Method for nondestructive ultrasonic testing of mechanical properties of solidified aeolian sand body

Abstract

The present application belongs to the technical field of geotechnical engineering detection, and particularly relates to a kind of solidified aeolian sand body mechanics performance nondestructive ultrasonic detection method, comprising: obtaining solidified aeolian sand body ultrasonic signal;Variational modal decomposition and highest frequency intrinsic mode function box fractal dimension are used to determine mode number;Energy proportion reconstruction signal is screened, and the intrinsic mode function with highest energy proportion is extracted;Based on effective instantaneous frequency, sliding time window is divided;The free cumulant sequence matrix of the local inverse function of normalized power spectral density is calculated;After principal component analysis resampling reconstruction phase space, the maximum Lyapunov index and correlation dimension are calculated;The characteristics of intrinsic mode function after non-linear energy operator processing are calculated;Input prediction model outputs mechanical performance parameters.The present application enhances the mapping stability between acoustic indicators and mechanical properties, and realizes high-precision nondestructive monitoring of solidified aeolian sand body mechanics performance.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering testing technology. More specifically, this invention relates to a non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand. Background Technology

[0002] The mechanical properties of solidified aeolian sand directly affect engineering quality evaluation and structural safety assessment, serving as key indicators in construction acceptance and service monitoring. Current testing methods largely rely on destructive tests such as core sampling and uniaxial compressive strength testing, which suffer from long cycles, high costs, and limited representativeness, and can cause irreversible damage to the engineering structure, making it difficult to meet the needs of large-area, continuous quality monitoring. Solidified aeolian sand is composed of sand grains, solidification products, pores, and microcracks, belonging to a typical multiphase, porous, heterogeneous medium. Ultrasonic waves propagating within it are prone to scattering, attenuation, and dispersion, and the received echoes typically exhibit significant nonlinear and non-stationary characteristics, accompanied by environmental and structural noise interference.

[0003] Existing signal processing methods tend to weaken weak structural responses or leave behind complex background noise during denoising. Conventional feature extraction methods often remain in the time and frequency domains or simple time-frequency indices, failing to characterize local transient changes and nonlinear dynamics in complex porous media. This results in an unstable mapping relationship between acoustic features and macroscopic mechanical properties, limiting prediction accuracy and generalization ability. Especially under different mix proportions, water content, and curing ages, echo signals vary significantly, making it difficult for single shallow indices to maintain stable applicability. Therefore, there is an urgent need for an ultrasonic nondestructive testing method suitable for solidified aeolian sand bodies that improves noise suppression capabilities while preserving effective structural responses, enhances local detail analysis and deep nonlinear feature extraction capabilities, and provides technical support for rapid and reliable prediction of mechanical properties. Summary of the Invention

[0004] To address the technical problems of existing technologies in the mechanical property testing of solidified aeolian sand bodies, such as strong destructiveness, complex signal noise, insufficient extraction of deep nonlinear features, and limited prediction accuracy, this invention provides a non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand bodies.

[0005] This invention provides a non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand bodies, comprising: S1: acquiring ultrasonic time-domain echo signals penetrating the solidified aeolian sand body; setting search variables to perform variational mode decomposition on the ultrasonic time-domain echo signals, comparing the box fractal dimension of the highest frequency intrinsic mode function obtained from the decomposition with a preset noise threshold, adaptively determining the target number of modes, and obtaining the intrinsic mode function set under the corresponding target number of modes; calculating the modal total energy ratio of each intrinsic mode function in the intrinsic mode function set, screening the intrinsic mode functions whose modal total energy ratio is greater than a preset energy ratio threshold to reconstruct the preprocessed signal, and extracting the intrinsic mode function with the highest energy ratio; S2: obtaining the effective intrinsic mode function with the highest energy ratio. Instantaneous frequency: Divide the preprocessed signal into sliding time windows based on the effective instantaneous frequency; calculate the normalized power spectral density of the signal segment within each sliding time window and solve for the corresponding local inverse function; expand the local inverse function to obtain the free cumulant, forming a free cumulant sequence matrix; S3: Perform principal component analysis on the free cumulant sequence matrix to obtain a uniformly spaced resampled sequence; reconstruct the phase space based on the uniformly spaced resampled sequence; calculate the maximum Lyapunov exponent and correlation dimension as the first performance feature; calculate the mean, variance, skewness, and kurtosis of the sequence of the eigenmode function with the highest energy proportion after processing by the nonlinear energy operator as the second performance feature; input the first and second performance features into the prediction model and output the mechanical performance parameters.

[0006] By adopting the above technical solutions, this invention achieves precise separation of high-frequency random noise and effective structural response through adaptive variational mode decomposition and box fractal dimension criterion linkage; it solves the problem of local feature ambiguity under the dispersion effect of heterogeneous media by utilizing a sliding time window based on the effective instantaneous frequency; it deeply explores the deep nonlinear modulation law of ultrasonic waves propagating in porous media by combining the free cumulative quantity of free probability theory and the chaotic dynamic characteristics of phase space reconstruction; and it enhances the mapping stability between acoustic indicators and mechanical properties by integrating local transient energy statistical characteristics to construct a multidimensional feature vector, thereby realizing high-precision, non-destructive real-time monitoring of the mechanical properties of solidified aeolian sand.

[0007] Preferably, the adaptive determination of the target number of modes includes: setting the search start point, search end point, and search step size of the search variables; using the search start point as the current number of modes, performing variational mode decomposition on the ultrasonic time-domain echo signal, extracting the intrinsic mode function with the largest center frequency as the highest frequency intrinsic mode function, and calculating the box fractal dimension of the highest frequency intrinsic mode function; if the box fractal dimension is greater than a preset noise threshold: when the current number of modes is greater than the search start point, subtracting the search step size from the current number of modes to obtain the target number of modes; when the current number of modes is equal to the search start point, using the search start point as the target number of modes; if the box fractal dimension is not greater than a preset noise threshold: if the search end point has not been reached, increasing the search step size of the current number of modes, and returning to perform variational mode decomposition on the ultrasonic time-domain echo signal and subsequent steps; if the search end point has been reached, using the search end point as the target number of modes.

[0008] By adopting the above technical solution, this invention introduces the box fractal dimension into the search mechanism of variational mode decomposition, and utilizes the fractal dimension's ability to describe the morphological complexity of signal waveforms to establish an objective over-decomposition identification criterion. This criterion can automatically distinguish between structural response components and random shot noise components based on the changes in the highest frequency intrinsic mode function. Therefore, without the need for manually preset parameters, it ensures that the decomposition process can fully preserve the effective modes reflecting the microscopic characteristics of the sand body while maximally eliminating spurious noise interference, thus improving the signal-to-noise ratio of the preprocessed signal.

[0009] Preferably, obtaining the effective instantaneous frequency of the intrinsic mode function with the highest energy proportion includes: extracting the analytic signal of the intrinsic mode function with the highest energy proportion through Hilbert transform; calculating the derivative of the instantaneous phase of the analytic signal to obtain an instantaneous frequency sequence; determining the current sliding time window start point of each sliding time window, and extracting the current instantaneous frequency value corresponding to the current sliding time window start point; obtaining the absolute value of the current instantaneous frequency value, and determining the maximum value of the absolute value compared with a preset frequency lower limit as the effective instantaneous frequency.

[0010] Preferably, dividing the preprocessed signal into sliding time windows based on the effective instantaneous frequency includes: setting a fixed number of periods for each sliding time window; dividing the fixed number of periods by the effective instantaneous frequency and multiplying by the system sampling rate and rounding down to calculate the number of sampling points of the sliding time window as the time window width; using the current sliding time window start point as a reference, intercepting the corresponding signal segment along the positive time axis according to the time window width; setting the sliding time step to a fixed proportion of the time window width and moving it along the positive time axis to obtain the starting position of the next sliding time window start point, repeating the division until the preprocessed signal is completely traversed.

[0011] By adopting the above technical solution, this invention achieves an adaptive balance of time-frequency analysis resolution by establishing an inversely proportional adjustment mechanism between the sliding time window width and the effective instantaneous frequency. In high-frequency dense signal regions, it automatically shrinks the window width to capture transient features related to minute cracks, while in low-frequency flat regions, it widens the window width to ensure sufficient spectral resolution to obtain macroscopic attenuation information. This effectively overcomes the smoothing and erasing effect of traditional fixed window widths on local details, enabling the extracted features to more sensitively reflect the local non-uniformity of the cementation state within aeolian sand bodies.

