Showing papers on "Model order reduction published in 2016"

Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum

Abstract: This contribution presents a novel homogenization technique for modeling heterogeneous materials with micro-inertia effects such as locally resonant acoustic metamaterials. Linear elastodynamics is used to model the micro and macro scale problems and an extended first order Computational Homogenization framework is used to establish the coupling. Craig Bampton Mode Synthesis is then applied to solve and eliminate the microscale problem, resulting in a compact closed form description of the microdynamics that accurately captures the Local Resonance phenomena. The resulting equations represent an enriched continuum in which additional kinematic degrees of freedom emerge to account for Local Resonance effects which would otherwise be absent in a classical continuum. Such an approach retains the accuracy and robustness offered by a standard Computational Homogenization implementation, whereby the problem and the computational time are reduced to the on-line solution of one scale only.

Frequency-Limited Balanced Truncation with Low-Rank Approximations

Abstract: In this article we investigate model order reduction of large-scale systems using frequency-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerical realization of this model reduction approach. We discuss numerical methods to take care of the involved matrix-valued functions. The occurring large-scale Lyapunov equations are solved for low-rank approximations for which we also establish results regarding the eigenvalues of their solutions. These results, and also numerical experiments, will show that the eigenvalues of the Lyapunov solutions in frequency-limited balanced truncation often decay faster than those in standard balanced truncation. Moreover, we show in further numerical examples that frequency-limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region.

Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization

Abstract: In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element. We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model can be built.

Randomized Dynamic Mode Decomposition for Non-Intrusive Reduced Order Modelling

Abstract: This paper focuses on a new framework for reduced order modelling of non-intrusive data with application to 2D flows. To overcome the shortcomings of intrusive model order reduction usually derived by combining the POD and the Galerkin projection methods, we developed a novel technique based on Randomized Dynamic Mode Decomposition as a fast and accurate option in model order reduction of non-intrusive data originating from Saint-Venant systems. Combining efficiently the Randomized Dynamic Mode Decomposition algorithm with Radial Basis Function interpolation, we produced an efficient tool in developing the linear model of a complex flow field described by non-intrusive (or experimental) data. The rank of the reduced DMD model is given as the unique solution of a constrained optimization problem. We emphasize the excellent behavior of the non-intrusive reduced order models by performing a qualitative analysis. In addition, we gain a significantly reduction of CPU time in computation of the reduced order models (ROMs) for non-intrusive numerical data.

On the use of modal derivatives for nonlinear model order reduction

Abstract: Summary Modal derivative is an approach to compute a reduced basis for model order reduction of large-scale nonlinear systems that typically stem from the discretization of partial differential equations. In this way, a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller, based on the resulting small-scale state-space model. We investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction, which has been missing so far. Concentrating on the application field of structural mechanics and structural dynamics, we show that the concept of modal derivatives can also be applied as nonlinear extension of the Craig–Bampton family of methods for substructuring. We furthermore generalize the approach from a pure projection scheme to a novel reduced-order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables. This complexity reduction leads to a frequency-preserving nonlinear quadratic state-space model. Numerical examples with carefully chosen nonlinear model problems and three-dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods, along with the current limitations. Copyright © 2016 John Wiley & Sons, Ltd.

Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations

Abstract: In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

Stability Analysis of the Two-level Orthogonal Arnoldi Procedure

Abstract: The second-order Arnoldi (SOAR) procedure is an algorithm for computing an orthonormal basis of the second-order Krylov subspace. It has found applications in solving quadratic eigenvalue problems and model order reduction of second-order dynamical systems among others. Unfortunately, the SOAR procedure can be numerically unstable. The two-level orthogonal Arnoldi (TOAR) procedure has been proposed as an alternative to SOAR to cure the numerical instability. In this paper, we provide a rigorous stability analysis of the TOAR procedure. We prove that under mild assumptions, the TOAR procedure is backward stable in computing an orthonormal basis of the associated linear Krylov subspace. The benefit of the backward stability of TOAR is demonstrated by its high accuracy in structure-preserving model order reduction of second-order dynamical systems.

Projection-based model reduction for contact problems

Abstract: Large scale finite element analysis requires model order reduction for computationally expensive applications such as optimization, parametric studies and control design. Although model reduction for nonlinear problems is an active area of research, a major hurdle is modeling and approximating contact problems. This manuscript introduces a projection-based model reduction approach for static and dynamic contact problems. In this approach, non-negative matrix factorization is utilized to optimally compress and strongly enforce positivity of contact forces in training simulation snapshots. Moreover, a greedy algorithm coupled with an error indicator is developed to efficiently construct parametrically robust low-order models. The proposed approach is successfully demonstrated for the model reduction of several two-dimensional elliptic and hyperbolic obstacle and self contact problems. ∗Corresponding author Email address: maciej.balajewicz@stanford.edu (Maciej Balajewicz) 1Postdoctoral Fellow 2Engineering Research Associate 3Vivian Church Hoff Professor of Aircraft Structures

pyMOR -- Generic Algorithms and Interfaces for Model Order Reduction

Abstract: Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator, and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach.

