# Mitigating high dimensionality in damage identification for plate-like structures through substructuring with interacting filtering-based approaches

O.A. Shereena, Eshwar Kuncham, Prakash Chand Jain, Laurent Mevel, Subhamoy Sen

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## To cite this version:

O.A. Shereena, Eshwar Kuncham, Prakash Chand Jain, Laurent Mevel, Subhamoy Sen. Mitigating high dimensionality in damage identification for plate-like structures through substructuring with interacting filtering-based approaches. Engineering Structures, 2025, 323 (A), pp.119190. 10.1016/j.engstruct.2024.119190. hal-04849379

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https://inria.hal.science/hal-04849379v1

Submitted on 19 Dec 2024

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# Mitigating high dimensionality in damage identification for plate-like structures through substructuring with interacting filtering-based approaches

O.A. Shereenaª, Eshwar Kunchamª, Prakash Chand Jainᵇ, Laurent Mevelᶜ, Subhamoy Senª,¹

ªi4S Laboratory, Indian Institute of Technology Mandi, Mandi, HP, India

ᵇDRDL Hyderabad India

ᶜUniv. Gustave Eiffel, Inria, Cosys-SII, I4S, Campus de Beaulieu, Rennes, France

## Abstract

Plate-like structures form the essential components of diverse engineering systems, such as building floors, aircraft wings, wind turbine blades, etc. Detection of early onset of damage/s and prevention of sudden catastrophic failures during operation are necessary for maintaining and enhancing the safety standards of such high-dimensional and intricate systems. To detect anomalies in high resolution within the structure, the employment of a high-dimensional support model parameterized with damage attributes becomes imperative within a model-based assessment framework. Further, to support the estimation of consequent high-dimensional parameter sets, the traditional structural health monitoring methods require dense instrumentation over the entire structure for accurate damage location and severity assessment. Targeting an overall cost reduction in the implementation and subsequently circumventing the augmented cost of computation, this article employs a robust Interacting Particle Kalman filtering (IPEnKF)-based approach for health estimation of plate-like structures that is capable of monitoring only a manageable subdomain of interest while being independent of the rest of the domain. Eventually, this approach allows stage-wise health estimation of complex plate-like structures under the limitation of instrumentation.

The proposed approach has been validated through both numerical simulations and real experiments on a scaled model of the plate roughly representing the NASA CRM wing, idealized as a trapezoidal Mindlin plate divided into independent substructures. The proposed method employs the IPEnKF algorithm to estimate each substructure independently by modifying the process model by injecting output towards achieving robustness against unknown substructure boundary forces. By using this subdomain estimation approach, the sensor requirement has been reduced by 60% while maintaining 95% accuracy in damage detection and assessment. Additionally, computational costs have been significantly reduced due to the algorithm's ability to independently monitor the system at its substructure level without depending on the rest of the structure. The method also exhibits excellent reliability, with minimal false alarms. The results emphasize the algorithm's robustness and the essential role of sensor placement in improving the accuracy of damage identification.

**Keywords:** Bayesian filters, Substructuring, Limited data, Aircraft wing, Plate-like structures

<footnote>*Corresponding author; E-mail address: subhamoy@iitmandi.ac.in</footnote>
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# 1. Introduction

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Plate-like structures serve as vital components in various engineering systems, including aircraft, ships, building floors, wind turbine blades, automobiles, and spacecraft. These structures endure complex aerodynamic forces and harsh environments during operation, potentially leading to significant damages such as burn-through, impact dents, corrosion, cuts, and cracks. The integration of sensors permanently installed on these structures enables the collection of damage-sensible data, facilitating real-time monitoring of their current state. SHM plays a crucial role in enhancing safety, reducing maintenance costs, and achieving lightweight design, as indicated by previous studies [1–3]. Early detection of damages is particularly important to prevent the intensification of the fatigue process, which could lead to catastrophic failure of the entire system. Therefore, the assessment of in-service structures for damages holds great significance. The specific focus of this study is on online SHM of complex aerostructures under unknown loading conditions, considering the poorly understood physics of operational scenarios. It is noteworthy that traditional damage detection techniques for aerostructural components often require numerous sensors distributed on their surface, along with a high-fidelity computational model. Despite these challenges, the motivation to pursue research in this field remains strong, as integrated SHM technology has the potential to significantly reduce the cost of maintenance and repair to the owner.

There are several established SHM techniques that are in practice in civil, mechanical, and aerospace industries [3–5]. Modal parameters (frequency, mode shapes, modal damping, etc.) of structures are functions of the physical properties of the structure (mass, stiffness, and damping). Accordingly, any damage-induced change in physical properties will eventually be reflected through variations in modal parameters. Based on this principle, vibration-based methods make use of the structure’s dynamic response characteristics to evaluate the integrity of the structure [6, 7]. The availability of affordable and reliable vibration sensors in the market adds to the widespread usage and applicability of such methods with natural frequency [7–10], mode shapes and their derivatives [11–14], flexibility [15, 16], and strain energy [17–19] as the damage-sensitive feature.

Vibration-based techniques have found applications across various structure types, encompassing rotating machinery, trusses, and frames, where the available instrumentation suffices for effective SHM. These methods have proven particularly useful for damage detection in 1-D continuous beam-type structures, where one-dimensional indices adequately localize damage. However, the application of similar techniques to 2-D plate-type structures is comparatively limited [20]. This limitation arises from the increased sensor requirements and the resulting computational complexity, which make mode-based approaches less effective for localizing or quantifying anomalies. In some studies, genetic algorithms and bat optimization algorithms have been employed, using objective functions based on natural frequencies and mode shapes, respectively, for damage identification in plates [21, 22]. The mode shape curvature method has also been adapted for plate-like structures by using two-dimensional mode shape curvatures and the Gapped Smoothing technique, which eliminates the need for a reference structure [23]. More recently, a coupled framework of artificial neural networks (ANN) and the antlion optimizer was developed, leveraging mode shape derivatives for damage
localization in plates [24].

However, these identification techniques, rooted in modal analysis, are generally incapable of real-time prediction and require high spatial and temporal resolution to maintain estimation accuracy. Furthermore, while vibration-based SHM enables damage detection, even in areas that are difficult to access due to intricate geometry, only a few parameters of lower modes exhibit sensitivity to structural damage. In cases where identifying modal parameters indicative of damage is challenging with a limited sensor array, deploying a substantial number of sensors strategically throughout the structure becomes necessary. This is particularly crucial when precise localization of damage with a narrow margin of uncertainty is required, necessitating the optimal deployment of a dense sensor network over the structure under consideration. For 2-D plates, however, this may sometimes become economically not viable. Consequently, while numerous vibration-based approaches exist for 1-D systems, such methods are less abundant for 2-D systems [24].

In short, monitoring plate-like structures suffers from the cost of instrumentation and computation, complex geometry, and the multidimensional nature of the problem. The identification process performed within a spatial subdomain of interest will alleviate the above-mentioned difficulties. Adopting a substructuring technique that maintains the internal equilibrium while accounting for the boundary forces, will allow stage-wise instrumentation, reducing huge at once sensor requirements considerably. Computational time will also be saved as a consequence, enabling parallel processing of the recorded data, as made possible by the substructuring strategy. Apart from the above, the identification process essentially needs to be within a probabilistic framework, wherein the uncertainty quantification surrounding the estimates inferred from real-life noise-contaminated data is feasible. This aspect is further discussed extensively in the following.

