Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

Apel, Thomas; Sändig, Anna‐Margarete; Solov’ev, S. I.

doi:10.1051/m2an:2003005

Cited by 33 publications

(24 citation statements)

References 33 publications

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“…The classical approach to solve k-th degree polynomial eigenvalue problems of the form (5) or rational eigenvalue problems of the forms (7)- (9) is to first perform a linearization [27], i.e. to transform the problem to an equivalent linear eigenvalue problem (λE − A)x = 0 with the same eigenvalues.…”

Section: Linearizationmentioning

confidence: 99%

“…A finite element approximation [5] leads to a finite-dimensional quadratic eigenvalue problem. After substitution λ = α + 0.5 this problem has the more convenient form (3).…”

Section: Singularity Exponentsmentioning

confidence: 99%

“…A quadratic problem of slightly different structure arises in the study of corner singularities in anisotropic elastic materials, [3,4,5,44,52,63,89], where the problem has the form…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

2004

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We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear eigenvalue problems arise and some numerical results.

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Section: Linearizationmentioning

confidence: 99%

“…A finite element approximation [5] leads to a finite-dimensional quadratic eigenvalue problem. After substitution λ = α + 0.5 this problem has the more convenient form (3).…”

Section: Singularity Exponentsmentioning

confidence: 99%

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Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

2004

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“…In the present case, the convergence rate was doubled and the error was reduced by orders of magnitude. In the authors' opinion, this makes the proposed implementation a promising tool for versatile and comprehensive analyses of 3D stress singularities and a proper alternative to former mesh grading strategies [1].…”

Section: Discussionmentioning

confidence: 94%

Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

Hell

Becker

2016

Proc Appl Math and Mech

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The SBFEM is employed for the extraction of 3D stress singularity exponents and their associated deformation modes. It makes use of a separation of variables approach, which is substituted into the virtual work equation so that only parts of the boundary need to be discretized while the 3D boundary value problem is considered analytically in a scaling variable. The enrichment of the standard separation of variables representation for the displacements with a decomposition of the analytical crack displacement field yields a substantial improvement of the method's accuracy and convergence properties.

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“…This approach has also been used for the FEM eigenanalysis (e.g. [2]) but still led to linearized eigenvalue problems so large that a complete solution with the QR-algorithm was avoided.…”

Section: Introductionmentioning

confidence: 99%

The Use of Enriched Base Functions in the Three-Dimensional Scaled Boundary Finite Element Method

Hell¹,

Becker²

2016

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

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Abstract. The Scaled Boundary Finite Element Method (SBFEM) is a semi-analytical method which combines the advantages of the Boundary Element Method (BEM) and the Finite Element Method (FEM). Like in the BEM, only the boundary needs to be discretized. On the other hand, the SBFEM is based on the virtual work principle and does not need any fundamental solutions. If the scalability condition is fulfilled, a separation of variables representation can be employed leading to a quadratic eigenvalue problem and a linear equation system which can be solved by standard methods. The SBFEM has proven its high efficiency and accuracy in the presence of stress singularities, especially in 2D fracture mechanics when the singularity is entirely located within the considered domain.However, in 3D elasticity problems, there can also be singularities on the discretized boundary itself. Then, the SBFEM suffers from drawbacks also well known from the standard FEM, i.e. moderate accuracy and bad convergence. To overcome these deficiencies in such 3D cases, we propose the enrichment of the standard separation of variables representation with analytical fields which are known to exactly fulfill the local boundary conditions:The examples of a single plane crack and two perpendicularly meeting cracks in an isotropic continuum are considered. It is demonstrated that the method's original excellent accuracy and convergence are regained, at a minimum cost of additional degrees of freedom (DOF). The normalized errors in the solution of the quadratic eigenvalue problem already become negligibly small for very coarse boundary meshes. The obtained convergence orders are often optimal and sometimes even superconvergence is observed.

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Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

Cited by 33 publications

References 33 publications

Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

Determination of singularity exponents in 3D elasticity problems using enriched base functions in the Scaled Boundary Finite Element Method

The Use of Enriched Base Functions in the Three-Dimensional Scaled Boundary Finite Element Method

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