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Article
Publication date: 8 September 2020

Yongliang Wang

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential…

Abstract

Purpose

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions.

Design/methodology/approach

This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error.

Findings

Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance.

Originality/value

The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

Details

Engineering Computations, vol. 38 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 1 March 1992

D.J. KIDGER and I.M. SMITH

The eigenvalues of element stiffness matrices K and the eigenvalues of the generalized problem Kx = λMx, where M is the element's mass matrix, are of fundamental…

Abstract

The eigenvalues of element stiffness matrices K and the eigenvalues of the generalized problem Kx = λMx, where M is the element's mass matrix, are of fundamental importance in finite element analysis. For instance, they may indicate the presence of ‘zero energy modes’, or control the critical timestep applicable in temporal integration of dynamic problems. Recently explicit formulae for the eigenvalues of the stiffness matrix of a plane, 4‐node rectangular element have been given, and the authors have extended this approach to deal with 8‐node solid brick elements as well. In the present paper, explicit eigenvalues are given for plane triangular elements and techniques for eigenmode visualization are applied to well‐known triangular and quadrilateral elements. In the companion paper (Part II), the stiffness matrices of solid tetrahedra and bricks are similarly treated.

Details

Engineering Computations, vol. 9 no. 3
Type: Research Article
ISSN: 0264-4401

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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…

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

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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…

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.

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Article
Publication date: 1 December 2001

Georg Hebermehl, Friedrich‐Karl Hübner, Rainer Schlundt, Thorsten Tischler, Horst Zscheile and Wolfgang Heinrich

The design of microwave circuits requires detailed knowledge on the electromagnetic properties of the transmission lines used. This can be obtained by applying Maxwell’s…

Abstract

The design of microwave circuits requires detailed knowledge on the electromagnetic properties of the transmission lines used. This can be obtained by applying Maxwell’s equations to a longitudinally homogeneous waveguide structure, which results in an eigenvalue problem for the propagation constant. Special attention is paid to the so‐called perfectly matched layer boundary conditions (PML). Using the finite integration technique we get an algebraic formulation. The finite volume of the PML introduces additional modes that are not an intrinsic property of the waveguide. In the presence of losses or absorbing boundary conditions the matrix of the eigenvalue problem is complex. A method which avoids the computation of all eigenvalues is presented in an effort to find the few propagating modes one is interested in. This method is an extension of a solver presented by the authors in a previous paper which analyses the lossless case. Using mapping relations between the planes of eigenvalues and propagation constants a strip in the complex plane is determined containing the desired propagation constants and some that correspond to the PML modes. In an additional step the PML modes are eliminated.The numerical effort of the presented method is reduced considerably compared to a full calculation of all eigenvalues.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 20 no. 4
Type: Research Article
ISSN: 0332-1649

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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…

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

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Article
Publication date: 1 June 2000

R.V.N. Melnik

The dynamics of coupling between spectrum and resolvent under ε‐perturbations of operator and matrix spectra are studied both theoretically and numerically. The phenomenon…

Abstract

The dynamics of coupling between spectrum and resolvent under ε‐perturbations of operator and matrix spectra are studied both theoretically and numerically. The phenomenon of non‐trivial pseudospectra encountered in these dynamics is treated by relating information in the complex plane to the behaviour of operators and matrices. On a number of numerical results we show how an intrinsic blend of theory with symbolic and numerical computations can be used effectively for the analysis of spectral problems arising from engineering applications.

Details

Engineering Computations, vol. 17 no. 4
Type: Research Article
ISSN: 0264-4401

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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…

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

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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…

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

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Article
Publication date: 3 May 2016

Renato M Cotta, Carolina Palma Naveira-Cotta and Diego C. Knupp

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary…

Abstract

Purpose

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

Design/methodology/approach

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

Findings

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

Originality/value

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 3/4
Type: Research Article
ISSN: 0961-5539

Keywords

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