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Quadratic Model Updating with No Spill-over and Incomplete Measured Data : Existence and Computation of Solution

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Quadratic Model Updating with No Spill-over

and Incomplete Measured Data : Existence

and Computation of Solution

Yueh-Cheng Kuo

a,

,1

, Biswa N. Datta

b,2

aDepartment of Applied Mathematics, National University of Kaohsiung,

Kaohsiung 811, Taiwan

bDepartment of Mathematical Sciences, Northern Illinois University, DeKalb, IL

60115, USA

Abstract

Quadratic Finite Element Model Updating Problem (QFEMUP), to be studied in this paper, is concerned with updating a symmetric nonsingular quadratic pencil in such a way that, a small set of measured eigenvalues and eigenvectors is reproduced by the updated model. If in addition, the updated model preserves the large number of unupdated eigenpairs of the original model, the model is said to be updated with no spill-over.

QFEMUP is, in general, a difficult and computationally challenging problem due to the practical constraint that only a very small number of eigenvalues and eigenvec-tors of the associated quadratic eigenvalue problem are available from computation or measurement. Additionally, for practical effectiveness, engineering concerns such as nonorthogonality and incompleteness of the measured eigenvectors must be con-sidered. Most of the existing methods, including those used in industrial settings, deal with updating a linear model only, ignoring damping. Only in the last few years a small number of papers been published on the quadratic model updating; several of the above issues have been dealt with both from theoretical and computational point of views. However, mathematical criterion for existence of solution has not been fully developed.

In this paper, we first (i) prove a set of necessary and sufficient conditions for the existence of a solution of the no spill-over QFEMUP, then (ii) present a parametric representation of the solution, assuming a solution exists and finally, (iii) propose an algorithm for QFEMUP with no spill-over and incomplete measured eigenvectors. Interestingly, it is shown that the parametric representation can be constructed with the knowledge of only the few eigenvalues and eigenvectors that are to be updated and the corresponding measured eigenvalues and eigenvectors - complete knowledge of eigenvalues and eigenvectors of the original pencil is not needed, which makes the solution readily applicable to real-life structures.

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1 Introduction

Given three square matrices M0, C0, K0, the quadratic matrix eigenvalue

problem (QEP) is to find the scalars λ, called the eigenvalues, and the vectors x, called the eigenvectors such that

(λ2M0+ λC0+ K0)x = 0. (1)

The matrix Q(λ) = λ2M

0+ λC0+ K0 is called the quadratic matrix pencil,

conveniently denoted by (M0, C0, K0). If each of the matrices M0, C0, and K0

is of the order n, and M0 is nonsingular, then there are 2n finite eigenvalues

and the corresponding 2n eigenvectors.

Because of the nonlinearity, the QEP is computationally difficult to solve, especially if the matrices are large, which is typically the case in many engi-neering applications.

Indeed, the state-of-the-art computational techniques, such as the Jacobi-Davidson method ( Sleijpen, et al [19], Datta [8], and Tisseur and Meerbergen [20]) and the second-order Arnoldi method (Bai and Su [1]), etc., are capable of computing only a few extremal eigenvalues and eigenvectors of the Q(λ). The quadratic inverse eigenvalue problem (QIEP) is to construct three matrices M0, C0, and K0 from the knowledge of complete or partial spectrum

and/or the eigenvectors. QIEP is equally important as QEP and arises in a wide variety of applications, such as in control theory, signal and image processing, fluid dynamics, and others.

In particular, partial quadratic inverse eigenvalue problem (QPIEP) which concerns solving the inverse problem from the knowledge of only par-tial spectrum and the corresponding eigenvectors, is a difficult and challenging problem. An important practical variation of QPIEP, called the finite ele-ment model updating problem (FEMUP), arises in vibration engineering. The problem is so called because it concerns with updating a finite-element generated model of a vibrating structure of the form:

∗ Corresponding author.

Email addresses: yckuo@nuk.edu.tw (Yueh-Cheng Kuo ), dattab@math.niu.edu (Biswa N. Datta).

1 The research of this author was partially supported by the National Science

Coun-cil and the National Center for Theoretical Sciences in Taiwan.

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M0q(t) + C¨ 0q(t) + K˙ 0q(t) = 0, (2)

where M0, C0, and K0 are respectively, known as mass, damping, and

stiff-ness matrices. The eigenvalues of the associated quadratic pencil Q(λ) are related to natural frequencies and the eigenvectors are the mode shapes of the vibrating system (2). The matrices M0, C0, and K0very often in practice

are structured−they are symmetric and in addition, the mass matrix is often positive definite and diagonal, and the stiffness matrix is tridiagonal and pos-itive semi-definite. If the matrices are symmetric and M is nonsingular, then we call the pencil, a symmetric nonsingular pencil.

