IMPULSE RESPONSE APPROXIMATIONS OF DISCRETE ... - Uni Trier

SIAM J. CONTROL OPTIM. Vol. 48, No. 4, pp. 2562–2580

c 2009 Society for Industrial and Applied Mathematics

IMPULSE RESPONSE APPROXIMATIONS OF DISCRETE SHAPE HESSIANS WITH APPLICATION IN CFD∗ STEPHAN SCHMIDT† AND VOLKER SCHULZ† Abstract. This paper discusses the symbol of the Hessian of a shape optimization problem in a viscid, incompressible flow. The symbol of the Hessian for the Stokes shape optimization problem is analytically approximated by tracking a Fourier mode through the operators involved. We propose a discrete method for finding the symbol and confirm the correctness by comparison with the analytical data. A preconditioner is constructed for both the Stokes and Navier–Stokes equation, which greatly accelerates the optimization. Key words. shape Hessian, operator symbol, preconditioning, Fourier mode, shape optimization, approximative Newton method, Stokes, Navier–Stokes AMS subject classifications. 65K10, 93C80, 35S99, 49M25, 76D55, 49Q10 DOI. 10.1137/080719844

1. Introduction. The majority of shape optimization problems in CFD are currently conducted by means of parameterized gradients. The shape of an obstacle in a flow is discretized by a certain parameterization, mainly b-splines or Hicks–Henne functions in 2D and freeform deformation in 3D. The gradients are computed with respect to these design parameters. This has several distinct disadvantages: First, the choice of parameterization limits the search space; e.g., it is next to impossible to compute an optimal shape with a sharp nose if the b-splines used cannot generate such a shape. Next, the computation of these gradients is usually not independent of the number of design parameters, because certain so-called “mesh sensitivities” must be computed, which is usually done via finite differences or automatic differentiation techniques. Lastly, the parameterized gradients camouflage the shape Hessian of the problem. The remedy for these disadvantages is the use of shape derivatives, as they have been proposed for aerodynamic optimization in [1, 2, 3, 4, 8]. The goal of this paper is to both analytically and discretely analyze the shape Hessian of a drag minimization problem in a viscous and incompressible flow and to use the information gathered to construct a preconditioner which greatly accelerates the optimization procedure by turning a steepest descent algorithm into an approximative Newton scheme. This procedure is often called “gradient smoothing,” and the smoothed gradients are also often referred to as “Sobolev gradients.” Usually, the gradients are smoothed by using a scaled tangential Laplace or Laplace–Beltrami operator as a preconditioning PDE [10, 11, 12] or some other ODE [9]. Other approaches use multilevel information [5]. However, the proper choice of the scaling or “smoothing” parameter and the PDE to use should be done by taking the Hessian operator of the problem into account. As such, the main focus of this paper is the shape Hessian and its relation to the preconditioning PDE. Interestingly, the problems discussed here will provide an example where the tangential Laplacian—which is mostly used—will result in an ∗ Received by the editors March 31, 2008; accepted for publication (in revised form) May 21, 2009; published electronically September 4, 2009. http://www.siam.org/journals/sicon/48-4/71984.html † Department of Mathematics, University of Trier, 54286 Trier, Germany ([email protected], [email protected]).

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APPROXIMATION OF DISCRETE SHAPE HESSIANS

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oversmoothing. We employ the approach of Arian [1], Arian and Ta’asan [2], and Arian and Vatsa [3] to first find the symbol of the Hessian operator. For an example where even more information of the Hessian can be extracted, see [7]. Next, we mimic this approach discretely to come up with a procedure to analyze the discrete shape Hessian even if the discretization, i.e., the flow solver, is treated like a black box. This is conducted by observing the effects the discrete Hessian has on a sinousidal input wave. By comparing the amplitude and phase of the output wave with the input wave, one can determine the symbol of the shape Hessian. The structure of the paper is as follows: First, we will derive the objective functions and the respective shape derivatives of a drag minimization problem for both the incompressible Navier–Stokes and Stokes equations similar to [14, 15, 16]. Next, we conduct an analysis for the Hessian of the Stokes shape optimization problem via its symbol similar to [3] which shows that the Hessian acts almost like the Dirichletto-Neumann map, sharing the same symbol. Next, we propose a discrete numerical method to be employed when either an analytic approach becomes infeasible due to the complexity of the problem or when one wants to take the effects of the discretization into account. We see that the results of our discrete approximation scheme match the analytic results very well. We then construct a preconditioner for both the Stokes and the Navier–Stokes equations, and using this preconditioner we greatly accelerate the optimization. 2. Modeling. Statement of the optimization problem. Before we come to the derivation of the objective function and the gradients, we would like to point out the following lemma. Lemma 2.1. Let u solve the incompressible Navier–Stokes equations u˙ − νΔ + ρu∇u + ∇p = 0

in Ω,

div u = 0, u = 0 on Γ. It then follows that

u(u∇u) dA = 0. Ω

Proof.

