We have reproduced our disputed simulation results from our original article as well as the simulations of the discussion article of Krack and Gross referenced above in the title. We have found several inaccuracies and omissions in the discussion article that invalidate its authors’ main claims. We also briefly address inherent performance differences between the SSMTool and the NLvib algorithms to the benefit of the former.

Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

Modeling in applied science and engineering targets increasingly ambitious objectives, which typically yield increasingly complex models. Despite major advances in computations, simulating such models with exceedingly high dimensions remains a challenge. Even if technically feasible, numerical simulations on such high-dimensional problems do not necessarily give the simplified insight into these phenomena that motivated their initial models. Reduced-order models hold more promise for a quick assessment of changes under parameters and uncertainties, as well as for effective prediction and control. Such models are also highly desirable for systems that are only known in the form of data sets. This focus issue will survey the latest trends in nonlinear model reduction for equations and data sets across various fields of applications, ranging from computational to theoretical aspects.

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system’s parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient.

A variety of reduced order modeling (ROM) methods for geometrically nonlinear structures have been developed over recent decades, each of which takes a distinct approach, and may have different advantages and disadvantages for a given application. This research challenge is motivated by the need for a consistent, reliable, and ongoing process for ROM comparison. In this chapter, seven state-of-the-art ROM methods are evaluated and compared in terms of accuracy and efficiency in capturing the nonlinear characteristics of a benchmark structure: a curved, perforated plate that is part of the exhaust system of a large diesel engine. Preliminary results comparing the full-order and ROM simulations are discussed. The predictions obtained by the various methods are compared to provide an understanding of the performance differences between the ROM methods participating in the challenge. Where possible, comments are provided on insight gained into how geometric nonlinearity contributes to the nonlinear behavior of the benchmark system.

Spectral submanifolds (SSMs) have emerged as accurate and predictive model reduction tools for dynamical systems defined either by equations or data sets. While finite-elements (FE) models belong to the equation-based class of problems, their implementations in commercial solvers do not generally provide information on the nonlinearities required for the analytical construction of SSMs. Here, we overcome this limitation by developing a data-driven construction of SSM-reduced models from a small number of unforced FE simulations. We then use these models to predict the forced response of the FE model without performing any costly forced simulation. This approach yields accurate forced response predictions even in the presence of internal resonances or quasi-periodic forcing, as we illustrate on several FE models. Our examples range from simple structures, such as beams and shells, to more complex geometries, such as a micro-resonator model containing more than a million degrees of freedom. In the latter case, our algorithm predicts accurate forced response curves in a small fraction of the time it takes to verify just a few points on those curves by simulating the full forced-response.

For mechanical systems subject to periodic excitation, forced response curves (FRCs) depict the relationship between the amplitude of the periodic response and the forcing frequency. For nonlinear systems, this functional relationship is different for different forcing amplitudes. Forced response surfaces (FRSs), which relate the response amplitude to both forcing frequency and forcing amplitude, are then required in such settings. Yet, FRSs have been rarely computed in the literature due to the higher numerical effort they require. Here, we use spectral submanifolds (SSMs) to construct reduced-order models (ROMs) for high-dimensional mechanical systems and then use multidimensional manifold continuation of fixed points of the SSM-based ROMs to efficiently extract the FRSs. Ridges and trenches in an FRS characterize the main features of the forced response. We show how to extract these ridges and trenches directly without computing the FRS via reduced optimization problems on the ROMs. We demonstrate the effectiveness and efficiency of the proposed approach by calculating the FRSs and their ridges and trenches for a plate with a 1:1 internal resonance and for a shallow shell with a 1:2 internal resonance.

A 1:2 internally resonant mechanical system can undergo secondary Hopf (Neimark-Sacker) bifurcations, resulting in a quasi-periodic response when the system is subject to harmonic excitation. While these quasi-periodic orbits have been observed in practice, their bifurcations are not well studied, especially in high-dimensional mechanical systems. This is mainly because of the challenges associated with the computation and bifurcation detection of these quasi-periodic motions. Here we present a computational framework to address these challenges via reductions on spectral submanifolds, which transforms quasi-periodic orbits of high-dimensional systems as limit cycles of four-dimensional reduced-order models. We apply the proposed framework to analyze bifurcations of quasi-periodic orbits in several mechanical systems exhibiting 1:2 internal resonance, including a finite element model of a shallow-curved shell. We uncover local bifurcations such as period-doubling and saddle-node, as well as global bifurcations such as homoclinic connections, isolas, and simple bifurcations of quasi-periodic orbits. We also observe cascades of period-doubling bifurcations of quasi-periodic orbits that eventually result in chaotic motions, as well as the coexistence of chaotic and quasi-periodic attractors. These findings elucidate the complex bifurcation mechanism of quasi-periodic orbits in 1:2 internally resonant systems.

