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Article
Publication date: 5 September 2018

Snehashish Chakraverty and Nisha Rani Mahato

In structural mechanics, systems with damping factor get converted to nonlinear eigenvalue problems (NEPs), namely, quadratic eigenvalue problems. Generally, the parameters of…

Abstract

Purpose

In structural mechanics, systems with damping factor get converted to nonlinear eigenvalue problems (NEPs), namely, quadratic eigenvalue problems. Generally, the parameters of NEPs are considered as crisp values but because of errors in measurement, observation or maintenance-induced errors, the parameters may have uncertain bounds of values, and such uncertain bounds may be considered in terms of closed intervals. As such, this paper aims to deal with solving nonlinear interval eigenvalue problems (NIEPs) with respect to damped spring-mass systems having interval parameters.

Design/methodology/approach

Two methods, namely, linear sufficient regularity perturbation (LSRP) and direct sufficient regularity perturbation (DSRP), have been proposed for solving NIEPs based on sufficient regularity perturbation method for intervals. LSRP may be used for solving NIEPs by linearizing the eigenvalue problems into generalized interval eigenvalue problems, and DSRP may be considered as a direct solution procedure for solving NIEPs.

Findings

LSRP and DSRP methods help in computing the lower and upper eigenvalue and eigenvector bounds for NIEPs which contain the crisp eigenvalues. Further, the DSRP method is computationally efficient compared to LSRP.

Originality/value

The efficiency of the proposed methods has been validated by example problems of NIEPs. Moreover, the procedures may be extended for other nonlinear interval eigenvalue application problems.

Article
Publication date: 1 February 1994

N. Kamiya and S.T. Wu

A new‐type eigenvalue formulation of the two‐dimensionalHelmholtz equation is presented in this paper. A boundary integral equationis derived using the T‐complete functions…

Abstract

A new‐type eigenvalue formulation of the two‐dimensional Helmholtz equation is presented in this paper. A boundary integral equation is derived using the T‐complete functions relevant to the Trefftz method, which is further transformed to the generalized eigenvalue problem. Boundary discretization and a standard eigenvalue computation routine, offered as a black box, are sufficient for the determination of the eigenvalues. The proposed method can reduce the users’ task in preprocessing and initial rough estimation when compared with the existing domain‐type solvers.

Details

Engineering Computations, vol. 11 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 11 March 2016

Nisha Rani Mahato and Snehashish Chakraverty

The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then…

Abstract

Purpose

The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then we get crisp eigenvalue problem. But in actual practice, instead of crisp material properties we may have only bounds of values as a result of errors in measurements, observations and calculations or it may be due to maintenance induced error etc. Such bounds of values may be considered in terms of interval or fuzzy numbers. The purpose of this paper is to develop a fuzzy filtering procedure for finding real eigenvalue bounds of different structural problems.

Design/methodology/approach

The proposed fuzzy filtering algorithm has been developed in terms of fuzzy number to solve the fuzzy eigenvalue problem. The initial bounds of fuzzy eigenvalues are filtered to obtain precise eigenvalue bounds which are depicted by fuzzy (Triangular Fuzzy Number) plots using α-cut.

Findings

Previously, bounds of eigenvalues of interval matrices have been investigated by few authors. But when the structural problem consists of fuzzy material properties, then the interval eigenvalue bounds may be obtained for each interval of the fuzzy number. The proposed algorithm has been applied for standard fuzzy eigenvalue problems which may be extended to generalised fuzzy eigenvalue problems for obtaining filtered fuzzy bounds.

Originality/value

The developed fuzzy filtering method is found to be efficient for different structural dynamics problems with fuzzy material properties.

Details

Engineering Computations, vol. 33 no. 3
Type: Research Article
ISSN: 0264-4401

Article
Publication date: 14 November 2008

L. Kolev

To suggest a polynomial complexity method for determining the range of real eigenvalues in the case of the generalized eigenvalue problem when the elements of the matrices…

Abstract

Purpose

To suggest a polynomial complexity method for determining the range of real eigenvalues in the case of the generalized eigenvalue problem when the elements of the matrices involved are independent intervals.