u(u∇u) dA = Ω

1 2

Ω

1 =− 2 = 0.

u∇ u2 dA −

div uu2 dA +

Ω

Ω

1 2

u (u × rot u) dA Γ

u2 u · n dS

We now derive the objective function of our drag minimization problem. As in [14, 15, 16], we are going to minimize the energy conversion from kinetic energy into heat. The kinetic energy Ev of the system is given by 3 1 1 Ev = mu2 = ρu2 dA 2 2 Ω i=1 i 3 ⇒ E˙ v = ui (ρu˙i ) dA. Ω i=1

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STEPHAN SCHMIDT AND VOLKER SCHULZ

Inserting the Navier–Stokes equations for ρu˙i , one arrives at E˙ v =

3 Ω i=1

ui (νΔui − ρu∇ui − ∇p)dA,

and with the Gauss theorem used on the pressure part the equation becomes E˙ v = =

3 (νui Δui − ui ρu∇ui − ui ∇p) dA Ω i=1

3 Ω i=1

νui Δui − ui ρu∇ui −

3 ∂ui p dA + pui ni dS. ∂xi Γ i=1

3 3 i Furthermore, i=1 ∂u i=1 pui ni = 0 ∂xi = 0 because of the divergence freedom and because of the no-slip boundary condition, which leads to E˙ v =

3 Ω i=1

[νui Δui − ui ρu∇ui ] dA.

With Lemma 2.1 this becomes E˙ v =

3 Ω i=1

νui Δui dA.

Swapping over Δ and eliminating the boundary integrals because of the no-slip boundary condition results in E˙ v = −ν

2 3 ∂ui dA. ∂xj i,j=1 Ω

Since we want to formulate this problem as a minimization problem, we change the signs and arrive at the following shape optimization problem: (2.1)

min J(u, Ω) :=

(u,Ω)

ν Ω

2 3 ∂ui dA ∂xj i,j=1

subject to −νΔu + ρu∇u + ∇p = 0 in Ω, div u = 0, (2.2)

u = 0

on Γ,

Vol(Ω) = V0 . The volume constraint Vol(Ω) = V0 prevents a degeneration of the shape, which would otherwise vanish. It is not part of the following gradient derivation but will be taken care of numerically by a discrete projection during each shape update. We would also like to emphasize that the existence of optimal shapes for the above problem is not the focus of this work. Such results can be found in [15, 16] for the Stokes equations and in [19] for the compressible Navier–Stokes equations.


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2.1. Hadamard formulation of the shape gradient. We refer to [17, 18] for the shape sensitivity analysis of the drag functional in the more general setting of the so-called approximate solutions of the stationary compressible Navier–Stokes equations. The perturbed domain is defined by perturbation of identity. Let V be some vector field; then our perturbed domain Ω is given by the boundary Γ = {x + V (x) : x ∈ Γ} , and a formal differentiation of (2.1) according to [6, 20] immediately yields ⎤ ⎡

2 3 3 ∂u ∂ui ∂ui [V ] i ⎦ dS + (2.3) dJ(u, Ω)[V ] = V, n ⎣ν 2ν dA, ∂xj ∂xj ∂xj Γ Ω i,j=1 i,j=1 where u [V ] and likewise p [V ] denote the perturbed state variables. The second part of the gradient is not yet in the so-called Hadamard form. The adjoint approach eliminates the remaining sensitivities. We define u and p as the solution of the incompressible Navier–Stokes equations in the domain Ω . Letting tend to zero, we arrive at the state equation for u [V ] and p [V ] on the original domain Ω: (2.4)

−νΔuk [V

3 ∂uk [V ] ∂uk ∂p [V ] ]+ + uj = 0 in Ω, ρ uj [V ] + ∂xj ∂xj ∂xk j=1 div u [V ] = 0,

(2.5)

where k = 1, 2, 3. It remains to determine the boundary condition for u [V ]. Lemma 2.2. The boundary condition for the change in the speed u [V ] on Γ is given by ui [V ] = −∇ui , V ,

(2.6)

which simplifies due to the no-slip boundary conditions for u: ∇ui , V =

∂ui ∂ui ∂ui V, n + V, τ = V, n. ∂n ∂τ ∂n

Note that on the nonvariable free stream boundary it automatically follows that ui [V ] = 0. Proof. Taylor expansion with respect to shape perturbations. Lemma 2.3. The shape derivative of the incompressible Navier–Stokes energy dissipation problem is given by 3 ∂uk 2 ∂λk ∂uk dS, dJ(u, Ω)[V ] = V, n ν − ∂n ∂n ∂n Γ k=1

where λ and λp solve the adjoint incompressible Navier–Stokes equations −νΔλi + ρ

3 ∂λj j=1

∂λi uj − uj ∂xi ∂xj

−

∂λp = −2νΔui ∂xi

−div λ = 0, with boundary conditions λ = 0 and λp free.

