Kailiang Xu

Mode separation of Lamb waves based on dispersion compensation method.

Ultrasonic Lamb modes typically propagate as a combination of multiple dispersive wave packets. Frequency components of each mode distribute widely in time domain due to dispersion and it is very challenging to separate individual modes by traditional signal processing methods. In the present study, a method of dispersion compensation is proposed for the purpose of mode separation. This numerical method compensates, i.e., compresses, the individual dispersive waveforms into temporal pulses, which thereby become nearly un-overlapped in time and frequency and can thus be extracted individually by rectangular time windows. It was further illustrated that the dispersion compensation also provided a method for predicting the plate thickness. Finally, based on reversibility of the numerical compensation method, an artificial dispersion technique was used to restore the original waveform of each mode from the separated compensated pulse. Performances of the compensation separation techniques were evaluated by processing synthetic and experimental signals which consisted of multiple Lamb modes with high dispersion. Individual modes were extracted with good accordance with the original waveforms and theoretical predictions.

Multiridge-based analysis for separating individual modes from multimodal guided wave signals in long bones

Quantitative ultrasound has great potential for assessing human bone quality. Considered as an elastic waveguide, long bone supports propagation of several guided modes, most of which carry useful information, individually, on different aspects of long bone properties. Therefore, precise knowledge of the behavior of each mode, such as velocity, attenuation, and amplitude, is important for bone quality assessment. However, because of the complicated characteristics of the guided waves, including dispersion and mode conversion, the measured signal often contains multiple wave modes, which yields the problem of mode separation. In this paper, some novel signal processing approaches were introduced to solve this problem. First, a crazy-climber algorithm was used to separate time-frequency ridges of individual modes from time-frequency representations (TFR) of multimodal signals. Next, corresponding time domain signals representing individual modes were reconstructed from the TFR ridges. It was found that the separated TFR ridges were in agreement with the theoretical dispersion, and the reconstructed signals were highly representative of the individual guided modes as well. The validations of this study were analyzed by simulated multimodal signals, with or without noise, and by in vitro experiments. Results of this study suggest that the ridge detection and individual reconstruction method are suitable for separating individual modes from multimodal signals. Such a method can improve the analysis of skeletal guided wave signals by providing accurate assessment of mode-specific ultrasonic parameters, such as group velocity, and indicate different bone quality properties.

Transmission analysis of ultrasonic Lamb mode conversion in a plate with partial-thickness notch.

Mode conversions of Lamb waves can occur upon encountering damage or defect such as a notch, leading to newly-converted modes apart from wave reflection and transmission. In this paper, the transmission of the fundamental Lamb modes symmetrical S0 and anti-symmetrical A0 with anti-symmetrical notches were investigated in steel plates within the relatively short propagation distance. The group velocity and modal energy of the converted modes were analyzed using simulations and experiments. Two-dimensional finite difference time domain (2D-FDTD) method was employed to calculate the scattering field and extract numerical trends for simulation study and experimental confirmation. Both simulations and experiments revealed that the apparent group velocities of the converted modes in the transmitted signals subject to the notch positions. To describe the mode conversion degree and evaluate the notch severity, wave packets of the originally-transmitted modes and newly-converted modes were separated and corresponding mode energy percentages were analyzed at different notch severities. Frequency-sweeping measurements illustrated that the modal energy percentages varied monotonically over the notch-depth increase with a statistically consistency (R=1.00, P<0.0004).

Wideband dispersion reversal of lamb waves

Ultrasonic guided waves have been widely acknowledged as the most promising tools for nondestructive evaluation (NDE). However, because of the multimodal dispersion, the received guided modes usually overlap in both time and frequency, which highly complicates the mode separation and signal interpretation. The time-reversal technique can be used to realize the time recompression of the Lamb waves, but because of the multimode excitation and reception, it still may not be able to remove the mode ambiguity and achieve the pure pulse compression. With the goal of overcoming this limitation, a wideband dispersion reversal (WDR) technique is proposed. The technique makes use of a priori knowledge of the guided dispersion characteristics to synthesize the corresponding dispersion reversal excitations, which are able to selectively excite the self-compensation pure mode pulse. The theoretical basis of the technique is thoroughly described. A two-dimensional finite-difference time-domain (2D-FDTD) method is employed to simulate the propagation of two fundamental Lamb modes, the symmetrical S0 and antisymmetrical A0 modes in a steel plate. The proposed method was verified through experimental investigation. Finally, the advantages and potential applications of the method are briefly discussed.

