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Computational Methods for Nonlinear Dynamical Systems Theory and Applications in Aerospace Engineering 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

Table of Contents Cover image Title page Copyright Preface Chapter 1. Introduction Abstract Summary 1.1 The weighted residual methods 1.2 Application of weighted residual methods 1.3 Finite difference methods 1.4 Asymptotic methods References

Chapter 2. Harmonic balance method and time domain collocation method Abstract Summary 2.1 Time Collocation in a period of oscillation 2.2 Relationship between collocation and harmonic balance 2.3 Initialization of Newton-Raphson method 2.4 Numerical examples Appendix A Appendix B References Chapter 3. Dealiasing for Harmonic Balance and Time Domain Collocation Methods Abstract Summary 3.1 Governing equations of the airfoil model 3.2 Formulation of the HB method 3.3 Formulation of the TDC method 3.4 Reconstruction harmonic balance method 3.5 Numerical examples

Appendix References Chapter 4. Application of time domain collocation in formation flying of satellites Abstract Summary 4.1 TDC searching scheme for periodic relative orbits 4.2 Initial values for TDC method 4.3 Evaluation of TDC search scheme 4.4 Numerical results Appendix References Chapter 5. Local Variational Iteration Method Abstract Summary 5.1 VIM and its relationship with PIM and ADM 5.2 Local variational iteration method 5.3 Conclusion References

Chapter 6. Collocation in Conjunction with the local variational iteration method Abstract Summary 6.1 Modifications of LVIM 6.2 Implementation of LVIM 6.3 Numerical examples 6.4 Conclusion References Chapter 7. Application of the Local Variational Iteration Method in Orbital Mechanics Abstract Summary 7.1 Local variational iteration method and quasi-linearization method 7.2 Perturbed orbit propagation 7.2.1 Comparison of local variational iteration method with the modified Chebyshev picard iteration method 7.2.2 Comparison of FAPI with Runge-Kutta 12(10) 7.3 Perturbed Lambert’s problem 7.4 Conclusion

References Chapter 8. Applications of the Local Variational Iteration Method in Structural Dynamics Abstract Summary 8.1 Elucidation of LVIM in structural dynamics 8.4 Conclusion Appendix A Appendix B Appendix C References Index

Copyright Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2023 China Science Publishing & Media Ltd. Published by Elsevier Inc. All rights reserved.

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Preface This book aims to introduce many recently developed highperformance 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 Runge–Kutta 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 Poincaré 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

p q g 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.

This book is a collection of recent work involving the collaborative research of the authors from Northwestern Polytechnical University and Texas Tech University. The contents are conveyed in eight chapters. Chapter 1 briefly depicts the outline of computational methods for linear and nonlinear, ordinary and partial differential equations. In Chapters 2–4 the global methods for solving periodic nonlinear dynamical behaviors are presented. Those methods are rooted in weighted residual methods and find their application in the space mission of formation flying. Chapters 5–8 concern local methods for solving transient nonlinear responses with an emphasis on the development of highly accurate and efficient numerical methods that can be used in real-life missions. The applications of those methods are revealed in orbital mechanics and structural dynamics.

We would like to express our deep gratitude to our colleagues and students who generously offered their help and advice in writing this book.

CHAPTER 1

Introduction Abstract Nonlinear dynamical systems are ubiquitous in science and engineering. Although linearization is often utilized to simplify the original problem, it is not applicable in some certain cases.

Some dynamical phenomena in nonlinear systems are very distinctive and cannot be revealed in linear systems. Sometimes, the consideration of nonlinearity is crucial to reveal the true dynamical behaviors that the nonlinear problem must be treated as it is. In this chapter the principal aim is to introduce the various computational methods for solving nonlinear problems.

Hence attention is focused on the developments of these techniques and their performance rather than investigating the unique and intriguing features of nonlinear dynamical systems.

The contents of this chapter are divided into four parts, in which we discuss the weighted residual methods, the application of weighted residual methods, the finite difference methods, and the asymptotic methods.

