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Structural Reliability Analysis solved through a SVM-based Response Surface
... Recently, many alternative response surface methods have been proposed to improve the FORM approximation [Alibrandi and Ricciardi (2005); Most and Bucher (2006); Bucher and Most (2008); Alibrandi and Ricciardi (2008); Alibrandi et al. (2010)]. Notably these methods include the response surface methods based on the SVM [Hurtado (2004[Hurtado ( , 2007; Hurtado and Alvarez (2010); Bourinet et al. (2011);Alibrandi and Ricciardi (2011)]. Using the SVM the structural reliability problem can be treated as a classification problem [Hurtado and Alvarez (2003)], since we are not interested in the exact value of the limit state function, but rather only its sign. ...
The geometry of high-dimensional spaces is very different from low dimensional spaces and possesses some counter-intuitive features. It is shown that, for high dimensions, the sampling points fall far away from the origin and concentrate within an intersection between a very thin shell and a suitable equatorial slab. The well-known First-Order Reliability Method (FORM), originally formulated for low dimensions, may work well in many engineering problems of high dimension. But it is not able to reveal the level of achieved accuracy. Considering the features of high-dimensional geometry, a novel linear response surface based on Support Vector Method (SVM) is proposed for structural reliability problems of high dimension. The method is shown to outperform FORM for structural reliability problems of high dimension in terms of robustness and accuracy.
... The RS models can be built to find the design point with reduced computational cost (Bucher & Burgound 1990, Alibrandi & Der Kiureghian 2012; recently, many alternative response surface methodologies have been proposed, whose aim is the improvement of the FORM approximation (Bucher & Most, 2008;Alibrandi & Ricciardi 2005, Alibrandi & Ricciardi 2008, Alibrandi, Impollonia & Ricciardi 2010. To the latter cathegory belong the RS approaches based on the SVM (Hurtado 2004;Alibrandi & Ricciardi 2011). Using the SVM the reliability problem is treated as a classification approach (Hurtado & Alvarez 2003), since we are not interested to the exact value of the LSF, but only to its sign. ...
The Structural Reliability theory allows the rational treatment of the uncertainties and gives the
methods for the evaluation of the safety of structures in presence of uncertain parameters. The main
challenge is the computational cost, since the failure probability with respect to an assigned limit
state is given as the solution of a very complicated multidimensional integral. The most robust
procedure is the Monte Carlo Simulation (MCS), but especially in its crude form is very
demanding. For this reason, wide popularity has been gained by the First Order Reliability Method
(FORM) by its simplicity and computational efficiency. However, for strongly nonlinear systems
the FORM approximation is not very close to the exact one. To this aim, in this paper we introduce
a novel Linear approximation of the limit state, based on the Support Vector Method (SVM), and
which allows to improve the FORM solution, starting from the knowledge of the design point.
In order to reduce computational costs in structural reliability analysis, it has been suggested to utilize approximate response functions for reliability assessment. One well-established class of methods to deal with suitable approximations is the Response Surface Method. The basic idea in utilizing the response surface method is to replace the true limit state function by an approximation, the so-called response surface, whose function values can be computed more easily. The functions are typically chosen to be first-or second-order polynomials. Higher-order polynomials on the one hand tend to show severe oscillations, and on the other hand they require too many support points. This may be overcome by applying smoothing techniques such as the moving least-squares method. An alternative approach is given by Artificial Neural Networks. In this approach, the input and output parameters are related by means of relatively simple yet flexible functions, such as linear, step, or sigmoid functions which are combined by adjustable weights. The main feature of this approach lies in the possibility of adapting the input–output relations very efficiently. A further possibility lies in the utilization of radial basis functions. This method also allows for a flexible adjustment of the interpolation scheme. In all approaches as presented it is essential to achieve high quality of approximation primarily in the region of the random variable space which contributes most significantly to the probability of failure. The paper presents an overview of these approximation methods and demonstrates their potential by application to several examples of nonlinear structural analysis.
The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.
The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.
An improved response surface method is proposed and applied to the reliability analysis of a stiffened plate structure. The response surface function is formed in a cumulative manner in order to account properly for the second order effects in the response surface with acceptable computational effort involved in the evaluation of the state function. First, a linear response surface is formed by searching for the design point by the first order reliability method. The vector projection technique is used to allocate the sampling points close to the response surface. Then, the linear response surface is improved by adding square terms, and the second order reliability method is employed to search for the design point. All the available sampling points except those generated in the very initial stage are used to obtain a well-conditioned system matrix for regression. Lastly, the response surface obtained is checked for the selected sampling points. If the response surface function is not satisfactory, it is improved adaptively by adding cross terms and removing some of the second order terms. In this way an appropriate incomplete second order response surface can be obtained and used for reliability analysis. A stiffened plate under combined load effects is analysed. The load components, plate thicknesses, and Young's modulus are treated as random variables. Structural imperfection in the form of critical buckling mode is also treated as a random variable. Sensitivity analyses are performed with respect to these random variables.
An efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state is proposed under the assumption that no imperfections are present and that the elastic parameters are uncertain and modeled as random variables. The approach allows a deeper investigation of structures which are not sensitive to imperfections. The procedure relies on a Response Surface Method adopting simple ratio of polynomials without cross-terms as performance function. Such a relationship approximates analytically the dependence between the buckling load and the basic variables furnishing a limit state equation which is very close to the exact one when a proper experimental design is adopted. In this way a Monte Carlo Simulation applied to the response surface leads to a good approximation with low computational effort. Several numerical examples show the accuracy and effectiveness of the method varying structural complexity, correlation between basic variables and their distribution.
The assessment of the reliability of structural systems is increasingly being estimated with regard to the spatial fluctuation of the mechanical properties as well as loads. This leads to a detailed probabilistic modeling known as stochastic finite elements (SFE). In this paper an approach that departs from the main stream of methods for the reliability analysis of SFE models is proposed. The difference lies in that the reliability problem is treated as a classification task and not as the computation of an integral. To this purpose use is made of a kernel method for classification, which is the object of intensive research in pattern recognition, image analysis, and other fields. A greedy sequential procedure requiring a minimal number of limit state evaluations is developed. The algorithm is based on the key concept of support vectors, which guarantee that only the points closest to the decision rule need to be evaluated. The numerical examples show that this algorithm allows obtaining a highly accurate approximation of the failure probability of SFE models with a minimal number of calls of the finite element solver and also a fast computation.
Under general conditions reliability analyses of elaborate structural models can be replaced by much simpler equivalent reliability analyses of corresponding idealized structural models such as those based on rigid-plastic theory. The equivalence is in the sense of being with respect to probability of failure and is obtained by applying a random factor correction of the strength variables of the simple model. This model-correction factor is called the effectivity factor, and it is approximated to the zeroth order by a constant or to the first order by an inhomogeneous linear function of all the basic variables in the realistic model. The coefficients of this function are calculated from the elaborate model (e.g.. by finite element calculations) with values of the input variables chosen on the basis of a first-order reliability analysis of the simple model (FORM). The model-correction-factor method is a special type of response-surface method based on mechanical principles. It is illustrated quantitatively on a yield hinge frame model with strain softening mutual rotation-moment behavior in the hinges upon the onset of yielding.
The present paper introduces and discusses a stochastic finite element method. It can be used for the analysis of structural and mechanical systems whose geometrical and material properties have spatial random variability. The method utilizes a polynomial expansion of the numerical nonlinear structural operator (for which actual analytical form is unknown). The expansion is made according to a response-surface approximation in terms of spatial averages of the design variables. The polynomial form is then modified by suitable error factors, one for each geometrical or mechanical property. Each error factor is due to the deviations, of the single property, from its spatial average in the different finite elements. The method demands an accurate design of the experiments to be conducted in order to identify the model parameters. A numerical example has been worked out. In this numerical example, the stresses and the strains in a light-water reactor pressurized vessel are computed by a stochastic three-dimensional finite element nonlinear analy
The reliability analysis of large and complex structural requires approximate techniques in order to reduce computational efforts to an acceptable level. Since it is, from an engineering point of view, desirable to make approximative assumptions at the level of the mechanical rather than the probabilistic modeling, simplifications should be carried out in the space of physically meaningful system- or loading variables.Within the context of this paper, a new adaptive interpolation scheme is suggested which enables fast and accurate representation of the system behavior by a response surface (RS). This response surface approach utilizes elementary statistical information on the basic variables (mean values and standard deviations) to increase the efficiency and accuracy. Thus the RS obtained is independent of the type of distribution or correlations among the basic variables which enables sensitivity studies with respect to these parameters without much computational effort.Subsequently, the response surface is utilized in conjunction with advanced Monte Carlo simulation techniques (importance sampling) to obtain the desired reliability estimates.Numerical examples are carried out in order to show the applicability of the suggested approach to structural systems reliability problems. The proposed method is shown to be superior both in efficiency and accuracy to existing approximate methods, i.e., the first order reliability methods.
In the framework of the structural reliability theory, the probabilistic limit analysis (PLA) represents a powerful tool for the evaluation of the safety of structures with stochastic strengths with respect to the ultimate limit state of plastic collapse. Aim of the PLA is to evaluate the conditional probability of collapse (CPC), that is, the probability of plastic collapse of the structural system for assigned value of the acting loads. In this paper the focus is on the static approach of the limit analysis theory, which is particularly attractive for engineers, because it gives upper bounds of the CPC, that is, safe bounds. The classical static approach, introduced more than 30 years ago, however, cannot in any case evaluate the exact CPC of the structural system, and moreover generally the bounds obtained are not very close, especially in the range of very small probabilities. In this paper an alternative static approach is proposed, which can obtain the exact CPC of the structural system considering a finite number of suitable chosen stochastic stress vectors through the use of the partial admissible domains; moreover, it gives good safe bounds of the CPC considering only a few stochastic stress vectors. Some simple numerical examples show the accuracy and effectiveness of the method. Copyright © 2007 John Wiley & Sons, Ltd.
