Algorithms for branch-following and critical point identification in the presence of symmetry

Abstract

The objective of this work is to present techniques that can be used to study steady state solutions to systems generically described by the equation F(x; #21;) = dx dt = 0, where x 2 RN is the set of degrees of freedom (DOFs) of the system and #21; 2 R is a loading parameter. In particular, the implications of the presence of discrete and continuous symmetry to systems described by F(x; #21;) = 0 will be explored. Additionally, numerical techniques that can be used to “path-follow” the evolution of the DOFs as the loading parameter #21; is varied will be presented along with schemes to increase the computationally efficiency of the path following algorithms. These techniques can be used to obtain solutions to F(x; #21;) = 0 around regular and singular points. Additionally, methods to detect, and compute singular points of F will be presented. Finally, the benefits of these techniques will be illustrated through the computation of solutions to two typical problems.