Hachmi Ben Dhia

Chokri Zammali
email: zammali@mssmat.ecp.fr

A velocity and Level-Sets based approach of dynamic contact problems with applications

Keywords: Level-Sets, contact, impact, Lagrange velocity formulation, Arlequin method

By introducing unknown Level-Sets fields defined on contact interfaces, the Signorini-Moreau contact conditions and the Coulomb friction laws are written as equations. From this, a continuous weak-strong formulation of dynamic frictional contact problems is derived; the Level-Sets being the intrinsic contact unknown fields. The problem is then discretized by time, space and collocation schemes. Some numerical tests are carried out, showing the effectiveness of our global approach. The paper is ended by showing a promising application of the multiscale Arlequin method [1] to contact-impact problems.

INTRODUCTION

Impact problems are nonlinear, irregular and multiscale. Their numerical approximation needs special numerical tools. Since the late seventies, Hughes et al. [START_REF] Hughes | A finite element method for a class of contact-impact problems[END_REF] have designed a posteriori numerical schemes to treat shocks due to impact and increasing efforts have been since furnished by the computational contact community to address this numerical issue (see e.g. [START_REF] Armero | Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems[END_REF][START_REF] Laursen | Improved implicit integrators for transient impact problems[END_REF][START_REF] Vola | Consistent time discretization for a dynamic frictional contact problem and complementary techniques[END_REF]). Very important advances have this way been done and some of the most pertinent schemes are now capitalized in commercial codes. But due to the aforementioned intrinsic difficulties of the impact problems (strong nonlinearities, both space and time irregularities and multiscale intrinsic characters), more effort is needed to address for instance dynamic contact problems such as those involved by vibrating structures under impact. This work is a contribution to this wide and complex theme. By applying the "viability lemma" of J.J. Moreau [START_REF] Moreau | Nonsmooth Mechanics and application[END_REF], the unilateral contact laws are written in terms of positions, velocities and contact pressions. Moreover, by introducing two unknown Level-Sets [START_REF] Sethian | Level-Set Methods evolving interfaces in geometry, fluid mechanics, computer vision, and materials science[END_REF] type fields standing for a location of the positions of contact surfaces with respect to each other and for the Sign of the normal velocity jump field on the contact interface, the obtained (Signorini-Moreau) contact model is converted to multi-valued equalities. The Coulomb's frictional laws are written similarly as equations by using a Level-Set field whose iso-1 values characterize the sticking contact regions. From these local equations, a seemingly new continuous weak-strong Lagrange formulation of a dynamic 3-D frictional contact problem is carried out in a straightforward manner. The associated discrete nonlinear systems are derived by using three discretization methods, namely a θ-time discretization scheme (e.g. [START_REF] Vola | Consistent time discretization for a dynamic frictional contact problem and complementary techniques[END_REF]), a Galerkin method and a collocation one. The numerical algorithm used to solve the nonlinear and irregular discrete problems is described. Some numerical dynamic (frictionless and frictional) contact tests exemplifying our methodology are given. It is shown in particular that our numerical solutions do not exhibit pathologies such as spurious oscillations of the numerical mechanical fields (velocities and contact loads). Finally a promising application of the multiscale Arlequin method [START_REF] Dhia | Multiscale mechanical problems: The Arlequin method[END_REF] to the (multiscale) impact problems is shown.

A DYNAMIC FRICTIONAL CONTACT MODEL

We consider the problem of dynamic frictional contact between two deformable solids contained in IR d (d=2 or 3). The Signorini-Moreau and Coulomb conditions are written as equations via the use of unknown Level-Sets like fields defined on known potential contact zones.

Signorini-Moreau contact model

The classical Signorini contact laws read:

d n ≤ 0, λ ≤ 0 and λ d n = 0, on I × Γ c ( 1
)
where c Γ , I, d n and λ denote the known potential "slave" contact surface, the time interval, the so-called signed distance and the normal contact pression, respectively.

