Assessment of an Optimal Design Method for a High-Energy Ultrasonic Igniter Based on Multi-Objective Robustness Optimization

Abstract

The current deterministic optimization design method ignores uncertainties in the material properties and potential machining error which could lead to unreliable or unstable designs. To improve the design efficiency and anti-jamming ability of a high-energy ultrasonic igniter, a Six Sigma multi-objective robustness optimization design method based on the response surface model and the design of the experiment has been proposed. In this paper, the initial structural dimensions of a high-energy ultrasonic igniter have been obtained by employing one-dimensional longitudinal vibration theory. The finite element simulation method of COMSOL Multiphysics software has been verified by the finite element simulation results of ANSYS Workbench software. The optimal igniter design has been achieved by using the proposed method, which is based on the finite element method, the Optimal Latin Hypercube Design method, Grey Relational Analysis, the response surface model, the non-dominated sorting genetic algorithm, and the mean value method. Considering the influence of manufacturing errors on the igniter’s performance, the Six Sigma method was used to optimize the robustness of the igniter. The Eigenfrequency analysis and the vibration velocity ratio calculation were conducted to verify the design’s effectiveness. The results reveal that the longitudinal resonant frequency of the deterministic optimization scheme and the robustness optimization scheme are closer to the design’s target frequency. The relative error is less than 0.1%. Compared with the deterministic optimization scheme, the vibration velocity ratio of the robustness optimization scheme is 2.8, which is about 15.7% higher than that of the deterministic optimization scheme, and the quality level of the design targets is raised to above Six Sigma. The proposed method can provide an efficient and accurate optimal design for developing a new special piezoelectric transducer.

1. Introduction

Traditional gasoline engines generally employ spark plug single-point ignition, which requires certain application conditions, including lean burn, in-cylinder direct injection, exhaust gas turbocharging, and exhaust gas recirculation to resolve problems such as ignition difficulty and poor flame stability [1,2]. Improving the spark plug ignition energy is an effective technical solution that achieves high efficiency and energy saving for the engine, but it will affect the lifetime of the spark plug to a certain extent [3,4]. Transient nonthermal plasma generated by laser [5], nanosecond pulse discharge [6], and dielectric barrier discharge [7] can ignite the lean fuel mixture and improve the ignition and combustion performance of the gas-liquid two-phase combustible working medium to a certain extent [8]. It cannot be widely used because of the high cost, narrow working range, and complexity of the required devices, and for other reasons [9]. Microwaves can induce nonequilibrium plasma and improve the ignition and combustion performance [10], which has the potential to achieve large-area multipoint ignition. However, the increment of the combustion chamber pressure will seriously affect the ignition performance [11]. Therefore, research and development of a high-energy igniter which can stably achieve the multipoint ignition of the fuel mixture in the combustion chamber is an important research direction for energy savings and emission reduction in internal combustion engines.

An ultrasonic wave is a high-frequency mechanical longitudinal wave greater than 20 kHz that displays large energy, strong penetration, weak diffraction, and good directionality. Its influence on chemical reactions mainly comes from ultrasonic mechanical action and the ultrasonic cavitation effect. High-frequency ultrasonic vibration and radiation pressure can form directional agitation and jet effects in the air and in liquid media. Due to the absorption of ultrasonic energy and the internal friction loss phenomenon of the media, a thermal effect temperature rise in the sound field area of the media under continuous ultrasonic action can be generated. In addition, at the moment of collapse, ultrasonic cavitation bubbles will produce extreme environments such as nanoscale transient high temperature, high pressure, and a high electric field, which can easily lead to complex physical, chemical, and biological effects [12,13,14]. According to international safety regulations, ultrasound is considered an ignition source [15]. The experiment and numerical simulation work by Ion et al. assessed the combustion characteristics of gas in an ultrasonic field. It was found that NO_x and CO emissions dropped, and combustion efficiency increased [16]. Di et al. used spherically focused ultrasound to carry out a spatially localized noncontact ignition study on a gas-liquid two-phase combustible working medium, and the results showed that the temperature of working medium reached the ignition threshold of traditional fuels such as gasoline and diesel when the sound source frequency was 300 kHz [17,18]. Nevertheless, there is a lack of studies on power ultrasound intervention in the ignition and combustion process, and its actual mechanisms of action and acoustic chemical effect are still unclear. Meanwhile, the design methods and performance parameters of a high-energy ultrasonic igniter attached to an internal combustion engine have not been studied systematically.

