Shaojing DONG, Yufan FANG, Wentong HU, Xiuli SHEN,Gaoxiang CHEN

a School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Beijing Key Laboratory of Aero-Engine Structure and Strength, Beihang University, Beijing 100083, China

c Jiangxi Research Institute, Beihang University, Nanchang 330096, China

d Navigation and Control Technology Research Institute of China Ordnance Industries, Beijing 100089, China

e Research Institute of Aero-engine, Beihang University, Beijing 100083, China

KEYWORDS Aero engine;Disruptive design;Flow pass multidisciplinary optimization;Pre-identification;Preliminary design phase

Abstract In aero engine design, determining whether the preliminary design will have disruptive effects on the detailed design is the key to multidisciplinary design optimization in the preliminary design stage. In order to adapt to the non-orthogonal parameter value range caused by the selfconstrained parametric modeling method, a non-orthogonal space mapping method that maps the optimal Latin hypercube sampling points of the traditional orthogonal design space to the non-orthogonal design space is proposed.Based on the logical regression method in machine learning field,a kind of feasible domain boundary identification method is employed to identify whether the sample spatial response meets the relevant criteria.The method proposed in this paper is used to identify and analyze the key technologies of the high-pressure turbine mortise joint structure. It is found that the preliminary design of the aero engine may lead to the failure to obtain a mortise joint structure meeting the design requirements in the detailed design stage. The mortise joint structure needs to be pre-optimized in the preliminary design stage.

1. Introduction

As a comprehensive system engineering, aero engine design involves multiple disciplines such as heat transfer,aerodynamics, structural strength, combustion, vibration, reliability and so on.1In the design process,there are complex coupling relationships among various disciplines, and multiple index requirements of each discipline restrict each other. It is necessary to constantly repeat,match,coordinate and balance in the design process. Therefore, the aero engine design is a process of long design cycle,high research investment,strong technical integration,high development cost and high development risk.

Traditional aero engine design is divided into three stages:conceptual design, preliminary design and detailed design. In the conceptual design stage, relatively independent thermal parameters such as flow rate, pressure ratio and bypass ratio are determined. In the preliminary design stage, the basic structure form of the aero-engine is determined. While in the detailed design stage, the basic structure is ‘‘frozen”.2Engine preliminary design stage requires the design work be carried out in detail and fully to avoid major repetitions or delays of the project to prevent project failure3. That means, in the preliminary design stage,the key disciplines and components that may affect the later detailed design stage should be fully considered, so as to avoid subversion of the preliminary or even the conceptual design schemes. In order to shorten the aero engine design cycle, find the optimal solution, and fully consider the coupling between disciplines, the researchers carried out a multidisciplinary design optimization study in the preliminary design stage. Nevertheless, with the continuous improvement of engine performance, there is still the possibility of subverting the preliminary design in the detailed design stage. How to pre-identify the key detailed design that may subvert the preliminary design and integrate them into the multidisciplinary design optimization of the preliminary design stage has become a question deserving research on.

The pre-identification method of key detailed design in the business field4and public service5is mainly divided into subjective judgment and objective derivation. However, in aero engine fields, subjective judgment requires designers have extensive experience. With the improvement of aero engine design requirements, especially higher performance requirements, some potential detailed design of the engine has been difficult to be discovered through design experience.

In terms of objective derivation, commercial enterprises extensively use patent data to carry out predictive research on technological innovations. Specifically, through the mining of patent text information, they can learn about potential disruptive technologies as soon as possible and make timely adjustments to research and development strategy.6–7Typically, UK Intellectual Property Office analyzed the initial patent data involved in related professional technical fields and used it as a continuous technical training set to develop a prototype tool for disruptive technology prediction,and successfully predicted potential disruptive design in the development of microwave heating, flash memory and digital-toanalog conversion.8In addition, Momeni and Rost7proposed a method based on patent-development paths,and predict that within the photovoltaic industry,thin-film technology is likely to replace the dominant technology,namely crystalline silicon.Cheng et al.9proposed a framework of application areas forecasting process for disruptive technology based on patent data.Radio frequency identification technology is selected as their case study and their method provides practical suggestion to firms and other stakeholders.Jia et al.10verified that in mobile communication and wireless, 5G and Internet of Things (IoT)technology are disruptive technology, using DeWinter Patent Database as the patent data source. Lyu et al.11made a study on the energy field using disruptive technology identification and determined the ranking of 45 potential disruptive technologies. At present, the disruptive technology preidentification of the business model has two inspirations for aero-engine designs:

1) Inspiration from document and patent text mining.

