A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

This paper presents a novel hybrid polynomial dimensional decomposition (PDD) method for stochastic computing in high-dimensional complex systems. When a stochastic response does not possess a strongly additive or a strongly multiplicative structure alone, then the existing additive and multiplicative PDD methods may not provide a sufficiently accurate probabilistic solution of such a system. To circumvent this problem, a new hybrid PDD method was developed that is based on a linear combination of an additive and a multiplicative PDD approximation, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. Two numerical problems involving mathematical functions or uncertain dynamic systems were solved to study how and when a hybrid PDD is more accurate and efficient than the additive or the multiplicative PDD. The results show that the univariate hybrid PDD method is slightly more expensive than the univariate additive or multiplicative PDD approximations, but it yields significantly more accurate stochastic solutions than the latter two methods. Therefore, the univariate truncation of the hybrid PDD is ideally suited to solving stochastic problems that may otherwise mandate expensive bivariate or higher-variate additive or multiplicative PDD approximations. Finally, a coupled acoustic-structural analysis of a pickup truck subjected to 46 random variables was performed, demonstrating the ability of the new method to solve large-scale engineering problems.     

Novel computational methods for high‐dimensional stochastic sensitivity analysis

This paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high‐dimensional complex systems subject to random input. The first method represents a novel integration of the polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean‐square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD‐saddlepoint approximation (SPA) or PDD‐SPA method, entailing SPA and score functions; and the PDD‐Monte Carlo simulation (MCS) or PDD‐MCS method, utilizing the embedded MCS of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100‐dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three‐hole bracket illustrates the usefulness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Global sensitivity analysis by polynomial dimensional decomposition

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.     

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.     

Stochastic sensitivity analysis by dimensional decomposition and score functions

This article presents a new class of computational methods, known as dimensional decomposition methods, for calculating stochastic sensitivities of mechanical systems with respect to probability distribution parameters. These methods involve a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions and score functions associated with probability distribution of a random input. The proposed decomposition facilitates univariate and bivariate approximations of stochastic sensitivity measures, lower-dimensional numerical integrations or Lagrange interpolations, and Monte Carlo simulation. Both the probabilistic response and its sensitivities can be estimated from a single stochastic analysis, without requiring performance function gradients. Numerical results indicate that the decomposition methods developed provide accurate and computationally efficient estimates of sensitivities of statistical moments or reliability, including stochastic design of mechanical systems. Future effort includes extending these decomposition methods to account for the performance function parameters in sensitivity analysis.     

Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.     

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.     

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

A dimensional decomposition method for stochastic fracture mechanics

This paper presents a new dimensional decomposition method for obtaining probabilistic characteristics of crack-driving forces and reliability analysis of general cracked structures subject to random loads, material properties, and crack geometry. The method involves a novel function decomposition permitting lower-variate approximations of a crack-driving force or a performance function, Lagrange interpolations for representing lower-variate component functions, and Monte Carlo simulation. The effort required by the proposed method can be viewed as performing deterministic fracture analyses at selected input defined by sample points. Compared with commonly-used first- and second-order reliability methods, no derivatives of fracture response are required by the new method developed. Results of three numerical examples involving both linear-elastic and nonlinear fracture mechanics of cracked structures indicate that the decomposition method provides accurate and computationally efficient estimates of probability density of the J-integral and probability of fracture initiation for various cases including material gradation characteristics and magnitudes of applied stresses and loads.     

A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

A new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S?N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional stochastic finite element method for elasto‐plastic bodies

A new stochastic finite element method (SFEM) is formulated for three‐dimensional softening elasto‐plastic bodies with random material properties. The method is based on the Karhunen–Loeve and polynomial chaos expansions, and able to efficiently estimate complete probabilistic characteristics of the response, such as moments or PDFs. To reduce the computational complexity in the three‐dimensional setting, two alterations are made with respect to the two‐dimensional SFEM proposed earlier by the authors. First, a variability preserving modification of the Karhunen–Loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties. Second, an efficient algorithm for parallel processing is developed, with time consumption being the same order as for an ordinary FEM, rendering the proposed SFEM an effective alternative to Monte‐Carlo simulation. The applicability of the proposed method to stochastic analysis of strain localization is examined using Monte‐Carlo simulation. Then, it is applied to a fault formation problem which is a recent concern of earthquake engineering. Ground surface layers are modelled by a softening elasto‐plastic body, and the evolution of probabilistic characteristics of the rupture process is analysed in detail. Some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint. Copyright © 2001 John Wiley & Sons, Ltd.     

Dimension reduction model for two-spatial dimensional stochastic wind field: Hybrid approach of spectral decomposition and wavenumber spectral representation

A renewed methodology for simulating two-spatial dimensional stochastic wind field is addressed in the present study. First, the concept of cross wavenumber spectral density (WSD) function is defined on the basis of power spectral density (PSD) function and spatial coherence function to characterize the spatial variability of the stochastic wind field in the two-spatial dimensions. Then, the hybrid approach of spectral representation and wavenumber spectral representation and that of proper orthogonal decomposition and wavenumber spectral representation are respectively derived from the Cholesky decomposition and eigen decomposition of the constructed WSD matrices. Immediately following that, the uniform hybrid expression of spectral decomposition and wavenumber spectral representation is obtained, which integrates the advantages of both the discrete and continuous methods of one-spatial dimensional stochastic field, allowing for reflecting the spatial characteristics of large-scale structures. Moreover, the dimension reduction model for two-spatial dimensional stochastic wind field is established via adopting random functions correlating the high-dimensional orthogonal random variables with merely 3 elementary random variables, such that this explicitly describes the probability information of stochastic wind field in probability density level. Finally, the numerical investigations of the two-spatial dimensional stochastic wind fields respectively acting on a long-span suspension bridge and a super high-rise building are implemented embedded in the FFT algorithm. The validity and engineering applicability of the proposed method are thus fully verified, providing a potentially effective approach for refined wind-resistance dynamic reliability analysis of large-scale complex engineering structures.     

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

A methodology is introduced for rapid reduced‐order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy. Copyright © 2005 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.     

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.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.) This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics

A priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The proper generalized decomposition method is used to construct a quasi‐optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure, which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. Because the dynamic response is highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension. Copyright © 2011 John Wiley & Sons, Ltd.     

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Engineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path‐wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees‐of‐freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path‐wise) solutions, it must be emphasized that sample realizations of weak solutions have no path‐wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is ‘close’ to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito–Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher‐level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis‐à‐vis a few other popularly used stochastic integration schemes as well as the path‐wise versions of the stochastic Newmark scheme. Copyright © 2006 John Wiley & Sons, Ltd.