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Highlights • A novel variant of the Newton-Raphson method within a time integration scheme tailored for nonlinear 3D mesoscale mechanical model of CNT nanocomposite. • An optimized numerical solution scheme suitable for damping material optimizations via Differential Evolution, usually requiring thousands of cyclic analyses. • Efficiency, while retaining numerical accuracy, obtained by a unique iteration (elastic stiness) matrix. • Both serial and parallel computations numerically tested to assess performance. • Performance further improved by a caching mechanism coupled with parallel execution of numerical simulations. Abstract The accuracy and efficiency of a numerical strategy for sequential nonlinear cyclic analyses of carbon nanotube nanocomposites are investigated. The computational approach resorts to a nonlinear 3D finite element implementation that seeks to solve the cyclic hysteretic response of the nanocomposite. A variant of the Newton-Raphson method within a time integration scheme is proposed whereby the elastic tangent matrix is chosen as iteration matrix without paying the price of its iterative update. This is especially rewarding in the context of the employed mechanical model which exhibits hysteresis manifested through a discontinuous change in the stiffness at the reversal points where the loading direction is reversed. Key implementation aspects – such as the integration of the nonlinear 3D equations of motion, the numerical accuracy/efficiency as a function of the time step or the mesh size – are discussed. In particular, efficiency is regarded as performing fast computations especially when the number of cyclic analyses becomes large. By making use of laptop CPU cores, a good speed of computations is achieved not only through parallelization but also employing a caching procedure for the iteration matrix. [ABSTRACT FROM AUTHOR]