2.6 Shape calculus and calculus on manifolds
2.6.2 Abstract view on shape calculus
Therefore, up next a very brief crash course in first and second order calculus on smooth (i. e. C∞-) manifolds, which is based on [1, Chp. 3 and 5]. It is far beyond the scope of this thesis either to give a sufficient review of this topic or to prove that the results thereof can be carried over to shape calculus directly. In particular, the manifoldOis notC∞if it is equipped with the structure, which was constructed inParagraph 2.6.1, since the elements ofΘare notC∞ (cf. the6thitem of the discussion above). The aim is to give some insight into the structure of calculus on manifolds and thereby to establish some understanding of shape calculus, without having to cope with technical details.
The first intermediate goal is the notion of a derivative of a smooth function on a manifold. The main difficulty is the fact that a direct generalization of directional derivatives in Banach spaces (i. e.Gateaux semiderivative, [44, Chp. 9 Def. 2.1(i)]) as
d f(x,ξ):=lim
t→0
f(x+tξ)−f(x) t
2.6.2 Abstract view on shape calculus 69 is not possible, since the argumentx+tξhas no meaning due to the lack of a linear vector space structure on a manifold. As a consequence, directional derivatives are constructed by means of curves on the manifold in manner ofHadamard semiderivatives; cf. [44, Chp. 9 Def. 2.1(ii)].
Definition 8(directional derivative, tangent vector, tangent space):
LetMbe a (real) manifold, letFx(M)be the set of real-valued, smooth functions defined in a neighbor-hood ofx∈ M. Furthermore, letα:R→Mbe a smooth mapping withα(0) =x, calledsmooth curve in Mtrough x. Then
1. dtd f(α(t))|t=0is called thedirectional derivative of f ∈Fx(M)in x
2. the map ˙α(0):Fx(M)→R, f 7→α˙(0)f := dtd f(α(t))|t=0is calledtangent vectororcanonical lifting of the curveαat t=0
3. atangent vectorξx to the manifoldMat point xis defined as mappingFx(M) →Rsuch that there exists a curveαtroughx(α(0) =x) with
ξx f =α˙(0)f
4. the setTxMof all tangent vectorsξxat pointxis calledtangent space toMat xand admits a linear vector space structure.
Definition 9(tangent bundle, vector field, derivation):
LetMbe a (real) manifold and letF(M)be the set of real-valued, smooth functions defined onM. Then 1. thetangent bundle TMis the collection of all tangent vectors toM27
TM:= [
x∈M
TxM
2. a smooth mappingξ:M →TM,x7→ξx∈TxMis calledvector field onM 3. the set of all vector fields onMis denoted byV(M)
4. aderivation at x∈ Mis defined as a mappingDx:F(M)→R, which fulfills a) R-linearity:∀a,b∈R, f,g∈F(M): Dx(a f+bg) =aDx(f) +bDx(g)and b) the product rule:∀f,g∈F(M): Dx(f g) =Dx(f)g+f Dx(g)
5. aderivation onF(M)is a mappingD:F(M)→F(M), which fulfills
a) R-linearity:∀a,b∈R, f,g∈F(M): D(a f +bg) =aD(f) +bD(g)and b) the product rule:∀f,g∈F(M): D(f g) =D(f)g+f D(g).
The notion of a derivation axiomizes the notion of vector fields and as a result of this it axiomizes the notion of the covariant derivative (cf.Definition 11). Each vector fieldξ∈V(M)defines a derivation
D:F(M)→F(M), f 7→D(f):=ξf, where (by means ofDefinition 8for a suitable curveα)
∀x∈ M: (ξf)(x):=ξxf =α˙(0)f = d
dtf(α(t))|t=0∈R.
Vice versa, each derivation onF(M)can be realized by a vector field. Consequently, it is sufficient to maintain the notion of a vector field and the notion of a derivation can be abandoned.
Definition 10(covector, cotangent space, cotangent bundle): LetMbe a (real) manifold. Then
1. acovector at x∈ Mis a linear functionalµx:TxM →R
2. the set of all covectors atx∈ Mform thecotangent space Tx∗MtoMat x, which is the dual space of the tangent spaceTxM
27Even if two tangent spacesTxMandTyM(x,y∈ M) are isomorphic, their elementsξxandξyare not identified with each other inTM. Consequently, each elementξx∈TMis characterized by the tangent vectorξx∈TxMitself and itsfoot x∈ M.
