lyamin y sloan, 2002b.pdf

8/11/2019 Lyamin y Sloan, 2002b.pdf

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Received 12 February 2001Copyright # 2001 John Wiley & Sons, Ltd. Revised 5 September 2001

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech ., 2002; 26 :181216 (DOI: 10.1002/nag.198)

Upper bound limit analysis using linear nite elements andnon-linear programming

A. V. Lyamin z and S. W. Sloan* ,y ,}

Department of Ci v il, Sur v eyin g and En v ironmental En g ineerin g , Uni v ersity of Newcastle, NSW 2308, Australia

SUMMARY

A new method for computing rigorous upper bounds on the limit loads for one-, two- and three-dimensional continua is described. The formulation is based on linear nite elements, permits kinematicallyadmissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissiblevelocity eld by solving a non-linear programming problem. In the latter, the objective functioncorresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equalityconstraints as well as linear and non-linear inequality constraints.

Provided the yield surface is convex, the optimization problem generated by the upper bound method isalso convex and can be solved efficiently by applying a two-stage, quasi-Newton scheme to thecorresponding KuhnTucker optimality conditions. A key advantage of this strategy is that its iterationcount is largely independent of the mesh size. Since the formulation permits non-linear constraints on theunknowns, no linearization of the yield surface is necessary and the modelling of three-dimensionalgeometries presents no special difficulties.

The utility of the proposed upper bound method is illustrated by applying it to a number of two- andthree-dimensional boundary value problems. For a variety of two-dimensional cases, the new scheme is upto two orders of magnitude faster than an equivalent linear programming scheme which uses yield surfacelinearization. Copyright # 2001 John Wiley & Sons, Ltd.

KEY WORDS : limit analysis; upper bound; kinematic; nite element; nonlinear programming

1. INTRODUCTION

The upper bound theorem states that the power dissipated by any kinematically admissiblevelocity eld can be equated to the power dissipated by the external loads to give a rigorousupper bound on the true limit load. A kinematically admissible velocity eld is one whichsatises compatibility, the ow rule and the velocity boundary conditions. The upper boundtheorem assumes that the deformations are small at incipient collapse and that the material canbe modelled with sufficient accuracy using perfect plasticity and an associated ow rule. Becausean upper bound calculation considers only velocity modes and energy dissipation, the

*Correspondence to. S. W. Sloan, Department of Civil, Surveying and Environmental Engineering, University of Newcastle, NSW 2308, Australia.

yE-mail: [email protected] Research Academic} Professor


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corresponding stress distribution (if one is computed) need not be in equilibrium. Such adistribution, however, must satisfy the yield criterion.

For problems involving complicated geometries, inhomogeneous materials, or complexloading, it is difficult to construct a kinematically admissible velocity eld analytically andnumerical methods are usually necessary. The most common numerical formulation of theupper bound theorem is based on a nite element discretization of the continuum. This leads toa standard non-linear optimization problem where the objective function corresponds to thedissipated power and the unknowns are subject to a highly sparse set of equality and inequalityconstraints. If linear nite elements are used in conjunction with a polyhedral approximation tothe yield surface, the resulting optimization problem is a classical linear program and can besolved using simplex, active set, or interior point algorithms. Although this type of approach hasbeen widely adopted in two-dimensions (see, e.g. References [13]), it is difficult to implement inthree dimensions because the linearization of the yield surface becomes non-trivial. Moreover,even if a suitable linearization can be performed, the process will usually generate a very largenumber of inequality constraints, thereby increasing the cost of solution.

This paper describes the steps involved in a general formulation of the upper bound theoremusing linear nite elements and non-linear programming. In two dimensions, each node has twounknown nodal velocities, each triangular element has three unknown stresses, and eachinterelement velocity discontinuity has four unknown variables. In three dimensions, each nodehas three unknown velocities, each tetrahedral element has six unknown stresses, and eachplanar interelement discontinuity has 12 unknowns. Velocity discontinuities are included in theformulation because they greatly enhance its ability to predict the true limit load, especially forfrictional materials.

In the nal optimization problem, the objective function corresponds to the power dissipatedby plastic deformation in the continuum as well as the power dissipated by plastic sliding alongthe velocity discontinuities. After imposing the ow rule conditions in the continuum, the owrule conditions in the discontinuities, the velocity boundary conditions, the yield conditions onthe stresses, and any loading constraints, the unknowns must satisfy a set of linear equalities as

well as a set of linear and non-linear inequalities. It is shown that the solution to this upperbound optimization problem can be found very efficiently by using a two-stage, quasi-Newtonalgorithm to solve the corresponding KuhnTucker optimality conditions.

The motivation for this work stems from the development of efficient non-linearprogramming schemes for mixed limit analysis formulations by Zouain et al . [4]. Theseprocedures obviate the need to linearize the yield constraints and thus can be used for any typeof yield function. The solution technique itself is based on the two-stage feasible point algorithmoriginating from the work of Herskovits [5]. The new algorithm for solving the upper boundoptimization problem is described in detail and its efficiency is illustrated by a number of examples. Timing comparisons for a variety of two-dimensional problems suggest that the newformulation is up to two orders of magnitude faster than an equivalent linear programmingformulation. Indeed, a key feature of the new solution scheme is that its iteration count is

typically constant, regardless of the mesh renement.

2. DISCRETE FORMULATION OF UPPER BOUND THEOREM

Consider a body with a volume V and surface area A, as shown in Figure 1. Let t and q denote,respectively, a set of xed tractions acting on the surface area A t and a set of unknown tractions

A. V. LYAMIN AND S. W. SLOAN182

Copyright # 2001 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech ., 2002; 26 :181216


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acting on the surface area Aq . Similarly, let g and h be a system of xed and unknown bodyforces which act, respectively, on the volume V . Under these conditions, the objective of anupper bound calculation is to nd a velocity distribution u which satises compatibility, the owrule, the velocity boundary conditions w on the surface area A w, and minimizes the dissipatedpower dened by

W int Z V r ee dV where r denotes the stresses and

ee denotes the plastic strain rates. Once W int has been minimizedto some value W intmin , an upper bound on the true collapse load can be found by equating it to thepower dissipated by the external loads according to

W intmin W ext Z At tT u dA Z Aq qT u dA Z V gT u dV Z V hT u dV

As this type of problem is difficult to solve analytically for cases with complex geometries,inhomogeneous material properties, or complicated loading patterns, we seek a numericalformulation which can model the velocity eld in a general manner. The most appropriatemethod for this task is the nite element method. There are a number of compelling reasons for

choosing linear nite elements, as opposed to higher-order nite elements, for an upper boundformulation. In summary, these are:

* Linear nite elements generate only linear equality constraints on the nodal velocities in thenal optimization problem. This simplies the design of the non-linear programming solverand allows us to devise a very effective solution strategy.

Figure 1. A body subject to a system of surface tractions and body forces.

UPPER BOUND LIMIT ANALYSIS 183



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* It is easier to incorporate kinematically admissible velocity discontinuities at interelementboundaries for linear elements. These discontinuities, which permit large velocity jumps tooccur over an innitesimal distance, allow accurate estimates of the collapse load to becomputed without using excessive numbers of elements. This feature, to a large extent,compensates for the low order of linear elements, especially for frictional materials. Afurther advantage of velocity discontinuities, rst reported by Sloan and Kleeman [3], isthat they prevent the well-known phenomenon of locking for incompressible plasticityproblems.

* Many problems in mechanics have boundaries which can be modelled, with sufficientaccuracy, using a piecewise linear approximation. For cases involving loaded curvedboundaries, it is possible to use a higher-order approximation for the geometry whencomputing the equivalent power dissipated by the external loads. This decreases the errorintroduced by the use of a piecewise linear boundary.

* All the mesh generation, assembly and solution software can be designed to be independentof the dimensionality of the problem. This means that the same code can deal with one-,two- and three-dimensional geometries and greatly simplies the software.

