Parametric Model Order Reduction Research Articles

Parametric Model Order Reduction for Structural Optimization of Fiber Composite Structures

This research addresses the inherent computational demand of metaheuristic optimization techniques by developing a novel optimization framework that uses parametric model order reduction (PMOR). The improvement in efficiency is demonstrated using the buckling optimization of variable-angle tow (VAT) composite panels, as navigating the complex design space of VAT composite structures with global optimization techniques requires a substantial number of function evaluations. A new affine summation relationship conducive to PMOR was derived for its finite element stiffness matrix using the concept of lamination parameters and material invariant matrices, and principal component analysis (PCA) was used to extract the reduced basis vectors. Particle swarm optimization was conducted using a full-order model (FOM) and a reduced-order model. PMOR-based optimization exhibited a 0.55% relative error in the optimal objective value compared to the FOM analysis and exhibited a similar convergence history. It was observed that the optimization time was reduced by 93% by the novel affine decomposition alone, but PMOR achieved a significant reduction of 99.8% in memory requirements compared to the FOM. As the affine-decomposition-based assembly of the FOM is not feasible for large problems, PMOR functions as an enabling tool for leveraging the improvement offered by it.

Direct Taylor Expansion of Substructures Constructed Parametric Reduced-Order Modeling Method

The parametric substructure modeling shows enormous potential for efficiently conducting dynamic reanalysis of large-scale structures with small geometric variability. Parametric substructure mode is a vital component in ensuring the accuracy of the modeling process. This work uses the Craig–Bampton Component Mode Synthesis method to calculate normal and constraint modes of substructures at interpolation points. Instead of using singular value decomposition of augmented fixed-interface modes, the presented approach performs direct Taylor expansion for substructure modes with randomly varying parameters. The parametric substructure modes are approximated by the modes at interpolation points. The global parametric reduced-order model is constructed through the synthesis of parametric substructures. The effectiveness of the proposed approach is validated through 1) a thickness-variable beam, 2) a cantilevered plate with varying thicknesses, and 3) a mistuned blisk. Numerical results demonstrate that the computational accuracy of the parametric reduced-order model aligns closely with a more time-consuming full-order model.

An efficient framework for design optimization using parametric reduced order modeling in a virtual prototyping phase

Virtual Prototype Assembly is a tool enabling the virtual assembly of components, the prediction of noise and vibration performance, and the evaluation of component modification scenarios. FRF-based substructuring is the technique used to assemble the components, either obtained from experimental measurements or numerical simulation models. However, numerical simulations of components can be time-consuming, especially when exploring several component design modifications of large models. The paper proposes a framework combining Virtual Prototype Assembly with a parametric Model Order Reduction technique for vehicle design optimization. The proposed parametric Model Order Reduction technique generates a Reduced Order Model of a component maintaining an explicit dependency on material parameters and properties, such as: Young's Modulus, density, Poisson coefficient, connection stiffness, etc.. This enables fast and efficient simulation of components designs without the need of recomputing numerical models, thus streamlining the component design optimization process. The effect of component design modifications will be assessed at the predicted target vibration levels on an academic application case.

Domain decomposition for physics-data combined neural network based parametric reduced order modelling

This paper proposes a novel domain decomposition (DD) method for the parametric reduced-order model based on the Physics-Data Combined Neural Network (DD-PDCNN). The computational domain is partitioned into a number of subdomains, and a Physics-Data Combined Neural Network (PDCNN) is constructed for each subdomain, which not only represents the dynamics within its region but also considers interactions with neighbouring subdomains. The interface conditions between subdomains are considered by averaging solutions at the neighbourhoods, incorporating averaged solutions, predicted solutions at current and next time levels into reduced governing equations. The implicit coupling among the subdomains ensures global continuity and establishes boundary conditions for each subdomain.The performance of this new DD-PDCNN is compared with that of PDCNN without domain decomposition, data-driven model and traditional Reduced Basis(RB) model. The capability of this model is tested using a number of parametric nonlinear problems, such as the Korteweg-de Vries (KdV) equation, the two-dimensional Kovasznay flow and the two-dimensional incompressible Navier–Stokes equation. The results indicate that the proposed method offers an effective way to construct an accurate reduced order model for parameterised PDEs through machine learning. In particular, in specific parameter ranges where distinct physical phenomenon regions are prominent, the method outperforms the model without domain decomposition, demonstrating excellent performance on several challenging problems.

Physics-aware neural network-based parametric model-order reduction of the electromagnetic analysis for a coated component

AbstractFinite element (FE) analysis is one of the most accurate methods for predicting electromagnetic field scatter; however, it presents a significant computational overhead. In this study, we propose a data-driven parametric model-order reduction (pMOR) framework to predict the scattered electromagnetic field of FE analysis. The surface impedance of a coated component is selected as parameter of analysis. A physics-aware (PA) neural network incorporated within a least-squares hierarchical-variational autoencoder (LSH-VAE) is selected for the data-driven pMOR method. The proposed PA-LSH-VAE framework directly accesses the scattered electromagnetic field represented by a large number of degrees of freedom (DOFs). Furthermore, it captures the behavior along with the variation of the complex-valued multi-parameters. A parallel computing approach is used to generate the training data efficiently. The PA-LSH-VAE framework is designed to handle over 2 million DOFs, providing satisfactory accuracy and exhibiting a second-order speed-up factor.

Greedy identification of latent dynamics from parametric flow data

Projection-based reduced-order models (ROMs) play a crucial role in simplifying the complex dynamics of fluid systems. Such models are achieved by projecting the Navier-Stokes equations onto a lower-dimensional subspace while preserving essential dynamics. However, this approach requires prior knowledge of the underlying high-fidelity model, limiting its effectiveness when applied to black-box data. This article introduces a novel, non-intrusive, data-driven method–Greedy Identification of Latent Dynamics (GILD)–for constructing parametric fluid ROMs. Unlike traditional methods, GILD constructs models directly from data, without relying on specific high-fidelity model information. It also employs interpolation within the manifold R∗N×q/Oq to accommodate parameter variability. Numerical experiments on various fluid dynamics scenarios, including lid-driven cavity flow, flow past a cylinder with varying Reynolds number, and Ahmed body flow with variable geometry, demonstrate GILD’s robust performance across both training and unseen parameter values. GILD’s ability to accurately capture system dynamics and its adaptability to diverse data sources highlight its potential as a powerful tool for constructing parametric reduced-order models in an easy and general way for complex fluid dynamics and beyond.

Data-driven surrogate modeling and optimization of supercritical jet into supersonic crossflow

For the design and optimization of advanced aero-engines, the prohibitively computational resources required for numerical simulations pose a significant challenge, due to the extensive exploration of design parameters across a vast design space. Surrogate modeling techniques offer a viable alternative for efficiently emulating numerical results within a notably compressed timeframe. This study introduces parametric Reduced-Order Models (ROMs) based on Convolutional AutoEncoders (CAE), Fully Connected AutoEncoders (FCAE), and Proper Orthogonal Decomposition (POD) to fast emulate spatial distributions of physical variables for a supercritical jet into a supersonic crossflow under different operating conditions. To further accelerate the decision-making process, an optimization model is developed to enhance fuel-oxidizer mixing efficiency while minimizing total pressure loss. Results indicate that CAE-based ROMs exhibit superior prediction accuracy while FCAE-based ROMs show inferior predictive accuracy but minimal uncertainty. The latter may be ascribed to the markedly greater number of hyperparameters. POD-based ROMs underperform in regions of strong nonlinear flow dynamics, coupled with higher overall prediction uncertainties. Both AE- and POD-based ROMs achieve online predictions approximately 9 orders of magnitude faster than conventional simulations. The established optimization model enables the attainment of Pareto-optimal frontiers for spatial mixing deficiencies and total pressure recovery coefficient.

Tensorial Parametric Model Order Reduction of Nonlinear Dynamical Systems

Tensorial Parametric Model Order Reduction of Nonlinear Dynamical Systems

A system-level interface sampling and reduction method for component mode synthesis with varying parameters

This paper presents a new interface sampling and reduction method for varying parameters within the framework of the component mode synthesis (CMS) method. In the conventional interpolation-based parametric reduced-order model (IB-PROM) based on CMS, reduced subsystems are synthesized, and interface degrees-of-freedom are reduced via the secondary eigenvalue analysis in the online stage, which is essential to obtain highly reduced system matrices. However, it inhibits rapid parameter updating and requires extra computational resources in the online stage. In the present study, the pre-reduced interface sampling method is proposed by synthesizing substructures and reducing the interface before beginning the online stage. The interface solution is obtained through offline and online adaptation, likewise the procedure of computing the interior solution of previous studies. Additionally, congruence transformations are applied to each reduced interface, which is mandatory for interpolating symmetric positive definite matrices with respect to a new input parameter. As a result, the online performance of IB-PROM is greatly enhanced, showing rapid and near real-time adaptations to the variations in system parameters.

Parameterized Reduced-Order Models for Probabilistic Analysis of Thermal Protection System Based on Proper Orthogonal Decomposition

The thermal protection system (TPS) represents one of the most critical subsystems for vehicle re-entry. However, due to uncertainties in thermal loads, material properties, and manufacturing deviations, the thermal response of the TPS exhibits significant randomness, posing considerable challenges in engineering design and reliability assessment. Given that uncertain aerodynamic heating loads manifest as a stochastic field over time, conventional surrogate models, typically accepting scalar random variables as inputs, face limitations in modeling them. Consequently, this paper introduces an effective characterization approach utilizing proper orthogonal decomposition (POD) to represent the uncertainties of aerodynamic heating. The augmented snapshots matrix is used to reduce the dimension of the random field by the decoupling method of independently spatial and temporal bases. The random variables describing material properties and geometric thickness are also employed as inputs for probabilistic analyses. An uncoupled POD Gaussian process regression (UPOD-GPR) model is then established to achieve highly accurate solutions for transient heat conduction. The model takes random heat flux fields as inputs and thermal response fields as outputs. Using a typical multi-layer TPS and thermal structure as two examples, probabilistic analyses are conducted. The mean square relative error of a typical multi-layer TPS is less than 4%. For the thermal structure, the averaged absolute error of the radiation and insulation layer is less than 25 °C and 6 °C when the maximum reaches 1200 °C and 150 °C, respectively. This approach can provide accurate and rapid predictions of thermal responses for TPS and thermal structures throughout their entire operating time when furnished with input heat flux fields and structural parameters.

Coupled thermal-mechanical analysis of power electronic modules with finite element method and parametric model order reduction

This work presents a new approach for performing a parametric study and examining nonlinear material behaviours of a coupled thermal-mechanical model of a Power Electronics Module (PEM) by integrating the Finite Element Method (ANSYS-FEM) with Parametric Model Order Reduction (pMOR). The considered coupling method solves the thermal and structural models concurrently compared to the widely practised sequential coupling method. Instead of constant parameter values, which are generally regarded for pMOR studies, the temperature-dependent material properties of the wire material have been parametrised in the work using the pMOR method. A generalised 2D model has been regarded here for thermal-mechanical analysis with the pMOR approach, parametrising temperature-dependent coefficient of thermal expansion (CTE) and Young’s modulus (E) of the wire material to explore their impact on wire bonds. The matrix interpolation method has been applied here for the pMOR study, and PRIMA, a Krylov subspace-based model order reduction (MOR) technique, has been exercised for local model order reductions. A new efficient process of matrix interpolation, based on the Lagrange interpolation technique, has been developed to implement matrix interpolation in the parametric reduced order model (pROM). The local reduced order models (ROMs) have a degree of freedom (DOF) of just 8, compared to the full-order models’ (FOMs) of 50,602. The pROM provides an excellent solution and reduces computational time by 84% for the presented case.

Thermal-Mechanical Analysis of a Power Module with Parametric Model Order Reduction

This paper presents parametric model order reduction (pMOR) by the Lagrange approach of matrix interpolation for the thermal-mechanical and reliability study of a power electronics module (PEM) with nonlinear behaviours. Most previous research in model order reduction (MOR) studies reports thermal-mechanical simulations using a sequentially coupled method. In this research, a direct-coupled thermal-mechanical analysis, which simultaneously solves the thermal and structural governing equations, has been used to obtain thermal and deformation results. Furthermore, for pMOR, the linear approach of matrix interpolation is limited to linear changes between sampled-parametric points. Hence, a new way of interpolating system matrices using the Lagrange interpolation method has been adopted to implement the matrix interpolation efficiently. The parametric reduced-order model (pROM) solution by the Lagrange approach of matrix interpolation agrees well with the full-order model (FOM) and takes similar computational time as the linear (bi-linear) approach of matrix interpolation. pROM simulations offer up to 85.5% reduction in computational time.

On reduced-order modeling of gas–solid flows using deep learning

Reduced-order models (ROMs) have been extensively employed to understand complex systems efficiently and adequately. In this study, a novel parametric ROM framework is developed to produce Eulerian–Lagrangian simulations. This study employs two typical parametric strategies to reproduce the physical phenomena of a gas–solid flow by predicting the adequate dynamics of modal coefficients in the ROM: (i) based on the radial-basis function (RBF) interpolation, termed ROM-RBF and (ii) based on a long–short term memory (LSTM) neural network, termed ROM-LSTM. In the ROM, an advanced technique, namely, Lanczos-based proper orthogonal decomposition (LPOD), is employed to efficiently transform numerical snapshots into the modal coefficients. Validation tests are conducted in a typical gas–solid flow system such as a spouted bed. The coherent structures of the gas–solid flows are shown to be captured by the LPOD technique. Besides, in comparison with the high-fidelity simulations, our proposed ROMs are shown to simulate the gas–solid flows by significantly reducing the calculation time by several orders of magnitude and faithfully predicting the macroscopic properties. In particular, compared to the ROM-RBF, the ROM-LSTM can capture the flow fields more accurately within the gas–solid flows.

Toward Digital Twin Development for Implant Placement Planning Using a Parametric Reduced-Order Model.

Real-time stress distribution data for implants and cortical bones can aid in determining appropriate implant placement plans and improving the post-placement success rate. This study aims to achieve these goals via a parametric reduced-order model (ROM) method based on stress distribution data obtained using finite element analysis. For the first time, the finite element analysis cases for six design variables related to implant placement were determined simultaneously via the design of experiments and a sensitivity analysis. The differences between the minimum and maximum stresses obtained for the six design variables confirm that the order of their influence is: Young's modulus of the cancellous bone > implant thickness > front-rear angle > left-right angle > implant length. Subsequently, a one-dimensional (1-D) CAE solver was created using the ROM with the highest coefficient of determination and prognosis accuracy. The proposed 1-D CAE solver was loaded into the Ondemand3D program and used to implement a digital twin that can aid with dentists' decision making by combining various tooth image data to evaluate and visualize the adequacy of the placement plan in real time. Because the proposed ROM method does not rely entirely on the doctor's judgment, it ensures objectivity.

Data-driven nonlinear parametric model order reduction framework using deep hierarchical variational autoencoder

A data-driven parametric model order reduction (MOR) method using a deep artificial neural network is proposed. The present network, a least-squares hierarchical variational autoencoder (LSH-VAE), is capable of performing nonlinear MOR for the parametric interpolation of a nonlinear dynamic system with a significant number of degrees of freedom. LSH-VAE differs from existing networks in two major respects: a deep hierarchical structure and a hybrid weighted, probabilistic loss function. The enhancements result in significantly improved accuracy and stability compared with conventional nonlinear MOR methods, autoencoders, and variational autoencoders. In LSH-VAE, the parametric MOR framework is based on the spherically linear interpolation of the latent manifold. The present framework is validated and evaluated on three nonlinear and multiphysics dynamic systems. First, the present framework is evaluated on the fluid–structure interaction benchmark problem to assess its efficiency and accuracy. Then, a highly nonlinear aeroelastic phenomenon, limit cycle oscillation, is analyzed. Finally, the framework is applied to three-dimensional fluid flow to demonstrate its capability to efficiently analyze a significantly large number of degrees of freedom. The superior performance of LSH-VAE is emphasized by comparing its results against those of widely used nonlinear MOR methods, a convolutional autoencoder, and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-VAE. The proposed framework exhibits significantly enhanced accuracy compared with that of conventional methods, while the computational efficiency continues to remain significantly higher.

Parametric Interpolation Model Order Reduction on Grassmann Manifolds by Parallelization

Parametric Interpolation Model Order Reduction on Grassmann Manifolds by Parallelization

Improving accuracy in parametric reduced-order models for classical guitars through data-driven discrepancy modeling

Recently developed high-fidelity finite element (FE) models represent a state-of-the-art approach for gaining a deeper understanding of the vibrational behavior of musical instruments. They can also be used as virtual prototypes. However, certain analyses, such as optimization or parameter identification, necessitate numerous model evaluations, resulting in long computation times when utilizing the FE model. Projection-based parametric model order reduction (PMOR) proves to be a powerful tool for enhancing the computational efficiency of FE models while retaining parameter dependencies. Despite their advantages, projection-based methods often require complete system matrices, which may have limited accessibility. Consequently, a systematic discrepancy is introduced in the reduced-order model compared to the original model. This contribution introduces a discrepancy modeling method designed to approximate the parameter-dependent effect of a radiating boundary condition in an FE model of a classical guitar that cannot be exported from the commercial FE software Abaqus. To achieve this, a projection-based reduced-order model is augmented by a data-driven model that captures the error in the approximation of eigenfrequencies and eigenmodes. Artificial neural networks account for the data-driven discrepancy models. This methodology offers significant computational savings and improved accuracy, making it highly suitable for far-reaching parametric studies and iterative processes. The combination of PMOR and neural networks demonstrate greater accuracy than using either approach alone. This paper extends our prior research presented in the proceedings of Forum Acusticum 2023, offering a more comprehensive examination and additional insights.

Parametric model reduction for a cantilevered pipe conveying fluid via parameter-dependent center and unstable manifolds

A cantilevered pipe conveying fluid can display highly nonlinear dynamical behaviors. High-resolution discrete models are used to accurately simulate the highly nonlinear dynamics of the pipe system. Such simulations need to be performed many times in order to study how the nonlinear responses are affected by the flow velocity. To facilitate and speed up the analysis of the nonlinear dynamics, we show how to use parameter-dependent center and unstable manifolds construct parametric reduced-order models (ROMs) that work for a range of flow velocities. Such ROMs enable efficient extraction of bifurcation diagrams characterizing how self-excited limit cycles of the pipe system in the post-flutter region are affected by the flow velocity. We find that the unstable-manifold-based ROMs outperform the center-manifold-based ROMs because the former work for a larger range of flow velocities. In addition, we demonstrate how to use the unstable-manifold-based ROMs predict forced nonlinear vibrations of the pipe, including periodic and quasi-periodic responses and their dependence on the flow velocity.

Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

Accurate error estimation is crucial in model order reduction, to obtain small reduced-order models as well as to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.

Admittance boundary conditions and sound pressure field estimation of vibro-acoustic systems using an extended Kalman filter and parametric model order reduction

Time-domain vibro-acoustic finite element simulations have been recently used in auralization and virtual sensing. However, when comparing the transient response based on a finite element model to actual measurements, often discrepancies are seen due to, amongst others, the presence of inaccurate admittance boundary conditions. To achieve accurate time-domain simulations, these uncertainties need to be included in the time-domain simulation. In this paper, the admittance boundary conditions are modelled using the superposition of several passive functions whose parameters are considered as the additional states in an extended Kalman filter scheme. Finite element models for vibro-acoustic systems including admittance boundary conditions have a high computational cost and result in frequency-dependent damping matrices. To get a smaller time-domain model, stability-preserving model order reduction is employed to reduce the number of degrees of freedom while maintaining a good accuracy and stability. Furthermore, to preserve the parameter dependency, parametric model order reduction based on a Taylor series expansion is employed. Experiments are performed on a vibro-acoustic system including a foam layer which can be represented by admittance boundary conditions. These conditions are identified and it is shown that the predicted sound field pressure is improved significantly as compared to the direct time-domain simulation.