High-Order Methods for Optimization and Control of...

IntroductionHigh-Order Numerical Scheme

Fully Discrete Perturbation MethodsConclusion

High-Order Methods for Optimization and Control ofConservation Laws on Deforming Domains

Matthew J. ZahrStanford University

Collaborators: Per-Olof Persson (UCB), Jon Wilkening (UCB)

FRG Seminar, Stanford UniversityTuesday, December 8, 2015

Zahr, Persson, Wilkening Optimization/Control Conservation Laws



Which motion ...

Has time-averaged x-force identically equal to 0?

Requires least energy to perform?

Energy = 9.4096x-force = -0.1766

Energy = 0.45695x-force = 0.000

Energy = 4.9475x-force = -2.500




Real-World Application: Micro Aerial Vehicles (MAV)

Autonomous flying vehicle with wingspanbetween 7.4cm and 15cm and speedbetween 15m/sMilitary applications

local reconnaissance and detection ofintrudersresemble small bird from distancetoo slow to be detected by radar

Commercial and civilian applications

Package delivery, crowd control, survivorsearch, pipeline inspection, high-riskindoor inspection

Difficulties

Thrust and lift requirementsStructural constraintsStability and control considerations

Micro Aerial Vehicle

Bumblebee MAV (USAF 2008)




Time-Dependent PDE-Constrained Optimization

Optimization of systems that are inherentlydynamic or without a steady-state solution

Introduction of fully discrete adjointmethod emanating from high-orderdiscretization of governing equations

Coupled with numerical optimization

Time-periodicity constraints

Vertical Axis Wind Turbines

Experimental design by G. Dahlbacka (LBNL) and collaborators

3kW unit, assembled unit (left), numerical simulation (right)

Micro Aerial Vehicle Vertical Windmill

Volkswagen Passat

LES Flow past Airfoil




Problem Formulation

Goal: Find the solution of the unsteady PDE-constrained optimization problem

minimizeU ,

J (U ,)

subject to C(U ,) 0U

t+ F (U ,U) = 0 in v(, t)

where

U(x, t) PDE solution

design/control parameters

J (U ,) = TfT0

j(U ,, t) dS dt objective function

C(U ,) =

TfT0

c(U ,, t) dS dt constraints




ALE Description of Conservation Law

Introduce map from fixed reference domain V to physical domain v(, t)

A point X V is mapped to x(, t) = G(X,, t) v(, t)

Introduce transformation

UX = gU

FX = gG1F UXG1vX

where

G = XG, g = detG, vX =Gt

X

g

t= X

(gG1vG

) X1X2

NdA

V

x1

x2

nda

vG, g, vX

Transformed conservation law1

UXt

X

+X FX(UX , XUX) = 0

1Geometric Conservation Law (GCL) satisfied by introduction of gZahr, Persson, Wilkening Optimization/Control Conservation Laws



Spatial Discretization: Discontinuous Galerkin

Re-write conservation law asfirst-order system

UXt

X

+X FX(UX , QX) = 0

QX XUX = 0

Discretize using DG

Roes method for inviscid flux

Compact DG (CDG) forviscous flux

Semi-discrete equations

Mu

t= r(u,, t)

u(0) = u0()The CDG Method Summary

1

1

2

2

3

3

4

4

andand

CDG :LDG :BR2 :

1

2 3

4

Element-wise compact stencil

Less connectivities than LDG/BR2/IP

More accurate than LDG and BR2

102 101 1001012

1010

108

106

104

102

100

p=1

p=2

p=3

p=4

p=5

x

L 2 er

ror

CDGLDGBR2

The CDG Method Summary

1

1

2

2

3

3

4

4

andand

CDG :LDG :BR2 :

1

2 3

4

Element-wise compact stencil

Less connectivities than LDG/BR2/IP

More accurate than LDG and BR2

102 101 1001012

1010

108

106

104

102

100

p=1

p=2

p=3

p=4

p=5

x

L 2 er

ror

CDGLDGBR2

Stencil for CDG, LDG, and BR2 fluxes




Temporal Discretization: Diagonally Implicit Runge-Kutta

Diagonally Implicit RK (DIRK) are implicit Runge-Kutta schemes definedby lower triangular Butcher tableau decoupled implicit stagesOvercomes 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) = u0()

u(n) = u(n1) +

si=1

bik(n)i

u(n)i = u

(n1) +

ij=1

aijk(n)j

Mk(n)i = tnr(u

(n)i , , tn1 + citn

)

c1 a11c2 a21 a22...

