Reliability Analysis of Pyrotechnic Igniter for Hydrogen-Oxygen Rocket Engine with Low Temperature Combustion Instability Failure Mode

Abstract

To evaluate the functional reliability of the pyrotechnic igniter in the failure mode of unstable combustion at low temperature, a reliability and reliability sensitivity analysis method based on the combination of an interior ballistic model and Kriging reliability method is proposed. Through the deterministic interior ballistic simulation, the failure mode of low temperature unstable combustion of the pyrotechnic igniter is examined, while the random variables are introduced to establish the ignition nonlinear implicit function of the pyrotechnic igniter. The ignition display function of the pyrotechnic igniter is established by the Kriging model, which avoids the repeated calculation of true limit state function values. This study provides an efficient approach to evaluate the ignition reliability of the pyrotechnic igniter and compared with the traditional Monte Carlo method to verify the accuracy of the results. Finally, reliability-based sensitivity indices are presented to quantify the significance of random parameters. It is shown that the influence of the uncertainties can be precisely described, and the diameter of the nozzle plays a dominant role in ignition reliability. Additionally, ignition experiments of nozzles with different diameters were performed to verify the result of sensitivity. This can further support the detailed design of the pyrotechnic igniter.

1. Introduction

The pyrotechnic igniter is an ignition device of the hydrogen-oxygen rocket engine, which is widely used in aerospace fields. The success or failure of ignition is crucial to the normal operation of the hydrogen-oxygen rocket engine [1,2,3]. A well-designed igniter will burn steadily under any condition. In order to improve the ignition reliability of the igniter, it is necessary to design the igniter in the low-temperature unstable combustion failure mode. In order to characterize the pyrotechnic igniter output, M. N. Skaggs et al. [4] used high-speed schlieren imaging to observe critical features of the post-combustion flow and designed a high-speed schlieren imaging diagnostic system for observing subsonic and supersonic multiphase material motion from the initiation of two types of titanium-based pyrotechnic igniters into the surrounding environment. Juyoung Oh et al. [5] introduced the effects of heating and humidity on ageing of a NASA standard pyrotechnic igniter. Calculations using Van’t Hoff equation indicate that moisture shortened the lifespan of the unaged zirconium potassium perchlorate (ZPP) up to about 85% under extreme relative humidity conditions, while significantly deteriorating the heat of reaction, sensitivity, and thus increased the risk of a misfire. Abdullah Ulas et al. [6] studied the performance of a boron/potassium-nitrate based pyrotechnic igniter. Kanagaraj Gnanaprakash et al. [7] studied ignition and combustion behavior of pyrotechnic materials based on zirconium as fuel and investigated potassium perchlorate or iron oxide as oxidant for various aging conditions. Qian Zhong et al. [8] fabricated the boron/potassium nitrate by electrostatic spraying to enhance their combustion efficiency and capability of igniting target materials and used the closed chamber tests to record the process of boron/potassium nitrate igniting guanidine nitrate/basic cupric nitrate gas generator. The aforementioned research studies on the pyrotechnic igniter are simply deterministic simulation analysis. However, ignition of the pyrotechnic igniter is a random process with the uncertainties of working loads, geometrical dimensions, as well as material properties, especially as low-temperature igniting is more prone to failure. Owing to the increased application of the pyrotechnic igniter in the field of aerospace, the reliability requirements are increasing. It is necessary to conduct an ignition reliability analysis of the pyrotechnic igniter to understand the influence of the material properties, structural size, machining error, and other factors on its igniting reliability, and appropriate design measures should be proposed to improve the ignition reliability level.