2.1. Linear nite elements

In our numerical formulation of the upper bound theorem, simplex nite elements are used todiscretize the continuum as shown in Figure 2. Unlike the usual form of the nite elementmethod, each node is unique to a particular element and more than one node can share the sameco-ordinates (Figure 3).

In an effort to provide the best possible solution, kinematically admissible velocitydiscontinuities are permitted at all interelement boundaries. If D is the dimensionality of theproblem (where D 1; 2; 3) then there are D 1 nodes in the element and each node isassociated with a D-dimensional vector of velocity variables

ul f ul i gT ; i 1; . . . ; D , ul 1 . . . ul D T

Figure 2. Simplex nite elements used in upper bound formulation.




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These, together with a D D 1/2-dimensional vector of elemental stresses

r e f s eij gT

; i 1; . . . ; D ; j i ; . . . ; D

or

r e f s e11 . . . seDD s

e12 . . . s

eD 1;D . . . s

e1D g

T

and a vector of discontinuity variables, vd , are taken as the problem variables. For convenience,the global vector of problem variables will be referred to as x u; v; r .

The variation of the velocities throughout each element may be written as

u XD 1

l 1

N l ul

where N l are linear shape functions. More formally, the shape functions N l can be written as

N l 1jC j XD

k0

1l k 1jC l kjxk 1

where x k are the co-ordinates of the point k at which the shape functions are to be computed(with the convention that x0 1), C is a D 1 D 1 matrix formed from the elementnodal co-ordinates according to

C

1 x11 ; . . . ; x1D1 x21 ; . . . ; x2D

. . . . . . ; . . . ; . . .

1 xD 11 ; . . . ; xD 1D

or C

x10 x11 ; . . . ; x1D

x20 x21 ; . . . ; x2D

. . . . . . ; . . . ; . . .

xD 10 xD 11 ; . . . ; x

D 1D

2

and C l k is a D D submatrix of C obtained by deleting the l th row and the kth column of C(the determinant jC l kj is the minor of the entry clk of C). In the expressions (2), the superscriptsare row numbers and correspond to the local node number of the element, while the subscriptsare column numbers and designate the co-ordinate index. Elements in the rst column of C are, by convention, allocated the subscript 0 and a value of unity. After grouping terms,

Figure 3. Mesh for upper bound analysis.




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Equation (1) can be written in the more compact form

N l XD

k0

a lk xk ; alk al ; k 1l k 1jC l kj

jC j 3

2.2. Plastic ow rule constraints for continuum

To be kinematically admissible, and thus provide a rigorous upper bound on the true collapseload, the velocity eld at any point in the body must satisfy the set of constraints imposed by anassociated ow rule

ee Bu l r f r ; l5 0; l f r 0 4

where

ee is a DD 1=2-dimensional vector of strain rates, B is the strain ratevelocitycompatibility operator, f is the yield function, l is a non-negative plastic multiplier rate, andr f @=@s 11 . . . @=@s DD @=@s 12 . . . @=@s D 1;D . . . @=@s 1D gT . Conditions (4) imply that the plasticstrains are normal to the yield surface. They also imply there is no plastic deformation inelements whose stresses lie inside the yield surface.

In terms of the notation of Section 2.1, the ow rule constraints (4) can be written for eachnite element e as

eee Beue l er f r e; l e5 0; l e f r e 0 5

where ue is a DD 1-dimensional vector of element nodal velocities, Be is a DD 1=2 DD 1 compatibility matrix composed of D 1 nodal compatibility matrices according to

Be B1; . . . ; Bl ; . . . ; BD 1 6

and

Bl T d111 al ; f 111 ; . . . ; d1DD al ; f 1DD ; . . . ; d11D al ; f 11D

. . . ; . . . ; . . . ; . . . ; . . .dD11 al ; f D11 ; . . . ; dDDD al ; f DDD ; . . . ; dD1D al ; f D1D

264375

7

with the coefficients al ; k given by (3) and the two auxiliary index functions dened as

di ; k; m dikm 1 if i k or i m0 otherwise(

f i ; k; m f ikm

m if i k

k if i m1 otherwise8>:Let B denote the global compatibility matrix (where each element block Be is multiplied by

the element volume V e), u the global velocity vector, r the global stress vector, and k the totalset of plastic multiplier rates (scaled by the element volume V e). We can then write the ow rule




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constraints for the mesh in matrix form as

Bu XE

j 1

l j r f j r 8

l j 5 0; j 1; . . . ; E 9

l j f j r 0; j 1; . . . ; E 10

where E is the number of elements. Thus, the continuum ow rule generates E ED D 1=2equality constraints and E inequality constraints on the problem variables.

2.3. Flow rule constraints for discontinuities

A three-dimensional velocity discontinuity is shown in Figure 4. These are introducedat all interelement boundaries, are dened by D pairs of adjacent nodes, and are of zero thickness. To be kinematically admissible, the normal and tangential velocity jumps acrossthe discontinuity must satisfy the ow rule. For a MohrCoulomb yield criterion, this is of the

formD un jD ut j tan f 11

where Dun and Dut are the normal and tangential velocity jumps, respectively, and f is thematerial friction angle. For a dilatational yield criterion whose shape is non-linear in themeridional plane, the ow rule constraints are applied using a local MohrCoulombapproximation to the yield surface (Figure 5). Non-dilatational yield criteria, such as theTresca and Von-Mises surfaces, with a cohesion c and f 0 are simpler to model as they imply

Figure 4. Three-dimensional velocity discontinuity.




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that Dun 0. The absolute value on the right-hand side of Equation (11) is necessary because,for a non-zero friction angle, dilation occurs regardless of the direction of tangential shearing.This feature makes it difficult to derive a set of ow rule constraints which are everywheredifferentiable. Our method for implementing kinematically admissible velocity discontinuities isa D-dimensional generalization of the formulation proposed originally by Sloan and Kleeman[3]. Each nodal pair ( l ; m) on a discontinuity is associated with D 1 pairs of non-negativevariables v

lm and v lm , as shown in Figure 4, and thus gives rise to 2 D 1 additionalunknowns. The tangential components D u0t

lm of the velocity jump Du0lm at each nodal pair aredened as the difference between these two sets of non-negative variables according to

D u0lmt v lm v lm 12

where

v lm v

lm1 ; . . . ; v

lmD 1

n oT

v lm v lm1 ; . . . ; vlmD 1n o

T

D u0t lm Du01lm ; . . . ; Du0lmD 1

T

D u0lm f D u0tlmgT ; Du0D

lm

T

The vector Du0 holds the velocity jumps in terms of the local co-ordinate system of Figure 4,and is related to the global velocity jumps Du by the expression

D u0 bD u

where b is the appropriate matrix of direction cosines. To remove the absolute value sign in theow rule relation (11), and thus avoid non-differentiable constraints, jD ut j is replaced by

XD 1

i 1v

i v i

Figure 5. MohrCoulomb approximation of a general yield function.




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so that the normal velocity jump is given by

D un D u0D XD 1

i 1v

i v i tan f 13Sloan and Kleeman [3] have proved that this substitution preserves the upper bound nature of

the solution, as it is equivalent to modelling a stronger discontinuity with properties c ; f where c 5 c and f 5f . Since the velocities vary linearly over each element, Equations (12) and(13) are satised for every point on the discontinuity if the ow rule constraints are imposed ateach pair of its nodes ( l ; m) according to

D u0i lm v

lmi v

lmi ; i 1; . . . ; D 1

D u0Dlm X

D 1

i 1v lm

i vlmi tan f

D u0lm b lm D ulm

D ulm um ul

v lm

i 5 0; vlmi 5 0; i 1; . . . ; D 1 14

In matrix notation, conditions (14) can be written for discontinuity d as

Ad u flow ud Ad v flow v

d bd flow 15

vd 5 0 16

where

Ad u flow

bl 1m1 bl 1m1 0

..

