Model Reduction Procedure Research Articles

A multi-scale simulation method for powder coating with moving powder guns

A mathematical model and numerical method, supported by experiments, to simulate industrial scale powder coating for moving coating guns is developed. The numerical method is based on a model reduction procedure that starts with a three species electrostatic model for the negative corona discharge at the nozzle exit of the gun. It is reduced to a one species electrostatic model away from the gun, which is coupled to a RANS fluid model and Lagrangian particle tracking and charging in a detailed stationary simulation. The final part consists of coating simulation with a moving gun, enabled by immersed boundary methods that do not require a body fitted mesh. In the coating simulation the detailed simulation results are reinjected, preserving the size and charge distribution of the powder particles. The method is validated with coating thickness measurements for horizontal and vertical lines on flat and two sets of bent panels: towards and away from the applicator. The bent panels are used to validate the performance on highly convex and concave objects.

Model reduction of Brownian oscillators: quantification of errors and long-time behavior

A procedure for model reduction of stochastic ordinary differential equations with additive noise was recently introduced in Colangeli et al (2022 J. Phys. A: Math. Theor. 55 505002), based on the Invariant Manifold method and on the Fluctuation–Dissipation relation. A general question thus arises as to whether one can rigorously quantify the error entailed by the use of the reduced dynamics in place of the original one. In this work we provide explicit formulae and estimates of the error in terms of the Wasserstein distance, both in the presence or in the absence of a sharp time-scale separation between the variables to be retained or eliminated from the description, as well as in the long-time behavior.

Structure-preserving model order reduction for K-power bilinear systems via Laguerre functions

ABSTRACT This paper presents a series of structure-preserving model order reduction algorithms for K-power bilinear systems via Laguerre functions. The method first aims to rewritten the K-power bilinear system as a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by combining the idea of Laguerre functions expansion of the matrix exponential function. After that, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. In order to achieve the purpose of model order reduction, the states with smaller singular values are then truncated, so as to further obtain the reduced order model. For this approach, there is a disadvantage that unstable systems may be generated although the original one is stable. To alleviate the inadequacies of this approach, we have improved the model reduction procedure, which is based upon the dominant subspace projection method. In addition, we also carried out a correlation analysis on the stability of improved algorithms. Finally, numerical experiments are employed to substantiate the effectiveness of the presented algorithms.

Hierarchical Model Reduction Driven by Machine Learning for Parametric Advection-Diffusion-Reaction Problems in the Presence of Noisy Data

We propose a new approach to generate a reliable reduced model for a parametric elliptic problem, in the presence of noisy data. The reference model reduction procedure is the directional HiPOD method, which combines Hierarchical Model reduction with a standard Proper Orthogonal Decomposition, according to an offline/online paradigm. In this paper we show that directional HiPOD looses in terms of accuracy when problem data are affected by noise. This is due to the interpolation driving the online phase, since it replicates, by definition, the noise trend. To overcome this limit, we replace interpolation with Machine Learning fitting models which better discriminate relevant physical features in the data from irrelevant unstructured noise. The numerical assessment, although preliminary, confirms the potentialities of the new approach.

Feedback control of transitional shear flows: sensor selection for performance recovery

The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier-Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control performance. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energy growth and suppressing transition; however, sensor-based output feedback controllers with comparable performance have been difficult to realize. In this study, we propose two methods for sensor selection that enable sensor-based output feedback controllers to recover full-information control performance: one based on a sparse controller synthesis approach, and one based on a balanced truncation procedure for model reduction. Both approaches are investigated within linear and nonlinear simulations of a sub-critical channel flow with blowing and suction actuation at the walls. We find that sensor configurations identified by both approaches allow sensor-based static output feedback LQR controllers to recover full-information LQR control performance, both in reducing transient energy growth and suppressing transition. Further, our results indicate that both the sensor selection methods and the resulting controllers exhibit robustness to Reynolds number variations.