Although the importance of reliability has been widely accepted, the reliability analysis is still a complicated task to reach. One of the main challenges is that, due to the highly nonlinear and coupling relations among variables, it is difficult for a complex structure to establish an explicit function between basic random variables and the structural response by using analytical methods [9]. Therefore, the ignition reliability analysis of the pyrotechnic igniter is an implicit function problem. As a result, numerical simulations have been introduced to establish the implicit function. Researchers have proposed many methods to improve the computational efficiency in solving this problem. According to probabilistic theory, the Monte Carlo Simulation (MCS) [10,11] has the highest accuracy for complex and highly nonlinear systems. However, the MCS requires a large sample size to achieve high simulation credibility, and it is inefficient, consuming extensive computational time. In order to improve the calculation efficiency of the MCS, the surrogate model is mainly used to establish a response model to improve the efficiency of reliability analysis. Commonly used surrogate models include response surfaces [12,13], support vector machines [14,15], neural networks [16,17], and the Kriging model [18,19,20]. This paper focuses on the Kriging model due to its validity being independent of the existence of random errors, and it has a good fitting effect for the problems of high nonlinearity and sudden changes in the local response. The Kriging surrogate model is used for global and local approximations [21,22,23,24,25]. When it comes to the pyrotechnic igniter, however, the current method to evaluate the reliability of the pyrotechnic igniter is mainly through ignition tests, which are time-consuming and expensive. To our knowledge, there are few numerical studies on the ignition reliability analysis of pyrotechnic igniters.

Therefore, it is essential to develop an efficient and accurate numerical way to conduct the ignition reliability analysis for the pyrotechnic igniter. Based on coupling the interior ballistic model and Kriging model, this paper proposes an ignition reliability analysis procedure of the pyrotechnic igniter concerning the failure mode of unstable combustion at low temperature. The method is applied to evaluate the reliability of a pyrotechnic igniter and is verified by comparing reliability results with that of MCS. The importance ranking of design variables extracted from propellant combustion parameters, geometric size, and material properties are obtained with sensitive analysis to assist the pyrotechnic igniter design and optimization.

The remainder of this paper is organized as follows. In Section 2, the working principle of the pyrotechnic igniter is introduced, and the function failure mode of a pyrotechnic igniter is defined to establish an implicit function. Section 3 conducts the function deterministic simulation and failure modes analysis of the pyrotechnic igniter based on the interior ballistic model. In Section 4, the Kriging model method is constructed to display the function, and the effects of independent random parameters on reliability and reliability sensitivity are developed. Then, by comparing with the results of MCS, the validity of this method is verified. To validate the result of reliability sensitivity, igniting experiments were performed. Finally, the conclusions are provided in Section 5.

2. Failure Analysis of Low-Temperature Unstable Combustion of the Pyrotechnic Igniter

2.1. Basic Structure and Working Principle

The structure of a pyrotechnic igniter is shown in Figure 1. The pyrotechnic igniter includes the shell, electric igniter, ignition powder, igniter case, propellant, diaphragm, nozzle, and other parts. The igniter case is filled with the boron potassium nitrate (BPN) and composite modified double-base propellant (CMDB). The main charge is the CMDB in main charge chamber. The ignition process of the pyrotechnic igniter can be divided into the combustion of the BPN and CMDB in the igniter case and the combustion of the CMDB in the main charge chamber.

During the combustion of the BPN and CMDB in the igniter case, the BPN is thermally initiated through contact with the electric igniter, and the BPN combustion releases high-energy particles to ignite CMDB. The energy released by the combustion of the BPN and CMDB breaks through the aluminum foil at the bottom of the igniter case, and then ignites the CMDB in the main charge chamber. During the combustion of the CMDB in the main charge chamber, the CMDB combustion generates a large amount of gas. After breaking the diaphragm, the flame enters the gas generator and ignites the liquid hydrogen and liquid oxygen mixture in the OHRM.

2.2. Failure Mode

This type of pyrotechnic igniter can be ignited normally at room temperature and high temperature. However, the pyrotechnic igniter shows unstable combustion at low temperatures, as shown in Figure 2. There are many factors that affect the combustion stability of the pyrotechnic igniter, mainly including propellant compositions, geometric size and other factors. However, these factors usually have inherent dispersion, so that the pyrotechnic igniter may not be able to achieve its specified functions. It is necessary to quantify the influence of the input random variables on the ignition performance of the pyrotechnic igniter. If the design parameters cannot meet the requirements, their reliability should be improved according to the sensitivity results.

