Numerical investigation of bandgap characteristics in integrated metallic metafilters for nonlinear guided wave applications

Structural damage in engineering structures and industrial equipment often originates at the microscopic level, a process known as nucleation. While imperceptible at first, this damage can escalate over time, eventually becoming visible and potentially compromising the integrity of the entire structure. Detecting these initial changes poses significant challenges as they occur internally and may not manifest externally until the damage has progressed. Timely diagnosis is crucial for preemptive action to prevent catastrophic failures, ensuring the prolonged safe operation of structures and equipment. This proactive approach not only saves costs but also mitigates risks associated with structural failures.^1–3 Guided-wave-based techniques, utilizing waves such as Lamb waves, have emerged as valuable tools in non-destructive testing and structural health monitoring (SHM). These methods offer extensive coverage over large areas, detecting defects or material changes with high sensitivity.^4,5 Notably, they boast low energy consumption, making them cost-effective and environmentally friendly. Their ability to detect even minor flaws enhances early detection, ensuring comprehensive SHM.

In thin-walled structures such as plates, ultrasonic guided waves, commonly used in SHM, penetrate deeply into materials and interact with internal features such as defects or boundaries, generating unique signatures that indicate damage or material alterations. Nevertheless, traditional ultrasonic methods encounter limitations in detecting minute features at the micro- or nanoscale due to the wavelength of the waves used, impeding early diagnosis.^6,7 To address this issue, gigahertz (GHz) frequencies are essential for improving resolution and detecting finer material features, although material attenuation poses challenges. Balancing sensitivity to small-scale features with penetration depth is key to developing effective early detection techniques and assessing material performance. In contrast, nonlinear guided waves, which exhibit a nonlinear behavior as they propagate through materials, are highly prized for their ability to penetrate deeply and detect microscopic defects within the material.^8–10 Offering advantages over traditional methods, they provide insights into internal structures and can detect small defects. However, practical deployment encounters challenges, particularly regarding accurate measurement and the presence of higher harmonics in signals. The intensity of harmonics typically falls one or two orders of magnitude below the excitation frequency and necessitates substantial amplification of transmitted signals (with typical input voltages exceeding 150 V). This high voltage amplification leads to the generation of “undesired” harmonic frequencies by the ultrasonic equipment, which are comparable in magnitude to those produced by internal defects such as microcracks.^11–13 Distinguishing between harmonics arising from defects and those stemming from external factors is crucial to avoid false positives in defect detection, underscoring the importance of harmonic suppression techniques for reliable and accurate nonlinear-based guided wave testing across diverse industries.

Recently, phononic crystals (PCs) and acoustic metamaterials (AMs), characterized by their spatially periodic geometry and tailored material properties, have emerged as promising avenues for wave manipulation and control.^14–16 A notable feature of these materials is their ability to be designed and manufactured to impede the propagation of ultrasonic waves within specific frequency ranges, known as bandgaps or stopbands. Beyond these bandgaps, elastic waves can propagate in any direction, rendering these materials as natural signal filters. They exhibit remarkable characteristics such as focusing, filtering, wave manipulation, guiding, attenuation, and bandgap creation, enabling a wide array of applications.^17–20 Both PC and AM are artificial structures with bandgap properties. PC relies on the Bragg effect (Bragg scattering), where bandgaps form due to the destructive interferences of waves scattered by inclusions, holes, and internal interfaces, orderly arranged in a spatially repeated fashion. Here, the bandgap frequency is dependent on the periodicity constant (lattice spacing). In contrast, AM consists of resonators (inclusions) coupled with the host medium that generate bandgaps through local resonance (hybridization bandgap), often at much lower (sub-wavelength) frequencies. AM structures can be periodic or aperiodic, with periodicity having a limited impact on bandgap boundaries. For more details on these differences, see Refs. 21–24. Studies have also shown that phononic crystal plates, which comprise a periodic arrangement of pillars placed on top of a thin homogeneous plate, and Bragg and hybridization bandgaps can be simultaneously observed in such phononic plates, and as such, they may be dually classified as PC and AM.²² Although several studies have explored the bandgap characteristics of specially designed PC and AM, their application in SHM remains relatively unexplored.

