Optimization of aero-engine combustion chambers with the assistance of Hierarchical-Kriging surrogate model based on POD downscaling method

doi:10.21203/rs.3.rs-2537920/v1

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Research Article

Optimization of aero-engine combustion chambers with the assistance of Hierarchical-Kriging surrogate model based on POD downscaling method

https://doi.org/10.21203/rs.3.rs-2537920/v1

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published 03 Jul, 2023

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In view of the long calculation cycle, high processing test and cost of the traditional aero-engine combustion chamber design process, which restricts the engine optimization design cycle, this paper innovatively proposes a surrogate model for the performance of aero-engine combustion chamber based on the POD-Hierarchical-Kriging method. Through experiments, the predicted results of the POD-Hierarchical-Kriging model are compared and analyzed with the calculated results of the one-dimensional program, and the root mean square error of the predicted values of combustion efficiency and total pressure loss are 0.0064% and 0.1995%, respectively. The accuracy of the POD-Hierarchical-Kriging model is compared with the cubic polynomials model, the basic Kriging model and the Hierarchical-Kriging model. It verifies the feasibility and accuracy of the POD-Hierarchical-Kriging model for the prediction of performance of aero-engine combustion chambers. The global sensitivity analysis method is applied to obtain the influence effect of the design variables on the performance. Then, a multi-objective optimization method based on NSGA-II algorithm is studied, and finally the optimal set of Pareto solutions is obtained and analyzed, which can be used to guide the optimal design of aero-engine combustion chamber and accelerate the progress of aero-engine development.

aero-engine combustion chamber

surrogate model

POD-Hierarchical-Kriging

multi-objective optimization

With the development and innovation of aero-engine technology, reducing the engine design cycle and calculation time has become an important part of current research. As one of the three core components of an aero-engine, the engine combustion chamber has higher requirements for design development cycle and design results. In the current combustion chamber design, it basically relies on CFD numerical simulation or wind tunnel test to calculate and evaluate. The high complexity of models and methods leads to long computational cycles and still has a large computational cost. In recent years, reducing the computational cost and improving the optimization efficiency by using surrogate models to approximate the target quantity has been widely studied by scholars. It mainly adopts a data-driven modeling approach to fit, regression, and feature learning based on a sample database composed of selected design variables and corresponding numerical calculations or real test results. Surrogate models that can highly approximate the real mathematical model are constructed, and the corresponding output parameters can be quickly and accurately obtained for the input parameters of the surrogate model, which can replace the complex and time-consuming calculation and analysis in the design process. The commonly used surrogate models in current research include polynomial response surfaces, moving least squares, radial basis functions, artificial neural networks, support vector machines, Kriging (Gaussian regression process), etc.

In 2012, ZH Han[1] et al. proposed the Hierarchical-Kriging model, using a sampled Kriging model with a low fidelity function as the model trend. A more accurate surrogate model of the high fidelity function is obtained by mapping the variation of the low fidelity data to the high fidelity data. It is also used to model the aerodynamic data of Rae2822 airfoil and industrial transport aircraft structures with efficiency, accuracy and robustness. In 2015, Hideaki Ogawa [2] et al. combined computational fluid dynamics and evolutionary algorithms to optimize multi-objective fuel injection at Mach 5.7 cross-flow after initial compression of Mach 7.6 inlet in scramjet with the assistance of various surrogate modeling including polynomial and Kriging. In 2019, M. Teraghavi [3] et al. used three popular structures -- nonlinear autoregressive network (narxnet), multilayer perceptron (MLP) and radial basis function (RBF) to build a multi-input single-output surrogate model to predict the SOC of HCCI engine operation. In 2019, Fernando Tejero [4] et al. developed a new framework for aero-engine nacelle design which combines surrogate modeling and numerical simulation. Compared with traditional methods, the algorithm successfully identifies the same optimal nacelle design and reduces the computational cost by 25%.In 2020, Du [13] et al. applied Hierarchical-Kriging to optimize the structural dynamics of rocket engines, constructed the relationship between the design variables of engine structural parameters and modal frequency characteristics, and used it to facilitate engine optimization design, achieving good accuracy. In 2021, Caterina Poggi [5] et al. developed a surrogate model suitable for simulating aerodynamic performance and acoustic emission, avoiding the need for computationally costly CFD/CAA predictions. Numerical studies have shown that the surrogate model is able to accurately reproduce the aerodynamic performance and radiated noise of the propeller. In 2021, Lu-Kai Song [6] et al. proposed the decomposed cooperative time-varying kriging surrogate model (DCTKS). By comparing several surrogate models, it is proved that the DCTKS has computational advantages in the efficiency and accuracy of probabilistic creep fatigue life evaluation, and improves the modeling accuracy and simulation efficiency. In 2021, Fernando Tejero[7] et al. designed and studied the non-axisymmetric aero-engine nacelle based on the surrogate model method. A series of surrogate-based adaptation methods are studied in order to reduce the computational cost. Compared to the conventional approach of full numerical simulation in the optimization cycle, the adaptation methods identified an acceptable design space and reduced the total computational cost by 65%.In 2023, Maotao Yang [8] et al. used Kriging surrogate model to predict the combustion efficiency and total pressure loss of the combustion chamber and compared it with ANN model to verify that the use of Kriging surrogate model can shorten the engine combustion chamber design cycle. In 2023 Yue Ma [9] et al, used a cubic polynomial surrogate model to predict the performance of combustion chambers. Based on this, a multi-objective optimization of the combustion chamber performance is carried out using PSO optimization algorithm. The surrogate model is proved to be useful for guiding the combustion chamber design. From the analysis of the above research status, the application of surrogate model to assist engine optimization design has achieved good development. However, the combustion chamber of aeroengine has the characteristics of large variation in working conditions, significant nonlinear characteristics and harsh working environment, so the traditional design method requires a large amount of calculation and a long design cycle, and can not meet the real-time prediction of performance parameters. Therefore, the application of surrogate model to assist aeroengine combustion chamber design has a good advantage, but there are still few relevant references.

