Xuechuan Wang - Computational Methods for Nonlinear Dynamical Systems: Theory and Applications in Aerospace Engineering

1. Tables in Chapter 2
2. Tables in Chapter 3
3. Tables in Chapter 4
4. Tables in Chapter 6
5. Tables in Chapter 7
6. Tables in Chapter 8

1. Figures in Chapter 2
2. Figures in Chapter 3
3. Figures in Chapter 4
4. Figures in Chapter 5
5. Figures in Chapter 6
6. Figures in Chapter 7
7. Figures in Chapter 8

Xuechuan Wang

School of Astronautics, Northwestern Polytechnical University, Xi'an, Shaanxi, China

Xiaokui Yue

School of Astronautics, Northwestern Polytechnical University, Xi'an, Shaanxi, China

Honghua Dai

School of Astronautics, Northwestern Polytechnical University, Xi'an, Shaanxi, China

Haoyang Feng

School of Astronautics, Northwestern Polytechnical University, Xi'an, Shaanxi, China

Satya N. Atluri

Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, United States

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This book aims to introduce many recently developed high-performance computational methods for strongly nonlinear dynamical systems and their applications in practical engineering problems for graduate students and researchers. The study of computational methods covers a broad range of mathematical problems emerging from science and engineering. Leveraging the wide usage of computers, it penetrates most, if not all, modern disciplines. Even sociology and politics are benefiting from computational methods in their state-of-the-art branches, such as social network dynamics. Nevertheless, the most significant developments of computational methods concentrate on applied mathematics and mechanics. The need for simulations in solid mechanics, fluid mechanics, multibody dynamics, and multiphysics has spawned off-the-shelf software that integrates ready-to-use computational methods. New methods are continuously being proposed, improved, and tested to achieve better performance.

The foundation of the computational methods that we use today was laid in the age of Isaac Newton and Leonhard Euler. Using the concepts of differentiation and the Taylor series, the finite difference method was developed. Its early history is marked by the Euler method. Hundreds of variants have been proposed, including the trapezoidal method, the RungeKutta method, and the Newmark method. Some of these methods have been used as standard numerical solvers in commercial software (MATLAB, ADAMS, ABAQUS, etc.). Moreover, it was found that the finite difference method can be introduced in a more general way via a weighted residual method, from which the finite element, finite volume, and boundary element methods can be derived. A practical framework of a weighted residual method was built in the 1950s via the development of the finite element method. It quickly stimulated the development of computer-aided engineering and design technology. Alongside that, the asymptotic methods were developed from the early works by Henri Poincar in his study of three-body problems. The asymptotic methods are very useful when the functional of the problem to be solved is available; therefore they are often applied in weakly nonlinear systems. The best-known asymptotic method for solving nonlinear differential equations is the perturbation method. In earlier times, the use of the perturbation method was limited to astronomical calculations. Scholars such as Laplace and Lagrange used it to investigate the perturbed motion of planets around the sun. Then Urbain Le Verrier successfully predicted the existence of the planet Neptune with this method. After this remarkable event, the application of the perturbation method was gradually broadened to the more general field of nonlinear mechanics.

By contrast, a strongly nonlinear dynamical system possesses complicated behaviors that cannot be found in linearized or weakly nonlinear systems. The difficulties in dealing with strongly nonlinear dynamical systems are twofold: first the simulation of dynamical responses and then the dynamical analysis. Both rely heavily on nonlinear computational methods to tackle the deterioration of convergence, accuracy, and efficiency, which arise from strong nonlinearities. It is a general and interdisciplinary problem to deal with nonlinearity, so we will not discuss it extensively in this book but will focus on dynamical systems that can be modeled or reduced to ordinary differential equations. Although there are numerous finite difference methods for solving ordinary differential equations, they are far from being satisfactory in theoretical analysis and engineering application in strongly nonlinear problems. It is a key point of this book to introduce useful tools to tackle computational difficulties in nonlinear dynamical systems. The solutions for nonlinear dynamical systems are broadly divided into two categories: periodic and transient (nonperiodic). For periodic problems the methodology of a new time domain collocation method is presented. Its equivalence to the high-dimensional harmonic balance method is explained in detail. It shows superiority to other methods in approximating subtle periodic responses of nonlinear dynamical systems. For transient responses the methodology of local variational iteration is presented. It provides possibilities for a new category of numerical methods by combining the principle of a weighted residual method and an asymptotic method. Compared with finite difference methods, the local variational iteration method can achieve much higher computational accuracy and efficiency by using very large step sizes. Applications of these methods are illustrated through examples in orbital mechanics and structural mechanics. The methodologies that are introduced in this book are expected to find applications in a broader range of nonlinear systems.