Numerical simulation of the viscoelastic Koiter shell model is challenging due to the complex coupling between spatial deformation and long-term memory effects. In this paper, we propose for the first time a finite-element method (FEM)-backward Euler (BE) scheme to solve the viscoelastic Koiter shell equation with a time-convolution term. The method employs conforming finite elements for spatial discretization, approximates the time-convolution term via a rectangular rule, and adopts the BE scheme for temporal discretization. Within this discretization framework, we rigorously prove convergence and derive optimal error estimates in the energy norm. Numerical experiments validate the theoretical analysis and highlight the influence of memory effects in the viscoelastic Koiter model by comparing its dynamic response to that of the elastic Koiter model under external loading.
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