Model order reduction of parameterized circuit equations based on interpolation

Sơn, Nguyễn Thanh; Stykel, Tatjana

doi:10.1007/s10444-015-9418-z

Cited by 9 publications

(6 citation statements)

References 48 publications

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“…For model order reduction of parametric systems, a variety of techniques, in particular based on interpolation, have been developed 16,18,19,27 and applied to structured systems. 26,28,29 In this paper, only the PMOR technique based on matrix interpolation 19,30 is considered. This technique was not developed for the simulation of moving loads, but the slow changes of the parameter for slowly moving loads enable the application of this method.…”

Section: Model Order Reduction Of An Elastic Body Subjected To Movingmentioning

confidence: 99%

Comparison of two model order reduction methods for elastic multibody systems with moving loads

Baumann

Vasilyev

Stykel

et al. 2016

Proceedings of the Institution of Mechanical Engineers, Part K:

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In this paper, two different model order reduction approaches for elastic multibody systems with moving loads are considered. The first approach is based on a parametric formulation of the input and output matrices and application of parametric model order reduction. Therefore, a quasi-static description without considering the time dependency of the parameter is proposed. In the second approach, the time-varying input matrix is approximated in a low-dimensional subspace and model order reduction of a time-invariant system is performed. To investigate the applicability of these methods for the generation of reduced elastic multibody systems, for the first time, both approaches are compared for a thin-walled cylinder model with a rotating force.

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Section: Model Order Reduction Of An Elastic Body Subjected To Movingmentioning

confidence: 99%

Comparison of two model order reduction methods for elastic multibody systems with moving loads

Baumann

Vasilyev

Stykel

et al. 2016

Proceedings of the Institution of Mechanical Engineers, Part K:

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“…For a given parameter the original FE model may be reduced by any suitable method. The only restriction is that the reduced models are again of type (1) and that they have the same number of states n and the same number of inputs m. In order to keep model interpolation meaningful, we also claim that the columns of L p refer to the same i/o channels for all p.…”

Section: Model Reduction For Fixed Parametermentioning

confidence: 99%

“…We consider reduced systems Σ p of type (1). Due to our assumptions on symmetry and definiteness we can always find a matrix U p composed of eigenvectors satisfying U t p M p U p = I and K p U p = M p U p Λ p , where Λ p is a diagonal matrix of non-negative eigenvalues.…”

Section: Matrix Matchingmentioning

confidence: 99%

“…The idea is to reduce FE models offline for selected parameter sets and to generate models for new parameters by cheap interpolation rather than expensive reduction. The different approaches to parametric linear model reduction may be divided into three classes [1]. Interpolation of transfer functions is well suited for parabolic or highly damped hyperbolic problems.…”

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confidence: 99%

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Parametric Reduction of FE Models with Variable Mesh Topology

Möhring¹

2014

Proc Appl Math and Mech

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Designing whole machines or processes you may need both, an integrated dynamic simulation of all components on system level and a detailed analysis of how the macroscopic behavior of a component depends on geometry and material parameters. The former analysis is usually based on systems of differential algebraic equations representing a component by not more than a few hundred states and requires tools like Matlab-Simulink R or Dymola R . The latter analysis solves discretized partial differential equations with several 100,000 degrees of freedom using finite element software like Ansys R or Comsol R . Model reduction bridges the gap between the two worlds providing small state space models with approximately the same input-output behavior as the original large finite element models. Building systems from generic components, e.g. a gas transport network from pipeline models with variable length, or optimizing the design of a device with respect to mechanical or thermal properties, we need parametric reduced models. The idea is to reduce FE models offline for selected parameter sets and to generate models for new parameters by cheap interpolation rather than expensive reduction. The different approaches to parametric linear model reduction may be divided into three classes [1]. Interpolation of transfer functions is well suited for parabolic or highly damped hyperbolic problems. However, poles are duplicated rather than shifted, which is unacceptable for weakly damped hyperbolic problems like in mechanics. The second class of methods look for a basis of state space covering system behavior over the full range of parameters. They share the critical assumption that number and meaning of states do not change with the parameters. In terms of finite elements this means that the meshes for different design parameters are morphed variants of the same reference mesh.This may become a severe restriction in practice when automatic meshing is to be applied to complicated geometries. Therefore, we propose a method from the third class, which is based on interpolating reduced system matrices [2]. Only those parts of the mesh need to share a constant topology where nodal inputs are applied and outputs are collected. The inner mesh, however, may change for different parameters. The main challenges arise from the fact that state space representations of a system are unique only up to a change of basis and that interpolating matrices which refer to non-fitting bases may cause arbitrary errors. In the article we will show how problems like leaving and entering modes or eigenvalue crossing can be overcome by using normal forms and eigenvalue tracking in parameter space. The method, which is implemented in the Fraunhofer Model Reduction Toolbox, is applied to a parametric model of a mechanical device the eigenfrequencies of which have to be kept away from some dominant excitation frequencies.

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“…Then they are interpolated using standard methods like Lagrange or spline interpolation. These approaches have been discussed intensively in many publications see, e.g., [6,15,16] for interpolating local reduced system matrices, [4,17] for interpolating projection subspaces, [5,18] for interpolating reduced transfer functions, and [19] for a detailed discussion on the use of manifold interpolation for model reduction. Each of them has its own strength and acts well in some specific applications but fails to be superior to the others in a general setting.…”

Section: Introductionmentioning

confidence: 99%

Balanced truncation for parametric linear systems using interpolation of Gramians: a comparison of algebraic and geometric approaches

Sơn¹,

Gousenbourger²,

Massart³

et al. 2020

Preprint

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When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetric positive semidefinite matrices, a property that should be preserved by the interpolation method. In this contribution, we propose and compare two approaches for Gramian interpolation. In the first approach, the interpolated Gramian is computed as a linear combination of the data Gramians with positive coefficients. Even though positive semidefiniteness is guaranteed in this method, the rank of the interpolated Gramian can be significantly larger than that of the data Gramians. The second approach aims to tackle this issue by performing the interpolation on the manifold of fixed-rank positive semidefinite matrices. The

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Model order reduction of parameterized circuit equations based on interpolation

Cited by 9 publications

References 48 publications

Comparison of two model order reduction methods for elastic multibody systems with moving loads

Comparison of two model order reduction methods for elastic multibody systems with moving loads

Parametric Reduction of FE Models with Variable Mesh Topology

Balanced truncation for parametric linear systems using interpolation of Gramians: a comparison of algebraic and geometric approaches

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