Suttmeier Numerical solution of Variational Inequalities by Adaptive Finite Elements

2008
ISBN: 978-3-8348-9546-2
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 162 Seiten, eBook

Reihe: Advances in Numerical Mathematics

The author presents a general approach to a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. The local weighted residuals, that result from an extension of the so-called Dual-Weighted-Residual method, are used in a feed-back process for generating economical meshes. Based on several model problems, a general concept is proposed, which provides a systematic way of adaptive error
control for problems stated in form of variational inequalities.



Dr. Franz-Theo Suttmeier is a professor of Scientific Computing at the Institute of Applied Analysis and Numerics at the University of Siegen.

Zielgruppe

Research

Autoren/Hrsg.

Suttmeier, Franz-Theo

Inhaltsverzeichnis

1;Summary;6
2;Contents;8
3;Chapter 1 Introduction;12
4;Chapter 2 Models in elasto-plasticity;24
4.1;2.1 Governing equations;25
4.2;2.2 Examples;31
5;Chapter 3 The dual-weighted-residual method;34
5.1;3.1 A model situation in plasticity;35
5.2;3.2 A posteriori error estimate;36
5.3;3.3 Evaluation of a posteriori error bounds;37
5.4;3.4 Strategies for mesh adaptation;39
5.5;3.5 Example;41
6;Chapter 4 Extensions to stabilised schemes;44
6.1;4.1 Discretisation for the membrane-problem;46
6.2;4.2 A posteriori error analysis;48
6.3;4.3 Numerical tests;53
7;Chapter 5 Obstacle problem;57
7.1;5.1 Energy norm;58
7.2;5.2 Duality argument;59
7.3;5.3 A posteriori estimates;61
7.4;5.4 Numerical results;64
8;Chapter 6 Signorini’s problem;67
8.1;6.1 A posteriori error bounds;68
8.2;6.2 Numerical results;72
8.3;6.3 A posteriori controlled boundary approximation;76
9;Chapter 7 Strang’s problem;79
9.1;7.1 A posteriori error bounds;80
9.2;7.2 Numerical results;83
10;Chapter 8 General concept;84
10.1;8.1 Orthogonality relation;84
10.2;8.2 Duality argument;85
10.3;8.3 Modification;86
11;Chapter 9 Lagrangian formalism;89
11.1;9.1 Torsion problem;89
11.2;9.2 A suboptimal error estimate;91
11.3;9.3 Saddle point problem;92
12;Chapter 10 Obstacle problem revisited;98
12.1;10.1 Weak formulation;98
12.2;10.2 A posteriori error estimates;99
12.3;10.3 Numerical results;101
13;Chapter 11 Variational inequalities of second kind;102
13.1;11.1 A flow problem;102
13.2;11.2 A friction problem;104
13.3;11.3 A posteriori error estimate;106
13.4;11.4 Numerical results;108
14;Chapter 12 Time-dependent problems;111
14.1;12.1 Discretisation;111
14.2;12.2 Error estimation;113
15;Chapter 13 Applications;114
15.1;13.1 Grinding;114
15.2;13.2 Milling;121
15.3;13.3 Elasto-plastic benchmark problem;122
16;Chapter 14 Iterative Algorithms;133
16.1;14.1 Introduction;133
16.2;14.2 A Smoothing Procedure;135
16.3;14.3 The Multilevel Procedure;136
16.4;14.4 A Conjugate Gradient Algorithm;138
16.5;14.5 Numerical Results;139
17;Chapter 15 Conclusion;144
18;Appendix A Algorithmic Aspects;145
19;Bibliography;156

Models in elasto-plasticity.- The dual-weighted-residual method.- Extensions to stabilised schemes.- Obstacle problem.- Signorini’s problem.- Strang’s problem.- General concept.- Lagrangian formalism.- Obstacle problem revisited.- Variational inequalities of second kind.- Time-dependent problems.- Applications.- Iterative Algorithms.- Conclusion.

Leseproben

Algorithmic Aspects (S. 143-144)

The idea of the finite element method is quite universal since problems with di.erent characteristic properties can be treated. As examples we mention the .elds of reactive .ow, radiative transfer and continuum mechanics. Therefore an approach to finite element code should re.ect this universality. Most parts of the implementation of this method can be done problem independent. So it is advisable to use a program library, which for example supports the very complex task of grid-handling required for each special application. In 1991 Guido Kanschat and the author started at Bonn with the development of such library, namely DEAL (Differential Equation Analysis Library) (Becker, Kanschat &, Suttmeier [21]).

In Kanschat [42] there are mentioned four development aims which are important in a complex software project, namely computing speed, memory requirements, veriflability of code and .exibility. We discuss the aspect of flexibility of our library at the example of the representation of a linear operator. DEAL ofiers the possibility to balance computing time and memory requirements with respect to the problem under consideration. Due to our idea to represent an operator independent of boundary conditions and hanging nodes, these points have to be handled separately. At the end of this chapter we demonstrate our method of filtering techniques to take these points into account.

The finite element package As mentioned above the universality is an important point in the finite element method. This means, .rst we look for properties all problems have in common. As an example we remark, that one main part of a finite element program consists of the geometric description of the domain and its discretisation in form of a triangulation. DEAL uses the object oriented concept for the gridhandling.

A triangulation is regarded as an object, which consists of cells. These cells itself are described by their vertices. We use a straightforward approach to organise the re.nement process. In detail this is explained in Kanschat [42]. A triangle, for example, is divided into four congruent ones. The hierarchy is stored, i.e. all cells know their father and children. This enables us to do coarsening, which is a very important feature, especially in time dependent problems.

As an example we refer to the sequence of grids at the end of Chapter 13. The moving transition zone between the elastic and the plastic part of the solution is resolved by the adaptive algorithm. The complex problem of implementation of a triangulation is a good example to demonstrate the advantages of our approach. In a natural way the whole task is split into several small steps. These single steps are treated separately and therefore it is easy to improve the code locally. Furthermore, if it is necessary to change a concept we can reuse the single parts and we do not have to do changes within the whole library.

Über Autor(innen)

Dr. Franz-Theo Suttmeier is a professor of Scientific Computing at the Institute of Applied Analysis and Numerics at the University of Siegen.