1 Introduction

As aerospace, microelectronics industry, the development of modern medical and biological engineering technology, three-dimensional microstructures (feature size in micron level to millimeter level) of the increasingly urgent demand, as well as the parts, material properties, structure, shape and function in the use of reliability requirements of more and more is also high [1]. There are many ways to measure microdefects, each with its advantages and disadvantages. In recent years, the university of Manchester institute of materials research and development of 3 d latest X-Ray can achieve a high resolution computed tomography (CT), but must cut the sample into small sample to test, destroyed the workpiece structure, unable to do NDT [2]. Atomic force microscopy (AFM) is only suitable for detecting surface defects in workpieces. Atomic force microscopy (AFM) is a method for studying a rigorous molecule's exterior down to the microscopic resolution. AFM creates 3-D pictures of the surface and employs a motorized probing to amplify physical properties up to 100,000,000 times. The topology of delicate biomolecules in their natural habitats may be imaged using the AFM. Additionally, it could be employed to examine the surface strength of cells as well as an external matrix, such as their inherent tensile properties and relationships between receptors and compounds. When studying biomolecules, the AFM offers a number of benefits versus transmission electron, along with the capacity to scan in solution with little sample processing (no labeling, fixing, or coating). Additionally, the AFM enables topographical material evaluation at sensitivities that electron microscopy is not capable of handling them. The recently studied atomic force acoustic microscopy (AFAM) can image subsurface defects with nanometer resolution, but it also has no penetration force and cannot detect defects inside the workpiece [3, 4]. A form of scanning probe microscope is atomic force acoustic microscopy (AFAM). It combines molecular force microscope with acoustic. The method is presently used to test the localized elasticity characteristics of substances both qualitatively and quantitatively. With this microscopy, users may analyze any kind of substance. Particularly, it is possible to quantify parameters at nanotechnology, including the shear modulus, Poisson ratio and the elastic modulus. In the conventional ultrasonic non-destructive testing, the defect is usually evaluated by analyzing the sound wave after the interaction between the sound field and the defect [5]. The ceramic particles are decent production wave resistors, which forms the foundation of the acoustic analytical technique. Therefore, the sound field characteristics obtained after the defect is the key to the analysis of the defect ultrasonic signal [6, 7]. In actual work, the thickness of the detected component is relatively thin, such as aerospace components [8]. According to the traditional ultrasound theory, within the near-field region of the probe, the sound pressure distribution on the single-frequency continuous-wave acoustic axis is fluctuating, so it is impossible to accurately quantify the defect. The discrepancy among the absolute pressures as well as median pressures at any given time at any place in the domain of an acoustic source. For example, using the direct probe to detect titanium alloy, and for the defects in the near field, it is necessary to conduct a comparative experiment with the buried test block, which brings many difficulties to the ultrasonic testing [9, 10]. For the sound field near the sound source, the phase and amplitude are different when the sound waves from different parts of the piston reach the observation point, so the interference image is more complicated [11]. It is difficult to calculate this kind of sound field in mathematics, and cannot get concise analytic expression. Initially, the testing is applied to optics by the angular spectrum method [12]. In the 1970s, it was introduced into the acoustic field, which provided a direction for the detection of ultra-diffraction limited resolution and attracted more and more attention [13, 14]. Based on the angular spectrum method to simulate the process of ultrasonic testing, according to the workpiece geometry algorithm model is established, through the programming to calculate ultrasonic micro defects and artifacts coupled acoustic field characteristics, and compared with the finite element simulation results, the corresponding conclusions [15].

