Research Article  Open Access
Analysis of Acoustic Characteristics of Arbitrary Triangular Prism and Quadrangular Prism Acoustic Cavities
Abstract
This paper proposes a method for the analysis of acoustic modals and steadystate responses of arbitrary triangular prism and quadrangular prism acoustic cavities based on the threedimensional improved Fourier series. First, the geometric models of arbitrary triangular prism and quadrangular prism acoustic cavities are established. To facilitate the calculation, the bottom and top surfaces of the irregular cavity are converted into the unit square domain by a coordinate transformation. Internal sound pressureadmissible functions are constructed, and energy expressions are derived after coordinate transformation based on the threedimensional improved Fourier series. The acoustic modals of arbitrary triangular prism and quadrangular prism acoustic cavities are obtained by the Rayleigh–Ritz technique. At the same time, a point sound source excitation is introduced into the cavity to further study the steadystate responses of prismatic acoustic cavities with different acoustic impedance boundary conditions. The reliability and universality of the method are verified by comparing with the finite element results. The method and results can provide some references and benchmarks for future research and application.
1. Introduction
In recent years, prismatic acoustic cavities with the complex acoustic walls are widely used in practical engineering, such as cabins, buildings, and automobiles. With the development of society and the improvement of people’s requirements for life quality, people’s requirements for vibration and noise control are also increasingly higher. At present, the research studies on this kind of acoustic field are a focus of topic. However, most research is carried out on the rectangular acoustic cavity, and there are still few studies on irregular acoustic cavities. It is difficult to meet the needs of complex acoustic space design and noise research in actual production and life. Therefore, it is of great theoretical significance and practical value to deeply study the acoustic modals and steadystate responses of arbitrary triangular prism and quadrangular prism acoustic cavities under impedance boundary conditions.
After many theoretical research studies and experiments, people have put forward many accurate and reliable methods for the study of acoustic cavities, for example, the finite element method [1–7], mixed numerical method [8, 9], boundary element method [10–12], statistical energy method [13, 14], and so on.
In practice, dissipative acoustic wall surfaces’ rectangular cavities are a common model, which have been paid much attention by many researchers. According to the firstorder shear deformation theory, Zhang et al. [15] studied vibration characteristics of a coupled system consisting of rectangular laminated plates under elastic boundary conditions and cavities under impedance walls. Franzoni and Dowell [16] put forward a modal coupling theory for solving acoustic cavity problems with acoustic absorbing walls. Pan [17] demonstrated that the method (modal analysis) used to describe the acoustic characteristics of an acoustic cavity with the sound absorption boundary is wrong when predicting the sound intensity distribution. Pan [18] further studied the convergence of rigid wall modal models with or without modal coupling in sound intensity domain predictions. On this basis, Pan [19] discussed a new method for sound pressure expansion using the extended modal shape function as the basic function of the sound pressure mode expansion. Martin and Bodrero [20] proposed a method to solve the position optimization problem of acoustic wall impedance with rigid walls. Their goal was to reduce the sound pressure level generated by the internal velocity source of acoustic cavities. Bistafa and Morrissey [21] studied two numerical methods for solving acoustic eigenvalues of rectangular acoustic cavities with arbitrary impedance walls. Naka et al. [22] studied the modal and acoustic response of rectangular acoustic cavities with arbitrary acoustic impedance boundaries by the interval Newton’s generalized dichotomy.
On the basis of the study of rectangular cavities, people gradually pay attention to irregular quasirectangular cavities. Sum and Pan [23] obtained shapes and resonance frequencies of trapezoidal cavities with rigid walls by the coupling of the rectangular cavity. They developed a method for identifying trapezoidal cavity modals. Chen et al. [24] proposed a general analysis method for the inherent characteristics and acoustic vibration characteristics of rectangular plates with arbitrary constraints supported by irregular acoustic cavities. Chen et al. [25] also proposed a domain decomposition method to predict the acoustic characteristics of an arbitrary shell composed of any number of subspaces. Missaoui and Cheng [26] put forward an integral modal method for calculating the acoustic characteristics of irregular cavities. The cavity was discretized into a series of subcavities whose sound pressure is decomposed based on regular or irregular modals of boundary cavities. Xie et al. [27] presented a method based on the weak variational principle to study the irregular cavity under arbitrary dip angle and impedance boundaries. Wang et al. [28] introduced the meshless method into the smallscale acoustic space cavity and derived the numerical calculation model for the acoustic characteristics of the smallscale acoustic space cavity with arbitrary shapes.
At the same time, many great achievements have been made in the study of rotary acoustic cavities. Chen et al. [29] put forward a new calculation method combining the Helmholtz internal integral equation to study the acoustic characteristics of circular and rectangular acoustic cavities. Choi et al. [30] proposed a theoretical method for studying natural frequencies and modal shapes of the annular cavity in local heterogeneous media. By solving the homogeneous wave equation in the elliptical cylindrical coordinate system, Hong and Kim [31] obtained the analytical solution of natural frequencies and mode shapes of elliptical cylinder acoustic cavities. Shao and Mechefske [32] proposed an approximate acoustic model of finite length cylindrical pipe for studying the acoustic radiation of MRI scanning. The sound field obtained by the analytical model was similar to that obtained by the boundary element method model. Lee [33] proposed a semianalytical method for solving the cavity eigenvalue problem with multiple elliptic boundaries. They gave the multipole expansion of sound pressure in the form of angular and radial Mathieu functions. Lee [34] proposed a semianalytical method for solving the eigenvalue problem of twodimensional smooth variable boundary cavity. They employed the Bessel function and Hankel function to express the multipole expansion of sound pressure. And on that basis, Lee [35] solved acoustic characteristics of threedimensional cylindrical cavities with multiple elliptical throughholes by a threedimensional semianalytical formula.
In this paper, on the basis of existing research and limitations, the threedimensional improved Fourier series method for analyzing acoustic field characteristics of arbitrary triangular prism and quadrangular prism acoustic cavities is proposed. First, the bottom and top surfaces of the irregular cavity are converted into the unit square domain. A threedimensional improved Fourier series is used to construct acoustic pressureadmissible functions of the cavity and derive energy expressions. The modal characteristics of acoustic cavities are obtained by the Rayleigh–Ritz technique. At the same time, a monopole point sound source is introduced into the cavity to further study the steadystate response of cavities. Compared with the FEM, it is found that this method can accurately calculate acoustic modal characteristics and steadystate responses of prismatic acoustic cavities.
2. Theory and Methods
2.1. Description of the Model
As shown in Figures 1 and 2, acoustic field analysis models of arbitrary triangular prism and quadrangular prism acoustic cavities are established. Figure 1(a) shows the triangular prism cavity model. The bottom surface of the cavity is a straight triangle, and its side lengths are a, b, and c. α and β are two angles. Figure 2(a) shows the quadrangular prism cavity model, in which the bottom surface of the cavity is a straight quadrilateral. The four sides are denoted by a, b, c, and d. The three internal angles are represented by α, β, and γ, respectively. The depth of cavities is expressed by h_{c}. The coordinate system of the whole cavity is established with the bottom surface of the cavity (z = 0) as the reference surface, and its height direction corresponds to the zdirection. Figures 1(b) and 2(b) show the shapes of the acoustic cavity after a coordinate transformation. In this paper, the acoustic field propagation medium inside cavities is air or water. In addition, a point sound source excitation Q is introduced into the acoustic cavity to analyze the steadystate response of acoustic pressure in the acoustic field.
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In engineering application, many acoustic wall conditions will be formed due to the difference of actual conditions. When acoustic pressure propagates in different acoustic walls, different energy dissipation situations will occur. Based on the current research results, some common definitions of acoustic wall conditions are given below:(1)Pressure release acoustic walls (also known as Dirichlet acoustic walls): these acoustic walls do not produce energy dissipation or energy reflection:(2)Rigid acoustic walls (also known as Neumann acoustic walls): specifically, the sound velocity or displacement of wall S_{1} is 0, and the sound wave is completely reflected at the acoustic wall:(3)Impedance acoustic walls: there is a certain impedance relation at the acoustic walls:where P is the sound pressure, the normal orientation of the acoustic wall is denoted by n, j represents the imaginary unit, denotes the mass density, and Z_{i} is the wall impedance.