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Proof. Introducing the adjoint states λ and λp , (2.3) can be linked with (2.4) and (2.5) and one arrives at dJ(u, Ω)[V ⎤ ⎡]

2 3 3 ∂u ∂ui ∂ui [V ] i ⎦ ⎣ dS + = V, n ν 2ν dA ∂xj ∂xj ∂xj Γ Ω i,j=1 i,j=1

3 3 ∂uk − νλk Δuk [V ] dA + λk uj [V ] dA ρ ∂xj Ω Ω k=1 k,j=1

3 3 ∂u [V ] ∂p [V ] + λk uj k dA + λk dA ρ ∂xj ∂xk Ω Ω k,j=1 k=1 + λp div u [V ] dA. Ω

All derivatives must now be removed from the state equation sensitivities u [V ] and p [V ] to which we will now use integration by parts: ⎤ ⎡

2 3 3 ∂ui ⎦ dJ(u, Ω)[V ] = V, n ⎣ν −2ν Δuk uk [V ] dA dS + ∂x j Γ Ω i,j=1 k=1 3 ∂u [V ] ∂λk − uk [V ] dS λk k − ν ∂n ∂n Γ k=1 3 − ν uk [V ]Δλk dA Ω

k=1

3

3 ∂(ρλi uj [V ]) ui dA ∂xj Γ i,j=1 Ω i,j=1 3 3 ∂(ρλi uj ) ρλi uj ui [V ]nj dS − ui [V ] dA + ∂xj Γ i,j=1 Ω i,j=1 3 3 ∂λi λi p [V ]ni dS − p [V ] dA + Γ i=1 Ω i=1 ∂xi 3 3 ∂λp + λp ui [V ]ni dS − ui [V ] dA. Γ Ω i,j=1 ∂xi i,j=1

+

ρλi uj [V

]ui nj dS −

Due to the divergence freedom of u and u [V ] one can see that 3 3 ∂(ρλi uj ) ∂λi ui [V ] dA = ρ uj ui [V ] dA, ∂x ∂x j j Ω i,j=1 Ω i,j=1 3 3 ∂(ρλi uj [V ]) ∂λi ui dA = ρ uj [V ]ui dA. ∂x ∂x j j Ω i,j=1 Ω i,j=1 Thus when λ and λp satisfy the adjoint incompressible Navier–Stokes equations in Ω

3 ∂λj ∂λi ∂λp −νΔλi + ρ uj − uj − = −2νΔui ∂x ∂x ∂xi i j j=1 −div λ = 0,


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all domain integrals will vanish and the preliminary gradient becomes ⎤ ⎡

2 3 ∂ui ⎦ dJ(u, Ω)[V ] = V, n ⎣ν dS ∂x j Γ i,j=1 3 ∂uk [V ] ∂λ k − ν − uk [V ] dS λk ∂n ∂n Γ k=1 3 3 + ρλj ui [V ]uj ni dS + ρλi uj ui [V ]nj dS Γ i,j=1

+

3 Γ i=1

λi p [V ]ni dS +

Γ i,j=1

Γ

λp

3

ui [V ]ni dS.

i=1

Since the divergence freedom is also valid at the boundary, we have 3 ∂ui i=1

∂n

ni = 0.

Using the above and setting λ = 0 on Γ, the gradient simplifies as follows: 3 ∂uk 2 ∂λk ∂uk dJ(u, Ω)[V ] = V, n ν dS − ∂n ∂n ∂n Γ k=1

which is of the desired structure. 3. Stokes problem. Since the Stokes problem can be considered a simplified Navier–Stokes equation, which is analytically much better accessible, we will now consider the same problem, but with the Stokes equation as a model for the fluid, resulting in (3.1)

min J(u, Ω) :=

ν

(u,Ω)

Ω

2 3 ∂ui dA ∂xj i,j=1

subject to −νΔu + ∇p = 0 in Ω, (3.2)

div u = 0, u = 0 on Γ, Vol(Ω) = V0 .

