In the thesis, the automation of stochastic finite element method to deal with stochastic fields is developed. The automation is based on hybrid symbolic-numeric approach. In this approach symbolic and algebraic computational system Mathematica, AceGen for the automatic code generation and the general-purpose finite element environment AceFEM are combined. The discretization of the stochastic field is done via Karhunen-Loève (K-L) expansion. Stochastic finite elements are developed to calculate the deterministic terms in K-L expansion. Further on, the approximation of exponential covariance function that reduces the cost of numerical calculation of K-L expansion is investigated. The thesis provides a proof that the approximation of covariance function that can often be found in the literature is not positive definite. The consequence is
possible numerical instability of the problem expressed as loss of positive definiteness of covariance matrix. The thesis presents two alternative modfications of the covariance function that reduce the computation cost and at the same time significantly improve the numerical stability of the procedure. For the calculation of the stochastic system response, higher-order perturbation method is chosen, based on the sensitivity analysis. For this step, the automatic differentiation based (ADB) formulation of higher-order sensitivity analysis is developed. The advantages of the developed formulation are calculation of
the exact derivatives, high numerical efficiency and abstract symbolic description of the finite element code to derive higher-order derivatives. The proposed formulation of stochastic finite element method is verified on all basic types of numerical examples that are important in the engineering practice, i.e. for various quantities modelled by stochastic field and for different failure modes: for excessive plastic deformations, as well as loss of structural stability.
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