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Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Numerical Optimizationof Partial Differential Equations
Part III: applications
Bartosz Protas
Department of Mathematics & StatisticsMcMaster University, Hamilton, Ontario, CanadaURL: http://www.math.mcmaster.ca/bprotas
Rencontres Normandes sur les aspects theoriques etnumeriques des EDP
5–9 November 2018, Rouen
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Optimal Open–Loop ControlPDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Inverse Problem of Vortex ReconstructionEuler System & Inverse FormulationSolution ApproachResults
Geometry Optimization in Heat TransferMotivation & Mathematical ModelOptimization ProblemResults
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
PART I
Optimal Open–Loop Control via
Adjoint–Based Optimization
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Motivation — Applications of Flow Control
I Wake Hazard
I Fluid–Structure Interaction
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Statement of the Problem (I)
I Flow Domain
infV
V(t)τ
(t)ϕ
Y
X
Ω
O
Γ
D
I Assumptions:I viscous, incompressible flowI plane, infinite domainI Re = 150
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Statement of the Problem (II)
I Find ϕopt = argminϕ J (ϕ) , where
J (ϕ) =1
2
∫ T
0
[power related to
the drag force
]+
[power needed tocontrol the flow
]dt
=1
2
∫ T
0
∮Γ0
[p(ϕ)n− µn ·D(v(ϕ))] · [ϕ (ez × r) + v∞] dσdt
I Subject to:[
∂v∂t + (v ·∇)v − µ∆v + ∇p
∇ · v
]=
[00
]in Ω× (0,T ),
v = 0 at t = 0,
v = ϕoptτ on Γ
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Abstract Framework (I)
I Constrained optimization problem min(x ,ϕ)J (x , ϕ)
S(x(ϕ), ϕ) = 0
I Equivalent unconstrained optimization problem (note thatx = x(ϕ) )
minϕJ (x(ϕ), ϕ) = min
ϕJ (ϕ)
I First–Order Optimality Conditions (U - Hilbert space ofcontrols)
∀ϕ′∈U J ′(ϕ;ϕ′) =(∇J , ϕ′
)U = 0,
with the Gateaux differentialJ ′(ϕ;ϕ′) = limε→0
1ε [J (ϕ+ εϕ′)− J (ϕ)].
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Abstract Framework (II)
I Minimization of J (ϕ) with a descent algorithm in U=⇒ solution to a steady state of the ODE in U
dϕ
dτ= −Q∇ϕJ (ϕ) on τ ∈ (0,∞) (pseudo–time),
ϕ = ϕ0 at τ = 0.
I Typically well–behaved (quadratic) cost functionalsI Typically ill–behaved constraints: the Navier–Stokes
systemI nonlinear, nonlocal, multiscale, evolutionary PDE,
I Dimensions:I state: 106 — 107 DoF × 102 — 103 time levelsI control: 104 — 105 DoF × 102 — 103 time levels
I No hope of using “matrix” formulation ...
I Formulation equivalent to Lagrange Multipliers
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Differential of the Cost Functional
I The cost functional:
J (ϕ) =1
2
∫ T
0
[power related to
the drag force
]+
[power needed tocontrol the flow
]dt
=1
2
∫ T
0
∮Γ0
[p(ϕ)n− µn ·D(v(ϕ))] · [ϕ (ez × r) + v∞] dσdt,
I Expression for the Gateaux differential:
J ′(ϕ; h) =1
2
∫ T
0
∮Γ0
[p′(h)n− µn ·D (v′(h))] · [ϕ (ez × r) + v∞] +
[p(ϕ)n− µn ·D(v(ϕ))] · (ez × r) hdσ dt = B1
= (∇J (t), h)L2([0,T ])
The fields v′(h), p′(h) solve the linearized perturbation system.
I How to calculate the Gradient ∇J ?
