# Computational Vibro-Acoustic Time Reversal for Source and Novelty Localization
## 3 Governing Equations of Structure and Acoustic Mediums and Their Coupling  
We consider an elastic solid at equilibrium occupying the domain \(\Omega_{s}\). The unknown displacement field of the solid is denoted by \(u^{s}\), while linearized associated stress and strain tensors are indicated by \(\varepsilon(u^{s})\) and \(\sigma(u^{s})\), respectively. The equation of motion for the  
Figure 1: A schematic representation of the computational time reversal experimental set-up. On the left, a schematic of the healthy configuration is shown. On the right the same configuration is presented, including the novelty on the structure indicated with the dashed-line beam. Red and blue dots are assumed to be pressure and kinematic response recorders, respectively. solid, fixed on some part \(\Gamma_{u}\) of the boundary and subject to a given force density \(F^{d}\) on the rest boundary part \(\Gamma_{t}\), is  
\[\mathrm{div}\sigma(u^{s})-\varrho_{s}\frac{\partial^{2}u^{s}}{ \partial t^{2}}=f\text{ in }\Omega_{s}, \tag{1a}\] \[u^{s}=0\hskip 56.905512 pt\text{ on }\Gamma_{u},\] (1b) \[\sigma(u^{s})n^{s}=F^{d}\hskip 56.905512 pt\text{ on }\Gamma_{t}, \tag{1c}\]  
where here we have also assumed silent initial conditions. The acoustic domain is considered to be a compressible inviscid and irrotational fluid, where using the pressure scalar field \(p\), the equation of motion is given as,  
\[\frac{\partial^{2}p}{\partial t^{2}}-c_{f}^{2}\
abla^{2}p=c_{f}^{ 2}\frac{\partial q_{f}}{\partial t}\text{ in }\Omega_{f}, \tag{2a}\] \[p=\bar{p}\hskip 56.905512 pt\text{ on }\Gamma_{p},\] (2b) \[\
abla p\,n^{f}=0\hskip 56.905512 pt\text{ on }\Gamma_{v}, \tag{2c}\]  
\(c_{f}\) being the speed of sound in the acoustic medium and \(q_{f}\) the added mass per unit volume. Similar to the solid, the fluid is assumed to be at rest with silent initial conditions. Now considering there is a part of the solid immersed or in touch with the fluid, we should appropriately consider the coupling of the domains and their responses. We denote that part as \(\partial\Omega_{c}\), where the fluid particles and the solid move together in the normal direction of \(\partial\Omega_{c}\). The normal unit vector external to the fluid is \(n^{f}\) and on the common part \(\partial\Omega_{c}\) is \(n^{f}\)\(=-\)\(n^{s}\), with \(n^{s}\) the normal unit vector external to the solid. We finally denote the unit normal as \(n\), which is equal to \(n^{f}\), while on \(\partial\Omega_{c}\), the following holds,  
\[u^{s}\,n=u^{f}\,n \tag{3}\]  
together with the continuity of pressure,  
\[\sigma(u^{s})n^{s}=-p. \tag{4}\]  
In order to couple the two different physics domains, a Neumann-Neumann approach [14] is followed, where Equation (4) is used for the realization of Equation (1c) and Equation (3) for Equation (2c), respectively. For the latter case, we use the linearized Euler equation, which because of Equation (3) takes the form,  
\[\
abla p\,n=-\varrho_{f}\frac{\partial^{2}u_{s}}{\partial t^{2}}\,n \tag{5}\]  
Without loss of generality, in this work we focus on structures consisting of beams or frame structures lying inside a two-dimensional acoustic field. Therefore, we consider here the case of a solid in the form of one-dimensional beam, as shown in \[\mathrm{div}\sigma(u^{s})-\varrho_{s}\frac{\partial^{2}u^{s}}{ \partial t^{2}}=f\text{ in }\Omega_{s}, \tag{1a}\]0  
The above has been implemented using the SDE, which is a Java-based environment with numerical procedure capabilities [15]. This java numerical methods framework is mostly created by one of the authors. The implementation follows mainly the fluid-structure interaction approach, as implemented in CALFEM [16]. Verification has been further accomplished by comparing with results obtained from commercial finite element multiphysics software [17], with very good agreement. We should mention that in our implementation, we have adopted the assumption of pressure continuity on the beam's line, which is a fair hypothesis for our problems, yet not appropriate to study problems of pressure gaps.