Quantifying Uncertainty
Nobile, F., Tempone, R., and Webster, C.G.. "A stochastic collocation method for elliptic partial differential equations with random input data" SIAM Journal on Numerical Analysis. 43
(3).
2007.
pp. 1005--1034.
Marzouk, Y.M., Najm, H.N., and Rahn, L.A.. "Stochastic spectral methods for efficient Bayesian solution of inverse problems" Journal of Computational Physics. 224
(2).
2007.
pp. 560--586.
We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling.
Ganapathysubramanian, B. and Zabaras, N.. "Sparse grid collocation schemes for stochastic natural convection problems" J. Comput. Phys.. 225
(1).
2007.
pp. 652--685.
In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are made.
Acharjee, S. and Zabaras, N.. "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes" Computers and Structures. 85
(5-6).
2007.
pp. 244--254.
Large deformation processes are inherently complex considering the non-linear phenomena that need to be accounted for. Stochastic analysis of these processes is a formidable task due to the numerous sources of uncertainty and the various random input parameters. As a result, uncertainty propagation using intrusive techniques requires tortuous analysis and overhaul of the internal structure of existing deterministic analysis codes. In this paper, we present an approach called non-intrusive stochastic Galerkin (NISG) method, which can be directly applied to presently available deterministic legacy software for modeling deformation processes with minimal effort for computing the complete probability distribution of the underlying stochastic processes. The method involves finite element discretization of the random support space and piecewise continuous interpolation of the probability distribution function over the support space with deterministic function evaluations at the element integration points. For the hyperelastic-viscoplastic large deformation problems considered here with varying levels of randomness in the input and boundary conditions, the NISG method provides highly accurate estimates of the statistical quantities of interest within a fraction of the time required using existing Monte Carlo methods.
Lin, G., Wan, X., Su, C.-H., and Karniadakis, G.E.. "Stochastic Computational Fluid Mechanics" Computing in Science and Engineering. 9
(2).
2007.
pp. 21--29.
Witteveen, J.A.S., Sarkar, S., and Bijl, H.. "Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos" Computers and Structures. 85
(11-14).
2007.
pp. 866--878.
A nonlinear dynamic problem of stall induced flutter oscillation subject to physical uncertainties is analyzed using arbitrary polynomial chaos. A single-degree-of-freedom stall flutter model with torsional oscillation is considered subject to nonlinear aerodynamic loads in the dynamic stall regime and nonlinear structural stiffness. The analysis of the deterministic aeroelastic response demonstrated that the problem is sensitive to variations in structural natural frequency and structural nonlinearity. The effect of uncertainties in these parameters is studied. Arbitrary polynomial chaos is employed in which appropriate expansion polynomials are constructed based on the statistical moments of the uncertain input. The arbitrary polynomial chaos results are compared with Monte Carlo simulations.
Ghanmi, S., Bouazizi, M.-L., and Bouhaddi, N.. "Robustness of mechanical systems against uncertainties" Finite Elements in Analysis and Design. 43
(9).
2007.
pp. 715--731.
In this paper, one can propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of stochastic response surface method (SRSM) and a genetic algorithm provided with a new robustness criterion. The SRSM is based on the use of stochastic finite element method (SFEM) via the use of the polynomial chaos expansion (PC). Thus, one can avoid the use of Monte Carlo simulation (MCS) whose costs become prohibitive in the optimization problems, especially when the finite element models are large and have a considerable number of design parameters. The objective of this study is on one hand to quantify efficiently the effects of these uncertainties on the responses variability or the cost functions which one wishes to optimize and on the other hand, to calculate solutions which are both optimal and robust with respect to the uncertainties of design parameters. In order to study the propagation of input uncertainties on the mechanical structure responses and the robust multi-objective optimization with respect to these uncertainty, two numerical examples were simulated. The results which relate to the quantification of the uncertainty effects on the responses variability were compared with those obtained by the reference method (REF) using MCS and with those of the deterministic response surfaces methodology (RSM). In the same way, the robust multi-objective optimization results resulting from the SRSM method were compared with those obtained by the direct optimization considered as reference (REF) and with RSM methodology. The SRSM method application to the response variability study and the robust multi-objective optimization gave convincing results.
Le Maitre, O.P., Najm, H.N., P\'ebay, P.P., Ghanem, R.G., and Knio, O.M.. "Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems" SIAM Journal on Scientific Computing. 29
(2).
2007.
pp. 864--889.
