# An Integration-Reduction Scheme for Simulation of Large Systems with Local Nonlinearity and Uncertainty - Application to moving load problems in railway mechanics

[Licentiatavhandling]

The focus of this research is mainly on computational efficiency, as future work is planned for reliability analysis that will be combined with design optimisation. Both these fields are known to be notoriously computationally demanding. Since the reliability-based optimization relies on fast simulations, the aim of this thesis has been on developing an efficient time-integration scheme for fast simulation of stiff FE models with local nonlinearity and uncertainty, such as railway tracks with sleepers put on a non-uniform ballast bed. Computational efficiency concerns that arise in simulation of large-scale systems are treated by using an approximation of the Jacobian matrix. This is achieved by combining the proposed exponential integrator with the developed methods for model reduction. A modal dominancy approach for modal reduction of dynamical systems is presented. Briefly stated, a quadratic dominancy metric, given in a closed form formulation, has been presented based on the modal contributions to the H_2 norm of the frequency response function (FRF) matrix and thus related to the r.m.s prediction. However, a main issue for many of the modal dominancy metrics is to detect the non-minimality and to handle systems with multiple eigenvalues, such as a track FE model with clusters of neighbouring eigenvalues, properly. In order to treat these systems, the problem of non-uniqueness of the proposed dominancy metric is studied in detail and two different methods to circumvent this problem are proposed. Another main concern for model reduction of large-scale systems, is to be more effective and to obtain as small order approximant as possible. However, typical input-output based reduction methods usually becomes ineffective, when they are applied to systems subjected to a moving/distributed loading. In this regard, the dominancy analysis procedure is improved to be able to use the information extracted from the spectral and internal structural properties of the external excitation. Moreover, the integration algorithm’s symplecticity, geometric properties and its global error are studied through Hamiltonian class examples and the efficiency and accuracy of the proposed integration-reduction scheme is investigated through a large-scale size problem that originates from a moving load problem in railway mechanics. It is demonstrated that the integrator is particularly effective in large-scale problems with local nonlinearity and/or uncertainty, when compared to the general purpose methods. Finally, the developed integration-reduction method is used in conjunction with a reliability approach using an adaptive two-stage procedure. In the first step it uses efficient methods such as First Order Reliability Form or Second Order Reliability Method, to find the approximate design point which constitutes the linearization point based on which the reduced model can be constructed. In the second step, an importance sampling technique constructed based on the obtained design point is used in conjunction with a Latin Hypercube Sampling to estimate the failure probability.

**Nyckelord: **time-integration, exponential integrator, symplectic integrator, modal dominancy analysis, reliability analysis, Monte-Carlo simulations, moving load problems

Denna post skapades 2014-01-29. Senast ändrad 2016-02-24.

CPL Pubid: 193247