The working performance output of the pyrotechnic igniter is a pressure-time parameter, i.e., P-t curve, as shown in Figure 3. In Figure 3, P_m is the peak pressure of combustion stage of the BPN and CMDB, P_i is the end pressure of combustion stage of the BPN and CMDB, t_s is the starting time of combustion stage of the BPN and CMDB, t_v is the working combustion time of the pyrotechnic igniter, I is the ignition impulse. The igniting impulse considers the pressure and working time in the combustion chamber, and is also an important monitoring index of propellant combustion energy. Therefore, the ignition impulse is taken as the sensitive parameter to evaluate the working performance of igniter. I is as follows [26]:

$$I={\displaystyle {\mathsf{\int }}_0^{t_s}p(\mathit{x})d\mathit{x}}$$

(1)

According to the stress–strength interference model [27], the ignition fails when the I is less than the I_min. In this model, the residual strength is defined as:

$$I(\mathit{x})=I-I_{\min }={\displaystyle {\mathsf{\int }}_0^{t_s}p(\mathit{x})}d\mathit{x}-I_{\min }$$

(2)

where, the I_min is the minimum ignition impulse and obtained from a normal ignition experiment at low temperature, I_min = 0.87.

3. Deterministic Simulation of Low Temperature of the Pyrotechnic Igniter

3.1. Simulation Model of the Pyrotechnic Igniter

According to the classical interior ballistic theory [28,29] and the working principle of the pyrotechnic igniter, the low-temperature ignition of the pyrotechnic igniter is simulated by using the Simulink module of MATLAB. The interior ballistic equations are as follows.

To simplify the combustion model, the shape functions of gun powder are constructed using Equations (3) and (4) [30].

$$\psi =\chi Z(1+\lambda Z+\mu Z^2)$$

(3)

$$\sigma =1+2\lambda Z+3\mu Z^2$$

(4)

where χ, λ, and μ are the shape characteristic quantities of the propellant, ψ is the combustible relative mass of the propellant, σ is the relative surface area of the combustible powder, Z = e/e_l represents the relative thickness, e is the arc thickness at any time, and e_l is the starting arc thickness.

Burning equation of the gun powder based on Saint Robert’s law is used to calculate the pressure of the gas inside the chamber [31]:

$$r_b=\frac{de}{dt}=u{P}^n$$

(5)

where P, n, and u are the pressure, burning rate-pressure exponent and burning rate coefficient.

The mass generation rate of the gas from the BPN and CMDB are estimated as follows [32,33]:

$${\dot{m}}_{gen(·)}={\eta }_{g(·)}{m}_{p(·)}\frac{d{\psi }_{(·)}}{dt}$$

(6)

where η_g is the mass fraction of gas in the combustion product of the propellant, m_p is the loading mass of the propellant and $(·)$ represent different types of gun powder.

The device has good sealing property and there is no mass exchange between the system and the outside world. Hence, the law of conservation of mass in a control volume of the combustion chamber is given by the following equation:

$${\dot{m}}_{IC}={\dot{m}}_{genIC}-{\dot{m}}_{portIC}$$

(7)

$${\dot{m}}_{MC}={\dot{m}}_{genMC}+{\dot{m}}_{portIC}-{\dot{m}}_{portMC}$$

(8)

where ${\dot{m}}_{IC}$ is the gas mass change rate in the igniter case. ${\dot{m}}_{MC}$ is the gas mass change rate in the main charge chamber. ${\dot{m}}_{genIC}$ is the mass production rate of propellant combustion products in igniter case, ${\dot{m}}_{genMC}$ is the mass production rate of propellant combustion products in the main charge chamber. ${\dot{m}}_{portIC}$ is the rate of gas flow from ignition chamber to main charge chamber. ${\dot{m}}_{portMC}$ is the rate of gas flow from main charge chamber to nozzle.

According to the gas flow theory [34], ${\dot{m}}_{port}$ is calculated by the following equation:

$${\dot{m}}_{port}=\{\begin{array}{lll}{P}_{IC}{A}_e\times \sqrt{\frac{2k}{R{T}_1(k-1)}\left[{(\frac{P_{IC}}{P_{MC}})}^{-\frac{2}{k}}-{(\frac{P_{IC}}{P_{MC}})}^{-\frac{k+1}{k}}\right]}& \mathrm{if}& {(\frac{P_{IC}}{P_{MC}})}^{-\frac{2}{k}}<{(\frac{k+1}{2})}^{\frac{k}{k-1}}\\ P_{IC}{A}_e\times \sqrt{\frac{k}{R{T}_1}{(\frac{2}{k+1})}^{\frac{k-1}{k+1}}}& \mathrm{if}& (\frac{P_{IC}}{P_{MC}})\ge {(\frac{k+1}{2})}^{\frac{k}{k-1}}\end{array}$$

(9)

where A_e is the cross-sectional area of the nozzle, k is the specific heat ratio of the fuel gas, R is a gas constant, P_IC is ignition case pressure, P_MC is the pressure of the main charge chamber.

