Chapter 1 Introduction 1
1.1 Engineering background 1
1.2 Engineering uncertainties 4
1.2.1 Unpredictable loading conditions 5
1.2.2 Randomness of material properties 7
1.2.3 Major developments in stochastic finite element methods 9
1.3 The aim and layout of book 11
1.3.1 The aim 11
1.3.2 Book layout 12
Chapter 2 Mathematical background 17
2.1 Probability theory 17
2.1.1 The development of probability theory 17
2.1.2 Probability space 18
2.1.3 Distribution 19
2.1.4 Characteristic function 20
2.1.5 Moments 22
2.1.6 Cumulants 24
2.1.7 Covariance matrix 26
2.1.8 Entropy 27
2.1.9 Stochastic convergences 29
2.1.10 Basic limit theorems and inequities 30
2.2 Stochastic field 32
2.2.1 Second-order random field 33
2.2.2 Mercer’s theorem 34
2.2.3 Stationary random fields 35
2.2.4 Spectral expansion of stationary random fields 37
2.2.5 Ergodicity 39
2.2.6 Gaussian random fields 40
2.3 Stochastic Analysis 41
2.3.1 Brownian motion 41
2.3.2 Stochastic differential equations 42
2.4 Statistics 43
2.4.1 Estimation 43
2.4.2 Preliminary nonparametric statistics 45
Chapter 3 Representation of random fields 49
3.1 Constitutive laws 49
3.2 Stochastic constitutive laws 51
3.2.1 The spectral representation method 53
3.2.2 Karhunen-Loève expansion method 54
3.2.3 Other representation methods 56
3.3 Representation of Non-Gaussian fields 58
Chapter 4 Solvers of stochastic linear equations 63
4.1 The discretization of the stochastic fields 63
4.2 Monte Carlo method 65
4.3 Perturbation method 66
4.4 The neumann expansion method 67
4.5 The polynomial chaos expansion method 68
4.6 Thejoint diagonalization strategy 70
Chapter 5 Aircraft engineering applications 75
5.1 Aircraft structural health monitoring with uncertainties 76
5.2 Aircraft composite materials 78
5.3 Analysis of typical aircraft structures 79
5.3.1 Aircraft composite material modeling with random properties 80
5.3.2 Stochastic modeling of commercial aircraft rudder 85
5.3.3 Stochastic modeling of reinforced composite floor under fire conditions 89
Chapter 6 Reliability analysis 94
6.1 Introduction of engineering reliability analysis 94
6.2 Codes of practice 96
6.3 Airworthiness regulations 97
6.4 Reliability analysis for a single structural element 100
6.4.1 Random properties of loading and resistance only 100
6.4.2 Single structural member with multi random variables 101
6.4.3 Reliability index for linear failure functions 102
6.4.4 Reliability index for nonlinear failure functions 102
6.5 Application to reliability analysis 105
6.5.1 Reliability analysis of simply supported beam 105
6.5.2 Reliability analysis of structural glass 107
Chapter 7 Uncertainties in aircraft big data and de-noising technologies 112
7 1 Uncertainties in aircraft big data 112
7.1.1 The“Cocktail party affect” 113
7.1.2 System model 114
7.1.3 Literature review 118
7.2 Blind source separation algorithm Ⅰ 119
7.2.1 Cost function 119
7.2.2 Minimizing the cost function via numerical procedures 120
7.2.3 Parameterization of A 121
7.2.4 The explicit form of matrix J 121
7.2.5 Numerical methods 122
7.2.6 Numerical simulation 127
7 3 Blind source separation algorithm Ⅱ 132
7.3.1 Cost function 132
7.3.2 Numerical simulations 136
Chapter 8 Future directions 142
Index Ⅰ General convention 146
Index Ⅱ Abbreviation 147
Index Ⅲ Symbolic notation 149