Abstract
The paper deals with a coupling algorithm using shape and topological derivatives of a given cost functional and a problem governed by nonstationary Maxwell’s equations in 3D. To establish the shape and topological derivatives, an adjoint method is used. For the topological asymptotic expansion, two examples of cost functionals are considered with the perturbation of the electric permittivity and magnetic permeability. We combine the shape derivative and topological one to propose an algorithm. The proposed algorithm allows to insert a small inhomogeneity (electric or magnetic) in a given shape.
1. Introduction
Shape optimization is a minimization problem where the unknown variables run over a class of admissible domains; then, every shape optimization problem can be written in the form where is the class of admissible domains and is the cost functional; generally, is the solution of a given PDE. Generally, the existence and uniqueness of the solution of such problems are not guaranteed even for simple cases. However, the literature about the subject is abundant, and the techniques used are numerous (homogenization, domain parametrization, geometric shape derivative, topological optimization, …), see [1] and references therein.
In this paper, we focus on geometric shape optimization and topological asymptotic expansion methods, which we briefly recall the principles to propose an algorithm coupling them that we apply to Maxwell’s equations.
Shape optimization problem is a minimization problem where the unknown variables run over a class of domains; then, every shape optimization problem can be written in the form where is the class of admissible domains and is the cost functional.
The underlying principle is the following. Let be an open bounded domain whose boundary is a manifold oriented by the normal vector field outgoing to , a family of open bounded domains of and the time interval. We note that is the cylinder evolution domain, and is the lateral boundary associated to any Let be the set of with for all we consider the follow mapping such that, for each as the form
For each is a one-to-one mapping from to such that (i)(ii) belongs to with (iii) belongs to
Such family is stable under the perturbation We denote by , the perturbed cylinder and , the perturbed lateral boundary.
To each element , we associate , the solution of (11). For any and we set that is said to be shape differentiable in if there exists such that
Definition 1. The shape derivative of to the direction in the unique element verifying
In the same way, we define the boundary shape derivative and the material derivative.
Definition 2. The element is the boundary shape derivative of , where such that
Definition 3. The material derivative of in the direction noted is defined as the limit in when goes to zeros of
For more details on this approach, see [2].
Remark 4. To compute the shape derivatives, we can also use the approach of Simon and Murat in [3], recalled by Henrot and Pierre in [1] and Allaire and Jouve in [4].
We consider a perturbation of the domain in the following sense, for . It is well known that for , sufficiently small is a diffeomorphism from
Definition 5. The shape derivative of at is defined as the Frechet derivative in at 0 of the application ; that is, where is a continuous and linear form on
In classical shape optimization, it is the boundary of the initial domain (or a part of the boundary) which moves for reaching the optimal shape. Thus, the optimal shape has the same topology as the initial one (for example, if the initial domain is simply connected, the optimal one will be also connected). Unlike the case of classical shape optimization, the topology of the design may change during the optimization process, for example, the inclusion of holes. The physical meaning of holes depends on the nature of the design. In the case of structural optimization, the insertion of the holes means simply removing some material, see [4]. In the case of fluid dynamics, creating a hole means inserting a small obstacle, see [5].
Topological sensitivity analysis aims at providing an asymptotic expansion of a shape functional acting on the neighborhood of a small hole created inside the domain. The underlying principle is the following, see also [6]: For a criterion and is the solution of a boundary value problem defined over (generally a PDE), the asymptotic expansion of the cost function can be generally written in the form:
is called topological derivative (or topological sensibility) and provides an information for creating small hole located at . Hence, the function can be used like a descent direction in the optimization process.
The algorithm that we propose combines the creation of holes (insertion of a dielectric object) using the topological gradient on the one hand and on the other hand the update of the edges by making evolve using the derivative of shape.
In Section 2, we present Maxwell’s equations and an existence and uniqueness result and partial regularity under some conditions. The calculous of shape derivatives and topological asymptotic expansion (under small perturbations of the magnetic or electric fields) of the adjoint state is presented in Section 3. Two examples of shape functionals are considered. The main contribution of this paper is the proposition of the algorithm coupling between shape and topological derivative, which is presented in Section 4. Section 5 gives a conclusion and some possible extensions.
2. Maxwell’s Equations and the Preliminary Results
Let be a fixed domain of and be an open bounded regular domain with Lipshitz boundary Let be the electric permittivity and be the magnetic permeability, which are both positive definite Hermitian matrices.
