Recovering added mass in nanoresonator sensors from finite axial eigenfrequency data

Recovering added mass in nanoresonator sensors from finite

axial eigenfrequency data

M. Dilena

a

, M. Fedele Dell’Oste

a

, J. Fernández-Sáez

b

, A. Morassi

a,⇑

, R. Zaera

b a

Polytechnic Department of Engineering and Architecture, University of Udine, via Cotonificio 114, 33100 Udine, Italy b

Department of Continuum Mechanics and Structural Analysis, Universidad Carlos III de Madrid, Av. de la Universidad 30, 28911 Leganés, Madrid, Spain

a r t i c l e i n f o

Article history:

Received 13 October 2018

Received in revised form 9 January 2019 Accepted 14 February 2019

Keywords:

Strain gradient theory Nanosensors Nanorods Mass identification Inverse problems Axial vibration

a b s t r a c t

In this paper we present a method for solving a finite inverse eigenvalue problem arising in the determination of added distributed mass in nanoresonator sensors by measurements of the first N natural frequencies of the free axial vibration under clamped end conditions.

The method is based on an iterative procedure that produces an approximation of the unknown mass density as a generalized Fourier partial sum of order N, whose coefficients are calculated from the first N eigenvalues. To avoid trivial non-uniqueness due to the sym-metry of the initial configuration of the nanorod, it is assumed that the mass variation has support contained in half of the axis interval. Moreover, the mass variation is supposed to be small with respect to the total mass of the initial nanorod.

An extended series of numerical examples shows that the method is efficient and gives excellent results in case of continuous mass variations. The determination of discontinuous coefficients exhibits no negligible oscillations near the discontinuity points, and requires more spectral data to obtain good reconstruction. A proof of local convergence of the iter-ation algorithm is provided for a family of finite dimensional mass coefficients. Surprisingly enough, in spite of its local character, the identification method performs well even for not necessarily small mass changes.

To the authors’ knowledge, this is the first quantitative study on the identification of dis-tributed mass attached on nanostructures modelled within generalized continuum mechanics theories by using finite eigenvalue data.

Ó 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In the last years, nanosensors based on Nanoelectromechanical systems (NEMS) have gained interest in different physical, chemical and biological applications[1,2]. Enhanced capabilities have been reached by scaling down the size of the nanosen-sors[3,4], moving the resolution of mass detection up to the zeptogram range[5]. A kind of nanosensor of great practical interest is the mechanical nanoresonator, for which the mass sensing principle is based on monitoring the variations of the resonant frequencies caused by (unknown) additional masses attached on the surface of the initial system.

The identification problem from minimal natural frequency data of a concentrated mass (modelled, for detection pur-poses, as Dirac-delta point mass) located on structural elements has been addressed by Morassi and Dilena[6]for the case

https://doi.org/10.1016/j.ymssp.2019.02.025

0888-3270/Ó 2019 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.

E-mail addresses:michele.dilena@uniud.it(M. Dilena),fedeledelloste.marta@spes.uniud.it(M. Fedele Dell’Oste),ppfer@ing.uc3m.es(J. Fernández-Sáez),

antonino.morassi@uniud.it(A. Morassi),rzaera@ing.uc3m.es(R. Zaera).

Contents lists available atScienceDirect

Mechanical Systems and Signal Processing

of rod or beams, and Rubio et al.[7]for the case of a simply supported, rectangular plate. In the above papers the classical elasticity theory was used and the added mass was considered small with respect to the total mass of the main structure. Moreover, Morassi and coworkers[8,9]have presented a procedure to solve the identification problem of a point mass located on the span of a rod[8]or beam[9], without any a priori assumption on the smallness of the attached mass. The latest analyses appear as auxiliary problems when a crack in a rod or beam is attempted to be identified.

Bouchaala et al.[10]presented a method to determine the position and intensity of a concentrated mass attached to the surface of an electrostatically actuated clamped-clamped microbeam used as a mass sensor. Although this last work is devoted to the study of a microsensor, the laws of classical elasticity are also used.

However, the experimental results of several authors[11–14]pointed out that the consideration of size effects in the nanoscale components could be relevant to predict their mechanical response. Therefore, since the classical continuum mechanics is a scale-free theory, other formulations based on generalized continuum mechanics approaches taking into account this size-dependent behavior, which present advantages in terms of computational cost in comparison to the molec-ular dynamics formulations, must be explored.

Among the generalized continuum theories, we cite here the strain gradient elasticity family including the couple stress theory[15], the first and second strain gradient theories of Mindlin[16,17], the modified couple stress theory[18]and the modified strain gradient theory[12]. The last approach, based on previous formulations of Mindlin[17] and Fleck and Hutchinson[19], needs new additional equilibrium equations to govern the behavior of higher-order stresses, and contains only three non-classical constants for isotropic linear elastic materials. This theory has been used by different authors to ana-lyze the mechanical response of different kind of nanostructures. The interested reader can see the very recent review by Thai et al.[20].

Using the modified strain gradient theory[12], the authors analyzed for the first time the vibrational behavior of nanor-ods[21], Euler–Bernoulli nanobeams[22], and rectangular simply-supported Kirchhoff nanoplates[23], carrying a single point mass. For the case of small intensity of the concentrated mass, and from the properties of the eigenvalue perturbative theory, a method for the identification of the attached point mass from minimal eigenfrequency data has been also devel-oped[21–23].

The above cited works consider concentrated masses attached to the base system. However, added distributed mass rep-resenting the adsorbed analyte seems to be more realistic in practice. In this respect much less effort to identify distributed mass on nanostructures is given until now. Nevertheless, it is worth to quote the works by Hanay et al.[24]and Bouchaala

[25]. Thus, Hanay et al.[24]proposed an inertial imaging methodology which enables simultaneous identification of position and shape of distributed masses through real time measurements of frequency shifts of the vibrational modes of the nanosensor. Bouchaala[25]analysed the effect of a distributed added mass on the natural frequencies of an electrostatically actuated resonator modelled as a classical clamped-clamped Euler-Bernoulli beam theory with geometrical nonlinearities. However, to the authors knowledge, a general formulation of the identification problem of distributed added mass attached on nanostructures described by generalized continuum mechanics theories is not given until now.

This paper deals with the inverse problem of determining the mass distribution of a nanorod from the knowledge of a finite number of lower natural frequencies of the axial vibration under clamped ends, assuming that the mass coefficient is a priori known on half of the nanorod and the added mass is a small perturbation of the total mass of the nanosensor.

