High-Order, Time-Dependent Aerodynamic Optimization using a - PowerPoint PPT Presentation

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions High-Order, Time-Dependent Aerodynamic Optimization using a Discontinuous Galerkin Discretization of the Navier-Stokes Equations Matthew J. Zahr Stanford University Collaborators: Per-Olof Persson (UCB), Jon Wilkening (UCB) AIAA SciTech Meeting and Exposition 54th AIAA Aerospace Sciences Meeting FD-04. CFD Applications and Design Monday, January 4, 2016 Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Thought Experiment: Which motion ... Has time-averaged x -force identically equal to 0? Requires least energy to perform? Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Thought Experiment: Which motion ... Has time-averaged x -force identically equal to 0? Requires least energy to perform? Energy = 9.4096 Energy = 0.45695 Energy = 4.9475 x -force = -0.8830 x -force = 0.000 x -force = -12.50 Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Real-World Application: Micro Aerial Vehicles (MAV) Unmanned flying vehicle usually flapping propulsion system wingspan between 7 . 4cm and 15cm speed between ≤ 15m/s Military applications surveillance, reconnaissance Micro Aerial Vehicle quiet, resemble small bird from distance Civilian applications Crowd monitoring, survivor search, pipeline inspection Di ffi culties Thrust and lift requirements Structural constraints Stability and control considerations Bumblebee MAV (USAF 2008) Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Time-Dependent PDE-Constrained Optimization Optimization of systems that are inherently dynamic or without a steady-state solution Introduction of fully discrete adjoint method emanating from high-order discretization of governing equations Coupled with numerical optimization Volkswagen Passat Time-periodicity constraints LES Flow past Airfoil Micro Aerial Vehicle Vertical Windmill Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Abstract Formulation of Problem of Interest 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 + r · F ( U , r U ) = 0 in v ( µ , t ) where U ( x , t ) PDE solution µ design/control parameters Z T f Z J ( U , µ ) = j ( U , µ , t ) dS dt objective function T 0 Γ Z T f Z C ( U , µ ) = c ( U , µ , t ) dS dt constraints T 0 Γ Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions 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 + r · F ( U , r U ) = 0 in v ( µ , t ) Fully discrete PDE-constrained optimization problem J ( u (0) , . . . , u ( N t ) , k (1) 1 , . . . , k ( N t ) minimize , µ ) s u (0) , ..., u ( Nt ) 2 R N u , k (1) , ..., k ( Nt ) 2 R N u , 1 s µ 2 R n µ C ( u (0) , . . . , u ( N t ) , k (1) 1 , . . . , k ( N t ) subject to , µ )  0 s u (0) � u 0 ( µ ) = 0 s u ( n ) � u ( n � 1) + b i k ( n ) X = 0 i i =1 ⇣ ⌘ M k ( n ) u ( n ) , µ , t ( n � 1) = 0 � ∆ t n r i i i Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Highlights of Globally High-Order Discretization n da Arbitrary Lagrangian-Eulerian Formulation: N dA G , g , v X Map, G ( · , µ , t ), from physical v ( µ , t ) to reference V v x 2 V � @ U X x 1 X 2 � + r X · F X ( U X , r X U X ) = 0 � @ t � X X 1 Space Discretization : Discontinuous Galerkin Mapping-Based ALE 1 2 3 4 M @ u @ t = r ( u , µ , t ) 1 2 4 3 3 2 Time Discretization : Diagonally Implicit RK 4 1 CDG : LDG : and BR2 : and s u ( n ) = u ( n � 1) + b i k ( n ) X DG Discretization i i =1 c 1 a 11 ⇣ ⌘ M k ( n ) u ( n ) , µ , t ( n � 1) = ∆ t n r c 2 a 21 a 22 i i i . . . ... . . . . . . Quantity of Interest : Solver-consistent 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 Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Generalized Reduced-Gradient Approach - Schematic Optimizer drives, Primal returns QoI values, Dual returns QoI gradients PRIMAL PDE OPTIMIZER MESH MOTION DUAL PDE Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Generalized Reduced-Gradient Approach - Schematic Optimizer drives, Primal returns QoI values, Dual returns QoI gradients PRIMAL PDE µ OPTIMIZER MESH MOTION DUAL PDE Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Generalized Reduced-Gradient Approach - Schematic Optimizer drives, Primal returns QoI values, Dual returns QoI gradients PRIMAL PDE x , ˙ x µ OPTIMIZER MESH MOTION DUAL PDE Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Generalized Reduced-Gradient Approach - Schematic Optimizer drives, Primal returns QoI values, Dual returns QoI gradients PRIMAL PDE x , ˙ x µ OPTIMIZER MESH MOTION k ( n ) u ( n ) , i x , ˙ x ∂ x ∂ µ , ∂ ˙ x ∂ µ DUAL PDE Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Generalized Reduced-Gradient Approach - Schematic Optimizer drives, Primal returns QoI values, Dual returns QoI gradients PRIMAL PDE J, C x , ˙ x µ OPTIMIZER MESH MOTION k ( n ) u ( n ) , i x , ˙ x d µ , d C d J ∂ µ , ∂ ˙ ∂ x x d µ ∂ µ DUAL PDE Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Adjoint Method to Compute QoI Gradients Consider the fully discrete output functional F ( u ( n ) , k ( n ) , µ ) i 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 @ k ( n ) @ u ( n ) d F d µ = @ F @ F @ F X X X i @ µ + + @ u ( n ) @ µ @ k ( n ) @ µ n =0 n =1 i =1 i and @ k ( n ) The sensitivities, @ u ( n ) i @ µ , are expensive to compute, requiring the @ µ 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 Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization

Adjoint Method for PDE-Constrained Optimization Adjoint Method with Periodicity Constraint Sensitivity Method for Time-Periodic Solutions Fully Discrete Adjoint Equations: Dissection Linear evolution equations solved backward in time Primal state/stage, u ( n ) required at each state/stage of dual problem i Heavily dependent on chosen ouput T @ F λ ( N t ) = @ u ( N t ) s T @ F @ r ⌘ T λ ( n � 1) = λ ( n ) + ⇣ u ( n ) κ ( n ) X + , µ , t n � 1 + c i ∆ t n ∆ t n i i @ u ( n � 1) @ u i =1 s T @ F @ r ⌘ T + b i λ ( n ) + M T κ ( n ) ⇣ X u ( n ) κ ( n ) = a ji ∆ t n , µ , t n � 1 + c j ∆ t n i j j @ u ( N t ) @ u j = i Gradient reconstruction via dual variables N t s T @ r d F d µ = @ F @ µ + λ (0) T @ u 0 κ ( n ) @ µ ( u ( n ) , µ , t ( n ) X X @ µ + ) ∆ t n i i i n =1 i =1 Zahr, Persson, Wilkening High-Order, Time-Dependent Aerodynamic Optimization