## 3 Imaging Method and Configuration  
### Topological Energy Imaging Computation  
The topological energy imaging method comes from the optimization for inverse problems [33] and was first introduced by [PERSON] [22]. This method is defined as a minimization problem of the mathematical distance between an unknown inspected medium and a simulated reference medium. One can assume that the reference undamaged medium has the same material properties as the damaged unknown one. In our case, the simulation is achievable and workable using a finite-element model with Comsol Multiphysics(r).  
The principle of the method is to minimize a cost function defined as the distance between the two media so that the reference medium, by inserting defects at the right locations, converges to the unknown one. The mathematical proofs are not presented in this paper and can be found in [22,33]. Nevertheless, a summary of the steps needed in the application of the topological energy method to obtain a map of the unknown medium is proposed (Figure 5 illustrates the steps in the application of the topological imaging method in a general case of bulk waves in a homogeneous material):  
1. Measurement in the experimental unknown medium of the acoustic field \(p_{exp}\left(  
\(p_{exp}\left(\overrightarrow{r},\ t\right)\) and \(p_{ref}\left(\overrightarrow{r},t\right)\). The acoustic field \(p_{adj}\left(\overrightarrow{r},\ T-t\right),\ t\in[0,T]\) is then measured in the whole medium and on the surface \(\Gamma_{mes}\).
4. Evaluation of the topological gradient defined by the limit conditions of the problem and the material properties, and \(p_{ref}\) and \(p_{adj}\) that are respectively the solutions of the direct and the adjoint problem.  
As said previously, the topological imaging method consists of an optimization; the optimization is iterative and can become very time consuming. A Fast Topological Imaging Method (FTIM) is used in [22; 29] to overcome this problem and gives reliable results. Only the first iteration of the algorithm is computed. The result of this method is an image of the medium and is given for each point \(\overrightarrow{r}\) by:  
\[ET\left(\overrightarrow{r}\right)=\int\left|p_{ref}\left(\overrightarrow{r},t \right)\right|^{2}\left|p_{adj}\left(\overrightarrow{r},\ t\right)\right|^{2}dt \tag{4}\]  
Figure 5: Diagram of the general principle of the topological imaging method in an isotropic medium [34]. Reproduced with permission from [PERSON]. [PERSON].  
In this equation, the square of the acoustic fields is used to enhance the high value of the topological energy on the final image.  
This Fast Topological Imaging Method is used here.  
One can interpret the topological energy as a temporal correlation between two simulated wave fields (direct and adjoint). The direct field corresponds to the propagation of the waves in the undamaged medium. The adjoint problem corresponds to the field induced by the backward propagation of the acoustic signature of the difference between the reference medium and the experimental medium (with defects). The topological energy between those fields takes maximum value at the location of the defects, where the two fields coincide.