The optimal design of a high-energy ultrasonic igniter is carried out based on piezoelectric transducers. The non-dominated sorting genetic algorithm (NSGA-II) is widely used in engineering applications [19,20]. Based on the electromechanical equivalence method, Li et al. accomplished the optimized design of an ultrasonic scalpel by using a response surface model and a multi-objective genetic algorithm [21]. Ji et al. completed structure optimization of the ultrasonic horn based on finite element simulation and NSGA-II [22,23]. Karl et al. designed an amplifier through shape optimization using genetic algorithms and verified the effectiveness of the methodology [24], but this method needs multiple finite element simulations, which increase the computing cost. The above methods ignore the influence of uncertainties in the material properties and of machining error, which usually results in unreliable designs and increases the risk of design failure.

This paper focuses on how to design and optimize a high-energy ultrasonic igniter which is applied to the field of internal combustion engines and special burners. Owning to the limitation of the installation space and the traditional design theory of the igniter, the design target frequency of the ultrasonic igniter is set as 35 kHz in this paper. Based on the design theory of the conventional piezoelectric transducer and the traditional spark plug scale of the internal combustion engine, the initial structural dimensions of a high-energy ultrasonic igniter are determined according to the one-dimensional longitudinal vibration theory. The grid size is determined through grid independence verification. Moreover, this paper proposes a new optimization method to develop the optimal igniter, which maximizes the ratio of the front-end vibration velocity to the back-end vibration velocity and minimizes the difference between the igniter’s longitudinal resonant frequency and the design’s target frequency. The approach is based on finite element (FE) analysis, the Optimal Latin Hypercube Design (OLHD) method, the response surface model, NSGA-II, and the Six Sigma robustness optimization method. The optimal structure dimensions of the igniter are obtained based on the proposed optimization method. Furthermore, the dynamic characteristics of the optimal igniter have been verified using finite element method (FEM)-based simulation.

2. Structure and Theoretical Analysis of High-Energy Ultrasonic Igniter

The high-energy ultrasonic igniter and the constant-volume combustion bomb are rigidly connected through a flange structure, as shown in Figure 1. The mounting flange of the spark plug and the mounting flange of the igniter have interchangeability, and the design of the experiment can be carried out by changing the installation location and numbers of igniters. Figure 2 shows that the igniter consists of a prestressed bolt, back mass, electrode slices, ring-shaped piezoceramics slices, and stepped-type horn, and that they are connected in a coaxial series.

In order to enhance the transmitting efficiencies of the igniter, according to the law of conservation of momentum, the back mass and stepped-type horn are made of 45Cr and 7075-T6. Their material properties are summarized in

Table 1. Material parameters for the ignition.

Components	Back Mass	Stepped-Type Horn
Material	40Cr	7075-T6
Density (kg/m3)	7850	2810
Poisson’s ratio	0.29	0.33
Young’s modulus (GPa)	206	71.7

.

PZT-4 is selected as the actuator, because it has a large mechanical quality factor, a high Curie temperature of 350 °C, and a low dissipation coefficient of 0.004, which is suitable for the field of high output ultrasonic intensity. The related parameters of PZT-4 are given as follows:

1.

Relative Permittivity Matrix

$$\left[\epsilon \right]=\left[\begin{array}{lll}762.5& 0& 0\\ 0& 762.5& 0\\ 0& 0& 663.2\end{array}\right]$$

2.

Elasticity Constant Matrix

$$\left[C^E\right]=\left[\begin{array}{cccccc}139& 77.8& 74.3& 0& 0& 0\\ 77.8& 139& 74.3& 0& 0& 0\\ 74.3& 74.3& 115& 0& 0& 0\\ 0& 0& 0& 25.6& 0& 0\\ 0& 0& 0& 0& 25.6& 0\\ 0& 0& 0& 0& 0& 25.6\end{array}\right]\mathrm{GPa}$$

3.