From a mathematical point of view, this problem falls into the class of mixed finite inverse problems for fourth-order dif-ferential operator of the Euler–Bernoulli type, since a finite number of eigenvalues belonging to a single spectrum is known, and partial knowledge of the unknown coefficient to be determined is available. A celebrated uniqueness result for this class of inverse problems can be traced back to Hochstadt and Lieberman[26]. The result holds for second-order differential oper-ators of Sturm–Liouville type governing the axial vibration of classical straight elastic rods, i.e., L

v

¼  1

qð Þx

v

00ð Þ, for a rodx

with unitary axial stiffness and linear mass density

q

ð Þ. Here,x

v

ð Þ expresses the longitudinal displacement at x of thex cross-section of the rod, x2 0; L½ , where L is the rod length. It is shown that if

_q

ð Þ is prescribed overx L

2; L



, then all the infi-nite eigenvalues under clamped end conditions

v

ð Þ ¼ 0 ¼0

v

ð Þ suffice to determine uniquelyL

q

ð Þ on 0;x L

2



. It should be remarked that there is another noticeable case in which the linear mass density of the rod can be uniquely determined from the single spectrum under clamped end conditions, namely when

q

ð Þ is symmetric about the midpoint of the interval 0; Lx ð Þ

[27].

In the case of fourth-order operators, such as the one that governs the axial vibration of a nanorod, e.g., L

v

¼1

q b

v

IV_{ a}

_v

00

 

, with a; b constant positive stiffness coefficients and

q

ð Þ unknown linear mass density function (seex Eq. (15) below), the results of uniqueness for

q

ð Þ are few and require the knowledge of an even larger infinite set of eigen-x values. For example, a classic result by Barcilon[28]shows that the unique determination of the coefficients p x_{ð Þ; q x}ð Þ of the Euler-Bernoulli operator L

v

¼

v

IV_{ p xð ð Þ}

v

0_Þ0_{þ q xð Þ}

_v

requires knowledge of three complete spectra associated with three dif-ferent boundary conditions, see the book of Gladwell[29]for a comprehensive analysis of this problem. We refer also to Schueller[30]for local uniqueness results related to an Euler-Bernoulli operator for a mixed-type inverse eigenvalue prob-lem with two even coefficients, and to Caudill et al.[31]for the first systematic study of isospectral coefficient sets for Euler-Bernoulli operators.

In the study of the fourth-order inverse eigenvalue problem with finite data further difficulties occur, mainly due to the non-uniqueness of the solution and to the difficulty in obtaining error estimates on the uniform approximation of the

unknown coefficient. It can be shown that these estimates require the knowledge of infinite eigenvalues or, at least, of an accurate asymptotic formula and of sufficient spectral data to make good approximation to the infinite data, see, for exam-ple,[32]. In real cases, neither of this is available. However, despite their importance and dissemination in practical appli-cations, general studies focused on inverse eigenvalue problems with finite data are relatively few. In this direction, the contribution by Barnes in[33]is illuminating. Barnes shows that, to deal with this class of inverse problems, it is vital to determine the weakest topology in which the available set of eigenvalues are continuous (with respect to the unknown coef-ficient) since, otherwise, one would attempting to extract more information from the spectral data than it contains. In[33], Barnes gives some refined approximation of the weakest topology.

With a view to these questions, and extending an idea developed in[34]for the identification of structural damage in classical full-scale rods, in this paper the finite inverse problem is investigated from a different point of view. Under the assumptions that the mass change is a small perturbation of the reference mass distribution of the axially vibrating nanorod, the inverse problem is linearized around the referential configuration, and frequency shifts caused by the mass change are correlated with generalized Fourier coefficients of the unknown mass variation. A numerical procedure based on an iterative first-order algorithm is proposed and a proof of local convergence of the reconstruction is provided. For the sake of complete-ness, it should be recalled that the idea of connecting Fourier coefficients of the unknown coefficient with frequency shifts is old and traces back to the cornerstone contribution in inverse eigenvalue theory given by Borg[35], see Hald[36]and Knobel and Lowe[37]for numerical applications.

The method has been tested on an extended class of coefficients, including smooth (e.g., continuous) and discontinuous mass variations, either with connected or disconnected support. Numerical simulations show good accuracy in approximat-ing smooth coefficients, even in the L1-norm, when the first 9 12 eigenfrequencies are used. The reconstruction of discon-tinuous coefficients turns out to be less accurate, especially because of non negligible oscillations of the reconstructed mass variation near the discontinuity points, and typically requires the first 15 20 eigenfrequencies to obtain reasonable accuracy.

The plan of the paper is as follows. The formulation of the mass identification problem is presented in Section2. Section3

is devoted to the description of the reconstruction method. This section includes the derivation of the eigenfrequency sen-sitivity to added masses (Section3.1), the linearization of the inverse problem (Section3.2), and its use in defining the iter-ative procedure (Section3.3). A proof of local convergence of the iterative method is presented in Section4. Results of a selected set of applications of the method are illustrated in Section5. The use of a posteriori physical filtering and the sta-bility of identification to noise are considered in Section5.3.4and Section5.3.5, respectively. Finally, for the sake of com-pleteness, a proof of the continuity of the eigenvalues and eigenfunctions with respect to perturbations of the mass density (Theorem 3.1) is included in the ‘‘Appendix”.

2. Formulation of the mass identification problem

The basis of the identification method proposed in this paper will be presented for a straight uniform nanorod in longi-tudinal vibration and under clamped end conditions. The spatial variation of the infinitesimal free longilongi-tudinal vibration of the unperturbed or referential nanorod of length L is governed by the following eigenvalue problem[38,39,21].

b

v

IV a

v

00_{¼ k}

_q

0

v

; x 2 0; Lð Þ; ð Þ1

v

ð Þ ¼ 0;0

v

00_{ð Þ ¼ 0;0 ð Þ2}

v

ð Þ ¼ 0;L

v

00_{ð Þ ¼ 0;L ð Þ3} 8 > < > :

k ¼

x

2_{being the eigenvalue, where}

_x

_{is the (radian) frequency, and}

v

_¼

v

_{ð Þ being the corresponding eigenfunction. Thex}

coefficient

_q

₀¼ const.,

q

0> 0, represents the mass density per unit length. The coefficient a ¼ const., a > 0, represents the

axial stiffness of the nanorod, and it can be expressed as a = EA, with E; E > 0, being the Young’s modulus, and A being a geo-metrical parameter that may be set to correspond with the cross-sectional area of the nanorod [38]. The coefficient b¼ const., b > 0, is determined as b¼ GA 2l2 0þ 4 5l 2 1   ; ð4Þ

where G¼ E= 2 1 þð ð

m

ÞÞ is the shear modulus defined in terms of E and of Poisson ratio

_m

;

m

> 0, and l0> 0; l1> 0 are length

scale parameters[12,38,21].

Among the possible non-classical boundary conditions for a clamped end, those selected correspond to a nil value of the higher order axial resultant, as it has been proposed by different authors, see[38,39]. Moreover, this condition ensures the self-adjointness of the eigenvalue problem and, then, the reality of the eigenvalues[21].

