Hierarchic models for laminated plates and shells

Actis, Ricardo L.; Szabó, Barna; Schwab, Christoph

doi:10.1016/s0045-7825(98)00226-6

Cited by 76 publications

(33 citation statements)

References 12 publications

(17 reference statements)

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“…to the cheapest one). In practice, at each time t j , according to the information provided by the modeling error estimator we are looking for, we compute the solution of an intermediate primal problem (1) with the trickier part d(u α )(ψ) "active" only in the subdomains Q i of Q where α(x, t j ) = 1, with x ∈ Ω. Notice that, even if we get rid of the semilinear form d(u α )(ψ) in some areas of the domain, we are neither changing the differential nature of problem (1) nor the associated boundary conditions.…”

Section: Modeling Error Analysis For Unsteady Problemsmentioning

confidence: 99%

“…The corresponding variational formulation is given by (1) with α = 1, and reads as: find u 1 ∈ V such that…”

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%

“…The corresponding variational formulation is given by (1) with α = 1, and reads as: find u 1 ∈ V such that…”

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 byproduct of these two issues is the necessity of estimating the error due to the coupling of the different models. Such estimates are usually called modeling error estimates and go back to the early works of Oden and coworkers [32,29,28,31] for heterogeneous materials, Stein and Ohnimus [30] for Solid Mechanics, and Actis, Szabo and Schwab [1] for laminated plates and shells. In [5], Braack and Ern develop a posteriori error estimators in a general setting in order to equilibrate the modeling error with the numerical error for a global adaptive method.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of this paper is to perform an error analysis between the solution (u a , v a ) of the model adaptation and the solution (u f , v f ) of the fine model (1) in order to fix the different parameters of the model adaptation algorithm (the time step ∆t a , the threshold Σ, and the buffer size δ) and obtain an error estimate between (u a , v a ) and (u f , v f ). Due to the possibility of exactly computing v f in the fine model (1), the different models rely on scalar conservation laws of the form…”

Section: Introductionmentioning

confidence: 99%

Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Cancès

Coquel

Godlewski

et al. 2016

Communications in Mathematical Sciences

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Abstract. We propose a dynamic model adaptation method for a nonlinear conservation law coupled with an ordinary differential equation. This model, called the fine model, involves a small time scale and setting this time scale to 0 leads to a classical conservation law, called the coarse model, with a flux which depends on the unknown and on space and time. The dynamic model adaptation consists in detecting the regions where the fine model can be replaced by the coarse one in an automatic way, without deteriorating the accuracy of the result. To do so, we provide an error estimate between the solution of the fine model and the solution of the adaptive method, enabling a sharp control of the different parameters. This estimate rests upon stability results for conservation laws with respect to the flux function. Numerical results are presented at the end and show that our estimate is optimal.

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Thep‐Version of the Finite Element Method

Szabó

Düster

Rank

2004

Encyclopedia of Computational Mechanics

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The basic algorithmic structure and performance characteristics of the p ‐version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p ‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that p ‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any data of interest. Given the growing demand for verified numerical solutions, the p ‐version is expected to play an increasingly important role in the development of future finite element software products.

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Hierarchic models for laminated plates and shells

Cited by 76 publications

References 12 publications

Adaptive modeling for free-surface flows

Adaptive modeling for free-surface flows

Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Thep‐Version of the Finite Element Method

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