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## Über dieses Buch

Over the past decades, the Boundary Element Method has emerged as a ver­ satile and powerful tool for the solution of engineering problems, presenting in many cases an alternative to the more widely used Finite Element Method. As with any numerical method, the engineer or scientist who applies it to a practical problem needs to be acquainted with, and understand, its basic principles to be able to apply it correctly and be aware of its limitations. It is with this intention that we have endeavoured to write this book: to give the student or practitioner an easy-to-understand introductory course to the method so as to enable him or her to apply it judiciously. As the title suggests, this book not only serves as an introductory course, but also cov­ ers some advanced topics that we consider important for the researcher who needs to be up-to-date with new developments. This book is the result of our teaching experiences with the Boundary Element Method, along with research and consulting activities carried out in the field. Its roots lie in a graduate course on the Boundary Element Method given by the authors at the university of Stuttgart. The experiences gained from teaching and the remarks and questions of the students have contributed to shaping the 'Introductory course' (Chapters 1-8) to the needs of the stu­ dents without assuming a background in numerical methods in general or the Boundary Element Method in particular.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
In the design of engineering structures, numerical simulations play an increasingly important role. This can be attributed to the rapid advances in computational power and software quality and the resulting decrease in the costs of computer simulations, as compared to the high costs and/or practical difficulties of experiments. However, to supplement or even replace experiments, numerical simulations have to fulfil strong requirements on efficiency, accuracy, and reliability. Whether these requirements can be met depends not only upon the physical and mathematical model that we choose for the real system we want to simulate, but also upon the choice of a proper simulation tool — e.g., the Boundary Element Method — and our skills in using it.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 2. Mathematical Preliminaries

Abstract
We assume that the reader is familiar with the basic concepts of vector calculus and matrix algebra as covered in any introductory course on engineering mathematics. In this chapter, we will outline some of the required concepts that the reader might not be familiar with.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 3. Continuum Physics

Abstract
In modern physics two theories are used for the description of physical phenomena: the particle theory and the continuum theory. The particle theory assumes the existence of discrete, indivisible units (the elementary particles). According to this theory, all natural phenomena can be described -at least theoretically - by interactions between these particles. On the other hand, continuum physics assumes a continuous distribution of field quantities, thereby disregarding the discrete nature of matter. Since the field theories of continuum physics are based upon observation and experience, they are also called phenomenological theories (Truesdell & Toupin (1960)).
Lothar Gaul, Martin Kögl, Marcus Wagner

### 4. Boundary Element Method for Potential Problems

Abstract
In Chapter 1, we introduced the Boundary Element Method by means of a simple one-dimensional example for which we sketched the basic steps from the differential equation to the boundary integral equation. However, since the boundary in a 1-D continuum degenerates to two points, we have neither encountered the special problems arising from the singularity of the fundamental solutions, nor could we describe the discretisation process with boundary elements. With the mathematical and physical background given in Chapters 2 and 3, we can now direct our attention to the more interesting and practically relevant two- and three-dimensional problems. In the present chapter, we will deal with simple potential problems in 2-D, so that we can concentrate on the basic features of the Boundary Element formulation. In Chapter 5, we then apply the method to 3-D problems of elastomechanics and static piezoelectricity.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 5. Boundary Element Method for Elastic Continua

Abstract
Having described the basic principles of the Boundary Element Method through the 2-D versions of Laplace’s and Poisson’s equations, we now extend the formulation to the consideration of 3-D problems in elastomechanics. In this context, we will also deal with anisotropic materials and give an introduction to the analysis of coupled field problems by means of static piezoelectricity.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 6. Numerical Integration

Abstract
In the Boundary Element Method we have to evaluate integrals of the type
$$I = \int\limits_{{\Gamma ^{(e)}}} {{{(\cdot )}^*}} (x,\xi )\phi (x)d\Gamma$$
(6.1)
over the boundary elements Γ (e) whose integrands are products of fundamental solutions (•)* and shape functions Φ, cf. (5.67). Depending on the relative position of the load point ξ to the element Γ (e) over which the integration is carried out, we have to distinguish between three types of integrals, see Figure 6.1.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 7. Dual Reciprocity Method for Potential Problems and Elastodynamics

Abstract
One of the cornerstones of the Boundary Element Method is its use of fundamental solutions to eliminate the domain integral over the adjoint operator. In the absence of source terms, the resulting formulation contains only boundary variables, so that only the boundary of the body has to be discre-tised and not its domain. This reduces the number of nodal unknowns and greatly simplifies the meshing and remeshing process as compared to domain discretisation methods such as Finite Elements or Finite Differences.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 8. Solution of the Equations of Motion

Abstract
When using static or stationary fundamental solutions in conjunction with the Dual Reciprocity Boundary Element Method, the resulting systems of equations contain time-independent system matrices. These systems are solved using different algorithms adapted to the particular kind of analysis to be performed, for example, transient analysis, solution of the eigenproblem, etc. The algorithms are similar to those employed in Finite Element analysis, but the system matrices in the Boundary Element Method are neither symmetric nor positive-definite; this can lead to difficulties in the solution of the eigenproblem or in transient analyses if the algorithms are not employed judiciously.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 9. Dynamic Piezoelectricity