3. thecotangent bundle T∗Mis the collection of all covectors toM T∗M:= [
x∈M
Tx∗M.
Definition 11(covector field, covariant derivative of functions):
LetMbe a (real) manifold. Then
1. acovector fieldis a smooth mapµ:M →T∗M,x7→µx
2. the set of all covector fields is referred to asV(M)∗
3. a covector fieldµacts on a vector fieldξ∈V(M)as follows
∀x∈ M: (µ[ξ])(x):=µx[ξx]∈R and consequentlyµ[ξ]∈F(M)
4. for each f ∈F(M)there exists one distinct covector field, thecovariant derivative of f, defined by Df : M →T∗M,x 7→(Df[.])(x), where∀ξ∈V(M): (Df[ξ])(x):=ξxf.
The covariant derivative is the generalization of the common concept of the first order derivative of a function f :RN → R, and thus the first intermediate goal is reached. The next step is to provide tools needed for a generalization of a second order derivative.
Second order derivatives are based on the notion of the first order derivative of vector fields. However, it is not possible to use the (manifold intrinsic) concept of curves to define directional derivatives of vector fields, as it was done inDefinition 8in order to introduce directional derivatives of real-valued functions:
limt→0
ξα(t)−ξα(0) t
is not well-defined since the tangent vectorsξα(t)andξα(0)live in different tangent spaces. Consequently, one has to choose an axiomatic approach to introduce the notion of a covariant derivative of a vector field, which is similar to the axiomatic definition of derivations inDefinition 9.
Definition 12(covariant derivative of (co-)vector fields): LetMbe a (real) manifold. Then
1. acovariant derivative of a vector fieldξ ∈V(M)with respect to the directionη ∈V(M)is a mapping
∇ηξ:M →TM,x7→(∇ηξ)(x)which fulfills
a) F(M)-linearity inη:∀f,g∈F(M),η,χ,ξ∈V(M):∇fη+gχξ= f∇ηξ+g∇χξ b) R-linearity inξ:∀a,b∈R,η,ξ,ζ∈V(M):∇η(aξ+bζ) =a∇ηξ+b∇ηζ
c) the product rule:∀f ∈F(M),η,ξ∈V(M):∇η(fξ) = (ηf)ξ+ f∇ηξ,28 2. acovariant derivative of a covector fieldµ∈V(M)∗can be defined by
∀η,ξ∈V(M): (∇ηµ)[ξ]:=η(µ[ξ])−µ[∇ηξ] = (D(µ[ξ]))[η]−µ[∇ηξ].
Remark:
The notion of a covariant derivative of a vector field is closely related to the notion of anaffine connection on the manifold; cf., for instance, [1] or [113, p. 101ff.]. In fact each affine connection defines a covariant derivative and vice versa. Furthermore, it is known, that there are infinitely many affine connections on a manifold [1, p. 94]. One or another of them may distinguish itself with respect to computational accessibility or some other properties, for instance, theRiemannian/Levi-Civita connection. All in all, there is some freedom for the choice of a covariant derivative∇on a manifold. In particular, one has to choose, since there is none of them given apriori.
28Note, that there is an essential difference betweenfξandξfforf ∈F(M)andξ∈V(M):(fξ)(x) =f(x)ξx∈TxMis a simple multiplication, whereas(ξf)(x) =ξxf = (Df[ξ])(x)∈Ris the application of a directional derivative.
2.6.2 Abstract view on shape calculus 71 Let M be a (real) manifold. Furthermore, let f ∈ F(M) and let η ∈ V(M). Hence, the covariant derivative of f applied to the vector fieldηis
F:=Df[η]∈F(M)
and it is possible to derive the covariant derivative ofFitself by repeated differentiation. By doing so, it becomes apparent, that the definition of the covariant derivative of a covector field is chosen in such a way that the expected rules for differentiation are fulfilled. Thus, letξ ∈ V(M)be another vector and compute
DF[ξ] =D(Df[η])[ξ] = (∇ξ(Df))[η] +Df[∇ξη]. (2.80) This equation can be understood as product rule for the differentiation of the “product” Df[η]. It accounts for some inherent, general properties of derivatives; cf. [1, p. 95f.]: A directional derivative depends locally on the object to be differentiated and pointwisely on the direction towards which is differentiated.