Let and be the electric density through and be the electric density through
The evolution of the electric field and the magnetic field in the space-time cylinder is given by the Maxwell equation:
Here, denotes the normal exterior vector along is the tangential component of and is a positive function. For the initial conditions are given by and
Let
The Maxwell system (15) can be written as
Furthermore, let where If is a Hermitian positive definite matrix, its transpose is denoted by
The well posedness of the Maxwell equations with given boundary and initial data is widely studies in the literature. We refer the literature to [7] for modeling and for the existence and unicity and regularity of the solution, see [8]. However, we recall here on a result which established the existence, unicity, and partial regularity of the Maxwell equations.
Theorem 6. Le and be such that almost everywhere on and almost everywhere on Given and with then there exists a unique weak solution Moreover, there exists a constant such that for
For the proof, see [2].
3. Shape Derivatives and Topological Asymptotic Expansion for Maxwell’s Equations
3.1. The Adjoint Method
To compute the shape derivative, we use the adjoint method. The fundamental propriety of the adjoint method is to provide the variation of a shape functional with respect to a parameter using the solution and an adjoint state which do not depend on the chosen parameter. Numerically, it means that the two systems must be solved for obtaining an approximation of the shape derivative The principle is as follows. We consider the minimization problem with constraints , , the Lagrangian of the system, is defined by
Its derivative with respect to is given by
We set as the adjoint state to obtain as an adjoint equation. It follows
3.2. Shape Derivative of the Cost Function
In the sequel, we turn our attention to the minimization of the cost functional defined by over a collection of open bounded sets with fixed boundaries. is the solution of the initial boundary value problem (11). The following result gives the shape derivative of (23) under constraints (11).
Theorem 7. Assume that are constants of Hermitian definite positive matrices, let be a positive constant, and and The shape functional (23) is Frechet differentiable at in the direction with the Frechet derivative where and are the surface divergences, is the surface curl, is the real part of a complex number, and is the solution of the adjoint initial boundary value problem: As , equation (23) is true for all and we have
Sketch of the proof. The Gâteaux derivation of in the direction is defined by where is the shape derivative of (15). Let be the adjoint state, and then we have To achieve the proof, we have to remove the derivative , by integration by parts. which finishes the proof.
For the complete proof, we refer the reader to [8].
3.3. Generalized Adjoint Method
Let be a complex Hilbert space. For all let be a sesquilinear and continuous form on and be a semilinear and continuous form on such that the following problem has one and only one solution:
Hypothesis 1. Assume that for there exists with and two complex numbers such that
Let be the cost functional. We denote by
Hypothesis 2. For all there exists a linear and continuous form, note , such that designs the norm on
For all let the solution of the so-called adjoint problem
is the solution of (30) for Under Hypotheses 1 and 2, we have the following result, which gives the asymptotic expansion for
Theorem 8. If Hypotheses 1 and 2 are satisfied, the asymptotic expansion to the cost function is given by where
The proof of this theorem is standard in topological optimization, see, for example, [6, 9].
The function is called topological derivative (or topological sensitivity) and provides an information for creating a small hole located at . Hence, the function can be used like a descent direction in the optimization process.
Remark 9. In the following, for all approximations, we use Laplace exterior problem
The function can be explicited by the help of a single layer potential with being the fundamental solution of the Laplace operator, which is given in by
3.4. Problem Formulation and Topological Asymptotic Expansion
The domain perturbation corresponds to the perturbation of electric permittivity and the magnetic permeability. The corresponding perturbed problem writes (for the electric field) where and
For , the corresponding problem is to find such that
The variational problem assisted with (38) writes the following: find such that where
And the variational problem associated to (43) is to find such that where
Lemma 10. Problems (47) and (50) admit one and only one solution. Moreover, there exists a function which goes to zeros with , such that
Proof. For the proof, we refer the reader to [6].
3.4.1. Variation of the Sesquilinear Form
Proposition 11. Let (resp ) solution of (47) (resp. (50)), and then there exists a real number , and a function tending to zero as tends to zero such that
Proof. Let and then is the solution of The rest of the proof follows from the following lemmas.
Lemma 12.
Proof. For the proof, see [6] for the example.
Lemma 13.
Proof. For the proof, see [6, 9] for the examples.
Remark 14. According to the definition of ,
3.4.2. Variation of the Cost Function
We consider two examples of cost functions. Each one corresponds to a specific analysis.
