Model Order Reduction Approach Research Articles

A bilinear mathcal {H}_{2} model order reduction approach to linear parameter-varying systems

This paper focuses on the model reduction problem for a special class of linear parameter-varying systems. This kind of systems can be reformulated as bilinear dynamical systems. Based on the bilinear system theory, we give a definition of the mathcal {H}_{2} norm in the generalized frequency domain. Then, a model reduction method is proposed based on the gradient descent on the Grassmann manifold. The merit of the method is that by utilizing the gradient flow analysis, the algorithm is guaranteed to converge, and further speedup of the convergence rate can be achieved as well. Two numerical examples are tested to demonstrate the proposed method.

An adaptive block tangential method for multi-input multi-output dynamical systems

Abstract In this paper, we present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named: Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.

Multi‐expansion modal reduction: A pragmatic semi–a priori model order reduction approach for nonlinear structural dynamics

SummaryA novel model order reduction scheme for nonlinear finite element (NL‐FE) structural dynamics models is proposed in this work, which enables an overall accelerated model evaluation. The proposed multi‐expansion modal (MEM) reduction employs a set of modal bases evaluated for a number of representative statically deformed configurations, combined with these static deformation patterns, to obtain a reduced order basis (ROB). This ROB is then exploited in an a priori element sampling to obtain a hyper‐reduced model based on the energy‐conserving sampling and weighting approach. The proposed element sampling scheme uses a nonnegative L1 optimization of the element weights where an equality between the projected and hyper‐reduced linearized stiffness matrix or static deformation forces is added. A procedure and several guidelines on the reduced order model setup and evaluation are presented, with specific attention to how the presented approach allows to exploit existing NL‐FE frameworks. The MEM approach is validated numerically on an academic cantilever beam model and is shown to provide good accuracy and consistent convergence properties.

A model order reduction approach to create patient-specific mechanical models of human liver in computational medicine applications

Background and objectiveThis paper focuses on computer simulation aspects of Digital Twin models in the medical framework. In particular, it addresses the need of fast and accurate simulators for the mechanical response at tissue and organ scale and the capability of integrating patient-specific anatomy from medical images to pinpoint the individual variations from standard anatomical models. MethodsWe propose an automated procedure to create mechanical models of the human liver with patient-specific geometry and real time capabilities. The method hinges on the use of Statistical Shape Analysis to extract the relevant anatomical features from a database of medical images and Model Order Reduction to compute an explicit parametric solution for the mechanical response as a function of such features. The Sparse Subspace Learning, coupled with a Finite Element solver, was chosen to create low-rank solutions using a non-intrusive sparse sampling of the feature space. ResultsIn the application presented in the paper, the statistical shape model was trained on a database of 385 three dimensional liver shapes, extracted from medical images, in order to create a parametrized representation of the liver anatomy. This parametrization and an additional parameter describing the breathing motion in linear elasticity were then used as input in the reduced order model. Results show a consistent agreement with the high fidelity Finite Element models built from liver images that were excluded from the training dataset. However, we evidence in the discussion the difficulty of having compact shape parametrizations arising from the extreme variability of the shapes found in the dataset and we propose potential strategies to tackle this issue. ConclusionsA method to represent patient-specific real-time liver deformations during breathing is proposed in linear elasticity. Since the proposed method does not require any adaptation to the direct Finite Element solver used in the training phase, the procedure can be easily extended to more complex non-linear constitutive behaviors - such as hyperelasticity - and more general load cases. Therefore it can be integrated with little intrusiveness to generic simulation software including more sophisticated and realistic models.

Balanced Truncation for Model Order Reduction of Linear Dynamical Systems with Quadratic Outputs

We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is defined as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs in approximate methods. To ameliorate this shortcoming we derive an equivalent quadratic-bilinear system with a single linear output and analyze the properties of this system. We examine MOR for this system via the technique of balanced truncation, which requires a stabilization of the system. Therein, the solution of two quadratic Lyapunov equations is traced back to the solution of just two linear Lyapunov equations. We present numerical results for several test examples comparing the two MOR approaches.

