Abstract

This study investigates the dispersive properties of ridge waves that travel circumferentially around piezoelectric circular ridge waveguides and investigates their resonant modes. Based on the variable separation method and Hamilton's principle, the displacement of ridge waveguides is represented as the product of a cross-sectional coordinate-dependent function and the propagator along the circumference of a circular ridge waveguide. The dispersion curves of the flexural waves and resonant frequencies corresponding to ridge waveguides are solved numerically by applying the bidimensional finite element method (Bi-d FEM) and using the three-dimensional (3D) ansys package. The estimated impedance curves are compared with the predicted dispersion curves of waves from ridge waveguides to validate the proposed numerical approach. The elastic constants of the circular piezoelectric ridge waveguide are determined through an inverse scheme that is based on the modified simplex method. The numerical and experimental results show that by using the modified simplex method to inverse calculate the elastic constants and geometric parameters of the piezoelectric circular ridge waveguides, a good degree of accuracy and sensitivity can be achieved.

1 Introduction

Compared with general traditional rectangular waveguides, under the same internal scale, ridge waveguides have a longer cutoff wavelength and a lower characteristic impedance. At the same time, ridge waveguides also have a wider free bandwidth under high modal interference. Because of these advantages, ridge waveguides have been used as matching or transition elements in waveguides-to-coaxial junctions, as filter elements, and as components for other special purposes in transmission lines in systems requiring a wide free range in the fundamental mode.

In 1957, Chen [1] proposed an algebraic expression that constitutes an approximation to Cohn's transcendental equation for the determination of the dominant-mode cutoff wavelength of ridge waveguides.

In 1972, Lagasse [2] and Maradudin et al. [3] utilized numerical calculation to prove that a ridge tip possessed waveguide functions within a specific range of angles. Moreover, they proved that the energy of an antisymmetrical flexural guided wave that is propagated using a ridge tip is primarily limited to a distance extending the equivalent of one wavelength from the ridge tip. In this case, the wave's phase velocity is lower than the velocity of a Rayleigh wave with the same wavelength.

In 1973, Lagasse et al. [4] stated that single-mode flexural vibrations are limited by the characteristics of ridges and produce traveling waves that propagate along the ridges. In this case, an elliptical motion curve of the flexural waves causes the sliding block above the rectangular structure to move in the opposite direction of the propagating waves.

In 1973, Lagasse [5] applied the variation principle and finite element method to examine the wave propagation behavior of linear acoustic waveguides. As the method proposed by Lagasse could be applied to various geometric cross sections, the method became a crucial tool for studying acoustic waveguides.

In 1991, Omar and Schunemann [6] developed the generalized spectral-domain technique to analyze rectangular waveguides with rectangular and circular metal inserts. In 1996, Balaji and Vahldieck [7] presented a radial mode-matching analysis to calculate a rigorous TE and TM mode propagation in single-, double-, triple-, and quadruple-ridged circular waveguide structures.

In 1997, Fontgalland et al. [8] presented a computation of the cutoff wave numbers using the boundary element method for the analysis of ridged rectangular waveguides and ridged circular waveguides. In 1997, Wang et al. [9] presented a rigorous analysis combining the method of orthogonal expansion with Galerkin method for higher-order eigenmodes in a circular–rectangular waveguide.

In 1998, Balaji and Vahldieck [10] describe a mode-matching algorithm for S-parameter computation of circular-ridged waveguide discontinuities. In 1999, Bornemann et al. [11] proposed a fast and efficient radial mode-matching technique to analyze and design components in circular ridge waveguide technology.

In 2000, Rong and Zaki [12] analyzed the generalized ridge waveguides using the mode-matching method. In 2004, Yang et al. [13] reported a high-performance 980-nm ridge waveguide quantum-well laser with an extremely low vertical beam divergence of 13 deg.

In 2005, Ruiz-Cruz et al. [14] proposed a canonical ridge waveguide filter with high selectivity, wide spurious free stop band, and compact size. In 2005, Tominaga et al. [15] proposed an ultrasonic linear motor using ridge-mode traveling waves.

In 2010, Pu et al. [16] presented an ultralow-loss coupler for interfacing a silicon-on-insulator ridge waveguide and a single-mode fiber in both polarizations. In 2018, Abdelaal et al. [17] utilized a nonstandard dual-mode, double-ridge waveguide to design a compact orthomode transducer. In 2018, Yu [18] proposed an optimal structural design of a circular cylindrical ridge wave ultrasonic motor.

In 2019, Bruske et al. [19] reported on efficient neodymium-doped titanium in-diffused ridge waveguide lasers in x-cut congruent LiNbO3 under excitation at 814 nm. In 2019, Nasr and Kishk [20] designed an ultrawideband vertical coaxial-to-single-ridge waveguide transition to cover Ku-, K-, and Ka-bands simultaneously.

In 2020, Khonina et al. [21] presented a significant improvement of the evanescent field ratio in a ridge waveguide modified into a dual hybrid plasmonic waveguide by tapering the middle section of the waveguide and placing a gold layer on both sides with a subwavelength gap. In 2020, Delcourt et al. [22] investigated third-order optical nonlinear effects relying on the instantaneous Kerr effect in a straight chalcogenide ridge waveguide.

To describe the wave propagation behavior of ridge waves propagated using a circular ridge waveguide, many mathematical models have been proposed; these models have yielded consistent experimental trends. However, the wave propagation behavior of ridge-guided waves entails complex geometric shapes and boundary-value problems. A precise analytical solution has not yet been proposed, and numerical methods are currently used to develop approximate solutions.

The simplex numerical algorithm is one of the methods of linear programing based on geometric relations, and it is still constantly being improved in terms of theory and practical applications [2325]. This study uses the natural resonant frequency of the piezoelectric circular ridge waveguide to inversely calculate material parameters. Before the material parameters can be inversely calculated, three procedures must first be employed: (i) the geometric parameters of the structure must be inversely calculated, (ii) the sensitivity of the structural parameter is discussed, and (iii) the convergence condition and convergence of the program need to be discussed. The results of the inverse calculation will be verified and integrated from multiple references such as literature review and theoretical framework with the expectation that material parameters are to be provided during the stress analysis of the piezoelectric circular ridge waveguide.

2 Basic Theoretical Analysis

2.1 Constitutive Equation of Piezoelectric Material.

Schematic of a circular ridge waveguide with the coordinate system presented in Fig. 1, the bottom was simply supported by a foam rubber.