The energy conservation relation of the internal gas and can be expressed as Equations (10) and (11) [35].

$$\frac{d(m_{IC}{c}_{v\left(IC\right)}{T}_{IC})}{dt}={\dot{m}}_{gen(·)}{c}_{p(·)}{T}_{f(·)}-{\dot{m}}_{port\left(IC\right)}{c}_{p\left(IC\right)}{T}_{IC}-{\dot{Q}}_{loss\left(IC\right)}$$

(10)

$$\frac{d(m_{MC}{c}_{v(MC)}{T}_{MC})}{dt}={\dot{m}}_{gen(·)}{c}_{p(·)}{T}_{f(·)}+{\dot{m}}_{port(IC)}{c}_{p(IC)}{T}_{IC}-{\dot{m}}_{port(MC)}{c}_{p(MC)}{T}_{MC}-{\dot{Q}}_{loss(MC)}$$

(11)

where c_v and c_p are the specific heats of constant volume and pressure of the gas, respectively, T_f is the explosion temperature at constant pressure, T_IC and T_MC are the temperature of ignition case and main charge chamber, respectively, ${\dot{Q}}_{loss\left(IC\right)}$ and ${\dot{Q}}_{loss\left(MC\right)}$ are the heat exchange rates of ignition case and main charge chamber to the outside world, respectively.

The intermolecular attraction of high temperature and high pressure gas products produced by propellant combustion is very small compared with the pressure in the chamber. The gas products in the ignition case and the main charge chamber obey the Nobel–Abel equation of state. The ideal gas state equations are given by Equations (12) and (13) [36,37].

$$P_{IC}\left(V_{IC}-{\alpha }_{IC}{m}_{IC}\right)=m_{IC}{R}_{IC}{T}_{IC}$$

(12)

$$P_{MC}\left(V_{MC}-{\alpha }_{MC}{m}_{MC}\right)=m_{MC}{R}_{MC}{T}_{MC}$$

(13)

where m_IC and m_MC are the mass of mixed gas in the igniter case and main charge chamber, respectively, α_(·) is the covolume.

Equation (12) is differentiated to obtain Equation (14).

$$\frac{d{T}_{IC}}{dt}=\frac{\frac{d{P}_{IC}}{dt}\left(V_{IC}-{\alpha }_{IC}{m}_{IC}\right)+P_{IC}{\dot{V}}_{IC}-\frac{P_{IC}{V}_{IC}}{m_{IC}}\frac{d{m}_{IC}}{dt}}{m_{IC}{R}_{IC}}$$

(14)

where R can be obtained by Equation (15).

$$R=c_v(k-1)$$

(15)

By substituting Equation (13) and Equation (14) into Equation (11), the change rate of pressure in the ignition cartridge is derived as Equation (16).

$$\frac{d{P}_{IC}}{dt}=\frac{(k-1)({\dot{m}}_{gen(·)}{c}_{p(·)}{T}_{f(·)}-{\dot{Q}}_{loss}-{\dot{m}}_{port\left(IC\right)}{c}_{p\left(IC\right)}{T}_{IC})+P_{IC}{\alpha }_{IC}{\dot{m}}_{IC}-P_{IC}{\dot{V}}_{IC}}{V_{IC}-{\alpha }_{IC}{m}_{IC}}$$

(16)

Similarly, the change rate of pressure in the main charge chamber can be obtained:

$$\frac{d{P}_{MC}}{dt}=\frac{(k-1)({\dot{m}}_{gen(·)}{c}_{p(·)}{T}_{f(·)}-{\dot{Q}}_{loss\left(IC\right)}-{\dot{m}}_{port\left(MC\right)}{c}_{p\left(MC\right)}{T}_{MC}+{\dot{m}}_{port\left(IC\right)}{c}_{p\left(IC\right)}{T}_{IC})+P_{IC}{\alpha }_{IC}{\dot{m}}_{IC}-P_{IC}{\dot{V}}_{IC}}{V_{IC}-{\alpha }_{IC}{m}_{IC}}$$

(17)

Due to the combustion of gun powder, the volume of the combustion chamber will change. The change rate of the volume of the ignition case and the main charge chamber can be calculated by Equation (18).

$${\dot{V}}_{(·)}={\dot{m}}_{gen(·)}/{\rho }_{(·)}$$

(18)

where ρ_(·) is the density of different types of solid gun powder.

