Large static problem in numerical limit analysis: A decomposition approach

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.887

Large static problem in numerical limit analysis:A decomposition approach

Zied Kammoun1, Franck Pastor2, Hichem Smaoui3 and Joseph Pastor4,∗,†

1Ecole Polytechnique de Tunisie, Laboratoire Systemes et Mecanique Appliquee, B. P. 743, 2078 La Marsa, Tunisia2Universite de Lille, Laboratoire LML, 59655 Villeneuve d’Ascq Cedex, France

3Ecole Nationale d’Ingenieurs de Tunis, B. P.37, Le Belvedere, 1002 Tunis, Tunisia4Universite de Savoie, POLYTECH’Savoie, Laboratoire LOCIE, 73376 Le Bourget du Lac, France

SUMMARY

A general decomposition approach for the static method of limit analysis is proposed. It is based onpiecewise linear stress fields, on a partition into finite element sub-problems and on a specific coordinationof the subproblem stress fields through auxiliary interface problems. The final convex optimizationproblems are solved using nonlinear interior point programming methods. As validated for the compressedbar with Tresca/von Mises materials in plane strain, this method appears rapidly convergent, so that verylarge problems with millions of constraints and variables can be solved. Then the method is applied tothe classical problem of the stability of a Tresca vertical cut: the static bound to the stability factor isimproved to 3.7752, a value to be compared with the recent best upper bound 3.7776. Copyright q 2010John Wiley & Sons, Ltd.

Received 13 October 2008; Revised 8 December 2009; Accepted 8 December 2009

KEY WORDS: limit analysis; static method; finite elements; convex optimization; decompositionapproach; vertical cut

1. INTRODUCTION

The Lysmer paper [1] was the first to consider the static method of limit analysis (LA) in geotechnicsusing a finite element method (FEM) for the Coulomb plasticity criterion. Based on this paper,an improved static approach was proposed in [2], also restricted to a bounded mechanical system;this drawback was eliminated in [3, 4] with the definition of infinite extension zones allowing thestress fields to remain admissible everywhere beyond the finite element mesh. More recently, in[5, 6], this earlier work was extended to reinforced soils via homogenization, with Coulomb’s soil

∗Correspondence to: Joseph Pastor, Universite de Savoie, POLYTECH’Savoie, Laboratoire LOCIE, 73376 Le Bourgetdu Lac, France.

†E-mail: [email protected]

Copyright q 2010 John Wiley & Sons, Ltd.

Z. KAMMOUN ET AL.

as a special case. All these studies used a polyhedral approximation of the plasticity criterion,resulting in linear programming problems.

Built upon the work of Herskovits [7], powerful nonlinear algorithms have recently beenproposed, such as in [8], for Tresca and Coulomb materials in 2D and 3D, using the previousextension elements. Another nonlinear, interior point formulation was proposed in [9], essentiallyvalid for von Mises or Tresca static plane problems.

For problems where the plasticity criterion is the von Mises or Drucker–Prager criterion, thenonlinear yield constraints give rise to conic programming problems to be solved by specificalgorithms and codes (cf. [10]) such as MOSEK [11]. They made it possible to solve various static(and kinematic) LA problems for predicting the macroscopic criterion of porous materials onthe basis of Gurson’s model [12, 13]. Conic programming was also used in [13–17] for variousproblems in geotechnics or homogenization of heterogeneous materials.

On the other hand, an interior point optimization solver was presented in [18–20] to solve thestatic problems for homogeneous Gurson materials—where conic programming does not apply—and the special case of von Mises or Tresca materials in plane strain. Henceforth, this optimizationsolver, fully detailed in [21], will be called IP-OPT.

The common drawback of both static and kinematic approaches to LA problems is that complexgeometry or pronounced heterogeneities give rise to very large optimization problems, particularlywhen high accuracy is required for the limit loading estimate. Despite the continuing progressin optimization algorithms and the ever-improving performance of computers, the increasingcomplexity of LA problems that one is attempted to tackle remains a challenge.

As a strategy to handle problem sizes beyond available machine capacities, it is common tosplit the original problem into subproblems that are smaller in size or simpler to solve. A classicalexample from structural analysis is substructuring. Moreover, in linear and nonlinear optimizationalgorithms numerous schemes, referred to as decomposition methods, have been proposed wherean original problem is converted into a sequence of smaller linear subproblems coordinated by analso small, evolutive master problem. To our knowledge, a decomposition approach was only usedin [22] for solving a kinematic LA problem. In that study the kinematic formulation was treatedusing a domain decomposition method with nonoverlapping subdomains. However, no details aregiven on how the variable values at the interfaces are updated from one iteration to another.

