INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 48:317–350

A numerical strategy for �nite element analysisof no-tension materials‡

G. Alfano, L. Rosati∗;† and N. Valoroso

Dipartimento di Scienza delle Construzioni; Universita di Napoli Federico II; Via Claudio 21;80125 Napoli; Italy

SUMMARY

We present an algorithmic procedure for the �nite element solution of structural problems for no-tensionmaterials. The approach is based upon a suitable modi�cation of the tangent strategy which is shown to becomputationally superior to conventional procedures for non-linear material models, namely the tangent strat-egy enhanced with line searches and the tangent-secant approach. The solution of the constitutive problem forno-tension materials is derived by an original path of reasoning and its formulation in a strain-driven format,directly amenable to a computer implementation, is presented. For completeness the existing expressions ofthe tangent and secant operators for the no-tension model are brie y recalled and an original formula forthe secant operator derived. The robustness of the proposed strategy is exempli�ed by the numerical resultsobtained for a masonry panel with openings. Remarkably, the solution is achieved by assigning a single loadstep and an asymptotically quadratic convergence rate is attained. Further, the numerical properties of theproposed solution strategy are practically una�ected by the adopted discretization. Copyright ? 2000 JohnWiley & Sons, Ltd.

KEY WORDS: masonry; �nite elements; no-tension models

1. INTRODUCTION

Over the last few decades masonry structures have been the object of a fairly abundant literaturein which both numerical techniques and constitutive modelling have been extensively addressed.An accurate description of the mechanical behaviour of masonry structures requires a large

amount of experimental data which are usually di�cult to obtain and present large uncertainties.Nonetheless, signi�cant advancements have been recently achieved in the constitutive charac-

terization of the masonry behaviour as well as of the con�ned masonry. This is the outcome ofthe comprehensive experimental tests which have been carried out by many researchers. We cite

∗Correspondence to: L. Rosati, Dipartimento di Scienza delle Costruzioni, Universita di Napoli Federico II, Via Claudio21, 80125 Napoli, Italy

†E-mail: rosati@unina.it‡The paper is dedicated to the memory of Professor Manfredi Romano, who passed away untimely in 1988

Contract=grant sponsor: CNR

Received 10 July 1998Copyright ? 2000 John Wiley & Sons, Ltd. Revised 20 July 1999

318 G. ALFANO, L. ROSATI AND N. VALOROSO

among others, without any claim of completeness, the papers by Page and co-workers [1–3] aswell as the one by Naraine [4].Sophisticated constitutive models have thus been presented in the literature [5–9], some of which

are based on homogeneization techniques [10–12].However, as observed by Heyman [13; 14] the crisis of old masonry structures is generally due

to crack openings rather than to the attainment of the limit compressive strength. The constitutivebehaviour of such structures can thus be conveniently described by the so-called perfectly no-tension (or masonry-like) model, i.e. as a material which can sustain unlimited compressive stressesbut having zero tensile strength.Although it represents a crude approximation of the real behaviour of the masonry, the no-tension

model has received a great attention by many authors [15–26] and its constitutive restrictions havebeen fully accounted for only recently [27–29].More precisely, the stress tensor is assumed to be negative semide�nite, due to the peculiar

properties of the no-tension model, and depends linearly and isotropically upon the elastic partof the total strain. In turn this is split into the sum of the elastic and inelastic part. A normalitylaw to the elastic domain is imposed for the inelastic strain so that it turns out to be positivesemide�nite.Therefore, the no-tension model de�nes a non-linear hyperelastic material whose e�ectiveness in

predicting the crack patterns in real structures has been recently validated from the experimentalpoint of view in Reference [30]. On the other hand, the model is inadequate to properly accountfor phenomena in which pronounced fractures or degradation of the material are involved.Due to the previous considerations, it is commonly accepted that the no-tension model can use-

fully describe the overall structural behaviour of old masonry structures, with particular referenceto the stress distribution. Hence, the model represents a valuable tool for a preliminary structuralanalysis provided that robust solution procedures can be exploited.Actually, numerical techniques must be necessarily resorted to since closed-form expressions of

the stress �eld are available only for geometrically simple structures subject to particular loadingconditions [31]. The two recent contributions by Alves and Alves [15] and Lucchesi [22] areonly the last ones of a long series of papers on the subject [7; 18–21; 23; 32–34] whose list isnecessarily not exhaustive.The various algorithmic approaches which have been exploited are motivated by the fact that, in

spite of its simplicity, the no-tension model poses serious numerical problems even for the solutionof two-dimensional structural problems. Actually, the rate of convergence is rather poor and thetolerance adopted to stop the iterative process by far lower than the one commonly employed inplasticity computations.Within a displacement-based �nite element approach, one of the �rst contributions has been

due to Romano and Sacco [34] who exploited a numerical strategy based upon the updating of asecant sti�ness operator at each iteration.It is worth noting that, in contrast to the signi�cance traditionally attributed to the secant moduli

of the so-called quasi-Newton methods, see e.g. Reference [35], the secant operator introduced inReference [34] provides an estimate of the total, non-iterative, stress increment in the load step.The secant strategy, originally applied by Romano and Sacco to two-dimensional problems and

subsequently extended by Sacco [26] to a three-dimensional context, exhibited remarkable stabilityproperties. According to the hyperelastic nature of the constitutive model, the load was assignedin a single step but the iterative process was rather slow and the adopted tolerance for the residualnorm not particularly narrow.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 319

A tangent approach for the no-tension model has been recently considered by Lucchesi et al.[22] and subsequently extended to more sophisticated models with bounded tensile and compressivestrength in References [8; 9]. It is based on the use of a tangent operator derived in closed formby ingeniously exploiting the isotropic nature of the non-linear law relating the stress tensor withthe total strain. However, the load history was incrementally applied and no information on theconvergence rate was reported by the authors.On the basis of the previous considerations it appears natural to look for a solution strategy

capable of ensuring an asymptotically quadratic convergence rate, as entailed by the adoption of afull Newton method, and to avoid time-marching solution schemes typical of dissipative constitutivemodels.At �rst sight, a natural candidate for such a strategy could appear to be a tangent procedure en-

hanced with line searches [36; 37] since it is routinely employed for satisfying analogous demandsarising in di�erent contexts such as plasticity and viscoplasticity. Additional strategies such as thetangent-secant one, proposed by the authors in References [38; 39] can be exploited as well.However such strategies do share the same drawback since they do not take into proper account

a peculiar feature which characterizes the structural models under investigation: the possibility that,due to the spread of the inelastic strains, zero-energy modes can be activated during the structuraliterations. This circumstance has been experienced to drastically reduce the convergence rate andoften preclude convergence of the algorithm itself.For this reason a detailed investigation has been carried out to assess the numerical properties

of some of the existing strategies for �nite element analysis of no-tension structural models: theconclusion which has been drawn is that they are quite ine�cient. Even the tangent-secant strategy,which exhibits excellent computational performances in elasto- and elasto=visco-plasticity, does notbehave analogously for the problem at hand.Therefore a new strategy has been formulated and its performances fully investigated in order

to assess its adequacy for practical calculations. The proposed approach, here termed enhancedtangent, is basically a tangent strategy supplemented by a �ctitious elastic sti�ness at the Gausspoints in which the elastic strains do vanish. Speci�cally, the sti�ness operator at such pointsis assumed to coincide with the elastic moduli tensor scaled by an energy-parametrized coe�-cient which decays during the iterative process till becoming zero when convergence has beenattained.The layout of the paper is as follows. The constitutive properties of the no-tension model

are brie y recalled and, with a view to computer implementation, the explicit expression of theinelastic strain reported by Romano and Sacco [34] is here derived on a di�erent basis and recastin a format similar to the so-called strain driven problems currently employed in elasto- and elasto=visco-plasticity [36].We then recall the expressions of the tangent operator for the no-tension model contributed by