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Sensitivities and Adjoint States
I The linearized perturbation systemN[
v′
p′
]=
[∂v′
∂t+ (v ·∇)v′ + (v′ ·∇)v − µ∆v′ + ∇p′
−∇ · v′
]=
[00
]in Ω× (0,T ),
v′ = 0 at t = 0,
v′ = hτ on Γ× (0,T )
I Duality pairing defining the adjoint operator⟨N[
v′
p′
],
[v∗
p∗
]⟩L2(0,T ;L2(Ω))
=
⟨[v′
p′
],N ∗
[v∗
p∗
]⟩L2(0,T ;L2(Ω))
+ B1 + B2
I The adjoint system ( terminal value problem !! )N ∗
[v∗
p∗
]=
[− ∂v∗
∂t− v ·
[∇v∗ + (∇v∗)T
]− µ∆v∗ + ∇p∗
−∇ · v∗
]=
[00
]in Ω× (0,T ),
v∗ = 0 at t = T ,
v∗ = r × (ϕez ) + v∞ on Γ× (0,T )
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Cost Functional Gradient
I The adjoint state and duality pairing can now be usedto re–express the cost functional differential as:
J ′(ϕ; h) =1
2
∫ T
0
∮ΓµRn ·D(v∗) · τ + µn ·D(v(ϕ)) · (ez × r) h dσ dt
I Identification of the cost functional gradient
J ′(ϕ; h) = (∇J (t), h)L2([0,T ]) =
∫ T
0
∇J (t) h dt
∇J (t) =1
2
∮Γ
µRn ·D(v∗) · τ + µn ·D(v(ϕ)) · (ez × r) dσ
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Optimality (KKT) system
I Complete optimality system for ϕopt , [vopt , popt ], and [v∗, p∗]
1
2
∮ΓµRn ·D(v∗) · τ + µn ·D(v(ϕopt )) · (ez × r) dσ = 0[
∂v∂t
+ (v ·∇)v − µ∆v + ∇p∇ · v
]=
[00
]in Ω× (0,T ),
v = 0 at t = 0,
v = ϕoptτ on ΓN ∗
[v∗
p∗
]=
[− ∂v∗
∂t− v ·
[∇v∗ + (∇v∗)T
]− µ∆v∗ + ∇p∗
−∇ · v∗
]=
[00
]in Ω× (0,T ),
v∗ = 0 at t = T ,
v∗ = r × (ϕopt ez ) + v∞ on Γ
I A counterpart of the Euler–Lagrange equation
I Solved with an iterative Gradient Algorithm(e.g., Conjugate Gradients, quasi–Newton, etc.)
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
An Iterative Optimization Procedure
0. provide initial guess ϕ0
1. Solve for
v(ϕi ); p(ϕi )
on [0,T ]
2. Solve for
v∗(ϕi ); p∗(ϕi )
on [0,T ]
3. Use
v(ϕi ); p(ϕi )
and
v∗(ϕi ); p∗(ϕi )
to compute ∇J i (t) on [0,T ]
4. update control according to ϕi+1(t) = ϕi (t)− αiγi (∇J (t))
5. iterate 1. through 4. until convergence, i.e. until ∇J i (t) ' 0
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Primal and Adjoint Simulationsfor Cylinder Rotation as Control
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults
Results
I No Control
I Flow Pattern Modifications due to Control (T = 6)
I Optimal Control ϕopt , drag coefficient cD , transverse velocity v
50 60 70 80 90 100 110 120
t
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Ropt/|V|
50 60 70 80 90 100 110 120
t
0.85
0.9
0.95
1.0
1.05
1.1
1.15
1.2
1.25
1.3
cD
T=6.0
Basic flow
No control
50 60 70 80 90 100 110 120
t
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
vNo control
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
PART II
Inverse Problem of Vortex
Reconstruction
joint work Ionut Danaila (Universite de Rouen)
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I Ubiquitous Vortex Rings
Lim & Nickels, 1995
t = 20t = 2
z
r
Danaila & Heiles, 2008
I Models of Vortex Rings:I based on linearized equations (Kaplanski & Rudi, 1999, 2005)
I obtained with perturbation techniques (Fukumoto, 2010)
I inviscid models: Hill’s and Norbury-Fraenkel’s vortices
I Present Approach:Optimal Vortex Rings via Inverse Formulation
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I Inviscid vortex ring in a moving frame of reference
zΩb
r(a) (b) r
Ωc
zψ=κ
ψ=
0Ωb
ω
r=
f (ψ) in Ωb,0 elsewhere,
f (ψ) — Vorticity Function(unspecified)
I 3D Axisymmetric Euler System
Lψ = −r f (ψ) in Ω,
ψ = 0 on γ.
where L := ∂∂z
(1r∂∂z
)+ ∂
∂r
(1r∂∂r
)= ∇ ·
(1r ∇)
and ∇ :=[∂∂z ,
∂∂r
]T.
