Optimization of a stabilized X-FEM formulation for frictional cracks

B. Trolléa,b, A. Gravouila,c, M.-C. Baiettoa, T.M.L. Nguyen-Tajanb

aUniversité de Lyon, CNRS INSA-Lyon, LaMCoS, UMR5259, 20 Avenue Albert Einstein, F69621 Villeurbanne Cedex,France

bSNCF, Direction Innovation et Recherche, 40 Avenue des Terroirs de France, 75611 Paris Cedex 12, FrancecInstitut Universitaire de France

Abstract

An efficient stabilized non-linear LATIN solver dedicated to frictional cracks in the X-FEM framework isproposed. The performance of such a solver has already been emphasized in detail in a previous paper ofA. Gravouil et. al : Stabilized global-local X-FEM for 3D non-planar frictional crack using relevant meshes,International Journal for Numerical Methods in Engineering, 88:1449-1475, 2011. Here, an optimization ofthe search direction of the LATIN solver is proposed. Intrinsic a priori formulas are proposed independentlyon the friction coefficient, the finite element mesh, the geometry and the boundary conditions. About onehundred thousand of calculations have been carried out in order to show the existence of a unique set ofparameters ensuring the optimal convergence rate. The number of iterations required to reach the givenlevel of accuracy is plotted versus two numerical parameters introduced, the search direction k and thestabilization operator ξ. A considerable decrease in the number of iterations and thus in the computingtime is obtained, an important condition to carry out with maximum of efficiency 3D simulations.

Keywords: three field weak formulation, frictional contact algorithm, X-FEM, LBB condition,optimization

1. Introduction

Modeling crack growth while encounting frictional contact at the interface is still a challenging problem.The remeshing issues can be solved using the extended element method [1, 2, 3, 4]. In the case of frictionalcontact, it still requires an accurate geometrical modeling of the crack and front shapes and a precisequantification of interface displacement and traction fields. Tribological fatigue like rolling contact fatigue[5], fretting fatigue [6, 7, 8, 9, 10] involve three-dimensional crack problems in which the interfacial crackbehavior is mainly governed by complex sequences of contact/friction states. In this context, enriched finiteelement methods (coupled for instance with a level set modeling of the possible non-planar crack shape) arevery well suited to model discontinuous physical behaviors [3, 4, 11]. The complications come from differentparts of the problem. One of them is the modelling of interface taking into account contact and friction.

Depending on the coupling methods between the bulk and the interface, oscillations in the frictionalcontact solution may arise. These numerical instabilities have been noticed in [12] and [13]. The interfaceconstraints may be prescribed on enriched finite elements using Lagrange multipliers or/and penalty meth-ods. When dealing with penalty methods, those numerical oscillations depend also on the penalty parametervalue.

Recently, many attempts have been proposed in the literature to study and improve the stability prop-erties of enriched finite element methods and more particularly the eXtended Finite Element Method withfrictional interfaces. Mortar methods and reduced Lagrange methods [3, 14, 15, 16, 17] have been used tosolve over-constrained contact problems by reducing the discrete interpolation space, i.e. the number of inte-gration points, on the interface. In a general point of view, it consists in defining adapted stabilized Lagrangespaces or heuristic rules for discretizing the interface in order to satisfy the Ladyzhenskaya-Babuska-Brezzicondition (LBB) [18, 19, 20]. Such stable Lagrange spaces can be complicated to define in the general caseand particularly in 3D.

Preprint submitted to Finite Element in Analysis and Design June 17, 2013

Previous successful attempts have already been performed using two field weak formulation with Lagrangemultipliers for frictionless contact within X-FEM, and using the penalty method for frictional contact [12, 21].More recently, Gravouil et al. [22] proposed a stabilization of a three-field weak formulation via a stabilizationoperator. The performance of the stabilization was demonstrated both on the 2D and 3D cases. To makesuch technique useful for three dimensional cases, it is necessary to obtain optimal selection of the (k,ξ)parameters governing the convergence rate of the method. These parameters are the search direction of theLATIN method and the value of the stabilization operator recently introduced. They are completely definedin the next section. The aim of this paper is to propose rules for their selection.