The viability lemma of J.J. Moreau [START_REF] Moreau | Nonsmooth Mechanics and application[END_REF] asserts that if the Signorini laws are satisfied at a given time in I, then they are satisfied at all future times as far as the following conditions are fulfilled:

λ=0 if d n < 0 (2.a) [[v n ]] ≤ 0, λ ≤ 0, λ [[v n ]] = 0 otherwise (2.b) with [[v n ]
] standing for the normal velocity jump field on the contact surface (in the classical contact mechanics sense (see e.g. [START_REF] Klarbring | Large displacement frictional contact: a continuum framework for finite element discretization[END_REF])).

By using two Sign-like functions (as introduced in [START_REF] Dhia | Modelling and solution by penalty duality method of unilateral contact problems[END_REF] for a penalized unilateral contact model), what could be labelled the Signorini-Moreau contact laws (2) are transformed to the following equalities:

( ) ]] [[ n n v u v h S S - = λ λ , on I × Γ c (3.a) ( ) n u d S - = - IR 1 , on I × Γ c (3.b) ( ) ]] [[ 1 IR n n v v S ρ λ - = - , on I × Γ c (3.c) with 0 ≠ n h , 0 > n ρ and K
1 denoting the characteristic function of the set K. We notice that S c =S u S v is a Level-Set field whose iso-1 values characterize the effective contact zone. This unknown field takes into account both of the relative positions of the contact surfaces and their relative velocities.

Coulomb friction laws

Similarly, the classical Coulomb's laws can be equivalently written as the following equations [START_REF] Dhia | Problèmes de contact frottant en grandes transformations: du continu au discret[END_REF]:

Λ R µλ τ =
, on

I × Γ c (4.a) with, ( ) 0 ]] [[ ) 1 , 0 ( = + - τ v Λ Λ τ h P B on I × Γ c if S u = 1 (4.b) 0 = Λ on I × Γ c otherwise (4.c)
where µ is the friction coefficient, τ h is a non zero real parameter, [[v τ ]] is the relative tangential velocity and P B(0,1) is the orthogonal projection on the unit ball, with respect to the Euclidian scalar product of IR d . Now, by introducing the following Level-Set field S f whose iso-1 values characterize the sticking zone:

( ) ]] [[ 1 ) 1 , 0 ( τ v Λ τ ρ + = B f S
, on

I × Γ c ( 0 > τ ρ ) (5)
one can express the Coulomb's laws as follows:

Λ R µλ τ =
, on

I × Γ c (6.a) with: ( ) ( ) 0 1 1 = - +                 - + - Λ G G G Λ τ τ τ u f f u S S S S
, on

I × Γ c (6.b) where ]] [[ τ τ v Λ G τ h + =
The new local settings of Signorini-Moreau contact conditions and Coulomb laws are interesting from a numerical point of view since they lead themselves to rather "natural" weak formulations, well-suited to numerical approximations.

WEAK-STRONG FORMULATION OF THE DYNAMIC FRICTIONAL CONTACT PROBLEM

By using the Virtual Work Principle (VWP) and the weak formulations of equations (3.a) and (6.b), keeping equations (3.b), (3.c) and ( 5) as local strong ones, the following (seemingly new) weak-strong mixed formulation of the dynamic frictional contact problem is derived: assuming that the displacement and velocity fields u i and v i (i=1,2) are known at a given instant t 0 ∈I, then for all t >t 0 , t ∈I, the problem to be solved is: (without explicit reference to time for clarity of notations)

Find ( ) ( ) * * , , ; , , ; , , , Λ w Λ u v λ λ i f v u i i S S S ∀ ( ) ( ) 0 ]] .[[ ]] [[ ]] [[ 1 ]] [[ ) ( : ) ( . 2 1 2 1 0 0 0 =         + + - + - - ∇ + ∫ ∫ ∑ ∫ ∑ ∫ Γ Γ = Ω = Ω τ τ τ w v Λ v Λ Λ w v Π w v c c i i h h S S S w S S f f u n v u i T i i i i i i i τ τ µλ λ ρ & (7.a) ( ) [ ] 0 ]] [[ 1 * = - - -∫ Γ c n n v u n v h S S h λ λ λ (7.b) ( ) ( ) ( ) 0 1 ]] [[ ]] [[ 1 ]] [[ 1 * * = - +                 + + - + + - ∫ ∫ Γ Γ c c u f f u S h h S h S S h ΛΛ Λ v Λ v Λ v Λ Λ τ τ τ τ τ τ τ µλ (7.c) ∫ + = t t i i i ds s t t 0 0 ) ( ) ( ) ( v u u in i 0 Ω (7.d) ( ) 0 1 IR = - -- n u d S , ( ) 0 ]] [[ 1 IR = - -- n n v v S ρ λ and ( ) 0 ]] [[ 1 ) 1 , 0 ( = + - τ v Λ τ ρ B f S on I × Γ c (7.e)
where i 0 ρ denotes the mass density in the reference state, an over-dot refers to partial time derivative, Π i is the first Piola-Kirchhoff stress tensor defined in i 0 Ω (given by a behaviour law) and where the initial conditions are known.