This paper takes an aero-engine combustion chamber as the research object, and creatively applies the POD-Hierarchical-Kriging method to establish a surrogate model of combustion performance. Firstly, different incoming flow conditions and main design parameters of the combustion chamber are selected as design variables. The sample collection in the input parameter space is completed by using the Latin hypercube sampling method. The numerical simulation program designed by Northwestern Polytechnic University is applied to pre-process the data and calculate the output values of combustion performance including combustion efficiency and total pressure loss. Then the samples are downscaled and reconstructed based on the POD algorithm, and the downscaled and reconstructed data are used as training datasets. Based on the Hierarchical-Kriging method, a surrogate model is developed to characterize the mapping relationship between the input and output performance parameters of the original complex model to simplify the complex and time-consuming computational process. Finally, the constructed data set is used to complete the training of the model, and the test data is used for verification. A global sensitivity analysis method based on variance is applied to rank the magnitude of design variables affecting the combustion performance. Based on POD-Hierarchical-Kriging model, a Parteo optimal solution set is obtained by NSGA-II multi-objective optimization algorithm, laying the foundation for the optimal design of aero-engine combustion chamber.

In the second part of this paper, the concentric graded combustion chamber model is introduced. The third part carries out the theoretical derivation of the POD algorithm, the basic Kriging and Hierarchical-Kriging surrogate models and the theory of the NSGA-II algorithm. In the fourth part, experiments are carried out, including surrogate model test prediction and error comparison analysis of various surrogate models, global sensitivity analysis, as well as multi-objective optimization based on NSGA-II algorithm. Pareto solution set obtained from optimization is analyzed. The fifth section provides a summary.

2.1 Reliability verification of high performance 1D design method

In order to quickly complete the calculation of combustor parameters, the one-dimensional calculation method of aero-engine combustor is studied jointly with Kazan Aviation University, and a set of one-dimensional calculation program of aero-engine combustor model is developed. The reliability of the three-dimensional numerical calculation is verified by comparing the experimental results with the three-dimensional numerical calculation results in a dual swirl combustor. Then, two different one-dimensional calculation models (Case1 and Case2) are used to compare the one-dimensional calculation results with the three-dimensional numerical results verified by reliability, so as to verify the consistency of the one-dimensional calculation results.

In the comparison of three-dimensional calculation and test in the combustion chamber, Fig. 1 shows the comparison cloud diagram between the calculation results of three-dimensional velocity field and the measured results of PIV test.

By comparing with the PIV measurement results, it can be seen that the velocity field distribution trend of the 3D numerical simulation cloud image is basically consistent with the experimental measurement results, and the error is within a reasonable range, which indicates that the 3D numerical simulation can reflect the changes and characteristics of the actual flow field to a certain extent.

Then, the one-dimensional calculation methods Case1, Case2 and CFD three-dimensional calculation are used to obtain the comparison for the air flow rate accumulated along the axial direction, as shown in Table 1. It can be seen that the results of Case1 calculation are closer to those of CFD three-dimensional calculation, and the inlet air volume in the main combustion zone is only slightly less than that of Case2 calculation.

Table 1

Comparison of cross-sectional flow distribution ratios between the main combustion hole and the mixing hole in the dual swirl combustor
Calculation method	Main combustion hole cross section	Mixing hole cross section
1D calculation (Case1)	0.488	0.809
1D calculation (Case2)	0.490	0.804
CFD 3D calculation	0.476	0.848

Based on the comparison and analysis of the flow parameters and performance parameters of the combustion chamber mentioned above, it can be seen that the one-dimensional calculation method adopted is in good agreement with the three-dimensional calculation. It is proved that the one-dimensional calculation method can calculate the performance parameters of the combustion chamber well, and the calculation results are accurate and feasible. A more detailed description of the method can be found in reference [10].

2.2 High performance combustion chamber design for aero engine

The object of this paper is a type of concentric graded combustion chamber. The design conditions met by this combustion chamber are shown in Table 2, and its structure schematic is shown in Fig. 2 According to the design requirements, the overall structural parameters and flow distribution of the combustion chamber are calculated, and on this basis, the design and calculation of each component of the combustion chamber are carried out. Finally, the preliminary design scheme of the combustion chamber is obtained. Through numerical simulation and test verification, it has been verified that this structure can meet certain performance index requirements. For more detailed design parameters, refer to literature [11].

Table 2

High temperature rise combustor design requirements
Parameters	Design Status	Slow state
Total inlet temperature(K)	850	492.9
Oil-gas ratio	0.0430	0.0106
Combustion efficiency	> 99%	> 98%
Total pressure loss	< 6%	< 7%
OTDF	< 0.20	-
RTDF	< 0.12	-

Based on the concentric graded combustion chamber model, the surrogate model construction is carried out according to the calculation results of the one-dimensional program of the self-designed aero-engine combustion chamber model. Combustion efficiency $\eta$(%) and total pressure loss $\varDelta p$(%) are selected as the target parameters. Design parameters are selected as total inlet temperature ${T}_{t}$(K), total inlet pressure ${P}_{t}$(kPa), inlet flow ${S}_{t}$(kg/s), oil-gas ratio ${R}_{OG}$, number of main combustion holes (single head and single wall) ${N}_{m}$(PCS), size of main combustion holes ${D}_{m}$(mm), number of cooling holes in each row ${N}_{c}$(PCS) and size of cooling holes ${D}_{c}$(mm). The design variables are detailed in Table 3.