In section 2 the three field weak formulation and the general stabilization technique are recalled. Thenin section 3 the stabilized non-linear solver based on the LATIN method is introduced. A priori formulas todefine the LATIN non-linear solver search direction and the stabilization term are presented. In section 4the existence of a unique couple of optimal parameters for the proposed solver is observed. This section isdevoted to an extensive parametric study performed with about one hundred thousands of calculations. Insection 5 we illustrate the non influence of the boundary conditions, the frictional contact conditions at theinterface, the crack orientation and the domain dimensions on the optimal k and ξ values. We also showthat the material properties and the cack length do influence the optimal couple (k,ξ). The reliabiliy of theformulas is then highlighted in section 6 when dealing with a kinked slant crack located at the center of thebulk. Finally section 7 deals with the influence of the value of the optimized parameters and the given levelof accuracy on the convergence rate of the method.

2. THE THREE FIELD WEAK FORMULATION, PROBLEM STATEMENT AND GLOBAL-LOCAL QUASI-STATIC FORMULATION

We consider a cracked body Ω where contact and friction can occur along the crack faces Γ+c and Γ−c .

Under small displacement and small strain assumptions, we assume the interface Γc = Γ+c ∪ Γ−c as an

autonomous entity with its own behavior possibly nonlinear.

Figure 1: Global-local definition of the cracked body problem

This fracture problem is divided in a global problem (structure scale Ω\Γc) and a local problem (crackscale Γc) (see figure (1)). The global problem is defined with its own primal and dual variables, u thedisplacement field and σ the Cauchy stress tensor respectively. The local problem is defined with its ownprimal and dual variables, w the interface displacement field and t the interface traction field respectively.Let n be the outward unit normal to Ω\Γc and nc the outward unit normal to Γc. We assume quasi-staticformulation and write the governing equations as follows for the global problem at a given time t ∈ [0;T ]:

divσ + f = 0 in Ω\Γc (1)σn = s on ∂σΩ (2)u = v on ∂uΩ (3)

where s and v are the prescribed load on ∂σΩ and displacement on ∂uΩ respectively, and f the bodyforce vector. Both global and local problems obey to a constitutive law possibly nonlinear in the bulk andon the interface respectively:

2

σ ≡ σ(u, ui, t) in Ω\Γc (4)t ≡ t(w, wi, t) on Γc (5)

where ui and wi are the internal variables of the global and local problem respectively and t a giventime. Furthermore, we add the coupling between the global and the local problems both on the primal anddual variables:

u = w on Γc (6)σn = t on Γc (7)

Equations (1,2,3,6,7) consist in the strong formulation of the crack model with contact and frictionbetween the crack faces. As already shown in [10], an equivalent three field weak formulation (the globaldisplacement field u, the local displacement field w and the Lagrange multipliers field λ) at a given timet ∈ [0;T ] can be introduced (see [22]).

0 = −∫

Ωσ : ε(u∗)dΩ +

∫Ω

f · u∗dΩ +∫∂σΩ

s · u∗dS +∫

Γcλ · u∗dS

+∫

Γc(t− λ) ·w∗dS +

∫Γc

(u−w) · λ∗dS

∀u∗ ∈ U0, ∀w∗ ∈ W, ∀λ∗ ∈ Λ, ∀t ∈ [0;T ]

(8)

with

u ∈ U ,U = u ∈ H1(Ω\Γc) / u = v on ∂uΩ (9)

u∗ ∈ U0,U0 = u ∈ H1(Ω\Γc) / u = 0 on ∂uΩ (10)

w ∈ W,w∗ ∈ W,W = w ∈ H1(Γc) (11)

λ ∈ Λ,λ∗ ∈ Λ,Λ = λ ∈ L2(Γc) (12)

and is equivalent to the previous strong formulation (1,2,3,6,7) (see [10]). This weak formulation (8)allows a global-local X-FEM formulation handling successfully the structure scale (global), the crack andthe interfacial non linear local scales. Here, an homogeneous, isotropic material with an elastic linearbehavior is considered. The stress strain law is then written according to the fourth order Hooke tensor H:

σ = H ε(u) in Ω\Γc (13)

Furthermore, interface quantities (w,t) obey to contact unilateral law and Coulomb’s friction law forthe normal and tangential problem respectively. w and t interface fields on the crack faces Γ+

c and Γ−c areexpressed in the local basis (nc,tc) relative to the crack (see figure (1)):

w = wNnc + wT tc and t = tNnc + tT tc (14)

where (wN ,tN ) and (wT ,tT ) are normal and tangential scalar interface fields, respectively. Furtherrelative interface displacements (opening and sliding) between the crack faces at a given interface point xare given by:

[wN (x, t)] = w+N (x, t)− w−N (x, t) and [wT (x, t)] = w+

T (x, t)− w−T (x, t) (15)

This yields for the local equations of the interface behavior:

3

• opening [wN (x, t)] > 0 → t+(x, t) = t−(x, t) = 0

• contact [wN (x, t)] = 0 → t+(x, t) = −t−(x, t)

• sticking |tT (x, t)| < µc |tN (x, t)| → ∆[wT (x, t)] = 0

• sliding |tT (x, t)| = µc |tN (x, t)| → ∃ γ > 0 / ∆[wT (x, t)] = −γ t+T (x, t)

(16)

where ∆[wT (x, t)] corresponds to a sliding increment for a given time step ∆t and µc is the frictioncoefficient. Contact calculation is performed in the initial configuration Ω(0).

The three field weak formulation (8) is the basis of the global-local X-FEM dedicated to frictional contact(see [10] for more details). The global and the local problems are considered as autonomous problems. Theyare glued together in a weak sense with continuity condition on interface quantities (w,t).

3. NON LINEAR STABILISED X-FEM FOR FRICTIONAL CRACKS

X-FEM is well adapted for modeling 2D or 3D crack growth. Indeed, the mesh does not necessarilyconform to the crack and both field interpolation and remeshing are not required during the possible crackpropagation [2, 3, 5, 23]. In this section, an efficient non-linear solver dedicated to a quasi-static formulationwith contact and friction is presented within the framework of the global-local X-FEM approach.

One assumes that the state vector Xn ≡ (un,σn,wn, tn) is known at time tn. Within a quasi-staticincremental framework, the next stage consists in calculating the unknown state vector Xn+1 at time steptn+1. Briefly, the LATIN method (see [2, 5, 24, 25, 26, 27]) consists in an iterative strategy between a localstage from X(i)

n+1 to X(i+ 12 )

n+1 and a global stage from X(i+ 12 )

n+1 to X(i+1)n+1 . The local stage corresponds to a

set of local equations (L) on figure 2, possibly non-linear SL (see equations (16)), and the global stage to aset of global linear equations SG (see (17)). This two-step approach requires search directions E+ and E−between the previous set of equations SL and SG (see figure 2). The process is repeated until convergenceis reached.

Figure 2: Schematic representation of the LATIN method

The main advantages of this solver are :

• the global matrix is symmetric

• there is no iteration on the local stepFrom the previous solution (wi,ti) we use a local contact indicator developped by [24] to determinethe area of contact, opening and sliding or sticking condition. Then we locally applied equations (16)and ti+

12 − ti = kL(wi+ 1

2 −wi). Thus we directly obtain (wi+ 12 ,ti+

12 ).

4

• there is no operator’s update during the non linear iterations

According to [10], the following global stage is obtained (combination of the three-field weak formulationand the LATIN method (8) (13)):

0 = −∫

Ωε(u(i+1)

n+1 ) : H : ε(u∗)dΩ +∫

Γcλ

(i+1)n+1 · u∗dS

+∫∂σΩ

sn+1 · u∗dS +∫

Ωfn+1 · u∗dΩ

+∫

Γc

(t(i+ 1

2 )n+1 + kLw

(i+ 12 )

n+1

)·w∗dS −

∫Γc

(λ

(i+1)n+1 + kGw(i+1)

n+1

)·w∗dS

+∫

Γc

(u(i+1)n+1 −w(i+1)

n+1

)· λ∗dS

∀u∗ ∈ U0, ∀w∗ ∈ W, ∀λ∗ ∈ Λ, ∀tn+1 ∈ [0;T ]

(17)

where the unknowns are u(i+1)n+1 ,w

(i+1)n+1 ,λ

(i+1)n+1 and kL and kG are strictly positive operators (Pa.m−1)

such that:

kL =

[kL,N 0

0 kL,T

]and kG =

[kG,N 0

0 kG,T

](18)

where (kL,N , kG,N ) and (kL,T , kG,T ) are penalty terms dedicated to normal and tangential problemsrespectively. These parameters do not modify the solution. They are defined in order to optimize theconvergence rate of the non linear solver [5, 25, 26]. In general kL is chosen equal to kG (respectively E+

and E−, see figure 2). Moreover, they are set equal on the normal and the tangential problem.

kL = kG =

[k 00 k

](19)

Introducing the X-FEM discretisation into (17) yields the following linear system (see [10]): K 0 −Kuλ

0 Kww Kwλ

−KTuλ KT

wλ 0

Ui+1

Wi+1

Λi+1

=

FKwλ ·Ti+ 1

2+ Kww ·Wi+ 1

2

0

(20)

where the set of discretized standard and enriched degrees of freedom U are defined according to thecrack (see [10, 22]).

