Numerical solution of Variational Inequalities by Adaptive Finite Elements (eBook)

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

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

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.