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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 384320, 19 pages
On the First-Order Shape Derivative of the Kohn-Vogelius Cost Functional of the Bernoulli Problem
1Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio City 2600, Philippines
2Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria
Received 18 August 2013; Accepted 1 November 2013
Academic Editor: Sergei V. Pereverzyev
Copyright © 2013 Jerico B. Bacani and Gunther Peichl. 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.
The exterior Bernoulli free boundary problem is being considered. The solution to the problem is studied via shape optimization techniques. The goal is to determine a domain having a specific regularity that gives a minimum value for the Kohn-Vogelius-type cost functional while simultaneously solving two PDE constraints: a pure Dirichlet boundary value problem and a Neumann boundary value problem. This paper focuses on the rigorous computation of the first-order shape derivative of the cost functional using the Hölder continuity of the state variables and not the usual approach which uses the shape derivatives of states.
The Bernoulli problem is the prototype of a stationary free boundary problem. It arises in various applications such as electrochemical machining, potential flow in fluid mechanics, tumor growth, optimal insulation, molecular diffusion, and steel and glass production [1–6]. A characteristic feature of free boundary problems is that not only the state variable is unknown but also the domain on which the state equation is posed. This represents a significant theoretical as well as numerical challenge. One can characterize the Bernoulli problem, at least along general lines, by finding a connected domain as well as a function which is harmonic on this domain. One component on the boundary is known. The other one is determined by a set of overdetermined boundary conditions (a Dirichlet condition and a Neumann condition) for the state. If the free boundary component is strictly exterior to the fixed part of the boundary, the problem is called exterior Bernoulli problem and interior Bernoulli problem otherwise. For more discussions related to interior and exterior Bernoulli problems, we refer the reader to [1, 4, 7–10].
Recent strategies to compute a numerical solution are based on reformulating the Bernoulli problem as a shape optimization problem. This can be achieved in several ways. 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 squares sense (cf. [11–13]). Alternatively, one can compute on a given domain two auxiliary states: which satisfies the Dirichlet condition and which satisfies the Neumann condition on the free boundary. The underlying domain is selected such that the difference is as small as possible. In fact, if for a domain then and is a solution of the Bernoulli problem. Sometimes is called Kohn-Vogelius functional since Kohn and Vogelius were among the first who used such a functional in the context of inverse problems . Standard algorithms to minimize require some gradient information. So in this paper, the first-order sensitivity analysis is carried out for the functional for the exterior Bernoulli problem. The main contribution in this paper is the application of a shape optimization technique that leads to the explicit expression for the shape derivative of the cost functional. This is done through variational means similar to the techniques developed in [9, 10, 13], wherein we use the Hölder continuity of the state variables satisfying the Dirichlet and Neumann problems but we do not introduce any adjoint variables. In our approach, we also bypass the use of the material derivatives of the states (which was done in ) and the use of states’ shape derivatives.
The rest of the paper is structured as follows. Section 2 presents the Bernoulli free boundary problem and its shape optimization formulations. Section 3 provides a list of shape optimization tools that are needed in the analysis for the shape derivatives of the Kohn-Vogelius cost functional . Section 4 presents an exhaustive discussion on the first-order shape derivative of . Finally, Section 5 draws conclusion and observation.
2. The Bernoulli Problem
The exterior Bernoulli free boundary problem is formulated as follows. Given a bounded and connected domain with a fixed boundary and a constant , one needs to find a bounded connected domain with a free boundary , containing the closure of , and an associated state function , where , such that the overdetermined conditions are satisfied:
On the other hand, the interior Bernoulli free boundary problem has the following formulation. Given a bounded and connected domain with a fixed boundary and a constant , one determines a bounded connected domain with a free boundary and an associated state function , where , subject to the following constraints: In both problems is the outward unit normal vector to . The difference in the domains of these two types of Bernoulli problems is depicted in Figure 1.
Methods of shape optimization can be employed in solving the exterior Bernoulli free boundary problem (1). As we observe, this boundary problem is ill-posed due to the fact that we have overdetermined conditions on the free boundary . So to overcome the difficulty of solving it, one can reformulate it as one of the following shape optimization problems which involves now a well-posed state equation.(1)Tracking Neumann data [11, 12] as where the state function is the solution to the Dirichlet problem (2)Tracking Dirichlet data [11, 13] as where the state function is the solution to the Neumann problem (3)Minimizing the Kohn-Vogelius type cost functional [12, 15] as where state functions and satisfy (4) and (6), respectively.