2.2. Coordinate Transformation
In the present research, the bottom and top surfaces of arbitrary triangular prism and quadrangular prism acoustic cavities are irregular shapes. In order to facilitate the integral calculation, these two surfaces need to be transformed into the unit square by a coordinate transformation. The coordinate transformation does not involve the zaxis direction, so the depth of the cavity remains the same before and after the coordinate transformation. The transformations are shown in Figures 1 and 2.
The equation of coordinate transformation iswhere xy is the coordinate system before the transformation, εη is the coordinate system after the transformation, and x_{i} and y_{i} (i = 1, 2, 3, 4) represent the coordinate value of the ith vertex in the original coordinate system. In addition, since the triangle has only three sides, the third vertex needs to be mapped twice. N_{i} (ε, η) is the coordinate transformation form function:
According to the aforementioned coordinate transformation equation and the chained derivation rule, the transformation of the first derivative between the original domain and the transformation domain is the following:
The specific expression of J iswhere
J^{−1} is the determinant of the Jacobian for the coordinate transformation.
For the sake of brevity, the above equation can be simplified as
2.3. Acoustic PressureAdmissible Functions
In order to eliminate the discontinuity or jump problem under impedance boundary conditions, an improved threedimensional Fourier series is proposed in the new coordinate system to construct the internal acoustic pressureadmissible functions of acoustic cavities under impedance walls:
These acoustic pressureadmissible functions can ensure the firstorder continuous derivability of the arbitrary point in the solution domain of the whole acoustic field, where P represents the acoustic pressure expression inside prismatic acoustic cavities, P_{i} represents the supplementary polynomial, and A_{mnl} is the unknown threedimensional Fourier coefficient matrix. Their specific expressions arewhere λ_{m} = mπ, λ_{n} = nπ, and λ_{l} = lπ/L_{z} and is the unknown Fourier series.
2.4. Energy Expressions
In order to obtain the acoustic characteristics of prismatic acoustic cavities, energy expressions of acoustic cavities should be deduced first according to the acoustic principle, and then acoustic pressure functions should be substituted into energy expressions. Finally, acoustic characteristics and steadystate responses of the acoustic field are solved by the Rayleigh–Ritz method.
The Lagrangian energy expression of the acoustic cavity is shown below:where U_{C} is the total acoustic potential energy of the acoustic field, T_{C} is the total kinetic energy of the acoustic cavity, W_{wall} is the dissipated energy of the impedance walls, and W_{ext} represents the work done by the point sound source. Their specific expressions arewhere ρ represents the density of the acoustic medium inside the acoustic cavity and c represents the propagation speed of the sound wave in the medium.where ω represents the circular frequency of the acoustic cavity and gradp represents the internal acoustic pressure gradient function within the acoustic cavity.
The triangular prism acoustic cavity has five impedance walls, and its dissipation energy can be expressed as
The quadrangular prism acoustic cavity has one more wall than the triangular prism cavity, so the dissipative energy of its impedance wall iswhere Z_{i} is the impedance function of the ith boundary wall surface. When Z_{i} is a pure imaginary number of the infinity, the rigid wall surface can be obtained. When Z_{i} is a pure imaginary number of the infinitesimal, the pressure release acoustic wall surface can be obtained.where Q represents the volumetric velocity amplitude of the point sound source, A is the amplitude of point sound source (kg/s^{2}), δ is the threedimensional Dirac delta function, k is the wave number of the sound wave and k = ω/c, and (ε_{0}, η_{0}, z_{0}) represents the position of the point sound source inside the acoustic cavity under the new coordinate system.
2.5. Solution Procedure
The energy expressions of acoustic cavities are substituted into Lagrangian energy equation (15). Setting the partial derivative of the Lagrangian equation L_{C} with respect to threedimensional unknown Fourier coefficient vectors as zero by the Rayleigh–Ritz method,
The acoustic pressureadmissible functions are substituted into the above formula and written in the matrix form:where K is the stiffness matrix of acoustic cavities, Z is the acoustic pressure dissipation energy matrix caused by impedance walls, M is the mass matrix, and F is the force vector of the internal point sound source. Their specific expressions can be written as
If the point sound source is not placed (F = 0), the free modal characteristics of acoustic cavities can be obtained. But, in this case, the equation contains both the primary terms and the square terms of frequencies at the same time. This is a nonlinear problem, which is not easy to solve directly.
