On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework

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1 On the control of the loa increments for a proper escription of multiple elamination in a omain ecomposition framework Olivier Allix, Pierre Kerfrien, Pierre Gosselet To cite this version: Olivier Allix, Pierre Kerfrien, Pierre Gosselet. On the control of the loa increments for a proper escription of multiple elamination in a omain ecomposition framework. International Journal for Numerical Methos in Engineering, Wiley, 2010, 83 11), pp < /nme.2884>. <hal > HAL I: hal Submitte on 27 Sep 2011 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 On the control of the loa increments for a proper escription of multiple elamination in a omain ecomposition framework. O. Allix, P. Kerfrien, P. Gosselet LMT-Cachan ENS Cachan/CNRS/UPMC/PRES UniverSu Paris), 61 av. u Présient Wilson, F Cachan, France January, 2010 In quasi-static nonlinear time-epenent analysis, the choice of the time iscretization is a complex issue. The most basic strategy consists in etermining a value of the loa increment that ensures the convergence of the solution with respect to time on the base of preliminary simulations. In more avance applications, the loa increments can be controlle for instance by prescribing the number of iterations of the nonlinear resolution proceure, or by using an arc-length algorithm. These techniques usually introuce a parameter whose correct value is not easy to obtain. In this paper, an alternative proceure is propose. It is base on the continuous control of the resiual of the reference problem over time, whose measure is easy to interpret. This iea is applie in the framework of a multiscale omain ecomposition strategy in orer to perform 3D elamination analysis. 1 INTRODUCTION The virtual testing of elamination is an objective wiely sprea among inustrialists especially in the aeronautical fiel. To achieve it, two research thematics which have unergone large evolution uring the last twenty years nee to be put in conjunction: the pertinent moeling of composites an the efficient computation of structures. Inee, there have been many avances towar a better unerstaning of the mechanics of laminate composites an of amage mechanisms. Two kins of moeling have prove their valiity: microscale an mesoscale moels. Microscale moels are strongly connecte to the physics of the material an thus provie a reliable framework for simulation [12, 22]. Unfortunately, the computation of moels efine at the micro scale require such a fine iscretization that only small test specimens can be simulate, structural computations being out of reach even on recent harware. Meso-moels [26, 4, 15, 7] are efine at a scale which enables both the introuction of physics-base ingreients an the simulation of small inustrial structures. They inee most often rely on the efinition of two meso-constituents, the ply 3D entity) an the interface 2D entity), which are moele using continuum amage) mechanics, their behavior being obtainable from the homogenization of micro-moels [21]. Anyhow, for reliable simulation, iscretizations still nee to be fine in orer, for instance, to represent correctly the graients of stresses ue to ege effects which are responsible for the initiation of many egraations) an associate systems thus remain very large in terms of number of egrees of freeom) an strongly nonlinear with potential instabilities). As a first approach of the reliable simulation of quasi-static simulations of the elamination in composite structures, we chose in [10] to neglect the effect of eterioration within the plies an to lump the egraations in the interfaces. We thus retaine the mesomoel presente in [3] where the elamination ability is localize in the interfaces an hanle through a cohesive behavior. The space iscretization is consiere sufficiently fine to represent accurately any evolution of multiple elamination cracks sufficient number of Gauss points in the length of the process zone [27, 1, 7]). 1

3 At each time step of an incremental time iscretization scheme, the associate large nonlinear system is solve using a three-scale omain ecomposition strategy. Base on the mixe LaTInbase omain ecomposition metho [14], this strategy has been given high numerical efficiency by aapting various ieas from the work of [23, 24, 25] to the computation of elamination. Threeimensional simulations of the elamination in realistic structural components have been performe on parallel computers without the nee to perform local space refinement. Though, a complex issue arises when choosing the loa increments: the solution to softening quasi-static problems epens on the time iscretization scheme parameters non-uniqueness of the solution an possible bifurcation paths). This remark brings us to the fiel of the valiation. In the literature, numerous error inicators have been evelope to control a posteriori the global error introuce in finite element schemes for linear problems [5, 28, 20]. These inicators have been extene to the valiation of nonlinear time-epenent problems [13, 6, 9]. One of the most avance criterion is the so-calle error in the constitutive law [13]. A solution to the nonlinear evolution problem being compute using a FE scheme an a classical time integration proceure, one constructs a solution which satisfies the kinematic an static amissibilities, an lump the resiual of the nonlinear evolution problem equations in the constitutive laws. A measure of this resiual permits to control at the same time the iscretization error in space, in time an the error introuce by the iterative solution proceures [16]. This iea has been formalize in [11] for materials escribe using internal variables. The state equations are satisfie by the reconstructe solutions, the measure of the non-verification of the evolution laws permits to erive a strict upper boun to the solution error. Though, this new amissible solution is not easily constructe in the case of softening behaviors. Specific evelopments in [17] meant to tackle this ifficulty, an the resulting proceure is use in [16] to erive an aaptive refinement proceure in space an time. Note that, at the present time, a link between the error in the constitutive law an the error in the solution is still to be establishe in the case of softening materials. The aim of the work presente in this paper is ouble. The first is to efine a comprehensive time iscretization error inicator inspire from the work of [13, 16] for elamination analysis an to ensure that its computation an use is numerically efficient within the LaTIn-base omain ecomposition strategy. Our secon goal is to use the evelope inicator to control on the fly the loa increment in quasi-static analysis in orer to ensure the convergence of the compute solution. The paper is organize as follows. The reference elamination problem is presente in Section 2. The epenency of the solution to this problem on the time iscretization scheme is emonstrate. In the following section, we present a time-epenent error inicator base on the error in the constitutive law an compute with respect to a continuous solution in time, constructe by interpolation over each time step. Although very general, this inicator is not irectly suitable for the LaTIn-base multiscale strategy use to perform the nonlinear resolutions. The main features of this strategy are recalle in Section 4. We focus in particular on the inicator base on the error in the constitutive law use to estimate the convergence of the iterative proceure. In Section 5, this convergence inicator is associate to the previous evelopments to erive an alternative an cheap time iscretization error inicator, which is the basis for the evelopment of an automatic time-step-control proceure. At last, this technique is valiate on multiscale an parallel elamination simulations in Section 6. Two ifferent problems are assesse: a simple an stable problem in which the time increments correspon to the increases in the prescribe loa, an a more complex an unstable problem, solve using an arc-length proceure, in which the time increments correspon to the value of the arc-lengths. 2 THE REFERENCE PROBLEM AND ITS DISCRETIZA- TION IN TIME 2.1 Reference problem at a given time of the analysis The elamination simulation is performe uner the assumptions of quasi-static, isothermal evolution over time an small perturbations. The laminate structure E occupying Domain Ω is mae out of N P ajacent plies occupying Domains Ω P ) P 1, NP of bounaries Ω P ) P 1, NP ), separate by N P 1) cohesive interfaces 2

