Generalized Jocobians and Newton method for solving the frictional contact problems.

Pop, Nicolae ; Cioban, Horia

1. INTRODUCTION

The theory of nonsmooth analysis shows that many results of standard analysis may be extended to a more general, nonsmooth framework. In this paper a new model based on nonsmooth equations is proposed for solving a nonlinear and nondifferential equation obtained by discretization of a quasivariational inequation that modeling the frictional contact problem. The main aim of this paper is to show that the Newton method based on the plenary hull of the Clarke generalized Jacobians (the nonsmooth damped Newton method) can be implemented for solving Lipschitz nonsmooth equation.

The theory and algorithms for solving nonsmooth equation have been developed by mathematicians such as Pang and Qi (1993), Xu (2001) etc. Recently, Hu et al. (2007) presented the employment of Newton's method to formulate B-differentiable equations involving projection for three dimensional contact problems. The contact constraints can be exposed as nonsmooth equation set in which the variables are related to the candidate contact nodes. For solving this nonsmooth equations can be propose the nonsmooth damped Newton method using the generalized Jacobians.

2. GENERALIZED GRADIENTS

Assume that F:[R.sup.n] [right arrow] [R.sup.n] is locally Lipschitzian but not necessary differentiable and nonsmooth equation

F(x) = 0 (1)

The most popular method for this solving equation (1) is the Newton method based on the Clarke generalization Jacobian,

[x.sub.k+i] = [x.sub.k] + [(V([x.sub.k])).sup.-1]F([x.sub.k])

where V([x.sub.k]) [member of] [partial derivative]F([x.sub.k]) is the Clarke generalized Jacobian of the F at [x.sub.k]. The Generalized Jacobian of F is defined as a convex hull of all n x n matrices obtained as the limit of a sequence [(V([x.sub.k])).sub.l], when [x.sub.k] [right arrow] x, k [right arrow] [infinity] and [x.sub.k] is a point at which F is differentiable. In case when F is nonsmooth, V([x.sub.k]) may not exist, and therefore, is should be replaced by a generealized derivative an using Rademarcher's Theorem, with F differentiable almost where.

Denote by DF the set of points where F is differentiable, and by [nabla]F(x), a n x n Jacobian matrix of partial derivative whenever [x.sub.k] is a point at which the partial derivative exist. By [partial derivative]F(x) we denote the generalized derivative, in sense Clarke

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

If F is a locally Lipschitzian and nonsmooth function and for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists, is said that F'(x,h) is the directional derivative of F.

The one fundamental distinction between the smooth (F-differentiable) and nonsmooth (B-differentiable) functions is the absence of linearity in the directional derivative for nonsmooth functions. Generalization of the classical theorems on the existence of inverse or implicit functions of locally Lipschitz continuous functions are given in Clarke (1983) and Kummer (1991).

3. NEWTON'S METHOD WITH GENERALIZED JACOBIAN

Definition 1. Let F: [R.sup.n] [right arrow] [R.sup.n] be a locally Lipschitz continuous function an the domain D, and let x [member of] D. The generalized Jacobian [partial derivative]F(x) is said to be nonsingular (as of maximal rank), if all matrices M [member of] [partial derivative]F(x) are nonsingulars in the usual sense, i.e. of maximal rank.

With the development of generalized derivatives one strives to extend the associated geometric concepts of tangents and normals to apply to nonsmooth sets. Under certain regularity conditions, these sets form locally Lipschitz continuous manifolds. Clarke's normal and tangential cones are only partly useful for geometrical characterization of such sets and of their images under smooth functions. The frictional contact problem, of an elastic body in contact with friction on a rigid obstacle, is discretized by means of the finite element method, and we obtain the following discrete problem

Ku + F(u) = Q (3)

where u is the vector of the nodal displacement, K is the stiffness matrix of the elastic body, Q is the vector of the loads and F(u) is the corresponding matrix which describes the influence of friction and the contact conditions. The next results leads to a sufficient condition such that the nonlinear and nondifferentiable system (3) does have one solution, obtained by Newton's method using generalized gradient.

