Uncertainty propagation in finite deformations––A spectral stochastic Lagrangian approach

Acharjee, Swagato; Zabaras, Nicholas

doi:10.1016/j.cma.2005.05.005

Cited by 58 publications

(45 citation statements)

References 18 publications

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“…The first application of the method to elasto-plastic problems can be found in [3] where plastic and failure analysis of earth faults is attempted by introducing some simplifying assumptions. Geometrical and material non-linearity cannot be taken into account efficiently since PCE has been found to perform poorly in problems involving sharp non-linearities, discontinuities, slope changes or bifurcations [1,55] (see also Section 2.2). In these cases of non-smooth solutions, the choice of other basis functions such as wavelets or the use of an adaptive multi-element generalized PCE has been suggested as a remedy to the problem [101,185,131].…”

Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

“…This approach is usually combined in the literature with the polynomial chaos (PC) approximation for the calculation of the response variability of uncertain finite element systems e.g. [1][2][3]19,22,28,29,41,66,67,94,108,113,187,189]. The combination is called the spectral stochastic finite element method (SSFEM).…”

Section: The Spectral Representation Methodsmentioning

confidence: 99%

“…Specifically, it has been demonstrated that the accuracy of the PC approximation is not always improved as additional terms are retained and the PC approximation of certain processes may become computationally demanding because of the large number of coefficients u j that need to be calculated. This is usually the case of problems involving sharp non-linearity and abrupt slope changes or bifurcations [1,55] (Fig. 11).…”

Section: Methods Based On Polynomial Chaos (Pc) Expansionmentioning

confidence: 99%

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The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

871

407

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a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

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Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

Section: The Spectral Representation Methodsmentioning

confidence: 99%

Section: Methods Based On Polynomial Chaos (Pc) Expansionmentioning

confidence: 99%

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The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

871

407

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“…A typical approach is to conduct a Galerkin projection to minimize the error of the finite-order gPC expansion, and the resulting set of equations for the expansion coefficients are deterministic and can be solved via conventional numerical techniques. This has been done in various applications and proved to be very effective (cf, [1,7,11,17,21,39,40,38]). …”

Section: Stochastic Modellingmentioning

confidence: 99%

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Xiu

Sherwin

2007

Journal of Computational Physics

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Reduced models of human arterial networks are an efficient approach to analyse 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 can not 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 discretisation in physical space, can successfully simulate this type of hyperbolic system subject to restrictions on the uncertainty interval. 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.

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“…Research areas for the stochastic modeling are: linear elasticity of solids and mechanics [4], plasticity of solids and mechanics [6,7], large deformations [8,9], fluid flow [10,11,12], flow-structure interactions [13,14] and linear convection problems [15]. An open task is the application of the modeling of rubber-like materials, such as natural rubber, which is constitutively represented by an Ogden model and is therefore the focus of this paper.…”

Section: Introductionmentioning

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

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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.

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Uncertainty propagation in finite deformations––A spectral stochastic Lagrangian approach

Cited by 58 publications

References 18 publications

The stochastic finite element method: Past, present and future

The stochastic finite element method: Past, present and future

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

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