Frequency-Domain Karhunen-Loeve Method and Its Application to Linear Dynamic Systems

Kim, Taehyoun

doi:10.2514/2.315

Cited by 117 publications

(48 citation statements)

References 8 publications

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“…In some applications of POD, the snapshots are centered to have zero mean, in which case K t is a scaled covariance matrix. Kim [141] develops the frequency domain POD method by showing that through a simple application of Parseval's theorem, in the single-input case one can write the kernel as…”

Section: Frequency Domain Podmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Frequency Domain Podmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…As discussed by Willcox et al [10], as the number of frequency points is increased and the number of moments matched at each point is reduced to one, the method becomes a frequency-domain POD approach, which uses SVD on a set of complex responses obtained at selected frequency sample points to construct a basis [15], [16], [17]. Figure 8 compares the accuracy provided by the multiplepoint Arnoldi method and the POD method.…”

Section: From Multiple-point Arnoldi To Proper Orthogonal Decomposmentioning

confidence: 99%

Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Lassaux

Willcox

2003

41st Aerospace Sciences Meeting and Exhibit

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I. ABSTRACT A multiple-point Arnoldi method is derived for model reduction of computational fluid dynamic systems. By choosing the number of frequency interpolation points and the number of Arnoldi vectors at each frequency point, the user can select the accuracy and range of validity of the resulting reduced-order model while balancing computational expense. The multiplepoint Arnoldi approach is combined with a singular value decomposition approach similar to that used in the proper orthogonal decomposition method. This additional processing of the basis allows a further reduction in the number of states to be obtained, while retaining a significant computational cost advantage over the proper orthogonal decomposition. Results are presented for a supersonic diffuser subject to mass flow bleed at the wall and perturbations in the incoming flow. The resulting reduced-order models capture the required dynamics accurately while providing a significant reduction in the number of states. The reduced-order models are used to generate transfer function data, which are then used to design a simple feedforward controller. The controller is shown to work effectively at maintaining the average diffuser throat Mach number. II. INTRODUCTIONComputational fluid dynamics (CFD) has reached a considerable level of maturity and is now routinely used in many applications for both external and internal flows. Euler and NavierStokes solvers enjoy widespread use for aerodynamic design and analysis, and provide accurate answers for a variety of complex flows. However, despite ever increasing computational power, unsteady problems are computationally very expensive and time-consuming. More efficient methods for time-varying flow can be obtained if the disturbances are small, and the unsteady solution can be considered to be a small perturbation about a steady-state flow [1]. In this case, a set of linearized equations is obtained which can be time-marched to obtain the flow solution at each instant.Even under the linearization assumption, any CFD-based technique will generate models with a prohibitively high number of states. For this reason, CFD models are not appropriate for many applications where model size and cost are issues. For example, when the aerodynamic solver must be coupled to another disciplinary model, as in aeroelastic analysis or multidisciplinary optimization, CFD models cannot be used. Another application which requires low-order models is control design.The concept of using active control to enhance the stability properties of an unsteady flow has been addressed for several applications [2], [3]. In order to derive control models that will be effective, it is vital that the relevant unsteady flow dynamics are captured accurately. A model is required that will capture not only the dynamics of the disturbance to be controlled, but also the visibility offered by the sensing and the effect on the flow of the actuation mechanism. A high-fidelity CFD code can offer the degree of flow resolution that is required; however, ...

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“…For linearized flows the frequency-domain POD technique was introduced in [69]. It replaces the time-domain representation of the Galerkin-projected equations with a Fourier-domain representation for each individual harmonic.…”

Section: Model Reduction Of Periodic Steady-state Solutionsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

164

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Frequency-Domain Karhunen-Loeve Method and Its Application to Linear Dynamic Systems

Cited by 117 publications

References 8 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

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