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Adaptive finite element strategies based on error assessment
10 Dec 1999-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 46, Iss: 10, pp 1803-1818
TL;DR: Two main ingredients are needed for adaptive finite element computations: the error of a given solution must be assessed, and a new spatial discretization must be defined via h-, p- or r-adaptivity.
Abstract: Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h-, p- or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics. Copyright © 1999 John Wiley & Sons, Ltd.
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15 Nov 2004
TL;DR: In this paper, the authors provide an in-depth survey of arbitrary Lagrangian-Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details.
Abstract: The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior.
Keywords:
ALE description;
convective transport;
finite elements;
stabilization techniques;
mesh regularization and adaptation;
fluid dynamics;
nonlinear solid mechanics;
stress-update procedures;
fluid–structure interaction
901 citations
Cites background or methods from "Adaptive finite element strategies ..."
...Studies on the use of ALE as a mesh adaptation technique in solid mechanics are reported, among others, by Pijaudier-Cabot et al. (1995), Huerta et al. (1999), Askes and Sluys (2000), Askes and Rodŕıguez-Ferran (2001), Askes et al. (2001) and Rodŕıguez-Ferran et al. (2002)....
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...The smoothed field ¤̃ can be obtained, for instance, by least-squares approximation, see Huétink et al. (1990). Another possibility is to retain the discontinuous field ¤ and devise appropriate algorithms that account for this fact....
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...Similarly, the crack propagation problems discussed by Koh and Haber (1986) and Koh et al. (1988), where the crack path is known a priori, also allow the use of this kind of mesh update procedure....
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...Similarly, the crack propagation problems discussed by Koh and Haber (1986) and Koh et al. (1988), where the crack path is known a priori, also allow the use of this kind of mesh update procedure. Other examples of prescribed mesh motion in nonlinear solid mechanics can be found in the works by Liu et al. (1986), Huétink et al....
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TL;DR: In this article, a method for characterisation of materials subjected to large strains beyond the levels when plastic instability occurs in standard tension tests is presented, where thin sheets of two types of hot-rolle...
248 citations
TL;DR: In this article, the application of the finite element method to predict the thermal, material and mechanical effects of welding is described, and some recent applications are reviewed and future developments are discussed.
241 citations
TL;DR: The basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem are reviewed and it is concluded that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care.
207 citations
TL;DR: In this paper, a mixed hierarchical approximation based on finite elements and meshless methods is presented, which couples regions where finite elements or meshless method are used to interpolate: continuity and consistency is preserved.
Abstract: A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.
199 citations
References
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TL;DR: It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error.
Abstract: This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283---301 (1985)].
411 citations
TL;DR: A very much improved recovery process yielding superconvergent values throughout the domain, named the superconversgent patch recovery, is presented with test results showing its efficiency.
382 citations
TL;DR: In this paper, an Arbitrary Lagrangian-Eulerian (ALE) finite element method for the simulation of fluid domains with moving structures is described, where the fluid is viscous, incompressible and unsteady and the fluid motion is solved by a fractional step discretization of the Navier-Stokes equations.
Abstract: This paper describes an Arbitrary Lagrangian- Eulerian (ALE) finite element method for the simulation of fluid domains with moving structures. The fluid is viscous, incompressible and unsteady and the fluid motion is solved by a fractional step discretization of the Navier-Stokes equations. The emphasis is on convection dominated flows, and a three-step method is used for the convection term. The moving structure causes the mesh of the fluid domain to move, and a new algorithm is proposed to solve the important and crucial problem of the calculation of the mesh velocities. Numerical calculations of the added mass and added damping of a vibrating two-dimensional circular cylinder in the frequency Reynolds number range Re
w
=20−2000 are performed to evaluate the proposed ALE finite element method. The numerically calculated added mass and added damping are compared to both analytical and numerical results. To further demonstrate the generality of the method, a numerical simulation of flow past an oscillating schematic sports car is presented.
343 citations
TL;DR: In this article, an arbitrary Lagrangian-Eulerian (ALE) Petrov-Galerkin finite element technique is developed to study nonlinear viscous fluids under large free surface wave motion.
Abstract: An arbitrary Lagrangian-Eulerian (ALE) Petrov-Galerkin finite element technique is developed to study nonlinear viscous fluids under large free surface wave motion. A review of the kinematics and field equations from an arbitrary reference is presented and since the major challenge of the ALE description lies in the mesh rezoning algorithm, various methods, including a new mixed formulation, are developed to update the mesh and map the moving domain in a more rational manner. Moreover, the streamline-upwind/Petrov-Galerkin formulation is implemented to accurately describe highly convective free surface flows. The effectiveness of the algorithm is demonstrated on a tsunami problem, the dam-break problem where the Reynolds number is taken as high as 3000, and a large-amplitude sloshing problem.
325 citations
TL;DR: In this article, a simple Lagrangian-Eulerian formulation of finite element programs is presented, where an operator split separates the Lagrangians and Eulerian processes, allowing a finite element program to be extended to this formulation with little difficulty.
Abstract: A simple arbitrary Lagrangian-Eulerian formulation is presented. An operator split separates the Lagrangian and Eulerian processes, allowing a Lagrangian finite element program to be extended to this formulation with little difficulty. Example problems illustrate the strengths and weaknesses of the formulation.
305 citations