It is well known that, under our assumptions, the sequence of eigenpairs kf n;

v

nð Þxg1n¼1of (1)–(3) is equal to

kn¼ n

_p

L  2 ₁

q

0 aþ b n

p

L  2    ; ð5Þ

v

nð Þ ¼x ffiffiffiffiffiffiffiffi 2

q

0L s sin n

p

x L   ; ð6Þ

where the following mass-normalization condition has been used

Z L

0

q

0

v

2

nð Þ ¼ 1; n P 1:x ð7Þ

Note that the sequence kf gn 1n¼1of the unperturbed nanorod is uniformly discrete, that is, there exists a separation constant

r

> 0, only depending on the parameters of the system, such that

jkn kmj P

r

; ð8Þ

for every m; n 2 N, with m – n. In particular, a direct calculation shows that

r

¼ 1

q

0

p

L  2 aþ 2b

p

L  2  : ð9Þ

This property will be useful in proving the continuity of the eigenfunctions of the nanorod with respect to perturbations of the linear mass density (see the proof ofTheorem 3.1inAppendix).

Let us assume that the mass per unit length of the nanorod changes, and let us denote by

q

ð Þ ¼x

q

0þ rð Þ; x 2 0; Lx ½ ; ð10Þ

the mass density per unit length of the perturbed nanorod. We shall assume the following hypotheses on the perturbation r_: i) (L2-perturbation and smallness)

1 L Z L 0 rð Þx ð Þ2 dx  1 2 ¼



q

0; ð11Þ

where the perturbation parameter



is a real number such that 0_<



_{6 b}



, with b



_{< 1 a small number to be chosen later on.} ii) (Regularity)

r_ð Þ 2 Lx 1_{ð½0; LÞ; ð12Þ}

where L1ð½0; LÞ is the space of (Lebesgue measurable) functions f : 0; L½  ! R such that kf k1¼ ess supx2 0;L½ jf xð Þj < 1 almost

everywhere in 0½ ; L.

iii) (Uniform lower and upper bound)

0<

q

₆

_q

_{ð Þ 6x}

_q

þ_{; x 2 0; L½ ; ð13Þ}

with

q

_;

_q

þ_;

_q

þ_P

_q

0þ krk1, given constants independent of



.

Remark 2.1. The smallness of the mass variation r_ð Þ expressed inx (11)allows to consider either perturbations of small amplitude given on large portions of the interval 0½ ; L (e.g., diffuse mass change) or perturbation having large value concentrated in small parts of 0½ ; L. Moreover, it should be noticed that the mass identification problem in nanorods involves positive variations of the mass density

q

0, that is

r_ð Þ P 0; x 2 0; Lx ½ : ð14Þ

However, it is difficult to include this constraint in our analysis, and condition(14)will be used in Section5.3.4to post-filtering the results of the proposed identification method.

Let us denote by kf nð Þ;

q

v

nðx;

q

Þg1n¼1the perturbed eigenpairs of (1)–(3) when the coefficient

q

0is replaced by

q

ð Þ, that isx

b

v

IV a

v

00_{¼ k}

_q

v

_{; x 2 0; Lð Þ; ð Þ15}

v

ð Þ ¼ 0;0

v

00_{ð Þ ¼ 0;0 ð Þ16}

v

ð Þ ¼ 0;L

v

00_{ð Þ ¼ 0:L ð Þ17} 8 > < > :

Under our assumptions i)-iii), for any



; 0 <



6 b



, the eigenvalue problem (15)–(17) still maintains the properties of the unperturbed eigenvalue problem, with 0< k1ð Þ < . . . < k

q

nð Þ < . . . ; lim

q

n!1knð Þ ¼ 1. The main goal of this paper is:

q

Given the unperturbed nanorod, to recover the added mass r_ð Þ from the knowledge of the finite eigenvalue data kx f nð Þ

q

gNn¼1.

Recalling that the knowledge of a single full spectrum is not enough to determine uniquely a general coefficient r_ð Þ (seex

(with finite data) in which the mass coefficient is known in half of the nanorod, namely the mass variation r_ð Þ has supportx contained in 0ð ; L=2Þ:

supp rð_ð ÞxÞ ¼ x 2 0; Lf ½ j r_ð Þ – 0x g  0;L₂  

: ð18Þ

3. An iterative first-order reconstruction procedure

In this section we shall present a reconstruction procedure which is inspired to the Generalized Fourier Coefficient Method introduced in[34]to deal with damage identification in (classical) beams and rods. In its essence, the method is based on the linearization of the inverse problem in a neighborhood of the referential configuration, and on using the eigen-value sensitivity to the unknown perturbation (see next two subsections). An iterative version of the procedure will be pre-sented in the third subsection. A formal study of the convergence is proposed in Section4.

3.1. Eigenfrequency sensitivity to added mass

Let us introduce some notation. For any integer mP 0 and for any real numbers ‘1; ‘2, with1 < ‘1< ‘2< þ1; Hmð‘1; ‘2Þ

denotes the real-valued Hilbert space of the Lebesgue measurable functions f: ‘ð1; ‘2Þ ! R, such that

R‘2 ‘1 f 2_þPm i¼1 d i_f dxi  2   < þ1, wheredi_f

dxiis the ith weak derivative of f (see, for example,[40]). We notice that, when m¼ 0, H0_‘

1; ‘2

ð Þ coincides with the space L2_‘

1; ‘2

ð Þ of the square integrable functions in ð‘1; ‘2Þ, with norm

kf k_L2_‘ 1;‘2 ð Þ¼ R‘2 ‘1f 2 x ð Þdx  1 2

. In order to simplify the notation, the L2 norm will be often denoted askf k2. Finally, let us set

kf k1¼ maxx2 0;L½ jf xð Þj for any bounded function f ; f : 0; L½  ! R. The characteristic function

v

I: R ! R of the closed interval

I, I R, is defined as

v

Ið Þ ¼ 1 if x 2 I,x

v

Ið Þ ¼ 0 if x 2 R n I.x

Let us rewrite the unperturbed eigenvalue problem (1)–(3) in the following form

Fð

v

ð ÞxÞ ¼ kMð

v

ð ÞxÞ; ð19Þ

where the operators F: H2

0; L ð Þ ! H2 0; L ð Þ; M : H2 0; L ð Þ ! H2 0; L ð Þ are defined as Fð

v

ð ÞxÞ ¼ b

v

IV x ð Þ  a

v

00_{ð Þ;x ð20Þ} Mð

v

ð ÞxÞ ¼

q

0

v

ð Þ:x ð21Þ

The perturbed eigenvalue problem (15)–(17) can be described similarly, that is

Fð

v

ðx;

q

ÞÞ ¼ kð ÞM

q

_ð

v

ðx;

q

ÞÞ; ð22Þ

where the operator M: H2ð0; LÞ ! H2

0; L

ð Þ is defined as

Mð

v

ðx;

q

ÞÞ ¼

q

v

ðx;

q

Þ;

q

¼

q

0þ r: ð23Þ The setH of admissible configurations of the clamped nanorod is given by

H ¼ f : 0; Lð Þ ! R j f 2 H2

0; L

ð Þ; f xð Þ ¼ 0 at x ¼ 0 and at x ¼ L

n o

: ð24Þ

We introduce the following operations fromH  H to R:

M fð Þ; g h i ¼ Z L 0

q

0fg; ð25Þ F fð Þ; g h i ¼ Z L 0 bfIV af00   g: ð26Þ

Integration by parts shows that both the operators M and F are self-adjoint in H0¼ f : 0; Lf ð Þ ! R j f 2 H;

f00ð Þ ¼ 0 at x ¼ 0 and at x ¼ Lg:, that is, M fx h ð Þ; gi ¼ fh ; M gð Þi and F fh ð Þ; gi ¼RL 0 bf

00

g00þ af0g0

 

¼ f ; F gh ð Þi for every f; g 2 H0.

