Numerical Simulation of Lamb Wave Propagation in Ice

Lamb wave is one of the most used ultrasonic guided waves, which has a wide application prospect in the field of aircraft icing detection. The platform of Lamb wave propagation based on piezoelectric wafer was used as the physical model, and COMSOL software was used as the numerical calculation tool to simulate the propagation of Lamb wave in different thickness, length and types of ice. The results showed that the attenuation of voltage amplitude increases with the thickness of ice in a certain range. The time delay of Lamb wave B1 mode at the receiving end increases linearly with the increase of ice length. As the mechanical parameters of ice layer decrease, the attenuation of voltage amplitude and the time delay of Lamb wave B1 mode increase gradually. The propagation characteristics of Lamb wave in ice are preliminarily proved, which provides a theoretical reference for the subsequent propagation experiments of Lamb wave in ice.


Introduction
When an aircraft flies at altitudes below 7000 meters, icing phenomena are frequent to occur at the leading edge of the wing, tail wing, engine intakes, and other positions, which seriously affect flight safety [1].Ice detection is one of the necessary means to ensure flight safety.Traditional ice detection methods include obstacle method, pressure difference method, resonance method, etc. [2,3].However, due to the high complexity of the ice layer formed on aircraft, both in terms of its external structure and material characteristics, traditional ice detection methods cannot meet practical usage requirements in terms of detection accuracy [4][5][6][7][8][9].
Lamb waves are formed by coupling transverse and longitudinal waves, with energy distributed throughout the entire cross-section layer of the medium.Lamb waves have the characteristics of wide propagation range, small attenuation along the path, and can propagate in solid and liquid environments.The team of University of Pennsylvania [10] used the global matrix method to analyse the propagation characteristics of ultrasonic guided waves in multi-layer flat structures, established a multi-layer model of icing on aircraft wings, and obtained Lamb wave dispersion curves for each model.The research in China started relatively late.Researchers from Ningbo University [11] used the potential function method to derive Lamb wave dispersion curves under different ice thickness conditions; Researchers from Nanjing University of Aeronautics and Astronautics [12,13] used global matrix method and semi analytical finite element method to obtain Lamb wave dispersion curves in different ice.By using numerical simulation and experimental research methods, the effects of ice with

Theory of Lamb Wave Propagation
Lamb waves are ultrasonic guided waves generated on a substrate with the same order of magnitude as the plate thickness and wavelength, as shown in figure 1.When the surface of the aluminium plate is covered with ice, Lamb waves will undergo complex reflection and interference phenomena when passing through the ice-covered area, causing changes in the waveform parameters of the receiving end.By monitoring these changes, the relevant parameters of the ice can be inferred.In order to facilitate the analysis of the propagation characteristics of Lamb waves in the ice, the received Lamb wave signal should be as pure as possible, with as few modes as possible.From the dispersion curve, it can be seen that at lower fd (the product of excitation frequency and substrate thickness), the excited Lamb waves only contain S0 and A0 modes, and the group velocity of S0 mode is always faster than that of A0 mode.Therefore, the article selects the S0 mode of Lamb waves as the research object.Based on the dispersion curve and the structural characteristics of S0 modal waves under different fd, 500KHz is determined as the excitation frequency.

Simulation Model
The Lamb wave propagation research platform based on piezoelectric ceramics consists of three parts: substrate, emitted piezoelectric wafer, and received piezoelectric wafer with the transmitter and receiver being piezoelectric ceramic sheets，as shown in figure 2. By applying alternating voltage to the two poles of the piezoelectric wafer, Lamb waves can be excited and generated on the aluminium plate.After excitation, Lamb waves will propagate forward in the aluminium plate.After arriving at the received piezoelectric wafer, the piezoelectric wafer converts the Lamb wave signal into an electrical signal through the piezoelectric effect and outputs it to the post-processing end.Piezoelectric conversion is a typical multi-physics coupling problem.The article will use the multi-physics coupling analysis software COMSOL Multiphysics to simulate Lamb wave propagation in different ice based on a two-dimensional Lamb wave propagation model.Select the "piezoelectric, solid" for the interface.The substrate is made of 1060 aluminium plate, with a thickness of 1mm and a length of 800mm.The left and right are fixed boundaries.The excitation and generation of Lamb waves require a certain excitation signal to be loaded onto the piezoelectric wafer.The article selects a sine signal modulated by a Hanning window as the excitation signal, with a Hanning window length of 5 signal cycles, as shown in equation ( 1): where f0 is the excitation frequency.

Result Analysis
After loading the excitation signal onto the two poles of the emitted piezoelectric wafer, Lamb wave will be excited on the aluminum plate.The piezoelectric wafer at the receiving end outputs Lamb waves in the form of electrical signals through the piezoelectric effect.Next, study the impact of ice on Lamb wave propagation through the time-domain waveform of the receiving signal.

