446 The Boundary Element Method with Programming Figure 16.8 Distribution of maximum compressive stress, comparison with theory 16.4 DYNAMICS Here we extend the coupling method to dynamics. The dynamic equilibrium equations which arise from finite element discretisation (see Bathe 4 ) can be written as (16.19) where > @ M , > @ C , > @ K are the assembled mass, damping and stiffness matrices and ^ ` u , ^ ` u , ^` u are the acceleration, velocity and displacement vectors. The time may be discretised into n time steps of size t' . Assuming an average acceleration within the time step the system of differential equations can be transformed into a system of algebraic equations (Newmark method 4 ) (16.20) where tnt 'and (1)tn t ' A nal y sis min V Theory > @ ^` > @ ^` > @ ^` ^` Mu Cu Ku F ^` ^` ()t ªº ¬¼ Ku F COUPLED BOUNDARY ELEMENT/FINITE ELEMENT ANALYSIS 447 The “dynamic stiffness matrix” is given by: (16.21) and (16.22) Since we have already worked out a “dynamic stiffness matrix” of the boundary element region in Chapter 14 the coupling procedure is now straightforward. For a fully coupled problem the system of equations is given by (16.23) 16.4.1 Example The example is that of a concrete column embedded in a semi-infinite soil mass. The description of the problem can be seen in Figure 16.9. The top of the column is subjected to a suddenly applied load p(t) of 1 MN/m 2 . The material properties for the column are: spec. weight= 2500 kg/m 3 , E=30 000 MN/m 2 , Q =0.2. For the soil we have: spec. weight= 2000 kg/m 3 , E=100 MN/m 2 , Q =0.2. Figure 16.9 Description of example Figure 16.10 shows the mesh used for the analysis it consists of a finite element region that describes the column and a boundary element region that describes the semi-infinite ground. The mesh has 1500 degrees of freedom. Figure 16.11 shows the time-dependent >@ >@> @ 2 42 t t ªº ¬¼ ' ' KMCK ^` >@ ^`^`^` >@ ^`^`^` 2 44 () () () 2 () () () ttt t t tt t t §· ¨¸ ' ' ©¹ §· ¨¸ ' ©¹ FM u u u Cu u F ^` ^` ^` () BE FE BE FE t ªº ªº ¬¼ ¬¼ NK K u F F () p t () p t t 448 The Boundary Element Method with Programming displacements at the top of the column obtained from the analysis. The results compare well with a reference solution with the FEM that used 1 Million elements. Figure 16.10 Coupled mesh Figure 16.11 Displacement at the top of the column 16.5 CONCLUSION In this chapter we have shown how the capability of a finite or boundary element program can be easily extended so that the advantages of both methods can be combined giving the user “the best of both worlds”. We have shown one example where the capability of the BEM in dealing with infinite domains was exploited. Many other such examples exist and we will show in the next chapter one industrial application that could (seconds) COUPLED BOUNDARY ELEMENT/FINITE ELEMENT ANALYSIS 449 not have been analysed with either method given the restrictions regarding time and computing resources. Although it is true that both methods can deal with almost any problem that arises in engineering (and comprehensive text books on the FEM and BEM assert this), it is also clear that they are more appropriate for some applications and less so for others. It should have become clear to the reader, for example, that the BEM is well suited for problems involving a small ratio of boundary surface to volume. Extreme cases of this are problems which can be considered as involving an infinite volume. Such problems exist, for example, in geomechanics 5 , where the earth’s crust has no lateral boundaries. Another extreme where the ratio boundary surface to volume is very large is the application to thin shell structures. Another aspect is the importance that is given to surface stresses. As we have seen in Chapter 9, stresses at the surface are computed more accurately with the BEM than with the FEM. We have shown that problems where “body forces” occur in the domain, as for example plasticity problems, etc., can be handled with the BEM but it has to be admitted that implementation is much more involved than with the FEM. A final aspect which is also gaining more importance, is the suitability of the methods for implementation with regards to computer hardware. The future seems to lie in massive parallel processing and we have seen in Chapter 8 that the BEM seems to lend itself to parallel programming. 16.6 REFERENCES 1. Zienkiewicz O.C. ,Kelly D.W. and Bettess P. (1979) Marriage a la mode- the best of both worlds (finite elements and boundary integrals) Chapter 5 of Energy Methods in Finite Element Analysis (ed. R.Glowinski, E.Y. Rodin and O.C.Zienkiewicz), pp. 82- 107, Wiley, London. 2. Beer G. (1977) Finite element, boundary element and coupled analysis of unbounded problems in elastostatics. Int. J. Numer. Methods Eng., 11, 355-376 3. Beer G. (1998) Marriage a la mode (finite and boundary elements) revisited. In Computational Mechanics New Trends and Applications (E.Onate and S.R.Idelsohn (eds). 4. Bathe K.J. (1982) Finite Element procedures in engineering analysis. Prentice Hall. 5. Beer G., Golser H., Jedlitschka G. and Zacher P. Coupled finite element/boundary element analysis in rock mechanics - industrial applications. Rock Mechanics for Industry, Amadei, Kranz, Scott & Smeallie(eds). Balkema,Rotterdam. 133-140. 17 Industrial Applications Grau ist alle Theorie (Grey is all theory ) J.W. Goethe 17.1 INTRODUCTION So far in this book we have developed software which can be applied to compute test examples. The purpose of this was to enable the reader to become familiar with the method, ascertain its accuracy and get a feel for the range of problems that can be solved. The emphasis in software development has been on an implementation that was concise and clear and could be well understood. As pointed out in the introduction to programming, this is not necessarily the most efficient code in terms of storage and computer resources. If one wants to tackle real engineering problems one is inevitably faced with the need to develop efficient code. The programs developed here would be unsuitable for such a task. Aspects of the software that need to be improved are: x Greater efficiency in the computation of coefficient matrices by rearranging DO loops, so that calculations that are independent of the DO loop variable are taken outside the loop. x Greater efficiency in data and memory management so that data are only stored in RAM when they are needed, use of hard disk storage to achieve this (see for example [1] ). It has been shown in Chapter 8 that a significant gain in efficiency can be achieved by using element by element techniques and parallel programming. Indeed, to solve 452 The Boundary Element Method with Programming problems at an industrial scale in a short time, special hardware, such as parallel computers may have to be used. In this chapter we attempt to show applications of the boundary element and coupled methods which have been compiled from a number of tasks that have been carried out over more than two decades using BEFE 2 , a combined finite element/boundary element program. The purpose of the chapter is twofold. Firstly, an attempt is made to demonstrate the applications for which the BEM may have a particular advantage over the FEM. These applications include: x Problems involving stress concentrations at the boundary, such as they occur in mechanical engineering x Problems consisting of infinite or semi-infinite domains, such as those occurring in geotechnical engineering x Problems involving slip and separation at material interfaces, such as they appear in mechanical and geotechnical engineering x Contact and crack propagation problems The second purpose of this chapter is to show how the very complex problems that invariably arise in industrial applications can be simplified, so that the analysis can be performed in a reasonable short time. It is very rarely the case that a problem can be modelled exactly as it is. In most cases we have to decrease its complexity. The process of modelling a given complex structure requires a lot of engineering ingenuity and experience. When we simplify a complex problem we must ensure that the important influences are retained neglecting other less important ones. For example, in a structural problem some parts of the structure may not contribute significantly to its load carrying capacity but are there because of design considerations. One very significant modelling decision is if a 3-D analysis needs to be carried out. Obviously this would result in much greater analysis effort. As an example in geotechnical engineering consider a tunnel which is very long compared to its diameter. If we are only interested in the displacements and stresses at a cross section far away from the tunnel face, then a plane strain analysis would obviously suffice. Another way of simplifying a problem is the introduction of planes of symmetry. As we have seen in some of the examples in Chapter 10, this results in considerable savings. Obviously if the prototype to be analysed is symmetric there is no loss in modelling accuracy. In some cases, however, symmetry planes can be assumed without significant loss in accuracy even if the prototype itself is not exactly symmetric. In the following we will present background information on each application, in some cases together with a story associated with it. We will start with the description of the problem and how it was simplified. We show the boundary element mesh generated and the results obtained. Comments are made on the quality of the results. The problem areas are divided into mechanical, geotechnical, geotechnical civil engineering and reservoir engineering. INDUSTRIAL APPLICATIONS 453 17.2 MECHANICAL ENGINEERING 17.2.1 A cracked extrusion press causes concern A small company in Austria manufactures rolled thin tubes by extrusion. The extrusion press in use was 35 years old and made of cast iron (see Figure 17.1). During a routine inspection cracks were detected on the surface of the cast iron casing, as indicated. The company was in the process of ordering a new press, however delivery was expected to take more than six months. There was some concern that something dramatic might happen during the extrusion process with the press suddenly breaking, meaning not only a danger to lives but also the possibility of losing the press. With full order books the latter was a very serious economic threat. Figure 17.1 35 year old drawing of extrusion press with location of cracks indicated The aim of the analysis was therefore to determine: x If the existing cracks would propagate x If this propagation would lead to a sudden collapse of the structure The geometry of the part to be analysed was fairly complicated and had to be reconstructed from the original plans. For the purpose of the analysis it was assumed that there were two planes of symmetry, as shown in Figure 17.2, although this was not strictly true. The cylindrical bar restraining the casing was not explicitly modelled but instead appropriate Dirichlet boundary conditions were applied. Each time a tube is extruded the casing is loaded with a force of 3700 tons (37 MN), as shown by the arrows. Although Cracks observed 454 The Boundary Element Method with Programming this load is actually applied dynamically it was assumed to be static for the purpose of the analysis. Figure 17.2 Boundary element model showing axes of symmetry and holding bar The drawing in Figure 17.2 actually looks like a finite element mesh but if viewed from the symmetry planes (Fig. 17.3) one can notice that, in contrast to a FEM discretisation, there are no elements inside the material. The mesh consists of a total of 1437 linear boundary elements and has 4520 degrees of freedom. There were two reasons why a boundary element analysis was chosen for this problem. Firstly, the generation of the mesh was found to be easier, since no internal INDUSTRIAL APPLICATIONS 455 elements and connection between surfaces had to be considered. Secondly, the task was to determine surface stresses and then to investigate crack propagation. As outlined previously, the BEM is well suited for this type of analysis. Figure 17.3 Boundary element mesh viewed from one of the symmetry planes Initially, an analysis with only one region was carried out without considering the presence of cracks. This was done in order to check that the analysis was able to predict crack initiation. The criteria chosen for this was the maximum tensile strength of the material, taking into consideration the dynamic nature of the loading and the number of cycles that the press had so far sustained (approx. 2 million cycles). This analysis was also carried out to see if the model was adequate and to enable the client to get confidence in the BEM analysis proposed. The contours of maximum stress obtained from the single region analysis, shown in Figure 17.4, clearly indicate a stress concentration at the locations where cracks were observed, of a magnitude which would cause crack initiation there after a number of cycles. After this verification of the model, a multi-region analysis was carried out. For this each of the flanges where the crack was observed was divided into two regions. For simplicity it was assumed that the crack path was known a priori and is in the diagonal direction, as observed. Along this assumed crack path an interface was assumed between regions and the interface was allowed to slip and separate. 456 The Boundary Element Method with Programming Figure 17.4 Contours of maximum principal stress Figure 17.5 Displaced shape showing crack opening [...]... ) The yield functions used were the Mohr-Coulomb and Hoek and Brown models The mesh used for the analysis for a 45° deviation is plotted in Figure 17.28 on the left Figure 17.28 Boundary element mesh (left) and results of the analysis (right) plotted on surface and dummy plane 472 The Boundary Element Method with Programming The mesh consists of about 100 0 linear boundary elements Plane strain elements... for the analysis and the displacement patterns, a 3-D Boundary Element analysis was first carried out Figure 17.21 Boundary element mesh of caverns and computed deformations Figure 17.22 Coupled mesh for the analysis of powerhouse cavern and concrete powerhouse structure 468 The Boundary Element Method with Programming However, this analysis does not consider the presence of geological features and. .. displacements the internal forces in the shell (bending moment and normal force) could be determined and used for designing the reinforcement The analysis shown here demonstrates that with limited resources available (time and computer), boundary element and coupled analysis offer an efficient alternative to the FEM 17.4 17.4.1 GEOLOGICAL ENGINEERING How to find gold with boundary elements The analysis was performed... stress boundary conditions applied 464 The Boundary Element Method with Programming „hot spot“ Figure 17.16 Contours of maximum compressive principal stress For the analysis a multi-region boundary element method was used with special contact/joint algorithms implemented on the interfaces between regions Figure 17.14 shows a view of the four regions considered Figure 17.15 shows the block analysed with. .. 17.21 shows the mesh with quadratic (8-node) boundary elements and the result for the case where both caverns are excavated, plotted as displacement contours on the excavation surface It can be seen that for a large portion of the cavern plane strain conditions can be observed It was therefore decided that the mesh could be reduced by the use of infinite plane strain boundary elements as they have been... a cross-section through the end of the powerhouse Figure 17.25 View showing the concrete powerhouse and the location of the cracks 470 The Boundary Element Method with Programming Figure 17.26 shows one result of the analysis namely the stress distribution in the concrete wall plotted as principal stress vectors It can be seen that the observed crack pattern on the right is perpendicular to the maximum... hydroelectric plant, the powerhouse cavern is inside the mountain on the left of the dam Figure 17.18 Layout of the Caverns indicating existing caverns and caverns being excavated 466 The Boundary Element Method with Programming Figure 17.17 shows a view of the hydroelectric facility and Figure 17.18 a plan of the layout showing the existing powerhouse cavern and the extension under construction The areas where... 2m) with an inclusion in centre (1m x 1m) under plane strain conditions The block is fixed at the bottom and loaded on the top surface with a constant pressure of p = 1N/m² The inclusion is assumed to be 10 times softer than the block The block is discretised with 4x12 quadratic boundary elements and the inclusion is discretised with 6x6 quadratic cells Figure 18.3 shows the deformed shape of the mesh... stress boundary conditions applied In this figure the deformation of the blocks and the movements on the Golden and Lucky faults can be seen The results of the analysis can be seen in Figure 17.16 as contours of the maximum (compressive) principal stress on the contact between regions I and II One can clearly see an anomaly of the compressive stress (“hot spot”) and this is near the location where the. .. to the rock mass along it’s length Figure 18.5 Example of a tunnel with a rock bolt The cross-section of the rock bolt is assumed to be small compared to its length and therefore the variation of the stress across the section can be assumed to be constant The approach is very similar to the previous one, i.e first an analysis is carried out without the rock bolt and then a correction made due to the . Quadratic boundary elements “plane strain” infinite boundary elements 460 The Boundary Element Method with Programming The overburden above crown is about 75 m. In the analysis therefore the. F () p t () p t t 448 The Boundary Element Method with Programming displacements at the top of the column obtained from the analysis. The results compare well with a reference solution with the FEM that. Figure 17.8. The aim of the analysis was to ascertain the range of validity of 2-D analyses carried out with a distinct element code. 458 The Boundary Element Method with Programming Figure