Model Reduction Techniques Research Articles

Simulation of melt film dynamics in laser fusion cutting using a boundary layer approximation

Process inherent instabilities in the dynamics of the thin melt film during laser fusion cutting reduce the quality of the cut surface with regards to mean profile height and angular tolerance. To enhance the process understanding and to identify and evaluate quality improvement measures, a 3D model of the melt film dynamics is developed and applied to experimental results from literature. To cope with the multi-scale character of the mathematical task, model reduction techniques are applied: the underlying transport equations are mapped to a conformal coordinate system and subjected to a scaling analysis. By using a perturbation series, the most significant contributions to the equations of motion are selected. An integral boundary layer approximation reduces the dimension of the numerical task to a two-dimensional problem that can be solved with very low computational effort. The resulting simulation is able to reflect the dynamics at a spatial resolution below Δx≤25μm and a temporal resolution below Δt≤2.5μs within a reasonable calculation time of a few days for a cut length of L=10mm and a cut thickness of a=3mm. Known correlations between process parameters and quality features are reproduced by this simulation. Results show that a concurrent improvement of mean profile height and angular tolerance cannot be obtained by straightforward measures like changing the beam profile from tophat to Gaussian and vice-versa. Future work will be focused on the extension of this model with regards to surface tension and evaporation.

Model Order Reduction of Cardiac Monodomain Model using Deep Autoencoder Based Neural Networks

The numerical study of electrocardiology involves prohibitive computational costs because of its complex and nonlinear dynamics. In this paper, a lowdimensional model of the cardiac monodomain formulation has been developed by using the deep learning method. The restricted Boltzmann machine and deep autoencoder machine learning techniques have been used to approximate the cardiac tissue’s full order dynamics. The proposed reduced order modeling begins with the development of the low-dimensional representations of the original system by implementing the neural networks from the numerical simulations of the full order monodomain system. Consequently, the reduced order representations have been utilized to construct the lower-dimensional model, and finally, it has been reconstructed back to the original system. Numerical results show that, the proposed deep learning MOR framework gained computational efficiency by a factor of 85 with acceptable accuracy. This paper compares the accuracy of the deep learning based model order reduction method with the two different techniques of model reduction: proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The model order reduction deploying the deep learning method outperforms the POD and DMD concerning the modeling accuracy.

Cascaded robust control of mechanical ventilator using fractional order sliding mode control.

A mechanical ventilator is an important medical equipment that assists patients who have breathing difficulties. In recent times a huge percentage of COVID-19 infected patients suffered from respiratory system failure. In order to ensure the abundant availability of mechanical ventilators during COVID-19 pandemic, most of the manufacturers around the globe utilized open source designs. Patients safety is of utmost importance while using mechanical ventilators for assisting them in breathing. Closed loop feedback control system plays vital role in ensuring the stability and reliability of dynamical systems such as mechanical ventilators. Ideal characteristics of mechanical ventilators include safety of patients, reliability, quick and smooth air pressure buildup and release.Unfortunately most of the open source designs and mechanical ventilator units with classical control loops cannot achieve the above mentioned ideal characteristics under system uncertainties. This article proposes a cascaded approach to formulate robust control system for regulating the states of ventilator unit using blower model reduction techniques. Model reduction allows to cascade the blower dynamics in the main controller design for airway pressure. The proposed controller is derived based on both integer and non integer calculus and the stability of the closed loop is ensured using Lyapunov theorems. The effectiveness of the proposed control method is demonstrated using extensive numerical simulations.