A simplified model of heat exchange between the propellant products and the outside through the cavity wall is established according to [38], ${\dot{Q}}_{loss(IC)}$ and ${\dot{Q}}_{loss(MC)}$ can be estimated using Equations (19) and (20).

$${\dot{Q}}_{loss\left(IC\right)}=h{A}_{W\left(IC\right)}(T_w-T_{IC})+\sigma A_{W(IC)}(A_r{T}_w^4-E_m{T}_{IC}^4)$$

(19)

$${\dot{Q}}_{loss(MC)}=h{A}_{W(IC)}(T_w-T_{MC})+\sigma A_{W(MC)}(A_r{T}_w^4-E_m{T}_{MC}^4)$$

(20)

where h is a constant convective heat transfer coefficient, T_w is the temperature of the vessel wall, σ_s is the Stefan–Boltzmann constant, A_r is the absorption rate of the vessel wall, E_m is the net emissivity of the product, and A_w is the instantaneous surface area of the vessel side wall in contact with the product.

The combustion parameters applied to this combustion model are listed in

Table 1. Parameters for the analytical combustion model.

Parameter	Value	Parameter	Value
Shape characteristic parameter of the CMDB (χ)	2.5	Shape characteristic parameter of the BPN (χ)	4.19
Shape characteristic parameter of the CMDB (λ)	−0.3	Shape characteristic parameter of the BPN (λ)	−1.28
Shape characteristic parameter of the CMDB (μ)	0	Shape characteristic parameter of the BPN (μ)	0.25
Density of the CMDB particles (ρCMDB)	1.589 g·cm−3	Density of the BPN particles (ρBPN)	1.679 g·cm−3
Explosion temperature of the CMDB (Tf)	3354 K	Explosion temperature of the BPN (Tf)	3321 K
Specific heat ratio of the CMDB (k)	1.87	Specific heat ratio of the BPN (k)	1.31
Burn rate exponent the CMDB (n)	0.22	Burn rate exponent the BPN (n)	0.31
Covolume of the CMDB (αg)	6.4 × 10−4 m3/kg	Covolume of the BPN (αg)	1.3 × 10−4 m3/kg
Convective heat transfer coefficient (h)	1050 W·m−2·K−1	Stefan–Boltzmann constant (σs)	5.67 × 10−8 W·m−2·K−1
Absorption rate of the vessel wall (αw)	0.24	Net emissivity of the product (Em)	0.24
Temperature of the ignition case (TIC) The diameter of the nozzle (d)	233.15 K 3.6 mm	Temperature of the main charge chamber (TIC)	233.15 K

.

3.2. Simulation Result Compared with Experimented Result

To validate the established simulation model, ignition experiments using the developed pyrotechnic igniter were conducted. The experiment concept is described in Figure 4. Figure 5 shows photographs of the experiment setup. The strain sensor is used to test the pressure of the ignition chamber. Three ignition experiments were performed. As seen from Figure 6, the simulation result is in accordance with the experimental result.

Table 2. Comparison between the Simulation model and the experiment characteristic parameters.

Parameter	Pm/MPa	Pi/MPa	ts/s	tv/s
Simulation model	11.07	6.17	0.16	2.14
Experiment 1	10.76	5.38	0.15	2.17
Experiment 2	10.48	5.60	0.14	2.23
Experiment 3	11.49	6.81	0.13	1.91
Difference with respect to the average value of exp. Results (%)	1.35	4.04	14.28	1.90

summarizes the characteristic parameter of the simulation model and experiment, showing quite small relative differences between the experimental and analytical results.