In the sequel, we shall use an explicit expression of the first-order perturbation (with respect to the parameter



) of the eigenvalues. Note that, the nth unperturbed and perturbed eigenpair will be denoted by kf n;

v

nð Þxg and kf nð Þ;

q

v

nðx;

q

Þg,

respectively. In order to find the first-order change of the nth eigenvalue, we find convenient to define

D

kn¼ knð Þ  k

q

n; ð27Þ

D

M¼ M_ M; ð29Þ

and rewrite the normalization condition(7)as

Mð Þ;

v

n

v

n

h i ¼ 1: ð30Þ

Let us multiply Eq.(19)(written for the nth unperturbed eigenfunction

v

nð Þ) and Eq.x (22)(written for the nth perturbed

eigenfunction

v

nðx;

q

Þ) by

v

nðx;

q

Þ and

v

nð Þ, respectively. Integrating, subtracting side by side, using the self-adjointnessx

property and condition(30), we deduce the fundamental identity between the states corresponding to mass coefficients

_q

₀ and

q

:

D

kn¼ knh

D

Mð Þ;

v

n

v

ni  knh

D

Mð

D

v

nÞ;

v

ni 

D

knhMð

D

v

nÞ;

v

ni 

D

knh

D

Mð Þ;

v

n

v

ni 

D

knh

D

Mð

D

v

nÞ;

v

ni: ð31Þ Next, we introduce the following theorem about the continuity of eigenvalues and eigenfunctions of the problem (15)–(17) with respect to L2-perturbations of the mass coefficient. This result will be also useful in studying the convergence of the iterative identification method presented in Section4.

Let us denote as a priori data the set_{A of quantities defining the unperturbed model (e.g., the coefficients a; b;}

_q

_{0; L), and} the uniform lower and upper bound of the mass density of the perturbed model

_q

ð Þ:x

A ¼ a; b;f

q

0; L;

q

;

q

þg: ð32Þ

Note that the separation constant

_r

of the unperturbed nanorod depends on the a priori quantities belonging toA, and there-fore it will not be explicitly included inA.

Theorem 3.1. Under the above notation, let

q

ið Þ ¼x

q

0þ r;ið Þ, where rx ;ið Þ satisfiesx (11)–(13), i¼ 1; 2. Let knð Þ;

q

i

v

nðx;

q

iÞ



; n P 1, be the nth eigenpair of (15)–(17) for i ¼ 1; 2. For every nP 1, there exists a constant Ck

n, Ckn> 0, only depending on the a priori data A and n, such that

jknð Þ  k

q

1 nð Þj 6 C

q

2 nkk

q

1

q

2k2: ð33Þ Let the eigenfunctions

v

nðx;

q

1Þ;

v

nðx;

q

2Þ be normalized such that

Z L 0

q

1

v

2 nðx;

q

1Þ ¼ Z L 0

q

1

v

2 nðx;

q

2Þ ¼ 1 ð34Þ and

v

0

nð0;

q

1Þ

v

0nð0;

q

2Þ > 0; n P 1. For every n P 1, there exist a number b



, 0< b



< 1, and a constant Cvn; Cvn> 0, both only

depending on the a priori dataA and n, such that

jj

v

nðx;

q

1Þ 

v

nðx;

q

2Þjj26 Cvnk

q

1

q

2k2; ð35Þ for every

q

1;

q

2satisfyingk

q

1

q

2k26 L

1 2b



_.

A proof ofTheorem 3.1is presented in theAppendix.

By using inequalities(33), (35)in identity(31), for 0<



6 b



, the first order change with respect to



of the nth eigenvalue is given by knð Þ ¼ k

q

n kn Z L 2 0 r_ð Þx

v

2 nð Þdx;x ð36Þ

for every nP 1 and for rsatisfying(18).

To prove(36), we first start by showing thatj

v

nð Þj is uniformly bounded in 0; Lx ½  in terms of the a priori data A and n. By

the definition of the eigenvalue problem for the unperturbed nanorod, by Hölder inequality and by using the normalization condition(7), for every x2 0; L½  and every n P 1, we have

j

v

nð Þj ¼x Z x 0

v

0 nð Þdss    6 L1 2 Z L 0

v

0 nð Þx  2 dx  1 2 6 L_a  1 2 Z L 0 b

v

00nð Þx  2 þ a

v

0 nð Þx  2   dx  1 2 6 L_a  1 2 k 1 2 n: ð37Þ

Next, we show that all the terms on the right hand side of(31), but the first, are of higher order with respect to



. Let us consider the second term. By using(37), Hölder inequality and estimate(35)inTheorem 3.1, for 0<



6 b



, we have

jknh

D

Mð

D

v

nÞ;

v

nij6 kn Z L 2 0 rð Þxð

v

nðx;

q

0þ rÞ 

v

nð ÞxÞ

v

nð Þdxx       6 knk

v

nk1krk2k

v

nðx;

q

0þ rÞ 

v

nð Þkx 2 6 knk

v

nk1Cvnkrk 2 26 C



2_{; ð38Þ}

where C> 0 is a constant only depending on the a priori data A and n.

The estimates of the third, fourth and fifth terms can be performed similarly, resulting, for 0<



6 b



,

where C> 0 is a constant only depending on the a priori data A and n. By using(38) and (39)in(31), and neglecting higher order terms with respect to



, we obtain(36).

Let us comment the expression(36). As expected by the general theory (see, for instance,[41]), expression(36)states that addition of mass causes decrease in all the eigenvalues. More precisely, the eigenvalue shift kð nð Þ  k

q

nÞ turns out to be

pro-portional to kn. This fact seems to have a certain importance in the inverse problem, since the relative variation of the

eigen-values appears to be significant also for large order n. Finally, it should be noticed that the expression(36)is independent of the boundary conditions of the eigenvalue problem and, therefore, the analysis could be extended also to other sets of boundary conditions of the nanorod.

3.2. The linearized inverse problem

In this section we shall use the eigenvalue sensitivity determined in(36)to formulate a linearized version of the inverse problem and to find an approximate solution.

Using the explicit expression of the unperturbed eigenfunctions(6)in(36), we have

dkn 1  kn

q

ð Þ kn ¼ Z L 2 0 r_ð Þx

U

nð Þdx;x ð40Þ where

U

nð Þ x ð

v

nð ÞxÞ2¼ 2

q

0L sin2 n

p

x L   ; n P 1: ð41Þ

Eq. (40) shows that the first-order relative shift of the nth eigenvalue coincides with the scalar product between the unknown mass variation r_ð Þ and the nth element of the familyx fUmð Þxg1m¼1.