Abstract
In Section 5.5, we described the solution of static piezoelectric field problems with the Boundary Element Method, where it was shown that the piezoelectric formulation could be developed through analogy with elastostatics when using a contracted notation to combine elastic and electric variables in a single expression.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 10. Coupled Thermoelasticity

Abstract
Having applied the Dual Reciprocity Method to dynamic piezoelectricity, we will now describe its formulation for another multi-field problem: dynamic coupled thermoelasticity. The first thermoelastic Boundary Element formulation was presented by Rizzo & Shippy (1977) for isotropic static thermoelasticity. Balas et al. (1989) and Tanaka et al. (1995) describe a number of different approaches for various thermoelastic theories including coupled thermoelasticity, but stop short of considering anisotropic materials. This was done for stationary thermoelasticity by Deb et al. (1991) and Deb (1996). Only recently, a general approach for anisotropic coupled thermoelasticity that comprises all previously discussed thermoelastic theories was presented by Kögl & Gaul (2000c,d). Owing to its versatility, this approach is described in the following.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 11. Variational Principles of Continuum Mechanics

Abstract
The governing differential equations of continuum mechanics can be derived by evaluating balance equations of vector fields, such as the momentum. A second approach for obtaining the governing differential equations, so-called analytical mechanics, is based on axioms defined in scalar quantities such as work and energy (Lanczos (1970), Riemer et al (1993), Szabó (1987)). We can obtain the governing vector equations by using variational calculus, outlined in Section 2.5. The two approaches are mechanically equivalent but when numerical solutions for complex systems are required, the two approaches provide significant differences in suitability and efficiency.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 12. The Hybrid Displacement Method

Abstract
In this chapter we use the Hybrid Displacement Boundary Element Method (HDBEM) to investigate several static and dynamic field problems. The method is based on variational principles, rather than on integral equations as the BEM from Part I and II, which is based on weighted residual statements. To avoid confusion, the weighted residual method will be referred to as Direct BEM for the remainder of the book.
Lothar Gaul, Martin Kögl, Marcus Wagner

### 13. The Hybrid Stress Method for Acoustics

Abstract
The strain energy is comprised of the product of the stress and strain tensor, which are connected by the elasticity tensor through a constitutive equation. In the case of linear elastic materials, the stresses depend linearly upon the strains, σ ij=C ijklεkl. Inverting this relation, we can express the strains in terms of the stresses. The Hybrid Displacement BEM uses the strains as the primary variables. On the other hand, the Hellinger-Reissner Principle is a mixed formulation and also employs the stress tensor as the primary unknown. Based on this principle, Dumont (1987a,1988,1989) derived the Hybrid Stress Boundary Element Method (HSBEM). The Hellinger-Reissner Principle is, by construction, a two-field principle that employs the derived field variable in the domain and the primary field variable (e.g. the displacement or the potential) on the boundary. In this chapter, we will derive the HSBEM for acoustics (see also Wagner (2000)).
Lothar Gaul, Martin Kögl, Marcus Wagner

### 14. The Hybrid Boundary Element Method in Time Domain

Abstract
In this chapter, we describe the Hybrid Displacement Boundary Element Method for the time domain and derive formulations for elastodynamics and acoustics. These formulations lead to symmetric systems of equations with time-invariant mass and stiffness matrices. The symmetry is exploited to formulate a symmetric coupled system of equations for fluid-structure interaction in time domain.
Lothar Gaul, Martin Kögl, Marcus Wagner

### A. Properties of Elastic Materials

Abstract
The classification, mass densities, and elasticity matrices C of four elastic materials are given in Table A.1 and A.2
Lothar Gaul, Martin Kögl, Marcus Wagner

### B. Fundamental Solutions

Abstract
The fundamental solution u* = u*(x, ζ) of a differential operator L is a fullspace solution of the differential equation
$$\mathcal{L}u*\left( {x,\xi } \right) = \delta \left( {x,\xi } \right)$$
(B.1)
and relates the effect of a point source at the load point ζ to the field point x. In all problems treated iu this book, the fundamental solution is a function of the distance vector r i = x i - ζ i between the field and the load point; when deriviug the fundamental solutions we can assume without loss of generality that the load poiut lies in the origin of the coordinate system, i.e., ζ=0. In this case, x i = r i.
Lothar Gaul, Martin Kögl, Marcus Wagner

### C. Particular Solutions

Abstract
In this appendix, we give interpolation functions and particular solutions for the field problem described in Part II.
Lothar Gaul, Martin Kögl, Marcus Wagner

### D. The Bott-Duffin Inverse

Abstract
To invert the singular matrices F 0 and H T 0 of the HSBEM in Chapter 13 we use a special generalised matrix inverse suitable for the equation system. This so-called restricted inverse was introduced by Bott & Duffin (1953) in the frame of the theory of electrical networks. The remainder of this section reformulates the matrix statement of the HSBEM to make it suitable to apply this type of inverse.
Lothar Gaul, Martin Kögl, Marcus Wagner

### Backmatter

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