Computational Mechanics Volume 37 issue 3 2006 [doi 10.1007_s00466-005-0706-1] A. Carpinteri; G. Ferro; G. Ventura -- An augmented lagrangian approach to material discontinuities in

ORIGINAL PAPER

A. Carpinteri Æ G. Ferro Æ G. Ventura

An augmented lagrangian approach to material discontinuitiesin meshless methods

Received: 21 January 2005 /Revised: 18 April 2005 / Published online: 6 July 2005� Springer-Verlag 2005

Abstract The purely nodal discretization, typical ofmeshless methods, turns out in the necessity of properlydefining both the external boundary and the materialinterfaces which may be present in the analysis ofmechanical problems. Each material domain is dicret-ized independently, and interface conditions are imposedinto the variational formulation by the augmentedLagrangian method. This allows for a numericallyefficient formulation where the number of approxima-tion variables is unchanged by the presence of anynumber of interface constraints.

1 Introduction

Mesh free methods have been demonstrated particularlyattractive for several reasons. The purely nodal discret-ization and the absence of nodal connectivity are clearadvantages in large deformation or moving interfaceproblems, as no remeshing or element distortion issuesarise. The high continuity of the approximated fieldsmakes possible an easy implementation of gradient andnon–local constitutive models. On the other hand, thecontinuity and differentiability of the solution in thediscretization domain turns out in the necessity ofan explicit modelling of discontinuities. Differentapproaches has been used in the literature for achievingthis result:

– Virtual subdivision of the domain into two distinctparts and enforcements of interface conditions. Thishas been proposed both for material interfaces [1] andfor crack discontinuity [2, 3];

– Introduction of discontinuities by modification of theweight function supports, e.g. the visibility, diffrac-

tion and transparency approaches [4, 5] for crackdiscontinuities;

– Basis enrichment [6, 7] with discontinuous functions.This approach has origin in the Partition of Unitymethod [8,9] and has been used both in the area ofmesh free and finite element methods. The discon-tinuous enrichment allows the discretization beingcompletely independent of the position and orienta-tion of discontinuities and requires the introductionof extra nodal variables in a neighbor of the discon-tinuity. Typical application of this approach havebeen modelling of holes, inclusions and cracks [10–15].

In the present paper attention is focused on the first ofthe above approaches. Even if, from the formulationviewpoint, the approach is close to the one suggested byCordes and Moran [1], the difference between a pureclassical Lagrangian multiplier approach and the aug-mented Lagrangian one [16] is substantial. In fact theshortcomings of the above mentioned techniques may bebriefly summarized as follows:

– The enforcement of interface conditions by a pureLagrangian method enlarges the approximationvariables space by the number of Lagrangian multi-pliers, and the resulting algebraic problem is notgoverned by a positive definite matrix even for linearelastic constitutive laws;

– Modification of weight functions is not suitable forgeneral interfaces modelling, and is basically limitedto crack tip discontinuities;

– Discontinuous enrichment requires the introductionof extra nodal variables. They are a very limitednumber in finite element formulations. On the otherhand, in meshless formulations, their number can besignificant, as each node whose support intersect thediscontinuity is to be enriched [7, 10, 11, 17].

The use of an augmented Lagrangian approach makesthe use of interface conditions attractive as a largenumber of constraints can be easily introduced into the

Comput Mech (2006) 37: 207–220DOI 10.1007/s00466-005-0706-1

A. Carpinteri Æ G. Ferro Æ G. Ventura (&)Department of Structural and Geotechnical Engineering, Politec-nico di Torino, Corso Duca degli Abruzzi, 24 - 10129 Torino, ItalyE-mail: [email protected]

Euler equations of the variational form throughLagrangian and penalty terms. However, neither theLagrangian multipliers are added to the approximationvariables, nor the penalty terms introduce ill condition-ing in the solution stage [16]. Moreover, the computingtime is not significantly affected by the total number ofconstraints. While the approximation properties of themodel and a comparison toward closed form results canbe found in [1], here the attention is focused on theconvergence properties of the augmented Lagrangianalgorithm and on the effect of the number of interfacepoints along the discontinuity. The outline of the paperis as follows: in Sect. 2, the Moving Least Squaresapproximation is recalled; in Sect. 3, the variationalformulation of the elastic problem with augmentedLagrangian boundary conditions enforcement is given;in Sect. 4, the material discontinuity interface equationsare stated and the relevant augmented Lagrangian termsare added to the variational problem; in Sect. 5, theaugmented Lagrangian solution algorithm is described,and Sect. 6, addresses the problem of the evaluation ofoptimal penalty parameters. Finally Sect. 7 investigatesa numerical test to illustrate the effect of the penaltyparameters initial values and interface discretization.The solution is then compared to finite element solutionswith similar and much denser discretization.

2 MLS approximation

In this section a short description of the moving leastsquares (MLS) approximation [18,19] is reported. Vec-tors will be indicated between curly braces fg, whilematrices will be between square braces ½�, with some oftheir dimensions reported underneath to improve read-ability.