Lucchesi et al. [22] and the one of the secant operator derived by Romano and Sacco [34]. For thesake of completeness, an original approach for deriving further expressions of the secant operatorhas also been proposed.The algorithmic implementation of the tangent–secant strategy and of the enhanced tangent

procedure is illustrated and a detailed numerical investigation is �nally carried out for a masonrypanel with openings. The e�ectiveness of the proposed approach is witnessed by the narrowtolerance which can be achieved during the equilibrium iterations, signi�cantly smaller than the oneconsidered thus far [15; 16; 18; 21; 22; 26; 34], and by the capability of obtaining an asymptoticallyquadratic convergence rate even by assigning a single load step.

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320 G. ALFANO, L. ROSATI AND N. VALOROSO

Finally, it is shown that the performances of the enhanced tangent strategy are to a great extentuna�ected by the adopted discretization, an issue which does not appear to have been adequatelyaddressed in the literature.

2. THE CONSTITUTIVE EQUATIONS OF NO-TENSION MATERIALS

The formulation of the constitutive equations for no-tension materials is substantially due toHeyman [13; 14] who �rst applied concepts and methods of Limit Analysis to masonry struc-tural models.To make the presentation self-consistent we brie y recall the constitutive assumption of the

no-tension material in the form presented by Romano and Sacco [34] and further investigated inReferences [23; 24; 26; 27].Let us denote by V a three-dimensional real vector space and by Lin the space of all linear

transformations on V endowed with the inner product:

A ·B = tr(AtB) ∀A;B∈Lin

wherew (·)t stands for transpose and tr denotes the trace operator. The symbols Sym; Sym+and Sym− indicate the subsets of Lin collecting symmetric, positive-semide�nite and negative-semide�nite tensors.The no-tension material has a linear elastic behaviour when subject to compressive normal

stresses but it cannot withstand tensile normal stresses. Accordingly, the admissible domain in thestress space S is the closed convex cone Q de�ned by

Q = Sym− = {b∈S: bn · n60} (1)

where b is the stress tensor and n is the unit normal.Stated equivalently the stress tensors belonging to Q are characterized by non-positive principal

values.Addressing the small strain case, the total strain U is decomposed in the elastic part e and in

the inelastic one T as follows:

U = e + T (2)

The inelastic strain is assumed to ful�l a normality rule to the admissible stress domain:

(c− b) · T60; b∈Q ∀c∈Q (3)

or equivalently,

b · T = 0; b∈Q; T∈Q− (4)

where Q− denotes the negative polar [40] of Q:

Q− = � = Sym+ = {W∈D: b · W60 ∀b∈Q}

and D is the strain space.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 321

The normality property of the inelastic strain can also be stated in the following equivalentform:

T∈NQ(b) = @t Q(b) ⇔ b∈N�(T) = @t�(T) (5)

where NC denotes the normal cone to the convex set C; t C is the convex indicator function ofC and @ stands for the subdi�erential operator [40]. The previous concepts of convex analysis,originally introduced by Romano and Romano [25; 29] in their studies on unilateral constitutivemodels, have been recently addressed in Reference [15].A linear isotropic elastic relation between the admissible stresses and the elastic strains is �nally

assumed:

b = Ee (6)

where

E = 2GI3 + �(13⊗ 13) (7)

G is the shear modulus, � is the second Lame’s constant while I3 and 13 denote the three-dimensional fourth- and second-order unit tensors, respectively.In conclusion, the constitutive equations of no-tension material are

U = e + Tb∈ @t�(T) (8)

b = Ee

Having in mind a displacement-based �nite element analysis of masonry structures, the total strainis known, typically at a generic Gauss point of the given mesh, and the remaining state variablesin (8) have to be determined.The explicit solution of the constitutive problem (8) to the best of the authors knowledge, has

been �rst provided by Romano and Sacco [34] for the two-dimensional case and by Sacco [26]for the three-dimensional one. It is based on the characteristic property of the inelastic strain, thesolution of (8), which is a minimum point of a suitable convex function.In order to derive the explicit expression of such function we exploit the potential theory for

monotone multivalued operators, recently contributed in Reference [41], since Equations (8) de�nea monotone multivalued law.To this end, Equations (8) are written in the symbolic form:

(0; 0; 0)∈M(b; T; e) + (U; 0; 0)or explicitly

0

0

0

∈

0 −I −I−I @t� 0−I 0 E

bTe

+

U00

(9)

The operator M de�nes a multivalued map which can be easily shown to be conservative [41].Accordingly, M is integrable and its potential is given by the convex function:

P(c; W; S) = 12ES · S − c · (S + W) + c · U+ t�(W) (10)

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322 G. ALFANO, L. ROSATI AND N. VALOROSO

For any given U a solution of the constitutive problem (8) can then be characterized as aminimum point of P:

(b; T; e) = argminP(c; W; S)

Enforcing the stationarity of P with respect to c we recover the additive decomposition of thetotal strain in the elastic and anelastic part U = S + W. Substituting this relation in (10) we thusinfer that the inelastic strain T, solution of (8), can be characterized as follows:

T= arg minW{12E(U− W) · (U− W) + t�(W)

}= arg minW

{12‖U− W‖E | W∈�

}(11)

i.e. as orthogonal projection, in the energy norm, of U onto � [29].Besides (11), the additional property which has been exploited in Reference [34] to derive an

explicit solution of the constitutive problem (8) is the coaxiality of the tensors U and T. The proofof such property, originally provided by Romano and Sacco for one- and two-dimensional cases,has been subsequently extended to the three-dimensional context in Reference [27] by followinga di�erent approach.The coaxiality of U and T allows one to greatly simplify the algebraic manipulation required by

the solution of (8) since only the principal directions of U can be taken into account.To this end let us �rst notice that the conditions

b∈Sym−; T∈Sym+

become

�i60; �i¿0; i=1; 2; 3 (12)

when expressed in principal components.The complementarity condition (4), written as

�1�1 + �2�2 + �3�3 = 0 (13)

implies that all the products �i�i; i=1; 2; 3, must necessarily be zero due to the sign constraints(12).