This paper presents a multi-resolution approach for the propagation of parametric uncertainty in chemical systems. It is motivated by previous studies where Galerkin formulations of Wiener-Hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates. The multi-resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi-wavelets, allowing for the control of the convergence in terms of polynomial order and resolution level. The resulting representation is shown to greatly improve the robustness of the Galerkin procedure in presence of steep dependences. However, this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates. To overcome this drawback an adaptive strategy is proposed to control locally (in the parameter space) and in time the resolution level. The efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters.
Canuto, C. and Kozubek, T.. "A fictitious domain approach to the numerical solution of PDEs in stochastic domains" Numer. Math.. 107
(2).
2007.
pp. 257--293.
We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is âspectralâ in the polynomial chaos order, in any subdomain which does not contain the random boundaries.
Doostan, A., Ghanem, R.G., and Red-Horse, J.R. "Stochastic model reduction for chaos representations" Computer Methods in Applied Mechanics and Engineering. 196
(37-40).
2007.
pp. 3951--3966.
This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galerkin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) in the Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms at hand, the solution of SPDE's becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided for the adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics.
Ghanem, R.G, Saad, G., and Doostan, A.. "Efficient solution of stochastic systems: Application to the embankment dam problem" Structural Safety. 29
(3).
2007.
pp. 238--251.
The embankment dam problem of the benchmark study is treated using the newly developed Stochastic Model Reduction for Polynomial Chaos Representations method. The elastic and shear moduli of the material, in the present problem, are modeled as two stochastic processes that are explicit functions of the same process possessing a relatively low correlation length. The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process. In such a setting, the spectral stochastic finite element method (SSFEM) for the specified spatial discretization is computationally prohibitive. The approach adopted in this paper enables the stochastic characterization of a fine mesh problem based on the high dimensional polynomial chaos solution of a coarse mesh analysis. After relatively reducing the dimensionality of the problem through a Karhunen-Loeve representation of the stochastic variables, the SSFEM solution consisting of a high dimensional polynomial in Gaussian independent variables is obtained for the coarse mesh problem. Then the attained solution is used to define a new basis for solving the fine mesh problem. The paper presents some new algorithms for the estimation of chaos coefficients in the presence of complex non-Gaussian dependencies. A numerical convergence study is presented together with a discussion of the results.
Ganapathysubramanian, B. and Zabaras, N.. "Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method" Journal of Computational Physics. 226
(1).
SEP 10 2007.
pp. 326--353.
In recent years, there has been intense interest in understanding various physical phenomena in random heterogeneous media. Any accurate description/simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties. An accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response. We propose a general methodology to construct a data-driven, reduced-order model to describe property variations in realistic heterogeneous media. This reduced-order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media. A decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties. A sparse-grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems. A variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems. An illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology. (C) 2007 Elsevier Inc. All rights reserved.
Xiu, D. and Shen, J.. "An efficient spectral method for acoustic scattering from rough surfaces" Communicationns in Computational Physcis. 2
(1).
FEB 2007.
pp. 54--72.
An efficient and accurate spectral method is presented for scattering problems with rough surfaces. A probabilistic framework is adopted by modeling the surface roughness as random process. An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain. The generalized polynomial chaos (gPC) is then used to discretize the random space; and a Fodrier-Legendre method to discretize the physical space. These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces. Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm.
Paffrath, M. and Wever, U.. "Adapted polynomial chaos expansion for failure detection" Journal of Computational Physics. 226
(1).
SEP 10 2007.
pp. 263--281.
In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem. (C) 2007 Elsevier Inc. All rights reserved.
Emery, A.F. and Bardot, D.. "Stochastic heat transfer in fins and transient cooling using polynomial chaos and wick products" Journal of Heat Transfer-Transactions of the AMSE. 129
(9).
SEP 2007.
pp. 1127--1133.
Stochastic heat transfer problems are often solved using a perturbation approach that yields estimates of mean values and standard deviations for properties and boundary conditions that are random variables. Methods based on polynomial chaos and Wick products can be used when the randomness is a random field or white noise to describe specific realizations and to determine the statistics of the response. Polynomial chaos is best suited for problems in which the properties are strongly correlated, while the Wick product approach is most effective for variables containing white noise components. A transient lumped capacitance cooling problem and a one-dimensional fin are analyzed by both methods to demonstrate their usefulness.
Ghanem, R.G. and Ghosh, D.. "Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition" International Journal for Numerical Methods is Engineering. 72
(4).
OCT 22 2007.
pp. 486--504.