Model reduction techniques for high-rise buildings andits reduced-order controller with an improved BT method

An AMD control system is usually built based on the original model of a target building. As a result, the fact leads a large calculation workload exists. Therefore, the orders of a structural model should be reduced appropriately. Among various model-reduction methods, a suitable reduced-order model is important to high-rise buildings. Meanwhile, a partial structural information is discarded directly in the model-reduction process, which leads to the accuracy reduction of its controller design. In this paper, an optimal technique is selected through comparing several common model-reduction methods. Then, considering the dynamic characteristics of a high-rise building, an improved balanced truncation (BT) method is proposed for establishing its reduced-order model. The abandoned structural information, including natural frequencies, damping ratios and modal information of the original model, is reconsidered. Based on the improved reduced-order model, a new reduced-order controller is designed by a regional pole-placement method. A high-rise building with an AMD system is regarded as an example, in which the energy distribution, the control effects and the control parameters are used as the indexes to analyze the performance of the improved reduced-order controller. To verify its effectiveness, the proposed methodology is also applied to a four-storey experimental frame. The results demonstrate that the new controller has a stable control performance and a relatively short calculation time, which provides good potential for structural vibration control of high-rise buildings.

Augmented phase reduction for periodic orbits near a homoclinic bifurcation and for relaxation oscillators

Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: periodic orbits born out of a homoclinic bifurcation, and relaxation oscillators. To validate our calculations, we simulate models in these dynamic regimes, and compare their numerically computed augmented phase reduction with the derived analytical expressions. These analytical and numerical calculations help us to understand conditions for which the use of augmented phase reduction over the standard phase reduction can be advantageous.

Dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation with periodic boundary conditions

The paper deals with the dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) with periodic boundary conditions. First, the dynamics of the MGKdVB equation is studied using the Fourier Galerkin and the Karhunen–Loeve (K–L) Galerkin methods. The Fourier Galerkin approach is used to generate a system of nine ordinary differential equations (ODEs) from a partial differential equation (PDE), and the K–L Galerkin method is used as a model reduction technique for nonlinear systems to derive a reduced-order system of two ODEs that imitates the dynamics of the MGKdVB equation. It is shown that the two-dimensional reduced-order ODE system based on the K–L Galerkin method is superior to the Fourier Galerkin n-dimensional ODE system for any dimension $$n<2N+1$$ , where $$N=2$$ is the number of unstable eigenvalues, and it is comparable to the traditional higher nine-dimensional Fourier Galerkin system. Then, in order to enhance the stability of the MGKdVB equation, we propose to use state feedback linearization control schemes for both systems and show that the reduced-order system based on K–L Galerkin method is less expensive to control as compared to the Fourier Galerkin system. Finally, numerical simulations of the controlled systems are shown to illustrate the developed theory.

Stein Variational Reduced Basis Bayesian Inversion

We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from the posterior distribution, we integrate an adaptive and goal-oriented model reduction technique with an optimization-based Stein variational gradient descent method. We present detailed analyses for the reduced basis approximation errors of the potential and its gradient, the induced errors of the posterior distribution measured by Kullback--Leibler divergence, as well as the induced errors of the Stein variational samples. To demonstrate the computational accuracy and efficiency of SVRB, we report results of numerical experiments on a Bayesian inverse problem governed by a diffusion PDE with random parameters with both uniform and Gaussian prior distributions. Over 100X speedups can be achieved while the accuracy of the approximation of the potential and its gradient is preserved.

Implicit higher-order moment matching technique for model reduction of quadratic-bilinear systems

We propose a projection based multi-moment matching method for model order reduction of quadratic-bilinear systems. The goal is to construct a reduced system that ensures higher-order moment matching for the multivariate transfer functions appearing in the input-output representation of the non-linear system. An existing technique achieves this for the first two multivariate transfer functions, in what is called the symmetric form of the multivariate transfer functions. We extend this framework to an equivalent and simplified form, the regular form, which allows us to show moment matching for the first three multivariate transfer functions. Numerical results for three benchmark examples of quadratic-bilinear systems show that the proposed framework exhibits better performance with reduced computational cost in comparison to existing techniques.

