Parameter optimization design of an aero-engine bearing chamber based on active learning Kriging

The air-oil two-phase flow in the aero-engine bearing chamber is highly complex and unsteady. To obtain the mapping relationship between the oil volume fraction in the chamber and the oil inlet flow rate and rotational speed for further design optimization, the air-oil two-phase flows are simulated using the level set method in COMSOL Multiphysics software. To avoid frequent calls to the simulation model during the optimization process, which consumes huge computational costs, the Kriging model is applied to approximate the true response of the computational fluid dynamics (CFD) simulations. The point addition process is assisted by introducing the active learning function EI to enhance the efficiency of constructing the Kriging model. Particle swarm optimization (PSO) is applied to optimize the Kriging-based optimization objective function. With the optimal parameters’ combination, the performance of the oil return in the aero-engine bearing chamber is improved. This study has guiding significance for the lubrication design of the aero-engine bearing chamber.


Introduction
The aero-engine bearing chamber is essential to the lubrication and sealing assembly.The rotational speed and oil inlet flow rate affect the air-oil two-phase flow in the chamber, further influencing the oil return performance.Thus, their setup is crucial to research [1,2].The air-oil two-phase flow in the aeroengine bearing chamber is highly complex.The highly frequent calls of simulation analysis of nonconstant flow fields make the computational costs skyrocket.Optimization based on the surrogate model has become necessary to reduce computation times.Researchers can apply the approximate solution of the surrogate model directly in the sample space, avoiding some complicated and time-consuming finite element calculations.
The Kriging model not only offers a predictive response at any given point but also provides an assessment of localized uncertainty via the Kriging variance on the response.The knowledge of the Kriging variance makes combining active learning methods possible.Active learning refers to the process of updating the Kriging model through the addition of a new data point to the existing DoE.This point is called "the best next point", which is selected based on its anticipated capacity to enhance the Kriging model's performance [3].The Kriging model is updated until a specific convergence condition is met, at which point the Kriging model is used instead of the actual function for the following works.This process will greatly improve the efficiency and accuracy of building models.Due to its unique characteristics and excellent performance, active learning kriging is widely used in optimization and reliability analysis problems involving implicit finite element models.
This study uses the level set method in COMSOL Multiphysics to simulate air-oil two-phase flow in an aero-engine bearing chamber.A kriging model was constructed to proximate the true response of CFD simulations.The active learning function EI [4] is introduced to assist the point addition process, thus further advancing the efficiency of constructing the Kriging model.After active learning iterations and error validation, the model is output and applied to the subsequent optimization design.Particle Swarm Optimization (PSO) was used to optimize the Kriging-based objective function within the range of variables to obtain an optimal parameters combination.The Parameter optimization design of an aeroengine bearing chamber based on active learning Kriging and air-oil two-phase flow in this study significantly improves the computational efficiency, a reference value for the lubrication design of aero engines.

The kriging model
The Kriging is a regression algorithm for predicting stochastic processes/random fields and spatial modeling based on covariance functions.It is an application of the Gaussian process regression method.The Kriging model gives the best linear unbiased Prediction (BLUP) for a given stochastic process.The model [5] can be expressed as Equation (1) ~ Equation ( 5): ( 5 ) where g k (X) is the predicted value.f i (X) is the base regression function.β i is a vector of regression coefficients estimated by generalized least squares.z(X) is a stochastic process.R(x (i) , x (j) ) correlation function with undetermined parameters.θ k A is a parameter vector determined by the maximum likelihood estimation method or the cross-validation method.g is the matrix consisting of the true responses of the interpolated points, and F represents the matrix comprising the regression model evaluated at m sample points.

Numerical model
Three-dimensional modeling was performed concerning the bearing chamber I in the Bearing Chamber test rig [6] at the University of Karlsruhe.Lubricating oil flows out through the internal cavity of the bearing and is thrown into the bearing chamber from the oil inlet.Sealing air injects the bearing chamber from the other side.Under the combined effects of forces, the oil and sealing air in the bearing chamber is discharged from the vent and the scavenge.The configuration of the bearing chamber's structural parameters is illustrated in Table 1.
Table 1.Geometry parameters of the bearing chamber.

