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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2006; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]

On the solution of generalized non-linear complex-symmetriceigenvalue problems

N. A. Dumont∗, †

Departamento de Engenharia CivilPontifıcia Universidade Catolica do Rio de Janeiro – PUC-Rio

22451-900 Rio de Janeiro, Brazil

SUMMARY

This paper brings an attempt toward the systematic solution of the generalized non-linear, complex-symmetric eigenproblem (K0 − iωC1 − ω2M1 − iω3C2 − ω4M2 − · · · )φ = 0, with real, symmetricmatrices K0,Cj ,Mj ∈ Rn×n, which are associated to the dynamic governing equations of a structuresubmitted to viscous damping, as laid out in the frame of an advanced mode superposition technique.The problem can be restated as (K(ω) − ωM(ω))φ = 0, where K(ω) = KT

(ω) and M(ω) = MT(ω) are

complex-symmetric matrices given as power series of the complex eigenfrequencies ω, such that, if(ω, φ) is a solution eigenpair, φTM(ω)φ = 1 and φTK(ω)φ = ω. The traditional Rayleigh quotientiteration and the more recent Jacobi-Davidson method are outlined for complex-symmetric linearproblems 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 linearanalysis, thus also presenting cubic convergence. Two examples help to visualize some of the basicconcepts developed. Three more examples illustrate the applicability of the proposed algorithm tosolve non-linear problems, in the general case of underdamping, but also for overdamping combinedwith multiple and close eigenvalues. Copyright c© 2006 John Wiley & Sons, Ltd.

key words: non-linear eigenproblems, advanced modal analysis, Rayleigh quotient iteration, Jacobi-

Davidson method, complex-symmetric matrices

1. INTRODUCTION

1.1. Problem justification

A variationally-based finite/boundary element formulation is in the background of the presentdevelopments. It was originally conceived for static problems [1, 2] as an extension of Pian’s

∗Correspondence to: N. A. Dumont, Departamento de Engenharia Civil, Pontifıcia Universidade Catolica doRio de Janeiro – PUC-Rio, 22451-900 Rio de Janeiro, Brazil.†E-mail: [email protected]

Contract/grant sponsor: Brazilian agency CNPq (Projects Nr. 475153/2003-0 and 301227/2003-9).

Received May 2006Copyright c© 2006 John Wiley & Sons, Ltd. Revised

2 N. A. DUMONT

hybrid finite element method [3, 4] and as a counterpart to the traditional boundary elementmethod [5]. Later on, the formulation was extended to the analysis of 2D and 3D transientproblems [6, 7, 8] built up on a proposition by Przemieniecki [9] intended for the free-vibrationanalysis of truss and beam structures.

The resulting advanced mode superposition technique [6, 7] starts with a frequency-domainformulation in terms of a power series expansion and requires the solution of a non-lineareigenvalue problem. General domain actions as well as general boundary and initial conditionsare almost as straightforward to deal with [6, 7, 10, 11, 12] as in classical dynamics [13, 14, 15].The inclusion of viscous damping makes the eigenproblem complex symmetric [10]. The non-linear eigenvalue problem was firstly solved using linearization, which leads to large, althoughsparse, matrices. The present paper is together with References [16, 11] an attempt to lay outthe theoretical basis of the proposed dynamic formulation.

1.2. Paper organization

A systematic review of the technical literature on the solution of linear and non-lineareigenproblems is beyond the scope of this paper. A general mathematical outline of the non-linear problem has been accomplished by Hadeler as early as in the year 1967 [17, 18]. SeeVoss [19] for an illuminating account of the subject including the historically most importantcontributions. Some basic references on eigenproblems are [13, 20, 21, 22, 23, 24].

Section 2 deals with the solution of linear, complex-symmetric eigenproblems. The Rayleighquotient iteration and the Jacobi-Davidson method are presented and compared with eachother, to make evident that they can be applied almost unchanged to the class of non-linearproblems one is actually concerned with.

This is the subject of Section 3, for an effective stiffness matrix, Equation (29), formulatedin terms of generalized damping and mass matrices. One firstly outlines the real-symmetriceigenproblem, in Section 3.2, to which the results of Section 2 can be directly extended. Theoutline makes use of sets of linearized eigenvalue problems in terms of augmented matrices withthe purpose of deriving generalized linear-algebra properties of the original non-linear problem.Some theorems are presented in order to (a) establish the equivalence between the augmented,linearized system and the original non-linear problem, (b) assess the positivity of the stiffnessand mass polynomial matrices, (c) determine the number of real solutions comprised in theformulation and (d) assess the existence range of the real eigenvalues.

The non-linear, complex-symmetric problem is formulated in Section 3.3. It is shown thatthe eigensolutions are given in complex-associated pairs, in the general case, as correspondingto underdamping. The eigenproblem related to overdamping (and to the unlikely case ofcritical damping) is also discussed. Finally, a perturbation analysis is presented to providethe mathematical justification for finding the complex solutions of interest starting from thesolutions of the underlying real-symmetric eigenproblem.

Five examples are given. The first example shows the stiffness and mass polynomial matricesof the simplest mathematical model one may conceive, namely a truss element with two degreesof freedom. The second example, for the heat conduction in a square plate, helps to visualizethe eigenvalue properties of a real-symmetric problem, as outlined in Section 3.2. The thirdexample makes use of the matrices of the first example applied to a fixed-free damped bar toillustrate the iterative eigenvalue solution with the Jacobi-Davidson algorithm of Section 3.5.The fourth example resources to the matrices of the second example to create an artificial,

Copyright c© 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 00:1–6Prepared using nmeauth.cls

NON-LINEAR COMPLEX-SYMMETRIC EIGENVALUE PROBLEMS 3

mathematically challenging, problem with overdamping and multiple eigenvalues. The lastexample deals with a rail-pad-sleeper-ballast dynamic interaction model of a railway track, inwhich several clusters of close eigenvalues have to be evaluated.

2. SOLUTION OF LINEAR, COMPLEX-SYMMETRIC EIGENPROBLEMS

2.1. Introduction

This Section deals with the linear eigenvalue problem

(K− λM) φ = 0 (1)

where K,M ∈ Cm×m are complex-symmetric matrices, KT = K, MT = M, and (λ,φ) is ageneric, complex eigenvalue and eigenvector solution pair. Bold, lower case is used for vectors;bold, upper case is used for matrices. An eigenvalue is generally characterized by the letterλ, thus encompassing both the real-symmetric formulation of Section 3.2, with real, positiveλ ≡ ω2, and the complex-symmetric formulation of Section 3.3, which leads to complex λ ≡ ω.Stiffness and mass matrix polynomials of a variable θ are denoted by K(θ) and M(θ).

Horn and Johnston [20] are the best reference on complex-symmetric matrices, whoseproperties are also explored by Arbenz and Hochstenbach [25] in their development of theJacobi-Davidson method. These two references, together with the classical book by Bathe andWilson for engineers [13], were of invaluable help for the outline of the following sections.

Definition 1 (Inner product for complex-symmetric eigenproblems) Given two com-plex vectors x,y ∈ Cm, and given a complex-symmetric matrix M ∈ Cm×m, one defines theinner product for the complex-symmetric eigenproblem as the indefinite bilinear form

〈x,y〉 = xTMy (2)

Strictly speaking, this is not an inner product definition, as it cannot be guaranteed that〈x,x〉 ≥ 0 for all x and that 〈x,x〉 = 0 only if x = 0, that is, x may be a quasi-null or isotropicvector (see Definition 5.1.3 in Reference [20]).

The eigenproblem of interest given by Equation (1) – as applied to structural dynamics –requires that K and M be simultaneously diagonalizable. In general, a matrix M ∈ Cm×m

is diagonalizable if it has m linearly independent eigenvectors. As complex symmetric, Mis diagonalizable if and only if M = QDQT, where D ∈ Cm×m is a diagonal matrix andQ ∈ Cm×m satisfies QTQ = I, as given in Theorem 4.4.13 of Reference [20]. (K and M arenot necessarily normal, whence Q is generally complex.) Accordingly, all eigenpair solutions(λ, φ) of Equation (1) may be gathered in the pair of matrices (Λ,Φ), where Λ is diagonaland Φ is a non-singular matrix that can be normalized such that

ΦTMΦ = I, ΦTKΦ = Λ, ΦTΦ = D−1 (3)

The use of the inner product Definition 1 is justified in the sense that, if convergence isoccurring in an iterative procedure for the solution of the eigenproblem of Equation (1), thenall approximations u of an eigenvector φ lay in a subspace of Cm for which 〈u,u〉 6= 0 as wellas uTu 6= 0. Even in the case of a real eigenproblem, for which the inner product of Definition1 has no restrictions, convergence of an iterative process is only asymptotical [13].

The following projector definitions will be helpful in the solution of the eigenvalue problem,as outlined in Section 2.2. The notation is in accord with Reference [26].


4 N. A. DUMONT

Definition 2 (Orthogonal projectors)

PMφ =MφφTMφTMMφ

(4)

is the orthogonal projector onto the subspace of Cm spanned by the non-isotropic vector Mφ:

PMφMφ = Mφ (5)

This is consistent with the inner product Definition 1, since, given two vectors x,y ∈ Cm,

〈x,y〉 = 0 ⇔ PMxy = 0 ⇔ PMyx = 0 (6)

As an orthogonal projector, PMφ is both symmetric and idempotent. Moreover,

P(Mφ)⊥ = I−PMφ (7)

Definition 3 (Oblique projectors)

Pφ,(Mφ)⊥ =φφTMφTMφ

(8)

is the oblique projector onto the subspace spanned by φ along the subspace spanned by (Mφ)⊥:

Pφ,(Mφ)⊥φ = φ ⇔ PTφ,(Mφ)⊥Mφ = Mφ (9)

Pφ,(Mφ)⊥ is idempotent and generally non-symmetric (for M 6= I). This definition is alsoconsistent with the inner product Definition 1, as, given two vectors x,y ∈ Cm,

〈x,y〉 = 0 ⇔ Px,(Mx)⊥y = 0 ⇔ Py,(My)⊥x = 0 (10)

Observe that PTφ,(Mφ)⊥

≡ P(Mφ),φ⊥ . Moreover,

PMφPφ,(Mφ)⊥ = PMφ, Pφ,(Mφ)⊥PMφ = Pφ,(Mφ)⊥ (11)

The clause on the normalization of u is not strictly required in the following definition, but isconvenient to simplify notation in the subsequent developments.

Definition 4 (Rayleigh quotient) Let u be a vector normalized according to the innerproduct Definition 1, provided that u is not quasi-null. The Rayleigh quotient is defined as

θ =uTKuuTMu

≡ uTKu (12)

The following Rayleigh quotient iteration [13, 21, 22] is referred to in the outline of the Jacobi-Davidson method of Section 2.2.

Algorithm 1 (Rayleigh quotient iteration) The procedure starts with an estimatedeigenvalue θ1 and an estimated iteration vector u1, normalized according to Definition 1. Defineb1 := Mu1. Then, for k = 1, 2, · · · :

(1) solve for uk+1 in (K− θkM) uk+1 = bk

(2) evaluate uk+1 by normalizing uk+1 according to Definition 1(3) evaluate bk+1 := Muk+1

(4) evaluate θk+1 := uTk+1bk + θk ≡ uT

k+1Kuk+1 as in Definition 4



If convergence occurs, bk+1 → Mφ and θk+1 → λ as k →∞. The iteration is repeated until

| (θk+1 − θk) /θk+1 |≤ tol (13)

provided that θk+1 > 0. If the system matrix in step (1) is singular, then θk is already theeigenvalue one is looking for and uk is the corresponding eigenvector.

Algorithm 1 follows a proposition in Reference [13] for a computationally efficient sequenceof calculations. The critical aspect of the algorithm – namely the solution of the equationsystem of step (1) – is not addressed here. See References [27, 28, 29, 30, 31, 23], for instance,which are mostly devoted to non-linear problems, but also present implementations in termsof inexact algorithms that might increase the overall efficiency of the method.

