Γ∗O
[p∗effδik−2µ∗effSik∗]
nkridΓ∗. (3.7) b) The force objective can also be examined by the momentum loss inside the compu-tational domain, cf. Fig. 3.1 (b). The latter is reformulated via integration by parts and evaluated along the far-field boundaries, namely
J∗Γ,E=−
∫
Γ∗O
[vi∗ρ∗vk∗+p∗effδik−2µ∗effSik∗]
nkridΓ∗. (3.8)
Inverse Concentration Objective
The inverse concentration objective aims at minimizing the deviation of the current con-centration fieldcw.r.t. a given target concentration distribution ctar
J∗Ω=
∫
Ω∗O
1 2
[c−ctar]2
dΩ∗. (3.9)
The deviation is measured in the least-square sense over a prescribed spatial region Ω∗O.
3.3 Constraint Shape Derivatives
The cost functionals of the previous section are controlled via parts of the walled flow domain. Fundamental relationships of fluid mechanical shape optimization are considered in the following, and the control is consistently considered as a vector-valued boundary coordinateu∗i from now on. Hence, an arbitrary shape Γ∗is described by an infinite number of surface points u∗i. Note that only the normal component of a surface perturbation influences a shape derivative, viz. δu∗i =δu∗ni=δn∗ni=δn∗i orδn∗ =δu∗i ni, also known as the structure theorem of Hadamard-Zol´esio, cf. Hadamard [1968], Delfour and Zol´esio [2011].
Two approaches to obtain and interpret a shape derivative are presented below. The first is based on a compareable intuitive engineering approach and follows the concept of material derivative. Subsequently, a somewhat more general approach based on the general shape calculus is discussed.
Concept of Material Derivative
A disturbance δu∗i is applied at any point along the surface of an Old Shape u∗i resulting in a New Shape u˜∗i =u∗i +δn∗i. The applied disturbance induces a change in the stateδφ∗ which in turn yields a New State φ˜∗ =φ∗+δφ∗ due to the New Shape. In line with Soto and L¨ohner [2004], Othmer [2008], Schmidt and Schulz [2009], St¨uck [2012], the latter can be approximated to second-order, viz.
˜
φ∗(˜u∗i) = φ∗+ [
δ+δn∗ ∂
∂ n∗ ]
φ∗+O(δn∗2). (3.10) To obtain a shape sensitivity, a quantification of the New State due to a New Shape is of interest. While direct methods explicitly evaluate theNew State due to (each)New Shape, the indirect adjoint approach allows the desired quantification of the New State on the Old Shape, since the variation of the flow state is not explicitly evaluated, cf. Sec. 3.1.1.
Accordingly, both direct and indirect methods evaluate the bracketed expression in (3.10) [
δ+δn∗ ∂
∂ n∗ ]
→ [
δl+δc+δg ]
(3.11) which is frequently denoted as a material derivative in terms of local (δl) as well as con-vective and –possibly– geometric (δn∗(∂/∂ n∗)) contributions, i.e. δl+δc +δg. A local perturbation follows from a local change in the state, a convective change accounts for a surface-normal displacements and a geometrical variation considers potential changes of the surface area and orientation, cf. St¨uck [2012]. Note that the surface perturbationδn∗ is zero everywhere except along the design boundaries Γ∗Dwhich likewise holds forconvective and geometric contributions. However, local variations have to be evaluated everywhere.
Expression (3.11) resembles the non-conservative, substantial derivative from Cha. 2, cf. Eqn. (2.2). In this analogy, the disturbance has velocity character, as discussed in the upcoming section. The outlined control perturbation based on the distinction between old/new shape/state allows the systematic application of the material derivative approach to the objective functional (δJ∗), its PDE constraints (δR∗φ), and their boundary condi-tions (δB∗φ). The former and the latter are relevant for this thesis and therefore performed in Sec. 3.4. Local variations of the field equations are efficiently circumvented via an ad-joint strategy. First, an alternative approach to determining shape derivatives is presented, which follows the mathematically more rigorous concept of shape calculus.
Shape Calculus
The transformationT∗ from an initial towards an alternative shape – e.g. for optimization purposes – can generally be performed in a design-time τ∗, viz.