Unlike conventional studies where waves propagate through materials, SHM necessitates surface-mounted devices to preserve structural integrity. The challenge lies in ensuring that these add-on devices interact effectively with the host structure, indirectly influencing wave propagation. Researchers must devise innovative design strategies tailored to surface-mounted devices to realize the concept’s potential in SHM applications. Recently, studies have also showcased their potential, including suppressing wave harmonics, filtering multiple frequencies, developing acoustic diodes, and enhancing sensitivity in damage detection during nonlinear ultrasonic testing.^25–29 Smith and Matlack³⁰ utilized additive manufacturing (AM) phononic materials with ultrasonic filtering capabilities to reduce extraneous nonlinearities, acting as low-pass filters where cutoff frequencies depend on unit cell geometry and AM defects. Sherwood et al.³¹ introduced 3D-printed acrylic PC waveguide transducers with sinusoidally corrugated profiles, demonstrating bandgaps from the destructive interference of Lamb waves. Shan et al.³² developed a metamaterial filter with T-shaped lead patches for SHM, highlighting the role of adhesive nonlinearity in refining nonlinear waves. Tian et al.³³ improved fatigue damage detection using an aluminum-lead metamaterial to enhance higher harmonic features by suppressing inherent nonlinearities around 100 kHz.

Sandeep Kumar et al.³⁴ investigated a metamaterial comprising periodic flat ridges on cylindrical rods, which effectively suppressed nonlinearity prior to signal inspection. Sun et al.³⁵ introduced multifunctional PC filters by modifying the plate geometry with additive bumps, designed to eliminate unnecessary modes and second harmonic waves. Liu et al.³⁶ proposed a metafilter using a topology optimization framework, tailored for wide bandgap control and validated through numerical simulations and experiments. Li et al.³⁷ explored a polylactic acid PC structure with connective layers that suppressed nonlinearity interference, enhancing micro-defect detection sensitivity in damaged aluminum plates at 100 kHz. Cao et al.³⁸ employed two acrylonitrile butadiene styrene PC filters to mitigate nonlinearity and enhance micro-crack detection, achieving significant second harmonic attenuation (20 and 12 dB) and accurate detection through combined sensor setups. Zhang et al.³⁹ used 1D PC in nonlinear ultrasonics to improve concrete damage detection by blocking frequencies above 100 kHz, enabling sub-wavelength defect identification while eliminating instrumentation-induced nonlinearity. Liu et al.⁴⁰ introduced a metamaterial optimized for second harmonic S₀ mode Lamb waves, facilitating easier measurement of these waves for detecting microstructural changes in thin-walled structures.

Several designs of PC and AM have been implemented to enhance the nonlinear-based ultrasonic testing. However, most of these studies focus on filtering second harmonics within a 100–300 kHz frequency range, primarily employing low-frequency excitation. Moreover, many of these designs are predominantly polymer-based, making them too small to filter high frequencies effectively. In contrast, metallic materials can be fabricated with millimeter-scale dimensions, making them well-suited for high-frequency applications and nonlinear ultrasonic testing of metal structures, such as thin plates or cylindrical pipes, where the AM can be matched to the structural component. In this study, we propose a ring-shaped meta-structure design to investigate high-order bandgaps above 500 kHz using guided waves. Three materials, namely aluminum, copper, and steel, are considered for the rings. This ring-type design offers ease of manufacture due to its simple geometry, offering a more practical approach compared to previous methods. To fully understand the bandgap characteristics, we begin by analyzing the mode shapes, providing insights into the underlying wave mechanics. This is followed by a steady-state dynamic analysis to assess the impact of non-periodic arrangements, an essential factor in practical applications. A detailed parametric study further investigates how metamaterial geometry, adhesive elastic properties, and its thickness affect the bandgap performance. Finally, the effectiveness of the proposed ring-shaped metafilter in suppressing harmonic components for nonlinear guided wave testing is evaluated using 3D time-domain numerical simulations.

The rest of this paper is organized as follows: Sec. II details the methodology for designing the metafilter, focusing on the principles of bandgap formation through mode shape and steady-state dynamic analysis. This section also examines how variations in dimensional properties, adhesive thickness, and material influence the bandgap behavior. In Sec. III, the performance of the proposed metafilter in refining signals for nonlinear guided wave testing in practical applications is emphasized. This section also explores the effects of the number of rings and loading cycles on harmonic suppression and illustrates mode conversion phenomena, further highlighting the metafilter’s impact on guided wave propagation. Finally, Sec. IV presents a comprehensive summary of the key conclusions derived from this study.

