The variational theory of complex rays for the calculation of  medium‐frequency vibrations

Ladevèze, Pierre; Arnaud, Lionel; Rouch, Philippe; Blanzé, Claude

doi:10.1108/02644400110365879

Cited by 75 publications

(66 citation statements)

References 38 publications

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“…Other Trefftz methods relying on discontinuous interpolations include the wave-based method [32] and the variational theory of complex rays (VTCR) [33], but these two methods cannot be formulated in the DG framework [12]. Results in [34] indicate that the UWVF tends to outperform the VTCR for simple Helmholtz problems.…”

Section: Introductionmentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.

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Section: Introductionmentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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“…This paper was written in French, though an abridged English version was included. In a recent paper written in English [165], Ladeveze et al covered VTCR in more detail and provided some numerical examples. They characterize the method by describing three basic features, which are summarized as follows:…”

Section: Variational Theory Of Complex Raysmentioning

confidence: 99%

“…A new variational formulation of the reference problem is used, in which substructure approximations are independent and thus do not necessarily satisfy the displacement and stress transmission conditions at the interfaces between substructures. In [165], the reference problem is an assembly of two connected structures subject to harmonic vibration at a fixed frequency, ui.…”

Section: Variational Theory Of Complex Raysmentioning

confidence: 99%

“…The authors refer to VTCR as a "true 'medium-frequency' method" because effective quantities are used for the time and space scales considered, rather than retaining "phenomena associated with small-length variations" that lead to "results which are very sensitive to data errors" [165]. In fact, the authors claim that the "spatial distribution of the solution has no 'physical meaning' from the mechanical point of view."…”

Section: Uik1 Yp) -W(x_y_p) ■ Exp(iujp ■ Y)mentioning

confidence: 99%

“…Three numerical examples are considered in [165]: a square plate, a triangular plate, and an assembly of 18 rectangular plates. For the square plate, the results at 600 Hz from modal analysis (1600 modes) and VTCR (64 DOF) are shown in Fig 11. The average displacement for the plate in the frequency range around 6000 Hz is plotted for each solution in Fig.…”

Section: Uik1 Yp) -W(x_y_p) ■ Exp(iujp ■ Y)mentioning

confidence: 99%

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Mid-Frequency Dynamics of Complex Structural Systems: Assessing the State of the Art and Defining Future Research Directions

Pierre¹,

Castanier²

2002

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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, searchi AUTHOR(S)PROFESSOR CHRISTOPHER PIERRE Approved for public rsleas3 ; distribution unlinätsd, PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)8 ABSTRACT (Maximum 200 words)One of the last frontiers of structural dynamics is mid-frequency vibration analysis of complex structures. In the low-frequency range, finite element analysis (FEA) is well established as the standard method, However, as the frequency of vibration increases, the cost of FEA becomes prohibitive due to the necessary refinement of the finite element mesh to capture the shorter wavelength of vibration. Furthermore, the system response becomes sensitive to small parameter variations at higher frequencies, which means that a statistical analysis should be employed to make confident response predictions. In the high-frequency range, statistical energy analysis (SEA) is popular. However, SEA provides only averaged response predictions and cannot capture the resonant behavior in the response that becomes evident as frequency decreases. ABSTRACTOne of the last frontiers of structural dynamics is mid-frequency vibration analysis of complex structures. In the low-frequency range, finite element analysis (FEA) is well established as the standard method. However, as the frequency of vibration increases, the cost of FEA becomes prohibitive due to the necessary refinement of the finite element mesh to capture the shorter wavelength of vibration. Furthermore, the system response becomes sensitive to small parameter variations at higher frequencies, which means that a statistical analysis should be employed to make confident response predictions. In the high-frequency range, statistical energy analysis (SEA) is popular. However, SEA provides only averaged response predictions and cannot capture the resonant behavior in the response that becomes evident as frequency decreases. Thus, there exists a midfrequency range in which there is no established analysis technique analogous to FEA or SEA. The goal of this research is to produce a review paper on mid-frequency vibration analysis that will provide a survey of the relevant literature, identify the key technical challenges, formulate an assessment of the state of the art, and propose directions for future research. OBJECTIVESThe main objectives of this research are as follows: 1. Perform a comprehensive literature search on mid-frequency vibration analysis and related topics, and collect an extensive set of references 2. Review the relevant literature and summarize the mid-frequency approaches and techniques that have been developed to date 3. Assess the state of the art by determining the major research challenges in the field and drawing conclusions on research progress to date, including applications to realistic complex structures 4. Based on these findings, propose general directions and specific topics for future research 5. Write a review paper that p...

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Multiscale Methods

Néron¹

2010

Encyclopedia of Aerospace Engineering

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The variational theory of complex rays for the calculation of medium‐frequency vibrations

Cited by 75 publications

References 38 publications

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Mid-Frequency Dynamics of Complex Structural Systems: Assessing the State of the Art and Defining Future Research Directions

Multiscale Methods

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