Narrow Results

This paper deals with traditional algorithms, Newton's method and a higher order generalization due to Euler. These iterations schemes and their Modifications have had a great success in solving nonlinear systems of equations. We give some understanding of this phenomenon by giving estimates of efficiency. The problem we focus on is that of finding a zero of a complex polynomial.

An inclusion under uniform eigenstrains is of fundamental importance in composite mechanics. Although it is well known that an ellipse is the only inclusion shape which achieves a uniform internal stress field under anti-plane shear, it remains unclear what inclusion shapes achieve a polynomial internal stress field. The present paper studies an anti-plane inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains. The inclusion shape which gives a polynomial internal stress field is determined and the relation between the degree of the non-uniformity of the internal polynomial stress field and the inclusion’s geometrical shape is derived.