[0012] Preferably, the step of calculating the normalized power spectral density of the signal segment within each sliding time window and solving for the corresponding local inverse function includes: performing a discrete Fourier transform on the signal segment within each sliding time window using the periodogram method to extract the frequency domain amplitude; calculating the square of the frequency domain amplitude and dividing it by the number of sampling points in the sliding time window to obtain the power spectral density function; calculating the area of ​​integration of the power spectral density function in the frequency domain interval; dividing the power spectral density function by the area of ​​integration to obtain the normalized power spectral density; solving the Cauchy transform of the normalized power spectral density and calculating the local inverse function of the Cauchy transform.

[0013] Preferably, the step of expanding the local inverse function to obtain the free cumulants and constructing a free cumulant sequence matrix includes: performing a Laurent series expansion on the local inverse function, removing the singular principal part, and obtaining the analytic part of the R-transform generating function; performing a power series expansion on the analytic part of the R-transform generating function, extracting the coefficients corresponding to the first k orders as free cumulants of orders 1 to k; and vertically concatenating the free cumulants of orders 1 to k obtained from each of the sliding time windows in chronological order to generate the free cumulant sequence matrix.

[0014] By employing the above technical solution, this invention utilizes the analytical expansion of the R-transform generating function to extract free cumulants, achieving a high-order statistical characterization of the frequency domain energy distribution pattern. The 1st to kth order free cumulants, by encoding fine structural information such as the center drift, discrete width, asymmetric skewness, and peak trend of the energy distribution, can deeply distinguish the nonlinear disturbances caused by pores and microcracks of different scales within solidified aeolian sand bodies to ultrasonic wave propagation from the perspective of frequency domain probability measures. Compared to traditional time-frequency indicators, it has higher identification accuracy and discriminative power for the damage state of the medium.

[0015] Preferably, the step of performing principal component analysis on the free cumulant sequence matrix to obtain a uniformly spaced resampled sequence includes: performing zero-mean processing on the free cumulant sequence matrix and calculating the covariance matrix of the zero-mean matrix; performing singular value decomposition on the covariance matrix, extracting the eigenvector corresponding to the largest eigenvalue, multiplying the zero-mean matrix and the eigenvector to form a first principal component time series; fitting the first principal component time series using cubic spline interpolation, and performing uniformly spaced resampling at a set reference frequency to obtain a uniformly spaced resampled sequence.

[0016] Preferably, the reconstructing of the phase space based on the uniformly spaced resampled sequence includes: calculating the mutual information function of the uniformly spaced resampled sequence using the mutual information method, and taking the time interval corresponding to the first minimum value of the mutual information function as the optimal delay time; calculating the proportion of false nearest neighbors under different embedding dimensions using the false nearest neighbor method, and determining the dimension corresponding to the first drop of the proportion of false nearest neighbors below a preset proportion as the optimal embedding dimension; and performing coordinate extension on the uniformly spaced resampled sequence based on the optimal delay time and the optimal embedding dimension to reconstruct a multidimensional mapped coordinate point set of the phase space.

[0017] Preferably, the calculation of the maximum Lyapunov exponent and the correlation dimension as the first performance feature includes: searching for the initial nearest neighbor of the reference phase point in the set of phase space multidimensional mapping coordinate points and recording the initial spatial distance; calculating the evolved spatial distance after time step evolution, obtaining the natural logarithm of the ratio of the evolved spatial distance to the initial spatial distance and dividing it by the evolution time step, and calculating the maximum Lyapunov exponent by ensemble averaging; counting the number of phase point pairs in the set of phase space multidimensional mapping coordinate points whose spatial distance is less than the correlation radius to obtain the correlation integral value, plotting a scatter plot of the natural logarithm of the correlation radius and the natural logarithm of the correlation integral value, performing linear regression fitting on the linear scale-free interval to extract the slope as the correlation dimension; and combining the maximum Lyapunov exponent and the correlation dimension as the first performance feature.

[0018] By employing the above technical solution, this invention quantitatively characterizes the global evolution characteristics of ultrasonic wave propagation from a nonlinear dynamics perspective by calculating the maximum Lyapunov exponent and correlation dimension in the reconstructed phase space. The maximum Lyapunov exponent, by calculating the sensitivity of the sound field evolution to the initial differences in the microstructure of the medium, and the correlation dimension, by characterizing the complexity of the attractor's geometry, together reveal the inherent nonlinear correlation between the internal pore layout and macroscopic mechanical strength degradation of solidified aeolian sand bodies. This provides characteristic support with clear dynamic and physical significance for the reliable prediction of mechanical performance parameters.

[0019] Preferably, the step of calculating the mean, variance, skewness, and kurtosis of the sequence of the intrinsic mode function with the highest energy proportion after processing by the nonlinear energy operator as the second performance feature includes: calculating the nonlinear characteristic energy of the intrinsic mode function with the highest energy proportion at each discrete sampling point by subtracting the product of the amplitude of the previous sampling point and the amplitude of the next sampling point from the square of the amplitude of the current sampling point, thereby generating a target sequence; calculating the mean, variance, and skewness of the target sequence respectively; calculating the kurtosis by performing a fourth-order central moment operation on the target sequence and subtracting a preset distribution benchmark constant; and combining the mean, variance, skewness, and kurtosis into a one-dimensional feature vector as the second performance feature.

[0020] The technical solution of the present invention has the following beneficial technical effects:

[0021] This invention utilizes a joint criterion of box fractal dimension and preset noise threshold to monitor the execution status of variational mode decomposition, thereby achieving adaptive determination of the target number of modes. Effective components are selected based on the proportion of total modal energy of each intrinsic mode function to reconstruct the preprocessed signal, accurately preserving the core acoustic response while suppressing background noise interference. The width of the sliding time window is dynamically adjusted by obtaining the effective instantaneous frequency of the intrinsic mode function with the highest energy proportion, and the free cumulant corresponding to the normalized power spectral density within each sliding time window is calculated to characterize the non-stationary frequency domain evolution law in different local time periods. The system further employs principal component analysis to eliminate redundant information in the free cumulant sequence, and combines the maximum Lyapunov exponent and correlation dimension extracted through phase space reconstruction, as well as the statistical features obtained after processing the intrinsic mode function with the highest energy proportion using a nonlinear energy operator, to construct a multi-dimensional fusion feature set. This fusion feature set enhances the characterization depth of ultrasonic echo signals for the evolution of internal pores, microcracks, and cementation states in solidified aeolian sand bodies, thereby improving the accuracy and robust stability of the non-destructive prediction process of mechanical performance parameters. Attached Figure Description

[0022] Figure 1 This is a flowchart of a non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand bodies according to the present invention; Figure 2 This is a schematic diagram illustrating the selection of the number of modes in variational mode decomposition based on the box fractal dimension; Figure 3 This is a schematic diagram of the ablation test results. Detailed Implementation

[0023] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0024] This invention discloses a non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand, referring to... Figure 1 This includes steps S1-S3: S1, Signal adaptive decomposition and reconstruction and intrinsic mode function extraction.

[0025] In an optional embodiment, an ultrasonic time-domain echo signal penetrating solidified aeolian sand is acquired. A search variable is set to perform variational mode decomposition on the ultrasonic time-domain echo signal. The target number of modes is adaptively determined by comparing the box fractal dimension of the highest frequency intrinsic mode function obtained by decomposition with a preset noise threshold, and the intrinsic mode function set under the corresponding target number of modes is obtained. The modal total energy ratio of each intrinsic mode function in the intrinsic mode function set is calculated. Intrinsic mode functions with a modal total energy ratio greater than a preset energy ratio threshold are screened to reconstruct the preprocessed signal, and the intrinsic mode function with the highest energy ratio is extracted.

[0026] In the actual execution process, the system first uses ultrasonic testing equipment to emit ultrasonic pulses to the solidified aeolian sand specimen and receives the time-domain echo signal after penetrating the specimen. The analog signal is then discretized into a one-dimensional time-domain echo signal sequence using an analog-to-digital converter at a system-preset sampling rate. The system sampling rate is determined based on the center frequency of the ultrasonic transducer and the Nyquist sampling theorem, and is usually a fixed parameter fixed at the factory when the equipment is shipped.

