# Boundary Conditions Of Discrete Adjoint Method For 2D NS Equations

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 May 30, 2014, 01:48 Boundary Conditions Of Discrete Adjoint Method For 2D NS Equations #1 New Member   Ashkaari Join Date: May 2014 Posts: 5 Rep Power: 11 Hello everyone, I've been working on the adjoint based grid adaptation for viscous flow.But my adjoint solver doesn't work out well. Here is the problem, back in the flow solver, no-slip and isothermal wall conditions were applied. At wall,u=v=0,E=ρT/(γ-1)=Kρ. And the residuals of momentum and energy equations were all zeroed out(R2=R3=0). The adjoint equations is, (әR/әW)Tψ=әI/әW. ψ=[ψ1,ψ2,ψ3,ψ4] is the adjoint solution. (әR/әW)T is the transposed Jacobian with W=[ρ,ρu,ρv,E], I is an output function like lift or drag coefficient. For a wall node i, by hand differentiation: әR1/әW=[0,nx,ny,0], әR2/әW=әR3/әW=[0,0,0,0], әR4/әW=K*әR1/әW. The transposed Jacobian becomes [0,0,0,0;nx,0,0,Knx;ny,0,0,Kny;0,0,0,0]. According to David Venditti, ψ2,ψ3 can still be known via special treatment. However, since the first and last row of jacobian matrix is zero vector, I can not get ψ1,ψ4 . There must be something wrong in the above deduction, could you kindly point it out for me? Many Thanks. Last edited by Ashkaari; May 30, 2014 at 04:50.

 May 30, 2014, 04:42 #2 New Member   Ashkaari Join Date: May 2014 Posts: 5 Rep Power: 11 The flow and adjoint solver utilize a median-dual finite volume scheme. According to Oliver Amoignon, discretization of the adjoint equations become: [(әR/әW)Tψ]i=∑(әRij/әWi)T*(ψj-ψi)/2+(әRib/әWi)ψi, and j stands for nodes share the same edge with node i. Rib will only be used for boundary nodes. For simplicity,artificial dissipation is not taken into consideration. Rij=(fc,fv)*(nx,ny). (nx,ny) is the outward normal vector on edge ij. The notation fc,fv stand for the convective and visous terms of node i.

September 5, 2014, 13:41
#3
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Francisco Palacios
Join Date: Jan 2013
Location: Long Beach, CA
Posts: 404
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Quote:
 Originally Posted by Ashkaari The flow and adjoint solver utilize a median-dual finite volume scheme. According to Oliver Amoignon, discretization of the adjoint equations become: [(әR/әW)Tψ]i=∑(әRij/әWi)T*(ψj-ψi)/2+(әRib/әWi)ψi, and j stands for nodes share the same edge with node i. Rib will only be used for boundary nodes. For simplicity,artificial dissipation is not taken into consideration. Rij=(fc,fv)*(nx,ny). (nx,ny) is the outward normal vector on edge ij. The notation fc,fv stand for the convective and visous terms of node i.
Maybe you can take a look at SU2 (su2.stanford.edu) and related articles.

Cheers,
Francisco

 September 6, 2014, 12:12 #4 New Member   Ashkaari Join Date: May 2014 Posts: 5 Rep Power: 11 I believe I have solved this problem. To my knowledge, the continuous adjoint method was applied in SU2, thank you all the same.

January 27, 2016, 21:05
#5
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Tommy Chen
Join Date: Mar 2011
Location: University of Michigan
Posts: 96
Rep Power: 14
Quote:
 Originally Posted by Ashkaari I believe I have solved this problem. To my knowledge, the continuous adjoint method was applied in SU2, thank you all the same.
Hi Ashkaari

How did you solve this problem ?

I'm also working on Ajoint method development

 Tags adjoint solver, boundary condition, jacobian matrix