The following error indicator is used (see [5]).The error indicator of convergence is computed after eachglobal solve as the distance between the current and previous approximations:

η = max(ηN ; ηT ) (21)

with:

ηN =‖ X(i+1)

N −X(i+ 12 )

N ‖2∞‖ X(i+1)

N ‖2∞ + ‖ X(i+ 12 )

N ‖2∞, ηT =

‖ X(i+1)T −X(i+ 1

2 )

T ‖2∞‖ X(i+1)

T ‖2∞ + ‖ X(i+ 12 )

T ‖2∞(22)

where the norm is defined by:

‖ X ‖2∞= max(kGT2 +1

kGW2) (23)

It allows to ensure the convergence of the state vector X both on the normal and tangential problemrespectively. At this point, a global-local X-FEM dedicated to 3D frictional cracks is proposed. Even if local

5

convergence can be ensured locally both on normal and tangential problems, it is well known that two fieldor three field formulations with Lagrange multipliers can involve spurious oscillations on the dual fields (see[12, 14, 16, 21]).

Then, a stabilization process is applied to the three-field weak formulation coupled with the LATIN non-linear solver (equations (17) and(20)). We introduce a stabilization operator on the weak coupling conditionbetween the global displacement field U and the interface displacement field W both on the left-hand sideand the right-hand side of the linearized system (see [22] for proofs and implementation details). Equation(20) becomes: K 0 −Kuλ

0 Kww Kwλ

−KTuλ KT

wλ Kλλ

Ui+1

Wi+1

Λi+1

=

FKwλ ·Ti+ 1

2+ Kww ·Wi+ 1

2

Kλλ ·Λi

(24)

where Kλλ is the stabilization operator defined by:

Kλλ = −ξ∫

Γc

Ψ′iΨ′j dS (25)

where ξ is the stabilization term and Ψ′ the shape function associated to the Lagrange multipliers field.This additional operator does not change the solution of the problem. Indeed, at convergence Kλλ ·Λi−

Kλλ ·Λi+1 tends then towards 0 for a prescribed level of accuracy.Finally a stabilized three field weak formulation respecting the LBB condition is proposed. The conver-

gence rate of this method depends on both the search direction k and the stabilization operator ξ. In [22], theefficiency of the stabilized formulation on the convergence rate was demonstrated: the algorithm convergesfaster towards a stabilized solution. A short study of the optimum parameters (k,ξ) was performed. In thispaper the aim is to obtain the best possible convergence rate. In previous uses of the LATIN method forcontact problems, authors have chosen to set the search direction according to the relationship [24, 28]:

k =E

L0(26)

where E is the Young’s modulus and L0 a characteristic lengh of the structure. For structure assemblyanalysis, L0 optimum values are chosen close to the structure maximum length [24, 28]. In the case offrictional contact at crack interface, L0 is the crack length [29]. An extensive parametric and numericalstudy is carried out to identify how the optimal k and ξ vary accorfing to the problem. They dependon the material property and a characteristic length, but not on the domain dimensions, the crack angle,the interfacial frictional coefficient and the boundary conditions. Hence the following a priori formulas areproposed:

k =E

l(27)

ξ =l

E(28)

where E and l are respectively the material Young’s modulus and the crack length. The existence of aunique optimum couple (k,ξ) ensuring the best possible convergence rate is first established.

4. EXISTENCE OF A UNIQUE OPTIMAL COUPLE OF PARAMETERS

The aim is to improve the understanding of the numerical behaviour of the method, stating that themechanical one was already tested and validated. Simple test cases are thus considered that do not requirea fine mesh. k and ξ ranging from 1.10−3 to 1.105 Pa/mm and from 1.10−5 to 1.103 mm/Pa, respectively,are considered. The requested precision (according to the indicator introduced with equations (21), (22)and (23))is set up to 10−4. The number of iterations required for this level of accuracy is plotted in the(k,ξ) space.