A contrario, an original decomposition approach was successfully achieved in the kinematiccase in [21] and presented in [23]. The purpose of the present study is to make this concept ofdecomposition operable for the finite element static formulation of LA problems. The proposedmethod also makes use of physical insight while remaining distinct from classical substructuring:this decomposition method proceeds iteratively with an auxiliary problem for each interface ateach decomposition level in order to upgrade the interface variables. Two example problems of astatic estimate of the limit load are presented to illustrate the proposed method.

2. LIMIT ANALYSIS AND THE STATIC METHOD

For the sake of clarity, without any loss in generality, we consider here that the velocity fields arecontinuous.

According to Salencon (see [24, 25]), a stress tensor field � is said to be statically admissible(SA) if equilibrium equations, stress vector continuity, and stress boundary conditions are verified.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2010)DOI: 10.1002/nag

LARGE STATIC PROBLEM IN NUMERICAL LIMIT ANALYSIS

It is said to be plastically admissible (PA) if f (�)�0, where f (�) is the (convex) plasticity criterionof the material. A field � that is SA and PA here will be said to be (fully) admissible.

Similarly, a strain rate tensor field v is kinematically admissible (KA) if it is derived from acontinuous velocity vector field u such that the velocity boundary conditions are verified. It is saidto be PA if the flow rule (1) is verified; the fields u and v which are KA and PA, will be calledadmissible in the following.

v=�� f

��, f (�)=0, ��0. (1)

The so-called associated flow rule (1) (or normality law) characterizes the standard material ofLA. Equivalently, a standard material satisfies Hill’s maximum work principle (MWP) [26], whichstates that:

(�−�∗) :v�0 ∀ PA �∗. (2)

A solution to the LA problem is a pair of fields (�,v) where � and v are both admissible andassociated by the normality law. Classically, these solutions can be found or approached usingtwo optimization methods. The first one, involving only the stresses as variables, is the static (orlower bound) method. The second one, involving only the displacement velocities as variables, isthe classical kinematic (or upper bound) method.

Let us assume that the virtual power of the external loads can be written as the scalar productof a loading vector Q ∈Rn and a generalized velocity vector q =q(u), linear in u. A loadingprocess linearly associated with an SA stress field �, Q = Q(�), is said to be admissible. The set ofthese admissible loadings forms a convex K in Rn and the n components of Q are called loadingparameters.

Finding the solution of the LA problem consists in determining an admissible field � togetherwith an admissible strain rate field associated with � by the normality law. In this case the loadingQ(�) is a limit loading of the mechanical system. The set of the limit loadings is the boundary �Kof the convex K : this boundary can be approached by solving the following family of optimizationproblems:

Qlim = (Qd1, . . .,�0 Qd

i , . . ., Qdn) (3a)

�0 = max{�, Q(�)= (Qd1, . . .,�Qd

i , . . ., Qdn)}, (3b)

or, alternatively the optimization problems

Qlim = �0(Qd1 , . . ., Qd

i , . . ., Qdn) (4a)

�0 = max{�, Q(�)=�(Qd1 , . . ., Qd

i , . . ., Qdn)}, (4b)

where � is an admissible stress field and Qd a given admissible loading. Then, by varying Qd itis possible to construct various points on �K : the smallest convex envelope of these points givesan approximation of �K from inside. This is the static, or lower bound method of LA, as will beused here.

It should be noted that for problems where some loading components are independent of theloading parameter the relevant formulation is that given by Equations (3). Nevertheless, it ispossible to obtain the limit load indirectly by solving problems in the form of Equations (4).


Z. KAMMOUN ET AL.

The idea is to first construct the whole approximation of �K by letting all the load componentsvary proportionally, then determination of the desired lower bound becomes straightforward.

3. FINITE ELEMENT FORMULATION OF THE STATIC PROBLEM

Here we use the numerical static method as it was defined and detailed in [3] in plane strainand recalled in [4] when considering the axisymmetrical case, including extension elements forsemi-infinite problems.

3.1. General formulation

Then, let us consider a triangular finite element discretization of the mechanical volume V in theglobal frame (x, y); the stress field is chosen as linearly varying in the x , y coordinates in eachtriangular element and it can be discontinuous through any element edge. In the plane strain case,the von Mises or Tresca criterion is written as

f (�)=√

(�x −�y)2+(2�xy)2−2c�0, (5)

where c is the cohesion of the material. It is worth noting here that the proposed decompositionmethod is valid for the Coulomb or Drucker–Prager criteria, or any other criterion provided thatthe final optimization problem can be solved by efficient mathematical programming techniques.

