Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

Ohnimus, S.; Stein, Erwin; Walhorn, E.

doi:10.1002/nme.228

Cited by 68 publications

(46 citation statements)

References 15 publications

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“…An adaptive mesh generation scheme attempts to automate the mesh generation and in the scheme, accuracy of error estimation of a given mesh is essential (Hwang, 1988;McFee and Giannacopoulos, 2001;Ohnimus et al, 2001;Stampfle et al, 2001;Yoon, 2005;Zhu et al, 1991). In general engineering problems, accurate solutions are not known and thus accurate error estimation is inherently a difficult problem.…”

Section: Error Estimation With Representative Strainsmentioning

confidence: 99%

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

Yoon¹

2012

Journal of Korean Society of Hazard Mitigation

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In structural design and analysis used for hazard mitigation systems, automation is becoming an essential feature. For analysis, the finite element method has proven to be an effective approximate method if proper element types and meshes are chosen. Recently, the method has been successfully applied to solve complex dynamic and nonlinear problems; and with a properly chosen element type and mesh, reliable results have been obtained. However, in automation and in complex analyses of a structures, using the initial mesh throughout the analysis may involve some elements to go through strains beyond the elements' reliable limits. Thus, the finite element mesh for these types of analyses must be dynamically adaptive, and considering the rapid process of analysis in real time, the dynamically adaptive finite element mesh generating schemes must be computationally efficient. In this paper, a computationally efficient dynamically adaptive finite element mesh generation scheme for dynamic analyses of planar problems is described. The concept of representative strain value is used for error estimates and the refinements of meshes use combinations of the h-method (node movement) and the r-method (element division). A coefficient that depends on shape of elements is used to correct overly distorted elements. The validity of the scheme is shown by a deep cantilever beam example under a dynamic concentrated load. The example shows reasonable accuracy and efficient computing time. Furthermore, the study shows the potential for the scheme's effective use in complex structural problems such as those under severe environmental hazards such as seismic or erratic wind loads.

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Section: Error Estimation With Representative Strainsmentioning

confidence: 99%

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

Yoon¹

2012

Journal of Korean Society of Hazard Mitigation

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“…We refer the reader to References [11][12][13][14] for a detailed description of this approach, and to [16,[28][29][30][31] for the extension to non-linear quantities of interest. For the sake of clarity we will omit the reference to the space of the random parameters Y.…”

Section: Estimation Of the Outputs Of Interestmentioning

confidence: 99%

“…The development of error estimators for such quantities was initiated in the 1980s [7,8]. Recently, numerous works have been published that provide error estimates and bounds for several quantities of interest [9][10][11][12][13][14][15][16]. However, these works have been developed for deterministic FEA and little research has been carried out on the verification of probabilistic models [17,18].…”

Section: Introductionmentioning

confidence: 99%

Error bounds for the reliability index in finite element reliability analysis

Gallimard¹

2011

Numerical Meth Engineering

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SUMMARYThis work presents an extension of the goal-oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first-order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index . The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented.

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“…Indeed methods based on the use of SA-field, like the error in constitutive equation (Ladevèze, 1975;Ladevèze and Leguillon, 1983) or like the equilibrated residuals (Babuška and Rheinboldt, 1978;Babuška and Miller, 1987), provide efficient error estimators, for global or local quantities (Ladevèze and Moës, 1999;Prudhomme and Oden, 1999;Ohnimus et al, 2001;Becker and Rannacher, 2001), even in some nonlinear contexts (Babuška and Rheinboldt, 1982;Ladevèze et al, 1986;Ladevèze, 2001;Louf et al, 2003) and domain decomposition methods (Parret-Fréaud et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

Rey¹,

Gosselet²,

Rey³

2014

Computer Methods in Applied Mechanics and Engineering

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This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward.

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Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

Cited by 68 publications

References 15 publications

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

Error bounds for the reliability index in finite element reliability analysis

Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

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