6

The existence of a unique optimal couple (k,ξ) leading to an important decrease in the number ofiterations is demonstrated for an inclined contacting frictional crack submitted to a compressive loading.The test case is detailed in figure 3. The interfacial frictional coefficient µc is equal to 0.1, the crack lengthl is 0.023mm, the crack angle θ is 15˚and the external applied pressure P is set equal to 0.0005Pa. Thedomain width and height are b = 0.2mm and h = 0.4mm, respectively. Young’s modulus E is set to theunity for simplicity.

X

Y

P

lh

b

θ

Figure 3: Test case and mesh used (30 * 60 elements, θ = 15˚, l = 0.023mm, b = 0.2mm, h = 0.4mm, E = 1Pa, P = 0.0005Pa,µc = 0.1)

The number of iterations required to solve the system using the LATIN solver with an imposed 10−4

accuracy is plotted in figure 4 in the bi-dimensional (k,ξ) space. Isovalues of the number of iterations are alsopresented. Two key points are highlighted, the first pointing out the obtention of a quasi-convexe surfaceresponse and the second emphasizing the existence of an optimal k value with a large range of ξ valuesover several decades for which the number of iterations is minimum and stable. The first result was alreadyobtained in [22]. The number of iterations decreases from 9000 to 46, illustrating the significant reductionin computing time and the need of relationships to predict the correct (k,ξ) couple. This is particularlyessential in dealing with three dimensional problems. A cross mark corresponding to (k,ξ) values obtainedaccording to relationships is added in figure 4. 83 iterations are necessary to converge towards the solutionaccording to the imposed accuracy. The difference is quite negligible compared to the worst 9000 iterationspossible and the estimated values obtained using these a priori relationships give a very good convergencerate. The next section is concerned with an analysis of the influence of different parameters on the optimalk and ξ values.

7

ξ (mm/Pa) k (Pa/mm)

Iterations

Figure 4: Response surface and iso-values, sliding contact conditions at the interface (µc = 0.1, θ = 15˚, l = 0.023mm,b = 0.2mm, h = 0.4mm, E = 1Pa, P = 0.0005Pa)

5. INFLUENCE OF DIFFERENT PARAMETERS ON THE OPTIMAL (k,ξ)

The previous section showed that there is a unique couple of values ensuring an optimal convergencerate. This section aims at identifying the factors influencing the best possible convergence rate. Factorslike, the domain dimensions (b and h), the interfacial frictional coefficient µc and the boundary conditionsare first analysed and their influence is shown to be negligible. But parameters like the crack geometry andthe material properties govern the convergence rate and define the optimal couple (k,ξ). All the calculationsare carried out using a precision of 10−4.

5.1. Parameters with no influence on (k, ξ)optimal

The test case corresponding to an inclined crack submitted to a compressive loading is considered again.All the data used previously are identical, except the one whose influence is analysed.

5.1.1. Interfacial frictional coefficient µcThe value of the frictional coefficient of friction, µc, is equal to 1 instead of 0.1. The response surface

plotted in figure 5 has a very similar shape to the one obtained for µc = 0.1. Identical optimum couple(k,ξ)=(20,0.003) and number of iterations (46) are obtained to reach the 10−4 relative error. It can beconcluded that the interfacial frictional coefficient has no significative effect on the value of the optimumparameters.

8

ξ (mm/Pa) k (Pa/mm)

Iterations

50

60

75

200

1000

4000

Figure 5: Response surface and iso-values, sticking contact conditions at the interface (µc = 1, θ = 15˚, l = 0.023mm,b = 0.2mm, h = 0.4mm, E = 1Pa, P = 0.0005Pa)

5.1.2. External loading PThe influence of the magnitude and its orientation with respect to the crack of the external loading are

analysed.

Magnitude of the external loading PThe external compressive loading is a thousand times higher than previously (P = 0.5Pa). Isovalues

corresponding to the reference and the modified magnitude loading are plotted in figure 6. The predicted(k,ξ) couple obtained according to the relationships is indicated with a cross mark in both cases. Theoptimum parameters are identical in both cases.

