  In this paper, we propose a novel one-dimensional (1D) discrete differential
geometry (DDG)-based numerical method for geometrically nonlinear mechanics
analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our
numerical model leverages differential geometry principles to accurately
capture the complex nonlinear deformation patterns exhibited by axisymmetric
shells. By discretizing the axisymmetric shell into interconnected 1D elements
along the meridional direction, the in-plane stretching and out-of-bending
potentials are formulated based on the geometric principles of 1D nodes and
edges under the Kirchhoff-Love hypothesis, and elastic force vector and
associated Hession matrix required by equations of motion are later derived
based on symbolic calculation. Through extensive validation with available
theoretical solutions and finite element method (FEM) simulations in
literature, our model demonstrates high accuracy in predicting the nonlinear
behavior of axisymmetric shells. Importantly, compared to the classical
theoretical model and three-dimensional (3D) FEM simulation, our model is
highly computationally efficient, making it suitable for large-scale real-time
simulations of nonlinear problems of shell structures such as instability and
snap-through phenomena. Moreover, our framework can easily incorporate complex
loading conditions, e.g., boundary nonlinear contact and multi-physics
actuation, which play an essential role in the use of engineering applications,
such as soft robots and flexible devices. This study demonstrates that the
simplicity and effectiveness of the 1D discrete differential geometry-based
approach render it a powerful tool for engineers and researchers interested in
nonlinear mechanics analysis of axisymmetric shells, with potential
applications in various engineering fields.