Establish the literature and patent database monitoring mechanism, and reveal the inherent relevance of technical research through vertical and horizontal analysis of the same keywords or the documents/patents citied, and further reveal the research hotspots. For example, the latest literature data can be obtained by monitoring the Web of Science or Essential Science Indicators database corresponding to the subject.Fig. 1 shows the number of SCI journal articles published in the field of multidisciplinary optimization in the past 10 years(data from the Web of Science, search keyword: Multidisciplinary Design and Optimization(MDO)).It can show the vertical development of this field.

However, the above-mentioned literature and patent mining analysis fundamentally belong to inform metrics, and will not be studied in this paper further.

2) Inspiration from continuous technology training set.

The aero engine development is a process of technology accumulation over the years, so the concept of continuous technology in the commercial field is also applicable to aero engine industry.Accumulated data and design experience play a vital role in all stages of engine development. However, as engine performance increase,the engines often work at the limits of technology and performance in all aspects,and the space for incremental improvements is extremely limited. In addition,limited by the specific requirements,models,and working environment of the engine,the data and design experience need to be carefully considered before generalized applications.Therefore,the relatively scarce continuous technology training set makes it difficult to imitate business models to pre-identify disruptive design.

Fig.1 Number of SCI journal documents related to MDO in the past 10 years.

In order to make up for the shortage of engine continuous training set, the Design of Experiment (DOE) method in multidisciplinary optimization can be used to generate training sample space, and the response of sample points can be obtained by numerical simulation. This process involves two problems:(A)Sampling in the non-orthogonal space.In traditional experimental design, the design variables are independent of each other, that is, the design space is orthogonal.12However, due to the non-orthogonal space formed by semiindependent design variables,the boundary of design variables is non-constant. There is no report on sampling method in non-orthogonal space so far. (B) Determine whether the response of samples in non-orthogonal space meets the relevant criteria. Feasible region boundary recognition method is essentially a binary classification. The boundary of feasible region can be quickly determined by machine learning classification method. Common classification methods include linear regression, support vector machine, decision tree and clustering algorithm. Logistic regression is an extension of linear regression. It is one of the fundamental classification algorithms that uses a weight vector to define and maximize a logarithmic probability in favor of one of the classes.13It is a widely used, well-understood, and often well-performing supervised learning technique, one of the most useful analytic tools in binary classification due to its ability to study the importance of individual features transparently.14

This paper draws on the pre-identification method of disruptive design of business innovation to develop the disruptive design pre-identification technology in the preliminary design stage of aero engine. Firstly,a non-orthogonal space mapping method that maps the optimal Latin hypercube sampling points of the traditional orthogonal design space to the nonorthogonal design space is proposed to adapt to the nonorthogonal parameters value range caused by the selfconstrained parametric modeling. Then, through a logistic regression method based on machine learning, the feasible region boundary is identified whether the sample spatial response meets the relevant criteria. Finally, the method proposed in this paper is used to identify and analyze the disruptive designs of the aero engine high-pressure turbine mortise joint structure.

2. Theoretical methods

2.1. Non-orthogonal spatial sampling mapping method based on experimental design method

Firstly, the concepts of dependent, independent and semiindependent parameters are explained as follows.For example,in the equation y=x+1,x is an independent parameter,y is a dependent parameter. In the inequation z < x + 1, x is an independent parameter, z is a semi-independent parameter.

In the process of parametric modeling, if the mutual constraint relationship between geometric dimensions is not fully considered, the local deformity or unreasonable geometric shape caused by body interference may occur in the constructed geometric model, or even it cannot be solved. Selfconstrained parametric modeling uses the modeling history playback function to gradually solve the geometric constraints under the new parameters when the model is updated,so as to effectively avoid the failure of automatic modeling.

Due to the introduction of the self-constrained parametric modeling method, some geometric parameters become semiindependent parameters (design variables), whose value boundary is no longer constant, resulting in the sample space is non-orthogonal space. As shown in Fig. 2, if the conventional experimental design method sampling at orthogonal boundaries is still used to obtain the sample space,the following contradictions will arise: (A) Sampling performed according to the maximum inscribed orthogonal boundary (the green boundary) will result in samples missing. (B) Sampling performed according to the minimum circumscribed orthogonal boundary (the blue boundary) will result in invalid samples.

Fig. 2 Two-factor logistic regression sample.