.. . . ..

.

0 bl D mD bl D mD

26664

37775

Ad v flow

1 . . . 1 . . . 0

..

.. . . ..

.0

tan f . . . tan f

..

.. . . ..

.

1 . . . 1 . . . 0

0 ...

. . . ...

tan f . . . tan f

26666666666666664

37777777777777775ud f um1 gT ; ful 1 gT ; . . . ; fumD gT ; ful D gT

T

vd v l 1m1n o

T; v l 1m1n o

T; . . . ; v

l D mDn oT

; v l D mDn oT

T

bd flow f 0g




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Thus each discontinuity gives rise to a total of D2 equality constraints and 2 DD 1inequality constraints on the nodal velocities and discontinuity variables.

2.4. Velocity boundary conditions

To be kinematically admissible, the computed velocity eld must satisfy the prescribedboundary conditions. Consider a distribution of prescribed velocities w p, p 2 P , where P isa set of N p prescribed components, over part of the boundary area Aw (Figure 6). If these velocities are specied in terms of global co-ordinates and distributed over thelinearized boundary area Adw, we can cast the velocity boundary conditions for every node l on this area as

ul p wl p; p 2 P 17

If the prescribed velocities are dened in terms of local co-ordinates, as shown in Figure 6,then the velocity boundary conditions may be written as

u0 pl b l pi u

l i w

l p; p 2 P 18

where bl pi are the direction cosines of the local Cartesian co-ordinate system for node l .Note that it is generally preferable to impose the constraints (18) on the nodal velocities with

respect to the original, and not the linearized, boundary. If the latter is used with complexsurface geometries, the surface normal is constant over each boundary segment and the velocityboundary conditions are represented less accurately. Considering (17) as a variant of (18) withb I , we can cast both of the above conditions in the general matrix form

Abbound ub bbbound 19

Figure 6. 2D velocity boundary conditions.




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where

ub ul

Abbound bl T p1 . . . b

l T pNp

h iT

bd bound f wl pg

T ; p 2 P

Every node that is subject to prescribed velocities generates a maximum of D equalityconstraints on the problem unknowns.

2.5. Objecti v e function

Plastic ow may occur in both the continuum and the velocity discontinuities. Consequently, thetotal power dissipated in these regions constitutes the objective function. For each continuumelement of volume V e, the power dissipated by the plastic stresses is given by

W ec Z V e rT ee dV Z V e r

T Bu dV V er eT Beue

where Be is dened in (6). For each velocity discontinuity of surface area S d , the powerdissipated by plastic shearing is given by an integral of the form

W sd Z S d jt D ut j s n D un dS which, using the ow rule (11) and the MohrCoulomb yield criterion jt j c s n tan f ,becomes

W sd Z S d cjD ut jdS Z S d c XD 1

i 1v i v i

2

" #1=2

dS

Unfortunately, W sd is non-differentiable with respect to the problem unknowns when D ut 0.To avoid this singularity, the proposed formulation uses the result that

jD ut j4 XD 1

i 1v

i v i so that an upper bound on the power dissipated in each discontinuity is given by

W sd

Z S d

c

X

D 1

i 1v

i v i

dS 20

Assuming the discontinuity variables and cohesion c vary linearly over the discontinuity,integrating (20) gives

W sd cT Z S d N sc eN sv dS vd 21




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where

c f cl 1 ; . . . ; cl D gT

N sc f N l 1 ; . . . ; N l D gT

N sv IN l 1 ; . . . ; IN l Dand e i s a 2D 1-dimensional unit vector, I i s a 2D 1 2D 1-dimensional identitymatrix, and the shape functions Nsc and N

sv are for ( D 1)-dimensional space.

When we equate the rate of work of the external forces

Z At tT u dA Z Aq qT u dA Z V gT u dV Z V hT u dV to the rate of internal dissipation

Z V e r T ee dV Z S d jt D ut j s n D un dS Z V e r T Bu dV Z S d cjD ut j dS then, if some of the external forces are xed, their work contributions need to be subtractedfrom the objective function. For side S t of an element with nodes l 1; . . . ; l D that is subject toxed surface tractions t we have

W st Z S t tT u dS Z S t N st sT N susdS t sT Z S t N sT N s dS us 22where

N s IN l 1 ; . . . ; IN l D

us f ul 1 gT ; . . . ; ful D gT

T

t s f t l 1 gT ; . . . ; f t l D gT T

Similarly, for an element e subjected to xed body forces g we can write

W eg Z V e gT u dV Z V e geT N eue dV geT Z V e N e dV ue 23where

N e IN l 1 ; . . . ; IN l D 1

ge f g 1; . . . ; g D gT

and I is now a D D dimensional identity matrix and the shape functions Ne are for D-

dimensional space. Thus, the nal objective function Q is dened by

Q XE

e1

W ec XDs

s1

W sd XN t

s1

W st XE

e1

W eg 24

where E is the total number of elements, Ds is the total number of discontinuities, and N t is thetotal number of element sides which are subject to prescribed surface tractions. Collecting




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coefficients, (24) can be written as

Q r T Bu cTu u cTv v 25

where B is the global compatibility matrix introduced in Section 2.2, and cu and cv are vectors of linear coefficients for velocity and discontinuity variables, respectively, accumulated from

expressions (21), (22) and (23).

2.6. Loadin g constraints

Additional constraints are often needed to specify the loading pattern and introduce relationsbetween the components of the velocity eld. For loading with a proportional displacementpattern, it is convenient to introduce the notion of a velocity prole. Figure 7 shows a boundaryelement segment, dened by the nodes 1 and 2, where the velocity eld is optimized subject tothe constraints of a known normal velocity prole Prf un . Under conditions of proportionaldisplacement, we have

u1n optPrf un 1

u2n opt

Prf un 2

where the precise values of Prf un 1 and Prf un 2 may be chosen arbitrarily, provided they t therequired distribution. Each velocity component may have its own prole and each node l on theprole has a sign, denoted si g nPrf un l , which is equal to 1 and determines the direction of the optimized component. In cases where the nal distribution of the optimized velocities isunknown, all nodes on the velocity prole are specied to have the same sign.

The most common types of loading constraints are load orientation and load proleconstraints. The former occur when the orientation of the velocities on a loaded part of the bodyis known, so that the ratio of velocity components at a particular point is xed. In Figure 8(a),for example, the ratio of the normal and tangential velocities at node 1 can be prescribed interms of local co-ordinates by choosing appropriate values for Prf u01

1 and Prf u021. Load

prole constraints are different to load orientation constraints in that they are used to specify

the distribution of a single component of the unknown velocities over the loaded segment. InFigure 8(b), the required distribution of the normal and tangential velocities can be obtained byprescribing suitable values, respectively, for Prf u01

1 and Prf u012 and Prf u02

1 , Prf u022 .

Restricting the normal velocities to be uniform over part of the boundary is one very commontype of load prole constraint.

Figure 7. Velocity prole for applied boundary traction.




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To specify the directions of the optimized deformations, we can introduce constraintswhich specify the ratio of the velocity components at all nodes along a loaded segment.Since the velocities are allowed to vary linearly within an element, this ratio need not beconstant along the segment. When the deformation pattern is specied with respect to theglobal co-ordinate system, the load orientation constraints for the velocities at node l can bewritten as

ul omPrf uom

l ul om 1

Prf uom 1 l ; om; om 1 2 O ; m 1; . . . ; N O 1

where O is the set and N O is the number of optimized velocity components. When theoptimization is conducted with respect to a local co-ordinate system, the above expressions takethe form

b l om i ul i

Prf u0om l

b l om 1 i ul i

Prf u0om 1 l

or

1

Prf u0om l b

l om

1

Prf u0om 1 l b

l om 1

" #ul

0

where

om; om 1 2 O ; m 1; . . . ; N O 1

Figure 8. 2D example of loading constraints: (a) velocity orientation constraints for node 1 and (b)velocity prole constraints for nodes 1 and 2.