4. Reliability and Reliability Sensitivity Analysis Method of the Igniter

4.1. Kriging Model of the Igniter

The limit state functions I(x) are highly nonlinear and implicit. The Kriging model with a three-level Full Factorial method (TLFF) [39] is employed to predict the limit state functions due to its high computational accuracy and efficiency. An initial Kriging model is established based on the Design of Experiments (DoE) and the initial DoE is constructed by a TLFF method in this paper. According to the Kriging theory, the output function, $\widehat{I}(\mathit{x})$, and random variable, x, are related as follows [40]:

$$\widehat{I}(\mathit{x})=f^T(\mathit{x})\beta +\epsilon (\mathit{x})$$

(21)

where $\beta ={\left\{{\beta }_1,{\beta }_2,\cdots ,{\beta }_{\mathrm{n}}\right\}}^{\mathrm{T}}$ is the regression coefficient, $f(\mathit{x})={\left\{f_1(x),f_2(x),\cdots ,f_n(x)\right\}}^{\mathrm{T}}$ is the basis function of the Kriging model. $\epsilon (\mathit{x})$ is a random correction process that provides local approximation and obeys the normal distribution, $N(0,{\sigma }^2)$, and the covariance is nonzero. Therefore, the covariance matrix can be expressed as:

$$Cov[\epsilon ({\mathit{x}}_i),\epsilon ({\mathit{x}}_i)]={\sigma }^2[R({\mathit{x}}_i,{\mathit{x}}_j)]$$

(22)

where $R({\mathit{x}}_i,{\mathit{x}}_j)$ represents the correlation function between the sampling points x_i and x_j. This plays a decisive role in the accuracy of the simulation. The correlation function applied in this paper is a Gaussian type expression, as follows:

$$R({\mathit{x}}_i,{\mathit{x}}_j)=\exp (-{\displaystyle \sum _{k=1}^m{\theta }_k{\left|x_k^{\left(i\right)}-x_k^{\left(j\right)}\right|}^2})$$

(23)

where ${\theta }_k(k=1,2,\cdots ,m)$ is an unknown related parameter.

The Kriging approximation model of $\widehat{I}(\mathit{x})$ can be expressed as Equation (24) [41].

$$\widehat{I}(\mathit{x})={\mathit{f}}^T(\mathit{x})\widehat{\mathsf{\beta }}+{\mathit{r}}^T(\mathit{x})R^{-1}(\mathit{I}-\mathit{F}\widehat{\mathsf{\beta }})$$

(24)

where $\widehat{\mathsf{\beta }}$ is the estimated value of $\mathsf{\beta }$, I is the column vector composed of the response value of the experimental design sample data, that is, $\mathit{I}={[I({\mathit{x}}_1),I({\mathit{x}}_2),\cdots ,I({\mathit{x}}_m)]}^T$, F is the m × n order matrix composed of the regression model at m sample points, and r(x) is the correlation function vector between the experimental data sample points and the prediction points and can be expressed as Equation (25).

$${\mathit{r}}^T(\mathit{x})=\left\{R(\mathit{x},{\mathit{x}}^{(1)}),R(\mathit{x},{\mathit{x}}^{(2)}),\cdots ,R(\mathit{x},{\mathit{x}}^{(m)})\right\}$$

(25)

$\widehat{\mathsf{\beta }}$ and variance estimate ${\widehat{\sigma }}^2$ are given as Equation (26).

$$\begin{array}{l}\widehat{\mathsf{\beta }}={({\mathit{F}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{F})}^{-1}{\mathit{F}}^T{\mathit{R}}^{-1}\mathit{I}\\ {\widehat{\sigma }}^2={(\mathit{I}-\mathit{F}\widehat{\mathit{\beta }})}^T{\mathit{R}}^{-1}(\mathit{I}-\mathit{F}\widehat{\mathsf{\beta }})/m\end{array}$$

(26)

The relevant parameter, $\mathsf{\theta }={\left\{{\theta }_1,{\theta }_2,\dots ,{\theta }_m\right\}}^T$, is obtained by calculating the maximum value of the maximum likelihood estimation, as shown in Equation (27).

$$\max F(\mathsf{\theta })=-\frac{m\ln ({\widehat{\sigma }}^2)+\ln \left|\mathit{R}\right|}{2},{\theta }_k\ge 0(k=1,2,\cdots ,m)$$

(27)

Equations (2) and (24) are combined to obtain the display function given by Equation (28):

$$\widehat{G}(\mathit{x})=\widehat{I}(\mathit{x})-I_{min}$$

(28)

when the approximated performance function Equation (28) is used for probabilistic evaluation, the failure probability can be calculated as:

$$P_f={\displaystyle {\mathsf{\int }}_{F}f(\mathit{x})}d\mathit{x}={\displaystyle {\mathsf{\int }}_{R^n}{I}_F(\mathit{x})f}(\mathit{x})d\mathit{x}=E\left[I_F(\mathit{x})\right]$$

(29)

$f(\mathit{x})$ is the joint probability density function of the random vector x. $I_F(\mathit{x})\{\begin{array}{l}1,\mathit{x}\in F\\ 0,\mathit{x}\notin F\end{array}$ is the indication function of the failure domain, and $F=\left\{\mathit{x}:\widehat{G}(x)\le 0\right\}$.

