Fast frequency sweep computations using a multi‐point Padé‐based reconstruction method and an efficient iterative solver

Avery, Philip; Farhat, Charbel; Reese, Garth M.

doi:10.1002/nme.1879

Cited by 45 publications

(84 citation statements)

References 21 publications

(25 reference statements)

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“…Thus, the Padé-reconstructed solutions are established with m f = m σ = 4, with the constraint n i = m i + 1 for the denominator truncations. This constraint echoes the choice made for univariate Padé approximants in recent publications [4,5]. Note that the choice of the best truncation orders is an important and intricate step of the method, which is still given much attention in the Padé approximants community.…”

Section: Resultsmentioning

confidence: 83%

“…In analogy to previous contributions for univariate FE problems [4,5], the partial derivatives of the solution vector can be calculated recursively by a method based on the use of the general Leibniz rule for multiple variables. For two independent variables, and applied to the left hand side of Eq.…”

Section: Solution Vector Partial Derivatives For the Bivariate Solutimentioning

confidence: 99%

“…As an example, recent research on performance optimization of sound absorbing porous materials [1], has highlighted the need for efficient methods, considering, e.g., the high computational cost of porous material modeling, the number of material parameters involved, and the uncertainty associated with these parameters, etc. Alternative approaches aiming at efficient multivariate solution strategies, have been studied, see e.g., reduced-order models for porous materials [2,3], efficient solution reconstruction for fast frequency sweeps using univariate Padé approximants [4,5], to name but a few.…”

Section: Introductionmentioning

confidence: 99%

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A finite element solution strategy based on Padé approximants for fast multiple frequency sweeps of multivariate problems

Rumpler

Göransson

2013

Proceedings of Meetings on Acoustics

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Analyses involving structural-acoustic finite element models including three-dimensional modelling of porous media are, in general, computationally costly. While being the most commonly used predictive tool in the context of noise and vibrations reduction, efficient solution strategies enabling the handling of large-size multiphysics industrial problems are still lacking, particularly in the context where multiple frequency response estimations are required, e.g. for topology optimization, multiple load cases analysis, etc. In this work, an original solution strategy is presented for the solution of multi-frequency structural-acoustic problems including poroelastic damping. Based on the use of Padé approximants, very accurate interpolations of multiple frequency sweeps are performed, allowing for substantial improvements in terms of computational ressources, i.e. time and memory allocation. The method is validated and will be demonstrated for its potential on 3D applications involving coupled elastic, poroelastic and internal acoustic domains.

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Section: Resultsmentioning

confidence: 83%

Section: Solution Vector Partial Derivatives For the Bivariate Solutimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A finite element solution strategy based on Padé approximants for fast multiple frequency sweeps of multivariate problems

Rumpler

Göransson

2013

Proceedings of Meetings on Acoustics

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“…It is said to be a multi-point Padé Approximation [13] if P frequency points are defined and interpolated in the frequency interval of interest, ∆ ω , where each frequency is denoted by ω p with p = 1, 2, 3, ..., P , using…”

Section: Multi-padé Approximationmentioning

confidence: 99%

Model Order Reduction in Structural Dynamics

Sanchez¹,

Buchschmid²,

Müller³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Frequency response analysis in structural dynamics usually requires solving large dynamical systems of the form (−ω 2 M + iωD + K)u(ω) = f (ω), which result from a FE discretization. A straightforward solution of big systems requires a high computational cost; therefore several Model Order Reduction (MOR) techniques have been developed in the last decades to obtain faster and efficient results. Between them interpolatory approaches have gained importance for solving second order dynamical systems. This work presents and compares ten MOR techniques which are suitable for structural dynamics problems. These are: Guyan-Irons Reduction, Improved Reduction System, Dynamic Reduction, Real Modal Analysis, Complex Modal Analysis, Craig-Bampton Method, and Interpolatory MOR methods like Multi-point Padé Approximation, the Krylov-based Galerkin Projection and the Derivativebased Galerkin Projection. A brief summary of the theoretical background is presented for each method. A first numerical example shows the applicability for damped systems and a second example shows suitability of the Interpolatory MOR methods for industrial applications, using data from a commercial FE software (ANSYS ® ).

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“…It has been shown that such a rational function can be accurately and efficiently approximated by an appropriate Padé expansion [5,6,7,4,10,1]. The numerical algorithm proposed in this paper for identifying the eigenvalues missed in a given range of interest builds on the aforementioned observations.…”

Section: Introductionmentioning

confidence: 99%

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

Avery

Farhat

Hetmaniuk

2009

Numerical Meth Engineering

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SUMMARYWhen computing the solution of a generalized symmetric eigenvalue problem of the form Ku = λMu, the Sturm sequence check is the most popular method for reporting the number of missed eigenvalues within a range [σL, σR]. This method requires the factorization of the matrices K − σLM and K − σRM. When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large-scale problems, the factorization of K − σM is not desired or feasible and therefore the Sturm sequence check cannot be performed. To this effect, the purpose of this paper is to present a factorization-free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σL, σR]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil (K, M), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large-scale structural vibrations problem.

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Fast frequency sweep computations using a multi‐point Padé‐based reconstruction method and an efficient iterative solver

Cited by 45 publications

References 21 publications

A finite element solution strategy based on Padé approximants for fast multiple frequency sweeps of multivariate problems

A finite element solution strategy based on Padé approximants for fast multiple frequency sweeps of multivariate problems

Model Order Reduction in Structural Dynamics

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

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