Estimation of Local Modeling Error and Goal-Oriented Adaptive Modeling of Heterogeneous Materials

Oden, J. Tinsley; Vemaganti, Kumar

doi:10.1006/jcph.2000.6585

Cited by 194 publications

(185 citation statements)

References 7 publications

(13 reference statements)

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“…Then, one can choose a coarse model simpler, for instance, from an analytical point of view (a coarse linear problem instead of a fine nonlinear one) or from a physical viewpoint (e.g., a mathematical model derived under simplifying physical hypotheses). For instance, in the elasticity framework, the most recurrent choice consists of substituting the elasticity tensor (usually a highly oscillatory function of the position) with a regularized elasticity tensor (see [27][28][29][30]). In Section 3 we specify the criterion adopted in the free-surface flows setting.…”

Section: Modeling Error Analysis For Unsteady Problemsmentioning

confidence: 99%

“…Thus, after neglecting the remainder term R, error estimates for e u and e z , in terms of computable quantities, should be found to make "operative" relation (20). This is the approach followed, for instance, in [27][28][29][30]. However estimates of this type cannot be easily derived for any differential problem.…”

Section: Modeling Error Analysis For Unsteady Problemsmentioning

confidence: 99%

“…One can make this choice a priori, moving, for instance, from physical considerations or, alternatively, driven by a suitable a posteriori modeling error estimator able to automatically detect the regions where each model can be more conveniently employed. However, if the a priori approach is rather widespread (just think about the classical domain decomposition theory [13,31,32], or the geometrical multiscale approach investigated in [14,15]), the a posteriori analysis represents a very recent area of interest, so far essentially confined to dimensionally homogeneous couplings and to the modeling of heterogeneous materials [1,[27][28][29][30]34].…”

Section: Introductionmentioning

confidence: 99%

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Adaptive modeling for free-surface flows

Perotto¹

2006

ESAIM: M2AN

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Abstract. This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such as a channel network or a river delta, by means of a suitable "combination" of different mathematical models. In this framework a wide spectrum of space and time scales is involved due to the presence of physical phenomena of different nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solve throughout the whole domain the most complex model (with solution u fine ) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional F, we aim to approximate, within a prescribed tolerance τ , the value F(u fine ) by means of the quantity F(u adapted ), u adapted being the so-called adapted solution solving the simpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving from the simplified setting where only two hydrodynamic models, fine and coarse, are considered, we provide an efficient tool able to automatically select the regions of the domain where the coarse model rather than the fine one are to be solved, while guaranteeing |F(u fine ) − F(u adapted )| below the tolerance τ . This goal is achieved via a suitable a posteriori modeling error analysis developed in the framework of a goal-oriented theory. We extend the dual-based approach provided in [Braack and Ern, Multiscale Model Sim. 1 (2003) 221-238], for steady equations to the case of a generic time-dependent problem. Then this analysis is specialized to the case we are interested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dual problems. Finally, in the last part of the paper a widespread numerical validation is carried out.Mathematics Subject Classification. 65J15, 65M15, 65M60.

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Section: Modeling Error Analysis For Unsteady Problemsmentioning

confidence: 99%

Section: Modeling Error Analysis For Unsteady Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive modeling for free-surface flows

Perotto¹

2006

ESAIM: M2AN

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“…A general theory for controling this type of modeling error in quantities of interest Q through a posteriori error estimation and adaptive modeling has been developed by Oden and Prudhomme [19,17], and Oden and Vemaganti [15,16]. Techniques for deriving a posteriori error estimates for ε approx and goal-oriented adaptive meshing have been advanced by Babuska and Strouboulis [3], Oden and Prudhomme [18,19], Rannacher, Becker and others [9].…”

Section: Coarse and Discrete Modelsmentioning

confidence: 99%

A Computational Infrastructure for Reliable Computer Simulations

Oden

Browne

Babuška

et al. 2003

Lecture Notes in Computer Science

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Abstract. The reliability of computer simulations of physical events or of engineering systems has emerged as the single most critical issue facing advancements in computational engineering and science. Without concrete and quantifiable measures of reliability, the confidence and usefulness of computer predictions are severely limited and the value of computer simulation, in general, is greatly diminished. This paper describes a mathematical and computational infrastructure for the systematic validation of computer simulations of complex physical systems and presents procedures for the integration of computer-based verification and validation processes into simulations. A general class of applications characterized by variational boundary-value problems in continuum mechanics is considered, but the approach is valid for virtually all types of models used in simulations. To simulate a particular feature or attribute of a physical event, four basic concepts are used: 1) hierarchical modeling and a posteriori estimation of modeling error, 2) a posteriori error estimation of approximation error, 3) quantification of uncertainty in the data and in the predicted response, and 4) the development of dynamic a data management paradigm, based on code composition of associated code interfaces and use of annotated source code. The result is a computational framework for computing bounds on, and estimating accuracy of, computer predictions of user-specified features of the response of physical systems.

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“…In all these works, the error is measured in terms of a userdefinable quantity of interest instead of a global norm. Goal-oriented error estimation in general is based on duality techniques and has already been used for a variety of applications such as mesh refinement in finite element methods [1,3] and control of the modeling error in homogenization [8].…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation

Arndt

Luskin

2007

Int J Mult Comp Eng

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Abstract. We give a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method, and we use this estimator to design an adaptive algorithm to compute a quantity of interest to a given tolerance by using a nearly minimal number of atomistic degrees of freedom. We present computational results that demonstrate the effectiveness of our algorithm for a periodic array of dislocations described by a Frenkel-Kontorova type model.

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Estimation of Local Modeling Error and Goal-Oriented Adaptive Modeling of Heterogeneous Materials

Cited by 194 publications

References 7 publications

Adaptive modeling for free-surface flows

Adaptive modeling for free-surface flows

A Computational Infrastructure for Reliable Computer Simulations

Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation

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