Systems and methods for damage localization in plate structures using the geometric phase of acoustic waves

Abstract

Examples of topological acoustic (TA) sensing techniques for localizing defects in plate structures utilize Lamb waves. TA sensing exploits changes in geometric phase of acoustic waves to detect perturbations in the supporting medium. This approach uses a geometric phase change – index (GPC-I), a measure of the geometry of the acoustic field averaged over a spectral domain, as detection metric in lieu of VD or AR. Calculations based on the finite element method (FEM) in Abaqus / CAE software verifies the effectiveness of the proposed GPC-I-based method. Randomly located defects on the surface of a plate are localized with higher sensitivity and accuracy, by the GPC-I method in comparison to VD or AR-based methods.

Description

Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) SYSTEMS AND METHODS FOR DAMAGE LOCALIZATION IN PLATE STRUCTURES USING THE GEOMETRIC PHASE OF ACOUSTIC WAVESCROSS REFERENCE TO RELATED APPLICATIONS

[0001] The present document is a PCT patent application that claims benefit to U. S. Provisional Patent Application Serial No. 63 / 723,901 filed on November 22, 2024, which is herein incorporated by reference in its entirety.GOVERNMENT SUPPORT

[0002] This invention was made with government support under Grant No.2242925 awarded by the National Science Foundation. The government has certain rights in the invention.FIELD

[0003] The present disclosure generally relates to acoustic technologies; and in particular to defect localization in plate structures by leveraging detected changes in geometric phase of acoustic waves.BACKGROUND

[0004] Defects or damages can occur in large-scale engineering structures, such as plate structures, due to foreign object impacts, local erosion, and manufacturing flaws, which can endanger the safety of structural components during operation. Localizing these defects or damages is essential to ensure the safe operations of the structures

[0005] Commonly used methods for defect localization in structures are based on velocity differences (VD) or amplitude ratio (AR) (or attenuation due to scattering) measured along different sensing paths between a reference system and a defective system. A high value on a sensing path indicates a higher probability of the presence of defect on that path. For localizing defects or damages in engineering1107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) structures, most research focused on predicting the acoustic sources generated by the defects. It is done by the popular passive acoustic emission (AE) technique and is commonly known as the acoustic source localization (ASL). Many researchers have investigated the ASL technique and have proposed many ASL methods and algorithms that are suitable in different scenarios. Passive AE technique is good for locating the damage initiation point but not for localizing and characterizing existing damages. Also, an already existing damage cannot be localized by the passive AE technique.Therefore, active defect localization method is needed to compensate for the limitations arising from passive ASL techniques.

[0006] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed.SUMMARY

[0007] Aspects of the present inventive concept can take the form of a method for defect localization in plate structures using Lamb waves. Topological acoustic (TA) sensing exploits changes in geometric phase of acoustic waves to detect perturbations in the supporting medium. This approach can include use of a geometric phase change - index (GPC-I), a measure of the geometry of the acoustic field averaged over a spectral domain, as a detection metric. Stated another way, aspects include methods for defect localization based on the change of the geometric phase of acoustic waves. Example steps of the method include providing a sensor array comprising a plurality of sensors along a structure; operating at least one sensor of the sensor array as a transmitter to launch acoustic waves into the structure and operating at least one other sensor of the sensor array as a receiver to obtain response signals corresponding to the acoustic waves; obtaining response signals from the sensor array for a reference state of the structure and for a perturbed state of the structure that includes the at least one defect; for a plurality of subsets of the plurality of sensors, each subset comprising at least two sensors that define at least one sensing path through the structure, processing the response signals to compute, for each subset, a geometric phase change between a reference-state acoustic field and a perturbed-state acoustic field and to calculate a geometric phase change index parameter (GPC-I) as a2107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) spectral average of the geometric phase change over a selected frequency band; and comparing the geometric phase change index parameters (GPC-I) for the plurality of subsets and identifying at least one region of the structure associated with one or more relatively larger geometric phase change index parameters (GPC-I) as a probable location of the at least one defect

[0008] In other aspects, the inventive concept can take the form of a system including a plurality of sensors positioned along a structure and configured to operate as transmitters and receivers of acoustic waves propagating in the plate structure. The system can further include one or more processors, configured to: control at least one sensor of the sensor array to operate as a transmitter so as to launch acoustic waves into the structure and control at least one other sensor of the sensor array to operate as a receiver so as to provide response signals corresponding to the acoustic waves; obtain response signals from the sensor array for a reference state of the structure and for a perturbed state of the structure that includes the at least one defect; for a plurality of subsets of the plurality of sensors, each subset comprising at least two sensors that define at least one sensing path through the structure, process the response signals to compute, for each subset, a geometric phase change between a reference-state acoustic field and a perturbed-state acoustic field and to calculate a geometric phase change index parameter (GPC-I) as a spectral average of the geometric phase change over a selected frequency band; and compare the geometric phase change index parameters (GPC-I) for the plurality of subsets and identify at least one region of the structure associated with one or more relatively larger geometric phase change index parameters (GPC-I) as a probable location of the at least one defect.

[0009] In other aspects, the inventive concept can take the form of a method for defect localization using acoustic waves, comprising steps of monitoring, by a plurality of sensors, a geometric phase change (GPC) defining a measure of changes in an acoustic wave’s spatial behavior relative to a structure; measuring changes in geometric phase of acoustic waves associated with a subset of the plurality of sensors at a time and continuously changing the subset to calculate a GPC index parameter (GPC-I) along various paths; and detecting at least one defect along the structure by3107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) utilizing the GPC index parameter (GPC-I) to identify perturbations caused by defects along the sensing paths between transmitters and receivers of the plurality of sensors.

[0010] The subject summary is provided to introduce examples of concepts in a simplified form. This summary is not intended to be exhaustive or to limit the present disclosure to the precise forms disclosed. Rather, the summary is intended to provide a brief overview of some of the concepts described herein as a prelude to the more detailed description that follows. Various non-limiting examples are described herein. The examples are provided merely to illustrate certain features and aspects of the disclosed systems and methods, and are not intended to limit the scope of the claims. Features of the examples described in connection with one embodiment may be used alone or in combination with features of other examples, and such combinations are contemplated as being within the scope of the present disclosure. It will be appreciated that the disclosed subject matter may be embodied in numerous different forms and that various substitutions, modifications, and alterations may be made without departing from the scope of the claims. Accordingly, the specific examples and configurations described herein are not intended to be limiting, but rather are provided to enable a person skilled in the art to make and use the claimed subject matter. In addition, while certain operations or elements may be described in a particular order or combination, such order or combination is not required unless explicitly recited in the claims, and alternative orders, combinations, and subcombinations are also contemplated.4107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) BRIEF DESCRIPTION OF THE DRAWINGS

[0011] FIG. 1 is a schematic illustration of a plate with array of sensors (Sj) for (a) reference plate and (b) perturbed state plate with a random defect.

[0012] FIG. 2 is a two-dimensional (2D) view of the problem geometry for numerical modeling - perturbed state with a random defect. The defect is absent in the reference state.

[0013] FIGS. 3A-3D are a series of Lamb wave displacement magnitudes for the reference plate (left column) and defective plate (right column) with transmitter Ti =1 at time 75 ps (top row) and T2=17 at time 35 ps (bottom row) - only the plate with the defect located in its first quadrant is shown.

[0014] FIGS. 4A-4D show time histories for both reference and perturbed states at a receiving sensor pair (FIG. 4A) 18, (FIG. 4B) 19, and sensor pair (FIG. 4C) 23 and (FIG. 4D) 24 when the transmitter is at location Ti =1.

[0015] FIGS. 5A-5B show calculated GPC for two sensing paths defined by sensor locations including at FIG. 5A sensor 18, and at FIG. 5B sensor 23 when the transmitter, Ti, is at location 1.

[0016] FIGS. 6A-6B show at FIG. 6A GPC-I calculated for all sensing paths (labelled by sensor number) with Ti =1, and at FIG. 6B the triangular region with maximum GPC-I value.

[0017] FIGS. 7A-7B show at FIG. 7A GPC-I value at each receiving sensor for the second transmitter location (T2= 17), and at FIG. 7B the predicted triangular region illustrating that the defect is more likely to be located within the triangular region enclosed by the transmitter T2=17 and the sensors 28 and 29.

[0018] FIG. 8 shows defect location in the parallelogram formed by the intersection of the two triangles with highest GPC-I.

[0019] FIGS. 9A-9C show at FIG. 9A distances from the transmitter 1 to all receiving sensors, at FIG. 9B time histories recorded at sensor 18 for the reference state (defect-free) and the perturbed state (with defect), and at FIG. 9C time histories recorded at sensor 23 for the reference state and the perturbed state.

[0020] FIGS. 10A-10C show defect localization result from transmitters 1 and 17 when the actual mass is located at coordinate (50 mm, 30 mm) including at FIG.5107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) 10A normalized velocity differences resulted from transmitter 1, at FIG. 10B normalized velocity differences resulted from transmitter 17, and at FIG. 10C predicted defect location and actual defect location.

[0021] FIGS. 11A-11C showVD based localization results for defect locations including at FIG. 11 A (-20 mm, 20 mm), FIG. 11 B (-30 mm, -50 mm) and FIG.11 C (20 mm, -40 mm).

[0022] FIGS. 12A-12B show amplitude plots including at FIG. 12A spectral amplitude plots for signals recorded at sensor 18 for the reference state and the perturbed state, and at FIG. 12B spectral amplitude plots for signals recorded at sensor 23 for the reference state and the perturbed state when the transmitter, Ti, is at location 1.

[0023] FIGS. 13A-13C show localization result from transmitters 1 and 17 when the extra mass is located at coordinate (50 mm, 30 mm) including at FIG. 13A amplitude ratio results for transmitter location 1, FIG. 13B amplitude ratio results for transmitter location 17, and at FIG. 13C predicted defect location and actual defect location.

[0024] FIGS. 14A-14C show AR based localization results for defect locations including at FIG. 15A (-20 mm, 20 mm), FIG. 15B (-30 mm, -50 mm) and FIG.15C (20 mm, -40 mm).

[0025] FIGS. 15A-15C show defect localization results from transmitters placed at locations 1 and 17 for the defect (extra mass) located at (-20 mm, 20 mm); including in FIG. 15AGPC-I results for transmitter 1; FIG. 15B GPC-I results for transmitter 17, and FIG. 15C predicted defect region and the actual defect location.

[0026] FIGS. 16A-16C show defect localization results from transmitters placed at locations 1 and 17 for the defect (extra mass) located at (-30 mm, -50 mm); including in FIG. 16A GPC-I results for transmitter 1, in FIG. 16B GPC-I results for transmitter 17, and FIG. 16C predicted defect region and the actual defect location.

[0027] FIGS. 17A-17C show defect localization results from transmitters placed at locations 1 and 17 for the defect (extra mass) located at (20 mm, -40 mm); illustrating in FIG. 17A GPC-I results for transmitter 1; illustrating in FIG. 17B GPC-I6107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) results for transmitter 17, and FIG. 17C showing predicted defect region and the actual defect location.

[0028] FIG. 18 is an example system diagram including non-limiting components that can be implemented for defect localization in plate structures as described herein.

[0029] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.7107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) DETAILED DESCRIPTION

[0030] The present disclosure relates to an inventive approach including a newly developed topological acoustic (TA) sensing technique for localizing defects in plate structures using, e.g., Lamb waves. TA sensing exploits changes in geometric phase of acoustic waves to detect perturbations in the supporting medium. This approach uses a geometric phase change - index (GPC-I), a measure of the geometry of the acoustic field averaged over a spectral domain, as detection metric in lieu of VD or AR. Calculations based on the finite element method (FEM) in Abaqus / CAE software verifies the effectiveness of the proposed GPC-l-based method. Randomly located defects on the surface of a plate are localized with higher sensitivity and accuracy, by the GPC-I method in comparison to VD or AR-based methods.

[0031] Stated another way, aspects include methods based on the change of the geometric phase of acoustic waves for localizing defects. Example steps include monitoring the geometric phase change (GPC), a measure of the changes in the acoustic wave’s spatial behavior by considering a sub-group of sensors at a time. The geometric phase is different from the dynamic phase which is related to the phase accumulated by a wave as it travels at some speed along some path. Example features include (1 ) consideration of a subset of sensors at a time and continuously changing the subset to calculate GPC-I (geometric phase change- index) along various paths for localizing the defect, and (2) localizing a defect successfully for the first time using the proposed TA based GPC-I parameter.

[0032] It is believed that the inventive concepts and various features described herein revolutionize flaw location by exploiting the geometric phase change of acoustic waves. It is shown that the geometric phase change can localize the defects more accurately than the conventional attenuation and velocity change based techniques.1. Introduction

[0033] Defects or damages can occur in large-scale engineering structures, such as plate structures, due to foreign object impacts, local erosion, and manufacturing flaws, which can endanger the safety of structural components during operation. Localizing these defects or damages is essential to ensure the safe8107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) operations of the structures. Ultrasonic nondestructive evaluation (UNDE) techniques are widely used for engineering structural health monitoring (SHM). In particular, ultrasonic Lamb waves have garnered significant attention due to their promising capabilities for long-distance propagation and wide-range sensing coverage, making them suitable for sensing in large-scale structures. Consequently, developing a sensitive and stable defect localization method based on Lamb waves remains highly desirable for SHM applications.

[0034] For localizing defects or damages in engineering structures, most research has focused on predicting the acoustic sources generated by the defects. It is done by the popular passive acoustic emission (AE) technique and is commonly known as the acoustic source localization (ASL). Many researchers investigated the ASL technique and have proposed many ASL methods and algorithms that are suitable in different scenarios. The passive AE technique is good for locating the damage initiation point but not for localizing and characterizing existing damages. Also, an already existing damage cannot be localized by the passive AE technique. Therefore, active defect localization method is needed to compensate for the limitations arising from passive ASL techniques. Unlike the vast majority of investigations on ASL technique, only a limited number of investigations have been reported on active defect localization methods in engineering structures. Ma et al. proposed a wave front shape-based method on time-difference-of-arrivals (TDOAs) for active damage localization in composite plates, and this wave front shape based method is an advanced tool adopted in ASL technique without knowing materials’ velocity profile. Shu et al. refined the popular conventional time-reversal method (TRM) for damage localization using Lamb waves. In the TRM as well as in its modified version, though reference signal and prior information is not necessary to extract damage features, the time of arrival (TOA) at the sensors is needed and the time difference of arrival (TDOA) at different sensors is used to identify the damage locations. Other defect localization methods using Lamb waves include analyzing envelope of wave modes characteristics, identical-group-velocity of different wave modes and combining these conventional TOA or envelope feature information with deep learning using neural network.9107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)

[0035] In general, defect localization using ultrasonic Lamb waves basically relies on the accurate determination of TOAto calculate TDOA between undamaged and damaged structures. It means that clear Lamb wave modes are required for accurate estimations of TOF or TDOA. However, damage index (DI) based acoustic parameters such as correlation coefficients, energy ratio or amplitude ratio, linear attenuation coefficients, and nonlinear acoustic parameters like sideband peak count - index (SPC-I) measures the overall differences between reference signals (healthy structures) and testing signals (damaged structures). Using Dis to characterize damages does not require to track specific Lamb wave modes thus can provide stable damage features. For example, the probability ellipse method with different weights of Dis has been adopted to localize and image damages in structures with active sensors network.

[0036] Recently, an emerging DI called geometric phase change (GPC) based on topological acoustic (TA) sensing was introduced. The GPC quantifies the variation in the geometric phase of an acoustic field (for example, Lamb waves propagating in structures) represented as a state vector in an abstract parameter space - Hilbert space. In the TA sensing technique with GPC, the state of the acoustic field in the unperturbed (damage-free) and perturbed (damaged) cases are mapped as multidimensional vectors in the same Hilbert space. As a global measure of the linear and nonlinear acoustic field a sensing approach based on the geometric phase can have higher sensitivity than magnitude-based orTOF-based sensing approaches.

[0037] With the TA sensing technique, originally, changes in complicated environments such as forests or the state of permafrost in the arctic were monitored using seismic waves. This method was further extended to monitoring perturbations due to a mass defect located on an array of coupled acoustic waveguides, mass defects in a nonlinear granular metamaterial and a small subwavelength object on a flat surface submerged in water. Taking the advantage of high sensitivity of GPC, damage growth in heterogenous topographical plate structures has been also successfully investigated, and GPC results showed superiority in comparison to the nonlinear ultrasonic based SPC-I technique for monitoring damage evolution in complex heterogeneous structures.10107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) However, like other Dis, at this time the GPC can only indicate the existence of defects in environment or engineering structures.

