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

Dumont, Ney Augusto

doi:10.1002/nme.1561

Cited by 5 publications

(8 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The non-linear eigenvalue problem was firstly solved using linearization, which leads to large, although sparse, matrices. The present paper is together with References [11,16] an attempt to lay out the theoretical basis of the proposed dynamic formulation.…”

Section: Problem Justificationmentioning

confidence: 99%

“…The expression of K eff( ) in terms of transcendental functions of is only possible when the finite element boundaries coalesce to points, as for trusses and beams. The general evaluation of mass and stiffness matrices as series expansions, according to Section 3.1 and as illustrated in the following, is conceptually straightforward and applicable to large finite/boundary element families [6,12,16].…”

Section: Jacobi-davidson Algorithm For Non-linear Complex-symmetric mentioning

confidence: 99%

“…However, this term is void in any variationally based finite/boundary element expansion one may obtain [6,7,[10][11][12]16], which is coherent with the developments in the classical books on dynamics that retain terms up to 2 : 13,14,43,44]. Except for M 0 , which corresponds to M 1 in the present paper, the mass matrix terms given in Equations (30) and (31) differ in physical meaning from the ones introduced in Equation (29).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Dumont

2007

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Problem Justificationmentioning

confidence: 99%

Section: Jacobi-davidson Algorithm For Non-linear Complex-symmetric mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Dumont

2007

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…The non-normalized matrixW of either constant potentials or rigid body displacements is straightforward to construct, as given below for a discrete model with nodal points of co-ordinates (x s , y s , z s ): W = 1 1 · · · 1 T for 2D and 3D potential problems (14.25) Table 14.1 summarizes some possibilities of finite element construction in the present hybrid formulation using non-singular fundamental solutions [34]. For the static case, or for the modal advanced analysis of Section 14.4, F * 0 is singular, according to (14.14), and one must resort to the theory of generalized inverses [41,42], which is well consolidated. For 3D problems, examples are given for 4-node and 10-node tetrahedrals (Te4 and Te10) as well as for 8-node and 20-node hexahedrals (H8 and H20).…”

Section: Non-singular Fundamental Solutions In the Frequency Domainmentioning

confidence: 99%

“…However, since the product V T F * 1 V is non-singular, the inverse of F * may be obtained by an algorithm whose implementation is straightforward [42]. However, since the product V T F * 1 V is non-singular, the inverse of F * may be obtained by an algorithm whose implementation is straightforward [42].…”

Section: Advanced Modal Analysis Of the Time-dependent Problemmentioning

confidence: 99%

Computational Modelling and Advanced Simulations

Muŕın¹,

Kompiš²,

Kutiš³

2011

Computational Methods in Applied Sciences

View full text Add to dashboard Cite

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) IntroductionThis book contains selected, extended papers presented in the thematic ECCOMAS conference on Computational Modelling and Advanced Simulations (CMAS2009) held in Bratislava, Slovakia, in June 30-July 3, 2009.Modelling and simulation of engineering problems play a very important role in the classic and new composite material sciences, and in design and computational prototyping of modern and advanced technologic parts and systems. According to this, the existing numerical methods have been improved and new numerical methods have been established for modelling and simulation of more and more complex and complicated engineering problems. The present book should contribute to the effort to make the modelling and simulation more effective and accurate.All the extended papers were reviewed by members of the Conference CMAS2009 Scientific Board, and these papers are arranged into following 18 separated chapters:In the first chapter a boundary element method is developed for the nonlinear dynamic analysis of beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross-section, partially supported on tensionless Winkler foundation, undergoing moderate large deflections under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy and the range of applications of the developed method.In the second chapter a new plate theory based on the direct approach is introduced and applied to plates composed of functionally graded materials (FGM). The material properties are changing in the thickness direction. Solving some problems of the global structural analysis it will be demonstrated that in some cases the results significantly differ from the results based on the Kirchhoff-type theory.In the third chapter the indirect Trefftz collocation method is applied to solve the inverse Cauchy problem of linear piezoelectricity. The static analysis of coupled fields in 2D piezoelectric solids is considered.In the forth chapter new phenomenological material model for solid foam materials is presented. This model describes the uniaxial compressive stress-strain curves of foam materials. Parameters of this model specifically control the shape of compressive stress-strain curve and they are function of foam density. v vi IntroductionThe fifth chapter is focused mainly on polypropylene based composites filled by mineral fillers, but some results can be generalized for...

show abstract

Solution of Time-Dependent Problems Using Exact Modal Analysis with Illustrative Application to One-Dimensional Truss Structures

Dumont

2024

Lecture Notes in Mechanical Engineering

View full text Add to dashboard Cite

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

Cited by 5 publications

References 41 publications

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

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

Computational Modelling and Advanced Simulations

Solution of Time-Dependent Problems Using Exact Modal Analysis with Illustrative Application to One-Dimensional Truss Structures

Contact Info

Product

Resources

About