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As each load orientation constraint is a simple linear equality, it can be stated in matrixnotation as

Aoorient u boorient 26

where the maximum number of constraints on the velocities is ( D 1) per node on Adq .If any component of the unknown velocity eld is to be optimized according to a particular

distribution (or prole), then the loading constraints need to include node-to-node relations toreect this. When the optimization is done with respect to global co-ordinates, the node-to-nodeload prole constraints for the oth component of the velocities can be written as

ul moPrf uol m

ul m 1o

Prf uol m 1; o 2 O ; l m; l m 1 2 Adq

When the load optimization is done in terms of local co-ordinates, the above expressionsbecome

b l moi ul mi

Prf uol m

bl m 1oi ul m 1i

Prf uol m 1

or

1

Prf uol mbl mo u

l m 1

Prf uol m 1bl m 1o u

l m 1 0

where

o 2 O; l m; l m 1 2 Adq

Each load prole constraint is a linear equality of the form

Aoprofile u boprofile 27

and adds a maximum of D equalities per pair of nodes on A dq .To ensure the velocity eld is unique we may, in some cases, need to impose an additional

scaling constraint on the velocity variables. For proportional loading this is typically of the form

Z Aq qT u dA Z V hT u dV 1Letting N q denote the number of element sides which are subject to a surface traction q, thisequation can be written in matrix notation as

XN q

s1

W sq XE

e1

W eh 1 28

where

W sq Z S t qT u dS Z S t N sqsT N susdS qsT Z S t N sT N s dS usW eh Z V e hT u dV Z V e heT N eue dV heT Z V e N e dV ue




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and

qs f ql 1 gT ; . . . ; fql D gT T

he h1; . . . ; hDf g T

ue f ul 1 gT ; . . . ; f ul D 1 gT T

As before, the shape functions N e are for D-dimensional space while the shape functions Ns

are for ( D 1)-dimensional space.Thus, for proportional loading, upper bound estimates on the applied forces can be expressed

as

qupper Q * q

hupper Q * h

where Q is the value of the objective function (24) at the optimum point x .When the external traction or body force distribution is unknown, the upper bound techniquecan be used to estimate the collapse load in terms of mean tractions or mean body forces. In thiscase, the prole of the mean load (

%

qq; %

hh) should be also specied so as to distinguish the forcecomponents. The scaling constraints now become

Z Aq %qqT u dA Z V %hhT u dV 1or

X

N q

s1

W sq

X

E

e1

W eh 1 29

where

W sq q

T Z S t u dS qT Z S t N susdS qT Z S t N s dS usW

eh h

T Z V e u dV hT Z V e N eue dV hT Z V e N e dV ueand the mean surface tractions and body forces are

q f q1; . . . ; qD gT

h f h1; . . . ; h

DgT

Thus, the mean upper bound estimates on the applied forces can be expressed as%

qqupper Q * %

qq

%

hhupper Q *

%

hh




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Finally, all the various loading constraints may be summarized by grouping Equations (26),(27) and (28) or (29) to give a matrix equation of the form

Aload u bload 30

2.7. Yield conditionThe only requirement for the stresses in the upper bound formulation is that they satisfy theyield condition. For a perfectly plastic solid, we thus have

f s ij 4 0 31

where f is a convex function which is dependent on the stresses and certain material parameters.The solution procedure presented here does not depend on a particular type of yield function,but does require it to be both convex and smooth. Convexity is necessary to ensure the solutionobtained from the optimization process is the global optimum, and is actually guaranteed by theassumptions of perfect plasticity. Smoothness is essential because the solution algorithm,described later, needs to be able to compute rst and second derivatives of the yield functionwith respect to the unknown stresses. For yield functions which have singularities in theirderivatives, such as the Tresca and MohrCoulomb criteria, it is necessary to adopt a smoothapproximation of the original yield surface.

As the element stresses are assumed to be constant, there is only one yield condition per niteelement. This implies that the stresses in the nite element model must satisfy the followinginequality:

f s eij 4 0; e 1; . . . ; E 32

or f j r 4 0; j 2 J s 33

for all E elements. Thus, in total, the yield conditions give rise to E non-linear inequalityconstraints on the element stresses. Because each element is associated with a unique set of stress

variables, it follows that each yield inequality is a function of an uncoupled set of stress variabless eij . This feature, discussed again later, can be exploited to give a very efficient solutionalgorithm.

2.8. Assembly of constraint equations

The steps necessary to formulate the upper bound theorem as an optimization problem havenow been covered, and all that remains is to group the linear constraints for the whole mesh.Using Equations (15), (19) and (30) the various equality constraints may be assembled to givethe linear equality constraint matrices according to

Au

X

Ds

d 1

Ad u flow

X

Bn

b1

Abbound

X

Ld

l 1

Al load

Av XDs

d 1

Ad v flow

where Ds is the total number of discontinuities, Bn is the total number of boundary nodessubject to prescribed surface tractions, Ld is the total number of loading constraints, and all




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coefficients are inserted into the appropriate rows and columns using the usual element assemblyrules.

Similarly, the corresponding right-hand side vector b is assembled according to

b

XDs

d 1

bd flow

XBn

b1

bbbound

XLd

l 1

bl load

Thus the problem of nding a kinematically admissible velocity eld which minimizes theinternal power dissipation may be stated as

minimize Q r T Bu cTu u cTv v on u; v 34

subject to Auu A v v b

Bu X j 2J s l j r f j r l j 5 0; j 2 J sl j f j r 0; j 2 J s

f j r 4 0; j 2 J sv5 0

u 2 R nu ; v 2 R nv ; r 2 R ns ; k 2 R E 35

where B is an ns nu global compatibility matrix, cu is an nu-dimensional vector of objectivefunction coefficients for the velocities, cv is an nv -dimensional vector of objective functioncoefficients for the discontinuity variables, ns ED D 1, nu DN , N is the number of nodesin the mesh, nv 2DD 1Ds , A u is an r nu matrix of equality constraint coefficients for thevelocities, Av is an r nv matrix of equality constraint coefficients for the discontinuityvariables, r D2Ds + number of boundary condition and loading constraints, f j r are yieldfunctions, l j are non-negative multipliers, and u, v and r are problem unknowns.

3. SOLUTION OF UPPER BOUND OPTIMIZATION PROBLEM

The numerical formulation of the upper bound theorem presented in the previous sectionsresults in a convex optimization problem. Standard optimization theory [6] indicates that, withan appropriate assumption of differentiability, the KuhnTucker optimality conditions are bothnecessary and sufficient for the solution to a such a problem. This fact means that we cantransform (34) into a set of KuhnTucker optimality conditions and explore the possibility of using a rapidly convergent Newton method to solve the resulting system.

Recently, Zouain et al . [4] have proposed a feasible point algorithm for solving problemsarising from their mixed limit analysis formulation. Although the optimization problem

considered by these authors is different from the one considered here, this algorithm can beadapted to suit our requirements. In brief, the Zouain et al . scheme uses a quasi-Newtoniteration formula for the set of all equalities in the optimality conditions. After each Newtoniteration, the resulting search direction is deected in order to preserve feasibility with respect tothe inequality conditions. In our algorithm, we use a new deection strategy, plus a number of other modications, to account for the nature of the optimization problem that is generated by




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the upper bound formulation. To proceed with the algorithm description, we rst write downthe KuhnTucker optimality conditions.

