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Showing papers on "Model order reduction published in 2017"


Journal ArticleDOI
TL;DR: In this article, the authors propose a reduced order modeling (ROM) approach to solve multiscale fracture problems through a FE2 approach, where a domain separation strategy is proposed as a first technique for model order reduction: unconventionally, the low-dimension space is spanned by a basis in terms of fluctuating strains, as primitive kinematic variables, instead of the conventional formulation in terms displacement fluctuations.

80 citations


Journal ArticleDOI
TL;DR: In this paper, the authors describe the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities, where the manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis.

76 citations


Book ChapterDOI
01 Jan 2017
TL;DR: This paper discusses the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations, and reviews efforts in extending popular methods related to balanced truncation and rational interpolation to descriptor systems.
Abstract: In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear state-space systems has been a topic of research for about 50 years at the time of writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the so-called stochastic balanced truncation method to descriptor systems which so far cannot be found in the literature.

75 citations


Journal ArticleDOI
TL;DR: The proposed local model order-reduction technique is applied to a hydraulic fracturing process described by a nonlinear parabolic PDE system with the time-dependent spatial domain and is shown to be more accurate and computationally efficient in approximating the original nonlinear system with fewer eigenfunctions.
Abstract: In this work, we present a temporally-local model order-reduction technique for nonlinear parabolic partial differential equation (PDE) systems with time-dependent spatial domains. In lieu of approximating the solution of interest using global (with respect to the time domain) empirical eigenfunctions, we derive low-dimensional models by constructing appropriate temporally-local eigenfunctions. Within this context, we partition the time domain into multiple clusters (i.e. subdomains) by using the framework known as global optimum search (GOS). This approach, a variant of Generalized Benders Decomposition (GBD), formulates clustering as a Mixed-Integer Nonlinear Programming problem and involves the iterative solution of a Linear Programming problem (primal problem) and a Mixed-Integer Linear Programming problem (master problem). Following the cluster generation, local (with respect to time) eigenfunctions are constructed by applying the proper orthogonal decomposition (POD) method to the snapshots contained within each cluster. Then, the Galerkin's projection method is employed to derive low-dimensional ordinary differential equation (ODE) systems for each cluster. The local ODE systems are subsequently used to compute approximate solutions to the original PDE system. The proposed local model order-reduction technique is applied to a hydraulic fracturing process described by a nonlinear parabolic PDE system with the time-dependent spatial domain. It is shown to be more accurate and computationally efficient in approximating the original nonlinear system with fewer eigenfunctions, compared to the model order-reduction technique with temporally-global eigenfunctions. This article is protected by copyright. All rights reserved.

75 citations


Journal ArticleDOI
TL;DR: The proposed IMC-PID design of reduced order model achieves good dynamic response and robustness against load disturbance with the original high order system.
Abstract: Load frequency controller has been designed for reduced order model of single area and two-area reheat hydro-thermal power system through internal model control - proportional integral derivative (IMC-PID) control techniques. The controller design method is based on two degree of freedom (2DOF) internal model control which combines with model order reduction technique. Here, in spite of taking full order system model a reduced order model has been considered for 2DOF-IMC-PID design and the designed controller is directly applied to full order system model. The Logarithmic based model order reduction technique is proposed to reduce the single and two-area high order power systems for the application of controller design.The proposed IMC-PID design of reduced order model achieves good dynamic response and robustness against load disturbance with the original high order system.

71 citations


Journal ArticleDOI
TL;DR: A Model Order Reduction technique for a system of nonlinear equations arising from the Finite Element Method (FEM) discretization of the three-dimensional quasistatic equilibrium equation equipped with a Perzyna viscoplasticity constitutive model is demonstrated.

64 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the required controller model complexity necessary to obtain optimal control performance for a given building, and showed that good MPC performances require controller models with a significantly higher number of states than the order used by most of the black and grey-box system identification techniques.