9

10−4

10−3

10−2

10−1

101

50

75

100

ξ (mm/Pa)

k (Pa/mm)50

10−4

50

10−3

10−2

10−1

100

101

ξ (mm/Pa)

k (Pa/mm)

50

75

100

Reference external loading External loading a thousand times higher

P = 0.0005 along -Y P = 0.5 along -Y

Figure 6: Non-influence of the external loading on (k,ξ) optimum couple - (P varies, θ = 15˚, l = 0.023mm, b = 0.2mm,h = 0.4mm, E = 1Pa, µc = 0.1)

Orientation of the external loading PThe compressive pressure was applied along the Y axis (see figure 3) in the reference case. Its direction

is modified. An external loading is superimposed along -X direction, on the same nodes. Reference andmodified isovalues are plotted in figure 7. (25,0.008) instead of (20,0.003) optimum couple is obtained. Therequested accuracy is reached after 67 iterations. The results are summarized in table 1. Changes are smalland may be due to modifications in the frictional conditions at the crack interface leading in turn to adifferent number of iterations required to reach the given level of accuracy. Therefore, It can be concludedthat the external loading has not a significative influence on the optimum parameters of the method.

10

10−4

10−3

10−2

10−1

101

50

75

100

ξ (mm/Pa)

k (Pa/mm)50

10−4

10−3

10−2

10−1

101

ξ (mm/Pa)

k (Pa/mm)50

100

75

Reference external loading External loading

P = 0.0005Pa along -Y P = 0.0005Pa along -X

Figure 7: Non-influence of the external loading on the optimum couple (k,ξ). (P = 0.0005, orientation varies, θ = 15˚,l = 0.023mm, b = 0.2mm, h = 0.4mm, E = 1Pa, µc = 0.1)

external loading reference magnitude a thousand times higher in the other direction

(figure 4) (figure 6) (figure 7)

Magnitude (Pa) 5.10−4 5.10−1 5.10−4

Orientation - Y -Y -X

k(Pa/mm) 20 20 25

ξ(mm/Pa) 0.003 0.003 0.008

Iterations for (k, ξ)optimal 46 46 67

Iterations for (43, 0.023)formulas 78 78 82

Table 1: Optimum parameters for different external loadings

5.1.3. Crack angle influence on the convergence rateTheta values are scanned from 0˚to 75˚using a step of 15˚. For each θ value, calculations are per-

formed for (k,ξ) ranging over the whole domains. The numerical optimum couples corresponding to theminimum of the response surfaces obtained are compared to the constant predicted couple obtained usingthe relationships, (k, ξ)formulas = (43, 0.023). It can be noted that the crack orientation has a negligibleinfluence on (k, ξ) predicted values as only the crack length is accounted for.

11

θ (˚) 0 15 30 45 60 75

k(Pa/mm) 15 20 30 35 25 20

ξ(mm/Pa) 0.005 0.003 0.3 0.09 0.4 0.02

Iterations with (k, ξ)optimal 47 46 89 81 83 81

Iterations with (43, 0.023)formulas 113 83 133 196 108 110

Table 2: Optimum parameters in term of convergence rate for a fixed crack length and a varying crack orientation(θ varies,l = 0.023mm, P = 0.0005Pa, b = 0.2mm, h = 0.4mm, E = 1Pa, µc = 0.1).

The first point to outline is that the crack inclination with respect to the loading has a slight influenceon the numerical optimum search direction value. k variations are comprised between 15 and 35 and arequite very small in comparison with possible variations when eight decades are scanned. Moreover, thecomparison with the number of iterations necessary if an unfavorable value is selected (more than 4000iterations) illustrates clearly the very good estimation of the optimum parameters. The second point is thatthe value of the stabilisation operator has a second order influence, allowing to select a value within a largerange of values around the optimal one without decreasing the convergence performance.

5.2. Parameters influencing the optimal (k,ξ) valuesThe next step of the analysis consists in identifying the most relevant parameters influencing the con-

vergence rate and to account for them in the a priori relationships.

5.2.1. Influence of b and h domain dimensionsThe influence of the domain dimensions, b and h, is first tested. A domaine of h = 0.4mm and b = 0.2mm

has been considered in the reference case. The optimum couple was (20,0.003). h = 0.8mm (figure 8) andb = 0.6mm (figure 9) are now considered. All the other parameters are unchanged (h or b varies, θ = 15˚,l = 0.023mm, P = 0.0005Pa, E = 1Pa, µc = 0.1). The corresponding optimum numerical couples are(20,0.002) and (45,0.001) respectively and the precision is reached after 121 and 52 iterations. The resultsare summarized in table 3.