Keywords Nonlinear problems; Computational methods; Asymptotic methods; Weighted residual methods

Summary Nonlinear dynamical systems are ubiquitous in science and engineering. Although the linearization is often utilized to simplify the original problem, it is not applicable in certain cases. Some

dynamical phenomena in nonlinear systems are very distinctive, and cannot be revealed in linear systems [1–4]. Sometimes, the consideration of nonlinearity is so crucial to reveal the true dynamical behaviors that the nonlinear problem must be treated as it is. In this chapter, the principal aim is to introduce the various computational methods for solving nonlinear problems. Hence the attention is focused on the developments of these techniques and their performance, rather than investigating the unique and intriguing features of nonlinear dynamical systems. The contents of this chapter are divided into four parts, including the weighted residual methods (Section 1.1), application of weighted residual methods (Section 1.2), finite difference methods (Section 1.3), and asymptotic methods (Section 1.4).

With the aid of modern computers, the most straightforward approach to investigate the solution trajectory of a nonlinear dynamical system could be the numerical integration methods.

There are literally hundreds of different numerical integration methods, of which the majority are finite difference methods such as the Runge-Kutta method [5,6], the Hilber-Hughes-Taylor method [7], and the Newmark method [8]. These methods directly use the definition of a differential and the Taylor’s theorem to discretize the original differential system and provide piecewise approximations in very small time segments. Depending on whether the approximation in each segment is forward or backward, the methods can normally be classified into explicit and implicit methods [9,10].

In general, the implicit methods are more stable than their explicit counterparts, but the former usually involves the solution of nonlinear algebraic equations.

For nonlinear dynamical systems, long-term responses are often desired. There is the well-known class of methods named Galerkin weak-form methods in the area of numerical analysis. As a special version of the Galerkin’s approach, the Harmonic Balance (HB) method [11–13] is well-developed to predict the periodic steady-state solutions for autonomous and non-autonomous nonlinear systems.

It is a global approximation method that approximates the steadystate responses with trial functions composed of harmonics or sub-

p p harmonics or super-harmonics. The HB method is a frequency domain method. Through the balance of the harmonics, the fundamental frequency and the amplitude of the periodic motion can be obtained. However, it is difficult to rewrite nonlinear terms in the form of harmonic series, even with the use of symbolic computation; thus, the resulting algebraic equations are usually very complex. To simplify the symbolic calculations of HB method, the alternate use of the High Dimensional Harmonic Balance (HDHB) method approximates the nonlinear terms using the relationship between its harmonic form and the values of it at a set of collocation points. This approach is much more convenient, but it is also found to be accompanied by the aliasing phenomenon because of the simplification. The HDHB method is later proved to be exactly the same as the Time Domain Collocation (TDC) method [14–16], which is a global collocation method or the spectral collocation method in the time domain.

Besides the periodic responses, the non-periodic responses, including chaotic motions, are more common in nonlinear dynamical systems. Due to the lack of periodicity, the accurate prediction of long term non-periodic responses for the whole domain is difficult. Normally, the approximation is made only in a finite time interval . Since the response is non-periodic, the trial function can be any appropriate function besides the harmonics.

Take the Radial Basis Functions (RBF) for example. By assuming the trial function in terms of RBFs, a RBF Collocation method [17,18] is proposed to solve the initial value and boundary value problems of nonlinear systems. Naturally, to achieve long-term prediction of non-periodic motion of nonlinear systems, the local methods are much more preferable than their global counterparts. By dividing the entire domain into finite but small segments and approximating the solution on each segment separately, the local methods can be more efficient and reliable [19,20].

To approximate the solution of a nonlinear dynamical system with arbitrary boundary conditions, the methods based on the weighted residual principle, which is more general than the idea of the global Galerkin’s method, can be used. In the works of Atluri [19] and

Dong, etc. [20], these methods are illustrated clearly and systematically, including the Finite Element Method (FEM), the Finite Element Method (FEM), the Boundary Element Method (BEM) and the Meshless Local Petrov Galerkin (MLPG) method, etc. These methods are developed using different test functions and trial functions. Among them, the collocation method is the simplest, using the Dirac Delta functions as the test functions. The selection of trial functions is flexible. Depending on the problem, one can use harmonics, polynomials, Radial Basis Functions (RBFs) and Moving Least Square (MLS) functions. But no matter which trial function is used, it will eventually lead to a system of nonlinear algebraic equations (NAEs). Therefore, the problem ultimately becomes how to solve the NAEs. For that, many NAE solvers are proposed. The most popular Newton-like methods mostly involve the inverse of the Jacobian matrix, and can be very sensitive to the initial guess of the solution. Many efforts have been made to solve the NAEs without inverting the Jacobian and without being sensitive to the initial guess. Among there are the scalar-homotopy iterative algorithms developed by Liu, etc. [21], and Liu and Atluri [22].

