Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
1
2
2011
FAST METHOD FOR HIGH-FREQUENCY ACOUSTIC SCATTERING FROM RANDOM SCATTERERS
99-117
Dongbin
Xiu
Mathematics and Scientific Computing and Imaging (SCI) Institute, The University of Utah, Utah, USA
Paul
Tsuji
ICES, University of Texas at Austin, Austin, TX 78712
Lexing
Ying
Department of Mathematics and ICES, University of Texas at Austin, TX 78712
This paper is concerned with the uncertainty quantification of
high-frequency acoustic scattering from objects with random shape in
two-dimensional space. Several new methods are introduced to efficiently
estimate the mean and variance of the random radar cross section
in all directions. In the physical domain, the scattering
problem is solved using the boundary integral formulation and
Nyström discretization; recently developed fast algorithms are
adapted to accelerate the computation of the integral operator and
the evaluation of the radar cross section. In the random domain,
it is discovered that due to the highly oscillatory nature of the solution,
the stochastic collocation method based on sparse grids does not perform
well. For this particular problem, satisfactory results are obtained by
using quasi-Monte Carlo methods. Numerical results
are given for several test cases to illustrate the properties of the
proposed approach.
DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS
119-146
Michael S.
Eldred
Sandia National Laboratories, P. O. Box 5800, Mail Stop: 1318, Org: 01411, Albuquerque, NM 87185-1318, USA
Howard C.
Elman
Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, USA
Nonintrusive polynomial chaos expansion (PCE) and stochastic
collocation (SC) methods are attractive techniques for uncertainty
quantification due to their fast convergence properties and ability to
produce functional representations of stochastic variability. PCE
estimates coefficients for known orthogonal polynomial basis functions
based on a set of response function evaluations, using sampling,
linear regression, tensor-product quadrature, cubature, or Smolyak sparse
grid approaches. SC, on the other hand, forms interpolation functions
for known coefficients and requires the use of structured collocation
point sets derived from tensor product or sparse grids.
Once PCE or SC representations have been obtained for a response
metric of interest, analytic expressions can be derived for the
moments of the expansion and for the design derivatives of these
moments, allowing for efficient design under uncertainty formulations
involving moment control (e.g., robust design). This paper presents
two approaches for moment design sensitivities, one involving a single
response function expansion over the full range of both the design and
uncertain variables and one involving response function and derivative
expansions over only the uncertain variables for each instance of the
design variables.
These two approaches present trade-offs involving expansion
dimensionality, global versus local validity, collocation point data
requirements, and L2 (mean, variance, probability) versus
L∞ (minima, maxima) interrogation requirements. Given this
capability for analytic moments and moment sensitivities, bilevel,
sequential, and multifidelity formulations for design under
uncertainty are explored. Computational results are presented for a
set of algebraic benchmark test problems, with attention to design
formulation, stochastic expansion type, stochastic sensitivity
approach, and numerical integration method.
ERROR AND UNCERTAINTY QUANTIFICATION AND SENSITIVITY ANALYSIS IN MECHANICS COMPUTATIONAL MODELS
147-161
Sankaran
Mahadevan
Civil and Environmental Engineering Department, Vanderbilt University, Nashville, Tennessee 37235, USA
Bin
Liang
Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235
Multiple sources of errors and uncertainty arise in mechanics computational models and contribute to the uncertainty in the final model prediction. This paper develops a systematic error quantification methodology for computational models. Some types of errors are deterministic, and some are stochastic. Appropriate procedures are developed to either correct the model prediction for deterministic errors or to account for the stochastic errors through sampling. First, input error, discretization error in finite element analysis (FEA), surrogate model error, and output measurement error are considered. Next, uncertainty quantification error, which arises due to the use of sampling-based methods, is also investigated. Model form error is estimated based on the comparison of corrected model prediction against physical observations and after accounting for solution approximation errors, uncertainty quantification errors, and experimental errors (input and output). Both local and global sensitivity measures are investigated to estimate and rank the contribution of each source of error to the uncertainty in the final result. Two numerical examples are used to demonstrate the proposed methodology by considering mechanical stress analysis and fatigue crack growth analysis.
ORTHOGONAL POLYNOMIAL EXPANSIONS FOR SOLVING RANDOM EIGENVALUE PROBLEMS
163-187
Sharif
Rahman
The University of Iowa
Vaibhav
Yadav
College of Engineering and Program of Applied Mathematical and Computational Sciences, The University of Iowa
This paper examines two stochastic methods stemming from polynomial dimensional decomposition (PDD) and polynomial chaos expansion (PCE) for solving random eigenvalue problems commonly encountered in dynamics of mechanical systems. Although the infinite series from PCE and PDD are equivalent, their truncations endow contrasting dimensional structures, creating significant differences between the resulting PDD and PCE approximations in terms of accuracy, efficiency, and convergence properties. When the cooperative effects of input variables on an eigenvalue attenuate rapidly or vanish altogether, the PDD approximation commits a smaller error than does the PCE approximation for identical expansion orders. Numerical analyses of mathematical functions or simple dynamic systems reveal markedly higher convergence rates of the PDD approximation than the PCE approximation. From the comparison of computational efforts, required to estimate with the same precision the frequency distributions of dynamic systems, including a piezoelectric transducer, the PDD approximation is significantly more efficient than the PCE approximation.