Example 1. The cost function defined by
Proposition 15. Let be the functional associated to the perturbed problem, and then there exists a real and a linear operator such that
Proof. We achieve the proof by setting
Example 2. Here, we focus on the function
Proposition 16. Let be the functional associated to the perturbed problem, and then there exists a real and a linear operator , such that
Proof.
In setting equation (63) writes
Consequently,
Setting
As we obtain the desired result.
The following result gives the asymptotic expansion of the cost function for the perturbation of the electric field.
Theorem 17. Let be the cost functional defined by (58), where is the solution of (47) and is the corresponding solution for . Let has the following asymptotic expansion: where is the solution of the so-called adjoint problem: find such that
Proof. The Lagrangian of the problem , , solution of (43) is defined by Its variation with respect to
It follows form Propositions 11 and 15 (or 16) that Hypotheses 1 and 2 are satisfied. We use the fact that the variation of the Lagrangian is equal to the variation of the cost function; that is, and we use Theorem 8 to conclude.
4. A Coupling Algorithm between Shape and Topological Derivative for Maxwell Equations
In this section, we propose an algorithm coupling the shape and topological derivative for a given shape functional under Maxwell equation as constraints. For the numerical approximation of Maxwell’s equation, one can use the finite differences method, the finite elements method, the Finite Element-Finite Difference Hybrid Methods, or the block pseudospectral method and XFEM method.
The optimization algorithm is summarized in Figure 1. (see also [10] in the case of linear elasticity).
Based on the boundary of the domain being an unknown of the problem, we introduce fixed (in general rectangular or parallelepiped) domain which includes all potential domains . The numerical method approximation requires the introduction of two finite element spaces and on the fictitious domain . As can be a rectangular or parallelepiped domain, the ones can be defined on the same structured mesh . Next, we shall suppose that where is a finite dimensional space of regular functions such that for some integer. The mesh parameter stands for where is the diameter of .
Since we use the topological gradient to create holes (inhomogeneities) during the optimization process, it is possible to start with a shape containing some initial holes or not. A very small penalization is used when solving the direct problem and the adjoint one to avoid the indeterminacy of the rigid motions of the eventual isolated part. Concerning step 4, a new hole of a given radius is created by the simple operation on the level-set function, which can be written on each finite element node where is the level-set function, is its new value, is the radius of the created hole, and is its center.
In step 6, the update of the level set is done directly thanks to the shape derivative applying the following evolution equation for the level-set function: where corresponds to the function in front of in the integral of (24). This evolution equation integrated on a small time interval. In our simulations, the gradient is extended by zero to the complementary of in . However, a smoother extension could be considered. This method is simpler than the classical way which is to integrate a Hamilton-Jacobi equation (see [4]). It seems also to be numerically more robust.
Note that it is convenient to apply a threshold on the gradient to avoid some incoherent values where the shape gradient may have a singularity (corners, transition from Dirichlet to Neumann condition).
To regularize the level-set function, the reinitialization step 7 is considered. It consists classically in solving whose stationary solution is a signed distance. This Hamilton-Jacobi equation is known to admit multiple nonsmooth solutions. Classically, a smooth solution is computed thanks to an upwind scheme. Since the fictitious domain can be a rectangular/parallelepiped domain, it is possible to use a classical upwind scheme on a Cartesian grid. However, to keep the possibility of having a nonstructured mesh, for instance, to proceed to a local refinement, we use a different strategy. Equation (78) is solved on a small time interval integrating the following equation where the nonlinearity is made explicit:
Here is the level-set function at the previous time step, and is given by . The problem (79) is a pure convection one. This problem can be solved, for instance, with the simple Galerkin-Characteristic scheme proposed in [11]. This scheme is unconditionally stable but rather dissipative. The effect is that the level sets are a little smoothed.
5. Conclusion and Extensions
In this paper, we used the adjoint method (respectively, generalized adjoint method) to compute shape derivatives (respectively, topological asymptotic expansions) associated with a given shape functional and Maxwell’s equations. The obtained derivatives (shape and topological) allow us to construct an algorithm, which can permit simultaneously to insert a small dielectric objects (holes) with topological derivative and to control its boundary by using the shape derivative and level set method.
In the forthcoming work, we will intent to apply this algorithm for some applications, for example, for the reconstruction of metallic buried objects.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.