FEMUP is to update the quadratic pencil Q(λ) to another quadratic pencil in such a way that a small number of measured eigenvalues and eigenvectors from a real-life structure or a vibration laboratory, are reproduced by the updated pencil. Besides these basic requirement of reproducing the measured data, there are certain other engineering issues that must be taken into account while solving the problem in practice. These include:

• Retention of the unupdated eigenvalues and eigenvectors (No spill-over phenomenon).

It is important that no spurious modes appear in the frequency range of interest after the model has been updated. Maintaining the no spill-over will guarantee that this will not happen.

• Incompleteness and orthogonality of the measured eigenvectors.

Due to the hardwire limitations, very often the measured eigenvectors fail to satisfy an orthogonality relation, which is an acceptable criterion of suitability of a set of given eigenvectors to be prospective candidates for updating (Friswell and Mottershead [12]). Similarly, for the same reason of hardwire limitations, the measured eigenvectors are usually of much shorter length than their counterparts of the finite-element model. Some measures much be taken to deal with nonorthogonality and incompleteness of the measured eigenvectors.

Because of its importance, FEMUP has been widely studied both by academic researchers and practicing engineers. A voluminous work exists. An excellent account of these before 1995 can be found in the book (Friswell and Motter-shead [12]).

Most of the earlier methods dealt with updating a symmetric-definite un-damped pencil; that is, a linear definite pencil of the form K − λM was updated. For such problems, very often the associated optimization problem has a unique explicit solution (Friswell and Mottershead [12], Wei [21]). The methods for updating damped models are rare. Only a very small number of methods for such models were available in the literature until a few years ago. (see Friswell and Mottershead [12], Friswell, Inman and Pilkey [11], Kuo, Lin

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and Xu [14]). Furthermore, these damped model updating methods did not consider the above mentioned engineering issues well.

The issue of the spill-over was hardly considered by these methods. As of sat-isfaction of orthogonality constraint by the measured eigenvectors, it has been implicitly assumed by most of these methods for undamped model updating that the measured eigenvectors satisfy the well-known mass normalized or-thogonality constraints: XTM X = I and XTKX ≡ D, where D is a diagonal

matrix and X is the matrix of eigenvectors (Datta [8]).

The problem of incompleteness of the measured eigenvectors was usually dealt with by expansion of the incomplete eigenvectors data so that they become of equal length as the finite-element eigenvectors (Friswell and Mottershead [12]).

Since damped model updating concerns updating of a quadratic matrix pencil, we will hereafter refer to the damped finite element model updating problem as the quadratic finite element model updating problem (QFEMUP) Some remarkable progress has been made, both on QIEP and QFEMUP, in the last few years. For an account of recent progress on QPIEP, see [2,15–18]. The progress on QFEMUP is summarized below.

In Carvalho [3], and Carvalho, et al [4], a parametric expression for updating the matrix K, of the symmetric model λ2M

0+ K0 has been derived to satisfy

the no spill-over property and it has been shown how to algorithmically choose the parametric matrix to complete the set of unmeasured eigenvector vectors so that the completed set satisfies the mass orthogonality constraint.

Carvalho, et al [5] have developed a low-rank updating algorithm to update a quadratic model so that only a set of measured eigenvalues are updated, the updating model remains symmetric, and the no spill-over is maintained. The no spill-over phenomenon itself has been studied in depth in two recent papers (Chu, et al. [6], [7]).

In Datta et al [9], a two-stage algorithm for QFEMUP has been developed. Stage I concerns updating a set of measured eigenvectors that satisfy the quadratic orthogonality relation. This relation was originally obtained by Datta, et al. [10] in 1997, but has been modified in [9] to deal exclusively with real data. In Stage II, the stiffness matrix is then updated (keeping M and C fixed), such that the updated model remains symmetric and the measured eigenvalues and updated measured eigenvectors from Stage I, are reproduced by the updated model. The role of Stage I in the solution of Stage II is also explained in this paper via a mathematical result on the quadratic partial inverse eigenvalue problem. The result says that Stage II has a solution if and only if Stage I is successful.

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These studies, however, did not fully consider the existence of solutions on different aspects of QFEMUP.

In this paper, we

• Prove a necessary and sufficient condition for the existence of a solution of QFEMUP with no spill-over.

• Derive parametric representations for a family of solutions of the no spill-over QFEMUP, when a solution exists, using a Sylvester matrix equation. Interestingly, such parametric solutions can be constructed using the knowl-edge of only a small number of p eigenvalues and the corresponding eigen-vectors that need to be updated.

• Based on the parametric expression of the solution, an algorithm is devel-oped for the no spill-over QFEMUP with incomplete measured eigenvectors. A numerical example is provided to illustrate the accuracy and validity of the algorithm.

2 Assumptions and Notations

We adopt the following notations and make some basic assumptions on the matrix pairs, to be used in this paper.

A1. Assume that {(λi, xi)}2ni=1are the eigenpairs of the original pencil Q0(λ) :=

λ2M

0+ λC0+ K0.

A2. Assume that the p eigenpairs {(λi, xi)}pi=1 are to be updated and p is less

than n.