Lemma 3.1. The shape derivative for the Stokes problem

2 3 ∂ui min J(u, Ω) := ν dA (u,Ω) Ω i,j=1 ∂xj

(3.3)

subject to −νΔu + ∇p = 0 in Ω, div u = 0, (3.4)

u = 0

on Γ

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is given by dJ(u, Ω)[V ] = −ν

Γ

V, n∇u2 dS,

which means that this problem is self-adjoint. Proof. Using the same argumentation as above, the shape derivative is given by dJ(u, Ω)[V ]

2

3 3 ∂ui ∂ui ∂ui [V ] = V, nν dS + ν 2 dA ∂xj ∂xj ∂xj Γ Ω i,j=1 i,j=1

=ν Γ

V, n∇u2 dS + 2ν

3 ∂ui − (Δui )ui [V ] dA + ui [V ] dS . ∂n Ω Γ i=1

Replacing Δui by the state equation results in dJ(u, Ω)[V ] 3 = ν V, n∇u2 dS + 2 Γ

i=1

Ω

∂p · u [V ] dA + ν ∂xi i

Γ

∂ui ∂n

ui [V ] dS .

Another integration by parts and using div u [V ] = 0 gives dJ(u, Ω)[V ] = ν V, n∇u2 dS Γ

3 ∂ui +2 − p div u [V ] dA + pui [V ]ni + ν ui [V ] dS ∂n Ω i=1 Γ

3 ∂ui 2 = ν V, n∇u dS + 2 pui [V ]ni + ν ui [V ] dS . ∂n Γ Γ i=1 Using Lemma 2.2 yields dJ(u, Ω)[V ] = −ν

Γ

2

V, n∇u dS + 2

∂ui ni dS . V, np ∂n Γ

3 i=1

Since the divergence freedom is also valid at the boundary, we can again drop the last term and arrive at dJ(u, Ω)[V ] = −ν V, n∇u2 dS. Γ

4. Discrete Fourier shape Hessian analysis. 4.1. General shape Hessian analysis. A detailed theoretical analysis of the shape Hessian for a given problem can be extremely tedious. Although one can formally derive the Hessian given the necessary regularity of the states and the boundary, the interpretation of the resulting expression is very difficult and its computation is even harder. While one can write down the shape Hessian using the appropriate symbols of the operators, their true nature, e.g., their (pseudo) differential operator order, usually remains hidden. For more details on this approach, see, for example, [8]. Also,

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this approach is highly problem dependent and requires extensive reevaluation if the objective function under consideration changes. Also, this approach does not capture effects the discretization might have. To reveal the nature of the differential operators involved, most authors usually consider a perturbation in the form of a single Fourier mode. Assuming q˜(x) = qêiωx is a perturbation of the input parameters, the angular frequency ω must now be tracked as it passes all operators involved. Thus, this approach requires some knowledge about the state equation PDE and how its linearization transforms the amplitude and frequency of the perturbation q˜. For the Navier–Stokes problem here this means we must find the symbol ω in the function u [V ] which solves the PDE −νΔuk [V

3 ∂uk [V ] ∂uk ∂p [V ] ]+ + uj =0 ρ uj [V ] + ∂xj ∂xj ∂xk j=1

in Ω(ˆ q ),

div u [V ] = 0, with the boundary conditions uk [V ] = −V, n

∂uk ∂n

on Γ(ˆ q ).

Even the linearized state equation PDE can become rather complex and an analytic solution on arbitrary domains can no longer be found. Thus this approach often requires some severe simplifications of the underlying PDEs. 4.2. The symbol of an operator. Considering q˜(x) = qêiωx one can approximate the pseudo-differential operator nature of the Hessian H by comparing the input q˜ with the output H q˜. For example, if H q˜ = iω qêiωx = iω q˜, then we call iω the symbol of the Hessian and this corresponds to a classical differential operator of order 1. If, for example, we have H q˜ = −ω 2 qêiωx = −ω 2 q˜, then we call −ω 2 the symbol of the Hessian and this corresponds to a classical differential operator of order 2. However, if we find that H q˜ = |ω|ˆ q eiωx = |ω|˜ q, then we know we are dealing with a pseudo-differential operator of order 1. 4.3. Discrete shape Hessian analysis. As a remedy, we will now mimic this approach discretely. The real-valued analogon to the complex-valued Fourier mode is a perturbation of the type q˜ω (x) = sin(2πωx). When we denote the perimeter of Γ with , we can see that a periodic wave on Γ can be expressed by ϕ q˜ω (ϕ) = sin 2πω ,

