11_11. 3d finite element analysis

11 . 3D finite element analysis

11.1 SynopsisAlthough there are many geotechnical problems that can be approximated to eitherplane strain or axi-symmetric conditions, some remain which are very threedimensional. Such problems will therefore require full three dimensional numericalanalysis. This chapter describes how the theory already presented can be extendedto account for such behaviour. However, as it will be shown, the computerresources required for such analysis are considerable. Two ways can be used toreduce these resources. One way is to use iterative techniques to invert the globalstiffness matrix, as opposed to the direct methods used so far. While such methodsare recommended in the literature, it will be shown that, although they can lead toeconomies for linear elastic analysis, they are unlikely to be helpful for nonlinearanalysis. The second way is to capitalise on any geometric symmetries that exist.In particular, if the geometry is axi-symmetric, but the loading or materialbehaviour is not, then the Fourier series aided finite element method can be used.A brief outline of this method and quantification of the savings in computerresources that can be achieved are described in Chapter 12.

11.2 IntroductionIn the presentation of the finite element theory given in previous chapters, attentionhas been restricted to either plane strain or axi-symmetric conditions. In both thesecases symmetry implies that displacements in one particular direction are zero. Bychoosing global axes such that one of the axes coincides with the direction of zerodisplacement, results in considerable simplifications. In finite element analysis itmeans that it is only necessary to account for two displacement degrees of freedomat each node, but more importantly, analysis can be performed over arepresentative two dimensional slice of the problem.

However, in reality most geotechnical problems are three dimensional, and,although in many, plane strain or axi-symmetric approximations are notunreasonable, there are some which must be treated as three dimensional. Thisimplies that three components of displacement must be accounted for and that thecomplete three dimensional geometry must be considered. While theoretically thisdoes not pose any insurmountable problems, in fact the basic theory presented inthe previous chapters still holds, the quantity of data and the size of the various

326 / Finite element analysis in geotechnical engineering: Theory

vectors and matrices increases enormously. This has severe implications forcomputer resources. For example, while an elasto-plastic finite element analysisof a strip footing (plane strain) may take a matter of a few minutes on a fastworkstation, a similar analysis of a square footing (three dimensional) may takemany hours. Most of this extra time is involved in inverting the global stiffnessmatrix. Consequently, mathematicians have investigated ways of improving theinversion process. One way of doing this is to use iterative techniques to solve thestiffness matrix equations, rather than direct methods. In this chapter we considerone of these iterative approaches and make comparisons with the direct approachdescribed in Chapter 2.

Another way of simplifying the analysis is to capitalise on any geometricsymmetries that exist. One approach that does this is the Fourier series aided finiteelement method. A brief description of this approach is given in Chapter 12 and theeconomies that can be obtained, quantified by comparing predictions with thoseobtained from conventional three dimensional analysis.

This chapter begins by describing the basis of conventional three dimensionalfinite element analysis.

11.3 Conventional 3D finite element analysisConventional three dimensional finiteelement analysis follows exactly thesame procedures as described inChapters 2, 9 and 10. The onlydifference is that instead of analysinga two dimensional slice of the originalboundary value problem, the wholedomain must be considered. Thisinvolves discretising the geometry ofthe problem into an assemblage ofthree dimensional finite elements. Themost popular elements are tetrahedraand hexahedra, see Figure 11.1. Theirgeometry is specified in terms of thecoordinates of the element nodes. Forelements with plane faces these nodesare usually located at the elementcorners. If the elements have curvedfaces, extra nodes, usually at the midpoint of each edge, must be introduced.

If the elements are isoparametric, see Section 2.5.1, global elements are derivedfrom a parent element which has the same number of nodes, but is defined withrespect to a natural coordinate system. Figure 11.2 shows the configuration of theparent element for a 20 noded hexahedron. For a global element derived from thisparent element, the global coordinates of a point in the element can be expressed

b) Tetrahedron

Figure 11.1: 3D finite elements

3D finite element analysis / 327

T=\

£/=-!

by coordinate interpolations of theform:

(11.1)

where xh y-t and z, are the globalcoordinates of the 20 nodes in theelement and Nf, / = 1, ..., 20, are theinterpolation functions. Theseinterpolation functions are expressed Rgure ; ; 2; Parent element for ain terms of the natural coordinates S, T 2Q noded hexahedronand U which vary from -1 to +1. Theytake the following form:

Mid - side nodes: .Ng = ,/

S=\

7= .