A right eigenvector of a complex-symmetric eigenproblem is also a left eigenvector. Then,provided that Definition 1 holds, the following theorem may be proved [25].

Theorem 1 (Quadratic approximation of the Rayleigh quotient) If a vector uk ap-proaches the eigenvector φ with error O (δk), then the corresponding Rayleigh quotient θk ofDefinition 4 approaches the eigenvalue λ with error O

(δ2k

).

In the case of a Hermitian eigenproblem, approximation of the Rayleigh quotient is linear.Convergence of the iterative process can be assessed as in the case of real-symmetriceigenproblems [13, 25] and stated, for convenience, in terms of the following theorem.

Theorem 2 (Cubic convergence of the Rayleigh quotient iteration) If a vector uk

approaches the eigenvector φ with error O (δk), then, as obtained in the frame of the Rayleighquotient iteration of Algorithm 1 and according to Theorem 1, uk+1 for k > 1 approaches φwith error O

(δ3k

).

As convergence is attained, K − θkM tends to become singular and the elements of thenon-normalized vector uk+1 in Algorithm 1 tend to become very large numbers. However, ifthe eigenvector φ is well conditioned, the error resulting from the solution of uk+1 is mainly inthe direction engendered by φ, which is the searched direction [32]. In such a case, machine-precision accuracy may be obtained for the numerical results. The process of proving thelatter theorem [13, 25] enables to generalize (for Hermitian eigenproblems and also for k = 1)that, if a vector uk approaches the eigenvector φ with error O (δk) and a value θk approachesthe eigenvalue λ with error O (εk), then uk+1 approaches φ with error O (εkδk). As a result,starting with θ1 equal to an eigenvalue λ within machine precision and with a vector u1 thatis not orthogonal to the searched eigenvector φ, only one iteration step is needed in Algorithm1 to achieve convergence within machine precision [32].

As shown in the following, the equation system given in Algorithm 1 may be rearranged, sothe increasing ill-conditioning of K− θkM is adequately dealt with.

2.2. Jacobi-Davidson method for complex-symmetric eigenproblems

2.2.1. Problem formulation. The Jacobi-Davidson method was introduced in 1994 by Sleijpenand van der Vorst [25, 33, 34] as an extension of Davidson’s method [35]. It is a member of alarge family of recently developed projection algorithms, as the Lanczo’s method, the Arnoldi’smethod and the rational Krylov method, which may be extended to non-linear problems. Afew references are [27, 28, 30, 31, 36, 37, 38]. Comprehensive problem formulation, literaturereview and algorithm outlines are found in References [29, 23, 39].


6 N. A. DUMONT

In the following, the Jacobi-Davidson method will be derived directly from – and willbe shown to be mathematically equivalent to – the Rayleigh quotient iteration, thereforewith asymptotically cubic convergence. Although not strictly original (see Reference [25],for instance), some didactic merit may be claimed for the present outline, particularly asa connection will be established to the Bott-Duffin inverse [26, 40].

Given an approximate eigenpair solution (θk,uk) of the problem introduced in Equation (1),where θk is the Rayleigh quotient of Definition 4, one may write the improved, non-normalized,solution of uk+1 in step (1) of Algorithm 1 as

uk+1 = (uk + δuk)βk (14)

where βk is a scaling factor – actually, an increasingly large number as convergence is attained.The vector increment δuk is necessarily orthogonal to uk in terms of Definition 1,

〈uk, δuk〉 = 0 (15)

and presents cubic convergence to zero in the frame of Theorem 2.The solution for uk+1 in Algorithm 1 is equivalent to solving the following restricted system

of equations for δuk, given Equation (15) and the orthogonal projector of Definition 2:

(K− θkM) δuk −Muk/βk = − (K− θkM)uk

restricted to PMukδuk = 0

given that PMukMuk = Muk

(16)

with subsequent evaluation of uk+1 from Equation (14).

2.2.2. Solution in terms of orthogonal projector. Assuming that (K− θkM)P(Muk)⊥+PMuk

is non-singular, the solution of Equation (16) may be uniquely expressed as the solution of thefollowing equation for the auxiliary vector δyk:

[(K− θkM)P(Muk)⊥ + PMuk

]δyk = − (K− θkM)uk (17)

with subsequent evaluation of δuk and βk as

δuk = P(Muk)⊥δyk, βk = − (uT

k PMukδyk

)−1(18)

In this procedure, one recognizes

A(−1)(Muk)⊥

= P(Muk)⊥

[(K− θkM)P(Muk)⊥ + PMuk

]−1 (19)

as the Bott-Duffin inverse of the restricted Equation (16) [26, 40].The system matrix of Equation (17) is well conditioned for θk converging to a simple

eigenvalue λ, in which case (K− θkM)P(Muk)⊥ and PMukare by construction complementary

matrices. In the case of multiple eigenvalues, the system matrix of Equation (17) tends tobecome ill conditioned. Nevertheless, δuk still tends cubically to zero, as the error resultingfrom the ill-conditioning is mainly in the direction engendered by φ, as one obtains fromChatelin’s proof for the Rayleigh quotient iteration [32]. In particular, if the starting value θ1

is equal (within machine precision) to a multiple eigenvalue λ, only one iteration is requiredfor convergence, provided that u1 is not orthogonal to the searched eigenvector φ.

One-step convergence is illustrated in Table I for the non-linear eigenproblem of Example2, as solved according to Algorithm 2, which is a generalization of the scheme proposed in



this Section. For large eigenproblems, round-off errors in the solution of the equation systemmake the occurrence of multiple eigenvalues within machine precision unlikely. However, theexistence of close eigenvalues generally contributes to the decrease of the number of iterationsrequired in the sequential solution of an eigenvalue problem. This is illustrated both in themodified case of Example 2 and in Example 5.

The development above was derived directly from the Rayleigh quotient iteration. However,this is not the only way of dealing with the present linear algebra problem.

2.2.3. Solution in terms of oblique projectors. Arbenz and Hochstenbach [25] outline asolution of δuk in the frame of a formulation that is conceptually equivalent to Equation(16), although contextually different, as it relies on the residual

rk = (K− θkM)uk (20)

which has the propertyuT

k rk = 0 (21)

for the Rayleigh quotient θk given as in Definition 4. This formulation is integrated into thefollowing elaboration of the results of Section 2.2.2, which makes use of the oblique projectorPuk,(Muk)⊥ ≡ ukuT

k M introduced in Definition 3, for 〈uk,uk〉 = 1.

Pre-multiplying both sides of the equation in step (1) of Algorithm 1 by(I−Puk,(Muk)⊥

)T

one obtains the consistency equation(I−Puk,(Muk)⊥

)T (K− θkM) uk+1 = 0 (22)

SincePMuk

δuk = 0 ⇔ Puk,(Muk)⊥δuk = 0 (23)

and in view of Equations (20) and (21), an expression equivalent to Equation (16) is (

I−Puk,(Muk)⊥

)T (K− θkM) δuk = − (K− θkM)uk

restricted to Puk,(Muk)⊥δuk = 0(24)

and δuk may be solved directly from[(I−Puk,(Muk)⊥

)T (K− θkM)(I−Puk,(Muk)⊥

)+ PMuk

]δuk = − (K− θkM)uk (25)

as an alternative to Equation (17).Another expression alternative to Equations (17) and (25) is

[(K− θkM)

(I−Puk,(Muk)⊥

)+ PMuk

]δyk = − (K− θkM)uk

δuk =(I−Puk,(Muk)⊥

)Tδyk

(26)

It is worth observing that the residual rk and the increment δuk are related by

δuTk rk = 1/βk (27)

This is obtained by first pre-multiplying both sides of Equation (20) by δuTk , with the result

δuTk rk = δuT

k Kuk, according to Equation (15). On the other hand, pre-multiplying both sidesof Equation (14) by uT

k (K− θkM) gives δuTk Kukβk = 1 after making use of the expression

of uk+1 in Algorithm 1 and of Equations (21) and (15). Since both rk and δuk have cubicconvergence to a zero vector as uk → φ, 1/βk converges cubically to zero with approximatelydouble the number of exact digits – see remark after Theorem 2. As a result, |1/βk| might beused as a convergence threshold, instead of Equation (13), which only applies for λ 6= 0.


8 N. A. DUMONT

2.3. On the numerical implementation of the Jacobi-Davidson method

The vector uk+1 obtained by normalizing uk + δuk, according to Equation (14), is unique(except for a plus/minus sign), whether δuk has been evaluated from Equations (17), (25)or (26), which is in either case mathematically equivalent to using the Rayleigh quotientiteration, Algorithm 1, as convergence is asymptotically cubic. The Jacobi-Davidson method,as formulated in terms of the vector increment δuk, is preferable for two main reasons: (a)the equation system is better conditioned; (b) the search subspace for δuk represented byI−P(Muk)⊥ may be further restricted with no lack of mathematical rigor, thus enabling eithersequential or simultaneous evaluation of a set of eigenpairs. The designation Jacobi refers tosubspace iteration [25, 33].

The projector PMukin Equations (17), (25) or (26) may be multiplied by any scalar γ 6= 0,

with no influence in the mathematical results. In fact, such a multiplication is advisable, asdone in step (4.1) of Algorithm 2, so the summands have approximately the same order ofmagnitude and round-off errors is minimized.

Once evaluated within the required tolerance, an eigenpair might in principle be removedfrom the original eigenproblem by matrix deflation [13, 20]. However, this introduces numericalerrors in the original system and is hardly applicable to non-linear eigenproblems. Analternative is to solve the problem sequentially, for the eigenpairs of a subspace of interest.

The algorithm of Section 3.5 outlines the numerical implementation of the Jacobi-Davidsonmethod in terms of the orthogonal projector of Section 2.2.2, for the evaluation of the completeset (Λ,Φ). The search of each eigenpair (λ, φ) starts with a first estimate u1 orthogonal toall previously evaluated eigenvectors. The incremental vector δuk is then evaluated iteratively,according to Equations (17) and (18), but using in place of PMuk

an orthogonal projectorthat encompasses all eigenvectors that have been already evaluated, besides uk. Algorithm 2is intended for non-linear problems, with mass and stiffness matrices given as functions of theeigenvalues, as detailed in Sections 3.2 and 3.3.

3. FORMULATION OF NON-LINEAR, COMPLEX-SYMMETRIC EIGENPROBLEMSOF THE STRUCTURAL ANALYSIS

3.1. Introduction

The dynamics problem subject of this paper is formulated in the frequency domain in termsof a generalized effective stiffness matrix Keff(ω) given as the frequency power series

Keff(ω) = K0−iωC1−ω2M1−iω3C2−ω4M2−· · ·−iω2n−1Cn−ω2nMn+ O(ω2n+1

)(28)

truncated after 2n + 1 terms, compactly expressed as

Keff(ω) = K0 −n∑

j=1

(iω2j−1Cj + ω2jMj

)+ O

(ω2n+1

)(29)

where K0 ∈ Rm×m is the stiffness matrix of the static case, Cj ,Mj ∈ Rm×m are generalizeddamping and mass matrices, and i =

√−1 is the imaginary number. Thus, Keff ∈ Cm×m

in Equation (29) is a complex-symmetric matrix for a problem with m degrees of freedom.Basis for this formulation is a frequency-dependent development of stiffness and mass matrices



proposed by Przemieniecki [9] in terms of a displacement approach for the free-vibrationanalysis of damping-free truss and beam structures, thus without the imaginary terms. Heobtained the real-symmetric, effective stiffness matrix Keff(ω2) of an element in the shape

Keff(ω2) = K0 − ω2M0 − ω4 (M2 −K4)− ω6 (M4 −K6) + O(ω8) (30)

which gave rise to a dynamic finite element formulation [27, 39, 41, 42], a not quite adequatedenomination, as it was only applied to free-vibration problems. It is worth remarking thatEquation (30) is sometimes represented as [27, 39]

Keff(ω2) = K0 − ω2 (M0 −K2)− ω4 (M2 −K4)− ω6 (M4 −K6) + O(ω8) (31)

with a coefficient matrix K2 in the term that multiplies ω2, probably as a consequenceof having been obtained in the frame of a displacement formulation that does not seekdynamic equilibrium satisfaction inside the finite element. However, this term is void in anyvariationally-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 termsup to ω2: Keff(ω2) = K0−ω2M0+O(ω4) [9, 13, 14, 43, 44]. Except for M0, which correspondsto M1 in the present paper, the mass matrix terms given in Equations (30) and (31) differ inphysical meaning from the ones introduced in Equation (29).