At each point of a domain X a scalar function gðxÞ isapproximated by a linear combination of basis functionswith non constants coefficients aðxÞf g, in the form

gðxÞ ¼ fpðxÞgTð1�mÞ

faðxÞgðm�1Þ

ð1Þ

m being the number of terms in the base. As an example,in the two dimensional case, the linear and quadraticbases are

pðxÞT ¼ 1; x1; x2f g ð2Þ

pðxÞT ¼ 1; x1; x2; x1x2; x21; x22

� �ð3Þ

The coefficients faðxÞg in (1) are obtained, at any pointx, by minimizing the square of the difference between theapproximation (1) and the value dI of the function at thediscretization node xI . Therefore, the following func-tional is minimized

JðxÞ ¼ 1

2

Xn

I¼1wIðx� xIÞ fpðxIÞgTfaðxÞg � dI

h i2ð4Þ

where wIðx� xIÞ is a weight function with compactsupport [5]. The presence of a weighting function withcompact support restricts the sum on I on the subsetWðxÞ of the n neighboring nodes given by

WðxÞ ¼ I 2 X : wðx� xIÞ > 0f g ð5ÞUpon minimization of (4) the following final shapefunction form is obtained [5]

gðxÞ ¼X

I2WðxÞ/IðxÞdI ¼ /ðxÞT

n o

ð1�nÞ

df gðn�1Þ

ð6Þ

f/ðxÞg being the vector of the MLS shape functions anddf g is the vector of the nodal values. Because of the localleast squares fit, the approximation gðxÞ is such thatgðxIÞ 6¼ dI . This results in the necessity to enforceexplicitly Dirichlet (essential) boundary conditions.

3 Variational formulation

The shape functions derived in the former section areintroduced into the minimum potential energy principlefor the elastic problem. If the problem is set in thend–dimensional space, let fgg be the displacement field,with nd components. The MLS approximation can beapplied to each component of the displacement field inthe form (6) and extended to all the discretization nodesby using the compact expression

fggnd�1ð Þ

¼ Ug� �

nd�Ndð ÞfdgNd�1ð Þ

ð7Þ

where Ug� �

is the matrix of the shape functions and fdgis redefined as the vector of all the nodal components ofthe approximation variables. To simplify the notationthe strains are expressed through the MLS shape func-tions by

fegðd�1Þ

¼ oe½ �d�ndð Þ

fggnd�1ð Þ

¼ oe½ �d�ndð Þ

½Ug�nd�Ndð Þ

df gNd�1ð Þ

¼ Ue½ �d�Ndð Þ

df gNd�1ð Þ

ð8Þ

with d ¼ 1 for nd ¼ 1, d ¼ 3� nd � 1ð Þ for nd ¼ 2; 3 andfeg is the strain vector, given in the plain case byfegT ¼ ex; ey ; cxy

� �. The matrix Ue½ � is determined as the

symmetric part of the gradient of the displacementthrough the differential operator oe½ �.

Let consider the domain X of a solid subjected tovolume forces bf g, surface forces qf g and prescribeddisplacements fgg, acting, respectively, on X, oXq andoXg

�oX¼oXq [ oXg; oXq \ oXg ¼ ;

�. The solution of

the Cauchy elastic problem is given by the minimum ofthe potential energy functional

PðfggÞ¼ 1

2

Z

X

fegTð1�dÞ

½C�ðd�dÞ

fegðd�1Þ

dX�Z

X

fbgTð1�ndÞ

fggðnd�1Þ

dX

�Z

oXq

fqgTð1�ndÞ

fggðnd�1Þ

dSþZ

oXu

ind f�ggðnd�1Þ

� fggðnd�1Þ

!

dS

ð9Þ

208

½C� being the elastic operator. The last part of (9) isadded as the MLS shape functions do not satisfy theessential boundary conditions. The indicator function isequal to

ind f�gg � fggð Þ ¼ 0 if f�gg � fgg ¼ f0gþ1 if f�gg � fgg 6¼ f0g

�ð10Þ

For the minimum of (9) to be actually computed, theindicator function is regularized by replacing it with itsaugmented Lagrangian counterpart [20, 16]Z

oXu

ind f�gg � fggð Þ dS

¼Z

oXu

suprfrgT f�grg � fgrgð Þh

þ 1

2a f�grg � fgrgð ÞT f�grg � fgrgð Þ

�dS ð11Þ

where the reactions rf g have the mathematical meaningof Lagrangian multipliers. Introducing the hypothesisthat the assigned displacements f�gg are interpolated by aset of Dirac Delta functions placed to correspond withprescribed points xi on oXg (which is equivalent toenforcing the essential boundary conditions only on adiscrete number of points), the following expression isobtainedZ

oXu

ind f�gg � fggð Þ dS ¼Xnr

i¼1supfrgi

frgTi f�grgi � fgrgi

� �h

þ 1

2a f�grgi � fgrgi

� �T f�grgi � fgrgi

� ��ð12Þ

where frgi and �grf gi are, respectively, the reactions andthe assigned displacements at the ith restrained point,while a > 0 is the penalty parameter. To improve theprecision in the discretization of the essential boundaryconditions, other points, not necessarily discretizationnodes, can be restrained along the boundary with thesame technique. Note also that, unlike penalty methods[21], augmented Lagrangian methods enforce the con-straints up to machine precision, independently of thevalue of the penalty parameter [16, 22].