Geometry parameters Specification
Shaft radius (mm) 64 Chamber height (mm) 10 Chamber width (mm) The simplified geometric model is schematically shown in Figure 1.The mesh division of the simplified geometric model is shown in Figure 2. The mesh is divided using a hexahedral structure, and the mesh quality can meet the simulation's accuracy requirements.Geometric modeling and meshing are performed in COMSOL Multiphysics.To balance the computational efficiency and simulation accuracy, the mesh division method with an element number of 24520 is used for the solution.The boundary conditions are as follows: The air and oil inlets are mass flow inlets.Both the vent and the scavenge are set up as pressure outlets with standard atmospheric outlet pressure values.For the accuracy of the numerical simulation, the backflow phenomenon is suppressed at all outlets, and hydrostatic pressure compensation is switched on.All the walls are set up to be no-slip, and the shaft wall is set as a moving wall.A wetting wall boundary condition is applied to all walls, excluding rotating walls, entrances, and exits.The contact angle between the wall and the lubricating oil is 3/8 π.In addition, the effect of gravity and surface tension are incorporated into the study.The temperature is set to a constant 330 K.The density of the lubricating oil is 931.88 kg/m 3 .The dynamic viscosity of the lubricating oil is 0.031 Pa⋅s.
The present calculation is founded on the assumption that the temperature remains unchanged and neglects to account for the vaporization of oil droplets and any fluid property variations.The level set method is used in COMSOL Multiphysics 6.0 to trace the air-oil two-phase medium flow.The 3D unsteady model is adopted.After phase initialization, the transient solver is applied to the solution.

Optimization design method
Directly calling numerous simulation calculations in the optimization design will make the computational cost too large to be acceptable.Building a Kriging model from a relatively small number of simulation results and then being used in the subsequent optimization process will significantly improve computational efficiency.The process of the optimization design method based on active learning Kriging is shown in Figure 3, which can be enumerated as follows: 1) Definition of the initial Design of Experiments (DoE): The initial DoE should be increased in size with the dimension of the problem.It is essential to prioritize a small initial design and incorporate only the most beneficial points to optimize the metamodel efficiently [3].For the dimensions in this study, it was sufficient to construct the initial Kriging model with 18 points.
2) Generation of a candidate sample pool in the design space: We generate a candidate sample pool of size 1e5 for the active learning function EI to choose the best next points.9) End of the method: We output the optimization result, and the method ends.

Application examples
The optimization design method proposed in Section 4 is applied to the numerical model mentioned in Section 3. To consider the interaction of the factors, 18 typical operating conditions were set up by the factorial design method.Three typical rotational speeds (4000 rpm, 8000 rpm,12000 rpm) and six typical oil inlet flow rates (0.04 kg/s, 0.06 kg/s, 0.08 kg/s, 0.10 kg/s, 0.12 kg/s, 0.14 kg/s) are combined to form 18 (3×6) classic operating conditions.The variation of the air inlet flow was not introduced into the study, so the only process parameters corresponding to these 18 classical conditions were the rotation speed and the inlet oil flow.In this study, the air inlet flow rate is a constant 0.005 kg/s in all cases.The oil inlet flow rate and rotational speed form the coordinates of the two-dimensional sample points used to construct the Kriging model and, at the same time, are used as independent variables for the objective function in the parameter optimization.CFD simulations of the sample points in the DoE were carried out by the level set method in COMSOL Multiphysics to obtain the response values corresponding to the sample points, i.e., the oil volume fraction in the chamber at 40 ms corresponding to these conditions.The stopping criteria of EI set in literation is max(E(I(x))) < 0.001.Due to the high complexity of the simulation, the stopping criterion needs to be more rigorous for the surrogate model constructed.The overly lenient stopping criterion will allow the active learning process to stop at a high value of EI, lacking the necessary learning times, making the surrogate model less accurate and misleading for the subsequent optimization process.A more conservative stopping criterion was used: max(E(I(x))) < 1.5×10 -6 .In this study, after three rounds of additive processes, the maximum EI meets the stopping criteria, and the search is terminated.
The LHS technique uses the multivariate standard normal probability density function to generate four error verification samples.The error between the Kriging prediction solution and the simulation solution for all error verification points is less than 0.1%, so the constructed Kriging model passes the error verification and is output for the following parameter optimization.The surface fitted by the Kriging model between the design variables and the response values is shown in Figure 4.
The lower oil volume fraction in the chamber means that the oil can be discharged more efficiently from the return line out of the chamber [8].The objective function based on the Kriging model is established, which takes the lowest oil volume fraction in the aero-engine bearing chamber as the optimization object.The shaft rotational speed x 1 and the inlet oil flow rate x 2 are selected as design variables.Equation ( 6) is the mathematical expression for this optimization problem.( , ) , 04 0.14 where f k is the Kriging model that fits the oil volume fraction in the chamber.The PSO was used for parameter optimization.Table 2 lists the main parameters to which the PSO was set.The values of c 1 and c 2 reflect the perception of the particle itself and the exchange of information between different particles, respectively.Particles are assumed to have a positive subjective tendency while at the same time exerting an influence on the judgments of other particles.The inertia weights ω are adaptively adjusted in the range [0.Finally, optimal parameters combination was obtained with an oil inlet flow rate of 0.04 kg/s and a rotational speed of 6120.8554rpm.An average of 7.8% reduces the oil volume fraction in the chamber under the optimal parameters' combination compared to the typical working conditions.The oil volume fraction in the bearing chamber of the aero-engine is reduced after optimization, and the performance of the oil return in the aero-engine bearing chamber is improved.

