A numerical method for fast and accurate determination of two-dimensional dispersion relation of elastic waves in acoustic resonators

Abstract

The present application provides a kind of fast and accurate numerical method for determining the two-dimensional dispersion relation of elastic wave in acoustic resonator, the two-dimensional dispersion relation is characterized by two-dimensional implicit complex algebraic equation f (k, ω) =0, the zero point of two-dimensional implicit complex algebraic equation f (k, ω) =0 is solved, and the two-dimensional dispersion relation of elastic wave in acoustic resonator is formed by zero point in frequency-wave number plane;The present application discloses a kind of fast and accurate numerical method for determining the two-dimensional dispersion relation of elastic wave in acoustic resonator, so as to determine the quantitative relationship between the frequency and propagation wave number of elastic wave when propagating in resonator structure, after obtaining the dispersion relation, the mode shape and resonance frequency of acoustic resonator can be further determined, so as to determine the mode and frequency selection of acoustic resonator when actually working, provide reference for the structure design and application of acoustic device.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of acoustic wave sensor, in particular, to a numerical method for quickly and accurately determining two-dimensional dispersion relation of elastic wave in acoustic wave resonator. BACKGROUND

[0002] Acoustic wave resonator is a kind of passive electronic device which utilizes the propagation and reflection characteristics of elastic wave in solid medium, and is suitable for velocity sensor, temperature sensor, mechanical sensor, and has been widely used in the fields of radio frequency communication, nondestructive testing, medical ultrasound, etc. The working principle of acoustic wave resonator is to utilize the piezoelectric effect of material, to excite bulk acoustic wave or surface acoustic wave in resonator structure by applying alternating current on the surface electrode of material, so that the acoustic wave resonator vibrates at a specific mode and frequency. Mastering the propagation characteristics of elastic wave in resonator structure is the basis for characterizing the working performance of acoustic wave resonator, and the propagation characteristics of elastic wave can be characterized by its dispersion relation, i.e. the relation between the frequency of elastic wave and its propagation wave number. By determining the dispersion relation of elastic wave in acoustic wave resonator, the resonant frequency, mode shape, acoustic wave velocity and other key performance indicators of acoustic wave resonator can be determined, thereby providing guidance for the design of acoustic wave resonator and the selection of working mode, frequency and wave velocity.

[0003] Generally speaking, the dispersion relation of elastic wave in elastic solid can be characterized by explicit algebraic relation, and the zero point of explicit algebraic relation can be determined by conventional numerical calculation method. However, due to the piezoelectric material force-electric coupling effect and the complexity of device structure, the dispersion relation of acoustic wave resonator cannot be characterized by explicit algebraic relation, and is often represented as implicit complex algebraic equation, so it is difficult to determine the zero point of equation by conventional numerical calculation method. Therefore, it is a difficult problem in the field of acoustic wave device research to develop an effective numerical method to quickly solve the implicit complex algebraic equation and determine the dispersion relation of elastic wave in acoustic wave resonator.

[0004] At present, some researchers have also invented some technical methods to determine the dispersion relation in acoustic wave device, for example, Nanjing University of Aeronautics and Astronautics has applied for an invention patent with publication number

CN105590025A

[0005] Although the above-mentioned numerical method can solve the dispersion curve in the acoustic wave sensor determined by the complex dispersion equation, practice shows that the calculation efficiency of the modulus convergence numerical algorithm is low, mainly reflected in: 1) When the implicit complex algebraic equation corresponding to the dispersion curve is solved by the technical method, whether there is a zero point in the search interval is judged by comparing the modulus of the complex dispersion equation in the search interval, so multiple sub-interval divisions of the search interval are needed, and the modulus of the complex dispersion equation at the end points of each sub-interval is calculated, so that the actual calculation amount required by each search interval is significantly increased; after determining that there may be a zero point in a step interval, the modulus of the complex dispersion equation is compared to converge, which also needs to divide a large number of sub-intervals for the step interval, and the modulus of the complex dispersion equation corresponding to each sub-step point is iteratively calculated, which greatly increases the actual calculation time required by the technical method, resulting in low efficiency of the technical method; 2) The number of sub-intervals divided by the search interval also affects the solving accuracy of the implicit complex algebraic equation, thereby affecting the accuracy of the determined dispersion relationship.

