Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

Apel, Thomas; Mehrmann, Volker; Watkins, David S.

doi:10.1016/s0045-7825(02)00390-0

Cited by 56 publications

(46 citation statements)

References 23 publications

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“…Hamiltonian eigenvalue problems arise in a variety of settings, including linearquadratic optimal control problems [17,26,68,70,71,74], determination of corner singularities in anisotropic elastic structures [6,75], and stability of gyroscopic systems [67]. Various algorithms of GR type for the Hamiltonian eigenvalue problem, as well as some background theory, are addressed in [3,4,13,26,78,82,93].…”

Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

“…The nonsymmetric Lanczos process [69] generates a pseudosymmetric tridiagonal matrix whose eigenvalues must then be computed. The Hamiltonian eigenvalue problem arises in several contexts, including linear-quadratic optimal control problems [17,26,68,70,71,74], determination of corner singularities in anisotropic elastic structures [6,75], and stability of gyroscopic systems [67]. The symplectic eigenvalue problem arises in discrete-time linear-quadratic control problems [58,68,74,79].…”

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confidence: 99%

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Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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Abstract. Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors:Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.

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Section: Pseudosymmetric Matrices a Real Tridiagonal Matrixmentioning

confidence: 99%

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confidence: 99%

Product Eigenvalue Problems

Watkins¹

2005

SIAM Rev.

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“…Modern methods to solve the algebraic eigenvalue problem (2) by exploiting the structure can be found, for example, in [3,7,38].…”

Section: Introductionmentioning

confidence: 99%

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

2002

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Abstract. This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.Mathematics Subject Classification. 65N25, 65N30, 74G70.

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“…Real and complex T -palindromic QEPs also arise in the numerical simulation of the behavior of periodic surface acoustic wave (SAW) filters [43,85]. Quadratic eigenproblems with T -alternating polynomials arise in the study of corner singularities in anisotropic elastic materials [7,8,70]. Gyroscopic systems [25,48,49] also lead to quadratic T -alternating matrix polynomials.…”

Section: Definition 8 (Adjoint Of Matrix Polynomials)mentioning

confidence: 99%

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey

Tisseur

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Matrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems. One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial by a matrix pencil with the same spectrum, and then computes with the pencil. Many matrix polynomials arising from applications have additional algebraic structure, leading to symmetries in the spectrum that are important for any computational method to respect. Thus it is useful to employ a structured linearization for a matrix polynomial with structure. This essay surveys the progress over the last decade in our understanding of linearizations and their construction, both with and without structure, and the impact this has had on numerical practice.

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Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

Cited by 56 publications

References 23 publications

Product Eigenvalue Problems

Product Eigenvalue Problems

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

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

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