The simplest illustration of this problem is the stiffness matrix of a truss element with twodegrees of freedom, as given in Example 1, which shows that the matrix terms Cj ,Mj , j > 1,when consistently obtained, contain damping, mass and stiffness contributions – in fact,the first mass matrix, M1, is already affected by viscous damping. In spite of the mixednature of Cj and Mj , they will be referred to as generalized damping and mass matrices.Equation (88) illustrates that, if the variational formulation makes use of fundamental solutionsgiven as real functions of complex arguments, the effective matrix Keff(ω) turns out to becomplex symmetric. Inherently complex formulations such as in References [45, 46] (the latterformulation is also non-variational) do not lead directly to complex-symmetric problems.

The linear algebra properties of these matrix terms are outlined in the followingdevelopments, which omit, for simplicity, the truncation error orders O(ω2n+1) and O(ω2n+2)(for complex and real problems). Observe the notation Keff(ω) as in Equation (29) for thecomplex-symmetric problem, and Keff(ω2) as in Equation (30) for the real-symmetric problem.One can always infer from context whether real or complex matrices are being dealt with.

This paper is concerned with the non-linear eigenvalue problem associated to Equation (29):

Keff(ω)φ ≡K0 −

n∑

j=1

(iω2j−1Cj + ω2jMj

) φ = 0 (32)

In order to adequately solve the complex eigenproblem of practical interest, it is advisableto start with the outline of the underlying real problem.

3.2. Non-linear, real-symmetric eigenproblems

3.2.1. Formulation. Equation (32), particularized to damping-free problems and writtenwith λ in place of ω2, for notation simplicity,

Keff(λ)φ ≡K0 −

n∑

j=1

λjMj

φ = 0 (33)


10 N. A. DUMONT

may be restated as the linearized, augmented set of equations [6],

K0 0 0 · · · 00 M2 M3 · · · Mn

0 M3 M4 · · · 0...

......

. . ....

0 Mn 0 · · · 0

− λ

M1 M2 M3 · · · Mn

M2 M3 · · · · · · 0

M3

.... . . · · · 0

......

.... . .

...Mn 0 0 · · · 0

φ0

φ1

φ2...

φn−1

= 0 (34)

whereφj = φ λj , j = 0, · · · , n− 1 (35)

The matrix product corresponding to the first row in Equation (34) is exactly the power seriesexpansion of Equation (33), with the remaining equations vanishing identically.

Equation (34) is a generalization of Duncan’s original proposition [47], as it is valid for anynumber n of matrix terms and is disposed in such a way that the augmented stiffness and massmatrices grow from left to right and from top down, while preserving symmetry. A similarstructure for complex-symmetric matrices is proposed in Section 3.3. There is an extensiveliterature on matrix arrangements similar to Equation (34), called matrix pencils [20, 48]. Asoutlined in the following, Equation (34) is key to the theoretical argumentation, but is notrequired for an algorithm implementation.

The linearized, augmented eigenproblem of Equation (34), compactly expressed as

(Kaug − λMaug)φaug = 0 (36)

leads to an enlarged set with m × n solution eigenpairs, which are in part complex evenin the case of a damping-free problem, as the augmented stiffness and mass matrices arenot positive definite. However, only m real eigensolutions

(λ, φaug

), corresponding to real

eigenpairs (λ, φ) ≡ (ω2,φ

), will be solutions of Equation (33) of practical interest in a vibration

or transient analysis. This is discussed and justified in the next Sections.Given a non-normalized eigenvector φaug, the corresponding normalized eigenvector φaug

of the augmented, linearized system is obtained as

φaug = φaug

(φ

T

augMaugφaug

)−1/2

(37)

in such a way thatφT

augMaugφaug = 1 φTaugKaugφaug = λ (38)

assuming that Maug works as positive definite for φaug, according to Definition 1.From the first matrix row in Equation (34), which is actually the only one of interest, one

infers that Equation (33) may be alternatively expressed as

Keff(λ)φ ≡ (K(λ) − λM(λ)

)φ = 0 (39)

in terms of frequency-dependent mass and stiffness matrices M(λ) and K(λ) defined as

M(λ) =n∑

j=1

jλj−1Mj = M1 + 2λM2 + 3λ2M3 + · · · (40)



K(λ) = K0 +n∑

j=2

(j − 1)λjMj = K0 + λ2M2 + 2λ3M3 + · · · (41)

According to Equation (37) for the augmented, linearized eigenproblem, one obtains for theeigenvectors of interest φ0 ≡ φ in Equation (39) that, given a non-normalized eigenvector φ,the corresponding normalized eigenvector φ is obtained as

φ = φ(φ

TM(λ)φ

)−1/2

(42)

in such a way thatφTM(λ)φ = 1 φTK(λ)φ = λ (43)

M(λ) and K(λ) must be positive definite and semidefinite, respectively, which depends on thephysical meaningfulness of the mathematical model, as outlined in Section 3.2.2.

Given two normalized eigenvectors φr and φs,

φTr M(λr,λs)φs = δrs (44)

where M(λr,λs) is the generalization of the frequency-dependent mass matrix of Equation (40)as referred to the eigenvalues λr, λs:

M(λr,λs) =n∑

j=1

j∑k=1

λk−1r λj−k

s Mj

≡ M1 + (λr + λs)M2 +(λ2

r + λrλs + λ2s

)M3 + · · ·

(45)

If all eigenpair solutions are gathered in the pair of matrices (Λ,Φ), where Λ is diagonal andΦ is the set of eigenvectors normalized according to Equation (42), one obtains for Equations(43) and (44), as a generalization of Equation (3) for the present non-linear problem [10]:

n∑

j=1

j∑

k=1

Λk−1ΦTMjΦΛj−k = I, ΦTK0Φ +n∑

j=2

j−1∑

k=1

ΛkΦTMjΦΛj−k = Λ (46)

3.2.2. Consistency of the non-linear, real-symmetric formulation. Equation (39) isequivalent to the original Equation (33), only rearranged in a convenient way in terms ofeigenvalue-dependent stiffness and mass matrices. Equation (34) – for an augmented, linearizedproblem – was laid out as an intermediary step between Equations (33) and (39). All threeequations lead to the same set of m × n eigensolutions, from which only a subset of m realeigensolutions are of interest in a practical application. The orthogonality statement 〈φr,φs〉arrived at in Equation (44) is consistent with both Definition 1 and the generalized innerproduct definition of Reference [17] for non-linear problems.

There are some key issues to be discussed about the present outline. The first one isformulated as the following Theorem.

Theorem 3 (Equivalence of Equations (34) and (39)) Equations (34) and (39) havethe same set of m× n solution pairs (λ, φ0) ≡ (λ, φ).

Proof: Equation (39), a non-linear system of degree n in λ and with coefficient matricesof order m, has exactly m× n generally complex solution eigenpairs (λ, φ), since thecharacteristic polynomial det

(Keff(λ)

)has m× n roots λ. Every solution of Equation (39)

satisfies Equation (34) identically, and this is a linear eigensystem in λ of order m× n.


12 N. A. DUMONT

The following Theorem, adapted from Observation 7.1.3 by Horn and Johnson [20], is ofinterest for the forthcoming developments.

Theorem 4 (Positivity of mass and stiffness matrices) M(λ) in Equation (40) ispositive definite for all nonnegative λ if and only if all matrices Mj , j = 1, · · · , n are positivedefinite. Moreover, given all Mj , j = 1, · · · , n as positive definite, K(λ) in Equation (41) ispositive definite or semidefinite if and only if K0 is positive definite or semidefinite.

A last concern are number and properties of the real solution eigenpairs one is able to obtainin any of the equivalent Equations (34) or (39).

3.2.3. Eigenvalue properties of the non-linear, real-symmetric formulation. The conclusionsof interest might be drawn from the minmax characterization of eigenvalues laid out by Hadelerfor non-linear eigenproblems [17, 18, 19]. The following independent development addressesthe non-linear eigenproblem as formulated in Equations (39-41), for n > 1. One starts byproving the following Theorem, in which K ←

kMk defines a matrix whose kth column is the

column Mk of M and whose remaining columns coincide with those of K [20].

Theorem 5 (Determinant derivatives and positive definite matrices) Let K,M ∈Cm×m be Hermitian matrices with constant elements, and suppose that K is positivesemidefinite and M is positive definite. Then,

m∑

k=1

det(K ←

kMk

)> 0 (47)

Proof: K + Mx is positive semidefinite for all x ≥ 0, according to Theorem 4. Also,K + Mx + M∆x is positive definite for all x ≥ 0 and a positive increment ∆x. Since M∆x =(K + Mx + M∆x) – (K + Mx) is positive definite, it follows from Corollary 7.7.4(b) byHorn and Johnson [20] that

det (K + Mx + M∆x) > det (K + Mx) (48)

As a result, the following determinant derivative [48] is positive for all x ≥ 0:

ddx

det (K + Mx) =m∑

k=1

det(K + Mx ←

kMk

)

= lim∆x→0

det (K + Mx + M∆x)− det (K + Mx)∆x

> 0 (49)

which proves the theorem when evaluated for x = 0.

The developments above apply to a negative matrix K, as well, by just replacingK with −K. Since det (−K) = (−1)m det (K), Equation (47) of Theorem 5 reads

(−1)m−1∑m

k=1 det(K ←

kMk

)> 0 in the case of a negative K (as there are m− 1 columns

of K in the summand). The proof given in Reference [20] for the item (b) of Corollary 7.7.4is also meant to apply to the case of the difference of two positive definite matrices A and Bbeing positive semidefinite, leading to det (A) ≥ det (B). However, this apparently involves a



mistake and the corollary is actually only applicable, except for the trivial case A ≡ B, in thecase of a positive definite matrix difference, as needed for arriving at Equation (48).

The eigenvalues λ of Equation (33) are the roots of

det(K(λ) − λM(λ)

)= 0 (50)

which has m×n complex solutions. However, if one is only interested in the subset of the realeigenvalue solutions, Equation (50) is more conveniently stated as the restricted linear system

det

(K(θ) − λM(θ)

)= 0 for all θ ≥ 0

such that λ = θ(51)

On the other hand, the set of isocurves

det(Keff(θ,λ)

) ≡ det(K(θ) − λM(θ)

)= C (52)

(with the simplifying notation Keff(θ,λ) used only in the following outline) may be stated inthe Cartesian system (θ, λ) as

∂

∂θdet

(Keff(θ,λ)

)+

∂

∂λdet

(Keff(θ,λ)

) dλ

dθ= 0 (53)

The determinant derivatives are conveniently expressed as

∂

∂θdet

(Keff(θ,λ)

)=

m∑

k=1

det(Keff(θ,λ) ←

k

∂

∂θKeff(θ,λ)k

)(54)

∂

∂λdet

(Keff(θ,λ)

)=

m∑

k=1

det(Keff(θ,λ) ←

k

∂

∂λKeff(θ,λ)k

)(55)

according to the notation introduced in Equation (47). The matrix derivatives needed inEquations (54) and (55) are obtained from Equations (40) and (41):

∂

∂θKeff(θ,λ) = (θ − λ)

n∑

j=2

j (j − 1) θj−2Mj = (θ − λ) (2M2 + 6θM3 + · · · ) (56)

∂

∂λKeff(θ,λ) = −M(θ) (57)

Then,

∂

∂θdet

(Keff(θ,λ)

)= (θ − λ)

m∑

k=1

det

Keff(θ,λ) ←

k

n∑

j=2

j (j − 1) θj−2Mjk

(58)

∂

∂λdet

(Keff(θ,λ)

)= −

m∑

k=1

det(Keff(θ,λ) ←

kM(θ)k

)(59)

Suppose Keff(θ,λ) is positive semidefinite, that is, C ≥ 0 in Equation (52). Then, for θ ≥ 0the sums given in Equations (58) and (59) are positive, according to Theorem 5, and it followsfrom Equation (53) that, necessarily,

signum(

dλ

dθ

)= signum (θ − λ) (60)


14 N. A. DUMONT

In the case of negative definite Keff(θ,λ), that is, C < 0 in Equation (52), Equation (60) stillholds true, according to the remark made after the proof of Theorem 5.