10−410

−2100

100

102

101

102

103

k (Pa/mm)ξ (mm/Pa)

Itera

tions

10−3

10−2

10−1

100

100

101

102

ξ (mm/Pa)

k (Pa/mm)

200 400

Figure 8: Response surface and iso-values (h/b = 4, h = 0.8mm, b = 0.2mm, µc = 0.1, E = 1Pa, l = 0.023mm, θ = 15˚,P = 0.0005Pa)

12

10−410

−2100

100

102

101

102

103

k (Pa/mm)ξ (mm/Pa)

Itera

tions

10−3

10−2

10−1

100

100

101

102

ξ (mm/Pa)

k (Pa/mm)

200 400

100

75

Figure 9: Response surface and iso-values (h/b = 2/3, h = 0.4mm, b = 0.6mm, µc = 0.1, E = 1Pa, l = 0.023mm, θ = 15˚,P = 0.0005Pa)

h(mm) 0.4 0.8 0.4

b(mm) 0.2 0.2 0.6

(figure 4) (figure 8) (figure 9)

k(Pa/mm) 20 20 45

ξ(mm/Pa) 0.003 0.002 0.001

Iterations for (k, ξ)optimal 46 121 52

Iterations for (43, 0.023)formulas 83 212 73

Table 3: Optimum parameters for different domain dimensions

With a longer domain (figure 8), the response surface shape becomes really smooth and flat in the opti-mum k direction. The optimum couple stays unchanged. Therefore the reference domain can be consideredas a semi-infinite domain and a longer domain does not change the optimum couple. In the same way, witha widder domain (figure 9) the optimum couple becomes closer to the one predicted by the formulas. Theseresults show that when the crack scale and the scale structure are separated, the formulas offer a very goodestimation of the optimum parameters.

5.2.2. Influence of the Young’s modulus and the crack lengthIn order to analyse the influence of the Young’s modulus, its value is set equal to 100Pa (E = 100Pa,

θ = 15˚, l = 0.023mm, P = 0.0005Pa, b = 0.2mm, h = 0.4mm, µc = 0.1). The number of iterationsrequired to reach a 10−4 accuracy versus (k,ξ) values is plotted in figure 10. The numerical and predictedoptimal couples are (2000,0.0005) and (4300,0.00023) instead of (20,0.003) and (43,0.023) using E = 1 Pa.The k search direction value has increased proportionally to the Young’s modulus.

13

200

100

75

50

Figure 10: Surface response with E = 100 Pa from above and iso-values (E = 100Pa, θ = 15˚, l = 0.023mm, P = 0.0005Pa,b = 0.2mm, h = 0.4mm, µc = 0.1)

To analyse the influence of the crack length, we consider a crack five times longer 0.115mm inclined atan angle of 15˚. We compare the results with the ones obtained in the reference case for a short crack. (k,ξ)numerical, predicted and reference optimum couples are (6,0.01), (8.5,0.115) and (20,0.003), respectively.

It is obvious that the crack length has a major influence on the search direction value. This influence iswell accounted for by the relationships.

75

100

200

Figure 11: Surface response with L=5Lreference from above and iso-values (l = 0.115mm, E = 100Pa, θ = 15˚, P = 0.0005Pa,b = 0.2mm, h = 0.4mm, µc = 0.1)

6. RESULTS WITH A KINKED SLANT CENTRAL CRACK

This section is concerned with the generalization of the a priori formulas to other problems. A kinkedslant central crack is considered (figure 12). To make the results meaningful, the same discretization of thestructure, the same external loading and boundary conditions than those of the previous case are considered.The requested level of accuracy remains 10−4. The optimal convergence rate is obtained with the couple

14

(9,0.4) and η is inferior to 10−4 after 90 iterations. In this case the total length of the crack is 0.12 mmwich gives according to the formulas (8.3,0.12). Using these parameters, the prescribed accuracy of 10−4 isreached after 149 iterations (see table 4). We can see that the response surface keeps the same shape as theone obtained in sections 4 and 5. Further, the formulas give a good estimation of the optimum parameters(figure 13). These results illustrate the robustness of the proposed formula as long as the scale of the bodyand the crack are different.

X

Y

P

h

b

l

c

c

θ

θ

45°

Figure 12: Boby with a kinked slant central crack (l = 0.1mm, c = 0.01mm, E = 1Pa, θ = 75˚, P = 0.0005Pa, b = 0.2mm,h = 0.4mm, µc = 0.3)

Iterations for (9, 0.4)optimal 90

Iterations for (8.3, 0.12)formulas 149

Table 4: Optimum parameters with a kinked slant central crack

15

100

200

300

400

k (Pa/mm)

ξ (mm/Pa)

10010

110

2

10-1

100

101

Figure 13: Surface response with the slant kinked crack from above and iso-values (l = 0.1mm, c = 0.01mm, E = 1Pa,θ = 75˚, P = 0.0005Pa, b = 0.2mm, h = 0.4mm, µc = 0.3)

7. INFLUENCE OF k AND ξ ON THE CONVERGENCE RATE

A synthesis of the convergence rate analysis carried out is proposed. Predicted (k,ξ) values are indicatedin figures 4, 5, 6,7, 8, 9, 10, 11 and 13) with a cross mark. The comparison with (k,ξ) numerical valuesshows an excellent agreement. A discussion is proposed.

