Smoothed finite element method with the node-based strain smoothing domains (NS-FEM) is remarkable for the upper-bound feature and insensitivity to the volumetric locking. As a mesh-based methodology, its application is limited by the burdensome meshing for which elements align with the physical domain. We extend the strain smoothing in NS-FEM to the numerical manifold method (NMM) with unfitted meshes, and propose a novel methodology named physical patch-based smoothed NMM. Benchmark problems demonstrate the optimal convergence, stability in term of the condition number, upper-bound property, suppression of the volumetric locking, as well as the stress stability.

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This work applies a novel geometric criterion for nonlinear autonomous differential equations developed by Lu and Lu (NARWA 36:20–43, 2017) to a nonlinear SEIVS epidemic model with temporary immunity and achieves its threshold dynamics. Specifically, global-stability problems for the SEIVS model of Cai and Li (AMM 33:2919–2926, 2009) are effectively solved. The corresponding optimal control system with vaccination, awareness campaigns and treatment is further established and four different control strategies are compared by numerical simulations to contain hepatitis B. It is concluded that joint implementation of these measures can minimize the numbers of exposed and infectious individuals in the shortest time, so it is the most efficient strategy to curb the hepatitis B epidemic. Moreover, vaccination for newborns plays the core role and maintains the high level of population immunity.