According to Equation (60), the eigensolutions λj(θ), j = 1, · · · ,m of Equation (51),interceptions with the straight line λ = θ, are determined as unique minimum values foreach isocurve λj(θ).

The developments above are illustrated in Figure 3 for Example 2, with a set of m = 12curves, showing the interception points with the line λ = θ as the eigenvalue solutions ofthe non-linear problem of Equation (50). The interceptions of the curves λj(θ) with the axisθ = 0 are the solutions of the linear problem given in Equation (51). As θ increases, each λj(θ)decreases continually until a minimum value, and then increases indefinitely again, according toEquation (60). Multiple eigenvalues cannot be perceived in a 2D plot as in Figure 3. Only a 3Danalysis, with a third axis corresponding to det

(K(θ) − λM(θ)

), would enable the visualization

of the effect of multiple eigenvalues, for the surface given by Equation (52) cut by the plane(θ, λ, 0). All curves λj(θ), j = 1, · · · ,m, converge to one single value, as, from Equation (50)together with Equations (40) and (41),

limθ→∞

λj

θ=

n− 1n

for all j (61)

The issues of intercepting or coinciding curves λj(θ) do not deserve a closer examination, as theimplemented iterative procedure follows a path given by the eigenpair (λ, φ) that is uniquelydetermined even in the case of multiple eigenvalues. It is worth observing in Figure 3 the caseof curves intercepting in the range between θ = 0 and λ = θ, for eigenvalues λ10 ≡ λ11 and λ12,more clearly seen by comparing the columns for n = 1 and n = 3 in Table I. Bifurcations cannotoccur in the present context. However, it is possible that two curves intercept at θ = 0, for K0

and M1 corresponding to a topological symmetry of the mathematical model, if, owing to someinappropriateness or numerical error in their evaluation, one or more of the remaining matricesMj in Equations (40) and (41), while still positive definite, fail to reflect the symmetry. Thisis illustrated in the modified case of Example 2.

The relevant conclusions of these developments are summarized in the following theorem.

Theorem 6 (Consistency of the non-linear symmetric eigenproblem) Given the non-linear, real-symmetric eigenvalue problem of Equation (50), where M(λ) and K(λ) are definedas in Equations (40) and (41), for K0 positive semidefinite and Mj , j = 1, · · · , n positivedefinite, (a) there are m real, non-negative eigenvalues and n ×m −m complex eigenvalues;(b) the non-zero real eigenvalues of the non-linear problem are smaller than the correspondingeigenvalues of the underlying linear problem.

Proof: The proof has already been given above. However, it is worth recalling that thefirst of Equation (51) admits exactly m sets λj(θ) of curves (including repeated eigenvalues),each one starting from λj(0) and decreasing until the global minimum value λj(θ) = θj isreached, which coincides with the restriction given by the second of Equation (51), andincreasing indefinitely again, according to Equation (61). The trivial linear case n = 1 isincluded in the theorem.

It may happen that some matrix Mj – whose elements usually decrease in magnitude as jincreases – are actually non-positive, as a consequence of some conceptual inappropriateness[see the remark after Equation (31)] or of round-off errors. Then, it will be impossible to



evaluate m eigenpairs as real solutions. In practice, solutions related to higher eigenvalues willbe missing, in such a case, as the relative contribution of the terms affected by the matrices Mj

increases with higher subscripts j. Then, although accuracy is expected to generally increasewith the number of power series terms, the quality of the evaluated eigenpairs is compromisedif inaccurate higher-order matrix terms are included.

3.3. Non-linear, complex-symmetric eigenproblems

3.3.1. Formulation. Although apparently repeated, the following developments are morethan a mere extension of the ones of Section 3.2. Equation (32) may also be expressed as alinearized, augmented set of equations, for φj = φωj , j = 0, · · · , n− 1 [10],

K0 0 0 0 · · · 00 M1 iC2 M2 · · · Mn

0 iC2 M2 iC3 · · · 0

0 M2 iC3. . . · · · 0

......

......

. . ....

0 Mn 0 0 · · · 0

− ω

iC1 M1 iC2 M2 · · · Mn

M1 iC2 M2 iC3 · · · 0iC2 M2 iC3 · · · · · · 0

M2 iC3

.... . . · · · 0

......

......

. . ....

Mn 0 0 0 · · · 0

φ0

φ1

φ2

φ3...

φn−1

= 0 (62)

This augmented complex-symmetric eigenproblem leads to an enlarged set of eigensolutionswith 2 × n × m mathematically possible solutions. However, the subvector solutions φ0 ≡φ of immediate interest correspond to a basic set of m eigenpairs (ω, φ). Section 3.3.2extends a generic solution (ω, φ) to its complex-associated eigenpair

(−ω, iφ), in the case

of underdamping. The case of overdamping is briefly assessed in Section 3.3.3.From the first matrix row in Equation (62), which is the only one of interest, one infers that

Equation (32) may be alternatively expressed as

Keff(ω)φ ≡ (K(ω) − ωM(ω)

)φ = 0 (63)

in terms of frequency-dependent mass and stiffness matrices M(ω) and K(ω):

M(ω) =n∑

j=1

(i(2j − 1)ω2j−2Cj + 2jω2j−1Mj

)

≡ iC1 + 2ωM1 + 3iω2C2 + 4ω3M2 + · · ·(64)

K(ω) = K0 +n∑

j=1

(i(2j − 2)ω2j−1Cj + (2j − 1)ω2jMj

)

≡ K0 + ω2M1 + 2iω3C2 + 3ω4M2 + · · ·(65)

Observe that, while Keff(λ≡ω2) in Equation (39) is a particular case of Keff(ω) in Equation(63), the same cannot be said about M(λ≡ω2) and K(λ≡ω2) in Equations (40) and (41) ascompared with M(ω) and K(ω) in Equations (64) and (65).

According to Equation (62) for the augmented, linearized eigenproblem, one normalizes aneigenvector φ ≡ φ0 of the non-linear problem, Equation (63), by proceeding as in Equation(42). However, this is only feasible if φTM(ω)φ 6= 0. For an exact solution eigenpair (ω, φ),such a condition is a premise that the proposed eigenproblem has a physical meaning.


16 N. A. DUMONT

Given two normalized eigenvectors φr and φs, the orthogonality expression of Equation (44)is valid for eigenvalues ωr, ωs and the frequency-dependent mass matrix of Equation (64):

M(ωr,ωs) =n∑

j=1

(i2j−2∑k=0

ωkr ω2j−k−2

s Cj +2j−1∑k=0

ωkr ω2j−k−1

s Mj

)

= iC1 + (ωr+ωs)M1 + i(ω2

r +ωrωs+ω2s

)C2 +

(ω3

r +ω2rωs+ωrω

2s +ω3

s

)M2 + · · ·

(66)

Matrix expressions similar to Equation (46) may be obtained for the complex case bygathering all eigenpair solutions in the pair of matrices (Ω,Φ) [10]:

n∑j=1

(i

2j∑k=2

Ωk−2ΦTCjΦΩ2j−k +2j∑

k=1

Ωk−1ΦTMjΦΩ2j−k

)= I

ΦTK0Φ +n∑

j=1

(i2j−1∑k=2

Ωk−1ΦTCjΦΩ2j−k +2j−1∑k=1

ΩkΦTMjΦΩ2j−k

)= Ω

(67)

3.3.2. Complex-associated eigenpairs in the general case of underdamping. The followingtheorem states that to every complex eigenpair (ω, φ) corresponds an eigenpair (−ω, iφ), asrequired in the modal analysis, which actually deals with real transient problems [10]. Thistheorem, although generally valid, is of practical relevance only in the case of underdamping.

Theorem 7 (Complex-associated solutions) If the complex eigenpair (ω, φ) is a solutionof Equation (32), normalized as in Equation (43) for M(ω) and K(ω) given in Equations (64)and (65), then the eigenpair (−ω, iφ) is also a normalized solution. Moreover, φr ≡ φ andφs ≡ iφ are orthogonal eigenvectors in the sense of the inner product of Equation (44) for thegeneralized mass matrix M(ωr,ωs) of Equation (66).

Proof: Only for the scope of this proof, write ω = r (cos θ + i sin θ), where r = |ω| is themodulus and θ is the amplitude of the complex number ω. According to Moivre’s theorem,ωj = rj (cos jθ + i sin jθ). Moreover, ω = r (cos θ − i sin θ) and(−ω)j = (−r)j (cos jθ − i sin jθ). Then, from the definitions of M(ω) and K(ω) in Equations(64) and (65), ωM(ω) ≡ −ωM(−ω), K(ω) ≡ K(−ω). As a result,

(K(ω) − ωM(ω)

)φ = 0 ⇔ (

K(ω) − ωM(ω)

)φ = 0 ⇔ (

K(−ω) − ωM(−ω)

)iφ = 0 (68)

Similarly,

φTM(ω)φ = 1 ⇔ iφTM(−ω) iφ = 1 (69)

φTK(ω)φ = ω ⇔ iφTK(−ω) iφ = −ω (70)

Also, it follows from the expression of M(ωr,ωs) in Equation (66) that

φTr M(ωr,ωs)φs = δrs ⇔ iφ

T

r M(−ωr,−ωs)iφs = δrs (71)

Finally, since (ω, φ) and (−ω, iφ) are solution eigenpairs of Equation (63), there must be bydefinition corresponding augmented, orthogonal eigenpairs of the augmented, linearizedeigenproblem of Equation (62). This proves that φTM(ω,−ω)iφ = 0.



3.3.3. Overdamping. The concepts of underdamping, critical damping and overdamping ina transient analysis are briefly introduced by Meirovitch [49]. The subject is dealt with in moredetail by Kolousek [43] and Warburton [44] in the frame of the dynamic analysis of engineeringstructures. In mathematics, the term ”overdamping” is used in a different context [50, 19, 51].

An undamped problem of structural dynamics has m real eigenpair solutions(ω2,φ

), as

outlined in Section 3.2, where the positive square root ω is the eigenfrequency of interest.As damping takes place, the solution evolves into m sets of complex eigeinpairs (ω, φ)and

(−ω, iφ), according to Theorem 7, with decreasing contribution of the real part and

increasing contribution of the (negative) imaginary part of ω. For a first critical dampingvalue, the associated complex eigenpairs corresponding to the smallest eigenvalues coalesceinto a single one, when ω becomes negative imaginary and φ may be made real. By increasingdamping the eigensolution branches again, with one of the imaginary eigenvalues tendingto zero and the other one tending to minus infinity. The eigenvectors remain real, butinassociated. (The denomination inassociated – in the case of overdamping – is used as opposedto the denomination complex-associated introduced for underdamping in Section 3.3.3.) Theeigenfrequency evolution described above may be followed analytically for the simple trusselement of Example 1 by solving Equation (87) for ω and then plotting the solutions forincreasing values of ζ, given a sequence of wave numbers k = (2j − 1)π/2/`, j = 1, 2, · · · . Asdamping increases, the described overdamping effect gradually affects all the eigenfrequencies.

