Strength Reliability Analysis of Turbine Blade Using Surrogate Models

There are many stochastic parameters that have an effect on the reliability of steam turbine blades performance in practical operation. In order to improve the reliability of blade design, it is necessary to take these stochastic parameters into account. In this study, a variable cross-section twisted blade is investigated and geometrical parameters, material parameters and load parameters are considered as random variables. A reliability analysis method as a combination of a Finite Element Method (FEM), a surrogate model and Monte Carlo Simulation (MCS), is applied to solve the blade reliability analysis. Based on the blade finite element parametrical model and the experimental design, two kinds of surrogate models, Polynomial Response Surface (PRS) and Artificial Neural Network (ANN), are applied to construct the approximation analytical expressions between the blade responses (including maximum stress and deflection) and random input variables, which act as a surrogate of finite element solver to drastically reduce the number of simulations required. Then the surrogate is used for most of the samples needed in the Monte Carlo method and the statistical parameters and cumulative distribution functions of the maximum stress and deflection are obtained by Monte Carlo simulation. Finally, the probabilistic sensitivities analysis, which combines the magnitude of the gradient and the width of the scatter range of the random input variables, is applied to evaluate how much the maximum stress and deflection of the blade are influenced by the random nature of input parameters.


INTRODUCTION
The turbine blade is one of the key components in a steam turbine.There are many unmeasurable and uncontrollable factors in the process of blade design, manufacturing, installation and operation that result in the randomness of structural responses.The traditional deterministic design methods (Yan et al., 2005;Liu and Meng, 1999) ignore these stochastic parameters effects, or make up the randomness through a conservative assumption (such as safety factor).So, it is difficult to explain why the blade is failed in normal operation as it is designed correctly by the traditional deterministic method and is also difficult to evaluate quantitatively how much the blade is safe.To realize the high reliability performance of the blade, it is necessary to consider these stochastic parameters and carry out reliability analysis based design.
In blade reliability analysis, structure responses of the blade (such as stress, deformation and frequencies) are obtained by a finite element method and the limit state functions are implicit with respect to basic random variables.Reliability analysis techniques, such as FORM and SORM (Choi et al., 2007;Grandhi and Wang, 1998), require limit state function gradients with respect to the basic random variables of finding most probable failure point at each iteration, but it is very difficult to obtain the gradients of the limit state function with respect to random variables when the limit state function is implicit.
Monte Carlo Simulation (MCS) can be applied to many practical problems, allowing the direct consideration of any type of probability distribution for the random variables; however, the computation time can be prohibitively high, especially when the structure exhibits non-linear behavior or the numerical model is rather complex.Although some variance reduction techniques (Park, 1994;Disciuva and Lomario, 2003), such as importance sampling and Latin hypercube sampling, have been proposed to reduce the number of samples and reduce the computational time to a certain extent, it is still not widely used in practical engineering.
Surrogate-based reliability analysis is considered to be an effective approximation approach for computationally expensive models with implicit limit functions (Queipo et al., 2005;Youn and Choi, 2004).
The basic idea of the approach consists in the substitution of the real limit state function by approximate simple functions in the neighborhood of the design points.As a surrogate model commonly presents a simple form and sometimes is represented by an explicit expression, the computational cost of the operation can be reduced with respect to the cost required when the real limit state function is used.Any of the classical procedures for structural reliability evaluation can be applied on the surrogate limit state function.Response Surface Method (RSM) (Duan and Zhao, 2009;Liu and Moses, 1994;Herbert and Armando, 2004), Artificial Neural Network techniques (ANN) (Duan and Wang, 2010;Deng et al., 2005;Elheewy et al., 2006), spline and kriging (Rao, 2002) are examples of methods used to generate surrogates.
Sensitivity analysis can quantitatively specify how the random input variables parameters influence the structural response.But in deterministic sensitivity analysis (Xie et al., 2005), a deterministic variation of an input parameter that is used to determine the gradient usually does not take the physical range of variability into account and gradient information is local information only.
A variable-section twisted blade, which usually locates in intermediate pressure stage and low pressure stage of steam turbine, has more complex geometrical shape than an equal-section blade.It operates in an extremely harsh environment, i.e., high temperature, high pressure, large centrifugal force, steam force and steam-excited vibration.So far, there is seldom discussion on the strength reliability analysis of a variable-section twisted blade considering stochastic parameters effects and there is no discussion on how to quantitatively evaluate the sensitivities of structural responses with respect to the random input parameters considering the physical range of variability of the input parameter.
In this study, a variable-section twisted steam turbine blade is investigated and a finite element model is built parametrically.The geometrical parameters, material parameters and load parameters of the blade are considered as random input variables, while the maximum deflection and maximum equivalent stress are stochastic outputs.Design of Experiments (DOE) is applied to create sample points.A quadratic polynomial with cross terms and a feed-forward back-propagation network (BP network) are separately selected to construct an approximation function as a surrogate of the finite element solver according to sample points.Then the Monte-Carlo method is used to obtain the statistical characteristics and cumulative distribution function of the maximum deflection and the maximum equivalent stress of the blade.Probabilistic sensitivities analysis, which not only takes the gradient at a particular location into account, but also all the values of the random input parameter, is considered to evaluate how much the output parameters are influenced by the random input parameters.Scatter plots of structural responses with respect to the random input variables are illustrated to analyze how to optimize the random input variables to improve the reliability of blade.