Locally means, that the object has to be known in a local neighborhood around the point of evaluation.
In particular, it is not enough if one has information about the vector fieldηat some pointx∈ Monly in order to compute D(Df[η])[ξ](x), since it is not possible to derive Df[∇ξη](x)then.
Definition 13(second covariant derivative):
LetMbe a (real) manifold. Then thesecond covariant derivative∇2of a function f ∈F(M)is defined as
∀η,ξ∈V(M): ∇2f[ξ,η]:= (∇ξ(Df))[η].
By means of the notion of the second covariant derivative one recognizes that the second directional derivative D(Df[η])[ξ]of a function f ∈F(M)decomposes into two parts: the second covariant deriva-tive and one term which contains the (first order) covariant derivaderiva-tive.
Herewith, the second intermediate goal of the notion of a second order derivative is reached. However, the termsshape gradientandshape Hessianare frequently used in the context of shape calculus. Thus, it is worthwhile to introduce the notions of(Riemannian) gradientand(Riemannian) Hessianin the calculus on (Riemannian) manifolds in a third intermediate step.
Definition 14(Riemannian metric, Riemannian manifold):
Let M be a (real) manifold and let all tangent spaces TxMbe Hilbert spaces with an inner product (symmetric positive definite)
gx(., .) =h. , .ix:TxM ×TxM →R.
Then
1. the mappingg:x7→gxis called aRiemannian metric onM, if for all vector fieldsξ,ζ∈V(M)the
map M →R, x7→ hξx,ζxix
is smooth,
2. a pair(M,g)of a manifold and a Riemannian metric is called aRiemannian manifold.
Definition 15(gradient, Hessian):
Let(M,g)be a Riemannian manifold and let f ∈F(M)be a smooth function.
1. The gradientgradf of f (with respect to the metric g)is the Riesz representative of the covariant derivative Df. In other words, it is defined as the unique vector field that fulfills
∀x∈ M, ∀ξ∈V(M):
(gradf)(x),ξx
x= (Df[ξ])(x)
2. Let∇be an affine connection onM. Then theHessianHessf of f (with respect to the metric g and the affine connection∇)is the mapping
Hessf :V(M)→V(M),ξ7→ ∇ξgradf. This is the linear operator induced by the second covariant derivative
∀ξ,η∈V(M):
(Hessf)[ξ],η
=∇ξgradf ,η
=∇2f[ξ,η].
By means of gradient and Hessian the second directional derivative (2.80) can be expressed as follows D(Df[η])[ξ] =∇2f[ξ,η] +Df[∇ξη] =(Hessf)[ξ],η
+gradf ,∇ξη .
The last intermediate step is concerned withvector bundles, see for example [83, Def. 10.2.2] or [113, Chp. III §1]. Vector bundles are a generalization of tangent bundles and help to understand the idea of function space parametrization in shape calculus and of shape dependent functions in general. The definition is given for the finite dimensional case in order to hide some technical overhead, although it is used in the infinite dimensional setting later on.
Definition 16(vector bundle):
LetMandEbe (real) manifolds and letBbe a Banach space.
Then
1. Etogether with a smooth mapπ : E → Mis called asmooth vector bundle onMif the following conditions are fulfilled
a) For eachx∈ M, there is an open neighborhoodUofxinMand a diffeomorphism ϕU:π−1(U)→U×B
commuting with the projection onU, prU : U×B → U, (x,v) 7→ x. That is, the following diagram is commutative
π−1(U) U×B
U ϕU
π prU
and in particular, by means of the projection prB : U×B → B,(x,v) 7→ v, one obtains an isomorphism for eachx∈U
ϕxU:=prB◦ϕU|π−1(x):π−1(x)→B.
b) For each x ∈ M the setπ−1(x)carries the structure of aBanachable space (i. e. a complete topological space whose topology can be defined by a norm) and the maps
ϕxU:π−1(x)→B are linear and continuous.