[0027] Subsequently, the system performs adaptive variational mode decomposition on the acquired time-domain echo signal. Since solidified aeolian sand is a multiphase porous heterogeneous medium composed of sand grains, solidification products, pores, and microcracks, ultrasonic waves propagate through it, resulting in scattering, attenuation, and dispersion. The received echo exhibits significant nonlinear and non-stationary characteristics, accompanied by environmental and structural noise interference. Traditional fixed-mode-number decomposition methods struggle to adapt to differences in echo signals under varying mix proportions and curing conditions. If the number of modes is too low, effective structural response components are merged and missed; if the number of modes is too high, spurious noise modes are introduced, interfering with subsequent feature extraction. Therefore, the system sets a search variable to perform cyclic variational mode decomposition on the time-domain echo signal. By comparing the box-fractal dimension of the highest-frequency intrinsic mode function obtained in each decomposition with a preset noise threshold, the target number of modes is adaptively determined, ensuring that the decomposition effectively separates the effective modes while avoiding over-decomposition that introduces noise. Among them, the preset noise threshold is a critical judgment value determined by statistical analysis of a large number of experimental tests on samples with different ratios and different maintenance ages based on the typical noise fractal characteristics of ultrasonic signals of aeolian sand bodies. It can effectively distinguish between high-frequency structural response signals and pure noise modal components caused by over-decomposition.

[0028] After determining the target number of modes, the system performs a final variational mode decomposition on the time-domain echo signal based on this number, obtaining a set of intrinsic mode functions (EMFs). Next, the system performs a Hilbert transform on each EMF to extract the corresponding analytic signal, and calculates the square of the magnitude of the analytic signal as the instantaneous energy sequence. The instantaneous energy sequences are accumulated along the time axis to obtain the total modal energy, and the total modal energy is divided by the sum of all modal energies to obtain the modal energy percentage. The magnitude of the energy percentage directly reflects the contribution weight of that modal component in the entire echo signal: modes with low energy percentages typically correspond to noise or weak scattering components, while modes with high energy percentages carry the main acoustic response information closely related to the internal structural state of solidified aeolian sand bodies. Therefore, the system selects EMFs with a total modal energy percentage greater than a preset energy percentage threshold, performs point-by-point linear addition in the time domain, and reconstructs the preprocessed signal, thereby suppressing background noise while retaining the main structural response components. Simultaneously, the system identifies the intrinsic mode function (IMF) with the highest energy proportion among all IMFs and extracts and records it separately. This IMF with the highest energy proportion contains the most representative propagation characteristic information during the process of ultrasonic waves penetrating solidified aeolian sand, and will serve as the core basis for subsequent time-frequency analysis, sliding time window adjustment, and nonlinear feature extraction. The preset energy proportion threshold is pre-set by the system software based on statistical analysis of multiple experiments; for example, it can be set to 5%, ensuring that noise modes with weak contributions are eliminated while all effective modes are retained.

[0029] In one possible embodiment, the specific process of adaptively determining the target number of modes includes: setting the search start point, search end point, and search step size of the search variables; using the search start point as the current number of modes, performing variational mode decomposition on the ultrasonic time-domain echo signal, extracting the intrinsic mode function with the largest center frequency as the highest frequency intrinsic mode function, and calculating the box fractal dimension of the highest frequency intrinsic mode function; if the box fractal dimension is greater than a preset noise threshold: when the current number of modes is greater than the search start point, subtracting the search step size from the current number of modes to obtain the target number of modes; when the current number of modes is equal to the search start point, using the search start point as the target number of modes; if the box fractal dimension is not greater than a preset noise threshold: if the search end point has not been reached, increasing the search step size of the current number of modes, and returning to perform variational mode decomposition on the ultrasonic time-domain echo signal and subsequent steps; if the search end point has been reached, using the search end point as the target number of modes.

[0030] More specifically, since variational mode decomposition requires pre-specifying the key parameter of mode number, and the complexity of ultrasonic signals in solidified aeolian sand varies with mix proportions, water content, and curing age, the system cannot predetermine the optimal mode number value. Therefore, a search mechanism based on the box-fractal dimension criterion is adopted to achieve adaptive selection. The search starting point is set as... The search endpoint is set to The search step size is set to 1. Setting the search starting point to 2 represents the minimum number of modes that are meaningful for variational mode decomposition, while setting the search ending point to 10 covers the entire range of modes that might be needed in actual aeolian sand echo signals. This is a reasonable upper limit derived from a large number of experimental samples. Initially, the system gradually increases the number of modes and performs decomposition. After each decomposition, the spectrum of each intrinsic mode function (EMF) is calculated using Fast Fourier Transform (FFT). The EMFs are sorted from highest to lowest center frequency, and the EMF with the highest center frequency is extracted as the highest frequency EMF. The box-fractal dimension algorithm is used to analyze the highest frequency EMF. Two-dimensional grids of different side lengths are generated to cover the time-domain curve, and the number of grids intersecting the signal curve is counted. A linear equation between the logarithmic number of grids and the reciprocal of the logarithmic grid side length is fitted using the least squares method; the slope of this equation is the box-fractal dimension. The highest frequency EMF typically carries high-frequency details or potential noise components of the signal, and the box-fractal dimension can quantitatively represent the waveform morphology complexity and irregularity of this mode component. For complex porous media such as solidified aeolian sand, the normal structural echo signal components are relatively smooth, and their box fractal dimension is generally between 1.1 and 1.4. This range is determined by experimental statistics based on the typical time-domain waveform complexity of ultrasonic signals from aeolian sand. In contrast, pure noise signals lack an inherent deterministic structure, and their time-domain waveforms exhibit highly irregular shapes, with their box fractal dimension usually approaching the theoretical upper limit of 2.

[0031] The preset noise threshold is set to 1.5. This value is the critical judgment value for distinguishing high-frequency structural signals from pure noise caused by over-decomposition. It is preset by the system after experimental calibration based on the noise characteristics of ultrasonic signals from aeolian sand bodies. When the box fractal dimension of the highest frequency intrinsic mode function first exceeds 1.5, it indicates that the current decomposition has crossed from the effective structural response region to the noise-dominated region, i.e., over-decomposition has occurred. At this time, the system immediately stops the search iteration. If the current number of modes is greater than the search starting point 2, the value after subtracting 1 from the current number of modes is determined as the optimal target number of modes, i.e., the state before over-decomposition is taken as the optimal solution. If the current number of modes is equal to the search starting point 2, it indicates that the signal itself has extremely high noise content. In this case, the search starting point 2 is directly taken as the target number of modes to ensure the basic effectiveness of the decomposition. Conversely, if the search reaches the endpoint... If no case occurs where the box fractal dimension exceeds the noise threshold, then the search endpoint of 10 is taken as the target number of modes, indicating that the signal structure is complex and requires a large number of modes for sufficient decomposition. This mechanism ensures that the determination of the number of modes does not rely on manual experience or presets, but automatically adapts to the signal characteristics of different samples through quantitative criteria. This effectively avoids the risks of introducing noise interference due to an excessive number of modes or missing effective structural components due to an insufficient number of modes. If this adaptive search mechanism is not adopted and a fixed number of modes is used for decomposition, under conditions where the signal complexity varies significantly, such as when there are large differences in echoes between solidified aeolian sand bodies with different ratios, either the insufficient number of modes leads to the merging of effective signal components and the loss of important microstructural response information, or the excessive number of modes generates false high-frequency shot noise, seriously interfering with the accuracy of subsequent feature extraction, thus increasing the deviation in mechanical performance prediction. For example, when K=3 or K=4, if the box-fractal dimension of the highest frequency intrinsic mode function is 1.32 and 1.45 respectively, neither exceeding 1.5, it indicates that the high-frequency signal still contains components that can represent the microscopic pores and interface reflections inside the solidified aeolian sand body. Therefore, K is incremented by 1 to continue iteration. When the search reaches K=5, if the highest frequency intrinsic mode function obtained from the decomposition is irregular high-frequency shot noise, the box-fractal dimension increases to 1.65, exceeding the noise threshold of 1.5 for the first time. This indicates that the signal has been over-decomposed, and the iteration is immediately stopped. The previous state before the over-decomposition, K=4, is determined as the target number of modes to reduce the characteristic interference caused by noise to subsequent mechanical performance evaluation. The mode number selection process based on the box-fractal dimension is as follows: Figure 2 As shown.

[0032] S2, Dynamic Time Window Division and Construction of Free Cumulative Sequence Matrix.

[0033] In an optional embodiment, the effective instantaneous frequency of the intrinsic mode function with the highest energy proportion is obtained, and the preprocessed signal is divided into sliding time windows according to the effective instantaneous frequency; the normalized power spectral density of the signal segment in each sliding time window is calculated and the corresponding local inverse function is solved; the local inverse function is expanded to obtain the free cumulant and form a free cumulant sequence matrix.

