Research Article | Open Access
Jerico B. Bacani, Julius Fergy T. Rabago, "On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method", International Journal of Differential Equations, vol. 2015, Article ID 954836, 10 pages, 2015. https://doi.org/10.1155/2015/954836
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.
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,  for some applications in fluid mechanics or aerodynamics, and  for other applications). A typical problem in this line of research is to find a domain, for instance, , in a set of admissible domains such that an objective functional achieves a minimum (or maximum) on it . 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 with the constraint . 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 aswhere the state is the solution to a partial differential equation (PDE) on the domain . For an extensive introduction to shape optimization problems, we refer to the book of Delfour and Zolésio  (see also ).
Recently, there has been an increasing interest in the applications of shape optimization in the study of Bernoulli problems. Abda et al.  rephrased the Bernoulli problem into a shape optimization problem and explicitly determined the shape derivative of the cost functional being studied. In , a framework for calculating the shape Hessian for the domain optimization problem with a PDE as the constraint was presented. In , a similar approach as in  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., [10–13]). 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 [13–15]).
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  used a second-order perturbation of the identity along the normal of the boundary for second-order elliptic problems in 1986. Simon  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  and a nonhomogeneous Dirichlet problem in .
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  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  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 with a fixed boundary , we need to find a bounded connected domain with a free boundary that contains the closure of , , and an associated real-valued (state) function defined on (where is the annulus formed by and ; refer to Figure 1) such that both unknowns and satisfy the following boundary value problem:where .
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 , which is defined asand minimizing this functional over a class of admissible domains, where the two state functions and solve a homogenous Dirichlet and nonhomogenous Neumann problem, respectively: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 , and the other one is by using the shape derivatives of states . The authors also computed its second-order shape derivative for general domains via the boundary differentiation scheme and via Tiihonen’s approach . The computation is found in .
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. ).
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.
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.
2.1. Perturbation of the Identity
Let and the universal or hold-all domain be smooth subdomains of , such that . A class of perturbations of the domain obtained from the perturbation of identity operator is defined as where the deformation field is in defined asThen, for sufficiently small , (i) is a homeomorphism, (ii) is a diffeomorphism and, in particular, is a diffeomorphism, (iii) , and , and (iv) (cf. [9, Theorem ]). 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 and the transformation (the perturbation of identity operator). Then, we can find a constant such that the functions defined above restricted to the interval have the following regularities and properties: (1),(2),(3),(4),(5),(6),(7)there are positive constants , , and such that and for ,(8),(9),(10),(11),(12), where ,(13), where the surface divergence is defined by
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 to denote time-dependent velocity fields in contrast to velocity fields denoted by which are not time-dependent.
Let be a given velocity field for some fixed . The map can be viewed as a family of nonautonomous velocity fields on defined by Let satisfy (V1) , (V2) , , ,where denotes the function . Associate with the solution of the ODE:
That is, we suppose that 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 (V2′) , , .
Remark 2. The above statement can be described equivalently as follows: Let belong to for some integer and a small real number . Let be a smooth bounded domain with boundary which is at least twice differentiable. The field is an element of which may depend on . It generates the transformations through the differential equation
As already described in the previous section, we denote the transformed domain at by .
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 on or a family of transformations of provided that the map , verifies (V1) and (V2) or the map , verifies assumptions (T1), (T2) and (T3) given below.
Theorem 3 (see ). (i) Under assumptions (V1) and (V2) on the map , the maps defined previously have the following properties: (T1), ,(T2) is bijective,(T3), .(ii) If there exists a real number and a map verifying assumptions (T1), (T2), and (T3), then the mapverifies assumptions (V1) and (V2), where is the inverse of .
In the above discussion, we see that the solution to the differential equation (10), , 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 , and determine the relationship of the autonomous field and the nonautonomous defined in (10). Differentiating with respect to , we getThis simply implies that . Now, replacing by , we get or, equivalently, . 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 . The relation also implies thatSo, in particular, we have
2.3. Domain and Boundary Transformation
We recall the following theorems on domain and boundary transformations.
Lemma 4 (see ). We have the following important transformations which will also be central to our investigation: (1)Let . Then and(2)Let . Then and
2.4. Material Derivatives
Let be defined in . An element , called the material derivative of , is defined as if the limit exists in .
Remark 5. As pointed out in , the material derivative of the state function can be written as and it, in fact, characterizes the behavior of the function at in the direction .
Now, on the other hand, an element is called the shape derivative of at at the direction of , if the following limit exists in :
Remark 6. We note that if and exist in , then the shape derivative can be expressed asIn general, if and both exist in , then so does .
