On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.


Introduction
Shape optimization is a key research topic with many applications in various fields of pure and applied sciences, especially in biomechanics and engineering (cf.[1,2] for applications in structural mechanics, [3] for some applications in fluid mechanics or aerodynamics, and [4] for other applications).A typical problem in this line of research is to find a domain, for instance, Ω, in a set of admissible domains A such that an objective functional  achieves a minimum (or maximum) on it [3].For instance, suppose, among all three-dimensional shapes of given volume, that we wish to find the one which has a minimal surface area.In this particular case, the problem can be described mathematically as finding the minimum of (Ω) = Area(Ω) with the constraint (Ω) = Volume(Ω) − constant.Obviously, the answer to this question would be the sphere.In general and in most cases of greater interest, shape optimization problems can be described mathematically as min (,Ω)  (, Ω) s.t. (, Ω) = 0, Ω ∈ A, (1) where the state  is the solution to a partial differential equation (PDE)  on the domain Ω.For an extensive introduction to shape optimization problems, we refer to the book of Delfour and Zolésio [5] (see also [6]).
Recently, there has been an increasing interest in the applications of shape optimization in the study of Bernoulli problems.Abda et al. [7] rephrased the Bernoulli problem into a shape optimization problem and explicitly determined the shape derivative of the cost functional being studied.In [8], a framework for calculating the shape Hessian for the domain optimization problem with a PDE as the constraint was presented.In [9], a similar approach as in [8] was applied in solving a shape optimization problem.
Another way to approach the solutions of shape optimization problems is through iterative methods.For the past few decades, several numerical methods have been developed to solve the two-dimensional Bernoulli problem (see, e.g., [10][11][12][13]).These strategies were also developed based on reformulating the Bernoulli problem as a shape optimization problem.This reformulation can be achieved in several ways.For instance, for a given domain, one can choose one of the boundary conditions on the free boundary to obtain a well-posed state equation.The domain is determined by the requirement that the other condition on the free boundary is satisfied in a least square sense (see [13][14][15]).

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Many authors have also studied the second variation of a cost functional for linear PDEs.Building on the shape optimization setting that is based on the perturbation of the identity method introduced by Murat and Simon (cf.[16,17]), Fujii [18] used a second-order perturbation of the identity along the normal of the boundary for second-order elliptic problems in 1986.Simon [19] computed the second variation via the first-order perturbation of the identity in 1988.A general approach via the velocity method (Figure 2) was systematically characterized by Delfour and Zolésio [20,21], and they computed the shape Hessian for a simple Neumann problem in [20] and a nonhomogeneous Dirichlet problem in [21].
However, a standard approach in dealing with the solution to (1) requires some information on gradients.So shape derivatives are essential in understanding the problem.
The recent paper focuses on the exterior Bernoulli free boundary problem (FBP).As far as the authors are concerned, the same functional was first studied by Eppler and Harbrecht and published in [22] wherein the first-order shape derivative, or equivalently the shape gradient, was derived for arbitrary variations in terms of the perturbation of the identity.Moreover, the second-order shape derivative, or equivalently the shape Hessian, has been computed and analyzed for the special cases of star-like domains.As a main result, by analyzing the shape Hessian at the optimal domain, Eppler and Harbrecht found out that the optimization problem is algebraically ill posed.In the present paper, the same functional is studied again but we focus on the application of velocity method in dealing with shape optimization problem.It would be a challenging research in the near future to study the ill-posedness of the shape optimization problem for general domains, as well as the comparison of the shape Hessians in this paper from [22] for the former uses Cartesian coordinates, while the latter used spherical/polar coordinates.The nice thing in the present paper is that the results attest to classical results in shape optimization problems.Now, the exterior Bernoulli FBP is formulated as follows.
Given a bounded and connected domain  ⊂ R 2 with a fixed boundary  fl Γ, we need to find a bounded connected domain  with a free boundary Σ that contains the closure of , , and an associated real-valued (state) function  defined on Ω (where Ω is the annulus formed by  and ; refer to Figure 1) such that both unknowns  and Ω satisfy the following boundary value problem: where  < 0.
In recent papers, Bacani and Peichl employed shape optimization methods to study the exterior Bernoulli FBP by reformulating it into Kohn-Vogelius-type cost functional  KV , which is defined as and Bacani and Peichl presented two strategies in computing the first-order shape derivative of the Kohn-Vogelius objective functional.One is by using the Hölder continuity of the two state variables involved [9], and the other one is by using the shape derivatives of states [23].The authors also computed its second-order shape derivative for general domains via the boundary differentiation scheme and via Tiihonen's approach [24].The computation is found in [25].
In this recent paper, we are going to solve the shape optimization problem using velocity method, wherein we consider nonautonomous velocity fields.The study is important since it confirms a classical result of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and the perturbation of identity technique (cf.[26]).
In the next section (Section 2), we present an overview of some concepts necessary for the understanding of the present study.The section also includes some important results of methods of shape optimization which will be useful in our investigation.We formally present our main contribution which pertains to the form of the second-order shape derivative of the Kohn-Vogelius objective functional in Section 3. Finally, we end our paper by summarizing our results in Section 4.

