Probabilistic Finite Element Model Updating Using Bayesian Statistics

Applications to Aeronautical and Mechanical Engineering

Tshilidzi Marwala, Ilyes Boulkaibet, Sondipon Adhikari

• Publisher: Wiley
• Year: 2016
• Print ISBN: 9781119153030
• eBook ISBN: 9781119153016
• Edition: 1
• Topic: Tecnología e ingeniería
• Subtopic: Ingeniería mecánica

Probabilistic Finite Element Model Updating Using Bayesian Statistics: Applications to Aeronautical and Mechanical Engineering 

Tshilidzi Marwala and Ilyes Boulkaibet, University of Johannesburg, South Africa

Sondipon Adhikari, Swansea University, UK

 

Covers the probabilistic finite element model based on Bayesian statistics with applications to aeronautical and mechanical engineering

 

Finite element models are used widely to model the dynamic behaviour of many systems including in electrical, aerospace and mechanical engineering.

The book covers probabilistic finite element model updating, achieved using Bayesian statistics. The Bayesian framework is employed to estimate the probabilistic finite element models which take into account of the uncertainties in the measurements and the modelling procedure. The Bayesian formulation achieves this by formulating the finite element model as the posterior distribution of the model given the measured data within the context of computational statistics and applies these in aeronautical and mechanical engineering.

Probabilistic Finite Element Model Updating Using Bayesian Statistics contains simple explanations of computational statistical techniques such as Metropolis-Hastings Algorithm, Slice sampling, Markov Chain Monte Carlo method, hybrid Monte Carlo as well as Shadow Hybrid Monte Carlo and their relevance in engineering.

 

Key features:

• Contains several contributions in the area of model updating using Bayesian techniques which are useful for graduate students.
• Explains in detail the use of Bayesian techniques to quantify uncertainties in mechanical structures as well as the use of Markov Chain Monte Carlo techniques to evaluate the Bayesian formulations.

 

The book is essential reading for researchers, practitioners and students in mechanical and aerospace engineering.

1
Introduction to Finite Element Model Updating

1.1 Introduction

Finite element model updating methods are intended to correct and improve a numerical model to match the dynamic behaviour of real structures (Marwala, 2010). Modern computers, with their ability to process large matrices at high speed, have facilitated the formulation of many large and complicated numerical models, including the boundary element method, the finite difference method and the finite element models. This book deals with the finite element model that was first applied in solving complex elasticity and structural analysis problems in aeronautical, mechanical and civil engineering. Finite element modelling was proposed by Hrennikoff (1941) and Courant and Robbins (1941). Courant applied the Ritz technique and variational calculus to solve vibration problems in structures (Hastings et al., 1985). Despite the fact that the approaches used by these researchers were different from conventional formulations, some important lessons are still relevant. These differences include mesh discretisation into elements (Babuska et al., 2004).

The Cooley–Turkey algorithms, which are used to speedily obtain Fourier transformations, have facilitated the development of complex techniques in vibration and experimental modal analysis. Conversely, the finite element model ordinarily predicts results that are different from the results obtained from experimental investigation. Among reasons for the discrepancy between finite element model prediction and experimentally measured data are as the following (Friswell and Mottershead, 1995; Marwala, 2010; Dhandole and Modak, 2011):

• model structure errors resulting from the difficulty in modelling damping and complex shapes such as joints, welds and edges;
• model order errors resulting from the difficulty in modelling non‐linearity and often assuming linearity;
• model parameter errors resulting in difficulty in identifying the correct material properties;
• errors in measurements and signal processing.

In finite element model updating, it is assumed that the measurements are correct within certain limits of uncertainty and, for that reason, a finite element model under consideration will need to be updated to better reflect the measured data. Additionally, finite element model updating assumes that the difficulty in modelling joints and other complicated boundary conditions can be compensated for by adjusting the material properties of the relevant elements. In this book, it is also assumed that a finite element model is linear and that damping is sufficiently low not to warrant complex modelling (Mottershead and Friswell, 1993; Friswell and Mottershead, 1995). Using data from experimental measurements, the initial finite element model is updated by correcting uncertain parameters so that the model is close to the measured data. Alternatively, finite element model updating is an inverse problem and the goal is to identify the system that generated the measured data (Brincker et al., 2001; Dhandole and Modak, 2010; Zhang et al., 2011; Boulkaibet, 2014; Fuellekrug et al., 2008; Cheung and Beck, 2009; Mottershead et al., 2000).

There are two main approaches to finite element model updating, namely, maximum likelihood and Bayesian methods (Marwala, 2010; Mottershead et al., 2011). In this book, we apply a Bayesian approach to finite element model updating.

1.2 Finite Element Modelling

Finite element models have been applied to aerospace, electrical, civil and mechanical engineering in designing and developing products such as aircraft wings and turbo‐machinery. Some of the applications of finite element modelling are (Marwala, 2010): thermal problems, electromagnetic problems, fluid problems and structural modelling. Finite element modelling typically entails choosing elements and basis functions (Chandrupatla and Belegudu, 2002; Marwala, 2010). Generally, there are two types of finite element analysis that are used: two‐dimensional and three‐dimensional modelling (Solin et al., 2004; Marwala, 2010).

Two‐dimensional modelling is simple and computationally efficient. Three‐dimensional modelling, on the other hand, is more accurate, though computationally expensive. Finite element analysis can be formulated in a linear or non‐linear fashion. Linear formulation is simple and usually does not consider plastic deformation, which non‐linear formulation does consider. This book only deals with linear finite element modelling, in the form of a second‐order ordinary differential equation of relations between mass, damping and stiffness matrices. A finite element model has nodes, with a grid called a mesh, as shown in Figure 1.1 (Marwala, 2001). The mesh has material and structural properties with particular loading and boundary conditions. Figure 1.1 shows the dynamics of a cylinder, and the mode shape of the first natural frequency occurring at 433 Hz.

Figure 1.1 A finite element model of a cylindrical shell

These loaded nodes are assigned a specific density all over the material, in accordance with the expected stress levels of that area (Baran, 1988). Sections which undergo more stress will then have a higher node density than those which experience less or no stress. Points of stress concentration may have fracture points of previously tested materials, joints, welds and high‐stress areas. The mesh may be imagined as a spider’s web so that, from each node, a mesh element extends to each of the neighbouring nodes. This web of vectors has the material properties of the object, resulting in a study of many elements.

On implementing finite element modelling, a choice of elements needs to be made and these include beam, plate, shell elements or solid elements. A question that needs to be answered when applying finite element analysis is whether the mater...

Table of contents

• Cover
• Title Page
• Table of Contents
• Acknowledgements
• Nomenclature
• 1 Introduction to Finite Element Model Updating
• 2 Model Selection in Finite Element Model Updating
• 3 Bayesian Statistics in Structural Dynamics
• 4 Metropolis–Hastings and Slice Sampling for Finite Element Updating
• 5 Dynamically Weighted Importance Sampling for Finite Element Updating
• 6 Adaptive Metropolis–Hastings for Finite Element Updating
• 7 Hybrid Monte Carlo Technique for Finite Element Model Updating
• 8 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating
• 9 Separable Shadow Hybrid Monte Carlo in Finite Element Updating
• 10 Evolutionary Approach to Finite Element Model Updating
• 11 Adaptive Markov Chain Monte Carlo Method for Finite Element Model Updating
• 12 Conclusions and Further Work
• Appendix A: Experimental Examples
• Appendix B: Markov Chain Monte Carlo
• Appendix C: Gaussian Distribution
• Index
• End User License Agreement