Design/methodology/approach

The basic approach is to make use of approximate interval solutions as regards the right and left eigenvectors of the eigenproblem considered, the so‐called outer solutions, in order to determine the range.

Findings

First, a new method for computing the outer solutions has been suggested. The main result of the paper, however, is the development of a simple method for determining the range of the real eigenvalues. Unlike the known general‐purpose methods that have exponential complexity, the present range determination method is much simpler as its complexity is only polynomial.

Research limitations/implications

The method is applicable if certain sufficient conditions reported in the paper are satisfied (an incomplete quadratic system is to have a positive solution and the signs of the outer solutions should satisfy a complete or partial invariance).

Practical implications

The method guarantees reliable numerical results when the original eigenproblems contain interval uncertainties as is, strictly speaking, most often the case in practice.

Originality/value

To the best of the author's knowledge, the present paper suggests, for the first time, a simple method of polynomial complexity for solving the problem considered which is inherently a NP‐hard problem (of exponential complexity).

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 27 no. 6
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 March 1992

JIAKANG ZHONG, LOUIS C. CHOW and WON SOON CHANG

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is divided into…

Abstract

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is divided into finite elements and at each finite element node, a closed‐form expression for the temperature as a function of time can be obtained. Three test problems which have exact solutions were solved in order to examine the merits of the eigenvalue method. It was found that this method yields accurate results even with a coarse mesh. It provides exact solution in the time domain and therefore has none of the time‐step restrictions of the conventional numerical techniques. The temperature field at any given time can be obtained directly from the initial condition and no time‐marching is necessary. For problems where the steady‐state solution is known, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed. These features lead to great savings in computation time, especially for problems with long time duration. Furthermore, the availability of the closed form expressions for the temperature field makes the present method very attractive for coupled problems such as solid—fluid and thermal—structure interactions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 July 2014

Yuying Xia and M. Friswell

Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving…

Abstract

Purpose

Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as fuzzy parameters. The requirement is to propagate the parameter uncertainty to the eigenvalues of the structure, specified as fuzzy eigenvalues. However, the usual approach is to transform the fuzzy problem into several interval eigenvalue problems by using the α-cuts method. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem.

Design/methodology/approach

Based on the fundamental perturbation principle and vertex theory, an efficient perturbation method is proposed, that gives the exact extrema of the first-order deviation of the structural eigenvalue. The fuzzy eigenvalue approach has also been improved by reusing the interval analysis results from previous α-cuts.

Findings

The proposed method was demonstrated on a simple cantilever beam with a pinned support, and produced very accurate fuzzy eigenvalues. The approach was also demonstrated on the model of a highway bridge with a large number of degrees of freedom.

Originality/value

This proposed Vertex-Perturbation method is more efficient than the standard perturbation method, and more general than interval arithmetic methods requiring the non-negative decomposition of the mass and stiffness matrices. The new increment method produces highly accurate solutions, even when the membership function for the fuzzy eigenvalues is complex.

Details

Engineering Computations, vol. 31 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 13 July 2012

Mehdi Dehghan and Masoud Hajarian

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Abstract

Purpose

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Design/methodology/approach

By extending the idea of conjugate gradient (CG) method, the authors present two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Findings

When the general coupled matrix equations are consistent over the generalized centro‐symmetric and central anti‐symmetric matrices, the generalized centro‐symmetric and central anti‐symmetric solutions can be obtained within nite iterative steps. Also the least Frobenius norm generalized centrosymmetric and central anti‐symmetric solutions can be derived by choosing a special kind of initial matrices. Furthermore, the optimal approximation generalized centrosymmetric and central anti‐symmetric solutions to given generalized centro‐symmetric and central anti‐symmetric matrices can be obtained by finding the least Frobenius norm generalized centro‐symmetric and central anti‐symmetric solutions of new matrix equations. The authors employ some numerical examples to support the theoretical results of this paper. Finally, the application of the presented methods is highlighted for solving the projected generalized continuous‐time algebraic Lyapunov equations (GCALE).