Assume that an eigenvalue solution of Equation (63) is negative imaginary. Then, one mayreplace ω with −iλ in Equations (63), (64) and (65), where λ is real positive, obtaining

(K(λ) − λM(λ)

)φ = 0 (72)

with a new definition of real-symmetric, generalized mass and stiffness matrices:

M(λ) =n∑

j=1

(−1)j(−(2j − 1)λ2j−2Cj + 2jλ2j−1Mj

)

≡ C1 − 2λM1 − 3λ2C2 + 4λ3M2 + · · ·(73)

K(λ) = K0 +n∑

j=1

(−1)j(−(2j − 2)λ2j−1Cj + (2j − 1)λ2jMj

)

≡ K0 − λ2M1 − 2λ3C2 + 3λ4M2 + · · ·(74)

The assumption λ ≥ 0 is consistent only if M(λ) is positive definite and K(λ) is positivesemidefinite, in which case φ can be made real. Not coincidentally, the eigenfrequency evolutiondescribed in the second paragraph of this Section resembles, for ω substituted with −iλ,the reverse of the bifurcation process that takes place in a structural stability problem [52].However, no matter how interesting these developments may look like from the mathematicalpoint of view, their applicability in terms of algorithm implementation according to Section 3.2remains to be demonstrated. In fact, convergence is unlikely to occur in the present framework,as an eigenvalue corresponding to overdamping may be of larger absolute value than some othereigenvalues corresponding to underdamped behavior. This is illustrated in Example 4.

On the other hand, the algorithm implemented for non-linear, complex-symmetriceigenproblems, as outlined in Section 3.5, has proved suited for the automatic evaluation of awhole set of m eigenpairs (ω, φ) unregarded underdamping, critical damping or overdampingeffects. As illustrated in Example 4 (for a problem with multiple eigenvalues and multiple


18 N. A. DUMONT

overdamping), the remaining set of inassociated eigenpairs can be obtained in a secondcomputer run by entering a new set of estimates. Although this last step may involve some trialand error, problems with overdamping are quite infrequent and, when existent, usually affectonly a few eigenpairs. In the present theoretical frame, the unlikely case of critical damping,when two eigenpairs coalesce into a single one (within machine precision), can only be inferredfrom the impossibility of finding a complementary (inassociated) solution.

3.4. Perturbation analysis

As given in Section 3.2, among all n × m solution eigenpairs (λ ≡ ω2, φ) of the non-linear,real-symmetric problem, only the subset of m real eigenpairs is of interest in an engineeringapplication, as there are m degrees of freedom in the discrete mathematical model. In the caseof a complex formulation, on the other hand, since all 2×n×m solutions (ω, φ) are complex,a perturbation analysis of the underlying real formulation is required in order to tell whichsubset of 2×m complex eigenpairs is of actual interest, as their mathematical patterns cannotdiffer too much from those of a corresponding real eigenproblem.

Although established for non-linear problems, both Equations (39) and (63) can be rewritten,for use in an iterative procedure, as the stepwise linear eigenproblem

(K(θk) − λkM(θk)

)φk = 0 (75)

where θk is the eigenvalue estimate for the kth step and the eigenpair (λk,φk) is thecorresponding exact solution. The matrices M(θk) and K(θk) do not depend on λk, as theyare evaluated at each step for either λ ≡ θk according to Equations (40) and (41), for thereal problem, or ω ≡ θk according to Equations (64) and (65), for the complex problem. As aresult, all definitions and theorems of Section 2 for the solution of the linear eigenproblem ofEquation (1) apply directly, if one just replaces λ, φ, M and K with λk, φk, M(θk) and K(θk).

It must be shown that the solution of Equation (75) corresponds to one step toward theactual solution of the non-linear problem given by either Equation (39) or (63):

(K(λ) − λM(λ)

)φ = 0 (76)

where the eigenpair (λ,φ) is the exact solution one is looking for. The scope of a perturbationanalysis is to investigate in what amount changes K(λ)−K(θk) and M(λ)−M(θk) in the stiffnessand mass matrices, as applied to the present problem, affect the eigenvalue solution [20].

Given the mass and stiffness definitions of Equations (40) and (41), for the real-symmetricproblem, or of Equations (64) and (65), for the complex-symmetric problem, one may writefrom Equations (75) and (76) the perturbation equation

K(λ) − λM(λ) = K(θk) − λM(θk) − (λ− θk)2 δM(θk) (77)

where, for real-symmetric and complex-symmetric problems, respectively,

δM(θk) =n−1∑j=1

Mj+1

j∑l=1

lλj−lθl−1k

≡ M2 + (2θk + λ)M3 +(3θ2

k + 2θkλ + λ2)M4 + · · ·

(78)

δM(θk) =n∑

j=1

(i2j−2∑l=1

lλ2j−l−2θl−1k Cj +

2j−1∑l=1

lλ2j−l−1θl−1k Mj

)

≡ M1 + i (2θk + λ)C2 +(3θ2

k + 2θkλ + λ2)M2 + · · ·

(79)



Next, one investigates the product ΦTk

(K(λ) − λM(λ)

)Φk, where K(λ)−λM(λ) from Equation

(76) is singular, and Φk is the m×m non-singular eigenvector matrix of the linear eigenproblemof Equation (75), normalized according to Equation (3) and with corresponding diagonaleigenvalue matrix Λk. One obtains from Equation (77):

ΦTk

(K(λ) − λM(λ)

)Φk = Λk − λI− (λ− θk)2 ΦT

k δM(θk)Φk (80)

If Λk − λI is singular, then λ = θk is the solution of the non-linear problem of Equation (76)and convergence has been achieved. If Λk−λI is non-singular, one may multiply the right-handside of Equation (80) by (Λk − λI)−1, obtaining

(Λk − λI)−1(Λk − λI− (λ− θk)2 ΦT

k δM(θk)Φk

)

= I− (λ− θk)2 (Λk − λI)−1 ΦTk δM(θk)Φk

(81)

Since this is a singular matrix,

1 ≤ (λ− θk)2 ‖ (Λk − λI)−1 ΦTk δM(θk)Φk‖ (82)

for a matrix norm ‖ · ‖ (Corollary 5.6.16 by Horn and Johnson [20]). Taking ‖ · ‖ such that‖ (Λk − λI)−1 ‖ = maximum diagonal value of (Λk − λI)−1, it results that

1 ≤ (λ− θk)2 |λk − λ|−1‖ΦTk ‖ ‖Φk‖ ‖δM(θk)‖ (83)

or|λk − λ| ≤ (λ− θk)2 ‖ΦT

k ‖ ‖Φk‖ ‖δM(θk)‖ (84)

where λk is the closest eigenvalue to the actual solution λ of the non-linear problem of Equation(76) one is attempting to solve. Equation (84) may also be written as

|λk − λ| ≤ (λ− θk)2 κ (Φk) ‖δM(θk)‖ (85)

where κ (Φk) is the condition number of the eigenproblem of Equation (75), referred to matricesK(θk) and M(θk), with respect to the matrix norm ‖ · ‖.

This perturbation analysis is mainly based on Horn and Johnson [20]. A closer investigationof the subject [22, 53, 54, 55] is beyond the scope of the present paper. However, one observesthat, for real-symmetric problems, all matrices are normal; moreover, the difference matrixδM(θk), as given in Equation (78), is positive definite and has elements of much smallermagnitude than M(θk) in Equation (40). In the case of a complex-symmetric problem, onthe other hand, not only are the matrices non-normal but also δM(θk), as given in Equation(79), has elements of magnitude comparable to M(θk) in Equation (64).

The main conclusion from Equation (85) is summarized in the following Theorem.

Theorem 8 (Perturbation Theorem) Let λ be a solution of the non-linear, complex-symmetric eigenproblem outlined in Equation (75) and let θk be an approximation. If K(λ)

and M(λ) are perturbed by an error O (|λ− θk|), then λ is perturbed by an error O(|λ− θk|2

).

Observe that, for the non-linear eigenproblem dealt with in this Section, Theorem 1 is restatedas |λk − θk| = O

(δ2k

). In Equation (85), θk is the Rayleigh quotient corresponding to the kth

step of an iterative procedure, λk is the exact eigenvalue solution of the perturbed problem withmatrices K(θk) and M(θk) – only required for the theoretical developments but not actuallyevaluated – and λ is the actual eigenvalue of the non-linear problem one is attempting to solve.


20 N. A. DUMONT

3.5. Jacobi-Davidson algorithm for non-linear, complex-symmetric eigenproblems

The algorithm outlined in Section 2.2 requires the input of the matrices and control variables:m := number of degrees of freedom (matrices order)n := number of summands in either Equation (32) or (33)K0,Cj ,Mj , j = 1, · · · , nnumber eig := number of eigenpairs of interest (as implemented, number eig = m)max it := maximum number of iterations for each eigenpairtol := convergence thresholdγ := scaling factor to prevent round-off errors (used in step (4.1))

Output is the eigenpair (Λ,Φ), where, for r = 1, · · · , number eig,Λ is a vector of the eigenvalues λr andΦ is a matrix whose columns are the eigenvectors φr

Following auxiliary matrices are needed in the evaluation of the rth eigenvalue problem:Keff(λr) := the effective stiffness matrixbs := a matrix with s = 1, · · · , r base vectorsM(λr,λs) := actually a row matrix, for s = 1, · · · , rrk and δuk := residual and incremental vectors, which may share the same allocation

For simplicity of writing the algorithm below, one uses λr and φr instead of θk and uk,which denote the current eigenvalue and eigenvector estimates of the developments in Sections2 and 3. The non-linear eigenproblem that can be either real symmetric or complex symmetric,with λ corresponding to either ω2 or ω, although two separate subroutines are required in theFORTRAN code, for the sake of adequately handling real and complex variables. A lineareigenproblem may be solved as a particular case, with no loss of computational efficiency.Step (1) produces an eigenvector estimate φr that is orthogonal to the previously evaluatedeigenvectors φs, s = 1, · · · , r−1, in terms of the orthogonal projector P(Mφ)⊥ of Equation (7),and at the same time creates a matrix with s = 1, · · · , r orthogonal base vectors bs that will beused in the Bott-Duffin solution for δuk in Step (4), according to Section 2.2.2. The matricesKeff(λr) and M(λr,λs) are referred to in Sections 2 and 3 as either Equations (39) and (45),for the real eigenproblem, or Equations (63) and (66), for the complex eigenproblem. Whensearching for an inassociated eigenpair, in the case of overdamping, the eigenvalue estimatemust be different from the one given in the algorithm, as illustrated in Example 4. Moreover,although not indicated, it is checked at every step whether φT

r Mθkφr is positive, for a real

eigenproblem, or non-null, for a complex eigenproblem.

Algorithm 2 (Jacobi-Davidson method) For r = 1, · · · , number eig:If this is a real-symmetric eigenproblem:

Input the eigenvector estimate φr := 〈1 1 1 · · · 〉TInput the eigenvalue estimate: If r = 1, then λr := 0, else λr := λr−1

If this is a complex-symmetric eigenproblem:Eigenvector estimate φr := eigenvector of the underlying real-symmetric problem.Eigenvalue estimate λr := eigenfrequency of the underlying real-symmetric problem.