[0034] In the specific execution process, the system first performs time-frequency analysis on the intrinsic mode function with the highest energy proportion recorded in stage S1 to obtain its instantaneous frequency characteristics. The system then constructs the analytic signal of this eigenmode function with the highest energy proportion using Hilbert transform, extracts the instantaneous phase sequence from the analytic signal, and performs unwinding processing to eliminate phase jumps. The system then calculates the first-order difference of the unwound instantaneous phase along the time direction and converts it into a discrete instantaneous frequency sequence in Hz. Next, the system extracts the effective instantaneous frequency corresponding to each moment from the instantaneous frequency sequence, i.e., taking the larger value between the absolute value of the instantaneous frequency and a preset lower frequency limit. This avoids the problem of time window calculation failure caused by zero frequency or extremely low frequencies. The preset lower frequency limit is pre-set by the system based on the lower limit of the ultrasonic transducer's operating bandwidth after factory calibration.

[0035] Subsequently, the system performs adaptive sliding time window division on the preprocessed signal based on the effective instantaneous frequency. When ultrasonic waves propagate in solidified aeolian sand, due to the spatial non-uniformity of the internal pore distribution, degree of cementation, and microcrack state, the echo signal exhibits significant non-stationary attenuation characteristics: the signal changes drastically and transient details are rich in the high-frequency dense region, while the signal changes slowly and reflects the overall attenuation trend in the low-frequency slowly varying region. A fixed-width analysis window cannot simultaneously accommodate both conditions: if the window width is too large, the transient details in the high-frequency region are smoothed out; if the window width is too small, the spectral resolution in the low-frequency region is insufficient. Therefore, the system designs the width of each sliding time window to be inversely proportional to the effective instantaneous frequency at the beginning of the current window, so that the high-frequency region window automatically narrows to capture transient damage details, and the low-frequency region window automatically widens to obtain sufficient spectral resolution. The system sets a fixed number of cycles in each sliding time window. The fixed number of cycles is multiplied by the system sampling rate, divided by the effective instantaneous frequency, and rounded down to calculate the number of sampling points in the current sliding time window as the time window width. The corresponding length of the signal segment is extracted from the starting point of the window and advanced along the time axis in the positive direction according to the preset sliding step size. The division is repeated until all preprocessed signals are completely traversed.

[0036] After dividing the time window, the system applies a Hanning window to the local signal segment within each time window to suppress spectral leakage. The corresponding power spectral density function is calculated using the periodogram method, and a normalized power spectral density with an integral area of ​​1 is obtained through frequency domain integral area normalization, which is then used as the frequency domain measure function. Next, based on free probability theory, the system calculates the Cauchy transform of the normalized power spectral density corresponding to each sliding time window. The local inverse function of the Cauchy transform is obtained using a series inversion method. A Laurent series expansion is performed on the local inverse function, and the singular principal part is removed to obtain the analytic part of the R-transform generating function. This analytic part is then expanded using a power series, and the first power is extracted in power order. The coefficients corresponding to each order are taken as 1 to... The free cumulative values ​​obtained from all sliding time windows are then concatenated vertically in chronological order to generate a multi-row, cumulative free cumulative value. The free cumulant sequence matrix, where rows correspond to the time axis and columns correspond to the order dimension, fully encodes the nonlinear evolution information of the local frequency domain energy distribution of the ultrasonic signal over time during propagation.

[0037] In one possible embodiment, the specific process of obtaining the effective instantaneous frequency of the intrinsic mode function with the highest energy proportion includes: the system extracts the analytic signal of the intrinsic mode function with the highest energy proportion through Hilbert transform; the instantaneous phase of the analytic signal is differentiated to obtain the instantaneous frequency sequence; the starting point of the current sliding time window of each sliding time window is determined, and the current instantaneous frequency value corresponding to the starting point of the current sliding time window is extracted; the absolute value of the current instantaneous frequency value is obtained, and the maximum value of the absolute value compared with the preset lower frequency limit is determined as the effective instantaneous frequency.

[0038] More specifically, the Hilbert transform maps real-valued time series to corresponding orthogonal components, and the combination of the two constitutes a complex analytic signal. The system obtains the instantaneous phase sequence by taking the amplitude of the analytic signal, unwinds the instantaneous phase sequence, calculates the first-order difference along the time direction, and multiplies the phase difference value by the system sampling rate. Divide by This yields a discrete instantaneous frequency sequence in Hz. In the formula This is a discrete-time index. The instantaneous frequency sequence describes the local oscillation rate of the intrinsic mode function with the highest energy content at different times, and is a direct indicator reflecting the propagation speed fluctuation and dispersion effect of ultrasound in a heterogeneous medium. In actual signals, due to noise interference or the influence of the signal envelope zero-crossing point, the instantaneous frequency may have a zero value or a negative value. If the time window width is directly calculated by its reciprocal, it will lead to numerical overflow or an infinite window width. Therefore, the system introduces a preset lower frequency limit. As a protection threshold, this lower limit is set based on a combination of the center frequency and the lower cutoff frequency of the effective operating bandwidth of the ultrasonic transducer. For example, for a center frequency of... The transducer can be set to a preset lower frequency limit. This ensures that the time window width below this frequency does not exceed a reasonable analysis range. Effective instantaneous frequency. Defined as This maximum value operation mathematically guarantees the numerical stability of subsequent division operations. Physically, it corresponds to the following: when the local oscillation frequency of the signal is lower than the lower limit of the effective bandwidth of the transducer, the system uses the lower limit frequency as the reference benchmark for window adjustment to maintain the baseline resolution of the spectrum analysis.

[0039] In another possible embodiment, the specific process of dividing the preprocessed signal into sliding time windows based on the effective instantaneous frequency includes: the system sets a fixed number of cycles in each sliding time window; the fixed number of cycles is divided by the effective instantaneous frequency and multiplied by the system sampling rate and rounded down to calculate the number of sampling points of the sliding time window as the time window width; the corresponding signal segment is intercepted along the positive time axis based on the current sliding time window starting point as a reference; the sliding time step is set to a fixed proportion of the time window width and moved along the positive time axis to obtain the starting position of the next sliding time window starting point, and the division is repeated until the preprocessed signal is completely traversed.

[0040] More specifically, the fixed number of periods is set to 5. This value ensures that each analysis window contains enough complete periods of signal oscillation characteristics to meet the statistical reliability requirements of spectral analysis. Too few periods result in excessively large variance in spectral estimation, while too many periods reduce time resolution. Therefore, 5 periods represent an experimentally verified optimal balance between time-domain resolution and frequency-domain accuracy. The formula for calculating the time window width is as follows: ,in The system sampling rate, This indicates a floor operation. The sliding time step is set to one-quarter of the time window width, meaning that adjacent windows maintain a 75% overlap. This ensures the continuity of time-frequency analysis along the time axis while avoiding computational redundancy caused by an excessively small step size. For example, assuming the system sampling rate... for If the effective instantaneous frequency at the starting point of the current window is The time span of a single period is After including 5 fixed periods, the corresponding time window span is Multiplying by the sampling rate yields a window width of 250 discrete sampling points, with a sliding step size of... Each sampling point has a large effective instantaneous frequency. In high-frequency, densely packed signal regions, the effective instantaneous frequency is large, and the window automatically narrows, allowing the system to precisely capture transient response details related to local microscopic defects. In low-frequency, slowly varying signal regions, the effective instantaneous frequency is small, and the window automatically widens, enabling the system to acquire spectral information that fully reflects the overall macroscopic attenuation characteristics of the medium. If this adaptive window mechanism is not used and a fixed-width analysis window is employed, under extreme conditions with significant signal frequency changes, such as severe local cracks within solidified aeolian sand causing drastic shifts in echo frequencies, the fixed window either fails to distinguish adjacent transient reflection events in the high-frequency region or suffers from severely insufficient spectral resolution in the low-frequency region. This leads to a decrease in the sensitivity of subsequent free accumulator features to local damage.