Nn =i/4(l-52)(1 +i/4(i-r2)(i-1/4(1-t/2)(i—

(11.2a)1/4 (l-U2

1/4(1 —C/2)(l — -, /4 ( l -5 2 ),/4(i-r2xi+5)(i+c/)./4(i-52)(i+r)(i+c/)

#» =Corner nodes:

N, =•T)(\-U) -

(11.2b)

N, =

I - 7)(1 + U)

,/8(l + 5)0-i/8(i-5)(i+:

,̂7 + ^20)


Since the element is isoparametric, the displacements within the element areinterpolated in the same way as the geometry. Therefore, the above interpolationfunctions TV,, N2, ..., N20, are used as the shape functions which express theincremental displacements at any point within the element in terms of the valuesat the nodes:

Au)(11.3)

nodesAw

where u, v and w are the displacements in the x, y and z directions respectively. For4 noded tetrahedron and 8 noded hexahedron the displacements vary linearly,whereas for 10 noded tetrahedron and 20 noded hexahedron the displacements varyquadratically.

The development of the finite element equations then follows the same processas described in Section 2.6, except that there are now six non-zero components ofstress and strain. When performing numerical integrations to evaluate the elementstiffness matrix (see Section 2.6.1), account must be taken of all three dimensionsand therefore more integration points are required. For example, for the 20 nodedhexahedron either 2x2x2 (reduced) or 3x3x3 (full) integration orders are used.

From the above discussion it is yclear that 3D analysis involvesconsiderably more elements, andtherefore nodes and integrationpoints, than comparable twodimensional plane strain or axi-symmetric analysis. It is therefore notsurprising that they require morecomputer resources. As an example,let us consider the analysis of strip,circular and square smooth rigidsurface footings subjected to verticalloading. As the strip and circularfootings are plane strain and axi-symmetric respectively, they can beanalysed using a two dimensionalfinite element mesh. Such a mesh isshown in Figure 11.3 and consists of145 eight noded elements, giving a

ill

A::

Ffs ••Q/j-

RS

a.k l •

:: = : +\/ \ /

\

\

\

y

20 m

a

Figure 11.3: Finite element meshfor strip and circular footing

total of 482 nodes and therefore 964 degrees of freedom (note: there are twodegrees of freedom per node). Because of the line of symmetry acting verticallythrough the centre line of the footing, the finite element mesh only considers halfthe problem. The square footing cannot be approximated to two dimensions andtherefore a full three dimensional analysis is required. There are, however, twovertical planes of symmetry acting through the footing and, consequently, only one


quarter of the problem needs to be discretised into finite elements. A mesh, whichis coarser than the two dimensional mesh shown in Figure 11.3, is shown in Figure11.4. This mesh has 416 twenty noded hexahedron elements, giving a total of 2201nodes and therefore 6603 degrees of freedom (note: there are three degrees offreedom per node).

" 1?~»

^> 1

tx A.;

I

ISO a*

i |i i

I J

\

y ioa4z& area ;S m) |

\ \x ;\ \ \ ;\ \ X :\ \ \ i

\ \ :\ Detail 'A{ \ ""'•\ \ :

Figure 11.4a: Plan view of 3D mesh for square footing(20 noded elements)

c5 m

HUT] Load

50 m

Figure 11.4b: Vertical cross-sectionof 3D mesh for square footing

(20 noded elements)

Analyses have been performed to obtain the undrained load displacementcurves for the three shapes of footing. In all cases the soil was assumed to beelasto-plastic, with a Tresca yield surface. The soil properties were: £=10000 kPa,^=0.45 and £?,= 100 kPa. For the strip and circular footings the mesh shown in


Figure 11.3 was used, whereas for the square footing the mesh shown in Figure11.4 was used. Reduced, 2x2 for the strip and circular footing analyses and 2x2x2for the square footing analysis, integration was used. Loading was simulated byapplying increments of vertical displacement to the footings.

