Combining arc‐length control and line searches in path following

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Abstract
When solving non‐linear algebraic equations resulting from a typical finite element discretization it is quite common that the popular Newton‐Raphson iteration fails to converge or requires an excessive number of iterations. To solve this problem, line searches have been proposed in conjunction with load control and quadratic arc‐length control, which has proved to be highly successful. In the paper we first review a number of the most popular arc‐length controls. Then we highlight the severe drawbacks associated with the coupling of the cylindrical arc‐length control and line searches. Next, various other forms of arc‐length controls are scrutinized in the light of this coupling. It is shown that for some of the arc‐length controls such a combination is extremely simple and straightforward, yet at the same time efficient and effective. For the rest the same complications incurred for the cylindrical arc‐length method remain.

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