On the arc-length and other quadratic control methods: Established, less known and new implementation procedures

The use of control methods to compute geometrically nonlinear equilibrium paths is now well established. This paper reviews the arc-length method and other quadratic control methods, discussing implementation issues like (i) the root selection at the predictor and corrector stages, (ii) the use of partial corrections and the conjugation of arc-length and line search procedures, (iii) switching from spherical predictions to cylindric corrections, (iv) adopting constraint surfaces not centred at the last computed equilibrium point, (v) aligning elipsoidal constraint surfaces with the equilibrium path and (vi) the use of contractions from a plane surface to the quadratic surface. While most of these issues are not new, it seems useful to revisit some valuable techniques that a large number of researchers are unaware of – thus, they are presented in an unified way and geometric interpretations are provided whenever possible. Some original contributions are also made, mainly concerning the elipsoidal constraint surfaces. The paper concludes with the presentation of a few numerical tests to assess and compare the efficiency of the various implementations addressed.