To ensure static and plastic admissibility of the stress solution field, the following conditionsare imposed:

• In each element, the equilibrium equations �i j, j +�i =0 expressed in the Cartesian frame,where � is the specific weight vector.

• Continuity conditions: the stress vector is continuous across a discontinuity line: for eachdiscontinuity segment of normal n, the continuity of the stress vector Ti =�i j n j is written atthe apices defining this discontinuity segment.

• Boundary conditions: the stress vector verifies �i j n j =T di at each apex of the boundary

element sides where the stress vector T d—linearly varying—is imposed.• Definition of the functional from the power of external loads: for example, the integral of the

normal stresses in the case of the footing under an imposed normal velocity.• Symmetry conditions, when applicable: the tangential stresses are null on the symmetry axis

when only half the domain is modeled.• Stress field PA: from the linear variation of the stress in a triangle and the convexity of the

criterion, (5) is imposed at each triangle apex.

The final problem is solved by the nonlinear code IP-OPT (see Section 4), or the criterion is directlywritten in the conic form V =2c�

√Y 2+ Z2 to use the commercial code MOSEK as in [13].

3.2. Extension elements

Defined first in [3], these extension elements, located at the boundary of the finite static mesh,make it possible to extend the stress field to infinity, giving rise by the way to true static boundsfor unbounded Tresca or Coulomb problems. They were extended to the axisymmetrical case in[4] and used 20 years later in [8], among others.



2 1

4 3I

2 1

32

1

3 V

III

IV VI VII

XI

X

XII

XIII

VIII

IX

x

y

2 1

4

II

3

Figure 1. A vertical-slope extended mesh.

These extension elements (Figure 1) consist in rectangular and triangular zones, the formerfeaturing two parallel directions of extension along which the criterion remains constantin the present von Mises/Tresca case, the latter where the criterion is constant over thezone.

The other conditions, i.e. the equilibrium, boundary conditions and continuity of the stress vectorbetween the zones, are imposed as in the inner elements, so that the stress field remains SA toinfinity, owing to the linearity of the stress variation in the zones.

3.3. Analysis of the redundancies

It should be noted that a meticulous analysis of the redundancies induced by the extension conditionswas necessary to make the final constraint matrix full row rank, as required by the optimizationalgorithm used in IP-OPT.

For example, let us consider the left rectangular extension zones in Figure 1, where the slope isweightless as described in a subsequent section. To keep the criterion constant along the direction1–2 we impose (�x −�y) and (2�xy ) to be equal at the vertices 1 and 2 and, independently, at 3and 4.

From the boundary condition at the soil surface and the linear variation of the stress field withinthe element, it follows that ��xy/�y =0 and ��x/�y =0. Using the boundary condition again andthe equilibrium equations, it results that �x =�x (y), �y =constant and �xy =0 over the entirerectangular extension element.

Thus, the first two extension conditions become equivalent. Moreover, as �xy =0 over theelement, the remaining two extension conditions are redundant. Therefore, among the four internalextension conditions, three are proven to be redundant and are therefore omitted from the problemformulation. For the other rectangular extension zones, analogous results are obtained by replacingthe boundary conditions by inter-element continuity equations.


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Similarly, it can be verified that, for the four triangular extension zones of Figure 1, a total ofsix extension conditions are only needed to formulate the problem without redundancy.

4. PRINCIPLE OF THE INTERIOR POINT METHOD FOR SOLVINGOPTIMIZATION PROBLEMS

In [19, 21], a general interior point algorithm for solving the LA static problem, hereafter calledIP-OPT, is detailed. The resulting optimization problems present a linear objective function and amix of linear and nonlinear convex constraints. For problems where the plasticity criterion is thevon Mises or Drucker–Prager criterion, the nonlinear constraints are convex quadratic inequalities,generating second-order conic programming problems for which specific codes exist. Unfortunately,these codes do not provide enough accuracy when post-analyzing the solutions in the secondexample presented herein, resulting in non-admissible solutions for our large-scale problems. TheIP-OPT code has overcome this drawback, as it appears to be limited only by the RAM capacityof the Apple Mac Pro used for the present calculations.

The general form of the optimization problems to be solved here is as follows:

max cTx

s.t. Ax =b,

g(x)+s =0, s�0,

(6)

where c, x ∈Rn , b∈Rm , A∈Rm×n is the matrix of the linear constraints, g = (g1, . . .,gp) is avector-valued function of p convex numerical functions gi and s ∈R

p+ is the vector of slack

variables associated with these convex constraints.The primal-dual interior point method consists in solving, instead of the previous problem, the

following one, parameterized by �>0, the barrier parameter:

max cTx +�p∑

i=1

ln(si )

s.t. Ax =b,

g(x)+s =0, s>0.