[0038] In the work associated with the present inventive concept described herein, the GPC sensing method is extended to active defect localization. A geometric phase change - index (GPC-I) derived from GPC spectral averages, serves as a metric for determining the probable location of defects along different sensing paths. Numerical modeling can be carried out with finite element method (FEM) in Abaqus / CAE software to verify the effectiveness of the GPC-I localization method. The method is then shown to compare very favorably with similar methods based on more common acoustic parameters, namely - velocity differences based on TOA and signal amplitude or attenuation variations based on recorded signal amplitudes.2. Theory and methodology

[0039] In this section, the notions of GPC and GPC - index (GPC-I) are briefly reviewed. A methodology is then introduced for actively locating defects in plate structures.2.1 Background on GPC plots and GPC-I

[0040] The frequency-dependent amplitude of an acoustic field supported by some medium can be represented as a complex vector in an infinite dimensional space whose basis components are associated with every point in the continuous medium. From a practical point of view, one can select a finite dimension subspace by considering the complex amplitude of the field at a discrete subset of “n” points in the medium. The finite set of complex amplitudes forms a complex vector in this n-dimensional subspace. This can be achieved by stimulating the medium at some location and recording the displacement or velocity field as a time series at the “n” chosen receiving locations. Each of these n time series are Fast Fourier transformed (FFT) to obtain complex amplitudes in the spectral domain. At a given frequency, f, these n complex amplitudes are used to form a normalized n-dimensional state vector. The n basis vectors correspond to the n receiving sensor locations in the physical space. This normalized state vector can be written as,11107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)C2ei<t>2C3ei<p2(1)

[0041] In equation (1), / is the imaginary unit. Ckand= l,2,3...n) are magnitude and spatial phase at each receiving point, respectively. Let us assume that Eq. (1) is representative of a reference medium without any damage. When damages are introduced, the perturbation in the physical space scatters the acoustic waves and modifies the spatial distribution of the acoustic field. The perturbations then change the normalized complex amplitude of the acoustic field to,C^2

[0042] The angle between the vector representation of the acoustic field along the n locations in the damage-free and damaged systems corresponds to a change in the geometric phase change of the acoustic field associated with the perturbation is then defined as the angle,A<p = arccos(Re(C* ■ £'))> < Pe[0, TT] (3)where C* denotes the complex conjugate of state vector C while Re stands for the real part of a complex quantity. The GPC, cp f measures the effect of the perturbation on the orientation of the field state vector.

[0043] The GPC - index (GPC-I) is defined as the average value of (p(f) in some frequency domain. This index provides a metric for quantifying the level of damage of the structure with larger damage levels corresponding to larger GPC-I values.12107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) 2.2 Defect localization methodology with GPC-I

[0044] Consider a damage-free plate structure equipped with multiple sensors distributed across its surface, as depicted by the reference plate in FIG. 1A. Given the practical dimensions of such plates, these sensors can be strategically placed near the plate’s edges to encompass a large sensing area (that may or may not have damage). Random defects that require localization may arise within this region due to impacts or local erosion, as shown in FIG. 1B.

[0045] We choose the sensor point Ti as the transmitter. All other sensors record displacement or velocity series generated from Ti. For the sake of simplicity, we chose the dimension of the field state vector to be n = 2. In the reference plate, a 2-D complex state vector can be constructed from time series received at two adjacent sensors (sensors j and j + 1). The normalized complex state vector at receiving sensor j can be written as,1 / \Jwhere j denotes the receiving sensor number, and i is the imaginary unit. A and are magnitude and spatial phase at each receiving point.

[0046] Accordingly, in the perturbed plate the normalized state vector at receiving sensor j can be expressed as,1 ( Aj'e^i' \ U; +A;+1Wie 7 /

[0047] The GPC at sensor number j is calculated in the two-dimensional subspace as,A<pj = arcos(Re(S;* ■ Sj')), A<p G [0, TT] (6) where S denotes the complex conjugate of state vector Sj at receiving sensor j while Re stands for the real part of a complex quantity.

[0048] The GPC-I at the receiving sensor number j is then calculated by averaging Pj(f) over a chosen frequency range. A large GPC-I value indicates that the presence of a defect is likely along the sensing paths from the transmitter to sensor 13107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) numbers j and j + 1. Subsequently, the defect is likely located in the triangular region formed by the transmitter Ti and the two sensors numbered j and j + 1. We now consider another transmitter location T2 and determine the GPC, (pj+nf), employing the two receiving sensors numbered j + n and j + n +1. The value of GPC-I at the receiving sensor j + n, is then used to potentially locate the defect in the triangle formed by T2, sensors j + n and j + n + 1. Large values of GPC-I for both triangles predicts the defected region as the intersection of these two triangles. The defective region forms a parallelogram whose four vertices are indexed as k = 1,2,3,4 with coordinates (xk,yk). We define the predicted loca (tion (xp,yp) of the defect by the following averages:max(xfc) + min(xfe)xp=2z(7) max(yk) + min(yfc)- ~2-3. Numerical model

[0049] In this section, numerical modeling analysis is carried out using Abaqus / CAE software. The numerical model includes multiple sensors, numbered from 1 to 32, distributed on the surface of a reference and defective plates. The dimension of both plates (reference and perturbed states) is 300 x 300 x 3mm3. The distance between two adjacent receiving points is set at 20 mm. These sensors enclose a square area with a side length of 160 mm as shown in FIG. 2. The defect is formed by attaching one block with dimension 10 x 10 x 6mm3on the surface of the plate structure. The center of the mass defect on the surface in the first quadrant is (50 mm, 30 mm). Then the mass is moved to three additional locations (-20 mm, 20 mm), (-30 mm, -50 mm) and (20 mm, -40 mm). The material properties for the plate structure and the mass defect are the same and are listed in Table 1.

[0050] First, we consider sensing paths by defining the sensor number 1 as the transmitter, T1, and the receiving sensors are 9 to 25. Similarly, we choose sensor 17 as the transmitter T2. In this case the signals are recorded at sensors 1 to 9 and 25 to 32. Note that there is no constraint on the selection of transmitters and receivers.14107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) Table 1. Material property parameters of plate structures and mass defects for FEM modeling.Parameter Young’s modulus (GPa) Poisson’s ratio Density (kg / m3) Values 71.50 0.33 2700.00

[0051] A Hanning window modulated excitation field of central frequency 200 kHz with two cycles is applied in the negative z direction first at Ti =1 and T2=17. At the selected receiving points, the out-of-plane velocity (in the z direction) is recorded for both the reference state and the perturbed state. The sampling frequency for recording the signals is 50 MSa / s (mega samples per second).4. Defect localization using GPC-I method

[0052] The model system can initially be investigated with the defect in the first quadrant of the plate, that is coordinates (50 mm, 30 mm) as shown in FIG. 2.FIGS. 3A-3D illustrate snapshots of Lamb wave displacement magnitude for both transmitter locations Ti =1 and T2=17.

