Spectral element methods for nonlinear spatio-temporal dynamics of an Euler-Bernoulli beam

Spectral element methods are high order accurate methods which have been successfully utilized for solving ordinary and partial differential equations. In this paper the space-time spectral element (STSE) method is employed to solve a simply supported modified Euler-Bernoulli nonlinear beam undergoing forced lateral vibrations. This system was chosen for analysis due to the availability of a reference solution of the form of a forced Duffing's equation. Two formulations were examined: i) a generalized Galerkin method with Hermitian polynomials as interpolants both in spatial and temporal discretization (HHSE), ii) a mixed discontinuous Galerkin formulation with Hermitian cubic polynomials as interpolants for spatial discretization and Lagrangian spectral polynomials as interpolants for temporal discretization (HLSE). The first method revealed severe stability problems while the second method exhibited unconditional stability and was selected for detailed analysis. The spatialh-convergence rate of the HLSE method is of order α=p _s+1 (wherep _s is the spatial polynomial order). Temporalp-convergence of the HLSE method is exponential and theh-convergence rate based on the end points (the points corresponding to the final time of each element) is of order 2p _T−1 ≤α≤2p _T+1 (wherep _T is the temporal polynomial order). Due to the high accuracy of the HLSE method, good results were achieved for the cases considered using a relatively large spatial grid size (4 elements for first mode solutions) and a large integration time step (1/4 of the system period for first mode solutions, withp _T=3). All the first mode solution features were detected including the onset of the first period doubling bifurcation, the onset of chaos and the return to periodic motion. Two examples of second mode excitation produced homogeneous second mode and coupled first and second mode periodic solutions. Consequently, the STSE method is shown to be an accurate numerical method for simulation of nonlinear spatio-temporal dynamical systems exhibiting chaotic response.     

Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C⁰ shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Monte Carlo filters for identification of nonlinear structural dynamical systems

The problem of identification of parameters of nonlinear structures using dynamic state estimation techniques is considered. The process equations are derived based on principles of mechanics and are augmented by mathematical models that relate a set of noisy observations to state variables of the system. The set of structural parameters to be identified is declared as an additional set of state variables. Both the process equation and the measurement equations are taken to be nonlinear in the state variables and contaminated by additive and (or) multiplicative Gaussian white noise processes. The problem of determining the posterior probability density function of the state variables conditioned on all available information is considered. The utility of three recursive Monte Carlo simulation-based filters, namely, a probability density function-based Monte Carlo filter, a Bayesian bootstrap filter and a filter based on sequential importance sampling, to solve this problem is explored. The state equations are discretized using certain variations of stochastic Taylor expansions enabling the incorporation of a class of non-smooth functions within the process equations. Illustrative examples on identification of the nonlinear stiffness parameter of a Duffing oscillator and the friction parameter in a Coulomb oscillator are presented. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

A nonlinear dynamical systems framework for structural diagnosis and prognosis

This work aims to establish a nonlinear dynamics framework for diagnosis and prognosis in structural dynamic systems. The objective is to develop an analytically sound means for extracting features, which can be used to characterize damage, from modal-based input-output data in complex hybrid structures with heterogeneous materials and many components. Although systems like this are complex in nature, the premise of the work here is that damage initiates and evolves in the same phenomenological way regardless of the physical system according to nonlinear dynamic processes. That is, bifurcations occur in healthy systems as a result of damage. By projecting a priori the equations of motion of high-dimensional structural dynamic systems onto lower dimensional center, or so-called ‘damage’, manifolds, it is demonstrated that model reduction near bifurcations might be a useful way to identify certain features in the input-output data that are helpful in identifying damage. Normal forms describing local co-dimension one and two bifurcations (e.g. transcritical, subcritical pitchfork, and asymmetric pitchfork bifurcations) are assumed to govern the initiation and evolution of damage in a low-order model. Real-world complications in damage prognosis involving spatial bifurcations, global bifurcation phenomena, and the sensitivity of damage to small changes in initial conditions are also briefly discussed.     

On the analysis of time-periodic nonlinear dynamical systems

In this study, a general technique for the analysis of time-period nonlinear dynamical systems is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (L-F) transformation. A general procedure for the computation of L-F transformation in terms of Chebyshev polynomials is outlined. Once the transformation has been applied, a periodic orbit in original coordinates has a fixed point representation in the transformed coordinates. The stability and bifurcation analysis of the transformed equations are studied by employing thetime-dependent normal form theory and time-dependent centre manifold reduction. For the two examples considered, the three generic codimension-one bifurcations, viz, Hopf, flip and tangent, are analysed. The methodology is semi-analytic in nature and provides a quantitative measure of stability even under critical conditions. Unlike the perturbation or averaging techniques, this method is applicable even to those systems where the periodic term in the linear part does not contain a small parameter or a generating solution does not exist due to the absence of the time-invariant term in the linear part.     

非线性随机动力系统的最优多项式控制

根据物理随机最优控制理论,发展了适用于一般非线性随机动力系统的最优多项式控制策略,考察了随机地震动作用下不同非线性水平Duffing系统的随机最优控制。结果表明,受控后系统反应的离散性大大降低、系统性态显著改善;采用能量均衡的超越概率准则,1阶线性控制器可以达到高阶非线性控制器的控制效果,这对于非线性控制器可能导致系统不稳定的场合具有重要意义。此外,相比较基于统计线性化的LQG控制,发展的非线性随机最优控制策略对Duffing系统的非线性水平不敏感、具有良好的鲁棒性,能实现系统的精细化控制,而采用名义白噪声输入的LQG控制则不具备这一特点。     

Application of stochastic averaging for nonlinear dynamical systems with high damping

The asymptotic behavior of coupled nonlinear dynamical systems in the presence of noise is studied using the method of stochastic averaging. It is shown that, for systems with rapidly oscillating and decaying components, the stochastic averaging technique yields a set of equations of considerably smaller dimension, and the resulting equations are simpler. General results of this method are applied to stochastically perturbed nonlinear nonconservative systems in R⁴. It is shown that in such systems the contribution of the stochastic components in the damped modes to the drift term of the critical mode may be beneficial in terms of stability in certain cases.     

Efficient path integration methods for nonlinear dynamic systems

New approaches for numerical implementation of the path integration (PI) method are described. In essence the PI method is a stepwise calculation of the joint probability density function (PDF) of a set of state space variables describing a white noise excited nonlinear dynamic system. The basic idea behind the proposed procedure is to apply a splines interpolation method to the logarithm of the calculated PDF to obtain an accurate representation of the PDF over the whole domain and not only at the chosen grid points. This exploits the fact that the logarithm of the PDF shows a more polynomial behaviour than the PDF itself, and therefore is better adapted to a splines representation. It is demonstrated that the proposed techniques may lead to significantly improved performance in calculating the response statistics of large classes of nonlinear oscillators excited by white or coloured noise when compared to other available implementations of the PI method. An advantage of the new approaches is that they allow time-variant dynamic systems to be analysed without significant increase in computer time. Numerical results for both 2D and 3D problems are presented.     

Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the proper orthogonal decomposition (POD)-Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the RB and the interpolation basis vectors, making the generation of a stable, compact and reliable ROM possible. This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm. An efficient output error indicator plays a key role in the adaptive scheme. We also propose an improved output error indicator based on previous work. Upon convergence of the POD-Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable ROM can be constructed. The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering.     

Analytic representation of dynamical systems using pseudo-inverse methods

Summary The term `analytic representation' in configuration space is often used for the representation of a physical system in terms of Lagrangians and/or Lagrange's equations. Such representations play a role in the methodological formulation for a wide variety of physical problems. We deal with two different approaches to construct Lagrangians for a number of equations. Examples cited cover both point and continuum mechanics. This work will be of special significance to those who would like to study problems of contemporary physics without being directly involved in the rigorous theory of Helmholtz for inverse variational problems. The first procedure chosen by us depends on the method of characteristics as used for solving first order partial differential equations while the second one exploits the symmetries of the Lagrangian and the equation of motion. For simple cases, both approaches are applicable without any modification. However, in more realistic situations the methods need to be supplemented by some ansatz.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Physics-integrated deep learning for uncertainty quantification and reliability estimation of nonlinear dynamical systems

Assumptions and approximations made while analyzing any physical system induce modeling uncertainty, which, if left unchecked, can result in the erroneous analysis of the system under consideration. Additionally, the discrepancy in the exact knowledge of system parameters can further result in deviation from the ground truth. This paper explores Physics-integrated Variational Auto-Encoder (PVAE) to account for modeling and parametric uncertainties in partially known nonlinear dynamical systems. The PVAE under consideration has three main parts: encoder, latent space, and decoder. The complete PVAE architecture is employed during the training stage of the machine learning model, while only the decoder is used to make the final predictions. The encoder determines the correct parameter values for the known part of the model (in the form of a known ODE). The decoder is augmented with an ODE solver that solves the known part of the system and the estimated discrepancy together to reconstruct the measurements. To test the efficacy of the PVAE architecture, three case studies are carried out, each presenting unique challenges. The probability density functions obtained for the various systems’ responses demonstrate the efficacy of the PVAE architecture. Furthermore, reliability analysis has been carried out, and the results produced have been compared against those obtained from a multi-layered, densely connected forward neural network.     

Boundary element methods for optimization of distributed parameter systems

A numerical method of solution is proposed for optimization problems of distributed parameter systems. Two model problems from continuum mechanics are investigated by means of constructing the problems as the steady-state optimal control problems governed by elliptic partial differential equations. The basis of the suggested method of solution lies in the space discretization of the necessary conditions for optimality by the boundary element method, and the minimization of the performance indices by the conjugate gradient method of optimization.     

高维非线性动力系统最简规范形的计算

运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵：相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助Mathematica语言,编制了计算任意高维非线性动力系统的最简规范形的通用程序。运行该程序,分别计算了4维、6维和7维非线性动力系统的直到4阶的最简规范形。     

Extended homotopy analysis method for multi-degree-of-freedom non-autonomous nonlinear dynamical systems and its application

In normal circumstances, numerous practical engineering problems are multi-degree-of-freedom (MDOF) nonlinear non-autonomous dynamical systems. Generally, exact solutions for MDOF dynamical systems are hardly obtained; thus, the development of analytical approximations becomes a robust and appealing avenue for an analysis of these systems. The homotopy analysis method (HAM) is one of the analytical methods, which can overcome the foregoing barriers of conventional asymptotic techniques. It has been widely used for solving various nonlinear problems in physical science and engineering. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for MDOF weakly damped non-autonomous dynamical systems. In terms of its flexibility and applicability, the EHAM is also applied to derive the approximate solutions of parametrically and externally excited thin plate systems. Besides, comparisons are performed between the results obtained by the EHAM and the numerical integration (i.e. Runge–Kutta) method. The present findings show that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions.     

Randomized low-rank approximation methods for projection-based model order reduction of large nonlinear dynamical problems

Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis to approximate high-dimensional quantities. However, the most popular methods for computing V , eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only or . We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems.     

Dual reciprocity boundary element analysis of nonlinear diffusion: temporal discretization

This article presents an application of higher order time integration schemes in the dual reciprocity boundary element analysis of nonlinear transient diffusion problems involving nonlinear boundary conditions as well as temperature-dependent material properties. Multistep θ-methods, cubic Hermitian schemes and one-step least squares method have been considered. An error estimate based on the conservation of thermal energy has been presented to assess the accuracy of the numerical solutions. Numerical results are presented for a set of representative test problems to demonstrate the usefulness of the presented time integration algorithms.     

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

A complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.