Shape Derivatives for General Objective Functions and the ...

Deutsche Forschungsgemeinschaft

Priority Program 1253 Optimization with Partial Differential Equations

Stephan Schmidt and Volker Schulz

Shape Derivatives for General Objective Functions and the Incompressible Navier-Stokes Equations February 2009

Preprint-Number SPP1253-10-05

http://www.am.uni-erlangen.de/home/spp1253

Shape Derivatives for General Objective Functions and the Incompressible Navier-Stokes Equations Stephan Schmidt∗ Volker Schulz† Abstract The aim of this paper is to present the shape derivative for a wide array of objective functions using the incompressible Navier-Stokes equations as a state constraint. Most real world applications of computational fluid dynamics are shape optimization problems in nature, yet special shape optimization techniques are seldomly used outside the field of elliptic partial differential equations and linear elasticity. As such, this work tries to be self contained, also presenting many useful results from the literature. We conclude with optimizing the shape of several flow channels, finding tubes which feature a greatly reduced pressure loss.

1

Introduction

Optimal control of fluids has seen much attention due to its importance for many technical, scientific, and engineering applications. A good overview on fluid control can for example be found in [6]. However, the problem is seldom treated from a pure shape optimization perspective. Notable exceptions are [9] and one chapter of [8]. When the problem is treated from a shape optimization approach, usually only one very specific type of objective function is considered: the volume dissipation of the kinetic energy of the fluid into heat. In the limit of the Stokes equations, this results in a self-adjoint problem, allowing for an elegant analysis. A more general volume objective function for the Navier-Stokes equations is considered in [7], where the existence is shown using surprisingly weak regularity assumptions. ∗ †

Dipl.-Math S. Schmidt, Univ. Trier, [email protected] Prof. Dr. V. Schulz, Univ. Trier, [email protected]

1

1 Introduction

2

Although many objective functions of practical relevance are defined on the surface of the flow obstacle alone, such an objective is seldom considered. On the one hand, the shape sensitivity analysis for surface functionals is much more complex in itself, but on the other hand, a surface functional often features a dependence on the geometry which adds to the difficulties deriving the correct gradient expression in Hadamard form. The Hadamard form enables a very efficient computation of the gradient without the need to compute the so called “mesh sensitivity” Jacobian. This is especially true for aerodynamic quantities such as drag or lift coefficient or matching a target surface pressure distribution. The shape derivative for such quantities can for example be found in [13] for a compressible fluid model. Since many auxiliary results from a special shape analysis background are needed to derive the Hadamard expression, this paper seeks to be self-contained, listing them from multiple literature sources such as [1, 2, 3, 15]. As such, the present work seeks to derive the expression for the shape derivative in an incompressible Navier-Stokes flow for an objective function that is as general as possible, combining both a volume part and a surface objective function. A dependence on the geometry is also included. The Hadamard form is extremely convenient for computational purposes, but must be re-derived for each problem, unless it is known for a general objective function on both surface and volume, which is the main purpose of this work. Since the pressure in an incompressible fluid has an artificial character and usually no boundary condition on the fluid obstacle, there will be some restriction on the surface part of the objective function such that the adjoint state exists. For more detailed existence results we would like to refer to [10, 11, 12]. We conclude with the optimization of internal flows where the pressure loss along the pipe could be reduced considerably.

2 The Optimization Problem

2

3

The Optimization Problem

The aim of this paper is to show the structure of the shape derivative of a general objective function over an incompressible Navier-Stokes fluid. Z Z (2.1) min J(u, Ω) := f (u, Du, p) dA + g(u, Dn u, p, n) dS (u,p,Ω)

Ω

Γ0

subject to −µ∆u + ρu∇u + ∇p = ρG in Ω div u = 0 u = u+ on Γ+ (2.2) u=0 on Γ0 ∂u pn − µ =0 on Γ− . ∂n Here, f : Rd ×Rd×d ×R → R and g : Rd ×Rd ×R×Rd → R are assumed to be continuously differentiable in each argument. Also, d is the dimension of the bounded domain Ω ⊂ Rd and in the following Γ := ∂Ω is used to denote the whole boundary of Ω. Furthermore, u : Ω → Rd is the velocity of the fluid and p : Ω → R is the pressure. The viscosity is given by µ and ρ denotes the density, which is constant in an incompressible fluid. The outflow boundary condition on Γ− is the finite element “do nothing” outflow condition that naturally arises due to integration by parts during the finite element matrix assembly. Additionally, Γ+ denotes the inflow boundary and Γ0 is the fluid obstacle or the channel wall, using the no-slip boundary condition. The shape Γ0 is the unknown to be found. Also, n is the normal vector with components ni and ρGi are the outside body forces, i.e. the forces per unit volume acting on the fluid. Due to the no-slip boundary condition on Γ0 it is sufficient to consider the derivative in normal direction Dn u := Du · n on Γ0 since the tangent derivative of the velocities is zero anyway. The control we are considering is the shape of the Dirichlet boundary Γ0 . As such, Γ0 is the unknown and we seek the total derivative of the above with respect to Γ0 . The inflow direction is considered constant and independent of the shape of the boundary. Since the outflow boundary has an artificial character anyway, we also consider the shape of the outflow boundary to be fixed. In order to keep the notation readable we refer to the Jacobian components as follows: Du =: [aij ]ij ∈ Rd×d ∂u =: [bi ]i ∈ Rd . Dn u = Du · n = ∂n

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4

Since the pressure has no explicit boundary condition on Γ0 , but is implicitly linked with the velocity, we need to impose the following restriction on g such that we can later arrive at a consistent adjoint boundary condition: We choose g such that there exists a functional λ satisfying the following conditions on Γ0 : 1 ∂g ∀i = 1, . . . , d µ ∂bi ∂g hλ, ni = − . ∂p λi =

This is less restrictive than it might appear. A consequence is that for a force minimization, the forces should be chosen in line with the state equation, i.e. since the state equation describes a viscous fluid, the objective function should also include the viscous forces. For a drag minimization at zero angle of attack, we have ∂u1 − pn1 g(u, Dn u, p, n) = µ ∂n which leads to ∂g = −n1 ∂p ∂g = µδ1,i ∂bi and the above is satisfied with λi = δ1,i . The inclusion of higher derivatives on the velocities is straight forward, but further limits the allowed surface functionals g and will not be considered here.

3

Shape Calculus

Since we are interested in a gradient with respect to the shape of Γ, we first define a one parametric family of bijective mappings Tt : (t, x) → 7 Tt (x). A deformed domain Ωt is then given by Ωt := {Tt (x0 ) : x0 ∈ Ω} . Usually, the mapping Tt is either given by the perturbation of identity Tt (x) = x + tV (x) or implicitly by the speed method dx = V (t, x), x(0) = x0 dt

3 Shape Calculus

5

where V is a vector field of appropriate smoothness. For first order calculus, it can be shown that both approaches are equivalent. Here, however, we will focus on the perturbation of identity. If (ut , pt ) denotes the solution of the Navier-Stokes equations in the perturbed domain Ωt , we seek to derive a formula for the shape derivative dJ(Ω)[V ] := lim+ t→0

J(ut , pt , Ωt ) − J(u, p, Ω) t

that can be computed very efficiently without taking into account the “mesh ∂c sensitivity” ∂q of the Navier-Stokes state equation c(u, p) = 0, which otherwise enters the calculation due to a formal Lagrangian approach dJ ∂c ∂J = − λT dq ∂q ∂q · ¸T ∂c ∂J λ= ∂(u, p) ∂(u, p) when the domain is parameterized by some finite design parameters q. The key ingredient here is the so-called Hadamard formula, a consequence from the Delfur-Zol´esio structure theorem. The consequence of the Hadamard formula is, that under some mild smoothness assumptions, the shape derivative dJ has the structure of a scalar product with the normal component hV, ni of the prescribed perturbation V of the domain Ω. One can thus use the shape gradient h directly as an update for the boundary which results in the steepest descent direction and can be ∂c . It also applied without knowledge of the costly mesh sensitivity Jacobian ∂q becomes obvious, that the choice V = αn is best in some sense, as then the projection hV, ni vanishes, i.e. Γt+1 = {x0 + ∆t · h(x0 )n(x0 ) : x0 ∈ Γt } is a very simple gradient marching optimization scheme with step length ∆t. In order to find the shape gradient h for the general Navier-Stokes problem we are considering, we need some additional results. Most of them are known from the literature [1, 2, 3, 15], but we think that listing them here will create a much more consistent derivation of the Navier-Stokes gradient.

