solution of 2d and 3d contact problems by means of lagrange

Solution of 2D and 3D contact problems by

means of Lagrange multipliers in the castem

2000 finite element program

Th. Charras, A. Millard, P. Verpeaux

CE /DMr/IAMS - CE7V 5Wa%/, Bf n*

Gif-sur-Yvette Cedex, France

1. INTRODUCTION

There are many ways to treat contact problems, withinthe frame of finite element programs. A firstclassification can be made between explicit andimplicit methods. In the case of explicit methods,one tries to determine the precise instant of contactbetween solids, and from it calculation goes on withnew boundary conditions. In the case of implicitmethods, contact generally occurs during a time stepand therefore boundary conditions have to be changedmeanwhile.

The second possible classification is associated withthe choice of unknowns, i.e. displacement variablesin the case of penalty methods like gap elements ordual variables in the case of duality methods likeLagrange multipliers.

These methods have been described in reference [1],within the frame of the implicit finite element codeCASTEM.

In our new generation finite element code CASTEM2000, we have chosen the latter technic. The aim ofthis paper is to present how the method can be usedfor the treatment of contact problems, involving ornot friction, and to show its capabilities on areference problem, which has been solved in 2D and 3Dconfigurations.

2. MAIN FEATURES OF CASTEM 2000

For the sake of completeness, we will start with abrief presentation of this new generation finiteelement program [2] [3]. Contrary to usual finite

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184 Contact Mechanics

element packages, which appear to the user more orless like "black boxes", CASTEMM 2000 has an openstructure.

CASTEM 2000 is built with totaly new concepts. As acomparison, it is possible to say that CASTEM 2000 isfor finite elements what Fortran is for numericalalgebra : on one hand, Fortran manipulates integers,real or complex variables, with basic algebraicoperators or mathematical operators like sinus,exponential, etc ..., in order to generate thedesired sequence of instructions. Fortran also offersa few language facilities like do loops, ifstatement, subroutines, etc ... On the other hand,CASTEM 2000 manipulates finite element objects likemeshes, stiffness matrices, nodal fields, elementfields, etc ..., as well as usual Fortran variables(integers, reals, etc ...) by means of more than 400basic operators like mesh generation of a line,computation of a stiffness matrix, computation of anelement stress field, etc ...

As well as Fortran, CASTEM 2000 does have a languagefacility called GIBIANE, which allows the user toorganize the sequence of instructions, with do loops,if statements and procedures, which are in facttreated as calculation operators. Within CASTEM 2000,there is no hierarchy between the various operators,which offers an extreme liberty to the user.

Therfore, thanks to this very important modularity,CASTEM 2000 can be used to solve many problems inmany disciplines (mechanics, thermics,magnetostatics, etc ...) leading to partialdifferential equations system, and it offers to theuser the possibility to program himself newapplications, simply by using the user's languageGIBIANE, without any need of compiling subroutines.

3. USE OF LAGRANGE MULTIPLIERS IN CASTEM 2000

The Lagrange multipliers technic has been firstintroduced in CASTEM 2000 for the treatment ofvarious Dirichlet boundary conditions like :

prescribed degree of freedom,

prescribed linear combination of degrees offreedom.


Contact Mechanics 185

Such conditions can be written in the general form :

A.U = b (1)

Where U denotes the vector of unknown degrees offreedom. A is a rectangular matrix, the number oflines of which being the number of conditions, and bis the vector of the corresponding prescribed values.

The above relations are enforced by means of Lagrangemultipliers, A, which have the physical meaning ofreaction forces if U are displacement unknowns, andare solution of the following problem :

Min Max -{- U*KU - U^F + A< (AU - b)U A ^

Where the first two terms correspond to the usualform of the total potential energy, in case of ausual displacement finite element approximation of amechanical problem.

the whole set of unknowns X* = (U* , A*) is found bysolution of the following algebraic system :

(2)

Whereas the K matrix is in general singular, theabove matrix is invertible provided sufficientDirichlet conditions have been prescribed.

In practice, the above system is solved using theGrout factorization algorithm. Care must be taken onone hand for the degrees of freedom numbering, and onthe other hand for the numerical conditionningarising from the differences between K et A matrices.

4. TREATMENT OF CONTACT IN CASTEM 2000

For sake of simplicity, we will restrict ourpresentation to the case of small displacements andquasi-static evolutions.

However, the method is general and can be extendedwithout major difficulty to other cases.



4.1. Problem statement

Let us consider two deformable solids, S^ and S^ ,which can interact on a part of their boundary, notedr .

FIGURE 1

These solids are subjected to various loadingconditions (body forces, point loads, etc ...) and toprescribed conditions on the boundaries complementaryto F^ . Both solids are modelized by finite elements.

For sake of simplicity, we will suppose that thenodes of the two solids, belonging to F^_ , arecorresponding two by two. If it is not the case, somecollision algorithm must be used to determine thenodes which will be in relation.

We will denote by t^, t^ two base vectors of thecommon tangent plane to S^ and 82 at a given node,and n the normal vector, oriented from S. to S^ .

FIGURE 2



The contact conditions at point P can be written as :

UJ2) _ u^l) ^ Q (3)

N < 0

N.(U(2) - U(D) = 0

Where U^ is the displacement of point P measured

along n.N represents the normal contact force appliedby Sg on S., at point P.

In CASTEM 2000, we associate to each of theinequalities (3) a Lagrange multiplier, which can bephysically interpreted as the normal contact force N.

The non linearity arising from the inequalities mustbe solved by an iterative technic. We use for thispurpose a fixed point algorithm.

We first begin by condensing the solids S,, and S^ onthe nodes belonging to T^, in order to limit thecomputational effort. Then the algorithm is asfollows :

initialization of the solution by supposing thatall contacts are active, which corresponds to thesatisfaction of equalities :

U^(2) . v(l) = o (4)

instead of inequalities,

for a given iteration, there is a subset of activecontacts, corresponding to the estimated solution.We then calculate the corresponding normal contactforce N-. If N- < 0, the contact is maintainedwhile if N- > 0, the contact is suppressed and anew iteration is performed,

for all non active contacts, we verify that thecontact is not yet achieved,

convergence is achieved when the same subsets ofactive contacts are obtained for two successiveiterations.

Then displacements can be calculated in the wholesolids S,, and S^ by usual back substitution.



In case of material non linearities, the contactiterations are performed for each of the iterationsrequired for the solution of equilibrium equations.

5. TREATMENT OF CONTACT WITH FRICTION

In the case of contact with friction between S-, andSg, some friction conditions must be added to theabove mentioned contact conditions (3) :

ITI < s

ITI < s => Uj. = 6

ITI = s =>

Where T stands for the tangent contact force atpoint P, applied by S^ on S-, .

s is a friction threshold.Uj. is the component of the displacement in

the tangent plane.a is a proportionality coefficient.

The usual Coulomb friction law corresponds to thefollowing dependance for s :

s = ,x INI

Where \.i is the friction coefficient.

As for the contact condition (4), Lagrangemultipliers are associated with the conditions (5) onthe tangential displacements.

The same solution algorithm is used, starting from aninitial solution for which all normal and tangentialconditions are supposed to be active.

Then, for a given iteration, the examination of thecontact forces leads to the following cases :

- the normal forces are first considered. If anormal constraint must be suppressed, then theassociated tangential constraints are alsosuppressed. If not, the threshold value isupdated,

then tests are performed on the tangentialforces : two subcases can be distinguished :a. case where the tangential constraints were

active at the previous iteration. Then, if



ITI < s, they are maintained. If not, they arereleased and tangential forces are created atpoint P,

b. case where the tangential constraints wereunactive at the previous iteration : then, ifITI < s, they are reactivated. If not, wecreate the tangential forces at point P onlyif the previous tangential forces had adirection opposite to the displacement one,otherwise we reactivate the tangentialconstraint.

Convergence is achieved when the state of activeconstraints is identical between two successiveiterations and when the relative variation offriction forces is less than a given tolerance.

6. EXAMPLE OF APPLICATION

We have analysed a test problem proposed by thefrench group of laboratoires "Greco - Mise en forme"[1]. The data are briefly reminded here.

A parallelipipedic piece of steel is lying on aninfinitely rigid plane and is loaded simultaneouslyby a vertical and a lateral pressure.

i i i i i i

Pz

FIGURE 3

The friction follows the Mohr-Coulomb law. The valuesadopted in the calculations are :

P* = 15

Pz = 5

|i = 1

For these values, the solution shows on T^ threedifferent zones :

a first zone without contact between the solid andthe rigid boundary,



a second zone where sliding occurs with friction,

a third zone without sliding, at the center of F^ .

In the calculation, only half of the structure ismodelized, for symetry reasons.

Using the present algorithm, we have found anexcellent agreement with this solution, as can beseen on figure 4 where the deformed body is displayedtogether with the contact forces on F^ . Figures 5 and6 show the plots of iso stresses a^ and o .

The same calculation has been performed in 3D, with acoarser mesh. Figures 7 and 8 show again the plots ofiso stresses o^ % and cr^ , which compare well with the2D prediction.

7. CONCLUSION

We have presented in this paper a simple and robustalgorithm for the treatment of 2D and 3D contactproblems, using Lagrange multipliers. It has beenimplemented in our new generation finite elementprogram CASTEM 2000. It has also been extended totreat contact problems with friction, and theperformance of the method has been illustrated by thecomputation in 2D and 3D of a test problem.

REFERENCES

[1] A. COMBESCURE - A. MILLARD - P. VERPEAUX."Numerical methods in the CASTEM system for thetreatment of contact problems involvingfriction". Journal of theoretical and appliedmechanics, supplement n° 1 to Vol. 7, 1988.

[2] P. VERPEAUX - A. MILLARD - Th. CHARRAS. "CASTEM2000, une approche moderne du calcul desstructures" in "Calcul des structures etintelligence artificielle". Ed. PLURALIS, 1989.

[3] P. VERPEAUX - A. MILLARD - Th. CHARRAS -A. COMBESCURE. "A modern approach of largecomputer codes for structural analysis".Proceedings of SMIRT 10 conference, LOS ANGELES,1989, Ed. HADJIAN, 1989.

[4] L. EBERSOLT - A. COMBESCURE - A. MILLARD -P. VERPEAUX. "Non-linear algorithms solved withthe help of the GIBIANE macro language".Proceeding of SMIRT 9 conference, LAUSANNE,1987, Ed. BALKEMA, 1987.



Figure 4. Deformed shape reaction forces.

ISO VALUES

-16.i -14.i -12.gll -9.40 -7.3i -5.1H -3.0

Figure 5. ISO stresses


-OO

(N


16.- U.- 11.- 9.1-6.9- 4.7- 2.5

no

£o

2

§•

Figure 7. ISO stresses o


I


solution of 2d and 3d contact problems by means of lagrange

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