Two-Level Condensation for Stochastic Dynamic Analysis: Enriched Karhunen-Loeve Basis and Extended Polynomial Chaos Expansion

The present study investigates the robust dynamic analysis of stochastic models in linear elastodynamics. A computational framework is developed by coupling the Stochastic Finite Element Method (SFEM) with a dynamic condensation technique, ensuring robustness with respect to stochastic parametric variations. The proposed approach combines SFEM with a Karhunen–Loeve–based dynamic condensation, enriched through random residual static responses that effectively capture stochastic alterations in the mechanical system. Uncertainties in design parameters are modelled using Polynomial Chaos Expansion (PCE) within the SFEM framework, enabling efficient variability analysis at a substantially lower computational cost than sampling-based methods such as Latin Hypercube Sampling (LHS). A key contribution of this work lies in extending the PCE formulation to cases where design parameters exhibit nonlinear dependence in the governing equations, including situations in which structural parameters, such as shell thickness or beam height, enter the finite element matrices in linear, quadratic, or cubic forms. Moreover, the integration of the stochastic formulation with robust condensation techniques allows for efficient analysis of complex structures, particularly large-scale finite element models or systems with nonlinear parameter dependence. Computational challenges related to execution time, the number of uncertain parameters, and the magnitude of parametric variations are examined, and the proposed methodology is shown to offer clear advantages over conventional approaches.