Probabilistic Finite Element Model Updating Using Bayesian Statistics (eBook)

Applications to Aeronautical and Mechanical Engineering

Tshilidzi Marwala, Ilyes Boulkaibet, Sondipon Adhikari (Autoren)

eBook Download: EPUB

2016
John Wiley & Sons (Verlag)
978-1-119-15301-6 (ISBN)

Infos

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.

Tshilidzi Marwala is a Professor of Mechanical and Electrical Engineering as well as Deputy Vice-Chancellor at the University of Johannesburg. He holds a Bachelor of Science in Mechanical Engineering from Case Western Reserve University, a Master of Mechanical Engineering from the University of Pretoria, a PhD in Engineering from Cambridge University and was a post-doctoral researcher at Imperial College (London). He is a Fellow of TWAS and a distinguished member of the ACM. His research interests are multi-disciplinary and include the applications of computational intelligence to engineering, computer science, finance, social science and medicine. He has supervised 45 Masters and 19 PhD students and has published 8 books and over 260 papers. He is an associate editor of the International Journal of Systems Science.

Dr. Ilyes Boulkaibet is currently a researcher at the University of Johannesburg. He received a PhD from the University of Johannesburg, a second MSc from Stellenbosch University, an MSc from the University of Constantine 1 Algeria, and a Bachelor of Engineering from University of Constantine 1 Algeria. Dr. Ilyes Boulkaibet has published papers in international journals and has participated in numerous conferences including the International Modal Analysis Conference. Dr. Boulkaibet's research areas are multidisciplinary in nature and include uncertainty quantification in computational mechanics, dynamics of complex systems, inverse problems for linear and non-linear dynamics and control systems.
Professor Adhikari is the chair of Aerospace Engineering in the College of Engineering of Swansea University. He received his MSc from the Indian Institute of Science and a PhD from the University of Cambridge. He was a lecturer at the Bristol University and a Junior Research Fellow in Fitzwilliam College, Cambridge. He has been a visiting Professor at the University of Johannesburg, Carleton University and the Los Alamos National Laboratory . Professor Adhikari's research areas are multidisciplinary in nature and include uncertainty quantification in computational mechanics, bio- and nano-mechanics (nanotubes, graphene, cell mechanics, nano-bio sensors), dynamics of complex systems, inverse problems for linear and non-linear dynamics and vibration energy harvesting.

Textauszug

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 material is isotropic (identical throughout the material), orthotropic (only identical at 90°) or anisotropic (different throughout the material) (Irons and Shrive, 1983; Zienkiewicz, 1986; Marwala, 2010).

Finite element analysis has been applied to model the following problems (Zienkiewicz, 1986; Marwala, 2010):

• vibration analysis for testing a structure for random vibrations, impact and shock;
• fatigue analysis to approximate the life cycle of a material or a structure due to cyclical loading;
• heat transfer analysis to model conductivity or thermal fluid dynamics of the material or structure.

Hlilou et al. (2009) successfully applied finite element analysis in softening material behaviour, while Zhang and Teo (2008) successfully applied it in the treatment of a lumbar degenerative disc disease. White et al. (2008) successfully applied finite element analysis for shallow‐water modelling, while Pepper and Wang (2007) successfully applied it in wind energy assessment of renewable energy in Nevada. Miao et al. (2009) successfully applied a three‐dimensional finite element analysis model in the simulation of shot peening. Bürg and Nazarov (2015) successfully applied goal‐oriented adaptive finite element methods in elliptic problems, while Amini et al. (2015) successfully applied finite element modelling in functionally graded piezoelectric harvesters. Haldar et al. (2015) successfully applied finite element modelling in the study of the flexural behaviour of singly curved sandwich composite structures, while Millar and Mora (2015) successfully applied finite element methods to study the buckling in simply supported Kirchhoff plates. Jung et al. (2015) successfully used finite element models and computed tomography to estimate cross‐sectional constants of composite blades, while Evans and Miller (2015) successfully applied a finite element model to predict the failure of pressure vessels. Other successful applications of finite element analysis are in the areas of metal powder compaction processing (Rahman et al., 2009), ferroelectric materials (Schrade et al., 2007), rock mechanics (Chen et al., 2009), orthopaedics (Easley et al., 2007), carbon nanotubes (Zuberi and Esat, 2015), nuclear reactors (Wadsworth et al., 2015) and elastic wave propagation (Gao et al., 2015; Gravenkamp et al., 2015).

1.3 Vibration Analysis

An important aspect to consider when implementing finite element analysis is the kind of data that the model is supposed to predict. It can predict data in many domains, such as the time, modal, frequency and time–frequency domains (Marwala, 2001, 2010). This book is concerned with constructing finite element models to predict measured data more accurately. Ideally, a finite element model is supposed to predict measured data irrespective of the domain in which the data are presented. However, this is not necessarily the case because models updated in the time domain will not necessarily predict data in the modal domain as accurately as they will for data in the time domain. To deal with this issue, Marwala and Heyns (1998) used data in the modal and frequency domains simultaneously to update the finite element model in a multi‐criteria optimisation fashion. Again, whichever domain is used, the updated model performs less well on data in a different domain than those used in the updating process. In this book, we use data in the modal domain. Raw data are measured in the time domain and Fourier analysis techniques...

Produkt-Details

• Erscheint lt. Verlag: 23.9.2016
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Fahrzeugbau / Schiffbau
• Themenwelt: Technik ► Luft- / Raumfahrttechnik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Bauingenieur- u. Bauwesen • Baustatik u. Baumechanik • Bayesian analysis • Bayesian theory • Bayes-Verfahren • Civil Engineering & Construction • Computational / Numerical Methods • Finite-Element-Methode • Finite Element Model • iterative methods • Markov Chain Monte Carlo • Maschinenbau • mechanical engineering • Model Selection • Model Updating • Optimization • Probabilistic model updating • Rechnergestützte / Numerische Verfahren im Maschinenbau • Rechnergestützte / Numerische Verfahren im Maschinenbau • Statistics • Statistik • structural dynamics • Structural Theory & Structural Mechanics • Uncertainty quantifications
• ISBN-10: 1-119-15301-8 / 1119153018
• ISBN-13: 978-1-119-15301-6 / 9781119153016