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This study is focused on a reduced-order model (ROM) based on proper orthogonal decomposition (POD) for unsteady flow around a stationary object, which allows prediction with different object geometry as a parameter. The conventional POD method is applicable only to data with the same computational grid for all snapshots. This study proposed a novel POD methodology that performs on flow snapshots, including some time-series data of flow fields around objects of different shapes and numerically computed by different computational grids. The concept of the proposed POD involved mapping the flow fields computed on different grids in computational space. Consequently, the optimal POD basis for minimizing reconstruction errors in physical space was obtained in the computational space. The proposed POD was applied to the flow around ellipses and airfoils generated via conformal mapping to a cylinder. The ROM formulated using the proposed POD bases reconstructed the flow fields around the ellipses with different aspect ratios and airfoils with varying shapes. Using the modes obtained by the proposed POD, the ROM was demonstrated to stably predict the time evolution of the flow around objects, which is not included in the snapshots. In the ROM, the difference between the frequency of the flow field in the POD snapshot and that of the reconstructed flow field resulted in a phase error owing to the time evolution. The mean squared error between the flow fields obtained via the ROM and the directly solved Navier–Stokes equations was under 1 0 − 7 when the reconstructed flow and the flow included in the snapshot had the same frequency as that of Kármán vorticities behind the objects. Based on these observations, the proposed POD is suitable for constructing an ROM to reconstruct the flow around various geometries. • Proper orthogonal decomposition (POD) is applied to flow around various geometries. • Reduced-order model (ROM) predicted flow around different geometry from the dataset. • Prediction error is related to the frequency difference between the flow in the dataset and the predicted. • Prediction error was under 1 0 − 7 when the frequency difference was less than 0.002. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O (1 0 2) to O (1 0 3) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction. [ABSTRACT FROM AUTHOR]

• A novel deep learning model (i.e., SKCAE) integrating physics and data is proposed to construct the nonlinear observation functions in Koopman models. • Four cases of nonlinear dynamical systems demonstrate the enhancements in accuracy and efficiency of SKCAE. • SKCAE identifies spatiotemporal intrinsic features of a dynamic system and hence retains physical interpretability. • SKCAE exhibits broad applicability in diverse cases without prior knowledge requirements. • SKCAE possesses remarkable capacity in scenarios with data noises and losses. One strategy for predicting the state of nonlinear dynamical systems (typically of high dimensionality) is global linearization, such as utilizing the Koopman analysis model to transform the system state into an invariant subspace that evolves linearly. A critical challenge in the Koopman model is designing or deriving observation functions, typically nonlinear, to linearize the dynamical systems. To address the challenge, this study proposes a model called SKCAE (Skip-connected Koopman Convolutional AutoEncoder) that combines guidance of physics and data to learn observation functions accurately and efficiently. The novelties of SKCAE are twofold: (1) a coordinate-transforming main network for the construction of nonlinear observation functions, enabling the explicit identification of the low-dimensional dynamics that dominate the system's dynamical evolution, and (2) a Koopman dynamics-oriented subnetwork for quantitatively interpreting the system's intrinsic mechanisms from the frequency perspective and efficiently predicting the system state. Four numerical cases with discrete/continuous spectra on Eulerian/Lagrangrian descriptions are studied, i.e., fixed-point attractor, Duffing oscillator, fluid flow past a cylinder, and channel turbulence. Results demonstrate that SKCAE achieves significant improvement (up to a hundred times) in accuracy compared to conventional models and possesses remarkable capability in handling scenarios with data noises and/or loss, owing to its intrinsic advantage in retaining a broad range of frequency components of a dynamical system. [ABSTRACT FROM AUTHOR]