The above formulation may recall available Lagrange ones of frictional contact problems [START_REF] Wrigger | Computational Contact Mechanics[END_REF]. It is however an new one (to our best knowledge) since it is a non augmented and non constrained Lagrange formulation of dynamic contact problems, based on equalities.

NUMERICAL SOLUTION STRATEGY

To be solved, the problem ( 7) is discretized by means of three numerical tools: i) A first-order finite time difference θ-scheme is used to approximate partial time derivatives. ii)

A Galerkin space discretisation method of the finite element type is used to approximate the continuous fields v i , (and u i ), λ and Λ while respecting the compatibility condition between approximating spaces (e.g. [START_REF] Dhia | Hybrid frictional contact particles-in elements[END_REF], in the quasistatic case). iii)

The Level-Sets like fields S u , S v and S f are approximated by means of a collocation method which consists in evaluating these fields only in a collection of selected points of the potential contact zone, corresponding to the contact numerical integration points.

The nonlinear systems are solved, at each time step, by fixed point strategies coupled with a Newton-Raphson method. The global algorithm is described by the following steps.

In this algorithm, k is an index referring to the time step and g indicates that the unknown fields are frozen (actually they are updated iteratively). The algorithm is stopped whenever all the loops reach convergence.

Remark:

The time discretization scheme used here is of implicit type in order to approximate friction and elastoplastic phenomena with high accuracy. The reader is referred to [START_REF] Hallquist | Sliding interfaces with contact-impact in large-scale Lagrangian computations[END_REF][START_REF] Belytschko | Contact-impact by the pinball algorithm with penalty and lagrangian methods[END_REF], for explicit-like approaches.

NUMERICAL RESULTS

To show the effectiveness of our global approach, we consider two frictionless and one frictional contact-impact examples.

Impact of two identical rods

We consider the classical test of impact of two elastic prismatic rods moving with equal speed in opposite directions (Fig. 2). The rods are initially undeformed and the problem is symmetric about the points at which the two rods experience impact. We mesh the two rods similarly with Hexa 3D-elements. The contact loads are approximated by a bilinear finite element space, defined at the contact interface. Two 3D-finite element solutions are plotted in figure 3, one with a dissipative Newmark scheme used for the time discretization of a classical displacement-based formulation and the other with the use of our continuous velocity-based formulation.

λ = λ g (in R τ )

A fixed point Level-Sets values loop

(S u , S v , S f )= (S u , S v , S f ) g Fig. 1 Global algorithm if converged GOTO 2.
if converged GOTO 3.

if converged GOTO 4. if converged GOTO 1.

A fixed point friction threshold loop 2. A fixed point geometry loop

Impact of a cylinder on a wall

The second example concerns the impact of an elastic cylinder on a quasi-rigid wall under plane strain conditions. The geometric and the material properties are given in figure 4. The velocity and contact fields are approximated by bilinear finite 2D and 1D elements and the nodes of the potential slave surface (belonging to the cylinder) have been taken as the collocation points for the Level-Sets fields. In figure 5, we show the time history of the displacement, velocity and contact pression of the bottom contacting point of the cylinder obtained by both of a dissipative Newmark scheme (β=0.4, γ=0.7) (for a displacement-based formulation) and the proposed method.