Therefore, the equation is transformed into the following eigenvalue equation by introducing E = ω A_{mnl}:
By solving generalized eigenvalues and eigenvectors of equation (29), the frequency parameters of acoustic cavities and the threedimensional Fourier unknown coefficient A_{mnl} of acoustic pressure functions can be obtained. It is worth noting that generalized eigenvalues and eigenvectors under the condition of impedance walls are complex numbers. At this time, the real part of the generalized eigenvalue corresponds to the natural frequency of the acoustic cavity, while the imaginary part is the attenuation coefficient of the acoustic mode. In other words, the introduction of the impedance wall does not change the modal characteristics of the acoustic field in the acoustic cavity.
3. Numerical Results
A series of calculating examples of arbitrary triangular prism and quadrangular prism acoustic cavities are presented in this section. The accuracy of the method is verified by comparing with FEM. The FEM model is established in ABAQUS v6.14, the element type adopts the threedimensional solid element, and the mesh size is set to 0.02 × 0.02 m. In this section, air and water are selected as the acoustic field mediums. The mass density of air and water are defined as ρ_{air} = 1.21 kg/m^{3} and ρ_{water} = 1000 kg/m^{3}. The speed of sound propagation in air and water are set as c_{air} = 340 m/s and c_{water} = 1480 m/s. In addition, if no special instructions are given, the cavity depth is h_{c} = 0.5 m.
3.1. Convergence Analysis
In Table 1, the convergence and precision of the first eight frequencies of the triangular prism acoustic cavity are studied. Geometric parameters are b/a = 1 and α = 75°. The truncation values of the Fourier series are expressed in terms of M, N, and L, which are used to compare and check the convergence of the solution. The truncated values are from 4 × 4 × 4 to 8 × 8 × 8. It can be seen from the table that the method has great convergence and precision. Under two different mediums, when the truncation value is M × N × L = 8 × 8 × 8, the data have been basically convergent. At the same time, in order to verify the accuracy of the calculation, the results obtained by FEM are also listed in the table. The maximum error between the calculated results and the FEM is not more than 0.032%. And the maximum deviation of calculated results under different truncation values is not more than 0.008%.

Next, in Table 2, the convergence and precision of the first eight frequency parameters of the quadrangular prism acoustic cavity are studied. Geometric parameters are b/a = 0.7, c/a = 0.8, α = 80°, and β = 75°. Under two different mediums, the maximum error between the calculated results and the FEM is not more than 0.039% when the truncation value is M × N × L = 8 × 8 × 8. The maximum deviation of calculated results under different truncation values is not more than 0.004%.

According to the above analysis of convergence and accuracy, the truncated value of sound pressureadmissible functions of the acoustic cavity calculated in this paper is uniformly set as M × N × L = 8 × 8 × 8.
3.2. Acoustic Modal Analysis of Prismatic Acoustic Cavities
After the analysis of convergence, more modal characteristics of arbitrary triangular prism and quadrangular prism acoustic cavities are studied.
First of all, Table 3 shows the comparison results of the first eight natural frequencies of the triangular prism acoustic cavities with FEM under different shapes. The bottom shape of acoustic cavities contains the isosceles triangle (b/a = 1, α = 75°), the equilateral triangle (b/a = 1, α = 60°), the right triangle (b/a = 2, α = 90°), the obtuse triangle (b/a = 2, α = 120°), and the acute triangle (b/a = 2, α = 75°). As can be seen from the results in the table, the calculated results of this method are matched well with the FEM. This method can accurately calculate natural frequencies of triangular prism acoustic cavities with different geometrical shapes. Figure 3 shows the first three modals of acoustic pressure distribution of the positive triangular prism acoustic cavity when the medium is air. The accuracy of this method is proved again by comparison.
 
^{a}Results from finite element software ABAQUS. 
