Modeling error and adaptivity in nonlinear continuum mechanics

Oden, J. T.; Prudhomme, Serge; Hammerand, Daniel C.; Kuczma, Mieczysław

doi:10.1016/s0045-7825(01)00256-0

Cited by 39 publications

(26 citation statements)

References 8 publications

(1 reference statement)

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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%

“…A third approach, alternative to the ones proposed in [9,[27][28][29][30], to make the quantity (I) + (II) in (20) computable, is now under investigation, even if it sounds not generalizable in a straightforward way to any differential problem, as well as the analysis in [27][28][29][30] (see also Sect. 6).…”

Section: Remark 25mentioning

confidence: 99%

“…i.e., to finding the solution U α ∈ W of the adapted primal problem (30) together with the solution Z α = (w α , κ α ) ∈ W of the associated dual problem: find Z α ∈ W such that…”

Section: Remark 32mentioning

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: Remark 25mentioning

confidence: 99%

Section: Remark 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive modeling for free-surface flows

Perotto¹

2006

ESAIM: M2AN

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Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Stein¹,

Rüter²

2004

Encyclopedia of Computational Mechanics

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The essential topics of the finite element method for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical point of view. As a starting point, the nonlinear, and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the finite element method in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated finite element methods. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

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Finite Element Methods for Elasticity with Error‐controlled Discretization and Model Adaptivity

Stein

Rüter

2004

Encyclopedia of Computational Mechanics

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The essential topics of the finite element method (FEM) for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical points of view. As a starting point, the nonlinear and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the FEM in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated FEMs. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

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Modeling error and adaptivity in nonlinear continuum mechanics

Cited by 39 publications

References 8 publications

Adaptive modeling for free-surface flows

Adaptive modeling for free-surface flows

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Finite Element Methods for Elasticity with Error‐controlled Discretization and Model Adaptivity

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