Conclusion
A numerical simulation of air-oil two-phase flow in an aero-engine bearing chamber is carried out.A factorial design method is applied to determine the initial DoE.The DoE constructs a Kriging model to approximate the oil volume fraction in the bearing chamber.The active learning function EI searches for the best next point in the candidate sample pool to update the DoE.The EI accelerates the efficiency of surrogate model construction, significantly reducing the computational cost.PSO was applied to optimize the objective function surrogated by the Kriging model.An optimal combination of parameters was obtained: Oil inlet flow rate of 0.04 kg/s and rotational speed of 6120.8554rpm.After optimization, the oil return performance of the aero-engine bearing chamber was improved.The parameter optimization design based on active learning kriging proposed in this paper effectively reduces arithmetic costs and improves computational efficiency.

3 ) 4 )
We construct or update the Kriging model to approximate the objective function: The initial Kriging model is constructed from the sample points in the DoE and their corresponding response values.When returning to this step due to not satisfying the stopping criterion of the learning function or not satisfying the error validation of the surrogate model, the best next point x* is searched through the learning function and added to the DoE, and the corresponding CFD simulations are performed.Finally, the Kriging model is updated.We select the best next point x* by active learning function: The active learning function EI is applied.In contrast to other active learning functions that concentrate solely on local areas of promise, EI can help balance local exploration and global exploitation, thus better finding the global optimal solution[7].In each iteration, the point with the highest EI value in the candidate sample pool is added to the DoE as the best next point x* for surrogate model updating.

Figure 3 . 5 ) 6 ) 7 ) 8 )
Figure 3. Flowchart of the optimization design method.5) Stopping criteria for learning: If the learning process has satisfied its stopping criteria, we go to step 7; otherwise, we go to step 6. 6) We obtain the simulation solution at point x* and update the DoE: Point x* is the next best searched by the active learning function EI.Its simulation solution is calculated and added to the DoE to bring maximum improvement to the model.7) Error verification: If the stopping criteria have been satisfied, the error validation of the surrogate model will be carried out.Latin hypercube sampling (LHS) is applied to extract four sample points for error validation, the predictions of the surrogate model at the sample points are calculated, and CFD simulations are carried out for the conditions corresponding to the sample points to obtain their simulated solutions.The predicted values of the surrogate model are compared with the CFD simulation solution.The results of this comparison are used to determine whether the model passes the error verification.We output the Kriging model if it passes the error verification; otherwise, we return to step 4.
Figure 5 shows the convergence curves.

Figure 4 .
Figure 4.The Kriging model fitting surfaces.Figure 5. Relation between iterations and fitness.

Figure 5 .
Figure 4.The Kriging model fitting surfaces.Figure 5. Relation between iterations and fitness.

Table 2 .
The main parameters of PSO.