Model Order Reduction Procedure Research Articles

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

A projection‐based reduced‐order model for parametric quasi‐static nonlinear mechanics using an open‐source industrial code

AbstractWe propose a projection‐based model order reduction procedure for a general class of parametric quasi‐static problems in nonlinear mechanics with internal variables. The methodology is integrated in the industrial finite element code codeaster. Model order reduction aims to lower the computational cost of engineering studies that involve the simulation to a costly high‐fidelity differential model for many different parameters, which correspond, for example to material properties or initial and boundary conditions. We develop an adaptive algorithm based on a POD‐Greedy strategy, and we develop an hyper‐reduction strategy based on an element‐wise empirical quadrature in order to speed up the assembly costs of the reduced‐order model by building an appropriate reduced mesh. We introduce a cost‐efficient error indicator which relies on the reconstruction of the stress field by a Gappy‐POD strategy. We present numerical results for a three‐dimensional elastoplastic system in order to illustrate and validate the methodology.

A model order reduction technique for the linearised compressible flow equations

AbstractViscous effects in acoustics are significant if the thickness of the oscillating viscous boundary layer (Stokes boundary layer) becomes comparable to characteristic problem dimensions. For air as an acoustic medium, this is frequently the case in small‐scale devices like micro‐electro‐mechanical systems or micro‐perforated panels for sound absorption. Accurate modelling of viscous effects can be done by the solution of the linearised compressible flow equations by the finite element method, which is computationally demanding due to the high number of unknowns arising in the problem: pressure, velocity and temperature. Furthermore, the discretisation must resolve the thin viscous boundary layers, creating a model with a high number of degrees of freedom. We suggest a projection‐based model order reduction procedure using the system eigenmodes as generalised coordinates. The procedure is tested based on a simple example problem in 2D. Here we show that the derived reduced order model is accurate and computationally highly efficient. The number of degrees of freedom can be reduced by several orders of magnitude, from around 30 000 to 100, thereby dramatically reducing the computation time for harmonic solutions.

A reduced basis method by means of transport maps for a fluid–structure interaction problem with slowly decaying Kolmogorov $ n $-width

The aim of this work is to present a Model Order Reduction (MOR) procedure that is carried out by means of a preprocessing of the snapshots in the offline phase, and to apply it to a Fluid–Structure Interaction (FSI) problem of interest, where the physical domain is two dimensional, the fluid is Newtonian and laminar, and the solid is one dimensional, linear and elastic. This problem exhibits a slow decay of the Kolmogorov $ n $-width: this is reflected, at the numerical level, by a slow decay in the magnitude of the eigenvalues returned by a Proper Orthogonal Decomposition on the solution manifold. By means of a preprocessing procedure, we show how we are able to control the decay of the Kolmogorov $ n $–width of the obtained solution manifold. The preprocessing employed in the manuscript is based on the composition of the snapshots with a map belonging to a family of smooth and invertible mappings from the physical domain into itself. In order to assess the capabilities and the performance of the proposed MOR strategy, we draw a comparison between the results of the novel offline stage and the standard one, as well as a comparison between the novel online phase and the standard one.

An adaptive projection‐based model reduction method for nonlinear mechanics with internal variables: Application to thermo‐hydro‐mechanical systems

AbstractWe propose a projection‐based monolithic model order reduction procedure for a class of problems in nonlinear mechanics with internal variables. The work is motivated by applications to thermo‐hydro‐mechanical (THM) systems for radioactive waste disposal. THM equations model the behavior of temperature, pore water pressure, and solid displacement in the neighborhood of geological repositories, which contain radioactive waste and are responsible for a significant thermal flux toward the Earth's surface. We develop an adaptive sampling strategy based on the POD‐Greedy method, and we develop an element‐wise empirical quadrature hyper‐reduction procedure to reduce assembling costs. We present numerical results for a two‐dimensional THM system to illustrate and validate the proposed methodology.