3.1

Shape Derivative for Volume Objectives

We start with recapitulating the Hadamard formula for objective functions which are defined over the whole domain Ω, such as the first part of the mixed objective function (2.1):

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Lemma 3.1 (Derivative of deformation determinant). The derivative of the determinant of the perturbation of identity approach is given by: d det DTt (x) = div V (x) dt t=0 Proof. For a matrix A(t) ∈ Rm×m where each entry is a differentiable function such that A(t)−1 exists for some interval I ⊂ R, the derivative of the determinant with respect to t is then given by µ ¶ d dA(t) −1 det A(t) = tr A(t) det A(t). dt dt Since DT0 (x) = I, we have µ ¶ dDTt (x) d det DTt (x) = tr dt t=0 dt t=0 = tr (DV (x)) = div V (x)

Lemma 3.2 (Hadamard formula for volume objective functions). For a general volume objective function f : Ω → R, not depending on a PDE constraint, i.e. Z J(Ω) =

f dA, Ω

the shape derivative and shape gradient are given by Z dJ(Ω)[V ] = hV, nif dS Γ

Proof. The proof requires the multi-dimensional substitution rule, the structure of the Jacobian DTt of the deformation, the derivative of the determinant

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operator, the multiplication rule backwards, and the divergence theorem: Z d d f (x) dA(x) J(Ωt ) := dJ(Ω)[V ] : = dt t=0 dt t=0 Ωt Z d = f (Tt (x)) det(DTt (x)) dA(x) dt t=0 Ω Z d f (Tt (x)) det(DTt (x)) dA(x) = dt t=0 Ω Z = h∇f (x), V (x)i + f (x)div V (x) dA(x) Ω Z

=

div (f (x)V (x)) dA(x) Ω Z

=

hV, nif dS Γ

3.2

Definitions and Lemmas

Before we recapitulate the Hadamard formula for objective functions which are defined on the boundary of the domain Ω, such as the second part of (2.1), some definitions and lemmas should be presented from the literature: Definition 3.3 (Submanifold). For k ∈ N and m ∈ {1, ..., d} a subset Ω ⊂ Rd is called m-dimensional submanifold in Rd of class C k if for each x ∈ Ω there exists an open neighborhood U (x) ⊂ Rd around x and a diffeomorphism h of class C k such that h−1 (U (x) ∩ Ω) = h−1 (U (x)) ∩ Rm where Rm = {(x1 , . . . , xm , 0, . . . , 0) ∈ Rd : (x1 , . . . , xm ) ∈ Rm } Definition 3.4 (Integral over submanifolds). Let Ω be an m-dimensional compact submanifold in Rd with finite open cover Ω⊂

l [ j=1

Mj

3 Shape Calculus

8

such that Ωj := h−1 j (Mj ) and a corresponding partition of unity l X

rj (x) = 1

j=1

with rj infinitely continuously differentiable with compact support. The integral over Ω is then defined by Z Ω

g dΩ :=

l Z X j=1

Ωj

grj dΩ :=

l Z X j=1

h−1 j (Mj )

q grj (hj (s)) det(Dhj T Dhj )(s) dS

where Dhj is the Jacobian of hj . Lemma 3.5 (Integral over the surface of submanifolds). Let Ω be as in the definition above. The integral over the surface of Ω is then given by Z Z g dS = g(h(s))| det Dh|k (Dh)−T ed k ds ∂Ω

B0

where B0 = {ξ ∈ Rd : kξk ≤ 1, ξd = 0} is the cut of the open d-dimensional unit ball with the ξd = 0 hyperplane and ed is the d-th unit vector. Proof. Let B := {ξ ∈ Rd : kξk ≤ 1} ⊂ Rd be the open unit Ball in Rd . The unit ball is segmented by a cut with the ξd = 0 hyperplane in B+ := {ξ ∈ B : ξd > 0} B− := {ξ ∈ B : ξd < 0} B0 := {ξ ∈ B : ξd = 0}. Without loss of generality, one can assume that the interior of Ωj is given by int Ωj = hj (B+ ) and consequently the boundary is given by ∂Ωj = hj (B0 ), i.e. ∂Ωj = {hj (ξ, 0) : (ξ, 0) := (ξ1 , · · · , ξd−1 , 0) ∈ B0 }. Hence, for a proper computation of the surface integral it is necessary to project the integration density det(Dhj T Dhj ) of the volume case above to the (ξ, 0)-hyperplane, i.e. dropping the last column and last row from the matrix, which is the dd-minor [Dhj T Dhj ]dd

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9

of Dhj T Dhj . By the definition of the cofactor-matrix, the determinant of the dd-minor is exactly the mdd -entry of the cofactor-matrix M (Dhj T Dhj ). Thus, the proper integration density for the surface integral is given by q √ mdd = eTd M (Dhj T Dhj )ed q = eTd M (Dhj T )M (Dhj )ed q = kM (Dhj )ed k22 = kM (Dhj )ed k2 = | det(Dhj )|kDh−T j e d k2 where in the last line the well known property M (A) = det(A)A−T was used. Hence, the corresponding boundary integral is given by Z ∂Ω

g dS :=

l Z X j=1

=

∂Ωj

l Z X j=1

Z

=:

B0

B0

grj dS

grj (hj (s))| det Dhj |k (Dhj )−T ed k ds

g(h(s))| det Dh|k (Dh)−T ed k ds

where s = (ξ, 0) = (ξ1 , · · · , ξd−1 , 0). Lemma 3.6 (Unit normal field on ∂Ω). For a regular surface ∂Ω, the unit normal field at x = h(ξ, 0) on ∂Ω is given by Pl −T ed Dh(ξ, 0)−T ed j=1 rj Dhj (ξ, 0) n(x) = Pl =: kDh(ξ, 0)−T ed k k j=1 rj Dhj (ξ, 0)−T ed k Proof. The tangent space of the regular surface ∂Ωj at x = hj (ξ, 0) is given by the span of the columns of the Jacobian Dhj (ξ, 0): Tx Ωj = span(Dhj (ξ, 0)ei , i = 1, · · · , d − 1), i.e. the (non-unit) tangent direction is given by τi := Dhj (ξ, 0)ei . Since hτi , Dhj (ξ, 0)−T ed i = hDhj (ξ, 0)ei , Dhj (ξ, 0)−T ed i = hDhj (ξ, 0)−1 Dhj (ξ, 0)ei , ed i = hei , ed i = 0 ∀i = 1, · · · , d − 1,