[0006] In summary, there is still a need to develop a numerical method to solve the problem that the elastic wave dispersion relationship in the acoustic wave resonator cannot be quickly and accurately solved, and to master the mode shape and resonance frequency of the working mode of the acoustic wave resonator. SUMMARY

[0007] In order to solve the problem of low efficiency of solving the elastic wave dispersion relationship in the existing acoustic wave resonator, the present application discloses a numerical method for quickly and accurately determining the two-dimensional dispersion relationship of elastic waves in an acoustic wave resonator.

[0008] The present application provides a numerical method for quickly and accurately determining the two-dimensional dispersion relationship of elastic waves in an acoustic wave resonator, the two-dimensional dispersion relationship is characterized by a two-dimensional implicit complex algebraic equation f(k, ω) = 0, the zero point of the two-dimensional implicit complex algebraic equation f(k, ω) = 0 is solved, and the zero point is used to form the two-dimensional dispersion relationship of the elastic wave in the acoustic wave resonator on the frequency-wave number plane, specifically comprising:

[0009] Step S1, determining the possible zero point interval of the two-dimensional implicit complex algebraic equation f(k, ω) = 0 based on the bisection method;

[0010] Step S2, iteratively converging the possible zero point interval based on the bisection method to determine the possible zero point;

[0011] Step S3, comparing the modulus of the complex dispersion equation function of the possible zero point and the left end point of the possible zero point interval in step S1 to determine whether it is a real zero point.

[0012] In some embodiments, in step S1, specifically comprising:

[0013] The root of the two-dimensional implicit complex algebraic equation exists in the plane space (k, ω), where the propagation wave number is a pure real number or a pure imaginary number;

[0014] The solution of the two-dimensional implicit complex algebraic equation is a curve in the plane space (k, ω), and a straight line with a fixed frequency or wave number can intersect the solution curve of the two-dimensional implicit complex algebraic equation at one point. The line element is used to search for the interval of the straight line with the fixed frequency or wave number.

[0015] When the line element is scanned for the root with the fixed frequency ω0, the function f(k, ω0) is expressed as f(k, ω0) = a + bi, where a and b represent the real part and the imaginary part of the function value respectively, and i represents an imaginary unit.

[0016] Let g(k, ω0) = a + b, and the zero point of f(k, ω0) = a + bi = 0 is solved, which is equivalent to solving the zero point of g(k, ω0) = a + b = 0.

[0017] The search interval is divided by selecting a step size, and the function value of the function g(k, ω0) = a + b at the two end points of each search interval is calculated. If the function values at the two end points are of opposite signs, the search interval has a possible zero point, which is referred to as a possible zero point interval. If the function values of the function g(k, ω0) = a + b at the two end points of the search interval are of the same sign, the function values at the two end points of the next search interval are calculated until the function values at the two end points of the search interval are of opposite signs, and the search interval is determined as the possible zero point interval.

[0018] In some embodiments, in step S2, specifically comprising:

[0019] Step S21, after the possible zero point interval is determined, the values of the function g(k, ω0) = a + b at the middle point and the left end point of the possible zero point interval are calculated.

[0020] Step S22, if the function value at the left end point and the function value at the middle point are of opposite signs, the left end point and the middle point are taken as a new iteration interval; if the function value at the left end point and the function value at the middle point are of the same sign, the middle point and the right end point of the original interval are taken as a new iteration interval.

[0021] Step S23, the function values at the middle point and the left end point of the new iteration interval are calculated, and the next new iteration interval is determined by repeating step S22; the above process is repeated until the difference between the middle point and the left end point of the new iteration interval is less than 10 -16 , which indicates that the iteration converges, and the middle point of the final iteration interval is the possible zero point of the function g(k, ω0) = a + b.

[0022] In some embodiments, in step S3, specifically comprising:

[0023] The possible zero points of g(k, ω0) = a + b = 0 include false zero points and real zero points;

[0024] The function f(k, ω0) at the left end point of the search interval is compared l The modulus value of the function f(k, ω0) at the left end point of the search interval is compared with the modulus value of the function f(k, ω0) at the possible zero point.

[0025] If the ratio tends to infinity, the possible zero point is a real zero point.

[0026] If the ratio is less than a real number M, the possible zero point is a false zero point.