(7)

The problem (7) has a solution if and only if the following conditions are satisfied:

−c+ ATw+(

�g

�x

)T

y = 0,

Ax −b = 0,

g(x)+s = 0,

Y Se = �e,

(8)

where w∈Rm , y ∈Rp , e=[1, . . .,1]T ∈Rp and Y , S are the diagonal matrices associated with yand s, respectively; from the last equation of (8) �>0 and s>0 imply y>0.



For each given �, the nonlinear system (8) is approximately solved by one iteration of theNewton method, thereby providing an approximate solution of the parameterized problem (7).Using a sequence of values for � decreasing to zero, we make the latter converge to the solutionof (6). Indeed, as � approaches zero, Equations (8) come close to the KKT conditions for theoriginal problem.

5. POSITION OF THE PROBLEM AND DECOMPOSITION PROCEDURE

The present decomposition is based on a partition of the FEM mesh, but with no connection withthe so-called domain decomposition methods, which are clearly presented in [27]. Moreover, thesemethods seem not to have been applied to optimization problems, as can be seen in the verycomplete reference [28] presenting the state of the art on the domain decomposition approach.

As regards the static approach of the LA problem, to our knowledge, a decomposition approachhas never been presented in the literature. The present method, as in [23] for the kinematic approach,proceeds iteratively with an auxiliary problem for each interface at each decomposition level inorder to upgrade their interface variables. Hereafter we present the method on the problem of abar punched between two rough rigid plates (Figure 2). This is an academic example presentinga relatively general combination of boundary and loading conditions.

5.1. The mechanical problem

Given the symmetries of the problem, only the left upper quarter of the bar is modeled. It ismeshed in 4×2 squares or rectangles, diagonally subdivided into four triangles each. This problem,called the target problem (Figure 3), will be solved here to give the convergence value for thedecomposition procedure.

Figure 2. The punched bar, b=2h, mesh 2N×N , here N =2.


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Figure 3. The target problem.

The rigid plate goes down with a uniform, vertical velocity U0, which is created through theaction of a force whose vertical component F is to be determined, the horizontal componentbeing free. The isotropic, weightless, homogeneous material obeys the von Mises (or Tresca)criterion with cohesion c0. The bar is also subjected to a uniform pressure p0 on its lateralsides, and the interface between the bar and the plates is a maximum friction one; conse-quently the stress vector at the interface is only limited by the plasticity criterion of the bar,and no additional interface constraints are needed. Then two loading parameter, QF = F/(bc0)

and Q p = p0/c0, can be defined from the expression of the external power. The solutionmethod is presented in the case of piecewise linear stress fields, i.e. with linear stresses inthe triangular finite element in order to ensure that the criterion is not violated inside theelements.

First of all, we consider the problem of finding the static lower bound to F with a given p0 ofFigure 2. Various direct methods can be used, among others.

• The F-methodMaximize F =�F0,��0, with fixed stress boundary conditions (p0,0) imposed on the lateralsides. This is the usual approach (3) to obtain the boundary of the set of admissible loads(F/(bc0), p/c0)stat, by changing the value of p0 appropriately. The resulting nonlinear convexproblem is solved by IP-OPT.

• The c1-methodMinimize the cohesion c=�c0, where � is an additional non-negative variable, while keepingF = F0 and p= p0; the minimum cohesion cmin gives the static solution (QF = F0/(bcmin),Q p = p0/(cmin)) and the corresponding stress field is obtained after multiplying the optimalstress field by the scaling factor c0/cmin, from the form of (5). This approach is equivalent tothe proportional loading case (4), and we need first to determine the boundary of the set ofadmissible loads before the desired point (F/(bc0), p0/c0)stat can be found.

• The c2-methodMinimize the cohesion c=�c0 as above, but after adding a new linear constraint such thatp=�p0 and imposing the lateral boundary condition (p,0) instead of (p0,0). After scalingthe stress field by c0/cmin we directly obtain the static load (QF = F0/(bcmin), Q p = p0/c0).Both c-methods induce typical conic programming problems, so that they are solved by usingthe MOSEK code [11].

Remark 1For geotechnical problems with body forces, such as the usual specific weight �, for example, thec2-method can also be used by considering � as a new variable which verifies another constraint



�=��0. After scaling, the stress field is compatible with the original �0, together with p0 (ifpresent) as above.

