Multigrid method applied to mixed formulation for ... · Multigrid method applied to mixed formulation for frictional contact problems P. Alart, F. Lebon Laboratoire de Mecanique

Multigrid method applied to mixed

formulation for frictional contact problems

P. Alart, F. Lebon

Laboratoire de Mecanique et Genie Civil,

Universite Montpellier II, URA 1214, CNRS, pi E.

Bataillon, cc 048, Montpellier 34095 cedex 05, France

ABSTRACT

Many structural analysis problems are concerned with friction contactphenomena. These problems are difficult to formulate and even more to solvebecause they are governed by multivalued tribological laws and some numericalresolutions can lead to unsymmetric operators. This last disadvantage becomescrucial for very large problems involving three dimensional discretization andtime evolution. This paper shows how to use a simple mixed formulationtogether with an efficient multigrid technic when dealing with frictional contactproblem. The augmented Lagrangian approach yields on non linear and nondifferentiable system the unknowns of which are node displacements andLagrangian multipliers identified with contact forces. So, for multigridframework, special restriction operators (from fine to coarse mesh) andinterpolation operators (from coarse to fine mesh) are defined. The tangentmatrix of the system being non symmetric, iterative methods of solving are usedwhen dealing with the fine mesh (generalized conjugate gradient acceleration)and direct methods when dealing with the coarse mesh. Test problem ispresented, feasibility is shown and efficiency is discussed.

1. AUGMENTED LAGRANGIAN AND NEWTON METHOD

1.1 Mixed contact operator by augmented Lagrangian approach

For convenience, we shall restrict our attention to the bidimensional contactfriction problem between a deform able elastic body and a rigid flat obstacle. Forour purpose, this assumption is not too hard; the reader is referred to [1,2] for

Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533

228 Contact Mechanics

different extensions of this approach to frictional contact problem between twobodies or to large sliding on strongly curved obstacles.

We consider a single particle a (or node) of the (discretized) deformable body in

the neighbourhood of the obstacle boundary. We note X its initial position, u itsdisplacement, x=X+u its current position.

Using the subdifferential of the indicator function of the positive half line(formalism of Convex Analysis [2]), the unilateral contact law can be written asa multivalued function between the signed normal contact distance d̂ (u) and the

normal component of the contact force X% (la). In the same way, the tangential

slip increment 8% is related to the tangential contact force X; by an inclusion (Ib)

which summarizes the friction criterium and the slip rule. The graphs of theserelationships are shown in figure 1.

Fig. 1 Unilateral contact and friction laws.

a) Xn 6 9Y( dn ) b) St e W( h )

The friction coefficient ji characterizes an iso tropic friction law by defining the

section of the Coulomb's cone C(̂ ) :

C(Xn)=(Xt: Kl<-^n} (̂ 0) (2)

Following a previous paper based on an augmented Lagrangian approach [1],the equilibrium of a discretized elastic body in frictional contact with an obstaclecan be given by the system of equations :


Contact Mechanics 229

Fint(u) - Fext notes the internal forces minus the external ones. F(u,X) defines a

continuous frictional contact operator which may be written for a single node asfollows :

R-

By extension, we coin the term 'augmented' multipliers to specify a^ and a :

(?n = ln + r dn(u) , a = A, + r 8 (u) (5)

For flat contact (fixed local frame) and isotropic friction, this operator isconewise linear in 2D and raywise linear in 3D discretisation, relevant notions tostudy uniqueness conditions [4].

1.2 Generalized Newton method.

The algorithms usually applied to augmented Lagrangian problems inoptimisation (e.g. Uzawa's scheme) are stable but very slow because they arebased on an alternate treatment of the primal and dual variables. We prefer to usea simultaneous treatment of both variables by Newton's method. Firstly it isuseful to resolve the system of equations (3) in two parts depending on the two

variables u and X, a differentiate part G and a non differentiate one 7,

(6)

Fint(u) -where G(x) = G(u,A,) =

A rigorous and straightforward extension of Newton method to non-differentiable but continuous equations such as (6) consists in

xi+1 = xi - ( Ki + Ji )-l ( G(xi) + f(xi) ), K; = 3G(xi) , J; <= 9f(xO (7)

where 9f(x*) is the generalized Jacobian of 7 at x* [5]. Each region of

differentiability (linearity in 2D discretisation) of the operator is identified to aglobal contact status composed of the local contact status of each contact node.This notion is illustrated in the next section. For a given local status (gap, stick,forward slip or backward slip), the generalized Jacobian is reduced to the singleclassical Jacobian matrix.



1.3 A simple example.

To illustrate the non symmetry of the tangent matrix, we present a simpleexample (see fig.2) which involves an elastic body discretized in two finiteelements. The contact with friction may be treated by introducing two specialcontact elements (by anology with finite elements for the solid). In a symbolicmanner, these elements are represented by two triangles containing four nodes :one belongs to deform able body, two draw the obstacle boundary in theneighbourhood of the first node, the last node with an arbitrary position in thecenter of the triangle contains the contact forces as its degrees of freedom.