2 Principle of angular spectrum analysis.

The essence of angular spectrum method is the spatial frequency domain representation of huygens principle. According to Huygens' concept, all positions on a waveform next to sound in a broadcasting media or of illumination in a vacuum or translucent material may be thought of as fresh generators of wavelet decomposition that spread out in all directions at a pace based on their velocity. Using information of the stress dielectric materials at a comparable sector, the approach may forecast the dispersion of the acoustic wave area above an aircraft. It is feasible to make forecasts for both onward as well as reverse transmission. Based on the basic idea of angular spectrum method, the sound pressure of a certain point in the circular piston radiation field in the semi-space can be expressed as the superposition of numerous different plane waves, expressed as the integral form as follows:

$$\begin{gathered} p(x,y,z) = \left( {\frac{1}{2\pi }} \right)^{2} \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {p(k_{x} ,k_{y} )} } \cdot \hfill \\ \exp \left[ {i(k_{x} x + k_{y} y + k_{z} z)} \right]dk_{x} dk_{y} \hfill \\ \end{gathered}$$
(1)

The trigonometry coefficients as well as the complicated coefficient of determination are fundamentally related, according to Euler's formula, a mathematical model used in multivariable calculus. The complicated exponentially is related to the cosine and sine curves by Euler's equation. The foremost crucial instrument in AC analysis is this formula. Because of this, electrical technicians must be able to work with complicated statistics. Similar to how we deconstruct a location in Euclidean space into the summation of its foundation vector elements, the Fourier transform explains a method of breaking down a function into a combination of orthogonal basis values. We can observe that the Fourier analysis consists of both real and fictitious elements. The frequency domain Euler formula and space Fourier transform theory can be obtained:

$$\begin{gathered} p(k_{x} ,k_{y} ) = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } { - \rho ckv_{z} (x,y,0)} } \cdot \hfill \\ \exp \left[ { - i(k_{x} x + k_{y} y)} \right]dxdy \hfill \\ \end{gathered}$$
(2)

Set the radius as the sound source of a in the plane, and its velocity field is defined as:

$$v_{z} \left( {x,y,0} \right) = \left\{ \begin{gathered} 0,\;\;\;\;\;x^{2} + y^{2} > a^{2} \hfill \\ v_{0} ,\;\;\;\;x^{2} + y^{2} \le a^{2} \hfill \\ \end{gathered} \right.$$
(3)

Substituting (2) (3) into Eq. (1), the sound field distribution of any plane in half space is obtained:

$$p(x,y,z) = V\int\limits_{ - \infty }^{ + \infty } {J_{1} (ak_{r} )J_{0} (rk_{r} )\exp (izk_{z} )dk_{r} }$$
(4)

Type,\(V = \rho ckv_{0} a\); \(k_{z} = \pm \sqrt {k^{2} - k_{x}^{2} - k_{y}^{2} }\); \(r = \sqrt {x^{2} + y^{2} }\); \(J_{0}\) Is the first class of zero order Bessel function; \(J_{1}\) Is the first class of first order Bessel function. Locating independent resolutions to Laplace's calculation as well as the Helmholtz solution in either cylindrical or spherical dimensions leads to the Bessel equations. Hence, Bessel functions are crucial for a variety of acoustic waves or dynamic possible problems. The typical Bessel function may be represented in terms of simple functions under these circumstances. It therefore, follows based on the initial probability function and so on. In that every Bessel function of the initial sort of half-integer value may be represented in various forms of linear equations. There are two conditions for the value of wave number, which respectively represent the propagation wave composition and evanescent wave components in the sound field:

$$k_{z1} = \sqrt {\left( {k_{x}^{2} + k_{y}^{2} } \right) - k^{2} } \;\left( {k^{2} < k_{x}^{2} + k_{y}^{2} } \right)$$
(5)
$$k_{z2} = \sqrt {k^{2} - \left( {k_{x}^{2} + k_{y}^{2} } \right)} \;\left( {k^{2} \ge k_{x}^{2} + k_{y}^{2} } \right)$$
(6)

