International Journal of Computer Mathematics

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Numerical solutions of fourth order variational inequalities a

Muhammad Aslam Noor & Eisa A. Al-Said

b

a

Department of Mathematics and Statistics , Dalhousie University , Halifax, Nova Scotia, B3H 3J5, Canada b

Department of Mathematics , College of Science, King Saud University , P.O. Box 2455, Riyadh, 11451, Saudi Arabia Published online: 19 Mar 2007.

To cite this article: Muhammad Aslam Noor & Eisa A. Al-Said (2000) Numerical solutions of fourth order variational inequalities, International Journal of Computer Mathematics, 75:1, 107-116, DOI: 10.1080/00207160008804968 To link to this article: http://dx.doi.org/10.1080/00207160008804968

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Intern. J. Computer Math., Vol. 75, pp. 107- 116 Reprints available directly from the publisher Photocopying permitted by license only

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NUMERICAL SOLUTIONS OF FOURTH ORDER VARIATIONAL INEQUALITIES MUHAMMAD ASLAM NOORa3*and EISA A. AL-SAID~ aDepartment of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; b~epartmentof Mathematics, College of Science, P.O. Box 2455, King Saud University, Ri-vadh 11451, Saudi Arabia (Received 13 April 1999) We use variational inequality theory along with finite difference technique to obtain an approximation for the solution of a class of obstacle problem in elasticity, like those describing the equilibrium configuration of an elastic stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the obstacle problems.

Keywords: Variational inequalities; obstacle problem; penalty function method; finite diffe~ence method 1991 AMS Subject Classifications: 49540, 65L12

C.R. Category: G.1.7

1. INTRODUCTION

Variational inequality theory has become an effective and powerful tool for studying obstacle and unilateral problems arising in mathematical and engineering sciences including fluid flow through porous media, elasticity, transportation and economics equilibrium, optimal control, nonlinear optimization, operations research, see, for example [3-7,101. The area of obstacle problems arising in fluid flow through porous media and elasticity forms an important foundation for the applications of variational inequalities. It has been shown by Kikuchi and Oden [9] that the *Corresponding author. e-mail: [email protected]


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M. A. NOOR AND E. A. AL-SAID

problem of equilibrium of elastic bodies in contact with a rigid frictionless foundation can be studied in the framework of variational inequalities. In a variational inequality formulation, the location of the free boundary (contact area) becomes an intrinsic part of the solution and no special technique are needed to locate it. Various numerical methods are being developed and applied to find the numerical solutions of the obstacle problems, see, for example [I, 2,4 - 8, 151 and the references therein. In principle, the finite difference methods cannot be applied directly to solve the obstacle problems. However, If the obstacle function is known, one can characterize the obstacle problem by a sequence of boundary value problems without constraints via the variational inequality and penalty function. The computational advantage of this method is its simple applicability for solving differential equations. Such types of penalty function have been used by Al-Said and Noor [I] and Khalifa and Noor [8] in solving a class of contact problems in elasticity in conjunction with finite difference and collocation methods techniques respectively. It is worth mentioning here that the standard finite difference and collocation methods do not give better than second order approximations for the solution of such problems, see [I, 2,8,15] for details. In this paper, we develop an accurate finite difference technique for solving a class of obstacle problem. The results obtained in this paper are much better than the previous ones and represent an improvement. 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. In Section 2, we consider a class of contact problems in elasticity and formulate them in terms of variational inequalities. Using the penalty function method of Lewy and Stampacchia [I I], we characterize the variational inequality by a sequence of variational equations. Numerical method for solving system of variational equations is discussed in Section 3.

2. FORMULATION

Let H be a real Hilbert space on which the inner product and norm are denoted by ( . , . ) and /(,I1respectively. Let K be a closed convex set in H. Let a(u. v) be a coercive continuous bilinear form on H, that is, there exist constants cr 2 0 and B 2 0 such that

VARIATIONAL INEQUALITIES FOR B.V. PROBLEMS

109

and


It is clear that a 5 p. Iff is a continuous linear functional on H, then it is well known [l 11 that there exists a unique solution u E K such that

The inequality (2.1) is known as variational inequality. If a(u, v) is symmetric and positive bilinear form, then problem (2.1) is equivalent to finding the minimum of Z[v] on K,where I [ v ] = a(u, v) - 2(f, v).

(2.4)

Following the penalty function technique of Lewy and Stampacchia [I I], the variational inequality (2.1) can be characterized by a sequence of variational equations as

Vv E H, where v(t) is the discontinuous function

is known as the penalty function and obstacle. For full details, see [9].

$J

1 0 on the boundary is an elastic

3. NUMERICAL RESULTS We consider the linear fourth order boundary value problem describing the equilibrium configuration of an elastic beam, pulled at the ends and lying over an elastic obstacle of constant height 114 and unit rigidity of the type:

where

+ is the given obstacle function.

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M. A. NOOR AND E. A. AL-SAID


We study Problem (3.1) via variational inequality formulation in the Sobolev space H ~ ( R ) ,which is a Hilbert space. We define the subspace H ; ( R ) of ~ ' ( 0as )

where the derivatives are considered in the generalized sense. The dual of H ~ ( R is) denoted by H g 2 ( R ) , see [3, 91. We define the set K by

which is a closed convex set in H ; ( R ) . It has been shown [9] that the energy functional associated with the Problem (3.1) can be given by

where

and

It is obvious that the form a(u, v) defined by Eq. (3.3) is positive and symmetric bilinear. One can easily show that the minimum of the energy functional I [ v ] defined by (3.2) on the closed convex set K can be characterized by the inequality of the type:

Similarly, one can show that a(u, v) defined by Eq. (3.3) is coercive and continuous. In fact, from the relation (3.3), it follows that

VARIATIONAL INEQUALITIES FOR B.V. PROBLEMS

is coercive with 0 5 CY 5 1 , and
