With increasing design demands of turbomachinery, stochastic flutter behavior has become more prominent and even appears a hazard to reliability and safety. Stochastic flutter assessment is an effective measure to quantify the failure risk and improve aeroelastic stability. However, for complex turbomachinery with multiple dynamic influencing factors (i.e., aeroengine compressor with time-variant loads), the stochastic flutter assessment is hard to be achieved effectively, since large deviations and inefficient computing will be incurred no matter considering influencing factors at a certain instant or the whole time domain. To improve the assessing efficiency and accuracy of stochastic flutter behavior, a dynamic meta-modeling approach (termed BA-DWTR) is presented with the integration of bat algorithm (BA) and dynamic wavelet tube regression (DWTR). The stochastic flutter assessment of a typical compressor blade is considered as one case to evaluate the proposed approach with respect to condition variabilities and load fluctuations. The evaluation results reveal that the compressor blade has 0.95% probability to induce flutter failure when operating 100% rotative rate at t = 170 s. The total temperature at rotor inlet and dynamic operating loads (vibrating frequency and rotative rate) are the primary sensitive parameters on flutter failure probability. By method comparisons, the presented approach is validated to possess high-accuracy and high-efficiency in assessing the stochastic flutter behavior for turbomachinery.
Stochastic flutter behavior is one of the critical issues widely existing in turbomachinery. With the increasing design demands of industry, it has become more prominent and even appears a hazard to turbomachinery safety [
To quantify the propagation of system uncertainty into the response, some progress [
In this case, as one viable alternative to the expensive Monte Carlo simulation, meta-model (or surrogate model, response surface model) methods were developed to release tremendous simulation burden and were widely employed in reliability prediction, sensitivity analysis and probabilistic evaluation [
To elevate the computational accuracy and efficiency of the traditional TR model, we first develop the DWTR meta-model by integrating the dynamic extremum thought and the wavelet transform technique into the traditional TR model. The dynamic extremum thought is employed to simplify the dynamic process by the focus on the extremum responses, which releases the heavy burden of computation, and the wavelet transform technique is utilized to elevate the nonlinear mapping ability by constructing the wavelet kernel with local precise description ability, which ensures the fitting fidelity. Nevertheless, it is still difficult for the DWTR meta-model directly applied to the stochastic flutter assessment as the overfitting and local optimization problems in DWTR training process always occur while time-varying and high nonlinear limit state function is fitted, which influences the estimation precision of DWTR. To resolve this issue, the BA-DWTR meta-model is further proposed. The bat algorithm is specially designed with global exploitation ability and local exploitation ability to search the optimal model parameters. Then the mathematical regression model of BA-DWTR is established.
The objective of this study is to develop a dynamic meta-modeling approach (BA-DWTR) to improve the computing accuracy and efficiency of stochastic flutter assessment. As for the presented approach, the dynamic extremum thought [
The rest of this paper is structured as follows.
For stochastic flutter assessment of turbomachinery, the dynamic behavior of fluid-structure system is a time-variant stochastic process with the transition between different working states. To precisely evaluate the flutter failure risk, a series of meta-models should be established at each state in the complete cycle, demanding large computational consumption. In this instance, as an important time-varying responses processing technique, the dynamic extremum thought [
As depicted in
With implicit mapping function
Obviously, the feasibility of DTR meta-model is dependent on the weight coefficient vector
To enhance the generalization ability of DTR meta-model, the quadratic tube error function is introduced as
Therefore, with the slack variable
In the optimization process, the kernel function
By solving the dual problem, the DTR meta-model function is retrieved with the optimal solution as
In DTR meta-modeling, the nonlinear fitting ability relies on the kernel function, as the scalar product in implicit feature space is directly expressed by the kernel trick. However, the Gaussian radial basis function (RBF) kernel function, as the conventional kernel function for TR meta-modeling, is insufficient when applied to approximate the extremum response in stochastic flutter assessment owing to high nonlinearity and strong interaction between fluid domain and structure domain. To improve the nonlinear fitting ability of DTR meta-model, a dynamic wavelet TR meta-model is further developed by employing the wavelet basis function as kernel function. The DWTR meta-modeling is introduced as follows.