A3. The eigenvalue and eigenvector matrices of Q0(λ), (Λc, Xc) ∈ R2n×2n ×

Rn×2n are partitioned as follows:

Λc= diag{ Λ1 |{z} p×p , Λ2 |{z} (n−p)×(n−p) , Λ3 |{z} n×n }, Xc = [ X1 |{z} n×p , X2 |{z} n×(n−p) , X3 |{z} n×n ].

Assume that both square matrices [X1, X2] and X3 are nonsingular and

σ(Λ1) ∩ σ(Λ2) = ∅, σ(Λ2) ∩ σ(Λ3) = ∅, σ(Λ1) ∩ σ(Λ3) = ∅, (3)

where σ(A) denotes the set of eigenvalues of A. Here (Λ1, X1) is the

matrix eigenpair that needs to be updated. A4. Set

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A5. Real-Valued Representation of the Eigenvalues and Eigenvectors: Λ1 = diag{λ [2] 1 , . . . , λ [2] l1 ; λ2l1+1, . . . , λp}, Λ2 = diag{λ [2] p+1, . . . , λ [2] p+l2; λp+2l1+1, . . . , λn}, Λ3 = diag{λ [2] n+1, . . . , λ [2] n+l3; λn+2l3+1, . . . , λ2n}, where λj ∈ R, λ [2] j =    αj βj −βj αj  

, αj, βj ∈ R, βj > 0 and that Λ1 is

invert-ible.

A6. Assume that the measured matrix eigenpair (Σ, Y ) is closed under con-jugation, that is Σ = diag{µ[2]1 , . . . , µ[2]˜l 1; µ2˜l1+1, . . . , µp}, where µj ∈ R, µ[2]j =     ˜ αj β˜j − ˜βj α˜j     , ˜αj, ˜βj ∈ R, ˜βj > 0.

It should be noted that the number of original and measured complex conjugate pairs of Λ1 and Σ may be different, i.e., we do not require

l1 = ˜l1.

A7. Let ˜Λc = diag{Σ, Λ2, Λ3}, ˜Xc = [Y, X2, X3]. Also assume that σ(Σ) ∩

σ(Λ2) = ∅, σ(Σ) ∩ σ(Λ3) = ∅ and    ˜ Xc ˜ XcΛ˜c    is invertible. A8. Denote D1 = n D ∈ Rp×p|DT = D, DΛ1 = ΛT1D o , (5a) D2 = n D ∈ R(n−p)×(n−p)|DT = D, DΛ 2 = ΛT2D o , (5b) D3 = n D ∈ Rn×n|DT = D, DΛ 3 = ΛT3D o , (5c) DΣ = n D ∈ Rp×p|DT = D, DΣ = ΣTDo. (5d) A9. For each i = 1, 2, 3, assume that λi,1, . . . , λi,ri are distinct eigenvalues of

Λi with multiplicities mi,1, . . . , mi,ri, respectively. Let ˆmi = maxjmi,j,

i = 1, 2, 3, and assume that max{ ˆm1, ˆm2, ˆm3} is less than n.

Remark 1 From A8, it is readily seen that D1, D2, D3 and DΣ are sets of

block diagonal matrices which are vector spaces. We analyze the dimension of Di, for i = 1, 2, 3. From A9, we have Prj=11 m1,j = p, Prj=12 m2,j = n − p and

Pr3

j=1m3,j = n. It is easy to check that

di ≡ dim(Di) = rj X j=1 mi,j(mi,j+ 1) 2 , (6)

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for i = 1, 2, 3. The upper bound of di can be estimated as follows d1 ≤ ˆ m1+ 1 2 p, d2 ≤ ˆ m2+ 1 2 (n − p), d3 ≤ ˆ m3+ 1 2 n. (7)

In particular, if the eigenvalues of Λ1, Λ2 and Λ3 are simple then, respectively,

d1 = p, d2 = n − p and d3 = n.

3 Problem Statement

Mathematically, the quadratic finite element model updating (QFEMUP) can be stated as follows:

QFEMUP: Given a symmetric quadratic model (M0, C0, K0) with M0

non-singular, and the measured matrix eigenpair (Σ, Y ), the problem is to update the model (M0, C0, K0) to the symmetric model (M, C, K) in such a way that

M remains nonsingular and

M Y Σ2+ CY Σ + KY = 0. (8)

If, in addition, the unupdated matrix eigenpair (Λ, X) satisfies :

M XΛ2+ CXΛ + KX = 0; (9) then it is referred to as the No Spill-over QFEMUP. In other words, the no spill-over QFEMUP is to update (M0, C0, K0) to (M, C, K) such that

M ˜XcΛ˜2c+ C ˜XcΛ˜c+ K ˜Xc= 0. (10)

4 Two Useful Lemmas

In the next section, we will prove our main result−necessary and sufficient conditions for solvability of the no spill-over QFEMUP. Before that, we prove two lemmas which will be needed in our proof.