The force generated at that particular wavelength is referred to be decreasing rate because a resolution to the wave function with an arbitrary wavelength range does not travel as a waveform but instead decays exponential. Each method by which waves move is known as wave propagation. Either the first order one-way wave function or the second order stationary wavefield wave calculation can be employed to compute individual acoustic waves. The distribution of propagation wave and evanescent wave is discussed by taking different distance z in the direction of space sound field axis. make \(k_{r} = \sqrt {k_{x}^{2} + k_{y}^{2} }\), The semi—space sound field described in Eq. (4) is regarded as the superposition of plane propagation wave and evanescent wave:

$$\begin{gathered} p(z) = V\int\limits_{0}^{k} {J_{1} (k_{r} a)\exp \left[ {i\left( {\sqrt {k^{2} - k_{r}^{2} } \cdot z} \right)} \right]dk_{r} } + \hfill \\ V\int\limits_{k}^{\infty } {J_{1} (k_{r} a)\exp \left[ { - 1\left( {\sqrt {k_{r}^{2} - k^{2} } \cdot z} \right)} \right]dk_{r} } \hfill \\ \end{gathered}$$
(7)

The formula (7) is used to calculate the acoustic pressure distribution on the central axis of the piston. The results are shown in Fig. 1. Comparing with the calculation results of Rayleigh integral, it can be seen that the angular spectrum analysis can not only describe the near-field and far-field sound pressure distributions of the piston axis, but also describe the distribution of different wavenumber components, namely the propagating wave and the evanescent wave.

Fig. 1
figure 1

Comparison of angle spectrum method and rayleigh integral method

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3 Wave number domain filtering

The filter window function of wave number domain is first proposed by Veronesi and Maynard, and its algorithm is simple and operable, and the application is most widely used. A framework for developing network filtering programs is offered by the Windows Filtering Platform (WFP), a collection of APIs and generally involves. The WFP API enables programmers to create code that communicates with the data transfer that occurs at various tiers of the operational program's protocol stacks. One might not even be aware of how they obstruct us. The information we're attempting to transmit can be filtered by anything that can remove, change, or generalize it. In each discussion, restrictions are in effect for both the transmitter and the recipient. A time series window, often known as boxcar filtering, is the most basic smoother technique. It substitutes every spectra value with the mean of the 2 m + 1 neighboring values inside the smoothing window. The computation is then performed with the frame moved forward by one degree. A good filtering effect is obtained by smoothing the cut-off wave number, and the expression of the window function is as follows:

$$\prod \left( {k_{x} ,k_{y} } \right) = \left\{ \begin{gathered} 1 - 1/2e^{{(k_{r} /k_{c} - 1)/\alpha }} ,\;\;k_{r} \le k_{c} \hfill \\ 1/2e^{{(1 - k_{r} /k_{c} )/\alpha }} ,\;\;\;\;\;\;k_{r} > k_{c} \hfill \\ \end{gathered} \right.$$
(8)

In the formula, KC is the space cut-off wave number;\(k_{r} = \sqrt {k_{x}^{2} + k_{y}^{2} }\); Alpha is an adjustable parameter, indicating the attenuation rate on the filter band.

3.1 Cut-off wave number selection

The value of the cutoff wave number kc determines the sound pressure Angle spectrum range of the plane participating in the calculation. The crucial wavelength among propagation and attenuation, or the wavelength at which the transverse amplitude of the wave is 0, is known as the cutoff frequency. Underneath the bandpass filter, when the wave propagation quantity is real, the waveform formulas are also applicable. The detail information about the rapid change of the evanescent wave component of the high wave number in the angular spectrum corresponding to the distance in the sound field while in order to obtain high resolution reconstruction images, it is necessary to include as many effective evanescent waves as possible in the calculation process, which requires the selection of larger kc; However, due to the evanescent wave attenuation, quickly through takanami number if the evanescent wave is not filter out, it is easy to be flooded by all kinds of noise error, in the process of rebuilding the evanescent wave together with all kinds of error would be sharp amplified, inverse transfer factor according to the index law to produce larger reconstruction error, obtained from this analysis kc is not too big. Therefore, it is necessary to determine a suitable kc to obtain the highest possible spatial resolution on the premise of ensuring the reconstruction accuracy. The value of kc is usually determined based on an empirical formula: \(k_{c} = 0.6\pi /\Delta\), The cut-off wave number is fixed in the formula as 60% of the maximum wave number component at the current sampling interval, Because the influence of factors such as signal-to-noise ratio, the distance between the calculation surface and the sound source surface, and the frequency of the sound wave are not taken into consideration, the filtering failure may occur due to improper selection during use. As a transverse wave, signal carries throughout air or even other media. The mechanical equipment that makes up the wave happens in the wave's movement direction.