2.1. Explicit solution for plane strain and plane stress

For the sake of completeness we here re-derive the solution originally obtained by Romano andSacco [24] in the two-dimensional case, by adopting a mechanically oriented path of reasoning.Further, the solution is presented in a format suitable for computer implementation, similar to

the so-called strain-driven problems currently employed in elasto- and visco-plasticity [37].For future reference we remind that the elastic operator (7) specializes to

E=2G[I + �(1⊗ 1)] (14)

where

�=�

1− 2� (plane strain); �=�

1− � (plane stress) (15)

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 323

� is the Poisson’s ratio, while I and 1 are two-dimensional counterparts of the analogous quantitiesde�ned in (7).In terms of principal components (6) and (14) yield

�1 = 2G[(1 + �)(�1 − �1) + �(�2 − �2)]�2 = 2G[�(�1 − �1) + (1 + �)(�2 − �2)]

(16)

The inverse relations

e1 =12G

[1 + �1 + 2�

�1 − �1 + 2�

�2

]

e2 =12G

[− �1 + 2�

�1 +1 + �1 + 2�

�2

] (17)

have been directly expressed as a function of ei= �i − �i for convenience.A further property can be proved in the plane strain case since the condition

�3 = 0= e3 + �3 (18)

yields �3 = 0 and e3 = 0. Actually, in the opposite case (�3¿0), it should be e3¡0 and hence�3¡0, in contrast to the hypothesis �3¿0.Conversely in the plane stress case (�3 = 0) we can only say that the no-tension model leaves

completely unde�ned the value �3. For this reason it is customary to set �3 = 0 in practical calcu-lations [22].Without loss of generality, we assume that the principal directions of the total strain tensors

have been de�ned so that �16�2.Accordingly, representing two-dimensional tensors with vectors collecting their principal com-

ponents, U cannot belong to the convex cone in the principal strain space (see Figure 1) de�nedby the conditions �1¿0 and �2¡0.Let us then detail the solution of the constitutive problem (8) when the vector U sweeps coun-

terclockwise the strain space starting from the quadrant �1¿0 and �2¿0.In this respect we shall denote by

intQ= {b∈S: �1¡0; �2¡0} (19)

the interior of the convex cone Q.In the discussion of the constitutive problem (8) three alternatives have to be taken into account.

(i) �1¿0: In this case it turns out to be T= U and b= 0.Actually, if it were b∈ intQ, it should be T= 0; e= U∈ intQ− and hence e1¿0 and e2¿0.We would then infer �1¿0 and �2¿0 from (16), in contrast to the assumption that b∈ int Q.It must then be b∈ bnd Q and we can set in this case �i=0, �j60 (i; j∈{1; 2}; i 6= j). Weshall prove, however, that the possibility �j¡0 must be ruled out. Actually

�i=0⇒ �i − �i= − �1 + �

(�j − �j)

which, substituted in (16)j, yields

�j =2G[− �2

1 + �(�j − �j) + (1 + �)(�j − �j)

]=2G

1 + 2�1 + �

(�j − �j)

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324 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 1. Principal strain space.

Hence, �j − �j60 since we assumed �j60. In particular, we must have �j =0 since, otherwise,�j =0 and �j60 in contrast to the original assumption on �1 and �2.

Due to the arbitrariness of the indices i and j, we conclude that b=(0; 0), e=(0; 0) andT= U.

(ii) �160; �2¿− ��1=(1 + �). This case is characterized by the solution T=(0; �2 + ��1=(1 + �))and b=(�1; 0) where �1 is given by

�1 = 2G1 + 2�1 + �

�1 (20)

To prove that �1 = 0 we �rst note that, in the opposite case, it should be �1 = 0. Sincenecessarily �260 we would infer from (17)1 that e1¿0. This possibility is however ruled outsince e1 + �1 could not be equal to the non-positive quantity �1. Hence �1 = 0 and e1 = �1.Let us further prove that �2 6= 0, by showing that, otherwise, an absurd would be arrivedat. Actually, setting �2 = 0 and hence �2 = e2, we infer from (17)2 and (16)1, after somemanipulations that

�2 +�

1 + ��1 =

12G(1 + �)

�2

The right-hand side is always non-positive and this is inconsistent with the assumption on thevalues of �1 and �2.In conclusion we must have, �1 = 0, �2 = 0 and from (16)2 we infer that

�2 = �2 +�

1 + ��1 (21)

which, substituted in (16)1 provides the correct value of �1.The value �2 can be graphically obtained by intersecting the second principal axis with the

segment drawn from the end point of the vector U parallel to the line whose angular coe�cientwith respect to the �rst principal axis is �=(1 + �), see Figure 1.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 325

(iii) �160; �26 − ��1=(1 + �). This case corresponds to a purely elastic solution T=(0; 0) andU= e.Let us �rst prove that �2 = 0. This is obviously true if �2¡0 but it can be easily proved alsowhen �2 = 0. Actually, in this case, we would derive from (16)2 that

�2 = �2 +�

1 + ��1 − �

1 + ��160

since �2 +��1=(1+�)60 by hypothesis and �1 is non-negative. Since �2¿0 we conclude that,necessarily, �2 = 0.We can now prove that �1 = 0. Ruling out the trivial case �1¡0, we infer from (16)1 the

condition �1 = 0 that

�1 = �1 +�

1 + ��26�1 +

�1 + �

(− �1 + �

�1

)=1 + 2�(1 + �)2

�1

where the inequality sign is a consequence of the hypothesis made on the values of �1 and �2.Since �160 the previous relation would imply �160, a condition clearly impossible.

To summarize the previous considerations we can conclude that the solution of the constitutiveproblem (8) leads to de�ne the following three subsets of Sym+, provided that the principal valuesof U have been ordered so that �16�2:

S1 = {U∈Sym: �1¿0} ⇒ �1 = �1; �2 = �2

S2 ={U∈Sym: �160; �2¿− �

1 + ��1

}⇒ �1 = 0; �2 = �2 +

�1 + �

�1

S3 ={U∈Sym: �160; �26− �

1 + ��1

}⇒ �1 = 0; �2 = 0

The previous sets will be referred to in the sequel to properly identify the sti�ness operators tobe used in the �nite element solution of structural problems endowed with the no-tension material.

3. SOLUTION STRATEGIES FOR THE STRUCTURAL PROBLEM

Although the constitutive model outlined in the previous section is conceptually simple, thedisplacement-based �nite element analysis of no-tension structural models has often been char-acterized by poor computational performances either in terms of convergence rate or accuracy ofthe numerical results.This has been partly due to the fact that, for particular loading conditions, no-tension structural

models can present indeterminate components of the displacement �eld even if the stress distri-bution can be shown to be unique [28]. Further, the possibility of having Gauss points in whichstresses are zero and the deformation is purely inelastic, see subset S1 in the previous section,can lead to numerical troubles in classical Newton methods since the relevant constitutive sti�nessoperator is not positive de�nite.When the load level is conveniently high, the number of such Gauss points can become so large

as to activate zero-energy modes.

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326 G. ALFANO, L. ROSATI AND N. VALOROSO

Over the past two decades, a series of sophisticated numerical techniques has thus been ex-ploited in order to overcome the above-mentioned shortcomings of the displacement approach.A penalty approach has been proposed in Reference [32] and subsequently elaborated inReference [33]. A complementary mixed approach has also been presented in [18] and its al-gorithmic implementation has been addressed later by the augmented lagrangian method [19; 20].A convex quadrating programming approach has been exploited in Reference [23].Quite recently, however, a renewed interest has been devoted to displacement-based �nite ele-

ment models for no-tension materials, (see [21]) and the set of papers contributed by Lucchesi andco-workers [8; 9; 22]. In particular, the last author has adopted a full Newton method for the so-lution of the non-linear �nite element equations based on the explicit derivation of the constitutivetangent operator for the no-tension model.As well known [35], any Newton method is conditionally convergent and, as a rule, the con-

vergence rate decreases as the stability of the solution process increases.The two extremes of the family of Newton iteration schemes are represented by the initial

sti�ness method, extremely robust but slow, and by the full Newton method [35], which providesthe highest convergence rate, i.e. asymptotically quadratic.Further numerical problems do, however, a�ect the solution of no-tension structural models.