[0041] In another possible embodiment, the specific process of calculating the normalized power spectral density of the signal segment within each sliding time window and solving the corresponding local inverse function, and expanding the local inverse function to obtain the free cumulants to form a free cumulants sequence matrix includes: the system uses the periodogram method to perform discrete Fourier transform on the signal segment within each sliding time window to extract the frequency domain amplitude, calculates the square of the frequency domain amplitude and divides it by the number of sampling points in the sliding time window to obtain the power spectral density function; calculates the integral area of ​​the power spectral density function in the frequency domain interval, and divides the power spectral density function by the integral area to obtain the normalized power spectral density; solves the Cauchy transform of the normalized power spectral density and calculates the local inverse function of the Cauchy transform; performs a Laurent series expansion on the local inverse function, removes the singular principal part, and obtains the analytic part of the R-transform generating function; performs a power series expansion on the analytic part of the R-transform generating function, and extracts the first... The coefficients corresponding to each order are taken as 1 to... The order of free cumulative quantity; the 1 to 2 times obtained from each sliding time window. The free accumulators are concatenated vertically in chronological order to generate a free accumulator sequence matrix.

[0042] More specifically, the system performs a Discrete Fourier Transform on the signal segment after applying a Hanning window within each sliding time window, mapping the time-domain signal to the frequency domain. Let the... The frequency domain amplitude of the signal segments within each sliding time window after discrete Fourier transform is The number of sampling points in this window is Then the power spectral density function is This function describes the distribution density of signal energy across its frequency components. To eliminate the impact of inconsistent absolute energy levels caused by differences in macroscopic acoustic attenuation among different solidified aeolian sand samples on subsequent feature extraction, the system calculates the power spectral density function across the entire Nyquist analysis band. Numerical integral area Dividing the original power spectral density by the integral area yields the normalized power spectral density. ,satisfy The normalized power spectral density can be equivalently regarded as a frequency domain probability density function, representing the relative residence probability of each frequency component in the total energy.

[0043] Next, the system normalizes the power spectral density. As a frequency domain measure function, its Cauchy transform, i.e., the Stieltjes transform, is defined as follows: ,in The variable is a complex independent variable. This integral transform maps the probability distribution on the real frequency axis to an analytic function in the complex upper half-plane, serving as a core bridge connecting frequency domain statistical characteristics and free cumulants in free probability theory. The system employs a series inversion mathematical method to calculate... Local inverse function After performing a Laurent series expansion on the local inverse function and removing the singular principal parts, the analytic part of the R-transform generating function is obtained. Expand the analytical part using a power series. ,in Let be the complex independent variable in the expansion, and be the coefficients of each power in the expansion. That is, the first First-order free cumulant. From a physical perspective, the first-order free cumulant... The second-order free cumulant reflects the center-mean shift of the frequency domain energy distribution. Discrete bandwidth characterizing the frequency band energy distribution, and third- and fourth-order free cumulants. , These represent the degree of asymmetric skewness and peak concentration tendency of the energy distribution, respectively; when When the optimal value is 10, to Further encoding of higher-order frequency-domain fine-grained jitter structure information enables it to distinguish the nonlinear modulation effects of pores and microcracks of different scales within solidified aeolian sand on ultrasonic wave propagation. The system processes the data calculated in each window according to the temporal progression of the sliding time window. The free cumulative quantities of each order are arranged in columns to form a structure with a time axis row dimension and a feature axis. free cumulant sequence matrix in column dimension ,in Represents the set of real numbers. This represents the total number of sliding time windows.

[0044] S3, Nonlinear Dynamics Feature Extraction and Performance Prediction.

[0045] In an optional embodiment, principal component analysis is performed on the free cumulant sequence matrix to obtain a uniformly spaced resampled sequence. The phase space is reconstructed based on the uniformly spaced resampled sequence, and the maximum Lyapunov exponent and correlation dimension are calculated as the first performance feature. The mean, variance, skewness, and kurtosis of the sequence of the intrinsic mode function with the highest energy proportion after processing by a nonlinear energy operator are calculated as the second performance feature. The first and second performance features are input into the prediction model, and the mechanical performance parameters are output.

[0046] In the specific execution process, the system first performs principal component analysis to reduce the dimensionality of the free cumulant sequence matrix generated in stage S2 along the order dimension. Due to the free cumulant sequence matrix... There is information redundancy among the different orders, meaning that the free cumulants of each order describe the statistical characteristics of the same frequency domain energy distribution from different perspectives and are not completely independent of each other. Principal component analysis can extract the orthogonal direction with the largest variance contribution as the main information carrier, reducing the feature dimension while retaining the most critical frequency domain evolution information. The system calculates the covariance matrix of the free cumulant sequence matrix along the order dimension and performs singular value decomposition, extracting the eigenvector corresponding to the largest eigenvalue as the projection direction. The original high-dimensional matrix is ​​projected onto this direction to obtain the first principal component time series.

[0047] Because the width of the aforementioned sliding time window adaptively changes with the instantaneous frequency, the sampling intervals when the center points of each window are mapped to the real time axis are not uniformly distributed. Furthermore, the phase space reconstruction method in nonlinear dynamics analysis requires the input sequence to have time intervals of equal step size. Non-uniform sampling would violate the fundamental premise of delayed coordinate embedding, leading to spurious dynamic structures. Therefore, the system uses cubic spline interpolation to fit the continuous trajectory of the first principal component time series, and then performs uniform interval resampling on the fitted curve at a set reference frequency to obtain a uniformly interval resampled sequence with consistent step size on the real time axis.

[0048] Subsequently, the system performs phase space reconstruction based on the uniformly spaced resampled sequence. The mutual information function between the resampled sequence and its delayed sequence is calculated using the mutual information method, and the time delay step corresponding to the first local minimum of the mutual information function is determined as the optimal delay time. The spurious nearest neighbor method was used to test the proportion of spurious nearest neighbors at different embedding dimensions, starting from embedding dimension 2. The dimension at which the proportion first dropped below a preset threshold was determined as the optimal embedding dimension. .based on and With two calibration parameters, the system maps the one-dimensional resampling sequence to a high-dimensional phase space through coordinate extension, obtaining a set of multi-dimensional mapped coordinate points in the phase space. Thus, under the meaning of topological equivalence, it recovers the dynamic attractor structure of the nonlinear propagation of ultrasonic waves in the porous medium of solidified aeolian sand.

[0049] In the reconstructed phase space, the system uses the Rosenstein algorithm to calculate the average exponential divergence rate of adjacent trajectories over time as the maximum Lyapunov exponent. This index quantitatively characterizes the sensitivity of ultrasonic wave propagation dynamics to initial conditions. This implies that the system exhibits chaotic divergent behavior. Simultaneously, the Grassberger-Procaccia algorithm is used to statistically analyze the distribution ratio of phase pairs under different correlation radii, and the slope of the linear scale-free interval is extracted as the correlation dimension in a double logarithmic coordinate system. This dimension characterizes the geometric compactness and dynamical degrees of freedom of the chaotic attractor in phase space. and Combined in a fixed order, a two-dimensional column vector is formed as the first performance characteristic, which is used to represent the nonlinear chaotic propagation characteristics of ultrasonic waves caused by the complex pore structure and microcrack distribution inside the solidified aeolian sand body.

[0050] Simultaneously, the system applies a nonlinear energy operator to perform discrete sequence transformation on the intrinsic mode function with the highest energy proportion recorded in stage S1 to extract local transient energy features. The nonlinear energy operator, by subtracting the product of the amplitudes of the previous and next sampling points from the square of the amplitude at the current sampling point, synchronously tracks the coupled fluctuations of instantaneous amplitude and instantaneous frequency within the same computational framework. This transforms the instantaneous phase shift and high-frequency local attenuation generated when ultrasound penetrates weakly cemented interfaces or micro-cracks into sharp energy jumps in the sequence. The system performs fourth-order statistical operations on the transformed target energy sequence, calculating the mean, variance, skewness, and kurtosis, and combining them sequentially into a one-dimensional feature vector as the second performance feature.

[0051] Finally, the system concatenates and merges the first and second performance features to form a comprehensive input feature vector. This comprehensive input feature vector contains six dimensions: maximum Lyapunov exponent, correlation dimension, mean of the nonlinear energy operator sequence, variance, skewness, and kurtosis. It is input into a pre-trained support vector regression prediction model, and the model outputs predicted values ​​of mechanical performance parameters of solidified aeolian sand bodies, including compressive strength.

[0052] In one possible embodiment, the specific process of performing principal component analysis on the free cumulant sequence matrix to obtain a uniformly spaced resampled sequence and reconstructing the phase space based on it includes: the system performs zero-mean processing on the free cumulant sequence matrix and calculates the covariance matrix of the zero-mean matrix; performs singular value decomposition on the covariance matrix, extracts the eigenvector corresponding to the largest eigenvalue, and multiplies the zero-mean matrix with the eigenvector to form the first principal component time series; fits the first principal component time series using cubic spline interpolation, and performs uniformly spaced resampling at a set reference frequency to obtain a uniformly spaced resampled sequence; calculates the mutual information function of the uniformly spaced resampled sequence using the mutual information method, and takes the time interval corresponding to the first minimum value of the mutual information function as the optimal delay time; calculates the proportion of false nearest neighbors under different embedding dimensions using the false nearest neighbor method, and determines the dimension corresponding to the first drop of the false nearest neighbor proportion below the preset proportion as the optimal embedding dimension; and performs coordinate extension on the uniformly spaced resampled sequence based on the optimal delay time and the optimal embedding dimension to reconstruct a multidimensional mapped coordinate point set of the phase space.

