A non-intrusive B-splines Bézier elements-based method for uncertainty propagation

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Abstract

A non-intrusive B-Splines Bézier Elements based Method (BSBEM) is proposed as an efficient tool for uncertainty propagation analysis in physical hyperbolic problems. The model’s output response is approximated using a surrogate model whose coefficients are obtained from a set of deterministic calls by means of a regression technique. The accuracy, efficiency and the generality of the proposed approach are assessed using benchmark numerical examples by comparing the convergence of the statistics moments with those of the Polynomial chaos-based point collocation method (Pcol) and the Monte Carlo (MC) method. The generic character of the proposed approach allows it to be implemented for several engineering fields. BSBEM is applied to quantify uncertainty propagation through dam break flows modelled by shallow water equations. The obtained results, which are depicted in terms of water depth and inundation line confidence intervals, show that with a meticulous exploitation of the multi-element aspect and the smoothness feature of the basis functions, the proposed method provides an accurate and smooth approximation of the stochastic output response.

Introduction

The increase of computational power has contributed to the development of advanced numerical models that are an essential and a powerful means to deepen the understanding and improve the prediction of physical phenomena in several fields of engineering sciences. The numerical results and the products derived from these models are usually performed with fixed values of the input parameters without taking into account the numerous uncertainties to which they may be subjected. The theme of uncertainty quantification (UQ) analysis has therefore attracted attention over the past few decades owing to its various engineering applications, including structural design [1], [2], [3], aerospace engineering [4], [5], nuclear safety analysis, thermal engineering [6] and hydraulic-environmental studies [7], [8].

The Monte Carlo method is among the most common and straightforward methods for uncertainty propagation due to its robustness and implementation simplicity [9]. However, the use of such a method for cases involving complex deterministic models can prove to be prohibitive in computational cost due to the large number of executions required to achieve an acceptable accuracy. Despite this drawback, the Monte Carlo method is often used as a benchmark solution to evaluate the accuracy and reliability of other sampling methods, such as Latin Hypercube Sampling (LHS) [10] and Multi-Level Monte Carlo (MLMC) [11], which were introduced to compensate for the low convergence rate of the Monte Carlo method.

As an alternative to sampling approaches, the polynomial chaos (PC) expansion proposed by Ghanem and Spanos [12], which belongs to the stochastic expansion methods, was developed to quantify uncertainty propagation in numerical models. This method, based on the homogeneous chaos theory of Wiener [13], uses Hermite polynomial basis functions to model the uncertainties of the output response resulting from a stochastic process with normally-distributed random input parameters. For non-Gaussian random input variables, an extension of the Wiener–Hermite PC expansion, known as the generalized polynomial chaos expansion (gPCE), was proposed by Xiu and Karniadakis [14]. In gPCE, orthogonal polynomials are selected from the Askey scheme in accordance with the corresponding probability distributions of the random input variables, thereby allowing for an exponential convergence rate of the statistical moments.

The coefficients of the PCE method are evaluated, in their earliest development, by the intrusive approach based on the Galerkin projection, which requires modification of the governing equations of the deterministic model, thus leading to a system of coupled equations whose unknowns are the coefficients of the PC expansion [15], [16]. This intrusive technique presents a real difficulty in some complex deterministic codes, requiring time-consuming cumbersome developments, which renders it much less attractive. As a result, other methods have been proposed, such as the so-called non-intrusive polynomial chaos expansion (NIPCE), which uses deterministic models as black boxes and thus require no modification of the governing equations. The coefficients of the NIPC expansion are evaluated either with a spectral projection technique that exploits the orthogonality of the polynomial basis, using the quadrature approach to estimate the resulting multidimensional integral [17], or with a regression technique, solving an oversized linear system of equations by using the least square minimization approach [18]. When the number of input random variables increases for a specified order of the polynomial basis, the number of model evaluations that the projection approach requires to evaluate the multidimensional integrals becomes prohibitively large, which leads to the so-called curse of dimensionality [19]. As an alternative, the sparse or Smolyak quadrature technique is introduced to alleviate the curse of dimensionality by reducing the number of the collocation points, as reported in [20], particularly in stochastic collocation methods [21]. Another scheme, the so-called probabilistic collocation method, which considers the collocations points as roots of the orthogonal polynomials, is proposed as a non-intrusive PCE and allows the number of model evaluations to be reduced [22], [23]. The point collocation method [24], [25], is also considered as an alternative approach to deal with uncertainty quantification with an appropriate accuracy. It combines the gPCE approach with well-known sampling methods (that are also associated with an adequate oversampling technique) to choose the collocation points.