By introducing the MLS approximation (7), (8) andthe augmented Lagrangian regularization (12) into thepotential energy functional (9), an augmented Lagrang-ian functional PALð df g; rf gÞ is obtained, whose saddlepoint gives the problem solution. This formulationallows for the elimination of all the drawbacks related tothe use of the Lagrangian and penalty methods, andrepresents a computationally effective approach for theintroduction of constraints on direct variables in theMLS approximation. Full details are reported in [16],while an extension to crack problems is given in [2].

After substitution of the MLS expressions (7), (8) forthe displacement and strain and the essential boundaryconditions being linear equality constraints, Eq. (12) canbe written in compact form

Z

oXg

ind f�gg � fggð Þ dS

¼ suprf g

1�nrndð Þ

frgT f�grg � Qg� �fdg

� �h

þ 1

2a f�grg � Qg

� �fdg

� �T f�grg � Qg� �fdg

� ��ð13Þ

where Qg� �

is the constraint matrix for the boundaryconditions. Note that in (13) and in the sequel rf g andf�grg will denote the reactions and prescribed displace-ments at all the nr restrained nodes, respectively. Letting

½K� ¼Z

X

½U�Te ½C�½U�e dX;

ff g ¼Z

X

½U�Tfbg dXþZ

oXq

½U�Tfqg dS ð14Þ

the discrete augmented Lagrangian functional can bewritten in the following compact form

PAL df g; frgð Þ ¼ 1

2df gT½K� df g � ff gT df g

þ frgT �grf g � Qg� �fdg

� �

þ 1

2a f�grg � Qg

� �df g

� �T

� f�grg � Qg� �

df g� �

ð15Þ

4 Material discontinuity

4.1 Statement of the problem

An elastic body X with a material discontinuity line C isconsidered, Fig. 1. The part X1 is assumed to be made bya material 1, and the part X2 is made of a material 2. Iffng denotes the outward normal along C to the regionX1, the continuity of the displacements fgg (congruence)

Fig. 1 Body with a line of material discontinuity

209

and of the traction vector ftg (equilibrium) must hold atevery point of C

fggð1Þ � fggð2Þ ¼ f0g ð16Þ

r½ �ð1Þ� r½ �ð2Þ

fng ¼ f0g ð17Þ

where the apex ðiÞ denotes the quantities belonging to Xi,r½ � is the stress tensor and ftgð1Þ ¼ r½ �ð1Þfng,ftgð2Þ ¼ � r½ �ð2Þfng. These expressions can be rewrittenconsidering the stress vector frg, given in 2–D byfrgT ¼ rx; ry ; sxy

� �and the normal projection operator

N½ � such that r½ �fng ¼ N½ �frg, whose expression in the2–D case is

N½ � ¼ nx 0 ny

0 ny nx

� �ð18Þ

being the unit normal fngT ¼ nx; ny� �

. Consequently theinterface conditions can be written in the form

fggð1Þ � fggð2Þ ¼ f0g ð19Þ

N½ � frgð1Þ � frgð2Þ

¼ f0g ð20Þ

4.2 Augmented Lagrangian approach

The augmented Lagrangian variational formulation hasbeen used in the previous treatment as a formal andnumerical tool to enforce the boundary conditions, butits use can be readily extended to the satisfaction of anyconstraint between the approximation variables. A typ-ical case is modelling discontinuous fields. In fact, thedifferentiability and continuity properties of the MLSsolution descends from the ones of the weight and basisfunctions, and discontinuities have to be introducedexplicitly. Here the case of the material discontinuity isaddressed. Other discontinuities, such as crack model-ling, have been treated using the same approach [2].

The introduction of the line of discontinuity as acomplete splitting of the solid in two parts with theintroduction of the constraints (16) and (17) is astraightforward extension of the visibility criterion forcracks first stated in [16, 17]. This approach has beenused in [1] for material discontinuity modeling usingLagrange multipliers. In this respect the augmentedLagrangian solution presented here can be seen as aunified way for constraint treatment in element–freeapproximations, whose advantages will be pointed outin Sect. 5.

The constraints (16), (17) can be included asEuler-Lagrange equations of the functional (9) byadding the termZ

C

ind fggð1Þ � fggð2Þ

dS

þZ

C

ind r½ �ð1Þ� r½ �ð2Þ

fng

dS ð21Þ

whose augmented Lagrangian regularisation is

suprCf g

Z

C

rCf gT fggð1Þ � fggð2Þ

þ 1

2a fggð1Þ � fggð2Þ 2

dS

þ supeCf g

Z

C

eCf gT r½ �ð1Þ� r½ �ð2Þ

fng

þ 1

2b r½ �ð1Þ� r½ �ð2Þ

fng 2

dS ð22Þ

Here two penalty coefficients a and b have been intro-duced as the displacements and the traction vectors arein general very differently scaled constraints.

According to the visibility criterion the two parts X1

and X2 are considered as two distinct solids with theboundary conditions (19), (20) along the discontinuityline C. Therefore the line C is conceptually treated aspart of the boundaries oX1 and oX2 and the supportsof the weight functions for the nodes belonging to X1

are not considered in the region X2 and viceversa,Fig. 2.

