10
COMMUNICATIONS 1N NUMERICAL METHODS IN ENGINEERING, VOl. 9, 815-824 (1993) APPLICATION OF AUGMENTED LAGRANGIAN TECHNIQUES FOR NON-LINEAR CONSTITUTIVE LAWS IN CONTACT INTERFACES P. WRIGGERS Insritut fur Mechanik, Technische Hochschule Darmstadt, Hochschulstr. I 0-6100 Darmstadt, Germany AND G. ZAVARISE Istituto di Scienza e Tecnica delle Costruzioni, Via Marzolo 9, 35131 Padova, Italy SUMMARY The use of micromechanically based constitutive equations for contact interfaces leads to technically relevant parameters to ill-conditioned finite-element equations. In the paper an augmented Lagrangian technique is employed to overcome this difficulty and to provide a good converging algorithm. INTRODUCTION Contact models which are based on micromechanical considerations within the interface often have very stiff characteristics. The contact stiffness can be several orders of magnitude higher than the stiffness of the surrounding elements. ' Thus this type of interface law can lead to ill- conditioning, which is also a well known problem in the context of penalty methods.' However, from the physical point of view these interface compliances are needed in case more sophisticated frictional laws have to be considered or thermomechanical contact behaviour has to be investigated. One way to overcome the problem of ill-conditioning is the use of very high-precision arithmetic throughout the computation.' This approach is motivated by the fact that there exist estimations for the magnitude of the penalty parameter c.~ These estimates lead to a choice of the following penalty parameter: where k is a characteristic stiffness parameter of the adjoint elements (e.g. the modulus of compression), N i s the total number of unknowns and t denotes the computer precision. Thus the penalty parameter is directly limited by the latter quantity. Since, however, the precision which has to be used for technical relevant interface laws is higher than the precision normally used for finite-element computations, this approach is not very advantageous. In this paper we employ a technique well known in optimization theory based on an augmentation of the Lagrangian multipliers. This technique has been considered within the context of incompressibility constraints, ' and has also been applied to contact problems for 0748-8025/93/ 1008 15- 10$10.00 0 1993 by John Wiley & Sons, Ltd. Received 8 July 1992 Revised 18 February 1993

Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

Embed Size (px)

Citation preview

Page 1: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

COMMUNICATIONS 1N NUMERICAL METHODS IN ENGINEERING, VOl. 9, 815-824 (1993)

APPLICATION OF AUGMENTED LAGRANGIAN TECHNIQUES FOR NON-LINEAR CONSTITUTIVE LAWS

IN CONTACT INTERFACES

P. WRIGGERS Insritut fur Mechanik, Technische Hochschule Darmstadt, Hochschulstr. I 0-6100 Darmstadt, Germany

AND

G. ZAVARISE Istituto di Scienza e Tecnica delle Costruzioni, Via Marzolo 9, 35131 Padova, Italy

SUMMARY The use of micromechanically based constitutive equations for contact interfaces leads to technically relevant parameters to ill-conditioned finite-element equations. In the paper an augmented Lagrangian technique is employed to overcome this difficulty and to provide a good converging algorithm.

INTRODUCTION

Contact models which are based on micromechanical considerations within the interface often have very stiff characteristics. The contact stiffness can be several orders of magnitude higher than the stiffness of the surrounding elements. ' Thus this type of interface law can lead to ill- conditioning, which is also a well known problem in the context of penalty methods.' However, from the physical point of view these interface compliances are needed in case more sophisticated frictional laws have to be considered or thermomechanical contact behaviour has to be investigated.

One way to overcome the problem of ill-conditioning is the use of very high-precision arithmetic throughout the computation.' This approach is motivated by the fact that there exist estimations for the magnitude of the penalty parameter c . ~ These estimates lead to a choice of the following penalty parameter:

where k is a characteristic stiffness parameter of the adjoint elements (e.g. the modulus of compression), N i s the total number of unknowns and t denotes the computer precision. Thus the penalty parameter is directly limited by the latter quantity. Since, however, the precision which has to be used for technical relevant interface laws is higher than the precision normally used for finite-element computations, this approach is not very advantageous.

In this paper we employ a technique well known in optimization theory based on an augmentation of the Lagrangian multipliers. This technique has been considered within the context of incompressibility constraints, ' and has also been applied to contact problems for

0748-8025/93/ 1008 15- 10$10.00 0 1993 by John Wiley & Sons, Ltd.