Figure 1 illustrates a schematic of the proposed metafilter attached to the plate waveguide. This figure also depicts the unit cell derived from the simplified 2D plane strain model of the 3D model, highlighting key dimensional parameters, including height (h), width (w), thickness (t), and lattice/unit cell length (l_c). The unit cell is attributed to the properties of aluminum. Moreover, this study also incorporates the effects of alternative materials such as steel and copper. The material properties used in this study are shown in Table I. A four-node linear plane strain element is employed to discretize the model. To ensure the precise representation of guided wave modes within the specified frequency range, the mesh size is selected to guarantee a minimum of 20 elements per wavelength, thereby achieving a high spatial resolution of waves. An eigenfrequency analysis is conducted by imposing Bloch–Floquet periodic boundary conditions on both lateral faces of the unit cell, marked by green lines. This approach considers that the wave behavior can be comprehensively characterized within the first Brillouin zone for periodic structures.

A parametric study is initially conducted to analyze the influence of geometric variations in the periodic cell on bandgap characteristics. Three lattice/unit cell lengths, l_c, are chosen, 3, 4, and 5 mm. Meanwhile, other dimensions, such as h, w, and t, are varied from 1 to 3, 1 to 4, and 1 to 5 mm, respectively. These parameters are systematically varied one at a time to examine the impact of each dimension on bandgap characteristics.

Figure 2 illustrates a dispersion curve for a typical case with l_c = 3 mm, and t, w, and h = 1 mm. The gray-shaded regions indicate the presence of bandgaps. For steel, mode shapes at specific points (marked by blue dots) on the curve are extracted, corresponding to the lower and upper edges of the bandgap, offering insights into the displacement fields and wave localization mechanisms responsible for bandgap formation. Within the frequency range of 0–1 MHz analyzed in this study, the number of extracted modes is limited, each sequentially numbered as 1, 2, 3, and so forth. Observations reveal three distinct bandgaps for aluminum and four distinct bandgaps for steel and copper. Upon comparing materials, it is evident that steel exhibits the widest bandgap (∼191 kHz) between modes 4 and 5.

The first bandgap for steel, located between modes 2 and 3, spans a frequency range from 199 kHz (lower edge) to 288 kHz (upper edge). Analyzing these mode shapes reveals that the top of the pillar shows maximum displacement, leading to strong local vibrations with a weak bending of the plate. For mode 3, since kl_c/π = 0, this suggests that the wavelength of the wave is very large relative to the lattice length. This is characteristic of a hybridization bandgap, where the coupling between the weak flexural modes of the plate and the local resonant modes of the pillar traps energy within the pillar. At the lower edge of the second bandgap, at 486 kHz, the plate shows significant bending deformation, while the pillar has weak displacement, indicating that wave propagation is primarily through the plate. In contrast, at the upper edge (535 kHz), both the plate and pillar resonate strongly in bending mode. Notably, the slight flattening of curve 3 from kl_c/π ≈ 0.7 to 1 suggests the onset of Bragg scattering, though not classical, as curve 4 does not show this behavior. This reflects a hybridization effect where the coupling between the plate’s flexural waves and the pillar’s resonances creates the bandgap.

The third bandgap, from 657 to 848 kHz, between curves 4 and 5, suggests a classical Bragg scattering bandgap, as both curves flatten near kl_c/π = 1. However, at the lower edge, the mode shape shows strong pillar resonance with moderate plate deformation, indicating the interaction between the pillar’s local resonance and the plate’s weak flexural waves. At the upper edge, the plate dominates the deformation, while the pillar shows minimal displacement, characteristic of a Bragg bandgap, with a symmetric distribution in the plate. This bandgap results from a combination of Bragg scattering and hybridization. Similar observations of simultaneous Bragg and hybridization mechanisms have been reported recently in certain phononic crystals.^22,41–43

To gain insights into the behavior of the meta-structure, a steady-state dynamic analysis is performed using two configurations: one with a periodic arrangement of the metafilter and the other with a non-periodic, randomly placed arrangement. Twelve pillar units are included, with a total wave propagation length of 36 mm in the plate. A sinusoidal excitation is applied to the left edge, sweeping frequencies from 75 to 1000 kHz in 1 kHz increments. The displacement response is measured at the right edge, allowing for the evaluation of the system’s transmission characteristics. In addition, a 40 mm absorbing region is modeled at both ends of the plate using the ALID (Absorbing Layers using Increasing Damping) method to mitigate the reflected wave effects.⁴⁴ 

The displacement transmission coefficient for both models is plotted on a dB scale in Fig. 3, providing a comparative analysis of the wave propagation characteristics. This figure highlights the key frequency points (1–5) along the curves, where distinct behaviors in the wave propagation are observed. For each point, the corresponding magnitude of real displacement contour plots is shown, with the top contour representing the periodic model and the bottom contour depicting the non-periodic arrangement. These visualizations offer a clear comparison of wave energy distribution and attenuation in both configurations, enhancing the interpretation of transmission characteristics across different frequency ranges.