4 I P ) P 1, NP 1 an see Figure 6), Page 9). An external traction fiel F respectively a isplacement fiel U ) is applie to the structure on Part Ω f respectively Ω u ) of the bounary Ω of Domain Ω. The volume force is enote f. Let u P be the isplacement fiel, σ P the Cauchy stress tensor an ɛ P the symmetric part of the isplacement graient in Ply P. At every time t [0 T ] of the analysis, the reference non-linear equilibrium problem reas: Fin s ref = s P ) P 1, NP ), where s P = u P, σ P ), which satisfies the following equations: Kinematic amissibility on Ω u : u P Ωu = U 1) Global equilibrium of Structure E: u P ) P 1, NP ) Tr σ P ɛu P ) Ω f.u P Ω F.u P Γ P Ω P P Ω P P Ω P Ω f + σ P n P.[u] Γ = 0 P P I P 2) where [u] P is the jump of isplacement of Interface I P : [u] P = u P +1 u P an n P is the outer normal to the bounary Ω P. Linear orthotropic behavior of the plies: σ P = K ɛu P ) 3) Constitutive law of the cohesive interfaces, local on any interface I P. The elastic amageable law propose in [4] is escribe using continuum amage mechanics. Three internal variables i ) i 1, 3 one for each elamination moe: traction along n P an shear along t 1 an t 2 on Figure 1)), ranging from 0 to 1 are introuce in the surface strain energy e in orer to take into account the irreversible amage mechanisms. n P P t 2 I PP P t 1 Figure 1: The mesomoel entities Thursay, 4 February 2010 Two state equations are erive from the expression of the free energy. The first one establishes a relation between the ual interface unknown σ P n P, an the primal interface unknown [u] P : σ P.n P = e ) ) which gives σ [u] P.n P = K P [u] P [u] τ [0 t] P 4) P where, in the basis n P, t 1, t 2 ), h + being the positive inicator function: ) ) ) 1 h + [u] P.n P ) 3 kn K P [u] P = 0 1 τ [0 t] 1 )kt )kt 0 3

5 The secon state equation links the thermoynamic forces Y i ) i 1, 3 to the primal interface unknown: Y 1 = 1 ) 2 Y i = e 2 k0 t [u] P.t 1 where Y 2 = 1 ) 2 i 2 k0 t [u] P.t 2 ] 5) Y 3 = 1 ) 2 2 k0 n h + [u].n P P ) The evolution laws are: 1 = 2 = 3 = min{1, wy )} ) n wy ) = n Y n+1 Y c where ) 1 α α α α Y = max τ t) Y 3 τ + γ 1 Y 1 τ + γ 2 Y 2 τ 6) Further etails on this cohesive zone moel an ientification issues can be foun in [4]. The issipate energy E issi will be use in this paper as a global measure of the elaminate area of the cohesive interfaces: E issi = t 3 ) Y i t Γ = A Γ 7) P I P 0 i=1 P I P where A is a scalar which only epens on the parameters of the interface moel. In the following evelopments, the investigations are restricte to simulations uner prescribe forces an isplacements following a unique loa function of time. In this context, the volume force will be assume negligible. These assumptions are not manatory to make use of the work presente in this paper, but simplify the construction of a continuous solution over time Section 3). 2.2 Time iscretization scheme An incremental proceure is use to solve the problem over time. It consists in iscretizing the time of the analysis [0 T ] in N intervals [t n t n+1 ] n 0, N 1. Successive nonlinear problems are solve at each computation time t n ) n 0, N. Hence, a solution to the iscretize problem in time is a set of N + 1 solutions satisfying the reference problem equations, the time epenency in the constitutive laws being iscretize. More precisely, at Computation time t n+1, the iscretization of Equations 4) an 6) reas: ) ) σ P.n P = K P [u] P [u] P 8) t [t 0, t n+1] In general, the time iscretization is chosen so that within each interval [t n t n+1 ] n 0, N 1, the evolution of the prescribe loa is monotonic, which will also be assume in the following. 2.3 Influence of the time increments on the solution to the iscretize elamination problem The solution to the iscretize reference problem reache at time T strongly epens on the time increments for two reasons: the iscretize cohesive law Equation 8)) epens on the iscrete history of the interface variables. Hence, the resiual stiffness of the cohesive interfaces epens on the time increments. This phenomenon is illustrate in the next section. structural problems involving softening materials may be unstable an may have multiple solutions. In those cases, the solution paths epen on both the algorithm use at each computation time step an the initialization of this algorithm i.e.: the previous converge solution). The resulting epenency on the time increments will be emonstrate in the last section of this paper. 4

