Energetically Optimal Flapping Flight via a Fully Discrete Adjoint Method with Explicit Treatment of Flapping Frequency Jingyi Wang, Matthew J. Zahr † , and Per-Olof Persson CFD-36, Optimization Techniques for CFD AIAA Aviation Sheraton Denver Downtown Hotel, Denver, CO June 9, 2017 † Luis W. Alvarez Postdoctoral Fellow Department of Mathematics Lawrence Berkeley National Laboratory University of California, Berkeley 1 / 28
Understand and design energetically optimal flapping motions Energetically optimal flapping flight critical to • understand biological systems • design Micro Aerial Vehicles (MAVs) Optimal flapping motion of micro aerial vehicle 2 / 28
Understand and design energetically optimal flapping motions Energetically optimal flapping flight critical to • understand biological systems • design Micro Aerial Vehicles (MAVs) Optimal flapping motion of micro aerial vehicle Flapping frequency critical consideration in energetically optimal flapping 2 / 28
Challenge: Parametrize frequency = ⇒ parametrize time domain • N t uniform timesteps per period required for accuracy • Flapping frequency (period) is parametrized f = f ( µ ) ( T = T ( µ ) ) T ( µ ) = N t ∆ t 3 / 28
Challenge: Parametrize frequency = ⇒ parametrize time domain • N t uniform timesteps per period required for accuracy • Flapping frequency (period) is parametrized f = f ( µ ) ( T = T ( µ ) ) T ( µ ) = N t ∆ t Fix N t , parametrize ∆ t = ∆ t ( µ ) 3 / 28
Generalization beyond flapping Generalization: PDE-constrained optimization with parametrized time domain • Optimal control • Determination of fundamental frequency, e.g., von Karman vortex shedding • Path/trajectory optimization : find motion that achieves desired final position in least amount of time 4 / 28
Unsteady PDE-constrained optimization formulation Goal : Find the solution of the unsteady PDE-constrained optimization problem minimize J ( U , µ ) U , µ subject to C ( U , µ ) ≤ 0 ∂ U ∂t + ∇ · F ( U , ∇ U ) = 0 in v ( µ , t ) where t ∈ [0 , T ( µ )] and • U ( x , t ) PDE solution design/control parameters • µ � T ( µ ) 1 � • J ( U , µ ) = j ( U , µ , t ) dS dt objective function T ( µ ) 0 Γ � T ( µ ) 1 � • C ( U , µ ) = c ( U , µ , t ) dS dt constraints T ( µ ) 0 Γ 5 / 28
Nested approach to PDE-constrained optimization PDE optimization requires repeated queries to primal and dual PDE Optimizer Primal PDE Dual PDE 6 / 28
Nested approach to PDE-constrained optimization PDE optimization requires repeated queries to primal and dual PDE Optimizer µ Primal PDE Dual PDE 6 / 28
Nested approach to PDE-constrained optimization PDE optimization requires repeated queries to primal and dual PDE Optimizer J ( U , µ ) Primal PDE Dual PDE 6 / 28
Nested approach to PDE-constrained optimization PDE optimization requires repeated queries to primal and dual PDE Optimizer µ J ( U , µ ) U Primal PDE Dual PDE 6 / 28
Nested approach to PDE-constrained optimization PDE optimization requires repeated queries to primal and dual PDE Optimizer d J d µ ( U , µ ) J ( U , µ ) Primal PDE Dual PDE 6 / 28
High-order discretization of PDE-constrained optimization • Continuous PDE-constrained optimization problem minimize J ( U , µ ) U , µ subject to C ( U , µ ) ≤ 0 ∂ U ∂t + ∇ · F ( U , ∇ U ) = 0 in v ( µ , t ) • Fully discrete PDE-constrained optimization problem minimize J ( u 0 , . . . , u N t , k 1 , 1 , . . . , k N t ,s , µ ) u 0 , ..., u Nt ∈ R N u , k 1 , 1 , ..., k Nt,s ∈ R N u , µ ∈ R n µ subject to C ( u 0 , . . . , u N t , k 1 , 1 , . . . , k N t ,s , µ ) ≤ 0 u 0 − ¯ u ( µ ) = 0 s � u n − u n − 1 − b i k n,i = 0 i =1 Mk n,i − ∆ t n ( µ ) r ( u n,i , µ , t n,i ( µ )) = 0 7 / 28
Highlights of globally high-order discretization n da • Arbitrary Lagrangian-Eulerian formulation: N dA G , g , v X v Map, G ( · , µ , t ) , from physical v ( µ , t ) to reference V x 2 V x 1 X 2 � ∂ U X � + ∇ X · F X ( U X , ∇ X U X ) = 0 � X 1 ∂t � X Mapping-Based ALE • Space discretization : discontinuous Galerkin M ∂ u ∂t = r ( u , µ , t ) • Time discretization : diagonally implicit RK s DG Discretization � u n = u n − 1 + b i k n,i i =1 c 1 a 11 Mk n,i = ∆ t n ( µ ) r ( u n,i , µ , t n,i ( µ )) c 2 a 21 a 22 . . . ... . . . . . . • Quantity of interest : solver-consistency c s a s 1 a s 2 · · · a ss · · · b 1 b 2 b s F ( u 0 , . . . , u N t , k 1 , 1 , . . . , k N t ,s ) Butcher Tableau for DIRK 8 / 28