Γ∗τ :={T∗τ(u∗i0) :u∗i0 ∈Γ∗0, τ∗ ≥0} with T∗0(u∗i0) =u∗i0. (3.12)
Compared to the physical time t∗, the design-time is a parent or superior quantity. The design-temporal change of all surface coordinates can be described via the so-called speed method:
du∗iτ
dτ∗ =Vi∗τ(u∗iτ) with τ∗ ≥0, (3.13) where the velocity field Vi∗τ is often denoted as design velocity, which therefore defines the units of Vi∗τ and τ∗. The mapping T∗ in Eqn. (3.12) can be interpreted as a forward integration following the gradientG∗τi in design time:
T∗τ(u∗0i ) :=u∗0i +τ∗G∗τi =u∗τi with τ∗ ≥0, (3.14) frequently labeled as perturbation of identity which is equivalent to the speed method for first-order calculus, cf. Schmidt et al. [2013]. Combining Eqns. (3.13) and (3.14) yields dT∗τ/dτ∗ =G∗τi = du∗τi /dτ∗ =Vi∗τ. The aim is now to connect the design velocity with derivative information of a volume J∗Ω or surface J∗Γ cost functional (cf. Eqn. (3.1) and Sec. 3.2). The shape derivative based on the introduced design velocity is obtained by Hadamard’s boundary variation method for volume and surface integrals, cf. App. B.2 and Hadamard [1968], Delfour and Zol´esio [2011], Delfour and Zol´esio [2011], Sokolowski and Zol´esio [1992], Allaire et al. [2021]. The determination of a shape derivative is individual and must be determined anew for each cost functional and state constraint. Hence, the following discussion is not exhaustive and aims at building a bridge towards the previous section.
In line with Allaire et al. [2021], a Lagrangian approach is employed L∗(φ∗(u∗τi ),φˆ∗, u∗τi ) = J∗(φ∗(u∗τi ), u∗τi ) +
∫ ˆ
φ∗R∗φ(φ∗(u∗τi ), u∗τi ) dΩ∗, (3.15) where R∗φ contains the general residual of the physics to be fulfilled and ˆφ∗ represents the associated adjoint variable that does not depend on the state and the design time, cf. its discrete analogue (3.5). The derivation of the shape sensitivity employs first-order optimality conditions to ensure the independence of (3.15) w.r.t. the primal and adjoint state. Subsequently, descent information can be obtained from remaining variational con-tributions. An exemplary laminar single-phase discussion is sketched in App. B.3 for the boundary-based force objectives (3.7)-(3.8) without particularly addressing the adjoint flow. The latter is intensively considered in the upcoming Cha. 4. The following shape derivative rule frequently appears (cf. Allaire et al. [2021]) and is therefore interpreted as a generalized total shape derivative expression, viz.
d
dτ∗L∗(φ∗(u∗iτ),φˆ∗, u∗iτ)
⏐⏐
⏐⏐
τ∗=0
=
∫
Γ∗D
[ s∗
]
δl
Vk∗0nkdΓ∗+
∫
Γ∗D∩Γ∗O
[ κ∗j∗Γ
]
δc+δg
Vk∗0nkdΓ∗, (3.16) where the three parts are denoted as local, convective and geometric derivative, respec-tively, cf. Schmidt [2010], St¨uck [2012], Schmidt et al. [2013]. The local component refers to the adjoint-based shape sensitivity derivative and typically inheres the inner product between primal and adjoint gradient normal to the controlled wall, i.e. s∗ = δlj∗Γ =
−(∂φˆ∗∂ n∗)(∂ φ∗/∂ n∗), cf. Allaire et al. [2021]. Convective and geometric changes are combined in a mean curvature expression, cf. Eqn. (B.6). The applied shape calculus allows the specification of boundary conditions for the design velocity Vk∗τ at τ∗ = 0. A natural choice refers to Vk∗0 = 0 along Γ∗\Γ∗D. Note that j∗Γ = 0 along Γ∗\Γ∗O so that a distinction between the habitat of the objective functional Γ∗O and the control Γ∗D is necessary: both the convective as well as the geometric contribution need to be evaluated along the intersected area Γ∗D∩Γ∗O ̸=∅. The crucial point now is: the complexity of the objective functional derivative expression is reduced by decoupling control and objective surfaces, as geometric as well as convective contributions to the total shape derivative dis-appear due to the unperturbed control, and only the local contribution remains. Section 3.4 outlines differences between an internal and external force evaluation approach from an adjoint perspective, which leads to a simplification of the total shape derivative (3.16).