Optimal Form for Compliance of Membrane Boundary Shift in Nonlinear Case

In this work, we search the existence shifting compliance optimal form of some boundary membrane, which is not elastic and not isotropic, generating nonlinear PDE. An optimal form of the elastic membrane described by the p-Laplacian is investigated. The boundary perturbation method due to Hadamard is applied in Sobolev spaces.


Introduction and Preliminaries
In this work we will study the geometric shape optimization of forms, where the main idea is to vary the edge position of a form, without changing its topology which remains the same. We use a membrane model as shown in Figure 1. At rest the membrane occupies a reference domain Ω whose edge is divided into three disjoint parts: where Γ is the free variable part, Γ is the fixed part of the boundary (Dirichlet boundary conditions), and Γ is also the free part of the boundary on which we apply the efforts ∈ (Γ ) (Neumann boundary condition). The three of parts of the boundary are supposed to be nonzero surface measurements, as we suppose that the free boundary variable Γ responds to homogenous Neumann condition. So the vertical displacement is the solution of the following membrane model: We want to minimize the compliance defined by (Ω) = ∫ Γ whenever ∈ Ω. The shape optimization problem is inf Ω∈ (Ω) where it remains to define the set of admissible forms.

Existence under a Condition of Regularity.
The main idea of this section is to apply a regularity constraint on all the admissible forms , to demonstrate a result of existence of optimal forms. The results and demonstrations are mainly due to F. Murat and J. Simons [1,2]. It rests on a very significant restriction of ; in other words, Ω is obtained by applying a regular diffeomorphism T to the reference domain Ω 0 . We first define a diffeomorphism set: Then we define a set of the admissible forms obtained by deformation of Ω: Thus, we introduce a condition of uniform regularity of the permissible forms, that is to say, Ω open sets close to Ω 0 in the sense of pseudo-distance; for each > 0 we pose The result is the following theorem.

Theorem 1. For all objective functions, the shape optimization problem inf Ω∈
(Ω) admits at least a minimum point.

1.2.
Derivation from the Domain. The boundary variation method that we study is a classical idea well known and used before by Hadamard [3] in 1907 and many others as [4][5][6][7][8][9][10][11][12]. We will adopt the same representation as F. Murat and J. Simons [1]. In fact, let Ω 0 be an open regular bounded referential domain of R N and the admissible form class (Ω 0 ) composed of the open sets such as Ω = (Ω 0 ) where is a Lipschitz diffeomorphism and where is the identity application, and we note Ω = ( + )(Ω 0 ) defined by Lemma 2 (See [13]). For all ∈ 1,∞ (R N , R N ) satisfying ‖ ‖ 1,∞ (R N ,R N ) < 1, the application = + is a bijection of R N and ∈ . Definition 3. Let be application from (Ω 0 ) to R. One says that it is differentiable with respect to the domain at Ω 0 if the function is Frechet differentiable at 0 in the Banach space 1,∞ (R N , R N ). i.e., ∃ , a linear continuous form on 1,∞ (R N , R N ), such that The linear form (Ω 0 ) depends only on the normal component of on the boundary of Ω 0 .

Proposition 4.
Let Ω 0 be a regular bounded open set of R N . Let be a differentiable application on Ω 0 .

Derivation of Integrals.
Since the compliance is defined by surface or volume integrals then its differentiation devotes the following tools.
International Journal of Mathematics and Mathematical Sciences 3 The surface integral derivative of a function with respect to the domain is given by the following proposition.
Proposition 8 (See [13]). Let Ω 0 be a regular bounded open set of R . Let ∈ 2,1 (R ) and be an application from where is the average curvature of Ω 0 defined by = V .

Derivation of a Domain Dependent Function.
In this section we try to derive a function depending on the domain; for this we use the Eulerian or Lagrangian derivative. The second is a more reliable concept than the first. Let ( , Ω) be a function defined for all ∈ Ω and depending on Ω.
It represents a solution of an PDE posed in Ω. In a point belonging to both Ω 0 and Ω = ( + )(Ω 0 ), we can calculate the differential (Ω, ): is a linear continuous form in ; it represents a directional derivative in the direction . This definition makes sense in the case where ∈ Ω, but it poses a problem if ∈ Ω 0 . Then in this case we use the Lagrangian derivative; for this we build the transported ( ) on Ω 0 .
To arrive at the derivative Lagrangian by drifting ( , ) with respect to is a linear continuous form in ; it represents a directional derivative in the direction .

Proposition 9.
Let Ω 0 be a regular bounded open set of . Let (Ω) be an application from (Ω 0 ) to 1 (R); one defines its transpose from 1,1 (R ) to 1 (R) ( , ) = (( + )(Ω 0 ) ( + )) which we suppose to be derivable in 0 and is considered as its derivative. So the application 1 from (Ω 0 ) to R defined by 1 In the same way, if ( ) is derivable as an application from Remark. For = 2 we obtain the linear operator "Laplacian". The variational formulation of problem (20) is as follows: Using the Green formula we obtain Proof. We consider a test function We remark that and are independent of . By a change of variable = ( ) and the Lemma 5, (23) becomes and since = + we have ∇ = + ∇ ; it implies that Then (23)⇐⇒ ∫ Ω 0 ( , )∇ = ∫ Ω 0 | ∘ |. | det ∇ | ; then we drift with respect to in 0.

Optimality Condition
To calculate the optimality conditions of the following problem inf Ω∈ (Ω) with = {Ω ∈ (Ω 0 ) and ∫ = 0 }, where (Ω 0 ) is the set of admissible forms obtained by diffeomorphism, the cost function (Ω) is the compliance defined by to reach a target displacement (Ω 0 ) ∈ (R N ) where the function (Ω) is solution of the boundary problem posed (resp., Dirichlet or Neumann boundary conditions).
We consider the following boundary value problems.

Neumann Boundary Condition
where ∈ 1, (R ) and ∈ 2, (R ). The problems admit a unique solution (Ω). Proof. By applying Proposition 4 for the compliance we obtain

Theorem 13. Let Ω 0 be a regular bounded open set. The cost function (Ω) is differentiable and its derivative is
where is the Lagrangian derivative, solution of the PDE.

Remark 14.
From extensive literature which deals with optimum condition calculus for problems as inf Ω∈U (Ω) we can cite So, to calculate the gradient of each compliance we use the same argument by the propositions [13].

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.