Fig. 1
Schematic of a circular ridge waveguide
Fig. 1
Schematic of a circular ridge waveguide
This study utilized the piezoelectric ceramic material PZT-4, a transversely isotropic material. Piezoelectric materials produce direct and inverse piezoelectric effects. Therefore, the application of positive pressure on piezoelectric materials causes the piezoelectric body to generate a voltage to maintain its original state by resisting the shortening and compression of the electric dipole moment. This phenomenon in which mechanical energy is converted to electrical energy is known as the direct piezoelectric effect. When piezoelectric materials are subjected to electric field effects, their electric dipole moment is elongated and the piezoelectric body changes based on the direction of the electric field. This phenomenon in which electrical energy is converted into mechanical energy is known as the inverse piezoelectric effect. When a piezoelectric structure is placed in electric fields or subjected to mechanical forces, the constitutive equation of the structure is as follows [26]:
T=cSeED=eSεsE
(1)

where T and S are the stress tensor and strain tensor, respectively; c is the elastic stiffness matrix; e is the tensor of the piezoelectric constants; D is the electric displacement; E is the electric field strength; and ɛS is the piezoelectric constant tensor of the constant strain. Furthermore, u and Φ are defined as the displacement vector and potential vector, respectively.

2.2 Hamilton's Principle.

Under boundary conditions in which the surface traction of the circular ridge acoustic waveguide is zero, Hamilton's principle states that the first variation in the total potential is zero when the elastomer approaches dynamic equilibrium
t1t2δ(TH)dt=0
(2)
Moreover, the first variation in all the field variables at the time intervals t1 and t2 is zero. Here, T and H are the elastomer's total kinetic energy and enthalpy, respectively. The kinetic energy in the material is expressed as
T=12volu̇Hρu̇dV=12θzru̇Hρu̇rdrdzdθ
(3)
The enthalpy H that is generated in the material due to electromechanical effects is defined as the difference between the strain energy and the electric potential energy
H=UEHD=SHTEHD=12θzr(SHcSSHeHEEHeSEHεSE)rdrdzdθ
(4)
We assume that ρ is a 3 × 3 diagonal matrix that represents the density of the flexible acoustic waveguide. Here, the superscripted H indicates the Hermitian matrix and represents the conjugate transpose. If the material stiffness matrix c and the density matrix ρ are symmetric matrices with real number values, then the strain and kinetic energies are positive and definite. The superscripted overdot represents the partial differentiation of time, and the Lagrangian function L of the circular ridge waveguide system is expressed as follows:
L=TH=12θzr(u̇Hρu̇SHcS+SHeHE+EHeS+EHεSE)rdrdzdθ
(5)
Hamilton's principle indicates that under the conditions of no external force and surface electric charge, integrating the Lagrangian function with time leads to
δt1t2Ldt=0
(6)

In the interval (t1, t2), the first derivative of all field variables is zero.

2.3 Bidimensional Finite Element Method.

The finite element method is applied to mesh the cross section of the circular acoustic waveguide into numerous elements that are continuous, small, and discrete. The displacement and potential at any point within an element are expressed as the product of the interpolation function and the node displacement and potential. The elements used in this study are isoparametric elements such as two-dimensional four-node Q4 elements.

Given that the flexible waves of a circular ridge waveguide are propagated circumferentially along the circumference, a state of resonance is maintained by the wave motion at the cross section perpendicular to the circumference. The displacement vector and potential vector are, respectively, assumed to be
u={uruzuθ}={ur(r,z,t))uz(r,z,t)juθ(r,z,t)}ejnθ
(7)
Φ=Φ=Φ(r,z,t)ejnθ
(8)
where the circumferential mode number n = kR, k is a wave number, j is the complex number, ejnθ is the wave propagation factor, and R is the distance from the circular ridge waveguide's center to the centerline of the circular ridge wall. For a traveling wave, n is a positive real number. However, for a standing wave, n is a positive integer. The displacement component at each point within an acoustic waveguide element is expressed using the nodal displacement and the interpolation function in the following matrix:
u=Nud
(9)
Φ=Nϕϕ
(10)
where
Nu=[N1N2N3N4]
(11)
Ni=[Ni000Ni000jNi]ejnθ
(12)
Nφ=Niejnθ
(13)
Ni=14(1+ξξi)(1+ηηi)
(14)
where d is the displacement vector of a node, ϕ is the potential vector of a node, and ξ and η are dumb valuables. The strain S and electrical potential energy Φ are then expressed using the following matrix:
S=Bud
(15)
E=Bφϕ
(16)
where
Bu=[Bu1Bu2Bu3Bu4]
(17)
Bui=[Ni/r000Ni/z0Ni/r0n(Ni/r)0jn(Ni/r)j(Ni/z)jn(Ni/r)0j(Ni/rNi/r)Ni/zNi/r0]ejnθ
(18)
Bφ=[Bφ1Bφ2Bφ3Bφ4]
(19)
Bφi=[Ni/rNi/zjn(Ni/r)]ejnθ
(20)

2.4 Dispersion Equation.

The expressions of the displacement u, strain S, electrical potential energy Φ, electric field E, and interpolation function of any point within a unit element in Eqs. (15) and (16) are substituted into Eqs. (3)(5). The values are summed to obtain the total kinetic energy T, enthalpy H, and Lagrangian function L for all the elements, as follows:
T=12θzrḋHNuHρNuḋrdrdzdθ=12ḋHmḋ
(21)
H=12θzrdHBuHcBud+dHBuHeHBφϕ+ϕHBφHeBudϕHBφHεSBφφ)rdrdzdθ=12(dHkuud+dHkuφϕ+ϕHkφudϕHkφφϕ
(22)
L=TH=12(ḋHmḋdHkuuddHkuφϕϕHkφud+ϕHkφφϕ)
(23)
By substituting Eq. (23) into Eq. (6), we obtain the following:
12{δdHt1t2(md̈+kuud+kuφϕ)dt+δϕHt1t2(kφud+kφφφ)dt+t1t2(d̈Hm+dHkuu+ϕHkφu)dtδd+t1t2(dHkuφ+ϕHkφφ)dtδϕ}=0
(24)
Hence, the equations of motion for the circular ridge waveguide can be expressed as follows:
{md̈+kuud+kuφϕ}=0
(25)
{kφud+kφφϕ}=0
(26)
{d̈Tm+dTkuu+ϕTkφu}=0
(27)
{dHkuφ+ϕTkφφ}=0
(28)
where m, kuu, k, kφu, and kφφ are symmetric matrices with real number values. Thus, Eqs. (25) and (27) are identical, as are Eqs. (26) and (28). Combining the discretized elements to form a global matrix, the equations of motion for the entire system are as follows:
[M000]{D̈Φ̈}+[KuuKuφKφuKφφ]{DΦ}={00}
(29)
If the nodal displacement has a time-harmonic factor eiωt, then D=D¯eiωt,D̈=ω2D¯eiωt,Φ=Φ¯eiωt, and Φ̈=ω2Φ¯eiωt is substituted into Eq. (24) to allow the formation of a dispersion equation that presents the relationship between the wave number k and the angular frequency ω
([KuuKuφKφuKφφ]ω2[M000]){D¯Φ¯}=0
(30)
Equation (30) presents an eigenvalue problem, and the sufficient condition for establishing a nontrivial solution of the system of equations is as follows:
det([KuuKuφKφuKφφ]ω2[M000])=0
(31)

