Advances in Materials Science and Engineering

Volume 2015 (2015), Article ID 680406, 9 pages

http://dx.doi.org/10.1155/2015/680406

## Optimum Control for Nonlinear Dynamic Radial Deformation of Turbine Casing with Time-Varying LSSVM

^{1}School of Computer Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China^{2}Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong^{3}School of Energy and Power Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Received 18 October 2014; Accepted 9 February 2015

Academic Editor: Peter Majewski

Copyright © 2015 Cheng-Wei Fei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the development of the high performance and high reliability of aeroengine, the blade-tip radial running clearance (BTRRC) of high pressure turbine seriously influences the reliability and performance of aeroengine, wherein the radial deformation control of turbine casing has to be concerned in BTRRC design. To improve BTRRC design, the optimum control-based probabilistic optimization of turbine casing radial deformation was implemented using time-varying least square support vector machine (T-LSSVM) by considering nonlinear material properties and dynamic thermal load. First the T-LSSVM method was proposed and its mathematical model was established. And then the nonlinear dynamic optimal control model of casing radial deformation was constructed with T-LSSVM. Thirdly, through the numerical experiments, the T-LSSVM method is demonstrated to be a promising approach in reducing additional design samples and improving computational efficiency with acceptable computational precision. Through the optimum control-based probabilistic optimization for nonlinear dynamic radial turbine casing deformation, the optimum radial deformation is 7.865 × 10^{−4} m with acceptable reliability degree 0.995 6, which is reduced by 7.86 × 10^{−5} m relative to that before optimization. These results validate the effectiveness and feasibility of the proposed T-LSSVM method, which provides a useful insight into casing radial deformation, BTRRC control, and the development of gas turbine with high performance and high reliability.

#### 1. Introduction

Blade-tip radial running clearance (BTRRC) of high pressure turbine (HPT) seriously influences the performance and reliability of gas turbine [1]. Large BTRRC causes the decline of aeroengine performance just like low working efficiency and thrust, high specific fuel consumption, and so forth, while small BTRRC results in the friction fault between blade-tip and casing [2, 3]. Hence, BTRRC has to hold a reasonable value under operating conditions. Hereinto, casing radial deformation is one important factor of directly effecting on BTRRC [4]. The optimum control of casing radial deformation should be done to make BTRRC more reasonable. In advanced propulsive systems, turbine casing bears vast thermal load and pressure difference under working conditions due to high temperature gradient and high rotational velocity [5, 6]. A significant objective of the probabilistic optimal control of casing radial deformation is to seek an optimum radial deformation to offset the radial deformations of turbine disk and blade under satisfying an acceptable reliability degree. Therefore, an effective optimization method-based probabilistic analysis is required for the optimum control of turbine casing radial deformation.

With the growth in the computing power of current computers, finite element (FE) method has become a common and important technique in product development process, which has been applied to the design analysis of aeroengine components like stress analysis, thermal analysis, vibration analysis, and fatigue life estimate [7–10]. Excessive computational cost and impractical runtime are confronted when FE method is adopted in the probabilistic optimal design of complex structure like turbine casing because of too large computational loads for nonlinear analysis, dynamic analysis, and multiple loop computations. To reduce the runtime, one alternative is to construct a simple surrogate model (also called as response surface method, RSM) to approximate the response of FE solver [5]. The surrogate model expresses the relationship between the objective or constraint functions and design variables by simple equations, which only cost a small number of FE analyses in order to significantly reduce computing time. The typical surrogate models include polynomial response surface method [11–16] and support vector machine (SVM) [17–21], and so forth. Polynomial response surface model uses least squares regression analysis to fit low-order polynomials to a set of experimental data [15, 16]. Currently, SVM as an intelligent statistical learning method and implicit performance function has been employed to the reliability analysis and optimal design of complex structure instead of the polynomial response surface method, because SVM holds the advantages of small training sample, high computational accuracy, and efficiency. To further improve the computing speed for building SVM model, the least square SVM (LSSVM) was developed and proved to be an effective method for structure reliability design-based steady analysis [18]. However, it is difficult that the LSSVM is directly applied to the nonlinear dynamic optimal analysis of complex structure with time-varying characteristics [22, 23].

To accomplish the optimum control for nonlinear dynamic radial deformation of turbine casing with LSSVM, the time-varying LSSVM (T-LSSVM) method with high efficiency and high precision is presented. And the mathematical model of T-LSSVM is established and verified to be effective and feasible by numerical experimentation. Finally, the optimum control-based probabilistic optimization for nonlinear dynamic radial deformation of turbine casing was performed with the proposed T-LSSVM, subject to constraints on reliability degree and other practical conditions with nonlinear material property and dynamic thermal load.

#### 2. Time-Varying LSSVM (T-LSSVM)

During aeroengine working, thermal stress and pressure difference are of variation. Hence, the probabilistic optimization of casing radial deformation needs a time-varying (or dynamic) optimization method. In this section, the T-LSSVM method is proposed based on LSSVM for dynamic probabilistic optimization.

