[...]...xi CONTENTS 16 Coupled Boundary Element/ Finite Element Analysis 435 16 .1 16.2 16 .5 16 .6 Introduction Coupling theory 16 .2 .1 Coupling to finite elements 16 .2.2 Coupling to boundary elements Example Dynamics 16 .4 .1 Example Conclusion References 435 436 436 443 444 446 447 447 449 17 Industrial Applications 4 51 17 .1 17.2 17 .7 17 .8 Introduction Mechanical engineering 17 .2 .1 A cracked extrusion... 17 .3 .1 CERN Caverns Geological engineering 17 .4 .1 How to find gold with boundary elements Civil engineering 17 .5 .1 Masjed-o-Soleiman underground power house Reservoir engineering 17 .6 .1 Borehole stability Conclusions References 4 51 453 453 457 457 4 61 4 61 464 464 470 470 472 473 18 Advanced topics 475 18 .1 18.2 Introduction Heterogeneous Domains 18 .2 .1 Theory 18 .2.2 Example Linear inclusions 18 .3 .1. .. inclusions 18 .3 .1 Theory 18 .3.2 Example Piezo-electricity 18 .4 .1 Changes required in General_Purpose_BEM Conclusions References 475 476 476 477 479 479 484 485 487 488 489 16 .3 16 .4 17 .3 17 .4 17 .5 17 .6 18 .3 18 .4 18 .5 18 .6 Appendix 4 91 xiii Preface This is a sequel to the book Programming the Boundary Element Method by G Beer published by Wiley in 20 01 The scope of this book is different however and this is... 13 also allows the solution of problems in elasto- and visco-plasticity Body forces of a different kind (mass forces) occur in the case of dynamics, but their treatment with the BEM is quite different to the FEM and this is discussed in Chapter 14 6 The Boundary Element Method with Programming In Chapter 15 we show that the solution of non-linear problems follows similar procedures as in the FEM and. .. u1 and u2 of element 3 into locations 3 and 4 of the global vector The ‘gather’ operation would do the opposite, i.e U_LOCAL= U_GLOBAL(CONNECTIVITY) would put the global values of u1 and u2 into the local positions 1 and 2 20 2.2.4 The Boundary Element Method with Programming Control Various features which can be used to control the flow of the program have been improved and new ones added With these... by Trefftz2), the assumed functions satisfy the DE exactly and the error in the satisfaction of the boundary conditions is minimised Most readers of this book will be familiar with the finite element method In the most common version of this method we subdivide the domain into elements and approximate 2 The Boundary Element Method with Programming the response to a specified loading with functions... between the tensor and matrix operations The index number of the location of the strain or stress components for matrix notation and tensor notation is given in Table 1. 1 Table 1. 1 Index numbering for strain and stress Notation Matrix Tensor 1 11 xx Index number 3 4 33 12 & 21 zz xy&yx 2 22 yy Similarly, the stress tensor ij 6 31& 13 zx&xz can be written as a pseudo-vector T x 5 23&32 yz&zy y z xy yz xz (2 .17 )... of the Boundary Element Method (BEM), that is easy for engineers and scientists to follow, is retained This is achieved by explaining some aspects of the method in an engineering rather than mathematical way Another new feature of the book is that it deals with the implementation of the method on parallel processing hardware I M Smith, who has been involved in programming the finite element method for. .. is the case with finite elements Figure 1. 1 Finite element mesh for the analysis of tunnel excavation Left side: mesh with contours of z-displacement; right side: detail with contours of maximum compressive stress Figure 1. 2 Boundary element mesh for the simulation of tunnel excavation with contours of maximum compressive stress plotted on excavation surface and result planes 4 The Boundary Element Method. .. 1 Ritz, W (19 09) Über eine Methode zur Lösung gewisser Variations-Probleme der mathematischen Physik Journal für reine und angewandte Mathematik, 13 5: 1- 6 1 2 Trefftz, E (19 26) Ein Gegenstück zum Ritz’schen Verfahren Proc 2nd int Congress in Applied Mechanics, Zurich, pp 13 1 3 Zienkiewicz O.C and Taylor R.L (2000) The Finite Element Method - Fifth Edition Butterworth-Heinemann, UK 4 Timoshenko, S.P and . 1. 1 Introduction 1 1. 2 Overview of book 4 1. 3 Mathematical preliminaries 6 1. 3 .1 Vector algebra 7 1. 3.2 Stress and strain 10 1. 4 Conclusions 11 1. 5 References 11 2 Programming 13 . regions 311 11 .4 .1 User’s manual 3 21 11 .4.2 Sample problem 323 11 .5 Conclusions 326 11 .6 Exercises 327 11 .7 References 328 12 Dealing with corners and changing geometry 329 12 .1 Introduction. 8 Element- by -element techniques and Parallel Programming 203 8 .1 Introduction 203 8 .1 The Element by Element Concept 204 8 .1. 1 Element- by -element storage requirements 206 8.2 PROGRAM 8.1