For instance, the tolerance which is usually adopted [22] to stop the iterations in the solutionof the non-linear system of equations is of several orders of magnitude greater than the onescommonly adopted in plasticity computations. Its value appears to be set so as to reach a reasonablecompromise between numerical accuracy of the results and the number of iterations required toachieve convergence.The main reason for the peculiar numerical behaviour of the no-tension model essentially lies in

the fact that the elastic properties of the structural model abruptly change during the iterations dueto the progressive failure of several Gauss points, in the sense that the relevant sti�ness vanishes,thus making the whole solution procedure extremely unstable.Further, it is worth noting that the no-tension model de�nes a non-linear hyperelastic material

[27] so that, in keeping with the principle, the solution can be achieved in a single step, in sharpcontrast to the dissipative models in which a time-stepping solution strategy must be necessarilyresorted to.We are thus interested to �nd out an algorithm for the solution of real scale no-tension structural

problems encompassing both robustness, in the sense of good stability and high convergence rate,and numerical accuracy.Motivated by the previous considerations we present the results of the numerical investigations

which have been carried out for several solution strategies. We will show in particular that atangent approach, even supplemented by line searches, turns out to be almost unreliable. Analogousproperties do however characterize a suitable modi�cation of the tangent-secant approach presentedin References [38; 39], although in elasto- and elasto=visco-plasticity it exhibits excellent numericalperformances.The ultimate goal of this strategy is to combine within the same load step the high convergence

rate ensured by the adoption of a tangent operator, conveniently updated at each iteration accordingto the full Newton method, and the remarkable stability properties associated with the use of asecant operator.Referring the interested reader to References [38; 39] for a full account, we shall provide in

the sequel the basic elements of the procedure in order to make the presentation reasonably self-consistent.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 327

Due to the experienced lack of e�ciency of the above-mentioned strategies we here proposea new solution algorithm. It is based on a suitable modi�cation of the tangent approach whichis accomplished by assuming for the inconsistent Gauss points, i.e. the ones exhibiting a purelyinelastic response, an energy-parametrized elastic sti�ness which reduces continuously during theiterative process. In this way the activation of zero-energy modes for the elements can be preventedduring structural iterations, though exactly ful�lling the constitutive relations.Within a displacement-based �nite element model, the structural problem leads to solving a

non-linear system of algebraic equations:

R(u)= l −∫BtE [Bu− T(u)] d= 0 (22)

in the unknown vector u of displacement parameters. In the previous equation l is the vector of theequivalent external loads, B denotes the discrete strain-displacements matrix and T(u) the inelasticstrain.The quantity R thus represents the out-of-balance forces obtained as the di�erence between the

applied load l and the external loads in equilibrium with the internal ones.The basic elements of the strategies exploited in the numerical calculations are reviewed here-

after.

3.1. The tangent approach

Linearization of (22) around the ith estimate u(i) of the solution leads to solving the linear systemof equations:

K(i)tanTu(i+1)(i) =R(u(i)) (23)

where Tu(i+1)(i) = u(i+1) − u(i) and

Ktan =∫BtEtanB d

is the tangent sti�ness operator.The explicit expression of Etan for the no-tension model has been �rst contributed by Lucchesi

et al. [22]. Speci�cally they observed that the coaxiality of U and b, and the fact that the eigenvaluesof b depend only upon the eigenvalues of U, de�ne a non-linear isotropic function f : U→ b.Assuming that U belongs to S2 and recalling that two-dimensional models are taken into account,

we can set

b= �01+ �1U (24)

where �0 and �1 are non-linear isotropic scalar functions of the principal invariants of U:

I1 = tr(U)= �1 + �2; I2 = detU= �1�2 (25)

where

�1 =I1 −

√I 21 − 4I22

; �2 =I1 +

√I 21 − 4I22

(26)

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328 G. ALFANO, L. ROSATI AND N. VALOROSO

Expressing (24) in principal values:

�0 + �1�1 =� �1�0 + �1�2 = 0

(27)

where

�=2G1− � (plane strain) and �=2G(1 + �) (plane stress)

we infer

�0 =��1�2�2 − �1 ; �1 =−� �1

�2 − �1 (28)

The previous expressions of � derive from (20) by virtue of (15).Di�erentiating (24) and taking into account (26 and (28) we �nally obtain

Etan = dUb = �[I1(3I3 − I 21 )

2 31⊗ 1− I3

3(1⊗ U+ U⊗ 1) + I1

3U⊗ U+ 1

2

(1− I1

)I]

(29)

where =√2I3 − I 21 and I3 = tr(U2)= I 21 − 2 I2.

3.2. The secant approach

The strategy presented in Reference [42], and subsequently enhanced in Reference [38], belongs tothe class of the so-called direct (or Picard) procedures [37]. Actually, a solution of the structuralproblem is iteratively sought for by de�ning a secant operator which associates the total, ratherthan iterative, displacement increment in the step with the relevant load increment.However, in order to combine the secant and the tangent strategies, it is extremely useful to

rewrite the linear system of equations, to be solved at each iteration of a pure secant strategy, interms of iterative increments of the unknowns. These are indeed the quantities which enter theclassical tangent solution scheme.Since the initial values of the variables, denoted by (·)0, satisfy the equilibrium equation (22)

we have

l0 −∫BtE[Bu0 − T0] d= 0 (30)

so that Equation (22) can be rewritten in terms of the step-increments:

R(�u)=�l −∫BtE(B�u − �T) d= 0 (31)

where �(·)= (·)− (·)0.Assuming that the solution �u of Equation (31) is known, we can de�ne a secant operator Esec,

function of the strain increment B�u, so as to satisfy the relation

EsecB�u=E(B�u − �T) (32)

Substituting this relation in (31) we get

R(�u)=�l −∫BtEsecB d�u= 0 (33)

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 329

Since �T is a function of �u, the non-linear system (33) must be solved iteratively. Accordingly,at the ith iteration, a trial value �u(i) = u(i) − u0 has been computed and we can evaluate a trialvalue E(i)sec =Esec(B�u(i)).In general, these values of �u(i) and E(i)sec will not satisfy relation (33) and so a new value

�u(i+1) can be computed by solving the linear system

K(i)sec�u(i+1) =�l; where K(i)sec =

∫BtE(i)secB d (34)

Expressing the previous relation in terms of the iterative increment Tu(i+1)(i) =�u(i+1) −�u(i) weget

K(i)secTu(i+1)(i) =�l − K(i)sec�u(i) (35)

By virtue of Equation (30),

�l − K(i)sec�u(i) = l − l0 −∫BtE[B(u(i) − u0)− (T(i) − T0)] d

= l −∫BtE(Bu(i) − T(i)) d=R(u(i)) (36)

and hence we �nally obtain

K(i)secTu(i+1)(i) =R(u(i)) (37)

Note that the tangent approach leads to a linear system of equations having the same unknowns,the same right-hand side vector and the tangent sti�ness matrix Ktan in place of Ksec, see Equa-tion (23). As emphasized at the beginning of this section, this basic property allows us to carry outan e�ective computer implementation of the tangent-secant strategy detailed in the next section.