Let M, C, and K be real symmetric matrices that satisfy equation (9). Define

D ≡ [XT, ΛTXT]    C M M 0       X XΛ    = XTCX + XTM XΛ + ΛTXTM X, (11)

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and E ≡ [XT, ΛTXT]    −K 0 0 M       X XΛ    = −XTKX + ΛTXTM XΛ, (12)

where the matrix pair (Λ, X) is given in (4). Since DT = D, ET = E and DΛ = E, we have DΛ = ΛTDT = ΛTD. Using the fact that Λ = diag{Λ

2, Λ3}

and σ(Λ2) ∩ σ(Λ3) = ∅, it is easily seen that D is a diagonal block matrix of

the form D = diag{D2, D3}, where D2 ∈ D2, D3 ∈ D3.

Lemma 2 Given the matrix pair (Λ, X) ∈ R(2n−p)×(2n−p)×Rn×(2n−p)as in (4)

satisfying A1-A5. If there exist real symmetric matrices M, C, and K with det(M ) 6= 0 satisfying the equation (9) and the set of remaining p eigenvalues of Q(λ) = λ2M + λC + K and σ(Λ) are disjoint then det(D) 6= 0 where D is

given in (11).

PROOF. Suppose that the triplet (M, C, K) with det(M ) 6= 0 is the solution of (9) such that the set of remaining p eigenvalues of Q(λ) and σ(Λ) are disjoint. Let (diag{Σ, Λ}, [Y, X]) be the complete eigenpair of the quadratic pencil Q(λ). Then we have σ(Σ) ∩ σ(Λ) = ∅ and the matrix

   Y X Y Σ XΛ    is

invertible. From (11), we obtain

   DΣ 0 0 D    =    Y X Y Σ XΛ    T    C M M 0       Y X Y Σ XΛ   .

Since M is invertible, and diag{DΣ, D} is invertible, it follows that D is

non-singular. 2

Suppose that (Λc, Xc) = (diag{Λ1, Λ2, Λ3}, [X1, X2, X3]) ∈ R2n×2n × Rn×2n

satisfies A1-A5. Since Λ1, Λ2 and Λ3 satisfy (3), we have

diag{D01, D02, D03} = [XT c , Λ T cX T c]    C0 M0 M0 0       Xc XcΛc   . (13)

It then follows that D0 1 = X1TC0X1+ X1TM0X1Λ1+ ΛT1X1TM0X1, D02 = X2TC0X2+ X2TM0X2Λ2+ ΛT2X2TM0X2, D0 3 = X3TC0X3+ X3TM0X3Λ3+ ΛT3X3TM0X3, (14)

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where D0

1 ∈ D1, D20 ∈ D2, and D03 ∈ D3. Let X12 = [X1, X2] ∈ Rn×n and

N12 =    I −X3−1X12   ∈ R 2n×n.

Then XcN12 = 0. Multiplying (13) by N12T and N12 from the left and right,

respectively, we have NT 12diag{D01, D02, D03}N12 = 0. This is equivalent to D01 = −X1TX3−TD30X3−1X1, (15a) D02 = −X2TX3−TD30X3−1X2, (15b) 0 = −X2TX3−TD30X3−1X1. (15c)

Since M0 is invertible, from Lemma 2 we have that D01, D02, and D30 are

non-singular. Hence we proved that

Lemma 3 If (Λc, Xc) ∈ R2n×2n × Rn×2n satisfies A1-A5. Then (15) holds

where D0

1 ∈ D1, D02 ∈ D2, and D03 ∈ D3 are nonsingular.

Remark 4 From Remark 1, we obtain that if D2 ∈ D2 and D3 ∈ D3 then

the degree of freedom of D2 and D3 are d2 and d3, respectively. Let ˆm =

max{ ˆm2, ˆm3}. By (7), we have d2 + d3 ≤ ( ˆm + 1)(2n − p)/2. Consider the

linear system

D2 = −X2TX −T

3 D3X3−1X2, D2 ∈ D2 and D3 ∈ D3. (16)

The number of equations of (16) are 12(n − p)(n − p + 1). Since p  n, ˆm  n and (D02, D03) is a solution of (16), generally, the dimension of solution set of the linear system (16) is one, i.e., D2 = cD20, D3 = cD30, where c ∈ R. In

the following, we assume that the linear system (16) only has solutions of one dimension.

5 Solvability of the No Spill-Over QFEMUP

The following theorem gives a necessary and sufficient condition for the exis-tence of solution of the no spill-over QFEMUP.

Theorem 5 Given the measured eigenpair (Σ, Y ) ∈ Rp×p, QFEMUP with no

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and a matrix DΣ ∈ DΣ such that Y = X1T, (17) and TTD01T = DΣ, (18) where D0 1 = −X1TX −T 3 D30X −1 3 X1.

