Model Reduction of Nonlinear Continuous Shallow Arch and Dynamic Buckling Simulations on Approximate Inertial Manifolds with Time Delay - Zhang, Jiazhong; Chen, Liying; Ren, Sheng (Journal of Applied Nonlinear Dynamics, 2026)
n comparison with Traditional Galerkin’s Method (TGM), an Approx- imate Inertial Manifolds with Time Delay (AIMTDs) is presented for the model reduction of shallow arch with large deformation, a kind of nonlinear continuous dynamic system governed by partial differ- ential equation with second order in time, and the numerical simulation is also presented for the dynamic buckling analysis of shallow arch under impact. By this method, the nonlinear governing equations, which is a kind of infinite dimensional dynamic system, are studied in the phase space, and the solutions of the governing equations are projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equations. With the introduction of AIMTDs, it decomposes the infinite-dimensional phase space into two complementary subspaces, namely, a finite-dimensional one and its infinite-dimensional complement, and the relation between these two subspaces is the one with a time delay, that is, the evolution of the high modes is not only relevant to the instantaneous low modes, but also to the past high modes. Hence, the system can be projected onto the subspace with finite-dimension, in combination with the relation between the two subspaces, implying the model is reduced. Then, the nonlinear Galerkin’s procedure is adapted to approach the solution on AIMTDs in which the final equilibrium positions are included. Finally, the method is applied to the numerical simulation of dynamic buck- ling of the shallow arch under impact, and some comparisons between Traditional Galerkin’s procedure and Approximate Inertial Manifolds with Time Delay are given. It can be concluded that the method pre- sented give a guarantee for the truncation of buckling modes as mode expansion is used, and are generally feasible for the model reduction of the nonlinear continuous dynamic systems with second order in time, and it is an efficient numerical method requiring less computing time and high accuracy. More, the AIMTDs can be easily approached by finite element method.