Error-controlled Adaptive Finite Elements in Solid Mechanics

Finite Element Methods are used for numerous engineering applications where numerical solutions of partial differential equations are needed. As computers can now deal with the millions of parameters used in these methods, automatic error estimation and automatic adaptation of the utilised method (according to this error estimation), has become a hot research topic.

This text offers comprehensive coverage of this new field of automatic adaptation and error estimation, bringing together the work of eight outstanding researchers in this field who have completed a six year national research project within the German Science Foundation. The result is a state-of-the-art work in true reference style. Each chapter is self-contained and covers theoretical, algorithmic and software presentations as well as solved problems. A main feature consists of several carefully elaborated benchmarks of 2D- and 3D- applications.

• First book to go beyond the Finite Element Method in itself
• Covers material from a new research area
• Presents benchmarks of 2D- and 3D- applications
• Fits with the new trend for genetic strategies in engineering

Zur numerischen Lösung partieller Differentialgleichungen greift man in der Technik häufig auf finite Elemente zurück. Seitdem Computer mit Millionen benötigter Variabler zurechtkommen, ist es zunehmend wichtig, Fehler automatisch abzuschätzen und die Methoden entsprechend anzupassen. Neun Wissenschaftler haben sich in einem sechsjährigen Forschungsprojekt der DFG mit diesem Thema befasst und geben in diesem Band ihre Erfahrungen an weiter. Jedes Kapitel ist in sich geschlossen mit theoretischen und algorithmischen Informationen, Angaben zur Software und durchgerechneten Übungsaufgaben. Besonders interessant sind mehrere sorgfältig erarbeitete Benchmarks von 2D- und 3D-Anwendungen. Preface.

Introduction (Stein).

Error Estimation and Adaptive Mesh Design for FE Models in Elasto-Plasticity Theory (Rannacher and Suttmeier).

Adaptive FEM for elasto-plastic Deformations (Stein and Schmidt).

Numerical Simulation of Localization Phenomena in Geomechanics by Entended Continuum Formulations (Wunderlich et al).

Adaptive Methods for Contact Problems (Rieger et al).

An Adaptive Boundary Element Method for Contact Problems (Eck et al).

Goal-Oriented Error Estimation in Solid Mechanics (Steeb,Maute and Ramm).

The p-version of the Finite Element Method for Structural Problems (Rank et al).

Adaptive Analysis of Plate and Shell Structures Under Transient Loading (Schweizerhof, Neumann and Riccius).

The Application of Adaptive Parallel Multigrid Methods to Problems in Nonlinear Solid Mechanics (Lang, Wieners and Wittum).

Benchmarks (Stein et al).

Erwin Stein, PhD is Professor Emeritus, Curator of the Leibniz Exhibition, Stein Institut für Baumechanik und Numerische Mechanik, Leibniz Universität, Hannover, Germany. Institute of Mechanics and Computational Mechanics.

This book contains the numerical methods, mathematical analysis, algorithms, software aspects and comparative results of a six year joint project of nine German University institutes, supported by the German National Science Foundation (DFG).

It covers new progress for efficient and reliable error estimates mainly of the Finite Element Method for non-linear problems in solid mechanics, as well as efficient re-meshing techniques and solution algorithms.

Topics covered include:
* Elastoplastic deformations and contact problems

* Shear localization of inelastic deformations

* Thin-walled structures

* Elastodynamics
The authors, all experts in computational mechanics, provide a wealth of new ideas for the next generation of software for adaptive finite elements, which will be of great interest to industry. Each chapter includes the state of the art, comparisons between different methods and the mechanical and mathematical background of the methods addressed. Practising engineers, researchers and students in engineering, applied mathematicians and computer scientists will find new material for developing, maintaining and applying finite element algorithms which reach new heights of approximation quality.