[0027] The present application has the following beneficial effects:

[0028] The present application discloses a numerical method for quickly and accurately determining the two-dimensional dispersion relation of elastic waves in an acoustic resonator, thereby determining the quantitative relation between the frequency and the propagation wave number of the elastic waves when the elastic waves propagate in the resonator structure. After the dispersion relation is obtained, the modal shape and the resonant frequency of the acoustic resonator can be further determined, thereby determining the mode and the frequency selection of the acoustic resonator in actual operation, providing a reference for the structure design and application of the acoustic device. BRIEF DESCRIPTION OF DRAWINGS

[0029] Figure 1 is a flow chart of the method of the present application;

[0030] Figure 2 is a schematic diagram of the linear unit scanning of the original dispersion equation with a fixed frequency and a wave number, and a schematic diagram of searching for zero points of the original function f(k, ω0) equivalent to linear algebraic equations g(k, ω0) provided by the present application;

[0031] Figure 3 is a zero point criterion diagram for judging real and false zero points of the present application;

[0032] Figure 4 is a dispersion curve of the thickness direction mode of Lamb waves in a film bulk acoustic resonator obtained by the present application and patent CN105590025A, which is symmetrical;

[0033] Figure 5 is a dispersion curve of the thickness direction mode of Lamb waves in a film bulk acoustic resonator obtained by the present application and patent CN105590025A, which is anti-symmetrical. DETAILED DESCRIPTION

[0034] Exemplary embodiments of the present disclosure are described herein below with reference to the accompanying drawings, in which various details are set forth to facilitate an understanding of the present disclosure. It should be appreciated that various embodiments of the present disclosure can be practiced with variation of details without departing from the scope and spirit of the present disclosure. Also, for the purpose of clarity and a concise description, descriptions of well-known functions and constructions can be omitted.

[0035] Referring to Figure 1 The embodiments of the present application disclose a numerical method for quickly and accurately determining the two-dimensional dispersion relation of elastic waves in an acoustic resonator, thereby determining the quantitative relation between the frequency and the propagation wave number of the elastic waves when propagating in the resonator structure, and the dispersion equation is represented by an implicit complex algebraic equation.

[0036] The numerical method provided by the present application can accurately solve the zero points of the implicit complex algebraic equation, and these zero points can form the two-dimensional dispersion relation of the elastic waves in the acoustic resonator in the frequency-wave number plane.

[0037] The two-dimensional dispersion relation of the acoustic resonator is represented by the implicit complex algebraic equation f(k, ω) = 0, so solving the two-dimensional dispersion relation of the elastic waves propagating in the acoustic resonator structure is to solve the implicit complex algebraic equation.

[0038] The present application discloses a numerical method for efficiently solving the two-dimensional dispersion relation of elastic waves in an acoustic resonator based on the bisection method, and the specific implementation manner comprises the following steps:

[0039] S1, determining a possible zero point interval of the two-dimensional implicit complex algebraic equation f(k, ω) = 0 based on the bisection method;

[0040] S2, iteratively converging the possible zero point interval based on the bisection method to determine a possible zero point;

[0041] S3, comparing the modulus values of the complex dispersion equation functions of the possible zero point and the left end point of the possible zero point interval in step 1 to determine whether it is a real zero point.

[0042] In the present embodiment, the root of the two-dimensional implicit complex algebraic equation exists in the plane space (k, ω), at this time the propagation wave number is a pure real number or a pure imaginary number, and the solution of the two-dimensional implicit complex algebraic equation is a curve on the plane space, so a straight line with a fixed frequency or wave number can intersect the solution curve of the two-dimensional implicit complex algebraic equation at a point, and then the straight line with a fixed frequency or wave number can be searched by the line unit.

[0043] Figure 2 (a) shows a schematic diagram of the line unit with a fixed frequency intersecting the zero point of the two-dimensional implicit complex algebraic equation f(k, ω) = 0.

[0044] For the complex algebraic equation, when the linear unit scanning search root is carried out at the fixed frequency ω0, the function f(k, ω0) can be expressed as f(k, ω0) = a + bi, wherein a and b respectively represent the real part and the imaginary part of the function value, and i represents an imaginary unit.

[0045] In order to improve the solving efficiency of the numerical method, the present application makes g(k, ω0) = a + b, and then the zero point of f(k, ω0) = a + bi = 0 can be solved, which is equivalent to solving the zero point of g(k, ω0) = a + b = 0.