PROOF. Suppose that the problem is solvable, that is, suppose that there exist matrices M, C, and K satisfying (10). Then, it has been proved in [2, Theorem 2.7] that the following equations hold:

   DΣ 0 0 D0 2   = −    YT XT 2   X −T 3 D 0 3X −1 3 [Y, X2], (19) and ˜ Xc†TD ˜˜N is nonsingular, (20) where DΣ ∈ DΣ, ˜D = diag{DΣ, D20, D30} and ˜N =

   I −X3−1[Y, X2]   . Here ˜X † c

denotes the pseudoinverse of ˜Xc. From Lemma 3, we have D30 ∈ D3is invertible

and using the fact [X1, X2] and X3 are nonsingular, it follows that the matrix

XT 2 X

−T 3 D30X

−1

3 ∈ R(n−p)×n and X1 ∈ Rn×p are of full rank. From (4), (19)

and (15c), it follows that there exists a matrix T ∈ Rp×p such that Y = X 1T .

Combing (15a) and the leading block of (19), we have DΣ = −YTX3−TD30X −1 3 Y = −TTXT 1X −T 3 D03X −1 3 X1T = TTD01T.

Next, we show that T is invertible.

The equation (19) is equivalent to ˜NTD ˜˜N = 0. From (20) and ˜NTD ˜˜N = 0

follows that     ˜ Xc†T ˜ NT     ˜ D[ ˜Xc†, ˜N ] =      ˜ Xc†TD ˜˜Xc† X˜c†TD ˜˜N ˜ NTD ˜˜X† c 0     

is invertible. Thus, ˜D = diag{DΣ, D20, D30} is invertible. Since DΣ = TTD10T ,

we see that T is nonsingular.

Conversely, suppose (17) and (18) hold. From [2, Theorem 2.7], we obtain that it suffices to show that the equations (19) and (20) hold. From (15), (17), and

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(18) we obtain     DΣ 0 0 D0 2     =     TT 0 0 I         D0 1 0 0 D0 2         T 0 0 I     = −     TTXT 1 XT 2     X3−TD0 3X −1 3 [X1T, X2], = −     YT XT 2     X3−TD0 3X −1 3 [Y, X2]. (21) Thus, (19) holds. Also, since D0

1 and T are invertible, we have DΣ is invertible. From Lemma 3,

it follows that ˜D is invertible. Since [ ˜Xc†, ˜N ] is invertible and ˜NTD ˜˜N = 0, we

have that ˜Xc†TD ˜˜N is nonsingular. 2

Remark 6 In matrix theory and linear algebra literature, it is customary to state the results in terms of Jordan pair or Jordan triple, whenever possible. It turns out that our main results can be also related to these terminologies as the following discussion shows:

First, we note that since the matrix

   ˜ Xc ˜ XcΛ˜c    is assumed to be invertible

(by our Assumption 7), and ( ˜Λc, ˜Xc) satisfy the relation (10), it follows that

( ˜Xc, ˜Λc) is a Jordan pair of the updated no spill-over quadratic pencil:

λ2M + λC + K.

Second, define the matrix ˜Yc by

˜ Yc≡    DΣ−1TTX1T D230 −1XT   ∈ R 2n×n ,

where X is as defined in (4) and D0

23 = diag{D20, D30}. Set now X12 = [X1, X2].

Then it is easy to see that ˜ XcY˜c= X1T DΣ−1TTX1T + XD 0 23 −1 XT = X12D120 −1 X12T + X3D03 −1 X3T = 0,

by virtue of the result that −D0 12 −1 = X12−1X3D30 −1 XT 3X −T 12 , where D120 = diag{D0

1, D02}. Thus, in view of our assumption that

   ˜ Xc ˜ XcΛ˜c    is nonsingular,

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we obtain    ˜ Xc ˜ XcΛ˜c    ˜ Yc=    0 W   ,

where W is an invertible matrix, showing that ( ˜Xc, ˜Λc, ˜Yc) is a Jordan triple.

Remark 7 For each real eigenvalue λ and eigenvector x of the quadratic model Q(λ) = λ2M + λC + K, we write the real number

xTCx + 2λ(xTM x)

in the form κ2, where  = ±1 and κ > 0. Then  is called the sign characteris-tic of the eigenvalue λ. From (11), it is easily seen that the sign characterischaracteris-tic of each real eigenvalue λi is the sign of ith diagonal entry of the diagonal block

matrix D.

6 Parametric Representation Solution of the No Spill-over QFEMUP

In this section, we show that when the no spill-over QFEMUP is solvable, there exists a parametric representation of the solution, which can be constructed using the knowledge of only (Λ1, X1) and the measured eigenpairs.

Theorem 8 If (M, C, K) is a solution of the no spill-over QFEMUP, then M = cM0 − M0X1ΦX1TM0, C = cC0+ M0X1ΦΛ−T1 X1TK0+ K0X1Λ−11 ΦX1TM0, K = cK0− K0X1Λ−11 ΦΛ −T 1 X1TK0, (22)

where c ∈ R is nonzero, and Φ = ΦT ∈ Rp×p satisfies the Sylvester equation:

(K1Λ−11 − Θ TM

1)ΦM1+ M1Φ(Λ−T1 K1− M1Θ) = c(Λ1− Θ)TM1+ cM1(Λ1− Θ).