Transported snapshot model order reduction approach for parametric, steady‐state fluid flows containing parameter‐dependent shocks

SummaryA new model order reduction approach is proposed for parametric steady‐state nonlinear fluid flows characterized by shocks and discontinuities whose spatial locations and orientations are strongly parameter dependent. In this method, solutions in the predictive regime are approximated using a linear superposition of parameter‐dependent basis. The sought‐after parametric reduced bases are obtained by transporting the snapshots in a spatially and parametrically dependent transport field. Key to the proposed approach is the observation that the transport fields are typically smooth and continuous, despite the solution themselves not being so. As a result, the transport fields can be accurately expressed using a low‐order polynomial expansion. Similar to traditional projection‐based model order reduction approaches, the proposed method is formulated mathematically as a residual minimization problem for the generalized coordinates. The proposed approach is also integrated with well‐known hyper‐reduction strategies to obtain significant computational speedups. The method is successfully applied to the reduction of a parametric one‐dimensional flow in a converging‐diverging nozzle, a parametric two‐dimensional supersonic flow over a forward‐facing step, and a parametric two‐dimensional jet diffusion flame in a combustor.

A data‐driven model order reduction approach for Stokes flow through random porous media

AbstractDirect numerical simulation of Stokes flow through a random, impermeable rigid body matrix by finite elements requires meshes fine enough to resolve the pore‐size scale and is thus a computationally expensive task. It is well known that on a larger scale, Stokes flow can accurately be approximated by Darcy's law [1] with an effective diffusivity field depending on viscosity and the matrix phase topology. We propose a fully probabilistic Darcy type reduced‐order model which, based on only a few tens of full‐order model runs, is capable to learn a fine‐scale topology to effective diffusion map which is maximally predictive for fine‐scale response. After training, the model provides sharp predictive distributions based on repeated evaluation of the much cheaper Darcy equation only.

Multiscale methods for model order reduction of non-linear multiphase flow problems

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence

We put forth a robust reduced-order modeling approach for near real-time prediction of mesoscale flows. In our hybrid-modeling framework, we combine physics-based projection methods with neural network closures to account for truncated modes. We introduce a weighting parameter between the Galerkin projection and extreme learning machine models and explore its effectiveness, accuracy and generalizability. To illustrate the success of the proposed modeling paradigm, we predict both the mean flow pattern and the time series response of a single-layer quasi-geostrophic ocean model, which is a simplified prototype for wind-driven general circulation models. We demonstrate that our approach yields significant improvements over both the standard Galerkin projection and fully non-intrusive neural network methods with a negligible computational overhead.

Model of induction brazing of nonmagnetic metals using model order reduction approach

PurposeA novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing.Design/methodology/approachThe complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM.FindingsThe proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability.Research limitations/implicationsThe quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated.Practical implicationsThe presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials.Originality/valueProposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

Data‐driven model order reduction of quadratic‐bilinear systems

SummaryWe introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments.

A new model order reduction method based on global kernelk-means clustering: Application in health monitoring of plate using Lamb wave propagation and impedance method

In this paper, a new method based on global kernel k-means clustering (GKKMC) has been developed as model order reduction to model electromechanical impedance and Lamb wave in a plate structure. First, a model, based on the spectral finite element method, is developed to simulate piezoelectric wafer active sensor (PWAS) induced electromechanical impedance and Lamb wave propagation. Second, the related coupled electromechanical field equations are solved in 3 dimensions, then an electric voltage signal is applied to PWAS actuator, and finally the produced voltage in PWAS sensor is calculated. In reality, high frequency impedance and Lamb wave simulation in a plate need high degrees of freedom, which leads to a very slow simulation in time and frequency domain calculations. To overcome this problem, GKKMC as a model order reduction approach is proposed and applied. For this purpose, in Lamb wave modeling, a time snapshot method based on the Rayleigh damping effect has been introduced. This proposed method not only has a high accuracy and less time to simulate the problem but also needs lower memory than a full order model. A comparison between the results obtained by GKKMC, the conventional balanced proper orthogonal decomposition method, and a full order method with experimental results have been carried out demonstrating a good agreement. The results show that using this new model order reduction based on GKKMC, simulation time speedups can be reached up to 6 times faster compared with applying a conventional balanced proper orthogonal decomposition method.