Table 3

design variable
Variable Symbols	Variable Name	Range of changes
${T}_{t}$/K	Total inlet temperature	427.55 ~ 1070.33
${P}_{t}$/kPa	Total inlet pressure	500 ~ 3200
${S}_{t}$/kg_*s^− 1	Total inlet flow	19.01 ~ 82.398
${R}_{OG}$	Oil-gas ratio	0.0106 ~ 0.0430
${N}_{m}$/PCS	Number of main combustion holes	2 ~ 4
${D}_{m}$/mm	size of main combustion holes	12.1 ~ 16.8
${N}_{c}$/PCS	Number of cooling holes per row	10 ~ 20
${D}_{c}$/mm	Cooling hole size	0.5 ~ 1.5

The combustor surrogate model design algorithm process mainly includes the following parts. First, Latin hypercube sampling is carried out in the 8-variable space to obtain sample data of input parameters. Then, the self-designed one-dimensional program of aero-engine combustion chamber model is used for calculation, and the corresponding combustion efficiency and total pressure loss of sample parameters are obtained. After data preprocessing, the sample data set is constructed and completed. Then, mapping relationships between eight input parameters and two output parameters ($\eta /\varDelta p$) are established based on POD-Hierarchical-Kriging model to replace the complex model among variables, that is, the surrogate model for aeroengine combustion chamber is constructed. Finally, based on the NSGA-II optimization algorithm and the surrogate model, the design parameters are optimized and the optimal configuration is obtained.

3.1 Design of POD-Hierarchical-Kriging combustion performance surrogate model

POD-Hierarchical-Kriging algorithm model is mainly based on Kriging model. Kriging model regards the unknown function as the concrete realization of a static stochastic process, including regression part and correlation part, among which the correlation part can be regarded as the realization of random Gaussian process, and has a good fitting effect for problems with a high degree of nonlinearity. Kriging model is based on the dynamic construction of sample information at known points and has both global and local statistical characteristics. The POD algorithm [12–14] is used to downscale and reconstruct the original samples, reduce the sample computational loss, extract the core information, filter out the marginal information. he data processed by the POD algorithm is used as the sample input for the Hierarchical-Kriging model. The above whole constitutes the POD-Hierarchical-Kriging algorithm model.

3.1.1 POD method

The core of the POD method is to find a set of "optimal" orthogonal bases $\{{{U}}_{1},{{U}}_{2},{{U}}_{3},\cdots ,{{U}}_{n}\}$ of the n-dimensional field space $\{{\mathbf{x}}_{\text{n}}\in {\Omega }\}$. Assuming that ${\mathbf{x}}_{\text{n}}$ can be approximated by a number of orthogonal bases, then ${\mathbf{x}}_{\text{n}}$ can be approximated by this set of "optimal" orthogonal bases $\{{{U}}_{1},{{U}}_{2},{{U}}_{3},\cdots ,{{U}}_{n}\}$:

$${{x}}_{n}=\sum _{i=1}^{n}{{\alpha }_{i}{U}}_{i}$$

Where ${{U}}_{i}$ is the eigenvector or basis vector of ${{x}}_{n}$; ${\alpha }_{i}$ is the correlation coefficient of ${{U}}_{i}$. If the dimension of the orthogonal space composed of "optimal" orthogonal bases is smaller than that of the original space, the above approximation process can be described as a downscaling of the sample space, followed by an approximate reconstruction of the sample, then:

$${{x}}_{n}=\sum _{i=1}^{r}{\alpha }_{i}{{U}}_{i} (r<n)$$

Therefore, the centralization of the sample data is performed. Then the new centralized sample data set is obtained:

$$\tilde{{X}}=\left\{{\tilde{{x}}}_{1},{\tilde{{x}}}_{2},{\tilde{{x}}}_{3},\cdots ,{\tilde{{x}}}_{n}\right\}$$

$$=\{{{x}}_{1}^{ }-\stackrel{-}{{{x}}_{1}},{{x}}_{2}^{ }-\stackrel{-}{{{x}}_{2}},{{x}}_{3}^{ }-\stackrel{-}{{{x}}_{3}},\cdots ,{{x}}_{n}^{ }-\stackrel{-}{{{x}}_{n}}\}$$

Followed by SVD singular value decomposition:

$$\tilde{{X}}={U}{\varSigma }{{V}}^{T}$$

For the value of $r$, $r$ should be less than n to reduce the scale of the feature vector space, but it should approximate the original data space as accurately as possible. The selection of r can be determined in the following ways:

$$I=\sum _{i=1}^{r}{\left({\lambda }_{i}\right)}^{2}/\sum _{i=1}^{n}{\left({\lambda }_{i}\right)}^{2}$$

$I$ is the energy information capacity or energy. The closer $I$ is to 1, the more complete the original information contained in the feature vector space is and the closer it is to the original function space. it is usually considered that $I$ is greater than 95% can be.Where $\lambda$ is the eigenvalue of the $\tilde{{X}}$ covariance matrix arranged from largest to smallest, satisfying ${\lambda }_{1}>{\lambda }_{2}>\cdots >{\lambda }_{n}$. Then the final POD model is:

$${{X}}_{RE}=\stackrel{-}{{X}}+\sum _{i=1}^{r}{{\alpha }_{i}{U}}_{i}$$

3.1.2 Basic Kriging model

The Kriging model is an unbiased estimation model. It has the characteristics of local estimation and has a good fit for problems with a high degree of nonlinearity. The following is a reference to the derivation process of Kriging method theory [16, 17]. In the analysis of regression model, the model assumes that the real relationship between the response value of the system and the independent variable can be expressed in the following form:

$${y}\left({x}\right)={f}({x}{)}^{T}{\beta }+{z}({x})$$

Where ${x}$ input parameter; ${f}\left({x}\right)={\left[{f}_{1}\left({x}\right),{f}_{2}\left({x}\right),\cdots ,{f}_{p}\left({x}\right)\right]}^{\text{T}}$; ${\beta }$ is the regression constant, expressed as ${\beta }={\left[{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{p}\right]}^{T}$, $p$ is the number of polynomial terms, the magnitude depends on the form of the polynomial; $y\left({x}\right)$ is the predicted value of combustion efficiency $\eta$(%) or total pressure loss $\varDelta p$(%), ${y}\left({x}\right)=[\eta ,\varDelta p]$; ${z}\left({x}\right)$ is a random process, which has the following statistical properties:

$$\left\{\begin{array}{c}E\left[z\right(x\left)\right]=0\\ Var\left[z\right(x\left)\right]={\sigma }^{2}\\ E\left[{z}\left({{x}}^{\text{i}}\right),{z}\left({x}\right)\right]={\sigma }^{2}R\end{array}\right.$$