The following steps can be used to map the orthogonal space samples generated by the experimental design to the non-orthogonal space generated during self-constrained parametric modeling to generate new samples in the nonorthogonal space:

In this paper, the elliptic differential equation is solved to obtain the mapping relationship between the non-orthogonal space and the experimental design space. The solution of the Laplace equation in the elliptic differential equation can only get the extreme value on the boundary which ensures the one-to-one correspondence between the non-orthogonal space and the experimental design space.15–17Therefore,the Laplace equation is beginning to solve the mapping relationship:

It is very difficult to solve the above equations with xias the independent variable directly.The way to solve the problem is to exchange the independent variable and dependent variable in the above formulas, and derive the transformation and inverse transformation from xito ξi.

The Laplace operator is

The first term on the right side transforms the equation set with x as the independent variable into one with ξ as the independent variable through coordinate transformation.

Suppose I is nth-order unit matrix, then

The results are shown in Fig. 3. It shows that the nonorthogonal space mapping algorithm can effectively solve the problem that the ordinary experimental design cannot completely fill the non-orthogonal boundary area.

2.2. Feasible region boundary recognition method based on machine learning logistic regression

After constructing the sampling space, the response of the sampling points,such as maximum stress,deformation,quality or efficiency is obtained through numerical simulation analysis. By judging whether the response meets the design criteria,the sample points can be classified as feasible and infeasible.Furthermore, the disruptive design pre-identification is completed.

In the sampling space, if a functional regression model z=θ（X) with X as the independent variable can be used to describe the relationship of the sampling point and its response. Then the regression model can be used to quickly calculate the response of the sampling points, and whether it is a disruptive design that needs to be considered in the preliminary design stage can be confirmed.

(1) Boundary recognition of logistic regression In machine learning, the Sigmoid function (Eq. (7)) is usually introduced to compress any real value z into the logical range {0，1 }.

As shown in Fig. 4, y=g（θ（X)) = 0.5 at z = 0. As z becomes smaller, y approaches 0, and when z becomes larger,y approaches 1.When 0.5 is used as the boundary between feasible and infeasible, the feasible region boundary function z=θ（X) can be inversely solved when y=g（θ（X))=0.5.

The functional regression model z=θ（X)can be set to any function type to describe the relationship between the experimental design factor and the logical output response. In this section, a polynomial model is selected as the functional regression model:

Fig. 3 Non-orthogonal mapping effect (m = 101).

Fig. 4 Sigmoid function image.

where X represents the multi-order feature item vector after mapping. If there are two experimental design factors x1and x2, both of which are first-order, their second-order feature items are x21，x1x2and x22. Therefore X=[1 x1x2x21x1x2x22], and then a polynomial model can be used to linearly sum the features of each order.The logistic regression function can be expressed as:

In order to minimize J（θ), the gradient descent method is used. The iterative formula is written as:

Where α is the search step of gradient descent method.θ can be calculated by the gradient descent method, so that the regression model is obtained.

(2) Regularization processing

The ultimate goal of identifying the boundary of the feasible region is to predict whether the new sampling point is feasible which needs the boundary have good generalization ability.

However,when the order of the selected polynomial regression model is too high,overfitting is likely to occur,which will affect the prediction accuracy.As shown in Fig.5,if a higherorder polynomial model is used to try to classify all the sampling points to improve the accuracy of the regression model,the overfitting(the green line)will reduce generalization ability of logistic regression model.

The first conceivable way to avoid overfitting is to manually reduce the polynomial order,but this method requires manual adjustment of the regression model, which is not conducive to the realization of modularity. The other way is regularization,which adds a penalty factor λ to the high-order term coefficients in the polynomial model in the cost function to make the high-order term coefficients in the final regression model smaller.

Fig. 5 Schematic diagram of logistic regression overfitting.

Then Eq. (11) becomes20:

3. Examples

3.1. Two-factor sample example

Draw the schematic diagrams of the boundary regression function with polynomial order from 2 to 13 respectively,some of which with representative boundary shapes are shown in Fig.7.The shape of the boundary regression function becomes more and more complex with the increasing order in order to better distinguish the sample points with different logical labels.

Fig. 6 Two-factor logistic regression sample.

The 2-5th order functions still maintain smooth boundary,while the 6-13th order ones gradually show different degrees of non-convex characteristics.

To remove the incidental infeasible points in the main boundary, the 6th-order and 11-13th-order use local voids,and the 7-8th order concave the main boundaries, while the 9-10th don’t form closed boundaries.

The following regression fitting accuracy function is set to evaluate the logistic regression model:

The mean function is to obtain the average value of the array.

