Interface aligned high order continuation for branch rich equilibria of sliding nonuniform Cosserat rods

Sliding nonuniform Cosserat rods can develop clustered branch events when stiffness jumps, notch boundaries, and sliding induced transitions interact with nearly singular tangent operators. In this setting, a satisfactory computation must preserve the organization of the equilibrium path, not only isolated critical loads. This work develops an interface aligned high order multidomain continuation method. Each rod is partitioned in its physical coordinate so that local discontinuities coincide with element interfaces; the coupled residual is pulled back to a reference coordinate and advanced with a consistent sparse tangent in pseudo arclength continuation. The path is assessed through descriptors of event onset, center, width, and multiplicity. Controlled fold and branch tests verify the detector orders, and nonlinear rod tests compare the multidomain discretization with low order finite element and single domain global baselines. In representative cases, the high order variants recover the tested cluster organization and four branch events, whereas the baselines compress the same response to one detected event. The results show that rod local interface alignment must precede continuation if the branch structure of sliding nonuniform slender systems is to be retained.