A circular ridge waveguide made from PZT-4 is considered an example. Table 1 presents the parameters of the PZT-4. The bidimensional finite element method (Bi-d FEM) allows precise and efficient numerical calculation for ridge waves propagation of the piezoelectric circular ridge waveguide for both the traveling wave (n is a noninteger) and the standing wave (n is an integer). The Bi-d FEM algorithm used in this study was written in the fortran programing language. The circular ridge waveguide was meshed by 240 elements and 279 nodes, as shown in Fig. 2(a). Figures 2(b)2(f) show the resonant mode shapes of the circular ridge waveguide as calculated by the Bi-d FEM.

Fig. 2
Sectional mode shapes of piezoelectric circular ridge waveguide as calculated by Bi-d FEM: (a) mesh map, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, and (f) m = 5
Fig. 2
Sectional mode shapes of piezoelectric circular ridge waveguide as calculated by Bi-d FEM: (a) mesh map, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, and (f) m = 5
Table 1

Parameters of the PZT-4 [26]

Elastic coefficient (GPa)
C11C12C13C33C44
13977.874.311525.6
Piezoelectric coefficient (C/m2)Dielectric coefficient
e15e31e33ε11/ε0ε33/ε0
12.7−5.615.1730635
Elastic coefficient (GPa)
C11C12C13C33C44
13977.874.311525.6
Piezoelectric coefficient (C/m2)Dielectric coefficient
e15e31e33ε11/ε0ε33/ε0
12.7−5.615.1730635

Note: ε0=8.854×1012 F/m.

Comparative numerical calculations of the ridge waveguide were carried out using the commercial code ansys [27] (ANSYS Inc., Canonsburg, PA). An element type of SOLID 5 was selected, and the ridge waveguide was meshed by 6044 elements with 7320 nodes, as shown in Fig. 3. Figure 4 illustrates that the mode shapes and resonant frequencies of a ridge waveguide calculated by ansys correspond to the axial modes m = 1–3. The F(m, n) stands for the flexural wave of a circular ridge waveguide, where m is the axial mode number and n is the circumferential mode number. Figure 5 presents the dispersion curves for the calculated wave number k (1/mm) and frequency ω (kHz). Comparisons of the simulation performed by using ansys and Bi-d FEM revealed that the two sets of results were consistent.

Fig. 3
Mesh view of a piezoelectric circular ridge waveguide by 3D ansys
Fig. 3
Mesh view of a piezoelectric circular ridge waveguide by 3D ansys
Fig. 4
Mode shapes and resonant frequencies of a piezoelectric circular ridge waveguide generated by 3D ansys. (a) F(1, 2), 6.039 kHz, (b) F(1, 3), 16.803 kHz, (c) F(1, 4), 31.445 kHz, (d) F(1, 5), 49.314 kHz, (e) F(1, 6), 69.817 kHz, (f) F(2, 2), 9.178 kHz, (g) F(2, 3), 22.988 kHz, (h) F(2, 4), 38.896 kHz, (i) F(2, 5), 56.912 kHz, (j) F(2, 6), 76.977 kHz, (k) F(3, 2), 64.967 kHz, (l) F(3, 3), 70.839 kHz, (m) F(3, 4), 80.764 kHz, (n) F(3, 5), 94.893 kHz, and (o) F(3, 6), 112.512 kHz.
Fig. 4
Mode shapes and resonant frequencies of a piezoelectric circular ridge waveguide generated by 3D ansys. (a) F(1, 2), 6.039 kHz, (b) F(1, 3), 16.803 kHz, (c) F(1, 4), 31.445 kHz, (d) F(1, 5), 49.314 kHz, (e) F(1, 6), 69.817 kHz, (f) F(2, 2), 9.178 kHz, (g) F(2, 3), 22.988 kHz, (h) F(2, 4), 38.896 kHz, (i) F(2, 5), 56.912 kHz, (j) F(2, 6), 76.977 kHz, (k) F(3, 2), 64.967 kHz, (l) F(3, 3), 70.839 kHz, (m) F(3, 4), 80.764 kHz, (n) F(3, 5), 94.893 kHz, and (o) F(3, 6), 112.512 kHz.
Fig. 5
Dispersion curves for the frequency and wave number of the flexural waves of the circular ridge waveguides
Fig. 5
Dispersion curves for the frequency and wave number of the flexural waves of the circular ridge waveguides

2.5 Correction of Ridge Tip Displacement.

The dispersion equation of the circular ridge waveguide is an eigenvalue problem, and the obtained eigenvectors are normalized to a characteristic length of one. Assuming that the corrected displacement is the product of the displacement obtained through the eigenvector corresponding to the eigenvalue and the proportionality constant A, the corrected displacement equation can be expressed as follows:
Ur=AUr¯
(32)
Uz=AUz¯
(33)
Uθ=AUθ¯
(34)

where Ur¯, Uz¯, and Uθ¯ are the displacement components prior to the correction, and Ur, Uz, and Uθ are the displacement components after the correction.

The kinetic energy and strain per unit time inside the circular ridge waveguide wall are summed up and expressed as 1 unit of energy. Given that the strain and kinetic energies of an elastic wave are equal within one cycle, the sum of the two is expressed as follows:
2(πA2ω2UTMU)=1
(35)
Here, the constant of proportionality A is as follows:
A=1ω12πUTMU
(36)

Thus, the real sizes of Ur, Uz, and Uθ corresponding to the unit time and energy can be obtained. Figures 6(a) and 6(b) illustrate the distribution of the radial displacement Ur at node “T” (as shown in Fig. 2(a)) before and after the correction. The horizontal axis h/Rin in the figures represents the circular ridge waveguide's height-to-radius ratio. Figures 7(a) and 7(b) illustrate the distribution of the circumferential displacement Uθ before and after the correction. The figures present a significant difference prior to and after conducting the correction.