MATERIALS AND METHODS
Response surface method: In the original conceptual form of the response surface technique, polynomials are used to approximate structural response functions.Polynomials employed in the response surface usually have a quadratic form.The response surface, approximated by a quadratic polynomial with cross terms, can be expressed as: where, ) (x f is the quadratic polynomial with the cross terms used to represent the real structural response function, x = (x 1 , x 2 , …, x i , …, x n ) (i = 1, 2, …, n) are design variables vector, n is the number of design variables, β 0 , β i , β ij are called regression coefficient and among them β 0 is the coefficient of the constant term, β i are the coefficient of the linear terms, β ij are the coefficient of the quadratic terms.
For N s sample points, the set of equations specified in Eq. ( 1) can be expressed in matrix form as: where, X = An N s ×N c matrix with the design variable values as the sampled points N c = The number of regression coefficients β = The regression coefficient column matrix ε = The error column matrix, the error expected value E (ε) is zero matrix The estimated parameters ࢼ (by least squares) are unbiased and have minimum variance.It can be found as: The predicted response function is given by: Artificial neural network: An Artificial Neural Network (ANN) includes nodes and connections which link the nodes.Before a neural network can act as a surrogate, it has to be trained by adjusting these weights.The most widely used network type for approximation problems is the multi-layer perception, which is also called a feed-forward Back-Propagation network (BP network).In this study, a typical threelayer BP network is used and its architecture is shown in Fig. 1.
There are m nodes in the input layer, q nodes in hidden layer and n nodes in output layer and w ji is the connecting weight from the j neuron in the hidden layer to the i neuron in the input layer, while θ j is the threshold of the j neuron in the hidden layer.w kj is the connecting weight from the k neuron in the output layer to the j neuron in the hidden layer and θ k is the threshold of the k neuron in the output layer.In the reliability, the m neurons of the input layer represent the m random input variables x i (i = 1, 2, … , m).These values are directly transmitted to the q neurons of the hidden layer affected linearly by weight w ji and threshold θ j .The total activation value of each neuron in this layer is: The output of each neuron in the hidden layer is a linear or nonlinear function of this activation value: where, f (.) is called the activation function.The most common non-linear activation function is the logistic sigmoidal function.It is given by: where, α is a parameter defining the slope of the function, usually α = 1.Finally, the information is transformed by the neuron of the output layers in the same way.The activation function in the output layers is a linear function, so the expectant output value O k is given by: Usually the predicted output value O k from the network will not be the same as the actual output value t k used in the training process.For each input-output pattern, the square of error E p is written as follows: where, k = The number of neurons in the output layer The average system error is given by: where, P = The number of training patterns The standard back-propagation algorithm is to adjust the different weights and thresholds as well as the derivatives of E p with respect to the input data to make the square of error least.More details of this process can be seen in the reference (Haykin, 1994).