55 citations


Journal ArticleDOI
TL;DR: Three different convex approximations that promote sparsity in the solution are considered for constructing reduced meshes that are suitable for hyper reduction and paired with appropriate active set algorithms for solving the resulting minimization problems.
Abstract: Summary In nonlinear model order reduction, hyper reduction designates the process of approximating a projection-based reduced-order operator on a reduced mesh, using a numerical algorithm whose computational complexity scales with the small size of the projection-based reduced-order model. Usually, the reduced mesh is constructed by sampling the large-scale mesh associated with the high-dimensional model underlying the projection-based reduced-order model. The sampling process itself is governed by the minimization of the size of the reduced mesh for which the hyper reduction method of interest delivers the desired accuracy for a chosen set of training reduced-order quantities. Because such a construction procedure is combinatorially hard, its key objective function is conveniently substituted with a convex approximation. Nevertheless, for large-scale meshes, the resulting mesh sampling procedure remains computationally intensive. In this paper, three different convex approximations that promote sparsity in the solution are considered for constructing reduced meshes that are suitable for hyper reduction and paired with appropriate active set algorithms for solving the resulting minimization problems. These algorithms are equipped with carefully designed parallel computational kernels in order to accelerate the overall process of mesh sampling for hyper reduction, and therefore achieve practicality for realistic, large-scale, nonlinear structural dynamics problems. Conclusions are also offered as to what algorithm is most suitable for constructing a reduced mesh for the purpose of hyper reduction. Copyright © 2016 John Wiley & Sons, Ltd.

54 citations



Journal ArticleDOI
TL;DR: This paper considers the Monodomain model and resorts to Proper Orthogonal Decomposition techniques to take advantage of an off-line step when solving iteratively the electrocardiological forward model online, and performs the Discrete Empirical Interpolation Method (DEIM) to tackle the nonlinearity of the model.

42 citations


Posted Content
TL;DR: It is illustrated that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region and it is shown that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time- limited balanced truncations can be smaller compared to standardbalanced truncation.
Abstract: In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.

Journal ArticleDOI
TL;DR: In this paper, a model order reduction strategy (MOR) based on the proper orthogonal decomposition (POD) method as well as its practical applicability to a realistic building structure is presented.
Abstract: Earthquake dynamic response analysis of large complex structures, especially in the presence of nonlinearities, usually turns out to be computationally expensive. In this paper, the methodical developments of a new model order reduction strategy (MOR) based on the proper orthogonal decomposition (POD) method as well as its practical applicability to a realistic building structure are presented. The seismic performance of the building structure, a medical complex, is to be improved by means of base isolation realized by frictional pendulum bearings. According to the new introduced MOR strategy, a set of deterministic POD modes (transformation matrix) is assembled, which is derived based on the information of parts of the response history, so-called snapshots, of the structure under a representative earthquake excitation. Subsequently, this transformation matrix is utilized to create reduced-order models of the structure subjected to different earthquake excitations. These sets of nonlinear low-order representations are now solved in a fractional amount of time in comparison with the computations of the full (non-reduced) systems. The results demonstrate accurate approximations of the physical (full) responses by means of this new MOR strategy if the probable behavior of the structure has already been captured in the POD snapshots. Copyright © 2016 The Authors. Earthquake Engineering & Structural Dynamics Published by John Wiley & Sons Ltd.

Journal ArticleDOI
TL;DR: In this article, a nonparametric probabilistic approach for modeling uncertainties in projection-based, nonlinear, reduced-order models is presented, which can also quantify uncertainties in the associated high-dimensional models.
Abstract: A nonparametric probabilistic approach for modeling uncertainties in projection-based, nonlinear, reduced-order models is presented. When experimental data is available, this approach can also quantify uncertainties in the associated high-dimensional models. The main underlying idea is twofold. First, to substitute the deterministic Reduced-Order Basis (ROB) with a stochastic counterpart. Second, to construct the probability measure of the Stochastic Reduced-Order Basis (SROB) on a subset of a compact Stiefel manifold in order to preserve some important properties of a ROB. The stochastic modeling is performed so that the probability distribution of the constructed SROB depends on a small number of hyperparameters. These are determined by solving a reduced-order statistical inverse problem. The mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations. Its potential is demonstrated through several example problems from computational structural dynamics.

Journal ArticleDOI
TL;DR: According to the error bounds, reduced-order models of both non-parametrizing and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably.
Abstract: We propose a posteriori error bounds for reduced-order models of non-parametrized linear time invariant (LTI) systems and parametrized LTI systems. The error bounds estimate the errors of the transfer functions of the reduced-order models, and are independent of the model reduction methods used. It is shown that for some special non-parametrized LTI systems, particularly efficiently computable error bounds can be derived. According to the error bounds, reduced-order models of both non-parametrized and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably. Simulations for several examples from engineering applications have demonstrated the robustness of the error bounds.

Journal ArticleDOI
TL;DR: This paper proposes a computationally efficient implementation of the full DFN battery model, which is convenient for real-time applications and based on applying model order reduction to a spatial and temporal discretisation of the governing model equations.

Journal ArticleDOI
TL;DR: 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, where the reduced-order model of a structure is obtained during an 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.
Abstract: 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.

