Abhishek Kundu

Transient Response of Structural Dynamic Systems with Parametric Uncertainty

AbstractThe time-domain response of a randomly parameterized structural dynamic system is investigated with a polynomial chaos expansion approach and a stochastic Krylov subspace projection, which has been proposed here. The latter uses time-adaptive stochastic spectral functions as weighting functions of the deterministic orthogonal basis onto which the solution is projected. The spectral functions are rational functions of the input random variables and depend on the spectral properties of the unperturbed system. The stochastic system response can be accurately resolved even when using low-order spectral functions, which are computationally advantageous. The time integration required for the resolution of the transient stochastic response has been performed with the unconditionally stable single-step implicit Newmark scheme using a stochastic integration operator. A semistatistical hybrid analytical and simulation-based computational approach has been utilized to obtain the moments and probability densi...

A Reduced Spectral Projection Method for Stochastic Finite Element Analysis

The stochastic finite element analysis of elliptic type partial differential equations are considered. An alternative approach by projecting the solution of the discretized equation into a finite dimensional orthonormal vector basis is investigated. It is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors. These functions, called as the spectral functions, can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation. An explicit relationship between these functions and polynomial chaos functions has been derived. Based on the projection in the orthonormal vector basis, a Galerkin error minimization approach is proposed. The constants appearing in the Galerkin method are solved from a system of linear equations which has the same dimension as the original discretized equation. A hybrid analytical and simulation based computational approach is proposed to obtain the moments and pdf of the solution. The method is illustrated using a stochastic beam problem. The results are compared with the direct Monte Carlo simulation results for different correlation lengths and strengths of randomness.

A Novel Reduced Spectral Function Approach for Finite Element Analysis of Stochastic Dynamical Systems

This work provides the theoretical development and simulation results of a novel Galerkin subspace projection scheme for damped dynamic systems with stochastic coefficients and homogeneous Dirichlet boundary conditions. The fundamental idea involved here is to solve the stochastic dynamic system in the frequency domain by projecting the solution into a reduced finite dimensional spatio-random vector basis spanning the stochastic Krylov subspace to approximate the response. Subsequently, Galerkin weighting coefficients have been employed to minimize the error induced due to the use of the reduced basis and a finite order of the spectral functions and hence to explicitly evaluate the stochastic system response. The statistical moments of the solution have been evaluated at all frequencies to illustrate and compare the stochastic system response with the deterministic case. The results have been validated with direct Monte-Carlo simulation for different correlation lengths and variability of randomness.

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.

Acoustic emission based damage localization in composites structures using Bayesian identification

Acoustic emission based damage detection in composite structures is based on detection of ultra high frequency packets of acoustic waves emitted from damage sources (such as fibre breakage, fatigue fracture, amongst others) with a network of distributed sensors. This non-destructive monitoring scheme requires solving an inverse problem where the measured signals are linked back to the location of the source. This in turn enables rapid deployment of mitigative measures. The presence of significant amount of uncertainty associated with the operating conditions and measurements makes the problem of damage identification quite challenging. The uncertainties stem from the fact that the measured signals are affected by the irregular geometries, manufacturing imprecision, imperfect boundary conditions, existing damages/structural degradation, amongst others. This work aims to tackle these uncertainties within a framework of automated probabilistic damage detection. The method trains a probabilistic model of the parametrized input and output model of the acoustic emission system with experimental data to give probabilistic descriptors of damage locations. A response surface modelling the acoustic emission as a function of parametrized damage signals collected from sensors would be calibrated with a training dataset using Bayesian inference. This is used to deduce damage locations in the online monitoring phase. During online monitoring, the spatially correlated time data is utilized in conjunction with the calibrated acoustic emissions model to infer the probabilistic description of the acoustic emission source within a hierarchical Bayesian inference framework. The methodology is tested on a composite structure consisting of carbon fibre panel with stiffeners and damage source behaviour has been experimentally simulated using standard H-N sources. The methodology presented in this study would be applicable in the current form to structural damage detection under varying operational loads and would be investigated in future studies.