At point 1 (around 255 kHz), a significant dip in both curves indicates a bandgap likely caused by the local resonance of the first pillar, as suggested by the similar displacement contour plots for both periodic and non-periodic models. At point 2 (around 426 kHz), the periodic model shows smooth wave transmission, while the non-periodic model exhibits energy localization between the eighth and ninth pillars. This localized resonance traps the wave energy, preventing propagation beyond this region and leading to the observed dip in transmission. At point 3 (around 505 kHz), the periodic structure shows a sharp dip in transmission, with the displacement contour indicating that wave propagation is confined to the first six pillars, beyond which displacement drops to nearly zero. This suggests strong wave reflection due to a combination of Bragg scattering (slight flattening of curve 3 in Fig. 2 for steel) and resonance effects. In contrast, the non-periodic model shows wave transmission across the entire structure, indicating minimal wave reflection and stronger transmission despite the irregular arrangement.

In the periodic arrangement, a narrow transmission band appears between 850 and 880 kHz, situated between the third and fourth bandgaps. In contrast, the non-periodic configuration exhibits a wider bandgap from 680 to 1000 kHz, with multiple peaks showing a maximum amplitude of less than −80 dB, indicating significant wave energy attenuation. The displacement contours at points 4 and 5 reveal localized resonance in the non-periodic model, between the sixth and seventh pillars for point 4 and between the third and fourth pillars for point 5. Similar localized resonances occur between other adjacent pillars at various peaks. This demonstrates the effectiveness of non-periodic arrangements in enhancing wave attenuation over a broader frequency range, likely due to multiple scattering and hybridization effects.

The following analysis investigates how variations in geometric dimensions and host plate thickness affect the metafilter’s bandgap characteristics. Specifically, this study explores the influence of metafilter height, width, and host plate thickness on bandgap width, intending to optimize these parameters for efficient wave filtering. Figure 4 illustrates the variation in bandgap width concerning the metafilter height, h, for widths w of 2, 3, and 4 mm. Other dimensions, such as t and l_c, are held constant at 1 and 5 mm, respectively. The first bandgap, marked between the red dashed and solid lines, represents the gap between modes 2 and 3, and the blue lines between modes 3 and 4. Only bandgaps with a minimum width of 10 kHz were considered for practicality. Analysis of Fig. 4 reveals that the width of the first bandgap (2–3) decreases with increasing height h across all values of w. However, this trend is not consistently observed in other bandgaps. In many cases, the width initially increases with h and then decreases. Notably, the widest gap, ∼300 kHz, is observed for steel at w = 4 mm and h = 3 mm, occurring close to 400 kHz. In addition, it is observed that as w increases, more bandgaps are detected at higher frequency ranges, albeit with a decreased width. However, it is important to note that the appearance of bandgaps is not consistent across all height ranges; bandgaps may appear or disappear depending on the specific parameters. Steel consistently exhibits the widest gaps across various frequency ranges.

Figure 5 illustrates the variation in bandgap width concerning metafilter height h for different host plate thicknesses t = 1, 3, and 5 mm. Here, the width w and lattice length l_c are constant at 1 and 5 mm, respectively. Observations reveal that the number of bandgaps and their widths decreases with an increase in t. For instance, it is noted that the first bandgap between modes 2 and 3 disappears for t = 5 mm for all materials. In addition, the second bandgap (between modes 3 and 4) disappears in aluminum, while it remains present for steel and copper. Conversely, the sixth bandgap (between modes 7 and 8) disappears for steel and copper but persists for aluminum. These observations highlight the significant influence of host plate thickness (t) on the emergence and characteristics of bandgaps in meta-structures.