Fully distributed time-varying formation tracking control of linear multi-agent systems with input delay and disturbances

Abstract This paper investigates the time-varying formation tracking (TVFT) control problem considering a constant input delay and unknown external disturbances for the general linear multi-agent system (MAS) under a directed communication graph containing a spanning tree. To achieve that, a new time-varying shape format is firstly proposed. Then, a disturbance observer (DO) is introduced to compensate the unknown disturbance effect. After that, the Artstein’s model reduction technique is adopted and modified to design a state predictor in order to transform the MAS with a delayed input into a delay-free system. Fourthly, an adaptive observer (AO), which is used to estimate the designed state predictor, is proposed by using the neighbors’ information such as the inputs, time-varying shapes, DOs, AOs and relative state measurements. The convergence of closed-loop system is guaranteed by the designed algebraic Riccati equation. The whole controller requires no eigenvalue information of the Laplacian matrix of communication graph, thus is fully distributed. Finally, the effectiveness of proposed fully distributed controller (FDC) is verified by numerical examples and factors influencing FDC performances are analyzed.

Laguerre Expansion Series Based Reduced Order Interval Systems

Model order reduction techniques have attracted researchers for the analysis of higher-order systems. The brief presents the order reduction of the interval system in state-space form using the Laguerre polynomial method with singular value decomposition. Laguerre - SVD model reduction technique is approved for the algorithms reduced computational complexity and stability perseverance. The proposed algorithm applies Laguerre approximation to interval system to compute an interval Krylov matrix using the Interval laboratory MATLAB toolbox. Then singular value decomposition is performed on the concatenated matrix with the upper limit and lower limit of the obtained interval Krylov matrix. The resultant column orthogonal matrix yields the reduced interval system through congruence transformation. The technique preserves the original system properties and is proved to be stable. Pertinent numerical results validate the proposed technique with examples from literature.

Nonlinear model predictive control of ultra-high-purity air separation units using transient wave propagation model

Model reduction techniques can be used to reduce the computational burden associated with nonlinear model predictive control (NMPC). In our recent work, we introduced the transient nonlinear wave propagation model (TWPM) for reduced dynamic modeling of multi-component distillation columns with variable holdup, and demonstrated its suitability for optimization and control of single-section distillation columns and simple air separation units [Caspari et al., J. Process Control, 2020]. We show here that the TWPM is well-suited for reduced modeling of multi-sectional ultra-high-purity distillation columns and enables real-time capable NMPC of complex process flowsheets with tight operational constraints. To demonstrate its performance and accuracy, we apply the TWPM for NMPC of an ultra-high-purity nitrogen air separation unit. We perform an in-silico closed-loop case study comprising a series of load changes. Our approach reduces CPU time by 84%, enabling NMPC in real time.

An Energy Closure Criterion for Model Reduction of a Kicked Euler–Bernoulli Beam

AbstractReduced order models (ROMs) can be simulated with lower computational cost while being more amenable to theoretical analysis. Here, we examine the performance of the proper orthogonal decomposition (POD), a data-driven model reduction technique. We show that the accuracy of ROMs obtained using POD depends on the type of data used and, more crucially, on the criterion used to select the number of proper orthogonal modes (POMs) used for the model. Simulations of a simply supported Euler–Bernoulli beam subjected to periodic impulsive loads are used to generate ROMs via POD, which are then simulated for comparison with the full system. We assess the accuracy of ROMs obtained using steady-state displacement, velocity, and strain fields, tuning the spatiotemporal localization of applied impulses to control the number of excited modes in, and hence the dimensionality of, the system’s response. We show that conventional variance-based mode selection leads to inaccurate models for sufficiently impulsive loading and that this poor performance is explained by the energy imbalance on the reduced subspace. Specifically, the subspace of POMs capturing a fixed amount (say, 99.9%) of the total variance underestimates the energy input and dissipated in the ROM, yielding inaccurate reduced-order simulations. This problem becomes more acute as the loading becomes more spatio-temporally localized (more impulsive). Thus, energy closure analysis provides an improved method for generating ROMs with energetics that properly reflect that of the full system, resulting in simulations that accurately represent the system’s true behavior.

BUILDING STRUCTURE OPTIMIZATION BASED ON A HIGH-DIMENSIONAL MODEL REPRESENTATION APPROACH

In this study, a model reduction technique based on a high-dimensional model representation (HDMR) approach was investigated and applied to design optimization of building structures. Those structures have long been designed using engineering intuition and an iterative trial-and-error method. In order to evaluate structural responses, a finite element (FE) analysis code is generally required. Gradient-based numerical algorithms and evolutionary algorithms are widely available and can be adopted to the design optimization of structures. An alternative category of optimization methods relies on approximate objective or constraint functions that can be created using various interpolation or regression techniques. In this work, the model reduction was achieved using augmented radial basis functions (RBFs) as component functions of HDMR. After sample points were generated along each variable axis, detailed FE analyses were conducted to evaluate building responses, which were used for constructing RBF-HDMR models of structural responses. The optimization was performed using a standard gradient-based numerical method. The accuracy of the RBF-HDMR could be improved if the optimal design point was added as an additional sample point. One advantage of the proposed optimization approach was that the interface programming with any existing FE code was not necessary. To illustrative the application of the method, a high-rise building was studied and optimized in order to reduce the building’s global torsional responses. The proposed optimization method worked well for the example.

ENGINEERING PROBABILISTIC ANALYSIS USING MODEL REDUCTION AND AUGMENTED RADIAL BASIS FUNCTIONS

Probabilistic analysis of practical engineering problems has long been based on traditional sampling-based approaches, such as Monte Carlo Simulations (MCS) and gradient-based first-order and second-order methods. Since the finite element (FE) or other numerical methods are required to evaluate engineering system responses, such as forces or displacements, it is not efficient to directly integrate FE and sampling-based analysis approaches. Over the years, various approximate methods have been developed and applied to the reliability analysis of engineering problems. In this study, an efficient model reduction technique based on high-dimensional model reduction (HDMR) method has been developed using augmented radial basis functions (RBFs). The basic idea is to use augmented RBFs to construct HDMR component functions. The first-order HDMR model only requires sample points along each variable axis. The HDMR model, once created and used to explicitly express a performance function, is further combined with MCS to perform probabilistic calculations. As test problems, a mathematical problem and a 10-bar truss example are studied using the proposed reliability analysis approach. The proposed method works well, and accurate reliability analysis results are found with a small number of original performance function evaluations, i.e., FE simulations.

Windowed least-squares model reduction for dynamical systems

This work proposes a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space–time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov–Galerkin (LSPG), and space–time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space–time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space–time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space–time residuals than Galerkin and LSPG projection.

Data-based model reduction for phase change problems with convective heat transfer

Latent heat thermal energy storages (LHTES) exploit the high energy density of phase change material (PCM). The typically low thermal conductivity of PCM limits the charging and discharging rates and poses considerable challenges for dynamic storage operation. To operate LHTES efficiently and to exploit their full potential, new methods are required to obtain accurate and fast models for state of charge estimation and control tasks. In LHTES the heat transfer in low viscosity PCM is driven by conduction and also significantly by convective transport. In previous works, various high-precision models have been developed which employ finite element, difference and volume methods to solve the coupled Navier–Stokes and energy equations, but they incur large computational effort. In the present work, a novel, high-fidelity model reduction technique is proposed to achieve real-time capability while preserving high model accuracy. The idea is to short-cut the laborious solution of the Navier–Stokes equations by an efficiently parametrized, data-based model which approximates the stream function of the typical convection flow pattern by singular value decomposition. To account for the complexity of the solution-dependent flow domain, a suitable transformation method is proposed. The efficiency and accuracy of the proposed reduction method is demonstrated in typical operating modes.

Dynamics models for identifying the key transmission parameters of the COVID-19 disease

After the analysis and forecast of COVID-19 spreading in China, Italy, and France the WHO has declared the COVID-19 a pandemic. There are around 100 research groups across the world trying to develop a vaccine for this coronavirus. Therefore, the quantitative and qualitative analysis of the COVID–19 pandemic is needed along with the effect of rapid test infection identification on controlling the spread of COVID-19. Mathematical models with computational simulations are the effective tools that help global efforts to estimate key transmission parameters and further improvements for controlling this disease. This is an infectious disease and can be modeled as a system of non-linear differential equations with reaction rates. In this paper, we develop the models for coronavirus disease at different stages with the addition of more parameters due to interactions among the individuals. Then, some key computational simulations and sensitivity analysis are investigated. Further, the local sensitivities for each model state concerning the model parameters are computed using the model reduction techniques: the dynamical models are eventually changed with the change of parameters are represented graphically.