With respect to the uncertainties surrounding ambient forcing, sensor noise, and computational models, the deterministic approaches [25, 26] suffer from estimation inaccuracies which may render them not useful for insightful inferences. In such a context, Bayesian filtering-based approaches are advantageous [27, 28] because of their capability to address the above-mentioned uncertainties in a systematic manner. For the linear state estimation problems, Kalman filter (KF) [29] is a perfect tool but not suitable for parameter estimation problems since such problems are inherently nonlinear in nature. Extended variants of KF could be employed when the state representation models (the system model formulations in system identification) are nonlinear, like in the case of typical parameter estimation frameworks. Such problem statements are typically non-linear, either with the damage parameters augmented state-vector or with a nonlinear observation model (typically high-fidelity Finite Element Method (FEM)) [30]. Hence, typically, any of the nonlinear variants, such as Extended Kalman filter (EKF) [31], Unscented Kalman filter(UKF) [32, 33], Ensemble Kalman filters (EnKF) [34] or Particle filters (PF) [35] have to be adopted for damage location and severity estimation frameworks. PF, despite being good at solving nonlinear formulations requires high computational expenditure. This cost requirement could, however, be resolved within an interacting filtering framework [36–39] wherein the parameters are identified by coupling PF and EnKF in a reduced state space [39, 40]. The parameters are estimated using PF, and EnKF has been used for state estimation depending on the nature of the state model (linear or nonlinear). EnKFs efficiently estimate high-dimensional response states, approximating state covariances through sample means,
which reduces computational complexity while retaining accuracy in linear contexts. By integrating the expensive PF for precision in managing complex (and probably non-Gaussian) distributions of small dimensional parameters, and the inexpensive EnKF for high-dimensional state estimation, the IPEnKF framework reduces overall computational demands without compromising accuracy. This hybrid approach is particularly advantageous for SHM, where computational intensity can affect the promptness of detection. The IPEnKF method ensures robust performance in diverse operational scenarios, enhancing the practicality and feasibility of monitoring and maintaining large, complex structures.

The interacting identification technique when performed on only a small subdomain of interest through substructuring provides a pragmatic solution to the challenge of reducing the usage of an enormous number of sensors otherwise required, and retrieving a probabilistic characterization of the uncertainties and the estimation inaccuracies [41]. The current article attempts to investigate the potential of such a framework in application to aircraft wing-like structures, which are complex and high-dimensional. Consequently, a cantilever plate-like structure is chosen as the apt numerical model for the extensive study.

In brief, this research addresses the gap in existing SHM methods that often require dense and costly instrumentation, particularly in complex plate-like structures. While filtering approaches with substructuring are already in existence for simple and less complex structures, this study specifically advances the field by demonstrating a significant reduction in sensor requirements while maintaining high accuracy in damage detection, in intricate systems. It also provides novel insights into the optimal placement of sensors and the impact of sensor malfunctions, offering a more cost-effective and practical solution for real-world applications. This work intends to bridge the gap between theoretical methods and their practical, scalable implementation in SHM. Following the above, the objective of this paper is to explore the effectiveness and success of Bayesian filtering in addressing challenging problems related to damage location and severity identification in plate-like structures, especially when dealing with a lower instrumentation density. Section 2 introduces the problem statement and outlines the plate model considered in this investigation. The methodology employed to address the problem is briefly explained in Section 3. Section 3.1 provides the derivation of process and measurement equations for the subdomain of interest in the state-space coordinates. These equations are needed for the implementation of the estimation algorithm. After arriving at the formulation expressions, Section 3.3 proceeds to outline the methodology - the joint state parameter estimation algorithm concerning the current application. The numerical experiments conducted on a trapezoidal Mindlin plate constructed from Aluminium, along with the corresponding results, are presented in Section 4. Additionally, Section 5 outlines the laboratory experimental setup used for benchmarking and further damage detection purposes. Finally, the main conclusions are discussed in Section 6.

## 2. Reissner-Mindlin plate

Lightweight structures such as the wing of an aircraft experience transverse loads such as that of the engines, predominantly leading to bending deformation. Consequently, an aircraft wing can be represented as a plate structure [42].
To model such structures, four-node quadrilateral shell elements, with each node possessing three degrees of freedom (DOF), are considered appropriate.

<figure><img src="image_3.png" /><figcaption>Figure 1: A representative plate element</figcaption></figure>

In thick plates, the impact of shear deformation and rotary inertia is noteworthy. The Reissner-Mindlin plate theory ([43]; [44]) is devised for analyzing thick plates, incorporating considerations for shear deformation and rotational inertia effects [42]. Figure 1 illustrates a thick plate element with a thickness of $h$, featuring coordinate axes xyz passing through the middle plane. The displacement components in the respective directions are denoted by $u$, $v$, and $w$. The Reissner-Mindlin theory posits that a straight line perpendicular to the undeformed middle plane remains straight, though not necessarily orthogonal to the middle plane after deformation [45], i.e., $\gamma_{xz} \neq 0$ and $\gamma_{yz} \neq 0$. Figure 2 illustrates the shear deformation in the plate and the rotation of fibers during deformation. The expressions for $u$ and $v$ at a distance $z$ from the middle plane are given by:

$$u(x, y, z) = z\theta_{y}(x, y) \quad \text{and} \quad v(x, y, z) = -z\theta_{x}(x, y)$$

where $\theta_x$ and $\theta_y$ are rotations of the fiber with respect to the X and Y axes.

<figure><img src="image_4.png" /><figcaption>Figure 2: Shear deformation in the plate, with rotation of the straight fibre</figcaption></figure>

## 2.1. Finite element model

The investigation utilizes the MITC4 (Mixed Interpolation of Tensorial Components) element, which was developed by E. N. Dvorkin and K. J. Bathe [46]. This four-node general quadrilateral non-linear shell element is formulated based on three-dimensional continuum mechanics theory. Notably, the element's application is not confined to a specific shell theory, and it proves effective in analyzing both thick and thin plates [46]. In the current context, MITC4
is a judicious selection as the subject of the study is uniformly flat, exhibiting minimal distortions and thus avoiding challenges associated with higher-order shear locking [47]. For future extensive investigations involving structures with curved geometries, advanced elements like MITC4+ can be considered based on the specific demands of the given problem.

## 3. Methodology

Monitoring a plate-like structure with limited sensor data is formulated as a state-parameter estimation problem, which is robust to inputs. The IPEnKF methodology is adopted to perform the estimation process on a series of stand-alone substructures that collectively constitute the entire structure. This approach involves modeling the substructure dynamics, transferring the models into interface force-free form, and then applying the IPEnKF. The state-space models of the substructures are discussed in Section 3.1. The resulting models have been transformed into interface-free form using an output injection technique, detailed in Section 3.2. The final process and measurement models for the substructure, given by Equations 9 and 10, are employed in the estimation framework based on the IPEnKF algorithm for the estimation of the state-parameter of the substructure. IPEnKF methodology is briefly detailed in Section 3.3, with Figure 3 presenting a detailed flowchart.