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the expression for the normal follows by summation over the partition of Ω and normalization. Remark 3.7 (Alternate representation of the normal). According to the above, a non-unit normal field m(x) at x = (ξ, 0) for the regular surface ∂Ω is given by m(x) = Dh(ξ, 0)−T ed . Due to the property M (A) = det(A)A−T , we have m(x) = det(Dh(ξ, 0))−1 M (Dh(ξ, 0))ed . The scalar determinant is non-zero due to the regularity of ∂Ω and can be dropped during the normalization, such that n(x) =

M (Dh(ξ, 0))ed kM (Dh(ξ, 0))ed k2

The structure of the normal can now be used in the definition of the surface integral. Using the above, the integral over the perturbed surface Γt can now be expressed with respect to the unperturbed surface Γ: Lemma 3.8 (Perturbed surface integral). The surface integral over the perturbed surface Γt is given by Z Z g dΓt = g(Tt (x))kM (DTt (x))n(x)k2 dΓ(x) Γ

Γt

where n is the unit normal to the unperturbed boundary Γ. Proof. The perturbed submanifold Γt can be described by ht (ξ, 0) := Tt (h(ξ, 0)).

(3.1)

The surface integral is then given by Z Z g(ht (s))kM (Dht (s))ed k2 ds g dSt = ∂Ωt

B0

The chain rule results in Dht (ξ, 0) = D[Tt (h(ξ, 0))] = DTt (h(ξ, 0))Dh(ξ, 0) and M (Dht (ξ, 0)) = M (DTt (h(ξ, 0)Dh(ξ, 0))) = M (DTt (h(ξ, 0)))M (Dh(ξ, 0)).

(3.2)

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Using the alternate representation of the normal: kM (Dht (s))ed k2 = kM (DTt (h(ξ, 0)))M (Dh(ξ, 0))ed k2 = kM (DTt (h(ξ, 0)))kM (Dh(ξ, 0))ed k2 n(h(ξ, 0))k2 = kM (Dh(ξ, 0))ed k2 kM (DTt (h(ξ, 0)))n(h(ξ, 0))k2 Thus, Z ∂Ωt

Z g dSt = =

B0

Z

∂Ω

g(Tt (h(s))kM (DTt (h(s)))n(h(s))k2 kM (Dh(s))ed k2 ds g(Tt (x))kM (DTt (x))n(x)k2 dΓ(x)

where again s = (ξ, 0) and x = h(s). Remark 3.9. Due to the definition of the cofactor matrix, we also have Z Z g(Tt (x))kM (DTt (x))n(x)k2 dΓ(x) g dSt = ∂Ω ∂Ωt Z g(Tt (x))| det DTt (x)|k(DTt (x))−T n(x)k2 dΓ(x). = ∂Ω

Since we assume the deformation mapping Tt does not change the orientation of Ωt relative to Ω, we can assume det DTt > 0 in subsequent considerations. Lemma 3.10 (Derivative through matrix inverse). Let A(t) ∈ Rm×m be a matrix where each entry is a differentiable function such that A(t)−1 exists for some interval I ⊂ R. The derivative of the matrix inverse with respect to t is then given by d dA(t) A(t)−1 = −A(t)−1 A(t)−1 dt dt Proof. Let aij (t) be the component functions of A(t) and let ajk (t) be the components of A(t)−1 , i.e.: δik = ⇒0=

m X j=1 m X j=1

aij (t)ajk (t) dajk (t) daij (t) jk a (t) + aij (t) dt dt

dA(t) dA(t)−1 −1 ⇒0= A(t) + A dt dt

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3.3

12

Shape Derivative for Surface Objectives

One can now form a preliminary shape derivative for objective functions which are defined on the boundary of the domain Ω alone, such as the second part of (2.1): Lemma 3.11 (Preliminary shape derivative). For g ∈ C(T (Γ)), where T (Γ) is a tubular neighborhood of Γ such that ∇g is defined on Γ, the preliminary shape derivative, not yet in Hadamard form, for the surface integral is given by Z Z d g dSt = h∇g, V i + g (div V − hDV n, ni) dS dt t=0 Γt Γ Proof. We start with the expression from remark 3.9: Z d g dSt dt t=0 ∂Ωt Z d = (g(Tt (x)) det DTt (x)k(DTt (x))−T n(x)k2 ) dΓ(x). ∂Ω dt t=0 Furthermore, γt := DTt−T n = ((I + tDV )T )−1 n gives

à d ! 21 µ ¶ X d 1 d d 2 T = kγ(t)k2 = γi (t) γ (0) γ(t) . dt t=0 dt t=0 i=1 kγ(0)k2 dt t=0

Especially γ(0) = n d d γ(t) = −I −1 (I + tDV )T I −1 n dt t=0 dt t=0 = −DV T n. Thus

d kγ(t)k2 = −nT DV T n = −hDV n, ni. dt t=0 Using det DT0 = det I = 1 and the product rule, the above results in ¸ Z Z · d d g dSt = (g(Tt ) det DTt ) n − ghDV n, ni dS dt t=0 ∂Ωt dt t=0 Z∂Ω = h∇g, V i + g (div V − hDV n, ni) dS, ∂Ω

where the formula for the determinant was again used.

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Definition 3.12 (Tangential gradient, tangential divergence, curvature). For a sufficiently smooth d-dimensional submanifold M ⊂ Rd and g ∈ C 2 (M ), the tangential gradient of g is defined by the projection of the ordinary gradient onto the tangent plane d−1 X ∂g τi ∈ Rd−1 ∇M g := PT (∇g) = ∂τ i i=1

where the τi form an orthonormal basis of the tangent space. For a differentiable vector field V , the tangential divergence is defined by À d−1 ¿ X ∂V divM V := tr(PT (DM V )) = , τi ∈ R. ∂τi i=1 It can be shown that this is well defined and independent of the particular choice of the orthonormal tangent basis. Furthermore, the mean curvature is defined by the tangential divergence of the unit normal: κ := divM n Remark 3.13. In the more general setting of manifolds Z Z p d g(h(ξ)) γ(ξ) dξ g dH = M

Ω

where the metric is given by γij = ∂i h∂j h and the area element by p det γij , the tangential gradient is usually defined by

√

γ =

∇M g := γ ij ∂j g∂i h where Einstein’s summation convention was used and γ ij denotes the inverse metric. The object is often called covariant derivative in this context. Thus, tangential gradient and tangential divergence are intrinsic objects. In the context we are primarily considering, i.e. the shape optimization under a PDE constraint, it is sufficient to assume that M is a submanifold in Rd of codimension 1, i.e. the set of unit normal and unit tangent vectors {n, τ1 , · · · , τd−1 } creates an orthonormal basis of Rd . The gradient ∇g can then be expressed in this basis, i.e. ∇g = h∇g, nin +

d−1 X h∇g, τi iτi . i=1

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Thus, for a function that extends into a tubular neighborhood of M such that ∂g exists, the tangential gradient can also be expressed as ∂n ∇M g = ∇g −

