## Appendix A Hybrid Method Description  
In this appendix, a quick description of the hybrid method is given; more theoretical details can be found in [42; 52]. The adopted strategy consists of coupling a semi-analytical propagation model to a scattering numerical model. The main principle of such coupling is to use the computation of the incident field from CIVA by applying the beam computation module [50], and to compute the interaction with the flaw using a dedicated scattering model. The echo-response from the defect is then given by the [PERSON]'s reciprocity principle [51], which links healthy (incident field) and damaged (defect scattering) components. Indeed, in the time domain, the elastodynamic response of a defect is expressed as:  
\[S_{\Gamma_{F}}=\int_{\partial\overline{B}_{\Gamma_{F}}}\underline{y}^{tot,E}* \left(\underline{\sigma},\underline{u}\right)^{inc,\ R}-\left(\underline{ \sigma},\underline{u}\right)^{tot,E}*\underline{y}^{inc,R}d\Gamma \tag{1}\]  
where the involved displacements \(\underline{y}\) and stresses \(\underline{\sigma}\) are defined below, \(\underline{u}\) refers to the normal integration surface (surrounding the flaw and depicted in dash blue in Figure 1) and indices \(E\) and \(R\) denote two states. \(E\) quantities correspond to the ultrasonic fields radiated by the emitting probe in the presence of the defect and \(R\) quantities are linked to the field radiated by the receiver (used as a fictitious emitter) in healthy components (without flaws). The total displacement field is defined as follows for each state as the sum of the incident and diffracted fields (similar notations/decompositions are employed for the stresses also involved in Equation (A1)):  
\[\underline{y}^{tot,\alpha}=\underline{y}^{inc,\alpha}+\underline{y}^{ diff,\alpha},\;\;\;\alpha\in\{E,R\}\] (A2)  
In the case of cracks, a stress-free boundary condition applies:  
\[\underline{\sigma}^{tot,E}\cdot\underline{n}|_{\Gamma}=0,\] (A3)  
and Equation (A1) reduces to:  
\[S_{\Gamma_{F}}=\int_{\Gamma_{F}}\left(\underline{y}^{inc,E}+\underline{y}^{ diff,E}\right)*\left(\underline{\sigma},\underline{n}\right)^{inc,\;R}d\Gamma\] (A4)  
The two incident fields involved in the previous Equation (A4) are calculated by the ray-based asymptotic model [50], whereas the diffracted field in state \(E\) is obtained by a numerical model. In the latter calculation, the input data of the scattering model is obtained in an area of interest associated with the flaw (around the defect) using the ray-based beam computation model in the healthy part. The response of the flaw to this incident field is then computed using the numerical dedicated model (high order spectral finite elements). The advantages of such an approach are that the flaw response does not disturb the incident field calculated by the ray method. In practice, the defect should be at sufficient distance from the component edges to not interact with them. Otherwise, it is necessary to consider the defect and the component edges in the diffraction pattern. Computational performances are significantly enhanced by the use of high-order spectral finite elements [43; 44] defined on hexahedral meshes. This numerical method is widely spread in the community of numerical solutions of transient high-frequency wave propagation problems since they combine the flexibility of finite element methods and the performances of standard finite differences by allowing a fully explicit numerical scheme thanks to a diagonal mass matrix--in the literature, this technique is referred to as the mass-lumping technique. By using a second-order leap-frog time discretization, a fully explicit numerical scheme is finally obtained and transparent boundary conditions are performed using the so-called perfectly matched layer (PML) formulation. Moreover, by allowing high-order polynomials to represent the solutions, the spectral finite elements require less discretization points to reach a given precision, which is a major asset when simulating 3D configurations.