3.3. The secant operator by Romano and Sacco

A general formulation of secant approaches to the solution of structural problems with unilateralconstitutive behaviour was �rst contributed by Romano and Romano [29].Within this framework, Romano and Sacco [34] presented a computational algorithm for the

analysis of masonry structures which required the calculation of a secant constitutive operatorEsec. The expression of Esec was obtained as follows.Ruling out the trivial cases, we assume that the total strain tensor U belongs to the region S2

de�ned in Section 2 and that the associated inelastic strain T=(0; �2) is known through formula(21).We also remind that the no-tension model de�nes a non-linear elastic material so that a single

load step can be taken into account. Accordingly we can set U0 = 0 and T0 = 0 in (32) to write

Esec(U)U=E(U− T(U)) (38)

In order to derive an expression for Esec(U) we can de�ne a linear operator

P :D→D such that PU= T(U) (39)

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330 G. ALFANO, L. ROSATI AND N. VALOROSO

In this way, the operator ful�lling condition (38) is given by

Esec =E− EP (40)

The de�nition (39) does not provide a unique expression of P. A natural choice is to assume Pas the linear projector onto the one-dimensional subspace L⊆Sym+ spanned by T(U). Actually,since T(U) is the orthogonal projection, in the energy norm, of U onto Sym+, it trivially representsalso the projection of U onto L.Considering the parametric representation of L in the form V= �T(U), � ∈ <, the linear projector

P is thus de�ned by the condition

PS= arg minV

{12||S − V||2E | V= �T(U)

}

or equivalently by

PS= �T where �= arg min�∈<

{12||S − �T||2E

}=ES · TET · T (41)

Accordingly

P=T⊗ETET · T (42)

Note that, coherently with the de�nition (39), �=1 in (41) when S= U. Actually, U and T(U)are the solution of the constitutive problem (8) so that the complementarity relation (4) yieldsb · T=E(U− T) · T=0.Recalling Equation (40) we �nally obtain the expression

Esec(U)=E− ET(U)⊗ET(U)ET(U) · T(U) (43)

originally provided by Romano and Sacco [34].By virtue of (14) the previous expression can also be written in the form [40]

Esec = 2GI +2G�1 + �

(1⊗ 1−D⊗ 1− 1⊗D)− 2G1 + �

D⊗D (44)

where D= d⊗ d and d is the unit eigenvector of the non-zero component of the inelastic straintensor.

3.4. An original expression of the secant operator

It is worth noting that the secant operator (43) has the general form

Esec =E− A (45)

where A is a suitable fourth-order tensor.In particular, the expression of A appearing in (43) is such that the secant operator ful�lls the

additional condition

Esec(U)T(U)= 0

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 331

Always retaining the basic meaning of the secant operator, according to which

Esec(U)U= b (46)

further expressions of A can however be obtained.Speci�cally, we are interested in an expression of Esec whose dependence upon U is made more

explicit than the one given by (44) where Esec depends upon U implicitly through the eigenvectorsof T.Setting therefore

A= a(1⊗ U+ U⊗ 1) + b(U⊗ U) (47)

we can substitute (45) into (46) to get, by recalling formulas (14) and (24), the following linearsystem in the two unknowns a and b:

2G�trU− a(U · U)= �02G − atrU− b(U · U)= �1

(48)

where �0 and �1 are given by (28).The solution is

a=1U · U [2G�trU− �0]

b=1U · U

[2G − �1 − trU

U · U (2G�trU− �0)] (49)

yielding the following formula, alternative to (43), for the secant operator in S2:

Esec = 2G[I + �(1⊗ 1)]− a(1⊗ U+ U⊗ 1)− b(U⊗ U) (50)

Note that the expression (50) of Esec is well posed.

4. ALGORITHMIC IMPLEMENTATION

Let us now focus on the the algorithmic details of the solution strategies discussed in the previoussections. In this respect one has to point out the following.First, it is mandatory to supplement the completely inconsistent Gauss points g (i.e. the ones such

that Ug ∈ S1), with some kind of constitutive sti�ness operator though this violates the constitutiverelations. Otherwise, during the structural iterations, zero-energy modes could be activated.Second, the most reliable choice for such a sti�ness operator appears to be the elastic moduli

tensor parametrized by a suitable coe�cient which will be referred to in the sequel as �0.Actually, there is no evident reason to assume a di�erent behaviour along the two principal

directions as it would be entailed by the adoption of a tangent or secant operator.The convergence of the iterative process is tested through the so-called incremental energy norm

of the residual:

D(i) =K(u(i))Tu(i+1)(i) · Tu(i+1)(i)

since it usually provides a very reliable and severe test [43]. Structural iterations are stopped whenD(i)6�D(0) where � represents the tolerance.

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332 G. ALFANO, L. ROSATI AND N. VALOROSO

Box 1. Scheme of the �nite element computations.

1. Initialization: u(0) = u0; i=0; set �; �o and �.2. To perform the enhanced tangent algorithm GOTO Box 3.3. Assemble sti�ness matrix and residual force vector:

Loop over elements

(i) Loop over Gauss points of the elementFor the given strain tensor evaluate:

the inelastic strain tensorthe stress tensorthe constitutive operator

(ii) Close the loop over Gauss points(iii) Evaluate the element residual force vector(iv) Evaluate the element sti�ness matrix

Close the loop over elements4. Solve equilibrium equations and evaluate the current energy norm:

K(u(i))Tu(i+1)(i) =R(i)

D(i) =R(i) · Tu(i+1)(i)

5. To perform the tangent–secant algorithm GOTO Box 2.6. Update displacements:

u(i+1) = u(i) + Tu(i+1)(i)

7. Convergence test:IF D(i)¡�D(0) THEN

EXITENDIF

8. Increment iteration counter:i= i + 1GOTO 2

The basic algorithmic steps involved in the classical displacement-based �nite element compu-tations are syntactically illustrated in Box 1. We underline that points 2 and 5 of Box 1 to referto di�erent solution strategies so that either the former or the latter has to be taken into accountdepending on the adopted strategy.

4.1. The tangent strategy with line searches

Since the strategy is well-known [36], we just recall that line search is activated whenever D(i)¿� �Dand it is continued until a scalar �� ful�lling the condition |D( ��)|6� �D is found, where

D(�)=R(u(i) + �Tu(i+1)(i) ) · Tu(i+1)(i) (51)

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 333

Numerical experiences [36; 37] have shown that the optimal values for � typically lie in the range0.5–0.8. Further, the maximum number nls of iterations is usually �xed for the solution of thenon-linear equation D(�)= 0.Although di�erent implementations are possible, for the numerical examples we shall refer to

the line search algorithm contained in the FEAP code rel. 7.1 [44] in which the condition ��61(i.e. no extrapolation) is enforced. However, even allowing ��¿1 no signi�cant improvements havebeen experienced in our numerical simulations.