Received 8 July 1992 Revised 18 February 1993

Page 2: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

816 P. WRIGGERS AND 0. ZAVARISE

Figure 1 . Physical approach in rc

Figure 2. Geometrical approach in rF

frictionless contact. 6*7 Recently this approach has also been extended successfully to large displacement contact problems including friction. *,* Yet for micromechanical interface laws this technique has not been applied and needs some additional considerations regarding the formulation of the constraint equation and the algorithmic treatment.

2. STATEMENT OF THE PROBLEM

Let II"(u") denote an elastic potential which is associated with a body B" depending on the displacement field u". We assume that the two bodies (a = 1,2) are coming into contact within the area rC. Within a micromechanically based contact law a non-linear relationship exists between the normal contact pressure p and the current mean plane distance d.

Let us assume the following general form of the constitutive law:

p = f ( d ) or d = h ( p ) (2) where f and h are non-linear functions. Since the explicit form of these relationships does not affect the algorithmic treatment we do not consider a specific relation here. A micromechanical interface law based on the considerations in Zavarise' is applied within the examples (see Section 5).

Now with the notation of Figure 1 we define the physical approach of the two surfaces in contact by

g N = [ - d (3) where 5 is the initial mean planes distance and g N denotes the approach of the surfaces in the contact area rc. Now we define the geometrical approach

where up are the displacements of the two bodies B" within the contact area and n represents

Page 3: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

AUGMENTED LAGRANGIAN TECHNIQUES 817

the normal to the surface rc. gN+ which is only non-zero for the case of a geometrically penetration is shown in Figure 2.

Thus the mathematical problem is defined by the minimization of a functional describing the bodies plus the constraint condition that the physical approach has to be equal to the geometrical one is expressed by

2

HA(U ") = C n"(uu) - MIN ( 5 )

subject to c+ (u",p) = gN+ (uu) - [t - d b ) ] = 0

With these preliminary definitions associated with the contact geometry and the contact interface law we are able to formulate the augmented Lagrangian functional which is used throughout the algorithmic treatment:

u=l

(6)

The sum on the right-hand side denotes the total energy of the two bodies coming into contact. The second term incorporates the constraint condition ( 5 ) via the Lagrangian multiplier method. As is well known, the Lagrangian multiplier p is related to the contact pressure. Finally, the third term is associated with the standard penalty contribution. In augmented Lagrangian techniques the quantity associated with the Lagrangian multiplier is held constant during the minimization of n ~ . Owing to equation (2) this functional is non-linear in p , but since this quantity is fixed we only have a quadratic dependency on gN+(u").

Note that we use here a linear penalty law even in the presence of a non-linear relation (2) for the approach. Thus the fulfilment of the non-linear interface law will be practically accounted for by the update of the Lagrangian multiplier p . We use the following update procedure:

with the known quantities (...In at the state n. Owing to the appearance of the non-linear function c+ the update is related, but different, from the standard update procedure for the Lagrangian multipliers. lo

For the finiteelement treatment we need the variation of IIA which yields, for fixed p , (e.g. d ( p ) = const)

& 2

H A ( u " , ~ ) = C na(uu) + 1 PC+ (u",P)dI' + S,, 2 [c+ (u",p)] d r Y MIN a=l rc

p n + l = p n + &C+(U,"+l,ijn) (7)

Thus we can combine the last two terms to

which represent the residual of the contact contributions due to the augmented Lagrangian formulation.

3. FINITE-ELEMENT FORMULATION

Throughout this paper we restrict ourselves to the case of small strains and displacements. Thus we can formulate a very simple element based on nodal contact (see also Reference 11). Considering a generic contact element with the normal n oriented in the y-direction (Figure 3),

Page 4: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

818 P. WRIGGERS AND G. ZAVARISE

cal node numbering

Figure 3. Generic contact element

the approach of the surfaces related to the current position of nodes 1 and 2 is

(10) g N , = y 2 - y 1 = ( Y2 + v 2 ) - (Y' + v ' ) where Y' and uu denote the initial position and the current displacement of the two bodies in the contact surface, respectively.