......

. . .

cs as1 as2 assb1 b2 bs

Butcher Tableau for DIRK scheme




Globally High-Order Discretization

Fully Discrete Conservation Law

u(0) = u0()

u(n) = u(n1) +

si=1

bik(n)i

u(n)i = u

(n1) +

ij=1

aijk(n)j

Mk(n)i = tnr(u

(n)i , , tn1 + citn

)Fully Discrete Output Functional

F (u(0), . . . ,u(Nt),k(1)1 , . . . ,k

(Nt)s ,)




Adjoint Method for PDE-Constrained OptimizationSensitivity Method for Time-Periodic SolutionsAdjoint Method with Periodicity Constraint

High-Order Discretization of PDE-Constrained Optimization

Continuous PDE-constrained optimization problem

minimizeU ,

J (U ,)

subject to C(U ,) 0U

t+ F (U ,U) = 0 in v(, t)

Fully discrete PDE-constrained optimization problem

minimizeu(0), ..., u(Nt)RNu ,k

(1)1 , ..., k

(Nt)s R

Nu ,Rn

J(u(0), . . . , u(Nt), k(1)1 , . . . , k

(Nt)s , )

subject to C(u(0), . . . , u(Nt), k(1)1 , . . . , k

(Nt)s , ) 0

u(0) u0() = 0

u(n) u(n1) +s

i=1

bik(n)i = 0

Mk(n)i tnr(u

(n)i , , t

(n1)i

)= 0





Generalized Reduced-Gradient Approach

Optimizer drives, PDE returns Quantity of Interest (QoI) values/gradients

OPTIMIZER

MESH MOTION PDE







OPTIMIZER

MESH MOTION PDE

x, x

x ,

x







OPTIMIZER

MESH MOTION PDE

x, x

x ,

x

J, dJd C,dCd





Generalized Reduced-Gradient Approach - Detailed

Optimizer drives, Primal returns QoI values, Dual returns QoI gradients

OPTIMIZER MESH MOTION

PRIMAL PDE

DUAL PDE








PRIMAL PDE

DUAL PDE

x, x








PRIMAL PDE

DUAL PDE

x, x

x, x

x ,

x

u(n), k(n)i








PRIMAL PDE

DUAL PDE

x, x

x, x

x ,

x

u(n), k(n)i

J,C

dJd ,

dCd





Adjoint Method to Compute QoI Gradients

Consider the fully discrete output functional F (u(n),k(n)i ,)

Represents either the objective function or a constraint

The total derivative with respect to the parameters , required in thecontext of gradient-based optimization, takes the form

dF

d=F

+

Ntn=0

F

u(n)u(n)

+

Ntn=1

si=1

F

k(n)i

k(n)i

The sensitivities,u(n)

and

k(n)i

, are expensive to compute, requiring the

solution of n linear evolution equations

Adjoint method: alternative method for computingdF

drequiring one

linear evolution evoluation equation for each quantity of interest, F





Adjoint Equation Derivation - Outline

Define auxiliary PDE-constrained optimization problem


(1)1 , ..., k

(Nt)s R

Nu

F (u(0), . . . , u(Nt), k(1)1 , . . . , k

(Nt)s , )

subject to r(0) = u(0) u0() = 0

r(n) = u(n) u(n1) +s

i=1

bik(n)i = 0

R(n)i = Mk

(n)i tnr

(u

(n)i , , t

(n1)i

)= 0

Define Lagrangian

L(u(n), k(n)i , (n),

(n)i ) = F

(0)T r(0)Ntn=1

(n)Tr(n)

Ntn=1

si=1

(n)i

TR

(n)i





Fully Discrete Adjoint Equations

The solution of the optimization problem is given by theKarush-Kuhn-Tucker (KKT) sytem

Lu(n)

= 0,Lk

(n)i

= 0,L(n)

= 0,L

(n)i

= 0

The derivatives w.r.t. the state variables,Lu(n)

= 0 andLk

(n)i

= 0, yield

the fully discrete adjoint equations

(Nt) =F

u(Nt)

T

(n1) = (n) +F

u(n1)

T

+

si=1

tnr

u

(u

(n)i , , tn1 + citn

)T

(n)i

MT(n)i =F

u(Nt)