Figure 6 illustrates the variation in bandgap width concerning metafilter height h for different lattice lengths l_c = 3, 4, and 5 mm. Here, the thickness t and width w are held constant at 1 and 2 mm, respectively. It is noted that there is a considerable shift of the bandgap to higher frequencies as l_c decreases. For example, when l_c = 5, the first bandgap for aluminum with h = 1 mm is close to 200 kHz, whereas, with l_c taken as 4 and 5 mm, the bandgap is observed close to 250 and 350 kHz, respectively. Similar observations are noted for other materials as well. Moreover, the bandgap width also appears to increase for most frequency ranges as l_c decreases. However, the number of bandgaps reduces as l_c decreases. A bandgap for steel with l_c = 3 and h between 1 and 2 mm is obtained close to 600 kHz with a maximum width of ∼300 kHz.

When metafilters are used as surface-mounted devices, they must be attached to the host structure using bonding agents such as silver epoxy, instant glue, thermoplastic tape, Loctite adhesive, and Araldite epoxy.⁴⁵ The choice of bonding agent is crucial for ensuring proper coupling between the metafilter and the host layer. In addition, the thickness of the adhesive layer impacts the metafilter’s behavior, as a shear lag effect occurs due to the adhesive, becoming more prominent as the layer thickness increases. Three different values of elastic modulus—1, 2, and 3 GPa—are considered, as most commercially available adhesives fall within this range.^45,46 The density and Poisson’s ratio of the adhesive are kept constant at 1200 kg/m³ and 0.4, respectively.

Figure 7 shows the comparison of the dispersion characteristics of models without an adhesive layer and those with adhesives of varying elastic moduli. When the elastic modulus is low (E = 1 GPa), more modes are present, with four bandgaps observed. This increase is likely due to the softer adhesive introducing more localized deformations and resonances, increasing the complexity of wave propagation. As the adhesive softens, the bandgap frequencies shift lower, indicating reduced overall system stiffness. For E = 2 GPa, the first, second, and third bandgaps narrow by ∼57%, 49%, and 16%, respectively. At E = 3 GPa, a similar trend is seen, although the third bandgap increases by around 7%.

Figure 8 illustrates the impact of increasing adhesive thickness on bandgap properties, with E = 3 GPa and ν = 0.4. As adhesive thickness increases, the bandgaps shift toward lower frequencies, and their width decreases. For instance, the first bandgap width reduces by 39%, 57%, and 68% for adhesive thicknesses of 20, 40, and 60 μm, respectively. This behavior can be attributed to the increased flexibility introduced by the thicker adhesive, which lowers the overall stiffness of the structure, reducing the resonance frequencies and narrowing the bandgaps. The appearance of a fourth bandgap at higher frequencies for the 60 μm adhesive could likely be due to localized resonances, allowing for more complex wave interactions and higher-order modes.

In ultrasonic-guided wave testing, where excitation signals are often chosen as tone burst signals with a specific number of cycles, ensuring a sufficient bandgap width becomes crucial. Tone burst signals exhibit a frequency response with a dominant main peak centered around the frequency of its sinusoidal component, and the energy spreads on either side of this peak, defining a bandwidth that characterizes the range of frequencies over which the signal exhibits a significant energy. Therefore, having a wider bandgap is essential for effectively filtering these tone burst signals and ensuring precise detection and analysis in nonlinear-based guided wave ultrasonic testing applications. Section III will demonstrate the influence of metafilters in refining the excitation signal in guided wave testing, the role of repeating units in filtering harmonic components, and how the number of tone burst cycles impacts the testing results.

The performance of the designed metafilter is evaluated through 3D elastodynamic simulations. The numerical model illustrated in Fig. 9 depicts the host plate with an attached metafilter near the excitation point. Both the plate and the metafilter are assumed to be made of aluminum. The ring’s dimensions are chosen to be 2 mm in width and 1.9 mm in height, and the center-to-center distance between the rings is set at 5 mm. The plate is modeled with dimensions of 500 × 330 × 1 mm³, featuring an absorbing region of 100 mm width attached to the plate boundary. This absorbing region is modeled based on the ALID method,⁴⁴ which utilizes materials with gradually increasing damping commonly employed in dynamic explicit analysis to define unbounded boundary conditions.

An eight-node linear brick element is used to mesh the plate and the metafilter. The finite element domain is discretized to accurately sample the propagating modes in the frequency range, ensuring 15–20 elements per wavelength and achieving an excellent spatial resolution of propagating waves.⁴⁷ The time increment (Δt) for explicit dynamics is chosen such that Δt < L_min/C, where L_min is the smallest element size and C is the wave speed.