Adjoint method to efficiently compute gradients of QoI • Consider the fully discrete output functional F ( u n , k n,i , µ ) • Represents either the objective function or a constraint • The total derivative with respect to the parameters µ , required in the context of gradient-based optimization, takes the form N t N t s d F d µ = ∂F ∂F ∂ u n ∂F ∂ k n,i � � � ∂ µ + ∂ µ + ∂ u n ∂ k n,i ∂ µ n =0 n =1 i =1 • The sensitivities, ∂ u n ∂ µ and ∂ k n,i ∂ µ , are expensive to compute, requiring the solution of n µ linear evolution equations • Adjoint method : alternative method for computing d F d µ that require one linear evolution evoluation equation for each quantity of interest, F 9 / 28
Adjoint equation derivation: outline • Define auxiliary PDE-constrained optimization problem minimize F ( u 0 , . . . , u N t , k 1 , 1 , . . . , k N t ,s , µ ) u 0 , ..., u Nt ∈ R N u , k 1 , 1 , ..., k Nt,s ∈ R N u subject to R 0 = u 0 − ¯ u ( µ ) = 0 s � R n = u n − u n − 1 − b i k n,i = 0 i =1 R n,i = Mk n,i − ∆ t n r ( u n,i , µ , t n,i ) = 0 • Define Lagrangian N t N t s T R 0 − T R n − T R n,i � � � L ( u n , k n,i , λ n , κ n,i ) = F − λ 0 λ n κ n,i n =1 n =1 i =1 • The solution of the optimization problem is given by the Karush-Kuhn-Tucker (KKT) sytem ∂ L ∂ L ∂ L ∂ L = 0 , = 0 , = 0 , = 0 ∂ u n ∂ k n,i ∂ λ n ∂ κ n,i 10 / 28
Dissection of fully discrete adjoint equations • Linear evolution equations solved backward in time • Primal state/stage, u n,i required at each state/stage of dual problem • Heavily dependent on chosen ouput T ∂F λ N t = ∂ u N t s T ∂F ∂ r ∂ u ( u n,i , µ , t n − 1 + c i ∆ t n ) T κ n,i � λ n − 1 = λ n + + ∆ t n ∂ u n − 1 i =1 s T ∂F ∂ r ∂ u ( u n,j , µ , t n − 1 + c j ∆ t n ) T κ n,j M T κ n,i = � + b i λ n + a ji ∆ t n ∂ k n,i j = i • Gradient reconstruction via dual variables N t s d F d µ = ∂F T ∂ ¯ u T ∂ r � � ∂ µ + λ 0 ∂ µ ( µ ) + ∆ t n κ n,i ∂ µ ( u n,i , µ , t n,i ) n =1 i =1 11 / 28
Dissection of fully discrete adjoint equations Parametrized time domain : modifies gradient reconstruction from adjoint solution, not adjoint equations themselves N t s d F d µ = ∂F ∂ ¯ u ∂ r � � ∂ µ + λ T κ T ∂ µ ( µ ) + ∆ t n ∂ µ ( u n,i , µ , t n,i ) 0 n,i n =1 i =1 N t s � ∂f h � ∂t ( u n,i , µ , t n,i ) ∂t n,i ∂ µ ( µ ) + f h ( u n,i , µ , t n,i ) ∂ ∆ t n � � + ∆ t n ∂ µ ( µ ) b i n =1 i =1 N t s � ∂ r ∂t ( u n,i , µ , t n,i ) ∂t n,i ∂ µ ( µ ) + r ( u n,i , µ , t n,i ) ∂ ∆ t n � � � κ T + ∆ t n ∂ µ ( µ ) n,i n =1 i =1 where f h ( u , µ , t ) is DG approximation to � Γ j ( U , µ , t ) dS and N t s � � b i f h ( u n,i , µ , t n,i ( µ )) F ( u 0 , . . . , u N t , k 1 , 1 , . . . , k N t ,s , µ ) = ∆ t n ( µ ) n =1 i =1 12 / 28
Implementation details • Implementation of the fully discrete adjoint method relies on the computation of the following terms from the spatial discretization ∂t , f h , ∂f h ∂ u , ∂f h ∂ µ , ∂f h M , r , ∂ r ∂ u , ∂ r ∂ µ , ∂ r ∂t , and terms from the temporal discretization ∂ µ , ∂ ∆ t n t n,i , ∆ t n , ∂t n,i ∂ µ . • In the case of deforming domain problems treated with ALE formulation: r = r ( u , x ( µ , t ) , ˙ x ( µ , t )) f h = f h ( u , x ( µ , t ) , ˙ x ( µ , t )) • Partial derivatives w.r.t. µ and t computed as: ∂ ˙ ∂ ˙ ∂ µ = ∂ r ∂ r ∂ x ∂ µ + ∂ r x ∂f h = ∂f h ∂ µ + ∂f h ∂ x x ∂ x ∂ ˙ ∂ µ ∂ µ ∂ x ∂ ˙ ∂ µ x x ∂ r ∂t = ∂ r ∂ x ∂t + ∂ r ∂ ˙ x ∂f h = ∂f h ∂ x ∂t + ∂f h ∂ ˙ x ∂ ˙ ∂ ˙ ∂ x x ∂t ∂t ∂ x x ∂t 13 / 28
Time-periodic solutions desired when optimizing cyclic motion • To properly optimize a cyclic, or periodic problem, need to simulate a representative period • Necessary to avoid transients that will impact quantity of interest and may cause simulation to crash • Task : Find initial condition, ¯ u , such that flow is periodic, i.e. u N t = ¯ u 14 / 28
Time-periodic solutions desired when optimizing cyclic motion Vorticity around airfoil with flow initialized from steady-state (left) and time-periodic flow (right) 0 0 power power − 20 − 2 − 40 − 4 − 60 0 2 4 0 2 4 time time Time history of power on airfoil of flow initialized from steady-state ( ) and from a time-periodic solution ( ) 15 / 28
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