∂g n ∂n

and likewise divM V = div V − hDV n, ni Lemma 3.14. Tangential gradient and tangential divergence are related to each other just like their ordinary counter parts, i.e. for a differentiable scalar function g and a differentiable vector field V divM gV = h∇M g, V i + gdivM V Proof. Since X ∂g j ∂g ∂g , τi i = Vj τi = h τi , V i = h∇M g, V i hV ∂τi ∂τi ∂τi j=1 d−1

where upper indices denote the vector component, we have divM gV =

d ¿ X ∂gV i=1

∂τi

À , τi

d−1 X

∂g , τi i + g = hV ∂τ i i=1

¿

∂V , τi ∂τi

À

= h∇M g, V i + gdivM V

Lemma 3.15. For a general surface objective function g : T (Γ) → R such ∂g that ∂n exists, the shape derivative and shape gradient are given by · ¸ Z ∂g dJ(Ω)[V ] = hV, ni + κg dS ∂n Γ

where κ = divΓ n is the tangential divergence of the normal, i.e. the additive mean curvature of Γ. Proof. Starting form the preliminary gradient of lemma 3.11, we have Z Z d g dSt = h∇g, V i + g (div V − hDV n, ni) dS dt t=0 ∂Ωt ∂Ω Z = h∇g, V i + gdivΓ V dS. ∂Ω

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Suppose V˜ := hV, nin. Since perturbations in tangent direction are mere reparametrizations, the shape derivative only depends on the normal component of V . Hence we have dJ(Ω)[V˜ ] = dJ(Ω)[V ] and d dt t=0

Z

·

Z

∂g + gdivΓ n g dSt = hV, ni ∂n ∂Ωt ∂Ω

¸ dS

where we have used that divΓ (hV, nin) = h∇M hV, ni, ni + hV, nidivΓ n = hV, nidivΓ n = hV, niκ

3.4

Shape Derivative of the Unit Normal

The derivative of the unit normal field with respect to shape perturbations is very often needed. Many objective functions stemming from physics, such as the fluid forces we are also considering, or any PDE constraint of the Neumann type require this knowledge. Lemma 3.16 (Unit normal on the perturbed domain). The unit normal on the perturbed domain Ωt is given by nt (Tt (x)) =

(DTt (x))−T n(x) k(DTt (x))−T n(x)k2

Proof. According to lemma 3.6, the unit normal on the perturbed domain is given by Dht (ξ, 0)−T ed nt (x) = kDht (ξ, 0)−T ed k Using equations (3.1) and (3.2), we have (DTt (h(ξ, 0)))−T (Dh(ξ, 0))−T ed k(DTt (h(ξ, 0)))−T (Dh(ξ, 0))−T ed k (DTt (x))−T n(x) = k(DTt (x))−T n(x)k

nt (Tt (x)) =

where lemma 3.6 was used again for the unperturbed domain.

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Lemma 3.17 (Preliminary shape derivative of the unit normal). The preliminary shape derivative of the unit normal is given by dn[V ](x) :=

d nt (Tt (x)) = hn, (DV (x))T n(x)in(x) − (DV (x))T n(x) dt t=0

Proof. Since DT0 (x) = I, the quotient rule simplifies to µ ¶ £ ¤ d −T (DTt (x)) n(x) dn[V ](x) := dt t=0 ¶ µ d −T k(DTt (x)) n(x)k2 . − n(x) dt t=0 Using lemma 3.10, the above transforms to µ ¶ d −T dn[V ](x) = n(x) k(DTt (x)) n(x)k2 − (DV (x))T n(x). dt t=0 For any vector v(t) where the components are differentiable functions, the chain rule gives d d kv(t)k2 = dt t=0 dt t=0

Ã

X

! 12 2

vi (t)

i

hv(0), v 0 (0)i = . kv(0)k2

Hence, for v(t) = (DTt (x))−T n(x) we have v(0) = n(x) and again due to lemma 3.10 we have v 0 (0) = (DV (x))T n(x), resulting in d kDTt (x)n(x)k2 = hn(x), (DV (x))T n(x)i, dt t=0 which gives the desired expression. Unfortunately, lemma 3.17 is not yet applicable to the Hadamard form and additional transformations are required for finding the Hadamard formula for the shape derivative when the objective function or the state constraint depend on the unit normal: Definition 3.18 (Tangential Jacobian). Similar to definition 3.12, the tangential Jacobian for a differentiable vector valued function V is defined as DM V = [∇M Vi ]Ti , that is, the rows of the matrix are given by the tangential gradient of the component functions.

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Remark 3.19. Similar to remark 3.13, there is also the equality " d−1 #T · ¸T X ∂Vi ∂Vi DM V = τk = ∇Vi − n = DV − DV nnT ∂τ ∂n k i k=1 i

should the required derivative in normal direction exist. Remark 3.20. The tangential Jacobian of the unit normal field n(x) at a point x lies in the tangent space Tx M , i. e. ¡ ¢ 0 = DM 1 = DM n(x)T n(x) = 2 (DM n(x))T n(x), thus DM n ⊥ n. Lemma 3.21. The shape derivative of the normal is equivalently given by dn[V ] = − (DΓ V )T n Proof. Assuming that the perturbation field V extends into a tubular neighborhood, we have DΓ V = DV − DV nnT and likewise (DΓ V )T n = (DV )T n − n (DV n)T n = −dn[V ] due to lemma 3.17 and the symmetry of DV . Lemma 3.22. For a perturbation normal to the boundary Γ, i.e. V˜ := hV, nin or equivalently hV˜ , τ i = 0 for a vector τ ∈ Tx Γ, we have dn[V˜ ] = −∇Γ hV˜ , ni Proof. Let {τi ∈ Tx Γ : 1 ≤ i ≤ d − 1} be an orthonormal basis of the tangent space and let the unit normal be given by n with components nk . By definition we have ∇Γ hV˜ , ni = = =

d−1 X ∂hV˜ , ni i=1 d−1 X

∂τi " d ∂ X

∂τi k=1 i=1 " d−1 X d X ∂ V˜k i=1

k=1

# V˜k nk τi

# ∂n k nk + V˜k τi . ∂τi ∂τi

3 Shape Calculus

18

According to remark 3.20, the variation of the normal in tangent directions is perpendicular to the normal and with the particular choice of V˜ , the second part vanishes. This results in ∇Γ hV˜ , ni = =

d−1 X d X ∂ V˜k

nk τi

∂τi i=1 k=1 (DΓ V˜ )T n =

−dn[V˜ ].

One can now use the tangential Stokes formula to perform an integration by parts on surfaces to arrive at a more convenient expression for the shape derivative of the normal: Lemma 3.23 (Tangential Stokes Formula). Let g be a real valued differentiable function on Γ and v be a differentiable vector valued function on Γ. Then the following relation holds: Z Z gdivΓ v + h∇Γ g, vi dS = κ g hv, ni dS Γ

Γ

Proof. Due to lemma 3.15 the following holds for a sufficiently smooth perturbation field V : · ¸ Z Z ∂g + g κ dS. h∇g, V i + gdivΓ V dS = hV, ni ∂n Γ Γ ∂g Using ∇Γ g = ∇g − ∂n n and V = v one arrives at the desired expression.

The Hadamard formula for the variation of the normal is thus given by Lemma 3.24 (Hadamard Formula of the Shape Derivative of the Normal). ∂g exists, let the For a sufficiently smooth vector valued function g such that ∂n objective be given by Z J(Γ) := hg, ni dS. Γ

Then the shape derivative of the above expression is · ¸ Z ∂g dJ(Γ)[V ] = hV, ni n + divΓ g dS, ∂n Γ i.e. the shape derivative of the normal fulfills hg, dn[V ]i = divΓ g − κhg, ni.

