Finite Elements in Fluids

General

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

Notices

The oral exams are planned for February 13 and March 16, 2009, starting at 12:00. Please contact Mike Nicolai to make an appointment if you haven't yet done so. Please bring your past graded exercises, as well as your last exercise.

Weekly Center for CES seminars take place Mondays, 17:00 in the graduate school AICES at Pauwelsstr. 12 near the University clinic, starting on October 20, 2008.

Lectures

• 2009.02.04 Lecture 14: compatibility conditions, pressure stabilization (6.5.3–8); steady and unsteady Navier-Stokes problem (6.6, 6.7)
• 2009.01.21 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); stationary Stokes problem (6.5.1, 6.5.2)

  Reading: 2D Unsteady Advection-Diffusion Element-Level Matrices and Vectors (PDF)

• 2009.01.14 Lecture 10: Space-time formulations (3.10); Fourier's error analysis for hyperbolic problems (3.5.1, 3.5.2)

  Reading: Space-time elasticity paper (RWTH only).

• 2009.01.07 Lecture 9: unsteady advection problems (3.1, 3.2, 3.4)
• 2008.12.17 Lecture 8: origins of stabilization: FIC (2.5.1), VMS (2.5.3), a priori estimates

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

  Stability Analysis of Scalar Advection-Diffusion Equation (PDF)

• 2007.12.10 Lecture N1: Galerkin FEM implementation of 1D advection-diffusion equation and SUPG stabilization.

  advdiff_1d.m, advdiff_1d_supg.m, func_advdiff_exact.m, func_poisson_exact.m

• 2008.12.03 Lecture 7: stabilized methods for transport problems (2.4)
• 2008.11.26 Lecture 6: Galerkin versus exact stencils and upwinding (2.3.1, 2.3.2); balancing diffusion (2.3.3)
• 2008.11.19 Lecture does not take place this week
• 2008.11.12 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

• 2008.11.05 Lecture 4: Galerkin form of Poisson equation; implementation (1.5.5)
• 2008.10.29 Lecture 3: conservation laws (1.4); strong and weak forms of Poisson equation (1.5.1, 1.5.2, 1.5.3, 1.5.4)
• 2008.10.22 Lecture 2: reference frames (1.3.1, 1.3.2, 1.3.3); Reynolds transport theorem (1.3.4)

  Reading: Huerta88a.pdf (RWTH only)

• 2008.10.15 Lecture 1: introduction; FE history

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

Steady advection-diffusion experiments using XNS

• Recreate Figure 2.7, without and with SUPG, at Peclet numbers 0.25, 0.90 and 5.0, using linear elements:
  1. Download XNS executable for Linux on Intel, Mac OS X on PowerPC, or Mac OS X on Intel (RWTH only; at CATS, use /usr/local/bin/xns) and input file xns.in.
  2. Examine the input file; run XNS simply with xns. You may need to add execute permission to XNS with chmod o+rx xns after downloading.
  3. Download Pager executable for Linux on Intel, Mac OS X on PowerPC, or Mac OS X on Intel (at CATS, use /usr/local/bin/pager) 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 may need to add execute permission to Pager with chmod o+rx 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 Figure 2.13 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 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

• 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?
• Send the solutions to Mike Nicolai by the time of the last lecture (2009.02.04). Since there are many figures, please arrange them using some word processing software, rather than sending raw Postscript files. Postscript can be converted to PDF using Preview on Mac OS or convert and many other utilities on Linux.

Stokes and Navier-Stokes experiments

• Compute steady Stokes flow in a lid-driven cavity 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.
• Please bring the plots to the oral exam.

Tools

• XNS simulation code: Linux/x86 executable xns (RWTH only)

  The Linux version of XNS still has the bug that requires right-hand side expressions for advection-diffusion cases to be divided by a factor equal to numer of element nodes (e.g., 4).

• Pager visualization code: Linux/x86 executable pager (RWTH only)
• 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