In order to determine rð Þ, a natural choice is to represent rx ð Þ on the familyx fUmð Þxg1m¼1as

rð Þ ¼x X 1 k¼1 bk

U

kð Þx

v

0;L 2 ½ ; ð42Þ

where the convergence of the series should be understood in the mean, that is

lim K!1 Z L 2 0 rð Þ x X K k¼1 bk

U

kð Þx !2 ¼ 0: ð43Þ The coefficients bð Þk 1

k¼1play the role of Generalized Fourier Coefficients of the unknown mass variation rð Þ evaluated on thex

familyf_Umð Þxg1m¼1. It should be noted that one cannot a priori exclude that the function rð Þ defined inx (42)may change sign

in the interval 0ð ; L=2Þ; see also the remark after condition(14).

It is at this point that the a priori condition(18)can be mathematically justified. In fact, the familyf_Umð Þxg1m¼1is a basis of

the square integrable functions defined on half span of the nanorod, e.g., the set L2 _0;L 2

 

. To prove this property, we notice that the functionsfUmð Þxg1m¼1are linearly independent and form a complete family in L2 0;L2

 

. This last property is satisfied if, for any rð Þ 2 Lx 2

0;L 2

 

, the conditionsR2L

0rð ÞxUmð Þdx ¼ 0 for every n P 1 imply rx ð Þ ¼ 0 in 0;x 2L

 

. The above conditions can be rewritten as 0¼ Z L 2 0 r_ð Þdx x Z L 2 0 r_ð Þ cosx 2m

p

x L   dx; ð44Þ

for every mP 1. Taking the limit in(44)as m! 1 and using the Riemann-Lebesgue Lemma (see, for instance,[42]), we have RL 2 0rð Þdx ¼ 0 and, then,x RL 2 0rð Þ cosx 2mLpx  

dx¼ 0 for every m P 1. Since the family cos 2mpx L

 

1

m¼1is a basis for the set of

func-tions belonging to L2 _0;L 2

 

and having zero-mean, we have r_ð Þ ¼ 0 in 0;x L 2

 

, and the thesis is proved. Replacing(42)in(40), we obtain the infinite linear system

dkn¼ X1 k¼1 Ankbk; n ¼ 1; 2; . . . ; ð45Þ where Ank¼ Z L 2 0

U

nð Þx

U

kð Þdx ¼x 4

q

0L ð Þ2 Z L 2 0 sin2 n

p

x L   sin2 k

p

x L   dx; ð46Þ

n; k ¼ 1; 2 . . .. The coefficients Ankcan be evaluated in closed form, and we have

Ank¼ 2 4

q

2 0L for k– n; Ann¼ 3 4

q

2 0L : ð47Þ

In real applications only a finite number of eigenvalues is available, typically the first 10 20. This leads us to consider the Nth finite dimensional approximation of the added mass, bN

k

 N

k¼1of(45), that is the N N linear system

dkn¼

XN k¼1

AnkbNk; n ¼ 1; . . . ; N: ð48Þ

A direct calculation shows that

det Að nkÞ ¼ 2N þ 1ð Þ 1 4

q

2 0L  N ; ð49Þ Ank ð Þ1_{¼ 4}

q

2 0L   2N  1 2Nþ 1 if n¼ k; ðAnkÞ1¼  4

q

20L   2 2Nþ 1 if n– k; ð50Þ

n; k ¼ 1; . . . ; M. Therefore, the system(48)has the closed-form solution

bNk ¼ 4

q

20L 2N 1 2Nþ 1 dkk 2 2Nþ 1 XN j¼1; j–k dkj ! ; k ¼ 1; . . . ; N; ð51Þ and, finally, r_ð Þ ¼ 8x

q

0 XN k¼1 2N 1 2Nþ 1 dkk 2 2Nþ 1 XN j¼1; j–k dkj ! sin2 k

p

x L   

_v

_0;L 2 ½ : ð52Þ

3.3. The reconstruction procedure

The estimation of rgiven in(52)can be improved by iterating the identification procedure illustrated in the previous section. In order to simplify the notation, here the index



has been omitted and kexp

n denotes the measured value of the

nth eigenvalue knð Þ of the perturbed nanorod. Moreover, we shall write b instead of b

q

N. The main steps of the reconstruction

procedure and the corresponding numerical algorithm are illustrated in the sequel. Let

q

ð Þ0_{ð Þ ¼x}

_q

0be the mass per unit length of the referential nanorod. The unknown mass per unit length is determined

on the interval 0;L 2



by the iteration

q

ðjþ1Þ_{ð Þ ¼x}

_q

ð Þj_{ð Þ þ rx} ð Þj_{ð Þ; j P 0;x ð53Þ} where the increment

rð Þj_{ð Þ ¼x} XN k¼1 bð Þkj

U

j ð Þ k ð Þx

v

0;L 2 ½  ¼ bð Þj 

U

ð Þjð Þx

v

0;L 2 ½  ð54Þ

is determined by solving the N N linear system

dkð Þnj  1  k exp n knð

q

ð ÞjÞ ¼X N k¼1 Að Þnkjb j ð Þ k ; ð55Þ

n¼ 1; . . . ; N, or, equivalently, in compact form

Að Þjbð Þj ¼ dkð Þj_{; ð56Þ} with bð Þj ¼ bð Þj 1; . . . ; b j ð Þ N   . Here, kn

q

ð Þj   ;

v

n x;

q

ð Þj   

is the nth (mass normalized) eigenpair of the problem

b

v

IV a

v

00_{¼ k}

_q

ð Þj

v

_{; x 2 0; Lð Þ; ð Þ57}

v

ð Þ ¼ 0;0

v

00_{ð Þ ¼ 0;0 ð Þ58}

v

ð Þ ¼ 0;L

v

00_{ð Þ ¼ 0:L ð Þ59} 8 > < > : Moreover,Uð Þ_kjð Þ ¼x

v

2 k x;

q

j ð Þ  

and the matrix A ð Þ_nkjis given by

Að Þ_nkj ¼ Z L 2 0

U

j ð Þ nð Þx

U

j ð Þ k ð Þdx; n; k ¼ 1; . . . ; N:x ð60Þ

rð Þj_{ð Þ ¼ Ax}  ð Þj1

dkð Þj 

U

ð Þjð Þx

_v

_0;L 2

½  ð61Þ

and, from(53), we have

q

ð Þj_{ð Þ ¼x}

_q

0þ

Xj1 i¼0

rð Þi_{ð Þ; j P 1:x ð62Þ}

In our application, the iterations go on until the updated mass coefficient satisfies the criterion

e1 N XN n¼1 kexpn  kn

q

jþ1 ð Þ kexpn  2!12 <

c

; 0 @ ð63Þ

for a small given number

c

.

