Simcenter STAR-CCM+ How to compute the average of the Surface Sensitivity in ad Adjoint Simulation

Let's imagine we are running an external aero case on a full car and we are using the Adjoint solution to morph the rear spoiler, to improve its efficiency. 
Due to the turbulent structures, the flow field might not be perfectly symmetric with respect to the longitudinal plane. As a consequence of this the sensitivity, the field computed on the component might be asymmetric. 
The displacement field used for the Morph Surface Mesh operation is a function of the sensitivity field and will then lead to an asymmetric shape of the rear spoiler, preventing these results to be used for an industrial case.  

In this article a very simple test case will show how to compute an average field starting from an asymmetric distribution of a generic quantity. The same procedure can be then applied to compute the mean Sensitivity field of an Adjoint simulation. 

Let's consider the asymmetric field function
 



Using a Data Mappers with the following settings:





Simcenter STAR-CCM+ will be able to create a new field function (mappedD) as mirror of the original one (_D):



At this point the average of mappedD and _D can be computed via the definition of the following field function:
 


_D and _avgD field functions can be converted into vectors via the definition of the following functions:
 
Now these functions can be used to define the Morph Surface Displacements. 

Images below show the differences between the displacements applied.



see also:
How to average fields on moving geometries?
Example of Using Surface Sensitivities to calculate displacements in an Adjoint Analysis