Theorem 1. (Clarke 1983) Let the operator F: [R.sup.n] [right arrow] [R.sup.n] be continuous, Lipschitz and affine on the number of the open cones with peak to [u.sub.0]. Then det M [not equal to] 0, ([for all]) M [member of] [partial derivative]F([u.sub.0]) is a sufficient condition for F to be globally homeomorphism from [R.sup.n] to [R.sup.n].

The generalized Newton's algorithm for solving problem (3), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The Theorem 1 assures a sufficient condition for the existence of the solutions of problem (3) busing the generalized Newton's algorithm (4).

4. AN EXAMPLE

Let [OMEGA] [subset] [R.sup.2] bean elastic body discretized by finite element method, in contact with friction on a rigid obstacle. We suppose there are NC nodal points on the contact boundary, to be placed last in those N nodal points of the partition [[OMEGA].sup.k]. The vector F having the form

[[0 ... 0; [F.sup.1]([u.sub.1.sup.1]); F1([u.sub.1.sup.2]); ... [F.sup.1]([u.sub.1.sup.1]); F1([u.sub.1.sup.2])].sup.T] (5)

Because F is piecewise differentiable, at this singularities, the Jacobian M will be replaced with the generalized Jacobian [partial derivative]F. We consider the operator F only at the nodal contact point "a" and we denote it by [F.sup.a], and the operator [F.sup.a], for the all possible situations: open contact, closed contact, forward slide and back slide, having the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

p is penalty coefficient which has the mechanical interpretation as contact force, [mu] is the friction coefficient, [Z.sub.i] is open cone and D is the initial state.

An easy verification shows that [F.sup.a] is piecewise linear, therefore F is continuous, Lipschitz and affine on a finite number of open cones with the peak to D. Consequently the generalized Jacobian [partial derivative]F, is the set of the matrices 2N x 2N block-diagonal.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

It remains to find out the condition on [mu] in order to guarantee the assumption from Theorem 1.

det M [not equal to] 0, ([for all]) M [member of] [partial derivative](K + F)(-D) (8)

A simple example with only one nodal contact point will be given. The stiffness will be a matrix 2 x 2 symmetric and positive defined

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

For simplicity we consider D = 0, and [partial derivative]F becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

For M [member of] [partial derivative]F(0) one obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, [gamma] [greater than or equal to] 0 and 0 [less than or equal to] [alpha] + [beta] + [gamma] [less than or equal to] 1, therefore all matrices M [member of] K + [partial derivative]F(0) have the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Hence [gamma]([alpha] + [beta] + [gamma]) > 0 and

det K > 0 [??] [mu] < [a.sub.11]/[absolute value of [a.sub.12]],

Consequently for the friction coefficient with [mu] < [a.sub.11]/[absolute value of [a.sub.12]], the contact problem (3) has one solution and it can be obtained using the generalized Newton's method.

5. CONCLUSION

For solving systems of nonlinear equations, one usually preconditions of the linearized equations is an approximate inverse of the center of the interval Jacobian.

Further research in this area of interest is to develop the solution presented in this paper on friction contact applied on gears. Other developments can be in quasistatic frictional contact problems in 3D modelling

6. REFERENCES

Clarke, F.H.(1983). Optimization and nonsmooth analysis, Wiley and Sons, ISBN: 0-471-87504-X, New York

Hu, Z.Q., Soh, A.K., Chen, W.J., Li, X.W. & Lin, G.(2007). Nonsmooth nonlinear equations methods for solving 3D elastoplastic frictional contact problems, Computational Mechanics, Vol.39, No.6, 849-858, ISSN 0178-7675

Kummer, B.(1991). Lipschitzian inverse functions, directional derivatives, and applications in [C.sup.1,1]-optimizatin, Journal of Optimization Theory and Applications Vol. 70, no. 3, 561-582, ISSN:0022-3239

Pang, J.S. & Qi, L.Q.(1993). Nonsmooth equations: motivation and algorithms, SIAM Journal of Optimization Vol.3, No.3, 443-465

Xu, H.(2001). Adaptive Smooting Method, Deterministically Computable Generalized Jacobians and the Newton Method, Journal of Optimization Theory and Applications, 109, no. 1, 215-224, ISSN:0022-3239