Reliability analysis of multi-site damage with failure dependency of the turbine based on flow-thermal-solid coupling analysis and the Monte Carlo validated simulations

▪ The stress distribution of the turbine under multiple loads is determined by the flow-thermal-solid coupling analysis method. ▪ The stress distribution with dispersion characteristics is obtained through the coupling analysis process and response surface method. ▪ This reliability analysis model considers the failure dependency between the failure sites. ▪ The accuracy of this reliability model is verified by Monte Carlo simulation. The harsh environmental loads may lead to strength failure in the turbine in an aero-engine. To accurately assess the strength reliability of the turbine under multiple loads, the stress distributions of 41 danger sites of a turbine under thermal, centrifugal, and pneumatic loads were determined by the flow-thermal-solid coupling analysis using ANSYS. Second, based on the flow-thermal-solid coupling analysis and response surface method, the probabilistic analysis model of stress at the danger site was established. And the probabilistic distribution of stress was determined by sampling and hypothesis testing. Finally, the reliability model of the turbine with multi-site damage and failure dependency was established, by which a reliability of 0.99802 was calculated. And the actual reliability of the turbine was 0.99626 determined by the Monte Carlo simulations, which verified the model in precision. The results indicated that the reliability model has a high efficiency and higher precision than the traditional reliability model with failure independence.


Introduction
Aero engines power the aircraft in flight and their reliability directly affects the safety of the aircraft. Current aircraft missions reflect greater diversity and specialization [24], which requires engines with lighter materials and greater performance [36]. The core of an aero engine is a highly complex structure that mainly contains a turbine, a combustion chamber, and a compressor. In recent years, numerous engine turbine rotors failed as a result of their broken and fractured rotor blades, fatigue of the metal of the turbine disk, excessive deformation due to high temperatures, and many other reasons, while triggering a continuously increasing number of discussions and studies [17,25,26].
Strength-based failure of a turbine is mainly caused by excessive load or insufficient strength [42]. The multi-field environmental loads on turbine rotors deserve focused attention in turbine failure analysis, including centrifugal load, aerodynamic load, and thermal load [19]. The performance of the turbine rotor significantly degrades in operation under high temperatures [16,20,39]. The large temperature gradient causes higher thermal stresses in vital sites [33]. And the high rotational speed leads to high centrifugal stresses in the turbine, which subsequently causes structural yielding [18]. Compared with other techniques such as the finite difference method (FDM) and method of moments (MOM), the fluid-thermal-solid coupling finite element analysis demonstrates greater efficiency and flexibility in structural analysis [5]. Therefore, it is extensively applied in industrial fields such as aviation, aerospace, marine, and nuclear engineering. Yazan discussed the influence of the heat pipe on the steady-state and transient temperature variations of the integral turbine in heat transfer analyses [35]. Balachandra. applied the finite element method in the failure analysis of turbine blades and found friction as a reason for the failure of cantilever blades [3]. Yang performed a transient coupling analysis of the turbine rotor considering thermal load, centrifugal load, and preload jointly to determine the temperature distribution and stress distribution of the structure [21]. According to related research, most of the highstress regions of turbine rotors under complex loading environments were distributed near the central holes of the turbine disks [10,18], the mortise, and the blades [49]. In general, these high-stress regions are often the first sites to fail, i.e., the danger sites.
During operation, loading parameters such as pre-turbine temperature and speed are usually not constant but random variables [40]. And the production and assembly are also random processes to some extent. Therefore, the performances of the turbine are not constant, but obey a probability distribution [23]. Due to these uncertainties, the failure of the turbine shows significant dispersion in nature, namely exposing reliability problems [6]. According to research on turbines, probabilistic analysis methods are widely used to investigate the problems such as probabilistic low cycle fatigue life prediction of turbine disks and their reliability assessment [13. 34, 50], the probabilistic response of high-pressure turbine tip clearance [9], creep-fatigue failure of turbine disks [7,37], and optimization of geometric parameters of turbine blades [14]. Probabilistic analysis is necessary for accurate failure prevention and reliability assessment of turbines [38]. Currently, probabilistic analysis methods based on surrogate models, such as response surface methods (RSM) [27,41], artificial neural networks (ANN) [1,46], Kriging [11], and radial basis functions (RBF), are extensively promoted and applied in engineering. Fan proposed a surrogate model based on local maximum entropy (LME) theory in reliability analysis and sensitivity analysis of turbine disks with random geometric parameters [8]. These surrogate t models demonstrated great efficiency and adaptability in reliability research on turbines compared to the traditional Monte Carlo method [2,4,28]. But the precisions of these methods were hard to guarantee, especially when the objects were complex structures or systems with multiple uncertainties and failure dependence.
For turbine structures, there are multiple sites in the structure that tend to damage due to the geometric complexity and loading diversity [31,32]. In systems with variable amplitude loads, there will be a statistical dependency between the failure of each danger site [30]. This situation is similar to the failure of a system with multi-site damage. Due to randomness in production, manufacturing, and operation, the stress and deformation of the turbine rotor obey a probability distribution. Therefore, the site where the failure occurs first is variable [15,22]. The stress-strength interference (SSI) model and its derivatives conveniently reflect the effects of random and common cause failure (CCF) on system failure [47,51].
However, it is difficult for the SSI model to be applied directly to the reliability analysis of complex systems in most situations because of the uncertain distributions of load and strength.
There are multiple danger sites on the turbine rotor. And the failure of any site will lead to the failure of the entire turbine rotor. In addition, the failures of danger sites are not simply independent due to the randomness of the loads but are dependent [29]. Traditional perspectives of independent failure consider the reliability of a system to be equal to the logical product of the reliability of the parts. For a mechanical structure, these views are not appropriate. and their evaluation of system reliability often deviates from the actual reliability. For systems with failure dependency, CCF is prevalent. And reliability analysis models with failure dependency are required. Xie et al developed a series of failure-dependent system reliability models through the minimum order statistics of strength based on the failure mechanism of the system [42][43][44][45]. On this basis, Gao investigated in depth the multi-state dynamic fuzzy reliability problem using generating function and explored the effect of failure dependency on the reliability level of the system [12]. Zhao proposed an adapted reliability analysis model of improved dynamic failure-dependent systems for rotor blade systems such as aero-engine turbines with severe degradation and catastrophic failures [48]. Currently, failure-dependent system reliability models are attracting increasing attention.
In this paper, the reliability of a short-life engine turbine rotor for missiles was investigated: the flow-thermal-solid coupling analysis method was applied to study the characteristics of all danger sites of the turbine under thermal, centrifugal, and pneumatic loads; and the response surface method was applied to determine the dispersion of the load; then the mathematical analysis model of the reliability of turbine strength failure was established with the failure dependency considered; finally, the Monte Carlo method was used to verify the feasibility of the model.