Let us define the displacement vector V, of one contact node and the vector N, as follows:

With this notation we obtain from (10) for the penetration g N ,

g,v+ = VTNs (12) Note that now the condition for penetration (4) reads ( Y2 + v 2 ) - (Y' + v ' ) < 0. With these definitions we arrive at a matrix formulation for the contact element which follows from the second term in (9). Assuming that the contact area rc is discretized by n, active contact elements, we obtain

where pS is now the constant pressure associated with the contact element s. We need for the solution of (9) also its linearization; from (13) the tangent matrix is easily deduced for the contribution of one contact element

(14)

By summing over all s nodes being in contact we obtain the finite-element discretization of the contact part of equation (9). Let v denote the nodal displacement of the two discretized bodies; then G(v) expresses the standard discretization of the continuum problem E:= an". Then (9) becomes

(15)

K T ~ = EN,N:

n,

G(v) + U @s + EC+ (Vs,js) l Ns = 0 s= 1

Page 5: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

AUGMENTED LAGRANGlAN TECHNIQUES 819

4. ALGORITHMIC TREATMENT

Augmented Lagrangian techniques are usually applied together with Uzawa-type algorithms (see Bertsekas," Glowinski and Le Tallec' or Laursen and Simo,2 which lead to an inner loop for the contact and an outer loop for the update of the Lagrangian parameters.

Let us remark that it is standard practice in augmented Lagrangian iterations also to update the penalty number E to obtain good convergence (see Bertsekas"). This is owing to the fact that a small penalty parameter leads to very slow convergence since the update formula (7) is of first order and the contact forces due to the penalty are small. Thus we will additionally increase the penalty parameter within a contact element s according to the following update scheme:

(1 6) 1OEsn for [c+ (Vs,@s)l n + l > [c+ (Vs,@s)l n and csn Q k/, /(Nt) t csn for [c+ (Vs,@s)I n + 1 Q tc+ (Vs,@s)I n

Esn+l =

In relation (16) a stopping criterion for the update of the penalty parameter has been introduced to avoid ill-conditioning. This is given by the estimate (1).

The global augmented Lagrangian algorithm is shown here:

initialize algorithm set: do=€, v"=O, p o = O , E = E O

LOOP over augmentations: n = 1, ..., convergence LOOP over iterations: i = 1, ..., convergence

Solve: Gc(vY, p n ) = G ( v ~ + LP= 1 m s n + EsnC + (Vsi, P s n ) ] Ns = 0 fulfilling the constraint condition (5) Check for convergence: 1) Gc(VrP:Pn) 1 ) Q TOL - END LOOP

END LOOP LOOP over contact nodes: s = 1, ..., nc

Update: psn + 1 = psn + EnC + W s i , psn)

Update: dsn + 1 = h@sn + 1) Update: Esn+ 1 according to (1 6 )

Check for convergence ;\I g,v+(Vs;) - 1E - d n + 1) 11 Q TOL * STOP END LOOP

END LOOP

5 . EXAMPLES

In this Section we show the performance of the approach developed in the preceding Section by means of two examples. For this purpose we consider a particular form of the micromechanical interface law (2). This law was developed in Zavarise, and Zavarise, Schrefler and Wriggers, and is based on a statistical model of the microgeometry proposed by Cooper, Mikic and Yovanovich, l 3 recently revisited in Song and Yovanovich: l4

Here CI and c2 are mechanical constants expressing the non-linear distribution of the surface hardness, u and rn are statistical parameters of the surface profile, representing, respectively, the RMS surface roughness and the mean asperity slope, and d is the current mean plane distance. Thus we have an exponential law of the form p = c3 e-C4dZ.

Page 6: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

820 P. WRIGGERS AND G. ZAVARISE

5.1. Contact between two elastic blocks

A simple example has been carried out to test the effectiveness of the proposed procedure. This benchmark has also been used to test other features of the contact a lg~ri thm.”~ We consider two stainless steel blocks in contact with a displacement imposed at the top (see Figure 4). The elastic modulus of the continuum is E = 2- 1 x lo5 MPa and the Poisson ratio is p = 0-3. Contact parameters available in Song and Yovanovich’4 have been used: RMS surface roughness, 17 = 0.478 x m, mean absolute asperity slope, m = 0.072, hardness distribution constants, c1 = 6271 MPa and cz = -0.229. The amplitude of the imposed displacement generates a high contact pressure, and hence in the solution process the non- linear behaviour is deeply involved.