Model order reduction framework for discrete-time systems with error bound via balanced structure

Two-dimensional models are challenging to deal with because of their complex structure. As the model's order grows larger, simulation, analysis, design, and control become more challenging. For one-dimensional and two-dimensional models, Enns' weighted (i.e. frequency and time) model reduction procedures result in an unstable reduced-order model. For one-dimensional reduced-order systems, researchers devised a number of frequency-weighted stability-preserving solutions; nevertheless, these solutions result in significant approximation errors. However, to the best of the authors' knowledge, no time-weighted stability-preserving model reduction framework is available in the literature for the discrete-time one-dimensional and two-dimensional systems. In this research, two-dimensional models are separated into two sub-models (i.e. two cascaded one-dimensional models) and reduced using time-weighted Gramians utilising minimal rank-decomposition. The approach applies to both one-dimensional and two-dimensional models. For one-dimensional and two-dimensional models, the suggested methodology additionally gives a priori error-bounds expressions. Numerical results and comparisons to existing methodologies illustrate the efficacy of the proposed methodology.

Static condensation of peridynamic heat conduction model

A model order reduction methodology for reducing the order of the peridynamic transient heat model is proposed. This methodology is based on the static condensation procedure. To set the development of the model reduction procedure on a sound mathematical setting, a nonlocal vector calculus was employed in the formulation of the heat transport problem. The model order reduction framework proposed in this study provides a technique to reduce the dimensionality of a peridynamic transport model while still maintaining accurate prediction of the model response. Moreover, the methodology can be adaptively applied to accommodate different resolution requirements for different sections of the model. Using numerical experiments, the proposed methodology is shown to be capable of accurately reproducing results of the full peridynamic transient heat transport problem.

Registration-based model reduction of parameterized two-dimensional conservation laws

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping Φ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

Imbalanced basal ganglia connectivity is associated with motor deficits and apathy in Huntington’s disease

The gating of movement depends on activity within the cortico-striato-thalamic loops. Within these loops, emerging from the cells of the striatum, run two opponent pathways—the direct and indirect basal ganglia pathways. Both are complex and polysynaptic, but the overall effect of activity within these pathways is thought to encourage and inhibit movement, respectively. In Huntington’s disease, the preferential early loss of striatal neurons forming the indirect pathway is thought to lead to disinhibition, giving rise to the characteristic motor features of the condition. But early Huntington’s disease is also associated with apathy, a loss of motivation and failure to engage in goal-directed movement. We hypothesized that in Huntington’s disease, motor signs and apathy may be selectively correlated with indirect and direct pathway dysfunction, respectively.We used spectral dynamic casual modelling of resting-state functional MRI data to model effective connectivity in a model of these cortico-striatal pathways. We tested both of these hypotheses in vivo for the first time in a large cohort of patients with prodromal Huntington’s disease. Using an advanced approach at the group level we combined parametric empirical Bayes and Bayesian model reduction procedures to generate a large number of competing models and compare them using Bayesian model comparison. With this automated Bayesian approach, associations between clinical measures and connectivity parameters emerge de novo from the data.We found very strong evidence (posterior probability > 0.99) to support both of our hypotheses. First, more severe motor signs in Huntington’s disease were associated with altered connectivity in the indirect pathway components of our model and, by comparison, loss of goal-direct behaviour or apathy, was associated with changes in the direct pathway component.The empirical evidence we provide here demonstrates that imbalanced basal ganglia connectivity may play an important role in the pathogenesis of some of commonest and disabling features of Huntington’s disease and may have important implications for therapeutics.

Sparse Bayesian learning with model reduction for probabilistic structural damage detection with limited measurements