6 U pre-cracke interface iffuse amage U crack front Figure 2: Definition of the four-ply DCB problem DCB ouble cantilever beam) test case The laminate structure that we consier is mae out of four isotropic plies Figure 3). One part of the meian cohesive interface is replace by a contact interface in orer to simulate an initial crack in the structure. Displacements are prescribe for the crack to propagate in a stable manner. The final prescribe isplacement is set to a preefine value, which fixes the propagation length. The initial stiffness of the cohesive interfaces is obtaine by integrating the Young an shear mouli of the matrix in the thickness of the interface 1/10 the thickness of the plies) [4]. The solution is not unique an epens on the loa increments. Figure 3 presents the amage state in the upper cohesive interface, four ifferent time iscretizations being applie these results will be fully etaile later on, for the values of the successive loa increments are obtaine by the aaptive time step proceure escribe in Section 5). ν time, is the criterion riving the time iscretization the largest ν time,, the coarser the iscretization). In cases 1 an 2, the number of time increment use in the propagation phase of the analysis are, respectively, 69 an 21. The ifferences in the amage state of the interfaces are not significant, the evolution of the crack front being sufficiently slow to capture the effects of the stress concentrations. Hence, both these solutions are converge with respect to the time. In case 3, obtaine with 9 coarse time increments, the solution is slightly ifferent from the previous reference cases. Finally, in case 4, using only 5 time increments to escribe the propagation of the crack clearly leas to the appearance of amage strips in the upper an lower interfaces. This is ue to the effect of the stress concentration at the tip of the crack which propagates in a iscrete manner with respect to time. Case 1 ν time, homogeneous amage state Case 2 increasing value of Case 3 amage strips Case 4 Figure 3: Influence of the prescribe value ν time, interface of the DCB problem on the amage state in the upper cohesive 5

7 3 A TIME DISCRETIZATION ERROR INDICATOR We suppose that two consecutive solutions to the reference problem, s n at Time t n an s n+1 at Time t n+1, have been compute using a nonlinear resolution strategy. The aim is to evaluate the relevancy of the solution compute at Time t n+1, the continuous evolution of the structure over the current time step [t n t n+1 ] being a priori unknown. We propose to construct an interpolate solution over the time step in orer to monitor the resiual of the nonlinear reference problem continuously. 3.1 Interpolation of the kinematic an static fiels over a time step Let us prescribe the continuous evolution of the prescribe bounary values over the time step: t [t n t n+1 ], { M Ωf, F t = α t) F tn + 1 α t)) F tn+1 M Ω u, U t = α t) U tn + 1 α t)) U tn+1 9) where the function α t) is the restriction of the loa function over [t n t n+1 ]. In the case of a linear evolution which will be the case in our applications), it simply reas: t [t n t n+1 ], α t) = t t n t n+1 t n 10) The evolution of the kinematic an static fiels over the current time is assume to follow the evolution of the prescribe loaing see Figure 4)), which writes: t [t n t n+1 ], P 1, N P, { up t = α t) u P tn + 1 α t)) u P tn+1 σ P t = α t) σ P tn + 1 α t)) σ P tn+1 11) s n+1 interpolate solution σ t,u t ) s n compute solutions t n t t n+1 Figure 4: Schematic representation of the interpolation performe over each time step s n an s n+1 are two solutions of the reference problem. In particular, they satisfy the following set of linear equations: kinematic amissibility, Equation 1) static amissibility, Equation 2), the volume force being assume negligible. constitutive law of the plies, Equation 3) As a consequence, the interpolate kinematic an static fiels over the current time step also satisfy this set of linear equations. Hence, the resiual of the reference problem at any time t [t n t n+1 ] is the resiual of the constitutive law of the cohesive interfaces, which remains the only non-satisfie equation. 6

8 3.2 Evolution of the amage variables over the current time step At any intermeiate time t [t n t n+1 ], the internal variables are calculate with respect to the continuous history of the interpolate isplacement fiel on Time interval [0 t]. Let us efine a new stress fiel σ which satisfies the nonlinear constitutive law of the interfaces: t [t n t n+1 ], P 1, N P 1, on I P, ) ) σ.n P t P = K [u] P P τ [0 t] [u] t 12) Alternatively, one can upate the amage variables with respect to the interpolate stress fiel, an efine a jump of isplacement fiel [u] satisfying the constitutive law of the cohesive interfaces. The amage variables initially compute at time t n+1 by the nonlinear resolution strategy are iscare. Inee, they may iffer from the ones obtaine at time t n+1 by the continuous construction over [t n t n+1 ], for solution s n+1 only satisfies the iscretize cohesive law 8). The resiual of the reference problem equations at Time t n+1 obtaine when upating the amage variables can be reuce by lowering the time increment t = t n+1 t n an performing new nonlinear resolutions at Time t n+1, which will be etaile in Section Definition of the time iscretization error inicator A measure ν interp interp stans for interpolation ) of the resiual of the reference problem equations at any time t [t n t n+1 ] can be obtaine by summing the local contributions to the error in the nonlinear constitutive laws: ) ν interp t = P σ σ P t n P t P IP ) where x σ + σ IP = x T x Γ 13) P t n P t P IP I P Or alternatively if the history is upate with respect to the interpolate stress fiel, ν interp t = [u] [u] ) P t P IP t P [u] + [u] ) 14) P t P IP t s n+1 σ t,u t ) ν time t n+1 ν interp t n+1 s n ν interp t σ t,u t ) t n t t n+1 Figure 5: Schematic representation of the time iscretization error inicator The time iscretization error inicator at Time t n+1 is efine as the maximum value of the previous measure over [t n t n+1 ] see Figure 5)), which reas: ν time t n+1 = max t [t n t νinterp n+1] t or alternatively ν t time n+1 = max t [t n t νinterp n+1] t 15) The concept introuce here fins its roots in the work of [13, 8], in which the sum over time of the prouct of Criteria 13) an 14) is use to measure the error in the constitutive law ue 7