Nonlinear model reduction by deep autoencoder of noise response data

Abstract: In this paper a novel model order reduction method for nonlinear systems is proposed. Differently from existing ones, the proposed method provides a suitable non-linear projection, which we refer to as control-oriented deep autoencoder (CoDA), in an easily implementable manner. This is done by combining noise response data based model reduction, whose control theoretic optimality was recently proven by the author, with stacked autoencoder design via deep learning.

PEBL-ROM: Projection-error based local reduced-order models

Abstract: Projection-based model order reduction (MOR) using local subspaces is becoming an increasingly important topic in the context of the fast simulation of complex nonlinear models. Most approaches rely on multiple local spaces constructed using parameter, time or state-space partitioning. State-space partitioning is usually based on Euclidean distances. This work highlights the fact that the Euclidean distance is suboptimal and that local MOR procedures can be improved by the use of a metric directly related to the projections underlying the reduction. More specifically, scale-invariances of the underlying model can be captured by the use of a true projection error as a dissimilarity criterion instead of the Euclidean distance. The capability of the proposed approach to construct local and compact reduced subspaces is illustrated by approximation experiments of several data sets and by the model reduction of two nonlinear systems.

Nonlinear Moment Matching-Based Model Order Reduction

Abstract: In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system. We also define a new notion of moment matching and present a family of (nonlinear) parametrized reduced order models that achieve moment matching. We establish relations with existing notions of moment for nonlinear systems, showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching, respectively, are equivalent. Furthermore, we compute the reduced order model that matches the moments at two chosen signal generators (one exciting the input of the system and another driven by the system), simultaneously. We also present a family of models computed on the basis of a nonlinear extension of the Petrov-Galerkin projection that achieve moment matching. Finally, we specialize the results to the case of nonlinear, input-affine systems.

Generation of Equivalent Circuit From Finite-Element Model Using Model Order Reduction

Abstract: This paper proposes an equivalent-circuit generation from the finite-element (FE) model of electromagnetic devices using a model order reduction (MOR). In this method, an equivalent circuit is directly generated from the reduced transfer function obtained using MOR based on Pade approximation via the Lanczos process. It is shown that the generated circuit yields sufficiently accurate results in both frequency and time domains. Moreover, the computational time of the present method is much shorter than that of the circuit generation based on frequency sweep using the conventional FE analysis.

Greedy Multipoint Model-Order Reduction Technique for Fast Computation of Scattering Parameters of Electromagnetic Systems

Abstract: This paper attempts to develop a new automated multipoint model-order reduction (MOR) technique, based on matching moments of the system input–output function, which would be suited for fast and accurate computation of scattering parameters for electromagnetic (EM) systems over a wide frequency band To this end, two questions are addressed Firstly, the cost of the wideband reduced model generation is optimized by automating a greedy multipoint MOR scheme This is achieved by introducing a new dual local-global model convergence scheme, which applies fast and reliable a posteriori error estimates to check both local model convergence, used to select the number of moments at a single expansion point, and global model convergence, used to optimally select the expansion points Secondly, the question of optimal convergence measure is addressed by proposing an enhanced a posteriori error estimator particularly suited for scattering parameter computations for lossy EM systems The effectiveness and efficiency of the proposed automated scheme is verified through numerical simulations using reduced-order models for examples of a bandstop dielectric resonator filter and a dielectric resonator antenna for a wide frequency band, and compared against the results obtained using the full-order model, a reduced model generated with the optimal greedy point selection algorithm, as well as the reduced-order models obtained using the reduced basis method (RBM) and the single-point second-order Arnoldi method for passive order reduction (SAPOR) method

Fast Finite-Element Analysis of Motors Using Block Model Order Reduction

Abstract: This paper proposes a model order reduction (MOR) based on the proper orthogonal decomposition (POD) to perform a fast analysis of motors. When a POD-based MOR is applied to the motor analysis, the number of basis vectors has to be increased to express the changes in the magnetic fields due to a rotational movement. Its computational efficiency is, thus, greatly deteriorated. To overcome this difficulty, the block-MOR is first applied to a motor analysis. In this method, a parameter space is subdivided into several blocks, which correspond to angular ranges in the motor analysis, in each of which the basis vectors are constructed from snapshotted fields. The computational time of the block-MOR is shown to be shorter than that of the conventional MOR, while the accuracy of both the methods is almost identical.

Structure Preserving Model Order Reduction of Large Sparse Second-Order Index-1 Systems and Application to a Mechatronics Model

Abstract: Nowadays, mechanical engineers heavily depend on mathematical models for simulation, optimization and controller design. In either of these tasks, reduced dimensional formulations are obligatory in order to achieve fast and accurate results. Usually, the structural mechanical systems of machine tools are described by systems of second-order differential equations. However, they become descriptor systems when extra constraints are imposed on the systems. This article discusses efficient techniques of Gramian-based model-order reduction for second-order index-1 descriptor systems. Unlike, our previous work, here we mainly focus on a second-order to second-order reduction technique for such systems, where the stability of the system is guaranteed to be preserved in contrast to the previous approaches. We show that a special choice of the first-order reformulation of the system allows us to solve only one Lyapuov equation instead of two. We also discuss improvements of the technique to solve the Lyapu...

