A theory of condensation model reduction

Flippen, L.D.

doi:10.1016/0898-1221(94)90034-5

Cited by 11 publications

(4 citation statements)

References 13 publications

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“…The idea of using a as in (3.35) and (3.36) has been presented before [52,20,49], but its use in connecting reduced basis methods with substructuring methods is emphasized here. There is some similarity in the use of (3.34) for a and the Modal Reduction method (see (10) of Ref.…”

Section: Substructuring Methodsmentioning

confidence: 99%

“…To maintain as general a setting as possible for the development of the reduced basis and substructuring methods, L of (2.1) is taken to be a square system matrix with operator components, as in Ref. [20]. In keeping with this, the definition of the multiplication of two arbitrary matrices is generalized to…”

Section: Classes Of Methods Encompassedmentioning

confidence: 99%

“…The first and still very popular such method is that due to Guyan [18]. Flippen [19] derived a number of such methods from a general time-derivative series solution [20]. The Modal Reduction method [21,22,23], popular for the synthesis of Test-Analysis-Models (TAMs) via elimination of (almost) all but the test-sensor-associated DOF, is an adaptation of modal reduced basis methods to substructuring.…”

Section: Introductionmentioning

confidence: 99%

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A Generic Bilevel Formalism for Unifying and Extending Model Reduction Methods

Flippen¹

2000

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. 12b. DISTRIBUTION CODE ABSTRACT (Maximum 200 words)An abstract, algebraic bilevel version of conventional multigrid methods has been developed that formally unifies and extends the reduced basis, substructuring, smoothing/homogenization, and frequency window methods of model reduction. Within the context of this formalism, a given model reduction method synthesizes a reduced model that plays the role of a "coarse grid" model. Another original-model approximation, conjugate to the model reduction method, plays the role of a "fine grid" model. Conventional multigrid methods can be thought of as an extension of the coarse grid model beyond the original scope of its accuracy through the alternating use of "fine-grid-error-smoothing" and "coarse-grid-correction" steps. Analogously fashioned, abstract versions of these steps extend the particular model reduction method used beyond the original scope of its accuracy. In addition to the development of the mathematical formalism, a generic continuation is proposed and developed as the conjugate approximation for use in the abstract version of the fine-grid-error-smoothing step. For generality, the nature of the parameter-embedding of the continuation is deliberately left unspecified; a particular parameter-embedding is chosen by the analyst to complement the particular model and model reduction method (or class thereof) in a given case. As an example, a particular submethod of this formalism is constructed by specializing the parameter-embedding to a temporal, biscale perturbational parameter-embedding for the case of a generic set of coupled ordinary differential equations with constant coefficients. The initial iteration of this submethod is a general, combined transient/frequency-window methodology for linear finite-element method applications. It is shown to encompass both the force derivative and Lanczos/Ritz vector methods for the limiting case of a zero frequency, singlepoint window for the "fast" time scale. To put this in perspective, the force derivative and Lanczos/Ritz vector methods are two leading approaches for extending/replacing modal reduced basis methods. Current frequency window methods, in turn, are very efficient for extensive time-harmonic «analysis of the system.

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Section: Substructuring Methodsmentioning

confidence: 99%

Section: Classes Of Methods Encompassedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Generic Bilevel Formalism for Unifying and Extending Model Reduction Methods

Flippen¹

2000

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“…An example is a method of dynamic condensation [20] in which the passive coordinates are represented by the active ones, and the eigenvalue problem is solved by a combined technique of Sturm sequence and subspace iteration. In the context of multi-scale analysis of eigenvalue problem, two model reduction methods, operator-function method and quantum scattering analog method, have also been developed [8][9][10][11]26]. In these approaches, a projection matrix that relates degrees of freedom at different scales is constructed and it enables the solution of macroscopic eigenvalue problem.…”

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confidence: 99%

An iterative asymptotic expansion method for elliptic eigenvalue problems with oscillating coefficients

Mehraeen

Chen

2009

Comput Mech

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In this work we propose a method for obtaining fine-scale eigensolution based on the coarse-scale eigensolution in elliptic eigenvalue problems with oscillating coefficient. This is achieved by introducing a 2-scale asymptotic expansion predictor in conjunction with an iterative corrector. The eigensolution predictor equation is formulated using the weak form of an auxiliary problem. It is shown that large errors exist in the higher eigenmodes when the 2-scale asymptotic expansion is used. The predictor solution is then corrected by the combined inverse iteration and Rayleigh quotient iteration. The numerical examples demonstrate the effectiveness of this approach.

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