(23) Here M1 = X1TM0X1, K1 = X1TK0X1 and Θ = T ΣT−1.

PROOF. Suppose that (M, C, K) is a solution of the no spill-over QFEMUP. Representation of (M, C, K) in the form (22) can then be obtained by com-bining Theorem 2.1 in [2], and Theorem 3.1 in [7]. Now, we show here the matrix Φ satisfies the Sylvester equation (23).

Let

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It is easily seen that DΣ ∈ DΣ. From (11), we have    DΣ 0 0 D    =    Y X Y Σ XΛ    T    C M M 0       Y X Y Σ XΛ   , (25)

where D = diag{D2, D3} and D2 ∈ D2,

D3 = X3TCX3 + X3TM X3Λ3+ ΛT3X T

3M X3 ∈ D3. (26)

Substituting the expressions of M, C, and K in (22) into (26) and using the fact that D30 = X3TC0X3 + X3TM0X3Λ3+ ΛT3X3TM0X3 and

diag{D10, D20, D30}Λc= [XcT, ΛTcXcT]    −K0 0 0 M0       Xc XcΛc   , we then have D3 =c(X3TC0X3+ X3TM0X3Λ3+ ΛT3X T 3 M0X3) + X3TM0X1Φ(Λ−T1 X T 1K0X3− X1TM0X3Λ3) + (X3TK0X1Λ−11 − Λ T 3X T 3M0X1)ΦX1TM0X3 = cD03.

From Lemma 2, we have D3being invertible. Hence c 6= 0. Let Z = [Ip, 0, −YTX3−T]T.

Multiplying (25) by ZT and Z from the left and right, respectively, we obtain

DΣ = −YTX3−TD3X3−1Y = −cY TX−T 3 D 0 3X −1 3 Y.

Since Y = X1T and T is invertible, it follows from (15a) that DΣ = cTTD10T .

Then (24) can be rewritten as

cD10 = T−TDΣT−1 = X1TCX1+ X1TM X1Σ + ΣTX1TM X1. (27)

Substituting the expressions of M, C, and K in (22) into (27), we have cD01 = c(X1TC0X1+ X1TM0X1Θ + ΘTX1TM0X1) +(XT 1 K0X1Λ−11 − ΘTX1TM0X1)ΦX1TM0X1 +X1TM0X1Φ(Λ−T1 X1TK0X1 − X1TM0X1Θ), (28) where Θ = T ΣT−1. Let M1 = X1TM0X1, C1 = X1TC0X1, K1 = X1TK0X1.

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The equation (28) can be rewritten as

(K1Λ−11 − ΘTM1) ΦM1+ M1Φ(Λ−T1 K1 − M1Θ)

= c(D10− C1− M1Θ − ΘTM1)

= c(Λ1− ΘT)M1+ cM1(Λ1− Θ). (From (14)).

This completes the proof. 2

7 Solution of the No Spill-over QFEMUP with Incomplete Mea-sured Data

In many practical situations, the measured eigenvector matrix Y is not com-pletely known. Suppose that Y =

   Y1 Y2    ∈ R

n×p and only the part Y 1 with

rank(Y1) = p has been measured and Y2 is unknown. In this section, we show

that if a certain computationally verifiable condition involving Y1 is satisfied,

then the problem still can be solved without knowing Y2, if the solution exists.

Combining this result with our earlier result on parametric solution (The-orem 8), we then state an algorithm for the no spill-over QFEMUP with incomplete measured data. Partition the eigenvector matrix X1 conformably

as: X1 =     X1 1 X1 2     }m }n − m . (29)

From Theorem 5 we know that if QFEMUP with no spill-over is solvable, then Y and X1 have the same column spaces, i.e., there exists an invertible matrix

T ∈ Rp×p such that Y = X

1T . From (29), we have

Y1 = X11T. (30)

Since rank(Y1) = p, and if rank(X11) < p, then (30) has no solution. This

implies that the problem is unsolvable. Assume that rank(X11) = p, then the QR factorization of X1

1 defines orthonormal matrices Q1 ∈ Rm×p and

Q2 ∈ Rm×(m−p), and an upper triangular matrix R ∈ Rp×p such that

X11 = [Q1, Q2]    R 0   , (31)

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where R is nonsingular. Let Q = [Q1, Q2]. Multiplying (30) by QT from the left, we obtain     QT 1Y1 QT 2Y1     =     RT 0     . (32) Hence, if QT

2Y1 = 0, then (30) has a unique solution T = R−1QT1Y1. Since

the solution of (30) is unique, from Theorem 5, we have that the problem is solvable if and only if TTD01T ∈ DΣ, where D01 is given by (14).