In light of the wavelet analysis theory [
Considering the rapid attenuation characteristics, the continuous Morlet wavelet function in
With the Morlet wavelet kernel function to replace the Gaussian RBF kernel function, the DWTR meta-model is structured as
The performance of aforementioned DWTR meta-model is largely reliant on three undetermined parameters (
BA is a metaheuristic searching method combining the standard particle swarm optimization and harmony search, which possesses fast-convergence and high-precision virtues [
In global exploitation, the bat position
In local exploration, once a candidate solution is selected among the current best solutions, the bat position is updated through random perturbation to the current best objective. With consideration of the time-variant characteristic and distribution of the current best objective, the shrinking factor-based Gaussian random walk is taken as the perturbation approach. The perturbation formula of the current best objective is shown as
During the global exploitation and local exploration, the loudness and pulse emission rate of bat algorithm would be updated when a bat approaches to the best solution, the corresponding update formula is presented as
On account of the flexibility of adaptive flight mechanism and time-variant random characteristic of shrinking Gaussian perturbation, the best meta-model parameters (
To enhance the computing accuracy and efficiency of stochastic flutter assessment, the BA-DWTR is developed by incorporating the simplified calculation ability of dynamic extremum thought, local precise description ability of wavelet TR and global searching ability of the improved BA. Based on the proposed BA-DWTR, the stochastic flutter assessment procedure is shown in
As illustrated in
Establish deterministic flutter model and configure input random variables (boundary conditions, dynamic loads, constraint conditions, material properties, model variations, etc.).
Acquire dynamic output responses by imposing input variables into deterministic flutter simulation.
Extract the extremum value of dynamic output responses to construct training & testing samples.
Perform BA-DWTR meta-modeling with the generated samples and meta-model function.
Accomplish the stochastic flutter assessment by the built BA-DWTR meta-model.
In Step (5), the stochastic flutter assessment mainly includes stochastic flutter failure probability assessment and sensitivity assessment, which is to quantify the flutter failure risk and to grade the influence degree of input random variables on flutter failure risk. With massive sampling based on the established BA-DWTR meta-model, the stochastic flutter assessment is implemented by the following expressions [
In this section, regarding the time-varying loads and boundary conditions, the stochastic flutter assessment of a typical aeroengine compressor blade [
As a typical turbomachinery, the aeroengine compressor is subjected to multiple time-varying loads and boundary conditions, which is prone to cause significant change on stochastic flutter behavior [
In one trial of deterministic flutter simulation, the rotative rate, vibrating frequency, total temperature at rotor inlet, total pressure at rotor inlet and static pressure at rotor outlet determine the output response, but these parameters have great fluctuation in operating condition, which results in the fluctuation of the output response. To measure the stochastic uncertainties of blade operating environment, the boundary conditions (i.e., total temperature at rotor inlet
Input variables | |||||
---|---|---|---|---|---|
Mean |
288.15 | 101136.91 | 135936 | 1799.9 | 1152.13 |
Standard deviation |
1.44 | 101.14 | 135.94 | 18 | 11.52 |
To acquire the time-varying flutter response, the deterministic analysis was conducted by steady flow simulation and unsteady flow simulation. Through the steady flow simulation, the steady flow field at each working condition was determined and transferred to the unsteady flow simulation. Then the unsteady flow simulation was performed by taking the steady flow field as the initial flow field. The aerodynamic modal damping ratio (AMDR) corresponding to the input variables was obtained as the output response to construct the input-output samples for subsequent BA-DWTR meta-modeling.