4.2. The tangent–secant strategy

The basic ingredient of the tangent–secant strategy is represented by the criterion to adopt in orderto choose the most convenient sti�ness operator, the tangent or the secant one, at each structuraliteration. The former is invoked to ensure a remarkable convergence rate while the latter is requiredto stabilize the solution procedure.Accordingly, we always try to adopt the tangent operator in our algorithm except for those

iterations in which the residual norm fails to meet a signi�cant reduction.Hence we have to address two main issues: the way to switch from the tangent to the secant

strategy and vice versa.The switch criterion we decided to adopt basically derived from the one successfully exploited

in the case of J2 elastoplasticity [38] and is detailed in Box 2. Starting from the �rst signi�cantiteration of the procedure (i¿1), the norm of the residual for the current iteration D(i) is comparedwith the threshold value �D which stores the minimum of the energy norms evaluated during theiterative process.

Box 2. The tangent–secant algorithm.

5.a Set initial value for �D:

IF i=0 THENSet STRATEGY= tangent�D=D(0)

GOTO 6ENDIF

5.b Switch algorithm

IF D(i)6� �D THEN�D=D(i)

Set STRATEGY= tangentELSE

IF STRATEGY= tangent�D=D(i)

END IFSet STRATEGY= secant

ENDIF

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334 G. ALFANO, L. ROSATI AND N. VALOROSO

Box 3. The enhanced tangent algorithm.

IF i=0 THENInitialization�(i) = 1:D0

ELSED(i−1)∗ =0:D0Loop over elements

Compute the element energy norm:

D(i−1)e =ng∑g=1E(i−1)g (��g)

(i)(i−1) · (��g)(i)(i−1)

IF e∈E∗ THEND(i−1)∗ =D(i−1)∗ + D(i−1)e

ENDIFClose loop over elements

�(i)∗ = minj∈{0;:::;i−1}

(D(j)∗D(0)

)1=mCompute the coe�cient:�(i) = min{�0; �(i)∗ }

ENDIF

Accordingly, if D(i)6� �D, the next iteration will be performed with the tangent strategy, sincea convenient reduction of the residual norm has been obtained.In the opposite case, whenever D(i)¿� �D, a secant approach will be carried out in the next

iteration and D(i) will become the new threshold value for all the subsequent ‘secant’ iterationsnecessary to conveniently reduce the norm prior to switch back again to the tangent strategy.In contrast to the elastoplastic case [38], here we allow �¿1. In other words, we do not neces-

sarily require that D(i) decrease at each iteration. Actually this condition would be too restrictiveespecially when, at a given iteration, the structural model is characterized by abrupt changes in itsmechanical properties due to the sudden failure of some elements.In this case a very rapid adoption of the secant operator, entailed by the choice �¡1, would

hopefully stabilize the procedure but could determine a further change of the elastic properties ofthe model incompatible with the operative need of ensuring a high convergence rate.On the contrary, when inelastic strains spread progressively and in a regular pattern, numerical

experiments have shown that it is more convenient to assume �¡1 in order to speed up theprocedure.

4.3. The enhanced tangent strategy

It will be shown in the section concerning numerical examples that the above-mentioned strategies,namely the tangent, the secant, the tangent enhanced with line searches and the tangent–secant ones,are computationally ine�cient for the solution of structural problems with the no-tension model.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 335

This is not surprising since the key point to be addressed in order to ful�ll the stopping criterionfor the structural iterations is the convenient reduction of the residual stresses generated at theinconsistent Gauss points by the �ctitious sti�ness which they have been equipped with.There is no other way to circumvent this problem since the adding of this �ctitious sti�ness

is made necessary for ensuring a positive-de�nite structural sti�ness operator, without which noother iteration could be performed at all.A pure tangent or secant approach in the form presented before does not allow any control of

the iterative displacement vector in order to alleviate this drawback.The line search technique fails as well since it amounts to scale the whole vector of the iterative

displacement parameters and no trivial way appears to be for selecting and properly correcting onlythe components which do generate the residual stresses at the inconsistent integration points.The same problem does a�ect the tangent-secant strategy since the change in the iterative search

direction concerns the whole vector as well.For this reason we here propose an original procedure (see Box 3) which represents a suitable

modi�cation of the tangent strategy. Namely, at the gth integration station �g, we evaluate theconstitutive sti�ness operator as follows:

E(i)g =E(i)(�g)=

�(i)E if U(i)g ∈ S1E(i)tan if U(i)g ∈ S2E if U(i)g ∈ S3

(52)

thus considering at the ith structural iteration, for the inconsistent integration points, the elastictensor moduli scaled by an energy-parametrized coe�cient �(i) which progressively decreases tillit becomes zero when convergence has been attained.The criterion which has been adopted to make �(i) decrease during structural iterations is based

on the following considerations.Let

D(i−1)∗ =∑e∈E∗

D(i−1)e =∑e∈E∗

ng∑g=1E(i−1)g (�Ug)(i)(i−1) · (�Ug)(i)(i−1) (53)

be the sum of the energy norms D(i−1)e corresponding to the set of elements E∗ which do containonly non-inconsistent Gauss points (i.e. the elements e such that U(i)g 6∈ S1 ∀g∈ e). Clearly, if E∗= ∅,it turns out to be D(i−1)∗ =D(i−1).It is worth noting that the quantity D∗ is associated with the iteration preceding the current one

since its computation requires the knowledge of the relevant solution.We further de�ne the coe�cient

�(i)∗ = minj∈{0; : : : : ; i−1}

(D( j)∗D(0)

)1=m(54)

where the minimum is taken over all the iterations preceding the current one. The coe�cient m isa quantity which is related to a characteristic dimension of the elements; speci�cally m decreasesas the mesh becomes coarser.Stated equivalently, we make �(i)∗ decrease slower as the number of elements increases since,

in this case, the possibility that zero-energy modes can be activated is by far greater.

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336 G. ALFANO, L. ROSATI AND N. VALOROSO

The value of �(i) in (52) is chosen as follows:

�(i) = min{�0; �(i)∗ }where �0¡1 is a given initial value.The reasons for de�ning �(i) as above are detailed as follows.At the early stages of the iteration process, the ‘reduced’ energy norm D(i−1)∗ is quite close to

D(0) so that the assigned value for �0 sets a conveniently upper bound for �(i) in order to speedup calculations.Starting from a certain point of the iterative process, see Tables I and II, only the re-distribution

of the residual stresses at the completely inconsistent Gauss points is needed.This is veri�ed by the fact that the main portion of D(i) is concentrated in the elements belonging

to E∗; indeed the ratio (D(i)−D(i)∗ )=D(i) is very close to 1 even if we are very far from convergence,i.e. D(i)=D(0)/�.

Table I. Masonry panel: energy norms for the tangent and enhanced tangent strategies; Mesh 1.