In the area of solving strongly nonlinear dynamical problems, the analytical asymptotic methods are also commonly used, such as the Modified Lindstedt-Poincare Method [23], the Homotopy Perturbation Method [24,25], the Variational Iteration Method (VIM) [26,27], the Picard Iteration Method (PIM) [28], the Adomian Decomposition Method (ADM) [29], and so on. These methods start from the solution of a linearized problem and iteratively correct the initial guess so that it approaches the real solution of the nonlinear problem. It implies is that the solution of the nonlinear problem can be obtained by constructing an iterative formula involving a functional of the original equations, instead of transforming the nonlinear differential equations into NAEs to be solved.

1.1 The weighted residual methods Before treating nonlinear dynamical systems, the static problem of a beam on elastic foundation [19] is considered in this section. With

this problem as an example, the various computational methods that are rooted in the idea of weighted residuals are introduced.

1.1.1 Problem description The deflection of a beam on elastic foundation is simply described by a 4th order ordinary differential equation in dimensionless form

(1.1) where is the normalized vertical displacement (deflection), is the normalized distributed load on the beam, and is the domain of interest. The boundary conditions are expressed as follows:

in which , , , and are the boundaries where (displacement), (rotation), (moment), and (shear) are prescribed, respectively.

Depending on the boundary conditions, the problem can be either well-posed or ill-posed. For the well-posed problem, the prescriptions of and are mutually disjoint, i.e.

(1.2) where being the boundary points . The well-posed problem is physically consistent with the solid mechanics used to describe the beam, in which the shear force reaction can be obtained once the deflection is prescribed, and the moment reaction can be obtained once the rotation is prescribed. Thus, the well-posed problem is “natural”, meaning that a unique solution exists and is not very sensitive to small variations of boundary values, although the

solution does change continuously with the boundary condition. A problem that is not well-posed, i.e. Eq. (1.2) does not hold, is termed ill-posed. The ill-posed problems are very common in various engineering applications such as structural health monitoring, system control, and medical imaging.

1.1.2 Primal methods 1.1.2.1 Global unsymmetric weak form I To solve Eq. (1.1), a trial function is used to approximate the true solution. It results to a residual error in the governing equation:

(1.3) Although it is usually difficult to make the residual error diminish in general nonlinear problems, it is relatively easier and often feasible to make the weighted residuals equal to zero. With a test function , the weighted residual form of the governing equation is written as follows, which is

(1.4) Conventionally, the weighted residual form is also referred to as the weak form.

Since Eq. (1.4) involves the fourth-order derivative of the trial function , it is required that the third order derivative of should be continuous, or briefly, that should be functions. Some commonly used functions include harmonics, polynomials, radial basis functions (RBFs), and moving least squares functions (MLSFs) [20].

As to the test function , there is no requirement on its continuity.

The selection of trial and test functions is very flexible, leading to various combinations. Among them, two cases are of certain

importance in the application to nonlinear dynamical problems and will be further explored in Section 1.2.

Case 1: test functions are selected as Dirac Delta functions, leading to the collocation method,

Case 2: test functions and trial functions are both selected as the same orthogonal basis functions, leading to the Galerkin method.

The collocation method is the simplest and the most straightforward among the various branches of weighted residual method. Based on the types of trial functions, the collocation methods are divided into different classes. Three kinds of commonly used trial functions are listed herein Harmonics:

(1.5) Polynomials (first kind of Chebyshev polynomials):

(1.6)

Radial Basis Function (Gaussian):

(1.7) Note that there are numerous types of trial functions [30–32]. Each of them possesses unique properties and behaves differently in application. From the perspective of computational accuracy and efficiency, the orthogonal functions are often preferred [33], in that they can bring great convenience in numerical computation as well as symbolic operation using their orthogonality property.

No matter which kind of trial function is used, we can always write the trial function in the following form:

(1.8) where each column of the row vector represent each of the basis functions, and the column vector is the vector of undetermined coefficients.