##### 2.1. Least Square Support Vector Machine

SVM is an important surrogate model based on intelligent statistical learning theory [24] and holds high computational precision and efficiency. LSSVM adopts linear equation in high-dimensional space replacing quadratic programming problem in low-dimensional space to improve computational efficiency by taking the term of squared error as the objective function and regarding an equation as the constraints [17, 18]. The mathematical model of LSSVM is summarized as follows.

From a certain kind of assumed distribution ( and ), the sampling points are generated, and a set of functions which maps a point in the space onto the space is denoted by in which is a set of parameters and is an undetermined parameter vector.

The regression object is to find a function which makes (1) have the lowest expected risk: in which is an error function [21] . Function can be determined by the following method.

For the nonlinear regression, each sampling point is mapped by a nonlinear function onto the high-dimensional space to conduct the linear regression, and then the original space nonlinear regression effect is attained. Thus, the function is rewritten as

Obviously, the problem of solving the regression function can be transformed into obtaining the following optimal solution: Considering the permissible errors, two slack variables and are introduced, and the optimization function is where is a penalty coefficient.

For obtaining the solution of this quadratic program, the Lagrange function is introduced: where , .

In the optimization process, a kernel function is used to replace the inner product in higher dimensional space, where the Lagrange duality problem is expressed by

After getting the optimized solution , , and , the regression estimating function (LSSVM model) is where SV is a set of support vectors for a given sample set; , is the number of sample values; is the number of random variables.

##### 2.2. Time-Varying LSSVM

For time-varying probabilistic optimization problem, the response of each calculation is a stochastic process so that it is difficult to finish the probabilistic analysis of complex structure with response surface method. In the face of this situation, conventional approaches fit a lot of response surface models in this time-varying process to select one reasonable response at one time point as the computational point of probabilistic optimization. However, in all of the loop computations of probabilistic optimization, it is difficult to make this computational point reasonable and feasible, so that the computational accuracy of probabilistic design is low. To resolve this issue using LSSVM, this paper advances the T-LSSVM method for the dynamic probabilistic optimization of turbine casing radial deformation. T-LSSVM is used to calculate a single extreme value rather than a series of dynamic output responses under different input samples within a time domain , which is equivalent to transform a stochastic process of output response into a random variable. In each stochastic analysis, the random variable as the response can be ensured to be the most reasonable and effective, which can pledge the analytical precision. And then the T-LSSVM model is established based on these data for probabilistic optimization. The procedure of probabilistic optimization with T-LSSVM is drawn as follows.(1)Establish the FE model of structure and select reasonable parameters (dynamic load, constraint conditions, time domain, etc.).(2)Extract the samples of input and output variables and define the maximum output responses for each stochastic analysis within time domain.(3)Fit the response surface function of T-LSSVM to these samples.(4)Apply the T-LSSVM function of to accomplish structural dynamic probabilistic optimization.

Obviously, T-LSSVM is promising to reduce computing cost and enhance calculation efficiency. For probabilistic analysis of turbine casing radial deformation, the maximum deformation may be considered as the output response of T-LSSVM for optimum control during the time domain , because the time-varying deformation of turbine casing is secure in as long as the maximum response is safe at the time point.

According to the basic thought of T-LSSVM, the data set with the maximum output response and input random variables is obtained through a number of stochastic analyses in time domain . With the th input samples , the extremum of output response is within time domain . The data set consisting of the maximum output responses is used to fit the extremum response curve : where is a response surface function, called T-LSSVM function (or model) here.

In fact, for dynamic probabilistic design the T-LSSVM function is a key factor because a valid T-LSSVM function is conductive to enhance the efficiency and precision of probabilistic design. If the T-LSSVM function replacing FE model is applied to structural dynamic probabilistic design, this method is called T-LSSVM method, which belongs to the global response surface method. Based on (8), the T-LSSVM function is built as follows: where , is the number of fitted LSSVM functions of in one stochastic analysis process within .

#### 3. Problem Formulation

Casing radial deformation under gas turbine operation is an important component influencing BTRRC variation [1–3, 25, 26]. In BTRRC control, small casing radial deformation produces a small BTRRC, which degrades the reliability of gas turbine, while large casing radial deformation increases BTRRC and lessens the performance of gas turbine. Clearly, casing radial deformation should be potentially controlled-based probabilistic optimization subject to a reasonable reliability degree.

##### 3.1. FE Model of Turbine Casing

An aeroengine high pressure turbine is studied to build its simplified model in Figure 1(a). Detailed studies of turbine stator system have been conducted [1, 3, 25–27]. The casing structure is modeled as a ring-like structure made up of a series of arcs joined together [27]. The lining ring of casing is a sensitive component so that its expansion and contraction lead to the variation of casing radial deformation and blade-tip clearance. Hence, the inner lining ring is only studied and simplified as a cylindrical structure. For the convenient calculation, an axial cross-section of the axisymmetric FE model is extracted as an interesting submodel as shown in Figure 1(b). The left axial degree of freedom of the submodel is restrained to prevent the axial motion. The inner surface of FE model is exposed to turbine inlet hot gas, and the outside surface is exposed to compressor discharge air.