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where ϕ is the parameterization of the curve. We will also call this perturbation the “input signal.” Thus we have a domain deformation of the type Γ (˜ qω ) = {x(ϕ) + ˜ qω (ϕ)n(ϕ) : ϕ ∈ [0, ]} . The shape Hessian in direction [˜ qω ] then is the limit dJ(u, Γ )[V ] − dJ(u, Γ)[V ] , →0

qω ] = lim d2 J(u, Γ)[V ][˜

which we replace by a finite difference. We call this finite difference the “output signal.” If we are dealing with a regular differential operator, we can interpret the output signal as 2

d J(u, Γ)[V ][˜ qω ](ϕ) =

2n

ck (ϕ)

k=1 n

=

dk q˜ω (ϕ) dϕk

2k

ϕ sin 2πω

k=0 n

2k−1 2πω ϕ . + (−1)k−1 c2k−1 (ϕ) cos 2πω

k

(−1) c2k (ϕ)

2πω

k=1

That means, we first have to split the output signal in a wave that oscillates in phase, which corresponds to the sin-part, and a wave that oscillates out of phase, which corresponds to the cos-part. Next, we must determine the value of each sum: For this, we evaluate the above discrete shape Hessian multiple times with different values for ω and observe the change in the amplitude of the output signal. Thus, if we, for example, compute the output signal for ω and 2ω and see a single wave in phase to our input signal with an amplitude that has changed by a factor of 4, we know that a differential operator of order 2 will best approximate the behavior of the discrete shape Hessian as this corresponds to the symbol of the Hessian being ω 2 . However, one can also observe the behavior of pseudo-differential operators nicely. If we observe, for example, an output signal that is oscillating in the same phase as the input signal but with an amplitude that scales linearly with the input frequency, one can conclude that the symbol of the discrete operator must be |ω| as our discrete approach will translate the imaginary part of the Fourier mode to phase shifts. We will see more of this later on. 4.4. Shape Hessian analysis for the Stokes problem. 4.4.1. Analysis of the analytic Hessian. We will now conduct an analytic analysis of the shape Hessian for the Stokes optimization problem as stated above, similar to [1, 2, 3]. This will give us a reliable result to gauge our numerical method against. Lemma 4.1. The reaction of the Hessian of the Stokes shape optimization problem to a Fourier mode α(x) := eiω1 x is given by

2 ∂ui ∂ui ∂ 2 ui |ω1 | + Hα = −2ν α, ∂y ∂y ∂y 2 i=1 meaning the shape Hessian is a pseudo-differential operator with the symbol |ω|.


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Proof. While the structure theorem for the shape Hessian of a Fréchet-differentiable shape functional is given in [13] for a piecewise smooth boundary of the optimized shape, it is sufficient for our considerations to observe a perturbation of the gradient in 2D, which is given by g˜[α] = − 2ν

2 ∂ui ∂ui [α] , ∂xj ∂xj i,j=1

where we assume our perturbation field to be V = αn where α(x) := eiω1 x . Additionally, u [α] again solves the linearized state equation with boundary conditions: −νΔu [α] + ∇p [α] = 0, div u [α] = 0, u [α] = −

∂ui α. ∂n

Now suppose Ω = {(x, y) : x ∈ R, y > 0} is the upper half-plane, which means that the perturbed gradient is now given by g˜[α] = − 2ν

2 ∂ui ∂u [α] i

i=1

∂y

∂y

due to the no-slip boundary conditions. We now need to describe the operator S given by ∂ui [α] = Sα. ∂y We also assume that the output perturbations of the states are of the form ui [α] = u î eiω1 x eω2 y , p [α] = pêiω1 x eω2 y .

Due to the boundary condition of the perturbed state ui [α] we see that (4.1)

u î =

∂ui . ∂y

Note that we are performing a local analysis of the Hessian. Therefore, we assume the coefficient uî to be constant in the vicinity of where we perform our analysis, ignoring terms which include derivatives of uî . As the output states must also satisfy the Stokes equation, we see that the coefficients uî and pˆ must satisfy the linear system ⎞ ⎛ ⎞ ⎤⎛ ⎡ 0 u ˆ1 −ν −ω12 + ω22 iω1 02 ⎣ ˆ2 ⎠ = ⎝ 0 ⎠ , (4.2) ω2 ⎦ ⎝ u 0 −ν −ω1 + ω22 0 pˆ iω1 ω2 0 which can be thought of as the Stokes equation in the Fourier space. The two equations (4.1) and (4.2) are only noncontradicting, if the matrix in (4.2) does not have full rank. We therefore see that this will limit the possibilities of how ω1 and ω2 are allowed to correlate to each other. The determinant of (4.2) is given by det = ν −ω12 + ω22 ω22 − ν −ω12 + ω22 ω12 .

