On the stability of the Rayleigh–Ritz method for eigenvalues

Gallistl, Dietmar; Huber, P.; Peterseim, Daniel

doi:10.1007/s00211-017-0876-8

Cited by 9 publications

(6 citation statements)

References 12 publications

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“…Following the idea of IGA, [2] which is known to have a nice spectral behavior compared to the FEM, [48] and to allow for a local refinement of the basis, the function space  is spanned by truncated hierarchical (TH) B-splines. [45,49] Alternatively, also T-splines could be considered.…”

Section: Adaptive Isogeometric Discretizationmentioning

confidence: 99%

A diffuse modeling approach for embedded interfaces in linear elasticity

et al. 2019

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In this contribution, we present a diffuse modeling approach to embed material interfaces into nonconforming meshes with a focus on linear elasticity. For this purpose, a regularized indicator function is employed that describes the distribution of the different materials by a scalar value. The material in the resulting diffuse interface region is redefined in terms of this indicator function and recomputed by a homogenization of the adjacent material parameters. The applied homogenization method fulfills the kinematic compatibility across the interface and the static equilibrium at the interface. In addition, an hℓ‐adaptive refinement strategy based on truncated hierarchical B‐spline is applied to provide an appropriate and efficient approximation of the diffuse interface region. We justify mathematically and demonstrate numerically that the applied approach leads to optimal convergence rates in the far field for one‐dimensional problems. A two‐dimensional example illustrates that the application of the hℓ‐adaptive refinement strategy allows for a clear reduction of the error in the near and far field and a good resolution of the local stress and strain fields at the interface. The use of a higher continuous B‐spline basis leads to efficient computations due to the higher continuity of the diffuse interface model.

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Section: Adaptive Isogeometric Discretizationmentioning

confidence: 99%

A diffuse modeling approach for embedded interfaces in linear elasticity

et al. 2019

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“…Compared to FEA, it was observed that IGA possesses superior approximation of eigenvalues, see [21,22,23,24]. Further studies show outliers appearing in the case of reduced continuity [25] and Neumann boundary conditions [26,27]. In our tests a major improvement was shown by a stronger enforcement of the Neumann boundary condition through a penalty approach, which can recover spectral results closer to the cases with no outliers.…”

Section: Introductionmentioning

confidence: 56%

A hybrid isogeometric approach on multi-patches with applications to Kirchhoff plates and eigenvalue problems

Horger

Reali

Wohlmuth

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We present a systematic study on higher-order penalty techniques for isogeometric mortar methods. In addition to the weak-continuity enforced by a mortar method, normal derivatives across the interface are penalized. The considered applications are fourth order problems as well as eigenvalue problems for second and fourth order equations. The hybrid coupling enables the discretization of fourth order problems in a multi-patch setting as well as a convenient implementation of natural boundary conditions. For second order eigenvalue problems, the pollution of the discrete spectrum -typically referred to as "outliers" -can be avoided.Numerical results illustrate the good behaviour of the proposed method in simple systematic studies as well as more complex multi-patch mapped geometries for linear elasticity and Kirchhoff plates.

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“…A modified Fourier version based on Rayleigh-Ritz method is performed. The Rayleigh-Ritz method is a powerful tool in the field of vibration analysis, in which the undetermined coefficients in the displacement can be obtained by minimizing the Lagrangian energy function expression or making them equal to zero [47]. Then, a series of equations can be summed up in matrix form as a standard characteristic equation, which can be easily solved to obtain the desired frequencies and modes.…”

Section: Stress-strain Relations and Stress Resultantsmentioning

confidence: 99%

Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

Wang

Yao

et al. 2019

Mathematical Problems in Engineering

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In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.

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On the stability of the Rayleigh–Ritz method for eigenvalues

Cited by 9 publications

References 12 publications

A diffuse modeling approach for embedded interfaces in linear elasticity

A diffuse modeling approach for embedded interfaces in linear elasticity

A hybrid isogeometric approach on multi-patches with applications to Kirchhoff plates and eigenvalue problems

Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

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