Transient response analysis of randomly parametrized finite element systems based on approximate balanced reduction

• We propose a model reduction technique for large scale stochastic finite element systems.
• The reduced basis spans the dominant eigenspace of the stochastic controllability Gramian.
• Computationally efficient iterative Arnoldi–Lyapunov basis building methods for large stochastic systems.
• Implicit restart scheme for Arnoldi–Lyapunov vector basis has been proposed.
• Transient response analysis of large dynamical systems illustrated with numerical examples.

A model order reduction scheme of the transient response of large-scale randomly parametrized linear finite element system in state space form has been proposed. The reduced order model realization is aimed at preserving the invariant properties of the dynamic system model based on the dominant coupling characteristics of the specified system inputs and outputs. An a-priori model reduction strategy based on the balanced truncation method has been proposed in conjunction with the stochastic spectral Galerkin finite element method. Approximation of the dominant modes of the controllability Gramian has been performed with iterative Arnoldi scheme applied to Lyapunov equations. The reduced order representation of the randomly parametrized dynamical system has been obtained with Arnoldi–Lyapunov vector basis using an implicit time stepping algorithm. The performance and the computational efficacy of the proposed scheme has been illustrated with examples of randomly parametrized advection–diffusion–reaction problem under the action of transient external forcing functions. The convergence of the proposed reduced order scheme has been shown with a-posteriori error estimates.