Tangent Enhanced tangent�0 = 1:E− 04Iteration D(i) D(i) − D(i)∗ D(i) D(i) − D(i)∗1 1:27E + 04 1:00E− 16 1:27E + 04 1:00E− 162 3:44E + 03 2:51E + 01 3:44E + 03 2:51E + 013 2:96E + 03 7:00E + 02 2:96E + 03 7:00E + 024 9:14E + 03 3:26E + 03 9:14E + 03 3:26E + 035 6:53E + 03 4:65E + 03 6:53E + 03 4:65E + 036 2:51E + 03 1:26E + 03 2:51E + 03 1:26E + 037 5:60E + 03 4:63E + 03 5:60E + 03 4:63E + 038 2:97E + 03 1:85E + 03 2:97E + 03 1:85E + 039 7:73E + 02 3:77E + 02 7:73E + 02 3:77E + 0210 6:08E + 02 3:01E + 02 6:08E + 02 3:01E + 0211 2:87E + 03 4:54E + 02 2:87E + 03 4:54E + 0212 2:31E + 02 1:77E + 02 2:31E + 02 1:77E + 0213 2:93E + 03 2:91E + 03 2:93E + 03 2:91E + 0314 3:83E + 02 3:76E + 02 3:83E + 02 3:76E + 0215 4:28E + 01 3:60E + 01 4:28E + 01 3:60E + 0116 1:28E + 01 1:22E + 01 1:28E + 01 1:22E + 0117 7:10E + 00 6:84E + 00 7:10E + 00 6:84E + 0018 1:60E + 00 1:55E + 00 1:82E + 00 1:74E + 0019 2:87E− 01 2:72E− 01 2:82E− 01 2:63E− 0120 5:77E− 02 5:51E− 02 4:66E− 02 4:61E− 0221 1:29E− 02 1:24E− 02 7:74E− 02 7:74E− 0222 4:32E− 03 4:23E− 03 8:97E− 02 8:94E− 0223 2:61E− 03 2:59E− 03 2:53E + 04 2:53E + 0424 2:24E− 03 2:23E− 03 1:99E + 00 1:97E + 0025 2:14E− 03 2:14E− 03 1:05E + 01 1:05E + 0126 2:09E− 03 2:09E− 03 7:33E− 03 7:26E− 0327 2:06E− 03 2:06E− 03 2:65E− 04 2:65E− 0428 2:03E− 03 2:03E− 03 2:19E− 09 2:18E− 0929 2:00E− 03 2:00E− 03 3:02E− 17 3:02E− 1730 1:97E− 03 1:97E− 03 Converged

— —40 1:70E− 03 1:70E− 03

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 337

Table I. Continued.

Tangent Enhanced tangent�0 = 1:E− 04Iteration D(i) D(i) − D(i)∗ D(i) D(i) − D(i)∗

— —50 1:47E− 03 1:47E− 03

— —60 1:27E− 03 1:27E− 03

— —70 1:09E− 03 1:09E− 03

— —80 9:45E− 04 9:45E− 04

— —90 8:18E− 04 8:18E− 04

— —100 8:97E− 06 7:51E− 06

— —110 5:76E− 10 4:36E− 10111 2:21E− 10 1:67E− 10112 8:49E− 11 6:42E− 11113 3:26E− 11 2:46E− 11114 1:25E− 11 9:45E− 12115 4:81E− 12 3:63E− 12116 1:85E− 12 1:39E− 12117 7:09E− 13 5:35E− 13

Converged

The �ctititious sti�ness will then decrease according to �(i)∗ .Subsequently, when convergence is deemed to occur, say D(i)=D(0) ' �1=2, the ratio D(i)∗ =D(0) is

close to ful�lling the stopping criterion so that, if the residual stresses at the completely inconsistentGauss points were not taken into account, convergence would have been already achieved.Accordingly, at the subsequent iterations, the �ctititious sti�ness must vanish.

5. NUMERICAL EXAMPLES

The proposed strategies have been implemented in the �nite element code FEAP, rel. 7.1, developedby Prof. R. L. Taylor as a research tool and partly documented in Reference [37].In order to illustrate the excellent performances of the proposed approach we here report on the

results of the numerical tests for the plane stress analysis of the unit-thickness masonry panel withopenings considered in Reference [23].The structural model and the relevant loading condition are shown in Figure 2. Young’s modulus

and Poisson’s ratio have been set equal to 1:0GPa and 0.2, respectively. The uniform horizontalload qo represents the live load which is controlled through the multiplier �.Three di�erent discretizations (see Figure 3) and eight-node isoparametric elements with nine

Gauss points have been considered in numerical experiments. A tolerance �=10−16 was alsoassumed.

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338 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 4 shows the load-displacement curve obtained for the three adopted discretizations byplotting the value of the load factor � versus the top right horizontal displacement V indicated inFigure 2.The numerical performances of the strategies discussed in the paper are illustrated in Figures

5–20. Speci�cally, for the meshes considered in Figure 3, in Figures 5–7 the plot of the energy

Table II. Masonry panel: energy norms for the tangent and enhanced tangent strate-gies; Mesh 2.

Tangent Enhanced tangent�0 = 1:E− 04Iteration D(i) D(i) − D(i)∗ D(i) D(i) − D(i)∗1 1:22E + 04 1:00E− 16 1:22E + 04 1:00E− 162 3:73E + 03 7:96E + 01 3:73E + 03 7:96E + 013 9:64E + 03 1:16E + 03 9:64E + 03 1:16E + 034 9:64E + 04 8:60E + 03 9:64E + 04 8:60E + 035 3:38E + 03 7:83E + 02 3:38E + 03 7:83E + 026 6:63E + 03 4:99E + 03 6:63E + 03 4:99E + 037 1:67E + 04 2:18E + 03 1:67E + 04 2:18E + 038 4:61E + 03 3:54E + 03 4:61E + 03 3:54E + 039 3:15E + 04 3:13E + 04 3:15E + 04 3:13E + 0410 8:13E + 03 1:68E + 03 8:13E + 03 1:68E + 0311 2:13E + 03 7:39E + 02 2:13E + 03 7:39E + 0212 4:38E + 03 1:86E + 03 4:38E + 03 1:86E + 0313 7:55E + 02 6:24E + 02 7:55E + 02 6:24E + 0214 7:87E + 02 6:22E + 02 7:87E + 02 6:22E + 0215 4:24E + 03 4:22E + 03 4:24E + 03 4:22E + 0316 6:40E + 03 5:56E + 03 6:40E + 03 5:56E + 0317 2:94E + 02 1:13E + 02 2:94E + 02 1:13E + 0218 1:11E + 02 1:08E + 02 1:11E + 02 1:08E + 0219 9:66E + 02 9:64E + 02 9:66E + 02 9:64E + 0220 6:95E + 02 6:93E + 02 6:95E + 02 6:93E + 0221 1:57E + 02 1:56E + 02 1:57E + 02 1:56E + 0222 2:16E + 01 2:14E + 01 2:16E + 01 2:14E + 0123 2:62E + 01 2:62E + 01 2:62E + 01 2:62E + 0124 1:28E + 01 1:28E + 01 1:28E + 01 1:28E + 0125 1:04E + 01 1:03E + 01 1:04E + 01 1:03E + 0126 5:28E + 00 5:27E + 00 5:28E + 00 5:27E + 0027 8:79E + 01 8:78E + 01 8:79E + 01 8:78E + 0128 1:83E + 02 1:83E + 02 1:83E + 02 1:83E + 0229 6:61E + 00 6:60E + 00 6:58E + 00 6:57E + 0030 1:08E + 00 1:08E + 00 1:10E + 00 1:10E + 0031 5:67E− 01 5:62E− 01 8:89E− 01 8:81E− 0132 3:63E + 00 3:63E + 00 2:20E + 01 1:87E + 0133 3:34E− 01 3:32E− 01 5:31E− 01 5:25E− 0134 8:91E− 01 8:89E− 01 8:29E− 01 8:25E− 0135 5:93E− 01 5:93E− 01 1:49E + 01 1:49E + 0136 1:73E− 01 1:73E− 01 8:66E− 02 8:57E− 0237 1:98E− 01 1:97E− 01 8:72E− 02 8:60E− 0238 1:85E + 01 1:85E + 01 8:39E− 02 8:27E− 0239 5:35E− 02 5:32E− 02 4:25E− 02 4:23E− 0240 4:37E− 02 4:34E− 02 2:51E− 02 2:49E− 02

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 339

Table II. Continued.

