NONLOCAL COMPUTATION METHODS APPLIED TO COMPOSITE STRUCTURES N. GERMAIN1, F. FEYEL1 and J. BESSON2 1ONERA/DMSE, 29 avenue de la Division Leclerc, F-92322 Châtillon Cedex, France 2CdM/ENSMP, BP 87, F-91003 Evry Cedex, FRANCE

ABSTRACT A new finite element method is presented to solve numerical problems appearing when a classical finite element method, coupled with a Newton-Raphson algorithm, is used to model the degradation of organic or ceramic matrix composites structures even under simple solicitations (traction, flexion, …). For instance, instabilities and localisation arise, which lead to a quick divergence of the solver. In order to overcome these difficulties, a new method is developed. It is composed of a set of tools which ensures a good balance between numerical conditioning, high performance computation and applicability to CMC or OMC structures. This method is based on a non local framework (with a damage implicit gradient) and an arclength algorithm. The final goal of our work is to be able to run simulations in order to follow the initiation and the propagation of fracture and study industrial problems. This kind of simulation requires a very thin mesh which leads to a large computational time. That is why all the procedures will be parallelized thanks to a FETI's like method.

1 INTRODUCTION Recent advances in mechanics give a better description of heterogeneous materials like organic or ceramic matrix composites. The finite elements method makes it possible to predict the evolution of damage in complex structures, and to prevent their failures. However within this framework, standard finite elements procedures rise a lot of numerical problems. For instance, instabilities and localisations appear, which lead to a quick divergence of the solver. Today's new sophisticated constitutive equation sets require more efficient methods, especially those taking into account damage phenomena. In order to overcome these difficulties, a non local model is used. This is not only a new finite element algorithm but set of tools which ensure a good balance between applicability to OMC structures (which exhibit an anisotropic (viscous-) elastic damageable behavior), high performance computation, and of course, a good numerical conditioning. The main parts of the method are:

• • •

A non local model including damage implicit gradient (instead of a local method), to solve the problem of mesh dependency while remaining applicable in anisotropic/heterogeneous problem, An optimized arc length algorithm, to solve the resulting nonlinear equations, A FETI's like method allowing non local parallel computation.

2 EXAMPLES OF NUMERICAL PROBLEMS In this section, we show some numerical examples to emphasize the need of such methods. Let us consider an industrial tube specimen made of ceramic matrix composite. The simulation consists of a three points bending test (fig. 1). The test specimen is modelled (taking into account the symmetry) using four different meshes. Figures 2 and 3 show the results. At the macroscopic level, the computations end when the damage value goes beyond 0.3. The damageable constitutive equations sets enters into the nonlinear stage: the Newton-Raphson algorithm fails to find the nonlinear solution. The slope of the curve increases as the elements size decreases (fig. 2). This effect comes either from the non convergence of the FE method or from a localisation problem. However, the third and fourth meshes are thin enough to ensure a good quality of the results. Figure 3 shows the horizontal stress. Mesh dependency is clear, the overstrain area is always one element thick. This simple example show that is necessary to develop advanced simulation methods to be able to take into account modern damageable constitutive equation sets.

figure 1: Tube specimen and boundary condition

figure 2: Force- Displacement curve

figure 3: isocontour of ε11 3 NON LOCAL METHOD Different ways have been explored to solve the localization problem. We decided to choose a non local method based on an implicit gradient formulation [1]. The delocalization is applied to the damage variable D. Let us consider the following local problem: in Ω ⎧div(σ ) = 0 ⎪σ = (1 − D )C : ε ⎪ ⎪ D = g (Y ) ⎪ ⎪Y = 1 (ε : C : ε ) ⎨ 2 ⎪ 1 ⎪ε = ∇ s (u ) 2 ⎪ on Γu ⎪u = ui ⎪σ : n = F on Γf ⎩

(1)

The nonlocal equivalent problem is defined by: in Ω ⎧div(σ ) = 0 ~ ⎪σ = 1 − D C :ε ⎪ ⎪ D = g (Y ) ⎪ ⎪Y = 1 (ε : C : ε ) ⎨ 2 ⎪ 1 ⎪ε = ∇ s (u ) 2 ⎪ on Γu ⎪u = ui ⎪σ : n = F on Γf ⎩ ~ A variational problem is added to drive the nonlocal damage variable D : ~ ~ ⎧ D − c∇ 2 D = D in Ω ⎪ ~ ⎨ ∂D on Γ ⎪ni ∂x = 0 i ⎩

(

)

(2)

(3)

This new formulation has been implemented and tested in a finite elements code. This model lacks some important features of composite materials, especially the initial anisotropy. A new formulation will be derived to take into account. 4. ARC LENGTH ALGORITHM To solve instabilities and divergence problems, an algorithm based on an arc length method is used [2]. A load parameter λ is introduced as a new unknown of the problem. It is assumed that the total load is proportional to λ and a load direction F. The problem to be solve is: ⎧⎪ g ( p, λ ) = qi ( p, λ ) − qext (λ ) = 0 (4) ⎨ 2 ⎪⎩h( p) = L