9 to both space an time iscretization for materials satisfying Drucker s stability equality. Three main ifferences shoul be outline here: In the case of softening materials, Drucker s stability equality is not satisfie. The mathematical properties which result from the efinition of the Drucker s law-base criterion o not apply. Hence, making use of this criterion is not relevant. In aition, computing ν time requires the monotony of the interface behavior uniqueness of the amissible isplacement jump for any arbitrary local stress state). In the following evelopments, we will use Criterion ν time to measure the resiual of the reference problem equations over the current time step. Our final goal being to provie an algorithm to control on-the-fly the time increments, ν time is not a norm over the whole time of the analysis, but it instea is evaluate locally over each time increment. To be consistent with [13, 8] the fiel σ P t shoul also be reconstructe with respect to the space variables so that it satisfies exactly the static amissibility conition 2). In this work we focus on the time iscretization an so we content ourselves with a weak iscrete) static amissibility. At Times t n an t n+1 solution fiels satisfy the constitutive law of the plies 3), the kinematic amissibility an the static amissibility in the finite element sense. Thus Criterion ν time which is introuce without reference to space iscretization) only accounts for the error ue to time iscretization. 3.4 Practical consierations Sub-intervals In practice, ν interp is compute at a given set of intermeiate times within the current time step. [t n t n+1 ] is subivie into N s subintervals [ t i t i+1 ] i 0, Ns 1, the time iscretization error inicator ν t time n+1 being compute as: Error in the cohesive law ν time t n+1 = max i 0, N s νinterp t i 16) Computing ν time requires to extract the transverse constraints σ P.n P ) P 1, NP which is not irectly available in finite element coes. Usually, cohesive interface elements are use to overcome this problem. Classical incremental Newton solvers can then be use to solve the elamination problem at each computation time t n ) n 0, N. The technique to control the time increment that we propose in Section can irectly be applie to such approaches. We focus on the insertion of the control technique within the framework escribe in [10]. The principle is to use an incremental LaTIn-base omain ecomposition strategy [18] to efficiently solve in parallel) the elamination problem at each computation time. In this case, the cohesive behavior is irectly escribe as a nonlinear joint between substructures. The mixe escription of the interface behavior makes the transverse constraints available naturally. As it shall be etaile in Section 5, the time iscretization error inicator can be efine as a time-epenent version of the convergence inicator use to stop the iterations of the LaTIn algorithm. 4 THE NONLINEAR RESOLUTION STRATEGY We propose an overview of the omain ecomposition strategy use to perform the successive nonlinear resolutions of the elamination analysis, first in the stable case, then in the unstable case, where it is combine with an arc-length proceure. We focus in a secon time on the evelopment of a convergence inicator base on the error in the constitutive law [13] to stop both of these iterative solvers. Further etails concerning the multiscale an parallel computing aspects can be foun in [10]. 8

10 4.1 Substructure formulation of the reference problem Laminates Moelling Ω U E F Ω F f U Substructuring P I PP P Cohesive interfaces E E Perfect interfaces Figure 6: Substructuring of the laminate composite structure The laminate structure E is ecompose into substructures an interfaces as represente in Figure 6). Each of these mechanical entities possesses its own kinematic an static unknown fiels linke by its behavior. The substructuring is riven by the will to match omain ecomposition interfaces with material cohesive interfaces, so that each substructure belongs to a unique ply an has a constant linear behavior. Each substructure is efine in a omain Ω E such that E 1, n E n E being the total number of substructures) an is connecte to a ajacent substructures through interfaces Γ EE = Ω E Ω E where E 1, n E Figure 7)). The surface entity Γ EE applies force istributions F E, F E as well as isplacement istributions W E, W E to Substructure E an Substructure E respectively. On Substructure E such that Ω E Ω, the bounary conition U, F ) is applie through a bounary interface Γ E. Let us efine Γ E = E 1, n E Γ EE Thursay, 4 February Γ E We finally introuce σ E, the Cauchy stress tensor, an ɛu E ), the symmetric part of the isplacement graient, in substructure E. The substructure quasi-static problem at any computation time t n+1 of the time iscretization scheme consists in fining s = s E ) E 1, ne, where s E = W E, F E ), which satisfies the following equations: Γ E F E,W E ) u E, σ E ) F E,W E ) u E, σ ) E E Γ EE E Figure 7: Substructuring of the laminate composite structure 9

11 Kinematic amissibility of Substructure E: u E ΓE = W E 17) Static amissibility of Substructure E: u E, W E ) U E W E / u E ΩE = W E, ) Tr σ E ɛu E ) Ω = F E.W E Γ 18) Ω E Γ E Linear orthotropic behavior of Substructure E: Behavior of the interfaces Γ EE Γ E : Behavior of the interfaces Γ E Γ E Ω): σ E = K ɛu E ) 19) R EE W E, W E, F E, F E ) = 0 20) R E W E, F E ) = 0 W E = u on Ω u an F E = F on Ω f ) 21) In elamination analysis, the formal relation R EE = 0 reas: { F For perfect interface: E + F E = 0 W E W E = 0 { F E + F E = 0 ) For cohesive interface: F E = K P W E W E ) t t0, t n+1 W E W E ) where Substructure E respectively E ) belongs to Ply P respectively P + 1). 4.2 Iterative resolution of the stable nonlinear substructure problem ŝ i+ 1 2 Γ E E + s ref s i+1 s i A Figure 8: Schematic representation of the LaTIn algorithm The equations of the problem can be split into the set of linear equations in substructures static an kinematic amissibility of the substructures, linear constitutive law of the substructures) an the set of local equations in interface variables behavior of the interfaces). The solutions s = s E ) E 1, ne = W E, F E ) E 1, ne to the first set of equations belong to Space A, while the solutions ŝ = ŝ E ) E 1, ne = Ŵ E, F E ) E 1, ne to the secon set of equations belong to Γ. Hence, the converge solution s ref is such that s ref A Γ. The LaTIn resolution scheme consists in searching for the solution s ref alternatively in these two spaces along search irections E + an E see Fig. 8): ) Fin ŝ i+ 1 2 Γ such that ŝ i+ 1 2 s i E + local stage) Fin s i+1 A such that In the following, the subscript i will be roppe. ) s i+1 ŝ i+ 1 2 E linear stage) 10