Introducing the MLS expressions for the displace-ments and the stresses, the interface conditions (19), (20)at a point xi on the discontinuity line C can be given theform

Ug� �ð1Þ

i � Ug� �ð2Þ

i

df g ¼ f0g ð23Þ

½N �i ½C�ð1Þ Ue½ �ð1Þi �½C�

ð2Þ Ue½ �ð2Þi df g

¼ f0g ð24Þ

Now suppose that a series of nC interface nodes areconsidered along C and that the constraints (23), (24)are imposed at these nodes. The treatment of theinterface constraints follow the same path outlined forthe boundary conditions and the discretised augmentedLagrangian functional is extended to this case byadding the Lagrangian and penalty term for bothconstraints

Fig. 2 Weight function supports for the regions 1 and 2

210

XnC

i¼1rCf gTi Ug

� �ð1Þi � Ug

� �ð2Þi

df g

þ 1

2a Ug

� �ð1Þi � Ug

� �ð2Þi

df g

h i2

þ eCf gTi ½N �i ½C�ð1Þ Ue½ �ð1Þi �½C�

ð2Þ Ue½ �ð2Þi

df g

h i

þ 1

2b ½N �i ½C�

ð1Þ Ue½ �ð1Þi � C½ �ð2Þ Ue½ �ð2Þi

df g

h i2ð25Þ

As the interface conditions are linear homogeneousconstraints, they can be written with reference to all thenC interface nodes in the form

XnC

i¼1Ug� �ð1Þ

i � Ug� �ð2Þ

i

df g ¼ f0g ) Qu½ � df g ¼ f0g ð26Þ

XnC

i¼1½N �i ½C�

ð1Þ Ue½ �ð1Þi �½C�ð2Þ Ue½ �ð2Þi

df g ¼ f0g

) Qt½ � df g ¼ f0g ð27Þ

where the constraint matrices Qu½ � and Qt½ � will bothhave ndimnC rows and Nd columns as at each interfacenode the traction and displacement vectors have ndimcomponents. Therefore (25) assumes the form

rCf gT Qu½ � df g þ 1

2a df gT Qu½ �T Qu½ � df g

þ eCf gT Qt½ � df g þ 1

2b df gT Qt½ �T Qt½ � df g ð28Þ

The augmented Lagrangian potential energy func-tional including the interface conditions will have thenthe final form

PCAL df g; frg; rCf g; eCf gð Þ ¼ 1

2df gT½K� df g � ff gT df g

þ frgT Q0½ � þ Q1½ � df gð Þ

þ 1

2a Q0½ � þ Q1½ � df gð Þ2

þ rCf gT Qu½ � df g

þ 1

2a df gT Qu½ �T Qu½ � df g

þ eCf gT Qt½ � df g

þ 1

2b df gT Qt½ �T Qt½ � df g ð29Þ

It is useful, to the end of the numerical implementa-tion of the augmented Lagrangian solution algorithm,noting that the Hessian of (29) w.r.t. the approximationvariables df g is given by

r2dd PC

AL ¼ ½K� þ a Q1½ �T Q1½ � þ Qu½ �T Qu½ �

þ b Qt½ �T Qt½ �

¼ ½K� þ a Hb½ � þ Hu½ �ð Þ þ b Ht½ � ð30Þ

where the Hessians Hb½ � of the boundary conditions andof the material discontinuity interface Hu½ � and Ht½ � areput in evidence. They are all positive semidefinite as theyare computed from the tensor products of the constraintmatrices Q½ �.

5 Augmented Lagrangian solution

The augmented Lagrangian methods are general meth-ods for the solution of nonlinear and non convex con-strained optimization problems. The typical feature ofthese methods is that they iterate alternatively on theapproximation variables and on the Lagrangian multi-pliers. This allows to leave the dimension of the directproblem unchanged for any number of constraints. Theconvergence rate on the Lagrangian multipliers is fast(see the examples in the following) and the constraintscan be satisfied with very precision. Unlike penaltymethods there is no ill conditioning of the governingfunctional and the solution is totally independent of thevalue of the penalty parameter. Unlike Lagrange mul-tipliers method, the presence of the Lagrangian multi-pliers does not turn a convex problem into a non convexone nor the sparsity of the governing matrices is affected.The reasons for that will be apparent in the following. Amore detailed discussion on augmented lagrangianmethods can be found in [22, 23], while their applicationinto an element free formulation has been first appearedin [16], and an extension to crack modelling has beenpresented in [2].

For simplifying the notation, df g and fkg will indi-cate in both cases the sets of the approximation variablesand Lagrangian multipliers and a single penaltyparameter is considered. Assuming this formalism, thegeneral scheme for the augmented Lagrangian iterationis as follows

df gðkþ1Þ¼ Ud df g; fkgðkþ1Þ; aðkÞ

ð31aÞ

fkgðkþ1Þ ¼ Uk df gðkþ1Þ; fkg; aðkÞ

ð31bÞ

aðkþ1Þ ¼ Up df gðkþ1Þ; fkgðkþ1Þ; aðkÞ

ð31cÞ

where Ud, Uk, Up are called respectively Lagrangianmultipliers update formula, direct variables update for-mula and penalty parameter increment scheme. Equa-tions (31) are evaluated in sequence until convergence onboth sets of variables is attained.

