Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems

Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower‐dimensional space than the sources themselves. In this work, we thus propose to use a dimension reduction technique for obtaining the representation of the exchanged information, and we propose a measure transformation technique that allows subproblem implementations to exploit this dimension reduction to achieve computational gains. The effectiveness of the proposed dimension reduction and measure transformation methodology is demonstrated through a multiphysics problem relevant to nuclear engineering. Copyright © 2012 John Wiley & Sons, Ltd.     

Dimension reduction in stochastic modeling of coupled problems

Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower dimensional space than the sources themselves. This work thus presents an investigation into the characterization of the exchanged information by a reduced‐dimensional representation and in particular by an adaptation of the Karhunen‐Loève decomposition. The effectiveness of the proposed dimension–reduction methodology is analyzed and demonstrated through a multiphysics problem relevant to nuclear engineering. Copyright © 2012 John Wiley & Sons, Ltd.     

Computation of higher‐order moments of generalized polynomial chaos expansions

Because of the complexity of fluid flow solvers, non‐intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher‐order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin‐type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third‐order, indeed fourth‐order moments of the polynomials are needed in this analysis. Besides, both intrusive and non‐intrusive approaches call for their computation provided that higher‐order moments of the quantities of interest need to be post‐processed. In most applications, they are evaluated by Gauss quadratures and eventually stored for use throughout the computations. In this paper, analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature‐free procedures instead. Matlab^© codes have been developed for this purpose and tested by comparisons with Gauss quadratures. Copyright © 2017 John Wiley & Sons, Ltd.     

Variational multiscale enrichment for modeling coupled mechano‐diffusion problems

In this paper, a new multiscale–multiphysics computational methodology is devised for the analysis of coupled diffusion–deformation problems. The proposed methodology is based on the variational multiscale principles. The basic premise of the approach is accurate fine‐scale representation at a small subdomain where it is known a priori that important physical phenomena are likely to occur. The response within the remainder of the problem domain is idealized on the basis of coarse‐scale representation. We apply this idea to evaluate a coupled mechano‐diffusion problem that idealizes the response of titanium structures subjected to a thermo–chemo–mechanical environment. The proposed methodology is used to devise a multiscale model in which the transport of oxygen into titanium is modeled as a diffusion process, whereas the mechanical response is idealized using concentration‐dependent elasticity equations. A coupled solution strategy based on operator split is formulated to evaluate the coupled multiphysics and multiscale problem. Numerical experiments are conducted to assess the accuracy and computational performance of the proposed methodology. Numerical simulations indicate that the variational multiscale enrichment has reasonable accuracy and is computationally efficient in modeling the coupled mechano‐diffusion response. Copyright © 2011 John Wiley & Sons, Ltd.     

Generalized polynomial chaos and random oscillators

We present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions, which are more efficient than the classical Wiener–Hermite expansions. The approach is general but here we present results for linear oscillators only with random forcing or random coefficients. In this context, we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution. We have also performed computational comparisons with Monte Carlo simulations which show that the new approach can be orders of magnitude faster, especially for compact distributions. Copyright © 2004 John Wiley & Sons, Ltd.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty quantification by crude Monte-Carlo simulation is not feasible. In this paper, we propose the use of sparse adaptive polynomial chaos expansions (PCE) as a surrogate of the full model. To overcome known limitations of PCE when applied to FRF simulation, we propose a frequency transformation strategy that maximizes the similarity between FRFs prior to the calculation of the PCE surrogate. This strategy results in lower-order PCEs for each frequency. Principal component analysis is then employed to reduce the number of random outputs. The proposed approach is applied to two case studies: a simple 2-DOF system and a 6-DOF system with 16 random inputs. The accuracy assessment of the results indicates that the proposed approach can predict single FRFs accurately. Besides, it is shown that the first two moments of the FRFs obtained by the PCE converge to the reference results faster than with the Monte-Carlo (MC) methods.     

A practical polynomial chaos Kalman filter implementation using nonlinear error projection on a reduced polynomial chaos expansion

The polynomial chaos Kalman filter (PCKF) has been gaining popularity as a computationally efficient and robust alternative to sampling methods in sequential data assimilation settings. The PCKF's sampling free scheme and attractive structure to represent non‐Gaussian uncertainties makes it a promising approach for data filtering techniques in nonlinear and non‐Gaussian frameworks. However, the accuracy of PCKF is dependent on the dimension and order of the polynomial chaos expansion used to represent all sources of uncertainty in the system. Thus, when independent sources of errors, like process noise and time independent sensors' errors are incorporated in the system, the curse of dimensionality hinders the efficiency and the applicability of PCKF. This study sheds light on this issue and presents a practical framework to maintain an acceptable accuracy of PCKF without scarifying the computational efficiency of the filter. The robustness and efficiency of the presented implementation is demonstrated on 3 typical numerical examples to illustrate its ability to achieve considerable accuracy at a low computational tax.     