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As this determinant must vanish, we find that ω1 = |ω2 | is the only frequency pairing which allows (4.1) and (4.2) to both be satisfied at the same time. Thus we have ui [α](x, y) =

∂ui iω1 x |ω1 |y e e ∂y

and we see that ∂ui [α] ∂y

∂ 2 ui iω1 x ∂ui |ω1 |eiω1 x e + ∂y 2 ∂y y=0

∂ui ∂ 2 ui |ω1 | + = α, ∂y ∂y 2 =

meaning that S is given by S=

∂ 2 ui ∂ui |ω1 | + ∂y ∂y 2

and the perturbation in the gradient is given by

2 ∂ui ∂ui ∂ 2 ui g˜[α] = −2ν |ω1 | + α, ∂y ∂y ∂y 2 i=1 which clearly is a pseudo-differential operator of order 1 with a symbol of |ω| and the two coefficients

2 2 ∂ui (4.3) , c2 = −2ν ∂y i=1 (4.4)

c1 = −2ν

2 ∂ui ∂ 2 ui i=1

∂y ∂y 2

.

Remark 4.2. Due to the analytic behavior of the Hessian, we will expect an output signal of our discrete shape Hessian analysis that is in phase to the input signal. We also expect the amplitude to scale linearly with the input angular frequency ω. Also note that the coefficient in front of the |ω| in the Hessian is essentially the gradient again. We therefore also expect the gradient to be modulated into the amplitude of the output signal. 4.4.2. Analysis of the discrete Hessian. We will now conduct the shape Hessian analysis for the minimization of the energy dissipation as described above. Our initial shape is a circle with 500 variable surface mesh nodes. As a flow solver we use a Taylor–Hood finite element solver. Due to the additional nodes of the Taylor–Hood finite elements, we also have 250 surface mesh nodes which are not part of the design variables. We use a parabolic inflow profile such that we have a Reynolds number of 80. The initial domain is shown in Figure 4.1. We found the very high number of surface nodes advantageous for properly resolving the sinousidal input wave even for high frequencies and to later have a very detailed output phase

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Fig. 4.1. Initial shape for the Stokes problem. 1 Input Signal

0.8

Output Signal

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

0.5

1

1.5

2

2.5

3

Fig. 4.2. Incoming and outgoing waves for the Stokes problem.

portrait. Also note that we conduct our Hessian analysis on the initial shape as we do not want to miss parts of the Hessian that vanish in the optimum. Since we are using the Hadamard form of the shape derivatives, the actual number of design parameters have no influence on the computational cost, meaning that taking as many as 500 design parameters is no burden at all. We also want to show the applicability of the method even when the optimum is not yet known. Now we modulate a sin-wave of frequency ω = 50 and with amplitude = 0.002. The phase portrait of incoming and outgoing wave is presented in Figure 4.2. One can clearly see how the output signal consists of a single wave with a phase that is precisely in phase to the input wave. Therefore the cos-part of the output signal must be zero, which precisely matches the analytical prediction. Next, we half the input frequency to 12 ω = 25 and observe that the maximum of the amplitude decreases by almost exactly a factor of 2, and thus the symbol of

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STEPHAN SCHMIDT AND VOLKER SCHULZ Output ω = 25

0.1

Output ω = 50

0.05

0

−0.05

−0.1

0

0.5

1

1.5

2

2.5

3

Fig. 4.3. Amplitude of the Stokes problem response scales linearly with the input frequency.

the discrete Hessian is |ω| as the output amplitude scales linearly with the input frequency. The actual waves are presented in Figure 4.3. Altogether, we see that the discrete method matches the analytical prediction extremely well. A comparison of the output signal with the gradient also confirms that the wave hidden in the amplitude modulation is indeed the gradient again, matching very well the highest coefficient of the Hessian as described above. 4.4.3. Construction of the preconditioner and optimization. As we have seen, we must mimic an operator with the symbol |ω|. Unfortunately, this is not trivial to reproduce discretely. Only operators of symbol 1, iω, and −ω 2 are easy to compute discretely, because operators with these symbols can easily be constructed by using the appropriate finite difference stencils on the submanifold of the flow obstacle. Thus the preconditioning PDE needs only to be solved on the flow obstacle Γ which has one less dimension than the flow domain Ω. Unfortunately, due to the pseudo-differential operator nature of the symbol |ω|, it seems impossible to construct an operator with this symbol without solving a PDE in Ω, i.e., the Dirichlet-to-Neumann map. An operator which can be constructed on Γ only and whose symbol is close to |ω| is the tangential Laplace operator ΔΓ . We therefore choose to approximate our discrete Hessian Hh by Hh ≈ αΔΓ + I,