In the following, we will present an algorithm to complete Y (given Y1) and

use this completed Y to compute the parametric matrix Φ with c = 1, and then compute the updated model (M, C, K) that reproduces the measured eigenvalues and eigenvectors.

Algorithm 1 An Algorithm for QFEMUP with no Spill-over and Incomplete Measured Data

Inputs: (i) The symmetric finite-element model (M0, C0, K0), (ii) the

mea-sured experimental data Σ ∈ Rp×p and Y1 ∈ Rm×pwith rank(Y1) = p, and (iii)

the corresponding finite-element matrices Λ1 ∈ Rp×p and X1 ∈ Rn×p.

Outputs: No solution or the complete measured eigenvector matrix Y and symmetric updated pencil (M, C, K) that reproduces the measured data with no spill-over. Partition X1 as X1 =     X1 1 X21     ;

Compute a QR-factorization of X11 with X11 = [Q1, Q2]

    R 0     ; If det(R) = 0, no solution, stop;

else if QT 2Y1 6= 0, no solution, stop; else Set T = R−1QT1Y1; Set M1 = X1TM0X1 and C1 = X1TC0X1; Compute D0 1 = C1+ M1Λ1+ ΛT1M1; If TTD01T 6∈ DΣ, no solution, stop;

else Set Y = X1T, Θ = T ΣT−1, and K1 = X1TK0X1;

Solve the Sylvester equation to obtain a symmetric matrix Φ ∈ Rp×p:

(K1Λ−11 −Θ TM

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Update: M = M0− M0X1ΦX1TM0, C = C0 + M0X1ΦΛ−T1 X1TK0+ K0X1Λ−11 ΦX1TM0, K = K0− K0X1Λ−11 ΦΛ −T 1 X1TK0.

8 An Illustrative Numerical Example

Consider the model Q0(λ) = λ2M0 + λC0 + K0, where the matrices M0, C0,

and K0 are given by

M0 =               1.2986 1.1333 1.1449 0.89678 0.59899 1.1333 1.6329 1.1301 1.6278 1.0328 1.1449 1.1301 1.4872 1.1390 0.93904 0.89678 1.6278 1.1390 1.8941 1.4627 0.59899 1.0328 0.93904 1.4627 1.6671               > 0, (Positive definite) C0 =               1.1958 1.0680 1.5349 1.1655 0.13457 1.0680 0.76364 0.66398 0.96843 0.98754 1.5349 0.66398 1.6590 1.1940 0.94584 1.1655 0.96843 1.1940 0.92860 0.62366 0.13457 0.98754 0.94584 0.62366 1.4520               , K0 =               0.70769 0.80205 0.78748 0.52588 0.50341 0.80205 2.2891 1.4690 1.4251 1.8186 0.78748 1.4690 2.0015 0.87745 1.1250 0.52588 1.4251 0.87745 1.1890 0.99863 0.50341 1.8186 1.1250 0.99863 1.5447               > 0. (Positive definite)

Because M0 > 0, the quadratic pencil Q0(λ) has 10 finite eigenvalues. We first

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matrices that need to be updated, Λ1 = 10−1     0.14084 0 0 −3.7868     , X1 =                  0.28663 0.57271 −1.0000 0.43241 −0.033441 −0.048644 0.36575 0.28873 0.86705 −1.0000                  ,

are chosen from those 10 computed eigenpairs of Q0(λ) so that the

eigenval-ues of Λ1 are close to the corresponding original eigenvalues and the other

8 eigenpairs are denoted by (Λ, X). The corresponding matrices of measured frequencies and mode shapes are taken as

Σ = 10−1     2.3957 8.5459 −8.5459 2.3957     , Y1 = 10−1         −2.7657 −4.7428 6.6470 3.3530 0.30023 0.45468         .

Using Algorithm 1, we get the complete eigenvector matrix Y and the sym-metric matrix Φ, as:

Y = 10−1                  −2.7657 −4.7428 6.6470 3.3530 0.30023 0.45468 −2.9831 −3.6522 −4.9913 0.48597                  and Φ =     0.31748 1.7738 1.7738 0.22030     .

Verification: The minimal eigenvalue of the symmetric matrix M is positive, i.e., the updated matrix M is positive definite. The relative residual of (Σ, Y ) and (Λ, X) are given by

kM Y Σ2+ CY Σ + KY k 2 kM Y Σ2k 2+ kCY Σk2+ kKY k2 = 1.2199 × 10−13 and kM XΛ2+ CXΛ + KXk 2 kM XΛ2k 2 + kCXΛk2+ kKXk2 = 5.9637 × 10−14.

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9 Conclusions

The finite element model updating problem routinely arises in vibration in-dustries. Certain practical engineering challenges make the problem difficult to solve, especially in the quadratic setting. These include, (i) maintaining the spill-over of the unupdated eigenvalues and eigenvectors, (ii) completing the set of incomplete measured data in a practical way, (iii) solving the problem with the help of only a small number of the eigenvalues and eigenvectors of the associated quadratic matrix pencil, which are computable or measurable, (iv) satisfaction of orthogonality constraints of the measured data, and (v) retaining certain desirable structures, such as the symmetry, positive definite-ness, and sparsity of the original model. Indeed, most of the existing methods, even those which are used in industrial settings, deal with the linear problem of updating only an unupdated model.