In this paper, we are just interested in the study of minimizing the Kohn-Vogelius functional .
3. Tools in Shape Optimization
3.1. Feasible Domain
In this work, we are interested in -domains, where . Aside from being we also assume that these are bounded and connected subsets of a bigger set which is also a bounded connected domain. This is called the universal or the hold-all domain. The smoothness of these domains can be defined in the following sense (cf. ).
Consider the standard unit orthonormal basis in . For a point , let so as to write . Consider the unit ball and introduce the subsets
Definition 1. A domain with a nonempty boundary is called a -domain, where , , if for every there exists a neighborhood of and a diffeomorphism such that , , and .
To illustrate this for and , see Figure 2.
Note that if is a bounded, open, connected set with a boundary, then . This was given in  and we prove it as follows.
Theorem 2 (see ). If is a bounded open connected subset of with Lipschitz continuous boundary, then .
Proof. The interior of is the largest open set contained in the set . Moreover, . It follows that . Next, we show that . Clearly, . We now show that if , then .
Suppose and . We need to show that any open set containing contains an element not in . We first note that by definition of domain, there exists a neighborhood of and a diffeomorphism . Let be an open set containing with . It follows that is an open set containing and this set is contained in . Hence, there exists such that . This implies that . Thus, contains an element not in , which is a contradiction. Therefore, . We have proven that if , then . Taking the contrapositive of this statement we get that if , then . Since but , we conclude . Thus, . We have shown that and . Therefore, .
3.2. The Perturbation of Identity Technique
Given bounded connected domains and of , where , and a linear space of vector fields , one can deform via the perturbation of identity operator where . For a given we denote the deformed domain to be , which is the image of under .
Throughout the paper, we use the usual infinity norms in the spaces , , and , where is a compact subset of . In addition to this, we also denote the Frobenius norm of to be This norm and the infinity norm of the matrix can be related as This can be shown easily. One can also show that if and , then the vector is bounded in . In fact, and the proof is trivial. Finally, the symbols or will refer to the usual Euclidean norm.
The Perturbed Domain . The domains that are considered in this work are of annulus type with boundary , which is the union of two disjoint sets and , referred to as the fixed and free boundaries, respectively. These domains are obtained through the operator defined in (9), where belongs to , which is defined as For , we obtain the reference domain , with a fixed boundary and a free boundary .
The main objective in this subsection is to show that is a diffeomorphism from to for sufficiently small . To verify this, we need the following results, which are given and proven in .
Theorem 3. If is a bounded, open, connected set in such that and is a continuous injective mapping from to , then
Theorem 4. Suppose(1) is a bounded, open, connected set in such that ,(2) where is injective,(3) such that Then (i) is a homeomorphism (i.e., is a bijection, is continuous, and is continuous),(ii) is a -diffeomorphism (i.e., is a bijection, , ),(iii), .
We also consider the following property of a domain, which is also found in [17, page 52].
Lemma 5. If is a bounded, open, connected subset of having a Lipschitz continuous boundary, then there is a number such that, for any given points , one can find a finite sequence of points , , satisfying the following properties: (a), for , ,(b) for ,(c).
We also recall the useful property of the determinant of the Jacobian of which is given in the next lemma. Here we use the notation
Proof. In general, for -dimensional case, the Jacobian of is given by , where if, , and . By definition of the determinant, we can write as
where refers to the set of all permutations of , is the identity permutation, , and is either (if the number of inversions is even) or (if the number of inversions is odd). We observe that the expression can be written as , where . We also observe that, for , each term of the expression has at least 2 factors that are of the form , . Hence we can write , where is in . All terms of have factors of the form , , and thus we have , which can be written as , where . Combining , , and , we get
with . In particular, for , the determinant is computed as follows:
This verifies . To show we first get the lower bound for . Take For , we obtain On the other hand, by triangle inequality we have Hence, we have shown that there are positive constants and such that for .
Considering the theorems and lemmas presented beforehand, we are now ready to prove the following theorem.
Theorem 7. Let and be nonempty bounded open connected subsets of with Lipschitz continuous boundaries, such that , and is the union of two disjoint boundaries and . Let be defined as in (9) where belongs to , defined as (13).