If the output response through the logistic regression model is consistent with the logical label of the sample point itself, it returns 1, otherwise 0. The regression fitting accuracy can be obtained by averaging the returned value array. As shown in Table 1, as the order of the polynomial increases, the coefficients of the polynomial to be obtained increase, but the regression fitting accuracy is not significantly improved, and even the 9-13th-order fitting accuracy is lower than the 8thorder’s. For low computational cost, it should be necessary to choose lower-order polynomials to reduce the number of polynomial coefficients while ensuring the fitting accuracy.

In addition,the fitting boundary function should have good generalization capability. Excessive pursuit of fitting accuracy may lead to the occurrence of over-fitting and hence the previous penalty factor λ is introduced.However,the penalty factor should ensure better fitting accuracy and prevent under fitting caused by the excessive punishment of high-order terms. The following feasible fitting accuracy is set:

where the Length function is used to count the number of array elements.The feasible fitting accuracy represents the proportion of all feasible sample points (the logical label is {y=1}) within the feasible boundary.If the number of sample points NSand the number of feasible sample points NFare known,the proportion of infeasible points outside the feasible boundary, that is, the infeasible fitting accuracy can be expressed as:

The effect of regularization is shown in Table 2 and Fig.8.

According to Table 2 and Fig. 8, it can be seen that the introduction of the penalty factor reduces the fitting accuracy.However, the penalty factor can avoid overfitting caused by the irregular boundary of high-order polynomials, and increase the generalization capability of the final fitting boundary. However, an excessively large penalty factor (λ = 1 to 100)will cause serious underfitting.Therefore,the penalty factor should not be too large.

Fig. 7 Boundary recognition of two-factor polynomial logistic regression with different orders (λ = 0).

Table 1 Fitting accuracy of two-factor polynomial logistic regression with different orders (λ = 0).

For the design sampling points to be identified,substituting into z=θ（X) to observe whether the value is positive or negative or substituting into g（θ（X)) to compare the values with 0.5 can identify whether they satisfy the design constraints. If not satisfied, they can be listed as disruptive design needingattention in the preliminary design phase and even conceptual design phase, and it will avoid subverting in the following design stage.

Table 2 Fitting accuracy of two-factor 8th-order polynomial with different penalty factors.

3.2. The disruptive design pre-identification of the high-pressure turbine mortise joint structure

Fig. 8 Fitting accuracy of two-factor 8th-order polynomial with different penalty factors.

This section takes the high-pressure turbine mortise joint structure as the research object to determine whether it is the disruptive design in the preliminary design stage of aero engine.It can determine whether the mortise joint structure can be simplified or ignored in the preliminary design stage. If it cannot be ignored,the mortise joint structure must be pre-studied during the preliminary design to prevent subverting in the detailed design stage.

In the preliminary design stage,the aero engine flow pass is a link between aerodynamics and strength,through which turbine disc outer diameter DiscROUT, rotor speed nSpeed and blade mass moment mBlade can be obtained.

The design of turbine mortise joint structure and the meridian surface of the turbine disc can be completed by the above three parameters. Then the above disruptive design preidentification is transformed into whether there is at least one set of mortise joint structure parameters can meet the requirements of established strength, and it is determined in turn whether the mortise joint structure should be prestudied in the preliminary design stage.

The value range of the upper-stage parameters should try to include all the parameter combination. The value range of upper-stage parameters are shown in Table 3.

Establish the mortise structure model and the meridian surface model of the turbine disc, as shown in Fig. 9.

To describe all possible combinations of mortise structure parameters, it contains 20 modeling parameters, as shown in Table 4, which includes 17 independent or semi-independent ones and 3 dependent ones. The independent parameters are sampled by OLHS. Since the value range of the semiindependent parameters is not constant, the non-orthogonal spatial sampling mapping method proposed in this paper is used for sampling. The dependent parameters are determined by the independent and semi-independent parameters. In the optimization or sampling process,the independent parameters are given in turn, and then the value of the dependent parameters and the value range of the semi-independent parameters can be gradually determined. And then the appropriate value of the semi-independent parameters should be selected,according to which the model expression file can be updated.

Table 3 Value range of the upper-level parameters.

Fig. 9 Parameterized models of mortise joint and turbine disc.

Table 4 Pre-identification parameters of disruptive design of mortise joint.

First, sample the upper-stage parameters with the experimental design levels m1and sample the lower-stage parameters with the experimental design levels of m2at each upper-stage sampling point. Therefore, it is necessary to perform m1× m2simulations to calculate the response of the sample combination. Considering the cost of simulation, m1= 161 and m2= 21 are selected in this section, and a total of 3381 sampling points are generated.