I Special solutions:I f (ψ) = C in Ωb =⇒ Hill’s vortexI f (ψ) = C ∀ψ > k and f (ψ) = 0 ∀ψ ≤ k =⇒ Norbury-Fraenkel’s vortex
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I Key Idea: determine vorticity function f (ψ) to match someobservation data =⇒ Inverse Problem
I Measurements of the tangential velocity component
+ n
r
γ b :
: ψ
= 0
Ωγ
z :: ψ = 0
z
B
C
O A
ψ = k
m := v · n⊥ =1
r
∂ψ
∂non boundary segments γz and γb
I Cost Functional
J (f ) :=αb
2
∫γb
(1
r
∂ψ
∂n
∣∣∣∣γb
−m
)2
dσ +αz
2
∫γz
(1
r
∂ψ
∂n
∣∣∣∣γz
−m
)2
dσ,
I Variational Minimization Problem: f := argminf ∈H1(I) J (f )
I nonnegativity constraint f (ψ) ≥ 0 ∀ψB. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I Inverse problem with unusual structure — reconstruction of a nonlinearsource term f (ψ)
I Assumptions
1. domain: f : I → R, I := [0, ψmax] — identifiability interval
2. smoothness: f ∈ H1(I) (square-integrable derivatives)
I Optimality condition: ∀f ′∈H1(I) J ′(f ; f ′) = 0
I Gradient iterations f = limk→∞ f (k)
f (k+1) = f (k) − τk∇J (f (k)), k = 1, 2, . . .
f (1) = f0,
f0 — initial guess, τk — step seize at k-th iteration
I Positivity enforcement via transformationf+ = (1/2)g2, Jg (g) := J ((1/2)g2)
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I Gradient Expression — sensitivity of cost functional J (f ) with respectto perturbations of the vorticity function f (ψ)
∇L2J (s) = −∫γs
ψ∗ r
(∂ψ
∂n
)−1
dσ, s ∈ [0, ψmax].
γs := x ∈ Ω : ψ(x) = s — streamfunction level sets
0.05
0.1
0.15
0.20.25
0.3
00
z
r
1 0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
γs
γz
γbΩ
γs
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
I ψ∗ — solution of adjoint system
∇ ·(
1
r∇ψ∗
)+ rfψ(ψ)ψ∗ = 0 in Ω,
ψ∗ = αb
(1
r
∂ψ
∂n
∣∣∣∣γb
−m
)on γb,
ψ∗ = αz
(1
r
∂ψ
∂n
∣∣∣∣γz
−m
)on γz ,
I Smoothness ensured via Sobolev gradients:
J ′(f ; f ′) =⟨∇L2J (f ), f ′
⟩L2(I)
=⟨∇H1
J (f ), f ′⟩
H1(I)
=⇒
(I − `2 d2
ds2
)∇H1
J = ∇L2J in I,
∇H1
J = 0 at s = 0,
d
ds∇H1
J = 0 at s = ψmax,
I Algorithm easily implemented in FreeFEM++
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
Reconstruction of Hill’s Vortex
1e-05
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6
J(f
(k) )
/ J(f
(0) )
k
1e-05
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6
|Γ(f
(k) )
/ Γ
Hill-1
|, |I(
f(k) )
/ I H
ill-1
|, |E
(f(k
) ) / E
Hill-1
|
k
CirculationImpulseEnergy
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25 0.3
f(ψ
)
ψ
fHill
ψmax
f0-f
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
Reconstruction of Vortex Rings from DNS Data(Re = 17, 000)
0
10
20
30
40
50
60
0 0.2 0.4 0.6 0.8 1
f(ψ
)
ψ
DNS
fDNS-f
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.05 0.1 0.15 0.2
f(ψ
)
ψ
DNS
fDNS-f
· · · DNS data - - - empirical fit fDNS —– optimal reconstruction f
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Euler System & Inverse FormulationSolution ApproachResults
Reconstruction of Vortex Rings from DNS Data(Re = 17, 000)
z
r
0.5 0 0.50
0.5
1
1.5
(a) DNS
z
r
0.5 0 0.50
0.5
1
1.5
(b) present reconstruction
z
r
0.5 0 0.50
0.5
1
1.5
(c) NorburyFraenkel model
z
r
0.5 0 0.50
0.5
1
1.5
(d) KaplanskiRudi model
Vorticity distribution in space
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
PART III
Geometry Optimization
in Heat Transfer
joint work Xiaohui Peng and Katya Niakhai(former Master’s students at McMaster)
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Problem: Efficient cooling of a battery system
I Goal: determine optimal shape of cooling channels for aprescribed heat distribution
I Few mathematically precise results in literature=⇒ need to develop new tools
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I 2D thermally isolated domain
I time–independent
I heat conduction only