On account of the cyclic symmetry of turbomachinery and the periodicity of blade vibration, the two blade flow passages are modeled for flow simulation. Considering the body-fitted hexahedron grid technique, the blade flow passage is discretized into 136,416 elements and 150,460 nodes, as shown in
Considering time-varying loads spectrum and steady flow simulation, the unsteady flow simulation was executed using the energy method in the most unstable vibration mode of 1 nodal diameter [
Based on the distribution traits of boundary conditions and time-varying loads in
Note: That the absolute error and relative error of each sample are calculated by
Error index | Meta-models | |||
---|---|---|---|---|
RS | DTR | DWTR | BA-DWTR | |
Mean relative error | 0.0565 | 0.0311 | 0.0255 | 0.0218 |
Root mean squared error | 0.0434 | 0.0185 | 0.0113 | 0.0084 |
By Latin hypercube sampling based on the distribution traits of boundary conditions and time-varying loads, the BA-DWTR meta-model replacing the deterministic flutter model was simulated 10000 times by MC simulation. The output responses are gained and the distribution features of output response are depicted in
In light of the stochastic flutter assessment model in
For the validation of the proposed BA-DWTR, the stochastic flutter assessment of aeroengine compressor blade is also investigated by direct MC simulation, RS, DTR and DWTR. Considering the rationale of methods comparison, all methods are assigned equal computing tasks and allocated the same computing resources. In view of the same input variables as shown in
As revealed in
Methods | Train meta-model | Computing time/(s) | |||
---|---|---|---|---|---|
Sample amount | Training time/(s) | 10^{2} | 10^{3} | 10^{4} | |
Direct MC | - | - | 2.601 × 10^{6} | 2.562 × 10^{7} | - |
RS | 147 | 3.823 × 10^{6} | 5.58 | 14.32 | 54.20 |
DTR | 89 | 2.315 × 10^{6} | 0.93 | 0.95 | 1.34 |
DWTR | 86 | 2.237 × 10^{6} | 0.61 | 0.89 | 1.32 |
BA-DWTR | 59 | 1.535 × 10^{6} | 0.44 | 0.50 | 1.03 |
Methods | Flutter failure probability | Computing accuracy/(%) | |||
---|---|---|---|---|---|
10^{2} | 10^{3} | 10^{4} | 10^{2} | 10^{3} | |
Direct MC | 0.01 | 0.010 | - | - | - |
RS | 0.03 | 0.021 | 0.0190 | 97.97 | 98.88 |
DTR | 0.02 | 0.005 | 0.0055 | 98.98 | 99.49 |
DWTR | 0.02 | 0.006 | 0.0063 | 98.98 | 99.59 |
BA-DWTR | 0.01 | 0.009 | 0.0095 | 100 | 99.89 |
Note: Note that the computing accuracy of each meta-model is calculated by 1 − [(|
As illustrated in
In summary, the comparison results validate that the proposed BA-DWTR can greatly enhance the computing efficiency with assured computing accuracy, and provide a feasible and effective way for the stochastic flutter assessment of turbomachinery.
In this paper, a dynamic meta-modeling approach (BA-DWTR) is developed to reveal the stochastic flutter behavior of turbomachinery. The dynamic extremum thought and the wavelet transform technique are employed to legitimately tackle with the large-dynamicity and high-nonlinearity issues induced by uncertain factors. To coordinate strong generalization ability and great nonlinear mapping ability, the bat algorithm is designed to search the optimal model parameters. The effectiveness of the presented approach has been validated by the stochastic flutter assessment of a typical aero-engine compressor blade. The distribution features and sensitivity factors of the aerodynamic modal damping ratio were achieved. Results show that there is 0.95% probability to trigger the flutter failure when the compressor reaches 100% rotative rate at about t = 170 s. The total temperature at rotor inlet and dynamic working loads (vibrating frequency and rotative rate) are the most critical design variables on the flutter failure probability since their effect probabilities of 64%, 19% and 14%, respectively. The negative control of total temperature at rotor inlet and the positive control of vibrating frequency and rotative rate can reduce the flutter failure probability. Through the comparison of methods (direct Monte Carlo simulation, response surface method, dynamic tube regression), the proposed BA-DWTR is demonstrated to possess high-accuracy and high-efficiency in stochastic flutter assessment.
Aerodynamic modal damping ratio
Bat algorithm
Bat algorithm optimized dynamic wavelet tube regression
Dynamic tube regression
Dynamic wavelet tube regression
Monte Carlo
Radial basis function
Response surface
Tube regression
Input sample
Time-varying response of
Extremum value of
Output extremum response
Meta-model function
Linear tube error function
Quadratic tube error function
Mapping function
Kernel function
Weight coefficient vector
Bias coefficient
Optimal solution of regression coefficients
Optimal solution of bias coefficient
Mother wavelet function
Loss insensitive degree
Penalty coefficient
Slack variable
Support vectors for a given sample set
Dilation factor
Frequency coefficient
Current bat velocity
Current bat position
Current best solution
Pulse frequency
Random vector drawn from the uniform distribution
Adaptive weight
Distance coefficient
Random number obeying Gaussian distribution
Current average loudness
Shrinking factor
Loudness of current bat
Pulse emission rate of current bat
Allowable minimum of extremum response
Limit state function
Stochastic flutter failure probability
Sensitivity degree of
Rotative rate
Vibrating frequency
Total temperature at rotor inlet
Total pressure at rotor inlet
Static pressure at rotor outlet