Recently, much work has been devoted into developing approaches that are more efficient for uncertainty propagation analysis and thereby reduce simulation costs. Some of these techniques have deeply investigated in terms of how to optimize sampling procedures and how to reduce the number of polynomial basis functions required to capture the main stochastic features of the output response. These efforts include the least angle regression technique (LAR) [18], [19], adaptive sparse polynomial chaos expansion [26], [27], sliced inverse regression-based sparse polynomial chaos expansions [28], multi-fidelity non-intrusive polynomial chaos [29], [30], [31], and more recently, using polynomial chaos decomposition with a differentiation approach [32].

Thanks to their multiple attractive features, non-intrusive polynomial chaos approximations are widely used in uncertainty quantification analysis to provide accurate statistical moments and to build an efficient surrogate for the complex original model of the output response. Nevertheless, the use of such basis functions presents some limitations when the output response presents hyperbolic behaviour [33]. In those cases, non-intrusive polynomial chaos approaches may fail to reproduce the stochastic behaviour of the model’s output response. Several methods are proposed in the literature to overcome this limitation, such as the Wiener–Har expansion based on wavelets, which belongs to a multi-element generalized polynomial chaos approach [34], [35], multi resolution analysis (MRA) [36] and a piecewise polynomial approximation based on a hybrid global and adaptive polynomial algorithm [37].

The basic concept motivating the present work is to introduce the B-Spline basis functions as an alternative approach to the existing schemes to overcome, as much as possible, the limitation of the polynomial chaos method to describe an output response with strong hyperbolic behaviour. It must be emphasized that B-Spline functions have only just been introduced in the field of uncertainty quantification analysis in recent studies [38], [39], [40], in which they were used as weighting functions in the projection phase to evaluate the coefficients of the approximations. More recently, B-spline functions have been introduced as basis functions to investigate the data-driven uncertainty quantification of structural systems [41].

In this paper, a non-intrusive regression stochastic method is proposed, wherein the expansion is defined on the cumulative probabilities domain that is decomposed into elements (called Bézier elements), and where the B-splines functions (with the input parameters’ sampling) are established locally in the Bézier elements. The use of such interpolations allows smooth statistics of the outputs (mean and variance) to be obtained in the case of discontinuous wave flows. The multi-element aspect of the method and the compact support feature of the BSBE method are explored to increase the efficiency of the sampling procedure and the smoothness of the approximation.

The paper is organized as follows. Section 2, presents a review of the polynomial chaos method. The proposed non-intrusive B-Spline Bézier elements method (BSBEM) is presented in Section 3. Section 4 evaluates the accuracy of the BSBE method using analytical test cases with two and three input random variables, and the proposed approach is applied to dam break flow test cases, first over a test bed and then over real topography. The accuracy of the method is compared to both to the PCE approach and to the Monte Carlo reference solutions. Finally, the conclusions of this study and some suggestions for future research are presented in Section 5.

Section snippets

Non-intrusive PCE for uncertainty propagation using stochastic collocation

The polynomial chaos expansion is based on the spectral representation of the scalar output response (Y=f(x)) of the deterministic model, which is approximated as a truncated sum of the orthonormal multivariate polynomials of a vector of uncertain input parameters x=x1,x2,,xmT: Ŷ=i=1MαiΨi(x)where m is the number of input random variables and αi are the deterministic weighting coefficients to be determined. Ψi(x) are the multivariate basis functions that describe the stochastic part of the

A non-intrusive B-splines Bézier elements’ expansion

The use of B-Splines functions has been steadily increasing due to the mathematical properties they provide. They have been coupled with the finite element method to set up a new concept called Isogeometric Analysis, introduced by Hughes et al. [43], [44]. The use of such basis functions in uncertainty quantification seems to be especially promising for phenomena with hyperbolic behaviour.

Bivariate numerical test case

The performance of the proposed method is assessed first by considering a benchmark test where an analytical function output is known as a deterministic model with two input uncertain parameters x1,x2 following a uniform distribution with given values of the mean and standard deviation μxi=0.4,σxi=0.16,i=1, 2 which is defined as [25]: Y=f(x1,x2)=e(x1+x2)

The evolution of the relative error with respect to the polynomial order and to the number of Bézier elements for the statistical moments (mean

Conclusion

This paper contributes to the uncertainty propagation analysis for non-linear hyperbolic problems. The proposed method belongs to the non-intrusive multi-element stochastic expansions that have been developed to overcome the limitations of classical approaches in the presence of discontinuities. The BSBEM combines the smoothness (Cp1 continuity) of the B-spline functions with the multi-element structure of the Bézier extraction operator. The expansion is combined with sampling methods to

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada and Hydro Québec, Canada . The financial support is gratefully acknowledged.

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