### 3.1. State-space modeling of the subdomain of interest

The current study considers a linear dynamic system given by the following governing differential equation (GDE):

$$M\ddot{u}(t) + C\dot{u}(t) + Ku(t) = f(t) \quad (1)$$

where $\ddot{u}(t)$, $\dot{u}(t)$, and $u(t)$ denote the acceleration, velocity, and displacement responses at the respective nodes. The system is driven by an external force $f(t)$, when the mass matrix $M$ is assumed to be time-invariant. Similarly, the global stiffness matrix $K$ and the global damping matrix $C$ are also ideally time-invariant.

Monitoring such a system of plate structures demands the identification of a high-dimensional estimable parameter set. Accordingly, aiming to reduce the dimension of estimable parameter sets significantly, this study focuses on an segment-wise health estimation concentrating on a subdomain of interest (say $\Omega^s$) at a time. This subdomain is indicated by the term “substructure”. The overall structural domain is divided into $N_s$ such independent, non-overlapping substructures. Within each substructure, the domain is further characterized based on the location as the interior elements and the exterior elements, resulting in an interior domain and an exterior domain. Specifically, depending on the location of the DOFs associated with these domains, the nodes may be termed as interior or exterior nodes. Consequently, the resulting substructural GDEs are numerically decoupled from the GDE of the entire domain/structure while balancing with the unmeasured boundary forces $g_e^s(t)$ on the exterior DOFs of a substructure.

Following [41, 48], the continuous-time dynamics of the substructure can be described by the following GDE:
$$ \begin{bmatrix} \mathcal{M}_{ii}^s & \mathcal{M}_{ie}^s \\ \mathcal{M}_{ei}^s & \mathcal{M}_{ee}^s \end{bmatrix} \begin{Bmatrix} \ddot{\mathbf{u}}_i^s \\ \ddot{\mathbf{u}}_e^s \end{Bmatrix} + \begin{bmatrix} \mathcal{C}_{ii}^s & \mathcal{C}_{ie}^s \\ \mathcal{C}_{ei}^s & \mathcal{C}_{ee}^s \end{bmatrix} \begin{Bmatrix} \ddot{\mathbf{u}}_i^s \\ \ddot{\mathbf{u}}_e^s \end{Bmatrix} + \begin{bmatrix} \mathcal{K}_{ii}^s & \mathcal{K}_{ie}^s \\ \mathcal{K}_{ei}^s & \mathcal{K}_{ee}^s \end{bmatrix} \begin{Bmatrix} \mathbf{u}_i^s \\ \mathbf{u}_e^s \end{Bmatrix} = \begin{Bmatrix} \mathbf{f}_i^s \\ \mathbf{f}_e^s \end{Bmatrix} + \begin{Bmatrix} 0 \\ \mathbf{g}_e^s \end{Bmatrix} \quad (2) $$

where the subscripts i and e refer to the interior and exterior DOFs respectively. The superscript s signifies the relevant substructure.

From Equation 2, the dynamics associated with the interior domain can further be isolated as:

$$ \mathcal{M}_{ii}^s \ddot{\mathbf{u}}_i^s + \mathcal{C}_{ii}^s \dot{\mathbf{u}}_i^s + \mathcal{K}_{ii}^s \mathbf{u}_i^s = \mathbf{f}_i^s - \mathcal{M}_{ie}^s \ddot{\mathbf{u}}_e^s - \mathcal{C}_{ie}^s \dot{\mathbf{u}}_e^s - \mathcal{K}_{ie}^s \mathbf{u}_e^s \quad (3) $$

The right hand side of Equation 3 represents the resultant external force exerted on the relevant interior DOFs of the substructure under consideration. The motion of these interior DOFs, denoted as $\mathbf{u}_i^s$ can further be decomposed into two components [41], i.e., the quasi-static component ($\mathbf{u}_i^{s,d}$) that represents the rigid body motion of the subdomain, and the relative component ($\mathbf{u}_i^{s,r}$) that signifies the flexible motion within the substructure. Assuming free boundaries, all the forces and the time-derivative terms in Equation (3) vanish, resulting in:

$$ \mathbf{u}_i^{s,d} = -\mathcal{K}_{ii}^{s-1} \mathcal{K}_{ie}^{s} \mathbf{u}_e^s = \tau^s \mathbf{u}_e^s $$

Here, the term $\tau^s(=\mathcal{K}_{ii}^{s-1}\mathcal{K}_{ie}^s)$ acts as a transmissibility constant that establishes a relationship between exterior and interior responses.

Next, substituting the relations $\mathbf{u}_i^s = \mathbf{u}_i^{s,d} + \mathbf{u}_i^{s,r}$, and $\ddot{\mathbf{u}}_i^{s,d} = \tau^s \ddot{\mathbf{u}}_e^s$ into Equation (3), and further ignoring the insignificant off-diagonal damping terms, the substructure GDE [41, 48] is simplified into:

$$ \mathcal{M}_{ii}^s \ddot{\mathbf{u}}_i^{s,r} + \mathcal{C}_{ii}^s \dot{\mathbf{u}}_i^{s,r} + \mathcal{K}_{ii}^s \mathbf{u}_i^{s,r} = \mathbf{f}_i^s - (\mathcal{M}_{ie}^s + \mathcal{M}_{ii}^s \tau^s) \ddot{\mathbf{u}}_e^s \quad (4) $$

Considering $\mathbf{x}^s(t) = \begin{Bmatrix} \mathbf{u}_i^{s,r} \\ \dot{\mathbf{u}}_i^{s,r} \end{Bmatrix}$ as the state vector, the state-space process model with an input $\mathbf{d}^s(t) = \mathbf{f}_i^s(t)$ could be written as:

$$ \dot{\mathbf{x}}^s(t) = \mathcal{F}^s \mathbf{x}^s(t) + \mathcal{B}^s \mathbf{d}^s(t) + \mathcal{E}^s \ddot{\mathbf{u}}_e^s(t) + \mathbf{v}^s(t) \quad (5) $$

where the co-efficient matrices are as follows:

$$ \mathcal{F}^s = \begin{bmatrix} \mathbf{0}_{n_i} & \mathbf{I}_{n_i} \\ -\mathcal{M}_{ii}^{s-1} \mathcal{K}_{ii}^s & -\mathcal{M}_{ii}^{s-1} \mathcal{C}_{ii}^s \end{bmatrix}; \quad \mathcal{B}^s = \begin{bmatrix} \mathbf{0}_{n_i} \\ \mathcal{M}_{ii}^{s-1} \end{bmatrix}; \quad \mathcal{E}^s = \begin{bmatrix} \mathbf{0}_{n_i} \\ -(\mathcal{M}_{ii}^{s-1} \mathcal{M}_{ie}^s + \tau^s) \end{bmatrix} $$