Fig. 6
Radial displacement distribution of the flexural waves of the circular ridge waveguides: (a) before the correction and (b) after the correction
Fig. 6
Radial displacement distribution of the flexural waves of the circular ridge waveguides: (a) before the correction and (b) after the correction
Fig. 7
Circumferential displacement distribution of the flexural waves of the circular ridge waveguides: (a) before the correction and (b) after the correction
Fig. 7
Circumferential displacement distribution of the flexural waves of the circular ridge waveguides: (a) before the correction and (b) after the correction

3 Experimental Results

3.1 Electrode Design.

This study utilized a PZT-4 piezoelectric circular ridge waveguide that was manufactured by Eleceram Inc., Taoyuan, Taiwan. The piezoelectric circular ridge waveguide's height, outer radius, and inner radius were 12.64 mm, 13 mm, and 11 mm, respectively, yielding a thickness of 2 mm. First, nitric acid was used to wash off the metal electrodes from the inner and outer section of the piezoelectric circular ridge waveguide. Subsequently, new electrodes were coated on the circular ridge waveguide surface based on the flexible wave mode that had to be excited according to the design. Using screen printing, silver glue was coated on the outer surface of the piezoelectric circular ridge waveguide. The printed silver lines served as driving electrodes for the transducer. Before the electrodes were coated, the printing screen had to be tightly affixed to the outer surface of the piezoelectric circular ridge waveguide, and a scraping knife was used to extrude the silver glue to allow the glue to pass through the electrode section of the screen. Finally, this process requires 10 min of heating in a furnace at 120 °C.

Figures 8 and 9 present the electrodes designed on autocad (Autodesk Inc., San Rafael, CA) and the final completed layout of the electrode, respectively. The thicker electrode in the center is the driving electrode, and the other electrodes divided into three segments are the sensing electrodes. Based on the different circumferential mode numbers n, the sensing electrodes are placed at corresponding wave valleys at intervals of 180 deg, 120 deg, 90 deg, 72 deg, and 60 deg. The sensing electrodes are designed in three stages to distinguish between the resonant frequencies corresponding to the different axial modes m. Figure 10 presents a piezoelectric circular ridge waveguide in which n = 2 electrodes have been arranged.

Fig. 8
Screen design drawing generated by using autocad. n = 2, n = 3, n = 4, n = 5, and n = 6.
Fig. 8
Screen design drawing generated by using autocad. n = 2, n = 3, n = 4, n = 5, and n = 6.
Fig. 9
Final screen layout
Fig. 9
Final screen layout
Fig. 10
Piezoelectric circular ridge waveguide in which the electrodes have been arranged for n = 2
Fig. 10
Piezoelectric circular ridge waveguide in which the electrodes have been arranged for n = 2

3.2 Resonant Frequency Measurement.

The impedance curve of the piezoelectric circular ridge waveguide was obtained using a network analyzer HP8751A (Agilent Technologies, Santa Clara, CA) on the resonance frequency measurement, as presented in Figs. 11 and 12. The network analyzer has three functions: function generation, signal acquisition, and analysis. During the measurement, the bandwidth is adjusted to the corresponding resonant frequency range based on the result of the simulation. First, the analyzer outputs a sine wave, amplified ten times by a broadband amplifier NF HAS-4051 (NF Corporation, Yokohama, Japan), which is connected to the driving electrode. Subsequently, the sensing electrode senses the voltage change caused by the deformation of the piezoelectric circular ridge waveguide. In this experiment, in order to reduce the influence of the boundary conditions, the piezoelectric circular ridge waveguide is placed on foam rubber, which is similar to the situation where the two ends are unconstrained.

Fig. 11
Experimental setup
Fig. 11
Experimental setup
Fig. 12
Photograph of the experimental setup
Fig. 12
Photograph of the experimental setup

The resonant modes of the circular ridge waveguide are longitudinal, torsional, and flexural modes. Longitudinal motion moves along the axial and radial directions. In this study, the resonant response of the flexural mode was used as the primary circumferential wave of the ridge waveguide. Two integers (m, n) were used to indicate the numbering and characteristics of the resonant modes. Here, n represents the number of circumferential modes, and m indicates the number of axial modes.

Figures 13(a)13(o) present the experimental impedance curves. In this study, F(m, n) is used to represent the corresponding flexural modes. Although the scanning frequency range is the same, different impedance curves are obtained due to the different positions of the sensing electrodes. Therefore, using the modal distribution, the sensing electrodes can be segmented to distinguish the frequencies of the different axial modes. Figure 14 presents a comparison between the experimental and theoretical values obtained from the dispersion curves. The theoretical dispersion curve is simulated by obtaining an average of the actual dimensions of the experimental and theoretical piezoelectric circular ridge waveguides. Table 2 presents the comparison of experimental and theoretical values. Table 3 presents the difference between the corresponding experimental and theoretical values before the inversion calculation of structural parameters.

Fig. 13
Impedance curves by measurement. (a) F(1, 2) mode, (b) F(2, 2) mode, (c) F(3, 2) mode, (d) F(1, 3) mode, (e) F(2, 3) mode, (f) F(3, 3) mode, (g) F(1, 4) mode, (h) F(2, 4) mode, (i) F(3, 4) mode, (j) F(1, 5) mode, (k) F(2, 5) mode, (l) F(3, 5) mode, (m) F(1, 6) mode, (n) F(2, 6) mode, and (o) F(3, 6) mode.
Fig. 13
Impedance curves by measurement. (a) F(1, 2) mode, (b) F(2, 2) mode, (c) F(3, 2) mode, (d) F(1, 3) mode, (e) F(2, 3) mode, (f) F(3, 3) mode, (g) F(1, 4) mode, (h) F(2, 4) mode, (i) F(3, 4) mode, (j) F(1, 5) mode, (k) F(2, 5) mode, (l) F(3, 5) mode, (m) F(1, 6) mode, (n) F(2, 6) mode, and (o) F(3, 6) mode.
Fig. 14
Comparison of the dispersion curves between experimental and theoretical values
Fig. 14
Comparison of the dispersion curves between experimental and theoretical values
Table 2

Comparison of experimental and theoretical values

m = 1m = 2m = 3
(kHz)Experimental valueTheoretical valueExperimental valueTheoretical valueExperimental valueTheoretical value
n = 25.9136.0398.9609.17863.92064.967
n = 316.26016.80322.26522.98868.90370.839
n = 430.55831.44538.17038.89679.01080.764
n = 548.90549.31456.62056.91294.07894.893
n = 669.48869.81776.92376.977112.093112.512
m = 1m = 2m = 3
(kHz)Experimental valueTheoretical valueExperimental valueTheoretical valueExperimental valueTheoretical value
n = 25.9136.0398.9609.17863.92064.967
n = 316.26016.80322.26522.98868.90370.839
n = 430.55831.44538.17038.89679.01080.764
n = 548.90549.31456.62056.91294.07894.893
n = 669.48869.81776.92376.977112.093112.512
Table 3