Parameters
The pre-turbine temperature, Ta

Thermal analysis and structural analysis of turbine
The turbine is made of the nickel-based high-temperature alloy

Dispersion of parameters about loads and material
Due to the prevalent randomness in the manufacturing, assembly, and operation of the turbine, the load and material parameters are not constant but obey a probabilistic distribution.
In the coupling analysis, the rotational speed, inlet temperature, material density, and the elastic modulus of the material were all set to specific values. However, it is not appropriate to neglect randomness in operation in reliability analysis for the turbine. Therefore, a probabilistic analysis of the turbine is necessary.
According to the coupling analysis, four key parameters including rotational speed, material density, turbine inlet temperature, and elastic modulus at room temperature were

Response surface models of the stress at the turbine danger points
The response surface methodology is a statistical method for

Probability distribution of stress at danger sites
Based on Eq. (4), the four factors were sampled 10000 times according to their distributions, with the sampling history shown in Fig. 9.a. The 10000 samples were fitted to a probability distribution of the stress at the danger site. After the KS test, the stress at the danger site at the blade root obeyed a normal distribution, i.e. ∼ (871.6876,26.602) MPa, as shown in Fig. 9.b. Since the simplified turbine model has a periodic symmetric structure, the stresses at danger sites at the blades' root can be considered to be independently and identically distributed.

Sensitivity analysis
Sensitivity analysis is an essential step in probabilistic modeling and reliability analysis. In probabilistic analysis, it identifies the factors that have a significant impact on the response, while being able to quantify the specific level of influence of all factors on the response. The partial derivatives of the factors of the response surface function are solved to know the rate of variation of the response with respect to the factors, which is also the sensitivity of the response with respect to the factors [34].
According to the results of sensitivity analysis, the preturbine temperature is the most significant factor affecting the stress at the danger site, while the modulus of elasticity has the least effect on the stress at the danger site, as shown in Fig. 10.
Therefore, the influence of pre-turbine temperature should be focused in the optimized design of the turbine.

The multi-site damage reliability model
There are 41 danger sites on this turbine that can be damaged, corresponding to each of the blades. And the failure of any one blade will lead to the failure of the whole turbine, which belongs to the series failure system. As there is a failure dependency among the blades, the turbine is a multi-site damage system with failure dependency. Failure of a mechanical structure is closely related to its strength and load. Theoretically, 'strength' and 'load' can be any pair of variables of the same magnitude but with a resistance relationship, such as stress, strain, temperature, etc.
This turbine belongs to a special class of aero-engine turbines with relatively short life and higher stress levels compared to others. In this case, the turbine is hardly subject to general fatigue failure. Therefore, it is generally considered that the turbine is safe once the stresses at the danger sites do not exceed their corresponding yield limits.
For any turbine danger site, when the strength is greater than the load, it is considered to be safe and reliable; conversely, it is dangerous. Stress is generally used to characterize the pair of interfering variables-strength and load. The load is the stress at that danger point, while the strength is the yield limit at that danger point. Then the failure state of the structure can be expressed by Eq. (3).

= − (3)
Where ≥ 0 represents that the structure is failed, and < 0 represents that the structure is safe and reliable.
where ′ and ′ donates the reliability and failure probability of the failure-independent system; 1 donates the reliability of only a failure site of the turbine; And the reliability and failure probability of the turbine was calculated to be 0.92197 and 0.07803 by the model with failure independence, respectively.

Carlo simulation
The Monte Carlo simulation is a method based on simulated tests in which the failure process of a structure with dispersed loads and strengths is simulated through sampling. This method is highly operational and simple in principle, but relatively inefficient [2].  Fig. 11 [4,28].  Table 6. Validation of three models in precision.