[0053] FIGS. 4A-4D illustrate the time histories in both reference and perturbed states recorded along two sensing paths defined by the sensor pair 18 and 19 (of FIG. 2) and the sensor pair 23 and 24 when the transmitter is at location Ti =1.

[0054] It is noted that the first dominant wave packet in FIGS. 4A-4D is a Ao mode while the So wave packet is very weak. In FIGS. 5A-5B, the calculated GPC is shown as a function of frequency for two different sensing paths.

[0055] The frequency-dependent GPC plots measure the effect of the perturbation on the acoustic field. Over a wide frequency range (from 0 to 800 kHz) the GPC show significantly larger values at receiving sensor 18 than that at sensor 23. This difference indicates that a defect is more likely to be present along the wave path between the transmitter 1 and sensor 18. The GPC-I is calculated by averaging the GPC over the frequency range 0 to 800 kHz. With the transmitter at location 1, the GPC-I as a function of the sensing path is shown in FIGS. 6A-6B.

[0056] FIG. 6B shows that the defect is more likely to be located within the triangular region enclosed by the transmitter Ti =1 and the sensors 18 and 19 (of FIG.15107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) 2). For the second transmitter location (T2= 17), GPC-I value at each receiving sensor is obtained and shown in FIG. 7Aand the predicted triangular region is shown in FIG. 7B.

[0057] FIG. 7B shows that the defect is more likely to be located within the triangular region enclosed by the transmitter T2=17 and the sensors 28 and 29. The intersection region between the two triangles with highest GPC-I in FIG. 8 indicated the most probable location of the defect.

[0058] This localization process is applied to all defect positions. Table 2 summarizes the predictions of the GPC-I localization method for all four example defective plates investigated.Table 2. Defect localization results based on the proposed GPC-I method and error analysis._,... Actual location (xa, ya) Predicted location (xp, Relative errors (mm) yp) (mm) (%)1 (50±5, 30±5) (58.42, 33.12) 2.44 2 (-20±5, 20±5) (-21.82, 21.82) 03 (-30±5, -50±5) (-33.12, -58.42) 2.44 4 (20±5, -40±5) (12.52, -38.64) 2.75

[0059] Furthermore, a defect localization error analysis is conducted by comparing the predicted location with the actual location. Since the mass defect has some dimensions, the scattered waves received by sensors are affected by the edges of the defect. If we can predict any boundary of the defect then it will be assumed that the localization of this defect is accurately determined. The relative error between the actual and the predicted locations of the defect is defined by equations 8 and 9.Equation (8) indicates that when the predicted location is inside the defect then we say it has accurately localized the defect and the relative error is defined as zero.Otherwise, we use equation (9) to find the values from the predicted location to one of the nearest edges of the defect to ca {lculate the error values.T'bx ^bx %a y — ^p — %a 1 T” L I / fry L I'by (8) Va — Tp — Ta 116107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)

[0060] In equation (8), (xp,yp) is the predicted localization coordinate obtained from equation (7). (xa,ya) is the actual center location of the defect. Lbxand Lbyare the side lengths of the mass defect in x and y directions, respectively.

[0061] In equation (9), Lxand Lyare side lengths in x and y directions, respectively of the enclosed region by those distributed sensors. The relative error is decomposed into the x and y directions, respectively, and then added to consider their combined effects. The relative errors are small, less than 3%. It is also interesting to see that for the predicted locations corresponding to defects No. 1 and 3, x and y values and their signs have been simply interchanged. This is because the transmitting sensors 1 and 17 here are symmetric about a diagonal line passing through sensors 9 and 25. All receiving sensors as well as the actual defect locations are also symmetric about this line. Hence, the prediction comes out to be symmetric about this line as well.5. Comparison between GPC-I localization method and other methods5.1. Localization based on velocity differences

[0062] In this section, velocity differences (VD) between reference state and perturbed state at each receiving sensor are obtained from the time of arrival (TOA) or time of flight (TOF) measurements. The VD parameter is then used to localize the defect following the same steps described above but GPC-I values are replaced by VD values. For the VD calculation from the TOF values, the distances from transmitter 1 to all receiving sensors are first calculated and shown in FIG. 9A. Time histories showing the So mode arrivals at sensor locations 18 and 23 in reference state and perturbed state are shown in FIGS. 9B and 9C, respectively.

[0063] First, the velocity value at each receiving sensor is calculated for both the reference state and the perturbed state. Then these velocity values are normalized with respect to the first velocity value (received at the first sensor, for transmitter at location 1 and the receiver at location 9) in each state (reference and17107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) perturbed). After taking the differences between the normalized velocity values of reference state and perturbed state the GPC-I values are replaced by the VD values for the damage localization. The normalized VD results from transmitter 1 at each receiving sensor are shown in FIG. 10A. It can be seen that there are three peaks (sharp peaks indicate defects appearing on the corresponding path) in the normalized VD plot which makes it difficult to determine the path along which defects occur. However, here we choose the peak with the largest magnitude as the indictor. Then the defect occurs on the path connecting transmitter 1 and sensor 19. Similarly, when transmitter at location 17 is the emitter, the largest peak appears at sensor 27 (shown in FIG. 10B). Then the predicted defect location is determined as the intersection of these two lines obtained from transmitters 1 and 17 as shown in FIG. 10C.

[0064] Following the same methodology and using the VD values, defects located for other three positions of the extra mass are shown in FIG. 11.

