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Direct/Adjoint Methods Lecture 12 ME EN 575 Andrew Ning - - PDF document
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 mL = 0 = 2 mL 4 Analytic Sensitivity Equations x i
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Analytic Sensitivity Equations
xi : design variables yj : state variables Rk : residuals fn : outputs (objectives and constraints)
Rk = 0 yj xi fn f
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Our end goal is to get d fn dxi for all i and n. fn = f(xi, yj(xi))
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R(xi, yj(xi)) = 0
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Analytic Sensitivity Equations
d fn dxi = ∂fn ∂xi − ∂fn ∂yj ∂Rk ∂yj −1 ∂Rk ∂xi
Rk = 0 yj xi fn f
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Direct/Adjoint
Two ways to solve. Partial derivatives are always the same, but order of operations is not. d fn dxi = ∂fn ∂xi − ∂fn ∂yj
Φ
- ∂Rk
∂yj −1 ∂Rk ∂xi
- Ψ
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Direct method: Adjoint method:
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Step Direct Adjoint Matrix Factorization same same Back-solve Nx times Nf times Multiplication same same
Number of Design Variables Normalized Time
500 1000 1500 2000 50 100 150 200 250 300 350 400 Finite Difference Adjoint
1.76 + 0.00004Nx 1.0 + 0.28Nx
- 2M CFD cells
- 300k CSM DOFs
- 56 processors
- 1 aerostructural
solution = 5.5 min
Kenway, Kennedy and Martins, AIAA Journal, 2012