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In this paper, a formulation of reduced-order finite element (FE) model is presented for geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation and harmonic balance method. The main focus is to formulate the nonlinear reduced-order models (ROMs) in the time and frequency domain without involving the corresponding full-order FE models, and it is carried out by means of a special factorization of the nonlinear strain–displacement matrix. Furthermore, a methodology for the enrichment of reduction basis (RB) over that obtained from conventional approaches is presented where the proper orthogonal decomposition method is utilized by comprising the correlation matrix as the union of stiffness-normalized reduction basis vectors and the corresponding static derivatives. The results reveal a significantly reduced computational time due to the formulation of the nonlinear ROMs without involving the full-order FE model. A good accuracy of the nonlinear ROMs of viscoelastic structures is also achieved through the present method of enrichment of RB.

We consider the dictionary-based ROM-net (Reduced Order Model) framework [Daniel et al., Adv. Model. Simul. Eng. Sci. 7 (2020) https://doi.org/10.1186/s40323-020-00153-6 ] and summarize the underlying methodologies and their recent improvements. The object of interest is a real-life industrial model of an elastoviscoplastic high-pressure turbine blade subjected to thermal, centrifugal and pressure loadings. The main contribution of this work is the application of the complete ROM-net workflow to the quantification of the uncertainty of dual quantities on this blade (such as the accumulated plastic strain and the stress tensor), generated by the uncertainty of the temperature loading field. The dictionary-based ROM-net computes predictions of dual quantities of interest for 1008 Monte Carlo draws of the temperature loading field in 2 h and 48 min, which corresponds to a speedup greater than 600 with respect to a reference parallel solver using domain decomposition, with a relative error in the order of 2%. Another contribution of this work consists in the derivation of a meta-model to reconstruct the dual quantities of interest over the complete mesh from their values on the reduced integration points.

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.

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

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.

This study focuses on reduced‐order dynamical modelling of droop‐controlled inverter‐based low‐voltage AC sub‐microgrid in a hybrid AC/DC microgrid. The authors aim to develop a comprehensive reduced‐order model for the low‐voltage AC side in this part. The reduced‐order models are preferred in real‐time calculations. In hybrid microgrids, electrical power is exchanged between the AC and DC sub‐microgrids by a bidirectional AC/DC converter. The distributed energy resources are connected to the main AC bus through power inverters. Voltage and frequency commands are generated by droop controllers for each inverter. For the main AC bus, equations describing voltage magnitude and frequency are derived. Steady‐state values of the phase angles and injected power of the inverters are calculated. The overall non‐linear dynamical and algebraic equations are derived for the low‐voltage AC side, and then linearised around the operating point. To validate the developed model, a hybrid microgrid is implemented in PSCAD. Then, the proposed model for the case study is implemented in Matlab/Simulink and the results are compared with PSCAD outputs. The comparative results show the validity of the developed comprehensive reduced‐order model which can be used in fault detection approaches.

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.

Aerodynamic reduced-order model (ROM) is a useful tool to predict nonlinear unsteady aerodynamics with reasonable accuracy and very low computational cost. The efficacy of this method has been validated by many recent studies. However, the generalization capability of aerodynamic ROMs with respect to different flow conditions and different aeroelastic parameters should be further improved. In order to enhance the predicting capability of ROM for varying operating conditions, this paper presents an unsteady aerodynamic model based on long short-term memory (LSTM) network from deep learning theory for large training dataset and sampling space. This type of network has attractive potential in modeling temporal sequence data, which is well suited for capturing the time-delayed effects of unsteady aerodynamics. Different from traditional reduced-order models, the current model based on LSTM network does not require the selection of delay orders. The performance of the proposed model is evaluated by a NACA 64A010 airfoil pitching and plunging in the transonic flow across multiple Mach numbers. It is demonstrated that the model can accurately capture the dynamic characteristics of aerodynamic and aeroelastic systems for varying flow and structural parameters.

A novel reduced order model (ROM) for unsteady hypersonic aerodynamics is developed, which is applicable for the variations of multi-parameters. The key to the developed ROM lies in the CFD-based model reduction of the steady aerodynamic component, which stems from the quasi-steady nature of aerodynamic forces in the hypersonic regime. Concretely, the proper orthogonal decomposition (POD) method, combined with Kriging interpolation, is used to construct the ROM for the steady aerodynamic component; meanwhile the unsteady part is directly obtained from Donov’s third-order piston theory. The new procedure is applied to a three-dimensional low aspect ratio wing (Lockheed F-104 Starfighter wing). It is shown that the developed ROM is able to accurately predict the unsteady hypersonic aerodynamic loads over a wide range of different flight conditions compared with the direct CFD computation.

A reduced-order model for an efficient analysis of cardiovascular hemodynamics problems using multiscale approach is presented in this work. Starting from a patient-specific computational mesh obtained by medical imaging techniques, an analysis methodology based on a two-step automatic procedure is proposed. First a coupled 1D-3D Finite Element Simulation is performed and the results are used to adjust a reduced-order model of the 3D patient-specific area of interest. Then, this reduced-order model is coupled with the 1D model. In this way, three-dimensional effects are accounted for in the 1D model in a cost effective manner, allowing fast computation under different scenarios. The methodology proposed is validated using a patient-specific aortic coarctation model under rest and non-rest conditions.