Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
1
4
2011
STATISTICAL STRENGTH OF HIERARCHICAL CARBON NANOTUBE COMPOSITES
279-295
10.1615/Int.J.UncertaintyQuantification.2011002456
X. Frank
Xu
Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
Keqiang
Hu
Department of Civil, Environmental & Ocean Engineering, Stevens Institute of Technology, Hoboken, New Jersey, 07307, USA
Irene J.
Beyerlein
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
George
Deodatis
Department of Civil Engineering and Engineering
Mechanics, Columbia University
500 West 120th St., 630 S.W. Mudd Bldg.
New York, NY 10027, USA
load sharing models
statistical size effect
statistical failure
carbon nanotubes
hierarchical structure
Monte Carlo simulation
In modeling and simulation of material failure, a major challenge lies in the computation of stress redistributions during the stochastic propagation of
localized failures. In this study, an nth-order generalized local load sharing (GLLS) model is introduced to account for the complexity of such local interactions in
an efficient way.The rule is flexible, covering
a wide range of load sharing mechanisms between the equal load sharing and local load sharing types. A Monte Carlo
simulation model employing various orders of this GLLS rule is used to study the effect of such load redistributions
on the failure of a micron-scale carbon nanotube (CNT) fiber. These CNT fibers exhibit a hierarchical structure. At
the lowest length scale are single- or multi-walled CNTs with nanoscale diameters (e.g., 1–10 nm), which are aligned
and clustered to form small bundles at the next higher length scale (15–60 nm in diameter). Thousands of these CNT
bundles aggregate and align to create CNT fibers with micron-scale diameters. The results of this study indicate that the
mean strength of the CNT fibers reduces by approximately two-thirds of an order of magnitude when up-scaling from
an individual CNT to a CNT fiber. This dramatic strength reduction occurs at three different stages of the up-scaling
process: (1) from individual CNTs of length lt to CNT bundles of the same length; (2) from a CNT bundle of length lt
to a CNT bundle of length lb(lb = 10lt); and (3) from CNT bundles of length lb
to CNT fibers of the same length.
The specific strength reductions during these three stages are provided in the paper. The computed fiber strengths are in
the same general range as corresponding experimental values reported in the literature. The ability of the GLLS model
to efficiently account for different mechanisms of load sharing, in combination with the multi-stage up-scaling Monte
Carlo simulation approach, is expected to benefit the design and optimization of robust structural composites built up
from carbon nanotubes.
ORTHOGONAL BASES FOR POLYNOMIAL REGRESSION WITH DERIVATIVE INFORMATION IN UNCERTAINTY QUANTIFICATION
297-320
10.1615/Int.J.UncertaintyQuantification.2011002790
Yiou
Li
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois, 60616, USA
Mihai
Anitescu
Mathematics and Computer Science Division, Argonne National Laboratory, USA
Oleg
Roderick
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 60439, USA
Fred
Hickernell
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois, 60616, USA
uncertainty quantification
representation of uncertainty
stochastic collocation
heat transfer
energy and the environment
We discuss the choice of polynomial basis for approximation of uncertainty propagation through complex simulation models with capability to output derivative information. Our work is part of a larger research effort in uncertainty quantification using sampling methods augmented with derivative information. The approach has new challenges compared with standard polynomial regression. In particular, we show that a tensor product multivariate orthogonal polynomial basis of an arbitrary degree may no longer be constructed. We provide sufficient conditions for an orthonormal set of this type to exist, a basis for the space it spans. We demonstrate the benefits of the basis in the propagation of material uncertainties through a simplified model of heat transport in a nuclear reactor core. Compared with the tensor product Hermite polynomial basis, the orthogonal basis results in a better numerical conditioning of the regression procedure, a modest improvement in approximation error when basis polynomials are chosen a priori, and a significant improvement when basis polynomials are chosen adaptively, using a stepwise fitting procedure.
BIAS MINIMIZATION IN GAUSSIAN PROCESS SURROGATE MODELING FOR UNCERTAINTY QUANTIFICATION
321-349
10.1615/Int.J.UncertaintyQuantification.2011003343
Vadiraj
Hombal
Vanderbilt University, Nashville, TN 37235
Sankaran
Mahadevan
Civil and Environmental Engineering Department, Vanderbilt University, Nashville, Tennessee 37235, USA
surrogate models
simulation
Gaussian processes
regression
interpolation
model error
Uncertainty quantification analyses often employ surrogate models as computationally efficient approximations of computer codes simulating the physical phenomena. The accuracy and economy in the construction of surrogate models depends on the quality and quantity of data collected from the computationally expensive system models. Computationally efficient methods for accurate surrogate model training are thus required. This paper develops a novel approach to surrogate model construction based on the hierarchical decomposition of the approximation error. The proposed algorithm employs sparse Gaussian processes on a hierarchical grid to achieve a sparse nonlinear approximation of the underlying function. In contrast to existing methods, which are based on minimizing prediction variance, the proposed approach focuses on model bias and aims to improve the quality of reconstruction represented by the model. The performance of the algorithm is compared to existing methods using several numerical examples. In the examples considered, the proposed method demonstrates significant improvement in the quality of reconstruction for the same sample size.
NUMERICAL SOLUTIONS FOR FORWARD BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS AND ZAKAI EQUATIONS
351-367
10.1615/Int.J.UncertaintyQuantification.2011003508
Feng
Bao
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849
Yanzhao
Cao
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849; School of Mathematics, Sun Yat Sun University, China
Weidong
Zhao
School of Mathematics, Shangdong University, Jinan, China
SPDEs
Zakai equation
backward SDEs
forward backward doubly stochastic differential equations
conditional expectation
The numerical solutions of decoupled forward backward doubly stochastic differential equations and the related stochastic partial differential equations (Zakai equations) are considered. Numerical algorithms are constructed using reference equations. Rate of convergence is obtained through rigorous error analysis. Numerical experiments are carried out to verify the rate of convergence results and to demonstrate the efficiency of the proposed numerical algorithms.