As is stated above, when the Dirac Delta functions are used as test functions, i.e. for a group of pre-selected points along the beam, the weighted residual formulation Eq. (1.4) leads to the point collocation method [34]:

(1.9) The boundary conditions can be incorporated in the formulation by collocating points on the boundaries:

(1.10)

By solving the algebraic equations consisting of Eqs. (1.9) and (1.10), the coefficients can be obtained, thus an approximated solution is provided by Eq. (1.8).

Other than the Dirac Delta function, we can also use Heaviside functions as test functions in Eq. (1.4), leading to the finite volume method [20].

1.1.2.2 Global symmetric weak form In Eq. (1.4) the continuity requirements for the admissible trial function and the admissible test function are different, hence the weak form in Eq. (1.4) is unsymmetric. For some methods, such as the Galerkin-based finite element method, the symmetric weak form is preferred. By integrating Eq. (1.4) by parts twice, the symmetric weak form is derived as following:

(1.11) For Eq. (1.11), both the trial and test function are required to be continuous. Therefore, it allows us to use simple functions, instead of complex continuous functions, to approximate the solution. As aforementioned, the trial and test function can be chosen as harmonics, polynomials and RBFs, etc., leading to the Galerkin method. However, these functions (harmonics, polynomials, RBFs) often lead to a dense, ill-conditioned system of algebraic equations.

Suppose that the trial function is

(1.12) The basis functions in Eq. (1.12) are defined in the whole domain and are mostly nonzero for an arbitrary in the domain . These kind of basis functions are classified as global approximation functions. In practice, it is more favorable to use local functions (which are nonzero only over elements of the global domain) instead of global ones.

For instance, we define the trial function as element based Hermite Interpolation as following

(1.13) Then only one local basis function need to be considered for , where are nonoverlapping subdomains in . By adopting the local trial functions, the weak form Eq. (1.11) leads to the finite element method [35–37], where the resulting system of algebraic equations is sparse.

1.1.2.3 Global unsymmetric weak form II Further integrating Eq. (1.11) by parts for twice yields another unsymmetric weak form:

(1.14) in which there is no continuity requirement for the trial function. If one can find the Green’s function for

(1.15) i.e. the fundamental solution of an infinite beam on an elastic foundation, Eq. (1.14) will reduce to the boundary integral equation

q

y

g

q (1.16)

where

In Eq. (1.16) the trial function only appears in the boundary terms, thus it leads to the boundary element method [38,39].

1.1.2.4 Local weak forms In the previously developed methods, the global weak form is considered in the whole domain . By dividing into subdomains (over-lapping or non-over-lapping), local weak forms are defined as follows:

(1.17) The corresponding local boundary conditions are set by the continuity requirement of on the local boundaries .

If are non-over-lapping subdomains, the local boundary conditions can be expressed as follows:

(1.18) In the local weak forms, the trial and test functions can be much simpler than those in the global weak forms Eq. (1.3). As a subdomain is much smaller than the original domain, it is relatively very easy to obtain highly accurate local solutions, even when the trial functions are composed of a small group of basis functions. In addition to that, the local weak forms are also more efficient in

computation, because the resulting system of algebraic equations is partitioned into blocks.

The local weak forms can also be rewritten into symmetric and unsymmetric forms, therefore it enables us to develop local collocation, local finite volume, local Galerkin, local finite element, and local boundary element method [40–43].

Herein, it should be emphasized that the local approximation methods are different from the global approximation methods using local trial functions. In the former, the weak forms are satisfied in each local subdomain, and the continuity of the solution is ensured by the local boundary conditions; while in the latter, the weak form is satisfied in the whole domain, and the local trial function needs to be selected carefully to satisfy the continuity requirements a-priori.

1.1.3 Mixed methods In the primal problem description, the governing Eq. (1.3) is of 4th order. This brings much inconvenience to the computation, due to the fact the trial function has to meet high order continuity requirements. It not only makes the trial function quite complex, but also easily leads to ill-conditioned matrix by including higher order derivatives of the trial function. To decrease the order of the governing Eq. (1.3), extra variables are introduced. By adding the moment , Eq. (1.3) is rewritten as

(1.19) or equivalently in matrix form as

(1.20) where

Considering a vector test function , the weak form of Eq. (1.20) can be obtained

(1.21) The required continuity of the trial function is reduced from to .

By introducing displacement , rotation , moment , and shear as independent variables, the original governing equation is expressed as

(1.22)

The corresponding matrix form is

(1.23) where

The weighted residual weak form can be written as follows: