numerical methods for a class of contact problems - Science Direct

0020-7225/91 $3.00+ 0.M) Copyright 0 1991Pergamon Press plc

Int. .I. Engng Sci. Vol. 29, No. 4, pp. 513-521, 1991 Printed in Great Britain. All rights reserved

NUMERICAL

METHODS FOR A CLASS OF CONTACT PROBLEMS MUHAMMAD

Mathematics Department,

ASLAM

NOOR

College of Science, King Saud University, Riyadh 11451, Saudi Arabia

S. I. A. TIRMIZI Department

of Mathematics and Computing, College of Science, Sultan Qaboos University, Muscat, Sultanate of Oman

Abet-It is well known that contact problems in elastostatic can be studied in the framework of variational inequalities. In this paper, we show that variational inequality related with a class of contact problems can be characterized by a sequence of variational equations, which can be solved by using the multiderivative methods. We describe numerical experience on the use of penalty function method for obtaining approximate solution of a class of unilateral problems in elasticity, like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the contact problem.

1. INTRODUCTION Variational inequality theory has experienced a vigorous development since its emergence in the mid-1960’s. The idea and approach of variational inequalities are being applied in a variety of diverse fields of mathematical and engineering sciences and prove to be productive and innovative. In brief, variational inequality theory has become not just another branch of mathematics, but an indispensible part of the modern pure and applied sciences. The area of contact problems in elasticity forms an important foundation for the applications of variational inequality theory, see Kikuchi and Oden [l], Crank [2], Fichera [3], Duvaut and Lions [4] and the references therein for mathematical and physical formulations: In a variational inequality formulation, the location of the free boundary (contact area) becomes an intrinsic part of the solution and no special devices are needed to locate it. This paper is concerned with the application of a penalty function method for solving a class of unilateral problems in elasticity and with their use as a basis of Pade approximants method of higher order for the numerical solution of problem of this type. Using the penalty function method of Lewy and Stampacchia [5], the variational inequality characterizing the contact problems can be formulated as a system of boundary value problems without constraints in the case of a known obstacle. This technique has been used by Noor and Tim&i [6] for solving unilateral problems. In this paper, we develop multiderivative methods of orders 6 and 8 based on the Pade approximants to compute the approximate solution of boundary value problems. We would like to emphasize that there does not exist any other method of order 8 in the literature for solving fourth order boundary value problems to compare with one. For the purpose of numerical experience, we consider an example of an elastic beam lying over an elastic obstacle. The formulation and the approximation of the elastic beam is very simple, however, it should be pointed out that the kind of numerical problems which occur for more complicated system will be the same. Our approach to these problems is to consider them in a general manner and specialize them later on. In Section 2, we consider the contact problem and formulate it in terms of variational inequality. Using the penalty function method, we characterize the variational inequality by a sequence of variational equations. The variational inequality formulation is used to discuss the uniqueness and the existence of the solution of the contact problems. In Section 3, we develop the multiderivative methods of orders 6 and 8 for solving the fourth order boundary value problems. Convergence criteria of these methods is considered in Section 4. Section 5 is devoted to the applications of the 513

M. A. NOOR and S. I. A. TIRMIZI

514

numerical methods for solving the problem of an elastic beam lying over an elastic obstacle. Our results indicate that our method of order 6 is much better than the previous methods of Jain, Iyenger and Saldanha [7] and Twizell and Tirmizi [8]. It is worthmentioning that the method of order 8 developed in this paper has no counterpart present in the literature for solving fourth order boundary value problem.

2. FORMULATION

AND

BASIC

RESULTS

For simplicity, we consider the obstacle problem of finding u such that

2f

Lu

24s3 (Lu -f)(u

- 111)= 0

g=o,u=o

on dG?I

(2.1)

where S2c R” is a simply-connected open domain with boundary X2, L = d4/dx4, a fourth order differential operator, f and @ are given functions. The problem (2.1) is studied via the variational inequality fo~ulation in the Sobolev space H’(Q), which is a Hilbert space. We define the subspace H%(Q) of H’(S2) as

H;(Q) =

{ufH2(Q):E =

0, u = 0 on an},

where the derivatives are considered in a generalized Ht2, see [l, 91 for definitions and notations. To do so, we defined the set K by K=(~EH~(Q):

sense. The dual of H:(R)

u%v

is denoted by

on Cl},

which is a closed convex set, see [4,5]. It has been shown in [l] that the functional associated with (2.1) can be given by

I[?.?]=

I, (Lu)u

dx - 26 fu dx,

for all 21E H;(S).

