Model Order Reduction Method Research Articles

H2model order reduction of bilinear systems via linear matrix inequality approach

AbstractThis paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods.

Reduced Order Based Active Disturbance Rejection Controller Design with Applications

Active disturbance rejection control (ADRC) has emerged as a well-addressed controller design technique in recent years. It is a suitable replacement for the error-based feedback PID controller design approaches. The ADRC constitutes two controller design techniques: linear active disturbance rejection control (LADRC) and generalized active disturbance rejection control (GADRC). The LADRC design approach requires minimal information about the plant, while in the GADRC approach, detailed information about the plant is needed. For higher-order plants, the design of ADRC controllers and extended state observers may be pretty complex and costlier. So, the performance analysis of the higher order plant becomes difficult. Therefore, to make the controller simpler, it is always advantageous to reduce the controller size. This paper proposes reduced order linear active disturbance rejection control (ROLADRC), and reduced order generalized active disturbance rejection control (ROGADRC) techniques instead of full-order LADRC and GADRC approaches. The stability equation method (SEM) and other widespread model order reduction (MOR) methods are utilized to reduce the order of the plant. Further, ROLADRC and ROGADRC are compared with existing control techniques in the literature. The effectiveness of the proposed scheme is tested on the sun tracker system (STS) for position control and power systems for load frequency control (LFC).

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Wasserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

A novel approach for computing washcoat diffusional effects in reduced order models of monolith reactors

Reduced order models (ROM) for catalytic reactors are usually expressed in terms of various internal and external transfer coefficients, which strongly depend on the local transport and reaction parameters, especially for fast reactions. The evaluation of these transfer coefficients requires the evaluation of functions of the local Thiele matrix, which could be cumbersome and computationally costly when the number of species or reactions is very large. In this work, we propose and validate a novel method based on the coupling of ROM and domain decomposition method for multi-component systems, which simplifies the evaluation of transfer coefficients, makes them independent of each other (diagonal matrices) and reaction parameters, as well as speed-up the simulations by orders of magnitude. This approach can be of significance in the real-time control of the reactors where speed of simulations is of critical importance.

Space‐local reduced‐order bases for accelerating reduced‐order models through sparsity

AbstractProjection‐based model order reduction (PMOR) methods based on linear or affine approximation subspaces accelerate numerical predictions by reducing the dimensionality of the underlying computational models. The state of the art of PMOR includes approximation methods based on state‐local subspaces—that is, subspaces associated with different regions of the solution manifold—and methods based on adaptive reduced‐order bases. For challenging applications such as those associated with highly nonlinear problems, such methods accelerate traditional PMOR by controlling the dimension of the reduced‐order basis (ROB) and associated projection‐based reduced‐order model (PROM). This article proposes an alternative as well as complementary approach for accelerating PMOR based on introducing sparsity into the ROB, in order to enhance the computational efficiency of the associated PROM. Specifically, the proposed approach introduces sparsity in PMOR by partitioning the computational domain rather than, or in addition to, the solution manifold, and therefore leads to the concept of a space‐local ROB. This concept is compatible with both concepts of a state‐local ROB and hyperreduction. It is demonstrated for two computational fluid dynamics problems in turbulent flow applications. Acceleration factors of the order of 1.5 relative to traditional PMOR and CPU time speedup factors of several orders of magnitude relative to high‐dimensional models are reported.

Stabilization of Generalized Empirical Interpolation Method (GEIM) in presence of noise: A novel approach based on Tikhonov regularization

The Empirical Interpolation Method (EIM), and its generalized version (GEIM), are non-intrusive, reduced-basis model order reduction methods hereby adopted and modified to address the problem of optimal placement of sensors and real-time estimation in thermo-hydraulics systems. These techniques have been used to extract the characteristic spatial modes of the system and select a set of points (or functionals) corresponding to the optimal locations for the sensors. Collecting experimental measurements in the available points allows the construction of an empirical interpolation of the fields employed to estimate the variable of interest. However, when these data are affected by noise, the (G)EIM loses its good convergence properties. In this context, stabilization techniques allow good field reconstruction even with noisy data. This work provides an alternative and effective solution to the problem of reconstructing the system state in the presence of experimental data affected by random noise by using the Tikhonov regularization technique. The developed methods have been tested on a simple thermo-fluid dynamics problem known as “two-sided lid-driven differentially heated square cavity”.

Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes

Repeatedly solving nonlinear partial differential equations with varying parameters is often an essential requirement to characterise the parametric dependences of dynamical systems. Reduced-order modelling (ROM) provides an economical way to construct low-dimensional parametric surrogates for rapid predictions of high-dimensional physical fields. This paper presents a physics-data combined machine learning (PDCML) method for non-intrusive parametric ROM in small-data regimes. Proper orthogonal decomposition (POD) is adopted for dimension reduction by deriving basis functions from a limited number of high-fidelity snapshots, and parametric ROM is thus transformed into establishing reliable mappings between the system parameters and the POD coefficients. To overcome labelled data scarcity, a physics-data combined ROM framework is developed to jointly integrate the physical principle and the small labelled data into feedforward neural networks (FNN) via a step-by-step training scheme. Specifically, a preliminary FNN model is firstly fitted via data-driven training, and then the governing physical rules are embedded into the loss function to improve the model interpolation and extrapolation performances through physics-guided training constrained by the labelled data. During the constrained optimisation procedure, dynamic weighting factors are used to adjust the physics-data proportion of the loss functions, aiming at continuously highlighting the physics loss as the primary optimisation objective and keeping the data loss as the constraint. This new PDCML method is tested on a series of nonlinear problems with different numbers of physical variables, and it is also compared with the data-driven ROM, the physics-guided ROM and the traditional projection-based ROM methods. The results demonstrate that the proposed method provides a cost-effective way for non-intrusive parametric ROM via machine learning, and it possesses good characteristics of high prediction accuracy, strong generalisation capability and small data requirement.

Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media

In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media. The Koopman operator has received extensive attention as an effective data-driven technique, which can transform the nonlinear dynamical systems into linear systems through acting on observation function spaces. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale media often vary in very different scales. In order to capture multiscale information well, the dimension of collected data may be high. To accurately construct the models of dynamical systems in multiscale media, we use high spatial dimension of observation data. It is challenging to compute the Koopman operators using the very high dimensional data. Thus, the strategy of reduced-order modeling is proposed to treat the difficulty. The proposed reduced-order modeling includes two stages: offline stage and online stage. In offline stage, a block-wise low rank decomposition is used to reduce the spatial dimension of initial snapshot data. For the nonautonomous dynamical systems, real-time observation data may be required to update the Koopman operators. The online reduced-order modeling is proposed to correct the offline reduced-order modeling. Three methods are developed for the online reduced-order modeling: fully online, semi-online and adaptive online. The adaptive online method automatically selects the fully online or semi-online and can achieve a good trade-off between modeling accuracy and efficiency. A few numerical examples are presented to illustrate the performance of the different reduced-order modeling methods.

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

MORPH-DSLAM: Model Order Reduction for Physics-Based Deformable SLAM.

We propose a new methodology to estimate the 3D displacement field of deformable objects from video sequences using standard monocular cameras. We solve in real time the complete (possibly visco-)hyperelasticity problem to properly describe the strain and stress fields that are consistent with the displacements captured by the images, constrained by real physics. We do not impose any ad-hoc prior or energy minimization in the external surface, since the real and complete mechanics problem is solved. This means that we can also estimate the internal state of the objects, even in occluded areas, just by observing the external surface and the knowledge of material properties and geometry. Solving this problem in real time using a realistic constitutive law, usually non-linear, is out of reach for current systems. To overcome this difficulty, we solve off-line a parametrized problem that considers each source of variability in the problem as a new parameter and, consequently, as a new dimension in the formulation. Model Order Reduction methods allow us to reduce the dimensionality of the problem, and therefore, its computational cost, while preserving the visualization of the solution in the high-dimensionality space. This allows an accurate estimation of the object deformations, improving also the robustness in the 3D points estimation.

Retraction Note to: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

Retraction Note to: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

Imperfection sensitivity study of the thermal–mechanical buckling of laminated composite cylinders using a novel reduced-order modeling method

Laminated composite cylinders subjected to axial compression exhibit the “snap-back” response, the buckling load of which is very sensitive to initial geometric imperfections. An operating environment with high temperatures influences both the buckling behavior and imperfection sensitivity of thin-walled cylinders. This paper proposes a novel reduced-order modeling method for nonlinear buckling and imperfection sensitivity analyses, considering the effects of the initial geometric imperfections and initial temperature field. The Koiter asymptotic theory is reformulated to construct a thermal–mechanical reduced-order model with initial geometric imperfections. The initial temperature field is converted to an additional degree of freedom during the construction of the reduced-order model. An a posteriori account of geometric imperfections is achieved for the thermomechanical reduced system of the perfect structure. A predictor–corrector process is developed to trace the thermoelastic geometrically nonlinear response of an axially compressed cylinder with geometric imperfections. Various numerical examples are designed to validate the high efficiency and satisfactory accuracy of the proposed method for imperfection sensitivity analyses. The optimization process based on the proposed method is applied to determine the most imperfection-sensitive laminate and the “worst” imperfection shape.

Fast-Dynamic Grey Wolf Optimizer for solving model order reduction of bilinear systems based on multi-moment matching technique

Grey Wolf Optimizer (GWO) has been known as one of the most popular and powerful metaheuristic search algorithms. It has great advantages such as simplicity, accuracy, and good convergence rate, but its performance degrades when the global optimum is not zero. To address these issues, in this paper, a new formulation is introduced to change its searching and hunting mechanisms, while the updating structure of wolves is also modified. With these changes, a proper compromise is achieved between exploitation and exploration. Therefore, the proposed algorithm, called Fast-Dynamic GWO (FDGWO), is more accurate with a faster convergence rate compared to GWO. Thirty-eight typical benchmark functions are optimized with FDGWO and compared with a set of efficient metaheuristic search algorithms. The Wilcoxon rank-sum test is used to statistically assess the results. With an overall efficacy of 84.2%, FDGWO is the most effective among competing algorithms. Then, the performance of FDGWO for ten real-world constrained engineering problems is compared with some highly accurate recent algorithms such as BP-ϵMAg-ES and COLSHADE. The results show that while FDGWO performs almost as accurate as these algorithms, but its computational complexity is quite less. Next, a new hybrid method based on FDGWO is proposed for model order reduction (MOR) of bilinear systems. First, the order of reduced system is determined via an iterative approach based on H2 norm of the error system. The unknown parameters of reduced model are then determined by minimizing a multi-objective fitness function aiming to match the multi-moments of original and reduced bilinear systems. Some stability constraints based on the Volterra series expansion of the states of the bilinear system are added to the optimization problem, which is solved by FDGWO algorithm. Finally, two bilinear test systems are reduced by the proposed hybrid method and compared with the most common bilinear MOR methods.

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier–Stokes Equations

This work investigates the use of sparse polynomial interpolation as a model order reduction method for the parametrized incompressible Navier–Stokes equations. Numerical results are presented underscoring the validity of sparse polynomial approximations and comparing with established reduced basis techniques. Two numerical models serve to assess the accuracy of the reduced order models (ROMs), in particular parametric nonlinearities arising from curved geometries are investigated in detail. Besides the accuracy of the ROMs, other important features of the method are covered, such as offline-online splitting, run time and ease of implementation. The findings provide a clear indication that sparse polynomial interpolation is a valid instrument in the toolbox of ROM methods.

Sector-model subspace iteration for vibration of multi-stage, cyclically symmetric systems

This work develops a computationally efficient sector-model subspace iteration to obtain accurate eigensolutions for the vibration of multi-stage systems where each stage is cyclically symmetric. The method fully captures the inter-stage coupling. The component stages are cyclically symmetric but the full multi-stage model is generally asymmetric because the number of sectors can differ among the stages. The sector-model subspace iteration is equivalent to the full-model subspace iteration in its accuracy, but it is more computationally efficient. Different from the full-model subspace iteration that relies on a full multi-stage system model, the sector-model subspace iteration only uses matrices of single-stage sector models. Parallel computation applies to most steps in each iteration and many matrices do not need to be recomputed for each iteration. The improved computational efficiency of the sector-model subspace iteration is more significant when the full model has a large number of stages, a large number of sectors within each stage, and/or many degrees of freedom in each single-stage sector. The iteration converges rapidly and can start with any initial trial vectors. Faster convergence is achieved using high-quality trial vectors from reduced-order modeling methods, such as single-stage analyses, multi-stage cyclic symmetry reduction, and component mode synthesis. In contrast to reduced-order modeling methods, the iteration provides eigensolutions with controllable accuracy based on user-selected convergence criteria. The application and advantages of the sector-model subspace iteration are demonstrated using two example multi-stage bladed disks.

Parallel model order reduction based on block discrete Fourier transform and Krylov subspace for parametric systems

This paper explores a time-domain parallel parametric model order reduction (PMOR) method for parametric systems based on the block discrete Fourier transform (DFT) and Krylov subspace. The proposed method is suitable for parametric systems with non-affine parametric dependence. With Taylor expansion, the expansion coefficients of the state variable are first obtained. Then, we show that the subspace spanned by the expansion coefficients belongs to a Krylov subspace. To speed up the PMOR process, a parallel strategy based on the block DFT and the structured matrices is proposed to compute the matrices involved in the Krylov subspace. This can avoid directly computing the inverse of the large-scale matrix. After that, the reduced parametric systems are constructed with the projection matrix obtained by the Arnoldi algorithm and orthonormalisation. Furthermore, we analyse the invertibility and the error estimations to guarantee the feasibility of the proposed PMOR method. Finally, the numerical experiments are given to demonstrate the effectiveness of the proposed method.

Inverse Identification of a Constitutive Model for High-Speed Forming Simulation: An Application to Electromagnetic Metal Forming

Forming simulation requires a constitutive model whose parameters are typically determined with tensile tests assumed static. However, this conventional approach is impractical for high-speed forming simulation characterized by high strain rates inducing transient effects. To identify constitutive parameters in relation to high-speed forming simulation, we formulated the problem of constitutive modeling as inverse parameter estimation addressed by regularized nonlinear least squares. Regarding the proposed inverse constitutive modeling, we adopted the L-curve method for proper regularization and model order reduction for rapid simulation. For demonstration, we corroborated the proposed strategy by identifying the modified Johnson–Cook model in the context of a free bulge test with electromagnetic metal forming simulation. The L-curve method allowed us to systematically choose a regularization parameter, and model order reduction brought enormous computational savings. After identifying constitutive parameters, we successfully verified and validated the reduced and original simulation models, respectively, with a manufactured workpiece. In addition, we validated the numerically identified constitutive model with a dynamic material test using a split Hopkinson pressure bar. Overall, we showed that inverse constitutive modeling for high-speed forming simulation can be effectively tackled by regularized nonlinear least squares with the help of an L-curve and a reduced-order model.

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

Comparing the effectiveness of structure-preserving model order reduction methods for vibro-acoustic problems

For the analysis of the vibration and dissipation behavior of structures, the evaluation of the systems' frequency response is essential. However, evaluating a frequency sweep on a high-fidelity model can be computationally very expensive. A remedy to this are model order reduction methods, which allow the computation of cheap-to-evaluate surrogate models. Additionally to the reduction of the internal system dimensions, model order reduction methods that preserve the matrix structure of the transfer function are preferred as the results may allow for physical reinterpretation and the use of similar analysis tools as for the original systems. In this work, we compare system-theoretic, structure-preserving model order reduction methods with a specific focus on their applicability to vibro-acoustic systems. We demonstrate their effectiveness in terms of accuracy and computational costs by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic-acoustic coupling. As a result, this comparative study allows us to derive some general recommendations for the choice of suitable model order reduction methods depending on the specific problem statement.