A new variant of damage detection algorithm is proposed based on Sparse Bayesian Learning (SBL) and model reduction for probabilistic structural damage detection. By exploring recent developments in SBL, we aim to produce reliable damage identification even for high-dimensional model parameter spaces for higher-resolution damage localization. By introducing the system modal parameters, the matching of model and experimental modes and solving the nonlinear eigenvalue problem of a structural model are not required. However, one inherent difficulty is that, because only a small number of degrees of freedom (DOFs) can be measured in practice due to limited instrumentation and unmeasurable rotational DOFs, the computation of the system mode shapes has a minimum chance of success. The proposed algorithm extends the applicability of SBL-based damage detection to cases with limited measurements by exploiting model reduction techniques to avoid computation of the system mode shapes for the full model. To effectively incorporate the model reduction procedure, a two-stage SBL-based damage detection algorithm is proposed, in which the first stage updates the reduced system modal properties employing the modal data, and the second stage learns the stiffness parameters and their associated hyper-parameters to produce model sparseness of stiffness losses. The performance of the proposed algorithm is investigated through a series of numerical and experimental studies involving two beam structures and a long-span cable-stayed bridge. Both modeling errors and measurement noises are considered in these studies. The results show that, despite limited measured DOFs in conjunction with modeling error and measurement noise, the proposed algorithm based on SBL and model reduction can successfully locate and quantify the damage along with their posterior uncertainties, which give a sense of identification confidence.

An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov–Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.

Static and Dynamic Modeling of the Electromagnets of the Maglev Vehicle Transrapid

This work provides a new approach for modeling the electromagnets of the magnetic levitation (Maglev) vehicle Transrapid intending to effectively and efficiently represent the statics and the dynamics in a frequency range relevant for Maglev control design and for the use in mechatronic simulation models. It includes the effects of magnetic reluctances, fringing and leakage flux, magnetic saturation, and eddy currents. The modeling follows a systematic approach by setting up the equivalent magnetic and electric circuits, and coupling the equations using Ampère's circuital law and Faraday's law of induction resulting in a system of differential algebraic equations. It allows to study effects occurring in the magnet design process or different operating points, resulting, e.g., from aerodynamic uplift, for which an extended validity range is needed. Since the computational effort of such models is significant, additionally, a numerical procedure for model reduction is presented yielding a simplified version, which possesses a nearly identical input-output behavior and is usable for control design or in large vehicle models. The proposed approach can be applied to any levitation or guidance magnet of the vehicle and is shown exemplarily for a Transrapid's levitation magnet. The model is validated for the parameters of the latest Transrapid vehicle, called TR09, showing good correspondence with the measured static force-current-gap characteristics and the inductance.

Modeling and parameter identification for real-time temperature controlled retinal laser therapies

Abstract Laser photocoagulation is a widely used treatment for a variety of retinal diseases. Temperature-controlled irradiation is a promising approach to enable uniform heating, reduce the risks of over- or undertreatment, and unburden the ophthalmologists from a time consuming manual power titration. In this paper, an approach is proposed for the development of models with different levels of detail, which serve as a basis for improved, more accurate observer and control designs. To this end, we employ a heat diffusion model and propose a suitable discretization and subsequent model reduction procedures. Since the absorption of the laser light can vary strongly at each irradiation site, a method for identifying the absorption coefficient is presented. To identify a parameter in a reduced order model, an optimal interpolatory projection method for parametric systems is used. In order to provide an online identification of the absorption coefficient, we prove and exploit monotonicity of the parameter influence.

Stochastic Modelling of Small-Scale Perturbation

In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics. Once large-scale and small-scale components of the dynamics have been identified, our method consists in modelling stochastically the small scales and, as a result, we obtain that a transport-type Stratonovich noise is sufficient to model the influence of the small scale structures on the large scales ones. This work aims to contribute to motivate the use of stochastic models in fluid mechanics and identifies examples of noise of interest for the reduction of complexity of the interaction between scales. The ideas are presented in full generality and applied to specific examples in the last section.

Comparative analysis of Reaction-Diffusion Manifold based reduced models for Head-On- and Side-Wall-Quenching flames