12 Local stage One searches for a solution ŝ = F E, Ŵ E) E 1, ne satisfying the local equations on the interfaces R EE = 0 or R E = 0), an search irection equation E +, introuce locally on the interfaces : F E F E ) k + E Ŵ E W E ) = 0 22) At this stage, variables F E et W E are known from the previous semi-iteration. In the case where R EE = 0 is a nonlinear equation, the local problem is solve by a quasi- Newton algorithm. Linear stage One searches for a solution s = F E, W E ) E 1, ne verifying the linear equations on each substructure an, at best, a search irection equation E, local on the interfaces, uner the constraint of average equilibrium of the interface forces : } { F E ΓE = arg min 1 Γ E uner the constraint: E, E), Π M Γ F EE E ΓEE + ΠM Γ F EE E Γ EE ) 2 k F E F E ) 2 + F E F E ).W E Ŵ E ) Γ E = 0 The macroscopic projectors Π M Γ extract an average of the interface forces, which is transfere EE into the whole structure. Technically, this stage consists in solving, in parallel, inepenent linear problems on the sub-structures using finite elements) an a small macroscopic linear problem which is global over the structure an iscrete by construction). 4.3 Iterative resolution of the unstable nonlinear problem When a snap-back appears in the global behavior of the simulate structure, the incremental LaTin algorithm is switche to a well-known local arc-length algorithm [27, 2, 10]. The algebraic nonlinear problem to solve at Time t n+1, in an unstable phase, reas: 23) q int U tn+1, U τ ) τ<tn+1 ) λ tn+1 q ext = 0 24) The amplitue of the loaing λ tn+1 is unknown. A control equation inspire from [2] is introuce so that the maximum local increment in the jump of isplacement over all the cohesive interfaces takes a preefine value l: c U tn+1 ) U tn+1 = l 25) where the. unknowns are the increments in the quantities over Time step [t n t n+1 ]. c is then the operator which extract the maximum of the positive) jump increment. Classically, the non-linear system 24, 25) is solve by a moifie Newton-Raphson scheme: The linearization of 24) an 25) aroun point U i, λ i ) leas to the system to solve at the preiction step of the i + 1) th iteration of this scheme: λ i+1 t n+1 = c U i t n+1 ) K U i+1 t n+1 = λ i+1 t n+1 K l + c U t i n+1 ) U tn ) 1 U t i n+1, U τ ) τ<tn+1 qext U i t n+1, U τ ) τ<tn+1 ) 1 qext 26) The inversion of the linearize stiffness operator i.e.: the resolution of the linear system Ū = KU i t n+1, U τ ) τ<tn+1 ) 1 q ext ) is performe by using the omain ecomposition metho escribe previously the internal variables of the interfaces are fixe uring the resolution) The correction step of the Newton scheme consists in upating Operators K an c with respect to the kinematic fiel U i+1 t n+1 foun at the preiction stage. 11

13 ŝ i+ 1 2 Γ s ref v v ν iter v ν s s i+1 E E + s i A Figure 9: Classical convergence inicator ν s of the LaTIn solver an inicator ν iter base on the error in the constitutive law 4.4 Stopping criterion Stable phase LaTIn algorithm) In orer to evaluate the convergence of the LaTIn algorithm, one classically measures the istance between spaces A an Γ along search irection E [11] criterion labele ν s on Figure 9), s staning for search irection ). In the work of [10, 19], a new criterion base on the error in the constitutive law has been successfully assesse in orer, originally, to get ri of the epenency of Convergence inicator ν s on the parameters of the LaTIn solver). The solutions resulting from a linear stage of the LaTIn algorithm satisfy all the equations of the substructure reference problem except the interfaces laws 20) an 21), whose resiuals can be easily compute Figure 9)). More precisely, a solution s i+1 A being reache, an inicator of the convergence of the algorithm is given by integrating the corresponing local resiuals of the interface behaviors over the structure resiuals of Equations 20) an 21) evaluate for s i+1 = s i+1 E ) E 1, n E = W i+1 E, F i+1 E ) E 1, n E ). Let us call q the number of interface relations being use i.e.: the number of istinct interface behaviors R EE = 0) E,E ) 1, n E 2 an R E = 0) E 1, ne use in the structure). In our case, q = 4 perfect interfaces, cohesive interfaces with homogeneous constants, Dirichlet an Neumann bounary conitions). Γi is the set interfaces of Behavior i, for all i 1, q. The vectorial relations R EE = 0 for i 1, q an Γ EE Γ i or Γ E Γ i are mae of p i vectorial equations Q ij = 0 2 equations for cohesive or perfect interfaces in 3D, 1 equation for bounary interfaces). Here, subscript j ranges from 1 to p. Convergence inicator ν iter iter stans for iterative ) reas: ν iter ) 2 q p i ) = ν iter 2 ij where i=1 j=1 ) ν iter 2 ij = Γ Γ i Γ Γ i Γ Γ Q ij.q ij Γ Q ij. Q ij Γ where, in the case of elamination i.e : involving perfect an cohesive LaTIn interfaces): on a perfect interface Γ EE Γ 1 : 27) Q 11 = F E + F E Q 11 = F E F E Q 12 = W E W E Q 12 = W E + W E 28) on a cohesive interface Γ EE Γ 2 : Q 21 = F E K P W E W E ) t {tn+1,[0 t n]}) W E W E ) Q 21 = F E + K P W E W E ) t {tn+1,[0 t n]}) W E W E ) Q 22 = F E K P W E W E ) t {tn+1,[0 t n]}) W E W E ) Q 22 = F E + K P W E W E ) t {tn+1,[0 t n]}) W E W E ) 29) 12