Fig. 3 Impact of two identical rods; histories of tip displacement, velocity and contact multiplier

Cylinder : E = 2 10 11 Pa , ν = 0, ρ = 7800kg.m -3 , R=30 mm. Initial gap= 20mm, V 0 =500m.s -1 , ∆t = 10 -6 s, θ=1.

Fig. 4 Impact of an elastic cylinder on a wall V 0

Elastoplastic impact of a plane rod

This example involves the frictional elastoplastic impact of a plane rod (E = 1.17 10 5 MPa ,ν = 0.35, ρ = 8930kg.m -3 , σ Y = 400MPa , E t = 100MPa) against a quasi-rigid wall. The rod has an initial velocity of 227m.s -1 . The friction coefficient µ is equal to 0.25. Bilinear 2D and 1D finite elements are used to approximate unknown fields. Plane strain hypotheses are used. This test is treated by two approaches. The first one is based on the proposed method within the classical monomodel framework and the second is carried out within the multi-model Arlequin framework.

the monomodel framework

We present in figure 6 both of computed deformed geometries and the equivalent plastic strain spread in the rod obtained by the application of the velocity based formulation in the classical modelling framework. The time step is equal to 2 10 -7 s. The histories of the displacement, velocity, contact pression and Lagrange semi-multiplier of friction at selected points of the impacting face of the rod are depicted in Figure 7. Notice that there are no numerical oscillations on the unknown fields such as velocities and contact pression. The values of the Lagrange semi-multiplier of friction indicate the states of the contacting points: ||Λ||∈]0,1[ whenever a contacting point is sticking and ||Λ||=1 when it slides.

the Arlequin framework

The previous mechanical test show that impact involves sharp localized variation of mechanical fields both in time and space. With coarse meshes and times steps, the numerical results can suffer precision. Adaptive methods have been designed to achieve the computations with a prescribed tolerance for the errors [START_REF] Lee | A posteriori error estimation of h-p finite element approximations of frictional contact problems[END_REF]. For a flexibility purpose we use here the multiscale approach named Arlequin [START_REF] Dhia | Multiscale mechanical problems: The Arlequin method[END_REF]. This method of global-local type consists in superimposing a local refined model to the global coarse one. The coexistence of the two different models allows to use. i) different formulations, ii) different time integration schemes, iii) different refinements in space and time.

In this framework, the impact problem can be written as given in appendix 1. The gluing forces field is denoted by F.

Fig. 2 1 . Loop over all time steps 5 .

 215 Fig. 2 Impact of two 3D elastic and identical rods

• p :

 p paired point to a potential contacting point p • (n,τ): local frame at p (p,n,τ)= (p,n,τ) g

Fig. 5 Fig. 6

 56 Fig. 5 Displacement, velocity and contact pression histories of the bottom contacting point of the cylinder

Fig. 8

 8 Fig. 8 Impact of a rod by the velocity and Level-Sets based formulation in (a) the monomodel framework (b) the Arlequin framework

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Acknowledgements

The support of Électricité de France (DRD Clamart) is gratefully acknowledged.

CONCLUSION

A velocity and Level-Sets based continuous Lagrange weak-strong formulation of dynamic frictional has been developed in this paper. The continuous formulation is derived from an equivalent setting of the Signorini-Moreau conditions and the Coulomb friction laws by using unknown Level-Sets fields. The problem is discretized by means of time, space and collocation method and solved by fixed-point strategies mixed with the Newton-Raphson method. Numerical examples show the effectiveness of our approach particularly for the treatment of spurious numerical oscillations. First promising results using the multiscale Arlequin framework are given. A mixing of time integration schemes (explicit/implicit and/or dissipative/conservative) in this framework is now in progress.

APPENDIX 1

In this appendix, we show how the Arlequin method can be applied to dynamic contact problems. For this, we consider the following domain decomposition of the elastoplastic frictionless impact of a plane rod on a quasi-rigid wall (paragraph 3. Find (v 1 ,u 1 ,u 2 ,λ,F ; S u ,S v ) such for all (v *1 ,w 1 ,w 2 ,λ * ,F * ):

where (.,.) is the duality bracket between the space of displacements defined in 1 s Ω and its dual space.