T

+ bi(n) +

sj=i

ajitnr

u

(u

(n)j , , tn1 + cjtn

)T

(n)j





Fully Discrete Adjoint Equations: Dissection

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

(Nt) =F

u(Nt)

T

(n1) = (n) +F

u(n1)

T

+

si=1

tnr

u

(u

(n)i , , tn1 + citn

)T

(n)i

MT(n)i =F

u(Nt)

T

+ bi(n) +

sj=i

ajitnr

u

(u

(n)j , , tn1 + cjtn

)T

(n)j





Gradient on Manifold of PDE Solutions via Dual Variables

Equipped with the solution to the primal problem, u(n) and k(n)i , and dual

problem, (n) and (n)i , the output gradient is reconstructed as

dF

d=F

(0)

T u0

Ntn=1

tn

si=1

(n)i

T r

(u

(n)i , , t

(n)i )

Independent of sensitivities,u(n)

and

k(n)i

Dependent on initial condition sensitivity,u0

Compute (0)T u0

directly if u0 is solution of steady-state equation

R(u0,) = 0

(0)T u0

=

[R

u

T(0)

]TR





Energetically Optimal Flapping under x-Force Constraint

minimize

3T

2T

f x dS dt

subject to

3T2T

f e1 dS dt = q

U(x, 0) = U(x)

U

t+ F (U ,U) = 0

Isentropic, compressible,Navier-Stokes

Re = 1000, M = 0.2

y(t), (t), c(t) parametrized viaperiodic cubic splines

Black-box optimizer: SNOPT

y(t)

(t)

ll/3

c(t)

Airfoil schematic, kinematic description





Adjoint Method Gradients Agree with Finite Differences

106

104

102

||dW d

W

|| 2

||

W|| 2

101103105107109108

105

102

||dJx

d

Jx

|| 2

||

Jx

|| 2

Comparison of adjoint gradients with those obtained with 2nd order finitedifference approximation with step





Optimal Control - Fixed Shape - Varied x-Force

Energy = 9.4096x-force = -0.1766

Energy = 0.45695x-force = 0.000

Energy = 4.9475x-force = -2.500





Optimal Time-Morphed Geometry and Control - Varied x-Force

Energy = 9.4096x-force = -0.1766

Energy = 0.45027x-force = 0.000

Energy = 4.6182x-force = -2.500





Optimal Time-Morphed Geometry and Control - x-Force = 2.5

Energy = 9.4096x-force = -0.1766

Energy = 4.9476x-force = -2.500

Energy = 4.6182x-force = -2.500





Trajectories of y(t), (t), and c(t)

10 12 14

1

0

1

t

y(t

)

10 12 1410.5

0

0.5

1

t

(t)

10 12 140.40.2

0

0.2

0.4

t

c(t)

Initial guess ( ), optimal control/fixed shape (q = 0.0: , q = 1.0: , q = 2.5:), and optimal control and time-morphed geometry (q = 0.0: , q = 1.0: ,

q = 2.5: ).





Instantaneous Power (Ph) and x-Force (Fhx ) Exerted on Airfoil

10 12 14

4

2

0

time

Ph

10 12 14

1

0.5

0

time

Fh x


q = 2.5: ).





Convergence of Total Work (W ) and x-Impulse (Fx) Exerted on Airfoil

SNOPT convergence history

0 20 40 6010

5

0

iteration

W

0 20 40 60

21

0

1

iteration

Jx


q = 2.5: ).





Time-Periodic Solutions Desired when Optimizing Cyclic Motion

To properly optimize a cyclic, or periodic problem, need to simulate arepresentative period

Necessary to avoid transients that will impact quantity of interest and maycause simulation to crash

Task: Find initial condition, u0, such that flow is periodic, i.e. u(Nt) = u0





Definition of Time-Periodic Solution of Fully Discrete PDE

Recall fully discrete conservation law

u(0) = u0()

u(n) = u(n1) +

si=1

bik(n)i

u(n)i = u

(n1) +

ij=1

aijk(n)j

Mk(n)i = tnr(u

(n)i , , tn1 + citn

)Discrete time-periodicity is defined as

u(Nt)(u0) = u0





Newton-Krylov Shooting Method for Time-Periodic Solutions

Apply Newtons method to solve nonlinear system of equations

R(u0) = u(Nt)(u0) u0 = 0

Nonlinear iteration defined as

u0 u0 J(u0)1R(u0)

where J(u0) =u(Nt)

u0 I

u(Nt)

