Abstract
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 -Laplacian is investigated. The boundary perturbation method due to Hadamard is applied in Sobolev spaces.
1. 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 whose edge is divided into three disjoint parts:where is the free variable part, is the fixed part of the boundary (Dirichlet boundary conditions), and is also the free part of the boundary on which we apply the efforts (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 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 whenever .
The shape optimization problem is where it remains to define the set of admissible forms.
1.1. 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, is obtained by applying a regular diffeomorphism T to the reference domain . We first define a diffeomorphism set:Then we define a set of the admissible forms obtained by deformation of :Finally we introduce a pseudo-distance on :Thus, we introduce a condition of uniform regularity of the permissible forms, that is to say, open sets close to in the sense of pseudo-distance; for each we pose = such that and where is an imposed volume. The result is the following theorem.
Theorem 1. For all objective functions, the shape optimization problem 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 [4–12]. We will adopt the same representation as F. Murat and J. Simons [1]. In fact, let be an open regular bounded referential domain of and the admissible form class composed of the open sets such as where is a Lipschitz diffeomorphism andwhere is the identity application, and we note defined by
Lemma 2 (See [13]). For all satisfying , the application is a bijection of and .
Definition 3. Let be application from to . One says that it is differentiable with respect to the domain at if the functionis Frechet differentiable at 0 in the Banach space . i.e., , a linear continuous form on , such thatThe linear form depends only on the normal component of on the boundary of .
Proposition 4. Let be a regular bounded open set of . Let be a differentiable application on . If are such that such that on , then the derivative is verifying:
1.3. Derivation of Integrals
Since the compliance is defined by surface or volume integrals then its differentiation devotes the following tools.
Lemma 5 (see [1, 7]). Let be an open set of . Let and .Then iff and one hasOn the other hand iff and one has
Proposition 6 (See [13]). Let be a regular bounded open set of . Let and be an application from to defined by . Then is differentiable in and
Now we move to a lemma on the change of variables in surfaces integrals.
Lemma 7 (See [1, 7]). Let be an open set of . Let be a class diffeomorphism of , and .Then iff and one haswhere is the external normal to .
The surface integral derivative of a function with respect to the domain is given by the following proposition.
Proposition 8 (See [13]). Let be a regular bounded open set of . Let and be an application from to defined by . Then is differentiable in andwhere is the average curvature of defined by .
1.4. 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 be a function defined for all and depending on . It represents a solution of an PDE posed in . In a point belonging to both and , we can calculate the differential : is a linear continuous form in ; it represents a directional derivative in the direction . This definition makes sense in the case where , but it poses a problem if . Then in this case we use the Lagrangian derivative; for this we build the transported on .
By changing variables we obtain .
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 .
There is a relation between these two derivatives .
Proposition 9. Let be a regular bounded open set of . Let be an application from to ; one defines its transpose from to which we suppose to be derivable in and is considered as its derivative. So the application from to defined by is differentiable in and for all one hasIn the same way, if is derivable as an application from to , so the application from to defined by is differentiable in and for all one has
2. Deriving an Equation with respect to the Domain
2.1. Dirichlet Conditions
We consider the following equation with Dirichlet boundary conditions:With a regular bounded open set in , and with
Equation (20) admits a unique solution in .
Remark. For we obtain the linear operator “Laplacian”.
The variational formulation of problem (20) is as follows: we have or ; it implies thatUsing the Green formula we obtainbut ; it implies that on . So the first term equals zero. Then
Proposition 10. Let ; is the solution of the problem (20). We define its transported on byThen the application from to is differentiable in and its directional derivative called Lagrangian derivative , is the unique solution ofwith .
Proof. We consider a test function . Let such that is a solution of problem (20) satisfying . We remark that and are independent of . By a change of variable and the Lemma 5, (23) becomesand since we have ; it implies thatThen (23); then we drift with respect to in 0.
On the other hand the application from to is differentiable in 0.
In fact with . Therefore because is small enough.By using [14] we findOn the other hand .
ThusTherefore, we haveThen afterwards we putThus the Lagrangian of is a solution of the following differential equation:
Remark 11. When we will have:(i)(ii)(iii) Then , the Lagrangian of , will be solution of the following reduced differential equation:with
2.2. Neumann Conditions
We consider the following equation with Neumann boundary conditions (see [15]):where a regular bounded open set in , , , and with .
The variational formulation of problem (35) is to find such that (v represents a test function) or .
It implies that :By using the Green formula we find
Proposition 12. Let ; is the solution of problem (35). We define its transport on by .
Then the application from to is differentiable in on the direction and its directional derivative called Lagrangian derivative of is where is the unique solution ofwhere .
Proof. We make a change of variable where and in the variational formulation (37). We pose noticing that does not depend on . Thus by drifting the variational formulation, we obtain by using Lemmas 5 and 7OrWe have by using [14]AndForSo the transported derivative of at 0 in the direction is the Lagrangian =<> which is the solution of the following equation:where .
3. Optimality Condition
To calculate the optimality conditions of the following problem with , where is the set of admissible forms obtained by diffeomorphism, the cost function is the compliance defined byto reach a target displacement where the function is solution of the boundary problem posed (resp., Dirichlet or Neumann boundary conditions).
We consider the following boundary value problems.
Dirichlet Boundary Conditionwhere .
Neumann Boundary Conditionwhere and .
The problems admit a unique solution .
Theorem 13. Let be a regular bounded open set. The cost function is differentiable and its derivative is is the Lagrangian derivative and also solution ofwith .
Proof. By applying Proposition 4 for the compliance we obtainwhere is the Lagrangian derivative, solution of the PDE.
Remark 14. From extensive literature which deals with optimum condition calculus for problems as we can cite(i)(ii)
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.