The term $\mathbf{v}^s(t)$ denotes the process noise that accounts for the uncertainties arising from modeling inaccuracies, unaccounted input factors and damping effects, and similar other sources of unaccounted uncertainties. Computationally, $\mathbf{v}^s(t)$ is modeled as a stationary white Gaussian noise (SWGN) characterized by the covariance matrix, $\mathcal{Q}^v$.
The measurement model employs the observation vector $\mathbf{y}^s(t)$ constituted by the measured acceleration data. The measured acceleration response, denoted by $\ddot{\mathbf{u}}_i^s$, could be expressed as a combination of the pseudo-static response component $(\ddot{\mathbf{u}}_i^{s,d})$ and the relative response component $(\ddot{\mathbf{u}}_i^{s,r})$. The measurement model is thus formulated in the following form:

$$ \mathbf{y}^s(t) = \mathbf{B}\{\ddot{\mathbf{u}}_i^{s,r}(t) + \tau^s \ddot{\mathbf{u}}_e^s(t)\} = \mathbf{B}\{\mathcal{H}^s \mathbf{x}^s(t) + \mathcal{D}^s \mathbf{d}^s(t) + \mathcal{L}^s \ddot{\mathbf{u}}_e^s(t) + \mathbf{w}^s(t)\} \quad (6) $$

where the measurement uncertainty is modeled as $\mathbf{w}^s(t) \sim \mathcal{N}(0, \mathbf{R})$. The matrix $\mathbf{B}$ functions as a Boolean selection matrix, delineating the measured DOFs. The coefficient matrices are further given as follows:

$$ \mathcal{H}^s = [-\mathcal{M}_{ii}^{s-1} \mathcal{K}_{ii}^s \quad -\mathcal{M}_{ii}^{s-1} \mathcal{C}_{ii}^s]; \quad \mathcal{D}^s = \mathcal{M}_{ii}^{s-1}; \quad \mathcal{L}^s = -\mathcal{M}_{ii}^{s-1} \mathcal{M}_{ie}^s $$

Considering that the responses are inherently subject to discrete sampling in practical scenarios, the process and measurement models described by Equations 5 and 6, respectively, are rewritten in discrete time [41].

When damage is introduced, the estimation process begins from the healthy state and is gradually updated to reflect the actual stiffness value through data assimilation in the filtering process. The time variation primarily appears through $\mathcal{K}$ and $\mathcal{C}$, and is accounted for in the discrete-time dynamics from now on, while the mass matrix $\mathcal{M}$ remains time-invariant, assuming no mass loss due to damage.

$$ \mathbf{x}_k^s = \mathcal{F}_k^s \mathbf{x}_{k-1}^s + \mathcal{B}_k^s \mathbf{d}_k^s + \mathcal{E}_k^s \ddot{\mathbf{u}}_{e,k}^s + \mathbf{v}_k^s \quad (7a) $$

$$ \mathbf{y}_k^s = \mathcal{H}_k^s \mathbf{x}_k^s + \mathcal{D}^s \mathbf{d}_k^s + \mathcal{L}^s \ddot{\mathbf{u}}_{e,k}^s + \mathbf{w}_k^s \quad (7b) $$

Equation 7 represents the discrete-time model corresponding to the continuous-time model described by the Equations 5 and 6. The time variation appears in the discrete-time dynamics by indexing the matrices, which are functions of $\mathcal{K}$ and $\mathcal{C}$, with respect to $k$. The selection matrix $\mathbf{B}$ has been omitted from the formulation, under the assumption that its influence has been integrated into the discrete-time matrices.

## 3.2. Interface force rejection using output injection technique

Construction of both the process (Equation (7a)) and the measurement model (Equation (7b)) requires the knowledge of the interface forces $\ddot{\mathbf{u}}_{e,k}^s$. Note that the term “interface” refers to the interface between two substructures.

To circumvent the requirement of interface measurements, the adopted approach integrates an output injection method wherein a part of observation data $\mathbf{y}_k^s$ is injected into the state model. For an arbitrary bounded injection matrix $\mathcal{I}_k$, the following relationship holds true:

$$ 0 = \mathcal{I}_k^s(\mathbf{y}_k^s - [\mathcal{H}_k^s \mathbf{x}_k^s + \mathcal{D}^s \mathbf{d}_k^s + \mathcal{L}^s \ddot{\mathbf{u}}_{e,k}^s + \mathbf{w}_k^s]) \quad (8) $$
220 Injecting Equation (8) into the process model (cf. Equation (7a)), and subsequently setting $L_k^s = I - I_k^s H_k^s$, will ultimately result in a null co-efficient matrix for the interface forces $\ddot{u}_{e,k}^s$, with a choice of $I_k^s$ as $I_k^s = E_k^s(H_k^s E_k^s + L^s)^*$, with * denoting Moore-Penrose Pseudo-inverse operation. The detailed transformation is outlined in [41, 49]. Thus, the final form robust to exterior boundary forces is as given in Equation 9.
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$$
\begin{equation}
\begin{aligned}
& \mathbf{x}_k^s = \tilde{\mathbf{F}}_k^s \mathbf{x}_{k-1}^s + \tilde{\mathbf{B}}_k^s \mathbf{d}_k^s + \mathcal{I}_k^s \mathbf{y}_k^s + \tilde{\mathbf{v}}_k^s \\
& \text{where } \tilde{\mathbf{F}}_k^s = \mathcal{I}_k^s \mathbf{F}_k^s, \quad \tilde{\mathbf{B}}_k^s = \mathcal{I}_k^s \mathbf{B}^s - \mathcal{I}_k^s \mathcal{D}^s, \text{ and } \tilde{\mathbf{v}}_k^s = \mathcal{I}_k^s \mathbf{v}_k^s - \mathcal{I}_k^s \mathbf{w}_k^s.
\end{aligned}
\tag{9}
\end{equation}
$$

Pre-multiplying the measurement equation (cf. Equation (7b)) by $\mathbf{T}^s$, computed as the left null space of $\mathcal{L}^s$ (the co-efficient matrix of the unmeasured interface forces, $\ddot{u}_{e,k}^s$ such that $\mathbf{T}^s \mathcal{L}^s = 0$, the measurement model is also transferred to an interface force-free form:
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$$
\begin{equation}
\begin{aligned}
& \mathbf{z}_k^s = \tilde{\mathbf{H}}_k^s \mathbf{x}_k^s + \tilde{\mathbf{D}}^s \mathbf{d}_k^s + \tilde{\mathbf{w}}_k^s \\
& \text{wherein } \mathbf{z}_k^s = \mathbf{T}^s \mathbf{y}_k^s, \quad \tilde{\mathbf{H}}_k^s = \mathbf{T}^s \mathcal{H}_k^s, \quad \tilde{\mathbf{D}}^s = \mathbf{T}^s \mathcal{D}^s \text{ and } \tilde{\mathbf{w}}_k^s = \mathbf{T}^s \mathbf{w}_k^s.
\end{aligned}
\tag{10}
\end{equation}
$$