Input conv := 1For k = 1, · · · , max it while conv > tol:

(1) Orthogonalize φr with respect to φs, s = 1, · · · , r − 1:(1.1) For s = 1, · · · , r

(1.1.1) Define bs := M(λr,λs)φs



(1.1.2) For l = 1, · · · , s− 1 project bs := bs − bl

(bT

l bs

)

(1.1.3) Normalize bs := bs/√

bTs bs

(1.2) For s = 1, · · · , r − 1 project φr := φr − bs

(bT

s φr

)(2) Evaluate the effective stiffness matrix Keff(λr) := K(λr) − λrM(λr)

(3) Evaluate the residual vector rk := Keff(λr)φr

(4) Solve for δuk in Keff(λr)δuk = −r such that PM(λs)φsδuk = 0, s = 1, · · · , r:(4.1) For s = 1, · · · , r

modify Keff(λr) := Keff(λr) +(γbs −Keff(λr)bs

)bT

s

(4.2) Solve Keff(λr)δuk = −rk

(4.3) For s = 1, · · · , r project δuk := δuk − bs

(bT

s δuk

)(5) Update φr := φr + δuk

(6) Evaluate the Rayleigh quotient θ :=(φT

r K(λr)φr

)/

(φT

r M(λr)φr

)

(7) Evaluate convergence conv := |θ − λr|/θ

(8) Normalize φr := φr/√

φTr M (λr)φr

(9) Update λr := θ

4. EXAMPLES

4.1. Example 1 – Frequency-domain effective stiffness matrix for a truss element

4.1.1. Problem formulation. To illustrate the central developments of this paper, one presentsthe simplest academic example conceivable, consisting of a truss element of constant cross-section A, length `, elasticity modulus E, specific mass density ρ and viscous damping µ = 2ζρ,submitted to harmonic vibration (Figure 1). The governing differential equation is:

∂2u∗(x)∂x2

+ k2u∗(x) = 0 (86)

wherek2 =

ρ

E(ω2 + 2iζω) (87)

One obtains the effective symmetric stiffness matrix for the truss element of Figure 1, in the

x

1G)b

2GW

1h2h

d1, p1 d2, p2)a

Figure 1. (a) Coordinate system for the stiffness matrix of a truss element; (b) definition of domainΩ, boundaries Γ1 and Γ2 and corresponding cosine directors η1 and η2.

frame of a hybrid finite element formulation [16], as

Keff(ω) =kEA

sin k`

[cos k` −1−1 cos k`

](88)

In the case of a damping-free structure, this is equivalent to

Keff(ω) = K(ω) − ω2M(ω) (89)


22 N. A. DUMONT

where K(ω) and M(ω) (here differently defined than in Section 3) are the split frequency-dependent stiffness and mass matrices obtained by Przemieniecki [9] in the frame of thedisplacement finite element formulation:

K(ω) =kEA

2 sin k`

[k` csc k` + cos k` −1− k` cot k`−1− k` cot k` k` csc k` + cos k`

](90)

M(ω) =kEA

2ω2 sin k`

[k` csc k`− cos k` 1− k` cot k`

1− k` cot k` k` csc k`− cos k`

](91)

These analytical expressions, as derived by Przemieniecki, can by no straightforward meansbe adapted for viscous damping.

The more compact expression of K as an effective stiffness matrix, Equation (88), is notonly simpler but also easier and more convenient to arrive at in the general frame of a hybridfinite element formulation including viscous damping. The expression of Keff(ω) in terms oftranscendental functions of ω is only possible when the finite element boundaries coalesce topoints, as for trusses and beams. The general evaluation of mass and stiffness matrices asseries expansions, according to Section 3.1 and as illustrated in the following, is conceptuallystraightforward and applicable to large finite/boundary element families [6, 12, 16].

4.1.2. Series expansions. The frequency power series expansion of the effective stiffnessmatrix of Equation (88) is, for a damping-free problem,

Keff(ω) =EA

`

[1 −1

−1 1

]− ω2

c2

`2

3`2

6

`2

6`2

3

− ω4

c4

`4

457`4

360

7`4

360`4

45

+ O(ω6) (92)

where c =√

E/ρ is the wave propagation velocity through the elastic medium. Thecorresponding frequency-dependent mass and stiffness matrices, as developed in Equations(40) and (41), coincide with the expansions of Przemieniecki’s Equations (90) and (91):

M(ω) =EA

`

1

c2

`2

3`2

6

`2

6`2

3

+

ω2

c4

2`4

457`4

180

7`4

1802`4

45

+ O(ω6) (93)

K(ω) =EA

`

[1 −1

−1 1

]+

ω4

c4

`4

457`4

360

7`4

360`4

45

+ O(ω6) (94)

In the case of viscous damping, one obtains the frequency power series expansion of theeffective stiffness matrix of Equation (88):

Keff(ω) = EA`

[1 −1

−1 1

]− iω`α

c

[23

13

13

23

]− ω2`2

c2

15−4α2

4515−7α2

90

15−7α2

9015−4α2

45

− iω3`3αc3

4(21−4α2)945

147−31α2

1890

147−31α2

18904(21−4α2)

945

− ω4`4

c4

16α4−120α2+1054725

127α4−930α2+73537800

127α4−930α2+73537800

16α4−120α2+1054725

+O(ω5)

(95)



where ζ is replaced with αc/`, in terms of a nondimensional viscosity parameter α, tosimplify notation. The corresponding expansions of the frequency-dependent mass and stiffnessmatrices, as introduced in Equations (64) and (65), are

M(ω) = EA`

i`α

c

[23

13

13

23

]+ 2ω`2

c2

15−4α2

4515−7α2

90

15−7α2

9015−4α2

45

+ iω2`3α

c3

4(21−4α2)315

147−31α2

630

147−31α2

6304(21−4α2)

315

+ω3`4

c4

64α4−480α2+4204725

127α4−930α2+7359450

127α4−930α2+7359450

64α4−480α2+4204725

+ O(ω5)

(96)

K(ω) = EA`

[1 −1

−1 1

]+ ω2`2

c2

15−4α2

4515−7α2

90

15−7α2

9015−4α2

45

+ iω3`3α

c3

8(21−4α2)945

147−31α2

945

147−31α2

9458(21−4α2)

945

+ω4`4

c4

16α4−120α2+1051575

127α4−930α2+73512600

127α4−930α2+73512600

16α4−120α2+1051575

+ O(ω5)

(97)

All matrix terms Mj in the expansions for the undamped problem are positive definite. Inthe case of damping, however, only C1 is unconditionally positive definite – see Example 3.

4.2. Example 2 – Non-linear eigenproblem for a two-dimensional transient heat conductionin a homogeneous square plate

Figure 2 represents a square domain for a homogeneous heat conduction problem with theindicated boundary conditions [56]. Isotropic thermal conductivity and specific heat areassumed as unity. The plate is discretized with 2 × 2 quadratic finite elements. A transientanalysis of this problem for homogeneous initial temperature condition in terms of advancedmode superposition is given in Reference [12]. At present, one is only concerned with the non-linear eigenvalue problem, that is, the solution of Equation (33) for heat conduction. Owingto imposed temperature along the edges x = 1.0 and y = 1.0, there is a total of 12 degreesof freedom. The first column of Table I presents, in order of magnitude, all 12 eigenvaluesevaluated for the linear case (n = 1), with the two subsequent columns indicating number ofiterations needed to achieve convergence and order of evaluation of the results as obtainedfrom Algorithm 2. The calculations are in double precision with error tolerance tol = 10−14,although only seven digits are displayed in the table. As a result of the plate’s symmetry aboutthe line y = x, the 2nd and 3rd, the 5th and 6th, and the 10th and 11th eigenvalues turn outto be equal within machine precision. A total of 41 iterations were necessary for the solutionof the linear eigenproblem. The next set of three columns in Table I gives the eigenvaluesfor the non-linear case of n = 3 generalized ”mass” matrices in Equation (33) together withnumber of iterations and order of evaluation, as obtained in the algorithm. A total number of55 iterations were required for the solution of the complete non-linear eigenproblem. As in thelinear case, once the first of a double eigenvalue is evaluated, only one iteration is necessaryto obtain the second eigenpair. An idea of the relative order of magnitude of the elements ofthe matrices involved may be obtained from the Frobenius norms of K0, M1, M2 and M3,which are approximately 8.5, 0.11, .73E-3 and .81E-5. In order to assess the performance ofthe algorithm in the case of close eigenvalues, the numerical model was also analyzed for a


24 N. A. DUMONT

small change in two elements of the matrix M3, to just break the model symmetry about theline y = x although still keeping M3 positive definite. The results are shown in the last threecolumns of Table I, indicating that, although more iterations have been required, in general,the evaluation of a neighbor eigenpair is easily taken care of with Algorithm 2. A consistencycheck of the calculated eigenpairs was carried out by constructing the matrix on the left-handside of Equation (44) and comparing with the identity matrix. In all three cases of Table I, theglobal error was of magnitude 10−17. Repeated evaluations for an error tolerance tol = 10−8

in the algorithm resulted in only one iteration saved for each eigenpair, in average, which is tobe expected for an algorithm with cubic convergence.

Figure 3 plots the curves λj(θ) given by the first of Equation (51) for all 12 eigenvalues ofthe present example, to illustrate that the solution of the non-linear eigenproblem correspondsto a minimum coinciding with the interception with the line λ = θ, as theoretically outlinedin Section 3.2.3. It is worth observing that λ12 is slightly larger than λ10 ≡ λ11, in the linearcase (θ = 0), but soon becomes smaller.

x

y(1, 1)(0, 1)

(0, 0) (0, 1)

u = 1.0

u = 1.0

u, = 0.0y

u, = 0.0x

Figure 2. Scheme for the homogeneous heat conduction in the square plate of Example 2, to illustratesome features of the solution of linear and non-linear eigenproblems.

4.3. Example 3 – Fixed-free bar with viscous damping

This example assesses the non-linear eigenvalues related to a fixed-free bar modeled with threetruss elements, as given in Figure 4, for the matrices introduced in Example 1. The damping-free case (µ = 0) is analyzed first, for n = 1, . . . , 4, according to Equation (33). The results forall three eigenfrequencies ω – the square roots of the eigenvalues – are given in Table II, withcorresponding numbers of iteration required to achieve a tolerance error tol = 10−14. Althoughonly seven digits are displayed, double precision was used in all evaluations. As in Example2, a tolerance error tol = 10−8 would require about one iteration less per eigenvalue. The lastrow shows the analytical eigenfrequencies for the fixed-free bar, given from k = (2j− 1)π/2/`,where k is the wave number introduced in Equation (87) and j = 1, 2, · · · . A second analysiswas run for a viscous damping coefficient µ = 10, which corresponds to a complex-symmetriceigenproblem with underdamping. An estimate of the relative magnitude of the elements of thematrices involved in the numerical discretization can be obtained from the Frobenius norms



Linear case (n = 1) Case with n = 3 Modified case with n = 3

λ#

iter.eval.order λ

#iter.

eval.order λ

#iter.

eval.order

5.149701 4 1 4.935737 5 1 4.935695 5 129.84521 4 2 24.98865 6 2 24.96592 3 329.84521 1 3 24.98865 1 3 24.97391 7 258.58476 5 6 46.04634 7 6 45.83252 7 6101.5537 7 4 70.85867 6 4 70.06585 6 4101.5537 1 5 70.85867 1 5 70.35938 4 5127.2303 4 11 89.37343 5 11 86.70345 6 12142.8023 5 7 100.9992 6 10 99.96489 6 10215.1464 3 12 148.4600 7 12 145.8409 5 11260.7451 2 9 180.4566 4 8 177.9188 8 7260.7451 1 10 180.4566 1 9 178.6309 5 8261.6948 4 8 177.8527 6 7 175.6848 4 9

Table I. Eigenvalues, number of iterations required for tol = 10−14 and order in which the eigenvalueswere found for Example 2, covering both linear (n = 1) and non-linear eigenproblems (n = 3), besides

a modified problem for the assessment of Algorithm 2 in the case of close eigenvalues.

