Dynamic symmetry breaking and structure-preserving analysis on the longitudinal wave in an elastic rod with a variable cross-section

The longitudinal wave propagating in an elastic rod with a variable cross-section owns wide engineering background, in which the longitudinal wave dissipation determines some important performances of the slender structure. To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section, a structure-preserving approach is developed based on the dynamic symmetry breaking theory. For the dynamic model controlling the longitudinal wave propagating in the elastic rod with the variable cross-section, the approximate multi-symplectic form is deduced based on the multi-symplectic method, and the expression of the local energy dissipation for the longitudinal wave propagating in the rod is presented, referring to the dynamic symmetry breaking theory. A structure-preserving method focusing on the residual of the multi-symplectic structure and the local energy dissipation of the dynamic model is constructed by using the midpoint difference discrete method. The longitudinal wave propagating in an elastic rod fixed at one end is simulated, and the local/total energy dissipations of the longitudinal wave are investigated by the constructed structure-preserving scheme in two typical cases in detail.