Tham khảo tài liệu 'the mems handbook introduction & fundamentals (2nd ed) - m. gad el hak part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Model-Based Flow Control for Distributed Architectures 15-31 for intermediate values of T 0 100 . The adjoint problem Equation though linear has complexity similar to that of the Navier-Stokes problem Equation and may be solved with similar numerical methods. Identification of Gradient The identity Equation is now simplified using the equations defining the state field Equation the perturbation field Equation and the adjoint field Equation . Due to the judicious choice of the forcing terms driving the adjoint problem the identity Equation reduces after some manipulation to Of . f . . . . Of . du . . rTr _. I I C C1u u dx dt I C C2u u t Tdx II vC ịr dx dt II P 0 dx dt. Jn Jq Jq J dn 0 J rT 0 0 lọ UH 0 lọ Using this equation the cost functional perturbation J0 may be rewritten as J0 0 0 i f p 20 0 dx dt A f f J r. J r. 0 1 2 0 1 2 PJUQ D 0 0 dx dt. Because 0 is arbitrary we may identify weakly the desired gradient as -PJ0 Q D ộ p C2Ộ. The desired gradient DJ0 0 D0 is a simple function of the solution of the adjoint problem proposed in Equation . Specifically in the present case of boundary forcing by wall-normal blowing and suction the gradient is a simple function of the adjoint pressure on the walls. In fact this simple result hints at the more fundamental physical interpretation of what the adjoint field actually represents The adjoint field q when properly defined is a measure of the sensitivity of the terms of the cost functional that appraise the state q to additional forcing of the state equation. There are exactly as many components of the adjoint field q as there are components of the state PDE on the interior of the domain. Also note that the adjoint field may take nontrivial values at the initial time t 0 and on the boundaries r 22. Depending upon where the control is applied to the state Equation . on the RHS of the mass or momentum equations on the interior of the domain on the boundary .