Originality/value

By the algorithms, the solvability of the general coupled matrix equations over generalized centro‐symmetric and central anti‐symmetric matrices can be determined automatically. The convergence results of the iterative algorithms are also proposed. Several examples and an application are given to show the efficiency of the presented methods.

Article
Publication date: 1 September 2004

Elizabeth A. Burroughs, Louis A. Romero, Richard B. Lehoucq and Andrew G. Salinger

Locates the onset of oscillatory instability in the fluid flow inside a differentially heated cavity with aspect ratio 2 by computing a steady‐state and analyzing the stability of…

Abstract

Locates the onset of oscillatory instability in the fluid flow inside a differentially heated cavity with aspect ratio 2 by computing a steady‐state and analyzing the stability of the system via eigenvalue approximation. Discusses the choice of parameters for the Cayley transformation so that the calculation of selected eigenvalues of the transformed system will reliably answer the question of stability. Also presents an argument that due to the symmetry of the problem, the first two unstable modes will have eigenvalues that are nearly identical, and the numerical experiments confirm this. Finally, locates a co‐dimension 2 bifurcation signifying where there is a switch in the mode of initial instability. The results were obtained using a parallel finite element CFD code (MPSalsa) along with an Arnoldi‐based eigensolver (ARPACK), a preconditioned Krylov method code for the necessary linear solves (Aztec), and a stability analysis library (LOCA).

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 14 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1989

Mingwu Yuan, Pu Chen, Shanji Xiong, Yuanneng Li and Edward L. Wilson

The advantages of a direct superposition of the Ritz vector in dynamic response analysis (developed by Wilson, Yuan, and Dickens in 1982 and termed the WYD method) are that: no…

Abstract

The advantages of a direct superposition of the Ritz vector in dynamic response analysis (developed by Wilson, Yuan, and Dickens in 1982 and termed the WYD method) are that: no iteration is involved; the method is at least four times faster than the subspace iteration method; and fewer Ritz vectors are necessary for the mode superposition of dynamic response analysis than exact eigenvectors are used. The major purpose of this paper is to illustrate that the WYD method can also be used as a general approximate algorithm to calculate eigenvalues and eigenvectors. The WYD and Lanczos algorithms are very similar and a formula that relates the two is given in this paper. Although the exact algebraic value of only a single eigenvector of a multi‐eigenvalue can be calculated using either the WYD or Lanczos methods, an artificial round‐off is presented that can be used to solve the eigenvalue problem. A method of estimating the error introduced by the WYD method is also developed. A dynamic substructuring technique, based on the WYD method, and which assumes that the connectivities on the interfaces among the substructures need not be considered is also presented.

Details

Engineering Computations, vol. 6 no. 1
Type: Research Article
ISSN: 0264-4401

Article
Publication date: 15 November 2011

Williams L. Nicomedes, Renato C. Mesquita and Fernando J.S. Moreira

The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods…

Abstract

Purpose

The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods, particularly the one known as meshless local Petrov‐Galerkin (MLPG).

Design/methodology/approach

Regarding eigenvalue problems, the authors show how to apply MLPG to the time‐independent Schrödinger equation stated in three dimensions. Through a special procedure, the numerical integration of weak forms is carried out only for internal nodes. The boundary conditions are enforced through a collocation method. The final result is a generalized eigenvalue problem, which is readily solved for the energy levels. An example of boundary value problem is described by the time‐dependent nonlinear Schrödinger equation. The weak forms are again stated only for internal nodes, whereas the same collocation scheme is employed to enforce the boundary conditions. The nonlinearity is dealt with by a predictor‐corrector scheme.

Findings

Results show that the combination of MLPG and a collocation scheme works very well. The numerical results are compared to those provided by analytical solutions, exhibiting good agreement.

Originality/value

The flexibility of MLPG is made explicit. There are different ways to obtain the weak forms, and the boundary conditions can be enforced through a number of ways, the collocation scheme being just one of them. The shape functions used to approximate the solution can incorporate modifications that reflect some feature of the problem, like periodic boundary conditions. The value of this work resides in the fact that problems from other areas of electromagnetism can be attacked by the very same ideas herein described.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 30 no. 6
Type: Research Article
ISSN: 0332-1649

Keywords

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