Remark 2In fact, both c1- and c2-methods can be used for any homogeneous criterion of degree one instress, i.e. for most of the classical criteria in geotechnics. For example, as regards the Drucker–Prager and Coulomb criteria, the constants �0 (Drucker–Prager) or the cohesion c (Coulomb) canbe minimized, keeping the friction angle fixed for Coulomb, as well as the cone angle of theDrucker–Prager criterion. The final problem is a conic programming one for the Drucker–Pragercriterion in 2D and 3D cases, and for Coulomb in plane strain or in axisymmetry (see [29]).Remark 3When, for example, the problem involves n several Coulomb materials (�0i ,c0i ,�0i , i =1,n), thec2-method can also be used by defining 2n new constraints �i =��0i and ci =�c0i , and minimizing� as in the homogeneous case.

5.2. The decomposition procedure

In this section we use the c2-method to present the decomposition process and we give a comparativeanalysis of the previous variants. Let us now consider the above problem but with a coarser mesh,called the starting problem, as in the following.

5.2.1. The starting problem. Solving first this reduced problem provides a good initialization forthe stress vector pattern at the separating interface AB of the target problem (Figure 3). Anotherinitial solution consisting in extracting the interface values from an analytical admissible solutioncould be used, as we will see later.

Let us consider now the 2×1 problem illustrated in Figure 4 solved by the c2-method. Thesolution of this starting problem gives—with no scaling—the initial values of the stress vector T atpoints A, C , B, defining a first distribution T0 acting on interface ACB in Figure 5; the stress vectorat the intermediate point C is calculated by linear interpolation, because of the linear variation ofthe stress tensor inside the triangles.

From the starting problem solution, the part of the compressive force on each of the followingleft and right subproblems is also calculated, if needed, in such a manner that the total force onthe whole plate remains equal to F0. In the present case, Figure 5, this distribution is trivial.

Figure 4. The starting problem.


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Figure 5. The ‘left’ and the ‘right’ problems I I� and I Ir .

5.3. The ‘left’ problem

The meshed problem is presented in Figure 5 in the left. The stress distribution T0 is applied on theACB side; the constraint p=�� p0 is added where the index � refers to the left problem, and thecorresponding boundary conditions modified as mentioned above. By minimizing the cohesion,in fact the factor ��, we obtain its optimal value �opt

� and in an external file we save the vertexvalues of the stress vector distribution T G H acting on the segment GH. When the left problemincludes a part of the plate, the normal component of the corresponding load is kept fixed to thevalue obtained from the post-analysis of the solution of the starting problem; doing the same forthe right problem, the total load on the global plate remains fixed to F0.

5.3.1. The ‘right’ problem. This time the stress vector distribution T0 is applied on the ACBside oriented to have the same normal as in the left problem, without adding any constraint. Byminimizing the cohesion, we obtain here �opt

r as the optimal value of the factor �r and the vertexvalues of the stress vector T IJ along the segment IJ are saved in the previous external file.

At this stage we select �opt =�optII =max(�opt

� ,�optr ), noting that copt =�optc0 is lower or equal

to the optimal cohesion of the starting problem, as the starting stress field is admissible in bothsubproblems by construction. Now, two cases occur, if �opt

� and �optr are different.

• If �opt =�opt� , copt =�optc0, the stress fields of both subproblems, fitted together, comply with

the boundary condition p0 and with the loading F0/�opt which is the load value at the end ofthe first iteration. Indeed, as copt

r is lower than copt, we are sure that a stress field admissiblewith (cr =copt,T0, F0) exists, and we can scale the load as in the case of a common �opt. Thisexplicit stress field, and a slightly improved T IJ could be calculated by re-running the rightproblem after lower bounding of the cohesion to copt, but this is not necessary in practice.

• If �opt =�optr , the same reasoning holds this time for the lateral boundary condition, and here

also we can conclude that F0/�opt is the admissible value of the load at the end of this firstiteration.

To progress by iterating the process, upgrading the stress vector at the ACB interface is obviouslynecessary, hence the idea of the following central phase III to improve these interface values.

5.3.2. The ‘central’ problem. Let us now consider the auxiliary problem III in Figure 6, withthe previously saved values imposed as boundary conditions on the lateral sides. The load on the



Figure 6. The ‘central’ problem III.

plate is calculated from the fields of both previous subproblems to keep F0 as the force acting onthe total plate. Solving this problem by minimizing the cohesion gives an updated stress vectordistribution T1 on the ACB interface. It is worth noting that the optimal cohesion is here lower orequal to the previously selected copt because the stress field extracted from the previous left andright subproblems is admissible with copt by construction.

Finally, by going back to phase II with the new values on ACB, the process can be resumed.Each subsequent global iteration consists in solving the problems III, II� and IIr . The process canbe terminated when no improvement is achieved in F/�opt from one iteration to another within aspecified tolerance.