Preliminaries
In this section we provide some important (but not exhaustive) background of some shape optimization techniques.We give an overview of these concepts to understand the present study.

Perturbation of the Identity.
Let Ω and the universal or hold-all domain  be smooth subdomains of R 2 , such that Ω ⊆ .A class of perturbations Ω  of the domain Ω obtained from the perturbation of identity operator   is defined as where the deformation field V is in Θ 1 defined as Then, for sufficiently small , (i) For convenience, we will use the following notations throughout the discussion: The next lemma provides some properties of the transformation   which are useful in accomplishing our main objective.
Lemma 1 (see [13,27]).Consider a fixed vector field V ∈ Θ 1 and the transformation   (the perturbation of identity operator).Then, we can find a constant   > 0 such that the functions defined above restricted to the interval   = (−  ,   ) have the following regularities and properties: ( 7) there are positive constants  1 ,  2 , and  such that 0 <  1 ≤   () ≤  2 and   () ≥  for  ∈ Ω, 2.2.The Velocity (Speed) Method (See [5,20,21]).In this paper, we are interested in solving the second-order shape derivative of the Kohn-Vogelius objective functional via the velocity method with nonautonomous velocity fields.Detailed discussions about this method can be seen in [5,20,21].We present some of the details here.As our primary interest is focused on nonautonomous velocity fields, we will use the notation V to denote time-dependent velocity fields in contrast to velocity fields denoted by V which are not timedependent.Let V : [0,   ] × R 2 → R 2 be a given velocity field for some fixed   > 0. The map V can be viewed as a family { V()} of nonautonomous velocity fields on R 2 defined by where V(⋅, ) denotes the function   → V(, ).Associate with V the solution (; ) of the ODE: That is, we suppose that V is continuous in  and at the same time Lipschitz in spatial variables.We remark that, in the case of autonomous velocity fields, the condition to be satisfied can be simplified as

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Remark 2. The above statement can be described equivalently as follows: Let V belong to   = ([0,   );   (R 2 , R 2 )) for some integer  ≥ 2 and a small real number   > 0. Let Ω ⊂ R 2 be a smooth bounded domain with boundary Γ which is at least twice differentiable.The field is an element of   (R 2 , R 2 ) which may depend on  ≥ 0. It generates the transformations through the differential equation As already described in the previous section, we denote the transformed domain   (Ω) at  ≥ 0 by Ω  .Now, the following theorem describes the relation between the perturbation of the identity method and the velocity method.The theorem basically tells us that we can start from either a family of velocity fields { V()} on R  or a family of transformations {  } of R  provided that the map V, V(, ) = V()(), verifies (V1) and (V2) or the map , (, ) =   (), verifies assumptions (T1), (T2) and (T3) given below.
In the above discussion, we see that the solution to the differential equation (10), V = V ∈ Θ 1 , is the perturbation of the identity operator   .Conversely, if   is the perturbation of the identity operator, then (10) is satisfied.Hence, we consider the special case, where () =  + V, and determine the relationship of the autonomous field V ∈ Θ 1 and the nonautonomous V defined in (10).Differentiating () =  + V with respect to , we get   (; ) = V () = V (,  (; )) ,  ∈ Ω.
This simply implies that V() = V(, ( + V)()).Now, replacing  by ( + V) −1 (), we get V( −1  ()) = V(, ) or, equivalently, V() = V∘ −1   .This is why many results obtained by the perturbation of identity technique can be acquired as well through the velocity method using nonautonomous velocity fields.As an immediate consequence, we note that So, in particular, we have

Domain and Boundary Transformation.
We recall the following theorems on domain and boundary transformations.
Let  be defined in [0,   ] × .An element u ∈   (Ω), called the material derivative of , is defined as if the limit exists in   (Ω).
Remark 5.As pointed out in [9], the material derivative of the state function  can be written as and it, in fact, characterizes the behavior of the function  at  ∈ Ω ⊂  in the direction V().Now, on the other hand, an element   ∈   (Ω) is called the shape derivative of  at Ω at the direction of V, if the following limit exists in   (Ω): Remark 6.We note that if u and ∇ ⋅ V exist in   (Ω), then the shape derivative can be expressed as In general, if u () and ∇ ⋅ V() both exist in  , (Ω), then so does   ().