Where:

$${R}=\left[\begin{array}{cccc}{\rho }_{11}& {\rho }_{12}& \cdots & {\rho }_{1m}\\ {\rho }_{21}& {\rho }_{22}& \cdots & {\rho }_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{m1}& {\rho }_{m2}& \cdots & {\rho }_{mm}\end{array}\right]$$

Where ${R}$ is "correlation matrix"; ${\rho }_{ij}$ is the correlation function value, representing the correlation between the i^th sample point and the j^th sample point; m is the size of the sample. Correlation function is artificially assumed, and Gauss function is commonly used. In this experiment, the Gauss function is used as the correlation function, and the specific form is as follows:

$${\rho }_{ij}=\text{e}\text{x}\text{p}\left(-\sum _{ℎ=1}^{n} {\theta }_{ℎ}{\left|{x}_{ℎ}^{i}-{x}_{ℎ}^{j}\right|}^{2}\right),i,j=\text{1,2},\dots ,m$$

Where the unknown parameter ${\theta }=\left[{\theta }_{1},{\theta }_{2},\dots ,{\theta }_{n}\right]$, the dimension n is the same size as the dimension of the sample point. ${x}_{ℎ}^{i}$ is the h^th variable of the i^th sample and can be expressed as any of the input parameters.

A prediction model formula (11) is given to approximate formula (7) :

$$\stackrel{\prime }{{y}}\left({x}\right)={{c}}^{T}\left({x}\right){Y}$$

Where ${Y}={\left[{\text{y}}_{1},{\text{y}}_{2},\dots ,{\text{y}}_{\text{m}}\right]}^{T}$, ${Y}$ is the known sample response vector; ${c}={\left[{c}_{1}\left({x}\right),{c}_{2}\left({x}\right),\dots ,{c}_{m}\left({x}\right)\right]}^{T}$, ${c}_{i}$ is related to a single sample point ${x}$. When the given sample points ${x}$ are different, the resulting ${c}_{i}$ is different.

Prediction error of the model is:

$$\stackrel{\prime }{{y}}\left({x}\right)-{y}\left({x}\right)={{c}}^{T}{z}-{z}+{\left({{F}}^{T}{c}-{f}\left({x}\right)\right)}^{T}{\beta }$$

Where ${z}={\left[{z}\left({{x}}_{1}\right),{z}\left({{x}}_{2}\right),\cdots ,{z}\left({{x}}_{m}\right)\right]}^{\text{T}}$, ${F}={\left[{{f}}^{T}\left({{x}}_{1}\right),{{f}}^{T}\left({{x}}_{2}\right),\dots ,{{f}}^{T}\left({{x}}_{m}\right)\right]}^{T}$.

Considering that $\stackrel{\prime }{{y}}\left({x}\right)$ is the optimal linear unbiased estimate of ${y}\left({x}\right)$, Eq. (13) is obtained from the unbiased estimate, and Eq. (14) is obtained from the minimum mean square error.

$${{F}}^{T}{c}={f}\left({x}\right)$$

$$MSE={{\sigma }}^{2}\left(1+{{c}}^{T}{R}{c}-2{{c}}^{T}{r}\right)$$

Where ${r}$ is the correlation vector composed of the correlation function between the point ${x}$ to be predicted and the original sample set.

By constructing Lagrange function, the response model expression can be obtained:

$$\stackrel{\prime }{{y}}\left({x}\right)={f}({x}{)}^{T}{{\beta }}^{\ast }+{r}({x}{)}^{T}{{\gamma }}^{\ast }$$

Where, ${{\beta }}^{\ast }={\left({{F}}^{T}{{R}}^{-1}{F}\right)}^{-1}{{F}}^{T}{{R}}^{-1}{Y}$, ${{\gamma }}^{\ast }={{R}}^{-1}\left({Y}-{{F}}^{T}{{\beta }}^{\ast }\right)$. It can be seen that ${F}$,${Y}$ can be obtained from the given sample, R contains only the parameter ${\theta }$. ${{\beta }}^{\ast }$, ${{\gamma }}^{\ast }$, ${r}\left({x}\right)$ can be obtained by finding the unknown parameter ${\theta }$.

Since each sample point is not independent, the joint probability density is:

$$L\left({\beta },{{\sigma }}^{2},{\theta }\right)=\frac{1}{\left(2\pi {)}^{m/2}{\left({{\sigma }}^{2}\right)}^{m/2}\right|\mathbf{R}{|}^{1/2}}\text{e}\text{x}\text{p}\left[-\frac{1}{2{{\sigma }}^{2}}{\left({Y}-{{F}}^{\text{T}}{\beta }\right)}^{T}{\mathbf{R}}^{-1}\left({Y}-{{F}}^{\text{T}}{\beta }\right)\right]$$

Taking the logarithm:

$$\text{l}\text{n}L\approx -\frac{1}{2}\left(m\text{l}\text{n}{\stackrel{\prime }{{\sigma }}}^{2}+\text{l}\text{n}\left(\right|\mathbf{R}\left({\theta }\right)\left|\right)\right)$$

Where $\text{l}\text{n}L$ only contains parameter ${\theta }$, the parameter training is transformed into the solution of a nonlinear optimization problem, which in turn yields R. Then ${{\beta }}^{\ast }$,${{\gamma }}^{\ast }$༌${r}\left({x}\right)$ are, si, ilarly available, so as to determine the final model.