Surrogate model-based Monte Carlo simulation:
It is time consuming to perform finite element analysis if Monte Carlo Simulation (MCS) is used directly.In contrast, evaluating a surrogate model requires only a fraction of a second.Hence, the Monte Carlo simulation samples can be produced by an approximation function from the surrogate model and the structural responses can be simulated for thousands and thousands of times.The basic idea of MCS based on a surrogate model is that according to the distribution of random input variables, the locations and values of sampling points of input variables are created by design of experiments, such as Central Composite Design (CCD), or Latin-Hypercube Sampling (LHS) (Haldar and Mahadevan, 2000).The values of sampling points of output variables are obtained by FEM and a quadratic polynomial or a BP network is employed to fit these sample points and obtain the approximate function between the output responses and the input variables.When the errors of the approximate function are less than the desired requirement, the approximate function (surrogate model) is substituted for the FEM model and is used to create the Monte Carlo simulation.Furthermore, the statistical characteristics and cumulative distribution function of output variables are obtained and the reliability analysis can be carried out according to the limit state function.Probabilistic sensitivity analysis: Sensitivity analysis can quantitatively show how the random input variables parameters influence the structural response and denote how to modify which random input parameters to improve the structure reliability.But deterministic sensitivity analysis has some disadvantages.For example, to evaluate deterministic sensitivities, each input parameter can be varied by ±10% (one at a time) while keeping all other input parameters constant and then seeing how the output parameters react to these variations.An output parameter would be considered very sensitive with respect to a certain input parameter if a large change of the output parameter value is observed.
A deterministic variation of an input parameter that is used to determine the gradient usually does not take the physical range of variability into account.An input parameter varied by ±10% is not meaningful for the analysis if ±10% is too large or too little compared with the actual range of physical variability and randomness.Moreover, the gradient information in deterministic sensitivities is local information.It does not take into account that the output parameter may react more or less with respect to variation of input parameters at other locations in the input parameter space.However, in the probabilistic sensitivities, the physical range of variability is inherently considered because of the distribution functions for input parameters.The probabilistic sensitivities approach not only takes the magnitude at a particular location into account, but also all the values the random output parameter can have within the space of the random input variables.Probabilistic sensitivities measure how much the range of scatter of an output response is influenced by the scatter of the random input variables.Hence, both effects have an influence on probabilistic sensitivities: the magnitude of the gradient, plus the width of the scatter range of the random input variables.This is illustrated in the Fig. 2. If a random input variable has a certain given range of scatter ∆X, then the scatter of the corresponding random output response is larger and the larger the magnitude of the output response curve is, ∆Y 1 >∆Y 2 , denoted in Fig. 2a.But remember that an output response with a moderate magnitude can have a significant scatter if the random input variables have a wider range of scatter, shown in Fig. 2b.
The probabilistic sensitivity analysis is based on the results of Monte Carlo Simulation.A statistical significant test is used to judge probabilistic sensitivities.Suppose the probabilistic sensitivity of an output response Y i with respect to random input variable X i is denoted as ߲ܻ /߲ܺ ; the hypothesis testing is: Given a confidence level γ, calculate the probability P i that H 0 is true based on MCS results.If P i >1-γ, accept this hypothesis test, which means the Y i is not sensitive to X i .The sensitivity of Y i with respect to the X i can be negligible.Otherwise, accept the X i and use 1-P i to express the probabilistic sensitivity of Y i with respect to X i .

Performance functions of blade:
When the steam turbine blade operates in a stable condition, the orientation and flow of steam remains almost unchanged and the steam flow is considered as a steady flow.According to the strength limit condition, when the maximum stress σ max of the blade is less than the yield strength σ s of the material, the blade can satisfy the static strength requirement, so the corresponding performance function G 1 (X) is: where, X = The random variables vector influencing σ max and σ s According to the deformation limit condition, when the maximum deflection of blade δ max is less than the allowable maximum deflection [δ max ], the blade can satisfy the deformation requirement, so the corresponding performance function G 2 (X) is: A case study: A variable-section twisted blade of the 24 th low pressure stage in steam path of some steam turbine is selected to be an example.The main parameters of the 24 th low pressure stage are shown in Table 1.In this study, the blade span l b , the blade chord B, stagger angular βy, nozzle steam angle α 1m , Young's modulus E, density ρ, rotation speed ω and material yield strength σ s are considered as random variables.We calculate the statistical characteristics and cumulative distribution functions of structural responses, including the maximum stress σ max and the maximum deflection δ max of the blade and their sensitivities with respect to random input parameters and evaluate the blade reliability.The material of the blade is chromium alloy 1Cr13.The random parameters and their statistical characteristics are shown in Table 2.

Finite element parametric model of blade:
The twisted blade is composed of blade profile and blade root.The blade profile is a complex shape assembled by several cross sections with molded lines, shown in Fig. 3.The blade root is a fork-type and is wedged tightly.The parameters l b , B, β y , α 1m , E, ρ and ω are selected as basic random input variables, which are also the variables in the process of finite element parametric model.The maximum stress σ max and the maximum deflection δ max are chosen as random output responses.The three-dimension solid element Solid 45 is used to mesh the blade, the whole model is composed of 2494 nodes and 7784 elements, shown in Fig. 4, here, the coordinate system x, y, z represent the blade peripheral direction, span direction and axial direction.

Steam force calculation:
The steam force acting on the blade can be expressed by the peripheral component force P x and axial component P z , shown in Fig. 5.The 1-1 axis and the 2-2 axis are the minimum principal axes of inertial and maximum principal axes of inertial respectively.
For a variable cross-section twisted blade, the variation of steam force along the blade span must be considered because the steam flow will be changed a lot along the blade span.It is difficult to find analytical expressions to calculate the steam parameters of blade, so the steam force of blade is usually calculated by an approximate method.The blade is divided into n segments along its span and the total number of the cross section is n+1.The length of every segment is denoted as ∆x j (j = 1, 2, …, n), the peripheral component of steam force P xj and the axial component of steam force P zj at the j th segment can be obtained by: = The steam flow of the j th segment and it can be calculated by: where, µ b is the blade flow coefficient, v 2t is the blade export isentropic specific volume and w 2t is the idea relative velocity of steam exporting from blade.In this study, the blade span is divided in to 10 segments and a total of 11 cross sections, shown in Fig. 6.
The steam flow parameters in every section, P xj and P zj in every segment are obtained according to the thermodynamic calculation, here, µ b is given as 0.96, v 2t is 3.645 m 3 /kg from p s1 expanding isentropic to p s2 .The details of process are presented in reference (Duan, 2009).
The loading boundaries of blade finite element model are: to put the steam force on every segment of blade to the corresponding loading surface and at the same time to load rotating speed to consider the role of centrifugal force.The displacement boundaries are: to constraint the x displacement and z displacement of fork-type blade root, i.e., u x = 0, u z = 0 and to constraint the all the degree of freedom in the pin holes, i.e., u x = u y = u z = 0.
When all the basic input random variables are equal to mean values, the equivalent stress and the    The regression result is the sum of every random input variables multiplying the corresponding coefficient, that is: Table 3 and 4 show the input variables and their corresponding coefficients for regressing maximum stress σ max and maximum deflection δ max , respectively.

BP network:
The input variables l b , B, β y , α 1m , E, ρ and ω act as network input corresponding to seven neurons of the input layer, while σ max and δ max are network outputs corresponding to two neurons of the output layer.The number of hidden neurons in the BP network is 25.The BP network of the blade is shown in Fig. 10.The mapping relationship between output responses and input variables is denoted as the following: The learning rate is given 0.05 and the minimum value of expected error is given 1×10 -5 .Seventy nine training samples are used to train the BP network by Levenberg-Marquardt rule (Haykin, 1994).The training errors attained 9.50634×10 -6 after 96 iterations.The weights and thresholds in the hidden layer and the output layer are also obtained.The 20 sample points produced by LHS are used as validation of surrogate model.Figure 11 and 12 show the values of R R of σ max and δ max obtained by Polynomial Response Surface (PRS) and BP Artificial Neural Network (ANN) respectively.For the σ max , the maximum values of R R from the two surrogate model are 1.6e-2 (PRS) and 4.95e-4 (ANN) respectively; for the δ max , the maximum values of R R are 2.8112e-3 (PRS) and 2.3065e-3 (ANN) respectively.The precision of ANN model is little higher than PRS model.

