Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system

We consider a conforming finite element approximation of the Reissner-Mindlin system. We propose a new robust a posteriori error estimator based on H(div) conforming finite elements and equilibrated fluxes. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests

A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate

We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Durán and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the H1 norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.Facultad de Ciencias Exacta

A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate

We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Durán and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the H1 norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.Facultad de Ciencias Exacta

Adaptive modeling of plate structures

V disertaciji se ukvarjamo z različnimi vidiki modeliranja ploskovnih konstrukcij s končnimi elementi. Modeliranje plošč je nekoliko specifično in je zaradi kompleksnosti in pojavov, ki jih opisuje, bistveno prispevalo k razvoju same metode končnih elementov. Danes je na voljo več uveljavljenih modelov plošč in pripadajočih končnih elementov, ki uporabniku nudijo široko množico možnosti, iz katere lahko izbira. Prav široka možnost izbire predstavlja tudi največjo težavo, saj je težje določiti, kateri model je primernejši in tudi, katera mreža končnih elementov je za dan problem optimalna. Glavni cilj disertacije je raziskati ključne korake v procesu prilagodljivega modeliranja plošč, ki omogoča samodejno določitev optimalnega modela za dan problem. Ker je prilagodljivo modeliranje odvisno od zanesljivih ocen napak, je večji del disertacije posvečen metodam za izračun diskretizacijske in modelske napake. Na praktičnih primerih smo preučili nekaj najbolj uveljavljenih metod za oceno napake. V nasprotju z ocenami napake diskretizacije, je modelsko napako mnogo težje določiti. Posebna pozornost je bila zato namenjena metodi uravnoteženja rezidualov, ki ima potencial tudi na področju ocene modelske napake. V tem smislu to delo predstavlja pomemben prispevek k področju računanja modelske napake za plošče. Koncept prilagodljivega modeliranja ploskovnih konstrukcij je bil preskušen na hierarhični družini končnih elementov za plošče - od tankih plošč do modelov višjega reda, ki upoštevajo deformacije po debelini. Ravno dobro vzpostavljena hierarhija v družini končnih elementov se je pokazala za ključno pri zanesljivi oceni modelske napake. Prilagodljivo modeliranje ploskovnih konstrukcije je bilo preskušeno na nekaj zahtevnejših primerih. Območje je bilo najprej modeliranjo z najbolj grobim modelom na sorazmerno redki mreži. Z uporabo informacije o napaki začetnega izračuna je bil zgrajen nov model. Primerjava izračuna na novem modelu z začetnim računom je pokazala, da je predlagan način prilagodljivega modeliranja sposoben nadzorovati porazdelitev napake, kakor tudi zajeti vse pomembnejše po- jave, ki so značilni za modeliranje plošč.The thesis deals with adaptive finite element modeling of plate structures. The finite element modeling of plates has grown to a mature research topic, which has contributed significantly to the development of the finite element method for structural analysis due to its complexity and inherently specific issues. At present, several validated plate models and corresponding families of working and efficient finite elements are available, offering a sound basis for an engineer to choose from. In our opinion, the main problems in the finite modeling of plates are nowadays related to the adaptive modeling. Adaptive modeling should reach much beyond standard discretization (finite element mesh) error estimates and related mesh (discretization) adaptivity. It should also include model error estimates and model adaptivity, which should provide the most appropriate mathematical model for a specific region of a structure. Thus in this work we study adaptive modeling for the case of plates. The primary goal of the thesis is to provide some answers to the questions related to the key steps in the process of adaptive modeling of plates. Since the adaptivity depends on reliable error estimates, a large part of the thesis is related to the derivation of computational procedures for discretization error estimates as well as model error estimates. A practical comparison of some of the established discretization error estimates is made. Special attention is paid to what is called equilibrated residuum method, which has a potential to be used both for discretization error and model error estimates. It should be emphasized that the model error estimates are quite hard to obtain, in contrast to the discretization error estimates. The concept of model adaptivity for plates is in this work implemented on the basis of equilibrated residuum method and hierarchic family of plate finite element models. The finite elements used in the thesis range from thin plate finite elements to thick plate finite elements. The latter are based on a newly derived higher order plate theory, which includes through the thickness stretching. The model error is estimated by local element-wise compu- tations. As all the finite elements, representing the chosen plate mathematical models, are re-derived in order to share the same interpolation bases, the difference between the local com- putations can be attributed mainly to the model error. This choice of finite elements enables effective computation of the model error estimate and improves the robustness of the adaptive modeling. Thus the discretization error can be computed by an independent procedure. Many numerical examples are provided as an illustration of performance of the derived plate elements, the derived discretization error procedures and the derived modeling error procedure. Since the basic goal of modeling in engineering is to produce an effective model, which will produce the most accurate results with the minimum input data, the need for the adaptive modeling will always be present. In this view, the present work is a contribution to the final goal of the finite element modeling of plate structures: a fully automatic adaptive procedure for the construction of an optimal computational model (an optimal finite element mesh and an optimal choice of a plate model for each element of the mesh) for a given plate structure. vii

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

Robust residual a posteriori error estimators for the Reissner-Mindlin eigenvalues system

International audienceWe consider a conforming finite element approximation of the Reissner-Mindlin eigenvalue system, for which a robust a posteriori error estimator for the eigenvector and the eigenvalue errors is proposed. For that purpose, we first perform a robust a priori error analysis without strong regularity assumption. Upper and lower bounds are then obtained up to higher order terms that are super convergent, provided that the eigenvalue is simple. The convergence rate of the proposed estimator is confirmed by a numerical test

Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy