the boundary element method with programming for engineers and scientists - phần 10 pot

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

Figure 16.8 Distribution of maximum compressive stress, comparison with theory

Here we extend the coupling method to dynamics The dynamic equilibrium equations which arise from finite element discretisation (see Bathe4) 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 method4)

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/m2 The material properties for the column are:

spec weight= 2500 kg/m3, E=30 000 MN/m2 , Q=0.2 For the soil we have: spec

weight= 2000 kg/m3, E=100 MN/m2 , 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

t t

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

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)

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 geomechanics5, 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

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

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

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 BEFE2, 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

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

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

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

Figure 17.4 Contours of maximum principal stress

Figure 17.5 Displaced shape showing crack opening

It was found that in the worst case (lowest parameters assumed for the material) the crack would tend to propagate to the corners of the flange (Figure 17.5) However, even with the crack propagated that far the model predicted that there would be no dramatic failure of the casing Instead, the deformations would become so large that the press would become inoperable

After half a year the new press arrived and was installed The old press gave service without any major problems prior to replacement

The advantages of the BEM over a FEM model may be summarised as:

x The fact that there are no elements inside and no connections were required between elements on opposing boundaries the mesh generation was simplified The number

of unknowns and elements was also reduced

x The stress concentrations were computed more accurately because they are not obtained using an extrapolation from inside the domain but from boundary results

x The method was well suited to model crack propagation

17.3.1 CERN Caverns

The European Laboratory for Particle Physics (CERN) is the world’s largest research laboratory for subatomic particle physics The laboratory occupies 602 hectars across the Franco-Swiss border and includes a series of linear and circular particle accelerators The main Large Electron Positron (LEP) accelerator has a circumference of 26.7 km and

a series of underground structures situated at eight access and detector points (Fig 17.6) The LEP accelerator has been operating since 1989 but in 2000 it has been shut down and replaced by the Large Hadron Collider (LHC) in 2005 This will use all existing LEP structures but will also require new surface and underground works Two new detectors will be installed in two separated cavern systems, called Point 1 and 5

Here we will present the three-dimensional analysis of the new caverns of Point 53,4(Fig 17.7) This is an interesting application because point 1 of the LHC was analysed using the finite element method and a picture of the results appear in the cover of the

book Programming the Finite Element Method 5 According to a report published on this study the mesh had approx 300 000 degrees of freedom and a supercomputer was required to solve the problem

Initially, an elastic analysis was carried out with the single region BE mesh shown in 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

Figure 17.6 Photo showing location of the CERN particle accelerator

Figure 17.7 Cavern system at Point 5, showing existing and new structures

Figure 17.8 Boundary element mesh, single region analysis

Figure 17.9 Results of single region analysis: contours of maximum compressive stress

Quadratic

boundary elements

“plane strain” infinite boundary elements

The overburden above crown is about 75 m In the analysis therefore the ground surface was assumed to be sufficiently far away so that its influence on the cavern was neglected In order to reduce the number of unknowns “plane strain” infinite elements were used, as introduced in section 3.7.2 and as indicated in Figure 17.8 The mesh has

a total of 4278 unknowns and the calculation took 10 minutes on a PC The results of the analysis are shown in Figure 17.9 Here the maximum compressive stress is plotted on two planes inside the rock mass Looking at the horizontal result plane it can be seen that

at a cross-section between the vertical shafts, nearly plane strain conditions are obtained, warranting a 2-D analysis there

Figure 17.10 Coupled boundary element / finite element mesh of USC55 cavern

Figure 17.11 Displacements of the concrete shell due to swelling

Infinite ‘plane

strain’ boundary

elements

Linear boundary elements Linear cells

Linear ‘brick’ finite elements Symmetry plane

Geologists found that a portion of the soil above the cavern could swell significantly

if subjected to moisture Therefore, an analysis had to be carried out to determine the effect of swelling on the final concrete lining Obviously, this cannot be simplified as a 2-D problem because the concrete lining acts as a 3-D shell structure For this analysis a coupled finite element/boundary element analysis was performed with the thin concrete shell modelled by finite elements The swelling zone was modelled by linear cells as explained in Chapter 13 In addition a symmetry plane was assumed between the large and the small cavern Even though in reality no symmetry exists this was thought to be acceptable since the assumption that the second cavern is the same size as the first one would give results that are on the safe side The main reason for the choice of this mesh was that due to time limitations the job had to be completed quickly and only standard PCs were available for performing the analysis The coupled mesh of cavern USC 55 is shown in Figure 17.10 The mesh has a total of 7575 degrees of freedom and the run took 45 minutes on a standard PC Most of the computing time was for computation of the stiffness matrix of the boundary element region

The displacements of the concrete lining due to swelling were determined from the analysis These are shown in figure 17.11 From these 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

17.4.1 How to find gold with boundary elements

The analysis was performed to test a theory of geologists that gold dust was originally suspended in water and was deposited in the ground in locations that had a significantly smaller amount of compressive stress than the surrounding rock6 This seems to make sense, since deposits would naturally occur in voids, i.e., areas where the compressive stress is zero

Since Australia is one of the richer countries in terms of gold resources the story takes place there In particular, the analysis concentrates on what is presumed to have occurred in a region of Western Australia (where a deposit was found) during the Precambrian period (about 800 million years ago) The geologists assume that the region was shortened in an approximate east/west direction and that the deposit was formed at approximately 2.5 km of depth below the surface On this basis it was suggested that a volume of rock of about 2000x2000x1000 m dimension with the geological structure as observed in that area should be analysed The geological structures are shown in Figures 17.12 and 17.13 Figure 17.12 shows contours of the contact between different rock types, whereas Fig.17.13 shows contours of two faults (termed Lucky and Golden faults)

Figure 17.12 Contours of contact between different rock types

Figure 17.13 Contours of Lucky and Golden faults

It was assumed that the block to be analysed was subjected to 2000 m of overburden (which was subsequently eroded) and to tectonic stresses which were estimated from the presumed shortening of the region

Figure 17.14 Definition of boundary element regions

Figure 17.15 Block analysed showing stress boundary conditions applied

132 MPa

50 MPa

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 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 gold deposit was assumed to be So the boundary element method was successfully applied to find gold deposits Note that an analysis with a domain type method would be feasible However, the mesh generation would be more complicated because of the presence of elements inside the regions and the necessity to assure proper connectivity

17.5.1 Masjed-o-Soleiman underground power house

The Masjed-o-Soleiman hydroelectric scheme is situated in the south of Iran The powerhouse is situated underground In 2002 the last of 4 turbines were installed in the existing powerhouse and an extension of the facility to house another 4 turbines was underway During the excavation of the extension, cracks were observed in the concrete walls of the existing powerhouse, which caused some concern In addition

„hot spot“

measurements from pressure cells installed behind the concrete walls recorded seasonally dependent pressure increases that showed an increasing tendency Following

a visit by the panel of experts it was decided to carry out a numerical analysis The aim

of the analysis was to determine the cause of the cracks and to predict if the cracking would get worse because of continuing excavation activity on the extension

Figure 17.17 View of 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

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 cracking was observed are shown in Figure 17.19 Special consideration was given to the circled area near the construction of the extension

Figure 17.19 Plan of powerhouse depicting areas where cracks were observed

The ground conditions in the vicinity of the caverns, as shown in Figure 17.20, are dominated by layers of very weak mudstone and sandstone

Figure 17.20 Geological conditions near the caverns

To ascertain the fineness of the mesh required 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

However, this analysis does not consider the presence of geological features and linear effects, which are important Figure 17.21 shows the mesh with quadratic (8-node) boundary elements and the result for the case where both caverns are excavated, plotted

non-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 introduced in Chapter 3

For a meaningful analysis, however, the effect of the geological features as well as the non-linear behaviour of the ground had to be considered For this purpose a coupled finite element/boundary element mesh was constructed as shown in Figure 17.22 Here the rock mass in the vicinity of the cavern, as well as the concrete structure of the powerhouse is discretised into finite elements Plane strain boundary elements were used

to shorten the mesh in the direction along the cavern, taking into consideration the displacement conditions, as depicted in Figure 17.21 This analysis allows to consider the geological features as well as the nonlinear behaviour of the ground (in particular the mudstone layers)

Figure 17.23 Two of the stages considered in the analysis

Several excavation stages were considered and two of these are shown in Figure 17.23 The mesh on the left models the complete excavation of cavern 1, the one on the right the construction of the powerhouse structure and the excavation stage of the extension as existed during the visit of the panel of experts Figure 17.24 shows the displaced shape on a section through the end of the existing cavern near the extension excavation The deformation of the FEM-BEM interface can be seen especially on top of the cavern, so an analysis without coupling to BEM would not have yielded meaningful

Existing powerhouse cavern

excavated

Powerhouse structure installed, excavation of extension, status Nov 2003

results A view of the finite element mesh of the powerhouse structure, indicating the location of the cracks near the extension is shown in Figure 17.25

Figure 17.24 Displaced shape in a cross-section through the end of the powerhouse

Figure 17.25 View showing the concrete powerhouse and the location of the cracks