14 where P 1, N P 1. Note that, in Equation 29), the history of the interface variables uring the current loa increment is not taken into account, for it is unknown at this stage of the resolution proceure. on an interface transmitting Neumann s bounary conition Γ E Γ 3 : Q 31 = F E F Q31 = F E + F 30) on an interface transmitting Dirichlet s bounary conition Γ E Γ 4 : Q 41 = W E W Q41 = W E + W 31) The computation of this criterion is very cheap as it simply requires local integration over each interface of the omain ecomposition metho, an a global sum of these local contributions over the structure Unstable phase arc-length proceure) The convergence of the algorithm is evaluate by computing Criterion ν iter after each preiction stage of the Newton scheme the resiual of the control equation, which has no physical meaning, is not accounte for) Typical values Our experiments of elamination analysis within the LaTIn framework have shown that a stopping criterion ν iter set to ν iter = stans here for esire ) is usually sufficient to ensure a global convergence of the iterative process at least, crack fronts are correctly localize, which means that the large wavelength piece of information is correctly capture). In our simulations, an in orer to force an accurate convergence of the local quantities equilibrium of the interface forces an verification of the cohesive law in the process zones), ν iter is set to AN AUTOMATIC PROCEDURE TO CONTROL THE LOAD INCREMENTS In this section, we combine the ieas etaile in Section 3 to estimate the time iscretization error, an the evelopments of the last section, eicate to the evaluation of the convergence of the iterative parallel resolutions to erive a new time iscretization error inicator, suite but not restricte) to the mixe omain ecomposition strategy. Base on this new inicator, an automatic proceure to control the loa increments is erive. 5.1 Time iscretization error criterion in a omain ecomposition framework Definition A sufficiently converge solution of the reference problem being reache at Current time t n+1, by making use of the LaTIn-base resolution strategy, a continuous solution is constructe over [t n t n+1 ], as escribe in Section 3. A new time iscretization error criterion ν time, stans for omain ecomposition ) is compute with respect to the interpolate solution: ν time, t n+1 = max i 0, N s νinterp, t i 32) where we recall that N s +1 is the number of intermeiate times t i ) i 0, Ns such that t n t i t n+1 at which intermeiate solutions are constructe, an Criterion ν interp, is evaluate. 13

15 is compute by the same formulas efining ν t iter i, except that the history of the interface variables is known continuously over Time interval [0 t i ] from the interpolation): ν interp, t i ν interp, t i ) 2 = q p i=1 j=1 ν interp, ij ti ) 2 where ν interp, ij ) 2 = Γ Γ i Γ Γ Γ i Γ Q ij ti.q ij Γ Q ij. Q ij Γ an, in Equation 29), the stiffness operator of the cohesive interfaces Γ EE E,E ) 1, n E 2 is replace by the reconstructe operator K P W E W E ) t [0 t i]). Note that Time iscretization error criteria ν time, an ν time are slightly ifferent. In Section 3, we assume that the solutions obtaine at Times t n an t n+1 satisfie the global static amissibility Equation 2)). This assumption cannot be mae anymore if the solver use is the mixe omain ecomposition strategy unless the convergence criterion threshol is set to a very low value, which woul be ineffective, from a numerical point of view). Inee, the equilibrium is only satisfie in terms of substructures an macroscopic interfaces variables. In aition, the kinematic amissibility is not fully satisfie, for each ply has been ecompose into substructures which are imperfectly bone uring the resolution process. Hence, the new time iscretization error criterion ν time, measures not only the cohesive law resiual, but also an interface microscopic equilibrium resiual which is small) an a jump of isplacement through perfect interfaces, both ue to a partial convergence of the iterative solver. 33)!& logν interp, ) logν time, )!&)!"!") logν time, t)) logν interp, t) t=11,5 )!' logν iter )!')! " # $ % &! &" &# &$ &% t t Figure 10: Grey curves : evolution of ν interp, as a function of t [t n t n+1 ] for ifferent values of t. Black curve: Evolution of ν time, with respect to t maximum values of the gray curves) Properties Figure 10) shows the evolution of ν interp, within a time interval [t n t n+1 ] for a given computation time t n of the unstable elamination simulation represente Figure 12) which will be etaile later on). The ifferent gray curves correspon to ifferent values of the time increment t value of the prescribe arc-length in this case). Note that the the value of ν interp, at Computation time t n is the value ν iter of ν iter which has been prescribe to ensure a sufficient convergence of the LaTIn iterative process. From this set of parametric stuies, the values of ν time, can be plotte with respect to t maximum values of ν interp, over [t n t n+1 ], black points on the figure). The resulting function black interpolate curve) is monotonic. One can also remark that even when a large time step is prescribe, the curve ν interp, as a function of t [t n t n+1 ] is smooth. Thus, a relatively small number of evaluation of this resiual 14