Propagation of Lamb Waves in Aluminum Plate
Firstly, analyze the rationality of the excitation frequency and piezoelectric wafers layout scheme through the propagation of Lamb waves in aluminum plate.When there is no ice cover on the surface of the aluminum plate, the time-domain waveform received by the receiving end is shown in figure 3. From figure 3, it can be observed that the receiving end can receive three wave packets, namely direct S0 wave packet, direct A0 wave packet, and reflected S0 wave packet, in order of arrival.The signal strength of the direct S0 wave packet is much stronger than that of the direct A0 wave packet, and even the signal strength of the reflected S0 wave packet after passing through the right boundary reflection is greater than that of the direct A0 wave packet.Due to the selection of S0 mode as the observation object in this article, the larger the signal strength of S0 mode, the more conducive it is to extract the propagation law of Lamb waves.From this point of view, selecting an excitation frequency of 500KHz is appropriate.In addition, as shown in figure 3, there is a certain time interval between each wave packet, and there is no case of wave packet mixing.The time period in which each wave packet appears can be clearly distinguished from the time domain.This indicates that the layout scheme of piezoelectric wafers is reasonable.

Influence of Ice Thickness on Lamb Wave Propagation
Ice with different geometric characteristics will have different effects on the propagation of Lamb waves.Firstly, study the influence of ice thickness on Lamb wave propagation.To extract the propagation patterns of Lamb waves in different thickness ice layers, numerical simulations were conducted to investigate the propagation of Lamb waves in ice with a thickness of 0-5mm.At this point, the thickness of the ice layer changes in steps of 0.5mm, the length of the ice layer is fixed at 30mm, and the ice type is clear ice.After extracting the trend of waveform parameters, the curve of waveform parameters changing with ice thickness can be obtained, as shown in figure 4. The iced aluminium plate belongs to an asymmetric system, and the modes of Lamb waves are usually marked with B0, B1, B2, etc.Among them, the B1 mode is similar to the S0 mode in symmetric systems.The definitions of voltage amplitude attenuation and time delay are shown in equation ( 3) and ( 4).In the formula, Val is the amplitude of the S0 mode voltage signal at the receiving end when Lamb wave propagates in a aluminum plate without ice, Vice is the amplitude of the B1 mode voltage signal at the receiving end when Lamb wave propagates in an ice covered aluminum plate, tal is the time when the receiving end receives the S0 mode voltage signal when Lamb wave propagates in a aluminum plate, and tice is the time when the receiving end receives the B1 mode voltage signal when Lamb wave propagates in an ice covered aluminum plate.
From figure 4, it can be seen that as the thickness of the ice changes, whether it is the attenuation of amplitude or delay of time, the trend of voltage, in-plane displacement, and out-plane displacement is basically consistent.The article mainly discusses the law of voltage change among them.When the thickness of the ice is less than 2.5mm, the attenuation of the voltage amplitude of the piezoelectric wafer at the receiving end and the delay of time of Lamb wave B1 mode increase with the thickness of the ice; After the thickness of the ice layer exceeds 2.5mm, the signal variation pattern is not strong, and a saturation state will appear when the thickness of the ice exceeds 5mm.Comparing the amplitude attenuation and time delay curves, it can be found that the thickness of the ice layer with inflection points on the time delay curve is smaller than the amplitude attenuation curve, and the linear region is relatively narrow.

Influence of Ice Length on Lamb Wave Propagation
The propagation characteristics of Lamb waves in ice of different lengths may also vary to some extent.It should be noted that in order to observe the impact of changes in ice length on Lamb wave propagation over a larger range, the position of the receiving piezoelectric wafer was moved 150mm towards the right boundary of the aluminium plate before conducting numerical simulations of Lamb wave propagation in ice of different lengths.
According to the dispersion curve, at an excitation frequency of 500KHz, the group velocity of the S0 (B1) mode wave packet in the aluminium plate can be obtained as 5373m/s, and the group velocity of the B1 mode wave packet in the iced aluminum plate is 4753m/s.Based on the layout scheme of piezoelectric plates in figure 2 Among them, ter_t is the theoretical propagation time of B1 mode between emitted and received piezoelectric wafers, Lice is the length of the ice on the surface of the aluminum plate, Vg_ Ice and Vg_ Al represents the group velocity of the B1 mode wave packet in the iced aluminum plate and the group velocity of the S0 (B1) mode wave packet in the aluminum plate.On this basis, the propagation of Lamb waves in ice with length of 0-100mm was numerically simulated.At this point, the length of the ice changes in steps of 10mm, the thickness of the ice layer is fixed at 1mm, and the ice type is clear ice.After extracting the trend of waveform changes, the curve of waveform parameters changing with the length of the ice can be obtained (figure 5).As the length of the ice increases, the attenuation of amplitude and time delay of voltage of B1 model, in-plane displacement, and out-plane displacement all show an increasing trend, and there is a good linear relationship between the time delay and the ice length.By fitting the time delay, the following relationship can be obtained: Among them, ter_ f is the propagation time of B1 mode obtained from numerical simulation between emitted and received piezoelectric wafers.Comparing equations ( 6) and ( 7), it was found that they have good consistency and can effectively identify the length of the ice layer on the iced aluminum plate through the arrival time of the B1 modal wave packet.