Many forms of the method are included in thisscheme. Here we will present the probably most usedone, originally proposed by Hestenes [24] and Powell[25]. The structure of the algorithm is as follows:

1. Let k ¼ 0 and tol some fixed tolerance. Set initialvalues for the penalty parameters and the Lagrangianmultipliers;

2. minimize the augmented Lagrangian functional w.r.t.the approximation variables df g

211

df g ¼ argmindf g

PCAL df g; frgðkþ1Þ; rCf gðkþ1Þ; eCf gðkþ1Þ

ð32Þ3. if the norm of the constraints is less than the fixed

tolerance tol the algorithm has converged;

k Q0½ � þ Q1½ � df gðkþ1Þk < tol ð33Þk Qu½ � df gðkþ1Þk < tol ð34Þk Qt½ � df gðkþ1Þk < tol ð35Þ4. compute updated values for the Lagrangian multi-

pliers by the formulas

rf gðkþ1Þ¼ rf gðkÞþaðkÞ Q0½ � þ Q1½ � df gðkþ1Þ

ð36Þ

rCf gðkþ1Þ¼ rCf gðkÞþaðkÞ Qu½ �fdgðkþ1Þ ð37ÞeCf gðkþ1Þ¼ eCf gðkÞþbðkÞ Qt½ � df gðkþ1Þ ð38Þ5. if the norm of all constraints has decreased (this is a

numerical stability test [16]) update the penaltyparameters [16, 22, 23] by the following formulas, setk ¼ k þ 1 and return to step 2.

if k Q0½ � þ Q1½ � df gðkþ1Þk > k Q0½ � þ Q1½ � df gðkÞk4

then aðkþ1Þ ¼ 10aðkÞ ð39aÞ

if k Qu½ � df gðkþ1Þk > k Qu½ � df gðkÞk4

then aðkþ1Þ ¼ 10aðkÞ ð39bÞ

if k Qt½ � df gðkþ1Þk > k Qt½ � df gðkÞk4

then bðkþ1Þ ¼ 10bðkÞ ð39cÞNote that the only modification to the augmented

Lagrangian algorithm in the interface model is treatingin parallel and in the same way all the Lagrange mul-tipliers. Only the penalty parameters a and b are keptdistinct due to the very different scaling betweendisplacement and traction constraints.

The method performs separate iterations onapproximation variables and Lagrangian multipliers.The optimization process w.r.t. a variables set is madeconsidering the others as constants. For this reason, ifthe original problem is convex, its augmentedLagrangian counterpart is convex as well, and the step(31a) is therefore a convex minimization problem.

In the present case it can be demonstrated that thealgorithm will theoretically converge for any positivevalue of the penalty parameters, since the potentialenergy functional is convex. To obtain satisfactory con-vergence rates the update formulas for the penaltyparameters (39) drive effectively the penalty parametersfrom an arbitrary initial low value to an optimal one forconvergence. This is in fact demonstrated by a theoremstating that if the solution exists and the Lagrangemultipliers are bounded, then the sequences generated

by (39) are bounded as well, and a linear convergencerate it obtained [23]. The situation turns out morecomplex in the interface model, where two independentpenalty parameters come into play. Numerical experi-ments point out that, although formulas (39) are effec-tive, the generated penalty parameters may leadsometimes to a non converging iteration. This aspect willbe addressed in Sect. 7, and is solved estimating theoptimal initial values of the penalty parameters priorstarting the iteration, as shown in the subsequent sec-tion.

The Hestenes and Powell iteration scheme is similarto some extent to the SUMT algorithm (SequentiallyUnconstrained Minimisation algoriThm) proposed byFiacco and McCormik [21], consisting in the solution ofa sequence of unconstrained minimization problemswith the quadratic penalty function methods fora! þ1. Note however that while for the SUMT toconverge a! þ1 is required (with the consequentnumerical problems), in the augmented Lagrangianmethod the penalty parameter value remains finite andno ill conditioning is introduced during the iteration. Inthis case the penalty parameter controls the rate ofconvergence in the dual variables and this motivates theupdate formulas (39).

6 Optimal selection of the penalty parameters

The importance of a correct selection of the penaltyparameters has been pointed out in the previous section.In fact, from a theoretical point of view, the algorithmwill converge for any positive value of the penaltyparameters a and b. In practice, numerical experimentshave shown that the selection of appropriate values isnot trivial. In fact:

– Low values of the penalty parameters are responsibleof slow convergence rate in the Lagrangian multi-pliers iteration, and even of numerical singularity ofthe Hessian matrix of the functional, being the rigidbody motions suppressed by the enforced constraints;

– High values of the penalty parameters cause, as in thepenalty method, ill conditioning of the Hessianmatrix and consequent non convergence in theapproximation variables.

As long as no interface conditions are considered (i.e.only one penalty parameter is present) a simple selectionstrategy could be the selection of an initial low value of athat will be automatically incremented to the optimumvalues by the scheme (39a). This leads to the solution buteach increment of the parameter imply a new factor-ization of the Hessian matrix for solving the minimiza-tion problem with the consequent numerical inefficiency(a Newton’s method has been used). The situationbecomes critical when the interface condition are pres-ent, as the two parameters a and b must be set. Thepractical experience has shown that the user selection of

212

these values is extremely inefficient and sometimes causenon convergence. For all these reasons an automaticpenalty selection algorithm has been adopted extendingthe concept reported in [16].