4. Convergence of the identification method

In this section we study the convergence of the iterative method shown in Section3.3. We prove that, under suitable assumptions, there exists

q

ð Þ 2 Cx 0

0;L 2   such that

q

ð Þ ¼ limx j!1

q

j ð Þ_{ð Þ ¼x}

q

0þ X1 i¼0 rð Þi_{ð Þ;x ð64Þ}

where the series is uniformly convergent in 0;L 2

 

. The proof is based on two main steps:

Step i) Existence of the inverse A ð Þj1and bound ofk A ð Þj1

k for every j P 1; Step ii) Bound ofjdkð Þj_{j to control kr}ð Þi_k

2for every jP 1.

In the sequel, we shall denote by C positive constants which may change from line to line. Moreover, for every N N matrix B2 RNN_{, we denote bykBk ¼} PN

i;j¼1 Bij 2

 1

2

the Frobenius norm.

Step i). Let us consider the set of coefficients

q

ð Þ given inx (10), e.g.,

q

ð Þ ¼x

q

0þ rð Þ in 0; Lx ½ , where rsatisfies conditions (10)–(13)and where



_{> 0 is a number small enough. We denote by A}ð Þ the matrix defined in

_q

(60)corresponding to the coefficient

q

ð Þ.x

We prove that there exists b



q, 0< b



q< 1, only depending on the a priori data A and N, such that, for every



< b



q, there

exists Að ð Þ

q

Þ1

and we have

k Að ð Þ

q

Þ1_{k 6 C; ð65Þ}

where C> 0 is a constant only depending on the a priori data A and N. To prove(65), we shall write

Að Þ ¼ A

q

ð Þ  P;

q

0 ð66Þ

where the inverse of Að Þ is given explicitly in

q

0 (50), and P¼ Pð Þ is a perturbation due to the change of mass r

q

ð Þ. Denotingx

D

v

kð Þ ¼x

v

kðx;

q

Þ 

v

kðx;

q

0Þ 

v

kðx;

q

Þ 

v

kð Þ, for every n; k ¼ 1; . . . N, byx (60)we have

Pnk¼ 2 Z L 2 0

v

k

v

2n

D

v

kþ

v

n

v

2k

D

v

n   þ Z L 2 0

v

2 nð

D

v

kÞ 2_þ

v

2 kð

D

v

nÞ 2_{þ 4}

v

n

v

k

D

v

n

D

v

k   þ 2 Z L 2 0

v

n

D

v

nð

D

v

kÞ 2_þ

v

k

D

v

kð

D

v

nÞ 2   þ Z L 2 0

D

v

n ð Þ2

D

v

k ð Þ2 : ð67Þ

In order to simplify the analysis, we shall control the L1norm of all the quantities inside the integrals in(67). The norm k

v

kk1(even in the whole interval 0ð ; LÞ) can be bound via inequality(37), which holds for unperturbed eigenfunctions with

a constant C> 0 only depending on the a priori data A. In order to bound kD

v

kk1, we use the Rellich-Kondrachov’s

Theo-rem (see[40], Theorem IX:26) and estimate(157): for every kP 1 we have

k

D

v

kkL1ð0;LÞ6 Ck

D

v

kkH2_ð0;LÞ6 Ck

q



q

₀k 1 2 L2_ð0;LÞ6 C



1 2; ð68Þ

for every



6 b



ð< 1Þ, where b



has been introduced inTheorem 3.1, and C> 0 is a constant only depending on the a priori dataA and k. Therefore, for every n; k ¼ 1; . . . ; N and for every



such that



6 b



, we have

jPnkj 6 C



1 2þ



þ



3 2þ



2   6 C



1 2 ð69Þ

and

kPk 6 CP



1

2; ð70Þ

where CP> 0 is a constant only depending on the a priori data A and N.

Let us introduce the number

b



q¼ min b



; 1 2CPk Að ð Þ

q

0 Þ 1_k !2 8 < : 9 = ;: ð71Þ

Note that, possibly increasing CP, the number _2C 1

Pk Að ðq0ÞÞ1k

 2

is less than 1. We now prove that if P satisfies(70)for every



6 b



q, then Að Þ is nonsingular, and

q

k Að ð Þ

q

Þ1_{k 6} k Að ð Þ

q

0Þ

1_{k  kIk}

1 k Að ð Þ

q

0Þ

1_{k  kPk}; ð72Þ

where I is the identity matrix in RNN_{. Let x2 R}N_{n 0f g. Then, since A}

_q

0

ð Þ is nonsingular, it is enough to prove that I Að ð Þ

q

0 Þ

1

P

 

x– 0 for every x 2 RN_{n 0f g. We have}

k I  Að ð Þ

q

0Þ 1_P   xk P 1  k Að ð Þ

q

0Þ 1_{k  kPk}   kxk > 0; ð73Þ

since, by(70) and (71), for every



; 0 <



6 b



q, 1 k Að ð Þ

q

0Þ 1_{k  kPk P 1  k A}

q

0 ð Þ ð Þ1_kC P



1 2P1 2> 0: ð74Þ Hence, if x– 0; I  Að ð Þ

q

0Þ 1 P  

x– 0 or, equivalently, Að ð Þ  P

q

0 Þx – 0 and Að Þ ¼ A

q

ð Þ  P is nonsingular.

q

0

Now, from the identity Að ð Þ  P

q

0 Þ Að ð Þ  P

q

0 Þ

1_{¼ I, it follows that} I Að ð Þ

_q

0 Þ 1_P  1 ¼ I þ Að ð Þ

_q

0Þ 1_{P I A}

q

0 ð Þ ð Þ1_P  1 ð75Þ and, then, k I  Að ð Þ

q

0Þ 1_P  1 k 6 kIk 1 k Að ð Þ

q

0Þ 1_{k  kPk}: ð76Þ

Inequalities(76) and (72)are equivalent, and imply(65).

Step ii). In order to controljdkð Þi_{j, we follow the steps of the iterative procedure shown in Section3.3.}

At step]1, we use the fundamental identity(31)between the unperturbed 0th state (corresponding to

q

ð Þ0_{ð Þ ¼x}

_q

0) and

the state to be identified (corresponding to the unknown mass coefficient, say

q

ð Þ, and target eigenvaluesx

kexpn  ; n ¼ 1; . . . ; N): kexpn ¼ kn

q

ð Þ0    kn

q

ð Þ0  

D

Mð Þ0

v

n

q

ð Þ0   ;

v

n

q

ð Þ0   D E þ eOnðEXP 0ð ÞÞ; ð77Þ n¼ 1; . . . ; N, whereDMð Þ0 _¼

q

ð Þ x

q

ð Þ0_{ð Þ and the higher order term ex O}

nðEXP 0ð ÞÞ depends on the initial state

q

ð Þ0 and on the

variationsDMð Þ0_;D

kð Þn0 ¼ k exp

n  kn

q

ð Þ0;D

v

nð Þ ¼x

v

nðx;

q

Þ 

v

nx;

q

ð Þ0. The mass coefficient

q

ð Þ0 is updated at step]1 as

q

ð Þ1_{ð Þ ¼x}

_q

ð Þ0_{ð Þ þ rx} ð Þ0_{ð Þ;x ð78Þ}

where rð Þ0_{ð Þ (given inx (54)for j¼ 0) is determined by neglecting the term eO}