In brief, the final formulations for process and measurement models described by Equations (9) and (10) are free of interface forces due to the output injection method. This approach eliminates the need for interface force computations. The transformed system of equations serves as the model utilized in IPEnKF for further processing and estimation. The algorithm and its application are detailed in the next section.
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3.3. Employing IPEnKF framework for joint state-parameter estimation

The structural health estimation is further carried out by the joint state-parameter estimation on the modified system robust against the exterior interface forces (Equations (9) and (10)). The parameter to be estimated is the reduction in the flexural rigidity ($EI$) of each element, and is termed the Health Index (HI). The algorithm employed for the above purpose is an interacting filtering technique known as Interacting Particle Ensemble Kalman Filter (IPEnKF) [40], wherein an Ensemble Kalman Filter (EnKF) is embedded inside a Particle Filter (PF). Within the PF environment, a collection of $N_p$ particles (denoted as $\zeta$) is generated corresponding to a set of $p_s$ number of location-based HIs considered as the estimable parameter set. The initial health state is indicated by $(EI)_k^0$, and the HI is given by $\zeta$ such that $(EI)_k^d = \zeta \cdot (EI)_k^0$, with $(EI)_k^d$ corresponds to the damaged health state.
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At every $k^{th}$ iteration, the initial estimates of parameters ($\zeta_{k-1}$) are propagated in time over the state evolution equation using particle filtering. This process involves the use of a set of $N_p$ parameter particles, $\zeta = [\zeta_{k-1}^1, \zeta_{k-1}^2, \dots, \zeta_{k-1}^{N_p}]_{p_s \times N_p}$. Each $j^{th}$ individual particle undergoes evolution by means of a random perturbation $\delta\zeta_k = (1-\alpha)\bar{\zeta}_{k-1}$ about its current position $\bar{\zeta}_{k-1}^j$. Here, $\alpha$ serves as a hyper-parameter that fine-tune the level of turbulence in the estimation of $\zeta$, for the best possible performance.
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Subsequently, the state particles go through further evolution within the EnKF computational loop. Particle evolution is conditioned on the respective estimated likelihood with the innovation term updated by EnKF. As a result, the trajectory of particle evolution becomes independent of the initial distribution of $\zeta$.
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For each $j^{th}$ particle, a total of $N_e$ ensembles are undergoing state evolution in the EnKF loop. The state prediction and the transformed measurement equations for the $i^{th}$ ensemble of the $j^{th}$ particle are as follows:

$$ 
\begin{aligned}
\mathbf{x}_{k|k-1}^{i,j} &= \tilde{\mathbf{F}}_{k}^{i,j} \mathbf{x}_{k-1|k-1}^{i,j} + \tilde{\mathbf{B}}_{k}^{i,j} \mathbf{d}_{k}^{i,j} + \mathbf{I}_{k}^{i,j} \mathbf{y}_{k} + \tilde{\mathbf{v}}_{k}^{i,j} \\
\mathbf{z}_{k|k-1}^{i,j} &= \tilde{\mathbf{H}}_{k}^{i,j} \mathbf{x}_{k|k-1}^{i,j} + \tilde{\mathbf{D}}_{k}^{i,j} \mathbf{d}_{k}^{i,j} + \tilde{\mathbf{w}}_{k}^{i,j}
\end{aligned}
\quad (11) $$

The innovation corresponding to the $i^{th}$ ensemble is indicated by $\epsilon_k^{i,j}$. It is computed as the discrepancy between the predicted value ($\mathbf{z}_{k|k-1}^{i,j}$) and the actual (transformed) measurement. Thereafter, the key quantities such as the ensemble mean ($\epsilon_k^j$) of the innovations, the error covariance ($\mathbf{C}_k^{j,xz}$) between the predicted state and the transformed measurements, and the error covariance ($\mathbf{S}_k^j$) of the transformed measurements are computed.

Subsequently, the EnKF gain ($\mathbf{G}_k^j$) is derived using the expressions $\mathbf{G}_k^j = (\mathbf{C}_k^{j,xz}) (\mathbf{S}_k^j)^{-1}$. The state ensembles are then updated to $\mathbf{x}_{k|k}^{i,j}$, and their ensemble mean $\mathbf{x}_{k|k}^j$ is calculated. These computations are carried out as detailed in [40].

With $\epsilon_k^j$ and $\mathbf{S}_k^j$ calculated, the likelihood of each particle is computed as $\mathcal{L}(\xi_k^j) = \frac{1}{(2\pi)^n \sqrt{|\mathbf{S}_k^j|}} e^{-0.5\epsilon_k^j T \mathbf{S}_k^j^{-1} \epsilon_k^j}$. This likelihood is then utilized to calculate the normalized weight, $w(\xi_k^j)$, for each particle. Finally, particle approximations for states and parameters are estimated as weighted means.

$$ \mathbf{x}_{k|k} = \sum_{j=1}^{N_p} w(\xi_k^j) \mathbf{x}_{k|k}^j \quad \text{and} \quad \xi_{k|k} = \sum_{j=1}^{N_p} w(\xi_k^j) \xi_k^j \quad (12) $$

A pseudo-code for the approach described above is presented in the article [41], with respect to subdomain level health monitoring.

## 4. Numerical experiments

The adaptability, capability, and robustness of the adopted algorithm for monitoring aerospace structures are investigated through numerical experiments. A cantilever Mindlin aluminum plate model is considered for numerical investigations. The material properties of a 10 mm thick plate used in this study are: Young's modulus ($E$) = 70 GPa, Density ($\rho$) = 1700 kg/m³, Poisson's ratio ($\mu$) = 0.3. Figure 4 presents the numerical model with relevant geometric properties. The details of the strategies followed in the numerical simulations are detailed in the following.

The plate is divided into a total of 10 equal-width trapezoidal MITC4 plate elements; each quadrilateral element consists of four nodes, each having three degrees of freedom (one translational and two rotational). As indicated in Figure 4, the substructure of the current study is chosen to be composed of the first five elements ($M_1, M_2, M_3, M_4, M_5$) of the cantilever beam. It should be noted that there are eight interior nodes ($N_3 - N_{10}$), and four exterior nodes ($N_1, N_2, N_{11}$, and $N_{12}$), in the substructure. A stationary white Gaussian noise ($\mathcal{N}(0, 1e^{-3})$) is applied to each nodal DOF for a time window of 30 s, and the responses are generated, which is then contaminated with 1% SNR (Signal-to-Noise-Ratio) noise to match the real field conditions. All eight interior nodes of the substructure are considered to be
<figure><img src="image_5.png" /><figcaption>Figure 3: Schematic representation of the IPEnKF-based state-parameter estimation framework</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4: Schematic diagram of a Mindlin aluminum plate model with its substructure</figcaption></figure>

instrumented, resulting in a total of 24 sensors. In other words, the acceleration data measured from the 24 sensors on the substructure is processed to estimate the parameters. The sampling frequency is chosen to be 50 Hz (with a time-step $\Delta t = 0.02$ s) based on an initial mesh convergence study.