In physical terms, sound is a disturbance that travels over a communication system like a gas, liquid, or substance as an interference pattern. A disruption or fluctuation that moves through space–time while power is transmitted is known as a wave. Transverse waves fluctuate in a plane that is perpendicular to the surface in which they travel. A waveform conveys electricity, not weight, in the plane of its transmission. It is possible to create a full-color, three-dimensional holographic of the human body. Both learners and professionals might explore three-dimensional models of complex organs that include the muscles, lungs, brain, nerves, heart, and liver. Surgery scheduling can potentially be aided by these technologies. In order to compensate for the shortcomings of the empirical formula, considering the factors such as signal-to-noise ratio, the distance between the holographic surface and the sound source plane, and the acoustic wave frequency, literature gives a selection formula for kc:

$$k_{c} = \left\{ {\begin{array}{*{20}c} {k_{q} } & {k_{q} \le \min \left( {k_{x\max } ,k_{y\max } } \right)} \\ {\min \left( {k_{x\max } ,k_{y\max } } \right)} & {k_{q} > \min \left( {k_{x\max } ,k_{y\max } } \right)} \\ \end{array} } \right.$$
(9)

among them:

$$k_{q} = \sqrt {k^{2} + \left[ {\frac{SNR\ln (10)}{{20d_{z} }}} \right]^{2} } \ge \sqrt {k_{x}^{2} + k_{y}^{2} }$$
(10)

According to the calculation with the sound source surface distance dz type (9) with type (10) to solve it out by the value of wave number, the filter can make the calculated values are arbitrary wave vector of the useful signals will not be submerged by noise. However, in the actual measurement, Eq. (10) is unknown to SNR, and the following is estimated by calculating the angular spectrum of the surface acoustic pressure signal.

3.2 SNR estimation

It is necessary to know the distribution of the effective signal components and noise in the information of the computing surface sound field by estimating the acoustic pressure signal SNR of the hologram surface. The range of the angular spectrum of the calculated surface sound pressure, as shown in Fig. 2, is transformed into three regions of Omega 1, Omega 2 and omega 3. Within the corresponding Omega 1 radiation source area, namely the wavenumber region of kr < k, the area usually contains a large amount of energy, which is not with the propagation wavenumber component amplitude wave propagation distance decay; annular between Omega 2 corresponds to two concentric circles, namely k < kr < km wavenumber region (denoted km = max (kxmax, kymax)). The number of components for the regional wave evanescent wave; Omega 3 as part of the four corners of the corresponding graph, namely the wavenumber region of km < kr, the same as the wavenumber component of evanescent wave, compared with the 2 wave number Omega higher wavenumber components, in the process of propagation attenuation more quickly. In general, the evanescent wave in Omega 3 almost all attenuates below the noise level of the measurement system and cannot be reused for reconstruction. Therefore, the noise error is dominant in Omega 3 and can be estimated by the evanescent wave components in Omega 3 to estimate the size of the noise energy in the acoustic pressure.

Fig. 2
figure 2

Calculate the surface angular spectrum analysis diagram

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The actual measurement normal vibration velocity including error interference in the design plane is \(v(x,y,z_{H} )\), Calculated surface theory normal velocity is \(v_{t} (x,y,z_{H} )\), The noise error component is \(v_{e} (x,y,z_{H} )\)° \(V(k_{x} ,k_{y} ,z_{H} )\), \(V_{t} (k_{x} ,k_{y} ,z_{H} )\) and \(V_{e} (k_{x} ,k_{y} ,z_{H} )\) Respectively \(v(x,y,z_{H} )\), \(v_{t} (x,y,z_{H} )\) and \(v_{e} (x,y,z_{H} )\) The discrete space Fourier transform. According to the above analysis, in the high wave number region,\(V_{t} (k_{x} ,k_{y} ,z_{H} )\) with the increase of the wave number kr rapidly attenuated, Useful signal in Ω 3 area \(V_{t} (k_{x} ,k_{y} ,z_{H} )\) Attenuation is exhausted, Almost entirely by the noise, so you can pass Ω 3 regional data on the estimated noise signal. The structure of a layer, its curving, the characteristics of different kinds of curvature on a ground, elements of distortion, the presence of an exterior with certain internally or exterior characteristics, etc. are all examined in the concept of materials. In combination with Euler's formula, the SNR of signal and noise can be approximated by the following formula:

