (see Tools section below) and input file xns.in.
  2. Examine the input file; run XNS simply with xns. You will probably need to add execute permission to XNS with chmod u+x xns after downloading.
  3. Download Pager executable (see Tools section below) and input file pin.e.
  4. Run Pager with pager pin.e, look at pager.ps with a Postscript viewer, e.g. Ghostscript gs or Ghostview gv. You will probably need to add execute permission to Pager with chmod u+x pager after downloading.

     If pager pin.e does nothing but display pin.e, that means that a Linux program pager is executed instead; to run our own Pager, specify the path, e.g., ./pager pin.e.

  5. Change tau_momentum_factor to 1.0 to enable SUPG; adjust viscosity to change Peclet number.
• Recreate parts of Figure 2.13 by computing and plotting FE solution with SUPG using linear elements:
  • Change bodyexp to something like:

    bodyexp 1 1.00 * 5 * exp(-100 * (x - 0.125) * (x - 0.125)) - ...

  • The first number after bodyexp is the degree of freedom; in our case, 1. The factor 1.00 needs to be adjusted to 0.25 in older buggy versions of XNS which may still be posted here and there.
• Recreate Figures 2.19 and 2.20 by computing and plotting FE solutions with Galerkin, SUPG and artificial diffusion:
  • Modify the input file to generate 10 × 10 mesh instead of 10 × 1 mesh.
  • Use expression

    rngdexp 4 1 heaviside(y - 0.2)

    to impose the step boundary condition on the left boundary.
  • Sample input is xns.in.2d. Use pin.e.2d input file for Pager.

Unsteady advection-diffusion experiments using XNS due 2011.01.26

• Recreate Figure 5.5 using given XNS input and Pager input. Show Pe = 0.5 and Pe = 5.0 at t = 0.1, 0.6 and 1.2, without stabilization.
• Recreate Figure 5.14 (SGS) using given XNS input and Pager input. Use high Pe (e.g., Pe = 500) and C = 1. Show SUPG (dtingls off) and GLS (dtingls on); skip SGS and LS. Always use dtintau on to get the τ in Remark 5.8.
• Recreate Figure 5.16 using given XNS input and Pager input. Show all 4 cases. You don't need to plot the exact solution at the final time.
• Recreate Figure 5.18 using given XNS input and Pager input. Show only Galerkin cases (left column); skip GLS, as stabilization is not important in this case. What happens when you use Crank-Nicholson time integration for the same parameters as space-time, i.e., Pe = 100 and C = 3?

Stokes and Navier-Stokes experiments using XNS due 2011.02.23

• Compute steady Stokes flow in a lid-driven cavity (Section 6.8.2) using XNS input and Pager input (pressure) or input (velocity) with and without (SUPG+PSPG) stabilization.
• Compute Reynolds number 400 flow in the same cavity using XNS input and Pager input (pressure) or input (velocity) with and without stabilization.
• If you take the oral exam on 2011.02.21, you should simply bring the printed completed assignment with you to the exam.

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, Finte Element Methods for Fluids, Wiley, 1989