Keywords: iterative, nonlinear, near-constrained, large-scale.

This paper considers a class of nonlinear problems which is of the form

(I) Minimise

where , with the properties
(a) is twice continuously differentiable and strictly convex on
(b) , and
(c)

and is a parameter. When is very large it can be shown that the vector constraint

is nearly satisfied and generally numerical difficulties are encountered when trying to solve problem (I) . For example, near incompressible problems often lead to numerical difficulties when a finite element displacement solution is sought. One way to overcome these difficulties is to formulate problem (I) in an alternative (often by approximating problem (I)) form. The constraint function is often approximated (for some problems this function is not approximated) by and when is very large the constraint 0 is replaced approximately by

where is an -vector variable. The function in is also replaced (often approximately) by , where is related to according to the given problem. The problem (I) is now replaced by the problem
(II) Minimise

subject to the constrained

where . An engineering example of problem (II) can be found in Shariff and Parker [1].

In this paper we develop an algorithm for problem (II) which is a generalization of the algorithm recently developed by Shariff [2]. The algorithm is suitable for large scale problems in the sense that it does not require a Hessian matrix. It is easily parallelised and suitable for both sparse and dense systems. The developed algorithm is given below:
1. Set , select and a penalty parameter .
2. Set    
3.
4. If : stop
5.        
6.    
7. Set ; go to 2
The function

and

where

Convergence analyses for the proposed algorithm and several test problems are given in this paper.

1

M.H.B.M. Shariff, D.F. Parker, "An Extension of Key's Principle to Nonlinear Elasticity", Jou. Engng. Maths., 37, 171-190, 2000. doi:10.1023/A:1004734311626

2

M.H.B.M. Shariff, "Fast and Large Scale Methods for Nonlinear Anisotropic Problems", Computational Mechanics using High Performance Computing, B.H.V. Topping (Editor), Saxe-Coburg Publications, Stirling, UK, 2002. doi:10.4203/csets.9.10

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