Posted Content
TL;DR: This paper proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems and investigates the Lyapunov stability of the reduced-order systems.
Abstract: We discuss balanced truncation model order reduction for large-scale quadratic-bilinear (QB) systems. Balanced truncation for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (1993), where it is shown that Gramians for nonlinear systems are the solutions of state-dependent nonlinear Hamilton-Jacobi equations. Therefore, they are not only difficult to compute for large-scale systems but also hard to utilize in the model reduction framework. In this paper, we propose algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relations with a certain type of generalized quadratic Lyapunov equation. Furthermore, we present how these algebraic Gramians and energy functionals relate to each other. Moreover, we characterize the reachability and observability of QB systems based on the proposed algebraic Gramians. This allows us to find those states that are hard to control and hard to observe via an appropriate transformation based on the Gramians. Truncating such states yields reduced-order systems. Additionally, we present a truncated version of the Gramians for QB systems and discuss their advantages in the model reduction framework. We also investigate the Lyapunov stability of the reduced-order systems. We finally illustrate the efficiency of the proposed balancing-type model reduction for QB systems by means of various semi-discretized nonlinear partial differential equations and show its competitiveness with the existing moment-matching methods for QB systems.

Journal ArticleDOI
TL;DR: An approximate robust formulation that employs linear and quadratic approximations to speed up the computation is proposed and is applied to the optimal placement of a permanent magnet in the rotor of a synchronous machine with moving rotor.
Abstract: We consider a nonlinear optimization problem governed by partial differential equations with uncertain parameters. It is addressed by a robust worst case formulation. The resulting optimization problem is of bilevel structure and is difficult to treat numerically. We propose an approximate robust formulation that employs linear and quadratic approximations. To speed up the computation, reduced order models based on proper orthogonal decomposition in combination with a posteriori error estimators are developed. The proposed strategy is then applied to the optimal placement of a permanent magnet in the rotor of a synchronous machine with moving rotor. Numerical results are presented to validate the presented approach.

Journal ArticleDOI
TL;DR: A novel order diminution method for linear time invariant continuous systems is proposed in this paper, where the reduced denominator polynomial coefficient is obtained by modified clustering algorithm and the reduced numerator coefficients are generated by an evolutionary algorithm as referred in this communication.
Abstract: A novel order diminution method for linear time invariant continuous systems is proposed in this paper. The reduced denominator polynomial is obtained by modified clustering algorithm and the reduced numerator polynomial coefficients are generated by an evolutionary algorithm as referred in this communication. Several numerical examples taken from the literature have been solved for the validation of the proposed method. The reduced system obtained by proposed method guarantees the stability if the original system is stable and also preserves all characteristics of given higher order system in the reduced one. The presented method is to be continued to multivariable systems and illustrates with numerical examples for proving the efficiency of the proposed method. Furthermore, the proposed method is also to be extended for order reduction of discrete time systems.

Book ChapterDOI
TL;DR: This paper presents a new method of approximation of linear time-invariant (LTI) discrete-time fractional-order state space systems by means of the Balanced Truncation Method, which obtains rational and relatively low- order state space system.
Abstract: This paper presents a new method of approximation of linear time-invariant (LTI) discrete-time fractional-order state space systems by means of the Balanced Truncation Method. This reduction method is applied to the rational form of fractional-order system in terms of expanded state equation. As an approximation result we obtain rational and relatively low-order state space system. Simulation experiments show very high accuracy of the introduced methodology.

Journal ArticleDOI
TL;DR: Reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied and three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness.
Abstract: Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model.

Journal ArticleDOI
TL;DR: In this paper, a new technique for model order reduction of large scale continuous linear time invariant descriptor systems in limited time interval is proposed, where the concept of time limited Gramians for descriptor systems is introduced for applications involving analysis, design or optimization in specific time interval.
Abstract: A new technique for model order reduction of large scale continuous linear time invariant descriptor systems in limited time interval is proposed. The concept of time limited Gramians for descriptor systems is introduced for applications involving analysis, design or optimization in specific time interval. Balanced truncation based on magnitudes of Hankel singular values of the computed Gramians is performed to obtain the reduced order models in limited time interval. An approximate solution of time limited projected Lyapunov equations is developed and Cholesky factorisation of time limited Gramians is performed. Moreover, stability conditions for reduced order system are stated and two algorithms for stability preservation of reduced order models are presented. Results certify the successful application of the proposed techniques.

Journal ArticleDOI
TL;DR: This work proposes the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning to efficiently solve the two sets of multiple shifted linear equations.
Abstract: We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear

Journal ArticleDOI
TL;DR: In this article, a reduced-order model based on the Galerkin projection and the proper orthogonal decomposition is proposed to improve the computational efficiency of ANCF for a large-scaled flexible multibody system.