The difference between the corresponding experimental and theoretical values before the inversion calculation of structural parameters

(Hz)m = 1m = 2m = 3
n = 21262181047
n = 35437231936
n = 48877261754
n = 5409292815
n = 632954419
(Hz)m = 1m = 2m = 3
n = 21262181047
n = 35437231936
n = 48877261754
n = 5409292815
n = 632954419

4 Inversion Calculation of Material Parameters

4.1 Modified Simplex Method.

The modified simplex algorithm is a method that can be used to find the optimized structural parameters based on the principles of geometry [2325]. The basic idea is to create N-dimensional spaces with N + 1 vertices to seek the most likely N parameters. For example, N = 2 represents that the structure is a triangle in space, and N = 3 represents a tetrahedron structure in space. The movement direction of the triangle or tetrahedron is determined by the magnitude of each vertex objective function value. When the value of the objective function reaches the minimum value, the parameter at the time is the optimal solution.

As shown in Fig. 15, taking N = 2 as an example, three vertices are established as two-dimensional function space, the W point with the largest objective function value, the B point with the smallest objective function value, and the O point with the value of the objective function between the two points. The process of searching for the best solution can be divided into the following four steps:

Fig. 15
Modified simplex algorithm
Fig. 15
Modified simplex algorithm
  • Step 1: Establish a reflection point R based on the highest point.

    Suppose that value d is the distance from the maximum value W to the midpoint Mp of the line connecting the remaining two points, and the WMp line extends d to the reflection point R. When the value of the point R is between the minimum and the maximum of the objective function value, a new vertex is established at reflection point R, and the maximum value of point W is replaced.

  • Step 2: Establish an expansion point E.

  • If the objective function value of the reflection point R is smaller than the lowest point B, the WMp line segment is extended by 2d to point E. A comparison of the objective function values between point R and point E is conducted to determine the smaller point at which the objective function value is accepted replacing the maximum point W.

  • Step 3: Establish a contraction point C.

    If the objective function value of reflection point R is larger than the maximum value of point W, a contraction point C is established at the midpoint of the WMp line segment. If the value of the objective function at point C is smaller than the highest point W, point W is replaced by point C.

  • Step 4: Shrink point O and point W to shrinkage point S.

    If the value of the objective function at point C is larger than the maximum value of point W, then point O and point W are contracted to point S along the line segments OB and WB, respectively, to establish a new triangular apex.

The modified simplex method will find a point with a smaller objective function value to replace the point with the largest objective function value to form a new triangle. Repeat these four steps until the smallest objective function value is found. The program used for the material parameter inversion calculation in this study was written using fortran.

4.2 Objective Function.

While performing an inversion calculation of the material parameters, the most important thing is to define the objective function. In addition to improving the accuracy of inverse calculation, a good objective function can save on computation time. Usually, the objective function is based on the difference between the experimental value and the numerical simulation result to find the parameter with the smallest error. In this paper, the objective function is defined as the sum of squares of the frequency difference between the experimental and theoretical value of each flexural mode, as follows:
FObject=1M*j=1M*[1N*i=1N*(fExperimentifTheoryi)2]
(37)

where FObject is the objective function, fExperimenti is the experimental value of the resonant frequency corresponding to each mode, fTheoryi is the theoretical value of the resonant frequency corresponding to the wave number k of each mode, N* is the experimental value of the circumferential mode number, and M* is the experimental value of the axial mode number.

4.3 Inversion Calculation of Structural Parameters.

Before inverse calculation of material parameters, the geometric parameters of the structure must be inverse calculated. Since the piezoelectric circular ridge waveguide used in this study is not perfect, there may be errors in the size of the geometric parameters. The distribution of the height (measured with a vernier caliper) of the ridge waveguide at different circumferences is about 12.55–12.73 mm, and the ridge waveguide wall thickness distribution is about 1.98–2.04 mm, which cannot completely match the simulated uniform geometry.

It can be found from Table 3 that the differences between the experimental values of the circumferential mode numbers n = 3 and 4 and the theoretical values were much larger than those of the other modes. Different circumferential modes correspond to different electrode positions, which are presumed to be caused by uneven size distribution. In the axial mode, the difference of m = 3 is the largest, assuming the theoretical value can infer the effect of uniform size. Therefore, during the first stage of this study, the structural parameters are inversely calculated to obtain the most likely uniform geometric size. After the actual geometric shape of the circular ridge waveguide is obtained, the second stage of the material parameters' inversion calculation is performed.

Before performing the structural parameter inverse calculation, the sensitivity of each structural parameter is discussed. Assuming that the outer diameter of the circular ridge waveguide is constant, the size parameters of the structure are the inner radius Rin and the height h. Figures 16 and 17 show the comparison of the variation of the circumferential flexural wave dispersion curve with the theoretical model curve after adding 0.5 mm to the two parameters. When Rin increases by 0.5 mm, the three dispersion curves are shifted to the low frequency, where m = 1 and 2 increases along with the increase of k, while the offset of m = 3 is similar to the horizontal offset. Increasing the height h results in an obvious shift of m = 3 curve to the low frequency, while the m = 2 curve shifts only slightly to the low frequency. The m = 1 curve has almost no change.

Fig. 16
Sensitivity test of inner radius Rin
Fig. 16
Sensitivity test of inner radius Rin
Fig. 17
Sensitivity test of height h
Fig. 17
Sensitivity test of height h
Before the inverse calculation of the experimental measurements of the resonance frequency, the convergence condition and convergence of the program need to be discussed. Convergence conditions are an important step in the whole process of the inversion calculation and an important factor in determining the parameter accuracy of the inversion calculation. The convergence condition in this study is defined as follows:
|XwXb|<1μm
(38)

where Xw represents the parameter corresponding to the maximum point of the difference of the objective function in each point, and Xb represents the parameter corresponding to the minimum point of the difference of the objective function in each point. Maximum and minimum parameter values of less than 1 μm indicate that the points are quite close; therefore, this can be judged as convergence.

As shown in Fig. 18, the solid points are the initial guess values, and the hollow point is the final convergence value. The initial guess value of each parameter is the theoretical model value × (1 ± random number (0.0–1.0)), and the theoretical model value is Rin = 11 mm and h = 12.5 mm. It can be seen from Fig. 18 that the ten sets of parameters that are randomly guessed converge to the theoretical value, excluding the possibility that the objective function has another local minimum.

