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Gruais Asymptotics of arbitrary order for a thin elastic clamped plate, I: Optimal error estimates. Asymptotic Analysis 13
, 1996
"... Abstract. This paper is the first of a series of two, where we study the asymptotics of the displacement in a thin clamped plate made of a rigid “monoclinic ” material, as the thickness of the plate tends to 0. The combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz ..."
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Abstract. This paper is the first of a series of two, where we study the asymptotics of the displacement in a thin clamped plate made of a rigid “monoclinic ” material, as the thickness of the plate tends to 0. The combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multiscale asymptotics of the displacement and leads to optimal error estimates in energy norm. We investigate the polynomial Ansatz in Part I, and the boundary layer Ansatz in Part II. If ε denotes the small parameter in the geometry, we first construct the algorithm for an infinite “even ” Ansatz involving only even powers of ε, which is a natural extension of the usual KirchhoffLove Ansatz. The boundary conditions of the clamped plate being only satisfied at the order 0, we try to compensate for them by boundary layer terms: we rely on a result proved in Part II giving necessary and sufficient conditions for the exponential decay of such terms. In order to fulfill these conditions, the constructive algorithm for the boundary layer terms has to be combined with an “odd ” polynomial Ansatz. The outcome is a twoscale asymptotics involving all non
Asymptotic Consistency Of The Polynomial Approximation In The Linearized Plate Theory. Application To The ReissnerMindlin Model
, 1997
"... . We establish a partial link between two standard methods for deriving plate models from linearized threedimensional elasticity: the asymptotic method, known to justify the KirchhoffLove model, and the polynomial reduction method. In the polynomial method, the reduced model is obtained by project ..."
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Cited by 9 (0 self)
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. We establish a partial link between two standard methods for deriving plate models from linearized threedimensional elasticity: the asymptotic method, known to justify the KirchhoffLove model, and the polynomial reduction method. In the polynomial method, the reduced model is obtained by projecting the threedimensional displacement on a closed subspace of admissible displacements, namely displacements that are polynomial with respect to the thickness variable. Our procedure characterizes minimal polynomial subspaces that are consistent with the KirchhoffLove model. In the same time, if a singular perturbation term is dropped in the equations of the lower degree model, we recover a ReissnerMindlin model. Key words: Plates, asymptotic method, projection method, polynomial basis, Reissner, Mindlin Mathematics subject classification: 73K10, 73V25, 35B25 1. Introduction Slender structures such as plates, shells, rods are of high technological interest. Numerous works have been devo...
Plates and shells: Asymptotic expansions and hierarchical models, Encyclopedia of Computational Mechanics. Edited by Erwin Stein, René de Borst and Thomas J.R. Hughes
, 2004
"... Concerning thin structures such as plates and shells, the idea of reducing the equations of elasticity to twodimensional models defined on the midsurface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the midsurface ..."
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Concerning thin structures such as plates and shells, the idea of reducing the equations of elasticity to twodimensional models defined on the midsurface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the midsurface. As nowadays, the asymptotic expansion of the displacement solution of the threedimensional linear model is fully known at least for plates and clamped elliptic shells, we start from a description of these expansions in order to introduce the twodimensional models known as hierarchical models: These models extend the classical models, and presuppose the displacement to be polynomial in the thickness variable, transverse to the midsurface. Because of the singularly perturbed character of the elasticity problem as the thickness approaches zero, boundary or internal layers may appear in the displacements and stresses, and so may numerical locking effects. The use of hierarchical models, discretized by higher degree polynomials (pversion of finite elements) may help to overcome these severe difficulties.
Investigation of twodimensional models of elastic prismatic shell
 GEORGIAN MATHEMATICAL JOURNAL
, 2003
"... Statical and dynamical twodimensional models of a prismatic elastic shell are constructed. The existence and uniqueness of solutions of the corresponding boundary and initial boundary value problems are proved, the rate of approximation of the solution of a threedimensional problem by the vecto ..."
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Cited by 5 (1 self)
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Statical and dynamical twodimensional models of a prismatic elastic shell are constructed. The existence and uniqueness of solutions of the corresponding boundary and initial boundary value problems are proved, the rate of approximation of the solution of a threedimensional problem by the vectorfunction restored from the solution of a twodimensional one is estimated.
Analysis of membrane locking in hp FEM for a cylindrical shell
"... In this paper we analyze the performance of the hp\GammaFinite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that boundary layers stemming from the edge effect are resolved. The numerical results illust ..."
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Cited by 2 (1 self)
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In this paper we analyze the performance of the hp\GammaFinite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that boundary layers stemming from the edge effect are resolved. The numerical results illustrate the mesh independence of the exponential convergence of the hp\GammaFEM. Mathematics Subject Classification (1991): Primary: 65N30, 73K15; Secondary: 65M70, 65L70. 1 Introduction The largest class of problems that are numerically solved in industrial FE analyses in the U.S. are linear, static problems for solids, and among these, again a major part is taken by thin structures, such as beams, plates and shells [3]. Owing to the singular perturbation character of these problems, it was found early [8] that conventional FEM do not converge satisfactorily due to locking effects. A large body of literature has been devoted to this phenomenon and to possible remedies. In the case of plate ...
On the investigation of static hierarchic model for elastic rods
 Appl. Math. Inf. Mech
"... In the present paper static onedimensional hierarchical model for elastic cusped rod is constructed. The corresponding boundary value problem is studied and the uniqueness and existence of its solution in suitable weighted Sobolev spaces is proved. The convergence of the sequence of approximate sol ..."
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Cited by 2 (2 self)
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In the present paper static onedimensional hierarchical model for elastic cusped rod is constructed. The corresponding boundary value problem is studied and the uniqueness and existence of its solution in suitable weighted Sobolev spaces is proved. The convergence of the sequence of approximate solutions restored from the solutions of onedimensional problems to the solution of original threedimensional problem is proved and under regularity conditions the rate of approximation is estimated. Key words and phrases: Mathematical modelling of linearly elastic cusped rods, a priori error estimation. AMS subject classification: 74K10, 65N30, 42C10. The construction and the intensive investigation of the lowerdimensional mathematical models of bodies with negligible thickness or width in comparison with the other geometric dimensions arise with the wide use of structures of such type in the practice ([1, 2]). One of the methods of constructing hierarchic models for elastic prismatic shells was proposed by
Some degenerate elliptic systems and applications to . . .
, 2004
"... The tensioncompression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given. ..."
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The tensioncompression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given.
Error Estimate for Elliptic Problems with Complicated Interfaces
, 2012
"... These lecture notes comprise the talks of the author on “A posteriori error estimates for modelling errors ” and on “A posteriori error estimates for highly indefinite problems” given at the Zürich Summerschool 2012. ..."
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These lecture notes comprise the talks of the author on “A posteriori error estimates for modelling errors ” and on “A posteriori error estimates for highly indefinite problems” given at the Zürich Summerschool 2012.