We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

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The Quadratic Eigenvalue Problem

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters. Copyright © 2006 John Wiley & Sons, Ltd.

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Recent computational developments in krylov subspace methods for linear systems

We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.

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NLEVP: A Collection of Nonlinear Eigenvalue Problems

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.

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Modified HSS iteration methods for a class of complex symmetric linear systems

Iterative system solvers for the frequency analysis of linear mechanical systems

https://www.tpbin.com/Uploads/Subjects/7269b478-4b86-4d25-ac03-1cac5d6f619e.pdf

A numerical procedure for simulating the multi-support seismic response of submerged floating tunnels anchored by cables

Modeling of Qiandao Lake submerged floating tunnel subject to multi-support seismic input

The paper deals with the representation of dissipative effects by means of equivalent viscous forces. A brief review of the classical analytical treatment of the subject is first presented devoting particular attention to the topics of hysteretic and modal damping. The problem of forming viscous matrices in the case of systems which are non-homogeneous from the point of view of dissipation is then addressed; soil-structure systems are first considered and some accepted techniques for forming the structure contribution to the viscous matrix are reviewed. A different technique is then proposed which avoids some of the drawbacks of the previously quoted methods. In the final section of the paper it is shown how this technique is easily applicable also in the case of systems having internal concentrated dampers of viscous type, this being a situation which is difficult to tackle with usual criteria.

The formation of viscous damping matrices for the dynamic analysis of mdof systems

The modeling of tethering elements of seabed anchored floating structures is addressed, with particular reference to the so-called Archimedes Bridge (submerged floating tunnel, SFT) solution for deep water crossing; attention is devoted to the design solution encompassing slender bars as anchor elements. Two numerical tools are proposed: firstly, a geometrically nonlinear finite element (NWB model), developed in previous work, has been refined in order to capture the effect of higher flexural modes of anchor bars. Secondly, a 3D beam element, based on the classical corotational formulation (CR model), has been developed and coded. Both elements are implemented in a numerical procedure for the dynamic time domain step-by-step analysis of nonlinear discretized systems; seismic loading is introduced by generating artificial time histories of spatially variable seismic motion. An example of application of the NWB element is shown regarding the behavior of the dynamic model of a complete SFT. The model was subjected to extreme multiple-support seismic loading. The seismic behavior is here illustrated and commented, especially in light of the effect of higher local vibration modes of the anchor bars. Finally, a comparison between the performances of the two modeling approaches is presented. Both harmonic and seismic excitations are considered in the test; the results justify the use of the simpler NWB approach, especially in the SFT design phase. Copyright © 2008 John Wiley & Sons, Ltd.

Anna Feriani

Papers

Iterative system solvers for the frequency analysis of linear mechanical systems

A numerical procedure for simulating the multi-support seismic response of submerged floating tunnels anchored by cables

Modeling of Qiandao Lake submerged floating tunnel subject to multi-support seismic input

The formation of viscous damping matrices for the dynamic analysis of mdof systems

Numerical models for the dynamic response of submerged floating tunnels under seismic loading