Tracing load-displacement paths in structural mechanics problems is complicated in the presence of critical points of instability where conventional load- or displacement control fails. To deal with this, arc-length methods have been developed since the 1970s, where control is taken over increments of load at these critical points, to allow full transit of the load-displacement path. However, despite their wide use and incorporation into commercial finite element software, conventional arc-length methods still struggle to cope with non-zero displacement constraints. In this paper we present a new displacement-controlled arc-length method that overcomes these shortcomings through a novel scheme of constraints on displacements and reaction forces. The new method is presented in a variety of serving suggestions, and is validated here on six very challenging problems involving truss and continuum finite elements. Despite this paper's focus on structural mechanics, the new procedure can be applied to any problems that involve nonhomogeneous Dirichlet constraints and challenging equilibrium paths.
Pretti, G., Coombs, W., & Augarde, C. (2022). A Displacement-controlled Arc-Length Solution Scheme. Computers and Structures, 258, Article 106674. https://doi.org/10.1016/j.compstruc.2021.106674