3.1.3 Hierarchical-Kriging model

The difference between Hierarchical-Kriging and basic Kriging is that Hierarchical-Kriging adopts multi-layer basic Kriging structure [1, 15]. Hierarchical-Kriging allows the output of the first layer to be the global approximate reference of the second layer, the second layer to be the global approximate reference of the third layer, and so on. Hierarchical-Kriging has a very great advantage in practical applications to make full use of the sample information [16, 17]. Basic Kriging is too sensitive to the noise of training samples and requires high reliability of training samples, while it is easy to overfit training samples with low reliability. However, Hierarchical-Kriging adopts a Hierarchical strategy, training the first layer with samples of low confidence and the second layer with samples of high confidence, so as to fully explore the potential of training samples. Take Hierarchical-Kriging with two layers as an example:

To construct two-layer Kriging, the first layer Kriging should be constructed with low confidence training sample points.

$${y}_{1\text{f}}\left({x}\right)={\beta }_{1\text{f}}+{z}_{1\text{f}}\left({x}\right)$$

The polynomial in the model of the first layer is a constant, because the reliability of the training sample is low, it is easy to fit the noise with the polynomial of higher order, resulting in the decline of the prediction accuracy of the final model.

$${\stackrel{\prime }{y}}_{1\text{f}}\left({x}\right)={\beta }_{1\text{f}}+{{r}}_{1\text{f}}^{T}\left({x}\right){\mathbf{R}}_{1\text{f}}^{-1}\left({\mathbf{y}}_{1\text{f}}-{\beta }_{1\text{f}}1\right)$$

Where ${\beta }_{1\text{f}}={\left({1}^{T}{\mathbf{R}}_{1\text{f}}^{-1}1\right)}^{-1}{1}^{T}{\mathbf{R}}_{1\text{f}}^{-1}{\mathbf{y}}_{1\text{f}}$,${\mathbf{R}}_{1\text{f}}\in {\mathbb{ℝ}}^{{n}_{1\text{f}}\times {n}_{1\text{f}}}$༌$1\in {\mathbb{ℝ}}^{{n}_{1\text{f}}}$༌${{r}}_{1\text{f}}\in {\mathbb{ℝ}}^{{n}_{1\text{f}}}$. This is the first l, yer through, he res, lt of the training sample points out.

The second layer structure is constructed on the basis of the first layer, and the high reliability sample points are used to train the second layer.

$${y}_{2\text{f}}\left({x}\right)=\beta {\stackrel{\prime }{y}}_{1\text{f}}\left({x}\right)+z\left({x}\right)$$

Using the same derivation as the first level, it can be derived that:

$${\stackrel{\prime }{y}}_{2\text{f}}\left({x}\right)=\beta {\stackrel{\prime }{y}}_{1\text{f}}\left({x}\right)+{{r}}_{2\text{f}}^{T}\left({x}\right){\mathbf{R}}_{2\text{f}}^{-1}\left({\mathbf{y}}_{2\text{f}}-\beta {F}\right)$$

Where ${F}={\left[{\stackrel{\prime }{y}}_{1\text{f}}\left({{x}}_{1}\right),\dots ,{\stackrel{\prime }{y}}_{1\text{f}}\left({{x}}_{m}\right)\right]}^{T}$; ${\mathbf{R}}_{2\text{f}}$ and ${{r}}_{2\text{f}}$ are both correlation matrices and correlation vectors composed with high confidence sample points (in the same specific form as basic kriging); ${\mathbf{y}}_{2\text{f}}$ is a vector composed of responses from sample points; $\beta ={\left({{F}}^{T}{\mathbf{R}}_{2\text{f}}{ }^{-1}{F}\right)}^{-1}{{F}}^{T}{\mathbf{R}}_{2\text{f}}{ }^{-1}{\mathbf{y}}_{2\text{f}}$.

3.2 NSGA-Ⅱ Multi-objective Optimization Design

Genetic algorithm is widely used in aero-engine [20, 21]. NSGA-II is a multi-objective optimization algorithm based on the optimal Pareto solution. It has low complexity, elite population strategy and degree of crowding comparison operator. The key steps of NSGA-II include fast non-dominated sorting, calculation of congestion, elite reservation policy and tournament selection.

The target parameters involved in the engine combustion chamber design are combustion efficiency $\eta$ and total pressure loss $\varDelta p$, where the combustion efficiency η should be as large as possible and ensure $\eta \ge 99\%$ and the total pressure loss $\varDelta p$ should be as small as possible and ensure $\varDelta p\le 4\%$. To meet the constraints, Circumferential Temperature Distribution Factor

(OTDF) is less than 0.2.

According to the multi-objective minimum optimization model, the optimization model can be described as:

$$obj.\left\{\begin{array}{c}max -\eta ({T}_{t},{P}_{t},{S}_{t},{R}_{OG},{N}_{m},{D}_{m},{N}_{c},{D}_{c})\\ min \varDelta p({T}_{t},{P}_{t},{S}_{t},{R}_{OG},{N}_{m},{D}_{m},{N}_{c},{D}_{c})\end{array}\right.$$

$$constraint：OTDF\in \left[\text{0,0.20}\right]$$

$$s.t.\left\{\begin{array}{c}{T}_{t}\in \left[\text{427.55,1070.33}\right]\\ {P}_{t}\in \left[\text{500,3200}\right]\\ {S}_{t}\in \left[\text{19.01,82.398}\right]\\ {R}_{OG}\in \left[\text{0.0106,0.0430}\right]\\ {N}_{m}\in \left[\text{2,4}\right]\\ {D}_{m}\in \left[\text{12.1,16.8}\right]\\ {N}_{c}\in \left[\text{10,20}\right]\\ {D}_{c}\in \left[\text{0.5,1.5}\right]\end{array}\right.$$

Different from the optimal selection of the single objective optimization problem, the multi-objective optimization problem [18–20] involved in this paper contains two conflicting optimization objectives. The objective parameters combustion efficiency $\eta$ and total pressure loss $\varDelta p$ are difficult to reach the optimal simultaneously. Therefore, in the optimization process, two conflicting objectives can only be weighed to reach the Pareto optimal. Finally, an optimal solution set (i.e. Pareto solution set) containing multiple elements is obtained.