2. The spacesEandBare calledtotal spaceandbase spaceof the vector bundle, respectively.
3. The setsBx:=π−1(x)are calledfibersof the bundle.
4. The prototype Banach spaceBis often calledstandard fiberof the bundle.
5. The mapsϕUare calledtrivializing mapsof the vector bundle.
6. A vector bundleEis calledtrivial, if it is isomorphic toM ×B. (Note, that a vector bundle is always trivial onU, i. e. a vector bundle is always locally trivial.)
Remark:
LetMbe a manifold andTMits tangent bundle. ThenTMis a vector bundle onMtogether with the natural projectionπ:TM → M,TxM 3ξx7→x.
The tight review of various notions known from the theory of manifolds is finished here. Following the approach of Delfour and Zolésio in [44, Chp. 9] one finds the following similarities among shape calculus and the differential calculus on manifolds, which is already slightly indicated in [42]:
1. The metric spaceH(Ω)plays the role of an infinite dimensional manifold; cf. the11thand the13th item of the discussion onpage 67.
2. Transformations f := F−Id ∈ Θ0and velocity fieldsV ∈ V define paths in H(Ω)by means of transformation- and flow approach, respectively (seepage 92). LetF ∈ H(Ω), let f ∈ Θ0and let
2.6.2 Abstract view on shape calculus 73 V∈ V then there exists an intervalI ⊂Rcontaining 0 such that one gets paths (straight lines and integral curves, cf. the5thitem) throughF
I→ H(Ω),t7→Tt(f):=F+t f
I→ H(Ω),t7→Tt(V):=x(t, .), the solution of d
dtx(t,X) =V(x(t,X)), x(0,X):=F(X), X∈Ω.
Consequently,f andVdefine tangent vectors f =∂t(Tt(f))|t=0andV=∂t(Tt(V))|t=0.
However, the spacesΘ0(see (2.79)) andV(see (2.47)) are not the same. In particular,Θ0⊂ V since each f ∈ Θ0 fulfills f ·n ≡ 0 onΓbut not everyV ∈ V is such thatV|Γ = 0. Hence, following Definition 8, the two approaches seem to induce different tangent spaces to H(Ω). Several facts should be noted in this respect:
• The difference betweenΘ0andVis concentrated in a neighborhood of the boundaryΓ, i. e.
∀K⊂⊂Ω,V∈ V ∃f ∈Θ0: V|K≡ f|K.
• As already discussed in the8thand the12thitem onpage 67f.one actually is interested in opti-mization within the setO(A)or at leastX(A)∼=H(Ω)/K(A). By means of the consequential fact that shape calculus is “set sensitive” only but not “pointwise sensitive”29(cf. the4thitem of the above mentioned discussion) and by means of the requirement that the distance be-tween the boundary of the active setB ∈ O and the boundary of the holdall Ω is positive dist(γ,Γ) ≥ δ > 0 (cf.Assumption 1, and the proof ofLemma 4), one can deduce that the constitution of the transformation nearΓ does not influence any result. Consequently, one should expect no disagreement when applying transformations f ∈ Θ0 instead of velocity fieldsV∈ V.
• These considerations find expression in two important results of shape calculus. For one thing themethod of perturbation of identity(which corresponds to the usage off ∈Θ0) and thevelocity methodyield the same shape derivatives [44, Chp. 9 Thm. 3.1] and for another thing the relevant part of the shape derivative is concentrated on the perturbed interface [44, Chp. 9 Thm. 3.6, Cor. 3.1].
• Nonetheless, it should be emphasized that the two approaches are not equivalent in general and that only the velocity method is straight forward applicable in presence of a constraining holdall (see [44, Chp. 9 Sec. 3.3]). A detailed analysis of the structure of shape derivatives in the presence of a holdall can be found in [43].
3. The tangent bundleTH(Ω)is trivial sinceH(Ω)is an open submanifold of the affine space Id+Θ0; cf. the reasoning onpage 95. Moreover, it is naturally isomorphic toH(Ω)×Θ0, in the sense, that one can use the identity as isomorphism.
