Data-driven computational framework for snap-through problems

The distance-minimizing data-driven computing is an emerging field of computational mechanics, which reformulates the classical boundary-value problem as a discrete-continuous optimization problem, seeking a minimum distance between the discrete material data space and the physical constraint space. However, the optimization problem becomes non-convex due to the limit equilibrium instability in snap-through phenomenon, which makes it challenging to obtain a desired convergent solution, especially near critical points. Towards this end, we propose a data-driven computational framework by virtue of the structural stability theory for analyzing snap-through problems. First, an auxiliary linearized perturbed system established at each equilibrium state (i.e., convergent data-driven solution), which can be solved with the stiffness-based method and data-based method. Then, a stability indicator is employed to detect the critical point and the corresponding buckling mode, which is utilized to construct a proper start point to trace the equilibrium path in the neighborhood of critical points. Several numerical examples are performed to evaluate the reliability of the proposed framework. It is proved that even for the complex problem with nonlinear constitutive behavior, the proposed framework can correctly trace the entire equilibrium path of snap-through behavior.