Equivalent-Circuit Generation From Finite-Element Solution Using Proper Orthogonal Decomposition

Abstract: This paper presents the generation of equivalent circuits from a finite-element (FE) model of electromagnetic devices using proper orthogonal decomposition (POD). This method effectively computes the frequency response of the reduced FE model constructed by the POD-based model order reduction. Then, the circuit parameters are determined so as to minimize the error between the frequency responses of the reduced FE model and equivalent circuit. The frequency characteristics of an inductor and the induction heating machine evaluated by the equivalent circuit are shown to be in good agreement with those computed from the original FE model.

Stability preserving post-processing methods applied in the Loewner framework

Abstract: We provide an overview of the Loewner methodology which is an interpolatory model order reduction technique. It uses measured or computed data (e.g. measurements of the frequency response of a to-be approximated system) instead of the system matrices, and constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. In some cases the resulting reduced systems may not be stable. This issue is addressed by developing reliable and robust post processing methods to yield a stable reduced model.

Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion

Abstract: We present new sparse-grid based algorithms for fast Bayesian estimation and inversion of parametric operator equations. We propose Reduced Basis (RB) acceleration of numerical integration based on Smolyak sparse grid quadrature. To tackle the curse-of-dimensionality in high-dimensional Bayesian inversion, we exploit sparsity of the parametric forward solution map as well as of the Bayesian posterior density with respect to the random parameters. We employ an dimension adaptive Sparse Grid method (aSG) for both, offline-training the reduced basis as well as for deterministic quadrature of the conditional expectations which arise in Bayesian estimates. For the forward problem with nonaffine dependence on the random variables, we perform further affine approximation based on the Empirical Interpolation Method (EIM) proposed in [1]. A novel combined algorithm to adaptively refine the sparse grid used for quadrature approximation of the Bayesian estimates, of the reduced basis approximation and to compress the parametric forward solutions by empirical interpolation is proposed. The theoretically predicted computational efficiency which is independent of the number of active parameters is demonstrated in numerical experiments for a model, nonaffine-parametric, stationary, elliptic diffusion problem, in two spacial and in parameter space dimensions up to 1024.

High-Fidelity Model Order Reduction for Microgrids Stability Assessment

Abstract: Proper modeling of inverter-based microgrids is crucial for accurate assessment of stability boundaries. It has been recently realized that the stability conditions for such microgrids are significantly different from those known for large- scale power systems. While detailed models are available, they are both computationally expensive and can not provide the insight into the instability mechanisms and factors. In this paper, a computationally efficient and accurate reduced-order model is proposed for modeling the inverter-based microgrids. The main factors affecting microgrid stability are analyzed using the developed reduced-order model and are shown to be unique for the microgrid-based network, which has no direct analogy to large-scale power systems. Particularly, it has been discovered that the stability limits for the conventional droop-based system (omega - P/V - Q) are determined by the ratio of inverter rating to network capacity, leading to a smaller stability region for microgrids with shorter lines. The theoretical derivation has been provided to verify the above investigation based on both the simplified and generalized network configurations. More impor- tantly, the proposed reduced-order model not only maintains the modeling accuracy but also enhances the computation efficiency. Finally, the results are verified with the detailed model via both frequency and time domain analyses.

Novel MOR approach for extracting dynamic compact thermal models with massive numbers of heat sources

Abstract: A novel Model Order Reduction approach for the construction of Dynamic Compact Thermal Models is presented. With respect to previous approaches, this methodology allows reducing the complexity of the constructed models, from quadratically to linearly dependent on the number of independent heat sources. In such a way, the approach allows constructing Dynamic Compact Thermal Models practically without limitations on the number of heat sources. The proposed methodology is validated through the application to two state-of-the-art electronic systems.

On the use of model order reduction for simulating automated fibre placement processes

Abstract: Automated fibre placement (AFP) is an incipient manufacturing process for composite structures. Despite its conceptual simplicity it involves many complexities related to the necessity of melting the thermoplastic at the interface tape-substrate, ensuring the consolidation that needs the diffusion of molecules and control the residual stresses installation responsible of the residual deformations of the formed parts. The optimisation of the process and the determination of the process window requires a plethora of simulations because there are many parameters involved in the characterization of the material and the process. The exploration of the design space cannot be envisaged by using standard simulation techniques. In this paper we propose the off-line calculation of rich parametric solutions that can be then explored on-line in real time in order to perform inverse analysis, process optimisation or on-line simulation-based control. In particular, in the present work, and in continuity with our former works, we consider two main extra-parameters, the first related to the line acceleration and the second to the number of plies laid-up.