3 Shape Calculus

19

Proof. The product rule and lemma 3.15 give · ¸ Z ∂g dJ(Γ)[V ] = hV, ni n + κhg, ni + hg, dn[V ]i dS ∂n Γ ¸ · Z ∂g n + κhg, ni + hg, − (DΓ V )T ni dS = hV, ni ∂n Γ Let V˜ := hV, nin be the perpendicular component of V . Applying lemma 3.22 results in · ¸ Z ∂g ˜ ˜ dJ(Γ)[V ] = hV , ni n + κhg, ni + hg, −∇Γ hV˜ , nii dS ∂n Γ and furthermore lemma 3.23 gives Z Z ˜ hg, −∇Γ hV , nii dS = hV˜ , ni [divΓ g − κhg, ni] dS. Γ

Γ

Using the same argumentation as in the proof of lemma 3.15, i.e. the tangential component of V does not change the shape and hV˜ , ni = hV, ni, one can see that the above also holds for arbitrary smooth perturbation fields V , which gives the desired formula.

3.5

Shape Derivatives under a State Constraint

In the presence of a state constraint, i.e. Z Z min J(u, Ω) := f (u) dA + g(u) dS (u,Ω)

Ω

subject to L(u) = uf in Ω Lb (u) = ub on Γ

Γ

where f and g do not depend on the geometry Ω (respective Γ), the chain rule immediately results in ¸ · Z ∂g(u) + κg(u) dS + dJ(u, Ω) := hV, ni f (u) + ∂n Γ Z Z ∂f (u) 0 ∂g(u) 0 + u [V ] dA + u [V ] dS ∂u ∂u Ω

subject to L(u) = uf in Ω ∂L(u) 0 u [V ] = 0 in Ω ∂u

Γ

3 Shape Calculus

20

which does not yet fulfill the Hadamard form. The Hadamard form for such a problem can now be found by the adjoint approach. Crucial for the adjoint approach is knowledge of the boundary conditions of the linearized problem which determines the local shape derivative u0 [V ] of the state. Definition 3.25 (Material Derivative, Local Derivative). Let ut solve the PDE constraint on the perturbed domain Ωt = Tt (Ω) and let xt := Tt (x) be a shifted boundary point. The material derivative is then defined as the total derivative du[V ](x) :=

d ut (xt ) dt t=0

and the local shape derivative is defined as the partial derivative u0 [V ](x) :=

d ut (x). dt t=0

A straight forward linearization of the PDE boundary conditions usually results in an expression for the material derivative. The general strategy when deriving shape derivatives is to first transfer the problem back to the original boundary before computing the limit, resulting in the need to compute the local shape derivative. The chain rule combines both by the relation: du[V ] = u0 [V ] + h∇u, V i. Thus, if the right hand side of the boundary condition does not depend on the geometry, one has dub [V ] = h∇ub , V i. Lemma 3.26 (Shape Derivative of the Dirichlet Boundary Condition). Suppose the state u is given as the solution of a PDE of the form L(u) = uf in Ω u = ub on ∂Ω, such that uf and ub do not depend on the geometry of Ω, e.g. the unit normal n, etc. The local shape derivative under the perturbation V is then given as the solution of the problem ∂L(u) 0 u [V ] = 0 in Ω ∂u ∂(ub − u) u0 [V ] = hV, ni on Γ ∂n where Γ is the variable part of the boundary of ∂Ω.

3 Shape Calculus

21

Proof. The linearization in Ω is straight forward. Taking the total derivative of the boundary condition results in du[V ] = dub [V ] on Γ. Using definition 3.25 the above can be transformed to u0 [V ] + h∇u, V i = du[V ] = dub [V ] = h∇ub , V i ⇒ u0 [V ] = h∇ (ub − u) , V i . The usual orthogonality argument gives the desired expression µ ¶ ∂ (ub − u) 0 u [V ] = hV, ni . ∂n

Lemma 3.27 (Shape Derivative of the Slip Boundary Condition). The slipboundary condition is often encountered in fluid dynamics, especially when an inviscid fluid is modeled: hu, ni(x) = 0 on Γ. The local shape derivative then satisfies the boundary condition hu0 [V ], ni = −hDuV, ni − hu, dn[V ]i À ¶ µ ¿ ∂u , n + hu, ∇Γ hV, nii = hV, ni − ∂n where the second part of the identity holds for a perturbation in normal direction only. Proof. The derivation is analog to lemma 3.26 using the product rule and the extension of definition 3.25 for a vector valued state u: du[V ] = u0 [V ] + hDu, V i.

Lemma 3.28 (Shape Derivative of the Neumann Boundary Condition). Suppose the state u is given as the solution of a PDE of the form L(u) = uf in Ω ∂u = ub on ∂Ω, ∂n

3 Shape Calculus

22

such that uf and ub do not depend on the geometry of Ω, e.g. the unit normal n, etc. The local shape derivative under the perturbation V is then given as the solution of the problem ∂L(u) 0 u [V ] = 0 in Ω ∂u ∂u0 [V ] = h∇ub , V i − hD2 uV, ni − h∇Γ u, dn[V ]i ∂n ¸ · ∂ub ∂ 2 u − 2 + h∇Γ u, ∇Γ hV, nii = hV, ni ∂n ∂n where the second identity holds for the orthogonal component of the perturbation field only. Proof. The Neumann boundary condition at xt = Tt (x) on the deformed domain Ωt reads ub ◦ xt = h∇ut , nt i ◦ xt = h∇ut , nt i ◦ Tt (x) = h(∇ut ) ◦ Tt (x), nt (xt )i. The chain rule results in ∇(ut ◦ Tt (x)) = ((∇ut ) ◦ Tt (x))T · DTt (x) = (DTt (x))T · [(∇ut ) ◦ Tt (x)] and the boundary condition becomes ub (xt ) = h(DTt (x))−T ∇(ut ◦ Tt (x)), nt (xt )i = (∇(ut (xt )))T DTt (x)−1 · nt (xt ). The total derivative with respect to t now yields the material derivative of ut (xt ). Using lemma 3.10 results in: dub [V ] = (∇du[V ])T n + (∇u)T (−DV ) n + h∇u, dn[V ]i, which results in ∂du[V ] = dub [V ] − h∇u, (−DV ) ni − h∇u, dn[V ]i. ∂n Using the relationship du[V ] = u0 [V ] + h∇u, V i dub [V ] = h∇ub , V i dn[V ] = −∇Γ hV, ni

3 Shape Calculus

23

we have ∂du[V ] ∂u0 [V ] = + hD2 uV, ni + h∇u, DV ni ∂n ∂n and the above can now be expressed in terms of the local shape derivatives: ∂u0 [V ] = h∇ub , V i − hD2 uV, ni − h∇u, dn[V ]i. ∂n Since h∇u, ni = 0, we have ∇u = ∇Γ u and with the usual orthogonality argument the boundary condition can be expressed as · ¸ ∂ub ∂ 2 u ∂u0 [V ] = hV, ni − 2 + h∇Γ u, ∇Γ hV, nii ∂n ∂n ∂n where the last part can be brought into Hadamard form using lemma 3.23. Remark 3.29. For the standard Laplace problem −∆u = uf in Ω ∂u = ub on ∂Ω, ∂n the local shape derivative can be further refined. The Laplace-Beltrami operator ∆Γ u := divΓ ∇Γ u = ∆u − κ

∂u ∂ 2 u − ∂n ∂n2

provides ∂ 2u = −∆Γ u − uf − κub ∂n2 which results in ∂u0 [V ] = divΓ (hV, ni∇Γ u) + hV, ni ∂n

µ

∂ub + κub + uf ∂n

¶ .