According to the basic principle of MCS [42], the estimated probability of failure is:

$${\widehat{P}}_f=\frac{1}{N}{\displaystyle \sum _{i=1}^N{I}_F({\mathit{x}}_i)}=\frac{{\mathit{N}}_F}{\mathit{N}}$$

(30)

where N is the number of sample points extracted from the joint probability density function of random variables, N_F is the number of sample points falling in the failure domain.

Therefore, the ignition reliability of the igniter under low temperature combustion instability failure mode is:

$$\widehat{R}=1-{\widehat{P}}_f$$

(31)

4.2. Reliability Sensitivity Analysis

According to [43], the parameter sensitivity calculation equation is defined as follows:

$$S_{\phi }=\lambda \frac{\partial \beta }{\partial \mathsf{\phi }}=\lambda {\left[\frac{\partial \mathit{u}}{\partial \mathsf{\phi }}\right]}^{\mathrm{T}}\frac{\partial \beta }{\partial \mathit{u}}=\lambda {\left[\frac{\partial T(\mathit{x})}{\partial \mathsf{\phi }}\right]}^T\frac{\partial \beta }{\partial \mathit{u}}$$

(32)

where $\beta$ is the reliability index according to [44], that is, $\mathit{R}=\varphi (\beta )$. T(x) is the transformation function of u and x [45]. $\mathsf{\phi }$ can be the mean values or standard deviations of the random variables, x and $\mathsf{\lambda }={\left[{\lambda }_{ij}\right]}_{n\times n}$, where ${\lambda }_{ij}$ represents the standard deviation of the i^th random variable when i = j, and its value is zero when i ≠ j. u is an independent standard normal random vector, given by Equation (33).

$$\mathit{u}={\displaystyle L_0^{-1}\mathit{y}}$$

(33)

where y is the random variable vector of the equivalent standard normal transformation of x, obtained based on the Nataf transformation [46]. $L_0$ is the lower triangular matrix obtained by the Choleskey decomposition of the correlation coefficient matrix of y.

4.3. Reliability Simulation Implementation of the Pyrotechnic Igniter

According to the above reliability analysis method, the performance reliability simulation of pyrotechnic igniter includes the following steps:

Step 1. Enter the random variable, x, and its distribution type, as shown in

Table 3. Key design parameters for the pyrotechnic igniter.

Symbol	Mean	Std.	Distribution Type
ρBPN (g·cm−3)	1.679	0.02	Normal
ρCMDB (g·cm−3)	1.589	0.01	Normal
nCMDB	0.22	0.01	Normal
d (mm)	3.6	0.007	Normal

;

Step 2. The sample data of m groups are obtained by a TLFF design to reduce the number of experiments. After obtaining m group samples through experimental optimization design, the samples are inputted into the ignition simulation model of the pyrotechnic igniter established by MATLAB to obtain the ignition impulse value I(x). Among them, all MATLAB software packages should be executed in a batch mode to automate the simulation process;

Step 3. The Kriging model, $\widehat{I}(\mathit{x})$, is fitted based on m groups of the sample data;

Step 4. The display function, $\widehat{G}(\mathit{x})$, is established based on the Kriging model of the ignition impulse $\widehat{I}(\mathit{x})$;

Step 5. The reliability and sensitivity are analyzed based on the display function, $\widehat{G}(\mathit{x})$.

The reliability simulation flow chart is shown in Figure 7.

The ignition simulation workflows of the pyrotechnic igniter are constructed by applying the aforementioned steps of the reliability analysis, as shown in Figure 8. The arrows in the figure represent the running order of the tasks. The details of the nodes in the simulation workflow are listed in

Table 4. Nodes in the simulation workflow.

