TOPOLOGICAL DERIVATIVE - THEORY AND APPLICATIONS

. The paper is devoted to present some mathematical aspects of the topological derivative and its applications in different fields of sciences such as shape optimization and inverse problems. First the definition of the topological derivative is given and the shape optimization problem is formulated. Next the form of the topological derivative is evaluated for a mixed boundary value problem defined in a geometrical domain. Finally, an example of an application of the topological derivative in the electric impedance tomography is presented.


Introduction
The Topological Derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbation [11,12]. It represents the variation of the shape functional when the domain is perturbed by holes, inclusions, defects or cracks. The form of the Topological Derivative is obtained by the asymptotic analysis of a solution to elliptic boundary value problem in singularly perturbed domain combined with the asymptotic analysis of the shape functional all together with respect to the small parameter which measures the size of the perturbation. The definition of the Topological Derivative was introduced by Sokołowski and Zochowski in 1999 [13,14]. Since then, the concept became extremely useful in the treatment of a wide range of problems [1,3,6]. Some tools of asymptotic analysis that allow to evaluate the form of the Topological Derivative was given in [8,9].
Over the last decade, topological asymptotic analysis has become a broad, rich and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields [2,5,6], such as shape and topology optimization, inverse problems [3,4], imaging processing [1], mechanical modeling including synthesis and/or optimal design of microstructures, fractures mechanics sensitivity analysis and damage evolution modeling.

Topological Derivative in Shape Optimization
The Topological Derivative evaluated for a given shape functional defined in a geometrical domain and dependent on a classical solution to elliptic boundary value problem is a principal tool in Shape Optimization. It represents the variation of the energy functional while the domain is singularly pertourbed by introducing a small hollow void. Let R n  , n = 2, 3 an open set with    local Lipschitz boundary of  , see Fig. 1 (left). Shape optimization problem consist in finding a boundary  of the geometrical domain  which minimizes a given (shape) functional J (  ) (e.g. weigh of the structure), and subject to some supplementary conditions on volume, energy or displacement on the boundary. The conditions imposed on  can concern the following properties:  regularity of the boundary -usually  is locally Lipschitz.

Sobolev Spaces
Let us introduce some notations for the functional spaces, which are necessary for the analysis of shape optimization problems. (i) We denote by D(  ) a space of test functions in  . Thus, for an open set R , 2,

Problem formulation
The shape optimization problem is usually defined as a minimization of a given shape functional. The shape functional can be written as an integral over the domain  or on the boundary  of a function which depends on a solution ( ; ) ux  of some boundary value problem.
Suppose that the function  is a solution to the following boundary value problem: and ND      boundary of the domain  . Consider the following shape functional: where :  × ℝℝ is a function of class 1 C defined in  × ℝ and depending on the solution ( ; ) ux  to the Boundary Value Problem (1). The shape optimization problem can be written as: In order to decrease the values of the shape functional ()  J we have two possibilities to change the shape of the domain  : (a) deformation of the boundary of the domain produced by changes at the boundary governed by shape derivative (cf. Fig. 2 (left)). (b) perforation of the domain by creating small holes inside  -topological changes governed by topological derivative (cf. Fig. 2 (right)).

Topological derivative
Changes of the topology of the domain  are made by introducing a small hole inside  . Such new domain is called a singular geometrical perturbation of the domain  and is defined as , where   is a (very small) subset of  created at point O (origin) and of size  such that [11,13]. In the modified domain ()   (see Fig. 2 (right)) the boundary value problem (1) is redefined in the following way: Note that in this problem, the boundary  is decomposed into two parts, one part is called D  on which the Dirichlet boundary condition is imposed, and the other part N  with the Neumann boundary condition.
Here, for 0   we have ()     and we define a funtion of parameter 0 is known O, then we can expand the function j in the Taylor series and get the following equality: (6) can be rewritten in the following form: TO takes place for the Neumann boundary conditions on    .

Shape derivative
Variations of the boundary    are made using the so-called shape gradient. The shape gradient defined as follows [12]:  is a deformation of the domain  .
 is a small perturbation of  and is defined as an image of  via the mapping t T . The mapping is given by the flux of the vector field : ℝ × ℝ 2 ℝ 2 , which is defined as: where (•,•) ∈ 1 ([0, ); 2 (ℝ ; ℝ )), Thus, in (10) we have the initial value ∈  ⊂ ℝ 2 and a position of a particle ( ) ( ) tt x t T X   such that we can write: The following algorithm can be applied in order to solve a traditional shape optimization problem. Here we present a scheme of the procedure that allows to find an optimal shape of a domain. Numerical method of finding a solution to a shape optimization problem which uses the topological derivative and the method of level set was presented in [2]. In [16] the numerical method of shape optimization for non-linear elliptic boundary value problem was described in details.
Step 1. Define a domain  for its shape optimization.
Step 2. Define a shape functional ( ; ) Ju  depending on the solution u to a boundary value problem given in the domain  .
Step 3. Solve the boundary value problem in the domain  .
Step 4. Create small holes in the domain  and/or deform the bounadry of the domain in order to minimize the shape function. In such modified domain solve the boundary value problem.
Step 5. Check the value of the shape function. If it is minimal, exit the process, if not return to Step 4.