Only in recent years, have some studies been devoted to solving the problem in a quadratic setting. This paper advances these studies a little further. In particular, necessary and sufficient conditions for existence of solution with no spill-over are developed, a parametric expression of the solution is derived, which enables one to solve the problem using only the small number of avail-able eigenvalues and eigenvectors, and a practical algorithm is developed to solve the problem with incomplete measured data and no spill-over.

Interestingly, some of our results can also be extended to the semi-simple case under certain assumptions on the eigenvalues and eigenvectors. Since semi-simple systems are not the main focus of this paper, we decided not to go into details here.

The aspect of structure preservation still needs close attention. Some attempts have been made in recent years. These include a recent manifold distance minimization method, developed by Halevi, et al. [13] for the undamped model But much needs to be done on this difficult but practical aspect of the problem.

Acknowledgement

The authors are grateful to Professor Wen-Wei Lin for his valuable insight into the solution of the problem considered in this paper. Numerous discussions the authors had with him were very helpful in development of this paper. The author, Y. C. Kuo, is specially indebted to Professor Lin for his guidance throughout this work. We also would like to thank the referee for his comments which led us to investigate if our result can be extended to the semi-simple case and how some of our results can be stated in terms of more familiar linear

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algebra terminologies, Jordan pair and Jordan triple.

References

[1] Z. Bai, Y. Su, SOAR : Second-order Arnoldi method for solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 26, (2005), pp. 640-659.

[2] Y.F. Cai, Y.C. Kuo, W.W. Lin, S.F. Xu, Solutions to a quadratic inverse eigenvalue problem, Lin. Alg. Appl., 430, (2009), pp. 1590-1606.

[3] J. Carvalho, State Estimation and Finite Element Model Updating for Vibrating Systems, Ph.D Dissertation, Northern Illinois University, DeKalb, IL, 2002.

[4] J. Carvalho, B.N. Datta, A. Gupta, M. Lagadupali, A direct method for matrix updating with incomplete measured data and without spurious modes, Mechanical Systems and Signal Processing, 21, (2007), pp. 2715-2731.

[5] J. Carvalho, B.N. Datta, W.W. Lin, C.S. Wang,, Symmetric preserving eigenvalue embedding in finite element model updating of vibrating structures, Journal of Sound and Vibration, 290(3-5) (2006), pp. 839-864.

[6] M.T. Chu, B.N. Datta, W.W. Lin, S.F. Xu, Spill-over phenomenon in quadratic model updating, AIAA Journal, 46(2), (2008), pp. 420-428.

[7] M.T. Chu, W.W. Lin, S.F. Xu, Updating quadratic models with no spill-over effect on unmeasured spectral data, Inverse Problem, 23, (2007), pp. 243-256. [8] B.N. Datta, Numerical Linear Algebra and Applications, Second Edition,

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[9] B.N. Datta, S. Deng, D. Sarkissian, V. Sokolov, An optimization technique for model updating with measured data satisfying quadratic orthogonality constraint, Mechanical Systems and Signal Processing, 23, (2009), pp. 1759-1772.

[10] B.N. Datta, S. Elhay, Y.M. Ram, Orthogonality and partial pole assignment for the symmetric definite quadratic pencil, Lin. Alg. Appl., 257, (1997), pp. 29-48.

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[13] Y. Halevi, D. Vilensky, B. N. Datta, Model updating via manifold distance minimization, Proc. Int. Conf. Noise and Vibration Engineering, 2010.

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[14] Y.C. Kuo, W.W. Lin, S.F. Xu, New methods for finite element model updating problems, AIAA Journal, 44(6), (2006), pp. 1310-1316.

[15] Y.C. Kuo, W.W. Lin, S.F. Xu, Solutions of the partially described inverse quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 29(1), (2006), pp. 33-53.

[16] P. Lancaster, Inverse spectral problems for semi-simple damped vibrating systems, SIAM J. Matrix Anal. Appl., 29, (2007), pp. 279-301.

[17] P. Lancaster, Isospectral vibrating systems, Part I: The spectral method, Linear Algebra and its Applications, 409, (2005), pp. 51-69.

[18] P. Lancaster, U. Prells, Inverse problems for damped vibrating systems, Journal of Sound and Vibration, 283, (2005), pp. 891-914.

[19] G.L.G. Sleijpen, H. A. van der Vorst, M. van Gijzen, Quadratic Eigenvalue Problems are no Problem, SIAM News, 29 (7) pp. 8-9.

[20] F. Tisseur, K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43, (2001), pp. 235-286.

[21] F.S. Wei, Mass and stiffness interaction effects in analytical model modification, AIAA Journal, 28(9), (1990), pp. 1686-1688.

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