PROBABILISTIC SENSITIVITY ANALYSIS OF CORRUGATED SKINS WITH RANDOM ELASTIC PARAMETERS AND SURFACE TOPOLOGY

The distinction between epistemic and aleatory uncertainty can sometimes be useful for practical purposes. In principle, epistemic uncertainty is reducible by obtaining more or better information. However, the boundaries between both types of uncertainties are often blurred. Fortunately, it is often the case that not all sources of uncertainty have the same impact on the predictive performance of numerical models, so that there is considerable computational benefit from investigating potential simplification in the number of parameters that should be modelled as being stochastic. In order to determine which parameters are the most important in terms of their contribution to the output’s uncertainty, probabilistic sensitivity analysis can be performed. We apply this idea to a numerical model of corrugated skins. Corrugated skins are particularly suitable for morphing applications in aerospace structures, largely due to the very high compliance they offer along the corrugation direction.

Uncertainty analysis of corrugated skin with random elastic parameters and surface topology

Uncertainty analysis of corrugated skins has been performed for random perturbations in geometric and elastic parameters of a chosen baseline model. The corrugated skins are particularly suitable for morphing applications in aerospace structures and their sensitivity to the various input uncertainties is a major concern in their design. The various sources of uncertainty include random perturbations of the geometrical parameters of the corrugation units, surface roughness and parametric uncertainty of the elastic parameters. These uncertainties are described here within the probabilistic framework and have been incorporated into the discretized stochastic finite element model used for their analysis. The propagation of these uncertainties to the dynamic response of the structure is a computationally intensive exercise especially for high dimensional stochastic models. Such high dimensional models have been resolved with statistical methods such as Gaussian Process Emulation and polynomial interpolation based sparse grid collocation techniques. The brute force Monte Carlo simulation technique results have been used as benchmark solutions. A global sensitivity analysis has been performed to identify the key uncertainty sources which affect the system response and the equivalent models using Sobol’s importance measure.

Stochastic Structural Dynamics Using Frequency Adaptive Basis Functions

A novel Galerkin subspace projection scheme for structural dynamic systems with stochastic parameters is developed in this chapter. The fundamental idea is to solve the discretised stochastic damped dynamical system in the frequency domain by projecting the solution into a reduced subspace of eigenvectors of the underlying deterministic operator. The associated complex random coefficients are obtained as frequency-dependent quantities, termed as spectral functions. Different orders of spectral functions are proposed depending on the order of the terms retained in the expression. Subsequently, Galerkin weighting coefficients are employed to minimise the error induced due to the reduced basis and finite order spectral functions. The complex response quantity is explicitly expressed in terms of the spectral functions, eigenvectors and the Galerkin weighting coefficients. The statistical moments of the solution are evaluated at all frequencies including the resonance and antiresonance frequencies for a fixed value of damping. Two examples involving a beam and a plate with stochastic properties subjected to harmonic excitations are considered. The results are compared to direct Monte Carlo simulation and polynomial chaos expansion for different correlation lengths and variability of randomness.

Transient Dynamics of Stochastic Structural Systems using a Reduced Order Spectral Function Approach

The transient dynamics of randomly parametrized structural systems have been considered in context of the stochastic finite element methods for handling a class of stochastic partial differential equations and homogeneous Dirichlet boundary conditions. An approach for solving the time domain equations using highly non-linear spectral functions approximating the solution in a reduced subspace has been proposed. The spectral functions of different orders can be expressed in terms of the spectral properties of the deterministic system matrices. It is shown that the solution can be obtained using a truncated finite series comprising of functions of random variables used to model the parametric uncertainty and the eigen-spectrum of the physical system. The solution at each time step has been computed iteratively with Newmark’s method of time integration. A semi-statical hybrid analytical and simulation based computational approach has been utilized to obtain the moments and probability density functions of the solution. The results have been compared with the Polynomial Chaos solution in terms of accuracy and computational efficiency. Direct Monte Carlo simulations, which serve as benchmark solutions, have been performed in the probability space for different degrees of variability to validate the results.