Then for sufficiently small , (1) is a homeomorphism,(2) is a diffeomorphism, and in particular, is a diffeomorphism,(3),(4).
Proof. First, because is a domain, it follows that by Theorem 2. Second, , and it is injective. Third, it is evident that is because is . For , because vanishes on . For , the determinant of the Jacobian of the perturbation of identity operator is given by (19). By Lemma 6, there exists a , given by (20), such that for all and for . Hence, by applying Theorem 4, we conclude that and for all , and is a homeomorphism. Furthermore, by Theorem 4, we find that is a diffeomorphism. To show that is a diffeomorphism, we are left to show that is Lipschitz continuous. To verify this we use Lemma 5.
Given any two points we choose such that properties (a)–(c) of Lemma 5 are satisfied. For fixed , differentiating the identities and will lead to for all . Thus,
Applying the infinity norm in the space we have
Since is Lipschitz continuous, we have where is the maximum of all Lipschitz constants of for all . Then finally, using the mean value theorem and property (c) in Lemma 5, we obtain Hence is Lipschitz continuous which shows that is a diffeomorphism for sufficiently small . Restricting to , this proves that is a diffeomorphism. (2) is clear because the fixed boundary is invariant under ; that is, since vanishes on . Lastly, using Theorem 3, definition of , (1), and (2), we obtain (3).
Corollary 8. Let and be two domains of with boundary. Then for , where is given by (20), the perturbed domain is also of class .
Proof. Given , we let . Then there exists a neighborhood of and a diffeomorphism such that , , and . We have also shown that defined in Theorem 7 is a diffeomorphism. Since is continuous, is a neighborhood of in . Define . This is bijective because and are bijective. because (hence ) and . Also, .
Next, we note that . Since is injective, we have . Thus by definition of we get We also observe the following:
This shows that is indeed of class .
Remark 9. Theorem 7 and Corollary 8 tell us that the reference and the perturbed domain have the same topological structure and regularity under the perturbation of identity operator for sufficiently small . See Figure 3 for illustration.
Properties of . In addition to (16) we also use the following notations throughout the work:
Remark 10. We note the following observations for fixed, sufficiently small . (1).(2).(3).(4).(5) implies that and are both finite.
We now provide several properties of .
Lemma 11 (see [9, 13, 16, 18]). Consider the transformation , where the fixed vector field belongs to , defined in (13). Then there exists such that and the functions in (16) and (31) restricted to the interval have the following regularity and properties.(1).(2).(3).(4).(5).(6)There is such that for .(7).(8).(9).(10).(11).(12).(13).(14),where the surface divergence is defined by
We provide proofs for properties (3) and (8). The rest can be seen in .
Proof. (3) Suppose , , and . Then
Using Lemma 5, we connect and by a chain , , satisfying (i), for , ,(ii) for ,(iii),and then we get
By reducing if necessary, we can assume without loss of generality that for . This allows us to represent as a Neumann series:
and its norm is estimated as follows:
This shows uniform convergence in and . Hence, for every one can choose a which implies that, for every , whenever . In other words, .
To show that is continuous from to , we only need to show that for every , whenever and . Let . Using (37), estimate as follows: Using the definition of Jacobian of a transformation and the regularity of , we further simplify (38) as follows: where is the Lipshitz constant for and is upper bound for . Taking the maximum of both sides of the inequality for all and using (34) we get where . Thus, for any , we choose , so that if , then . Therefore, .
Proof of property in Lemma 11 is as follows. Given , we have . This implies that Manipulating the left hand side of (41), we get We first work on . Applying the definition of , we get Thus, Similarly, we can write as follows: Hence, we have Suppose is a coordinate function of . By the mean value theorem, we observe that where is a point on the segment joining and , and as tends to infinity, (47) tends to . Thus, Combining (44) and (48), we get which implies that Evaluating (50) at , we get .
3.3. The Method of Mapping
If is defined in and is defined in , then the direct comparison of with is generally not possible since the functions are defined on different domains. To overcome this difficulty, one maps back to by composing it with ; that is, one defines . With this new mapping one can define the material and the shape derivatives of states, the domain and boundary integral transformations, and derivatives of integrals, as well as the Eulerian derivative of the shape functional. This technique is called the method of mapping.
Definition 12. Let be defined in . An element , called the material derivative of , is defined as
if the limit exists in ().
