Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Unsteady CFD Optimization using High-Order Discontinuous Galerkin Finite Element Methods Matthew J. Zahr and Per-Olof Persson Stanford University and University of California, Berkeley 13th US National Congress on Computational Mechanics, San Diego, CA TS4, Tuesday, July 28, 2015 Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Optimal Control: Flapping Flight Optimal control of a body immersed in a fluid leads to unsteady PDE-constrained optimization Goal: Determine kinematics of the body that minimizes some cost functional subject to constraints Dragonfly Experiment Steady-state analysis insufficient (A. Song, Brown U) Example: Energetically-optimal flapping at constant thrust Biology Micro Aerial Vehicles Micro Aerial Vehicle Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Shape Optimization: Turbulence Shape optimization of a static or moving body in turbulent flow also leads to unsteady PDE-constrained optimization Non-existence of steady-state necessitates unsteady analysis Goal: Determine shape (and possibly kinematics) that minimizes a cost Vertical Windmill functional, subject to constraints Applications Shape of windmill blade for maximum energy harvesting Maximum lift airfoil LES Flow past Airfoil Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Problem 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 design/control parameters µ � 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 Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Approach to Unsteady Optimization Recast conservation law on deforming domain into one on fixed , reference domain (Arbitrary Lagrangian-Euler formulation) Globally high-order numerical discretization of transformed equations Spatial Discretization: Discontinuous Galerkin FEM Temporal Discretization: Diagonally-Implicit Runge-Kutta Solver-consistent discretization of output quantities Fully-discrete adjoint method for high-order numerical discretization Gradient-based optimization Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion ALE Description of Conservation Law Map from fixed reference domain V to physical, deformable (parametrized) domain v ( µ , t ) A point X ∈ V is mapped to x ( µ , t ) = G ( X , µ , t ) ∈ v ( µ , t ) n da Introduce transformation U X = g U N dA G , g , v X v F X = g G − 1 F − U X G − 1 v X x 2 V where x 1 X 2 � G = ∇ X G , g = det G , v X = ∂ G � � ∂t X 1 � X Transformed conservation law � ∂ U X � + ∇ X · F X ( U X , ∇ X U X ) = 0 � ∂t � X Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Domain Deformation ∂ Require mapping x = G ( X , t ) to obtain derivatives ∇ X G , ∂t G Shape deformation, via Radial Basis Functions (RBFs), and translational kinematic motion, v , applied to reference domain X ′ = X + v + � w i Φ( || X − c i || ) Shape Deformation, Translation Undeformed Mesh Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Domain Deformation ∂ Require mapping x = G ( X , t ) to obtain derivatives ∇ X G , ∂t G Rotational kinematic motion, Q , applied via blending map x = b ( d R ( X )) X ′ + (1 − b ( d R ( X ))) QX b : R → R is a polynomial on [0 , 1] with ( n − 1) / 2 vanishing derivatives at 0 , 1 d R ( X ) is signed distance between X and circle of radius R Shape Deformation, Translation Rigid Rotation of Mesh Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Domain Deformation ∂ Require mapping x = G ( X , t ) to obtain derivatives ∇ X G , ∂t G Rotational kinematic motion, Q , applied via blending map x = b ( d R ( X )) X ′ + (1 − b ( d R ( X ))) QX b : R → R is a polynomial on [0 , 1] with ( n − 1) / 2 vanishing derivatives at 0 , 1 d R ( X ) is signed distance between X and circle of radius R Blended Mesh Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Domain Deformation Blended Mesh Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Spatial Discretization: Discontinuous Galerkin Re-write conservation law as Roe’s method for inviscid flux first-order system Compact DG (CDG) for viscous flux � ∂ U X � + ∇ X · F X ( U X , Q X ) = 0 Semi-discrete equations � ∂t � X Q X − ∇ X U X = 0 M ∂ u ∂t = r ( u , µ , t ) Discretize using DG u (0) = u 0 ( µ ) 1 2 3 4 1 2 4 3 3 2 4 1 CDG : LDG : and BR2 : and Stencil for CDG, LDG, and BR2 fluxes Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Temporal Discretization: Diagonally-Implicit Runge-Kutta Diagonally-Implicit RK (DIRK) are implicit Runge-Kutta schemes defined by lower triangular Butcher tableau → decoupled implicit stages Overcomes issues with high-order BDF and IRK Limited accuracy of A-stable BDF schemes (2nd order) High cost of general implicit RK schemes (coupled stages) u (0) = u 0 ( µ ) c 1 a 11 s c 2 a 21 a 22 u ( n ) = u ( n − 1) + b i k ( n ) � . . . i ... . . . . . . i =1 i c s a s 1 a s 2 · · · a ss = u ( n − 1) + u ( n ) a ij k ( n ) � · · · b 1 b 2 b s i j j =1 Butcher Tableau for DIRK scheme � � M k ( n ) u ( n ) = ∆ t n r , µ , t n − 1 + c i ∆ t n i i Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Consistent Discretization of Output Quantities Consider any output functional of the form � T f � F ( U , µ ) = f ( U , µ , t ) dS dt T 0 Γ Define f h as the high-order approximation of the spatial integral via the DG shape functions � � � f h ( u ( t ) , µ , t ) = w i f ( u ei ( t ) , µ , t ) ≈ f ( U , µ , t ) dS Γ T e ∈T Γ Q i ∈Q T e Then, the output functional becomes � T f F ( U , µ ) ≈ F h ( u , µ ) = f h ( u ( t ) , µ , t ) dt T 0 Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Consistent Discretization of Output Quantities Semi-discretized output Apply DIRK scheme to obtain functional s u ( n ) = u ( n − 1) + � b i k ( n ) � t i F h ( u , µ , t ) = f h ( u ( t ) , µ , t ) dt i =1 T 0 s � � F ( n ) = F ( n − 1) u ( n ) , µ , t ( n − 1) � + b i f h Differentiation w.r.t. time leads i i h h i =1 to the i = u ( n − 1) + u ( n ) � a ij k ( n ) ˙ F h ( u , µ , t ) = f h ( u ( t ) , µ , t ) i j j =1 Write semi-discretized output � � M k ( n ) u ( n ) , µ , t ( n − 1) = ∆ t n r i i i functional and conservation law as monolithic system where t ( n − 1) = t n − 1 + c i ∆ t n � � ˙ i � � � � 0 u r ( u , µ , t ) M Only interested in final time = ˙ 0 1 F h f h ( u , µ , t ) F ( u ( n ) , k ( n ) , µ ) = F ( N t ) i h Zahr and Persson Unsteady CFD Optimization
Introduction High-Order Numerical Scheme Fully-Discrete Adjoint Method Applications Conclusion Adjoint Method Consider the fully-discrete output functional F ( u ( n ) , k ( n ) , µ ) corresponding i � T f � to the continuous output functional F ( U , µ ) = f ( U , µ , t ) dS dt T 0 Γ May correspond to 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 µ = ∂F d F ∂F ∂F � � � 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 requiring one linear evolution evoluation equation for each output functional, F Zahr and Persson Unsteady CFD Optimization
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