On the number of reliable finite‐element eigenmodes

Givoli, Dan

doi:10.1002/cnm.1088

Cited by 3 publications

(6 citation statements)

References 11 publications

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“…Several works in the literature deal with the correct determination of eigenmodes in elliptical problems. An a priori error estimator for the so‐called h version of the FEM, see Hughes, 1987 and more recently Givoli, 2007, is available. This error estimator is: Equation 18 where λ n is the exact n th eigenvalue ( λ n is ordered by increasing magnitude), λ n h is the FEM approximation eigenvalue, h is a mesh parameter, i.e.…”

Section: Free‐vibrations In Axisymmetric Plates and Shellsmentioning

confidence: 99%

“…Equation (39), presented by Givoli (2007), determines the number of eigenvalues, B, that can be obtained with an error, ε , from a linear system associated with equation (38) for N dof: Equation 40 In equation (39), r o is a constant that depends on the material properties and type of element employed, d is the problem dimension (1 for 1D, 2 for 2D, 3 for 3D) and p is the approximation space polynomial order.…”

Section: Free‐vibrations In Axisymmetric Plates and Shellsmentioning

confidence: 99%

“…Results in terms of the properties of the GFEM approximation space, enrichment and regularity, are provided. To evaluate the “accuracy” of the method, we establish a comparison between the numerical solution of the axisymmetric modes, calculated using the FEM and GFEM, to a reference solution given by the “rule of thumb” see Givoli, 2007.…”

Section: Introductionmentioning

confidence: 99%

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Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…

Suarez

Rossi

Silva

2012

Engineering Computations

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PurposeThe purpose of this paper is to investigate the approximation performance of a family of piecewise rational polynomial shape functions, which are enriched by a set of monomials of order p to obtain high order approximations. To numerically demonstrate the features of the enriched approximation some examples on the mechanical elastic response and free‐vibration of axisymmetric plates and shells are carried out.Design/methodology/approachThe global approximation is based on a particular family of weight function, which is defined on the parametric domain of the element, ξ∈[−1,1], resulting in shape functions with compact support, which have regularity C0k,k=0,2,4… in the global domain Σ. The PU shape functions are enriched by a set of monomials of order p to obtain high order approximation spaces.FindingsBased on the numerical results of elastic axisymmetric plates and shells, it is demonstrated that the proposed methodology produces satisfactory results in terms of keeping the ill‐conditioning of the system of equations under accepted levels. Comparisons are established between linear and Hermitian shape functions showing similar results. The observed results for the free‐vibration problem of plates and shells show the potential of the proposed approximation space.Research limitations/implicationsIn this paper the formulation is limited to the modeling of axisymmetric plate and shell problems. However, it can be applied to model other problems where the high regularity of the approximation is required.Originality/valueThe paper presents an alternative approach to construct partition of unity shape functions based on a particular family of weight function.

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Section: Free‐vibrations In Axisymmetric Plates and Shellsmentioning

confidence: 99%

Section: Free‐vibrations In Axisymmetric Plates and Shellsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…

Suarez

Rossi

Silva

2012

Engineering Computations

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“…i , criterion D with ε q = 10 −3 ; (c) (4) i , criterion D with ε q = 10 −4 ; (d) (5) i , criterion D with ε q = 10 −5 . The calculated eigenvalues i under criterion B with ε = 10 −8 are of high precision and are selected as reference eigenvalues.…”

Section: Finite Element Analysis Of a Cofferdammentioning

confidence: 99%

“…Error bound estimation is indispensable for both computational efficiency and reliability of large scale engineering problems [1][2][3][4][5][6][7][8][9][10][11]. In particular, the posteriori error estimation and related adaptive techniques have been the major research focus to develop effective adaptive algorithms for problems involving large distortion, contact, fracture and various material interfaces.…”

Section: Introductionmentioning

confidence: 99%

An enhanced formulation of error bound in subspace iteration method

Chen

Gong

Chen

et al. 2010

Numer Methods Biomed Eng

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SUMMARYIn subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM.

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On the numerical determination of eigenvalues/eigenvectors using a high regularity finite element method

Suarez¹,

Rossi²,

Altafini³

et al. 2015

Applied Mathematical Modelling

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On the number of reliable finite‐element eigenmodes

Cited by 3 publications

References 11 publications

Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…

Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…

An enhanced formulation of error bound in subspace iteration method

On the numerical determination of eigenvalues/eigenvectors using a high regularity finite element method

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On the number of reliable finite‐element eigenmodes

Cited by 3 publications

References 11 publications

Enrichment of a rational polynomial family of shape functions with regularity Ck0, k=0,2,4…

Enrichment of a rational polynomial family of shape functions with regularity Ck0, k=0,2,4…

An enhanced formulation of error bound in subspace iteration method

On the numerical determination of eigenvalues/eigenvectors using a high regularity finite element method

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Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…

Enrichment of a rational polynomial family of shape functions with regularity C^k₀, k=0,2,4…