The resulting load-displacementcurves are shown in Figure 11.5.These will be discussed in more detailin Volume 2 of this book. For thepresent discussion we are interested inthe computer resources required foreach analysis. In this respect, it shouldbe noted that all analyses were run onthe same work station. The run timeswere 10 min, 60 min and 18 hours forthe strip, circular and square footingsrespectively. The memory requiredfor the strip and circular footinganalyses was the same, 0.5 Mbytes.For the square footing a much larger

Strip (A/; =5.19)Square (JV/=5.72)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09hlr, b/B

Figure 11.5: Load-displacementcurves for square, circular and strip

footingsrequirement of 38 Mbytes wasneeded. Although this is substantially larger than that required for the strip andsquare footings, the workstation had sufficient memory and it was able to run theanalysis without having to resort to time consuming swapping between memoryand disc.

These results clearly show the massive increase in computer resources that arerequired for three dimensional analyses. Comparing the strip and square footinganalyses, the latter required a 76 fold increase in memory and an 108 fold increasein time. As most practical three dimensional geotechnical problems involve a morecomplex geometry and soil stratigraphy than the simple footing problems analysedabove, they would require considerably more finite elements. The demand oncomputer resources would be such that it would not be possible to performanalyses unless a super computer is used, which would be so costly that analysescould only be justified for a minority of practical problems.

One way of reducing the computer resources is to use 8 noded instead of 20noded hexahedron elements. A mesh of such elements for the square footingproblem is shown in Figure 11.6. This consists of 1740 elements giving a total of2236 nodes and 6708 degrees of freedom. The number of degrees of freedom istherefore slightly higher than that of the analysis described above, using the meshshown in Figure 11.4. The analysis required 28 Mbytes of memory and a run timeof 14.5 hours. The reason for the reduction in memory, compared to the analysisusing 20 noded elements, is due to the reduction in band width of the globalstiffness matrix that arises by using 8 noded elements. This is evident from Figure11.7, which compares the connectivities of the centre node of a block of eight 8noded with a block of eight 20 noded elements. For the 8 noded elements the


centre node is connected, via its eight surrounding elements, to 26 other nodes,whereas for the 20 noded elements the centre node is connected to 80 other nodes.

Figure 11.6a: Plan view of 3D mesh for square footing(8 noded elements)

5m

I III ^oaa

50 m

8

Figure 11.6b: Vertical cross-sectionof 3D mesh for square footing

(8 noded elements)

The load displacement curves obtained from the two square footing analysesare shown in Figure 11.8. While the analysis with the 20 noded hexahedronelements gives a limit load, the analysis with 8 noded hexahedron elementscontinues to show increasing load with displacement. Therefore, although the latteranalysis uses less computer resources, it fails to provide an accurate prediction.Similar behaviour is observed in plane strain analysis when analyses performed


a)Blockof8 8-nodedelements

b)Blockof8 20-nodedelements

Figure 11.7: Node connectivities

3D mesh with 20-noded elements

with 4 and 8 noded quadrilateral Central pointelements are compared. The analysiswith 4 noded elements fails to give anaccurate prediction of the collapseload. The stiff behaviour of the linearelements (4 noded for 2D and 8noded for 3D analysis) is wellrecognised in the literature and it istherefore recommended that thehigher order quadratic elements areused.

Alternative ways of reducing thecomputer resources are thereforerequired. Two such methods aredescribed in this book: iterativetechniques for the solution of theglobal equations are presented in thenext section, while the pseudo 3Danalysis, in the form of the Fourierseries aided finite element method, isthe subject of Chapter 12.

11.4 Iterative solutions11.4.1 IntroductionClose examination of 3D finiteelement analyses reveals that most ofthe computer resources are used in inverting the global stiffness matrix. Thereforeconsiderable effort has been made in trying to find ways to streamline this process.One approach that has recently been recommended in the literature is to use aniterative solution technique, as opposed to the Gaussian elimination approachoutlined in Chapter 2.

In this section, the general process by which iterative solutions invert thestiffness matrix is outlined. The conjugate gradient method, which is one of themost popular iterative methods in current use, is then described. Finally, acomparison of the performance of this method with that of Gaussian eliminationis given.

