Discrete Korn’s inequality for shells

Zhang, Sheng

doi:10.1007/s10092-016-0185-0

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…The equivalence constant C could be dependent on the shell midsurface and shape regularity K of the triangulation T h , but otherwise independent of the triangulation. A proof of this result can be found in [20].…”

Section: A Discrete Korn's Inequality For Naghdi Shellmentioning

confidence: 77%

A linear finite element procedure for the Naghdi shell model

Zhang¹

2014

Preprint

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We prove the accuracy of a mixed finite element method for bending dominated shells in which a major part of the membrane/shear strain is reduced, to free up membrane/shear locking. When no part of the membrane/shear strain is reduced, the method becomes a consistent discontinuous Galerkin method that is proven accurate for membrane/shear dominated shells and intermediate shells. The two methods can be coded in a single program by using a parameter. We propose a procedure of numerically detecting the asymptotic behavior of a shell, choosing the parameter value in the method, and producing accurate approximation for a given shell problem. The method uses piecewise linear functions to approximate all the variables. The analysis is carried out for shells whose middle surfaces have the most general geometries, which shows that the method has the optimal order of accuracy for general shells and the accuracy is robust with respect to the shell thickness. In the particular case that the geometrical coefficients of the shell middle surface are piecewise constants the accuracy is uniform with respect to the shell thickness.

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Section: A Discrete Korn's Inequality For Naghdi Shellmentioning

confidence: 77%

A linear finite element procedure for the Naghdi shell model

Zhang¹

2014

Preprint

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“…In order to obtain the desired generalized inequalities, we will follow the approach of Brenner and coworkers [6,8,5]. We note that other researchers have performed extensive work on constructing discrete standard Korn's inequalities, including Knobloch and Tobiska [27], Attia and Starke [3], Mardal and Winther [30], Zhang [35], and Lee [29]; however, we choose to primarily focus on the work of Brenner in this paper. The interested reader is encouraged to consult [22] for a compact review of the other work on this topic.…”

Section: Introductionmentioning

confidence: 99%

Generalized Korn's Inequalities for Piecewise $H^2$ Vector Fields

Williams¹,

Hong²

2022

Preprint

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The purpose of this paper is to construct a new class of discrete generalized Korn's inequalities for piecewise H 2 vector fields in threedimensional space. The resulting Korn's inequalities are different from the standard Korn's inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. It is anticipated that the new generalized Korn's inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.

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Generalized Korn’s inequalities for piecewise 𝐻¹ and 𝐻² vector fields

Williams,

Hong

2024

Math. Comp.

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The purpose of this paper is to construct a new class of discrete generalized Korn’s inequalities for piecewise H 1 H^1 vector fields and piecewise H 2 H^2 vector fields in three-dimensional space. The resulting Korn’s inequalities are different from the standard Korn’s inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. It is anticipated that the new generalized Korn’s inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.

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Discrete Korn’s inequality for shells

Abstract: We prove discrete Korn's inequalities for Naghdi and Koiter shell models, which are applicable to discontinuous piecewise functions. They are useful in study of discontinuous finite element methods for shells.

Cited by 3 publications

References 18 publications

A linear finite element procedure for the Naghdi shell model

A linear finite element procedure for the Naghdi shell model

Generalized Korn's Inequalities for Piecewise $H^2$ Vector Fields

Generalized Korn’s inequalities for piecewise 𝐻¹ and 𝐻² vector fields

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