Journal ArticleDOI
TL;DR: In this article, the authors presented a generalized methodology based on the switching instants to obtain large-signal EHD models of VSC-based power systems, and three model order reduction approaches were also proposed to address the increased size of the resulting EHD model.
Abstract: Averaged modeling is a commonly used approach used to obtain mathematical representations of VSC-based systems. However, essential characteristics mainly related to the modulation process and the harmonic distortion of the signals are not able to be accurately captured and analyzed. The extended harmonic domain (EHD) has recently been seen as an alternative modeling framework since it allows us to consider the harmonic interaction explicitly. However, there is not a clearly established methodology to derive the EHD models in the presence of power electronic switches. This paper presents a generalized methodology based on the switching instants to obtain large-signal EHD models of VSC-based power systems. Three model order reduction approaches are also proposed to address the increased size of the resulting EHD models. Analytic formulas of three modulation techniques: sinusoidal pulse-width modulation, third harmonic injection pulse-width modulation, and space vector pulse-width modulation are provided to obtain the open-loop large signal EHD models. A performance assessment of the proposed modeling approach in respect to model size, the computational time and the accuracy is presented based on simulations and experimental case studies. The obtained results show that the resulting EHD models are accurate and reliable, while the memory and computation time are improved with the proposed model order reductions.

Journal ArticleDOI
TL;DR: Using the proposed method a realistic number of muscle fibres and motor units can be considered in numerical EMG simulations, which is not feasible using full models due to their high computational cost.

Posted Content
TL;DR: This work constructs the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space.
Abstract: The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing Proper Orthogonal Decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model / basis using the eigensystem of the correlation matrix (snapshot gramian), which is motivated from a continuous perspective and is set up explicitly e.g. without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembling of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility our approach, we present a test case of the Cahn-Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.

Journal ArticleDOI
TL;DR: One of the main contribution of this paper is showing that differential balancing has close relationships with the Fréchet derivative of the nonlinear Hankel operator.
Abstract: In this paper, we construct balancing theory for nonlinear systems in the contraction framework. First, we define two novel controllability and observability functions via prolonged systems. We analyze their properties in relation to controllability and observability, and use them for so-called differential balancing, and its application to model order reduction. One of the main contribution of this paper is showing that differential balancing has close relationships with the Frechet derivative of the nonlinear Hankel operator. Inspired by [3] , we provide a generalization in order to have a computationally more feasible method. Moreover, error bounds for model reduction by generalized balancing are provided.

Journal ArticleDOI
TL;DR: A novel approach tailored to approximate the Navier-Stokes equations to simulate fluid flow in three-dimensional tubular domains of arbitrary cross-sectional shape to fill the gap between (cheap) one-dimensional and (expensive) three- dimensional models, featuring descriptive capabilities comparable with the full and accurate 3D description of the problem at a low computational cost.
Abstract: In this work, we present a novel approach tailored to approximate the Navier-Stokes equations to simulate fluid flow in three-dimensional tubular domains of arbitrary cross-sectional shape. The proposed methodology is aimed at filling the gap between (cheap) one-dimensional and (expensive) three-dimensional models, featuring descriptive capabilities comparable with the full and accurate 3D description of the problem at a low computational cost. In addition, this methodology can easily be tuned or even adapted to address local features demanding more accuracy. The numerical strategy employs finite (pipe-type) elements that take advantage of the pipe structure of the spatial domain under analysis. While low order approximation is used for the longitudinal description of the physical fields, transverse approximation is enriched using high order polynomials. Although our application of interest is computational hemodynamics and its relevance to pathological dynamics like atherosclerosis, the approach is quite general and can be applied in any internal fluid dynamics problem in pipe-like domains. Numerical examples covering academic cases as well as patient-specific coronary arterial geometries demonstrate the potentialities of the developed methodology and its performance when compared against traditional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: This work shows stability and proves a priori error estimates of the space-time discrete scheme and the fully discrete IGA-$$\theta $$θ-POD Galerkin scheme for model order reduction of linear parabolic partial differential equations.
Abstract: We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discretization with the $$\theta $$ź-scheme, and eigenvalue truncation by POD. First, we prove a priori error estimates of the spatial IGA semi-discrete scheme. Then, we show stability and prove a priori error estimates of the space-time discrete scheme and the fully discrete IGA-$$\theta $$ź-POD Galerkin scheme. Numerical tests are provided to show efficiency and accuracy of NURBS-based IGA for model order reduction in comparison with standard finite element-based POD techniques.