Fig. 18
Convergence test of structural parameters Rin and h
Fig. 18
Convergence test of structural parameters Rin and h

Finally, the inversion calculation is performed using the experimentally measured resonance frequency. The convergence is shown in Fig. 19, value is Rin = 11.01 mm and h = 12.77 mm. It can be seen from Fig. 20 that the dispersion curve obtained by the modified shape has significantly reduced the error between the experimental values and the simulation results, specifically for the dispersion curve of m = 3. The corrected frequency and experimental values are summarized in Table 4, while in Table 5 the differences between the frequency and experimental values are listed.

Fig. 19
Inversion calculation results of structural parameters Rin and h
Fig. 19
Inversion calculation results of structural parameters Rin and h
Fig. 20
Comparison of the experimental results of the piezoelectric circular ridge waveguide dispersion curves after the structural parameters' inversion calculation
Fig. 20
Comparison of the experimental results of the piezoelectric circular ridge waveguide dispersion curves after the structural parameters' inversion calculation
Table 4

Theoretical values and experimental values after the inversion calculation of structural parameters

m = 1m = 2m = 3
(kHz)Experimental valueTheoretical valueExperimental valueTheoretical valueExperimental valueTheoretical value
n = 25.9136.0118.9609.11063.92064.022
n = 316.26016.72522.26522.80468.90369.769
n = 430.55831.30238.17038.60979.01079.634
n = 548.90549.10156.62056.54294.07893.754
n = 669.48869.53276.92376.537112.093111.378
m = 1m = 2m = 3
(kHz)Experimental valueTheoretical valueExperimental valueTheoretical valueExperimental valueTheoretical value
n = 25.9136.0118.9609.11063.92064.022
n = 316.26016.72522.26522.80468.90369.769
n = 430.55831.30238.17038.60979.01079.634
n = 548.90549.10156.62056.54294.07893.754
n = 669.48869.53276.92376.537112.093111.378
Table 5

The differences between the theoretical values and experimental values after the inversion calculation of structural parameters

(Hz)m = 1m = 2m = 3
n = 298150102
n = 3465539866
n = 4744439624
n = 5196−78−324
n = 644−386−715
(Hz)m = 1m = 2m = 3
n = 298150102
n = 3465539866
n = 4744439624
n = 5196−78−324
n = 644−386−715

4.4 Sensitivity Analysis of Material Parameters.

The independent coefficient of piezoelectric material contains five elastic coefficients, three piezoelectric coefficients, and two dielectric coefficients. Before the material coefficient is inversely calculated, the sensitivity of each coefficient is discussed.

4.4.1 The Sensitivity of Elastic Coefficients.

In the case of the elastic coefficient, the first aspect that must be discussed is the influence of the change in C11 on the dispersion curves. Figure 21(a) shows the effect of C11 on the dispersion curve when it is changed to 120% of the target value. When C11 increases, the frequency of m = 1 and 2 curves increases, the frequency is further increased as a result from the increase of wave number k. The frequency of the m = 3 curve is shifted to the higher frequency in an approximately horizontal translation. The three dispersion curves all depict significant changes.

Fig. 21
Sensitivity test of the elastic coefficients: (a) C11, (b) C12, (c) C13, (d) C33, (e) C44, (f) e15, (g) e31, (h) e33, (i) ε11, and (j) ε33Sensitivity test of the elastic coefficients: (a) C11, (b) C12, (c) C13, (d) C33, (e) C44, (f) e15, (g) e31, (h) e33, (i) ε11, and (j) ε33
Fig. 21
Sensitivity test of the elastic coefficients: (a) C11, (b) C12, (c) C13, (d) C33, (e) C44, (f) e15, (g) e31, (h) e33, (i) ε11, and (j) ε33Sensitivity test of the elastic coefficients: (a) C11, (b) C12, (c) C13, (d) C33, (e) C44, (f) e15, (g) e31, (h) e33, (i) ε11, and (j) ε33

Next, we discuss the effect of C12 changes on the dispersion curves, as shown in Fig. 21(b). When C12 becomes larger, the three dispersion curves are shifted to the low frequency. The resonance frequency of m = 1 and 2 increases with k, and there is a tendency to shift toward lower frequencies. The resonance frequency of the m = 3 curve is shifted parallel to the low frequency.

We next observe the effect of C13 changes, as shown in Fig. 21(c). After an increase in C13, the variation in the dispersion curve is shifted to the lower frequency, to the same extent as the shift with the change in C12. The amount of change caused by an increase of C13, however, is significantly smaller than the effect of an increase in C12, especially with regard to the dispersion curves at m = 1 and 2. Although the amount of m = 3 is reduced, the change is more apparent as compared to m = 1 and 2, suggesting that the dispersion curve of m = 3 has a higher sensitivity to C13.

We then discuss the sensitivity of C33, as shown in Fig. 21(d). When C33 is increased to 120%, there is almost no change in the corresponding three dispersion curves. Only the dispersion curve of m = 3 rises slightly with the increase of k, while the offset to the high frequency has a slight decrease.

Finally, the effects of the changes in C44 on the dispersion curve are discussed, as shown in Fig. 21(e). When C44 is changed to 120%, the three dispersion curves shift slightly to high frequencies. When k increases, the frequency also increases slightly; however, the variation of the three curves is not particularly obvious. The degree of influence of C44 and C13 is different, while the latter has a greater influence on the m = 3 curve. As a result, C44 is ranked after C13 when calculating priority inversions.

4.4.2 The Sensitivity of Piezoelectric Coefficients.

With respect to the piezoelectric coefficients, the effects e15, e31, and e33 are discussed. Figures 21(f)21(h) depict corresponding dispersion curves shifted slightly to higher frequencies when e15, e31, and e33 increase to 120%. However, the offset of each dispersion curve is not obvious; in particular, the dispersion curve corresponding to e33 exhibits almost no change.

4.4.3 The Sensitivity of Dielectric Coefficients.

With respect to the dielectric coefficients, the dispersion curves corresponding to the increase of ε11 and ε33 to 120% are shown in Figs. 21(i) and 21(j), respectively. The change for ε11 results in almost no change in the dispersion curve. The change for ε33 results in an unnoticeable change in the dispersion curve; however, all three dispersion curves shift slightly to low frequencies.

The above discussions of the effects of the elastic coefficients, piezoelectric coefficients, and dielectric coefficients indicate that the elastic coefficients affect the variation of the dispersion curves to a greater degree than the piezoelectric coefficients and the dielectric coefficients. This indicates a greater sensitivity by the elastic coefficients, especially C11 and C12, followed by C13. The dispersion curve of m = 3 has a high sensitivity to C13, while it has lower sensitivity to the other coefficients. As a result, in this study, the only material coefficients discussed are the elastic coefficients, and the priority of the inversion calculation is as follows: C11 > C12 > C13 > C44 > C33.

