Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

Debusschere, Bert; Najm, Habib N.; Pébaÿ, Philippe Pierre; Knio, Omar M.; Ghanem, Roger; Maître, Olivier Le

doi:10.1137/s1064827503427741

Cited by 394 publications

(271 citation statements)

References 24 publications

Supporting

Mentioning

260

Contrasting

Order By: Relevance

“…The system of equations can be solved by using PC-arithmetics, such as product and addition in combination with an optimization algorithms. Basic operations of the PC arithmetic can be found in [26].…”

Section: A Stochastic Variation Of Ogden's Materials Modelmentioning

confidence: 99%

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Penner¹,

Caylak²,

Mahnken³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

View full text Add to dashboard Cite

Abstract. In many engineering applications the material behavior, e.g. hyper-elasticity and plasticity, is described by appropriate mathematical models. However, these become uncertain due to different types of uncertainty such as variation in the manufacturing process, measurement errors and missing or incomplete information on material properties. This contribution presents a framework for nonlinear elastic stochastic material model at large deformations. As a key idea uncertainty of the material is described by parameters, which are modeled as stochastic variables.To this end, 150 specimens for three different rubber materials are experimentally investigated in tensile tests. Based on experimental results 1000 artificial data are generated by aid of an ARMA process [1]. The artificial data are used for parameter identification of an Ogden material model for rubber materials . Furthermore, statistical analysis of material parameters including their correlations is studied. The number of material parameters define the dimension of the stochastic space. Usually, the stochastic material parameters are considered as stochastically independent. However, in our work we consider the dependency including the correlation obtained from experimental data.The hyperelastic stochastic material parameters are expanded with the multivariate PCE. In this context, the stresses depend on stochastic variables. To determine the corresponding PC coefficients for non-independent stochastic material parameters we use a Cholesky decomposition. As a numerical example we consider the static problem for uniaxial tension of the rectangular plate. This structure is investigated in order to represent the experimental setup conditions.

show abstract

Section: A Stochastic Variation Of Ogden's Materials Modelmentioning

confidence: 99%

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Penner¹,

Caylak²,

Mahnken³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

View full text Add to dashboard Cite

show abstract

“…. , φ i p (ζ i )} associated to ζ i applying the Gram-Schmidt orthonormalization method and considering the scalar product ζ i defined in (9) to the canonical basis…”

Section: Adaptive Polynomial Chaos Algorithmmentioning

confidence: 99%

“…Here, we are specifically interested in the generalized Polynomial Chaos method (gPC) since it has been shown to be particularly effective for a number of problems [12,26,9]. gPC was proposed by Xiu and Karniadakis [26] as a remarkable generalization of the pioneering work by N. Wiener [25], which is usually referred to as the homogeneous chaos or Hermite-PC.…”

Section: Introductionmentioning

confidence: 99%

Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach

Chen-Charpentier

López

Licea

et al. 2015

Mathematics and Computers in Simulation

View full text Add to dashboard Cite

ElsevierChen Charpentier, BM.; Cortés, J.; Licea Salazar, JA.; Romero, J.; Roselló Ferragud, MD.; Santonja, F.; Villanueva Micó, RJ. (2015). Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach. AbstractDue to errors in measurements and inherent variability in the quantities of interest, models based on random differential equations give more realistic results than their deterministic counterpart. The generalized polynomial chaos (gPC) is a powerful technique used to approximate the solution of these equations when the random inputs follow standard probability distributions. But in many cases these random inputs do not have a standard probability distribution. In this paper, we present a step-by-step constructive methodology to implement directly a useful version of adaptive gPC for arbitrary distributions, extending the applicability of the gPC. The paper mainly focuses on the computational aspects, on the implementation of the method and on the creation of a useful software tool. This tool allows the user to easily change the types of distributions and the order of the expansions, and to study their the usefulness of the method are included.

show abstract

“…For more details, see [6]. Let u P denote the vector of coefficients of the MW expansion of u of order P = P (N p , N r ).…”

Section: Stochastic Galerkin Formulation Of the Euler Equationsmentioning

confidence: 99%

A stochastic Galerkin method for the Euler equations with Roe variable transformation

Pettersson

Iaccarino

Nordström

2014

Journal of Computational Physics

View full text Add to dashboard Cite

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables and no auxiliary quantities are needed, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

show abstract

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

Cited by 394 publications

References 24 publications

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach

A stochastic Galerkin method for the Euler equations with Roe variable transformation

Contact Info

Product

Resources

About