Reduced Order Modeling of Deformable Tire-Soil Interaction With Proper Orthogonal Decomposition

AbstractIn this study, model order reduction of high-fidelity off-road mobility models is explored to address the computational intensity of nonlinear finite element deformable tire–soil interaction models. To this end, a model order reduction procedure for the tire–soil interaction model is developed with the proper orthogonal decomposition (POD), and it is integrated into the off-road mobility simulation framework, leveraging high-performance computing. The POD is, however, limited in that the modes are dependent on snapshot data collected during the running of a full order model, limiting the modes to being accurate only for the specific scenario from which they were collected. Due to this limitation, a method of mode adaptation through interpolation on a tangent space of the Grassmann manifold is investigated to allow modes to be predicted for cases in which a full order model has not been run. It is demonstrated by several numerical examples that the POD modes are effective at retaining predictive accuracy while reducing computational time. The results show that adapted POD modes are more capable of characterizing the behavior of the model than modes produced at a different value of the simulation parameter. The POD-based reduced order modeling approach is further extended to the full vehicle simulation on deformable terrain through the co-simulation coupling algorithm by leveraging the high-performance computing technique.

Registration-Based Model Reduction in Complex Two-Dimensional Geometries

We present a general—i.e., independent of the underlying equation—egistration procedure for parameterized model order reduction. Given the spatial domain \(\varOmega \subset \mathbb {R}^2\) and the manifold \(\mathcal {M}= \{ u_{\mu } : \mu \in \mathcal {P} \}\) associated with the parameter domain \(\mathcal {P} \subset \mathbb {R}^P\) and the parametric field \(\mu \mapsto u_{\mu } \in L^2(\varOmega )\), our approach takes as input a set of snapshots \(\{ u^k \}_{k=1}^{n_\mathrm{train}} \subset \mathcal {M}\) and returns a parameter-dependent bijective mapping \({\varPhi }: \varOmega \times \mathcal {P} \rightarrow \mathbb {R}^2\): the mapping is designed to make the mapped manifold \(\{ u_{\mu } \circ {\varPhi }_{\mu }: \, \mu \in \mathcal {P} \}\) more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition \(\{ \varOmega _{q} \}_{q=1} ^{N_\mathrm{dd}}\) of the domain \(\varOmega \) such that \(\varOmega _1,\ldots ,\varOmega _{N_\mathrm{dd}}\) are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

Control-Oriented Modeling for Managed Pressure Drilling Automation Using Model Order Reduction

Automation of managed pressure drilling (MPD) enables fast and accurate pressure control in drilling operations. The performance that can be achieved by automated MPD is determined by, first, the controller design and, second, the hydraulics model that is used as a basis for controller design. On the one hand, such hydraulics model should be able to accurately capture essential flow dynamics, for example, wave propagation effects, for which typically complex models are needed. On the other hand, a suitable model should be simple enough to allow for extensive simulation studies supporting scenario analysis and high-performance controller design well. In this paper, we develop a model order reduction approach for the derivation of such a control-oriented model for single-phase flow MPD operations. In particular, a nonlinear model order reduction procedure is presented that preserves key system properties such as stability and provides guaranteed (accuracy) bounds on the reduction error. To demonstrate the quality of the derived control-oriented model, comparisons with field data and both open-loop and closed-loop simulation-based case studies are presented.

Design of Low Frequency Suitable Fractional Order Differintegrators

Design of low frequency applicable fractional order digital differentiators and integrators using direct discretization is the main objective of this paper. Model order reduction procedure is used to obtain the novel first order s-to-z transform. Continued fraction expansion (CFE) technique is used to discretize that transform. The design of one-half digital differintegrators based on proposed transforms will be compared with the well known existing first order transforms namely, Bi-linear and Al-Alaoui transforms. The efficacy of the designed methods presented interms of normalized magnitude error (NME). The designed fractional order digital filter coefficients are tabulated and all simulation results are carried out by using MATLAB software.