The First-and Second-Order Eulerian Shape Derivatives.
We first recall the definition of directional Eulerian shape derivative or simply shape derivative of a shape functional.Suppose that the shape functional  : Ω → R is well-defined.Given the deformation field V, the directional Eulerian shape derivative of  at Ω in the direction of deformation field V is defined as if the limit exists.The objective functional  is shape differentiable at Ω provided that (Ω; V) exists for all V and if (Ω; V) is linear and continuous with respect to V. On the other hand, the second-order Eulerian shape derivative of a well-defined shape functional  at Ω in the direction of the deformation fields V and Ŵ is defined to be if the limit exists.The functional is said to be twice shape differentiable if, for all V and Ŵ,  2 (Ω; V; Ŵ) exists and if  2 (Ω; V; Ŵ) is bilinear and continuous with respect to V and Ŵ.Following these definitions, the second-order shape derivative of the functional being studied can also be computed as follows: In this case, the transported domain Ω , which is a result of two deformation fields V and Ŵ is illustrated in Figure 3.
Remark 7.Under appropriate hypothesis on the map V  → (Ω; V(0)), one can show that (Ω; V) = (Ω; V(0)) (cf.[21]).Therefore, if V is associated with the deformation field V in the perturbation of the identity, then (Ω; V(0)) coincides with (Ω; V) by V(0) = V.Hence, the firstorder shape derivative of a cost functional obtained via the velocity method coincides with the one obtained from the perturbation of the identity technique.

Analysis for the Nonautonomous Case
In this section, we will see how important is the expression is no other than V().Since we are interested in the second-order shape derivative of the Kohn-Vogelius objective functional, we should first recall (without proof) the first-order shape derivative of the Kohn-Vogelius objective functional which is stated in the following theorem.
Theorem 8 (see [9,22,23]).For a  1,1 -bounded domain, the first-order shape derivative of the Kohn-Vogelius (KV) cost functional in the direction of a perturbation field V ∈ Θ, where V ∈ Θ = { V ∈  1,1 (, R 2 ) : V| Γ∪ = 0} and the state functions   and   satisfy the Dirichlet problem () and the Neumann problem (), respectively, is given by where n is the unit exterior normal vector to Σ,  is a unit tangent vector to Σ, and  is the mean curvature of Σ.
The above result was first proven by Eppler and Harbrecht in [22].Other proofs were also given by Bacani and Peichl [9,23], by using two different approaches.Now, we give our main result of the second-order shape derivative of the Kohn-Vogelius objective functional via velocity method.
and let V and Ŵ be any two velocity fields from the set Assume that, for some sufficiently small , (Ω  ( Ŵ); V()) exists at Ω  ( Ŵ) =   ( Ŵ)(Ω) in the direction V().Then, for a where n is the unit exterior normal vector to Σ,  is a unit tangent vector to Σ, and  is the mean curvature of Σ.
Before we proceed in the formal computation of the second-order shape derivative of the Kohn-Vogelius objective functional, we first prove the following auxilliary result.

Lemma 10. Let 𝐹 and Ḟ
be, respectively, defined as follows: Then, we have Proof.We first note the following: Now, for the expression (/)(  ∘   )| =0 , we note that and since  is constant,  ∘   = .Hence, we get by applying product rule.Furthermore, by substitution and distributing the (partial) differential operator, we obtain Using the chain rule and again the product rule twice, we have Now, we simplify each of the expressions  1 ,  2 , and  3 .In the sequel, we will be needing the material and shape derivatives of the vectors n and  and the mean curvature .For their corresponding forms, we refer the readers to [25, Theorems 4, 5, and 6] which are proven in [5,28].So we proceed as follows.
First, using the product rule twice and then the chain rule, we get Interchanging the gradient and the differential operator and upon evaluation of  at 0, we get Here we note that u , is the material derivative of   in the direction W. By definition, u , can be written in terms of the shape derivative   , : u , = The expression κ  represents the material derivative of the mean curvature  and it can be shown that κ Finally, for the last expression  3 , we have the following: where τ  is the material derivative of  in the direction W and is given by τ = [(W)  n ⋅ ]n.
Combining all of these simplification expressions and by the relations in (28), we get the desired result.Now, we are in the position to prove our main result.In the sequel, we suppose V and Ŵ to be nonautonomous velocity fields and proceed for the computation as follows.
3.1.On the Boundary Transformation Approach.Now, in this section we compute for the second-order shape derivative of the Kohn-Vogelius objective functional through boundary transformation approach.Again, we assume V and Ŵ to be nonautonomous velocity fields.So we proceed as follows.
First, using the definition of the second-order shape derivative and by the relation V = V ∘  −