Assessment of the Fundamental Flexural Guided Wave in Cortical Bone by an Ultrasonic Axial-Transmission Array Transducer

The fundamental flexural guided wave (FFGW), as modeled, for example, by the A0 Lamb mode, is a clinically useful indicator of cortical bone thickness. In the work described in this article, we tested so-called multiridge-based analysis, based on the crazy climber algorithm and short-time Fourier transform, for assessment of the FFGW component recorded by a clinical array transducer featuring a limited number of elements. Methods included numerical finite-element simulations and experiments in bone phantoms and human radius specimens (n = 41). The proposed approach enabled extraction of the FFGW component and determination of its group velocity. This group velocity was in good agreement with theoretical predictions and possessed reasonable sensitivity to cortical width (r(2) = 0.51, p < 0.001) in the in vitro experiments. It is expected that the proposed approach enables related clinical application. Further work is still needed to analyze in more detail the challenges related to the impact of the overlying soft tissue.

Coded excitation of ultrasonic guided waves in long bone fracture assessment

Ultrasonic guided wave (GW) assessment of long bone fracture have conventionally been based on pulse excitation. However, the high attenuation during propagation diminishes the amplitude of received GWs and results in low signal-to-noise ratio (SNR). The Barker code (BC) excitation and the optimal binary code (OBC) excitation were utilized in this study to overcome this limitation. Both simulations and in vitro experiments were performed on the fractured cortical bone plate model, and measured signals from both the BC and OBC excitations were decoded using the finite impulse response least squares inverse filter (FIR-LSIF) and then compared with sine pulse (SP) excited signals. The results suggest the efficiency of coded excitation for amplitude and SNR improvement. Furthermore, time-frequency representation (TFR) analysis was applied to experimental signals; with increasing fracture depth, energy transformation between predominate GW modes A1 and S2 was confirmed. These results show the potential of using BC and OBC excitations to evaluate the depth of long bone fracture.

Sparse SVD Method for High-Resolution Extraction of the Dispersion Curves of Ultrasonic Guided Waves

The 2-D Fourier transform analysis of multichannel signals is a straightforward method to extract the dispersion curves of guided modes. Basically, the time signals recorded at several positions along the waveguide are converted to the wavenumber-frequency space, so that the dispersion curves (i.e., the frequency-dependent wavenumbers) of the guided modes can be extracted by detecting peaks of energy trajectories. In order to improve the dispersion curve extraction of low-amplitude modes propagating in a cortical bone, a multiemitter and multireceiver transducer array has been developed together with an effective singular vector decomposition (SVD)-based signal processing method. However, in practice, the limited number of positions where these signals are recorded results in a much lower resolution in the wavenumber axis than in the frequency axis. This prevents a clear identification of overlapping dispersion curves. In this paper, a sparse SVD (S-SVD) method, which combines the signal-to-noise ratio improvement of the SVD-based approach with the high wavenumber resolution advantage of the sparse optimization, is presented to overcome the above-mentioned limitation. Different penalty constraints, i.e.,

Quantification of guided mode propagation in fractured long bones.

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Axial Transmission Method for Long Bone Fracture Evaluation by Ultrasonic Guided Waves: Simulation, Phantom and in Vitro Experiments

-norm, Frobenius norm, and revised Cauchy norm, are compared with the sparse characteristics. The regularization parameters are investigated with respect to the convergence property and wavenumber resolution. The proposed S-SVD method is investigated using synthetic wideband signals and experimental data obtained from a bone-mimicking phantom and from an ex-vivo human radius. The analysis of the results suggests that the S-SVD method has the potential to significantly enhance the wavenumber resolution and to improve the extraction of the dispersion curves.

Multichannel processing for dispersion curves extraction of ultrasonic axial-transmission signals: Comparisons and case studies

Guided modes propagation in intact, fractured and healing long bone has drawn significant research interests. However, mode quantifications for the direct comparison are still necessary to address. The aim of the study is to analyze the mode interaction with a notch-fracture in the long bone and find quantitative ultrasound parameters sensitive to depth and width variation of the fracture. We analyzed the impacts of the partially and completely diaphyseal osteotomy on fundamental guided modes propagation using the two-dimension finite-difference time-domain (2D-FDTD) simulations. The long bones were built as three layer models by a cortical plate embedded between overlying soft tissue and inner-coated marrow. Narrowband low-frequency sinusoids (100 kHz) were employed to only excite two fundamental guided modes. The mode amplitude variations were investigated as functions of the gap-breakage width and depth. It is found that the transverse fractures have strong influences on the anti-symmetric mode A0 transmission and reflection, whereas amplitudes of the symmetric mode S0 are not sensitive to the fracture degree. The quantitative results consistently indicate that reflection energy and transmission coefficients of the S0 and A0 modes can be used to quantify the mode interaction in the fractured long bone and further to evaluate long bone fracture status. Future study is needed to investigate the physical experiments on realistic fractured long bone and to insure that the proposed ultrasound parameters can be used to quantitatively evaluate the long bone fracture in clinical application.