Piezoelectric Stress Matrix

$$\left[e\right]=\left[\begin{array}{cccccc}0& 0& 0& 0& 12.7& 0\\ 0& 0& 0& 12.7& 0& 0\\ -5.2& -5.2& 15.1& 0& 0& 0\end{array}\right]{\mathrm{C}/\mathrm{m}}^2$$

The back mass and the stepped-type horn of the high-energy ultrasonic igniter are composed of equal-section cylinders. The longitudinal vibration model of the equal-section cylinder is shown in Figure 3. The transverse dimension of the equal-section cylinder is far less than the one-fourth wavelength of the material corresponding to its working frequency. Therefore, the vibration of the equal-section cylinder can be regarded as one-dimensional longitudinal vibration [25], and the one-dimensional longitudinal vibration wave equation is shown in Equation (1).

$$\frac{{\partial }^2\xi }{\partial x^2}+k^2\xi =0$$

(1)

where $\xi$ is the particle displacement function, $k=\omega /c$ is the circular wave number, $c$ is the longitudinal vibration speed in the cylindrical rod, and $\omega$ is the vibration frequency.

Figure 4 is a schematic diagram of the ultrasonic igniter structure with the node-plane located in the front mass, where $L_1$ is the length of the front mass, $L_2$ is the thickness of the piezoceramics stack, $L_3$ is the length of the rear mass, and $L_4$ and $L_5$ are the lengths of the thick section and thin section of the stepped-type horn. Generally, the front mass and the stepped-type horn are made as one, namely, $L_1=0$. Because the nodal plane location displacement is close to 0 mm, the flange of the igniter is set at the nodal plane location to make the thin section of the stepped-type horn enter the combustion chamber. Therefore, the flange is set at the interface between the piezoceramics stack and the thick section of the stepped-type horn. According to the quarter-wavelength theory, the igniter is divided into a quarter-wavelength vibrator and a quarter-wavelength horn. Ignoring the influence of prestressed bolt and electrode slices, the frequency equations of a quarter-wavelength vibrator and a quarter-wavelength horn are shown in Equations (2) and (3). Through theoretical calculation and analysis, the igniter is designed in

Table 2. The initial dimensions of igniter.

Parameter	Value
F (kHz)	35
L3 (mm)	4.54
L4 (mm)	3.13
L5 (mm)	39.39
R (mm)	1.66
D (mm)	7.44

.

$$(Z_3/Z_2)\tan k_2{L}_2\tan k_3{L}_3+(Z_3/Z_1)\tan k_1{L}_1\tan k_3{L}_3+(Z_2/Z_1)\tan k_1{L}_1\tan k_2{L}_2=1$$

(2)

$$\tan k_4{L}_4\tan k_5{L}_5=Z_4/Z_5$$

(3)

where Z₁, Z₂, Z₃, Z₄, and Z₅ are, respectively, the equivalent impedances of the front mass, the piezoceramics stack, the rear mass, the thick section, and the thin section of the stepped-type horn, and $k_i(i=1, 2, 3, 4, 5)$ are respectively the circular wave number of the front mass, the piezoceramics stack, the rear mass, the thick section, and the thin section of the stepped-type horn.

3. Verification of Finite Element Model

The high-energy ultrasonic igniter is designed based on a piezoelectric transducer. The FEM of the latter is also suitable for the igniter. ANSYS has been regarded as the authoritative software for the finite element simulation design and analysis of piezoelectric transducers [26,27]. Because COMSOL Multiphysics has unique advantages in the multi-field coupling and structure parameters optimization of piezoelectric transducers, the longitudinal resonant frequency obtained from ANSYS modal analysis has been used as the basis for evaluation. To evaluate the accuracy of the COMSOL Multiphysics finite element simulation calculation, Figure 5 shows the thin end face vibration amplitude (TEVA)-frequency curve of the stepped-type horn obtained by the simulations of the two pieces of software, and the frequency corresponding to the maximum amplitude is the longitudinal resonant frequency of the igniter.

The longitudinal vibrational modal shapes of the igniter at a frequency of 30 kHz are calculated by using the simulation method of reference [28]. As shown in Figure 6, its longitudinal resonant frequency is 29.98 kHz. Figure 7 shows the longitudinal vibrational modal shapes obtained by simulation with COMSOL Multiphysics for an igniter with the same structural dimensions and grid size: its longitudinal resonant frequency is 29.906 kHz. The longitudinal resonant frequency relative error of the two pieces of software is less than 1%, and both of them coincide well with the design frequency of 30 kHz. Therefore, COMSOL Multiphysics can ensure the scientificity and accuracy of the FE analysis.

