Stochastic Finite Element Methods : An Introduction.

Intro -- Preface -- Acknowledgements -- Contents -- List of Figures -- List of Tables -- 1 Stochastic Processes -- 1.1 Moments of Random Processes -- 1.1.1 Autocorrelation and Autocovariance Function -- 1.1.2 Stationary Stochastic Processes -- 1.1.3 Ergodic Stochastic Processes -- 1.2 Fourier Integrals and Transforms -- 1.2.1 Power Spectral Density Function -- 1.2.2 The Fourier Transform of the Autocorrelation Function -- 1.3 Common Stochastic Processes -- 1.3.1 Gaussian Processes -- 1.3.2 Markov Processes -- 1.3.3 Brownian Process -- 1.3.4 Stationary White Noise -- 1.3.5 Random Variable Case -- 1.3.6 Narrow and Wideband Random Processes -- 1.3.7 Kanai--Tajimi Power Spectrum -- 1.4 Solved Numerical Examples -- 1.5 Exercises -- 2 Representation of a Stochastic Process -- 2.1 Point Discretization Methods -- 2.1.1 Midpoint Method -- 2.1.2 Integration Point Method -- 2.1.3 Average Discretization Method -- 2.1.4 Interpolation Method -- 2.2 Series Expansion Methods -- 2.2.1 The Karhunen--Loève Expansion -- 2.2.2 Spectral Representation Method -- 2.2.3 Simulation Formula for Stationary Stochastic Fields -- 2.3 Non-Gaussian Stochastic Processes -- 2.4 Solved Numerical Examples -- 2.5 Exercises -- 3 Stochastic Finite Element Method -- 3.1 Stochastic Principle of Virtual Work -- 3.2 Nonintrusive Monte Carlo Simulation -- 3.2.1 Neumann Series Expansion Method -- 3.2.2 The Weighted Integral Method -- 3.3 Perturbation-Taylor Series Expansion Method -- 3.4 Intrusive Spectral Stochastic Finite Element Method (SSFEM) -- 3.4.1 Homogeneous Chaos -- 3.4.2 Galerkin Minimization -- 3.5 Closed Forms and Analytical Solutions with Variability Response Functions (VRFs) -- 3.5.1 Exact VRF for Statically Determinate Beams -- 3.5.2 VRF Approximation for General Stochastic FEM Systems -- 3.5.3 Fast Monte Carlo Simulation -- 3.5.4 Extension to Two-Dimensional FEM Problems.

3.6 Solved Numerical Examples -- 3.7 Exercises -- 4 Reliability Analysis -- 4.1 Definition -- 4.1.1 Linear Limit-State Functions -- 4.1.2 Nonlinear Limit-State Functions -- 4.1.3 First- and Second-Order Approximation Methods -- 4.2 Monte Carlo Simulation (MCS) -- 4.2.1 The Law of Large Numbers -- 4.2.2 Random Number Generators -- 4.2.3 Crude Monte Carlo Simulation -- 4.3 Variance Reduction Methods -- 4.3.1 Importance Sampling -- 4.3.2 Latin Hypercube Sampling (LHS) -- 4.4 Monte Carlo Methods in Reliability Analysis -- 4.4.1 Crude Monte Carlo Simulation -- 4.4.2 Importance Sampling -- 4.4.3 The Subset Simulation (SS) -- 4.5 Artificial Neural Networks (ANN) -- 4.5.1 Structure of an Artificial Neuron -- 4.5.2 Architecture of Neural Networks -- 4.5.3 Training of Neural Networks -- 4.5.4 ANN in the Framework of Reliability Analysis -- 4.6 Numerical Examples -- 4.7 Exercises -- Appendix A Probability Theory -- A.1 Axiomatic Probability Theory -- A.1.1 Basic Set Operations -- A.1.2 Set Equality Theorems -- A.1.3 Venn Diagrams -- A.2 Definitions of Probability -- A.2.1 Classical Definition -- A.2.2 Geometric Definition -- A.2.3 Frequentist Definition -- A.2.4 Probability Space -- A.2.5 The Axiomatic Approach to Probability -- A.3 Conditional Probability -- A.3.1 Multiplication Rule of Probability -- A.3.2 Law of Total Probability -- A.3.3 Bayes Theorem -- Appendix B Random Variables -- B.1 Distribution Functions of Random Variables -- B.1.1 Cumulative Distribution Function (cdf) -- B.1.2 Probability Density Function (pdf) -- B.2 Moments of Random Variables -- B.2.1 Central Moments -- B.3 Functions of Random Variables -- B.3.1 One to One (1--1) Mappings -- B.3.2 Not One to One (1--1) Mappings -- B.3.3 Moments of Functions of Random Variables -- B.4 Jointly Distributed Random Variables.

B.4.1 Moments of Jointly Distributed Random Variables -- B.4.2 Binomial Distribution and Bernoulli Trials -- B.5 Gaussian (Normal) Random Variables -- B.5.1 Jointly Distributed Gaussian Random Variables -- B.5.2 Gaussian Random Vectors -- B.6 Transformation to the Standard Normal Space -- B.6.1 Nataf Transformation -- B.7 Solved Numerical Examples -- B.8 Exercises -- Appendix C Subset Simulation Aspects -- C.1 Modified Metropolis- Hastings Algorithm -- C.2 SS Conditional Probability Estimators -- References.

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