I cooling channel — line heat sinkmodelled with Newton’s law of cooling
S = γ(u − u0)
u0 — temperature of the coolant fluidin the channel modelled by the coil C,
u0(s) = Ta +Tb − Ta
Ls, s ∈ [0, L],
I want to maintain prescribedtemperature u in the subdomain A(revised optimization objective)
Ω2
Ω1
δΩ
n
c
A
n
Ω1
Ω2
δΩ
A
nn
c
s
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Governing System
−k∆u1 = q in Ω1,
−k∆u2 = q in Ω2,
u1 = u2 on C∂u2
∂n− ∂u1
∂n= γ (u1 − u0) on C
∂u2
∂n= 0 on ∂Ω,
whereI Ω1 — the interior of the curve C,I Ω2 — the exterior of the curve C,I ui (x) is the temperature distribution u restricted in the domain
Ωi , for i = 1, 2,I k is the heat conductivity coefficient (a known material
property),I q is the distribution of heat sources (battery heating),I n are the unit outer normal on C and ∂ Ω
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Assuming:I a given distribution of heat sources q(x),I heat transfer described by governing equation,I a fixed length L =
∮C ds of the cooling channel C,
find the shape of the curve C which ensures that over thesubdomain A the actual temperature u(x , y) is as close aspossible to the prescribed temperature u
I Define
J (C) =
∫A
(u − u)2 dΩ
I Formal statement of optimization problem
maxCJ (C),
subject to: Governing System,∮Cds = L
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Optimal shape C characterized by the condition
J ′(C,Z) = 0 for all shape perturbations Z
I Gradient descent algorithm
x(n+1)C = x
(n)C − τn n∇J (C(n)), n = 1, 2, . . . ,
x(0)C = xC0 ,
where ∇J (C(n)) is the gradient of the cost functional
c
δΩ
Ω
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Problem of Shape Optimization (contour geometry),I Shape Calculus: parametrization of geometry
x(t,Z) = x + tZ for x ∈ ΓSL(0),
where Z : ΩSL → R2 is the perturbation “velocity” field.I Gateaux Shape Differential
J ′(ΓSL(0); Z) , limt→0
J (ΓSL(t,Z))− J (ΓSL(0))
t.
I Main Theorem [shape–differentiation of integrals w.r.t. theshape of the domain]: ∫
Ω(t,Z)
f dΩ +
∫∂Ω(t,Z)
g ds
′
=
∫Ω(0)
f ′ dΩ +
∫∂Ω(0)
g ′ ds+
+
∫∂Ω(0)
(f + κ g +
∂g
∂n
)Z · n ds,
I How to compute the gradient ∇J ?
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I L2 Gradient ∇L2J (C(n)) computed as follows
∇L2J (C(n)) =γ
k(u1 − u0)
(∂u∗1∂n− κ u∗1
)− γ
k
∂u2
∂nu∗1 − λκ on C(n)
where u∗1 and u∗2 are solutions of the following adjointsystem
k∆u∗1 = (u − u)χA1 in Ω1,
k∆u∗2 = (u − u)χA2 in Ω2,
u∗1 − u∗2 = 0 on C(n),
k
(∂u∗2∂n− ∂u∗1
∂n
)= −γ u∗1 on C(n),
∂u∗2∂n
= 0 on ∂Ω2
I Optimal step size τn computed via line–minimization (usingBrent’s method)
τn = argminτ>0J (C(n) − τ∇J (C(n))B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Incorporation of the Length Constraint∮Cds = L0
I Modified (augmented) cost functional:
Jα(C) := J (C) +α
2
(∮Cds − L0
)2
,
where α ∈ R is a parameter
I After shape–differentiating the constraint, modified gradient
∇L2Jα(C) = ∇L2J (C) + α
(∮C(m)
ds − L0
)κ
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Gradients obtained using Riesz Representation Theorem
J ′(C; ζn) =⟨∇XJ , ζ
⟩X (C)
X — selected Hilbert space
I What is the required regularity of the gradients ∇J ?I xC(s) must be (at least) continuous
I L2 gradients ∇L2J (C) [X = L2(C)] may be discontinuous ...
I Need Sobolev Gradients [X = H1(C)]⟨∇H1
J , ζ⟩
H1(C)=
∫ L
0
∇H1
J ζ + `2 ∂∇H1
J∂s
∂ζ
∂sds, ∀ζ∈H1(C)
=⇒
(1− `2 ∂2
∂s2
)∇H1
J = ∇L2J on (0, L),
Periodic boundary conditions (P1),
∂
∂s∇H1
J∣∣∣
s=0,L= 0 (P2).
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Reformulation of the Governing System:
u = up + uh in Ω,
where ∀x∈Ω\C uh(x) = − 12π
∮C ln |x− xC |µ(xC) dσ.
I The new dependent variables up(x), x ∈ Ω;µ(x), x ∈ C satisfy
−k ∆ up = q in Ω,
µ(x) +γ
2π k
∮C
ln |x− xC |µ(xC) dσ =γ
k(up + uh − u0) on C,
∂up
∂n= −∂uh
∂non ∂Ω.