4. Vibration Characteristic Analysis

“Solid Mechanics” physics and “Electrostatic” physics in COMSOL Multiphysics are chosen to analyze the dynamic characteristics of the igniter. “Piezoelectric Effect” is used to couple the “Solid Mechanics” and “Electrostatics” equations solved in the PZT-4 domains via the linear constitutive equations that model the piezoelectric effect by coupling stresses and strains with the electric field and electric displacement. When setting the properties of the piezoelectric material in the solution domain, the piezoelectric parameters based on the stress-charge material are selected, namely the E-type piezoelectric equation shown in Equations (4) and (5). The electrostatic interface solves the PZT-4 according to Equations (6) and (7). The piezoceramics are stacked alternately, the ground potential is equal to 0 V, and the terminal potential is set to 30 V.

$$T=c_E\cdot S-e^t\cdot E$$

(4)

$$D=e\cdot S+{\epsilon }_s\cdot E$$

(5)

$$\nabla \cdot D=p_v$$

(6)

$$E=-\nabla V$$

(7)

where $T$ is the stress vector and $D$ denotes the electric flux density vector, $S$ expresses the strain vector, and $E$ is the electric field intensity vector; $c_E$ represents the elasticity matrix for constant electric field, $e$ is the piezoelectric stress matrix, and ${\epsilon }_S$ is the dielectric matrix; and $\nabla \cdot D$ is the electric charge density, $p_v$ is the electric charge concentration, and $E$ is the electric field due to the electric potential $V$.

Due to the significant impact of bolt pretension on the performance of the igniter, 12.9-grade alloy steel bolts have been selected in the simulation model. The range of piezoceramics pretension is generally 3000–3500 N/cm². A pretension force of about 3600 N has been applied. As a result of the prestress in the igniter, the harmonic variation of stress and other physical quantities during vibration takes place on top of the static bias stress. Hence, we need to solve this model using a two-step approach, where the first step involves solving for the static stress distribution using a “Stationary Study” step. The solution from this step is then used as a linearization point for solving the vibration problem in the “Frequency Domain Perturbation Study” step and the “Eigenfrequency Study” step [29]. With the COMSOL Multiphysics software, grid independence verification has been performed by contrasting the longitudinal resonant frequency under different grid scales. Base grid sizes from 0.8 mm to 1.5 mm are selected. The results of the grid independence verification are shown in Figure 8. The maximum relative error of the longitudinal resonant frequency is less than 1%, which indicates the grid independence is well verified. Considering the calculation accuracy and time cost, 1.3 mm is used as the grid size of the finite element model of igniter in this study. This model has 38,623 elements in total, and the minimum element quality of the mesh is 0.24. Figure 9 depicts the mesh model of the igniter.

The longitudinal vibration modal and TEVA-frequency curve can be obtained by Eigenfrequency analysis and Frequency Domain Perturbation analysis using COMSOL Multiphysics, as shown in Figure 10 and Figure 11. It can be seen that the longitudinal resonant frequency of the igniter is 35.668 kHz, slightly higher than the theoretical design frequency of 35 kHz. Therefore, it needs to be modified to achieve optimal performance.

5. Establishment of Approximate Model

5.1. Calculation Method of Vibration Velocity Ratio

The practice has proved that the simplified theoretical analysis model can reflect the working state of the igniter to a large extent. To simplify the structure of the igniter and reduce the energy loss, the stepped-type horn and front mass are designed as a whole. Based on one-dimensional longitudinal vibration theory and the longitudinal boundary conditions and the influence of the prestressed bolt, electrodes slices, rounded corners, and flange, the electromechanical equivalent circuit of the igniter can be obtained, as shown in Figure 12 [30].

For simplicity, the forces on the front end and back end of the igniter can be neglected, $F_f=F_b=0$. In Figure 12, C and N are the clamped capacitance and electromechanical transformation coefficient of the piezoceramics stack, as shown in Equations (8) and (9).

$$C=p^2{S}_2{\epsilon }_{33}^T(1-k_{33}^2)/L_2$$

(8)

$$N=p{d}_{33}{S}_2/(s_{33}^E{L}_2)$$

(9)

