Isogeometric generalized n th order perturbation-based stochastic method for exact geometric modeling of (composite) structures: Static and dynamic analysis with random material parameters

Abstract

The contribution herein proposes an isogeometric generalized n th order perturbation-based stochastic method for exactly modeling/representing composite structures comprising of different materials with particular attention to both static and dynamic analysis of structures with random material characteristics. Herein, we exactly represent the geometric model via isogeometric analysis (IGA), such as exactly modeling the interfaces in composite structures with dissimilar materials and also continuous variable thicknesses that cannot be achieved by existing traditional methods such as FEM. Besides, only a limited or scant work has been conducted thus far with IGA and stochastic methods. We consider the uncertainties of elastic modulus and mass density into account as stochastic inputs in static and dynamic isogeometric stochastic analyses. Moreover, we derive and expand the IGA based random-input parameter and all state functions included in static and dynamic equilibrium equations around their expectations via a generalized n th order Taylor series using a small perturbation parameter ε. In addition, we determine the probabilistic moments of the stochastic solution that satisfy the given accuracy requirement by expanding to n th order. The results obtained by the proposed method, the finite element method (wherever feasible), and Monte Carlo simulations for both benchmark and engineering applications verify the following: (a) the proposed methodology can achieve more accurate deterministic solutions with improved efficiency, thereby strengthening the effectiveness and efficiency brought by the stochastic method. This is in contrast to the FEM based method which weakens them, (b) on the other hand, it can more efficiently acquire reliable and more accurate stochastic results, both for expected values and standard deviations in the static and dynamic (free vibration) analyses. Note that the larger the problem size is, the more efficient the proposed method will be.