[0046] The specific principle of the equivalence is that because the function f(k, ω0) = a + bi is a continuous function, g(k, ω0) = a + b is also a continuous function, when f(k, ω0) = a + bi = 0, a and b are both zero, therefore, g(k, ω0) = a + b is also equal to zero, but the case of g(k, ω0) = a + b = 0 also includes that a and b are opposite numbers not equal to zero; therefore, it is necessary to finally remove such false zero points, and the zero points of g(k, ω0) = a + b = 0 obtained are the same as the zero points of f(k, ω0) = a + bi = 0, and a zero point equivalence schematic diagram is shown as Figure 2 .

[0047] For the fixed frequency straight line, a suitable step is selected to divide the search interval, and the present application adopts the bisection method to judge the possible zero point interval, as shown in Figure 2 (b).

[0048] Specifically, the function value of the function g(k, ω0) = a + b at the two end points of each search interval is calculated, if the function values at the two end points are of opposite signs, the search interval has a possible zero point, which is called a possible zero point interval.

[0049] If the function values of the function g(k, ω0) = a + b at the two end points of the search interval are of the same sign, the function values at the two end points of the next search interval are calculated, until the function values at the two end points of the search interval are of opposite signs, and the search interval is determined as the possible zero point interval.

[0050] After the possible zero point interval is determined, the bisection method is adopted to converge to the possible zero point.

[0051] Among them, the values of the function g(k, ω0) = a + b at the middle point and the left end point of the possible zero point interval are calculated, if the function value at the left end point and the function value at the middle point are of opposite signs, the left end point and the middle point are taken as a new iteration interval; if the function value at the left end point and the function value at the middle point are of the same sign, the middle point and the right end point of the original interval are taken as a new iteration interval.

[0052] Further, the function values of the middle point and the left end point of the new iteration interval are calculated, and the next new iteration interval is determined according to the above judging rule. -16 , indicating that the iteration convergence ends, and the middle point of the last obtained iteration interval is the possible zero point of the function g(k, ω0) = a + b.

[0053] The present application equates the zero point of the original complex dispersion equation f(k, ω0) = a + bi = 0 to the zero point of the equation g(k, ω0) = a + b = 0, but the zero point of the equation g(k, ω0) = a + b = 0 can be the opposite number of a and b.

[0054] In the fixed frequency scanning root search, at the false zero point (k0, ω0), a and b are c and -c respectively, and c is not equal to zero, so at the false zero point, the function value of the original complex dispersion equation is f(k0, ω0) = c - ci, and the modulus value is

[0055] Since the function f(k, ω) is continuous, at the false zero point (k0, ω0), even if it is constantly converging to the false zero point, the modulus value of the function value of the function f(k, ω) must tend to Therefore, when comparing the modulus value of the function f(k0, ω0) at the false zero point with the modulus value of the function f(k l , ω0) at the left end point of the interval, due to the continuity of the function, there must be a real number M, such that the ratio of the modulus value of the function value at the left end point to the modulus value of the function value at the false zero point is less than M, as shown in (a); Figure 3

[0056] And for the true zero point (k0, ω0), as it is constantly converging to the true zero point, a and b tend to zero more and more, so the modulus value of the function value of the original dispersion equation at the true zero point also tends to zero more and more, so the ratio of the modulus value of the function f(k l , ω0) at the left end point of the search interval to the modulus value of the function at the true zero point must tend to infinity, as shown in (b); Figure 3

[0057] Taking the difference as the zero point judging criterion can filter out the true zero point.

[0058] The present application takes the Lamb wave propagation in a film bulk acoustic resonator as an example to illustrate the application principle of the present application and the effect comparison with the prior art scheme:

[0059] The dispersion relation of the Lamb wave propagation in the film bulk acoustic resonator is represented by a two-dimensional implicit complex algebraic equation, i.e. there is no explicit expression about the frequency ω and the wave number k;

[0060] ​​The present application uses a function f(k, omega) to represent the two-dimensional implicit complex algebraic equation, and the numerical method for solving the two-dimensional complex dispersion equation based on the bisection method of the present application is used to obtain a dispersion relation of the thickness direction mode symmetry as shown in Figure 4 (a);

[0061] The corresponding dispersion relation obtained by using the numerical method disclosed in the prior application

CN105590025A

[0062] Figure 5 (a) and Figure 5 (b) are respectively the dispersion relation diagrams of the thickness direction mode antisymmetry of the film bulk acoustic resonator obtained by using the bisection method and the modulus value convergence numerical method;