where $p=4$ is the number of piezoceramics rings; $S_2$ and $L_2$ are the cross-sectional area and length of the piezoceramics stack; $d_{33}$, ${\epsilon }_{33}^T$, $s_{33}^E$, and $k_{33}$ are piezoelectric constant, free dielectric constant, elastic compliance constant, and electromechanical coupling coefficient; $V$ is the voltage applied to the piezoceramics stack; $V_b$ is the longitudinal vibration speed at the outer surface of the back mass; $V_f$ is the longitudinal vibration velocity of the outer surface of the thin section of the stepped-type horn; $V_1$ is the longitudinal vibration velocity at the interface between the back mass and the piezoceramics stack; $V_2$ is the longitudinal vibration velocity of the interface between the thick section of stepped-type horn and the piezoceramics stack; $V_3$ is the longitudinal vibration velocity of the interface between the thick section and the thin section of the stepped-type horn; $Z_{p1}$, $Z_{p2}$, and $Z_{p3}$ are the equivalent impedances of the piezoceramics stack; $Z_{fl}$ and $Z_{bl}$ are the load impedances of the igniter, namely $Z_{fl}=Z_{bl}=0$; $Z_{i1}$, $Z_{i2}$, and $Z_{i3}$ ($i=3, 4, 5$) are the back mass, thick section, and thin section of the stepped-type horn; and the equivalent impedances of the thick section and the thin section of the stepped-type horn are calculated as shown in Equations (10) and (11).

$$Z_{i1}=Z_{i2}=j{\rho }_i{c}_i{S}_i\tan (k_i{L}_i/2)$$

(10)

$$Z_{i3}=-j{\rho }_i{c}_i{S}_i\sin k_i{L}_i$$

(11)

where ${\rho }_i(i=2, 3, 4, 5)$ are the densities of the piezoceramics stack, back mass, thick section, and thin section of the stepped-type horn. $c_i(i=2, 3, 4, 5)$ are the sound speeds of the piezoceramics stack, back mass, thick section, and thin section of the stepped-type horn. $S_i(i=2, 3, 4, 5)$ are the cross-sectional areas of the piezoceramics stack, back mass, thick section, and thin section of the stepped-type horn.

The impedances of each igniter component are expressed in Equations (12)–(14). The calculation of the longitudinal vibration velocity is shown in Equations (15)–(18), and the calculation of the vibration velocity ratio of the front and back mass of the igniter is shown in Equation (19).

$$Z_3=\frac{Z_{31}{Z}_{33}}{Z_{31}+Z_{33}}+Z_{32}$$

(12)

$$Z_4=\frac{Z_{43}(Z_5+Z_{42})}{Z_{43}+Z_5+Z_{42}}+Z_{41}$$

(13)

$$V_3\cdot (Z_{42}+Z_5)=(V_2-V_3)\cdot Z_{53}$$

(14)

$$V_f\cdot Z_{52}=(V_3-V_f)\cdot Z_{53}$$

(15)

$$V_3\cdot (Z_{42}+Z_5)=(V_2-V_3)\cdot Z_{53}$$

(16)

$$V_1\cdot (Z_{p1}+Z_3)=V_2\cdot (Z_{p2}+Z_4)$$

(17)

$$V_b\cdot Z_{31}=(V_1-V_b)\cdot Z_{33}$$

(18)

$$M=\left|\frac{V_f}{V_b}\right|=\left|\frac{Z_{53}\cdot Z_{43}\cdot (Z_{p1}+Z_3)\cdot Z_3}{(Z_{52}+Z_{53})\cdot (Z_{42}+Z_5+Z_{43})\cdot (Z_{p1}+Z_4)\cdot (Z_3+Z_{33})}\right|$$

(19)

5.2. Selection of Design Variables

The optimization of the igniter has been preceded by a sensitivity analysis. The major purpose of the analysis is to identify the correlation between the values of the variable and the objective variable. The closer the correlation coefficient value is to 1, the stronger the correlation degree is with the objective variables. The sensitivity values are based on the Grey Relational Analysis method [21]. The dimensions of the igniter have an important influence on its performance [31,32]. In this paper, $L_3$, $L_4$, $L_5$, R, and D are selected as the design variables, with 30 simulations and calculations, and the results of the sensitivity analysis are presented in Figure 13. Therefore, it can be concluded that $L_3$, $L_4$, $L_5$, R, and D are the major variables. In Figure 13, M is the ratio of the igniter’s front-end vibration velocity to its back-end vibration velocity, and f denotes the longitudinal resonant frequency.