Three numerical models are considered for this study: the first comprises a bare plate without considering the metafilter, the second involves a plate with an attached metafilter, and the third incorporates a plate with an attached metafilter and the presence of damage. The damage is modeled by introducing a seam crack of length 5 mm located at a distance of 100 mm from the excitation point. The surface-to-surface contact interaction is modeled using the Lagrange multiplier method, effectively preventing the interpenetration of crack surfaces during wave propagation. As guided waves approach a crack, they cause the crack surface to open and close alternately. This phenomenon allows the compressive part of the wave to pass through while scattering the tensile part, resulting in contact acoustic nonlinearity. Twelve repeating ring units were considered, resulting in a total diameter of 125 mm for the metafilter. Only half of the metafilter model is considered to simplify the numerical model and reduce computational costs, as depicted in Fig. 9. 

A nonlinear transformation, such as exponentiation, is applied to the original excitation signal, altering its characteristics and introducing complexity. This transformation results in a signal that deviates from linearity, a crucial consideration when assessing nonlinearities stemming from the measurement system. Furthermore, the excitation signal considered for this study comprises two frequencies. This choice aims to generate multiple harmonics, including combinational harmonics. The use of mixed frequency signal results in nonlinear guided wave mixing, which offers numerous advantages, such as broadband sensitivity and the ability to selectively excite specific modes or wave types within a structure. It provides greater flexibility in designing wave interactions such as codirectional, counter-propagating, and non-collinear setups. It has been demonstrated that the combinational harmonic components exhibit a high sensitivity to microstructural defects, and mixing generates additional harmonic components that are sensitive to certain defect types and can be correlated with the degree of damage.

One key advantage of wave mixing is the elimination of higher harmonics induced by instrumentation.⁴⁸ This can be achieved by employing two transducers, each connected to separate channels of the signal generator. However, when designing wave interactions, such as codirectional interactions, a single transducer can be utilized to generate signals consisting of two frequencies. Moreover, in cases where the signal generator lacks two channels, a single channel can suffice. Previous studies have demonstrated the feasibility of using a mixed signal with a single actuator connected to a single channel.^49–52 However, in such setups, instrument nonlinearity may even induce combinational harmonics in addition to second harmonics. Figure 10 illustrates the excitation signal in both the time and frequency domains, comprising two Hanning windowed tone burst signals with central frequencies of 230 kHz (f₁) and 360 kHz (f₂). The second harmonics and combinational harmonics generated by these primary frequencies fall within the bandgap as determined by the selected dimensions of the rings.

Figures 11(a) and 11(b) depict the comparison of response signals from different models, detected at a distance of 175 mm from the source, in both the time domain and frequency domain, respectively. Notably, in the time-domain signal, the cases with the metafilter (with and without damage) exhibit a slight reduction in signal amplitude and a small delay in signal arrival time. In addition, these signals manifest several trailing wave components following the main wave packet, attributable to the influence of the metafilter. These aspects will be further discussed in Sec. III C–III D. From Fig. 11(b), it is evident that the metafilter effectively suppresses all harmonic components. Conversely, the case with both metafilter and damage exhibits harmonic components solely attributable to the damage, and any nonlinearity arising from the input signal is effectively filtered out.

To examine the effect of the number of rings on filtering the harmonic components, we varied the number of rings as 3, 6, 9, and 12. Figure 12(a) illustrates the frequency response under different cases. It can be observed that a minimum number of repeating ring units is necessary to suppress all the harmonic components. To gain insights into how these peaks reduce with the number of rings, the percentage reduction in amplitudes of different harmonic components with respect to the number of rings is presented in Fig. 12(b). Notably, the reduction in amplitude is not consistent for all harmonics. For instance, a considerable reduction is observed for the second harmonic component (2f₂) and the sum frequency component (⁠$f1+f2$⁠. However, the rate of reduction is lower for the other second harmonic (2f₁) and the difference frequency component (⁠$f2−f1$⁠. In addition, it can be observed that the reduction in amplitudes corresponding to fundamental frequencies is negligible. Furthermore, it should be noted that although peaks may be observable for higher numbers of rings, in reality, the peaks may not be distinguishable for nine rings and above, as the signal may be submerged in noise due to electronic interference, environmental effects, and other factors. These findings underscore the importance of optimizing the number of rings to effectively attenuate harmonic components, thus enhancing the overall performance of the metafilters.

