Konrad,

VectorTools::project_boundary_ 
> <https://www.dealii.org/8.5.1/doxygen/deal.II/group__constraints.html#ga86427a4e8b3a7e580dabb4e473005288>values_div_conforming
>   
>
> <https://www.dealii.org/8.5.1/doxygen/deal.II/group__constraints.html#ga86427a4e8b3a7e580dabb4e473005288>is
>  
> then according to my understanding a pure projection (as the name suggests) 
> of vector valued functions to the Raviart-Thomas space. This is clearly a 
> helpful function (for even more than just boundaries) but not in 
> prescribing flux or traction boundary conditions since, as you mentioned, 
> this implicitly requires knowledge of the face normal but this is imo not 
> desired from the user perspective (complicated or changing domains?). For 
> traction BCs it is even more complicated since that means you need to 
> prescribe stress tensors. This unfortunately makes it difficult to 
> prescribe such BCs. 
>
> The strategy that I chose works for the lowest order Raviart-Thomas 
> element but what about higher order (because the dofs are edge moment?)?
>
Actually, it should be quite simple to make modify 
VectorTools::project_boundary_values_div_conforming such that 
(scalar-valued) flux boundary conditions can be applied.
The only places where the boundary function currently enters is in 
VectorTools::internal::compute_face_projection_div_conforming(https://github.com/dealii/dealii/blob/master/include/deal.II/numerics/vector_tools.templates.h#L6400).
In particular, it is only used for multiplying with the normal vector 
immediately after.
You could try to copy the relevant parts out of 
include/deal.II/numerics/vector_tools.templates.h. We are of course also 
happy to take contributions in that direction.
If you need help with that, don't hesitate to ask.

Best,
Daniel

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