4.1 Analysis of combustion performance surrogate model based on POD-Hierarchical-Kriging

In this paper, based on the model and method deduced above, the experimental process is mainly carried out to collect samples of combustion performance parameters of aero-engine combustor model, and the POD-Hierarchical-Kriging surrogate model is established to predict. Firstly, based on 91 sets of sample data after Latin hypercube sampling and pre-processing, the one-dimensional procedure of the aero-engine combustion chamber model designed by Northwestern Polytechnic University is used to obtain the calculation results of the target optimization parameters corresponding to the sample data, and then the construction of the sample data set is completed.

After that, parameter estimation of POD-Hierarchical-Kriging surrogate model is performed to obtain surrogate models of combustion efficiency $\eta$ and total pressure loss $\varDelta p$.

Using the above established model, 20 groups of untrained test data are used to test the prediction accuracy of the established surrogate model, and root mean square error (RMSE) is used as the measurement of error magnitude. The prediction errors of the model for combustion efficiency and total pressure loss are 0.0064% and 0.2523%, respectively. The relative prediction errors for the combustion efficiency and total pressure loss are shown in Fig. 3 We performed cross-validation of the POD-Hierarchical-Kriging model to calculate the fitness between the model and the flow solver. The results of the cross-validation of the predicted data points are depicted in Fig. 4 The horizontal axis represents the original values of the test data and the vertical axis represents the values predicted by the POD-Hierarchical-Kriging model. If the sample points are closer to the symmetrical diagonal, the model is considered accurate [22].

The accuracy of POD-Hierarchical-Kriging model is preliminarily verified based on the prediction error relative graph and cross validation graph. In order to fully validate the advantages of the POD-Hierarchical-Kriging model as a prediction of combustion chamber performance, a comparison with other methods is necessary. The POD-Hierarchical-Kriging model is compared with the commonly used cubic polynomial response surface model, the basic Kriging model, and the Hierarchical-Kriging model. The basic Kriging model is trained using two schemes. The first scheme is trained with 91 sample points(Notated as A-Kriging model) and the second scheme is trained with only 12 high confidence samples(Notated as OH-Kriging model). The RMSE is used as the criterion to judge the prediction error of each model. The smaller the root mean square error of the model, the better the model prediction is.

As shown in Fig. 5, for the combustion efficiency, the RMSE of the predictions of the four models using the new data set are 0.0115% for the cubic polynomial model, 0.018% for the A-Kriging model, 0.0128% for the OH-Kriging model, 0.0083% for the Hierarchical -Kriging model, and 0.0083% for the POD-Hierarchical-Kriging model 0.0064%. It is shown that all four surrogate models can well characterize the original model. For the combustion efficiency parameters, the POD-Hierarchical-Kriging model predicts the smallest RMSE, and the Hierarchical -Kriging model is the next best, both better than the cubic polynomial model and the basic Kriging model. Therefore, the POD-Hierarchical-Kriging surrogate model is accurate in predicting the combustion efficiency.

As shown in Fig. 6, for the total pressure loss, the RMSE of the predictions of the 4 models using 20 test data sets are 0.2002% for the cubic polynomial model, 0.1942% for the A-Kriging model, 0.4576% for the OH-Kriging model, 0.1833% for the Hierarchical -Kriging model, and 0.1995% for the POD- Hierarchical -Kriging model 0.1995%. It shows that the four surrogate models can well characterize the original model. However, for the total pressure loss, the RMSE of the values predicted by the Hierarchical-Kriging model is the smallest. The POD-Hierarchical-Kriging model has similar predictions as the A-Kriging model. Their RMSE are larger than those predicted by the Hierarchical -Kriging model and smaller than those of the cubic polynomial model. The RMSE of OH-Kriging model is the largest, and its prediction is the worst.

By comparing with common surrogate models, POD-Hierarchical-Kriging model shows its advantage as a model for predicting combustion chamber performance. Compared with the OH-Kriging model, the POD-Hierarchical-Kriging model performs better in both the prediction of combustion efficiency and the prediction of total pressure loss. This indicates that the POD-Hierarchical-Kriging model compensates the problem that the basic kriging model requires high confidence in the training sample points. It makes full use of the low confidence training samples in the sample dataset, thus improving the prediction accuracy. For comparison experiments on combustion efficiency prediction, POD-Hierarchical-Kriging model has the best prediction effect and the RMSE of POD-Hierarchical-Kriging model is 22.8% lower than the RMSE of the next best Hierarchical-Kriging model. It shows that the POD algorithm can better characterize the original data space, extract more critical sample information, and filter out marginal information after dimensionality reduction and reconstruction of sample data set. For the prediction of total pressure loss, the prediction accuracy of POD-Hierarchical-Kriging model is slightly lower than that of optimal Hierarchical-Kriging model and higher than that of cubic polynomial model. It is shown that combining POD algorithm with Hierarchical-Kriging model avoids overfitting and reduces computing loss to a certain extent. In the face of a large number of parameter prediction, the rapidity of the POD-Hierarchical-Kriging model can be demonstrated, and the prediction accuracy has not been greatly reduced.