5.4. Decomposition into more than two subproblems

The decomposition method can also be used in a recursive manner by applying, for example, theiterative process on each previous subproblem separately (as if they are target problems themselves),before considering central problem III. Another possibility consists in decomposing the targetproblem into more than two subproblems and proceeding as illustrated in Figure 7. This is thedecomposition mode we will use in the second test problem.

6. NUMERICAL EXAMPLES

In this section, two LA problems are treated to illustrate the proposed decomposition technique.The distinctive feature among these problems lies in the nature of the applied load. In the first,a rectangular domain is compressed between two rigid plates, a problem whose exact solution isknown. The second is the until now open problem of the stability of a vertical cut where the loadis defined by the specific weight of the soil. In all problems, the material is assumed to be isotropicand homogeneous, governed by a von Mises/Tresca criterion with cohesion c, in plane strain.

6.1. The compressed bar problem

Let us consider the problem, depicted in Figure 8, of a bar compressed between rough, rigid plateswith a width-to-height ratio of 2, whose exact solution was given in [30]. The rigid bar is subjectto the compressive action of a plate descending with a fixed vertical velocity U0. The goal is tofind the maximum compressive average load QF = F/(bc), where the load F , vertical here, isassociated with U0 through the expression of the external power. Because of the symmetries of


Z. KAMMOUN ET AL.

Figure 7. Second decomposition of the 64×32 plate.

Figure 8. The compressed bar, b=2h, mesh 2N × N , here N =2.

the problem, only the upper left quarter of the bar is considered, as in the previous punched barproblem.

With a target discretization of 32 triangular elements, the process is started by solving a coarsermodel problem P0 with only eight triangular elements (Figure 9). The two variants of the decompo-sition scheme defined in the previous section have been implemented and tested, i.e. the F-methodand the c-method, given that the c1-method is equivalent here to the c2-method. In both methods,the interface values are left as given by the subproblem solutions. An illustration of the first steps



Figure 9. Decomposition flow diagram.

of the decomposition procedure is given in Figure 9. These steps are described next for eachvariant.

6.1.1. Using the F-method. Here the cohesion c remains set to the value c0 for all tests, and thevertical load component F is maximized.

First iteration.Starting problem. Solving the reduced size problem P0 provides an initial estimate of the stress

field and a corresponding lower bound QF0 =2.31611.


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Figure 10. Iteration history of load parameter Q = F/(bc) for the bar problem (target mesh: 4×2).

Phase II. The model domain is partitioned into two subdomains with 16 elements each, which aresubject at the top to loadings whose vertical components FII� and FIIr are optimized under the stressboundary condition T0. The new lower bound at this stage is QF = (Fopt

II�+ Fopt

IIr)/(bc0)=2.35074.

Second iteration. In this iteration, which is representative of all subsequent iterations, the vectorT is updated to T1 by solving problem III with as boundary conditions the stress vectors TG H andTIJ obtained from the solutions of problems II at the previous iteration. Solving subproblems PIIagain gives the new lower bound QF =2.36152.

Ninth iteration. At this stage, the lower bound solution to problem P is QF =2.36225. It canbe seen in Figure10 that the first iterations give the major part of the increase, followed by a veryslow convergence to the direct solution of the target problem QFstat =2.37064.

6.1.2. Using the c-method. The problem is now subject to the vertical load F0, with F0/b=1 forthe sake of clarity, and the cohesion c is minimized.

First iteration.Starting problem. The minimum obtained is copt =0.43176, corresponding as expected to the

previous value of QF0 .Phase II. The stress field resulting from the starting problem gives the stress vector T0 and the

vertical load components FII� =0.48540F0 and FIIr =0.51460F0 verifying FII� + FIIr = F0. Withthe interface stresses set to T0 and the vertical load components maintained at their values, thecohesion is minimized in each subproblem. The solution stress fields are PA with respect to theirrespective new minimum cohesions cIIr =0.42298 and cII� =0.42760=copt . The correspondingvalue of QF is F0/(bcopt)=2.33863.



Second iteration. As above, the vector T is updated to T1 by solving problem III with boundarystress vectors TG H , TIJ ; the vertical component of the load is fixed to the sum of the correspondingvalues on segments HB and BI calculated from the solutions of problems II. The result for thecohesion is cIII =0.42363. Then, using T1 and the new values of corresponding parts of the load(here FII� =0.49237F0, FIIr =0.50763F0, including the respective modifications from the PhaseIII solution), Phase II is repeated. The cohesions are now cIIr =0.42332 and cII� =0.42336=copt;the new lower bound is then QF2 =2.36206.