3.1. KuhnTucker optimality conditions

To begin, we note that Equations (2)(5) of (35) are the KuhnTucker optimality conditions forthe following problem:

maximize Q r T Bu cTu u cTv v on r

subject to f j r 4 0; j 2 J su 2 R nu ; v 2 R nv ; r 2 R ns

Incorporating this optimization problem in the original problem (34), and dropping thecorresponding optimality conditions, leads to

maximize Q r T Bu cTu u cTv v on r

minimize Q r T Bu cTu

u cTv v on u; v

subject to Auu A v v b f j r 4 0; j 2 J s f j v4 0; j 2 J vu 2 R nu ; v 2 R nv ; r 2 R ns 36

where the original conditions v5 0 are, for the sake of convenience, written as f j v4 0; j 2 J v .Since the objective function and the functions f j are convex, and all equality constraints arelinear, the solution to (34), ( u ; v ; r ), must satisfy the KuhnTucker optimality conditions

Bu *

X j 2J s

l s j r f j r * 0

cv ATv l X j 2J s l v j r f j v* 0BT r * cu ATu l 0Auu A v v bl s j f j r * 0; j 2 J sl v j f j v* 0; j 2 J v f j r * 4 0; j 2 J s f j v* 4 0; j 2 J vl s j 5 0; j 2 J s

l v j 5 0; j 2 J vu * 2 R nu ; v* 2 R nv ; r * 2 R ns

k s 2 R E ; kv 2 R nv ; l 2 R r 37

A detailed description of the two stage quasi-Newton strategy for solving the system (37) isgiven in the following sections.




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where

H s P j 2J s lks j r

2 f j r k

H v

P j 2J v

l kv j r2 f j vk

G Ts r f 1rk . . . r f E r k

G Tv r f 1vk. . . r f nv v

k

F s diag f j r k=l ks j 50

F v diag f j vk=l kv j 51

r Buk

a BT r k cu

and

ds r k 1 r k ; dv vk 1 vk ; du uk 1 uk

l l k 1; ks kk 1s ; kv kk 1v

If we denote

T

H s 0 B 0 G Ts 0

0 H v 0 ATv 0 G

Tv

BT 0 0 ATu 0 0

0 Av Au 0 0 0

G s 0 0 0 F s 0

0 G v 0 0 0 Fv

26666666664

37777777775

; y

dsdvdul

k s

k v

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

; w

r

cva

0

0

0

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;then the system dened by Equations (44)(49) may be rewritten as

Ty w 52

Now we have a highly sparse, symmetric system of ns nv nu r E nv linear equationsthat can be solved for the unknowns y.

Since our formulation employs an uncoupled set of stress variables for each element, H s isblock diagonal and G s also consists of disjoint blocks. The size of each block in H s is equal tothe number of stress variables in the corresponding yield constraint (with a maximum of six inthe 3D case) and H 1s can be computed very quickly element-by-element. The matrices H v andG v are diagonal. All entries in H v are articially set to a small value to allow H 1v and the partialfactorization of the matrix T to be computed.

For some yield functions, the Hessian H 1s is not a positive denite matrix. In these cases, weapply a small perturbation eI to restore positive deniteness according to

H s X j 2J s l ks j r2 f j r k e j I j where e 2 10 6; 10 4 .




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3.2.1. Sta g e 1: estimatin g the increment . In the rst stage of the algorithm, estimates for ds , dv ,du, l , k s and k v are found by exploiting the above-mentioned features of the matrices H s , G s , H vand G v . From (44) we have

ds 0 H 1s B%

uu du0 GTs k s 0 53

where the subscript 0 signies that the solution is from stage one of the algorithm and uk isreplaced by

%

uu. Substituting (53) into (48) gives

k s 0 W 1s QTs B

%

uu du0 54

where the diagonal matrix W s and the matrix Q s are computed from

W s G s H 1s GTs Fs

Q s H1

s GTs

Next we can eliminate ks 0 from (53) using (54) to give

ds 0 D s B%

uu du0 55with

D s H 1s Q s W1

s QTs

Multiplying both sides of (55) by BT and taking into account (46) we obtain

Ks du0 ATu l 0 a Ks%

uu 56

where

Ks BT D s B

is an nu nu symmetric positive semi-denite matrix [4] whose density is roughly double thatof B.

Repeating the above steps for Equations (45), (49) and (47) we nally have

k v 0 W 1v QTv A

Tv l 0 cv 57

dv 0 D v ATv l 0 cv 58

ATu du0 Kv l 0 A v D v cv 59

where

W v G v H 1v GTv Fv

Q v H 1v GTvD v H 1v Q v W

1v Q

Tv

and

Kv A v D v ATv




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is an r r symmetric positive semi-denite matrix [4]. To start the computational sequence, du0and l 0 are rst found using (56) and (59)

du0l 0( ) E 1 a Ks

%

uu

Av D v cv( ) 60where

E Ks ATuAu Kv" #

The quantities ds 0 , dv0 , ks 0 and kv 0 are then computed, respectively, using (55), (58), (54)and (57).

For large 3D problems, this computational strategy is up to ten times faster than solving (52)by direct factorization of the matrix T . It should be noted, however, that the condition numberof the matrix E is equal to the square of the condition number of the matrix T as the optimumsolution is approached. Fortunately, experience suggests that double precision arithmetic issufficient to get an accurate solution using (60) before the matrix E becomes badly ill-

conditioned or numerically singular.

3.2.2. Sta g e 2: computin g a deected feasible direction . From (48) we see that the incrementvector ds 0 found in the rst stage of the algorithm is tangential to the active yield constraints f j r k 0. To preserve feasibility, this search direction must be adjusted by setting the right sideof every row in (48) to some negative value which is known as the deection factor y. Lyamin [7]and Lyamin and Sloan [8] have described such a deection technique for lower bound limitanalysis. This strategy, which is applicable to the upper bound formulation as well, has a simplegeometrical interpretation and works for all common yield criteria.

Consider the component-wise form of Equation (48). Letting ds j denote the part of the vectords which corresponds to the set of variables of the function f j , and using the notation

% ll j insteadof l k j , we can rewrite (48) in the form

r f Ts j ds j f s j =% ll s j l s j 0 61

Using the geometrical denition of the scalar product, this may also be expressed as

kr f s j kkds j kcos c j f s j =% ll s j l s j 0

where c j is the angle between the normal to the j th constraint and the vector ds j . If we nowintroduce the deection factor as the product kr f s j kkds 0 j kys j , then ys j is equal to cos c j for allactive yield constraints. When an equal value of ys j is used for these cases, this implies that theangle between the vector ds j and the tangential plane to the yield surface at the point r k j , whichwe term the deection angle ys , is identical for all active constraints. For many yield surfacesused in limit analysis, the curvature at a given point r k j is strongly dependent on the directionds j . Therefore, it is better to make y proportional not only to the length of the vector ds j , butalso to the curvature ws j of the function f s j in the direction ds j . The latter may be estimated asthe norm of the derivative of r f s j in the direction ds 0 j multiplied by Rs j , where Rs j is thedistance from the point r k j to the axis of the yield surface in the octahedral plane. Combining allthe above factors suggests that (61) should be replaced in stage 2 of the algorithm by

r f Ts j ds j f s j =% ll s j l s j ys kr f s j kkds 0 j kws j 62




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where

ys 2 0; 1ws j max w0s j =ws max ; wmin

w0s j

kr 2 f s j r k ds 0 j kR s j kr f s j kkds 0 j k

ws max max j 2J sw0s j

wmin 2 0; 1

Note that ys is equal to the sine of the maximum deection angle ys max and can be relatedanalytically to the increment in the objective function dTs 0Bu0 (as will be shown later).