$$SNR \approx 10\lg \frac{{\left\| {\rho ckV_{t} (k_{x} ,k_{y} ,z_{H} )} \right\|_{2}^{2} }}{{\left\| {\rho ckV_{e} (k_{x} ,k_{y} ,z_{H} )} \right\|_{2}^{2} }} = 10\lg \frac{{||P_{H} (\Omega_{1} \Omega_{2} )||_{2}^{2} }}{{||P_{H} (\Omega_{3} )||_{2}^{2} }}$$
(11)

In the formula ||·||2 and 2 norm; Said to calculate surface sound pressure Angle spectrum in Ω Ω 1 + 2 area of wave number component, namely the useful signal; Said sound pressure Angle spectrum in Ω wave number component 3 area, namely the noise signal. It is relatively difficult to directly solve \(P_{H} (\Omega_{3} )\), we can use the sound pressure information \(P_{H} (k_{x} ,k_{y} ,z_{H} )\) in all areas to remove \(P_{H} (\Omega_{1} \Omega_{2} )\) so (11) can be rewritten as:

$$\begin{gathered} SNR \approx 10\lg \frac{{\left\| {\rho ckV_{t} (k_{x} ,k_{y} ,z_{H} )} \right\|_{2}^{2} }}{{\left\| {\rho ck\left[ {V(k_{x} ,k_{y} ,z_{H} ) - V_{t} (k_{x} ,k_{y} ,z_{H} )} \right]} \right\|_{2}^{2} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} = 10\lg \frac{{||P_{H} (\Omega_{1} \Omega_{2} )||_{2}^{2} }}{{||P_{H} (k_{x} ,k_{y} ,z_{H} ) - P_{H} (\Omega_{1} \Omega_{2} )||_{2}^{2} }} \hfill \\ \end{gathered}$$
(12)

4 Ultrasound and micro-defect coupling sound field calculation

4.1 Calculation model

Under the assumption of low-amplitude sound waves, the propagation law of sound waves in the medium can be seen as linear. As shown in Fig. 3, there is a circular hole defect of radius r in the near surface of an aluminum thin plate of thickness h. Let the sound source radius of the disk be a, the distance from the sound source surface Z0 to the plane Z1 (tangent to the front of the round hole) is d1, and the distance from the plane Z2 (tangent to the rear end surface of the round hole) to the back end surface Z3 of the thin plate is d2.

Fig. 3
figure 3

Ultrasound and micro-defect coupling sound field model

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Give an excitation signal to the transducer, assuming that the acoustic wave propagates within the isotropic, unattenuated workpiece. The sound pressure p1 at Z1 was calculated by angular spectrum method. Sound waves by the Z1 continue to spread to planar Zi, this time with the angular spectrum method of direct calculation (regardless of the acoustic coupling) and defects of the sound pressure of PI ", but because of the existence of round hole defect makes the acoustic scattering after encountered defects. The sound pressure on the end surface of the hole is PI ', so the actual sound pressure PI of Zi is PI 'and PI'. And so on, a series of plane interception from Z1 to Z2, P2 to Z2 of the sound propagation can be obtained through the coupling of acoustic waves and defects of P2 iterative algorithm, again using the angular spectrum method is used to calculate the acoustic propagation to the end surface of the sample by Z2 Z3 pressure P3, thus can obtain the specimen surface acoustic information containing Z3 micro defects. Since it offers reasonable magnitude precision, adequate frequency response, and permeability prevention, Hanning appears to be the best window function for multiple wavelengths.