Name	Type
ρBPN	Input variable
d	Input variable
ρCMDB	Input variable
nCMDB	Input variable
Imin	Input variable
MATLAB_input.m	MATLAB input script file
MATLAB_output	MATLAB output file
MATLAB	MATLAB execution commands
I(x)	Output variable
$\widehat{G}(\mathit{x})$	Display function of the ignition

.

5. Reliability and Sensitivity Analysis Results

From the above procedure, the reliability is analyzed based on the Kriging + TLFF + MCS, and the correctness of the reliability analysis results is verified by using the MCS method. The reliability results are presented in

Table 5. Results for the ignition reliability of the pyrotechnic igniter.

Method	Reliability	Reliability Index
Kriging + TLFF + MCS (81 samples) MCS (106 samples)	0.8383 0.8281	0.0719 0.0706

. The results, which are gained by the above methods, and the relative error, are 1.2%. The Kriging + TLFF + MCS method requires 81 samples to construct the initial Kriging model. However, the MCS sampling requires about 10⁶ performance function evaluations. Therefore, the Kriging + TLFF + MCS method is suitable for the functional reliability analysis of the pyrotechnic igniter.

According to the ignition reliability simulation program of the pyrotechnic igniter, the Kriging response surface model of some random variables, x, is shown in Figure 9.

Table 6. Sensitivity results for the pyrotechnic igniter.

Variable	Mean Sensitivity	Standard Deviation Sensitivity
ρBPN	0.03	−0.001
nCMDB	0.411	−0.011
d	−0.900	−0.006
ρCMDB	0.136	−0.001

shows the results of sensitivity analysis of ignition of the pyrotechnic igniter based on Equation (32). Among these, ρ_BPN, n_CMDB and ρ_CMDB have a positive influence on the results of the mean sensitivity analysis; that is, increasing the mean value of these parameters improves the ignition reliability of the pyrotechnic igniter. Parameter d has a negative effect on the mean sensitivity; in other words, reducing the mean value of its parameters improves the ignition reliability of the igniter. The sensitivity results of the standard deviation indicate that reducing the standard deviation of each variable aids in improving the reliability. From the perspective of the design, the decrease in d is what mainly leads to the higher ignition reliability.

As for the pyrotechnic igniter, it has a high reliability requirement, usually with 0.999 at least for the igniter device. Then, it should take some measures for reliability improvement according to the sensitivity results in Table 6. The straightest way towards improvement may be by decreasing the mean of the nozzle diameter d. The reliability results with different mean values of nozzle diameter d are shown in Figure 10, which reveals that the igniter can meet its reliability requirement once the mean values of nozzle diameter d are controlled to be lower than 3.5 mm.

According to the above analysis, igniters with different nozzles are used for the test, and the test results are shown in Figure 11.

Figure 11 shows that with the increase in nozzle diameter, the failure mode of unstable combustion is more likely to appear, which verifies the results of sensitivity analysis and provides valuable reference for the nozzle structure design of the igniter.

6. Conclusions

Aiming at evaluating the ignition reliability of the pyrotechnic igniter under the failure mode of low temperature unstable combustion, a framework and corresponding approach were proposed based on the combination of the ignition simulation model of the pyrotechnic igniter and Kriging + TLFF + MCS, and its effectiveness was verified:

(1)

The ignition simulation model of the pyrotechnic igniter was established, and the accuracy of the simulation model was verified experimentally. The results show the experimental and analytical results are in good agreement;

(2)

The nonlinear implicit function of pyrotechnic igniter is established with ignition impulse as performance parameter;

(3)

The Kriging + TLFF + MCS method is verified by analyzing the ignition reliability of the pyrotechnic igniter. The results show that the relative deviations between the Kriging + TLFF + MCS with 81 samples and the MCS are less than 1.2%;

(4)

Finally, the sensitivity analysis is performed to quantify the importance ranking of random variables, which offers a valuable insight into reliability-based design and optimization process for the pyrotechnic igniter. The results indicate parameter d is the main factor affecting the ignition reliability of the pyrotechnic igniter, followed by n_MLR, ρ_CMDB and ρ_BPN. Further improvement measures are presented based on reliability and sensitivity results for the pyrotechnic igniter and the correctness of the improved measures was verified by experiment.

Compared to the reliability assessment by experiment, the reliability analysis method based on numerical simulations proposed in this paper provides a valuable insight into the reliability-based design and optimization process for a pyrotechnic igniter.