u0is a large, dense matrix and expensive to construct

Krylov method to solve J(u0)1R(u0) only requires matrix-vector products

J(u0)v =u(Nt)

u0v v





Fully Discrete Sensitivity Method to Compute u(Nt)

u0v

Direct differentiation of fully discrete conservation law, and multiplicationby v, leads to the fully discrete sensitivity equations

u(0)

u0v = v

u(n)

u0v =

u(n1)

u0v +

si=1

bik

(n)i

u0v

Mk

(n)i

u0v = tn

r

u

(u

(n)i , , t

(n1)i

)u(n1)u0

v +

ij=1

aijk

(n)j

u0v

Sensitivity variables:

u(n)

u0v, and

k(n)i

u0v





Fully Discrete Sensitivity Equations: Dissection

Linear evolution equations solved forward in time

Primal state/stage, u(n)i required at each state/stage of sensitivity problem

Heavily dependent on chosen vector

u(0)

u0v = v

u(n)

u0v =

u(n1)

u0v +

si=1

bik

(n)i

u0v

Mk

(n)i

u0v = tn

r

u

(u

(n)i , , t

(n1)i

)u(n1)u0

v +

ij=1

aijk

(n)j

u0v





Time-Periodic Flow: Flapping Foil

Initial GuessSolution of

Newton-Krylov





Nonlinear Solver Convergence

100 101 102

109

107

105

103

101

101

103

iterations (primal solves)

||u(N

t)u

0|| 2

Fixed Point Iteration

Newton-GMRES: = 102

Newton-GMRES: = 103

Newton-GMRES: = 104





Linear Solver Convergence

0 10 20 30 40 501010

108

106

104

102

100

102

iterations (linearized solves)

||Jxr|| 2





Time-Periodicity Constraints in PDE-Constrained Optimization

Recall fully discrete PDE-constrained optimization problem


(1)1 , ..., k

(Nt)s R

Nu ,Rn

J(u(0), . . . , u(Nt), k(1)1 , . . . , k

(Nt)s , )


(Nt)s , ) 0

u(0) u0() = 0

u(n) u(n1) +s

i=1

bik(n)i = 0

Mk(n)i tnr(u

(n)i , , t

(n1)i

)= 0





Time-Periodicity Constraints in PDE-Constrained Optimization

Slight modification leads to fully discrete periodic PDE-constrained optimizationproblem


(1)1 , ..., k

(Nt)s R

Nu ,Rn

J(u(0), . . . , u(Nt), k(1)1 , . . . , k

(Nt)s , )


(Nt)s , ) 0

u(0) u(Nt) = 0

u(n) u(n1) +s

i=1

bik(n)i = 0

Mk(n)i tnr(u

(n)i , , t

(n1)i

)= 0





Adjoint Method for Periodic PDE-Constrained Optimization

Following identical procedure as for non-periodic case, the adjoint equationscorresponding to the periodic conservation law are

(Nt) = (0) +F

u(Nt)

T

(n1) = (n) +F

u(n1)

T

+

si=1

tnr

u

(u

(n)i , , tn1 + citn

)T

(n)i

MT(n)i =F

u(Nt)

T

+ bi(n) +

sj=i

ajitnr

u

(u

(n)j , , tn1 + cjtn

)T

(n)j

Dual problem is also periodic

Solve linear, periodic problem using Krylov shooting method







PRIMAL PERIODIC PDE

DUAL PERIODIC PDE

x, x

x, x

x ,

x

u(n), k(n)i

J,C

dJd ,

dCd





Energetically Optimal Flapping under x-Force, Time-Periodicity Constraint

minimize

T

0

f x dS dt

subject to

T0

f e1 dS dt = q

U(x, 0) = U(x, T )

U

t+ F (U ,U) = 0

Isentropic, compressible,Navier-Stokes

Re = 1000, M = 0.2

y(t), (t), c(t) parametrized viaperiodic cubic splines

Black-box optimizer: SNOPT

y(t)

(t)

ll/3

Airfoil schematic, kinematic description





Adjoint Method Gradients Agree with Finite Differences

108 106 104 102

106

105

104

103

||dW d

W

|| 2/||

W|| 2

108 106 104 102

106

105

104

||dJx

d

Jx

|| 2/||

Jx

|| 2

Comparison of adjoint gradients with those obtained with 2nd order finitedifference approximation with step