2.5. 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 is well-defined. Given the deformation field , the directional Eulerian shape derivative of at in the direction of deformation field is defined asif the limit exists. The objective functional is shape differentiable at provided that exists for all and if is linear and continuous with respect to . On the other hand, the second-order Eulerian shape derivative of a well-defined shape functional at in the direction of the deformation fields and is defined to beif the limit exists. The functional is said to be twice shape differentiable if, for all and , exists and if is bilinear and continuous with respect to and . 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 which is a result of two deformation fields and is illustrated in Figure 3.
Remark 7. Under appropriate hypothesis on the map , one can show that (cf. ). Therefore, if is associated with the deformation field in the perturbation of the identity, then coincides with by . Hence, the first-order shape derivative of a cost functional obtained via the velocity method coincides with the one obtained from the perturbation of the identity technique.
3. Analysis for the Nonautonomous Case
In this section, we will see how important is the expression in considering nonautonomous velocity fields . Of course if is autonomous, is no other than . 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 -bounded domain, the first-order shape derivative of the Kohn-Vogelius cost functional in the direction of a perturbation field , where and the state functions and satisfy the Dirichlet problem and the Neumann problem , respectively, is given bywhere is the unit exterior normal vector to , is a unit tangent vector to , and is the mean curvature of .
Now, we give our main result of the second-order shape derivative of the Kohn-Vogelius objective functional via velocity method.
Theorem 9. Denote and let and be any two velocity fields from the setAssume that, for some sufficiently small , exists at in the direction . Then, for a -bounded domain, the second-order shape derivative of the Kohn-Vogelius cost functional in the directions of the two perturbation fields and and the state functions and which satisfy the Dirichlet problem and the Neumann problem , respectively, is given by where is the unit exterior normal vector to , is a unit tangent vector to , and is the mean curvature of .
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 and be, respectively, defined as follows: Then, we have
Proof. We first note the following:Now, for the expression , we note thatand since is constant, . Hence, we get by applying product rule. Furthermore, by substitution and distributing the (partial) differential operator, we obtainUsing the chain rule and again the product rule twice, we haveNow, we simplify each of the expressions , , and . In the sequel, we will be needing the material and shape derivatives of the vectors and and the mean curvature . For their corresponding forms, we refer the readers to [25, Theorems , , and ] which are proven in [5, 28]. So we proceed as follows. First, using the product rule twice and then the chain rule, we getInterchanging the gradient and the differential operator and upon evaluation of at 0, we getHere we note that is the material derivative of in the direction . By definition, can be written in terms of the shape derivative : , where satisfiesNow, for the second expression , we haveHere we stress that is the material derivative of in the direction . Also, , where satisfiesThe expression represents the material derivative of the mean curvature and it can be shown that .
Finally, for the last expression , we have the following: where is the material derivative of in the direction and is given by .
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 and to be nonautonomous velocity fields and proceed for the computation as follows. Letting , we have
Note that we already have a simplified form for from Lemma 10, so we only need to simplify the expressions , , and . For , we have need for and, for , we easily get , where is the same with what is given in Lemma 10. Now, for the last expression , we have Here, we stress that and that the relation implies the identitywhere
With these identities, we now have
Now, we note that, for autonomous velocity fields, we have the relation (i.e., for , we have ). Thus, we now have from which we see that and here we remark that this result coincides with the one presented by Delfour and Zolésio in [5, page 1420, Equation ]. This proves the main Theorem 9.
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 and to be nonautonomous velocity fields. So we proceed as follows.
First, using the definition of the second-order shape derivative and by the relation , we have Invoking the boundary differentiation rule, we get
Remark 11. Here, is the bilinear term which Delfour and Zolésio refer to as the shape Hessian.
Remark 12. We remark that, as compared to the form of the shape Hessian presented in , the result we have established here in Theorem 9 clearly shows the relation pointed in  by Delfour and Zolésio about the form of shape Hessians obtained through nonautonomous velocity fields.
We have computed the second-order shape derivative of the Kohn-Vogelius objective functional in the direction of two (nonautonomous) deformation fields and via velocity method. The computed expression for is given by where , is the unit exterior normal vector to , is a unit tangent vector to , and is the mean curvature of . We stress out that the expression computed for the second-order shape derivative of is not yet in its explicit form. The explicit form, however, could easily be obtained by substituting all those computed values for each of the expressions found on the integrand or the interested reader may refer directly to  for the explicit form of . It is apparent in our first result that the computed expression for the shape derivative coincides with the one presented by Delfour and Zolésio in . Interestingly, the result obtained through boundary transformation approach which again uses nonautonomous velocity fields also verifies the result found by Delfour and Zolésio in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank UP Baguio for giving research load credits to write this paper. This work was funded by the UP System Emerging Interdisciplinary Research Program (OVPAA-EIDR-C05-015).
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Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.