It comes from observations on the results that k has a strong influence on the convergence rate. k beingset close to its optimal value, ξ can be roughly estimated (see figures 4, 5, 6, 7, 8, 9, 10, 11 and 13). Theflatness of the response surfaces in this direction authorizes to consider a large range of ξ values that stillensures a very good convergence rate.

This is for instance illustrated in the first numerical example presented in section 4. The accuracy isreached after 46 iterations using (20,0.004) couple. k value being imposed according to the optimum value,figure 14 clearly shows that ξ has a small influence on the convergence rate. Indeed, if ξ fluctuates of afactor 10 near its optimum value, we obtain a small difference of iterations required (60 instead of 46) toreach the given level of accuracy. Therefore we must first select k search direction value and then impose ξstabilization operator one.

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Figure 14: Convergence rate for different values of ξ with koptimal (µc = 0.1, b = 0.2mm, h = 0.4mm, E = 1Pa, l = 0.023mm,θ = 15˚, F = 0.0005Pa)

Finally, to conclude this analysis, the surface responses are plotted for different imposed level of accuracy(see figure 15). The case described in figure 3 is considered. The aim is to verify that the level of accuracy,defining the acceptable numerical errors and thus the number of iterations to reach the approximatedsolution, has a limited influence on (k,ξ) numerical values and that the predicted values obtained accordingto the proposed relationships are still valid. Results are plotted in figure 15. Depending on the level ofaccuracy requested we obtain a different location for the optimum couple (k,ξ). Nevertheless, it seemsthat with a higher precision, the numerical optimum couple comes closer to the one predicted according tothe formulas (see figures 15). The results are summarized in table 5. The maximum number of iterationsobtained without any optimization to reach the convergence over (k,ξ) space considered is indicated in eachcase for comparison. For instance, a one hundred ratio between the optimized and worst number of iterationsis obtained for a requested 10−4 level of acccuracy. This result emphasizes the improvements in computingtime and cost due to (k,ξ) optimization based on the a priori relationships.

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k (Pa/mm)

ξ (mm/Pa)

η 10-2

η 10-3

η 10-4

<<<

25

20

10

20

50

60

75

10

10

10

10

10

100101102

0

-1

-2

-3

-4

Figure 15: Three iso-valuess obtained according to different level of accuracy (10−2, 10−3 and 10−4)

accuracy 10−2 10−3 10−4

k(Pa/mm) 6 10 20

ξ(mm/Pa) 0.02 0.01 0.007

Iterations for (k, ξ)optimal 10 20 46

Iterations for (43, 0.023)formulas 25 50 83

Iterations for (k, ξ)worst 360 2420 8722

Table 5: Optimum parameters for different level of accuracy

8. CONCLUSION AND PERSPECTIVES

This paper presents an intensive numerical study based on a recent X-FEM stabilization technique ofa three field weak formulation of frictional crack problems. The stabilization operator introduced ensuresto fulfill the LBB condition independently on the mesh size. Thousands of calculation have been carriedout in order to study the influence of the search direction and the stabilization operator on the convergencerate. We have seen that the response surfaces are quasi-convex and flat close to the optimum couple (k,ξ).Changes in the external loading, the contact condition at the interfaces and the domain shape have acceptableconsequences on the values of the optimum couple (k,ξ). The proposed a priori formulas have been validatedprovided the crack scale and the structrure scale are separated. They depend on a caracteristic value (thecrack length) and the material properties. Moreover it has been noticed that the search direction shouldbe first set to its optimum value. Then the relationship between the requested level of accuracy and theoptimum parameters (k,ξ) have been emphasized. Finally we have demonstrated the robustness of themethod. We now can control the convergence rate of the presented algorithm. Future work includes theupdate of the search direction and the stabilization operator with evolving interfaces.

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Aknowledgement

The support of ANRT and SNCF is gratefully acknowledged.

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