[0063] Table 1

[0064] Calculation time Bisection numerical method Modular convergence numerical method Symmetric in thickness direction mode 65 seconds (1.08 minutes) 530 seconds (8.8 minutes) Anti-symmetric in thickness direction mode 66 seconds (1.1 minutes) 504 seconds (8.4 minutes)

[0065] Table 1 is a comparison of the time for calculating the dispersion curves of different modes of elastic waves in the film bulk acoustic resonator according to the present application and the patent CN105590025A;

[0066] Among them, the computer is configured as CPU: Inter i9-14900KF 3.2GHz, memory: 64G.

[0067] The above specific embodiments do not constitute a limitation on the protection scope of the present disclosure. Those skilled in the art should understand that various modifications, combinations, sub-combinations and substitutions can be made according to design requirements and other factors. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present disclosure should be included in the protection scope of the present disclosure.

Claims

1. A numerical method for fast and accurate determination of the two-dimensional dispersion relation of elastic waves in acoustic resonators, characterized in that, The two-dimensional dispersion relation is represented by a two-dimensional implicit complex algebraic equation f(k, ω)=0, and a zero point of the two-dimensional implicit complex algebraic equation f(k, ω)=0 is solved, and the two-dimensional dispersion relation of the elastic wave in the acoustic resonator is formed by the zero point in the frequency-wave number plane, and specifically comprises: In step S1, a possible zero point interval of the two-dimensional implicit complex algebraic equation f(k, ω)=0 is determined based on the bisection method; In step S2, the possible zero point interval is iteratively converged based on the bisection method to determine the possible zero point; In step S3, the modulus values of the complex dispersion equation functions of the possible zero point and the left end point of the possible zero point interval in step S1 are compared to determine whether it is a real zero point; In step S1, specifically comprising: The root of the two-dimensional implicit complex algebraic equation exists in the plane space (k, ω), at this time the propagation wave number is a pure real number or a pure imaginary number; The solution of the two-dimensional implicit complex algebraic equation is a curve in the plane space (k, ω), a straight line of fixed frequency or wave number can intersect the solution curve of the two-dimensional implicit complex algebraic equation at one point, and the straight line of fixed frequency or wave number is searched by interval by the line unit; When the fixed frequency is ω0 and the line unit is scanned to search for the root, the function f(k, ω0) is represented as f(k, ω0)=a+bi, wherein a and b represent the real part and the imaginary part of the function value respectively, and i represents the imaginary unit; Let g(k, ω0)=a+b, solve the zero point of f(k, ω0)=a+bi=0, which is equivalent to solving the zero point of g(k, ω0)=a+b=0; Select a step to divide the search interval, and calculate the function values of the function g(k, ω0)=a+b at both ends of each search interval, if the function values at both ends are of opposite signs, then the search interval has a possible zero point, which is called a possible zero point interval; if the function values of the function g(k, ω0)=a+b at both ends of the search interval are of the same sign, then the function values at both ends of the next search interval are calculated until the function values at both ends of the search interval are of opposite signs, and the search interval is determined as the possible zero point interval; In step S2, specifically comprising: In step S21, after the possible zero point interval is determined, the values of the function g(k, ω0)=a+b at the middle point and the left end point of the possible zero point interval are calculated; In step S22, if the function value at the left end point and the function value at the middle point are of opposite signs, then the left end point and the middle point are taken as a new iteration interval; if the function value at the left end point and the function value at the middle point are of the same sign, then the middle point and the right end point of the original interval are taken as a new iteration interval; Step S23, calculate the function value of the middle point and the left end point of the new iteration interval, repeat step S22 to determine the next new iteration interval; repeat the above process until the difference between the middle point and the left end point of the new iteration interval is less than 10 -16 , indicating that the iteration converges, and the middle point of the last obtained iteration interval is the possible zero point of the function g(k, ω0) = a + b; In step S3, specifically comprising: The possible zero point of g(k, ω0)=a+b=0 includes a false zero point and a real zero point; The function f(k) that compares the left endpoints of the search interval l The ratio of the magnitude of f(k0,ω0) to the magnitude of the function f(k0,ω0) at possible zeros; If the ratio tends to infinity, then the possible zero point is a real zero point; If the ratio is less than a real number M, then the possible zero point is a false zero point.

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