nðEXP 0ð ÞÞ in(77), e.g., for every n¼ 1; . . . ; N,

kexpn  kn

q

ð Þ0   ¼ kn

q

ð Þ0   rð Þ0

_v

n

q

ð Þ0   ;

v

n

q

ð Þ0     : ð79Þ

By using the updated value of the mass coefficient

q

ð Þ1_{ð Þ, we now estimate kx}

n

q

ð Þ1; n ¼ 1; . . . ; N. By applying the

fundamen-tal identity(31)between the states

q

ð Þ0_{ð Þ andx}

_q

ð Þ1_{ð Þ, we havex}

kn

q

ð Þ1    kn

q

ð Þ0   ¼ kn

q

ð Þ0   rð Þ0

_v

n

q

ð Þ0   ;

v

n

q

ð Þ0     þ eOnðð Þ  01 ð ÞÞ: ð80Þ The first term on the right hand side of(80)can be determined by(79), namely

kn

q

ð Þ1    kn

q

ð Þ0   ¼ kexp n  kn

q

ð Þ0   þ eOnðð Þ  01 ð ÞÞ; ð81Þ that is kn

q

ð Þ1    kexp n ¼ eOnðð Þ  01 ð ÞÞ; n ¼ 1; . . . ; N: ð82Þ

Similarly, at step]2, the mass coefficient is updated as

q

ð Þ2_{ð Þ ¼x}

_q

ð Þ1_{ð Þ þ rx} ð Þ1_{ð Þ;x ð83Þ} where rð Þ1_{ð Þ satisfiesx} kexpn  kn

q

ð Þ1   ¼ kn

q

ð Þ1   rð Þ1

v

n

q

ð Þ1   ;

v

n

q

ð Þ1     ; ð84Þ

n¼ 1; . . . ; N. The identity(31)between the states

q

ð Þ1_{ð Þ andx}

_q

ð Þ2_{ð Þ reads asx}

kn

q

ð Þ2    kn

q

ð Þ1   ¼ kn

q

ð Þ1   rð Þ1

v

n

q

ð Þ1   ;

v

n

q

ð Þ1     þ eOnðð Þ  12 ð ÞÞ; ð85Þ and, by(84) and (85), we obtain

kn

q

ð Þ2

   kexp

n ¼ eOnðð Þ  12 ð ÞÞ; n ¼ 1; . . . ; N: ð86Þ By iteration, at step] i þ 1ð Þ we have

kn

q

ðiþ1Þ

   kexp

n ¼ eOnððiþ 1Þ  ið ÞÞ; n ¼ 1; . . . ; N: ð87Þ At this point we need to bound the high order term eOnððiþ 1Þ  ið ÞÞ appearing in(87).

Let us first estimate eOnðð Þ  01 ð ÞÞ. By(31)(written between the unperturbed state

q

ð Þ0 and the first updated state

q

ð Þ1) we

have

jeOnðð Þ  01 ð ÞÞj 6 knjh

D

Mð

D

v

nÞ;

v

nij þ j

D

knj  j Mh ð

D

v

nÞ;

v

nij þ j

D

knj  jh

D

Mð Þ;

v

n

v

nij þ j

D

knj  jh

D

Mð

D

v

nÞ;

v

nij

 Anþ Bnþ Cnþ Dn: ð88Þ

Let us consider the term An. By using Hölder inequality, we have

An6 knjj

v

njj1 Z L 0 j

D

M xð Þj  j

v

n x;

q

ð Þ1   

v

n x;

q

ð Þ0   jdx 6 knjj

v

njj1jj

D

Mjj2 jj

v

n x;

q

ð Þ1   

v

nð Þjjx 2: ð89Þ Recalling thatDM¼ rð Þ0_{ð Þ ¼x} PN k¼1bð Þk0Uk x;

q

ð Þ0  

v

0;L 2 ½ , withUk x;

q

ð Þ0   ¼

v

2 k x;

q

ð Þ0  

, and noticing thatk

v

nk1can be estimated

via(37), we have jj

D

Mjj26 L32 a

K

jb 0 ð Þ_{j; ð90Þ}

where the quantity

K

ð Þ0 _¼ XN k¼1 k2k !1 2 ð91Þ

can be calculated in terms of the unperturbed nanorod and, therefore, it depends on the a priori data setA only. By using

(37),(90) and (35)in(89), we have An6 kð Þn 3 2L L a  5 2

K

ð Þ0  2 Cvnjb 0 ð Þ_j2_{: ð92Þ}

The other terms in(88)can be evaluated similarly, namely

Bn6

q

1 2 0L L a  2

K

ð Þ0  2 Ck nCvnjb 0 ð Þ_j2 ; ð93Þ Cn6 knL 3 2 L a  3

K

ð Þ0  2 Ck njb 0 ð Þ_j2 ; ð94Þ Dn6 kð Þn 1 2_L32 L a  7 2

K

ð Þ0  3 Ck nCvnjb 0 ð Þ_j3_{: ð95Þ}

Therefore, collecting the estimates(92)–(95), we have

jeOnðð Þ  01 ð ÞÞj 6 Cjbð Þ0j 2

1þ Cjbð Þ0_j

 

; ð96Þ

where C> 0 is a constant only depending on the a priori data A and n; n ¼ 1; . . . ; N. By using(65)in(56)(for j¼ 0), we have

and(96)becomes

jeOnðð Þ  01 ð ÞÞj 6 Cjdkð Þ0j2 1þ Cjdkð Þ0j

 

6 Cjdkð Þ0_j2_{; ð98Þ}

provided thatjdkð Þ0

j 6 1, where C > 0 is a constant only depending on the a priori data A and n. Therefore, using(98)in(82), for every n¼ 1; . . . ; N, we have

jkn

q

ð Þ1    kexp n j 6 Cjdk 0 ð Þ_j2 ; ð99Þ

forjdkð Þ0_{j 6 1, where C > 0 is a constant only depending on the a priori data A and n.}

We proceed similarly in estimating eOnðð Þ  12 ð ÞÞ. By adapting the previous analysis to the states

q

ð Þ2 and

q

ð Þ1, we obtain

jeOnðð Þ  12 ð ÞÞj 6 Cjbð Þ1j2 1þ Cjbð Þ1j

 

; ð100Þ

and, recalling(65) and (56),

jeOnðð Þ  12 ð ÞÞj 6 Cjdkð Þ1j2; ð101Þ that is, by(86), jkn

q

ð Þ2    kexp n j 6 Cjdk 1 ð Þ_j2 ; ð102Þ

forjdkð Þ1_{j 6 1, where C > 0 is a constant only depending on the a priori data A and n. Using(99)in(102)(and noticing that}

kn

q

ð Þ1can be estimated from below by a positive quantity only depending on the a priori dataA and n), we have

jkn

q

ð Þ2    kexp n j 6 Cjdk 0 ð Þ_j22 : ð103Þ

By iteration, for every iP 1 and n ¼ 1; . . . ; N, we obtain

jkn

q

ð Þi    kexp n j 6 Cjdk 0 ð Þ_j2i ; ð104Þ

where C> 0 is a constant only depending on the a priori data A and n, and jdkð Þ0_{j 6 1.}

Conclusion of the proof. By using the estimates(65) and (104), the seriesP1_i¼0rð Þi_{ð Þ inx (64)is L}2_{-uniformly convergent to a}

continuous function in 0;L 2



, since the ith term can be bounded as

krð Þi_{ð Þkx} L2_0;L 2 ½  6 Cjdkð Þ0j2 i ; i P 0; ð105Þ

where C> 0 is a constant only depending on the a priori data A, and the numerical seriesP1 i¼0jdkð Þ0j2

i

is convergent provided thatjdkð Þ0

j 6 b



k< 1 (Weierstrass’s criterion).