Ninth iteration and comparison. The final stress field is SA with respect to the load F0 and PAwith respect to the final cohesions cII� =0.42332 and cIIr =0.42332. Setting then copt =cII� =cIIrthe lower bound solution to problem P is given as F0/(bcopt)=2.36228. The iteration history ofthe load parameter F/(bc) is plotted in Figure 10.

Comparison on a more refined mesh. Now the problem is solved for a more realistic discretizationalso using the optimization code MOSEK. The exact solution for the continuum problem, F/(bc)=2.42768, makes numerical solution error estimation straightforward.

Here also the iterations are conducted using both variants for a (60×40) target problem (9600elements) whose direct solution is F/(bc)=2.42494. As for the small problem, the two methodsconverge, the c-method being slower as can be seen in Figure 11.

Influence of the starting solution. Figure 12 gives the iteration history to the 17th one, using thec-method with three different starting fields for a (60×40) target problem. As above, a startingproblem (30×20) that is four times smaller is tested (lozenge line), then the smallest mesh forthe starting problem (square line), and finally the zero vector T0 (triangle line) given by the exactsolution (�xx =�xy =0,�yy =−2c) of the same problem but with smooth interfaces. As expected,

Figure 11. Iteration history of load parameter F/(bc) for the bar problem (target mesh: 60×40).


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Figure 12. Influence of the starting solutions for the bar problem (target mesh: 60×40).

Table I. Static bounds to QF for the compressed bar problem.

Direct DecompositionElement Errors (%) vs

Mesh number Solution CPU (s) Solution CPU (s) direct/exact sol.

36×24 3456 2.42325 25 2.42270 31 0.023/0.20560×40 9600 2.42497 112 2.42456 105 0.017/0.129120×80 38400 2.42629 835 2.42605 748 0.010/0.067180×120 86400 2.42674 3085 2.42658 2773 0.007/0.045240×160 153600 — — 2.42684 6731 —/0.035264×176 185856 — — 2.42690 8160 —/0.032288×192 221184 — — 2.42697 13326 —/0.029312×208 259584 — — 2.42701 18053 —/0.028

the convergence speed depends on the starting point, and this convergence is obtained at roughly15 iterations in the worst case.

Final results. With a four times smaller starting problem, the steep rise at the first iteration ofthe F-method is typical and was observed in all runs with degrees of discretization ranging from3456 (=4×(36×24)) to 259 584 elements.

Table I shows the static bounds obtained with the MOSEK optimizer for different mesh sizeswith the F-method. The first are the results of the complete computation by the direct method.The second are those of the first iteration of the decomposition.



Also reported in Table I are the relative errors between the lower bounds and the correspondingCPU times in seconds taken on a Mac mini equipped with an Intel Core Duo processor (1.66 GHz)and 2 GB of RAM.

It should be noted that the last example of Table I is the largest problem that could be solved bydecomposition, with the chosen domain partition, because of the memory limitation of the MOSEK

optimizer. Larger mesh sizes could therefore be treated by adopting a partition with more subdo-mains.

It is also noted that the finer the discretization is, the faster the convergence of the decompositionalgorithm in terms of the number of iterations (Table I). This can be seen from the relative errorto the direct solution, which drops from 0.023% for the 3456-elements mesh to 0.007% for the86 400-element mesh, even though only one iteration was performed.

Finally, the exact solution (2.42768) is approximated with greater precision than 0.028%, provingthat the proposed decomposition is a powerful tool to determine the limit load.

The merit of the decomposition algorithm is clearly seen from its ability to solve problems withmuch larger dimensions than the direct method can handle, by using a first-level decompositionas above, or in a recursive manner if necessary.

6.2. The vertical slope problem

6.2.1. Position of the problem. In this example, the classical vertical cut problem is examined. Thevon Mises-Tresca soil, with cohesion c, is subject to its sole weight �. The loading parameter isdefined by the stability factor �H/c, where H denotes the height of the cut, as shown in Figure 13.As in the compressed bar problem, the domain is discretized into a mesh of rectangles, each madeup of four triangular elements, in a global M ×N grid, as in the 3×4 mesh of Figure 1.

The c1-method was tested first; the resultant conic programming problems were solved usingthe MOSEK solver. However, this strategy failed to achieve convergence to admissible solutionsfrom medium-size finite-elements meshes.

In a more successful formulation, the problem is replaced in a preliminary phase with anequivalent one, as first suggested in [31]. In this equivalent problem, the soil is weightless and theboundary conditions are defined by (�n =�h,�nt =0) on the surface of the soil, where �n denotesthe stress vector component normal to the boundary and h the depth measured from the upper

nh

nt=0

H

x

y

Figure 13. Equivalent problem for the vertical cut.