Let qi be the internal reactions and qext the external reactions (equal to λF). h is a control function. The solution scheme is based on an iterative Newton-Raphson algorithm; therefore, the problem is to find δλ and δu in order to satisfy at the jth iteration of the ith increment: ⎧ p = p0 + δ p (5) ⎨ ⎩λ = λ0 + δλ Such as: (6) δp = − K 0−1 g ( p0 , λ0 ) - δλK 0−1q01 Where:

⎧ ∂q K = i p ,λ (7) ⎪⎪ 0 ∂p 0 0 ⎨ ⎪q1 = ∂qi p , λ − ∂qext λ 0 ⎪⎩ 0 ∂λ 0 0 ∂λ In the literature, a lot of methods have been developed ([3], [4], [5]) but none of them is always satisfying. It has been decided to develop an oriented object implementation of this class of algorithms, in order to test existing method and implement new ones easily.

(

)

(

)

( )

First, two algorithm has been tested where h is equal to: h( p ) = ΔpT Δp [3]

(8)

(9) h( p) = Δp0T Δp [4] The choosen control function can lead to an order two polynomial system (eq. 8): another criterion is needed to choose between the two roots. Thus, different methods could be used: • Choose the one which minimizes the angle between the previous approximation and the new one [3], • Choose the one which minimizes the residual [5]. It appears through different tests that the "better" solution seems to be the second order equation coupled with an angle minimization technique. That's why for the nonlocal method, we will develop a method based on this one. The originality of our ~ to evaluate the non linearity of the algorithm is in the use of the new unknown D behaviour and to weight the influence of watched points. We hope that this method gives a good balance between step size and numerical conditioning. 5. PARALLEL COMPUTATION The increase of the structures and behaviours complexity increase the simulation time. The parallelization of the finite elements code becomes necessary. This is especially true for the nonlocal model and the arc-length method: the nonlocal model needs more unknowns. Moreover, it is difficult to predict how many time steps the computation will run with the arc-length algorithm. Therefore, we have decided to parallelize the nonlocal finite elements method. First, we shave to parallelize the arc-length algorithm. Inside each sub-domain Ωi of Ω ( Ω = U Ω i ), the value of the load parameters λi, will be computed and the i =1

minimum value for λ will be chosen. Possibly, it will be necessary to find another method in order to increase the time step.

Next, we have to parallelize the nonlocal method. The method will be based on a FETI's algorithm (Farhat & Roux, [6]), which consists, for local problems, in giving new unknowns: the forces αi to be enforced on domain boundaries to ensure displacement continuity across sub-domains. The forces αi are estimated with an iterative method. We have indeed the following equivalence: ⎧ K i qi = Fi + Bitα i local equilibriu m of each sub - domain Ω i ⎪ (10) Kq = F ⇔ S ⎨ B q =0 i i ⎪⎩∑ i =1

displacement continuity

In our method, we have to make the equilibrium of the interface reaction and of the interface local damage. The difficulty is that these entities don't have the same dimension. Thus, the parallelization of the nonlocal equations will need the use of a FETI's extension to multifields problem. CONCLUSION The first results given by the nonlocal model and the arc-length algorithm seem to be very interesting. The problems of localisation and instabilities have been overcame. The parallelization and the generalization of these methods to organic or ceramic matrix composite will allow simulation of industrial structures (Onera, [7]) in order to predict the evolution of damage and to prevent their failures. REFERENCES [1] Peerlings, R.H.J., "Enhanced damage modelling for fracture and fatigue – PhD these", Technische Universiteit Eindhoven, pp. 31-65, 1999. [2] Zienkiewicz, O.C., TAYLOR R.L., "The finite element method: Volume 2 Solid Mechanics, fifth edition", Butterworth Heinemann, (2000), pp. 22-37 [3] Alfano, G. & Crisfield, M.A., "Solution strategies for the delamination analysis based n a combination of local-control arc-length and line searches", Int. J. Numer. Meth. Engng., vol. 58, pp. 999-1048, 2003. [4] de BORST, R., "Computation of post-bifurcation and post failure behavior of strain-softening solids", Computers & Structures, 25-2, (1987), pp. 211-224. [5] HELLWEG, H.-B., "A new arc-length method for handling sharp snap-backs", Computers & Structures, 66-5, (1998), pp. 705-709. [6] Farhat, C. and Roux, F.-X., "A method of finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Meth. Engrg., 32, (1991), pp. 1205–1227. [7] ONERA, "Rapport final de la tranche 1", PEA AMERICO, pp. 141-143, 2003 => je dois voir JF pour la référence accessible