Let recall the Hessian expression (30). It is formed bythe positive semidefinite matrix K½ �, accounting for thestrain energy of the unconstrained body and the positivesemidefinite constraints matrices Hb½ �, Hu½ �, Ht½ �. If thesolid is properly constrained, the Hessian matrix issymmetric and positive definite as it represents the totalstrain energy.

The matrices K½ �, Hb½ �, Hu½ �, Ht½ � are summed upweighted by the penalty parameters. To ensure goodnumerical conditioning of the problem the matricesmust be properly scaled. As these matrices are all posi-tive semidefinite, their minimum eigenvalue can beassumed in general equal to zero. Before starting theaugmented Lagrangian iteration the following maxi-mum eigenvalues are estimated by forward iteration [26]with very large convergence tolerance (only the order ofmagnitude is needed)

kK ¼ max eig K½ � ð40Þkb ¼ max eig Hb½ � ð41Þku ¼ max eig Hu½ � ð42Þkt ¼ max eig Ht½ � ð43Þand the penalty coefficients are set as follows

a ¼ kKmin kb; kuð Þ : ð44aÞ

b ¼ kKkt

ð44bÞ

Equations (44) fix the augmented parameter values insuch a way the spectral radii of the matrices K½ �, Hb½ �,Hu½ �, Ht½ �, giving by (30) the total Hessian, are all com-parable. As noticed before a controls the constraints onboundary and material interface displacements, so that,in general, the eigenvalues kb and ku will be the sameorder of magnitude. Some results about the convergencebehavior of the augmented Lagrangian algorithm fordifferent initial values of the penalty parameters will beshown, see Sect. 7.

As a final comment to the selection of the initialvalues, it has been found speeding up the computationto assume a rounded value of the initial penaltyparameters in the form 10n, n being an integer power.This simplifies the multiplication operation of the pen-alty parameters times the elements of the constraintsHessians H½ �.

7 Numerical results

The material discontinuity interface model is tested inthis section with reference to a typical example. Theability of the interface model to reproduce the materialdiscontinuity has been previously analyzed in [1].

Consequently here the aspects involving the augmentedLagrangian iteration are focused, discussing the influ-ence of the penalty parameters and the influence of the

Fig. 3 Bimaterial rectangular region with prescribed end displace-ments

Fig. 4 Considered discretizations for the bimaterial problem withm ¼ 0. The discretization nodes are the small full dots along theperimeter, while hollow circles are virtual nodes. The interface pointsare pairs along the interface, as evidenced by the nodal numbering in(b). The interface points are represented by the little squares, while thetwo large circles are the starting and ending points of the interface

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interface discretization. The computed solution is com-pared to a finite element one.

Note in fact that the Lagrangian multipliers deter-mined by the Augmented Lagrangian method are coin-cident with the ones determined by the classicalLagrangian functional, while the penalty term is zero atthe converged solution (the penalty parameter beingsmall and finite in contrast to penalty methods). Con-sequently, the proposed approach yields the same solu-tion as [1] and has the same convergence properties.

The test problem is illustrated in Fig. 3. It is a rect-angular bimaterial region with prescribed displacementsat the left and right ends. The Young modulus is equalto Eð1Þ in the left part ð1Þ and to Eð2Þ in the right part ð2Þ.The Poisson coefficient m is assumed to be equal in bothparts and plane stress condition is considered.

In the following the linear basis

pðxÞT ¼ f1; x; yg ð45Þand the cubic spline weight

wðsÞ ¼23� 4s2 þ 4s3 s � 1

243� 4sþ 4s2 � 4

3 s3 12 < s � 1

0 s > 1

8<

:ð51Þ

will be used in all the examples. In (46) it is s ¼kx� xIk=dm where dm is the radius of the support of theweight function.

The first simple test is to assume Eð1Þ ¼ Eð2Þ ¼ 10000and m ¼ 0. In this case, basically a patch test, a uniformuniaxial stress state must be recovered. As illustrated inFig. 4, eight discretization nodes are used, with radius ofthe support of the weight function dm ¼ 50. On each

Fig. 5 Deformed and undeformed geometry for Eð2Þ ¼ 2Eð1Þ, m ¼ 0

Fig. 6 Bimaterial bar problem. Computed finite element stress rxalong the left part of the interface

Fig. 7 Bimaterial bar problem. Computed finite element stress ryalong the left part of the interface

Fig. 8 Bimaterial bar problem. Computed finite element stress sxyalong the left part of the interface

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material domain a single quadrature cell with a 10� 10Gauss quadrature rule has been considered, so that atotal of 200 Gauss points were assumed. Boundary andinterface conditions are enforced at the discretizationnodes and at some extra points. The extra points arerepresented by hollow circles (at the boundary) orsquares (at the interface) in the figures. We call thesepoints virtual nodes, as they are used only to enforceboundary or interface conditions but they do not haveany associated discretization variable. The considerednumber of interface points varies from 2, Fig. 4(a), to 10,Fig. 4(d). Consequently the number of discretizationvariables, given by the number of discretization nodestimes the number of spatial dimensions, is equal to8� 2 ¼ 16 in all cases (a). . .(d). The number of con-straints is computed as follows:

– Vertical and horizontal displacement at the left lowernode;

– Horizontal displacement for all left and right endpoints;

– Four scalar constraints (vertical and horizontal dis-placement and traction vector) at each interfacepoint.