[0065] Defect localization results using velocity difference are summarized in Table 3. The error is calculated using the same equations presented in section 4. Table 3. Defect localization results and relative errors obtained from the analysis using velocity differences._ „. Actual location (xa, ya) Predicted location (xp, Relative errorsDefect N0(mm) yp)(mm)(%)1 (50±5, 3O±5) (67.69, 30.77) 8.36 2 (-20±5, 20±5) (1.63, 50.61) 19.09 3 (-30±5, -50±5) (-30.77, -67.69) 8.36 4 (20±5, -40±5) (62.22, 8.89) 35.97

[0066] It can be seen in these localization results that for some defect locations (defects 2 and 4) there are large deviations between the predicted defect location and its actual location (shown in FIGS. 11 A and 11 C). Therefore, we can say that the VD parameter is unstable for defect localization.5.2. Localization based on amplitude ratio

[0067] In this section, the amplitude ratio (AR) parameter is utilized to localize the defect for comparison. The AR value at each receiving sensor is defined as the ratio between the maximum peak value in spectral amplitude plots between the18107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) reference state and the perturbed state. Spectral amplitude plots at receiving sensors 18 and 23 for both the reference state and the perturbed state are presented in FIGS.12Aand 12B, respectively, as an example.

[0068] The amplitude ratio at each receiving sensor is calculated and used to localize the defect. The amplitude ratio results for transmitter 1 at each receiving sensor are shown in FIG. 13A. It can be seen that there is a sharp dip at receiving sensor 18 and multiple small peaks and dips (oscillations) at other sensors. Here we choose the largest deviation of a peak or a dip value as the indictor that defects appear on the sensing path which is from transmitter 1 to receiving sensor 18. Similarly, when the transmitter is at location 17 the largest deviation appears at receiving sensor 32 (shown in FIG. 13B). Then the predicted defect location is determined as the intersection of these two lines going through transmitters 1 and 17 as shown in FIG. 13C.

[0069] Defect localization results using AR parameter for other positions of the defect are shown in FIGS. 14A-14C. These defect localization results are summarized in Table 4.Table 4. Defect localization results and relative errors obtained from the analysis using amplitude ratio._ r.x,TActual location (xa, ya) Predicted location (xp, yp) Relative errors (mm) (mm) (%)1 (50±5, 3O±5) (5.33, -5.33) 31.21 2 (-20±5, 20±5) (-26.67, 26.67) 1.47 3 (-30±5, -50±5) (5.33, -5.33) 31.21 4 (20±5, -40±5) (5.33, -5.33) 19.50

[0070] It can be seen in these localization results that for some defects (defects 1, 3 and 4) there are large deviations between predicted defect location and actual location (shown in FIGS. 13C, 14B and 14C). The relative errors varied from less than 2% to more than 30%; therefore, the AR parameter is also not reliable for defect localization.19107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) 6. Conclusion

[0071] The GPC-l-based method, introduced here, demonstrates superior accuracy in locating defects on plates compared to the more commonly used approaches based on the wave velocity or the amplitude difference. The velocity differences show very low sensitivity (< 2% difference in velocity) between reference and perturbed states. Such small difference can be recorded through numerical modeling but can be difficult to detect experimentally or for in situ applications. The amplitude ratio parameter exhibits higher sensitivity to defects than velocity difference. However, amplitude ratio finds difficulty in identifying paths most likely to contain defects due to strong spatial variations. The GPC-I is a global measure of the acoustic field and does not suffer from the inherent locality of the velocity and amplitude. This investigation was performed using Abaqus / CAE simulation software. In numerical modeling investigations without any background noise, defect locations can be predicted in some cases (defects 1 and 3) with the VD parameter. However, in real life structural health monitoring (SHM) applications with high background noise, obtaining accurate time-of-flight (TOF) estimations can become nearly impossible. GPC or GPC-I, on the other hand, exhibit stable and clear damage features with high sensitivity. The GPC-I is a global measure of the acoustic field which does not require precise TOF or time-difference-of-arrival (TDOA) information, nor any specific Lamb wave modes.Consequently, the GPC-I method holds significant promise for advancing the defect localization technique in large engineering structures, thereby offering substantial benefits to the SHM community. The advantages of the GPC-I based sensing method make it particularly beneficial for monitoring damages in complex structures, such as heterogeneous or topographical structures containing different types of materials.Future research on the proposed defect localization method will focus on experimental validation and extending applicability to complex three-dimensional structures and multiple defect localizations.

[0072] Referring to FIG. 18, one example of a system 100 for defect localization and implementing other aspects herein is illustrated. As indicated, the system 100 includes a sensor array 120 that can include a plurality of sensors configured for positioning and / or implementation along a target structure such as a plate20107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) structure 110. The sensors of the sensor array 120 can be configured to operate as transmitters and receivers of acoustic waves, such as Lamb waves, in the plate structure 110 for defect localization according to methods described herein. In some examples, the system includes an excitation module 130 that can include a waveform generator that can be coupled to (or in operable communication with via a wired or wireless medium) the sensor array 120 and configured to drive a selected one of the sensors as a transmitter to launch an excitation waveform into the plate structure 110. The system 100 can include a switching network 140 that selectively engages, groups, and / or activates subsets of the sensors of the sensor array 120 to the excitation module 130 as transmitters and to an analog front-end 150 as receivers. The analog front-end 150, which may include one or more preamplifiers and filters, conditions time-domain response signals from the receiving sensors and provides corresponding conditioned signals to a data acquisition subsystem 160.

[0073] In some examples, the data acquisition subsystem 160 digitizes the conditioned signals and supplies digitized response data to a processing unit 170 of the system 100. The processing unit 170 can include one or more processors configured to store reference-state and perturbed-state response data, to process the data to compute geometric phase change and a geometric phase change index (GPC-I) for multiple sensing paths defined by different subsets of the sensors in the sensor array 120, and to detect and localize at least one defect in the plate structure 110 based on relative values of the GPC-I along the sensing paths. The system 100 can further include a user interface 180 in operable communication with the processing unit 170 and configured to receive control inputs to initiate measurements, select transmitter and receiver configurations, and display defect-localization results. The system 100 can further include a power supply 190 that provides operating power to the excitation module 130, switching network 140, analog front-end 150, data acquisition subsystem 160, processing unit 170, user interface 180, and other components of the system 100 as contemplated.