0 50 100 150 200 250 3000

50

100

150

200

250

300

θ

λ j

λ1

λ4

λ2 ≡ λ

3

λ12

λ9

λ8

λ7

λ5 ≡ λ

6

λ10

≡ λ11

λ = θ

Figure 3. Curves λj(θ) given by the first of Equation (51) for all 12 eigenvalues of Example 2, toillustrate that the solution of the non-linear eigenproblem corresponds to a minimum coinciding with

the interception with the line λ = θ.


26 N. A. DUMONT

of K0, C1, . . . ,C4 and M1, . . . ,M4, which are respectively 360., 10.5, 0.147E-1, 0.197E-4,0.232E-7, 0.978, 0.553E-3, 0.337E-6 and 0.103E-9. The numerical results are given in TableIII in blocks that contain the complex eigenvalue and its absolute value, besides the startingreal eigenvalue used in the solution of the complex-symmetric eigenproblem, as described inthe perturbation analysis of Section 3.4 and in Algorithm 2. The analytical eigenfrequencyresults, obtained from k = (2j − 1)π/2/`, are displayed in the bottom row of Table IIItogether with their absolute values. Since the starting eigenpair estimates for the complex-symmetric case (ω, φ) are obtained in a first run of Algorithm 2 for a real-symmetric problemresulted from disregarding the generalized damping matrices Cj , these estimates correspondonly approximately to the results of the damping-free analysis, as given in Table II. In fact,as shown in the developments of Section 4.1.1, the generalized mass matrices are affected bythe damping coefficient µ when obtained in the frame of a consistent formulation. In thisand in other examples analyzed, the use of the damping-free results as starting values forthe complex-symmetric analysis ended up with approximately the same number of iterationsto achieve convergence. In the FORTRAN code implemented, Algorithm 2 is used in twosteps, first by calling a routine for real numbers that disregards Cj and produces the realeigenpair estimates, and then by calling a version of the same routine for complex numbersthat solves the complete complex-symmetric eigenproblem. In the present example, the firststep has required the same iteration numbers to achieve convergence as given in Table II.The second step required a total of 20 iterations for each value of n. The present strategy ofrunning a first step for the corresponding real-symmetric problem obtained by disregarding thedamping matrices Cj only works if the mass Mj are all positive definite even in the presenceof damping. As given in Equation (96) and assembled for three truss elements, this is trueonly if α < 1.748474, which corresponds to µ < 34.96949. For larger damping factors in thepresent truss problem, the real-symmetric problem of the first step has to correspond to theactual damping-free case.

Figure 4. Scheme of a fixed-free bar for Example 3, modeled with three truss elements as describedin Example 1. In consistent units: total length ` = 3, elasticity modulus E = 100, cross area A = 1,

inertia ρ = 1 and viscous damping µ = 10.

ω1#

iter. ω2#

iter. ω3#

iter.n = 1 5.295986 4 17.32050 4 31.42192 2n = 2 5.237565 5 16.00720 6 28.34557 6n = 3 5.236031 5 15.77676 6 27.39721 6n = 4 5.235988 5 15.72472 6 26.92935 6

analytical 5.235987 – 15.70796 – 26.17993 –

Table II. Eigenfrequencies for the fixed-free bar of Example 3 modeled as a real-symmetriceigenproblem for µ = 0.



ω1

|ω1| ω1(start)

ω2

|ω2| ω2(start)

ω3

|ω3| ω3(start)

n = 10.9455901− 5.445289i5.526781 5.526781

17.10934− 5.357142i17.92842 17.92842

31.46100− 5.146581i31.87918 31.87918

n = 2 1.556697− 4.977728i5.215466 5.475053

15.34120− 5.262882i16.21883 16.75392

28.25366− 5.245177i28.73641 29.07573

n = 3 1.554601− 5.000691i5.236764 5.474121

14.96868− 5.111293i15.81729 16.60472

27.18905− 5.311572i27.70301 28.44241

n = 41.554187− 4.999989i5.235971 5.474112

14.89600− 5.034719i15.72384 16.59288

26.59903− 5.316633i27.12518 28.31762

analytical1.554209− 5.000000i5.235987 −−−

14.89094− 5.000000i15.70796 −−−

25.69803− 5.000000i26.17993 −−−

Table III. Eigenfrequencies for the fixed-free bar of Example 3 modeled as a complex-symmetriceigenproblem for µ = 10. Each block contains ω, its absolute value and the starting real estimate

obtained by disregarding all generalized damping matrices Cj in a first run of Algorithm 2.

4.4. Example 4 – Two artificial problems with overdamping

The application of Algorithm 2 to overdamping is demonstrated for two artificial problems:In the first problem, one uses matrices K0, M1 and M2 from Example 2 as the entries K0,C1 and M1 of a new mathematical model that corresponds to n = 1 in Equation (32), whichis just non-linear; In the second problem, one uses matrix M3 of Example 2 twice, as entriesC2 and M2, in order to create a generalization (n = 2) of the first problem. Although theseartificial problems correspond to no actual modeling of a mechanical phenomenon, it is worthinvestigating them from the mathematical point of view, as the Frobenius norm of the firstdamping matrix, C1, is much larger than that of the first mass matrix, M1, which are .11 and.73E-3, respectively, according to the data of Example 2. As in the previous example, Algorithm2 is run first for the corresponding damping-free problem, with results used subsequently forthe complex case, as outlined in Section 3.5. Tolerance error tol = 10−14 is adopted in bothsteps. The eigenvalue results are displayed in Table IV with seven digits and in order ofevaluation. The two problems analyzed, as cases n = 1 and n = 2, are actually unrelated andare presented side by side only for the sake of brevity. In brackets are indicated the numbersof iterations required for convergence in the real and complex steps. As the problem topologyof Example 2 has been preserved, three double eigenvalues are also obtained. The problemturns out to be overdamped for four out of the 12 eigenvalues in the case of n = 1, and foronly one eigenvalue in the case of n = 2. In the case of underdamping, the complex-associatedeigenpairs are given directly as (−ω, iφ). For overdamping, however, Algorithm 2 has to be runonce more in order to evaluate the inassociated eigenpairs (Section 3.3.3). New starting valuesfor the complex step, in the present examples, were taken as shifts of the negative square rootsof the eigenvalues obtained in the real step, −ω − 100, while keeping the initial eigenvectorsas the same ones of the real step. The results are shown in Table IV side by side with thecorresponding eigenvalues found previously and with numbers of iterations as the second valuesin brackets. The eigenvectors obtained in the first complex run are approximately proportionalto the eigenvectors of the initial real step. The eigenvectors corresponding to the inassociatedeigenpairs, on the other hand, are of completely different pattern, which makes them very


28 N. A. DUMONT

elusive in terms of numerical evaluation. In the present theoretical frame it is not expectedthat a code can run without human interference to obtain all complex eigenpairs, in the caseof overdamping, or else to infer that the unlikely case of critical damping has occurred.

Artificial problem with n = 1 Artificial problem with n = 2ω(start) ω (# iter.) ω(start) ω (# iter.)

24.89241 (4) −5.390892i,−121.1726i (6,5) 24.86960 (7) 25.56068− 5.380976i (6)68.16300 (4) −39.15214i,−231.4377i (7,9) 24.86960 (1) 25.56068− 5.380976i (6)68.16300 (1) −39.15214i,−231.4377i (7,9) 13.81117 (7) −5.321045i,−84.61692i (7,11)112.9713 (5) −96.92565i,−281.6820i (9,8) 34.73903 (10) 35.18777− 5.165071i (6)165.2637 (4) 72.19122− 102.4555i (6) 48.22291 (6) 48.66042− 7.541742i (6)165.2637 (1) 72.19122− 102.4555i (6) 48.22291 (1) 48.66042− 7.541742i (6)240.1926 (4) 89.63278− 137.6027i (7) 91.40394 (6) 91.79706− 9.275463i (6)406.6003 (5) 213.1366− 333.7571i (7) 93.80572 (5) 94.19215− 9.682346i (6)395.7893 (2) 224.1970− 298.9793i (7) 93.80572 (1) 94.19215− 9.682346i (6)395.7893 (1) 224.1970− 298.9793i (7) 65.51588 (6) 66.05254− 9.764081i (6)193.2663 (5) 125.7180− 146.7885i (7) 52.99561 (6) 53.25906− 5.983618i (6)316.8612 (3) 214.3764− 233.3319i (7) 76.40455 (6) 76.68535− 7.235495i (6)

Table IV. Eigenfrequency results for two overdamped problems of Example 4 obtained artificially byusing the matrices of Example 2.

4.5. Example 5 – A rail-pad-sleeper-ballast model

A railway track comprising 20 sleepers is modeled as in Figure 5 [57]. Owing to longitudinalsymmetry, only half of a complete railway track is represented. Each rail segment betweensleepers is modeled with one Timoshenko beam element. Truss elements simulate the padsthat connect rail and sleepers. A half sleeper consists of two Timoshenko beam elements oflengths 0.76 m and 0.50 m on a visco-elastic foundation to take into account the ballast (thedistance between rails is 1.52 m) [58]. Table V shows the geometrical and mechanical propertiesused in the model: cross-section area A, moment of inertia I, length l, cross-section factor κrelated to shear force, Poisson’s ratio ν, elasticity modulus E, mass m per unit length; theballast has viscosity µ and stiffness w, both defined per unit length.

The eigenproblem is assessed for n = 2 generalized damping and mass matrices, accordingto Equation (32), thus dealing with a total of five highly sparse matrices of order 140. In theinitial run without the damping matrices, an average of 5.99 iterations was necessary to solvefor each eigenpair with tolerance error tol = 10−14 and using double precision. As shown inFigure 6, there are several clusters of close eigenvalues (equal within two to six digits), althoughno machine-precision multiplicity is obtained (as in the case of Example 2). The structure’ssymmetry is reflected in the eigenvectors, as illustrated schematically in the left plot of Figure7 for values corresponding to vertical displacements of the rail nodes. Results corresponding tothe four lowest and the two highest eigenvalues (φ1, φ85, φ2, φ84 and φ94, φ95) are displayed.The symmetry/antisymmetry patterns of these eigenvectors are 12 digits accurate. The ploton the right of Figure 7 shows real versus imaginary parts of the complex eigenvalues obtainedin the second run of the algorithm, starting from the results without damping. An average of4.47 iterations was necessary to solve for each complex eigenpair starting from the real results



A(m2) I(m4) l(m) κ ν E(N/m2) m(kg/m) µ(Ns/m) w(N/m2)Sleeper 0.05126 2.31×10−4 1.26 5/6 0.25 2.1×1010 99.603 4.667×104 1.1×108

Rail 0.007686 3.217×10−5 0.545 1 0.25 2.059×1011 60.640 – –Pad 0.04 – 0.02 – – 3.25×108 39.2 3.75×106 –

Table V. Geometric and physical properties used in Example 5 [57].

of the first run, also with tolerance error tol = 10−14. In the consistency check of the calculatedeigenpairs, according to Equation (44), the global errors were of magnitude 10−17 in the firstrun and 10−15 for the complex eigenproblem.

Figure 5. Rail-pad-sleeper-ballast model with a total of 20 × 7 degrees of freedom.

0 20 40 60 80 100 120 140

106

107

108

109

λ j

0 20 40 60 80 100 120 140

106

107

108

109

λ j

Figure 6. Real eigenvalues of Example 5 in the sequence of evaluation (left) and of magnitude (right).