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Table 4 shows the comparison results of the first eight natural frequencies of the quadrangular prism acoustic cavities with different shapes. The bottom shape contains the quadrilateral (b/a = 0.7, c/a = 0.8, α = 70°, β = 75°), the square (b/a = 1, c/a = 1, α = 90°, β = 90°), the rectangle (d/a = 2, c/a = 2, α = 90°, β = 90°), the rhombus (b/a = 1, c/a = 1, α = 60°, β = 120°), and the trapezoid (b/a = 1, c/a = 1.064, α = 70°, β = 90°). Table 4 illustrates the accuracy of this method in calculating quadrangular prism acoustic cavities with different geometric shapes by comparing with the FEM. Figure 4 shows the first three modals of acoustic pressure distribution of the quadrangular prism acoustic cavity when the medium is water.
 
^{a}Results from finite element software ABAQUS. 
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For prismatic cavities, cavity depth is also a very important parameter. Tables 5 and 6 list the first eight frequency parameters of the triangular prism (b/a = 1, α = 75°) and the quadrangular prism (b/a = 0.7, c/a = 0.8, α = 80°, β = 75°) acoustic cavities at different cavity depths. As can be seen from two tables, the variation trend of frequency parameters is basically the same in two mediums. The first three natural frequency parameters do not change significantly when the depth is 0.2–0.4 m. Subsequently, natural frequency parameters of cavities decrease with the increase of the depth, and the higherorder frequency parameters are sensitive to the change of the depth.


On the basis of above results, the effects of geometric parameters on the natural frequencies of acoustic cavities are further studied. Figures 5 and 6 show the variation of the first three natural frequency parameters of the triangular prism and the quadrangular prism acoustic cavities under different angles (α). The bottom surface of the triangular prism is an isosceles triangle. The bottom surface of the quadrangular prism is a rhombus. It can be seen from the figure that in the two mediums, frequency change trends of cavities are similar, but the amplitude is different. For the triangular prism acoustic cavity, the firstorder frequency first increases slowly with the increase of α and reaches the maximum at 60° and then decreases with the increase of the angle. The secondorder frequency first decreases with the increase of α and reaches the minimum at 60° and then increases with the increase of α and decreases again after 90°. The thirdorder frequency first decreases with the increase of α and remains unchanged between 40° and 110° and finally decreases again. For the rhombus prism acoustic cavity, the firstorder frequency increases with the increase of α. The secondorder frequency first increases with the increase of α and reaches the maximum at 60° and then decreases suddenly. The thirdorder frequency first increases with the increase of α and reaches the maximum at 40°, then decreases with the increase of α and reaches the minimum at 60°, and then increases slowly again. At 60°, the secondorder frequency and the thirdorder frequency are equal. We can also find that when the angle is equal to 60°, there will be a phenomenon in which the two natural frequencies are equal. Such phenomenon of the same frequency and different modes (Figure 3) due to the symmetry of the structure is called the doublemode phenomenon.
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Figures 7 and 8 show the variation of the first three natural frequencies of the triangular prism and the quadrangular prism acoustic cavities under different aspect ratios (b/a). The bottom surface of the triangular prism is a right triangle. The bottom surface of the quadrangular prism is a rectangular. In two different mediums, variation trends of frequencies are very close. But, the frequency parameters of the water cavity are much higher than those of the air cavity. For the triangular acoustic cavity, the first three frequencies all decrease with the increase of b/a. For the rectangular acoustic cavity, the firstorder frequency decreases with the increase of b/a. The secondorder frequency remains constant between 1.0 and 2.0 and then gradually decreases. The thirdorder frequency first decreases with the increase of b/a, then remains the same between 2.0 and 3.0, and finally decreases again.
