Direct/Adjoint Methods Lecture 12 ME EN 575 Andrew Ning - PDF document

Direct/Adjoint Methods Lecture 12 ME EN 575 Andrew Ning aning@byu.edu Outline Motivating Example Analytic Sensitivity Equations Direct/Adjoint

Motivating Example 1 + cos λ − λ mL = 0 Ω = λ 2 mL 4

Analytic Sensitivity Equations x i : design variables y j : state variables R k : residuals f n : outputs (objectives and constraints) x i y j R k = 0 f n f

Our end goal is to get d f n dx i for all i and n . f n = f ( x i , y j ( x i ))

R ( x i , y j ( x i )) = 0

Analytic Sensitivity Equations � − 1 ∂ R k � ∂ R k d f n = ∂f n − ∂f n dx i ∂x i ∂y j ∂y j ∂x i x i y j R k = 0 f n f

Direct/Adjoint Two ways to solve. Partial derivatives are always the same, but order of operations is not. Φ � �� � � − 1 ∂ R k � ∂ R k d f n = ∂f n − ∂f n dx i ∂x i ∂y j ∂y j ∂x i � �� � Ψ

Direct method: Adjoint method:

Step Direct Adjoint Matrix Factorization same same Back-solve N x times N f times Multiplication same same 400 350 • 2M CFD cells 300 • 300k CSM DOFs Normalized Time 250 Finite Difference • 56 processors Adjoint 200 1.0 + 0.28N x 150 • 1 aerostructural 100 solution = 5.5 min 50 1.76 + 0.00004N x 0 0 500 1000 1500 2000 Number of Design Variables Kenway, Kennedy and Martins, AIAA Journal, 2012