CONCLUDING REMARKS

The solution Algorithm 2 is based on the Rayleigh quotient iteration and on the Jacobi-Davidson method. It was implemented in FORTRAN for the complete in-core solution. Tomake sure that convergence is attained to the complex solutions of interest, a first analysisis run for the underlying real-symmetric problem, where K0 is positive semidefinite andMj , j = 1, 2, · · · are positive definite. The solution of a problem with (machine-precision)


30 N. A. DUMONT

2 4 6 8 10 12 14 16 18 20

−4

−3

−2

−1

0

1

2

3

4

x 10−3

φ1

φ2

φ94

φ95

φ84

φ85

103

104

100

101

102

103

Re (ωj)

− Im

(ωj)

Figure 7. Real-eigenmode schemes of the four lowest (φ1, φ85, φ2, φ84) and the two highest (φ94, φ95)eigenvalues of Example 5 (left). On the right: real versus imaginary parts of the complex eigenvalues.

multiple and close real eigenvalues is illustrated in Example 2. In a built-in second run ofthe algorithm, the results of the real analysis are input as starting estimates for the actualcomplex problem. The damping matrix C1 must be non-singular, although Mj , j = 1, 2, · · · , aseventually modified by the presence of viscous damping, are no longer required to be positivedefinite (Example 1). In a problem with m degrees of freedom, a maximum of m complexsolutions can be found directly. In the case of underdamping, there is a complementary complexeigenpair (−ω, iφ) associated to every primary solution (ω, φ) found with the algorithm(Example 3), as required in the numerical simulation of the transient problem using modalanalysis [10, 11]. If overdamping occurs, some eigenvalues are negative imaginary – withcorresponding eigenvectors that can be made real by properly scaling (although they aregenerally multiplied by a complex factor when normalized). In such a case, the remainingeigenpairs (here called inassociated) must be evaluated in an additional run of the code. Asthese eigenpairs are very elusive to obtain in the iterative process, the analyst’s interferenceis required to choose new, more adequate starting estimates. As shown in Example 4, thesolution of problems with multiple and close eigenvalues poses no additional difficulties evenin the case of overdamping. Example 5 illustrates the application of the algorithm to a largereigenproblem, in which several clusters of close eigenvalues are obtained. Given the smalltolerance error of 10−14, round-off errors have been kept very low.

The algorithm always finds the smallest eigenvalue as the first solution. However, theremaining eigenvalues are found in ascending order of magnitude, even in the case of linearproblems. To be applicable to large problems, the code must be modified, so that, aftersolution of the complete real eigenproblem, a set of eigenpairs of interest is selected andthan input as estimates of the complex case. Adequate storage allocation of large matricesand efficient solution for δuk in step (4.2) of Algorithm 2 also should be a concern. Iterationon an a priori given subspace as well as matrix deflation might be conceived to improvethe algorithm. However, these techniques themselves are not exempt from theoretical andnumerical difficulties in the non-linear context.

The author is indebted to both reviewers for their invaluable contribution by pointing out



several mistakes in the initial version of the paper and by suggesting some relevant literature.

REFERENCES

1. Dumont NA. The hybrid boundary element method: an alliance between mechanical consistency andsimplicity. Applied Mechanics Reviews 1989; 42(11 Part 2):S54-S63.

2. Dumont NA. Variationally-based, hybrid boundary element methods. Computer Assisted Mechanics andEngineering Sciences (CAMES) 2003; 10:407-430.

3. Pian THH. Derivation of element stiffness matrices by assumed stress distribution. AIAA Journal 1964;2:1333-1336.

4. Pian THH. Reflections and remarks on hybrid and mixed finite element methods. In Hybrid and MixedFinite Element Methods, Atluri, SN, Gallagher, RH & Zienkiewicz OC (eds). John Wiley & Sons, 1983;565-570.

5. Brebbia CA, Telles JFC, Wrobel LC. Boundary Element Techniques. Springer-Verlag: Berlin and NewYork, 1984.

6. Dumont NA, Oliveira R. From frequency-dependent mass and stiffness matrices to the dynamic responseof elastic systems. International Journal of Solids and Structures 2001; 38(10-13):1813-1830.

7. Dumont NA, Chaves RAP. General time-dependent analysis with the frequency-domain hybrid boundaryelement method. Computer Assisted Mechanics and Engineering Sciences (CAMES) 2003; 10:431-452.

8. Dumont NA, Oliveira R. On exact and approximate frequency-domain formulations of the hybrid boundaryelement method. In Procs. EURODINAME’99 – Dynamic Problems in Mechanics and Mechatronics,Gunzburg, Germany, 1999;219-224.

9. Przemieniecki J S. Theory of Matrix Structural Analysis. McGraw-Hill Book Company: New York, 1968.10. Dumont NA. An advanced mode superposition technique for the general analysis of time-dependent

problems. Advances in Boundary Element Techniques VI 2005, eds. Selvadurai APS, Tan CL, AliabadiMH; CL Ltd.: England, 333-344.

11. Dumont NA. Advanced mode superposition and hybrid variational technique for the general analysisof time-dependent problems. International Journal for Numerical Methods in Engineering 2006, to besubmitted.

12. Dumont NA, Prazeres PGC. A family of advanced hybrid finite elements for the general analysis oftime-dependent problems and non-homogeneous materials. In Procs. XXV CILAMCE – XXV IberianLatin-American Congress on Computational Methods in Engineering 2004, Recife, Brazil; 15 pp on CD.

13. Bathe K-J, Wilson EL. Numerical Methods in Finite Element Analysis. Prentice-Hall: New Jersey, 1976.14. Clough RW, Penzien J. Dynamics of Structures. McGraw-Hill: New York, 1975.15. Petyt M. Introduction to Finite Element Vibration Analysis. Cambridge Univ. Press: Cambridge, 1990.16. Dumont NA. On the inverse of generalized λ-matrices with singular leading term. International Journal

for Numerical Methods in Engineering 2006; 66(4):571-603.17. Hadeler KP. Mehrparametrige und nichtlineare Eigenwertaufgaben. Arch. Rat. Mech. Anal 1967; 27:306-

328.18. Hadeler KP. Variationsprinzipien bei nichtlinearen Eigenwertaufgaben. Arch. Rat. Mech. Anal 1968;

30:297-307.19. Voss H. Variational characterization of eigenvalues of nonlinear eigenproblems. Proceedings of the

International Conference on Mathematical and Computer Modelling in Science and Engineering 2003,eds. Kocandrlova M, Kelar V; Czech Technical University in Prague, 379-383.

20. Horn RA, Johnson CR. Matrix Analysis. Cambridge University Press: Cambridge, 1985.21. Wilkinson JH, The Algebraic Eigenvalue Problem. Oxford University Press: Oxford, 1977.22. Golub GH, Van Loan CF. Matrix Computations. The Johns Hopkings University Press: Baltimore, MD,

3rd ed., 1996.23. Mehrmann V, Voss H. Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods.

Report 83, Arbeitsbereich Mathematik, TU Hamburg-Harburg, 2004.24. Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst H, editors. Templates for the Solution of Algebraic

Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000.25. Arbenz P, Hochstenbach ME. A Jacobi-Davidson method for solving complex-symmetric eigenvalue

problems. SIAM Journal of Scientific Computation 2004; 25(5): 1655-1673.26. Ben-Israel A, Greville TNE. Generalized Inverses: Theory and Applications (2nd edn). Springer-Verlag:

New York, 2003.27. Voss H. A new justification of finite dynamic element methods. International Series of Numerical

Mathematics 1987; 83:232-242.


32 N. A. DUMONT

28. Jarlebring E, Voss H. Rational Krylov for nonlinear eigenproblems, an iterative projection method .Applications of Mathematics 2005; 50(6):543-554.

29. Betcke T. Efficient Methods for Nonlinear Eigenvalue Problems. Diploma Thesis, Technical University ofHamburg-Harburg, 2002.

30. Betcke T, Voss H. A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems. FutureGeneration Computer Systems 2004; 20(3):363-372.

31. Mehrmann V, Watkins D. Polynomial eigenvalue problems with Hamiltonian structure. ElectronicTransactions on Numerical Analysis 2002; 13:106-118.

32. Chatelin F.Valeurs Propres de Matrices. Masson: Paris, 1988.33. Sleijpen GLG, Booten AGL, Fokkema DR, Van der Vorst HA. Jacobi-Davidson type methods for

generalized eigenproblems and polynomial eigenproblems: part I. Internal Report Universiteit Utrecht,1995.

34. Sleijpen GLG, Van der Vorst HA. A Jacobi-Davidson iteration method for linear eigenvalue problems.SIAM Journal of Matrix Analysis Applications 1996; 17:401-425.

35. Davidson ER. The iterative calculation of a few or the lowest eigenvalues and corresponding eigenvectorsof large real-symmetric matrices. Journal of Computational Physics 1975; 17:87-94.

36. Tisseur F, Higham NJ. Structured Pseudospectra for polynomial eigenvalue problems, with applications.SIAM Journal of Matrix Analysis 2001; 23(1):187-208.

37. Casciaro R. A fast iterative solver for the nonlinear eigenvalue problem. Report 39, Laboratorio diMeccanica Computazionale, Universita della Calabria, 2004.

38. Hwang TM, Lin WW, Liu JL, Wang W. Jacobi-Davidson methods for cubic eigenvalue problems.NumericalLinear Algebra Applicattions 2004; 12(7):605-624

39. Triebsch F. Eigenwertalgorithmen fur symmetrische lambda-Matrizen. Ph.D. Thesis, TechnischeUniversitat Chemnitz-Zwickau, Germany, 1995.

40. Bott R, Duffin RJ. On the algebra of networks. Trans. Amer. Math. Soc. 1953; 74:99-109.41. Gupta KK. On a finite dynamic element method for free vibration analysis of structures. Computational

Methods in Applied Mechanics and Engineering 1976; 9:105-120.42. Paz M, Dung L. Power series expansion of the general stiffness matrix for beam elements. International

Journal for Numerical Methods in Engineering 1975; 9:449-459.43. Kolousek V. Dynamics in Engineering Structures. Butterworths: London, 1973.44. Warburton GB. The Dynamical Behaviour of Structures. Pergamon Press, 2nd edition, 1976.45. Gaul L, Wagner M, Wenzel W, Dumont NA. On the treatment of acoustical problems with the hybrid

boundary element method. International Journal of Solids and Structures 2001; 38(10-13):1871-1888.46. Sato K. Complex variable boundary element method for potential flow with thin objects. Computer

Methods in Applied Mechanics and Engineering 2003; 192:1421-1433.47. Duncan WJ. Some devices for the solution of large sets of simultaneous linear equations. Philosofic

Magazine 1944; 35(7):660-670.48. Lancaster P. Lambda-Matrices and Vibrating Systems. Pergamon Press: Oxford, 1966.49. Meirovitch L. Elements of Vibration Analysis. McGraw-Hill: Tokyo, 1975.50. Rogers EH. A minimax theory for overdamped systems. Arch. Rat. Mech. Anal 1964; 16:89-96.51. Guo C-H, Lancaster P. Algorithms for hyperbolic quadratic eigenvalue problems. Mathematics of

Computation 2005; 74(252):1777-1791.52. Croll JGA, Walker AC. Elements of Structural Stability. Butter & Tanner Ltd.: London, 1972.53. Stewart GW. Perturbation theory for the generalized eigenvalue problem. Recent Advances in Numerical

Analysis 1978, eds. de Boor C, Golub GH, Academic Press: New York54. Bathia R, Li R-C. On perturbations of matrix pencils with real spectra. II. Mathematics of Computation

1996; 65(214):637-645.55. Fassbender H, Kressner D. Structured eigenvalue problems. GAMM Mitteilungen, Themenheft Applied

and Numerical Linear Algebra, Part II, 2005.56. Bruch JC, Zyvoloski G. Transient two dimensional heat conduction problems solved by the finite element

method. International Journal for Numerical Methods in Engineering 1974; 8:481-494.57. Oliveira AC. Um Modelo de Analise da Interacao Dinamica entre os Elementos de uma Via Ferrea. M.Sc.

Thesis, PUC-Rio, Brazil, 2006.58. Dumont, NA, Oliveira AC. A dynamic interaction model of railway track structural elements. In Procs.

XXVII CILAMCE – XXVII Iberian Latin-American Congress on Computational Methods in Engineering2006, Belem, Brazil; 16 pp on CD.