Closed-form mechanical models to predict the behaviour of complex structural systems often are unavailable. Although reliability analysis of such systems can be carried out by Monte Carlo simulations, the large number of structural analyses required results in prohibitively high computational costs. By using polynomial approximations of actual limit states in the reliability analysis, the number of analyses required can be minimized. Such approximations are referred to as Response Surfaces. This paper briefly describes the response surface methodology and critically evaluates existing approaches for choosing the experimental points at which the structural analyses must be performed. Methods are investigated to incorporate information on probability distributions of random variables in selecting the experimental points and to ensure that the response surface fits the actual limit state in the region of maximum likelihood. A criterion for reduction in the number of experiments after the first iteration is suggested. Two numerical examples show the application of the approach.
A method is developed to successively find the multiple design points of a component reliability problem, when they exist on the limit-state surface. FORM or SORM approximations at each design point followed by a series system reliability analysis is shown to lead to improved estimates of the failure probability. Three example applications show the generality and robustness of the method.
In structural reliability, simulation methods are oriented to the estimation of the probability integral over the failure domain, while solver-surrogate methods are intended to approximate such a domain before carrying out the simulation. A method combining these two purposes at a time and intended to obtain a drastic reduction of the computational labor implied by simulation techniques is proposed. The method is based on the concept that linear or nonlinear transformations of the performance function that do not affect the boundary between safe and failure classes lead to the same failure probability than the original function. Useful transformations that imply reducing the number of performance function calls can be built with several kinds of squashing functions. A most practical of them is provided by the pattern recognition technique known as support vector machines. An algorithm for estimating the failure probability combining this method with importance sampling is developed. The method takes advantage of the guidance offered by the main principles of each of these techniques to assist the other. The illustrative examples show that the method is very powerful. For instance, a classical series problem solved with O(1000) importance sampling solver calls by several authors is solved in this paper with less than 40 calls with similar accuracy.
In the reliability estimation of complex structures, response surface methodology has been suggested as a way to estimate the actual but implicit limit state function. Typically the response surface is constructed from a polynomial function and fitted to the implicit function at a number of points. The location of these points has been noted as being an issue but the effect of varying their location has had little attention in the literature. In the present paper some simple examples are used to indicate possible effects. It is noted that the probability can be both under- and over-estimated, depending on the choice of points, but that no clear guidance for point selection can be given in any one case. A particularly disturbing feature is that for some types of problems there can be instability in the probability estimate as the location of the points is changed. This is demonstrated through a previously well-discussed example.
The reliability analysis of complex structures is hindered by the implicit nature of the limit-state function. For their approximation use has been made of the Response Surface Method (RSM) and, more recently, of Neural Networks. From the statistical viewpoint this corresponds to a regression approach. In the structural reliability literature little attention has been paid, however, to the possibility of treating the problem as a classification task. This enlarges the list of methods that are eventually useful to the purpose at hand and justifies an overall examination of their distinguishing features. This task is performed in this paper from the point of view of the Theory of Statistical Learning, which provides a unified framework for all regression, classification and probability density estimation. The classification methods are grouped into three categories and it is shown that only one group is useful for structural reliability, according to some specific criteria. In this category are the Multi-Layer Perceptrons and the Support Vector Machines, which are the recommended methods because (a) they can estimate the function on the basis of a few samples, (b) they use flexible and adaptive models and (c) they can overcome the curse of dimensionality. The paper also includes an in-depth analysis of the RSM from the point of view of statistical learning. It is shown that the empirically found instability of this method is explained with statistical learning concepts.
: The model correction factor method (MCFM) is used in conjunction with the first-order reliability method (FORM) to solve structural reliability problems involving integrals of non-Gaussian random fields. The approach replaces the limit-state function with an idealized one, in which the integrals are considered to be Gaussian. Conventional FORM analysis yields the linearization point of the idealized limit-state surface. A model correction factor is then introduced to push the idealized limit-state surface onto the actual limit-state surface. A few iterations yield a good approximation of the reliability index for the original problem. This method has application to many civil engineering problems that involve random fields of material properties or loads. An application to reliability analysis of foundation piles illustrates the proposed method. 1 INTRODUCTION Many problems in structural reliability involve integrals of random fields in the definition of the limit state function. T...
- R Courant
- D Hilbert
Courant, R. & Hilbert, D. 1953. Methods of Mathematical
Physics. Interscience.
- O Ditlevsen
- H O Madsen
Ditlevsen, O. & Madsen, H.O. 1996. Structural Reliability
Methods, Wiley.