4.2 Optimization results and analysis

Sensitivity analysis is used to study the influence of eight design variables on the two objective functions, and to explore which decision variables have a greater or lesser influence on the test results. Figure 7 shows the sensitivity index analysis of design variables on combustion efficiency and total pressure loss respectively and the influence ratio of each design variable. As shown in the figure, no matter the sensitivity index of combustion efficiency or the sensitivity index of total pressure loss, the design variables with great influence are: total inlet temperature, total inlet pressure,total inlet flow and oil-gas ratio. The total influence of the other four design variables is less than 0.006% and can be ignored. Among them, the total inlet pressure has a great influence in the combustion efficiency and total pressure loss, so it can be seen that the design of total inlet pressure is in the dominant position in the combustion chamber design. In addition, the oil-gas ratio has the greatest influence on the combustion efficiency of the combustion chamber, and the total inlet flow has the greatest influence on the total pressure loss of the combustion chamber. Therefore, when doing the optimization design of the combustion chamber, it is advisable to focus on the four aspects of total inlet temperature, total inlet pressure, total inlet flow and oil-gas ratio.

Based on the POD-Hierarchical-Kriging surrogate model, the optimization of aero-engine combustion chamber configuration parameters is carried out for the target optimization parameters of combustion efficiency $\eta$ and total pressure loss $\varDelta p$. NSGA-II algorithm is used to solve the non-dominant solutions of two objective functions constructed by POD-Hierarchical-Kriging surrogate model. The population size and iteration size are 200 and 50, respectively. The crossover rate and mutation rate are fixed at 1 and 0.03. Crossover parameters and mutation parameters are fixed at 100 and 100.

Pareto front is shown in Fig. 8 The horizontal and vertical axes are total pressure loss and combustion efficiency respectively. As shown in the figure, the overall trend of the Pareto solution set is that as the combustion efficiency increases, the total pressure loss also increases.

The constraint OTDF is predicted by using the basic Kriging surrogate model during the iterative process. Figure 9 illustrates the constraints to satisfy OTDF in the process of optimizing the two objective functions. The dotted line indicates the boundary of the design space, and the shaded area indicates the area outside the design constraints.

It can be seen from Fig. 9 that Pareto fronts all meet the restriction of OTDF. It is also observed that most of the solutions in the optimal solution set have OTDFs greater than 0.1, indicating that most of the solutions in the Pareto solution set obtained by solving with the multi-objective of combustion efficiency and total pressure loss will have larger OTDFs. Based on the sensitivity analysis of combustion efficiency and total pressure loss, the influence of these design variables on combustion efficiency and total pressure loss is further explored by using four design variables with great influence. According to the Pareto solution set of the POD-Hierarchical-Kriging model, the influence of the four variables on the two objective functions is shown in Fig. 10 and Fig. 11.

The relationship between total pressure loss and combustion efficiency with design variables is analyzed according to the four influential design variables using the pareto solution set of the POD-Hierarchical-Kriging model. Figure 10(a) plots the relationship between combustion efficiency and total inlet temperature. Figure 10(b) plots the relationship between combustion efficiency and total inlet pressure. Figure 10(c) plots the relationship between combustion efficiency and total inlet temperature. Figure 10(d) plots the relationship between combustion efficiency and oil-gas ratio. As shown in Fig. 10(a), in the Pareto solution set, the combustion efficiency of the optimal solution with total temperature between 550K and 650K is small, and that of the optimal solution between 650K and 750K is large. The iteration of the genetic algorithm produced a large number of individuals with total inlet temperatures between 650 K and 750 K, indicating that the superior property of the total inlet temperature between 650 K and 750 K in the design of the combustion chamber. As shown in Fig. 10(b), the combustion efficiency of the optimal solution between 2000kpa and 3000kpa at the inlet total pressure is relatively high, and the individuals tend to first aggregate and then disperse with the increase of the inlet total pressure. It reflects that increasing the total inlet pressure within a certain range will be beneficial to improve the combustion efficiency, while when the total inlet pressure is too large, the property of improving the combustion efficiency will be lost. As shown in Fig. 10(c), two optimal solutions with minimum combustion efficiency occurr between the total inlet flow of 40kg/s and 45kg/s. As for combustion efficiency, the design variable of total inlet flow in the whole population has no obvious aggregation, which confirms that the influence of total inlet flow on combustion efficiency is smaller than that of total inlet pressure and oil-gas ratio in sensitivity analysis. As shown in Fig. 10(d), a large number of optimal solutions in the Pareto solution set are clustered between 0.03 and 0.035, reflecting the excellent property of the oil-gas ratio of 0.03 to 0.035 on the combustion efficiency. After iteration the oil-gas ratio shows a significant aggregation between 0.025 and 0.035, which is consistent with the largest influence of oil-gas ratio on combustion efficiency in sensitivity analysis.