As can be seen from Fig. 3, the key to assigning the sound pressure pi'' to the inner face of the defect is the need to know the size of the cross-section at the defect (the shaded part in Fig. 3). According to the geometric relationship in the figure, the diameter of the shaded circle cross section is the chord length l corresponding to the center angle θ. Face to make use of Matlab numerical calculation, the meshing, a defect in the shadow part of the (Fig. 3) sound pressure amplitude value is 0, the matrix of the dimension of all 0 as the chord length l, and other location except in the context of the shadow of section sound pressure amplitude is set to 1, namely to fill 1 all around zero matrix operations, make up the final dimension of the matrix is equal to the dimension of PI '. The actual sound pressure is of representation as Zi. The mathematical modeling of ultrasonic wave propagation in the workpiece, coupled with the defect, and the final transmission to the surface of the workpiece, the algorithm flow is shown in Fig. 4.

Fig. 4
figure 4

Ultrasound and micro-defect coupling sound field model

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4.2 Calculation of sound pressure distribution

According to the above theory, it is calculated that the ultrasonic wave propagates inside the micro-defective workpiece and reaches the sound field at the surface of the workpiece. The thickness of the aluminum sheet is 3 mm, the diameter of the inner circular hole defect is 0.2 mm, d1 = 1.4 mm from the front surface of the workpiece, and d2 = 0.4 mm from the rear surface. The diameter of the circular transducer is 12.7 mm (0.5in). The excitation signal adopts a frequency of 2.5 MHz. The duration is 3 cycles of a 1.2us sine wave. A quick Fourier transform is used to filter signals or images employing the window operator HANNING. The outcomes can be enhanced by analyzing information with HANNING preceding performing FFT. The window determined by HANNING is essentially the positive cosine frequencies or the initial part of a cosine. The Hanning window is modulated to approximate the real signal. The form of the Hanning window, named for its creator Von Hann, is that of a cosine wave with 1 to make it always optimistic. The outcome of multiplying the collected transmitted signal through the Hanning algorithm is seen in the image. In the context of the Hanning window, the maximum side lobe has a frequency of -32 dB. The ultrasonic excitation is carried out on the surface of the workpiece on the surface of the workpiece, and the distribution of the sound field on the back surface of the workpiece is calculated by the angle spectrum method, and the calculated results are filtered in the wavenumber domain Z0. Sound Pressure Degree Equation: 20 log(p/po), wherein po is the sources based intensity of 0.00002 pascals or 0 dB, the minimum of perception in the atmosphere at 1 kHz, while p is the noise intensity. The perceived loudness is a measure of sound pressure standard (dB). In Fig. 5, (a)–(c) is transmitted to the top view of the sound pressure distribution at Z1, Z2 and Z3, as shown in Fig. 3, (d). It can be seen from (a)–(c) that the ultrasonic wave propagates in the workpiece, and when the defect coupling is finally propagated to the surface Z3 of the workpiece, the sound pressure at the center of the workpiece is attenuated. The sound pressure in the Z3 center of the undefective workpiece surface was compared with the sound pressure at the center of the workpiece surface with defects. The speed in decibels per second (dB/s) in which the audio quality in a space falls off once a generator stops producing information at a certain wavelength. Defining the sound pressure decay rate \(k = \frac{{\left\| {p_{3} - p_{3}^{\prime } } \right\|_{2}^{2} }}{{\left\| {p_{3} } \right\|_{2}^{2} }} \times 100\% = 47.29\%\). Therefore, the small defects in the workpiece can be judged by the change of the amplitude of the central sound pressure of the workpiece surface. The magnitude of the fluctuation in air pressure brought on by an acoustic source is quantified as force components. There is ever-present air pressure in complete quiet. The pressure gradient, which is expressed in newtons/meter2, is around 105 N/m2.

