The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

A Priori and a Posteriori Error Analysis of Finite Element Methods for Plate Models

Jarkko Niiranen

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium E at Helsinki University of Technology (Espoo, Finland) on the 9th of November, 2007, at 12 noon.

Overview in PDF format (ISBN 978-951-22-9004-8)   [220 KB]
Dissertation is also available in print (ISBN 978-951-22-9003-1)

Abstract

The focus of this dissertation is in the theoretical and computational analysis of the discretization error indused by finite element methods for plate problems. For the Reissner–Mindlin plate model, regularity results with respect to the loading and a priori convergence estimates for the MITC finite elements are presented. The convergence results are valid uniformly with respect to the thickness parameter. In addition, we prove a local superconvergence result for the deflection approximation of MITC elements, and introduce a postprocessing method improving the accuracy of the approximation. The convergence results are confirmed by numerical computations. For the Kirchhoff–Love plate model, a new family of C0-continuous, optimally convergent finite elements is introduced. Furthermore, we derive a reliable and efficient a posteriori error indicator and verify the results by benchmark computations. Another a posteriori error analysis is performed for the Morley plate element.

This thesis consists of an overview and of the following 5 publications:

  1. Mikko Lyly, Jarkko Niiranen, Rolf Stenberg. A refined error analysis of MITC plate elements. Mathematical Models and Methods in Applied Sciences, 16 (2006), 967-977.
  2. Mikko Lyly, Jarkko Niiranen, Rolf Stenberg. Superconvergence and postprocessing of MITC plate elements. Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3110-3126.
  3. Lourenço Beirão da Veiga, Jarkko Niiranen, Rolf Stenberg. A family of C0 finite elements for Kirchhoff plates I: Error analysis. SIAM Journal on Numerical Analysis, 45 (2007), 2047-2071. © 2007 Society for Industrial and Applied Mathematics (SIAM). By permission.
  4. Lourenço Beirão da Veiga, Jarkko Niiranen, Rolf Stenberg. A family of C0 finite elements for Kirchhoff plates II: Numerical results. Helsinki University of Technology, Institute of Mathematics, Research Reports, A526 (2007), 29 pages. Computer Methods in Applied Mechanics and Engineering, accepted for publication. © 2007 by authors and © 2007 Elsevier Science. By permission.
  5. Lourenço Beirão da Veiga, Jarkko Niiranen, Rolf Stenberg. A posteriori error estimates for the Morley plate bending element. Numerische Mathematik, 106 (2007), 165-179. © 2007 Springer Science+Business Media. By permission.

Errata of publications 1, 3 and 4

Keywords: finite elements, a priori error analysis, a posteriori error analysis, plate models, MITC-elements, adaptivity

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© 2007 Helsinki University of Technology

Last update 2011-05-26