To investigate the influence of the loading cycle, we considered different cycle numbers. The cycle numbers of the excitation signal were varied while keeping the excitation frequency fixed. Figure 13(a) illustrates the displacement response in the frequency domain measured at 375 mm from the actuation point with different signal lengths for the model with six rings. Generally, increasing the cycle number results in better frequency resolution and higher peak magnitudes at the fundamental frequency and its harmonics. However, this trend is not observed here. Figure 13(b) shows how these peak amplitudes vary with signal duration. It is observed that an increase in peak is noted in the sum frequency component and one of the second harmonics (2f₂), whereas the peaks corresponding to the difference frequency component and the other second harmonic (2f₁) decrease with the increase in signal duration. This phenomenon could be attributed to the variation and position of the bandgap width with respect to each harmonic component. As observed in Fig. 6, based on the dimensions chosen for the metafilter, the difference frequency component and the second harmonic 2f₁ are approximately located at the center of the bandgap width, whereas the sum frequency component and the second harmonics 2f₂ lie close to the boundary. Since both these components lie close to the boundary, some part of the energy of the frequency content lies outside the bandgap width, which does not have an effect on increasing the cycle number of the signal. These observations highlight the importance of signal duration and bandgap width, which need careful consideration when designing metafilters for optimal performance.

A more detailed analysis is conducted to identify the propagating modes in all models through a 2D fast Fourier transform. Figure 14 illustrates the wavenumber–frequency spectrum obtained through simulation, superimposed with the analytical dispersion curves. In Fig. 14(a), it can be observed that all generated harmonic components belong to the S₀ guided wave mode. However, when the plate is excited with the integration of metafilters [Fig. 14(b)], mode conversion occurs, leading to the conversion of part of the wave energy to the A₀ guided wave mode. In addition, it conveys that the metafilter is capable of filtering all harmonic components induced by the input excitation signal. In the presence of damage, as depicted in Fig. 14(c), higher harmonics, including combinational harmonics, are generated, consisting of both S₀ and A₀ guided wave modes. This highlights that after the integration of the metafilter to monitor a damaged plate, harmonic components solely attributable to the damage are exhibited, effectively suppressing any nonlinearity arising from the input signal. Furthermore, mode identification through this analysis reveals that the several trailing wave packets following the main wave packet seen in the time-domain signals [Fig. 11(a)] are due to the generation of A₀ guided wave modes influenced by the metafilters. This phenomenon is also noticeable in the displacement (magnitude) contour extracted at 90 μs, as illustrated in Fig. 15.

Several approaches and algorithms have been developed to localize defects using sensor data in guided wave-based SHM systems. Common methods include beamforming, triangulation, and time-of-flight (TOF) analysis, each relying on different aspects of wave propagation.^53–55 These techniques aim to identify the location of discontinuities or cracks by interpreting wave reflections, phase shifts, or arrival times. In the current study, we employ the Time Difference of Arrival (TDOA) method to determine the defect location. TDOA is conventionally used in fields such as telecommunications, acoustics, and radar systems to locate signal sources based on the arrival time differences at multiple sensors.⁵⁶ It is also employed in guided wave-based SHM techniques, where reflected or scattered wave signals are used for defect localization.^57,58 Here, we adapt the same principle to nonlinear guided wave-based SHM, where, instead of locating a signal source, we determine the defect location using the time delays of harmonic wave components recorded at different sensors. These harmonic components are generated when the primary wave interacts with a crack and propagates at different velocities due to the dispersive nature of guided waves. By measuring the arrival times of specific wave modes at particular frequencies, we can precisely localize the defect.

Figure 16 illustrates the sensor configuration and defect location used in this study. The three sensors, labeled S1–S3, are deliberately positioned to ensure that they are non-collinear, a design choice that significantly improves the accuracy of defect localization. By calculating the time differences between these arrival times, we can determine the corresponding differences in distance from the defect to each sensor. These distance differences are then used to formulate the hyperbolic curves, which ultimately converge to provide an estimate of the defect’s location.

Figure 17 shows the raw signal extracted from sensor S1, along with its corresponding harmonic components at the sum frequency (590 kHz) and the difference frequency (130 kHz). These harmonic components are obtained by applying a bandpass filter to the raw signal, isolating the desired frequency bands. The sum and difference frequency components provide insights into the nonlinear interactions occurring during wave propagation, offering valuable information for defect detection and localization. Similarly, other harmonic components, such as second or third harmonics, can also be utilized, each exhibiting different sensitivities to various types of damage.

Figure 18 shows the time-domain response of the sum frequency component obtained through bandpass filtering at sensors S1–S3. In addition, the envelope of each signal is computed and plotted using the Hilbert transform to more clearly identify the distinct S₀ and A₀ wave packets. By analyzing the envelopes, the peak locations corresponding to the first and second wave packets are identified, enabling a precise determination of the arrival times of the sum frequency component at each sensor. Table II provides a summary of the arrival times, group velocities, and frequencies of the second-order wave modes recorded at sensors S1–S3. These arrival times are then used to compute the time differences between sensor pairs, forming the basis for constructing hyperbolic curves for defect localization.

Figure 19 illustrates the hyperbolic curves generated from the time-of-arrival differences between sensor pairs S1–S2, S1–S3, and S2–S3 for three distinct second-order wave modes. Each curve represents the loci of possible defect locations based on the measured time differences, with the intersection of these hyperbolas providing an estimate of the defect location. From Fig. 19(a) corresponding to the S₀ wave mode at 590 kHz, it can be observed that the curves do not converge at a single point. However, the zoomed inset highlights a small triangular region of uncertainty around the intersection. Figures 19(b) and 19(c) show similar results for the A₀ wave mode at 590 and 130 kHz, respectively. The predicted defect locations in each case are approximately (0.1748, −0.0009 m), (0.1751, 0.0001 m), and (0.1747, −0.0004 m) in the x and y coordinates, respectively, with a maximum error of less than 0.2% in the x-coordinate from the actual defect position. This demonstrates the high precision of the localization method across different wave modes and frequencies.

The present results demonstrate the feasibility of integrating metallic metafilters for suppressing harmonic components in measurement systems during nonlinear guided wave testing, significantly improving defect detection accuracy and sensitivity. Notably, the defect localization using TDOA is illustrated, demonstrating that incorporating various harmonic components such as second, third, sum, and difference frequencies can enhance the accuracy of defect location estimation. However, the small dimensions of these metafilters pose practical manufacturing challenges, particularly in terms of precision and scalability. Advanced techniques such as micro-fabrication, laser cutting, or high-resolution 3D printing may be necessary to meet these demands. Furthermore, environmental and operational factors, including temperature and mechanical stress, could affect the filter performance by altering material properties and bandgap characteristics. Future studies should explore these environmental influences to ensure the reliability and effectiveness of metafilters in real-world applications. In addition, integrating advanced materials such as shape memory alloys and piezoelectric materials could enable multifunctional metafilters that respond to environmental stimuli, maintaining high accuracy while broadening the scope for adaptive wave filtering.

This study proposes and implements a metal metafilter aimed at enhancing the detection of material microstructural changes in high-frequency, nonlinear, guided wave-based techniques. Mode shape and steady-state analysis revealed that the bandgap characteristics of the metafilter are induced by Bragg scattering, local resonance, or a combination of Bragg scattering and hybridization effects. A parametric study on the metafilter geometry showed that bandgaps vary depending on specific parameters, highlighting the need for careful design considerations. The filter’s effectiveness was further evaluated using time-domain numerical simulations, emphasizing the importance of factors such as the number of rings, signal duration, and bandgap width in optimizing performance. In addition, the influence of bandgap characteristics on the peak magnitudes of harmonic components underscores the complexity of metafilter design. The observed mode conversion from S₀ to A₀ guided wave modes further demonstrates the filter’s ability to influence wave propagation. The defect localization method, based on the time difference of arrival of second-order wave modes, successfully predicted the defect location with less than 0.2% error, further validating the accuracy and effectiveness of the approach. These findings highlight the importance of systematic design approaches and comprehensive analyses in achieving optimal metafilter performance for nonlinear-based structural health monitoring and related applications.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (Grant No. 2019R1A5A8083201) and the Ministry of Education (Grant Nos. 2022R1I1A3069291 and 2021RIS-003).

The authors have no conflicts to disclose.

Mohammed Aslam: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Boris I: Data curation (equal); Software (equal); Visualization (equal); Writing – original draft (equal). Jaesun Lee: Conceptualization (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.