Applying the standard deection strategy to the linear inequality constraints for thediscontinuity variables leads to (49) being replaced for each component by

r f Tv j dv j f v j =% ll v j l v j yv kr f v j kkdv0 j k 63

where

yv wmin

Replacing (48) and (49) by, respectively, the matrix form of (62) and (63) furnishes the systemof equations for computing the new feasible direction as

H s ds Bdu GTs k s r 64

H v dv ATv l G

Tv k v cv 65

BT ds ATu l a 66

Av dv Audu 0 67

G s ds Fs k s ys x s 68

G v dv Fv k v yv x v 69

where x s and x v are, respectively, E - and nv -dimensional vectors with components

o s j kr f s j kkds 0 j kws j

o v j kr f v j kkdv 0 j k

Applying the same computational sequence that was used to solve the system (44)(49), thesolution of Equations (64)(69) can be obtained using the steps

ds D s B%

uu du ys Q s W1

s x s 70

k s W 1s QTs B

%

uu du ys W 1s x s

dv D v ATv l cv yv Q v W

1v x v

k v W 1v QTv A

Tv l cv yv W

1v x v




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where du and l are rst found from

dul( ) E 1 a Ks

%

uu ys BT Q s W1

s x s

Av D v cv yv Av Q v W1

v x v( )To derive an expression for ys we multiply both sides of (70) by

%

uu du0BT

, then use (54) and(55) to obtain

ys dTs 0Bu dTs Bu0=k

T0 x

whereu0

%

uu du0; u %

uu du

Assuming u u0 and imposing the condition that the Stage 1 and Stage 2 changes in theelement dissipated power are related by a xed parameter b 2 0; 1 , such that dTs Bu bdTs 0Bu0; ys can be computed as

ys 1 bdTs 0Bu0=kTs 0x

This gives the required deection angle for each iteration. A typical value for b, which is usedfor all the runs in this paper, is 0.7.

3.3. Initialization

So far we have assumed that the current solution ( uk ; vk ; r k ; kks ; kkv ) is a feasible point. To start

the iterations, we therefore need a procedure which will furnish an initial feasible point(u0; v0; r 0; k0s ; k

0v ). For r

0; k0s ; k0v we can start with

r 0 0; k0s 1; k0v 1

but the pair ( u0; v0) must satisfy the equality constraints

Auu0 A v v0 b 71

In order to convert (71) to a linear system with a square symmetric coefficient matrix, it can beaugmented by a set of dummy variables g to become

I A T

A 0" # z0g( ) 0b( )where

A AuAv ; z0 u0T ; v0T T

Writing the equations in this form allows us to use the same solver as for the main system of equations (44)(49). They can be readily solved using the relations

z0 AT g

AA T g b

When factorizing the symmetric matrix AAT , however, we need to be aware of possibleredundancies in the matrix of equalities A . These are detected when a zero (or near zero) entryoccurs on the diagonal of the factored form of AAT , and are marked in a single factorizationpass. If detected, the redundant equalities are deleted from further consideration in thealgorithm (except when checking the feasibility of the current solution) and AAT is factorized




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afresh to obtain values of g and the initial feasible point z0 . Once z0 has been obtained, we thensubstitute it into the inequalities and compute the scalar

a max j 2J v

f v j z0

If a4 0; z0

satises all the inequalities and the initial search direction d0 f d0Tu ; d

0Tv g

T

can befound by solving

I A T

A 0" # d0g( ) c0( )where c f cTu ; cTv g

T and g is again a set of dummy variables that are introduced for computationalconvenience. We can then start the iteration process. If, on the other hand, a > 0, this signies thatz0 does not satisfy all the inequalities and it is necessary to solve the so-called phase one problem

minimize a r 1subject to A a za b

a j 1 a j 0; j 2 J v f v j za a j 4 0; j 2 J v

a r 14 0

za 2 R nu nv r 1

72

where zTa f zT ; a1; . . . ; ar 1g and A a is the matrix A augmented by r 1 columns. Because of the

similarity of problem (72) with problem (34), it can be solved using the same two-stage algorithmbut dropping the yield inequalities. If the optimal solution to the phase one problem (72) is such thata a1 a r 1 0, then z0 is found by discarding the last r 1 entries in za . Otherwise, theoriginal upper bound optimization problem has no feasible solution and the process is halted.

3.4. Con v er g ence criteria

At the end of each Stage 1 iteration, various convergence tests are performed on the set of optimalityconditions (37). Noting that a11 constraints of the optimization problem (36) are satised at everystage in the solution process, our dimensionless convergence checks are of the form

Bu X j 2J s l s 0 j r f j r 4e ukBuk 73l s 0 j 5 el l s max ; j 2 J s 74

l s 0 j 4e l l s max if f j r 5 e f ; j 2 J s 75

cv ATv l X j 2J v l

v0 j r f j v 4e ukcv k 76

l v 0 j 5 el l v max ; j 2 J v 77

l v 0 j 4e l l v max if f j v5 e f ; j 2 J v 78




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in expressions (50), (51) and we wish to maintain negative entries on the diagonals of F s and F vso that the deection strategy of (62), (63) is able to maintain feasibility. The rule for updatingthe Lagrange multipliers is

l s j max l s 0 j ; dsl l s max ; j 2 J s

l v j max l v 0 j ; dv l l v max ; j 2 J v

with

dsl min d0l ; kds k=kr k

dv l min d0l ; kdv k=kvk

where d0l 2 104; 10 2 is a prescribed tolerance. This rule allows the l j to converge to a positive

value if l 0 j is positive, but sets l j to a small positive value if l 0 j is tending to zero.

4. NUMERICAL RESULTS

We now study some typical two- and three-dimensional soil stability problems to demonstratethe efficiency of the new upper bound algorithm. All of the examples are for Tresca or Mohr Coulomb criteria. The latter, which is widely used to model the failure of cohesivefrictionalsoils, can be troublesome for non-linear optimization techniques because of its apex and cornerswhich generate gradient singularities. In our implementation, these are smoothed using thehyperbolic approximation of Abbo and Sloan [9]. The yield surface curvatures which result fromthis model vary strongly, but are well controlled by the special deection technique described inSection 3.4. The inuence of the smoothing approximation on the resulting bounds has beenevaluated by adjusting the values of the smoothing parameters. Although problem dependent, itwas found that this inuence is of the same order of magnitude as the smoothing parametersthemselves. Therefore, a value of 10 6 was used to round the apex and a transition angle of 29.5 8was used to round the corners (see Reference [9]).

For the two-dimensional cases, results are presented for three different meshes which areclassied as coarse, medium and ne. Because of the improved upper bounds that they generate,velocity discontinuities are used at all interelement boundaries.

To solve the sparse symmetric system of equations (60) efficiently, we use the MA27multifrontal code developed at Harwell. This is a direct solution method and employs thresholdpivoting to maintain stability and sparsity in the factorization.

All CPU times given in the tables are for a Dell Precision 220 workstation with a Pentium III800EB MHz processor operating under Windows 98 second edition. The compiler used wasVisual Fortran Standard Edition 6.1.0 and full optimization was employed.

4.1. Ri g id strip footin g on cohesi vefrictional soil

The exact collapse pressure for a rigid strip footing on a weightless cohesivefrictional soil withno surcharge is given by the Prandtl [10] solution

q=c0 exp p tan f 0tan 245 f 0=2 1cot f 0

where c0 and f 0 are, respectively, the effective cohesion and the effective friction angle. For a soilwith a friction angle of f 0 358 it gives q=c0 46:14. The velocity boundary conditions and




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material properties used in the upper bound analysis, together with one of the meshes, areshown in Figure 9.

The results presented in Table I compare the performance of the new non-linear formulationwith that of the linear programming formulation described by Sloan [2,11] and Sloan andKleeman [3]. The new algorithm demonstrates fast convergence to the optimum solution and,

most importantly, the number of iterations required is essentially independent of the problemsize.For the coarsest mesh, the non-linear formulation is nearly four times faster than the linear

programming formulation and gives an upper bound limit pressure which is 1 per cent better.For the nest mesh the speed advantage of the non-linear procedure is more dramatic, with a155-fold reduction in the CPU time. In this case, the upper bound limit pressure is 2.5 per cent

Figure 9. Upper bound mesh and collapse pattern for strip footing problem, (a) velocities,(b) deformations, (c) plastic zones.