5 (m)

Figure 11.8: Load-displacementcurves for a square footing

11.4.2 General iterative solutionThe formulation of a set of global stiffness equations in the form of Equation(11.4) was presented in Chapter 2:

nG(11.4)


where [KG] is the global stiffness matrix, {Ad}nG is the vector containing theincremental nodal displacements and {ARG} is the global right hand side vectorcontaining equivalent incremental nodal forces. In finite element analysis it isnecessary to solve this equation for the unknown incremental nodal displacements.Efficient direct solution techiques, described in Chapter 2, require elements of thestiffness matrix to form a narrow band along the main diagonal, see Figure 2.19.This narrow band entraps some zero terms, which must not be ignored becausethey can become non-zero during the process of elimination. As the number ofelements increases, particularly in 3D problems (because of nodal connectivities,see Figure 11.7), a large number of zeros have to be stored within the band, whichautomatically increases the requirements for computer memory. The advantage ofiterative methods is that they use only the non-zero terms in a solution process andtherefore none of the zero terms need to be stored.

Writing Equation (11.4) in an open form gives a system of simultaneousEquations (11.5):

KG]Ad]

nG + KX2Ad2nG + ... + K]

GnAdn

nG = A ^K2]Adl

nG + K22Ad2G + ... + K2nAd"nG = AR2

+ ... + ... + ... = ...K^AdlG + Kn

G2Ad2

G + ... + KnG

nAdnnG =

from which the unknown displacements can be expressed as:

Ad1 - G Ad2 - G Ad" +ACl -—*d " ~W^d +

AdKG AG

01.6)

A<4 A4; ... +G KG KG

Iterative solution methods arrive at a solution to Equations (11.6) by essentiallymaking an initial guess at the answer:

dlnG(0) Ad2

G(0) ... A<G(0)} (11.7)

and then iterating until the correct solution is obtained:

KG

where: Ad'tlG - unknown incremental displacementk - number of iteration

Clearly, the success of such an approach depends on the initial guess and on howthis is modified on successive iterations. Equation (11.8) is usually known as a


Jacobi iteration, or the 'method of simultaneous displacements', because everyelement of the solution vector is changed before any of the new elements are usedin the iteration. It is the simplest procedure, but very often diverges andconsequently there are several different manipulations which can be applied toensure and accelerate convergence.

Adding and subtracting Ad'nG(k) to the right hand side of Equation (11.8) gives:

A<G(* + 1) A<G(*) +KG

From here, the usual iterative form of Equation (11.9), for the total vector {Ad}ttG,can be written as:

where {g}k is a residual, or out of balance, vector generated at each iteration:

For equilibrium, i.e. the correct solution, {g}k should equal zero.The second part of the right hand side of Equation (11.10) is generally written

as an iterative vector, {S}k, and Equation (11.10) becomes:

where, generally:

{<=-[*«]»* (1U3)

The matrix [Ka] in the iterative vector {S}k should be easily invertible in order tospeed up the iterative process. Equation (11.10) indicates that [Ka] should equal thematrix of diagonal stiffness terms. However, sometimes an alternative matrix isused to accelerate convergence. The simplest form of iteration is obtained for [Ka]= [/] (i.e. the identity matrix). If [A«]*[/] the process is known as scaled ox pre-conditioned.

11.4.3 The gradient methodThis group of iterative methods is based on the minimisation of the followingquadratic form (Ralston (1965)):

Q({Ad}knG) = ±«M}k

nG)T[KG]{M}knG -«M}k

nG)r{ARG} (11.14)

Minimising this expression with respect to {Arf},,G gives:


- 0 < • • • " >

Equation (11.15) shows that the minimum of Q is the solution of Equation (11.4).The general technique to minimise Q is to choose an initial vector {Ad}°n(h a

direction of minimisation v° and a step size rf to get to {Ad} lnG. Further iterations

can be written as:

The basis of the method is to choose {v}* to be in the direction of the greatestrate of change of Q, namely in the direction of the gradient. However, because thegradient reflects the direction in which the system grows, and here the opposite isrequired (i.e. minimisation of g), {v}* should be set in the direction opposite to thedirection of the gradient. Since Equation (11.15) shows that the gradient of Q is theresidual vector {g}k, then {v}* = - {g}k. Generally:

where: |<5J =~{g\To find rf it is necessary to rewrite Q in terms of rf:

nG { )nO ' \ ) C l l 17")

[KG]

)T {ARG}k (11.18)

*c)T{g}* -±({Ad}"nG)T{ARG}k

If {Ad}kn(! and {S}k are fixed from the previous iteration, Q has a minimum

dependent on rf:

where from:

* _ mk)J{g}k

It is sometimes useful to express Equation (11.20) in an alternative form:suppose that the out of balance forces {g}k+l are computed from the first 'trial'expression {Ad}k£ which has the step rj = 1. Then:

Subtracting Equation (11.11) from (11.21) gives:


( 1 L 2 2 )

so that the new expression for rf becomes:

a _ «Slk)T{g}k

7

To modify {g}k+l(rj = 1) to the new {g}k+l(rjk) a linear relationship is used, suchthat:

without having to resort to the computationally more expensive Equation (11.11).The gradient method described here is based on an iterative vector, {S}, being

equal to the residual vector, - {g}, see Equation (11.17). However, it can also beapplied to the scaled, or pre-conditioned, gradient direction so that:

( 1 L 2 5 )

11.4.4 The conjugate gradient methodThis is a two parameter extension of the gradient method, which involves aniterative vector, {S}, of the form:

or{ } k { } k * k { } k - ' (12.27)

The parameter pk is found from the condition that:

\ =0 > when l<k (11.28)

which essentially means that the process involves generating a set of [KG]-orthogonal (i.e. [!TG]-conjugate) vectors and then minimising successively in thedirection of each of them, until reaching the final solution. It can be shown (seeCrisfield (1986)) that the condition (11.28) leads to the orthogonality of theresidual vectors, such that:

M)T 0,...,k (11.29)

which is the main 'conjugate gradient' condition. From here, multiplying Equation(11.27) by [KG]{d}"-] gives:


and bearing in mind condition (11.28) leads to:

nk ({8}k*)T[KG]{S}k~l n n n

P ~ k-\ FT (11.31)

As for the parameter if, the above expression can be simplified, in order to avoidcostly calculations with the [KG] matrix, to give the following expression for fik\

The flow chart for the conjugate gradient method is given in Figure 11.9. Asbefore, if [#„]=[/] the method is called 'basic', otherwise it is 'scaled' or 'pre-conditioned'.

The iteration procedure normally terminates when a convergence criterion, p,reaches a pre-set limit, tol. This convergence criterion is usually calculated as theratio of the norm of the residual vector and the norm of the right hand side loadvector:

\\{g}k\\4A (H.33)

11.4.5 Comparison of the conjugate gradient and bandedsolution techniques

This section compares the performance of the conjugate gradient (CG) and banded(BD) solution techniques. As explained in Section 11.4.2, the advantage of the CGapproach is that it is only necessary to store the non-zero terms of the globalstiffness matrix, while for the banded solution some of the zero terms within theband width must be stored. However, the drawback of the CG method is that theinverted global stiffness matrix is never calculated. While this is not a disadvantagein linear problems, it is for nonlinear problems, where the solution strategyinvolves several solutions with the same global stiffness matrix, but different righthand side load vectors (i.e. in the modified Newton-Raphson approach (MNR), seeSection 9.6).

First of all, the requirements for computer memory are investigated, byconsidering blocks of elements with an equal number of elements on each face.These are classified as n x n x n 3D meshes, where n is the number of elements ineach coordinate direction. Figure 11.10 shows how the memory requirementincreases with the 'number of degrees of freedom' (NDF), based on 3D meshes oflxlxl , 2x2x2, 5x5x5, 8x8x8, 10x10x10 and 12x12x12 twenty noded hexahedronelements. Memory is plotted against the NDF, which varies from 60 (lxlxl) to24,843 (12x12x12). It can be seen that the CG solver is more beneficial once theNDF exceeds approximately 1000.


0-th iteration:£ = 0;P° = 0set {Ad}° = 0then {g}° = -{ARG}

Calculate 'trial' solution for r| = 1

Calculate 'trial' residual:

Adjust {Ad}^ and {^}^' to allow forthe new step length:

Calculate tolerance:

tol

YES ,NO

End of iteration

Figure 1 7.9: Flow chart for the conjugategradient algorithm


For lower numbers of degrees offreedom the memory requirement forthe CG method exceeds that for theBD method. This occurs because twostorage vectors are required for theCG method: one containing the non-zero terms of the stiffness matrix andthe other containing the positions ofthe terms in the stiffness matrix.

Solution times for a singleinversion of the global stiffnessmatrix are considered next. This isdone by applying a point load to thecorner of the n x n x n meshes, seeFigure 11.11. The soil is assumed tobe isotropic linear elastic (Young'smodulus, £=1000 kPa) and thereforethe analyses are all linear. Two sets ofanalysis were performed, one setdrained, with a Poisson's ratio //=0.3,and the other set undrained, withju=0A99S. For the CG solutions pre-conditioning was performed, withmatrix [Kn] equal to the diagonalmatrix, [DM], of the global stiffnessmatrix, [KG]. The convergencetolerance was set to 0.001%. Theresults are presented in Figure 11.12for both the CG and BD analyses. Itshould be noted that the solutiontimes for the BD analysis are notdependent on the Poisson's ratio,whereas for the CG analysis thenumber of iterations, and hence thecomputer time, is. Inspection ofFigure 11.12 indicates that forundrained analysis, the CG solver isbeneficial only for the very largenumber of degrees of freedom (over10,000), i.e. for over 1000 twentynoded 3D elements. However, for thedrained analysis (ju=03) the CGsolver is always superior to the BDsolver.

- Banded solution (BD)- Conjugate gradient solution (CG)

100 1000 10000 100000Number of degrees of freedom

Figure 11.10: Memory requirementsfor CG and BD solvers

Point load

nxnxn mesh

Figure 11.11: Boundary conditionsfor nxnxn meshes

- Banded solution- Conjugate gradient - undrained- Conjugate gradient - \k = 0.3

10 100 1000 10000 100000Number of degrees of freedom

Figure 11.12: Times for a singleinversion of the stiffness matrix


100 1000 10000

Number of degrees of freedom

Figure 11.13: Influence of pre-conditioning on the number of

iterations for the CG solver

BD- CG: [Kd\ = [DM]-CG: [Ka] = [I]

The number of iterations requiredfor the drained (//=0.3) analyses areplotted against NDF in Figure 11.13.Also shown on this Figure areequivalent results obtained using [Ka]= [/]. Comparison shows that pre-conditioning with [DM] reduces thenumber of iterations by about 20%.

From the results presented inFigure 11.12 it is clear that the CGsolver is highly dependent on thecompressibility of the soil (i.e. thevalue of Poisson's ratio). Toinvestigate this further, a series ofanalyses using the 8x8x8 mesh, butwith different Poisson's ratios (e.g. juof 0.3,0.4,0.45,0.48,0.49,0.495 and0.499) have been performed. Theresults are presented in Figure 11.14.While the BD solver is insensitive tothe value of ju, the number ofiterations, and therefore computertime, for the CG solver, both with[DM\ and [/] matrices, and with a0.001% convergence tolerance,increases rapidly once |u > 0.48.

As noted above, the drawbackwith the CG (or any iterative) methodis that the global stiffness matrix isnever actually inverted. While this isnot a disadvantage for linear analyses, where only one solution of the stiffnessmatrix is required, it can be a serious drawback for nonlinear analyses in which thesame global stiffness matrix needs to be solved for several different right hand sideload vectors. Such a situation occurs if the visco-plastic or modified Newton-Raphson solution algorithms are used. In these methods the same stiffness matrixis used for all iterations of a particular increment of the analysis (see Sections 9.5and 9.6). If a BD solver is used, the global stiffness matrix can be inverted on thefirst iteration and stored to be re-used on subsequent iterations. If the CG solver isused, the same procedure for solving the global stiffness matrix must be repeatedfor each new right hand side load vector and therefore for each iteration. This canbe very uneconomical and can result in the situation that although the CG solvercan solve the equations quicker, the BD solver wins because it does not have toresolve the equations for each iteration of an increment. As an example, theundrained bearing capacity of a smooth rigid square footing is calculated using

550

500

450

^ 4 0 0

1,350g 300

£ 250

200

150

1000.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Poisson's ratio

Figure 11.14: Sensitivity of asolution to the value of Poisson 's

ratio


both solvers. The mesh is given in Figure 11.4 and consists of 416 twenty nodedelements with 6603 degrees of freedom. The soil is modelled as a Tresca typematerial, with properties as given in Section 11.3 (i.e. £=10000 kPa, ^=0.45,£,,=100 kPa). For the CG analysis the convergence tolerance for the CG algorithmwas set at 0.01%. Both analyses gave the same results, but the CG analysisrequired 7 days of computer time, whereas the BD analysis only required 18 hours,for the same number of load increments in each analysis. Clearly, the BD analyseswas far more economical for this particular situation.