The number of constraints in Fig. 4(a) to (d) variestherefore as follows: 13 (a), 25 (b), 37 (c), 61 (d). Theaugmented Lagrangian iteration converges to the exactsolution up to machine precision in all cases. Table 1summarizes the results with the relevant data. Note thehigh difference of order of magnitude in the penalty

Fig. 9 Bimaterial bar problem. Typical mesh free discretizationshowing 4� 8 MLS approximation nodes (small full dots) and virtualnodes for essential boundary and interface conditions (hollow circlesor squares)

Fig. 10 Computed horizontal stress along the left part of the interfacein 3� 4 element free discretizations

Fig. 11 Computed vertical stress along the left part of the interface in3� 4 element free discretizations

Fig. 12 Computed tangential stress along the left part of the interfacein 3� 4 element free discretizations

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parameters a (displacement constraints) and b (tractionvector constraints).

A similar behavior is observed by setting Eð1Þ ¼10000, Eð2Þ ¼ 20000, m ¼ 0, and considering the samesupport radius and quadrature cells as in the previousexample. Here a constant uniaxial stress and a piecewiseconstant strain is expected at the solution. The resultsare summarized in Table 2, while Fig. 5 reports theundeformed and deformed geometry, illustrating thedifferent deformation in the two parts.

When Eð1Þ ¼ 10000, Eð2Þ ¼ 20000, m ¼ 0:2 the solu-tion of the problem is no longer a uniaxial stress state,but vertical and tangential stresses develop at the inter-face.

Figures 6, 7 and 8 plot the finite element solutionalong the interface. The stresses are computed on the leftpart of the interface. No smoothing is considered tocancel the discontinuities in between the elements. Thesolution has been obtained with a mesh of 10000 fournode rectangular elements. Along the interface 100 ele-ments have been used, while the mesh density in the

Fig. 13 Computed horizontal stress along the left part of the interfacein 4� 8 element free discretizations

Table 1 Results for the bimaterial problem with Eð1Þ ¼ Eð2Þ ¼10000, m ¼ 0. The reported penalty parameters are the optimalones, computed as described in Sect. 6

Example a b Directiterations

Dualiterations

rmaxx rmin

x

Fig. 4(a) 1.0E + 5 1.0E)2 16 5 1000.0 1000.0Fig. 4(b) 1.0E + 4 1.0E)2 25 8 1000.0 1000.0Fig. 4(c) 1.0E + 4 1.0E)3 22 7 1000.0 1000.0Fig. 4(d) 1.0E + 4 1.0E)2 19 6 1000.0 1000.0

Table 2 Results for the bimaterial problem with Eð1Þ ¼ 10000,Eð2Þ ¼ 20000, m ¼ 0. The reported penalty parameters are theoptimal ones, computed as described in Sect. 6

Example a b Directiterations

Dualiterations

rmaxx rmin

x

Fig. 4(a) 1.0E + 5 1.0E)2 16 5 1333.3 1333.3Fig. 4(b) 1.0E + 5 1.0E)2 13 4 1333.3 1333.3Fig. 4(c) 1.0E + 4 1.0E)3 25 8 1333.3 1333.3Fig. 4(d) 1.0E + 4 1.0E)2 19 6 1333.3 1333.3

Table 3 Mesh free discretizations adopted in the bimaterial barproblem

MLSNodes

Interfacenodes

Number ofapproximationvariables

Number of constraints(Lagrangian multipliers)

3� 4 4 48 253� 4 10 48 613� 4 16 48 974� 8 8 128 494� 8 22 128 1334� 8 36 128 2178� 16 16 512 978� 16 46 512 2778� 16 76 512 457

Table 4 Quadrature and support radii for the bimaterial bar pro-blem (data referred to a single material domain)

MLS Nodes Quadrature cells Gauss rule Support radiusdm

3� 4 2� 3 10� 10 504� 8 3� 7 10� 10 508� 16 7� 15 10� 10 25

Fig. 14 Computed vertical stress along the left part of the interface in4� 8 element free discretizations

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horizontal direction has been increased as the interface isapproached. This will be assumed as a ‘‘reference’’solution, even though some edge effect is present, asreported in the figures.

The mesh free simulation has been run with threedifferent discretizations of 3� 4, 4� 8 and 8� 16 nodesin each material domain (at the left and at the right ofthe interface), and varying the number of interfacenodes, as reported in Table 3. Figure 9 illustrates the

typical discretization adopted. Table 4 reports theinfluence radii and quadrature details.

Figures 10, 11, 12 illustrate computed the stressesalong the left part of the interface in the 3� 4 discreti-zations. Figures 13, 14, 15 illustrate the same stresses inthe 4� 8 discretizations. Finally Figs. 16, 17, 18 com-pares the stresses of the element free discretizations8� 16 to a finite element similar discretization, having15 rectangular four node finite elements along the

Fig. 15 Computed tangential stress along the left part of the interfacein 4� 8 element free discretizations

Fig. 16 Mesh free versus finite element horizontal stress along the leftpart of the interface. The discretization is 8� 16 nodes in eachmaterial domain

Fig. 17 Mesh free versus finite element vertical stress along the leftpart of the interface. The discretization is 8� 16 nodes in eachmaterial domain

Fig. 18 Mesh free versus finite element tangential stress along the leftpart of the interface. The discretization is 8� 16 nodes in eachmaterial domain

217

interface and 7 elements in the horizontal direction onboth the left and right part of the interface. The finiteelement stresses are not smoothed and exhibit the typicaldiscontinuous behavior. In fact the finite element con-nectivity at the interface satisfies the compatibility con-

dition, while the continuity of the traction vector is notexplicitly enforced and in general may not hold.