Table I. Upper bounds for smooth strip footing on weightless cohesivefrictional soil ( c0 1, f 0 358,g 0, NSID =number of sides in linearized yield surface).

Mesh Quantity LP ( NSID =24) NLP ( e 0:01) LP/NLP

Coarse q =c0 50.48 49.82 1.01459 nodes No. of iterations 1396 28 49.9153 elements CPU (s) 6.76 1.82 3.7219 disc Error (%) +9.4 +8.0 1.18

Medium q =c0 48.77 48.00 1.021365 nodes No. of iterations 6875 29 237455 elements CPU (s) 133.7 5.5 24.3665 disc Error (%) +5.7 +4.0 1.43

Fine q =c0 48.03 47.30 1.022751 nodes No. of iterations 16771 30 559917 elements CPU (s) 2105 13.6 1551351 disc Error (%) +4.1 +2.5 1.65




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above the exact limit pressure and the analysis uses just 13.6 s of CPU time. Because the numberof iterations with the new algorithm is essentially constant for all cases, its CPU time growsalmost linearly with the problem size. This is in contrast to the linear programming formulation,where the iterations and CPU time grow at a much faster rate. The CPU time savings from thenon-linear method become larger as the problem size increases.

The tests in Table I were all run with the convergence tolerances (dened in Section 3.6) set toec el e f e 0:01. The inuence of changing these values, for the coarse mesh illustrated inFigure 9, is shown in Table II. Clearly, a uniform tolerance value of 0.01 is sufficiently accuratefor practical calculations with the non-linear algorithm. Indeed, tightening the tolerances from0.05 to 0.001 affects the collapse pressure error only marginally, reducing it from +9.4 to +8.0per cent. For the linear programming formulation, at least 24 sides ( NSID ) need to be used inthe yield surface linearization, but this value may increase with increasing friction angles.Increasing NSID , however, dramatically increases the solution cost. To obtain a solution withthe same accuracy as the non-linear programming method using e 0:01, the linearprogramming formulation requires 100 sides in the yield surface linearization and over a 16-fold increase in CPU time. As expected, the linear and non-linear programming methods givethe same collapse pressure when the former is used with an accurate linearization and the latteris used with stringent convergence tolerances.

4.2. Thick cylinder expansion in cohesi v efrictional soil

We now consider the collapse of a thick cylinder of cohesive-frictional soil subject to a uniforminternal pressure. The exact solution to this problem has been obtained by Yu [12] and is givenby the formula

p=c0 Y a 1 p0

c0a 1 b=aa 1=a 1 p0=c0

where p is the collapse pressure, p0 is the initial hydrostatic pressure acting throughout the soil, aand b are the inner and outer radii of the cylinder, and Y 2c0 cos f 0=1 sin f 0 and a

tan245 f

0

=2 are material constants. For p0 0; b=a 1:5; c0

1 and f0

308, the exactcollapse pressure is p 0:5376. One of the meshes used for the upper bound analyses is shown inFigure 10, together with the assumed loading and boundary conditions. Because of symmetry,only a 90 8 sector of the cylinder is discretized. As in all numerical examples considered in thepaper, velocity discontinuities are included between all adjacent elements to minimize the upperbound.

Table II. Results for smooth strip footing problem with different convergence tolerances e ec el e f .

LP NLP

CPU Iterations NSID q =c0 q=c0 e Iterations CPU

6.8 1396 24 50.48 50.49 0.05 20 1.0430.4 3537 100 49.84 49.82 0.01 28 1.82

260 5990 200 49.82 49.80 0.005 30 1.84405 8722 300 49.81 49.80 0.001 48 2.47




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The results for the three different meshes, shown in Table III, demonstrate similar trends tothe footing problem in regard to the performance of the non-linear and linear programmingtechniques. The former needs, respectively, just 16, 23 and 20 iterations to obtain the solutionfor the coarse, medium and ne meshes, which suggests that its iteration counts are again largely

independent of the problem size. This characteristic makes the non-linear programmingprocedure very efficient and, for the ne mesh, it takes only 11.2 s of CPU time to obtain anupper bound with 0.04 per cent error. In contrast, the iterations required by the linearprogramming formulation grow rapidly as the mesh is rened. Indeed, for the ne mesh, thisprocedure requires 22 325 iterations and a CPU time which is roughly 1120 times greater thanthat of the non-linear programming method.

Figure 10. Upper bound mesh and collapse pattern for expansion of thick cylinder,(a) velocities, (b) deformations, (c) plastic zones.

Table III. Upper bounds for expansion of thick cylinder of cohesivefrictional soil ( p0 0, b=a 1:5,c0 1, f 0 308).

Mesh Quantity LP( NSID =36) NLP( e 0:01) LP/NLP

Coarse q =c0 0.5535 0.5527 1.00136 nodes No. of iterations 79 16 3.4412 elements CPU (s) 0.06 0.28 0.2114 disc Error (%) +2.96 +2.80 1.06

Medium q =c0 0.5394 0.5384 1.002900 nodes No. of iterations 4221 23 183.5300 elements CPU (s) 60.0 2.59 23.2430 disc Error (%) +0.33 +0.15 2.2Fine q =c0 0.5386 0.5378 1.0013600 nodes No. of iterations 22325 20 11161200 elements CPU (s) 12553 11.2 11211760 disc Error (%) +0.18 +0.04 4.5




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4.3. Stability of an unsupported vertical cut under plane strain

All the problems considered so far have been concerned with the optimization of surfacetractions applied along a specied part of the domain boundary. We now consider the case of along unsupported vertical cut in a purely cohesive Tresca soil, where the unit weight g is

optimized for a given cohesion cu and height of cut H . Since the failure of this problem isgoverned by the stability number N s H g=cu , the analysis may also be interpreted as nding themaximum height of a cut for a given unit weight and undrained cohesion. One of the threemeshes used to analyse the cut is shown in Figure 11, together with the imposed velocityboundary conditions. Although this is a classical stability problem for construction in undrainedclays, its exact solution remains unknown. To date, the best upper and lower bounds on thestability number for a plane strain vertical cut are N s =3.785864 and N s =3.77200, and wereobtained, respectively, by Pastor et al . [13] using a linear programming formulation with theIBM Optimization Subroutine Library and by Lyamin [7] using a quasi-Newton two-stage non-linear programming algorithm. Pastor et al . [13] report their CPU times for a processor which isabout 3 times slower than the one used for our analyses, and it is thus interesting to compare theperformance of their linear programming upper bound formulation against the performance of

the new non-linear programming formulation presented here.According to Pastor et al . [13], their linear programming formulation required about 1000 h

( 60 000 min) of CPU time to solve for a mesh with 3700 triangular elements. In comparison,our non-linear programming scheme required only 2 min 23 s of CPU time for a mesh with 4230elements (not shown here) to give a slightly better upper bound of N s =3.78445. This solutionwould now appear to be the best known upper bound for the vertical cut problem, andhighlights the outstanding speed and accuracy advantages to be gained from the non-linearprogramming formulation.

The results for the various meshes, shown in Table IV, again compare the performance of thenew non-linear programming method with that of the active set linear programming methoddescribed in [2,3,11]. As in all previous examples, the iteration counts for the linearprogramming formulation grow rapidly as the mesh is rened, while the iteration counts for

Figure 11. Upper bound mesh and collapse pattern for vertical cut problem (a) velocities,(b) deformations, (c) plastic zones.




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the non-linear programming method are essentially constant. For the nest mesh, thesetwo solution schemes give upper bounds of N s 3:808 and N s =3.794, respectively, butthe non-linear programming method is over 170 times faster. The best upper bound of N s =3.794, differs by just 0.59 per cent from Lyamins [7] lower bound of N s =3.772, and usesonly 13.5 s of CPU time. In this example, the slight discrepancy between the linear programmingand non-linear programming solutions is caused by the yield surface linearization used in theformer.

