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## Über dieses Buch

1. 1 The Hybrid Displacement Boundary Element Model This work is concerned with the derivation of a numerical model for the solution of boundary-value problems in potential theory and linear elasticity. It is considered a boundary element model because the final integral equation involves some boundary integrals, whose evaluation requires a boundary discretization. Furthermore, all the unknowns are boundary vari­ ables. The model is completely new; it differs from the classical boundary element formulation ·in the way it is generated and consequently in the fi­ nal equations. A generalized variational principle is used as a basis for its derivation, whereas the conventional boundary element formulation is based on Green's formula (potential problems) and on Somigliana's identity (elas­ ticity), or alternatively through the weighted residual technique. 2 The multi-field variational principle which generates the formulation in­ volves three independent variables. For potential problems, these are the potential in the domain and the potential and its normal derivative on the boundary. In the case of elasticity, these variables are displacements in the domain and displacements and tractions on the boundary. For this reason, by analogy with the assumed displacement hybrid finite element model, ini­ tially proposed by Tong [1] in 1970, it can be called a hybrid displacement model. The final system of equations to be solved is similar to that found in a stiffness formulation. The stiffness matrix for this model is symmetric and can be evaluated by only performing integrations along the boundary.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This work is concerned with the derivation of a numerical model for the solution of boundary-value problems in potential theory and linear elasticity.
Tania G. B. DeFigueiredo

### Chapter 2. Potential Problems

Abstract
Potential problems are here defined as those that can be expressed in terms of a potential function and are governed by either a Laplace or a Poisson equation. Sometimes these problems are more specifically referred to as scalar potential problems [61], as opposed to vector potential problems, which can be used to represent other physical phenomena such as those treated in the theory of elasticity.
Tania G. B. DeFigueiredo

### Chapter 3. Numerical Aspects in Potential Problems

Abstract
The procedures required for the numerical solution of the equations presented in the previous chapter will be explained in the following sections. Although only the two dimensional case will be considered throughout this chapter, similar considerations could be used for three dimensional problems.
Tania G. B. DeFigueiredo

### Chapter 4. Elastostatics

Abstract
In this chapter, the hybrid displacement boundary element formulation for the solution of linear elastostatic problems will be presented. The existing similarities between the potential theory (also called scalar potential theory) and the theory of elasticity (also called vector potential theory [61]) results in the same format for this chapter as for chapter 2.
Tania G. B. DeFigueiredo

### Chapter 5. Numerical Aspects in Elastostatics Problems

Abstract
This chapter is concerned with the computer implementation of the hybrid displacement boundary element formulation presented in chapter 4. The case of two-dimensional problems will be considered, for simplicity. Similar considerations could be used in the implementation of the formulation for the three-dimensional case, although some additional work would be required. Special attention is dedicated to the treatment of some singular integrals which appear in the generation of the matrices F and G.
Tania G. B. DeFigueiredo

### Chapter 6. Numerical Applications

Abstract
The hybrid-displacement boundary element formulation, which has been proposed in the earlier parts of this work, is a novel numerical technique. Several problems in two-dimensional potential and elasticity theories have been solved to demonstrate its feasibility and accuracy. Some of these solutions will be presented in this chapter and their accuracy will then be assessed by comparing these numerical results to analytical solutions. Hybrid boundary elements results are also compared to numerical results obtained by the conventional boundary element method. The performance of the new approach can then be compared against the performance of the conventional BEM, which has already been proved to be efficient and accurate. The convergence of the hybrid-displacement method has also been demonstrated numerically and the results are shown for some of the cases studied in function of an error norm.
Tania G. B. DeFigueiredo

### Chapter 7. Conclusions

Abstract
In this work, a novel boundary element formulation has been presented. It has been derived and implemented computationally to solve problems in potential theory and linear elastostatics.
Tania G. B. DeFigueiredo

### 8. Bibliography

Without Abstract
Tania G. B. DeFigueiredo

### Backmatter

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