Instead of conducting the adjoint calculus in a general setting, we now return to the Navier-Stokes problem.

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

4

24

Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

According to section 3, a formal shape differentiation of (2.1) immediately yields dJ(u, p, Ω)[V ] = Z hV, nif (u, Du, p) dS

(4.1)

Γ0

! Ã d ! Z ÃX d X ∂f ∂u0 [V ] ∂f 0 ∂f 0 i + ui [V ] + + p [V ] dA ∂ui ∂aij ∂xj ∂p i=1 i,j=1 Ω Z + hV, ni [h∇g(u, Dn u, p, n), ni + κg(u, Dn u, p, n)] dS

(4.2) (4.3)

Γ0

! ! Ã d Z ÃX d X ∂g ∂u0 [V ] ∂g 0 ∂g i + ui [V ] + nj + p0 [V ] dS ∂ui ∂bi ∂xj ∂p i=1 i,j=1

(4.4)

Z X d ∂g + dni [V ] dS ∂n i i=1

(4.5)

Γ0

Γ0

where u0 [V ] and p0 [V ] are given as the solution of the linearized Navier-Stokes equations −µ∆u0 [V ] + ρ (u0 [V ]∇u + u∇u0 [V ]) + ∇p0 [V ] = 0 in Ω div u0 [V ] = 0. The boundary condition on Γ0 is given by lemma 3.26. Since the other boundaries are considered fixed, one does not have to consider differences between the material and the local shape derivative and a linearization is straight forward: u0i [V ] = −hV, ni

∂ui on Γ0 ∂n

u0i [V ] = 0 on Γ+ p0 [V ]ni − µh∇u0i [V ], ni = 0 on Γ−

(4.6) (4.7) (4.8)

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

25

Lemma 4.1. For a sufficiently smooth, arbitrary λ and λp the relation " Ã d ! # Z X d X ∂λj ∂λi ∂λp 0 (4.9) 0= −µ∆λi − ρ uj + uj − ui [V ] dA ∂xi ∂xj ∂xi i=1 i,j=1 Ω

Z X d ∂λi 0 − p [V ] dA ∂xi i=1 Ω " # Z X d d X ∂λi + µ +ρ (λj uj ni + λi uj nj ) u0i [V ] dS ∂n i=1 j=1 Γ

Z λp

+

d X

u0i [V

]ni dS +

i=1

Γ

Z X d Γ

0

λi ni p [V ] dS +

i=1

Z X d Γ

i=1

(4.10) (4.11) −µλi

∂u0i [V ] dS ∂n (4.12)

holds. Proof. Multiplying the volume part of the linearized Navier-Stokes equations with an arbitrary λ and λp results in " Ã d ! # Z X d 0 0 X ∂ui ∂u [V ] ∂p [V ] λi −µ∆u0i [V ] + ρ u0j [V ] 0= + uj i + dA ∂xj ∂xj ∂xi i=1 j=1 Ω Z + λp div u0 [V ] dA. Ω

Integration by parts gives Z X d Ω

−µλi ∆u0i [V

] dA =

i=1

Z X d Γ

+

i=1

µ ¶ ∂u0i [V ] ∂λi 0 −µ λi − ui [V ] dS ∂n ∂n

Z X d Ω

−µu0i [V ]∆λi dA

i=1

and likewise due to div u0 [V ] = 0: Z X d Ω i,j=1

λi u0j [V

∂ui dA = ] ∂xj

Z X d Γ

i,j=1

λi u0j [V

Z X d ∂λi 0 ]ui nj dS − uj [V ]ui dA. ∂x j i,j=1 Ω

Note that in the above equation the index on the local shape derivative is j and not i. To derive the desired expression, the indices i and j must be

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

26

switched. Integration by parts on the second part of the linearized convection results in Z X Z X Z X d d d ∂λi ∂u0i [V ] 0 λi uj λi uj ui [V ]nj dS − dA = uj u0i [V ] dA. ∂xj ∂xj i,j=1 i,j=1 i,j=1 Ω

Ω

Γ

The pressure variation provides Z X Z Z X d d ∂λi 0 ∂p0 [V ] 0 dA = λi ni p [V ] dS − p [V ] dA λi ∂x ∂x i i i=1 i=1 Γ

Ω

Ω

and the divergence constraint provides Z Z X Z d d d X X ∂u0i [V ] ∂λp 0 0 λp ui [V ]ni dS − dA = λp u [V ] dA. ∂xi ∂xi i i=1 i=1 i=1 Ω

Ω

Γ

Summarizing the above creates the desired expression. It is now possible to state the adjoint right hand side in the volume Lemma 4.2 (Adjoint Right Hand Side, Volume). The adjoint equation must fulfill in the domain Ω: ¶ d d µ X X ∂λj ∂λi ∂λp ∂f ∂ ∂f uj + uj − = − µ∆λi − ρ ∂xi ∂xj ∂xi ∂ui j=1 ∂xj ∂aij j=1 div λp =

∂f . ∂p

Proof. Due to equations (4.9) - (4.12) summing to zero, they can be added to the preliminary gradient (4.1) - (4.5). Integration by parts on equation (4.2) yields ! Ã d ! Z ÃX d X ∂f ∂u0 [V ] ∂f 0 ∂f 0 i ui [V ] + + p [V ] dA ∂ui ∂aij ∂xj ∂p i,j=1 i=1 Ω

Z X d ∂f 0 = u [V ]nj dS ∂aij i i,j=1 Γ Ã ! Z X Z d d X ∂f ∂f 0 ∂ ∂f 0 + − ui [V ] dA + p [V ] dA ∂ui j=1 ∂xj ∂aij ∂p i=1 Ω

(4.13) (4.14)

Ω

and a direct comparison between the above and equations (4.9) and (4.10) reveals the required adjoint right hand side in Ω. Note that this has introduced a new boundary term.

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

27

Lemma 4.3 (Adjoint Boundary Condition at Inflow). The adjoint boundary condition on the inflow boundary Γ+ is given by λ=0 λp free. Proof. Since the inflow velocity is fixed and independent of the shape of the fluid obstacle, we have u0 [V ] = 0 on Γ+ . Hence, the only term appearing on Γ+ is the normal variation of u0 [V ] and the pressure variation p0 [V ] from equation (4.12): Z X d

0

λi ni p [V ] dS +

Z X d

−µλi

Γ+ i=1

Γ+ i=1

∂u0i [V ] dS ∂n

which is removed by λ = 0 on Γ+ . Lemma 4.4 (Adjoint Boundary Condition at No-Slip). The adjoint boundary condition on the no-slip boundary Γ0 is given by 1 ∂g ∀i = 1, . . . , d µ ∂bi ∂g hλ, ni = − ∂p λp free. λi =

Proof. The sensitivities on Γ0 are equations (4.13), (4.4), (4.11), and (4.12): Z X d ∂f 0 ui [V ]nj dS ∂a ij Γ0 i,j=1 ! Ã d ! Z ÃX d 0 X ∂g 0 ∂g ∂ui [V ] ∂g + ui [V ] + + p0 [V ] dS ∂ui ∂aij ∂xj ∂p i=1 i,j=1 Γ0 # " Z X d d X ∂λi + µ +ρ (λj uj ni + λi uj nj ) u0i [V ] dS ∂n i=1 j=1 Γ0

Z

+

λp Γ0

d X i=1

u0i [V

]ni dS +

Z X d Γ0

i=1

0

λi ni p [V ] dS +

Z X d Γ0

i=1

−µλi

∂u0i [V ] dS. ∂n

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

28

Using the no-slip boundary condition and the boundary condition for the local shape derivative, the above transforms to " d à ! # Z d X ∂g X ∂λi ∂ui ∂f hV, ni − +µ + λp ni + nj dS ∂ui ∂n ∂aij ∂n i=1 j=1 Γ0 ! à ! Z ÃX d d ∂g ∂u0i [V ] ∂g X + + + λi ni p0 [V ] dS ∂b ∂n ∂p i i=1 i=1 Γ0

+

Z X d

Γ0

i=1

−µλi

∂u0i [V ] dS, ∂n

where the first part now also enters the gradient (4.1) - (4.5). Expressing ∇ui in local coordinates on the boundary results in d X ∇ui = h∇ui , nin + h∇ui , τj iτj , j=1

hence X ∂ui ∂ui ∂ui nj ⇒ 0 = λp ni = ∂xj ∂n ∂n i=1 d

due to the mass conservation on Γ0 . Hence, λp does not receive a boundary condition. The remaining sensitivities can be eliminated by 1 ∂g ∀i = 1, . . . , d µ ∂bi ∂g hλ, ni = − . ∂p λi =

Using lemma 4.2, 4.3, and lemma 4.4, the preliminary gradient simplifies

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

29

to dJ(u, p, Ω)[V ] = Z hV, nif (u, Du, p) dS

(4.15)

Γ0

Z

hV, ni [h∇g(u, Dn u, p, n), ni + κg(u, Dn u, p, n)] dS

+ Γ0

"

Z

hV, ni −

+

d X i=1

Γ0

Ã

∂g ∂λi X ∂f +µ + nj ∂ui ∂n ∂aij j=1 d

!

∂ui ∂n

(4.16)

#

Z X d ∂g dni [V ] dS. + ∂n i i=1

dS

(4.17)

(4.18)

Γ0

Lemma 4.5 (Adjoint Boundary Condition at Outflow). The adjoint boundary condition on the outflow boundary Γ− is given by à d ! X ∂λi +ρ µ λj uj ni + λi uj nj + λp ni = 0. ∂n j=1 Proof. Inserting equation (4.8) into equations (4.9) - (4.12), the remaining sensitivity is " # Z X d d X ∂λi µ +ρ (λj uj ni + λi uj nj ) u0i [V ] dS ∂n i=1 j=1 Γ−

Z

+ Γ−

λp

d X

u0i [V ]ni dS.

i=1

Hence, the required boundary condition is ! Ã d X ∂λi µ λj uj ni + λi uj nj + λp ni = 0. +ρ ∂n j=1

4 Shape Derivative and Adjoint Calculus for the general Navier-Stokes Problem

30

Lemma 4.6 (Shape Derivative for the General Navier-Stokes Problem). The shape derivative in Hadamard form for the problem under consideration is given by dJ(u, p, Ω)[V ] = Z hV, nif (u, Du, p) dS Γ0

Z

hV, ni [h∇g(u, Dn u, p, n), ni + κg(u, Dn u, p, n)] dS

+ Γ0

"

Z

hV, ni −

+

i=1

Γ0

Z

+

d X

Ã

∂g ∂λi X ∂f +µ + nj ∂ui ∂n ∂a ij j=1 d

!

∂ui ∂n

# dS

hV, ni [divΓ ∇n g − κh∇n g, ni] dS Γ0

where ∇n denotes the vector consisting of components and p solve the incompressible Navier-Stokes equations −µ∆u + ρu∇u + ∇p = ρG div u = 0 u = u+ u=0 ∂u pn − µ =0 ∂n

∂g . ∂ni

in

Ω

on on

Γ+ Γ0

on

Γ−

Furthermore, u

and λ and λp solve the adjoint incompressible Navier-Stokes equations µ∆λi − ρ

d µ X ∂λj j=1

∂λi uj + uj ∂xi ∂xj

¶

X ∂ ∂f ∂f ∂λp = − ∂xi ∂ui j=1 ∂xj ∂aij d

−

div λp =

∂f ∂p

in Ω

5 Example Application

31

with boundary conditions λ=0 on Γ+ 1 ∂g λi = on Γ0 µ ∂bi ∂g hλ, ni = − on Γ0 ∂p

à d ! X ∂λi µ +ρ λj uj ni + λi uj nj + λp ni = 0 ∂n j=1

on Γ− .

Proof. The adjoint boundary conditions are derived in lemma 4.3, 4.4, and 4.5. The adjoint right hand side is derived in lemma 4.2 and removing the shape derivative of the normal follows exactly lemma 3.24.

5

Example Application

As an example application we consider the optimization of a channel filled with water in two dimensions. The objective is to minimize the dissipation of kinetic energy into heat, given by f (u, Du, p) = µ

¶2 2 µ X ∂ui i,j=1

∂xj

g(u, Dn u, p, n) = 0 which results in ∂f ∂ui = 2µaij = 2µ ∂aij ∂xj ∂f = 0. ∂ui According to lemma 4.6, the adjoint equation is given by µ∆λi − ρ

d µ X ∂λj j=1

∂λi uj + uj ∂xi ∂xj

¶

X ∂ ∂f ∂λp ∂f − = − ∂xi ∂ui j=1 ∂xj ∂aij d

= −2µ∆ui ∂f div λp = =0 ∂p

in Ω

5 Example Application

32

with boundary conditions λ=0 on Γ+ 1 ∂g λi = = 0 on Γ0 µ ∂bi ∂g hλ, ni = − = 0 on Γ0 ∂p

à 2 ! X ∂λi +ρ µ λj uj ni + λi uj nj + λp ni = 0 ∂n j=1

on Γ− .

Both conditions on Γ0 are satisfied by λ = 0 and consequently the gradient is given by # " 2 µ Z X ∂ui ¶2 dS dJ(u, p, Ω)[V ] = hV, ni µ ∂x j i,j=1 Γ0 " 2 Ã # ! Z 2 X ∂λi X ∂f ∂ui + hV, ni − µ + dS nj ∂n ∂a ∂n ij i=1 j=1 Γ0 " # ¶2 Z 2 µ X ∂ui = hV, ni µ dS ∂n i=1 Γ0 ! # " 2 Ã Z 2 X ∂ui ∂λi X ∂ui + 2µ nj dS + hV, ni − µ ∂n ∂x ∂n j j=1 i=1 Γ0 " µ ¶2 # Z 2 X ∂ui ∂λi ∂ui + dS. = hV, ni −µ ∂n ∂n ∂n i=1 Γ0

5.1

Flow Solver

The Navier-Stokes equations are discretized by mixed Taylor-Hood finite elements. The velocities are discretized using a 6 node triangular element with quadratic ansatz functions while the pressure exists on standard 3 node triangular elements using linear ansatz functions. The non-linear convection operator in the momentum equation creates a set of non-linear discrete equations which are solved by an exact Newton method. Although a Picard linearization provides a larger radius of convergence for high Reynolds numbers, the Newton method has the advantage of easily creating a discrete exact adjoint solver by solving the Newton system once in transpose mode.