16 over the time step is sufficient to obtain an accurate value of the time iscretization error criterion ν time,. In aition, as the computation of Criterion ν interp, is cheap, even a large number of intermeiate time steps woul not alter the numerical efficiency of the strategy. In practice, we choose N s = Aaptive loa increment proceure Our aim is to solve the elamination problem at Computation time t n+1 uner the constraint of given value ν time, of the time iscretization error inicator, the current time increment t = t n+1 t n i.e.: the prescribe arc-length or the loa increment) being set as a new unknown. A quasi-newton technique is use to solve the nonlinear constraint equation: ν time, t) ν time, = 0 34) Basically, each iteration of this scheme is ecompose in three steps: a linear step, where a value of the time increment is preicte see formulas 35,36) in next paragraph). a correction stage where the full elamination problem is solve, at the current time step t n+1, until convergence of the unerlying nonlinear solver. The time increment t is here fixe to its preicte value. the computation of a convergence inicator norm of the resiual of Equation 34)) The linear preiction stage at Iteration k + 1 of Computation time t n+1 consists in solving the linearize relation linking ν time, to the time increment t see Figure 11)). This preiction is one by the following formula: t k+1 = t k 1 + νtime, ν time,k 1 t k t k 1) 35) ν time,k ν time,k 1 Previous formula oes not warranty the preiction of a positive arc-length the function to linearize is concave). If a negative time increment is preicte, equation 35) is replace by the following relation, which has empirically shown goo convergence properties: t k+1 ν time, = ν time,k tk 36) ν time,k ν time, ν time,k 1 ν iter t k 1 t k+1 t k Figure 11: Preiction step of the Newton algorithm esigne to solve the elamination problem uner the constraint of given time iscretization error criterion 15

17 6 VALIDATION OF THE STRATEGY 6.1 First numerical stuies of the time step control proceure on the stable DCB case Let us fully etail the results obtaine on the stable DCB case presente in Section 2, Figure 2. This problem is globally stable an solve using, at each computation time, the LaTIn-base omain ecomposition strategy. In the four simulations presente Figure 3, the time increments have been obtaine by prescribing increasing values of ν time,, respectively , , an The resulting average time increment increases, the total number of computation times N being respectively equal to 69, 21, 9 an 5. As explaine in Section 3, the amage state in the cohesive interfaces tens to the one obtaine for very small loa increments case 1) when the value of ν time, ecreases. More precisely, the elaminate area of the cohesive interfaces i.e.: the issipate energy) converges in a monotonic manner with ecreasing values of threshol of the time iscretization inicator. When this threshol is set to a value smaller than , the error mae in terms of issipate energy is not significant. Though, this test case is too specific stable, only one crack front) to give a reliable threshol value ν time, which shoul be applie in the general case in orer to insure a sufficient convergence of the solution with respect to time. 6.2 Unstable hole-plate elamination problem U U plane of symmetry Figure 12: Definition of the hole plate problem o.f.) We consier a eight-plies hole-plate structure, uner traction prescribe isplacements). The plies are orthotropic stiffness ratio: 1/20) an the stacking sequence is [0 ± 45 90] s, which leas to the initiation of elamination ue to ege effects. The initial stiffness properties of the cohesive interfaces are obtaine by the same homogenization in the thickness of the interfaces one tenth of the thickness of the plies) which has been escribe for the DCB problem in Section 2. Due to the material an structural symmetries, only the top half of the structure is simulate. The unstable quasi-static time analysis is performe by making use of the arc-length proceure escribe in Monay, 25 January 2010 Section 4. The global response curve plotte in Figure 13), Case 3) of this case shows two main zones of instability. The first one correspons to a an unstable propagation of the elamination in the [ ] interface while the secon one is a crack propagation in the [0 45] interface, both mainly in shear moe Prescribe time step Cases numbere 1, 2 an 3 in the whole analysis of the results) The first set of simulations is performe by successively prescribing three ifferent fixe arc-lengths. The arc-length which has been arbitrarily chosen in Case 1 is ivie by three in Case 2, an by nine in Case 3. Instabilities appear in the global response of the structure Figure 13)). Figure 16

18 14) shows the amage maps in the [0 45] [ ] an [ 45 90] cohesive interfaces in a monotonic phase of the global behavior limit point after which the elamination front evolves in an unstable manner in the [0 45] interface, which correspons to the circle point in all graphs of Figure 13)). This particular point of interest has been reache respectively in 8, 63 an 237 time increments. Note that we o not aim at iscussing the valiity of the solutions reache but at ensuring that the incremental strategy follows the equilibrium path of the converge solution with respect to the time. Case 1 Case 2 Case 3 reaction forces N.mm -2 ) reaction forces N.mm -2 ) reaction forces N.mm -2 ) prescribe isplacement mm) prescribe isplacement mm) prescribe isplacement mm) Figure 13: Global reaction force versus prescribe isplacement in the hole plate problem uner traction, three ifferent preefine arc-lengths being applie Cases 1,2 an 3) Case 1 [0-45] Case 2 Monay, 25 January 2010 [ ] [+45 90] Figure 14: Damage state in the interfaces of the hole plate at the beginning of a global instability in the case of a coarse time gri Case 1), an in a converge case Case 2). A Fixe arc-length is prescribe in both cases. In the first case, the amage in the [ ] interface is unerestimate. No significant ifference can be observe in the amage maps an global response curves corresponing to the two finest analysis in time, which means that the solutions are sufficiently converge with respect to time in Cases 2 an 3. In Case 1, the time increments are too coarse, which results in the incremental resolution proceure to follow a ifferent equilibrium path see the amage maps in Figure 14)). This phenomenon can impair the global response of the structure, as it can be seen on Figure 13). The instability phases frame on the converge solutions Cases 2 an 3) are wrongly preicte in Case 1. These ifferences appear even more clearly on the issipate energy versus prescribe isplacement curves plotte on Figure 15) the curves labele reference an coarse gri refer respectively to Cases 3 an 1), corresponing to the first global instability an 17

19 to the following stable phases of the time analysis reference case 3) coarse gri case 1) ν time, = case 4) ν time, = case 5) issipation mj) prescribe isplacement mm) Figure 15: Dissipation versus loaing curves for ifferent resolution strategies: explicit fine an coarse time steps Cases 1 an 3, fixe arc-length) or automatically controlle time increments Cases 4 an 5, fixe time iscretization error) logν time, ) Figure 16: Time iscretization error criterion as a function of the Computation times in Case 1 circles, coarse time steps) an Case 2 crosses, small time steps). A Fixe arc-length is prescribe in both cases. Figure 16) presents the values of ν time, t n ) n 1, N as a function of the successive computation times in Cases 1 an 2, from the beginning of the analysis to the starting point of the secon global instability. One can see that in Case 2, in which the time increments are sufficiently small to let the iterative algorithm follow the correct equilibrium path, the values of the iscretization error inicator ν time, range from to Conversely, we show in the next set of stuies that setting the threshol value ν time, of the time control proceure to the maximum of the values ν time, obtaine in Case 2 permits to obtain a correctly preicte solution. t 18

