A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

Acharjee, Swagato; Zabaras, Nicholas

doi:10.1002/nme.1611

Cited by 22 publications

(15 citation statements)

References 38 publications

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“…Its horizon has also been vastly extended in various aspects, e.g. the Wiener-Askey scheme for non-Gaussian random variables [24] and sparse grid-based stochastic collocation methods [28] and model reduction [29] for mitigating the curse of dimensionality. For the numerical analysis of the SSFEM for solving elliptic problems, we refer to the recent works [25,18].…”

Section: Spectral Stochastic Finite Element Methodsmentioning

confidence: 99%

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Inversion of Robin coefficient by a spectral stochastic finite element approach

Jin

Zou

2008

Journal of Computational Physics

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This paper investigates a variational approach to the nonlinear stochastic inverse problem of probabilistically calibrating the Robin coefficient from boundary measurements for the steady-state heat conduction. The problem is formulated into an optimization problem, and mathematical properties relevant to its numerical computations are investigated. The spectral stochastic finite element method using polynomial chaos is utilized for the discretization of the optimization problem, and its convergence is analyzed. The nonlinear conjugate gradient method is derived for the optimization system. Numerical results for several two-dimensional problems are presented to illustrate the accuracy and efficiency of the stochastic finite element method.

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Section: Spectral Stochastic Finite Element Methodsmentioning

confidence: 99%

“…Recent algorithm innovations of the SFEM, e.g. the sparse grid technique [28,37], model reduction [29,38] and the adaptive multi-element technique [39], may provide viable means for accelerating the algorithm.…”

mentioning

confidence: 99%

Inversion of Robin coefficient by a spectral stochastic finite element approach

Jin

Zou

2008

Journal of Computational Physics

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“…A 9x9 grid was used for computing the statistics. The mean underfill was estimated to be 0.046976mm 3 with a standard deviation of 0.0022mm 3 . Using a 10xlO support space grid the mean underfill was found to be 0.046187mm 3 with a standard deviation of 0.…”

Section: < Xw•y(w >= Fe Xyf (C)dc (230)mentioning

confidence: 97%

“…Development of spectral stochastic finite element method and non-intrusive stochastic Galerkin method for robust modeling and design of deformation processes [1,2,3] 2. Development of multi-scale sensitivity analysis for designing microstructure-sensitive properties in deformation processes [4,5,6].…”

Section: Status Of Effortmentioning

confidence: 99%

Robust Multi-Length Scale Deformation Process Design for the Control of Microstructure-Sensitive Material Properties

Zabaras¹

2007

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information , 1215 Jefferson Davis Highway, Suite 1204. Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YYYY)2 SUPPLEMENTARY NOTES14. ABSTRACT The objective of this work was to develop a robust design methodology for optimizing microstructure-sensitive properties in aircraft components manufactured using metal forming processes. The multi-scale forming design simulator developed provides means to select the sequence of deformation processes, design the dies and preforms for each process stage as well as the process conditions such that a product is obtained with desired shape and microstructure. Modeling of uncertainty propagation in such multi-scale models of deformation is extremely complex considering the nonlinear coupled phenomena that need to be accounted for. The work addresses key mathematical and computational issues related to robust multi-scale design of deformation processes. Our research accomplishments include development of new mathematical models based on spectral polynomial chaos, support space, and entropy maximization techniques for modeling sources of uncertainties in material deformation processes. These models, in conjunction with multi-scale homogenization models, allow simulations of the effect of microstructural variability on the reliability of macro-scale systems. We have developed the first stochastic variational multi-scale simulator with an explicit sub-grid model, a robust deformation process simulator using spectral and collocation methods for simulating uncertainties in metal forming processes. Finally, recent developments including an information theoretic framework for modeling microstructural uncertainties is summarized. EXECUTIVE SUMMARYThe objective of this work is to develop a robust design methodology for optimizing microstructuresensitive properties in aircraft components manufactured using metal forming processes. The multi-scale forming design simulator developed as a part of this project provides means to select the sequence of deformation processes, design the dies and preforms for each process stage as well as the process conditions such that a product is obtained with desired shape and microstructure. Modeling of uncertainty propagation in such multi-scale models of deformation is ex...

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“…When the exchanged information has a low effective stochastic dimension, this representation allows a reduction in the number of requisite stochastic DOF to be achieved while maintaining accuracy, thus paving the way for a solution in a reduced‐dimensional space, which in turn reduces the computational cost. It should be noted that in references , the integration of dimension–reduction techniques in algorithms for solving stochastic partial differential equations has already been demonstrated; however, this paper contributes by highlighting the role that dimension–reduction techniques can play in solving coupled problems.…”

Section: Introductionmentioning

confidence: 94%

Dimension reduction in stochastic modeling of coupled problems

Arnst

Ghanem

Phipps

et al. 2012

Numerical Meth Engineering

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SUMMARY Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower dimensional space than the sources themselves. This work thus presents an investigation into the characterization of the exchanged information by a reduced‐dimensional representation and in particular by an adaptation of the Karhunen‐Loève decomposition. The effectiveness of the proposed dimension–reduction methodology is analyzed and demonstrated through a multiphysics problem relevant to nuclear engineering. Copyright © 2012 John Wiley & Sons, Ltd.

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A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

Cited by 22 publications

References 38 publications

Inversion of Robin coefficient by a spectral stochastic finite element approach

Inversion of Robin coefficient by a spectral stochastic finite element approach

Robust Multi-Length Scale Deformation Process Design for the Control of Microstructure-Sensitive Material Properties

Dimension reduction in stochastic modeling of coupled problems

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