11.4.6 Normalisation of the stiffness matrixAnother problem when using iterative procedures arises when terms within thestiffness matrix differ by several orders of magnitude. The iterative process thenfinds it difficult to converge to a very tight tolerance. Such a situation arises if aconsolidation analysis is performed, because part of the stiffness matrix consistsof the structural stiffness, [KG], (which are large numbers), while the other part hasterms defined from the permeability coefficients, [&G] (which are small numbers),see Equation (10.27). One possible way of overcoming this problem is to normalisethe stiffness matrix in some way, so that it brings all terms within the matrix to asimilar order of magnitude. Naylor (1997) suggested a very elegant form ofnormalisation which retains the symmetry of the stiffness matrix, but does notrequire large computing resources. It consists of the following steps:

- The assembled stiffness equations of the form (11.5) are:

, i=l,...n (11.34).7=1

The normalisation involves the following substitutions:

KG

by AdiG=AdJGylKi! (11.35)

by ~~'IKG

which gives a new system of equations:

" ~ ~AdiG=~ARG , / = !,...« (11.36)7=1

This type of normalisation has been implemented, with pre-conditioning usingthe diagonal terms of the normalised stiffness matrix to obtain [Ka]. As thenormalised diagonal terms are equal to unity, this is equivalent to [#„]=[/].Analyses have been performed for the simple example of n x n x n 3D meshespresented above. The number of iterations for a single inversion of the stiffness


matrix in drained analyses ispresented in Figure 11.15. Alsoshown in Figure 11.15 are resultsfrom Figure 11.13, which are basedon the original (un-normalised)stiffness matrix. Comparison of theseresults indicates that the normalisationprocedure reduces the number ofiterations and is therefore moreefficient. However, its effect is onlymarginal compared to the analyseswith the original stiffness matrix andpre-conditioning with [Ka]=[DM\. Asimilar conclusion arises forundrained analyses of the sameproblem.

The Authors experience with the normalisation procedure when applied to otherboundary value problems is not conclusive. In some cases it is more beneficial thanthe un-normalised approach, whereas in others it is not. In particular, it does notalways overcome the problems involved in coupled consolidation analysishighlighted above.

100 1000 10000 100(

Number of degrees of freedom

Figure 11.15: Influence ofnormalisation on the number of

iterations for the CG solver

11.4.7 CommentsIt can be concluded from the above comparisons that the CG method becomesmore economical than the BD method at solving the global stiffness equations asthe number of degrees of freedom increases and/or the soil is compressible.However, because the global stiffness matrix is never actually inverted, the CGmethod is not ideally suited to nonlinear finite element analysis. The Authors'experience is that for nonlinear problems that are solvable in a realistic time (i.e.with a run time less than a week), the BD solver is always more economical thanthe CG solver. Of course, the situation may well change as computer resourcesimprove and it is possible to analyse larger 3D meshes, in which case the CGapproach may then become more effective.

11.5 Summary1. Full 3D finite element analysis does not involve any major developments to the

theory presented for 2D plane strain and axi-symmetric analyses. The maindifferences are that the full 3D geometry must be discretised and that there arenow three, as opposed to two, displacement degrees of freedom at each node.

2. Full 3D finite element analyses require large amounts of computer resources,both memory and time. With present day computers only very simple nonlinear3D problems can be analysed.


3. It is not advisable to use linear displacement elements in order to reduce thecomputer resources required, because these elements are not able to accuratelyreproduce limit loads. For geotechnical problems higher order (at leastquadratic) displacement elements should be used.

4. In the literature it is claimed that, by using an iterative method to solve theglobal stiffness matrix equations, the computer resources required for full 3Danalysis can be reduced. This is certainly true if the analysis is linear, there isa large number of degrees of freedom and the material is compressible. If thisis not the case, it is questionable whether there is any advantage in using sucha technique. For nonlinear problems, which can be solved in a reasonable time,it appears that direct solution techniques are faster than the iterative methods.This arises because the iterative techniques do not actually evaluate the inverseof the global stiffness matrix and therefore a full solution is required even ifonly the right hand side load vector changes.