Finite Element Model Reduction Applied to Nonlinear Impact Simulation for Squeak and Rattle Prediction

&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;Increasing demand for simulation accuracy often leads to increased finite element model complexity, which in turn, results in higher computational costs. As a provision, component mode synthesis approaches are employed to approximate the system response by using dynamic substructuring and model reduction techniques in linear systems. However, the use of available model reduction techniques in nonlinear problems containing the contact type of nonlinearities remains an interesting topic. In this paper, the application of a component mode synthesis method in squeak and rattle nonlinear simulation has been investigated. Critical regions for squeak and rattle of the side door model of a passenger car were modelled by nonlinear contact definition in finite element simulation. Craig-Bampton model reduction method was employed to substructure the finite element model while keeping the nonlinear contacts in the model. The model response was evaluated using the modal assurance criterion, frequency response analysis and contact force magnitude in comparison with the baseline model. Results showed that a great reduction in computational time (about 98%) can be achieved while the accuracy of the system response was maintained at an acceptable range for the intended application for squeak and rattle simulation. Although the prediction of impact events in time was done accurately, the contact force magnitude was estimated with average error of 2.5% to 22%, compared with the baseline results. The outcomes of the study show that to empower squeak and rattle prediction by including contact interfaces in finite element simulations, implementation of the model reduction approach can compensate the simulation cost.&lt;/div&gt;&lt;/div&gt;

Hybrid Modeling of Nonlinear-Jointed Structures via Finite-Element Model Reduction and Deep Learning Techniques

In engineering practice many structures are assembled by several linear components through nonlinear joints. A novel hybrid modeling method based on finite element model reduction and deep learning techniques is proposed to meet the ever-increasing requirements of efficient and accurate modeling for nonlinear jointed structures. The main idea of the hybrid modeling method for nonlinear jointed structures is summarized as follows: Firstly, finite element models of linear components are reduced to improve the computing efficiency using the free-interface mode synthesis method, as numerical integration of governing equations of nonlinear structures with large numbers of degrees-of-freedom is always time-consuming. Secondly, deep neural networks are used to equivalently represent the nonlinear joints which are difficult to describe by accurate and physically-motivated models, so as to avoid the errors caused by traditional mechanism modeling or system identification. Nonlinear joints are finally replaced with their equivalent neural networks and connected with the substructure models of linear components through the compatibility of displacements and equilibrium of forces at the interfaces. The performance of the proposed hybrid modeling method is tested and assessed via a case study focused on a cantilever plate with nonlinear joints. Comparative results demonstrate the capability of the proposed method for efficient and accurate modeling of nonlinear jointed structures and predicting their intrinsic nonlinear behavior.

Numerical simulations for the equilibrium state and solution behaviors in the multi-phase catalytic reaction mechanism

According to Horiuti rule, the higher-dimensional catalytic complex chemical problems are split down in different possible reaction routes. This will remove the stiffness of the system and allows us to analyze the distinct behavior of the catalytic and non-catalytic species during the reaction as well as the whole reaction mechanism. The construction and comparison of the multi-invariant region and 1D and 2D curves (manifold) require detailed analysis of the system and have never been reported so far.The purpose of writing this article is, to find and compare the transition period of the chemical species and solution trajectories of the multi-phase reaction mechanism through the model reduction technique (MRT), i.e., Spectral Quasi Equilibrium Manifold (SQEM) for multi-route reaction mechanisms. For convenience, a multi-step chemical reaction mechanism over the closed system is considered in detail and the results are obtained through programming in MATLAB.