RESULTS AND DISCUSSION
Substitute the surrogate model for the FEM model and create the 100000 Monte-Carlo simulation samples to carry out the statistical analysis for these simulation results, the following quantities can be obtained: • Statistical characteristics of the maximum stress σ max and the maximum deflection δ max , shown in Table 5.It can be seen that the statistical parameters of σ max and δ max from the two methods are approached; the data discreteness from ANN-MCS is larger.Table 6 shows the results of β 1 , β 2 and P 1 , P 2 from the different methods PRS-MCS, ANN-MCS and direct MCS.For G 1 (X), the reliability P 1 from the three methods is the same, but β 1 from ANN-MCS is more approach to that from direct MCS.For G 2 (X), the value β 2 from ANN-MCS is also more approach to that from direct MCS.• Probability sensitivities of σ max and δ max with respect to the random input variables: The confidence level γ is given to 0.025 which means the incorrect probability of hypothesis testing is 2.5%.According to the results of statistical significant test, the input random variables are divided into two groups: important group and unimportant group.The variables which accept the statistical significant test belong to unimportant group and the other variables refusing the test belong to important group.The variables in the important group are sorted by the absolute value of their probability sensitivity (1-P i ), shown in Fig. 14.The sequence of sensitivity of σ max with respect to random input variables from high to low is l b , β y , ρ and ω, while the sequence of the sensitivity of δ max with respect to random input variables is l b , E, β y , ω and ρ.

CONCLUSION
• This study combines the finite element, surrogate model (including quadratic polynomial response surface and BP artificial neural network) and Monte Carlo simulation method to obtain the statistical characteristics and cumulative distribution functions of maximum stress and maximum deflection of a variable-section twisted blade of a steam turbine and carries out strength reliability analysis in the presence of random geometrical parameters, material parameters and load parameters.
• Comparison of PRS-MCS and ANN-MCS with direct MCS shows the surrogate-based reliability analysis approach can act as an ideal tool for the reliability analysis and design of a steam turbine blade.The approach is applied to complex structure reliability analysis with an implicit limit state function.
• Probability sensitivities analysis is used to quantitatively specify the degree of influence of random input variables on maximum deflection and maximum stress of the blade.

Fig. 3 :
Fig. 3: Blade profile Fig. 5: Steam force on blade Fig. 7: Equivalent stress and deformation of blade deformation distribution of the blade are shown in Fig. 7.The value of the maximum stress is 125 MPa and it is located near the middle of steam-out edge because of the twisted recovery produced by centrifugal force, which makes the distribution of maximum stress access to the middle profile.The value of the maximum deflection is 1.65 mm and it occurs on the tip of the blade.Design of experiments: As there are 7 random input variables, 79 sample points are created by CCD method and perform looping of finite element model for 79 times.The sample points to regress the approximate function are shown in Fig. 8.In order to validate the BP network, 20 validation sample points, which are different from the former sample points created by CCD, are created by Latin-Hypercube Sampling method and corresponding sample points of σ max and δ max are obtained by performing loop of finite element model for 20 times.The validation sample points are shown in Fig. 9.
Fig. 8: Sample points for constructing surrogate

Fig. 10 :
Fig. 10: BP model of blade Surrogate model evaluation:The relative residual error R R is used to describe the error of predicted values with actual values in validation set, R R is defined as: actual value obtained from FEM calculation ‫ݕ‬ ො = The predicted value obtained from the surrogate model • The CDF of σ max and the CDF of δ max from the PRS-MCS and ANN-MCS are shown in Fig. 13.• Reliability evaluation of blade: As the mean ߤ ఙ ౣ౮ and the standard deviation ߪ ఙ ౣ౮ of the maximum stress σ max are obtained in the above step and the statistical characteristic of σ s is also given, the reliability index β 1 corresponding to G 1 (X) is: = The mean value of σ s ߪ ఙ ೞ = The standard deviations of σ sThe reliability of blade corresponding to G 1 (X) is:

Table 1 :
Main parameters of the 24 th low pressure stage α1, β2 and Ω are the values at the blade average diameter; B: The chord of the first cross-section at the blade root

Table 2 :
Random parameters and statistical characteristics

Table 3 :
Regression terms and coefficients for σmax

Table 6 :
Strength reliability of blade