The test has been solved using the original formulation without augmentation, but with quadruple precision instead of double precision. The displacement was imposed in a single step. The solution requires seven iterations to converge (see Table I).

m. Different solutions with the augmented Lagrangian technique have been obtained, to check the best strategy in the definition of the initial and maximum penalty value. The number of augmentation cycles is strongly dependent on both parameters. The test is carried out with an initial penalty EO= 5.0 x 10’ and a maximum permitted penalty emax = 5 - 0 x 1013, where the update of the penalty parameter is performed according to (16). A very accurate solution ($11 gN+ (Vsj) - (t - & + I ) 1) < requires ten augmentation cycles (see Table 11). For most

The computed approach of the two surfaces is gN = 0.11671 1 x

Figure 4. Mesh of contact between two blocks

Table I. Results, quadruple precision

Residual norm Energy norm

0.8142e + 8 0-1097e+ 1 1 0-7501e + 7 0.9717e + 6 0.248Oe + 5 0 - 1746e + 2 0.8668e - 5 0.2137e - 17 0-5758e - 23

0.1 135e + 5 0.9616e + 6 0-8101e + 0 0.1814e- 1 0-1245e - 4 0.6176e - 11 0.1523e - 23 0.9253e - 49 0-7392e - 58

Page 7: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

AUGMENTED LAGRANGIAN TECHNIQUES 82 1

practical purposes an accuracy of 511 g N , (Vsi) - (E - - dn+ I ) 11 G is sufficient, which will limit the number of augmentations to six in this example.

Inside each cycle two Newton iterations are needed to balance the system. We note that in this geometrically linear problem the number of contact nodes does not change after the first iteration. Thus the inner loop can be solved most efficiently by a modified Newton-Raphson scheme which only uses backsolves and one triangularization.

We remark further that this displacement-driven example is more difficult to solve than the associated force-driven problem. This is owing to the fact that even very small penalty numbers lead in the latter problem to contact forces which balance the applied load. Thus the first update (7) is already a very good estimate for the contact pressures p . In a displacement-driven problem the forces needed to balance the reaction forces of the final state are only known at the end of the iterative process in the augmented Lagrangian algorithm. For illustration, we show in Table I11 the convergence behaviour for the same problem, now force-driven. We observe that the final solution is achieved now with only one augmentation step.

5.2. Contact between a ring and plane surface

A more complicated example, proposed in Papadopoulos and Taylor, shows the effectiveness of the procedure. Here a deformable cylindrical ring is pressed between two elastic planes (see Figure 5) .

The symmetry permits to discretize only one-quarter of the model. A downward displacement w is imposed at the top of the ring. The displacement is applied in five steps of 0.5 m and two steps of 0.25 m. At the start only the first element is in contact. With increasing load the contact zone also increases but also changes its shape.

Table 11. Results of augmented Lagrangian iteration, displacement-driven

It. i Augm. n gN+ Normal force En (€ - dn+ 1 )

2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10

0.599287e - 4 0.592269e - 4 0.529843e - 4 0.246400e - 4 0.290473e - 5 0.129733e - 5 0.117739e - 5 0.116844e- 5 0.116777e - 5 0~116772e - 5 0.116771e - 5

0.2%643e + 4 0-324688e + 5 0.294659e + 6 0- 148512e + 7 0.239800e + 7 0-246551e + 7 0.247055e + 7 0.247093e + 7 0-247O95e + 7 0.247096e + 7 0-247O96e + 7

O.5e + 9 O.5e + 10 0.5e+ 11 0.5e + 12 0.5e + 13 0.5e + 13 0.5e + 13 0.5e + 13 0.5e + 13 0-5e + 13 0.5e + 13

0.282210e - 6 0.546287e - 6 0.830800e - 6 0.107897e - 5 0.116231e-5 0.116732e- 5 0- 116768e - 5 0.116771e - 5 0.116771e-5 0.116771e - 5 0.116771e-5

Table 111. Results of augmented Lagrangian iteration, force-driven

It. i Augm. n gN+ Normal force En (€ - dn+ I )

2 0 0-4942e - 1 0.247O96e + 7 O*Se + 9 0.116771e - 5 2 1 0-116771e- 5 0.247096e + 7 0-5e + 9 0.116771e - 5

Page 8: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

822 P. WRIGGERS AND G. ZAVARISE

Finite Deformation of an 0-Ring UYY

.... = 40.0

- = 10.0

-- = 5.0

"" I

h 20

10

b I I I

0 1 2 3 4

X-coordinate (rn) (b)

Figure 5. Mesh, deformed state at w = 3.0 m with distribution of uyy stresses

The iterative process is documented in Table IV. Here we have monitored for each load step the number of augmentations, the final accuracy ($llgN+(Vsi) - (t - dn+l)ll), the number of iterations per augmentation step and the number of active contact elements. We observe that, despite the non-linearity of the problem, only two or three iteration steps are needed for one augmentation cycle once the first penalty solution is established. As expected, this behaviour is independent of the number of nodes in contact.

