Non-linear partial least squares response surface method for structural reliability analysis
Introduction
Structural reliability analysis has received a great deal of attention due to the uncertainties [1] and a number of methods have been developed and deployed, such as the Monte Carlo simulation (MCS) [2], the first order reliability method (FORM) [3] and the second order reliability method (SORM) [4], [5], [6]. However, assessing the reliability of a complex structure presents two main challenges. First, failure is a small probability event, so the simulation requires a large number of model computations; and the need for this huge computational effort impedes the efficiency of the numerical simulation [7]. Second, the performance functions are often non-linear and unknown, and require a large computation time when a large number of random variables is involved when using FORM and SORM [8]. Recently, approximation methods [9], [10], [11], [12], [13], [14], [15] have been gaining attention for current engineering analyses which have complex and expensive computer analysis codes, such as the structure performance function, which is often highly complicated, and leads to expensive and difficult problems for the structural reliability computation. One of the most common methods is the surrogate model (Metamodel) method [16], [17], and there are three key problems (or steps) to deal with in this kind of method.
- 1.
The first problem is how to get the sample points. Many experimental designs have been used for the construction of the model, such as the Bucher's design, Central Composite Design (CCD), the orthogonal design and the uniform design (UD).
- 2.
Another problem is the format of the surrogate model. The first-order polynomial,the quadratic polynomial and high-order polynomials, all can be used as the response surface function, but the one most widely used is the quadratic polynomial without the cross terms.
Recently, several surrogate models have been proposed such as the Kriging [18], [19], [20], Artificial Neural Networks [21], [22], [23], and Support Vector Machines [24], [25]. These models can improve the accuracy of reliability analysis to a certain extent; however, the complex calculation process and the influence of too many parameters hinder the widespread application of the methods.
- 3.
In addition, the fitting method for the surrogate model is a key issue. Most of the existing methods are mainly based on LS regressions or an improvement of them [8], [18]. However, surrogate models based on the LS method have several key limitations, such as the difficulty of dealing with the problem of small samples and sample correlation. Various problems are often encountered, such as low accuracy and inefficiency when considering realistic engineering problems, especially with high-dimensional systems.
In fact, sample correlation in reliability analysis is widespread. The first step, i.e. the design of the experiment (DoE), is very important, because the accuracy of a surrogate model depends greatly on the number of sample data points used and their locations in multidimensional space. But when the limit state function is high-dimensional, this sampling process can be very costly, and even impractical, in terms of design and computation. In high-dimensional problem, the number of sample points is often not related to the dimensions of the independent variables, so there must be correlation between some sample points. Also, in the second step, i.e. the process of constructing the surrogate model, the model of the structural performance function is often non-linear; however, the fitting method is linear, e.g. the least square procedure, so the idea of a quasi-linear and variable substitution is a common method for dealing with the problems. The variable substitution can increase the dimension of the space, thus strengthen the correlation of the data. Therefore, in the final step of fitting the model, correlation cannot be neglected especially in the multi-dimensional reliability problem. In this condition, the models obtained by the original LS regression are not precise, and the results of the reliability analysis cannot satisfy the accuracy requirements.
This paper explores a new method for multi-dimensional non-linear structural reliability problems called the UD-BP-PLS surrogate model method. It seeks to minimize the number of performance function computations and to guarantee an accurate estimation of the failure probability. First, based on traditional polynomial least square (P-LS) surrogate model methods, a partial least square (PLS) technique, which is a substitute for original least square method, is used to build a surrogate model for the analysis of structural reliability; and this method called P-PLS, is presented in Section 3. It is well adapted to deal with few samples in multi-dimensional structural reliability, but the assessment of high dimensional and high non-linear structural reliability problems remains an issue. Then, an extension of the algorithm previously used for complex structural reliability problems, i.e. the UD-BP-PLS surrogate model method, is proposed, which is detailed in Section 4. This improvement is validated with academic applications in Section 5. Section 6 concludes with a summary of the main advantages of the proposed method. The partial least square technique has the ability of solving with small samples with some correlation; therefore, the new method has better performance and can achieve more accurate solutions for structural reliability analysis when compared with conventional response surface methods based on least square regression.
