Henry von Wahl & Thomas Richter & Jan Heiland

Christoph Lehrenfeld & Piotr Minakowski

GAMM CSE -- 21 November 2019

I haven't found a clear definition of what a benchmark is. However, here is what I think makes a *numerical example* a benchmark

**Common acceptance as a benchmark**-- there are other publications that discuss the same setup.**Practical relevance**-- either in applications or as a testing field for numerical algorithms.**Reliable reference data**-- so that others can test their codes and methods against it.

Basically, everything that would motivate a fellow researchers to use the provided setup and data to

benchmarktheir code.

Quantitative assessments, evaluate performance:

We know that this computation

correct, but howefficientis it?

Examples:

Qualitive assessments, evaluate confidence:

Are the computations

correct?

Example: FEATFLOW CFD Benchmarking project

Note that:

The more complex the model is, the more necessary are benchmarks

butthe more difficult are benchmark definitions.

- Changing domain.
- Coupling of Models (and scales).

- The domain is fixed.
- Can concentrate on the coupling of the models.
- Accessible to standard CFD solvers.

- Relevant as a testing field for algorithms and
- reliable test data.

(1.) General acceptance as a benchmark may come later.

- A fluid flows through a channel with a sphere that can rotate freely.
- The stresses at the sphere/fluid interface induce rotation.
- The
*no-slip*condition induces motion of the flow at the interface.

where \(v_s\) is the solid's velocity at the fluid-solid interface.

with the body's centre of mass \(\mathbf c\).

- 2D and 3D
- stationary -- where there is no torque (low
*Re*-number) - periodic -- a limit cycle (moderate
*Re*-number) - time dependent -- a start-up period

To assess *the truth* the reported data should be

- independent of numerical setup (like the mesh or the scheme),
- dimensionless and suitably parametrized (like through the
*Reynolds number*), - characteristic for the setup, and
- meaningful for, say, applications.

Variable | Definition |
---|---|

\(C_L\) | lift coefficient(s) |

\(C_D\) | drag coefficient(s) |

\(C_T\) | torque coefficient(s) |

\(\Delta_p\) | pressure difference at the cylinder |

\({\omega}^{ * }\) | dimensionless rotation |

We used

- the
*Strouhal number*to characterize the frequency, - minima, maxima of \(C_D\), \(C_L\), \(C_T\), and \(\omega^{ * }\),
- and \(\Delta_p(t^ * )\) -- at the middle of a period.

There were 5 independent implementations using established libraries:

- Netgen/NGSolve
- FEniCS/dolfin
- Gascoigne
- SciPy

*inf-sup*stable and stabilized equal order elements.- High-order and standard
*Taylor-Hood*(\(P_2-P_1)\) elements. - Divergence conforming elements.
- Hybrid Discontinous Galerkin methods.
- Implicit/Explicit time integration.
- Most critical: Evaluation of the boundary integrals.

The reported (converged) characteristic outputs ly within certain confidence intervals \(\Delta_I\):

Test case | Relative size of \(\Delta_I\) | Critical value |
---|---|---|

stationary-2D | \(10^{-5}\) | \(C_L\) |

periodic-2D | \(10^{-3}\) | \(C_T\) |

time-dep-2D | \(10^{-3}\) | \(C_L\) |

stationary-3D | (\(1\)), \(10^{-1}\) | \(\omega^ { * }\) |

- The 2D simulations gave reliable results.
- Also the stationary 3D case (at least in absolute numbers).
- The time-dependent 3D results were inconclusive.
- We did not include them in the report.
- Conflict with another benchmark property:
*the numerical setup should be challenging.*

Full data sets for the results as well as all implementations can be found at

Benchmarks are valuable for the assessment of numerical models.

- The proposed benchmark may qualify as such because of
- reliable data,
- setup accessible to standard solvers.

I learned: Numerical analysis matters.

von Wahl, Henry, Thomas Richter, Christoph Lehrenfeld, Jan Heiland, and Piotr Minakowski. 2019. “Numerical Benchmarking of Fluid-Rigid Body Interactions.” *Computers & Fluids*. doi:10.1016/j.compfluid.2019.104290.