[0074] In various embodiments, the sensor array 120 may include any suitable devices capable of transducing acoustic waves in the plate structure 110 into electrical signals and, in some cases, vice versa. For example, the sensor array 12021107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) may comprise a plurality of piezoelectric transducers, such as piezoelectric wafer active sensors (PWAS) or disk-type lead-zirconate-titanate (PZT) elements adhesively bonded to a surface of the plate structure 110. In other embodiments, the sensor array 120 may include polyvinylidene fluoride (PVDF) film transducers, piezoelectric stack actuators, or micro-electromechanical system (MEMS) ultrasonic transducers. The sensor array 120 may additionally or alternatively incorporate strain-sensitive devices such as resistive strain gauges or fiber-optic sensors, for example fiber Bragg grating (FBG) sensors bonded or embedded in proximity to the surface of the plate structure 110. In still other embodiments, the sensor array 120 may include non-contact sensing devices, such as one or more laser Doppler vibrometer heads positioned to measure surface motion at discrete locations that function as virtual sensor positions. Each sensor of the sensor array 120 may be configured to operate selectively as a transmitter, a receiver, or both, under control of the switching network 140 and processing unit 170.

[0075] As indicated, the excitation module 130 is configured to stimulate the plate structure 110 at one or more transmitter locations so as to launch controlled acoustic waves, for example Lamb waves, that propagate along the plate structure 110 and are sensed by other sensors of the sensor array 120 operating as receivers. In some embodiments, the excitation module 130 comprises an arbitrary waveform generator or function generator configured to synthesize a prescribed excitation waveform, such as a single-frequency tone burst, a Hanning-windowed tone burst, a chirp, or another broadband or narrowband excitation within a Lamb-wave frequency band. The waveform generator may be implemented as a benchtop instrument or as an integrated circuit or module mounted on a printed circuit board within the system 100.

[0076] In certain implementations, the excitation module 130 further includes a power amplifier or piezoelectric driver stage configured to receive a low-voltage excitation signal from the waveform generator and to amplify the excitation signal to a voltage and / or current level suitable for driving piezoelectric sensors of the sensor array 120. The power amplifier may comprise a high-voltage linear amplifier, a power operational amplifier, a class-D switching amplifier, an H-bridge driver, or another driver topology suitable for energizing piezoelectric transducers.22107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)

[0077] In some embodiments, the excitation module 130 comprises an ultrasonic pulser or pulser-receiver instrument commonly used in nondestructive evaluation. The ultrasonic pulser may be configured to generate a high-voltage spike pulse, square pulse, or tone-burst waveform and to apply the waveform to a selected sensor of the sensor array 120 operating as a transmitter. In such embodiments, the pulser-receiver may additionally provide receive-side amplification and filtering, while the processing unit 170 performs the geometric phase change calculations described herein.

[0078] In other embodiments, the excitation module 130 is implemented as an embedded waveform generator and driver circuit controlled by the processing unit 170. For example, the processing unit 170 may include a digital-to-analog converter (DAC) or a pulse-width-modulation (PWM) output configured to generate the excitation waveform digitally. The DAC or PWM output may be coupled to an on-board power amplifier or high-side driver that directly drives one of the sensors in the sensor array 120. In still other embodiments, the excitation module 130 may comprise a mechanical actuator, such as an impact hammer, an electrodynamic shaker, or a piezoelectric stack actuator, mechanically coupled to the plate structure 110 and configured to impart a controlled mechanical excitation at a transmitter location.

[0079] Example cooperation of excitation module and other components:

[0080] During operation, the processing unit 170 can control the excitation module 130 and a switching network 140 to sequentially designate different sensors of the sensor array 120 as transmitters and receivers. In one example sequence, the processing unit 170 issues a command to the switching network 140 to couple a first sensor of the sensor array 120 to the excitation module 130 as a transmitter and to couple a plurality of remaining sensors to an analog front-end 150 as receivers. The processing unit 170 then triggers the excitation module 130 to generate the prescribed excitation waveform and apply the waveform to the selected transmitter sensor.Application of the excitation waveform causes the transmitter sensor to mechanically stimulate the plate structure 110 and launch acoustic waves, such as Lamb waves, which propagate through the plate structure 110 and are sensed by the receiver sensors of the sensor array 120.23107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)

[0081] The receiver sensors of the sensor array 120 generate corresponding electrical response signals that represent time-domain motion, displacement, velocity, or strain at the respective sensor locations. These response signals are provided to the analog front-end 150, which may include one or more preamplifiers, charge amplifiers, or voltage amplifiers and one or more analog filters. The analog front-end 150 conditions the response signals, for example by amplifying the signals to a desired level and filtering the signals to a bandwidth appropriate for subsequent digitization.

[0082] The conditioned response signals can then be supplied to the data acquisition subsystem 160, which includes at least one analog-to-digital converter and, in some embodiments, a timing and synchronization unit configured to coordinate sampling across multiple receive channels. The data acquisition subsystem 160 digitizes the time-domain response signals to produce digitized response data for each receiver sensor. The digitized response data may be stored temporarily in a local data buffer and then transferred to the processing unit 170 for further analysis.

[0083] The processing unit 170, which may include one or more processors, digital signal processors, or programmable logic devices, is configured to process the digitized response data for a reference state of the plate structure 110 and for a perturbed state of the plate structure 110 that includes at least one defect. For a given transmitter configuration and a given subset of sensors of the sensor array 120, the processing unit 170 may transform the time-domain response data to a frequency domain to obtain complex spectral amplitudes at a plurality of frequencies. Using the complex amplitudes, the processing unit 170 forms normalized complex state vectors representing acoustic field states in a Hilbert space for the reference state and for the perturbed state, computes geometric phase change as an angle between the state vectors at each frequency, and calculates a geometric phase change index (GPC-I) as a spectral average of the geometric phase change over a selected frequency band.