20 6.2.2 Control of the time step Cases numbere 4 an 5) The secon set of simulations is performe by making use of the proceure escribe in Section 5 to control the successive prescribe arc-length. ν time, is successively set to Case 4) an Case 5). The amage state in the cohesive interfaces at the beginning of the secon global instability phase preicte by prescribing ν time, = is very close to the one obtaine in Case 2 of our first set of simulations see Figure 17)). The total number of time increments rops from 63 to 40. Case 2 [0-45] Case 4 [ ] [+45 90] Figure 17: Damage state in the interfaces of the hole plate at the beginning of an instability obtaine in a converge case prescribe arc-length, Case 2), an by using the time control proceure fixe iscretization error, Case 4) As explaine previously, the issipate energy versus prescribe isplacement curves Figure 15)) obtaine in the reference case 3 very small prescribe arc-length) an in Test case 1 coarse prescribe arc-length) are very ifferent incorrect equilibrium path in the secon case). When using the time control strategy, ν time, being successively set to an , the correct equilibrium path is followe. In aition, the issipate energy versus prescribe isplacement curves gets closer to the one obtaine in the reference simulation when the value of ν time, ecreases. 6.3 Discussion The threshol value foun numerically here can be compare to the one prescribe to ensure the convergence of the iterative resolution strategy at each computation time, ν iter. As explaine in Section 4, the value ν iter which ensures a sufficient convergence of the LaTIn solver can be obtaine empirically by performing time inepenent benchmark tests for instance the first time step of a elamination analysis). The time control strategy evelope in this paper consists in monitoring the resiual of the reference problem equations continuously uring the time analysis, the measure use at any time being a time inepenent version of ν iter. Hence, it is not surprising to fin out in the numerical examples that the higher value ν time, permitting to follow the correct equilibrium path is the value of ν iter which permits to obtain the convergence of the global informations position of the crack fronts) at each computation time. Hence, applying the time control proceure only requires the prior knowlege of inicator ν iter. 7 CONCLUSION In this paper, we presente a strategy to aapt automatically the time increment in quasi-static elamination problems to the very sharp non-linearities which are involve. This strategy is base a continuous monitoring of the resiual of the reference problem equations with respect to time. This has been achieve by calculating the error in the constitutive law on amissible solutions 19

21 interpolate over each time steps, which enables to efine a time iscretization error criterion evaluating the relevancy of the nonlinear computations performe at each time increment. Base on this inicator, a simple proceure to control the time step has been erive. The main parameter of this technique is easy to obtain as it only requires to perform time-inepenent benchmark tests prior to the elamination simulations. The valiity of this proceure has been emonstrate on elamination problems unergoing global instabilities. Our current interest being to perform buckling-riven elamination analysis, the valiity of this strategy shall be verifie, in the future, on computations involving geometrical nonlinearities. References [1] G. Alfano an M. A. Crisfiel. Finite element interface moels for the elamination analysis of laminate composites: mechanical an computational issus. International Journal for Numerical Methos in Engineering, 50: , [2] O. Allix an A. Corigliano. Moeling an simulation of crack propagation in mixe-moes interlaminar fracture specimens. International Journal of Fracture, 77:11 140, [3] O. Allix an P. Laevèze. Interlaminar interface moelling for the preiction of elamination. Computers an Structures, 22: , [4] O. Allix, D. Lévêque, an L. Perret. Ientification an forecast of elamination in composite laminates by an interlaminar interface moel. Composites Science an Technology, 58: , [5] I. Babuška an W.C. Rheinbolt. A posteriori error estimates for the finite element metho. International Journal for Numerical Methos in Engineering, 12: , [6] J.M. Bass an J.T. Oen. Aaptative finite element methos for a class of evolution problems in viscoplasticity. International Journal of Engineering Science, 256): , [7] R. De Borst an J. C. Remmers. Computational moelling of elamination. Composites Science an Technology, 66: , [8] L. Gallimar, P. Laevèze, an J. P. Pelle. Error estimation an aaptivity in elastoplasticity. International Journal for Numerical Methos in Engineering, 39: , [9] C. Johnson an P. Hansbo. Aaptative finite element methos in computational mechanics. Computer Methos in Applie Mechanics an Engineering, 101: , [10] P. Kerfrien, O. Allix, an P. Gosselet. A three-scale omain ecomposition metho for the 3 analysis of eboning in laminates. Computational Mechanics, 443): , [11] P. Laevèze. Nonlinear Computational Structural Mechanics - New Approaches an Non- Incremental Methos of Calculation. Springer Verlag, [12] P. Laevèze. Multiscale moelling of amage an fracture processes in composite materials, chapter Multiscale computational amage moelling of laminate composites. Springer-Verlag, [13] P. Laevèze, G. Coffignal, an J. P. Pelle. Accuracy of elastoplastic an ynamic analysis. In I. Babuška, J. Gago, E. Oliveira, an O. C. Zienkiewicz, eitors, Accuracy Estimates an Aaptative Refinement in Finite Element Computations., pages Wiley, [14] P. Laevèze an D. Dureisseix. A micro/macro approch for parallel computing of heterogeneous structures. International Journal for computational Civil an Structural Engineering, 1:18 28, [15] P. Laevèze an G. Lubineau. An enhance mesomoel for laminates base on micromechanics. Composites Science an Technology, 624): ,