[0053] More specifically, let the free cumulant sequence matrix be... The dimension is ,Right now Each time window Free cumulative quantity. The system calculates the arithmetic mean of each column of data. In the formula Free cumulant sequence matrix The Middle The first time window, the first The elements corresponding to the order free cumulant, where Subtract the mean from each column element to perform zero-mean normalization, resulting in a zero-mean matrix. Then calculate covariance matrix superscript in the formula This represents the transpose operation of a matrix that encodes pairwise cross-variations between ten-order free cumulants. Perform singular value decomposition, extract all eigenvalues ​​and sort them in descending order. The largest eigenvalue Corresponding feature vector The direction of maximum variance of the data in high-dimensional space is defined. The zero-mean matrix is ​​multiplied by the principal eigenvectors. The length is obtained as First principal component time series Its variance cumulative contribution rate can usually reach over 90%, and it can effectively represent the joint dynamic evolution information of ten orders of free cumulative quantities in a single dimension.

[0054] Because the width of the sliding time window varies with the instantaneous frequency in the preliminary step, the intervals between the sampling points of the first principal component time series on the real time axis are not uniform. The system uses cubic spline interpolation to fit a piecewise cubic polynomial. The continuous trajectory ensures that the first and second derivatives of the interpolation curve are continuous at the original data points, avoiding corner artifacts introduced by linear interpolation. The fitting curve is based on a reference frequency. Perform uniform interval resampling, i.e., at a fixed time interval. This yields a uniformly spaced resampling sequence that meets the equidistant requirements for phase space reconstruction. Reference frequency. The selection of the frequency comprehensively considers the effective bandwidth of the ultrasonic signal and the sampling density requirements of subsequent nonlinear dynamic analysis: if the frequency is too low, information about rapid signal changes will be lost; if the frequency is too high, computational redundancy will occur and interpolation artifacts may be introduced. correspond The sampling interval is an optimal value determined experimentally between information fidelity and computational efficiency.

[0055] During the phase space reconstruction stage, the system uses the mutual information method to calculate the uniformly spaced resampled sequence. Its delayed sequence Mutual information function between For the number of delay steps Increase and gradually decrease The curve, when the curve first drops to a local minimum, for example when That is, the delay time is At this point, the information redundancy between delayed coordinates reaches a low level, and the independence between each coordinate component satisfies the requirements for phase space reconstruction. Synchronously, in conjunction with the spurious nearest neighbor method, the system improves the embedded dimension. Begin incremental testing, for each Value statistics represent the proportion of points incorrectly projected as adjacent pairs in a high-dimensional embedding space due to insufficient dimension: when The proportion of false nearest neighbors can be as high as 60%. As the ratio increases, it gradually decreases; when The proportion of false nearest neighbors dropped to the preset threshold for the first time. The following demonstrates that the high-dimensional dynamical manifold has been fully unfolded, and there are no longer any spurious adjacency relationships caused by projection. Step length, The dimension is determined by the delay coordinate method for resampling one-dimensional sequences. Coordinate extension was performed to construct a set of multidimensional phase space mapping coordinate points representing the nonlinear scattering evolution of ultrasonic waves in the porous medium of solidified sand. A preset threshold of 5% was used as a general criterion for the false nearest neighbor method. When the proportion of false nearest neighbors drops below 5%, the high-dimensional attractor structure has been basically unfolded, and further increasing the embedding dimension will not significantly change the value of the dynamic metric, while avoiding the data sparsity problem caused by excessively high embedding dimensions.

[0056] In another possible embodiment, the specific process of calculating the maximum Lyapunov exponent and the correlation dimension as the first performance feature includes: the system searches for the initial nearest neighbor of the reference phase point in the set of phase space multidimensional mapping coordinate points and records the initial spatial distance; calculates the evolved spatial distance after time step evolution, calculates the natural logarithm of the ratio of the evolved spatial distance to the initial spatial distance and divides it by the total evolution time, and calculates the maximum Lyapunov exponent by ensemble averaging; counts the number of phase point pairs in the set of phase space multidimensional mapping coordinate points whose spatial distance is less than the correlation radius to obtain the correlation integral value, plots a scatter plot of the natural logarithm of the correlation radius and the natural logarithm of the correlation integral value, performs linear regression fitting on the linear scale-free interval to extract the slope as the correlation dimension; and combines the maximum Lyapunov exponent and the correlation dimension as the first performance feature.

[0057] More specifically, the maximum Lyapunov index and correlation dimension These are two global invariants that characterize the nonlinear chaotic propagation properties of ultrasonic waves in the porous medium of solidified aeolian sand from different perspectives. In the calculation... At that time, the system was already in the deployed phase. Traversing each reference phase point in the phase space trajectory Within the entire phase space, the nearest phase point is searched by traversing the Euclidean distance. Record the initial spatial distance as the initial nearest neighbor. ,in The Euclidean distance norm in phase space is typically on the order of 1 / 2. Order of magnitude. To avoid underestimating the divergence rate due to the reference point and nearest neighbor being located adjacent to each other on the same orbit, the system imposes a minimum time interval constraint to ensure that the index distance between the two points is greater than the average period of the orbit. Subsequently, the system evolves the reference phase point and the nearest neighbor forward along their respective orbits. A time step, for example Calculate the evolved spatial distance Calculate the natural logarithm of the ratio of evolutionary distance to initial distance and divide it by the total evolution time. The local exponential divergence rate of the phase point pair over the evolution time is obtained. By traversing all reference phase points in the phase space and performing an ensemble average of the local divergence rates, the global maximum Lyapunov exponent is obtained. In the formula The total number of reference phase points involved in the calculation. This represents the time step for resampling. If the calculation result... For example, typical value This indicates that ultrasound propagation in porous media exhibits a chaotic divergence effect that is sensitive to initial acoustic field conditions. In other words, minute differences in the medium's structure are continuously amplified during propagation. This characteristic can sensitively reflect subtle changes in the microstructure inside solidified aeolian sand bodies.

[0058] In calculating the correlation dimension At that time, the system constructs the associated radius. An increasing sequence, for example from From Stop, the step size is distributed at logarithmic intervals. For each in the sequence The system iterates through all phase point pairs in the phase space to find the value. ,in, and Let each represent a phase point at any two different evolution times in the reconstructed phase space, and let the spatial distance be calculated. Number of phase pairs And divide by the total number of all possible phase pairs in phase space. Calculate the related integral value This value indicates that two randomly selected phase points fall within the range of [missing information]. The probability within a hypersphere of radius . (The rest of the text appears to be incomplete and requires further context.) and vertical axis In the constructed double logarithmic coordinate system, Follow The change in the x-axis exhibits a linear relationship within a certain interval, which is called the scale-free interval. For example, the x-axis coordinate... to The existence of the interval indicates that the attractor possesses a self-similar fractal structure within this scale range. The system employs least squares regression to fit the scattered points within the scale-free interval to first-order linear regression, extracting the regression slope as the correlation dimension. Typical values ​​are as follows: As the strength of solidified aeolian sand bodies deteriorates and the internal porosity increases, the ultrasonic propagation path becomes more complex, the topological dimension of the attractor typically increases accordingly, and the correlation dimension value rises accordingly. It can serve as an effective indicator for assessing the internal damage state of a medium. The system will... and Concatenate them into a two-dimensional column vector in a fixed order. This vector constitutes the first performance characteristic representing the nonlinear chaotic dynamics of solidified aeolian sand bodies.

[0059] In another possible embodiment, the specific process of using the mean, variance, skewness, and kurtosis of the sequence after processing by the nonlinear energy operator for calculating the intrinsic mode function with the highest energy proportion as the second performance feature includes: the system calculates the nonlinear characteristic energy of the intrinsic mode function with the highest energy proportion at each discrete sampling point by subtracting the product of the amplitude of the previous sampling point and the amplitude of the next sampling point from the square of the amplitude of the current sampling point, and generates the target sequence; the mean, variance, and skewness of the target sequence are calculated respectively; the kurtosis is calculated by performing a fourth-order central moment operation on the target sequence and subtracting a preset distribution reference constant; and the mean, variance, skewness, and kurtosis are combined into a one-dimensional feature vector as the second performance feature.