Fig. 5
figure 5

Distribution of the acoustic pressure inside and outside the workpiece

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5 Finite element analysis

In order to better analyze scattering sound field situation of round hole defects on the ultrasonic propagation in containing the blowhole defects has carried on the finite element simulation, the transient acoustic field distribution of solid, and verified based on the angular spectrum method to calculate the reliability of the scattering wave sound field distribution. The simulation was performed using the transient dynamics module of Ansys software. The stiffness, durability, flexibility, surface temperature, magnetization, flow properties, and other characteristics of computer-simulated simulations of constructions, devices, or industrial machinery are examined using the Ansys Mechanical analysis of finite elements program. It solves issues by using numerical methods that are computer-based. It is frequently employed in the sector to analyze possibilities. The model size was 25 mm*2 mm, and the middle of the model was defective. The defect depth is 1.5 mm, and the hole defect diameter is 0.2 mm. The model material is aluminum and the acoustic wave propagating longitudinal wave velocity is cL = 6320 m/s. The material parameters are: Young's modulus E = 72 Gpa, density ρ = 2700 kg/m3, Poisson's ratio v = 0.33. The grid is divided into quadrilaterals with a side length of 0.1 mm. Using vertical wave vertical incidence, the excitation source is a Hanning window modulated signal with an excitation frequency of 2.5 MHz. Figures 6 and 7 show the propagation of ultrasonic longitudinal waves in medium aluminum at different times. It can be seen from Fig. 7 that scattering occurs when the incident wave passes through the hole defect.

Fig. 6
figure 6

The phase of initial incentive

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Fig. 7
figure 7

The phase of incentive reaching the circular hole defect

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Figure 8(a) shows the sound pressure distribution when the ultrasonic wave front reaches the top surface of the workpiece Z3, and Fig. 8(b) shows the instantaneous sound pressure distribution of the ultrasonic wave propagated in the non-defective workpiece at the same time. Can be seen from the Fig. 8 also, wave propagation in homogeneous medium aluminum, encountered when there is no hole defect, at normal speed along a fixed route forward propagation and without interference, when faced with round of holes, the wave propagation route changed, i.e. without interference of the incident wave, under the action of circular hole defects have played an important role in a secondary waves, formed a wave scattering, and wave and it has deviated the original path, the diffraction.

Fig. 8
figure 8

Transient sound pressure distribution in the workpiece

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Similarly, the sound pressure amplitudes of the central node of the workpiece surface Z3 in Fig. 8 (a) and (b) are extracted and compared to define the sound pressure attenuation ratio: \(k^{\prime} = \left| {\frac{{p_{n} - p_{n}^{\prime } }}{{p_{n} }}} \right| \times 100\% = 43.87\%\). This has some error with the attenuation ratio calculated by the angle spectrum method. The analysis error may be due to the following: (1) When using finite element calculations, the hardware configuration of the computer determines the accuracy of the meshing, resulting in certain errors in the calculation results; (2) In the spectrum analysis method, an approximate calculation is performed when calculating the actual sound pressure amplitude of the defect section. It will also lead to some deviations in the calculation results. However, comparison of K and K, we can see that the calculation results of finite element calculation results and the angular spectrum shows basically consistent that are also described respectively with and without ultrasonic defect cases, after the workpiece internal to the surface of the workpiece, the surface acoustic field characteristics of different. According to the ratio of attenuation, it can be used to determine whether there are small defects in thin workpiece, and lay a theoretical foundation for the position and size of quantitative analysis of defects in the next step.

6 Conclusion

Based on the angle spectrum method, this paper proposes an iterative calculation method for ultrasonic field propagation in a thin plate structure coupled with a small defect, and compares it with the finite element simulation to verify each other. The calculation results show that the iterative algorithm based on angular spectrum analysis can accurately reflect the distribution law of near-field sound pressure propagation in the microstructure and coupling with defects. The method provides a brand-new idea for establishing the mathematical model of the coupling of ultrasonic and micro-defects, and facilitates people to more deeply understand the process of ultrasonic wave propagation in the medium, and thus can more accurately and reliably evaluate the ultrasonic test results. In the future, the process of ultrasonic wave propagation in the medium will be much efficient for computing the ultrasonic outcomes by using the novel technologies.