# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 1 Introduction  
The dynamic analysis of damaged domains is essential in several engineering applications, such as the study of impacts [1; 2], seismic and hydraulic engineering [3; 4], and structural health monitoring with guided waves [5; 6]. Despite the progress accomplished in the last few decades, these technologies still involve a substantial computational cost, which limits their reach [5]. Typically, the necessity of representing high-frequency eigenmodes of a structure calls for high-resolution models, i.e., a fine discretization in both time and space. The discontinuous and/or singular character of features such as cracks or voids exacerbates these issues by complicating the modeling process, impacting the performance of traditional finite element (FE) models and severely affecting the conditional stability of explicit solvers [7; 8; 9]. In tackling these challenges, the engineer must adopt strategies pertaining to three main aspects of a model, which, as expressed by the underlying physics of wave propagation, are inevitably interlinked. These are: (i) a solution space appropriate for the dynamic phenomena of interest (space discretization); (ii) an efficient and versatilemethod to represent localized features such as defects; and (iii) an efficient and stable solution to the equations of motion (time discretization/integration). For time integration, explicit solvers [10; 11; 12] can often outperform unconditionally stable, implicit [13; 14; 15] ones, due to their higher efficiency in terms of both memory storage usage and the number of operations. They require, however, the enforcement of stability conditions [16], as well as a diagonal mass matrix [17]. Although stability conditions in the form of a critical time step are coherent with the need for a fine time discretization, severe restrictions leading to excessive computational costs can arise from the introduction of very small elements in the meshing of details, as well as from discontinuities related to damage or interfaces. Since these features are often localized, possible remedies involve the adoption of leap-frog [18; 19] or implicit-explicit (IMEX) [20; 21; 22] solvers. These strategies allow one to restrict the use of smaller time steps and implicit algorithms, respectively, to selected regions of the mesh, thus maintaining good performance for the majority of the domain, where explicit solvers with larger time steps can still be used. Secondly, mass lumping is a key aspect of explicit algorithms, as analysts are often willing to dispense with the variationally consistent formulation of the mass matrix in favor of accelerated solutions. With the exception of optimal lumping [23; 24; 25], it should be kept in mind that these methods cannot guarantee the convergence properties associated with the variational formulation and must therefore be intensively scrutinized. Concerning space discretization, efficient methods for problems with smooth solutions can be achieved via high-order shape functions, as in the p-version of the FEM (p-FEM) [26; 27; 28], isogeometric analysis (IGA) [29; 30; 31], or the time-domain spectral element method (SEM) [25; 32; 33; 34]. In the context of elastodynamics, these methods demonstrate similar capabilities in terms of evaluating the weak form in a variationally consistent sense [35]; however, only the SEM in a Gauss-Lobatto-Legendre (GLL) nodal configuration [36; 37; 38] offers a diagonal mass matrix within this formulation: an instance of optimal lumping, which is achieved via the nodal quadrature method [17; 23; 24]. This enables the deployment of explicit solvers without compromising accuracy, thus resulting in substantial performance gains. For these reasons, the SEM in conjunction with an explicit solver has seen widespread adoption in dynamic analysis [25; 34; 37; 39; 40].