Dynamical systems are often subject to algebraic constraints in conjunction with their governing ordinary differential equations. In particular, multibody systems are commonly subject to configuration constraints that define kinematic compatibility between the motion of different bodies. A full-scale numerical simulation of such constrained problems is challenging, making reduced-order models (ROMs) of paramount importance. In this work, we show how to use spectral submanifolds (SSMs) to construct rigorous ROMs for mechanical systems with configuration constraints. These SSM-based ROMs enable the direct extraction of backbone curves and forced response curves and facilitate efficient bifurcation analysis. We demonstrate the effectiveness of this SSM-based reduction procedure on several examples of varying complexity, including nonlinear finite-element models of multibody systems. We also provide an open-source implementation of the proposed method that also contains all details of our numerical examples.

We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the reduced-order models are discussed in this paper and its companion, Part II, respectively. The dimension of a reduced-order model is determined by the number of modes involved in the internal resonance, independently of the dimension of the full system. The periodic responses of the full system are obtained as equilibria of the reduced-order model on spectral submanifolds. The forced response curve of periodic orbits then becomes a manifold of equilibria, which can be easily extracted using parameter continuation. To demonstrate the effectiveness and efficiency of the reduction, we compute the forced response curves of several high-dimensional nonlinear mechanical systems, including the finite-element models of a von Kármán beam and a plate.

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Very high dimensional nonlinear systems arise in many engineering problems due to semi-discretization of the governing partial differential equations, e.g. through finite element methods. The complexity of these systems present computational challenges for direct application to automatic control. While model reduction has seen ubiquitous applications in control, the use of nonlinear model reduction methods in this setting remains difficult. The problem lies in preserving the structure of the nonlinear dynamics in the reduced order model for high-fidelity control. In this work, we leverage recent advances in Spectral Submanifold (SSM) theory to enable model reduction under well-defined assumptions for the purpose of efficiently synthesizing feedback controllers.

Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of spectral submanifolds (SSMs), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulae for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite-element models.

We propose a reformulation for a recent integral equations approach to steady-state response computation for periodically forced nonlinear mechanical systems. This reformulation results in additional speed-up and better convergence. We show that the solutions of the reformulated equations are in one-to-one correspondence with those of the original integral equations and derive conditions under which a collocation-type approximation converges to the exact solution in the reformulated setting. Furthermore, we observe that model reduction using a selected set of vibration modes of the linearized system substantially enhances the computational performance. Finally, we discuss an open-source implementation of this approach and demonstrate the gains in computational performance using three examples that also include nonlinear finite-element models.

We show how spectral submanifold (SSM) theory can be used to extract forced-response curves without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems. We use multivariate recurrence relations to construct the SSMs, achieving a major speed-up relative to earlier autonomous SSM algorithms. The increase in computational efficiency promises to close the current gap between studying lower-dimensional academic examples and analyzing larger systems obtained from finite-element modeling, as we illustrate on two different discretized damped-forced beam models. Using the exact reduction procedure via SSMs for obtaining forced response curves, we further demonstrate speed gains of several orders in magnitude relative to the available state-of-the-art continuation packages, while retaining accuracy.

The thermal dynamics in thermo-mechanical systems exhibits a much slower time scale compared to the structural dynamics. In this work, we use the method of multiple scales to reduce the thermo-mechanical structural models with a slowly-varying temperature distribution in a systematic manner. In the process, we construct a reduction basis that adapts according to the instantaneous temperature distribution of the structure, facilitating an efficient reduction in the number of unknowns. As a proof of concept, we demonstrate the method on a range of linear and nonlinear beam examples and obtain a consistently better accuracy and reduction in the number of unknowns than the standard Galerkin projection using a constant basis.

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green’s function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton– Raphson iteration instead, obtaining robust convergence. We further show that this integral equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.

Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.