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.

This paper describes the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities. The manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis of the eigenvalue problem, or its static approximation, along the vibration modes. The construction of the quadratic manifold requires minimal computational effort once the vibration modes are known. The reduced order model is then obtained by Galerkin projection, where the configurationdependent tangent space of the manifold is used to project the discretized equations of motion.

In this paper, a generalization of a quadratic manifold approach for the reduction of geometrically nonlinear structural dynamics problems is presented. This generalization is constructed by a linearization of the static force with respect to the generalized coordinates, resulting in a shift of the quadratic behavior from the force to the manifold. In this framework, static derivatives emerge as natural extensions to modal derivatives for displacement fields other than the vibration modes, such as the Krylov subspace vectors. Here the dynamic problem is projected onto the tangent space of the quadratic manifold, allowing for a much less number of generalized coordinates compared to linear basis methods. The potential of the quadratic manifold approach is investigated in a numerical study, where several variations of the approach are compared on different examples, indicating a clear pattern where the proposed approach is applicable.

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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. arXiv:1810.10103v1 [math.DS] 23 Oct 2018 ods such as the Craig-Bampton method [54] (cf. Theodosiou et al. [47]), proper orthogonal decomposition [55] (cf. Kerchen et al. [48]), reduction using natural modes (cf. Amabili [49], Touzé et al. [50]) and the modal-derivative method of Idelsohn & Cardona [53] (cf. Sombroek et al. [51], Jain et al. [52]).A common feature of these methods is their local nature: they seek to approximate nonlinear steadystate response in the vicinity of an equilibrium. Thus, high-amplitude oscillations are generally missed by these approaches.On the analytic side, perturbation techniques relying on a small parameter have been widely used to approximate the steady-state response of nonlinear systems. Nayfeh et al. [24,25] give a formal multiple-scales expansion applied to a system with small damping, small nonlinearities and small forcing. Their results are detailed amplitude equations to be worked out on a case-by-case basis. Mitropolskii and Van Dao [23] apply the method of averaging (cf. Bogoliubov and Mitropolsky [7] or, more recently, Sanders and Verhulst [30]) after a transformation to amplitude-phase coordinates in the case of small damping, nonlinearities and forcing. They consider single as well as multi-harmonic forcing of multi degree of freedom systems and obtain the solution in terms of a multi-frequency Fourier expansion. Their formulas become involved even for a single oscillator, thus condensed formulas or algorithms are unavailable for general systems. As conceded by Mitroposkii and Van Dao [23], the series expansion is formal, as no attention is given to the actual existence of a periodic response. Existence is indeed a subtle question in this context, since the envisioned periodic orbits would perturb from a non-hyperbolic fixed point.Vakakis [36] relaxes the small nonlinearity assumption and describes a perturbation approach for obtaining the periodic response of a single-degree-of-freedom Duffing oscillator subject to small forcing and small damping. A formal series expansion is performed around a conservative limit, where periodic solutions are explicitly known (elliptic Duffing oscillator). This approach only works for perturbatio...