Finite Elements in Fluids

General

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

Notices

• The oral exam dates are set for 5.03.2007 and 19.03.2007. Please contact Mike Nicolai to make an appointment.
• Results of unsteady advection-diffusion experiments and Stokes and Navier-Stokes experiments (see below) will be due one week before the exam date.
• Because of the mistake in bodyexp expression below (this affects efforts to reproduce Fig. 2.13) I will be accepting the results of steady advection-diffusion experiments for one more week, until 24.01.
• The first lecture will be an organizational meeting, taking place on 18.10.2006 in Fo 7 at 15:45.

Lectures

• 2007.02.07 Lecture 13: compatibility conditions, pressure stabilization (rest of 6.5); steady and unsteady Navier-Stokes problem (6.6, 6.7)
• 2007.01.31 Lecture 12: 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, 6.5.3, 6.5.4)

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

• 2007.01.24 Lecture 11: Modified equation method (3.5.3); third-order explicit Taylor-Galerkin method (3.6.2); unsteady advection-diffusion-reaction problems (5.1, 5.2); linear multistep methods (5.3.1); fractional-step methods (5.3.2); Galerkin formulation (5.4.1); error analysis of Galerkin formulation of θ family methods (5.4.2)
• 2007.01.17 Lecture 10: Fourier's error analysis for hyperbolic problems (3.5.1, 3.5.2)
• 2007.01.10 Lecture 9: unsteady advection problems (3.1, 3.2, 3.4); space-time formulations (3.10)
• 2006.12.20 Lecture 8: origins of stabilization: FIC (2.5.1), VMS (2.5.3)

  Onate2000a.pdf, Hughes95a.pdf (RWTH only)

• 2006.12.13 Lecture 7: a priori error estimates

  Stability Analysis of Scalar Advection-Diffusion Equation (PDF)

• 2006.12.06 Lecture 6: Upwinding (2.3.1, 2.3.2); balancing diffusion (2.3.3); stabilized methods for transport problems (2.4).
• 2006.11.29 Lecture 5: strong, weak and Galerkin form of advection-diffusion equation (2.1, 2.2); Galerkin versus exact stencils.
• 2006.11.22 Lecture N1: Galerkin FEM implementation of 1D advection-diffusion equation.

  advdiff_1d.m, func_advdiff_exact.m, func_poisson_exact.m

• 2006.11.15 Lecture 4: Galerkin form of Poisson equation; implementation (1.5.5)
• 2006.11.08 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)
• 2006.10.25 Lecture 2: reference frames (1.3.1, 1.3.2, 1.3.3); Reynolds transport theorem (1.3.4)
• 2006.10.18 Lecture 1: introduction; FE history

  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 Linux executable xns (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 Linux executable pager (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 0.25 ∗ 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 of 0.25 is surprising, but necessary.
• 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 .
• Recreate Figure 5.14 (SGS) using given XNS input and Pager input .
• Recreate Figure 5.16 using given XNS input and Pager input .
• Recreate Figure 5.18 using given XNS input and Pager input . What happens when you use Clark-Nicholson time integration for the same parameters as space-time, i.e., Pe = 100 and C = 3?

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.

Tools

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