Comparing the finite element to the element free nonsmoothed solutions it is apparent, as expected, that theelement free stresses are close to the finite element onesbut they basically doesn’t require smoothing. A partic-ular care should be exerted in enforcing the interfaceconditions at extra points in between the discretizationnodes. Apart the single case reported in the 3� 4 dis-cretization, Figs. 10, 11, this has affected adversely thesolution. Spurious oscillations in the horizontal andvertical stresses has been detected, as reported in thefigures. Now it is known in meshless methods thatenforcing the essential boundary conditions at extrapoints, beside the discretization nodes, improves theprecision of the solution. A natural question is why inthe examined example these oscillation arise. For thisreason the interface model has been changed, and onlydisplacement continuity as been enforced, likewise thefinite element model. The computed stresses, comparedto the finite element ones, are reported in Figs. 19, 20,21. In this case including extra points does not cause thesolution to oscillate, but has a smoothing effect. This canbe explained by observing that, as equilibrium is locallysatisfied in a weak sense in the discrete model, the twoconstraints, compatibility and traction continuity, whenenforced at several points tend to be incompatible andcause the solution to oscillate. These oscillations shouldnot take place if a mixed formulation is used.

As stressed in Sect. 6 an important aspect of theaugmented Lagrangian iteration lies in the selection ofthe initial penalty parameters. To illustrate this aspect

Fig. 19 Mesh free versus finite element horizontal stress along the leftpart of the interface. In the meshless solution no traction continuityhas been enforced. The discretization is 8� 16 nodes in each materialdomain

Fig. 20 Mesh free versus finite element vertical stress along the leftpart of the interface. In the meshless solution no traction continuityhas been enforced. The discretization is 8� 16 nodes in each materialdomain

Fig. 21 Mesh free versus finite element tangential stress along the leftpart of the interface. In the meshless solution no traction continuityhas been enforced. The discretization is 8� 16 nodes in each materialdomain

218

the 4� 8 dicretization with 8 interface nodes has beenconsidered, and the penalty parameters a and b arevaried from their optimal values aopt ¼ 10þ5 andbopt ¼ 10�3 by three orders of magnitude. The optimalvalues are computed according to what reported inSect. 6. Table 5 reports the number of direct/dual iter-ations as well as the non convergence of the iterationand the number of Hessian factorizations required inbetween parentheses. The automatic penalty parameterincrement scheme reported in Sect. 5 has been alwaysused as it enlarges significantly the convergence region ofthe algorithm [16]. The symbol NC denotes non con-vergence due to an ill conditioned Hessian. It can beobserved in the Table that the optimal values guaranteeone of the fastest iteration without being too close tovalues leading to non convergent behavior. In this sensethe automatic value selection for the penalty parametershas been observed guaranteing convergence in all sim-ulations [16, 2].

8 Conclusions

In the paper it has been show that the AugmentedLagrangian Element Free approach [16, 2] can be effi-ciently used to enforce material interface constraints.This confirms the ability of the ALEF to handle a largenumber of very differently scaled constraints withoutany difficulty by using a straightforward extension of thevariational formulation. In this sense the ALEF can beconsidered a natural candidate in the implementation ofnonlinear interfaces and constitutive models. Theinvestigated material interface model is solved bygrouping the displacement and traction continuity con-straints. To each group a different penalty parameter isassociated, whose optimal magnitude can be efficientlyestimated. Finally the influence of the number of pointswhere the constraints are enforced has been shown. Theresults are compared to finite element ones and to asimplified interface model, where only displacementcontinuity is set.

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Table 5 Influence of the initialpenalty parameter on the aug-mented Lagrangian iterationconvergence. The adoptedsymbology is Newton itera-tions/Multipliers updates(Hessian factorizations)

10�3aopt 10�2aopt 10�1aopt aopt ¼ 10þ5 10þ1aopt 10þ2aopt 10þ3aopt

10�3bopt 31/11 (10) 28/9 (7) 31/10 (7) NC NC NC NC10�2bopt 28/9 (7) 25/8 (6) 28/9 (8) 25/8 (5) NC NC NC10�1bopt 28/9 (8) 25/8 (5) 22/7 (5) 22/7 (5) NC NC NCbopt ¼ 10�3 28/9 (8) 22/7 (6) 16/5 (4) 13/4 (3) NC 32/5 (4) NC10þ1bopt 25/8 (8) 25/8 (5) 19/6 (4) 13/4 (3) 13/4 (4) 10/3 (2) NC10þ2bopt NC 22/7 (6) 19/6 (5) 10/3 (2) 7/2 (1) 7/2 (2) NC10þ3bopt 25/8 (5) 22/7 (6) 16/5 (4) 10/3 (2) 7/2 (2) 7/2 (2) 5/1 (1)

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