During the computation with the interface model (17) we made several observations which are discussed in the following.

When the process is near the solution, the geometrical approach can became less than the physical one. Hence c+ becomes negative, and also the penalty forces. If the augmented forces are small, then the sum of penalty and augmented forces is negative. This is clearly unrealistic; hence a check on the sign of the total forces has to be carried out inside the iterations. If the

Page 9: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

AUGMENTED LAGRANGIAN TECHNIQUES 823

Table IV. Number of iterations, example 5.2

Step Displ. No. augm.

0.5 1-0 1-5 2.0 2.5 2-75 3.0

iIIgN+ - (€ - d ) II 0.4320e - 7 0 4 2 9 e - 7 0.1386e-9 04052e - 6 0.1534e - 9

0.3068e - 8 0.1 156e - 8

~~ ~

No. iter. Elements closed

5-3-2 1 5-2 1

5-3-2-2 1-2 5-2 1-2

7-3-2-2 1-2-3-10-1 1 6-3-2-2 1-2-1 1-12-1 3-14 6-3-2-2 1-2- 12- 13- 14-1 5- 16-1 7

total force becomes negative the gap should be considered as open and no contributions of stiffness matrix and residuum vector should be added to the global system.

Owing to the possibility of negative penalty forces, solutions can also be achieved if one starts with a penalty parameter which is stiffer than the real physical stiffness. However, our experience suggests to start with a penalty stiffness less than the physical one to obtain better convergence.

To avoid problems during the iterative process the increase of the penalty parameter should be stopped when c+ changes its sign for the first time. At that point the system starts to oscillate around the solution. Then the solution converges faster with a constant penalty parameter.

REFERENCES 1. G. Zavarise, B. Schrefler and P. Wriggers, ‘Consistent formulation for thermomechanical contact

based on microscopic interface laws’, Proc. 3rd Int. Conf. on Computational Plasticity, E. Onate, R. Owen and E. Hinton (Eds), Barcelona, Spain, 1992.

2. T. A. Laursen and J. C. Simo, ‘On the formulation and numerical treatment of finite deformation frictional contact problems’, in Computational Methods in Nonlinear Mechanics, P. Wriggers and W. Wagner (Eds), Springer, Berlin, 1991.

3. G. Zavarise, P. Wriggers, E. Stein and B. A. Schrefler, ‘A numerical model for thermomechanical contact based on microscopic interface laws’. Mech. Res. Commun. (1992). to be published.

4. B. Nour-Omid and P. Wriggers, ‘A note on the optimum choice for penalty parameters’, Commun. appl. numer. methods, 3. 581-585 (1987).

5. R. Glowinski and P. Le Tallec, ‘Finite element analysis in nonlinear incompressible elasticity’, in Finite Element, Vol. V: Special Problems in Solid Mechanics, J. T. Oden and G. F. Carey, (Eds), Prentice-Hall, Englewood Cliffs, New Jersey, 1984.

6. P. Wriggers, J. C. Simo and R. L. Taylor, ‘Penalty and augmented Lagrangian formulations for contact problems’, in Proceedings of NUMETA Conference, J. Middleton and G. N. Pande (Eds), Balkema, Rotterdam, 1985.

7. N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988.

8. J. C. Simo and T. A. Laursen, ‘An augmented Lagragian treatment of contact problems involving friction’, Comput. Struct., 37, 319-331 (1992).

9. G. Zavarise, ‘Problemi termomeccanici di contatto - aspetti fisici e computazionali, Ph.D. thesis, 1st. di Scienza Tecnica delle Costruzioni, University of Padua, Italy, 1991.

10. D. P. Bersekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1984.

11. P. Wriggers and G. Zavarise, ‘Thermomechanical contact - A rigorous but simple numerical approach’, Comput. Struct., 46, 47-53 (1993).

12. 0. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4 edn, McGraw-Hill, London, 1989.

Page 10: Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces

824 P. WRIGGERS AND G. ZAVARISE

13. M. G. Cooper, B. B. Mikic and M. M. Yovanovich, ‘Thermal contact conductance’, fnt . J. Heat Mass Transfer, 12, 279-300 (1969).

14. S. Song and M. M. Yovanovich, ‘Explicit relative contact pressure expression: dependence upon surface roughness parameters and Vickers micro-hardness coefficients’, A I M Paper 87-0152, 1987.

15. P. Papadopoulos and R. L. Taylor, ‘A mixed formulation for the finite element solution of contact problems’, Comput. Methods Appl. Mech. Eng., 94, 373-389 (1992).