4.5 Inversion Calculation of Material Parameters.

Before the inversion calculation of the material coefficients is performed, the convergence condition and convergence of the program are discussed. The material coefficient inverse calculation convergence condition is defined as follows:
|CijwCijb|<1MPa
(39)

where Cijw represents the material coefficient corresponding to the maximum point of the difference between the objective functions in each point, and Cijb represents the material coefficient corresponding to the minimum point of the objective function difference in each point. When the difference between the two material coefficients is less than 1 MPa, it is judged to be convergent.

In the convergence test of the inverse calculation program, three methods m = 1, m = 1 and 2, and m = 1, 2, and 3 are used to discuss the number of coefficients that can reach convergence. The initial value in the inverse calculation process is the material coefficients × (1 ± random number (0.0–0.3)) in Table 1.

4.5.1 Inverse Calculation Using m = 1 Dispersion Curve.

Figures 22(a)22(d) show the convergence of two coefficients and three coefficients using only the m = 1 curve. Figure 22(a) shows that if only two coefficients are inversely calculated using the m = 1 curve, the ten sets of randomly given material coefficients will converge to the theoretical value. In the case of the inverse calculation of the three coefficients, Figs. 22(b)22(d) show that only seven of the ten sets of material coefficients randomly given will converge to the theoretical value. The final convergence of C11 is between 135.28 and 144.15 GPa, that of C12 is between 71.63 and 85.45 GPa, and that of C13 is between 73.65 and 75.89 GPa, so the determination does not converge.

Fig. 22
Convergence test results of inversely calculated by m = 1 dispersion curve. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), and (d) three elastic coefficients (C12–C13 plane).
Fig. 22
Convergence test results of inversely calculated by m = 1 dispersion curve. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), and (d) three elastic coefficients (C12–C13 plane).

4.5.2 Inverse Calculation Using m = 1 and 2 Dispersion Curves.

Figures 23(a)23(j) show the convergence of two to four coefficients using the m = 1 and 2 curves. Figure 23(a) shows the convergence of the two coefficients in the inverse calculation. As in the previous state where only the m = 1 curve was used, the ten sets of coefficients all converge to the theoretical value. Figures 23(b)23(d) show the convergence of the three coefficients. The effect is better than when only m = 1, and the ten sets of coefficients all converge to the theoretical value. Figures 23(e)23(j) show the convergence of the four coefficients. Only seven sets of the randomly given coefficients converge to the theoretical value. The convergence results of the remaining three points establish final convergence of C11 between 135.27 and 140.39 GPa, C12 is between 74.19 and 79.14 GPa, C13 is between 68.88 and 76.18 GPa, while C44 is between 25.48 and 25.97 GPa. Although the distribution range of C44 is quite close to the theoretical value of C44, the convergence values of the other three material coefficients are quite different from the theoretical values, so it is determined that there is no convergence.

Fig. 23
Convergence test results of inversely calculated by m = 1 and 2 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C44 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), and (j) four elastic coefficients (C13–C44 plane).Convergence test results of inversely calculated by m = 1 and 2 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C44 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), and (j) four elastic coefficients (C13–C44 plane).
Fig. 23
Convergence test results of inversely calculated by m = 1 and 2 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C44 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), and (j) four elastic coefficients (C13–C44 plane).Convergence test results of inversely calculated by m = 1 and 2 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C44 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), and (j) four elastic coefficients (C13–C44 plane).

4.5.3 Inverse Calculation Using m = 1, 2, and 3 Dispersion Curves.

If the m = 1, 2, and 3 curves are used, the convergence of two to five coefficients is discussed separately, and the test results are shown in Figs. 24(a)24(t). Figures 24(a)24(j) show the convergence results of the two to four coefficients of the inverse calculation and show that the ten sets of random coefficients can converge to the theoretical value. Figures 24(k)24(t) show the convergence of the five coefficients. C11 is between 13.18 and 145.14 GPa, C12 is between 70.23 and 83.82 GPa, C13 is between 57.51 and 87.09 GPa, C44 is between 25.24 and 26.30 GPa, and C33 is between 83.29 and 139.46 GPa. The results of each convergence point are widely dispersed, and there are only two points close to the theoretical value. The reason for this situation is that the sensitivity of C33 is very low, and it is not suitable for inverse calculation. In summary, the method of inversion calculation using three dispersion curves is deemed superior as it is capable of triggering the largest number of the coefficients to converge. As a result, when the material coefficient is inversely calculated, three dispersion curves will be used for the inversion calculation.

Fig. 24
Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).
Fig. 24
Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).Convergence test results of inversely calculated by m = 1, 2, and 3 dispersion curves. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), (d) three elastic coefficients (C12–C13 plane), (e) four elastic coefficients (C11–C12 plane), (f) four elastic coefficients (C11–C13 plane), (g) four elastic coefficients (C11–C14 plane), (h) four elastic coefficients (C12–C13 plane), (i) four elastic coefficients (C12–C44 plane), (j) four elastic coefficients (C13–C44 plane), (k) five elastic coefficients (C11–C12 plane), (l) five elastic coefficients (C11–C13 plane), (m) five elastic coefficients (C11–C44 plane), (n) five elastic coefficients (C11–C33 plane), (o) five elastic coefficients (C12–C13 plane), (p) five elastic coefficients (C12–C44 plane), (q) five elastic coefficients (C12–C33 plane), (r) five elastic coefficients (C13–C44 plane), (s) five elastic coefficients (C13–C33 plane), and (t) five elastic coefficients (C44–C33 plane).

Based on the measured values of the resonant frequency, the two elastic coefficients and the three elastic coefficients are, respectively, inversely calculated, and the results are shown in Figs. 25(a)25(d). The differences between the convergence values and the reference values are listed in Tables 6 and 7. The reference value used is the material coefficient of PZT-4 [26], and the inversion calculation result is very close to the reference value. Additionally, the results of the inversion calculation without structural parameters are shown in Fig. 26 and Table 8. Clearly, it is necessary to first inversely calculate the structural coefficients and then inversely calculate the material coefficients.