Solution of Time-Periodic, Energetically Optimal Flapping




Conclusion

Derived adjoint equations for DG-DIRK discretization of generalconservation laws on deforming domainIntroduced fully discrete adjoint method for computing gradients ofquantities of interest

Framework demonstrated on the computation of energetically optimalmotions of a 2D airfoil in a flow field with constraints

Introduced fully discrete sensitivity equations and used Newton-Krylovshooting method to compute time-periodic flows

Framework and solver introduced for incorporating time-periodicityconstraints in optimization problem

Next steps: 3D, multiphysics, model reduction


Domain Deformation

Require mapping x = G(X,, t) to obtain derivatives XG, tGShape deformation, via Radial Basis Functions (RBFs), applied to referencedomain

X = X +

wi(||X ci||)

Undeformed Mesh Shape Deformation


Domain Deformation

Rigid body translation, v, and rotation, Q, applied to deformedconfiguration

X = v +QX

Shape Deformation Shape Deformation, Rigid Motion


Domain Deformation

Spatial blending between deformation with and without rigid body motionto avoid large velocities at far-field

x = b(X)X + (1 b(X))X

b : Rnsd R is a function that smoothly transitions from 0 inside a circle ofradius R1 to 1 outside circle of radius R2

Blended Mesh


Domain Deformation

Blended Mesh


Consistent Discretization of Output Quantities

Consider any quantity of interest of the form

F(U ,) = TfT0

f(U ,, t) dS dt

Define fh as the high-order approximation of the spatial integral via the DGshape functions

fh(u(t),, t) =TeT

QiQTe

wif(uei(t),, t)

f(U ,, t) dS

Then, the quantity of interest becomes

F(U ,) Fh(u,) = TfT0

fh(u(t),, t) dt


Consistent Discretization of Output Quantities

Semi-discretized outputfunctional

Fh(u,, t) = tT0

fh(u(t),, t) dt

Differentiation w.r.t. time leadsto the

Fh(u,, t) = fh(u(t),, t)

Write semi-discretized outputfunctional and conservation lawas monolithic system[

M 00 1

] [u

Fh

]=

[r(u,, t)fh(u,, t)

]

Apply DIRK scheme to obtain

u(n) = u(n1) +

si=1

bik(n)i

F (n)h = F(n1)h +

si=1

bifh

(u

(n)i , , t

(n1)i

)u

(n)i = u

(n1) +

ij=1

aijk(n)j

Mk(n)i = tnr(u

(n)i , , t

(n1)i

)where t

(n1)i = tn1 + citn

Only interested in final time

F (u(n),k(n)i ,) = F

(Nt)h


Isentropic, Compressible Navier-Stokes Equations

Applications in this work focused on compressible Navier-Stokes equations

t+

xi(ui) = 0

t(ui) +

xi(uiuj + p) = +

ijxj

for i = 1, 2, 3

t(E) +

xi(uj(E + p)) =

qjxj

+

xj(ujij)

Isentropic assumption (entropy constant) made to reduce dimension of PDEsystem from nsd + 2 to nsd + 1


Stability of Periodic Orbits of Fully Discrete PDE

Let u0() be a fully discrete time-periodic solution of the PDE

Define the operator

u(nNt)(u0; ) = u(Nt)(; ) u(Nt)(u0; )

A Taylor expansion of u(Nt) about the periodic solution leads to

u(Nt)(u0(); ) = u0() +

u(Nt)

u0(u0(); ) u+O(||u||2)

where time-periodicity of u0() was used

Repeated application of leads to

u(nNt)(u0()+u; ) = u0()+

[u(Nt)

u0(u0(); )

]nu+O(||u||n+1)

Periodic orbit is stable if eigenvalues ofu(Nt)

u0(u0(); ) have magnitude

less than unity


IntroductionHigh-Order Numerical SchemeFully Discrete Perturbation MethodsAdjoint Method for PDE-Constrained OptimizationSensitivity Method for Time-Periodic SolutionsAdjoint Method with Periodicity Constraint

ConclusionAppendix

fd@vidPitchMesh0: fd@vidPitchMesh0: fd@init_flap01-thrust0_flap01-thrust2p5: fd@init_flap01morph01-thrust0_flap01morph01-thrust2p5: fd@init_flap01-thrust2p5_flap01morph01-thrust2p5: fd@rm@5: fd@rm@6: fd@periodic_opt:

High-Order Methods for Optimization and Control of...

Documents

Transcript of High-Order Methods for Optimization and Control of...