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Quadratic Eigenvalue Problem (quadratic + eigenvalue_problem)
Selected AbstractsOn singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with cornersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004A. Dimitrov Abstract In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin,Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+,Q+,2R)d=0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix,vector products for finding all eigenvalues in the interval ,,(,) , (,0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd. [source] Efficient computation of order and mode of corner singularities in 3D-elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001A. Dimitrov Abstract A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin,Petrov finite element method. A quadratic eigenvalue problem (P+,Q+,2R) u=0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have optimal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ,e(,),(,0.5,1.0) (no eigenpairs can be ,lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd. [source] A Quadratic Eigenproblem in the Analysis of a Time Delay SystemPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006Elias JarlebringArticle first published online: 4 DEC 200 In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problem where ,1, ,, ,m ,1 , , are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation, the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift-invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation. The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theoryNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2006F. Chaitin-Chatelin Abstract This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ,. The admittance t = 1/, is the classical homotopy parameter. In particular, we study the spectrum when t , ,. We show that in the limit part of the eigenvalues remain bounded and converge to the so-called kernel points. We also show that there exist the so-called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| , , enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright © 2006 John Wiley & Sons, Ltd. [source] | ||||||||||