5 Example Application

33

In each Newton step for the forward problem, a system of the type ¸µ ¶ · δu A B∗ = −c(u, p) B 0 δp has to be solved, where A is the diffusion and linearized convection operator and B is the discrete divergence operator. Both A and B ∗ also contain the boundary conditions but not B, as the divergence freedom must be valid everywhere. Thus, the analytical correct boundary condition for λ is not created when the system is transposed, which makes manual intervention necessary to ensure the adjoint solver enforces the adjoint boundary condition λ = 0 precisely on Γ0 . Using the shape derivative appears to be somewhat sensitive at this point to a continuous versus discrete adjoint approach. To prevent a loss of smoothness in the shape, the discrete adjoint had to be adapted to satisfy the analytic boundary conditions precisely.

5.2

Flow Through a Pipe

As a first test case, we optimize the shape of a tube connecting two points. The geometry is shown in figure 5.1. The channel has a cross section of 1.0

fixed variable

variable fixed

Fig. 5.1: The initial tube. The area inside the stylized compartment is the unknown to be found, the remainder outside is considered fixed. Color denotes pressure. 1 and we set the viscosity to µ = 400 . The inflow velocity profile is parabolic up to a maximum value of u1 = 1.5. The fluid enters the channel on the bottom left edge. With a constant density of ρ = 1.0, the average mass influx results in a Reynolds number of 400 using the channel width as reference length. The Reynolds number is close to the maximum such that the Newton iteration

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converges, with the flow probably becoming instationary for higher Reynolds numbers. Each of the sharp bends create a strong stationary vortex in the

Fig. 5.2: Magnification oft the initial tube. Strong counter vortices develop after the bends. Color denotes speed. flow, with the streamlines shown in figure 5.2. As a consequence, the initial tube has a comparatively strong pressure gradient, resulting in a distinct pressure loss of the flow, see figure 5.1. As an optimization procedure, we

Fig. 5.3: Pressure distribution optimized tube. The pressure loss is almost completely removed. employ a straight forward steepest descent algorithm using a fixed step length and the shape derivative. The perturbation direction is chosen as V = αn, i.e. in each iteration each boundary node is shifted into the direction of the normal at that node. Thus, the boundary nodes follow a curved path during

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the optimization. In order to prevent a degeneration of the tube, there is also the constraint that the volume must be preserved, which is enforced by a projection step after each shape update. The optimized tube is shown in figure 5.3. Without a line search, the steepest descent algorithm manages to reduce the objective function very quickly at first, but slows down considerably afterwards. Since the main focus of this paper was to derive the general shape derivative for the Navier-Stokes equations, we would like to refer to [4, 5, 14] for higher order optimization schemes such as approximative SQP methods. The initial channel has a net loss of kinetic energy into heat of J = 0.9077, which is reduced to J = 0.4308, a reduction by 52.54%.

5.3

Flow through a T-Connection

We conclude with optimizing a T-junction. Inlet and outlet are positioned such that the flow enters on the bottom and must be redirected by 90◦ , leaving the pipe system on the left and right. The initial geometry is shown in figure 5.4 and the optimized in figure 5.5. A considerable amount of the channel after the actual fork is fixed, such that the optimization cannot circumvent a net turning of 90◦ after the fork and the flow has to exit parallel to the x-axis. The inflow profile is quadratic and the average mass influx results in a Reynolds number of Re = 100 with the channel cross-section as reference area. The initial junction has a net loss of kinetic energy into heat

Fig. 5.4: Inital T-connection. The fluid is entering on the bottom. The stylized box contains the variable part of the boundary. Its upper side is unbounded. Color denotes speed. of J = 0.9208, which is reduced to J = 0.6900, a reduction by 25.05%. The total area occupied by the fluid was enforced to stay the same during the whole optimization.

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Fig. 5.5: Optimized T-connection.

6

Summary

The main purpose of this work was to derive the Hadamard form of the shape derivative for a wide class of objective functions in a Navier-Stokes flow, especially also considering boundary integrals as the objective. The Hadamard form enables a very efficient computation of the gradient, since knowledge of the “mesh sensitivity” Jacobian is not required. Being an analytic expression, the Hadamard form must be re-derived for each problem unless a generic objective is considered. Due to the artificial nature of the pressure in an incompressible flow, some restrictions appear on the surface part of the objective function. Otherwise one cannot formulate a consistent adjoint equation, since the pressure does not have a boundary condition but is implicitly given such that mass is conserved. We also list many important literature results from shape analysis and geometry, such that the paper is self contained and can easily be adapted to other kind of problems. We conclude with an application of the shape derivative to the optimization of internal channel flows, where the energy loss of the flow - and consequently the pressure gradient along the pipe - could be reduced by more than 50%.

References ˇ Neˇcasov´a, and J. Sokolowski. Shape sensitivity anal[1] C. Amrouche, S. ysis of the neumann problem of the laplace equation in the half-space. Technical Report preprint 2007-12-20, Institute of Mathematics, AS CR, Prague, 2007. [2] S. Boisg´erault and J.-P. Zol´esio. Shape derivative of sharp functionals governed by navier-stokes flow. In W. J¨ager, J. Necas, O. John, K. Najzar, and J. Star´a, editors, Partial Differential Equations: Theory and

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Numerical Solution, pages 49–63. Chapman & Hall/CRC Reseach Notes in Mathematics, 1993. [3] M. C. Delfour and J.-P. Zol´esio. Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM Philadelphia, 2001. [4] K. Eppler and H. Harbrecht. A regularized newton method in electrical impedance tomography using shape hessian information. Control and Cybernetics, 34(1):203–225, 2005. [5] K. Eppler, S. Schmidt, V. Schulz, and C. Ilic. Preconditioning the pressure tracking in fluid dynamics by shape Hessian information. Technical Report Preprint-Nr.: SPP1253-10-01, DFG-SPP 1253, 2007. to appear in JOTA, April 2009. [6] M.D. Gunzburger. Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM Philadelphia, 2003. [7] K. Ito, K. Kunisch, G. Gunther, and H. Peichl. Variational approach to shape derivatives. Control, Optimisation and Calculus of Variations, 14:517–539, 2008. [8] B. Mohammadi and O. Pironneau. Applied Shape Optimization for Fluids. Numerical Mathematics and Scientific Computation. Clarendon Press Oxford, 2001. [9] O. Pironneau. On optimum profiles in stokes flow. Journal of Fluid Mechanics, 59(1):117–128, 1973. [10] P. I. Plotnikov, E. V. Ruban, and J. Sokolowski. Inhomogeneous boundary value problems for compressible navier-stokes equations: Wellposedness and sensitivity analysis. Journal on Mathematical Analysis, 40(3):1152–1200, 2008. [11] P. I. Plotnikov and J. Sokolowski. On compactness, domain dependence and existence of steady state solutions to compressible isothermal navierstokes equations. Journal of Mathematical Fluid Mechanics, 7(4):529– 573, 2005. [12] P. I. Plotnikov and J. Sokolowski. Stationary boundary value problems for compressible navier-stokes equations. Handbook of Dierential Equations, 6:313–410, 2008.

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[13] S. Schmidt, C. Ilic, N. Gauger, and V. Schulz. Shape gradients and their smoothness for practical aerodynamic design optimization. Technical Report Preprint-Nr.: SPP1253-10-03, DFG-SPP 1253, 2008. submitted (OPTE). [14] S. Schmidt and V. Schulz. Impulse response approximations of discrete shape Hessians with application in cfd. Technical Report Preprint-Nr.: SPP1253-10-02, DFG-SPP 1253, 2008. submitted (SICON). [15] J. Sokolowski and J.-P. Zol´esio. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, 1992.