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.

Model order reduction of Mikota’s vibration chain including damping effects by means of proper orthogonal decomposition

In engineering disciplines, both in scientific and practical applications, systems with a tremendous number of degrees of freedom occur. Hence, there is a need for reducing the computational effort in investigating these systems. If the system behaviour has to be calculated for many time instances and/or load scenarios, the need for efficient calculations further increases. Model order reduction is a common procedure in order to cope with such large systems. The aim of model order reduction is to reduce the (computational) effort in solving the given task while still keeping main features of the respective system. One approach of model order reduction uses the proper orthogonal decomposition. This approach is applied to Mikota’s vibration chain, a linear vibration chain with remarkable properties, where two cases of an undamped and a damped structure are investigated.

A meta-heuristic cuckoo search and eigen permutation approach for model order reduction

In this paper, a new model order reduction technique is presented by combining the benefits of the meta-heuristic cuckoo search optimization and Eigen permutation methods for order reduction of higher order continuous-time systems. In the proposed approach, the numerator and the denominator polynomials of reduced order model are determined by Cuckoo search and Eigen permutation approaches, respectively. The proposed approach preserves the stability of the original system into the lower order model as the Eigen permutation retains the dominant pole with simultaneous cluster formation of the remaining real and complex poles. The effectiveness of the proposed method is validated by single-input single-output and multiple-inputs multiple-outputs numerical examples.

Fourier‐series approach to model order reduction and controller interaction analysis of large‐scale power system models

In this study, a comprehensive approach for model order reduction based on a Fourier series of a discrete system representation is proposed. The developed method represents an alternative to model reduction of large-scale dynamical systems and can be used for the analysis of unstable dynamic systems and the study of interactions among power system controls. Drawing on the principle of conformal mapping, the linearised system model is first transformed into its discrete equivalent. Then, the Fourier series of the discrete-time transfer function is used to obtain a reduced-order model (ROM). Using this model, approximate Hankel-based interaction measures are proposed to efficiently analyse large, unstable linear system representations and determine pairs of input outputs to design controller. The high degree of applicability and accuracy offered by the method, and its ability to extract ROMs from stable and unstable systems is demonstrated on three test systems.

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.

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.

Searching the weakest link: Demagnetizing fields and magnetization reversal in permanent magnets

Magnetization reversal in permanent magnets occurs by the nucleation and expansion of reversed domains. Micromagnetic theory offers the possibility to localize the spots within the complex structure of the magnet where magnetization reversal starts. We compute maps of the local nucleation field in a Nd2Fe14B permanent magnet using a model order reduction approach. Considering thermal fluctuations in numerical micromagnetics we can also quantify the reduction of the coercive field due to thermal activation. However, the major reduction of the coercive field is caused by the soft magnetic grain boundary phases and misorientation if there is no surface damage.

A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities

The SCRBE (Static-Condensation Reduced-Basis-Element) method is a component-to-system model order reduction approach for efficient many-query and real-time treatment of linear partial differential equations characterized by many spatially distributed constitutive, geometry, and topology parameters. In this paper we incorporate the SCRBE approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities. In particular, we exploit a linear–nonlinear domain decomposition to develop a hybrid formulation: we consider a SCRBE approximation over the (assumed predominant) part of the domain associated to a linear elasticity model; we revert to a full finite element (FE) approximation over the part of the domain associated to the locally nonlinear model. We adapt the SCRBE port training procedures to anticipate the behavior of the field over the linear–nonlinear interface and hence ensure an accurate solution over the entire domain. We choose a globally conforming approximation which exploits the intrinsic port structure of the SCRBE method and yields a decoupled “non-invasive” formulation of the respective linear and nonlinear blocks of the residual vector and (Newton) Jacobian matrix. We present numerical results for several local nonlinearities – elastic contact with and without friction, plastic (contact) – which demonstrate substantial savings relative to standard FE approximation over the entire domain, in particular in the regime in which the linear part of the domain represents the majority of the (FE) degrees of freedom.