I'm going to answer from the point of view of an engineer (as I am), mathematicians will probably disagree.

The shape derivative is not a Fréchet derivative, but a Gateaux derivative. The difference between the two can be understood by comparison with classical calculus. If the Fréchet derivative can be thought of as the extension of the derivative of a function to functionals, the Gateaux derivative is the extension of the directional derivative of functions over vector field to functionals over vector fields. In summary:

                                FUNCTIONS                                                   FUNCTIONALS

Domain                 real numbers                                  Banach spaces (sets of functions)

                                 derivative                                                         Fréchet derivative

                         directional derivative                                            Gateaux derivative

For details:

http://www.math.ttu.edu/~klong/5311-spr09/diff.pdf

http://download.springer.com/static/pdf/39/bbm%253A978-3-642-16286-2%252F1.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fbook%2Fbbm%3A978-3-642-16286-2%2F1&token2=exp=1470126459~acl=%2Fstatic%2Fpdf%2F39%2Fbbm%25253A978-3-642-16286-2%25252F1.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fbook%252Fbbm%253A978-3-642-16286-2%252F1*~hmac=d302869395e72b060bbd64e5052f4b097af97687074da4022535b451f4b66d90

So, why is the shape derivative a  Gateaux derivative and not a Frechet derivative? Because our objective is to find the optimal configuration of a domain, i.e. the best shape among a collection of them.

In other words, the shape changes and we want to find the (hopefully) one state of this dynamical system (the shape changes according to some dynamics) where the functional J is optimal. As the shape change is described by some dynamics, the transition from one state to the another is described by a vector field V describing the shape change, which can be thought as a velocity field. When the shape is governed by a physical process, it is actually the velocity field.

To find the optimal state, we have to take the derivative of the functional over the flow of shapes and not on the single shape, thus over the vector field V. Given that, the shape derivative is not a Fréchet derivative but a Gateaux derivative.

The origin of the dynamics governing the shape change is not restricted to physical processes: you might compute the shape derivatives of a functional defined on a manifold which is changing following a Ricci flow or some other geometric flow.

For an example of the use of shape derivatives, its relation with the Gateaux derivative and thorough derivations of them, you can have at this paper (on electromagnetics, but the concepts are the same):

https://perso.univ-rennes1.fr/martin.costabel/publis/CoFLL_shape_derivatives_20100206.pdf

I hope I have been clear (and useful).

Best,

Luca

 everybody talks about shape optimization, it is more general, easy and and gives the reader more insight into the optimization area (Topology, Shape, Size....,etc.). Fréchet-derivative can be derivied from the shape derivative general form. I belive mathematician working in Optimization Models deals with it more than  Engineers.

Warm Regards,

Tamer Qaimari

I'm going to answer from the point of view of an engineer (as I am), mathematicians will probably disagree.

The shape derivative is not a Fréchet derivative, but a Gateaux derivative. The difference between the two can be understood by comparison with classical calculus. If the Fréchet derivative can be thought of as the extension of the derivative of a function to functionals, the Gateaux derivative is the extension of the directional derivative of functions over vector field to functionals over vector fields. In summary:

                                FUNCTIONS                                                   FUNCTIONALS

Domain                 real numbers                                  Banach spaces (sets of functions)

                                 derivative                                                         Fréchet derivative

                         directional derivative                                            Gateaux derivative

For details:

http://www.math.ttu.edu/~klong/5311-spr09/diff.pdf

http://download.springer.com/static/pdf/39/bbm%253A978-3-642-16286-2%252F1.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fbook%2Fbbm%3A978-3-642-16286-2%2F1&token2=exp=1470126459~acl=%2Fstatic%2Fpdf%2F39%2Fbbm%25253A978-3-642-16286-2%25252F1.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fbook%252Fbbm%253A978-3-642-16286-2%252F1*~hmac=d302869395e72b060bbd64e5052f4b097af97687074da4022535b451f4b66d90

So, why is the shape derivative a  Gateaux derivative and not a Frechet derivative? Because our objective is to find the optimal configuration of a domain, i.e. the best shape among a collection of them.

In other words, the shape changes and we want to find the (hopefully) one state of this dynamical system (the shape changes according to some dynamics) where the functional J is optimal. As the shape change is described by some dynamics, the transition from one state to the another is described by a vector field V describing the shape change, which can be thought as a velocity field. When the shape is governed by a physical process, it is actually the velocity field.

To find the optimal state, we have to take the derivative of the functional over the flow of shapes and not on the single shape, thus over the vector field V. Given that, the shape derivative is not a Fréchet derivative but a Gateaux derivative.

The origin of the dynamics governing the shape change is not restricted to physical processes: you might compute the shape derivatives of a functional defined on a manifold which is changing following a Ricci flow or some other geometric flow.

For an example of the use of shape derivatives, its relation with the Gateaux derivative and thorough derivations of them, you can have at this paper (on electromagnetics, but the concepts are the same):

https://perso.univ-rennes1.fr/martin.costabel/publis/CoFLL_shape_derivatives_20100206.pdf

I hope I have been clear (and useful).

Best,

Luca

The question is interesting  and  I will try ony to outline a reply.

Luca Di Stasio  gave   an  answer from the point of view of an engineer.

The shape derivative is   a Gateaux derivative. We can say  ''The difference between the two can be understood by comparison with classical calculus. If the Fréchet derivative can be thought of as the extension of the derivative of a function to functionals, the Gateaux derivative is the extension of the directional derivative of functions over vector field to functionals over vector fields''.

We will give a few comments:

The Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.

Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the classical directional derivative.

Let V and W be Banach spaces, and $ U\subset V$ be an open subset of V. A function $f : U \rightarrow W $ is called Gâteaux differentiable at $x \in U$ if f has a directional derivative along all directions at x. This means that there exists a function g from V into W and that for any chosen vector h in V the limit g(h) =[f(x+th)-f(x)]/h exists if t tends to 0.

If f is Fréchet differentiable at x, it is also Gâteaux differentiable there, and g is just the linear operator A = Df(x). However, not every Gâteaux differentiable function is Fréchet differentiable. This is analogous to the fact that the existence of all directional derivatives at a point does not guarantee total differentiability (or even continuity) at that point.

There is approach  to think of the shape evolution as of a flow problem. That is, one can imagine that the shape is made of a plastic material gradually deforming such that any point inside or on the boundary of the shape can be always traced back to a point of the original shape in a one-to-one fashion. Mathematically, if  \Omega0  is the initial shape, and  \Omegat  is the shape at time t, one considers the diffeomorphisms ft between \Omega0 and \Omegat.

The idea is again that shapes are difficult entities to be dealt with directly, so manipulate them by means of a function.

Consider a functional $F(\Omega)$. Mathematically, shape optimization can be posed as the problem of finding a bounded set $\Omega$, minimizing a functional

$F(\Omega)$ possibly subject to a constraint of the form $G(\Omega)=0$.

Consider a smooth velocity field V.

For given $x_0$ in $\Omega _{0}$ we consider trajectory defined by $x(0)=x_{0}$ and $x'(s)=V(x(s))$ and define $T_{s}(x_{0})=x(s)$.

Then the Gateaux or shape derivative of F at $\Omega _{0}$ with respect to the shape is the limit of (F(\Omega _s) -F(\Omega _0))/s

if this limit exists.

Here, it is specific that we work with shapes.

At first glance Fréchet-derivative works in more general settings.