I Analogously for the Adjoint System with u∗p(x), x ∈ Ω;µ∗(x), x ∈ C
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Two coupled subproblems:
I Poisson equation for up (resp., u∗p )
I Singular Boundary Integral Equation for µ (resp., µ∗)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
y
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I Optimal discretization for each subproblem:
I spectral Chebyshev method for up (resp., u∗p ) in Ω
∆NU = f + q,
I spectral boundary-integral method with an analytic treatment of thesingular kernel for µ (resp., µ∗) on C(
I +γ
kK1 +
γ
kK2
)m +
γ
kPU =
γ
ku01,
I spectral interpolation P to couple up and µ (resp., u∗p and µ∗)[−∆N Bγk P I + γ
k K1 + γk K2
] [Um
]=
1
k
[q
γ u0 1
].
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
CASE I: α = 0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
10
20
30
40
50
60
70
80
90
q(x , y)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
18.5
19
19.5
20
20.5
u0 10 20 30 40 50 60 70 80 90 100
100
101
102
J (C(m))
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
C(m)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
20
25
30
35
40
45
50
Initial Solution u
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
16
16.5
17
17.5
18
18.5
19
19.5
20
20.5
21
21.5
Final Solution u
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
CASE II: α = 0
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
38
40
42
44
46
48
50
52
54
56
58
60
q(x , y)
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
u0 5 10 15 20 25 30
10−1
100
101
102
103
104
J (C(m))
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
C(m)
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
25
30
35
40
45
50
Initial Solution u
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
13.8
14
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Final Solution u
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
CASE III: α = 0, 1, 10, 102, 103; L0 = 2.3
x
y
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
−80
−60
−40
−20
0
20
40
60
80
100
120
140
q(x , y)x
y
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
30
40
50
60
70
80
90
100
110
120
130
140
u 1 2 3 4 5 6 7 810
0
101
102
103
104
n
J
J (C(m))
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
x
y
C1 2 3 4 5 6 7 8
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
n
L(C
(n) )
Length of C(m)x
y
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
27
28
29
30
31
32
33
34
35
36
37
Final Solution u (α = 103)
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
Conclusions
I Formulation of PDE control and estimation problems as constrainedoptimization
I PDE–constrained gradients via Adjoint EquationsI Vorticity form of the adjoint equationsI Optimization of free boundary problems via shape–differential calculus
I Inverse Problem of Vortex ReconstructionI Nonintuitive insights revealed by reconstruction from DNS dataI Big Question: what are the fundamental accuracy limits for
representation of real flows in terms of inviscid models?
I Shape-optimization approach for a model of 2D steady heat transferI Shape calculusI Spectrally-accurate solution of the governing and adjoint PDE systems
B. Protas Numerical Optimization of PDEs
Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction
Geometry Optimization in Heat Transfer
Motivation & Mathematical ModelOptimization ProblemResults
I References Part I — Open–Loop Control:I B. Protas, T. R. Bewley and G. Hagen, “A comprehensive framework for the
regularization of adjoint analysis in multiscale PDE systems”, Journal ofComputational Physics 195(1), 49-89, 2004.
I B. Protas, “On the “Vorticity” Formulation of the Adjoint Equations and itsSolution Using Vortex Method”, Journal of Turbulence 3, 048, 2002.
I B. Protas and A. Styczek, “Optimal Rotary Control of the Cylinder Wake in theLaminar Regime”, Physics of Fluids 14(7), 2073–2087, 2002.
I B. Protas, “Vortex Design Problem”, Journal of Computational and AppliedMathematics 236, 1926–1946, 2012.
I References Part II — Inverse Problem of Vortex Reconstruction:I V. Bukshtynov, O. Volkov, and B. Protas, “On Optimal Reconstruction of
Constitutive Relations”, Physica D 240, 1228–1244, 2011.I V. Bukshtynov and B. Protas, “Optimal Reconstruction of Material Properties in
Complex Multiphysics Phenomena”, Journal of Computational Physics 242,889–914, 2013.
I I. Danaila and B. Protas, “Optimal Reconstruction of Inviscid Vortices”,Proceedings of the Royal Society A (in press), 2015.
I References Part III — Shape Optimization:I X. Peng, K. Niakhai and B. Protas, “A Method for Geometry Optimization in a
Simple Model of Two-Dimensional Heat Transfer”, SIAM Journal on ScientificComputing 35, B1105–B1131, 2013.
B. Protas Numerical Optimization of PDEs
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