Question about Continuous Adjoint Method
 

Hi,

I'm studying the "continuous" adjoint method by reading Dr.Soto's article
titled "ON THE COMPUTATION OF FLOW SENSITIVITIES FROM BOUNDARY
INTEGRALS".
http://www.scs.gmu.edu/~rlohner/page.../reno04adj.pdf

I have a question regarding this article. Could someone answer the
following question?

Q)Question about the derivation of the adjoint boundary conditions
My understanding is that the adjoint boundary conditions are derived so
that all the boundary terms of equation (13) are exactly canceled at all
boundaries. I think we need the value of δu and δp at boundaries when
deriving the conditions.

Could you tell me how to know these values?

It says in this article that "δu=0 on S and on the outflow (Γout)(δu is the
test function associated to the adjoint velocity, then it is zero wherever
ψu is prescribed)" but I can't understand the meaning of this description at
all.

Thanks.

If on some part of boundary, u is fixed by the boundary condition, then du=0 on that part.

If u is not specified on some part S of boundary, and you have a term like

int(S)(X du)

then set X=0 on S. The idea is simply to get rid of all the variations in the primal quantities.

Thanks for reply. I understand your comment.

It says in this article that "δu=0 on S(solid boundaries to be optimized) and
on the outflow (Γout)". I think that δu=0 is valid on S because of the
dirichlet (no-slip) boundary condition but I can't understand why δu=0 on
the outlet boundary. I think dirichlet conditions are not imposed on the
velocity at the outlet when solving the flow.

Q) Is δu=0 valid on the outlet boundary?

I have another question.
Q) Could someone understand the meaning of the following statement?

"δu is the test function associated to the adjoint velocity, then it is zero
wherever ψu(adjoint velocity) is prescribed"

Can you indicate page/paragraph in that paper about which you have the question ?

Please look at the paragraph under the equation (19) on page three.
In this paragraph the adjoint boundary conditions are discussed.

On outflow boundary, u must come from the solution, it is not prescribed. Hence we dont know du. So to get rid of du on outflow boundary, set Psi_u=0. Thats how I look at it.

Could someone tell me why the following statement is valid?

"δu is the test function associated to the adjoint velocity, then it is zero
wherever ψu(adjoint velocity) is prescribed"