A generalized elastic coordinate method (GECM) is proposed for the vibrational (elastic) response analysis of unconstrained structural dynamics. Firstly, the generalized rigid body and elastic coordinates are defined to describe the overall motion of a discrete unconstrained structure. Secondly, the corresponding equations of motion for generalized elastic coordinates are established with a tiny computational cost, and the stiffness matrix is still sparse. Finally, the elastic deformation in physical space can be exactly caught by the proposed equations of motion after determining the rigid body modes and the coordinate system of the rigid body motion. Different from directly solving the original semi-positive definite system in physical space, in which the overall displacement can only obtain, and the direct modal projection, the modal truncation error problem can be effectively overcome based on the established positive definite system when the modal order reduction is carried out. In light of this, the modal truncation error correction of the elastic displacement for the transient time domain and steady-state frequency domain of unconstrained structures is directly realized, summed the above as GECM, and the validity and practicability are illustrated through the numerical examples. • GECM is proposed for the elastic deformation of unconstrained structural dynamics. • The equations of motion for generalized elastic coordinates are established. • A positive definite system is provided. • The transient elastic response analysis is effectively realized. • The elastic frequency response analysis is effectively carried out. [ABSTRACT FROM AUTHOR]

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of a stochastic reduced-order computational model in computational linear dynamics for linear viscoelastic composite structures in the presence of uncertainties. The computational framework proposed is based on a recent theoretical work that allows for constructing the stochastic reduced-order model using the nonparametric probabilistic approach. In the frequency domain, the generalized damping matrix and the generalized stiffness matrix of the stochastic computational reduced-order model are random matrices. Due to the causality of the dynamical system, these two frequency-dependent random matrices are statistically dependent and are linked by a compatibility equation induced by the causality of the system, involving a Hilbert transform. The computational aspects related to the nonparametric stochastic modeling of the reduced stiffness matrix and the reduced damping matrix that are frequency-dependent random matrices are presented. A dedicated numerical approach is developed for obtaining an efficient computation of the Cauchy principal value integrals involved in those equations for which an integration over a broad frequency domain is required. A computational analysis of the propagation of uncertainties is conducted for a composite viscoelastic structure in the frequency range. It is shown that the uncertainties on the damping matrix have a strong influence on the observed statistical dispersion of the stiffness matrix. [ABSTRACT FROM AUTHOR]

Fractional flow reserve (FFR) is the "gold standard" for diagnosis of myocardial ischemia. The existing 3D computational fluid dynamics (CFD) algorithm based on coronary computed tomography angiography (CTA) has realized the non-invasive calculation of FFR (FFR CT), but the CFD simulation in 3D coronary arteries is time consuming. The purpose of this study was to develop a reduced-order (0D) cardiovascular model for rapid computation of pressure and flow through coronary arteries and stenosis and to ensure accuracy in prediction of FFR (FFR 0D). One hundred patients (118 lesions) who had undergone invasive FFR were enrolled retrospectively. Based on coronary CTA, a patient-specific lumped parameter model of the cardiovascular system and coronary system was constructed. In addition, an experimentally validated, theoretical calculation model of stenosis resistance was introduced. Stenosis resistance and flow rate were quantified by coupling the two models, and FFR 0D was calculated according to the coronary pressure distribution. The overall modeling time of FFR 0D was approximately 41.2 min while FFR CT was reported to be 3.5 h. Using the clinical cut-off value of FFR (0.8), the accuracy, sensitivity, specificity, positive predictive value and negative predictive value of FFR 0D were 89.8%, 79.2%, 92.6%, 73.1% and 94.6%, respectively, while that of FFR CT were reported to be 84.3%, 87.9%, 82.2%, 73.9% and 92.2%, respectively. We successfully proposed a reduced-order cardiovascular model for rapid and accurate prediction of FFR. Compared with FFR CT , FFR 0D reduced computational cost and ensured computational accuracy in the case of our limited samples. These advantages represent an advance in the diagnosis and treatment of myocardial ischemia. • Feasible technology for noninvasive diagnosis of myocardial ischemia. • Physiologically realistic reduced-order (0D) closed-loop cardiovascular model. • The coupling algorithm determines individual stenosis resistance and flow rate. • Major advance in rapid and accurate prediction of FFR. [ABSTRACT FROM AUTHOR]

A weakly-invasive version of the LATIN-PGD method is introduced to compute reduced-order models in solid mechanics for time-dependent non-linear parametrized problems under quasi-static assumption. Its main interest is to be easily implemented in any general-purpose industrial finite element software while taking advantage of all facilities offered, including the ability to handle any kind of non-linearities. This way of proceeding provides unified tools – for the construction of reduced-order models in non-linear context – all integrated in one certified product in consistency with the purpose of having end-to-end processes. Possibilities of our approach are illustrated thanks to the first implementation conducted within Simcenter Samcef™ software developed by Siemens Digital Industries Software without ever redeveloping any non-linear part of the software. Finally, interesting performance gains are highlighted on a non-linear time-dependent test-case involving some parameters. [ABSTRACT FROM AUTHOR]

Prediction of complex fluid flows from sparse and noisy sensor measurements is widely applied to many engineering fields. In the present study, a novel compressed sensing reduced-order modeling framework has been proposed to predict unsteady flow fields from sparse and noisy sensor observations, which includes offline learning and online forecasting. In the offline learning stage, the Long Short Term Memory (LSTM) model is used for modeling the dynamic evolution of the sensor signals. Moreover, the sparsity-promoting Dynamic Mode Decomposition (DMD) algorithm is applied to extract spatio-temporal coherent structures from high-dimensional fluid dynamic system and choose optimal DMD modes for flow reconstruction. In order to establish the correlation between sensor space and feature-based coherent structures space, the Deep Neural Network (DNN) is employed. In the online forecasting stage, the trained framework is applied to predict the flow fields from limited sensor measurements in the actual experiment setup. In the current work, the proposed framework is applied in two numerical examples, including the flow past a circle cylinder at R e = 100 characterized by laminar flow and at R e = 3900 characterized by turbulent flow. It is shown that the proposed framework performs accurately and robustly in both prediction of sensor measurements and unsteady flow fields. Especially, the important physical features and quantities can be well maintained for both laminar and turbulent flow, even if high level noise is involved. Therefore, the proposed framework is a promising tool for sparse representation from complex flow fields in realistic experiments. [ABSTRACT FROM AUTHOR]

Abstract: The construction of a nonlinear reduced-order model for fluid–structure interaction problems is investigated in this paper for unsteady compressible flows excited by the rigid body motion of a structure. The reduction is achieved by means of a Galerkin projection of the Navier–Stokes equations on the first POD modes resulting from the proper orthogonal decomposition. In the first part of the paper, the projection technique is carried out on a purely aerodynamic case in order (i) to validate an efficient iterative technique based on an updated QR decomposition to compute the POD modes, and (ii) to discuss the merits of different correction methods introduced to improve the long-term stability of the reduced-order model. The second and most original part of the paper deals with the construction of the reduced set of equations which arise from the projection of the compressible Navier–Stokes equations formulated in a suitable moving frame representing the rigid body motion. The expressions of the resulting non-autonomous terms appearing in the reduced-order model have also been optimized to reduce the computational costs. [Copyright &y& Elsevier]

This paper proposes an efficient proper orthogonal decomposition with interpolation (PODI) method for real-time simulation of indenter contact problems. In the offline stage, POD models are constructed from displacement and von-Mises stress snapshots of full-order finite element solutions of indenter contact problems with training contact locations and indentation depths. Indentation depth and contact location are respectively used to interpolate POD basis vectors and POD coefficients by Lagrange interpolation with a congruence transformation and radial basis function (RBF) interpolation. In the online stage, displacements and von-Mises stress of solids under a new indentation loading are obtained very quickly by using the interpolated POD matrix and POD coefficients. A clustering-based contact domain decomposition (CDD) scheme is proposed to further improve the computational performance of the present PODI–RBF​ method. Numerical results show that the PODI–RBF method using the CDD scheme can be used to obtain finite element solutions of indenter contact problems at 400 ∼ 900 Hz on a personal computer for finite element models with 8000 ∼ 16000 nodes, which is promising to real-time simulation of indenter contact problems. • An efficient PODI method is proposed to solve indenter contact problems in real time. • Indentation depth and contact location are used to interpolate POD basis vectors and POD coefficients. • RBF interpolation is employed to quickly obtain FE solutions of indenter contact problems. • A contact domain decomposition scheme is proposed to improve the performance of PODI–RBF method. • PODI–RBF method can be applicable to real-time simulation of indenter contact problems. [ABSTRACT FROM AUTHOR]

Design and implementation of reduced-order optimal controller for flow separation are investigated. The reduced-order controller design is based on proper orthogonal decomposition (POD) and Galerkin projection. Detailed finite element simulations are performed, and POD is applied to the resulting data to extract the most energetic eigenmodes. These global eigenmodes are then used in conjunction with the Galerkin projection to obtain a reduced-order (low-dimensional) model. Reduced-order models are not only attractive for real-time control computation but also crucial for detailed stability and bifurcation analysis. We investigate the design of optimal controller for flow separation in a channel with this model where actuation is performed on a small part of the boundary. Two different types of surface actuation are considered––tangential blowing and suction through a single slot. Best locations for actuation and adaptive procedure for reduced-order model are explored. The proposed approaches are evaluated by investigating the control of flow separation over a forward-facing step channel. Our methods are found to be efficient and fast, and our methods demonstrate a significant reduction in computational time and feasibility. Dramatic separation delays are observed on all cases. It is found that the tangential blowing is more efficient in mitigating flow separation and reducing wake spread. [Copyright &y& Elsevier]

We present a globally convergent method to accelerate density-based topology optimization using projection-based reduced-order models (ROMs) and trust-region methods. To accelerate topology optimization, we replace the large-scale finite element simulation, which dominates the computational cost, with ROMs that reduce the cost of objective function and gradient evaluations by orders of magnitude. To guarantee convergence, we first introduce a trust-region method that employs generalized trust-region constraints and prove it is globally convergent. We then devise a class of globally convergent ROM-accelerated topology optimization methods informed by two theories: the aforementioned trust-region theory, which identifies the ROM accuracy conditions required to guarantee the method converges to a critical point of the original topology optimization problem; a posteriori error estimation theory for projection-based ROMs, which informs ROM construction procedure to meet the accuracy conditions. This leads to trust-region methods that construct and update the ROM on-the-fly during optimization; the methods are guaranteed to converge to a critical point of the original, unreduced topology optimization problem, regardless of starting point. Numerical experiments on three different structural topology optimization problems demonstrate the proposed reduced topology optimization methods accelerate convergence to the optimal design by up to an order of magnitude. [ABSTRACT FROM AUTHOR]

In order to efficiently solve nonlinear transient heat conduction problem, a method combining the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) is proposed to establish the nonlinear reduced-order model, and the implementation of the reduced-order model makes fast numerical simulation possible in engineering field. Transient heat conduction problems with temperature-dependent and temperature-independent thermal conductivity are firstly solved by the RIBEM respectively, and two types of snapshots are composed of these solved transient temperature fields. Subsequently, the POD is applied in the analysis of the snapshots, and then a set of optimal orthogonal basis can be obtained. In the procedure of establishing the reduced-order model, we need to redistribute the nonlinear equations. The strategy is to separate the boundary nodes defined as Dirichlet condition, and then the orthogonal basis is used to derive a reduced-order expression for the remaining unknowns. Therefore, the degree of freedom that needs to be solved can be greatly reduced. Numerical examples show that different reduced-order models based on nonlinear basis and linear basis are all consistent with the full-order model. What is more, the computational efficiency is greatly improved due to the introduction of the reduced-order model. [ABSTRACT FROM AUTHOR]