An adaptive MOR method for vibro-acoustic analysis of dynamic systems with viscoelastic damping

Viscoelastic materials have the capability of dissipating vibrational energy which makes them very effective in the passive control of structural vibration and noise radiation. Efficient modeling-and simulation strategies for vibro-acoustic systems with these frequency-dependent damping materials are necessary. One way of doing that is to utilize the finite element (FE) method supported by a damping model that describes the frequency-dependency of the complex elastic modulus. However, in many applications, very large-scale FE models are required to obtain reliable predictions, which makes full-order system evaluation often intractable. Model order reduction (MOR), however, cannot be directly applied to models including viscoelastic materials due to the frequency-dependent property which exactly makes that the equations of motion are not of a standard second-order form as that for regular FE models. In this paper, a transformation technique based on Taylor’s theorem is introduced to treat the non-standard form of the equations of motion. After transforming the problem into a second-order system equation with an associated remainder term, an adaptive MOR procedure is applied to reduce the computational complexity of the vibro-acoustic systems. A numerical example is provided to demonstrate the simplicity and efficiency of the proposed approach.

An adaptive sampling procedure for parametric model order reduction by matrix interpolation

An adaptive sampling approach for parametric model order reduction by matrix interpolation is developed. This approach is based on an efficient exploration of the candidate parameter sets and identification of the points with maximum errors. An error indicator is defined and used for fast evaluation of the parameter points in the configuration space. Furthermore, the exact error of the model with maximum error indicator is calculated to determine whether the adaptive sampling procedure reaches a desired error tolerance. To improve the accuracy, the orthogonal eigenvectors are utilized as the reduced-order basis. The proposed adaptive sampling procedure is then illustrated by application in the moving coil of electrical-dynamic shaker. It is shown that the new method can sample the parameter space adaptively and efficiently with the assurance of the resulting reduced-order models’ accuracy.

A Review of the Application of Machine Learning and Data Mining Approaches in Continuum Materials Mechanics

Machine learning tools represent key enablers for empowering material scientists and engineers to accelerate the development of novel materials, processes and techniques. One of the aims of using such approaches in the field of materials science is to achieve high-throughput identification and quantification of essential features along the process-structure-property-performance chain. In this contribution, machine learning and statistical learning approaches are reviewed in terms of their successful application to specific problems in the field of continuum materials mechanics. They are categorized with respect to their type of task designated to be either descriptive, predictive or prescriptive; thus to ultimately achieve identification, prediction or even optimization of essential characteristics. The respective choice of the most appropriate machine learning approach highly depends on the specific use-case, type of material, kind of data involved, spatial and temporal scales, formats, and desired knowledge gain as well as affordable computational costs. Different examples are reviewed involving case-by-case dependent application of different types of artificial neural networks and other data-driven approaches such as support vector machines, decision trees and random forests as well as Bayesian learning, and model order reduction procedures such as principal component analysis, among others. These techniques are applied to accelerate the identification of material parameters or salient features for materials characterization, to support rapid design and optimization of novel materials or manufacturing methods, to improve and correct complex measurement devices, or to better understand and predict fatigue behavior, among other examples. Besides experimentally obtained datasets, numerous studies draw required information from simulation-based data mining. Altogether, it is shown that experiment- and simulation-based data mining in combination with machine leaning tools provide exceptional opportunities to enable highly reliant identification of fundamental interrelations within materials for characterization and optimization in a scale-bridging manner. Potentials of further utilizing applied machine learning in materials science and empowering significant acceleration of knowledge output are pointed out.

Stable model order reduction method for fractional-order systems based on unsymmetric Lanczos algorithm

This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.