Thus, many notions concerning manifolds, which have to be distinguished in general denote the same object in the context ofH(Ω), since one rather is in framework of vector spaces than in those of manifolds. Especially the handling of vector fields and many therefrom deduced notions is considerably simplified. This finding is amplified by the fact that the manifoldH(Ω)can be covered by a single chart. Unfortunately, it is not possible to give a complete overview of simplifications here, and many interesting consequences are left to the reader. He is referred to the extensive textbook of Lang [113] once again.
4. One outcome of the (natural) triviality of the tangent bundle TH(Ω) = H(Ω)×Θ0 is, that an f ∈Θ0(as well as anV∈ V) is a tangent vector to allF∈ H(Ω)simultaneously. In particular, there is a natural notion ofparallel transport(see [108, Chp. II Sec. 3]) of tangent vectors. Two elements
f ∈ TFH(Ω) =Θ0andg∈TGH(Ω) =Θ0are parallel if and only if f =g∈Θ0.
This canonical parallel transport induces a canonical choice for a covariant derivative of vector fields onH(Ω), cf. the7thitem.
5. Vector fields onH(Ω)in the sense ofDefinition 8are smooth maps
H(Ω)3F7→ξF∈TFH(Ω) =Θ0 or respectively H(Ω)3F7→ξF ∈ V.
29All functions considered in this thesis are dependent on specific sets only (F = F(B),uJ = uJ(J),σ = σ(β), . . . ) and are invariant with respect to pointwise reparametrization of the sets. This means the functions fulfill a so calledcompatibility condition; cf. [44, p. 202]
They are not common in shape calculus, but it is usual to work with nonautonomous velocity fields (these aretime-dependent vector fields, cf. [113, Chp. IV §1 and §2]) and their corresponding paths (these areintegral curves, ibidem). A nonautonomous velocity fieldV ∈ C1([0;τ],V)induces an integral curve in the manifoldH(Ω)through Id∈ H(Ω)by means of
T.(V):[0;τ]→ H(Ω), t7→Tt(V):=x(t, .), where d
dtx(t,X) =V(t,x(t,X)), x(0,X):=X∈Ω.
This finding can be generalized to an integral curve inH(Ω)through arbitrary elementsF∈ H(Ω) by imposing the initial conditionx(0,X):=F(X)forX∈Ω, as it was done in the2nditem.
In contrast, the paths defined via the transformation approacht 7→ F+t f define straight lines in H(Ω)regarded as subset of the affine space Id+Θ0.
6. There are two different canonical identifications of the tangent bundle of the manifoldH(Ω). One of them is induced by the underlying Banach spaceΘ0, sinceH(Ω)is an open subset of the affine space Id+Θ0. The other one is due to the group structure ofH(Ω)with respect to composition.
It will be come apparent in the following (in particular, cf. the18thitem), that the first one gives the notion of a shape derivative (cf. [151, Sec. 2.30], whereas the second one yields theEulerianor material derivative(cf. [151, Sec. 2.11 and 2.25]).
Letξ∈V(H(Ω))be a vector field. It is a smooth map
ξ:H(Ω)→TH(Ω), F7→ξF ∈TFH(Ω) =Θ0.
The underlying Banach space structure permits the canonical identification of the tangent bundle.
That is, for arbitraryF∈ H(Ω)identifyTFH(Ω)with the standard fiberTIdH(Ω) =Θ0by means of the trivializing map30
ϕId:TH(Ω)→ H(Ω)×Θ0, which is defined fiber-wise by means of the identity
ϕIdF :TFH(Ω)→Θ0, ξF 7→ξF.
This results in the identification of different tangent spaces by means of the identity (ϕGId)−1◦ϕIdF :TFH(Ω)→TGH(Ω), ξF 7→(ϕGId)−1◦ϕFId(ξF) =Id(ξF) =ξF.
The group structure permits another canonical identification of tangent bundle. That is, for arbi-traryF∈ H(Ω)identifyTFH(Ω)with the standard fiberTIdH(Ω) =Θ0by means of the trivializ-ing map
ϕ◦ :TH(Ω)→ H(Ω)×Θ0, which is defined fiber-wise by means of the pull-back to Id
ϕF◦ :TFH(Ω)→Θ0, ξF 7→ξF◦F.
This results in the identification of different tangent spaces by means of the composition of trans-formations
(ϕG◦)−1◦ϕF◦ :TFH(Ω)→TGH(Ω), ξF 7→(ϕG◦)−1◦ϕF◦(ξF) =ξF◦F◦G−1.