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.     

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.     

Linear random vibration by stochastic reduced‐order models

A practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non‐Gaussian noise and systems with uncertain properties subjected to Gaussian and non‐Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non‐Gaussian. The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced‐order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM‐based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

Galerkin reduced‐order modeling scheme for time‐dependent randomly parametrized linear partial differential equations

In this paper, we consider the problem of constructing reduced‐order models of a class of time‐dependent randomly parametrized linear partial differential equations. Our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates, parameter space, and time. The proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions. The unknown basis functions in the decomposition are estimated using an alternating iterative Galerkin projection scheme. Numerical studies on the time‐dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos‐based Galerkin projection schemes. Comparison studies are also made against Nouy's generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages. Copyright © 2012 John Wiley & Sons, Ltd.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient uncertainty propagation for network multiphysics systems

We consider a multiphysics system with multiple component PDE models coupled together through network coupling interfaces, that is, a handful of scalars. If each component model contains uncertainties represented by a set of parameters, a straightforward uncertainty quantification study would collect all uncertainties into a single set and treat the multiphysics model as a black box. Such an approach ignores the rich structure of the multiphysics system, and the combined space of uncertainties can have a large dimension that prohibits the use of polynomial surrogate models. We propose an intrusive methodology that exploits the structure of the network coupled multiphysics system to efficiently construct a polynomial surrogate of the model output as a function of uncertain inputs. Using a nonlinear elimination strategy, we treat the solution as a composite function: the model outputs are functions of the coupling terms, which are functions of the uncertain parameters. The composite structure allows us to construct and employ a reduced polynomial basis that depends on the coupling terms. The basis can be constructed with many fewer PDE solves than the naive approach, which results in substantial computational savings. We demonstrate the method on an idealized model of a nuclear reactor. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions

Given their mathematical structure, methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory. This is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition. This enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non‐continuous dependence of the solution on the stochastic parameters. In particular, given the polynomial structure of these approximations, the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated. An enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non‐smooth functions by integrating a priori knowledge about their behaviour. The proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues. It is observed that through judicious selection of enrichment functions, the spectrum of such a random system can be more efficiently characterized, even for systems with closely spaced eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e.of Galerkin type) or non-intrusive) unaffordable when the deterministic finite element model is expensive to evaluate.     

Uncertainty‐based robust aerodynamic optimization of rotor blades

The aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

Reliability‐based topology optimization against geometric imperfections with random threshold model

This paper develops a new reliability‐based topology optimization framework considering spatially varying geometric uncertainties. Geometric imperfections arising from manufacturing errors are modeled with a random threshold model. The projection threshold is represented by a memoryless transformation of a Gaussian random field, which is then discretized by means of the expansion optimal linear estimation. The structural response and their sensitivities are evaluated with the polynomial chaos expansion, and the accuracy of the proposed method is verified by Monte Carlo simulations. The performance measure approach is adopted to tackle the reliability constraints in the reliability‐based topology optimization problem. The optimized designs obtained with the present method are compared with the deterministic solutions and the reliability‐based design considering random variables. Numerical examples demonstrate the efficiency of the proposed method.     

A new probabilistic distribution matching PATC formulation using polynomial chaos expansion

In the existing probabilistic hierarchical optimization approaches, such as probabilistic analytical target cascading (PATC), all the stochastic interrelated responses are characterized only by the first two statistical moments. However, due to the high nonlinear relation between the inputs and outputs, the interrelated responses are not necessarily normally distributed. The existing approaches, therefore, may not accurately quantify the probabilistic characteristics of the interrelated responses, and would further prevent achieving the real optimal solution. To overcome this deficiency, a novel PATC approach, named PATC-PCE is developed. By employing the polynomial chaos expansion (PCE) technique, the entire distribution of interrelated response can be characterized by a PCE coefficients vector, and then matched and propagated in the hierarchy. Comparative studies show that PATC-PCE outperforms PATC in terms of yielding more accurate optimal solutions and fewer design cycles when the interrelated response are random non-normal quantities, while at a sacrifice of extra function evaluations.     

Time response of structures with uncertain parameters under stochastic inputs based on coupled polynomial chaos expansion and component mode synthesis methods

This article presents a numerical procedure to compute the stochastic dynamic response of large finite element models with uncertain parameters based on polynomial chaos and component mode synthesis methods. Polynomial chaos expansions at higher orders are used to derive the statistical solution of the dynamic response as well as the Monte Carlo simulation procedure. Based on various component mode synthesis methods, the size of the model is reduced. These methods are coupled with polynomial chaos expansion and the explicit mathematical formulations are given. Numerical results illustrating the accuracy and efficiency of the proposed coupled methodological procedures are presented.