Dominant Pole Based Approximation for Discrete Time System

This paper presents an effective procedure for model order reduction of discrete time control system. The exact model derived from complex dynamic systems proves to be very complicated for analysis, control and design. This necessity brings about using a tool known as model order reduction technique or model simplification. A novel mixed method has been implemented in this paper for reducing the order of the large scale dynamic discrete system. Dominant pole based pole clustering method has been used to derive the coefficients of denominator polynomial while Padé approximation has been applied to obtain the coefficients of numerator polynomial of the reduced order model. The proposed method is quite simple and able to generate a stable reduced order model from high order stable discrete systems. The dominancy of poles has been decided by values of the ratio of residue to its pole. The pole is considered dominant which have larger ratio value. An illustrative example has been considered to show the various reduction steps. The result obtained confirms the effectiveness of the approach.

Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

ABSTRACTIn this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.

A Krylov subspace method based on multi-moment matching for model order reduction of large-scale second order bilinear systems

• A new second order Krylov subspace method is proposed for model order reduction of large-scale bilinear systems. • The corresponding multi-moment matching for model order reduction of large-scale second order bilinear systems is derived. • The main characteristics such as symmetry and positive definiteness of the mass and stiffness matrices are preserved. • To show the efficacy of the presented method, an electrostatically actuated micro-device is considered as a case study. In this paper, a Krylov subspace method based on multi-moment matching is utilized for model order reduction of large-scale second order bilinear systems. Accordingly, model order reduction procedure will be directly applied to second order systems which avoids converting them into first order ones. In this way, the main characteristics of the second order system such as symmetry and positive definiteness of the mass and stiffness matrices will be preserved. Furthermore, an electrostatically actuated micro-electro-mechanical system device will be considered as a case study to show the effectiveness of the presented method. Simulation results indicate the excellent performance of the proposed model order reduction method.

Randomized Local Model Order Reduction

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.

Model Order Reduction for Managed Pressure Drilling Systems based on a Model with Local Nonlinearities

Automated Managed Pressure Drilling (MPD) is a method for fast and accurate pressure control in drilling operations. The achievable performance of automated MPD is limited, firstly, by the control system and, secondly, by the hydraulics model based on which this control system is designed. Hence, an accurate hydraulics model is needed that, at the same time, is simple enough to allow for the use of high performance controller design methods. This paper presents an approach for nonlinear Model Order Reduction (MOR) for MPD systems. For a single-phase flow MPD system, a nonlinear model is derived that can be decomposed into a feedback interconnection of a high-order linear subsystem and low-order nonlinear subsystem. This structure, under certain conditions, allows for a nonlinear MOR procedure that preserves key system properties such as stability and provides a computable error bound. The effectiveness of this MOR method for MPD systems is illustrated through simulations.

Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters

In this paper, an approach based on the synergistic use of proper orthogonal decomposition and Kalman filtering is proposed for the online health monitoring of damaged structures. The reduced-order model of a structure is obtained during an (offline) initial training stage of monitoring; afterward, effective estimations of a possible structural damage are provided online by tracking the evolution in time of stiffness parameters and projection bases handled in the model order reduction procedure. Such tracking is accomplished via two Kalman filters: a first (extended) one to deal with the time evolution of a joint state vector, gathering the reduced-order state and the stiffness terms degraded by damage; a second one to deal with the update of the reduced-order model in case of damage evolution. Both filters exploit the information conveyed by measurements of the structural response to the external excitations. Results are reported for a (pseudo-experimental) benchmark test on an eight-story shear building. Capability and performance of the proposed approach are assessed in terms of tracked variation of the stiffness terms of the reduced-order model, identified damage location and speed-up of the whole health monitoring procedure.

A Method for the Reduction of Linear Higher Order Discrete-Time SISO System

ABSTRACTThis paper presents a method for the reduction of higher order linear discrete time system into a desired reduced order model. The proposed algorithm uses the concept of pattern recognition which is adopted here for clustering the poles of the system. The centres of the clusters are treated as poles of the reduced order model and the overall transfer function of the reduced order model is obtained by both clustering algorithm and matching the ratio of coefficients of original and reduced order models. The original algorithm for c-means clustering is modified in order to avoid some limitations in model order reduction procedure. This new method makes the procedure simple, in addition to retaining the stability of the original higher order system in their reduced order models.