The mesh convergence study is carried out for the first two dominant modes of the numerical model considered. The natural frequencies ($f_1 \le f_2$) are plotted against the number of elements considered along the Y-direction as shown in Figure 5. Based on the above study, it can be observed that any choice for the number of elements greater than 6 will be suitable. Furthermore, a discretization of 10 elements is chosen for optimum stability and convenience when applying and demonstrating the substructuring technique. Apart from the computational cost consideration, it can be remarked that a greater number of elements (say up to 20 elements) could be chosen for a detailed investigation that seeks accuracy improvement. The first five natural frequencies of the 10-element FE numerical model obtained are as follows: [2.36 Hz 9.32 Hz 14.91 Hz 24.80 Hz 37.39 Hz]. The third dominant mode has a frequency value 14.91 Hz, and hence the Nyquist sampling frequency is approximately 30 Hz. The Nyquist rate of 30 Hz allows to choose a sampling frequency of 50 Hz for the numerical experiment. Further, the choice still keeps an optimum number of time points in the purview of computational efforts. Hence, the measurement data is typically sampled at 50 Hz, for the numerical experiments conducted, on a time window of 30 s.

Various parametric studies are conducted to ensure the broad applicability of the framework to complex, multidimensional structural components. Initially, sensitivity to different levels of damage (e.g., 80%, 60%, and 40% damage) is assessed at specified locations on the plate. While *EI* might be assumed uniform throughout an undamaged plate, a change in the parameter dependent on *EI* at a specific location or element indicates damage at that location or element. The parameter estimation problem thus provides an estimate of damage characterized by a change in *EI*, with the corresponding element identified as the damage location. To simulate damage, a percentage reduction in the *EI*
<figure><img src="image_7.png" /><figcaption>Figure 5: Mesh convergence study for the first two modes</figcaption></figure>

of the constituent element is applied. For instance, 80% damage corresponds to an 80% reduction in EI, which results in a health index of 0.2. This health index quantifies the remaining healthy fraction of EI, ranging from 0 to 1.

Secondly, it is essentially required to check the damage identification accuracy at higher measurement noise levels, and at limited instrumentation/missing data scenarios, despite the element complexity. Accordingly, a noise sensitivity study is also carried out. Another relevant concern is sensor malfunction during operation that affects estimation accuracy and yields false positives. Therefore, a third parametric investigation is conducted that focuses on studying the influence of sensor fallouts at varying locations.

In the current study, the **HI**s are considered to be the unknown parameters to be estimated. The initial distribution of the **HI** is assumed to be Gaussian ($N(1, 0.2)$). The tuning parameter, $\alpha$ is set to 0.99 while 1000 particles are used in PF, along with 200 ensembles in EnKF. The internal substructural nodal accelerations alone are considered in the measurement model, for the following numerical investigations, in conjunction with the substructure state-prediction model.

## 4.1. Base-case: 60% damage in a single element

60% damage is induced in the substructural element $M_3$, by reducing EI by 60%, i.e. $EI_{M_3} = 0.4EI_0$ with $EI_0$ denoting undamaged EI. The state estimate for acceleration time history at Node $N_{10}$, corresponding to the translational DOF is shown in Figure 6a in comparison with the measurement data. Figure 6b shows that the estimation error is of the order of $10^{-3}$, which is satisfactorily low and, therefore, acceptable. Despite the high level of damage in an element as high as 60%, the framework adopted here identified the location and the quantity of damage promptly and distinctively, as demonstrated in Figure 7a.

## 4.2. Parametric study: Varying the damage intensities

The damage intensities are further varied by reducing the EI by 40%, 60%, and 80%, representing moderate to severe damage levels, relative to the original value. It is important to note that these extreme values, such as
<figure><img src="image_8.png" /><figcaption>Figure 6: Performance of the substructure algorithm in the base case for predicting the states (response from the translational DOF of node $N_{10}$ is shown; the inset diagram illustrates the magnitude of error.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7: Performance of the substructure algorithm in the base case and under various damage level scenarios (dashed lines represent the respective true values)</figcaption></figure>

80%, are applied to individual elements rather than to the entire structure. Applying such a reduction to the whole structure would imply that it has lost 80% of its strength or capacity, which is practically infeasible. However, testing the algorithm's capability at such extreme levels of damage at the element level is crucial. In a separate study, the algorithm's sensitivity to smaller damage levels is evaluated with appropriate parameter tuning.

As observed from the results presented in Figure 7b, the algorithm accurately identifies both the location and extent of damage for all considered scenarios, without producing false positives. The parameter estimates converge to the true values, demonstrating the robust performance of the algorithm and its applicability to plate-like structures.
### 4.3. Stability test over an extended time span

The stability of the algorithm has been checked by performing the analysis over an extended time window of 60 s. The parameter estimation results are presented in Figure 8a, showing satisfactory and consistent progress over the extended time span. The health parameter estimation has been observed to be nondivergent and smooth even for a longer observation period, without error accumulation, estimation divergence, or instability. Figure 8a presents the temporal variation of particles (HIs) with a 95% confidence interval with respect to the computed mean estimates, corresponding to both the respective damaged element, and a representative undamaged element, in order to provide a better understanding of the proposed estimation approach.

<figure><img src="image_10.png" /><figcaption>Figure 8: (a) Inset diagrams show the estimation variance at different time points for damaged and undamaged elements; (b) Performance of the substructure algorithm under different operating scenarios on the base case with 60% damage in $M_3$ (dashed lines represent the respective true values; regrading the legends, 'undam' stands for the representative undamaged element, and 'dam' stands for the representative damaged element).</figcaption></figure>

### 4.4. Performance against different noise severity levels

The sensitivity of the algorithm to various noise levels, specifically 1%, 2%, 5%, and 10% SNR, was examined using a base case where a 60% reduction in the EI of element $M_3$ was assumed. The results, as illustrated in Figure 8b, demonstrate the robustness of the framework against different levels of noise in the measurement data. The performance of the algorithm exhibits notable fluctuations in the estimates as noise levels increase. The overall accuracy tends to decrease with rising noise levels, reflecting the general trend observed. An initial inconsistency at 10% SNR may be attributed to random variations in the noise sequence. Despite reduced accuracy in damage quantification at 10% SNR, the results for damage localization remain precise, with no false positives detected. This underscores the framework's reliability in identifying damage locations even in high noise conditions.

### 4.5. Performance at small localized damage intensity

The effectiveness of vibration-based SHM approaches is fundamentally tied to how much damage affects a structure's dynamic properties. These methods typically detect damage by identifying changes in these properties. However, when dealing with minor or localized damage that doesn't significantly alter vibrational responses, traditional
non-destructive testing (NDT) methods might be more suitable. Vibration-based SHM techniques are effective for localized damage only if the damage causes perceivable changes in vibration characteristics that are not masked by noise. If the damage doesn't produce a significant change, it can go undetected amidst the noise.

Taking this practicality into consideration, a numerical study was conducted to assess the detection capability for localized damage of 20% in element $M_3$, with measurement noise levels ranging from 0.5% to 5% SNR. The results, presented in Figure 9, demonstrate that higher noise levels progressively obscure the accuracy of damage estimates. This highlights the importance of evaluating an algorithm's ability to mitigate noise-induced masking of damage signatures, particularly when dealing with small, localized damages under higher noise conditions. The study emphasizes how significant noise can obscure the impact of damage on the sampled response, making it undetectable even with an otherwise efficient algorithm.