A new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented. The eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis. An efficient method for estimating the coefficients with respect to this basis is proposed. The method uses a Galerkin-based approach by orthogonalizing the residual in the eigenvalue-eigenvector equation to the subspace spanned by the basis functions used for approximation. In this way, the stochastic problem is framed as a system of deterministic non-linear algebraic equations. This system of equations is solved using a Newton-Raphson algorithm. Although the proposed approach is not based on statistical sampling, the efficiency of the proposed method can be significantly enhanced by initializing the non-linear iterative process with a small statistical sample synthesized through a Monte Carlo sampling scheme. The proposed method offers a number of advantages over existing methods based on statistical sampling. First, it provides an approximation to the complete probabilistic description of the eigensolution. Second, it reduces the computational overhead associated with solving the statistical eigenvalue problem. Finally, it circumvents the dependence of the statistical solution on the quality of the underlying random number generator. Copyright (C) 2007 John Wiley & Sons, Ltd.
Rupert, C.P. and Miller, C.T.. "An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems" Journal of Computational Physics. 226
(2).
OCT 2007.
pp. 2175--2205.
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. (c) 2007 Elsevier Inc. All rights reserved.
Li, H. and Zhang, D.. "Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods" Water Resources Research. 43
(9).
SEP 2007.
An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen-Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen-Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment-equation approach based on Karhunen-Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.
Lucor, D., Meyers, J., and Sagaut, P.. "Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos" Journal of Fluid Mechanics.
vol. 585.
AUG 25 2007.
pp. 255--279.
We address the sensitivity of large-eddy simulations (LES) to parametric uncertainty in the subgrid-scale model. More specifically, we investigate the sensitivity of the LES statistical moments of decaying homogeneous isotropic turbulence to the uncertainty in the Smagorinsky model free parameter C-s (i.e. the Smagorinsky constant). Our sensitivity methodology relies on the non-intrusive approach of the generalized Polynomial Chaos (gPC) method; the gPC is a spectral non-statistical numerical method well-suited to representing random processes not restricted to Gaussian fields. The analysis is carried out at Re-lambda=100 and for different grid resolutions and C-s distributions. Numerical predictions are also compared to direct numerical simulation evidence. We have shown that the different turbulent scales of the LES solution respond differently to the variability in C-s. In particular, the study of the relative turbulent kinetic energy distributions for different C-s distributions indicates that small scales are mainly affected by changes in the subgrid-model parametric uncertainty.
Huang, S., Mahadevan, S., and Rebba, R.. "Collocation-based stochastic finite element analysis for random field problems" Probabilstic Engineering Mechanics. 22
(2).
APR 2007.
pp. 194--205.
A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen-Loeve (K-L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box. SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM. (C) 2006 Elsevier Ltd. All rights reserved.
Todor, R.A. and Schwab, C.. "Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients" IMA J. of Numer. Anal.. 27
(2).
APR 2007.
pp. 232--261.
A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, omega) in a bounded domain D subset of < Ropf >(d) is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x is an element of D) and stochastic (omega is an element of Omega) variables in a(x, omega) via Karhunen-Loeve or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, omega) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.
Kim, D., Debusschere, B.J., and Najm, H.N.. "Spectral methods for parametric sensitivity in stochastic dynamical systems" Biophyscial Journal. 92
(2).
JAN 2007.
pp. 379--393.
Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.
Babuska, I.M., Nobile, F., and Tempone, R.. "A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data" SIAM J. Numer. Anal.. 45
(3).
2007.
pp. 1005--1034.
In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825 and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the ``probability error'' with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.
Xiu, D.. "Efficient Collocation approach for parametric uncertainty" Communications in Compuatational Physics. 2
(2).
APR 2007.
pp. 293--309.
A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting ``pseudo-spectral'' type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations.
Xiu, D. and Sherwin, S.J.. "Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network" Journal of Computational Physics. 226
(2).
2007.
pp. 1385--1407.
Reduced models of human arterial networks are an efficient approach to analyze quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour cannot be made unless the effect of such data uncertainty is quantified. In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a high-order stochastic collocation method based on the generalized polynomial chaos expansion, combined with a discontinuous Galerkin spectral/hp element discretization in physical space, can successfully simulate this type of hyperbolic system subject to uncertain inputs with bounds. Building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation, a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network. Using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations, thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks.
Huang, S. and Kou, X.. "An extended stochastic response surface method for random field problems" Acta Mechanica Sinica. 23
(4).
AUG 2007.
pp. 445--450.
An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.