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3.3. SteadyState Response Analysis of Prismatic Acoustic Cavities
The steadystate acoustic pressure responses of prismatic cavities under an internal point sound source are studied in this section. Figures 9 and 10 show acoustic pressure response curves of the triangular prism and the quadrangular prism cavities with rigid walls under an internal unit point sound source excitation. The bottom surface area unit is equilateral triangle and isosceles trapezoid (b/a = 1, c/a = 1, α = β = 80°). For the triangular prism acoustic cavity, the loading point is (0.5, 0.5, 0.2). Observation point 1 is (0.82, 0.219, 0.2), and observation point 2 is (0.5, 0.566, 0.4). For the quadrangular prism acoustic cavity, the loading point is (0.5, 0.492, 0.25). Observation point 1 is (0.21, 0.2, 0.1), and observation point 2 is (0.8, 0.59, 0.4). It can be seen that the difference in the physical properties of the medium does not affect the variation trend of the sound pressure response by comparing the response curve trend of two different mediums in a certain frequency band. But, the response amplitude of the water cavity is slightly larger than that of the air cavity. At the same time, the sound pressure response curves calculated by this method are consistent with the FEM basically, which proves the correctness of the acoustic field analysis model again in this paper. The reason for the difference is mainly because the error between the paper and the finite element results is greater at high frequencies. At the same time, the damping is not introduced when calculating the response, so the simulation error for values of resonance peak is larger.
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Figures 11 and 12 show acoustic pressure response curves of the triangular prism and the quadrangular prism acoustic cavities with impedance walls under an internal unit point sound source excitation. The impedance value is Z = ρc (200–10j). We set the bottom surface (z = 0) and the top surface (z = h_{c}) of prismatic cavities as impedance walls and the other surfaces as rigid walls. For the triangular cavity, the bottom surface is an isosceles right triangle. The loading point is (0.2, 0.453, 0.1), and the observation point is (0.4, 0.45, 0.3). For the quadrangular acoustic cavity, the bottom surface is a rhombus (α = 60°, β = 120°). The loading point is (0.7, 0.432, 0.25), and the observation point is (0.4, 0.173, 0.15). It can be seen from the figure that the trend of the acoustic pressure response curves calculated by this method in this paper and the FEM is still basically consistent under impedance walls. This indicates that prismatic acoustic cavity analysis models established in this paper can also accurately predict the steadystate response of the acoustic field in prismatic acoustic cavities under impedance walls.
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On the basis of correctness, this section studies the effect of impedance walls on the steadystate response of the prismatic acoustic cavity. Figure 13 shows the acoustic pressure response of prismatic acoustic cavities with various impedance walls under the unit point sound source excitation. The impedance value is Z = ρc (200–10j). Since the cavity medium does not obviously affect the sound pressure response curve, the cavity medium is set as air uniformly. For the triangular cavity, the bottom surface is an isosceles triangle (α = 120°). The loading point is (0.3, 0.15, 0.2), and the observation point is (0.5, 0.2, 0.3). For the quadrangular acoustic cavity, the bottom surface is a rhombus (α = 55°, β = 125°). The loading point is (0.45, 0.35, 0.15), and the observation point is (0.8, 0.3, 0.35). It can be seen from figures that the introduction of impedance walls can suppress the sound pressure level of the steadystate response of acoustic cavities, and the inhibition effect increases with the number of impedance walls. However, the introduction of impedance walls does not affect the trend of sound pressure response.
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4. Conclusions
In this paper, an acoustic field characteristics analysis method of arbitrary triangular prism and quadrangular prism acoustic cavities is firstly presented based on the threedimensional improved Fourier series. The modal and response results of the prism acoustic cavity with various impendence walls are obtained by transforming the xyz coordinate system into the εη one. After verifying the correctness of this method, the influence of different geometric parameters on the acoustic field characteristics is analyzed.
The examples and numerical results in this study indicate some implications and conclusions:(1)It is found that this method can calculate acoustic characteristics and steadystate responses of arbitrary triangular prism and quadrangular prism acoustic cavities exactly. This method can be applied to the spatial acoustic characteristics analysis and noise control research in this study field.(2)From the theoretical part of the research process, this method does not involve any complicated theories, equations, or programs. This method has great reference values for the study of acoustic characteristics and steadystate responses of similar structures.(3)From the numerical results, it can be seen that geometric parameters, medium properties, and boundary conditions all have effects on acoustic characteristics and steadystate responses of prismatic acoustic cavities. Therefore, special attention should be paid to these factors when designing and applying cavity structures.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no potential conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (grant no. 51679056) and Natural Science Foundation of Heilongjiang Province of China (E2016024).
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Copyright © 2019 Dongyan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.