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to Boundary Elements Theory and Applications With 194 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Dr.-Ing. Friedel Hartmann University of Dortmund Department of Civil Engineering 4600 Dortmund 50 FRG ISBN-13: 978-3-642-48875-7 e-ISBN-13: 978-3-642-48873-3 001: 10.1007/978-3-642-48873-3 Library of Congress Cataloging-in-Publication Data Hartmann, F. (Friedel) Introduction to boundary elements: theory and applications/Friedel Hartmann. ISBN-13: 978-3-642-48875-7 1. Boundary value problems. I. Title. TA347.B69H371989 515.3'5--dc19 89-4160 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provision of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1 st edition 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.




If you hold a ruler to the end points of a linear function you can draw then the function with a pencil, see Fig. 1.
Friedel Hartmann

1. Fundamentals

This chapter is a concise summary of the fundamentals of the boundary element method. The experienced reader may prefer to look over this material rather casually and then refer to it again when need arises. The novice reader is advised to begin with chapter 2 where the method is explained in detail by applying it to one-dimensional problems.
Friedel Hartmann

2. One-dimensional problems

This chapter is intended as a boundary element primer. The method is explained by applying it to the one-dimensional problems of rods and beams.
Friedel Hartmann

3. Membranes

A membrane is assumed to be a perfectly flexible, thin elastic fabric, which is uniformly stretched in all directions by a tension which has a constant value N per unit length along any section or boundary. The deflection u (= u3) satisfies the differential equation
$$-N\left( u,11+u,22 \right)=-N\Delta u=p$$
where p is the lateral pressure. The traction across a cut is the product of the tension N and the derivative in the direction of the normal vector n = {n1, n2} T of the cut,
$$t=N\frac{\partial u}{\partial n}=N\left( u,1{{n}_{1}}+u{{,}_{2}}{{n}_{2}} \right),$$
that is the N-fold normal derivative or N-fold slope. The close connection between the slope and the traction expresses Fig. 3.1. The greater the pressure the more the membrane will deflect and the greater the slope on the boundary and, therefore, also the traction t on the boundary.
Friedel Hartmann

4. Elastic plates and bodies

In this chapter we use the boundary element method to calculate the displacements and stresses within elastic plates and bodies.
Friedel Hartmann

5. Nonlinear problems

Influence functions, as the influence function for the longitudinal displacement u(x)of a rod
$${{u}_{1}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right){{p}_{1}}\left( y \right)dy,$$
are L2-scalar products between the Green’s function and the exterior load p1 and because the scalar product is distributive
$$underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( {{p}_{1}}+{{p}_{2}} \right)dy=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}{{p}_{1}}dy+\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}{{p}_{2}}dy,$$
the influence function for a nonlinear equation cannot be of the form (5.1). If Eq.(5.1) were the solution of the problem
$${{u}_{2}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right){{p}_{2}}\left( y \right)dy$$
the solution of the problem
then the functionld
$$u\left( x \right)={{u}_{1}}\left( x \right)+{{u}_{2}}\left( x \right)=\underset{0}{\overset{1}{\mathop{\int }}}\,{{G}_{0}}\left( y,x \right)\left( {{p}_{1}}\left( y \right)+{{p}_{2}}\left( y \right) \right)dy$$
Would be the solution of the problem
$${{D}^{NL}}\left( {{u}_{1}}+{{u}_{2}} \right)={{p}_{1}}+{{p}_{2}}.$$
But this is a contradiction.
Friedel Hartmann

6. Plates

In this chapter we apply the boundary element method to Kirchhoff plates. These plates are governed by a fourth-order equation, the bi-harmonic equation.
Friedel Hartmann

7. Boundary elements and finite elements

Both methods have their strong points
element library reduction of dimension
robust higher precision
variable coefficients exterior problems
so that a coupling of the two methods or, as it was phrased, a Marriage à la Mode,[77], should benefit from both. The coupling will usually be done by reformulating the coupling conditions of the boundary data of the BE-domain as a stiffness matrix and to couple this stiffness matrix with the stiffness matrices of the neighboring finite elements.
Friedel Hartmann

8. Harmonic oscillations

Dynamical loads cause inertial forces ρü in a structure. These forces appear on the left-hand side of the differential equation
$$Du+\rho \ddot{u}=p(x,t)$$
If the excitation is harmonic
$$p(x,t)=p(x)\cos (\omega t+\varphi )$$
, then the response of the structure is also harmonic. This important case is the topic of this chapter.
Friedel Hartmann

9. Transient problems

Transient vibrations are aperiodic vibrations. A separation of the variables is therefore no longer possible. The time t becomes a further variable.
Friedel Hartmann

10. Computer programs

As a supplement to this book we offer a package of three programs
which run on the IBM-PC, PS/2 and compatible computers. Hardware requirement are 640 K RAM, a coprocessor 80×87, a hard disk and one of the following graphics adapters: Hercules card, Color Graphics Adapter, Enhanced Graphics Adapter or the Olivetti Graphics card.
Friedel Hartmann


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