Influence of Ice Type on Lamb Wave Propagation
In addition to thickness and length, ice type is also an important variable describing the ice layer.Different types of ice layers are essentially ice with different material properties, so to some extent, different types of ice layers can be described through material properties.The material properties of the ice can be divided into mechanical properties, optical properties, acoustic properties, electrical properties, etc. Due to the fact that the propagation of Lamb waves in the ice is essentially a transfer of mechanical vibrations of particles, the differences in their propagation process in the ice are caused by the mechanical properties of the ice layer.Therefore, mechanical parameters can be used to describe different types of ice layers.For the Lamb wave propagation problem in ice covered panels that the article focuses on, the mechanical properties of the ice can be described by density, Poisson's ratio, and Young's modulus.Firstly, provide the mechanical parameters of several ice forms, as shown in table 1.Among them, as the number increases, the ice type gradually changes from clear ice to frost ice, and the mechanical parameters of the ice ecrease, namely the ice layer density, Poisson's ratio, and Young's modulus gradually decrease.At this point, the thickness of the ice is fixed at 1mm and the length is 30mm.By using numerical simulation methods, the time-domain waveform of the voltage at the receiving end can be obtained.After extracting the trend of waveform changes, the curve of waveform parameters changing with ice type can be obtained, as shown in figure 6.
As the density, Poisson's ratio, and Young's modulus of the ice decrease, the amplitude attenuation of the receiving terminal voltage, in-plane displacement, and in-plane displacement gradually increases.When Lamb waves propagate in ice with lower density, Poisson's ratio, and Young's modulus, the energy attenuation is greater, and the looser the ice layer structure is more likely to cause energy leakage; The time delay of Lamb wave B1 mode at the receiving end shows an increasing trend with the decrease of ice layer density, Poisson's ratio, and Young's modulus, that is, the slower the propagation speed of Lamb waves in the more loose ice layer structure, which is consistent with the theoretically derived dispersion curve results.In addition, observing the trend of the curve in figure 6, it can be observed that the curve is relatively flat during the transition from ice type 1 to ice type 3, while the curve is relatively steep during the transition from ice type 3 to ice type 5. From this perspective, when the mechanical properties of the ice are close to that of clear ice, the changes in ice density, Poisson's ratio, and Young's modulus have a relatively small impact on Lamb wave propagation; When the mechanical properties of the ice layer approach frost ice, the changes in ice density, Poisson's ratio, and Young's modulus have a more significant impact on Lamb wave propagation.

Conclusion
Considering the material and placement of piezoelectric wafers, the article conducted numerical simulation of propagation characteristics of Lamb wave in ice, and preliminarily explored the propagation characteristics of Lamb waves in complex ice.The conclusions are drawn as follows: 1) When the thickness of the ice is less than 2.5mm, the attenuation of the voltage amplitude of the piezoelectric warfer at the receiving end and the delay time of Lamb wave B1 mode increase with the thickness of the ice; After the thickness of the ice layer exceeds 2.5mm, as the thickness of the layer increases, the signal variation pattern is not strong, and a saturation state occurs when the thickness of the ice layer exceeds 5mm.Monitoring the attenuation of voltage amplitude of piezoelectric wafers can detect ice within a certain thickness range.2) When the length of the ice is within the range of 0-100mm, the attenuation of the voltage amplitude of the piezoelectric wafer at the receiving end and the time delay of Lamb wave B1 mode both show an increasing trend, and there is a good linear relationship between the time delay and the length of the ice.By monitoring the arrival time of the B1 mode wave packet, the length of the ice layer on the iced aluminum plate can be better identified.3) As the density, Poisson's ratio, and Young's modulus of ice decrease, the attenuation of the voltage amplitude of the piezoelectric wafer at the receiving end and the time delay of Lamb wave B1 mode gradually increase.When the mechanical properties of the ice layer are close to that of clear ice, the changes in ice layer density, Poisson's ratio, and Young's modulus have little effect on Lamb wave propagation; When the mechanical properties of the ice layer approach frost ice, the changes in ice density, Poisson's ratio, and Young's modulus have a more significant impact on Lamb wave propagation.

Figure 1 .
Figure 1.The schematic diagram of Lamb wave propagation.

Figure 2 .
Figure 2. The Lamb wave propagation model.

Figure 3 .
Figure 3.The time domain waveform of receiving end in aluminium plate.

Figure 4 .
Figure 4.The variation trend of waveform parameters of signal at receiving end with ice thickness.

Figure 5 .
Figure 5.The variation trend of waveform parameters of signal at receiving end with ice length.

Figure 6 .
Figure 6.The variation trend of waveform parameters of signal at receiving end with ice type.
, the theoretical propagation time of B1 mode wave packet between transmitting and receiving piezoelectric plates can be obtained as follows:

Table 1 .
The mechanical parameters of several types of ice.