Eulerian Formulation of Elastic Rods

In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff canal surface. This reformulation hinges on describing the rod deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parameterizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this approach not only trivializes the detection of new contacts but also suppresses the isoperimetric constraints resulting from the self-feeding feature of these elementary problems and the imposition of the rod position at the extremities of each rod segments