7. There is a canonical covariant derivative of vector fields onH(Ω)which is induced by the underly-ing Banach spaceΘ0, sinceH(Ω)is an open subset of the affine space Id+Θ0. Letξ,η∈V(H(Ω) be two vector fields. They are smooth mapsξ,η :H(Ω)→TH(Ω),F7→ξF,ηF ∈TFH(Ω) = Θ0. By means of the (global) chartφ : H(Ω) → Θ0, F 7→ φ(F) := F−Id, the vector fields can be uniquely identified with smooth vector fields on (a subset of) the Banach spaceΘ0:
φ∗:V(H(Ω))→V(Θ0), ξ7→φ∗(ξ), where for f ∈ H(Ω)−Id=φ(H(Ω))⊂Θ0
(φ∗(ξ))f := ϕφ
−1(f)
Id (ξφ−1(f)) =ξId+f. The following commutative diagram illustrates the situation.
30Note, that in defiance of the notation ofDefinition 16the subscript of the trivializing map does not indicate the local chart here, but the connection to the identity in order to distinguish it from the trivializing map to be introduced below.
2.6.2 Abstract view on shape calculus 75
TH(Ω) H(Ω)
H(Ω)×Θ0
TΘ0=Θ0×Θ0 φ(H(Ω))⊂Θ0 ξ
ϕId
φ×Id
φ∗(ξ)
φ=.−Id
A canonical choice for a covariant derivative ofξwith respect to directionηatFis then given by the directional derivative of the projectionφ∗(ξ)with respect toηFat the projected pointφ(F) =F−Id.
In order to mark that this choice it related to the trivializing map ϕId of 6thitem, the covariant derivatives gets the superscript Id
∇ηIdξ:H(Ω)→TH(Ω), F7→(∇ηIdξ)F∈ TFH(Ω), where forF=Id+ f ∈ H(Ω)
(∇ηIdξ)F :=lim
t→0
(φ∗(ξ))f+t(φ∗(η))f −(φ∗(ξ))f
t =lim
t→0
ξF+tηF−ξF
t .
Consequently, the definition of the covariant derivative corresponds to the usual directional deriva-tive in the Banach spaceΘ0. It is important here, thatF+tηF = (Id+tηF◦F−1)◦Fis an element ofH(Ω)for sufficiently smallt≥0. Fortunately this is true, cf. the3rditem onpage 66.
8. In order to see the relation of the covariant derivative from the7thitem to the derivative of nonau-tonomous velocity fields (cf. [44, Chp. 9 Sec. 6.3]), letα : I ⊂R → H(Ω)be anintegral curvetoη throughF. That is to say,
∀t∈I: α˙(t) =ηα(t)andα(0) =F (in particular 0∈ I).
( ˙α:I→TH(Ω)is calledcanonical liftingofα(see [114, Chp. IV §3]).) This defines a nonautonomous velocity field (cf. [113, Chp. IV §2])
W :I×Ω→R2, (t,x)7→W(t,x):=ηα(t)(x). In particular, there holds
∀t∈ I: W(t, .)∈Tα(t)H(Ω)
∀X∈Ω,∀t∈ I: α˙(t)(X) =W(t,α(t)(X)), α(0)(X) =F(X). The second vector fieldξdefines a nonautonomous velocity fieldV(lifting) alongα
∀x∈Ω,∀t∈ I: V(t,x):=ξα(t)(x). Now one recognizes, that there holds31
(∇ηIdξ)F(x) =lim
t→0
ξF+tηF(x)−ξF(x) t
=lim
t→0
ξα(t)(x)−ξα(0)(x) t
=lim
t→0
V(t,x)−V(0,x) t
= ∂
∂tV(t,x)
t=0
=V0(0,x).
31A detailed analysis for the justification for the transition from Gateaux to Hadamard semiderivatives is left open here and the reader is referred to [44, Chp. 9 Sec. 3.1 and 3.3].
In other words, the derivative is given by
(∇ηIdξ)F = d dtξIdα (t)
t=0,
whereξIdα is the by means ofϕIdto the standard fiberΘ0transported liftingξα ξIdα :I→Θ0, t7→ ϕα(t)Id (ξα(t)) =ξα(t).