In this study, multi-dimensional molecular transport phenomena during Flame-Wall-Interactions (FWI) and their effects on model reduction strategies are investigated. In order to access the problem, the standard configurations of a two-dimensional Side-Wall Quenching (SWQ) flame and a one-dimensional Head-On Quenching (HOQ) flame are used and compared. In the case of the SWQ configuration it is shown that the gradients of the species scatter significantly both in the physical space and in the state space. Moreover, the gradient vector of the specific enthalpy describing energy losses towards the wall is not aligned with the gradient vectors of the species, which can be considered as a typical case while a flame in application might approach to the wall at any arbitrary transversal direction. This observation motivates to take the gradients’ scattering and multi-dimensional transport phenomena into account during model reduction to describe reliably the quenching process.The Reaction-Diffusion Manifold (REDIM) method is applied in this work. The method allows to take into account multi-dimensional transport in a very generic way. In order to generate the REDIM, gradient estimates are approximated by using a Singular-Value Decomposition (SVD) of SWQ detailed gradients fields. Two-dimensional REDIMs for both cases are constructed and compared to each other. Different transport (diffusion) models are implemented to compare quantitatively the manifolds with HOQ and SOQ gradients estimates. The comparison shows that the differences between reduced models with varying transport models is significantly larger than the differences for varying configurations (multidimensional gradient estimates). This justifies the use of a relatively simple REDIM for more complicated geometries and configurations. This simplifies the treatment and model reduction procedure significantly for such complicated transient phenomena.

Multiscale model reduction for the Allen–Cahn problem in perforated domains

In this paper, we consider a class of multiscale methods for the solution of nonlinear problem in perforated domains. These problems are of multiscale nature and their discretizations lead to large nonlinear systems. To discretize these problems, we construct a fine grid approximation using the finite element method with implicit time approximation and the Newton’s method. In order to solve these large systems efficiently, we will develop a model reduction procedure. To perform the model reduction, we construct a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM). The GMsFEM consists of an offline and online stages. In the offline stage, we construct multiscale basis functions based on the solution of some local spectral problems defined in the snapshot space. Then, we enrich the offline multiscale space by additional multiscale basis that handle non-homogeneous boundary condition. For the accurate solution of the nonlinear problem, we use two techniques on the online stage: (1) residual based multiscale basis functions and (2) residual based local correction. We will present numerical results for two-dimensional Allen–Cahn problems in perforated domains.

Size-dependent dynamics of a FG Nanobeam near nonlinear resonances induced by heat

This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester.

An improved model reduction-modal based method for model updating and health monitoring of an offshore jacket-type platform

Numerous constructed jacket platforms in the Persian Gulf require continuous monitoring due to aging which poses a significant hazard to the environment. The main aim of this study is to sketch a health monitoring system to address the problem of limitations and shortcomings of sensor networks and also to solve the localization problem. In order to achieve this goal, experimental modal analysis (EMA) is performed on a physical model of an offshore jacket platform to develop a numerically updated finite element model. Both traditional and improved Cross-Model Cross-Mode (CMCM) iteration algorithms are employed for this purpose and different model order reduction methods are compared in terms of their effects on the structural dynamic behavior of the case-study (SPD8 jacket platform). A new formulation is also proposed for the first time to compare the results of damage scenarios using both iteration algorithms. The effectiveness of the methodology in practical cases is also assessed by considering various numbers of sensors. According to the results, the improved iteration technique eliminates the detrimental impacts of the model reduction procedure on the proposed methodology, and physical parameters such as length, type, and location of a member play significant roles in the success of a damage detection scenario.

A three-scale index for flow in karst conduits in carbonate rocks

A three-scale model for flow in karst conduit networks in fractured carbonates is rigorously constructed based on a reiterated homogenization procedure. The first upscaling, performed from the high-fidelity flow model, is based on sequential partially and fully topological model reduction procedures considering two discrete networks of solution-enlarged fractures and conduits. The subsequent macroscopization procedure projects the reduced model into the cells of a coarse computational grid, where homogenized equivalent properties are numerically constructed. Such a two-level upscaling gives rise to a macroscopic flow model characterized by mass-transfer functions between the geological structures. A notable consequence of the approach proposed herein is the appearance of a new karst index concept, whose underlying physics relies on the generalization of the traditional Peaceman’s theory of well index. Such a concept rules the mass exchange between conduits and matrix and can be extended to the general scenario of coupled flow in multi-branch karst conduit systems, displaying general non-circular cross sections and surrounding damage zones. The downscaling representation for the karst index can be further explored to improve accuracy of the exchange coefficient between the geological objects. Numerical experiments are carried-out showing the magnitude of the index for certain conduit and fracture arrangements, along with illustrating the impact upon flow patterns.