2.2 Reliability analysis
2.2.3 Stochastic expansion methods
“Stochastic expansion methods” aim to represent uncertainties of structural responses by using a set of polynomials targeting to represent the stochastic system. These methods are efficient tools for reliability analysis because the direct use of polyno- mial expansions, which are based on the concept of random processes, offer appealing convergence properties for the stochastic analysis (Cameron and Martin [30]).
Stochastic expansions require the evaluation of the structural responses at several collo- cation points to identify the coefficients of the polynomial expansion and consequently a sampling is required. According to Choi et al [39], stochastic expansions approaches can be classified into the “non-intrusive” and “intrusive” formulations. The intrusive formulation requires that the representation of the uncertainty is expressed explicitly within the analysis of the system. Otherwise in non-intrusive formulations the uncer- tainty is not expressed explicitly. Practically, this means that intrusive formulations require the modification of the analysis code in order to represent uncertainty whereas non-intrusive formulations may treat them as “black-box”. In non-intrusive formula- tions stochastic expansions are used to built the response surface without intersecting with the FE simulations, while intrusive formulations use the stochastic expansions to modify the stiffness matrix in the FE analyses. In this research only the non-intrusive
Define µ and σ of the Normal/equivalent Normal distribution For k = 0: xk= µx Transformation: xk → uk Evaluation of G(uk) and ∇uG(uk)
Obtention of
αk = −
∇G(uk)
k∇G(uk)k
Obtention of β
Next design point
uk+1 = β · αk Convergence of β? Transformation: uk+1 → xk+1 x∗ = x k+1 k = k + 1 k = k + 1 yes no no HL/RF (µx0, σx0) HL
Figure 2.4: Flowchart of the HL/HL-RF algorithm
formulation is used, being the most popular methods the Polynomial Chaos Expansion (PCE) and the Stochastic Collocation (SC).
2.2.3.1 Polynomial Chaos Expansion (PCE)
The original Hermite Polynomial Chaos Expansion (PCE), also known as homoge- neous chaos, was first derived in Wiener [211] for the spectral representation of any stochastic response in terms of Gaussian random variables. Hermite polynomials are a subset of the hypergeometrical orthogonal polynomials known as the Askey scheme (Askey and Wilson [13]), which have the orthogonal property of having as weighting functions the probability density functions (PDF) of some well known continuous ran- dom distributions (Gamma, Uniform, Beta, ...). In the study of Xiu and Karniadakis [215] the method was extended under the Wiener-Askey scheme to different random distributions. In Table 2.1 the set of polynomials that provide an optimal basis for different Probability Density Functions (PDFs) is presented.
Table 2.1: Correspondence between orthogonal polynomials and PDF
Orthogonal polynomial Weight function PDF Density function
Hermite He(x) e −x2 2 Normal √1 2πe −x2 2 Legendre Pn(x) 1 Uniform 12 Laguerre Ln(x) e−x Exponential e−x Jacobi Pnα,β(x) xαe−x Beta Γ(α+1)xαe−x
The PCE uses the orthogonal polynomials of the random variables to represent the re- lation between the stochastic response output and each of its random inputs. Moreover, PCE is convergent in the mean-square sense and any order PCE consists of orthogo- nal polynomials, which greatly simplify the calculation of statistical moments (Choi et al [39]). Hence, PCE can be used to approximate both Gaussian and non-Gaussian distributions using a least square scheme.
The PCE for a response where n random variables are involved is an infinite polynomial expansion which in practice is truncated at a certain order p, being the number of coefficients Nc of the expansion given by:
Nc =
(n + p)!
n!p! (2.16)
From this expression it can be noted that increasing the number of random variables or the order of the polynomial will cause a substantial grow in the number of terms Nc of the PCE. The main interests of this method are that offers relevant statistical information of the stochastic response througout its coefficients and that requires a relatively low number of sample points. The basis of the PCE lies in obtaining the unknown coefficients of the regression. The details of the PCE formulation and the calculation of the coefficients will be discussed in Section 3.5.2.1.
As exposed above, PCE requires a sampling of collocation points to determine their responses through FE simulations and built an accurate metamodel. One of the lim- itations of the PCE is that as the number of inputs and the order or the expansion increases, the number of available collocation points increase exponentially (Isukapalli [102]) and consequently many of them are not sampled. This leads to an incomplete information about the whole design space as the collocation points selected for the sampling usually do not guarantee a complete space filling, leaving generally unrepre- sented regions of very low probability (tail regions of the PDF) as shown in Figure2.5. Thus this method requires a stratified sampling where all intervals of the PDF includ- ing tail regions must be represented (Figure 2.6). In Section 2.2.4 a deeper study of sampling methods is carried out.
x P DF Highest probability Lowest probability x1 x2
Figure 2.5: Collocation points of a PDF.
x P DF Highest probability Lowest probability x1 x2
Figure 2.6: Stratified sampling of a PDF.
2.2.3.2 Stochastic Collocation (SC)
The Stochastic Collocation (SC) expansion is built as a sum of Lagrange polynomials Lj, one per collocation point. Thanks to the properties of these polynomials (are equal to 1 in the collocation points and 0 at the rest of the points), the coefficients of the regression can be obtained as the response values at the collocation points.
To maximize the performance of this method is essential to use as collocation points the Gaussian points defined from the optimal orthogonal polynomials, which are their roots, as exposed in Eldred [64]. An advantage of SC expansion is that no tailoring of the expansion form is required as there is for the PCE, since the polynomials of the expansion are Lagrange polynomials. The main drawback is that it only works if the Gaussian points are known a priori, being any other set of points useless.