[0060] More specifically, the system uses discrete sampling sequences of the eigenmode functions with the highest energy percentage. Applying the Teager-Kaiser nonlinear energy operator at each discrete sampling point Calculate the nonlinear characteristic energy This generates a target energy sequence of the same length as the original sequence. This operator differs from traditional squared energy operators, which rely solely on the instantaneous amplitude of a single sampling point. By simultaneously introducing the product of forward and backward sampling points, it synchronously tracks the coupled fluctuation effect of the instantaneous amplitude and frequency of the signal within a compact differential framework. When ultrasonic waves pass through weakly cemented interfaces, microcracks, or pore clusters within solidified aeolian sand, the echo signal undergoes instantaneous phase shift and high-frequency local attenuation. The nonlinear energy operator outputs a spike-like high-amplitude response at the corresponding location, thus explicitly mapping the microscopic medium damage effect onto the target energy sequence on a continuous time axis. If such nonlinear energy features are not extracted and only global chaotic features reconstructed from phase space are relied upon, the influence of local micro-defects, such as isolated microcracks, on the global chaotic attractor structure may be diluted by a large amount of signal from normal propagation paths when encountering such conditions, leading to insufficient sensitivity of the prediction model in detecting this type of early damage.

[0061] The system generates the target sequence Perform a full fourth-order statistical operation to characterize its probability distribution. Let the length of the target sequence be... The mean is , representing the overall residual energy level preserved in the intrinsic mode function with the highest energy proportion after ultrasonic wave transmission through the sandy matrix. The variance is... Its arithmetic square root The standard deviation is denoted as , which quantitatively represents the amplitude of local energy fluctuations induced by the randomness and porosity inhomogeneity of the aeolian sand skeleton. The skewness is . The asymmetry of energy jump distribution is described by the third-order central moments: a skewness greater than zero indicates a right-skewed energy distribution, meaning the medium contains special structures capable of producing long-tailed high-energy transient reflections, such as large-sized fracture surfaces; a skewness close to zero indicates a symmetrical energy distribution. Kurtosis, or excess kurtosis, is calculated as follows: The system subtracts the normal distribution reference constant of 3 from the fourth-order central moments to identify the steepness of the spike pulse distribution: an excess kurtosis greater than zero indicates the presence of a more concentrated cluster of spike pulses in the energy sequence than in a normal distribution, corresponding to concentrated defects within the sand body, such as locally loose, weakly cemented regions; an excess kurtosis close to zero indicates an energy distribution close to a normal morphology. , , and They are combined in a fixed order to form a one-dimensional feature vector. As a second performance characteristic.

[0062] After extracting the first and second performance features, the system concatenates and merges them into a six-dimensional comprehensive input feature vector. This feature vector simultaneously encodes the global nonlinear chaotic dynamics of the ultrasonic wave propagation process. and And the statistical characteristics of the local transient energy distribution of the intrinsic mode function with the highest energy proportion, i.e. , , and The internal structural state of solidified aeolian sand bodies is characterized in a multidimensional way from two complementary levels.

[0063] The system inputs the comprehensive input feature vectors into a pre-trained support vector regression model. This support vector regression model employs a radial basis function kernel structure. In the formula and Let each represent any two distinct feature vector samples from the input. The width parameter of the radial basis function (RBF) kernel is used. The input to this model is the aforementioned six-dimensional integrated input feature vector, and the output is the predicted mechanical property parameters of the solidified aeolian sand, such as the predicted compressive strength, in MPa. The RBF kernel is chosen based on its good generalization performance in high-dimensional nonlinear regression problems and its adaptability to small sample datasets. If multiple mechanical property parameters, such as compressive strength and elastic modulus, need to be predicted simultaneously, independent support vector regression models are established for different mechanical property parameters. Each model shares the same six-dimensional input features but has its own trained model parameters.

[0064] The model training process is as follows: The system collects ultrasonic echo signals from solidified aeolian sand samples with different mix ratios and curing ages. The number of samples needs to cover the expected mix ratio range and curing age span to ensure the model's generalization ability. Corresponding real mechanical performance labels are obtained through mechanical testing methods such as standard uniaxial compressive strength tests. Following all the aforementioned steps, a six-dimensional comprehensive input feature vector is extracted for each training sample. Z-score standardization is performed on each dimension feature, i.e., subtracting the mean of that dimension on the training set and dividing by the standard deviation, so that the mean of each standardized feature dimension is 0 and the variance is 1, to eliminate the interference of differences in the dimensions and numerical magnitudes between different feature dimensions on model training. The standardized feature vector is used as the training input, and the real mechanical performance labels are used as the supervision output. A radial basis function kernel function support vector regression algorithm is used for fitting training. During the training process, through... Cross-validation, for example Adjust the penalty coefficient within the preset parameter search space. Kernel width parameter and insensitive loss parameters The parameter combination that minimizes the mean absolute error on the validation set is selected as the final model parameters. After training, the mean and standard deviation parameters of each dimension used in the standardization process, as well as the decision function parameters of the support vector regression model, are fixed and used for subsequent prediction of the mechanical properties of the solidified aeolian sand specimens.

[0065] This ablation experiment used ultrasonic testing on 150 solidified aeolian sand specimens with different mix proportions and curing ages. Time-domain echo signals from each specimen were collected to form a test dataset. The hardware platform used a graphics workstation equipped with a ten-core processor. The experiment included a baseline model containing all signal processing modules and three ablation comparison schemes. Comparison Model 1 used a fixed number of modes (5) instead of the variational mode decomposition search mechanism based on box-fractal dimension determination; Comparison Model 2 used a window with a fixed width of 50 sampling points instead of a sliding time window based on the absolute value of instantaneous frequency; Comparison Model 3 extracted only the maximum Lyapunov exponent and correlation dimension as performance features, eliminating the feature vector fusion step based on nonlinear energy operators.

[0066] Under the same testing environment and experimental conditions, the mechanical properties of various solidified aeolian sand specimens were predicted, and the prediction errors and correlation indices of different schemes on the test set were statistically analyzed. Experimental results show that the mean absolute error of the baseline model fully encompassing all core modules is [value missing]. The coefficient of determination is The mean absolute error of model 1 is... The coefficient of determination is The mean absolute error of model 2 is... The coefficient of determination is The mean absolute error of model 3 is... The coefficient of determination is ,like Figure 3 As shown.

[0067] Compared to Comparison Model 1, the baseline model adaptively selects the number of modes through a box-fractal dimension determination mechanism, reducing the impact of high-frequency shot noise generated by over-decomposition on feature extraction. The coefficient of determination is changed from... Upgraded to Compared to Comparison Model 2, the baseline model employs a sliding time window based on instantaneous frequency. This allows for adjustment of the analysis window width according to local frequency changes, balancing local transient information with overall attenuation characteristics, thus reducing the mean absolute error. Reduce to Compared to the comparative model 3, the baseline model further integrates the statistical features of the nonlinear energy operator, supplementing the energy information of local high-frequency abrupt changes, thus further reducing the prediction error. The above results demonstrate that jointly representing the nonlinear dynamic features obtained from phase space reconstruction with the statistical features of the nonlinear energy operator enhances the ability of ultrasonic echo features to characterize the internal structural state and mechanical property changes of solidified aeolian sand bodies, thereby improving the stability and accuracy of non-destructive prediction of mechanical properties.

[0068] Figure 2This diagram illustrates the selection of the number of modes in variational mode decomposition based on box-fractal dimension. The broken line formed by square sampling points represents the box-fractal dimension of the highest-frequency intrinsic mode function obtained under different modes. The graph shows that the broken line increases monotonically with the number of modes. Within the range of 2 to 4 modes, the sampling points are all below the horizontal dashed line, indicating that the extracted high-frequency components are still dominated by the internal structural response of the solidified aeolian sand body. When the number of modes reaches 5, the sampling points cross the dashed threshold for the first time, proving that the algorithm successfully captured the critical point where the effective signal transforms into random noise, thus achieving adaptive selection of the target number of modes. Observing the end of the curve, it can be seen that the sampling point value eventually approaches 2, which corresponds to the physical characteristic of extremely high waveform irregularity of the random shot noise generated by over-decomposition as described in the specific implementation.