I⊂R H(Ω) TH(Ω)
H(Ω)×Θ0
α ξ
ϕId ξα
ξαId
9. The construction of the8thitem can be repeated with use of the trivializing mapϕ◦ from the6th item instead ofϕId. The transport of the liftingξαto the standard fiberΘ0now reads
I⊂R H(Ω) TH(Ω)
H(Ω)×Θ0
α ξ
ϕ◦
ξα
ξα◦
ξα◦:I→Θ0, t7→ϕα(t)◦ (ξα(t)) =ξα(t)◦α(t). This gives rise to a second canonical covariant derivative
(∇η◦ξ)F(x) := d dtξα◦(t)
t=0
= d
dtξα(t) α(t)(X)
t=0
= d
dtV t,α(t)(X)
t=0
= ∂
∂tV t,α(t)(X)
t=0+DxV t,α(t)(X)α˙(t)(X)
t=0
=V0 0,α(0)(X)+DxV 0,α(0)(X)W 0,α(0)(X)
=V0(0,x) +DxV(0,x)W(0,x)
= (∇ηIdξ)F(x) +DxξF(x)ηF(x).
As it becomes apparent during the course of the18thitem, the approach via ϕ◦ is more general, since covariant derivatives in more general vector bundles onH(Ω)can be constructed that way.
Nonetheless, the covariant derivative∇Idis useful in order to understand the differences between material and shape derivatives, from the perspective of calculus on the manifoldH(Ω).
10. To the best of the author’s knowledge, the usual shape calculus avoids the introduction of a co-variant derivative of vector fields and consequently of covector fields, too. One confines oneself with the computation of first and second order directional derivatives of shape functionals. This approach is general enough to be able to extract the essence of shape calculus in terms of differential calculus on manifolds at least up to second order derivatives. In other words, one recognizes that these objects possess some intrinsic structure afterwards.
11. The notion of theHadamard semiderivative[44, Chp. 9 Rem. 2.1] corresponds to the notion of the directional derivative fromDefinition 8.
2.6.2 Abstract view on shape calculus 77 12. In contrast, the directional derivative which corresponds to the method of perturbation of identity, is aGateaux derivative[44, Chp. 9 Def. 2.1(i)]. Moreover, the transformation approach is essentially based on the fact, that the manifoldH(Ω)is embedded to the affine space Id+Θ0.32 Hence, it is possible (by means of the notation of theRemarkonpage 63and ofLemma 13) to define
FB :B1(0)⊂Θ0→R,f 7→ FB(f):=F([Id+f]B(B)) =F(B[Id+f])
for eachB ∈ O. In other words, it is possible to locally transform the shape functionalF defined on the manifoldX(B) ⊂ Ointo a functionalFBdefined on the Banach spaceΘ0. This notation is also used by Delfour and Zolésio; cf. [44, Chp. 9 Sec. 3.3].
However, it is important to notice that this kind of notation is not perfectly adapted to the situation of shape calculus, where the considered objects usually are set dependent only. The shape func-tionalF is defined onOor at least on a the subset X(B) (for an arbitrarily chosenB ∈ O), but the local definition ofFB is based on a subset ofΘ0which corresponds toH(Ω). In particular, all f ∈ [F]B−Id := {G−Id ∈ Θ|G ∈ [F]B}yield the same set B[F] ∈ X(B) ⊂ O and hence they evaluateF at the same point (i. e. setB[F]).
A remedy would be to transport the equivalence classes[F]B ⊂ H(Ω)down to the Banach spaceΘ0
– that is to define the quotient ofΘ0and the equivalence relation induced by the classes[F]B−Id.
But the equivalence classes are no (affine) linear subspaces and consequently the quotient is no linear space. Hence, this reasoning contradicts the original goal of transporting shape calculus into a Banach space.
This aim can only be achieved by the introduction of a retraction (cf.Definition 20) or charts on the manifoldX(B). However, this topic goes beyond the scope of this thesis.
13. The notion ofshape derivativeof a real-valued shape functional corresponds to the notion of a
13. The notion ofshape derivativeof a real-valued shape functional corresponds to the notion of a