Path Integrals in Stochastic Engineering Dynamics

This book organizes and explains, in a systematic and pedagogically effective manner, recent advances in path integral solution techniques with applications in stochastic engineering dynamics. It fills a gap in the literature by introducing to the engineering mechanics community, for the first time in the form of a book, the Wiener path integral as a potent uncertainty quantification tool. Since the path integral flourished within the realm of quantum mechanics and theoretical physics applications, most books on the topic have focused on the complex-valued Feynman integral with only few exceptions, which present path integrals from a stochastic processes perspective. Remarkably, there are only few papers, and no books, dedicated to path integral as a solution technique in stochastic engineering dynamics. Summarizing recently developed techniques, this volume is ideal for engineering analysts interested in further establishing path integrals as an alternative potent conceptual and computational vehicle in stochastic engineering dynamics. Preface Contents List of Abbreviations 1 Introduction 1.1 Stochastic Engineering Dynamics 1.1.1 Historical Aspects 1.1.2 Persistent and Current Challenges due to Increasingly Complex Modeling 1.1.3 State-of-the-Art Solution Techniques and Limitations 1.2 Auxiliary Concepts 1.2.1 Markov Processes and Stochastic Differential equations 1.2.2 Discretized Chapman-Kolmogorov Equation Perspective 1.3 Scope of the Book and Outline 2 Wiener Path Integral Formalism 2.1 Current and Potential Impact of Path Integrals 2.2 Functional Integral over the Space of Paths: A General Formalism 2.3 Functional Integral over the Space of Paths: The Special Case of Markovian Response Processes 2.4 Functional Series Expansion and Most Probable Path Approximation 2.5 WPI Formalism Encapsulation 3 Linear Systems Under Gaussian White Noise Excitation: Exact Closed-Form Solutions 3.1 Preliminary Remarks 3.2 SDOF Linear Oscillator 3.3 MDOF Linear Oscillator 3.4 Examples 3.4.1 2-DOF Linear Oscillator 3.4.2 Beam Bending Problem with Young's Modulus Modeled as a Stochastic Field: An Alternative Perspective 4 Nonlinear Systems Under Gaussian White Noise Excitation 4.1 Preliminary Remarks 4.2 WPI Formulation 4.3 A Brute-Force Numerical Solution Treatment 4.3.1 Variational Problem Numerical Treatment 4.3.2 Mechanization of the Numerical Implementation 4.4 Numerical Examples 4.4.1 Duffing Nonlinear Oscillator 4.4.2 2-DOF System with Cubic Damping and Stiffness Nonlinearities 5 Nonlinear Systems Under Non-White, Non-Gaussian, and Nonstationary Excitation 5.1 Preliminary Remarks 5.2 Accounting for Non-White and Non-Gaussian Excitation Modeling 5.3 Accounting for Nonstationary Excitation Modeling 5.4 Numerical Examples 5.4.1 Structural System Subject to Nonlinear Flow-Induced Forces 5.4.2 Duffing Nonlinear Oscillator Subject to Time-Modulated Non-White Excitation Process 6 Nonlinear Systems with Singular Diffusion Matrices: A Broad Perspective Including Hysteresis Modeling 6.1 Preliminary Remarks 6.2 Generalization of the WPI Formulation to Account for Singular Diffusion Matrices: A Constrained Variational Problem 6.3 Constrained Variational Problem Solution Treatment 6.3.1 A Lagrange Multipliers Approach 6.3.2 A Rayleigh-Ritz Approach 6.3.2.1 Linear Constraints 6.3.2.2 Nonlinear Constraints 6.4 Numerical Examples 6.4.1 Oscillator with Only a Subset of Its DOFs Stochastically Excited 6.4.1.1 Linear Oscillator with Linear Constraints 6.4.1.2 Linear Oscillator Under Kanai-Tajimi Earthquake Excitation 6.4.1.3 Nonlinear Oscillator with Linear Constraints 6.4.1.4 Nonlinear Oscillator with Nonlinear Constraints 6.4.2 Bouc-Wen Hysteretic Oscillator 6.4.3 Nonlinear Electromechanical Energy Harvester 7 High-Dimensional Nonlinear Systems: Circumventing the Curse of Dimensionality via a Reduced-Order Formulation 7.1 Preliminary Remarks 7.2 Generalization of the WPI Formulation to Account for Mixed Fixed/Free Boundary Conditions 7.2.1 WPI Representation of Lower-Dimensional Joint PDF 7.2.2 Most Probable Path with Mixed Fixed/Free Boundaries 7.2.3 Generalization to Treat MDOF Systems Governed by Second-Order SDEs 7.2.4 Computational Aspects 7.2.5 Mechanization of the Numerical Implementation 7.3 Numerical Examples 7.3.1 Structural System Subject to Nonlinear Flow-Induced Forces 7.3.2 100-DOF System Modeling an Array of Coupled Nonlinear Nano-mechanical Oscillators 8 Efficient Numerical Implementation Strategies via Sparse Representations and Compressive Sampling 8.1 Preliminary Remarks 8.2 Response Joint PDF Representations: Multidimensional Global Bases 8.2.1 Kronecker Product Representation 8.2.2 Positive Definite Functions Representation 8.2.3 Mechanization of the Numerical Implementation 8.3 Response Joint PDF Representations: Sparse Expansions and Compressive Sampling 8.3.1 Sparse Polynomial Approximations and Group Sparsity 8.3.2 Optimization Algorithm Aspects 8.3.3 Performance Analysis 8.3.4 Quantification of Computational Efficiency Enhancement 8.3.5 Mechanization of the Numerical Implementation 8.4 Numerical Examples 8.4.1 Duffing Nonlinear Oscillator with a Hardening Restoring Force: A Kronecker Product Numerical Implementation 8.4.2 Duffing Nonlinear Oscillator with a Bimodal Response PDF: A Kronecker Product Numerical Implementation 8.4.3 Beam Bending Problem with a Non-Gaussian and Non-homogeneous Stochastic Young's Modulus: A Positive Definite Functions Numerical Implementation 8.4.4 10-DOF System with Cubic Damping and Stiffness Nonlinearites: A Compressive Sampling Numerical Implementation 9 An Enhanced Quadratic WPI Approximation with Applications to Nonlinear System Reliability Assessment 9.1 Preliminary Remarks 9.2 Nonlinear System Response Joint PDF Determination: A Quadratic WPI Approximation 9.2.1 Truncated Functional Series Expansion and WPI Approximation 9.2.2 A Closed-Form Expression for Efficient Calculation of the Fluctuation Factor 9.2.3 Computational Aspects 9.2.4 Advantages of the Quadratic WPI Approximation for Reliability Assessment Applications 9.2.5 Mechanization of the Numerical Implementation 9.3 Numerical Examples 9.3.1 SDOF Examples 9.3.1.1 Duffing Nonlinear Oscillator with a Bimodal Response PDF 9.3.1.2 Nonlinear Oscillator with an Asymmetric Response PDF 9.3.2 2-DOF Example 10 Epilogue Appendix A A.1 Appendix to the Numerical Example of Sect.3.4.1: Comparisons of Exact Solutions Obtained by the WPI Technique and by Alternative Solution Methodologies References Index