Evaluation of the topological derivative for a mixed boundary value problem
The form of the topological derivative is obtained via asymptotic analysis of a boundary value problem and of an energy functional. The definition of the topological derivative was introduced in [13,15]. Some notations of the asymptotic analysis that allow to evaluate the form of the topological derivative was given in [9,10]. In this paper a mixed boundary value problem is considered, the domain decomposition is introduced in order to find the explicit form of the topological derivative.

Problem formulation
Let ℝ , N=2, 3 be a bounded domain with a smooth boundary    , see Fig. 4 (left). In such domain we define the following elliptic boundary value problem: Next, we define the energy functional ⟼ ℝ as: where u:  ⟼ ℝ is a solution to the boundary value problem (12): a Hilbert space, then the elliptic equation can be written in the following variational formulation with a test function Taking u   we get from (13) We can also suppose that the solution u  is free on the boundary   of a small inclusion; thus the corresponding Neumann boundary value problem has the following form: For the Neumann condition 0 n u   on the boundary   of a hole, we have: on the boundary   of a small hole.

Domain decomposition
According to [13], in order to get the form of the topological derivative we introduce the so-called domain decomposition of   In the ring ( , ) CR (cf. Fig. 7 (right)) we consider: ( , ) in ( , ) , 0 on , on .
By the Green formulae 1 , the variational formulation of the problem is the following (taking w  as a test function): Thus: The term on the left hand side is the energy (if we know the asymptotic expansion of the energy functional with respect to  , then we know also the norm of A  and the first term of the asymptotic expansion of the energy functional).
From (30) we have the following properties of the operator  The operator A  is linear and symmetric since: Let us consider now the term From the properties of Hilbert space (1/ 2) () R H  we have that: and from (37) and (38) with w  a solution of the boundary value problem (27). Let us now find the asymptotic expansion of the energy function (42) in order to determine the first term of the expansion. We suppose that the harmonic function w has the following Fourier series: and then: Let us observe that the second integral in (53) disappear according to the ortogonality of polynomials and for the first integral we have Let us consider the third integral, we get:

Example: Electrical Impedance Tomography
Electrical Impedance Tomography (EIT) is a non-destructive imaging technique which has various applications in medical imaging, geophysics and other fields.
Its purpose is to reconstruct the electric conductivity and permittivity of hidden objects inside a medium with the help of boundary field measurements. This part was prepared based on the papers of M. Hintermuller, A. Laurain and A. Novotny [3,4].

Fig. 9. Signal propagation in Electrical Impedance Tomography
Let us denote by  a bounded domain in ℝ , ≥ 2 being the background medium with  its smooth boundary where the currents are applied. Assume that  contains material with electrical conductivity 0 ( ) 0 q x q  . Then the electrical potential u(x) satisfies: and conductivities q 1 and q 2 , respectively, so that      with 1    (see Fig. 8). We then have = 1 ⃗  1 + 2 ⃗  2 . Due to the particular form of q the regularization term becomes: where ()  P stands for the perimeter of 1  .
Therefore, the problem is reduced to solving the following problem which depends only on 2  and the scalar values q 1 , q 2 : where u i is the solution of: Further, m i is the boundary measurement corresponding to f i . In order to fulfill the compatibility conditions required for the Neumann boundary conditio (7), the measurements must satisfy: Since the solution of the Neumann problem (62)-(63) is not uniquely defined, we impose the condition: in order to obtain uniqueness. We also introduce the functional: where q 1 , q 2 are now assumed to be fixed. Referring back to (61) we clearly see that it contains 2  as an unknown quantity. Hence, (61) represents a shape optimization problem.

Topological derivative
Now we assume that the domain 1  is a small ball of radius  to get uniqueness.
The following form of the topological derivative was obtained in the EIT problem [3]: with Some numerical results are presented below. For more details we refer reader to [3], where the asymptotic analysis was provided for the EIT problem and the analytical for of the topological derivative was developed.