<figure><img src="image_11.png" /><figcaption>Figure 9: Sensitivity to small, localized damage under varying measurement noise levels in the case of 20% damage in $M_3$ (dashed lines represent the respective true value).</figcaption></figure>

## 4.6. Effect of instrumentation density

In real-world applications, sensor malfunctions, such as failures to function, cessation of operation, or missing sensors, can critically affect the performance and accuracy of structural health monitoring (SHM) frameworks. To address these challenges, it is crucial to conduct a comprehensive numerical investigation to evaluate the effects of sensor failures. This study explores the impact of missing sensors located near internal boundaries, free boundaries, and central nodes, using a base case of 60% damage to the element $M_3$.

The investigation reveals that omitting sensors at nodes near the internal boundary (e.g., $N_9$ and $N_{10}$) has a minimal effect on the performance of the identification framework, as shown in Figure 10a and Figure 10b. In contrast, when sensors near the free end (e.g., $N_3$ and $N_4$) are left uninstrumented, the estimation accuracy deteriorates, leading to false positives, as illustrated in Figure 10c and Figure 10d. This suggests that the operational integrity of sensors located at the free end is critical for accurate damage detection. When the sensor at an interior node, such as Node $N_7$,
<figure><img src="image_12.png" /><figcaption>(a) $N_9$ uninstrumented - 3 sensors missing</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) $N_3$ uninstrumented - 3 sensors missing</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(e) $N_7$ uninstrumented - 3 sensors missing</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) $N_{10}$ uninstrumented - 3 sensors missing</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(d) $N_4$ uninstrumented - 3 sensors missing</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(f) $N_9$ and $N_{10}$ uninstrumented - 6 sensors missing</figcaption></figure>

<caption>Figure 10: Estimation robustness with 60% damage in $M_3$ and cases of sensor malfunctioning at different nodes (dashed lines represent respective true values)(cont.)</caption>

malfunctions, the accuracy of the estimation remains consistent with that observed when all 24 sensors are functioning (cf. Figure 10e).

Further analysis of scenarios with six malfunctioning sensors reveals additional insights. When sensors at two nodes near the internal boundary (e.g., $N_9$ and $N_{10}$) are missing, the estimates become unstable, as depicted in Figure 10f. In another scenario, where two consecutive nodes on the same side of the plate (e.g., $N_7$ and $N_9$) are uninstrumented, there is a failure in accurately estimating the damage, as shown in Figure 10h. Additionally, when sensors at interior nodes (e.g., $N_7$ and $N_8$) malfunction simultaneously, false alarms of damage are observed, as illustrated in Figure 10g.
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Figure 10: Estimation robustness with 60% damage in $M_3$ and cases of sensor malfunctioning at different nodes (dashed lines represent respective true values)</figcaption></figure>

Eventually, it has been perceived that, while the malfunction of sensors near internal boundaries is less critical, maintaining functional sensors at the free end is essential for achieving accurate damage estimation. The presence of multiple uninstrumented nodes can lead to inaccurate estimates and false alarms. For a detailed overview of these findings, please refer to the results summarized in Table 1.

<table><caption>Table 1: Sensor density study for the proposed algorithm (cf. Figure 10)</caption><thead><tr><td>Case No.</td><td>Uninstrumented Nodes</td><td>Accuracy of HI estimate</td><td>False positive occurrence</td></tr></thead><tbody><tr><td>1</td><td>None</td><td>95.11%</td><td>×</td></tr><tr><td>2</td><td>$N_3$</td><td>86.53%</td><td>✓</td></tr><tr><td>3</td><td>$N_7$</td><td>80.94%</td><td>×</td></tr><tr><td>4</td><td>$N_2$</td><td>54.36%</td><td>✓</td></tr><tr><td>5</td><td>$N_8$</td><td>78.87%</td><td>×</td></tr><tr><td>6</td><td>$N_9$</td><td>90.04%</td><td>×</td></tr><tr><td>7</td><td>$N_6$ & $N_8$</td><td>79.79%</td><td>✓</td></tr><tr><td>8</td><td>$N_7$ & $N_8$</td><td>49.78%</td><td>×</td></tr><tr><td>9</td><td>$N_8$ & $N_9$</td><td>72.94%</td><td>✓</td></tr></tbody></table>

Finally, the computational benefits of the proposed substructure estimation approach are also compared to its traditional approach of application, where the entire structure is estimated as a whole. The implementation of the IPEnKF algorithm on a single substructure required 1 hour and 23 minutes using MATLAB R2022b on a 12th Gen Intel Core i5-12400 processor. Accordingly, processing the entire structure, composed of two substructures, took 2 hours and 46 minutes — 40% less time than traditional IPEnKF methods. This reduction could be further enhanced by parallel estimation, where multiple substructures are estimated simultaneously as well as independently rather than sequentially, significantly improving the promptness of detection. Further time reductions could be achieved through parallel processing and GPU acceleration. The proposed method, unlike other Bayesian filter-based SHM approaches, estimates smaller subdomains with fewer sensors and simpler models, maintaining accuracy and precision while benefiting from the parallelization of PF/EnKF processes.
## 5. Experimental validation

A laboratory experiment was further conducted on a trapezoidal steel plate, with one end fixed, aiming to assess the performance of the substructure algorithm in the context of real-life plate structures. The relevant geometric properties used in the experiment are as follows: a span of 1 m, and a thickness of 2 mm. Thickness is measured at different points along the plate, yielding a mean value of 2.006 mm, with a standard deviation of 0.082 mm upon processing. The initial material properties are considered as follows: mass density (ρ) of 7850 kg/m³, elastic modulus (E) of 190 GPa, and Poisson ratio (ν) of 0.303. The theoretical mass of the plate is computed as 3.394 kg. The actual mass of the steel plate specimen (without base plate) is measured to be 3.515 kg. The plate was securely held in position by clamping one end (Figure 11a), and this boundary condition was replicated numerically. The substructure-based model for health assessment replicates the physical plate using a specialized Mindlin-Reissner plate modeled with bilinear quad elements (MITC4). Both the experimental plate and its numerical model were discretized into ten equal parts, denoted as $M_1, M_2, \dots, M_{10}$. Specifically, a component substructure consisting of elements from $M_1$ to $M_5$ was selected for independent health monitoring, resulting in the introduction of eight interior nodes within this substructure. Comprehensive details of the experimental setup and a schematic representation of the substructured domain, are provided in Figure 11a and Figure 11b, respectively.

The plate tip is excited using an impact hammer (Model Number: 086C03) fitted with a piezoelectric tip, from which the forcing history can be retrieved. Eight uniaxial ICP (Integrated Circuit Piezoelectric) accelerometers (Model Number: 7104A-0050) are attached at different locations of the structure and sampled using a KRYPTON-4xACC and 8xACC channel Data Acquisition (DAQ) [50] system at a constant sampling frequency of 500 Hz. The DAQ is connected to the KRYPTON CPU, and DewesoftX software is used to view and record the experimental data. Figures 12a and 12b present the impact force generated by the hammer and the response from the accelerometers on the plate. Impact force sequences over four subsequent applications of excitation are shown in the Figure 12a, with the last set used to generate the responses shown in Figure 12b. The raw data is then used to calculate the frequency (refer to Table 2) through frequency domain decomposition (FDD).