Fig.2 Elementary discretized example

The table 3 shows the system to solve for the following global status : backwardslip for the first contact element and forward slip for the second element.

kn Ann

An 21

- 1/r + ja/r

-1/r

Table 3 Linear system Ax = b in the case (slip-, slip+)



1.4 Conjugate Gradient Squared method (CGS)

The linear system involved in (7) can be solved using a conjugate gradientsquared method [6,8]. The aim of the generalized conjugate gradient method fornon symmetric problems is to associate a function to minimizeJ(res) = (Hres,res), where His a symmetric positive definite matrix and res isthe residium b - Ax. In the case of the CGS method,

H =0

0 (8)

The iterative process is written : x%+l = CGS(x̂ ).

2. MULTIGRID METHOD

The coupling of the mixed formulation and multigrid method [71 has to beintroduced at two levels, at first in order to improve the convergence of theNewton method, afterwards to accelerate the convergence of the CGS method(fig. 4).

CDECD

Initialisation

Coarse grid solution

Interpolation on fine grid

Newton loop on fine grid

Tangent matrix computation

CD

E 18 §/5c ®

Solver

Multigrid process with coarse grid correction

or

CGS with multigrid preconditionner

Fig. 4 MULTIgrid LAGrangian algorithm (MULTILAG).



For using multigrid method, we need to define two grids, one of level h andanother of level H. Let pj^ an operator from the fine grid to the coarse grid

(restriction) and p^j an operator from the coarse grid to the fine one

(interpolation). Different kinds of restriction and interpolation for classical PIelements and for contact elements (a new kind of operators) are presented onfigures 5, 6 and 7.

• coarse node gg fine node

Fig. 5 Interpolation and restriction for elastic finite elements.

\*^ -i/o -\/O \*^

Fig. 6 Restriction for contact elements (four nodes).

1

1/2 (7

1/2 1/2

Fig. 7 Interpolation for contact elements (four nodes).



The multigrid algorithm is written (v the number of smoothing iterations is equalto 2 or 3):

(9)

3. NUMERICAL RESULTS

Fig. 8 The Inland test (coarse mesh)

We test the previous approach on a classical elastostatic problem (Inland test,see figure 8), which has the advantage of being very elementary and that ofgiving complicated contact solutions according to the loadings and the frictioncoefficient values. So, the distribution of contact status area (gap, stick andslip) is unknown a priori.

This application involves a long bar (32 contact nodes on coarse mesh and 64contact nodes on fine mesh) pressed on a plane. We present results with thefriction coefficient equal to 0.2.

The solution of the problem on the coarse grid gives a good approximateddistribution (first improvement level). However, displacements and forces(multipliers) are interpolated (see section 2) and the contact status is evaluatedautomatically with respect to these new values for all contact elements (fine andcoarse).



automatically with respect to these new values for all contact elements (fine andcoarse).

The non symmetric multigrid solver leads to improve the COS method. Theseresults confirm the necessity to couple the CGS method with a preconditionner[8] (second improvement level).

REFERENCES

1. Alart, P. and Curnier, A. 'A mixed formulation for frictional contactproblems prone to Newton like solution methods', Comp. Met. in AppL Mec. &Eng., Vol.92, no 3, 1991.

2. Moreau, J.J. 'Application of convex analysis to some problems of dryfriction', Trends of Pure Mathematics Applied to Mechanics, Zorski ed., 1979.

3. Alart, P. 'A simple contact algorithm applied to large sliding and anisotropicfriction', in CMIS (Ed. Curnier, A.), pp 321-336, Proceeding of the ContactMechanics International Symposium, Lausanne, Switzerland, 1992. PressePolytechniques et Universitaires Romandes.

4. Alart, P. 'Criteres d'injectivite et de surjectivite pour certaines applications deRn dans lui-meme : application a la mecanique du contact' RAIRO, ModelisationMathematique et Analyse Numeriqiie, Vol 27, n°2, pp ()()()-()()(), 1993.

5. Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley Ed., New York,1983.

6. Joly, P., Mise en oeuvre de la methode des elements finis, Ellipse Ed., Paris,1990.

7. Lebon, F., Raous, M., Latil, J.C. and Grego, L.'Multigrid method in nonlinear structure mechanics', Proceedings of European Conference of newadvances in structural analysis, Giens, France, 1991.

8. Sonnenveld, P., Wesseling, P.and de Zeeuw P.M. 'Multigrid and conjugategradient methods as convergence acceleration techniques', Short course onmultignd method, Bristol, 1983.


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Multigrid method applied to mixed formulation for ... · Multigrid method applied to mixed formulation for frictional contact problems P. Alart, F. Lebon Laboratoire de Mecanique