=j-(g)‘dx-2~fidx,

=u(v, v) -

2(f, 4,

(2.2)

where 2 a(u,

v)=

2

du.dl)& I Q&2

dr2

is obtained by integration by parts. It is clear that the form a(u, V) is bilinear, positive and symmetric, so one can easily show that the minimum of the functional I[v] defined by (2.2) on the closed convex set K can be characterized by the variational inequality of the type: a(u,v--u)>V,v--u),

for all v E K.

From the relation

it follows that for all v E Hg(S2) and a(u, v> s

lb42

Il7Jll2,

for all zf, 2, E H;(Q),

(2.3)

Numerical methods for a class of contact problems

515

On the basis of the above observations, we conclude that the bilinear form a(u, u) associated (2.1) is in fact coercive and continuous with 0 < (Y6 1 and 0 C @< 1. Thus it follows that

with

there does exist a unique solution u E K satisfying the variational inequality (2.3), see Lions and Stampacchia [lo] and Noor [ll]. REMARK.2.1.

Many authors including Crank [2], Baiocchi and Capelo 1121and Kikuchi and Oden [l] have shown that a large number of moving and free boundary value problems arising in fluid flow through porous media; elasticity, trans~~ation and economics can be form~ated in terms of (2.2) and (2.3). Here, the variational inequality (2.3) characterizes the Si~o~ni problem in elasto-statics. If Q is an open bounded domain in R” with smooth boundary dQ, representing the elastic beam subject to the external forces; and if part of the boundary may come into contact with a rigid foundation, then inequality (2.3) is simply a statement of virtual work for an elastic beam. The strain energy of the elastic beam corresponding to an admissible displacement v is a(~, v). Thus a(u, u) is the work produced by the stresses through strains caused by the virtual displacement TV- u. The linear continuous functional f represents the work done by the external forces. We now consider a simple, but a powerful technique for solving variational inequalities of type (2.3). Such type of technique was used by Lewy and Stampacchia [S] to study the existence and regularity of the solution of variational inequalities. The computational advantage of the technique is its simple applicability for solving unilateral problems. Following the idea of Lewy and Stampacchia [5], the variational inequality (2.3) can be characterized by a sequence of variational equations such as: a(u, v) + (v(u - w)(u - lu), u) = V, t.Q, for all u E H;(B),

where v(t) is the discontinuous

(2.4)

function defined by ts0

4, v(t) = i o ,

t>O

is known as the penalty function and @ < 0 on dS2 is an elastic obstacle. penalty techniques, see Kikuchi and Oden [l].

3. NUMERICAL

(2.5) For other types of

METHODS

In order to solve (2.4) by higher order methods based on the Pade approximants, we first consider the problem of an elastic beam which satisfies the linear two-point boundary value problem &4’“(X) + Ku = 4 with boundary conditions u

(*fz>=o,

II”

(3.1)

(*;z>=o,

Here D is the flextural rigidity of the beam, K is the spring constant of the elastic foundation and q is the uniform load applied to the surface of the elastic beam. For the mathematical

formulation of (3.1), see [13,14]. The analytical solution of the above system cannot be determined always if D, K and q are variables. Therefore, we resort to numerical methods. Various authors, including Fox [15], Varga [16], Usmani [17,18], Jain et al. [17] and Chawla and Katti [19] have used finite difference methods to solve this system, while Usmani and Warsi [20] and Papamichael and Worsey [21] used the spline techniques. More recently Twizell and Tirmizi [8] and Noor and Tirmizi f6] have used multiderivative methods based on the Pade approximants. Noor and Tirmizi [6] reported methods of order two and four, whereas in this paper, we develop higher order methods of orders six and eight based on the (2,4) and the (0,8) Pade approximant respectively.

516

M. A. NOOR

and S. I. A. TIRMIZI

For our purpose, it is enough to consider the following system. a