Adjoint-Based Optimization of Time-Dependent Fluid-Structure Systems - PowerPoint PPT Presentation

Adjoint-Based Optimization of Time-Dependent Fluid-Structure Systems using a High-Order Discontinuous Galerkin Discretization Matthew J. Zahr † and Per-Olof Persson 19h International Conference on Finite Elements in Flow Problems (FEF) La Sapienza University, Rome, Italy April 5-7, 2017 † Luis W. Alvarez Postdoctoral Fellow Department of Mathematics Lawrence Berkeley National Laboratory University of California, Berkeley 1 / 28

PDE optimization is ubiquitous in science and engineering Design : Find system that optimizes performance metric, satisfies constraints Aerodynamic shape design of automobile Optimal flapping motion of micro aerial vehicle 2 / 28

PDE optimization is ubiquitous in science and engineering Control : Drive system to a desired state Boundary flow control Metamaterial cloaking – electromagnetic invisibility 3 / 28

PDE optimization is ubiquitous in science and engineering Inverse problems : Infer the problem setup given solution observations Left : Material inversion – find inclusions from acoustic, structural measurements Right : Source inversion – find source of airborne contaminant from downstream measurements Full waveform inversion – estimate subsurface of Earth’s crust from acoustic 4 / 28 measurements

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 • U ( x , t ) PDE solution • µ design/control parameters � T f � • J ( U , µ ) = j ( U , µ , t ) dS dt objective function T 0 Γ � T f � • C ( U , µ ) = c ( U , µ , t ) dS dt constraints T 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 − g ( µ ) = 0 s � u n − u n − 1 − b i k n,i = 0 i =1 Mk n,i − ∆ t n r ( u n,i , µ , t n,i ) = 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 r ( u n,i , µ , t n,i ) 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 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 − g ( 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 ∂ u N t j = i • Gradient reconstruction via dual variables N t s d F d µ = ∂F T ∂ g T ∂ r � � ∂ µ + λ 0 ∂ µ ( µ ) + ∆ t n κ n,i ∂ µ ( u n,i , µ , t n,i ) n =1 i =1 [Zahr and Persson, 2016] 11 / 28

Energetically optimal flapping under x -impulse constraint � 3 T � • Isentropic, compressible, minimize f · v dS dt − µ 2 T Γ Navier-Stokes � 3 T � subject to f · e 1 dS dt = q • Re = 1000, M = 0.2 2 T Γ • y ( t ) , θ ( t ) , c ( t ) parametrized via U ( x , 0) = g ( x ) periodic cubic splines ∂ U ∂t + ∇ · F ( U , ∇ U ) = 0 • Black-box optimizer: SNOPT l l/3 θ (t) c(t) y(t) Airfoil schematic, kinematic description 12 / 28

Optimal control, time-morphed geometry Optimal Rigid Body Motion (RBM) and Time-Morphed Geometry (TMG), x -impulse = − 2 . 5 Energy = 9.4096 Energy = 4.9476 Energy = 4.6182 x -impulse = -0.1766 x -impulse = -2.500 x -impulse = -2.500 Optimal RBM Optimal RBM/TMG Initial Guess J x = − 2 . 5 J x = − 2 . 5 13 / 28

Energetically optimal flapping in three-dimensions Energy = 1.4459e-01 Energy = 3.1378e-01 Thrust = -1.1192e-01 Thrust = 0.0000e+00 14 / 28

Structure: semi-discretization, first-order form M s ∂ u s ∂t = r s ( u s ; t ) = r ss ( u s ) + r sf · t • Semidiscretization (CG-FEM) of continuum (hyperelasticity) ∂ p ∂t − ∇ · P ( G ) = b in Ω 0 P ( G ) · N = t on Γ N x = x D on Γ D • Force balance on rigid body M ∂ 2 q ∂t 2 + C ∂ q ∂t + Kq = t θ ( µ , t ) h ( u s ) c 15 / 28

Coupled fluid-structure formulation • Write discretized fluid and structure equations as ODEs M f ˙ u f = r f ( u f ; x ) M s ˙ u s = r s ( u s ; t ) = r ss ( u s ) + r sf · t in the fluid u f and structure u s variables • Apply couplings • Structure-to-fluid: deform fluid domain x = x ( u s ) • Fluid-to-structure: apply boundary traction t = t ( u f ) • Write coupled system as M ˙ u = r ( u ) � � � � � � M f u f r f ( u f ; x ( u s )) u = r ( u ) = M = M s u s r s ( u s ; t ( u f )) • Structure of linearized residual   ∂ r f ∂ r f ∂ x ∂ r ∂ u f ∂ u s  ∂ x  ∂ u ( u ) =   ∂ r s ∂ r s  ∂ t  ∂ u f ∂ u s ∂ t 16 / 28

High-order partitioned FSI solver: IMEX Runge-Kutta 1 • Exploit linear dependence of structure residual ( r s ) on traction ( t ) � � � � � � r f ( u f ; x ( u s ) r f ( u f ; x ( u s )) r ( u ) = = + r sf · ( t ( u f ) − ˜ r s ( u s ; ˜ r s ( u s ; t ( u f )) t ) t ) � �� � � �� � f ( u ) g ( u ) • Apply high-order implicit-explicit Runge-Kutta scheme to discretize M ∂ u ∂t = r ( u ) = f ( u ) + g ( u ) � �� � ���� explicit implicit c, ˆ A, ˆ • Explicit Runge-Kutta scheme ˆ b for f ( u ) • Diagonally implicit scheme c, A, b for g ( u ) s s � ˆ b i ˆ � u n = u n − 1 + k n,i + b i k n,i i =1 i =1 � i − 1 i � � a ij ˆ � Mk n,i = ∆ t n g u n − 1 + ˆ k n,j + a ij k n,j j =1 j =1 � i − 1 i � M ˆ � a ij ˆ � k n,i = ∆ t n f u n − 1 ˆ k n,j + a ij k n,j j =1 j =1 1 [van Zuijlen and Bijl, 2005, Froehle and Persson, 2014] 17 / 28