Remark 13. The material derivative can be written as It characterizes the behavior of the function at in the direction .
Definition 14. Let be defined in . An element is called the shape derivative of at in the direction , if the following limit exists in :
Remark 15. The shape derivative of is also defined as follows:
We note that if and exist in , then the shape derivative can be written as
In general, if and both exist in , then also exists in that space.
Domain and Boundary Transformations
Domain and Boundary Differentiation. We recall some results concerning the derivative of integrals with respect to the domain of integration. For the first theorem, it is sufficient to have domains while the second theorem requires domains. For proofs, see .
Theorem 17 (domain differentiation formula). Let and suppose exists in . Then
Theorem 18 (boundary differentiation formula). Let be defined in a neighborhood of . If and , then
where is the mean curvature of the free boundary .
The First-Order Eulerian Derivative
Definition 19. The Eulerian derivative of the shape functional defined in (7) at the domain in the direction of the deformation field is given by if the limit exists.
Remark 20. is said to be shape differentiable at if exists for all and is linear and continuous with respect to .
4. Main Result
In this section we derive in a rigorous manner the first-order shape derivative of the Kohn-Vogelius functional , defined by (7), subject to the Dirichlet and Neumann boundary value problems (BVPs) (4) and (6), respectively. Our strategy bypasses the material or shape derivatives of states. In the derivation, we have employed techniques used in [9, 10, 13] but there is no need to use adjoint variables.
This section discusses the variational forms of the PDEs, the state variables in the perturbed domains, the Hölder continuity of the state variables, and the higher regularity of the solutions to the BVPs. The rest of the proof is presented in the last part of this section.
4.1. Variational Forms of the Dirichlet and Neumann Problems
We recall that we are considering the shape optimization problem (7) where solves the pure Dirichlet problem (4) and solves the Neumann problem (6). As in , we consider the Hilbert space which is endowed with the norm and a linear manifold defined by for .
First, we determine the variational equations for the Dirichlet and the Neumann problems. The variational form of the Dirichlet problem (4) is given by the following.
Find such that It is also well known that (65) has a unique solution.
4.2. Analysis of State Variables in Deformed Domains
We now consider the class of perturbed problems: where solves the pure Dirichlet problem and solves the Neumann problem Here, is the outward unit normal to the deformed free boundary . The variational form of (67) is formulated as follows.
Find such that It is known that (69) has a unique solution.
Remark 21. The function can be referred to as the reference domain by composing with ; that is, and by chain rule of differentiation, we get
Let be the solution of (69). Applying Lemma 16 for all we have Applying (71) and noting that because , we obtain where . Hence, if solves the variational equation (69), then satisfies the variational equation, for all , on , and on .
Now we show that is the unique solution of (74) in . First, we show that is the unique solution to for all . The bilinear form defined by is continuous, because The bilinear form is also coercive. To show this we recall that uniformly on . This is equivalent to the statement Let . So for sufficiently small , , and So is coercive.
Next, we show that the functional is bounded: Therefore, by the Lax-Milgram lemma, is the unique solution to the variational equation This implies the existence of a unique solution of (74) as verified below.
Let . Using (81) we obtain Thus (74) is satisfied. The boundary conditions are also satisfied because on , and and on , both and are zero. To show uniqueness, we let and be solutions of (74). This implies that there exist and such that and , where and are solutions to (81). Taking the difference of and and considering that solution to (81) is unique, we get .
Next, we consider (68) whose variational form is formulated as follows.
Find such that Similarly, if solves the variational problem (83), then solves the variational equation where on .
As shown before, the bilinear form defined by is coercive and continuous. The linear functional defined by is continuous on because By the Lax-Milgram lemma, is the unique solution in of Let . Then, by (87), we get Since , on . Uniqueness of follows from the uniqueness of . Therefore, is the unique solution of the variational problem (84) in .
4.3. Hölder Continuity of the States
We show that and are Hölder continuous on .
Proof. We first prove the uniform boundedness of in for . Since , by using coercivity of we get
Also, by applying (81), we have
Now we take the difference between the weak form of (4) and the variational equation (74), to get Note that . This implies that is in . Note also that . Let . So for sufficiently small , . Now choosing as a test function and by the uniform coercivity of one obtains
If , then the inequality above holds. For we have Hence, Squaring and multiplying on both sides of the inequality give us