[0084] By repeating the excitation and acquisition sequence for different transmitter locations and for multiple subsets of sensors, the processing unit 170 obtains GPC-I values associated with a plurality of sensing paths through the plate structure 110. The processing unit 170 then compares the GPC-I values across the24107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) sensing paths and identifies one or more sensing paths or regions of the plate structure 110 associated with relatively larger GPC-I values as probable locations of defects. In some embodiments, the processing unit 170 maps the GPC-I values to triangular or other geometric regions defined by the transmitter and receiver sensor locations and determines an intersection region of regions having the largest GPC-I values as an estimated defect region.

[0085] Additional aspects of this disclosure are set out in the independent claims and preferred features are set out in the dependent claims. Features of one aspect may be applied to each aspect alone or in combination with other aspects. In addition, while certain operations in the claims are provided in a particular order, it is appreciated that such order is not required unless the context otherwise indicates.25107332941.1

Claims

Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) CLAIMSWhat is claimed is:

1. A method for defect localization using acoustic waves, comprising:providing a sensor array comprising a plurality of sensors along a structure;operating at least one sensor of the sensor array as a transmitter to launch acoustic waves into the structure and operating at least one other sensor of the sensor array as a receiver to obtain response signals corresponding to the acoustic waves;obtaining response signals from the sensor array for a reference state of the structure and for a perturbed state of the structure that includes the at least one defect;for a plurality of subsets of the plurality of sensors, each subset comprising at least two sensors that define at least one sensing path through the structure, processing the response signals to compute, for each subset, a geometric phase change between a reference-state acoustic field and a perturbed-state acoustic field and to calculate a geometric phase change index parameter (GPC- I) as a spectral average of the geometric phase change over a selected frequency band; andcomparing the geometric phase change index parameters (GPC-I) for the plurality of subsets and identifying at least one region of the structure associated with one or more relatively larger geometric phase change index parameters (GPC-I) as a probable location of the at least one defect.

2. The method of claim 1, wherein the GPC-I parameter is derived from spectral averages of the geometric phase change using topological acoustic (TA) sensing.26107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)3. The method of claim 1, further comprising identifying a set of largest GPC-I values for various sensing paths associated with the plurality of sensors, wherein a higher GPC-I value on a sensing path implies a higher probability of having a defect on that path.

4. The method of claim 1, further comprising exciting the structure by at least one transmitter of the plurality of sensors to launch the acoustic waves and recording, at one or more receiving sensors of the plurality of sensors, time-domain response signals for a reference state of the structure and for a perturbed state of the structure that includes the at least one defect.

5. The method of claim 1, wherein operating at least one sensor as the transmitter and at least one other sensor as the receiver comprises sequentially operating different sensors of the sensor array as transmitters in a plurality of measurement cycles and, in each measurement cycle, operating a plurality of remaining sensors of the sensor array as receivers to obtain response signals for a plurality of sensing paths.

6. The method of claim 1, wherein each subset of the plurality of sensors comprises at least two receiving sensors that define, together with a transmitter of the plurality of sensors, at least one sensing path along which the GPC-I is calculated.

7. The method of claim 1, further comprising identifying a set of largest GPC-I values along the structure including:for each transmitter of at least two transmitters:(a) associating each GPC-I value with a triangular region of the structure having vertices at the transmitter and a corresponding pair of receiving sensors of a subset; and(b) selecting at least one triangular region associated with one of the27107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056) largest GPC-I values as a candidate region in which the at least one defect is more likely to be located.

8. The method of claim 7, wherein the at least two transmitters and the subsets of receiving sensors define first and second sets of triangular regions, and wherein localizing the at least one defect comprises identifying an intersection region between a first triangular region associated with a largest GPC-I value of the first set and a second triangular region associated with a largest GPC-I value of the second set.

9. A method for defect localization, comprising:monitoring, by a plurality of sensors, a geometric phase change (GPC) defining a measure of changes in an acoustic wave’s spatial behavior associated with a structure;measuring changes in geometric phase of acoustic waves associated with a subset of the plurality of sensors at a time and continuously changing the subset to calculate a GPC index parameter (GPC-I) along various paths; anddetecting at least one defect by utilizing the GPC index parameter (GPC-I) to identify perturbations caused by defects along sensing paths between transmitters and receivers of the plurality of sensors.

10. The method of claim 9, wherein calculating the GPC index parameter (GPC-I) along the various paths comprises deriving the GPC index parameter (GPC-I) from spectral averages of the geometric phase change (GPC) over a selected frequency band.

11. The method of claim 9, wherein measuring changes in geometric phase of acoustic waves associated with a subset of the plurality of sensors comprises selecting, as the subset, at least two sensors that define at least one sensing path between a transmitter and the at least two sensors.28107332941.1Attorney’s Docket No.: 085067-859030Client’s Ref.: (UA25-056)12. The method of claim 9, wherein measuring changes in geometric phase of acoustic waves associated with a subset of the plurality of sensors at a time and continuously changing the subset comprises selecting pairs of adjacent sensors of the plurality of sensors as the subset and successively shifting a pairwise window along the plurality of sensors to define the various paths.

13. The method of claim 9, wherein detecting the at least one defect comprises identifying a set of largest GPC index parameters (GPC-I) among the various paths and associating the at least one defect with at least one region along the sensing paths corresponding to the largest GPC index parameters (GPC-I).

14. The method of claim 9, wherein detecting the at least one defect further comprises operating at least two different sensors of the plurality of sensors as transmitters to define different sets of sensing paths and determining an intersection region between regions associated with largest GPC index parameters (GPC-I) for the different transmitters as a probable location of the at least one defect.

15. The method of claim 9, wherein the GPC index parameter is a measure of a geometry of an acoustic field of the structure averaged over a spectral domain.29107332941.1

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