Calibrating the fundamental material properties of the numerical FEM support model to match those of the physical structure is a strongly recommended practice. During this calibration attempt, it was found that the material density remained consistent and did not require significant adjustments. However, the frequencies estimated through FDD of the recorded response from the plate structure did not match those obtained numerically, assuming uniform material elasticity along the plate's length. To resolve this disparity, the material elasticity was perturbed to approximately align (within acceptable tolerances) the numerical frequencies with the experimental ones (refer to Table 2). The calibrated uniform elasticity throughout the length of the plate was determined to be approximately 192.1 GPa.

Subsequently, the numerical model of the experimental setup was further refined to account for the non-uniform elasticity present in real-life structures. This refinement was achieved through the implementation of the proposed health assessment algorithm. As a result, the modulus of elasticity for each element was re-calibrated. For this purpose,
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Figure 11: Experimental setup - trapezoidal steel plate with one end fixed</figcaption></figure>

<table><caption>Table 2: Comparison between frequencies obtained experimentally and numerically.</caption><thead><tr><th colspan="2">Data</th><th>Experimental</th><th>Calibrated numerical</th><th>Relative error (%)</th></tr></thead><tbody><tr><td rowspan="3">Frequency (Hz)</td><td>ω<sub>1</sub></td><td>2.32</td><td>2.37</td><td>2.28</td></tr><tr><td>ω<sub>2</sub></td><td>31.57</td><td>32.48</td><td>2.93</td></tr><tr><td>ω<sub>3</sub></td><td>60.73</td><td>57.71</td><td>4.98</td></tr></tbody></table>

the substructure was equipped with four uniaxial accelerometers placed equidistantly on each side of the plate, which exclusively captured structural vibration responses in the horizontal DOFs. Data was collected at a sampling frequency of 500 Hz. The algorithm employed 250 ensembles and 2000 particles for IPEnKF, with $\alpha = 0.99$.

The initial elasticity of each element was set to 192.1 GPa, a value determined through the aforementioned calibra-
<figure><img src="image_22.png" /><figcaption>(a) Impact hammer force</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(b) Experimentally recorded accelerations</figcaption></figure>

<caption>Figure 12: Experimental data on excitation and responses</caption>

tion process. The health status of the system was estimated by initially assuming prior estimates for all **HIs** as 1, with a standard deviation of 0.02. Subsequently, the algorithm proceeded to update these estimates based on information drawn from the time domain response data collected when the test structure was in a known healthy state.

The material property calibration procedure as detailed above can be regarded as a benchmarking process for health indices when dealing with a structure of unknown health status. The results of **HI** estimation are depicted in Figure 13a, revealing that the **HIs** have converged to different values, predominantly below 1. The quality of the updated model is validated by comparing the measurement predictions of the model with the actual recorded measurements retrieved from the sensors as shown in Figure 13b.

<figure><img src="image_24.png" /><figcaption>(a) Benchmarking of **HIs** for the experimentally tested undamaged structure</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) Comparison of sensor measurements and numerical model predictions</figcaption></figure>

<caption>Figure 13: Evaluation of substructure measurement data comparison and health indices of a trapezoidal steel structure</caption>

The efficiency of the proposed algorithm was tested on the same plate after inducing damage in the form of a cut with a depth of 7 cm and a width of 1 mm (see Figure 11) in the third element ($M_3$) of the first substructure, leading to a reduction in localized stiffness. Using the benchmarked stiffness values as initial estimates for **HIs**, the proposed approach was employed to initiate damage localization and estimate the updated **HIs** for the damaged state of the
structure. The results, shown in Figure 14, demonstrate the algorithm's ability to accurately estimate and localize damage within the test structure without generating any false alarms. Specifically, the elasticity estimate for sub-structural element $M_3$ converges to approximately 0.86, which can be perceived to be logical [51]. Additionally, the HI estimation is highly responsive, with the affected area being detected immediately upon damage occurrence. The health indicators for undamaged elements exhibit minor fluctuations below 1, corresponding to non-significant damage (around 5%), which can be disregarded from a practical perspective and is attributed to the inherent uncertainty of the statistical algorithm.

<figure><img src="image_26.png" /><figcaption>(a) Estimated HI's for the experimentally tested damaged structure</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(b) Comparison of sensor measurements and numerical model predictions</figcaption></figure>

<caption>Figure 14: Estimation of the health index and comparison of measurement data for the damaged structure</caption>

## 6. Conclusions

Monitoring complex structures like building floors, aircraft wings, or turbine blades involves overseeing intricate geometries with inaccessible areas for instrumentation. Ensuring high-resolution monitoring with a model-based assessment framework requires cost-intensive detailed models and dense instrumentation, which are imperative in conventional model-based SHM approaches. Despite these efforts, the problem still suffers from the curse of high dimensionality, often leading to ill-posed problems that yield divergent or impractical solutions. This situation challenges the reliability of the SHM algorithm and the safety of such structures in operation especially for structures that are represented with high-dimensional models.

This proposed methodology addresses these issues by introducing a substructure monitoring approach that estimates only a manageable part of the entire structure independently of the rest. The proposal efficiently handles the unknown forces at the substructure boundaries through an output injection approach, making the substructure estimation completely standalone. This enables conceptualizing a large structural system as an assemblage of several independent low-dimensional components. With plates being one such system, this method is highly beneficial as it allows the estimation of plate substructures with comparatively less instrumentation and computation, which can subsequently be integrated to realize a digital replica of the entire plate. The presented work, therefore, streamlines
the otherwise cumbersome process of comprehensive monitoring of high-dimensional structures, making it achievable
with reduced instrumentation demands and associated costs.

The thorough parametric study substantiates the practicality of stage-wise instrumentation, supported by crucial
insights from numerical investigations. Significantly accurate parameter estimates were achieved under diverse noise
levels and damage intensities, demonstrating commendable performance in various operational scenarios despite mea-
surement uncertainty. The proposed method has been observed to reduce the required instrumentation by 60%, while
maintaining 95% accuracy in detecting and assessing damage in the Reissner-Mindlin plate. Evaluation of sensor
malfunction scenarios revealed satisfactory damage estimation when sensors at the internal nodes failed, with occa-
sional false alarms observed in specific cases. Experimental validation further confirms the feasibility of the proposed
stage-wise monitoring approach for real-life structures. Ultimately, the suggested component-wise estimation approach
establishes its merit for practical application in real-world scenarios.

**Funding:** This study was funded by Aeronautics Research and Development Board, New Delhi, India, through grant file no. ARDB/01/1052042/M/I.

**Declaration of generative AI and AI-assisted technologies in the writing process:** During the preparation of this work, the authors used OpenAI's GPT-4 in order to partially assist in drafting and refining the text. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

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