(4.5)

where α is a coefficient used to correct the discrepancies between the correct symbol |ω| and the symbol of our approximation ω 2 . We must now determine α for which we utilize the fact that we want to construct a preconditioner for the discrete case. Suppose the analytic Hessian has a symbol of β|ω| with some coefficient β varying along the obstacle. Since the number of discretization points of the obstacle limits the highest frequency ω which can occur discretely, we choose α such that the amount of under- and oversmoothing roughly cancel each other out for all frequencies ω within [0, ωmax ], where ωmax is the maximum frequency that can be expressed with a given discretization of the obstacle. We arrive at the equation ωmax ωmax αx2 + 1 dx = βx dx, 0

0

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APPROXIMATION OF DISCRETE SHAPE HESSIANS 47

Preconditioned Unpreconditioned

46.5

46

45.5

45

44.5

44 0

20

40

60

80

100

120

140

160

180

200

220

Fig. 4.4. Convergence rates of the preconditioned Stokes problem.

which when solved for α gives

(4.6)

α=

3 2 βωmax − 2 ωmax

3

.

Assuming that the proper discretization of a wave requires at least 2 to 4 points, we see that given 200 variable surface nodes here, a proper choice for ωmax lies between 50 and 100. For the following optimization we set ωmax to 75 and choose β as shown in (4.3). Note that the choice of β was made because of the discrete observations of the response to our input wave. The true coefficient β of the operator is certainly more complex; however due to the assumptions made in (4.2), it is not possible to derive the proper coefficient analytically using the method presented above. We compare this preconditioned approximate SQP method with an unpreconditioned one, i.e., standard steepest descent. Since the unpreconditioned steepest descent algorithm is not level independent, we reduce the number of variable boundary nodes to 200 as with the 500 variable nodes of the previous chapter a direct comparison of the preconditioned and the nonpreconditioned iterations would not have been possible. The steplength d for the unpreconditioned iteration was constant with d = 0.02, which we found was the maximum steplength useable with 200 variable boundary nodes. The steplength for the preconditioned iteration was chosen to be d = 0.25, and we used a backtracking linesearch to determine convergence. The backtracking linesearch has reported optimality in iteration 12. This means a saving of 94% of the iterations if we accept iteration 200 as optimal in the unpreconditioned iteration. The history of the optimization is presented in Figure 4.4. According to [15] the optimal shape is a wedge with a 60 deg front and back angle. As shown in Figure 4.5, we found our computation with preconditioning to very well match this shape, whereas without preconditioning or an improper choice of ωmax we observe the rear end to be computed slightly too round which can also be seen in the value of the unpreconditioned objective function shown in Figure 4.4, which is slightly higher than the preconditioned one.

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Fig. 4.5. Final shape for the preconditioned Stokes problem.

5. Shape Hessian and optimization for the Navier–Stokes problem. 5.1. Discrete shape Hessian analysis. We have seen in (4.2) how the analytical determination of the symbol requires explicit knowledge of the roots of the determinant of the state equation PDE in the Fourier space. For the Navier–Stokes problem, the equivalent of the determinant of (4.2) becomes a polynomial of degree 4 with complex coefficients and the roots of this polynomial can no longer be elegantly given. Thus, we perform only our discrete analysis. For the Navier–Stokes problem we slightly change the geometry by numerically removing the walls of our virtual wind tunnel. This reduces the effects the walls have on the flow. The Reynolds number is kept at 80, resulting in a steady state laminar flow. We again start from a circle with 1000 surface mesh nodes with variable positions and modulate a sin-wave of amplitude 0.002 and an angular frequency of ω = 50 onto this circle. The comparison of the input and output signal is given in Figure 5.1. One can again see that both input and 6

Input Signal Output Signal

4 2 0 −2 −4 −6 −8 −10 0

0.5

1

1.5

2

2.5

Fig. 5.1. Incoming and outgoing waves for the Navier–Stokes problem.

