Projection Methods for Stochastic Structural Dynamics

A set of novel hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. An optimal basis for the response of a stochastic system has been computed from the eigen modes of the parametrized structural dynamic system. The hybrid projection methods are obtained by applying appropriate approximations and by reducing the modal basis. These methods have been further improved by an implementation of a sample based Galerkin error minimization approach. In total four methods are presented and compared for numerical accuracy and efficiency by analysing the bending of a Euler-Bernoulli cantilever beam

Projection methods for stochastic dynamic systems: A frequency domain approach

A collection of hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. In this study, an optimal basis for the approximation of the response of a stochastically parametrized structural dynamic system has been computed from its generalized eigenmodes. By applying appropriate approximations in conjunction with a reduced set of modal basis functions, a collection of hybrid projection methods are obtained. These methods have been further improved by the implementation of a sample based Galerkin error minimization approach. In total six methods are presented and compared for numerical accuracy and computational efficiency. Expressions for the lower order statistical moments of the hybrid projection methods have been derived and discussed. The proposed methods have been implemented to solve two numerical examples: the bending of a Euler–Bernoulli cantilever beam and the bending of a Kirchhoff–Love plate where both structures have stochastic elastic parameters. The response and accuracy of the proposed methods are subsequently discussed and compared with the benchmark solution obtained using an expensive Monte Carlo method

Sample-based and sample-aggregated based Galerkin projection schemes for structural dynamics

A comparative study of two new Galerkin projection schemes to compute the response of discretized stochastic partial differential equations is presented for discretized structures subjected to static and dynamic loads. By applying an eigen-decomposition of a discretized system, the response of a discretized system can be expressed with a reduced basis of eigen-components. Computational reduction is subsequently achieved by approximating the random eigensolutions, and by only including dominant terms. Two novel error minimisation techniques have been proposed in order to lower the error introduced by the approximations and the truncations: (a) Sample-based Galerkin projection scheme, (b) Sample-aggregated based Galerkin projection scheme. These have been applied through introducing unknown multiplicative scalars into the expressions of the response. The proposed methods are applied to analyse the bending of a cantilever beam with stochastic parameters undergoing both a static and a dynamic load. For the static case the response is real, however the response for the case of a dynamic loading is complex and frequency-dependent. The results obtained through the proposed approaches are compared with those obtained by utilising a direct Monte Carlo approach

Stochastic finite element response analysis using random eigenfunction expansion

A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system’s stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos

Multi-instrument probing of the polar ionosphere under steady northward IMF

International audienceObservations are presented of the polar ionosphere under steady, northward IMF. The measurements, made by six complementary experimental techniques, including radio tomography, all-sky and meridian scanning photometer optical imaging, incoherent and coherent scatter radars and satellite particle detection, reveal plasma parameters consistent with ionospheric signatures of lobe reconnection. The optical green-line footprint of the reconnection site is seen to lie in the sunward plasma convection of the lobe cells. Downstream in the region of softer precipitation the reverse energy dispersion of the incoming ions can be identified. A steep latitudinal density gradient at the equatorward edge of the precipitation identifies the general location of an adiaroic boundary, separating the open field lines of polar lobe cells from the closed field of viscous-driven cells. Enhancements in plasma density to the south of the gradient are interpreted as ionisation being reconfigured as it is thrust against the boundary by the antisunward flow of the viscous cells near noon. Each of the instruments individually provides valuable information on certain aspects of the ionosphere, but the paper demonstrates that taken together the different experiments complement each other to give a consistent and comprehensive picture of the dayside polar ionosphere