Table 5 Simulation cost of disruptive design preidentification.

Fig. 10 Samples distribution of disruptive design pre-identification of the mortise joint structure.

Fig. 11 Different order identification results of disruptive design pre-identification of the mortise joint structure (λ = 0).

Table 6 Different orders of logistic regression fitting accuracy for disruptive design pre-identification of the mortise joint structure.

Table 7 Fitting accuracy of 4-7th order polynomials for disruptive design pre-identification of the mortise joint structure with different λ.

Fig. 12 Logistic regression boundary of disruptive design preidentification of the mortise joint structure(4th-order,λ=0.001).

Then the axisymmetric model and the three-dimensional model are used to analyze the strength of each sample point.Use 4 meridian plane strength reserve coefficients, the maximum single-point radial and circumferential stress of the disc,as well as the maximum radial and circumferential stress of the mortise joint structure,a total of 8 evaluation criteria,to make logical judgment on the response of the sample points. Under each upper-stage parameter sampling combination, if all sampling points of the lower-stage parameters meet the evaluation criteria, the upper-stage parameter sampling combination is assigned{1},if any lower-stage parameter sampling point does not meet the evaluation criteria, {0} is assigned.

The simulation cost is shown in Table 5.

It is worth noting that a total of 65 sample points(accounting for 1.923%)failed to output the response in the process of obtaining the response of sampling points. All of them were terminated forcibly because their simulation execution time exceeded the maximum execution time (20 min) and the simulation analysis of the next sample point was carried out. And these 65 non-response sample points are not concentrated in one upper-stage parameters combination. There are at most 3 non-response sample points of one upper-stage parameter combination, so it does not affect the accuracy of the logical judgment of the upper-stage parameters. The sample point response for logical judgment is shown in Fig. 10. Use the logistic regression boundary judgment method proposed in this paper to describe the logical boundary. Because m1= 161, a highest 7th-order polynomial (C37+3=120<161，C38+3=165>161) can be selected. When λ = 0, the logical boundary fitting results are shown in Fig. 11.

Since there are only a few infeasible({y=0})sample points invading the feasible area,the logistic boundary regression surface of each order is smooth.The fitting accuracy for different order polynomials shown in Table 6 show that 3th-order polynomial has the highest fitting accuracy.Therefore, a 3th-order polynomial can be selected to describe the boundary of logistic regression when λ = 0.

Since the penalty factor can reduce the coefficients of higher-order terms and thereby weaken the influence ofhigher-order terms, regularization is carried out on the 4-7th order polynomials. The fitting accuracy are shown in Table 7.

Table 8 Logistic regression boundary function of disruptive design pre-identification of the mortise joint structure.

Table 9 Disruptive design pre-identification of mortise joint structure.

With the improvement of λ, the feasible fitting accuracy of each order polynomial has been improved, and the ability of the logistic regression function to determine the logical feasible points is further enhanced. But the regression fitting accuracy reaches peak at λ=0.001–0.01,which affects the infeasible fitting accuracy reach peak at the same time.

Synthesizing the results shown in Table 7, the 4th-order polynomial and λ = 0.001 are selected to depict the logistic regression boundary as shown in Fig. 12.

The feature items and coefficients of the logistic regression boundary function are shown in Table 8.

Next,a preliminary multi-disciplinary optimization scheme of the aero engine flow pass is used as the initial value to determine whether it is necessary to conduct a pre- study on the mortise joint structure in the preliminary design stage. The results are shown in Table 9.It can be seen that after the judgment of the logistic regression boundary function, the upperstage parameter combination of the preliminary scheme is outside the feasible boundary (θ（X)<0 or g（θ（X))<0.5). That means the mortise joint structure cannot be ignored in the preliminary design stage.

4. Conclusions

Inspired by the method of identifying disruptive technologies in business models, this paper developed the disruptive design pre-identification methods in the preliminary design stage of the aero engine. The following conclusions can be obtained:

(1) Based on the experimental design method in the field of multidisciplinary optimization, the non-orthogonal spatial sampling and mapping method is proposed for the non-orthogonal geometric parameters caused by the self-constrained parameterized modeling. It can form sample points that can be identified.

(2) A feasible region boundary identification method for disruptive design pre-identification of the aero engine is obtained based on the logistic regression method in machine learning. According to it, the disruptive design needing detailed structural design in the preliminary design stage can be picked out.

(3) The disruptive design pre-identification of mortise joint structure is carried out by the methods mentioned above. It is concluded that the mortise joint structure must be detailed studied in the preliminary design, so as to prevent the large repeated work.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.