5.3. Objective Functions and Constraints

The longitudinal resonant frequency f is an important parameter of the igniter. Only when igniter is driven at its longitudinal resonant frequency can the maximum amplitude of the output end achieve the maximum values. The vibration velocity ratio M reflects the transmitted sound power of the igniter, and the transmitted sound power has a positive ratio with the vibration velocity ratio. In order to improve the comprehensive performance of the igniter, f and M are taken as the optimization objectives. The optimization of igniter can be viewed as a multi-objective optimization problem and shown in Equation (20).

$$\begin{array}{l}\mathrm{Min}{\left[\frac{1}{\left|M\right|},\left|f-35000\right|\right]}^{\mathrm{T}}\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{c}4.05\le L_3\le 4.95\\ 3.03\le L_4\le 3.25\\ 38.32\le L_5\le 39.88\\ \begin{array}{l}7.2\le D\le 8.8\\ 1\le R\le 3\end{array}\end{array}\end{array}$$

(20)

where $M$ is the ratio of the front-end vibration velocity to the back-end vibration velocity of the igniter; $f$ denotes the longitudinal resonant frequency; R is the radius of the rounded corners between the stepped-type horn’s thick section and thin section. D is the diameter of the stepped-type horn’s thin section.

5.4. Approximate Model Validation

The design of the experiment for design points is implemented based on the OLHD method. Thirty groups of design points were set in COMSOL Multiphysics [32]. In addition, for the convenience of expression, the diameter of the stepped-type horn thin section, the length of the back mass, the length of stepped-type horn thick and thin section, and the radius of rounded corners were replaced by x₁, x₂, x₃, x₄, and x₅. The approximate second-order response surface model was established by calculating their corresponding longitudinal resonant frequencies and vibration velocity ratios, giving the following expressions:

$$\begin{array}{l}f=-227.19736+1.8921x_1-1.4866x_2+13.2063x_3+12.5087x_4+12.4241x_5\\ -0.1368x_1^2-0.4803x_2^2-0.29406x_3^2-8.13305x_4^2-0.16808x_5^2-0.003628x_1{x}_2\\ -0.09705x_1{x}_3+0.01087x_1{x}_4-0.012x_1{x}_5-0.215x_2{x}_3+0.6418x_2{x}_4+0.1345x_2{x}_5\\ -1.564x_3{x}_4-0.205x_3{x}_5+0.2898x_4{x}_5\end{array}$$

$$\begin{array}{l}M=-1131.63+3.8615x_1+20.8285x_2+18.977x_3+83.4454x_4+51.1719x_5\\ -0.4153x_1^2-2.0981x_2^2-2.005x_3^2-29.6261x_4^2-0.6379x_5^2-0.0279x_1{x}_2\\ -0.04614x_1{x}_3-0.7383x_1{x}_4-0.02687x_1{x}_5+0.06317x_2{x}_3+1.0165x_2{x}_4\\ -0.16617x_2{x}_5-0.4283x_3{x}_4-0.0152x_3{x}_5-0.42154x_4{x}_5\end{array}$$

Cross-validation was used to test the prediction accuracy of the second-order response surface model [33]. Figure 14a,b shows the simulated value of the 10 verification points versus the predicted values from the developed response surface model, where $f_{pre}$ is the predicted value of longitudinal resonant frequency and $f_{act}$ is the simulated value of the longitudinal resonant frequency. $M_{pre}$ is the predicted value of the vibration velocity ratio, and $M_{act}$ is the simulated value of the vibration velocity ratio. The closer the verification points are to the diagonal line, the better the response surface model fits the points. The accuracy of the response surface model was evaluated using the coefficient of determination (R²), average error, maximum error, and root mean square error. The error analysis results are listed in

Table 3. Error analysis results of the response surface model.

	Average Error	Maximum Error	Root Mean Square Error	Determination Coefficient R2
f	0.05693	0.09602	0.06052	0.9602
M	0.03989	0.0701	0.04311	0.9757
Acceptance level	<0.2	<0.3	<0.2	>0.9

. Thus, it can be concluded that the response surface model has sufficient accuracy, demonstrating that all variations can be explained with the developed response model.

6. Multi-Objective Deterministic Optimization Analysis

The NSGA-II algorithm has advantages in finding Pareto optimal solutions to multi-objective optimization problems [34,35]. The multi-objective optimization processes of the igniter have been illustrated in Figure 15. The initial population number of the NSGA-II algorithm was set as 12, and the multi-objective Pareto optimal solution was obtained after 20 iterations. The red dots represent the Pareto front of the deterministic optimization in Figure 16. A comprehensive optimum is selected from the Pareto front. A group of the optimal dimensions of the igniter is listed in

Table 4. The optimal dimensions of the igniter.

