Curl

The curl operator calculates the curl $\mathbf{r}$ of a source vector $\mathbf{s}$.

$\mathbf{r} = \nabla \times \mathbf{s}$

Therefore the derivative is performed on radial basis functions (see Radial Basis Funtions). The theoretical background of the implemented RBF interpolation scheme is published in [^1]. When using the conservative filters please provide a citation in you publication:

Schoder, Stefan, et al. "Aeroacoustic source term computation based on radial basis functions." International Journal for Numerical Methods in Engineering (2020).

The filter is defined as following in the CFSdat xml input:

<differentiation type="SpaceDifferentiation_Curl" inputFilterIds="input" id="curl">
  <RBF_Settings epsilonScaling="..." betaScaling="..." kScaling="..." logEps="..."/>
  <targetMesh>
    <hdf5 fileName="targetMesh.h5"/>
  </targetMesh>
  <singleResult>
    <inputQuantity resultName="..."/>
    <outputQuantity resultName="..."/>
  </singleResult>
  <regions>
    <sourceRegions>
      <region name="..."/>
    </sourceRegions>
    <targetRegions>
      <region name="..."/>
    </targetRegions>
  </regions>
</differentiation>

In RBF_Settings, the following (optional and mandatory) attributes may be adjusted:

• epsilonScaling (mandatory): Controls the "smoothness" of the basis function. The smoother the Gauss-like surface is, the better the results will be BUT only until a certain number, when the matrix becomes so ill-conditioned, which will either result in an exception or very bad results. Typical values ~0.1

• betaScaling: Slope of the linear term that was added to the radial basic functions

• kScaling: constant term that was added to the radial basic functions

• logEps: Console output of [minimal distance, maximal distance, optimized epsilon]. NOTE: Should only be used for investigating the quality of the derivative, because it produces a LOT of console output.

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Acknowledgement

Please provide an acknowledgement at the end of your publication using this software part for simulations

The computational results presented have been achieved [in part] using the software openCFS [Curl].

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References

[^1]: Stefan Schoder, Klaus Roppert, Michael Weitz, Clemens Junger, and Manfred Kaltenbacher. Aeroacoustic source term computation based on radial basis functions. International Journal for Numerical Methods in Engineering, 121$9$:2051–2067, 2020.