Tangent Enhanced tangent�0 = 1:E− 04Iteration D(i) D(i) − D(i)∗ D(i) D(i) − D(i)∗41 3:66E− 02 3:64E− 02 6:44E− 02 6:43E− 0242 3:11E− 02 3:09E− 02 2:96E− 02 2:95E− 0243 2:67E− 02 2:66E− 02 1:96E + 00 1:96E + 0044 2:31E− 02 2:30E− 02 5:77E− 02 5:77E− 0245 2:06E− 02 2:05E− 02 1:32E− 02 1:32E− 0246 1:55E− 02 1:55E− 02 2:95E− 02 2:95E− 0247 1:33E− 02 1:32E− 02 1:97E− 02 1:96E− 0248 1:14E− 02 1:14E− 02 6:38E− 03 6:36E− 0349 9:96E− 03 9:93E− 03 2:48E− 03 2:47E− 0350 8:73E− 03 8:71E− 03 1:34E− 03 1:34E− 0351 7:65E− 03 7:63E− 03 1:08E− 03 1:07E− 0352 6:82E− 03 6:80E− 03 1:09E− 03 1:08E− 0353 6:12E− 03 6:11E− 03 7:41E− 04 7:34E− 0454 5:52E− 03 5:51E− 03 3:36E− 05 3:36E− 0555 5:01E− 03 5:01E− 03 4:07E− 07 4:04E− 0756 4:57E− 03 4:56E− 03 1:46E− 08 1:45E− 0857 4:18E− 03 4:18E− 03 2:83E− 15 2:82E− 1558 3:84E− 03 3:84E− 03 Converged

— —100 6:07E− 04 6:07E− 04

— —200 3:11E− 04 3:11E− 04

— —300 1:28E− 04 1:28E− 04

— —400 3:96E− 05 3:96E− 05

— —500 1:76E− 05 1:76E− 05

Figure 2. Masonry panel: model problem.

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340 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 3. Masonry panel: �nite element meshes.

Figure 4. Masonry panel: load-displacement curves.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 341

Figure 5. Energy norm vs. number of iterations. Mesh 1: tangent strategy.

Figure 6. Energy norm vs. number of iterations. Mesh 2: tangent strategy.

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342 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 7. Energy norm vs. number of iterations. Mesh 3: tangent strategy.

norm versus the number of iterations is given. The di�erent curves refer to the di�erent values ofthe coe�cient �0, see Section 4.3.It is worth noting that convergence becomes more and more troublesome as the number of

elements increases and that an erratic behaviour of the energy norm is likely to be experiencedwith lower values of �0.Further, lower values of �0 do not necessarily improve the performances of the solution strategy

since they can make it extremely unstable, see in this respect the curve of Figure 7 correspondingto �0 = 10−5.Figures 8–10, which refer to the tangent strategy endowed with line searches, highlight the

absence of any signi�cant improvement with respect to a pure tangent approach.Essentially the same behaviour is observed by adopting the tangent–secant strategy, see Figures

11 and 12, which refer to the case �=1.The erratic behaviour of the energy norm (see Figure 12) is essentially due to the strong

interaction between the activation of zero-energy modes and the change in the direction of theiterative displacement vector Tu(i+1)(i) entailed by the switch between the tangent and the secantoperator.The same behaviour, not documented for brevity, has however been experienced by adopting

di�erent values for the switch parameter � and either the secant operator (43) or (50); it can bethus reasonably argued that this is due to an intrinsic de�ciency of the strategy.The dramatic e�ects associated with the activation of zero-energy modes are shown in Figures 13

and 14.For each structural iteration we further report in Tables I and II the values of the incremental

energy D(i) and the di�erence D(i) − D(i)∗, i.e. the incremental energy pertaining to the elementswhich possess at least one completely inconsistent Gauss point. It is here evident how, starting

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 343

Figure 8. Energy norm vs. number of iterations. Mesh 1: tangent strategy + line search.

Figure 9. Energy norm vs. number of iterations. Mesh 2: tangent strategy + line search.

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344 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 10. Energy norm vs. number of iterations. Mesh 3: tangent strategy + line search.

Figure 11. Energy norm vs. number of iterations. Mesh 1: tangent–secant strategy.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 345

Figure 12. Energy norm vs. number of iterations. Mesh 2: tangent–secant strategy.

Figure 13. Activation of zero-energy modes. Mesh1: iteration 5; �=7:0; �0 = 1:0E − 06. Tangent

strategy.

Figure 14. Activation of zero-energy modes. Mesh1: iteration 5; �=7:0; �0 = 1:0E − 07. Tangent

strategy.

from a certain point of the iterative process, the incremental energy D(i) is almost completelylocated in these elements.Hence, all subsequent iterations are basically required to reduce the relevant residuals.This is accomplished in a very e�ective way by the enhanced tangent strategy, see Figures 15–17.Actually, the modi�cation of the parameter �(i) during the structural iterations, according to the

criterion illustrated in Section 4.3, allows one to obtain a very robust solution procedure whichcan be a�ordably used for practical calculations.In the numerical simulations with the enhanced tangent strategy m=1 has been assumed for

the coarser mesh and m=1:5 for others.For completeness we also show in Figures 18 and 19, with reference to the �rst mesh, the

contour plots of the minimum principal stress and of the equivalent inelastic strain.

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346 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 15. Energy norm vs. number of iterations. Mesh 1: enhanced tangent strategy.

Figure 16. Energy norm vs. number of iterations. Mesh 2: enhanced tangent strategy.

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NUMERICAL STRATEGY FOR FINITE ELEMENT ANALYSIS 347

Figure 17. Energy norm vs. number of iterations. Mesh 3: enhanced tangent strategy.

Figure 18. Masonry panel: contour plot of the minimum principal stress at solution.

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348 G. ALFANO, L. ROSATI AND N. VALOROSO

Figure 19. Masonry panel: contour plot of the equivalent inelastic strain at solution.

6. CONCLUDING REMARKS

The numerical investigation which has been carried out on some existing solution strategies adoptedfor structural models with no-tension materials has allowed us to state their lack of robustness.Basically this is due to the fact that such strategies are simply the specialization to the case at

hand of the ones usually conceived for constitutive models in elasto- and elasto=visco-plasticity.Actually, they fail to fully take into account the peculiar feature of structural problems with

the no-tension material model for which zero-energy modes can be activated, especially for �nermeshes.For this reason an original strategy, termed enhanced tangent, has been presented and its per-

formances fully investigated. In the authors’ opinion it represents a robust algorithm for carryingout the analysis of �nite element models of old masonry structures whose constitutive behaviouris conveniently described by the no-tension model.

ACKNOWLEDGEMENTS

The �nancial support of the CNR (National Research Council) is gratefully acknowledged.

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