Section snippets
P-LS surrogate model method
One of the most widely used approximation methods, the polynomial surrogate model method, has been used successfully in many deterministic engineering design problems. Bucher and Bourgand [9] applied a two-stage method with a quadratic polynomial surrogate model. In the first stage, the experimental points required to form the first model are generated along the axis with the coordinates , where and are the mean and standard deviation of the random variable . Then the
Introduction to the partial least square technique
Partial Least Squares [27], [28] covers a wide class of methods for modeling relations between sets of observed variables by means of latent variables. It comprises regression and classification tasks, as well as dimension reduction techniques and modeling tools. The underlying assumption of the PLS method is that the observed data is generated by a system, or process, which is driven by a small number of latent (not directly observed or measured) variables. Projections of the observed data to
The UD-BP-PLS surrogate model method
Due to the restriction created by the polynomial functions, the traditional response surface method cannot achieve satisfactory accuracy for multi-dimensional variables and high non-linearity problems. This paper presents the UD-BP-PLS non-linear surrogate model method for the calculation of reliability. The principal idea of this method is stated as follows:
The implicit performance function can be written as a non-linear additive model:
Then, several different spline
Numerical examples
In order to demonstrate the effectiveness of the proposed method, several examples are illustrated which cover a wide variety of limit states: first, an example of dimension 3 is tested to observe the method's behavior. It is selected to test applicability of the proposed method. Following this, a non-linear multi-dimension analytical function is tested. To finish, the method is performed on a real engineering example. The results of these examples from the two PLS-based surrogate model methods
Conclusions
This paper proposes an original and easily implementable method called UD-BP-PLS method, combining uniform design, B-spline functions and partial least squares techniques to perform complex structural reliability analysis. The proposed strategy is found to be economic in the number of calls to the expensive performance function and its results are accurate for the probability of failure. The method is particularly designed to deal with the correlation of the reliability analysis especially in
Acknowledgments
This research was supported by the National Natural Science Foundation of China (NSFC, 51308158), and the China Postdoctoral Science Foundation (CPSF, 2013M541390), which are gratefully acknowledged by the authors. The support from the Department of Civil Engineering in the University of Manchester is also gratefully acknowledged.
References (42)
- et al.
Uncertainties in probabilistic numerical analysis of structures and solids - stochastic finite elements
Struct Saf
(1997) - et al.
First-order concepts in system reliability
Struct Saf
(1983) - et al.
Optimization algorithms for structural reliability
Struct Saf
(1991) - et al.
A combined importance sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models
Reliab Eng Syst Saf
(2013) - et al.
A fast and efficient response surface approach for structural reliability problems
Struct Saf
(1990) - et al.
A new look at the response surface approach for reliability analysis
Struct Saf
(1993) - et al.
Response surface method using vector projected sampling points
Struct Saf
(1997) - et al.
Cumulative formation of response surface and its use in reliability analysis
Probabilistic Eng Mech
(2000) - et al.
CQ2RS: a new statistical approach to the response surface method for reliability analysis
Struct Saf
(2003) - et al.
A response surface method based on weighted regression for structural reliability analysis
Probabilistic Eng Mech
(2005)
A comparison of approximate response functions in structural reliability analysis
Probabilistic Eng Mech
Efficient surrogate models for reliability analysis of systems with multiple failure modes
Reliab Eng Syst Saf
An improvement of response surface method
Struct Saf
Application of kriging method to structural reliability problems
Struct Saf
Assessment of the efficiency of Kriging surrogate models for structural reliability analysis
Probabilistic Eng Mech
AK-SYS: an adaptation of the AK-MCS method for system reliability
Reliab Eng Syst Saf
Performance-based design and seismic reliability analysis using designed experiments and neural networks
Probabilistic Eng Mech
Structural reliability analysis for implicit performance functions using artificial neural network[J]
Struct Saf
Review and application of artificial neural networks models in reliability analysis of steel structures
Struct Saf
Rare-event probability estimation with adaptive support vector regression surrogates
Reliab Eng Syst Saf
Response surface augmented moment method for efficient reliability analysis[J]
Struct Saf
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