[0069] Figure 3 This is a schematic diagram of the ablation test results. In the paired bars in the figure, the bars on the left represent the mean absolute error of the prediction results, and the bars on the right represent the coefficient of determination of the prediction results. The image shows that the height of the left bar corresponding to the baseline model is significantly lower than that of the other three variants, while the height of the right bar is closest to the top of the axis of 1. This proves that the algorithm successfully eliminates the interference of heterogeneous media and improves the prediction accuracy by fusing nonlinear dynamic characteristics and local transient energy characteristics. Observing the evolution of the indicators from Comparative Model 1 to Comparative Model 3, it can be found that after removing the adaptive search mechanism or dynamic time window, the error bars all show varying degrees of increase. This corresponds to the irreplaceable synergistic gain effect of each signal processing module described in the specific implementation when extracting the acoustic features of complex porous media.

[0070] It should be noted that those skilled in the art can make various modifications and improvements without departing from the inventive concept, and these all fall within the scope of protection of this invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand, characterized in that, include: S1: Acquire ultrasonic time-domain echo signals that penetrate solidified aeolian sand bodies; The ultrasonic time-domain echo signal is subjected to variational mode decomposition by setting search variables. The box fractal dimension of the highest frequency intrinsic mode function obtained by decomposition is compared with the preset noise threshold to adaptively determine the target number of modes and obtain the intrinsic mode function set under the corresponding target number of modes. The modal total energy ratio of each intrinsic mode function in the intrinsic mode function set is calculated. The intrinsic mode functions with a modal total energy ratio greater than the preset energy ratio threshold are selected to reconstruct the preprocessed signal, and the intrinsic mode function with the highest energy ratio is extracted. S2: The effective instantaneous frequency of the intrinsic mode function with the highest energy ratio is obtained. The preprocessed signal is divided into sliding time windows according to the effective instantaneous frequency. The normalized power spectral density of the signal segment in each sliding time window is calculated and the corresponding local inverse function is solved. The local inverse function is expanded to obtain the free cumulant and form a free cumulant sequence matrix. S3: Principal component analysis is performed on the free cumulant sequence matrix to obtain a uniform interval resampling sequence. The phase space is reconstructed based on the uniform interval resampling sequence, and the maximum Lyapunov exponent and correlation dimension are calculated as the first performance feature. The mean, variance, skewness, and kurtosis of the sequence of intrinsic mode functions with the highest energy proportion after processing by a nonlinear energy operator are used as the second performance feature. The first and second performance features are input into the prediction model to output mechanical performance parameters.

2. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The adaptive determination of the target modal number includes: setting the search start point, search end point, and search step size of the search variables; using the search start point as the current modal number, performing variational mode decomposition on the ultrasonic time-domain echo signal, extracting the intrinsic mode function with the largest center frequency as the highest frequency intrinsic mode function, and calculating the box fractal dimension of the highest frequency intrinsic mode function; if the box fractal dimension is greater than a preset noise threshold: when the current modal number is greater than the search start point, subtracting the search step size from the current modal number to obtain the target modal number; when the current modal number is equal to the search start point, using the search start point as the target modal number; if the box fractal dimension is not greater than a preset noise threshold: if the search end point has not been reached, increasing the search step size of the current modal number, and returning to perform variational mode decomposition on the ultrasonic time-domain echo signal and subsequent steps; if the search end point has been reached, using the search end point as the target modal number.

3. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The step of obtaining the effective instantaneous frequency of the intrinsic mode function with the highest energy proportion includes: extracting the analytic signal of the intrinsic mode function with the highest energy proportion through Hilbert transform; calculating the derivative of the instantaneous phase of the analytic signal to obtain an instantaneous frequency sequence; determining the current starting point of each sliding time window and extracting the current instantaneous frequency value corresponding to the current starting point of the sliding time window; obtaining the absolute value of the current instantaneous frequency value, and determining the maximum value of the absolute value compared with a preset lower frequency limit as the effective instantaneous frequency.

4. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The step of dividing the preprocessed signal into sliding time windows based on the effective instantaneous frequency includes: setting a fixed number of periods for each sliding time window; dividing the fixed number of periods by the effective instantaneous frequency and multiplying by the system sampling rate and rounding down to calculate the number of sampling points of the sliding time window as the time window width; using the current starting point of the sliding time window as a reference, intercepting the corresponding signal segment along the positive time axis according to the time window width; setting the sliding time step to a fixed proportion of the time window width and moving it along the positive time axis to obtain the starting position of the next sliding time window, and repeating the division until the preprocessed signal is completely traversed.

5. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The step of calculating the normalized power spectral density of the signal segment within each sliding time window and solving for the corresponding local inverse function includes: performing a discrete Fourier transform on the signal segment within each sliding time window using the periodogram method to extract the frequency domain amplitude; calculating the square of the frequency domain amplitude and dividing it by the number of sampling points in the sliding time window to obtain the power spectral density function; calculating the area of ​​integration of the power spectral density function in the frequency domain interval; dividing the power spectral density function by the area of ​​integration to obtain the normalized power spectral density; solving the Cauchy transform of the normalized power spectral density and calculating the local inverse function of the Cauchy transform.

6. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The step of expanding the local inverse function to obtain the free cumulants and constructing a free cumulant sequence matrix includes: performing a Laurent series expansion on the local inverse function, removing the singular principal part, and obtaining the analytic part of the R-transform generating function; performing a power series expansion on the analytic part of the R-transform generating function, extracting the coefficients corresponding to the first k orders as free cumulants of orders 1 to k; and vertically concatenating the free cumulants of orders 1 to k obtained from each of the sliding time windows in chronological order to generate the free cumulant sequence matrix.

7. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The step of performing principal component analysis on the free cumulant sequence matrix to obtain a uniformly spaced resampled sequence includes: performing zero-mean processing on the free cumulant sequence matrix and calculating the covariance matrix of the zero-mean matrix; performing singular value decomposition on the covariance matrix, extracting the eigenvector corresponding to the largest eigenvalue, multiplying the zero-mean matrix and the eigenvector to form a first principal component time series; fitting the first principal component time series using cubic spline interpolation, and performing uniformly spaced resampling at a set reference frequency to obtain a uniformly spaced resampled sequence.

8. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 1, characterized in that, The reconstructing of the phase space based on the uniformly spaced resampled sequence includes: calculating the mutual information function of the uniformly spaced resampled sequence using the mutual information method, and taking the time interval corresponding to the first minimum value of the mutual information function as the optimal delay time; calculating the proportion of false nearest neighbors under different embedding dimensions using the false nearest neighbor method, and determining the dimension corresponding to the first drop of the proportion of false nearest neighbors below a preset proportion as the optimal embedding dimension; and performing coordinate extension on the uniformly spaced resampled sequence based on the optimal delay time and the optimal embedding dimension to reconstruct a multidimensional mapped coordinate point set of the phase space.

9. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sand as described in claim 8, characterized in that, The calculation of the maximum Lyapunov exponent and the correlation dimension as the first performance feature includes: searching for the initial nearest neighbor of the reference phase point in the set of phase space multidimensional mapping coordinate points and recording the initial spatial distance; calculating the evolved spatial distance after the time step, obtaining the natural logarithm of the ratio of the evolved spatial distance to the initial spatial distance and dividing it by the evolution time step, and calculating the maximum Lyapunov exponent by ensemble averaging; counting the number of phase point pairs in the set of phase space multidimensional mapping coordinate points whose spatial distance is less than the correlation radius to obtain the correlation integral value, plotting a scatter plot of the natural logarithm of the correlation radius and the natural logarithm of the correlation integral value, performing linear regression fitting on the linear scale-free interval to extract the slope as the correlation dimension; and combining the maximum Lyapunov exponent and the correlation dimension as the first performance feature.

10. The non-destructive ultrasonic testing method for the mechanical properties of solidified aeolian sandstone according to claim 1, characterized in that, The calculation of the mean, variance, skewness, and kurtosis of the sequence of the intrinsic mode function with the highest energy proportion after processing by the nonlinear energy operator as the second performance feature includes: calculating the nonlinear characteristic energy of the intrinsic mode function with the highest energy proportion at each discrete sampling point by subtracting the product of the amplitude of the previous sampling point and the amplitude of the next sampling point from the square of the amplitude of the current sampling point, and generating a target sequence; calculating the mean, variance, and skewness of the target sequence respectively; calculating the kurtosis by performing a fourth-order central moment operation on the target sequence and subtracting a preset distribution benchmark constant; and combining the mean, variance, skewness, and kurtosis into a one-dimensional feature vector as the second performance feature.

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