4.4. Stability of an unsupported circular exca v ation

The stability of an unsupported circular excavation is another important practical problemwhose exact solution remains unknown. For the case of undrained clays with a Tresca yieldcriterion, solutions have been obtained by Bjerrum and Eide [14], Prater [15], Pastor andTurgeman [16], Pastor [17], Britto and Kusakabe [18], and Sloan [19]. No rigorous results,however, appear to be available for the case of cohesivefrictional soils.

We now consider the stability of a circular cut, of height H and radius R, in a cohesive-frictional soil whose behaviour is governed by the MohrCoulomb yield criterion. Solutionsfrom the new upper bound formulation are combined with solutions from a recently developedlower bound technique [7,8] to bracket the exact stability number from above and below. Toexploit its inherent symmetry we consider only a 15 8 sector of the problem, as shown inFigure 12, and restrict our attention to the case of wall failure. As discussed by Britto and

Kusakabe [18], this mode of failure is critical for circular cuts in clay provided H =R does notexceed 7.The computed stability numbers are presented in Table V and Figure 13 for a range of soil

friction angles and height to radius ratios H =R. For the special case of f 08, whichcorresponds to a cut in a (Tresca) undrained clay, the new upper bounds improve substantiallyon the upper bounds of Britto and Kusakabe [18]. For all cases, including those with non-zero

Table IV. Upper bounds for plane strain vertical cut in purely cohesive soil ( cu 1; f u 08).

Mesh Quantity LP( NSID =24) NLP( e 0:01) LP/NLP

Coarse N s 3.867 3.855 1.003156 nodes No. of iterations 402 26 15.552 elements CPU (s) 0.93 0.55 1.6972 disc Error (%)* +2.5 +2.2 1.14

Medium N s 3.818 3.804 1.0041110 nodes No. of iterations 8272 30 278370 elements CPU (s) 132.3 4.34 30.5540 disc Error (%)* +1.2 +0.85 1.41

Fine N s 3.808 3.794 1.0042928 nodes No. of iterations 31910 37 862976 elements CPU (s) 2303 13.5 1711440 disc Error (%)* +0.96 +0.59 1.63

*Measured with respect to Lyamin [7] lower bound of N s =3.77200.




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friction angles, the new upper and lower bounds bracket the exact stability number to 7 per centor better. For H =R 1 and f 08, Pastor and Turgemans [16] lower bound of N s =3.464 hasbeen raised to N s =5.09. For the same H =R ratio Pastors [17] upper bound has been reducedfrom N s =5.298 to N

s =5.26. No rigorous results could be found for frictional soils, so the

solutions presented in Table V are new. The gap between the upper and lower bound estimatesof the computed stability numbers tends to widen as the friction angle increases, with an averagediscrepancy of 6.8 per cent for f 208 compared to 3.3 per cent for f 08.

The nest upper bound mesh for the circular excavation comprised 26 688 nodes, 6672elements and 12 168 discontinuities. Bearing in mind that each node has three unknownvelocities, each element has six unknown stresses, and each discontinuity has 12 unknownvariables, this grid generates a very large optimization problem and it comes as no surprise thatthe method is computationally demanding. On average, the CPU time for the analyses shown inFigure 12 was 5000 s for a Dell Precision 220 workstation with a Pentium III 800EB MHz

Figure 12. Typical upper bound mesh for circular excavation.

Table V. Upper and lower bounds for circular excavation in cohesivefrictional soil.

f 8 Stability number gH =c0

H =R 1 H =R 2 H =R 3

LB UB LB UB LB UB

0 5.09 5.26 5.95 6.14 6.59 6.8310 6.48 6.73 7.82 8.20 9.00 9.4220 8.30 8.85 10.49 11.26 12.54 13.43




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processor operating under Windows 98 second edition. Although large, these timings arecertainly competitive with those that would be needed for a three-dimensional incrementalanalysis with the displacement nite element method. Moreover, the difference between theupper and lower bounds provides a direct estimate of the error in the computed collapse load.

5. CONCLUSIONS

A new formulation for performing upper bound limit analysis has been presented. Detailednumerical results show that the approach is fast and accurate for a broad spectrum of two- andthree-dimensional stability problems. In terms of CPU time, it is vastly superior to a widely usedlinear programming formulation, especially for large scale applications where the speedup canbe several orders of magnitude. Indeed, the new upper bound scheme is so efficient that largethree-dimensional problems can now be analysed on a desktop PC.

REFERENCES

1. Bottero A, Negre R, Pastor J, Turgeman S. Finite element method and limit analysis theory for soil mechanicsproblems. Computer Methods in Applied Mechanics and En g ineerin g 1980; 22: 131149.

2. Sloan SW. Upper bound limit analysis using nite elements and linear programming. International Journal forNumerical and Analytical Methods in Geomechanics 1989; 13: 263282.3. Sloan SW, Kleeman PW. Upper bound limit analysis using discontinuous velocity elds. Computer Methods in

Applied Mechanics and En g ineerin g 1995; 127:293314.4. Zouain N, Herskovits J, Borges LA, Feij

!

ooo RA. An iterative algorithm for limit analysis with nonlinear yieldfunctions. International Journal of Solids and Structures 1993; 30 (10):13971417.

5. Herskovits J. A two-stage feasible directions algorithm for nonlinearly constrained optimization. Mathematical Pro g rammin g 1986; 36: 1938.

Figure 13. Upper and lower bounds for circular excavation.




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6. Avriel L. Nonlinear Pro g rammin g , Analysis and Methods . Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1976.7. Lyamin AV. Three-dimensional lower bound limit analysis using nonlinear programming. Ph.D. Thesis , Department

of Civil, Surveying and Environmental Engineering, University of Newcastle, Australia, 1999.8. Lyamin AV, Sloan SW. A comparison of linear and nonlinear programming formulations for lower bound limit

analysis. Proceedin g s of the 6th International Symposium on Numerical Models in Geomechanics , Pietruszczak, Pande(eds), Balkema: Rotterdam, 1997; 367373.

9. Abbo AJ, Sloan SW. A smooth hyperbolic approximation to the MohrCoulomb yield criterion. Computers and Structures 1995; 54: 427441.10. Prandtl L.

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UUber die H.

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oorper. Nachrichten von der Gesellschaft der Wissenschaften zu G.

oottin g en,Mathematisch -Physikalische Klasse 1920; 12: 7485.

11. Sloan SW. A steepest edge active set algorithm for solving sparse linear programming problems. International Journal for Numerical Methods in En g ineerin g 1988; 26: 26712685.

12. Yu HS. Expansion of a thick cylinder of soils. Computers and Geotechnics 1992; 14: 2141.13. Pastor J, Thai T-H, Francescato P. New bounds for the height limit of a vertical slope. International Journal for

Numerical and Analytical Methods in Geomechanics 2000; 24: 165182.14. Bjerrum L, Eide O. Stability of strutted excavations in clay. Geotechnique , 1956; 6(1):3247.15. Prater EG. An examination of some theories of earth pressure on shaft linings. Canadian Geotechnical Journal 1977;

14(1):91106.16. Pastor J, Turgeman S. Formulation lin

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eeaire des m!

eethodes de lanalys limite en sym!

eetrie axiale. Fourth Con g r!

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eechanique , Nancy, France, 1979.17. Pastor J. Analyse limite et stabilite des fouilles. Proceedin g s of the 10th International Conference on Soil Mechanics

and Foundation En g ineerin g , vol. 3, Stockholm, Sweden 1981; 505508.

18. Britto AM, Kusakabe O. Stability of unsupported axisymmetric excavations in soft clay. Geotechnique 1982;32(3):261270.

19. Sloan SW. Numerical analysis of incompressible and plastic solids using nite elements. Ph.D. Thesis, Uni v ersity of Cambrid g e, 1981.



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