3

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Output ω = 25 Output ω = 50

4 2 0 −2 −4 −6 −8 −10 0

0.5

1

1.5

2

2.5

3

Fig. 5.2. Amplitude scaling with frequency doubling for the Navier–Stokes problem.

output signals stay in phase, which again points to either a differential operator of even order, or a pseudo-differential operator very similar to the Stokes problem. Note that the apparent flipping of the in-phase and out-of-phase parts of the output signal in Figure 5.1 simply correspond to the sign change of the gradient. Next, we again observe the scaling of the amplitude when we half the input frequency to 12 ω = 25. Similar to the Stokes problem, we observe that the amplitude of the output signal scales linearly with the frequency of the input signal. The corresponding waves are shown in Figure 5.2. Thus, we come to the conclusion that the discrete Hessian for the Navier–Stokes problem is also a pseudo-differential operator of symbol |ω| just as in the Stokes problem. Remark 5.1. Unfortunately, the discrete approach cannot reveal the dependence of the symbol of the Hessian and the Reynolds number. It is thus entirely possible that the behavior of the Hessian for the Navier–Stokes equation changes significantly with the occurrence of turbulence. However, the treatment of turbulent flow is beyond the scope of this paper. 5.2. Preconditioning and optimization. We again construct a preconditioner based on the available information concerning the Hessian. In the Stokes problem, we have used the outer derivative of the gradient (4.3) as the variable coefficient β in (4.6). In the Navier–Stokes case, this outer derivative is more complex, as it now involves the costate also. We therefore use the constant factor (5.1)

α=

3 2 ωmax − 2 ωmax

3

in our preconditioner (4.5). In order to again be able to compare the speedup with the unpreconditioned iteration, we now reduce the number of variable surface mesh nodes to 100. This results in a value of ωmax between 25 and 50 which we kept fixed at 30. The initial and optimized shapes are shown in Figures 5.3 and 5.4. The preconditioned optimization requires 71 steps till convergence using a steplength of d = 0.06 and the unpreconditioned optimization requires around 350 iterations using a steplength of d = 0.005, which is the longest steplength possible. We found the resulting optimal ogive shape to be astonishing ship-like. Also, the preconditioned iteration requires only 20% of the unpreconditioned gradient steps. That is, for the

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Fig. 5.3. Initial shape of the Navier–Stokes problem.

Fig. 5.4. Optimized shape of the preconditioned Navier–Stokes problem.

Navier–Stokes problem we found our preconditioning to reduce the computational effort needed by 80%. The precise comparison of the convergence history is plotted in Figure 5.5.

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Preconditioned Unpreconditioned

60

55

50

45

40 0

50

100

150

200

250

300

350

Fig. 5.5. Convergence rates of the preconditioned Navier–Stokes problem.

6. Conclusion. The goal of this paper was to find both discrete and analytic approximations of the Hessian in viscous incompressible flows. This can be achieved by tracking a disturbance of the shape in the form of a single Fourier mode through all operators involved. We proposed a discrete method where we observe the influence of the Hessian on a sin-input wave. By comparison of the phase and the amplitude of the output wave, we compute the symbol of the Hessian discretely. After deriving the objective function and the respective shape derivatives for the Stokes and Navier–Stokes shape optimization problem, we first conducted an analysis of the analytic Hessian of the Stokes problem. This has revealed that the operator acts like a pseudo-differential operator of order 1 similar to the Dirichlet-to-Neumann map, sharing the same symbol. Next we conducted our discrete method on the same problem and found the results to match very well. Using the new information about the Hessian for the Stokes problem, we propose an approximation of the symbol using the discrete Laplace–Beltrami operator and the L1 norm. This has achieved a major speedup in the optimization. Next, we focused on the Navier–Stokes problem. We again conducted our discrete Hessian analysis and again found this operator to be a pseudo-differential operator of order 1. Using a preconditioner similar to the Stokes problem, we again were able to achieve major speedups during the optimization. REFERENCES [1] E. Arian, Analysis of the Hessian for aeroelastic optimization, Technical report 95-84, ICASE, NASA Langley Research Center, Hampton, VA, 1995. [2] E. Arian and S. Ta’asan, Analysis of the Hessian for aerodynamic optimization: Inviscid flow, Technical report 96-28, ICASE, NASA Langley Research Center, Hampton, VA, 1996. [3] E. Arian and V. N. Vatsa, A preconditioning method for shape optimization governed by the Euler equations, Int. J. Comput. Fluid Dyn., 12 (1999), pp. 17–27. [4] C. Castro, C. Lozano, F. Palacios, and E. Zuazua, Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids, AIAA J., 45 (2007), pp. 2125–2139. [5] F. Courty and A. Dervieux, Multilevel functional preconditioning for shape optimisation, Int. J. Comput. Fluid Dyn., 20 (2006), pp. 481–490. [6] M. Delfour and J.-P. Zol´ esio, Shapes and Geometries, SIAM, Philadelphia, 2001.

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