On the solution of generalized non‐linear complex‐symmetric eigenvalue problems

Abstract

SUMMARYThis paper brings an attempt toward the systematic solution of the generalized non-linear, complexsymmetric eigenproblem, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as (are complex-symmetric matrices given as power series of the complex eigenfrequencies , such that, if ( , /) is a solution eigenpair,The traditional Rayleigh quotient iteration and the more recent JacobiDavidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi-Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues.

On the inverse of generalized λ‐matrices with singular leading term

“…An issue of paramount importance is the cost-effective solution of the non-linear eigenvalue problem associated with -matrices [2][3][4][5][6][7] (see References [52] for a thorough outline of the general non-linear eigenvalue problem). For the present context, a promising procedure has been recently developed [28] on the basis of the Jacobi-Davidson iteration [53][54][55] and making use of the generalized eigenvalue properties introduced in Reference [9].…”

“…The author and his collaborators found no case of finite element development [9,20,24], no matter how complicated the underlying physical problem was, in which splitting K as in Equations (A14)-(A16) would yield some mechanical interpretation that renders it more useful than Equation (A13)-see a discussion on the subject in Reference [28].…”

“…In the more general case of viscous damping, on the other hand, the expression of the stiffness matrix K, as illustrated in Equation (A25) for a simple truss element, cannot be easily separated into stiffness, mass and damping contributions. However, although this might become a concern in the frame of a displacement-based variational formulation, as proposed by Przemieniecki, it is completely irrelevant for K obtained in the context of a hybrid formulation (this subject is discussed in some detail in Reference [28]). Moreover, the linear algebra outline of this paper deals with general complex coefficient matrices and no special care has to be taken with respect to the way these matrices are obtained.…”

“…• In case of an elasticity problem with underlying real fundamental solutions but taking into account viscous damping in the term ( 2 + 2i )-as in Equation (A3)-the resultant stiffness matrix is complex symmetric [28] and all coefficient matrices A j in the summand (−i ) j A j of Equation (1) are real (and ≡ ), which means that only real linear algebra operations need to be performed in the evaluation of the coefficient matrices of the inverse X, with subsequent multiplication by (−i ) j .…”

On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems

“…The NEP (1.1), including the typical linear and quadratic eigenvalue problems as special cases, is of great importance in a large number of disciplines of scientific computing and engineering applications such as density functional theory calculations [4], vibration of viscoelastic structures [9], dynamic finite element method [11], photonic band structure calculations [38], vibration of fluid-solid structures [40], and so on. When A(λ) is a matrix polynomial, the NEP (1.1) can be reformulated as a linear eigenvalue problem [24,31] and, hence, it may be effectively solved by the Jacobi-Davidson method; see, e.g., [19,36].…”