Z. KAMMOUN ET AL.

Figure 14. Domain partition for the vertical cut. In the left side, from the top: the subproblems PI I−1 toPI I−4. In the right side: PI I I−1 to PI I I−3.

surface (Figure 13). The decomposition is carried out according to the F-method on the partitionshown in Figure 14 as the load is applied on all the subproblems, and the weight � is maximizedusing the optimizer IP-OPT. At both steps in each iteration the global loading parameter is chosenas the smallest value among the subproblem solutions.

6.2.2. Results. Direct approach. For meshes composed of more than 60 000 triangles, commercialconic codes fail to give full optimal and admissible post-analyzed solutions. Beyond this limit,only the IP-OPT code converges, using up to a 94 400-element mesh in roughly 3 h and 20 min onthe Mac Pro 3 GHz (and 4 GB of RAM), giving 3.7742 as a direct optimal value for �H/c. Thenlarger problems need to be solved by decomposition.

Using decomposition. To analyze the convergence of the numerical solution, the problem issolved for various levels of discretization. As shown in Figure 15, for all mesh sizes a like, up tonine iterations of the decomposition algorithm were needed to achieve convergence.

Table II shows the lower bounds obtained using the IP-OPT code to solve the final convexproblems. The best lower bound, �H/c=3.7752 (i.e. the smallest value, here obtained for thesubproblem PII-3, Figure 14), is computed with a 351 824-element global model. This is animproved value with respect to the best known estimate of 3.772 [8]. All admissibility conditionsare checked a posteriori with a tolerance of 10−8 or better in all the tests.

Moreover, to obtain a global admissible stress field for the original problem P from the lastiteration, the three other subproblems are solved again while imposing the previous �min value asa common bound on the load parameters in order to obtain the same final specific weight for allfour subproblems. By adding the isotropic field �i j =�min×x�i j to the final four fields, we obtainthe global solution for the real soil. In this case also, the field is verified to be admissible as aboveby post-analysis.

From the recent upper bound given in [21] and improved in [23, 32], we thus achieve the bestbracketing to date for this problem:

3.7752��H/c�3.7776.



Figure 15. Iteration histories for load parameter �H/c.

Table II. Vertical slope and decomposition: lower boundresults after nine iterations.

M × N Number of FE �(H/c)stat

260×184 179 408 3.77452270×200 200 336 3.77466300×224 249 600 3.77482316×240 281 104 3.77500344×280 351 824 3.77522

Table III. Iteration evolution of the lower bound (344×280 mesh).

Sub-pb Iter. 1 Iter. 2 Iter. 3 Iter. 8 Iter. 9 Time iter. 9 (s)

PIII−13.77413

3.77470 3.77523 3.77523 3.7752288 9841PIII−2 3.77420 3.77468 3.77516 3.7752158 7341PIII−3 3.77442 3.77440 3.77512 3.7753132 4986PII−1 3.77413 3.77519 3.77525 3.77524 3.7752299 12342PII−2 3.77413 3.77491 3.77525 3.77522 3.7752250 5700PII−3 3.77413 3.77429 3.77457 3.77514 3.7752248 4629PII−4 3.77486 3.77460 3.77451 3.77539 3.7755297 5373Result 3.77413 3.77429 3.77451 3.77514 3.77522

Since the decomposition makes it possible to solve problems that are impossible otherwise, theissue of computational effort becomes secondary: in fact, the CPU times for problems PII−1to PII−4 vary from 1.3 to 3.4 h only (see Table III). It is also worth noting that the structureof the decomposition scheme reveals inherent parallelism resulting from the decoupling of thesubproblems, which can be exploited in a parallel processing environment.


Z. KAMMOUN ET AL.

7. CONCLUSION

The proposed decomposition method fully preserves the specific features of the static approachby finite elements, solving problems that could not be solved directly until now, at least to ourknowledge. Its remarkable efficiency lies in fact in the constant robustness and the rapidity ofthe IP-OPT solver under MATLAB. In all cases, the stress fields were controlled a posteriori inorder to guarantee the lower bound character of the solutions. Hence, we obtain the best numericalsolution for the compressed bar problem whose exact solution is known. Ending with the caseof the vertical cut problem, which is still open, we were able to improve the static bound up to3.7752. Thus, the exact solution is now bounded by 3.7752��H/c�3.7776, i.e. the best boundsuntil now. Furthermore, the proposed decomposition exhibits inherent parallelism, opening newperspectives for solving highly heterogeneous problems such as the stability of a footing restingon a micropile net.

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Large static problem in numerical limit analysis: A decomposition approach

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