# Divergence

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

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

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_Divergence" inputFilterIds="input" id="divergence">
<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.

## 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 [Divergence].

[^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. [^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.