An arc‐length method including line searches and accelerations

TLDR

The paper proposes incorporating line searches into the arc‑length solution procedure to enhance convergence. The approach employs line searches to compute an optimal scalar step length, skips them on many iterations using a loose tolerance, adds a single‑parameter acceleration, and is applied to geometrically nonlinear shallow shells and materially nonlinear reinforced concrete beams and slabs. On difficult iterations, the line searches markedly improve convergence, and the new arc‑length method demonstrates significant performance gains over the standard method.

Abstract

Abstract This paper describes a method for introducing line searches into the arc‐length solution procedure. Such line searches may be used at each iteration to calculate an optimum scalar step‐length which scales the normal iterative vector. In practice, a loose tolerance is provided so that on many iterations the line searches are avoided. However on ‘difficult iterations’, the line searches are shown to lead to a substantial improvement in the convergence characteristics. A simple single‐parameter acceleration is also developed using line search concepts. The new arc‐length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs. Significant improvements are demonstrated in relation to the standard arc‐length method.