	Initial Value	Optimal Value
L3 (mm)	4.54	4.47
L4 (mm)	1.13	1.13
L5 (mm)	39.39	39.57
R (mm)	1.66	1.5
D (mm)	7.44	7.42

.

An Eigenfrequency analysis of the optimized igniter has been conducted to obtain the longitudinal vibration frequency and the corresponding vibration modal. The results show that the longitudinal resonant frequency is 35.01 kHz, which coincides well with the objective frequency of 35 kHz. Frequency Domain Perturbation analysis has been conducted to study the dynamic characteristics of the igniter under a sinusoidal excitation voltage of 30 V. Figure 17, Figure 18 and Figure 19 indicate the longitudinal vibration modal, the displacement distribution, and the TEVA-frequency curve of the igniter, respectively. It can be included from Figure 18 that the wave node is close to the flange location, which can be used to fix without affecting the working of the igniter. Using Matlab to calculate the vibration velocity ratio yields a ratio of 2.42, which is about 1.2% higher than that before optimization, ensuring that most of the ultrasonic energy is transmitted to the combustion chamber.

7. Multi-Objective Robustness Optimization Analysis

Owning to the limitations of the levels of processing and manufacturing, the sizes of the igniter’s structures will fluctuate, leading to deviations in its performance. However, deterministic optimization ignores these uncertain factors. It is necessary to conduct a robustness analysis. The Isight software integrated response surface approximation model was used, and the deterministic optimization results were used as input conditions for the Mean Value Method Reliability analysis. The probability distribution of the design variables and the objective variables was set as the normal distribution, and the coefficient of variation was set as 0.01. The results show that the Sigma level of the longitudinal resonant frequency is 4.015, as shown in Figure 20. The failure risk of the product is high, and a robustness optimization design is needed.

The NSGA-II algorithm was used to optimize the Six Sigma robustness model, and the optimized design scheme is shown in

Table 5. Six Sigma analysis results.

Parameter	Initial Value	Deterministic Optimization		Robust Optimization
		Result	Sigma Level	Result	Sigma Level
L3 (mm)	4.54	4.47	1.23	4.25	3.76
L4 (mm)	3.13	3.13	8	3.14	8
L5 (mm)	39.39	39.57	8	38.93	8
R (mm)	1.66	1.5	8	1	8
D (mm)	7.44	7.42	8	7.88	8
f (kHz)	35.668	35.01	4.015	34.976	6.088
M	1.935	2.084	6.58	2.086	6.188

. Figure 21, Figure 22 and Figure 23 show the results for the robustness optimization design. Figure 24 shows the Sigma level of the longitudinal resonant frequency in the robustness optimization. Compared with the deterministic optimization design results, the relative difference of the longitudinal resonant frequency has little variation, which is better than the optimization results of literature [21,36], the vibration velocity ratio is raised by 15.7%, and the Sigma levels of both are above Six Sigma. Therefore, the results indicate that the proposed approach could effectively solve igniter optimization problems.

8. Conclusions

Using a Six Sigma multi-objective reliability-based optimization design method, a novel high-energy ultrasonic igniter has been designed. The proposed method can provide the necessary guidance for the optimization design of the high-energy ultrasonic igniter. Simultaneously, it has an important reference value for a new special piezoelectric transducer to improve research and development efficiency.

The initial dimensions of different components of the high-energy ultrasonic igniter have been determined using the one-dimensional longitudinal vibration theory combined with the boundary conditions. The multi-field coupling prestressed modal analysis of the igniter was carried out using COMSOL Multiphysics. The longitudinal resonant frequency of the igniter is 35.668 kHz, and the relative error with the design target frequency is 1.4%. The length of the thin section of the stepped-type horn has the biggest impact on the resonant frequency and vibration velocity ratio by means of Grey Relational Analysis. An approach based on the OLHD method, FEM, the response surface model, and NSGA-II is used to complete the deterministic optimization design and the Six Sigma robustness optimization design. The robustness optimization results show that the dynamic characteristics of the igniter have been greatly improved. The relative difference of the longitudinal resonant frequency is just 0.07%. At the same time, the vibration velocity ratio is increased by 15.7%. The optimal igniter has better reliability and quality levels, a much smaller resonant frequency shift, and a higher vibration velocity ratio.

Future work will be focused on developing the direct ultrasonic ignition and auxiliary catalytic ignition control methods in which ultrasound actively feeds into the internal engines and the constant-volume combustion bomb.