Fig. 25
The results after the structural parameters' inversion calculation. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), and (d) three elastic coefficients (C12–C13 plane).
Fig. 25
The results after the structural parameters' inversion calculation. (a) Two elastic coefficients C11 and C12, (b) three elastic coefficients (C11–C12 plane), (c) three elastic coefficients (C11–C13 plane), and (d) three elastic coefficients (C12–C13 plane).
Fig. 26
Inversion calculation results of the two elastic coefficients C11 and C12 before the structural parameters' inversion calculation
Fig. 26
Inversion calculation results of the two elastic coefficients C11 and C12 before the structural parameters' inversion calculation
Table 6

Inversion calculation results for the two elastic coefficients after the structural parameters' inversion calculation

C11C12
Theoretical value (GPa)13977.8
Inversion calculation value (GPa)138.72377.458
Difference (%)0.199%0.440%
C11C12
Theoretical value (GPa)13977.8
Inversion calculation value (GPa)138.72377.458
Difference (%)0.199%0.440%
Table 7

Inversion calculation results for the three elastic coefficients after the structural parameters' inversion calculation

C11C12C13
Theoretical value (GPa)13977.874.3
Inversion calculation value (GPa)137.46675.83373.004
Difference (%)1.104%2.528%1.744%
C11C12C13
Theoretical value (GPa)13977.874.3
Inversion calculation value (GPa)137.46675.83373.004
Difference (%)1.104%2.528%1.744%
Table 8

Inversion calculation results for the two elastic coefficients before the structural parameters' inversion calculation

C11C12
Theoretical value (GPa)13977.8
Inversion calculation value (GPa)133.39372.146
Difference (%)4.034%7.267%
C11C12
Theoretical value (GPa)13977.8
Inversion calculation value (GPa)133.39372.146
Difference (%)4.034%7.267%

5 Discussion

The resonant modes of a circular ridge waveguide include the number of circumferential modes n and the number of axial modes m. For elastic waves, when n is an integer, the flexible waves of the circular ridge waveguide that are propagating in the clockwise and counterclockwise directions combine to form standing waves. However, when n is not an integer, the progressive waves are transmitted bidirectionally. The basic concepts examined in this study are based on the above. The data from a wave propagation analysis conducted by using a Bi-d FEM and vibration value analysis conducted by using a three-dimensional (3D) ansys were compared, and the results obtained were optimal for both isotropic and piezoelectric circular ridge waveguides.

To analyze the issues posed by the circular ridge waveguide, the resonance frequencies of the axial and circumferential modes were designed in an interlaced manner and were examined based on the number of modes (sequentially from high to low). As a result, mode identification was a time-consuming process. By analyzing the 3D ansys results, we observed that the resonant modes were present from low to high frequencies. The deformation diagram for each mode was illustrated to select a specific mode. A Bi-d FEM analysis provided the resonance frequency for the continuously changing circumferential and axial modes of a circular ridge waveguide. This saved on the time spent in the identification and enabled the identification of modes even when the structure was geometrically complex. The other advantage of the Bi-d FEM was that the method enabled the use of the wave propagation factor ejnθ as the interpolation function for the circumferential direction. It is a benefit when the number of circumferential modes is high and, consequently, there is no need to be concerned about large analysis errors due to an insufficient number of circumferential elements.

For experimental measurements, the position of the sensing electrodes can be designed on the basis of the different resonance modes to achieve the function of enhancing or lowering a specific modal signal. However, the effect on the circumferential mode number or the circumferential hoop mode is not significant. In addition to segmenting the original sensing electrodes in the electrode design, the driving electrode has the same number of circumferential modes, which allows the impedance curve to obtain the best signal-to-noise ratio. Because of the anisotropic sintering characteristics of piezoelectric ceramics, the geometrical dimensions of piezoelectric circular ridge waveguides are uneven. The differences between the experimental values and values obtained by numerical analysis of the resonant frequencies are high for the various circumferential modes. This may be as a result of the sensing points corresponding to different circumferential modal numbers; thus, the influence of the uneven shape is more apparent.

For that reason, before the inversion calculation of material coefficients, the inverse of the structural geometric coefficients is calculated first. If the structural geometric coefficients with the smallest error are obtained, the error of the material coefficient inverse calculation can be reduced. After the inversion calculation, the development trends of the dispersion curves can be more obviously corrected, specifically for the dispersion curve of m = 3.

In the inversion calculation of material coefficients, since the experimental data that can be compared consist of only a few modal resonant frequencies, strategic choices must be made. Therefore, only the coefficients with higher sensitivity among the coefficients can be inversely calculated, and the influence of piezoelectric and dielectric coefficients cannot be discussed. The inversion calculation results show that the high sensitivity C11 and C12 values are quite accurate. However, if the material coefficient of the lower sensitivity is inversely calculated at the same time, the error increases. If the comparison of the experimental data can be increased, it should be possible to solve the issue.

The numerical simulation results showed that it is advantageous to analyze the vibration problem using the concept of elastic wave propagation. In particular, in a structure that has two dimensions, the resonance frequency cannot be arranged in the order of the corresponding modal numbers, which results in difficulties when identifying and pairing the resonance frequency and the resonance mode. Therefore, the dispersion curve can be used to sequentially calculate the resonance frequency of each mode, and the dispersion curve obtained by experimental measurement can be used to establish a method different from the traditional vibration measurement.

This study has attempted to measure the dispersion curves of the flexural waves of the piezoelectric circular ridge waveguide using the laser ultrasonic method. However, the measurement technology for mastering transient signals is not yet sufficiently sophisticated, and it is impossible to avoid the influence of the return wave in the opposite direction of the circular ridge waveguide. Therefore, the structural geometric parameters and material coefficients are inversely calculated from the measured data of the structural resonance frequency. For measurement of the flexural wave dispersion curve of the circular ridge waveguide, the use of a circular ridge waveguide with a larger radius is recommended for technical discussion. The first objective is to improve the separation of each axial mode dispersion curve. The second is to increase the recognition of the dispersion curve, while the third is to improve the resolution of the signal by optical techniques.

6 Conclusion

This study successfully used the Bi-d FEM to derive the dispersion equation of the piezoelectric circular ridge waveguide and calculated the dispersion curves and resonance modes of the piezoelectric circular ridge waveguide's flexural ridge wave. The simulation yielded Bi-d FEM results that were almost identical to the numerical results of 3D ansys.

In the experimental parts of the study, the resonant frequency of the piezoelectric circular ridge waveguide was measured with a network analyzer, and the experimental data were used to inversely calculate the structural geometry parameters and material coefficients using the modified simplex method.

The numerical analysis, experimental measurement, and inverse calculation of material coefficients and structural parameters prove that the modified simplex method used in this study can improve the accuracy and sensitivity of models of the ridge wave propagation behavior of the piezoelectric circular ridge waveguides. The systematic investigations of the Bi-d FEM and the modified simplex method will provide an effective reference and application for future research on piezoelectric circular ridge waveguides.

Acknowledgment

We would like to sincerely thank the Ministry of Science and Technology of Taiwan (MOST 110-2221-E-239-015) for funding this research.

Funding Data

  • Ministry of Science and Technology of Taiwan (MOST 110-2221-E-239-015; Funder ID: 10.13039/501100004663).

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