Figure 11(a) plots the relationship between the total pressure loss and the total temperature at the inlet. Figure 11(b) plots the relationship between the total pressure loss and the total pressure at the inlet. Figure 11(c) plots the relationship between the total pressure loss and the total temperature at the inlet. Figure 11(d) plots the relationship between total pressure loss and oil-gas ratio. As shown in Fig. 11(a), more optimal solutions with low total pressure loss occur in the Pareto solution set between 550K and 650K for the total inlet temperature, while more optimal solutions with high total pressure loss are obtained for the most between 650K and 750K. This corresponds exactly to the influence of the total inlet temperature on the combustion efficiency in Fig. 10(a), where the combustion efficiency is smaller for the optimal solutions with the total inlet temperature between 550 K and 650 K and larger for the optimal solutions between 650 K and 750 K. It indicates that the total inlet temperature needs to be designed with a trade-off between high combustion efficiency and low total pressure loss in the optimization process with combustion efficiency and total pressure loss as the optimization objective. As shown in Fig. 11(b), more optimal solutions with low total pressure loss occur between 1500kpa and 2500kpa of the total inlet pressure. In Fig. 10(b), the optimal solution for the total inlet pressure between 2000kpa and 3000kpa has a larger combustion efficiency. It is inferred that the design variables in the combustion chamber design process will have better performance in the combustion chamber with the total inlet pressure between 2000kpa and 2500kpa. As shown in Fig. 11(c), many of the optimal solutions in the Pareto solution set aggregate between 60 kg/s and 75 kg/s of total inlet flow. As shown in Fig. 11(d), aggregation occurs at oil-gas ratios of 0.03 to 0.035, however many optimal solutions with oil-gas ratio between 0.03 and 0.035 have total pressure loss greater than 1%. In the non-aggregated oil-gas ratio between 0.02 and 0.025, there are more optimal solutions with low total pressure loss. In Fig. 10(d), the optimal solutions clustered between 0.03 and 0.035 oil-gas ratio have higher combustion efficiency. It shows that the aggregation of individuals for the design variable of oil-gas ratio produced during the iterative process is mainly used to seek higher combustion efficiency. Therefore, the change in oil-gas ratio during the design of the combustion chamber will have a greater influence on the change in combustion efficiency. It corresponds to the fact that the oil-gas ratio has the greatest influence on the combustion efficiency during the sensitivity analysis, while the oil-gas ratio has less influence on the total pressure loss than the total inlet pressure and the total inlet flow.

1) Firstly, we apply a concentric graded combustion chamber of an aero-engine independently developed by Northwestern Polytechnic University, innovatively merge the POD algorithm with the Hierarchical-Kriging model, and propose a POD-Hierarchical-Kriging based combustion performance surrogate model method. The sample data are reconstructed by POD method to downscale, and then the design variables are performed by using the downscaled data. A study on the design and construction of a surrogate model for the performance of the combustion chamber is carried out. Through experiments, the predicted results of the POD-Hierarchical-Kriging model are compared and analyzed with the calculated results of the one-dimensional program, and the root mean square error of the predicted values of combustion efficiency and total pressure loss are 0.0064% and 0.1995%, respectively.

2) Subsequently, the POD-Hierarchical-Kriging model is compared with the cubic polynomial response surface model, the basic Kriging model and the Hierarchical-Kriging model. It is found that the POD-Hierarchical-Kriging model combines the advantages of the POD method and the Hierarchical-Kriging model. The POD-Hierarchical-Kriging model can better characterize the original data space, extract more critical sample information, and make full use of low confidence training sample points combined with a small number of high-precision training sample points to achieve high-precision prediction of unknown points. Compared with traditional simulations and numerical calculations, the POD-Hierarchical-Kriging model has a faster prediction speed.

3) On this basis, the multi-objective NSGA-II optimization method is applied to further carry out a parametric optimization study on eight design variables. Under the constraints, high combustion efficiency and low total pressure loss are the optimization objectives. Through sensitivity analysis, it is known that the design variables of total inlet temperature, total inlet pressure,total inlet flow and oil-gas ratio have a greater influence on the combustion efficiency and total pressure loss. Among them, the oil-gas ratio has the greatest influence on the combustion efficiency and the total inlet flow has the greatest influence on the total pressure loss. A Pareto solution set is obtained by NSGA-II optimization method. By analyzing Pareto solution set, NSGA-II multi-objective optimization results based on POD-Hierarchical-Kriging surrogate model can correspond to sensitivity analysis. The critical design factors and the specific influence states of the design parameters on the target parameters that have a large impact on the design objectives are obtained to guide the optimal design of the combustion chamber.

Competing interests

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ contributions

Conceptualization: Jialing Le, Hua Zhang; methodology: Mingming Guo; simulation: Shuhong Tong, Yue Ma; data acquisition: Wenyan Song; review and editing: Ye Tian. All authors have read and agreed to the published version of the manuscript.

Funding

This project was supported by Sichuan Science and Technology Program(Grant No. 2023YFG0336).

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request

Acknowledgements

Not applicable.

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published 03 Jul, 2023

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Editorial decision: Minor revision
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Optimization of aero-engine combustion chambers with the assistance of Hierarchical-Kriging surrogate model based on POD downscaling method

Status:

Journal Publication

Version 1

Abstract

Figures

1 Introduction

2 Combustion Chamber Model And Data Acquisition

2.1 Reliability verification of high performance 1D design method

2.2 High performance combustion chamber design for aero engine

3 Method

3.1 Design of POD-Hierarchical-Kriging combustion performance surrogate model

3.1.1 POD method

3.1.2 Basic Kriging model

3.1.3 Hierarchical-Kriging model

3.2 NSGA-Ⅱ Multi-objective Optimization Design

4 Experiments And Analysis Of The Results

4.1 Analysis of combustion performance surrogate model based on POD-Hierarchical-Kriging

4.2 Optimization results and analysis

5 Conclusions

Declarations

References

Status:

Journal Publication

Version 1

Variable Symbols	Variable Name	Range of changes
\({T}_{t}\)/K	Total inlet temperature	427.55 ~ 1070.33
\({P}_{t}\)/kPa	Total inlet pressure	500 ~ 3200
\({S}_{t}\)/kg_*s^− 1	Total inlet flow	19.01 ~ 82.398
\({R}_{OG}\)	Oil-gas ratio	0.0106 ~ 0.0430
\({N}_{m}\)/PCS	Number of main combustion holes	2 ~ 4
\({D}_{m}\)/mm	size of main combustion holes	12.1 ~ 16.8
\({N}_{c}\)/PCS	Number of cooling holes per row	10 ~ 20
\({D}_{c}\)/mm	Cooling hole size	0.5 ~ 1.5