Summarizing, in this section we have proved the following result:

If



6 b



q, where b



qis given in(71), andjdkð Þ0j 6 b



k< 1, then the iterative procedure of identification converges uniformly to a

continuous function in 0;L 2



.

Our convergence result has local character, since the proof is built on the assumption that the mass variation is a small perturbation of the total mass of the unperturbed nanorod. It would be interesting, and also useful for practical applications, to determine or, at least, to find good approximation of the maximum value allowed for b



q. Moreover, the local character is

also reflected on the condition b



k< 1, which means that the first N eigenvalues of the unperturbed nanorod must be close

enough to the corresponding target eigenvalues.

5. Application of the Generalized Fourier Coefficient method 5.1. Specimen

In order to illustrate the application of the identification method, reference is made to the geometrical and material prop-erties of the nanorod used in[43]. In particular, the diameter D of the circular equivalent cross-section is equal to 100

_l

m (¼ 100  106_{m) and the length L is taken equal to 20D; the material length scale parameters are assumed to be equal, and}

‘0¼ ‘1¼ ‘2¼ ‘ ¼ 17:6

l

m; the Young’s modulus E is equal to 1:44 GPa; the Poisson’s coefficient is

m

¼ 0:38; and the volume

mass density is equal to

q

vol¼ 1000 kg=m3. The coefficients a; b;

q

0corresponding to the above parameters take the value

a¼ 11:310 N; b ¼ 3:554  109 Nm2_;

_q

0¼

q

vol

p

D 2 4¼ 7:854  10 6 kg=m, respectively.

5.2. Discrete model and eigenvalue problem

The practical application of the mass identification method requires the development of a specific numerical code. The weak formulation of the eigenvalue problem (15)–(17) consists in determining a non-trivial function

v

2 H (see definition

(24)), and a positive real number k such that

ZL 0 b

v

00

_u

00þ a

v

0

u

0   ¼ k Z L 0

q

v

u

; ð106Þ

for every

u

2 H. To find a finite element model of the weak formulation(106), we work on the finite-dimensional subspace ofH formed by three-degree polynomial spline approximation of the axial displacement of the nanorod in each element. More precisely, let xf 0¼ 0 < x1< x2< . . . < xNe¼ Lg be the nodes of a mesh of the interval 0; L½ , with xiþ1 xi¼Dx¼NLe, for every i¼ 0; 1; . . . ; Ne. Therefore, the discrete version of(106) consists in finding the approximating eigenpair ek; e

v

 

, e

v

2 R2Nen 0f g, solution to

eF e

_v

¼ ekfMe

v

; ð107Þ

where eF; fM is the 2Ne 2Nereal symmetric matrix of the stiffness and of the inertia of the nanorod, respectively.

The mass coefficient

_q

¼

q

ð Þx is approximated by a continuous piecewise-linear function, that is,

e

q

ð Þ ¼x

q

ð Þ þxi fð

q

ðxiþ1Þ 

q

ð Þxi Þ=ðDxÞgx for x 2 x½ i; xiþ1. Therefore, the n; kð Þ entry of the local mass, Menki

 

, and stiffness, eFnk

i

 

, matrix are given by

e Mnk i ¼ Z xiþ1 xi e

q

ð Þx

u

nð Þx

u

kð Þdx; eFx nk i ¼ Z xiþ1 xi b

u

00 nð Þx

u

00kð Þ þ ax

u

0nð Þx

u

0kð Þx   dx; ð108Þ

n; k ¼ 1; . . . ; 4, and are evaluated in exact form. Numerical integration of the quantities(60)was developed by means of a standard trapezoidal method. The discrete eigenvalue problem(107)was solved by using the Implicit Restarted Lanczos’s method[44]. The identification procedure was built in Scilab environment (version 5_{:5:2) by developing a specific numerical} code. Regarding the computational burden, the time needed to complete a single iteration of the identification algorithm (for Ne¼ 200) was about 1 s.

5.3. Results

In this section we present a selection of the results obtained in an extended series of applications of the Generalized Four-ier Coefficient Method. Among other parameters, the procedure has been tested with respect to the number N of the first eigenfrequencies used and to the geometry of the mass variation (e.g., position, intensity, regularity). In particular, the effi-ciency of the method has been evaluated on two main classes of mass variations corresponding to smooth (Section5.3.1) or discontinuous (Section5.3.2) function r_. In both cases, the support of the mass variation coincides with an interval contained in 0;L

2

 

. We have first considered free-error data, i.e., the measurement errors are null with the exception of the errors induced by the numerical approximation via the finite element method. The analysis of overlapping mass variations, possibly having disconnected support, is considered in Section5.3.3. Section5.3.4is devoted to the application of suitable physical filtering to improve the accuracy of identification. Finally, the stability of the reconstruction in presence of noise is investi-gated in Section5.3.5.

5.3.1. Identification of smooth mass densities

In this series of simulations, the mass density of the perturbed nanorod is defined as

q

ð Þ ¼x

q

0þ

q

0t cos 2

p

ðx sÞ c  

v

sc 2;sþ2c ½ ; ð109Þ

where s is the central point of the support of the mass variation, c is the length of the support,

q

0t is the maximum amplitude

of variation, seeFig. 1(a). Note that

_q

2 C1₀

; L ½ .

The definition(109)allows to consider a wide family of coefficients, including either localized or diffuse mass variations. In the sequel, reference is made mainly to positionss

L¼ 0:15; 0:25; 0:35, extensionscL¼ 0:10; 0:20; 0:30, and intensities

t¼ 0:10; 0:20; 0:50; 1:00, for all the possible combinations of the parameters. Therefore, the global mass change ranges from 0_{:5 percent to 15 percent of the initial mass}

_q

₀L, for c

L¼ 0:10; t ¼ 0:10   and c L¼ 0:30; t ¼ 1:0   , respectively. In order to select a suitable mesh for the numerical solution of the eigenvalue problem, preliminary tests have been car-ried out on the uniform nanorod, for which analytical closed form expressions of the eigenpairs are available. The analysis suggests to assume a mesh with Ne¼ 200 equally spaced finite elements, which turns out to be a good compromise between

including the reconstruction procedure, seeTable 1. Moreover, this set of preliminary tests suggests to choose

_c

¼ 105_{in the}

stopping criterion(63).

Regarding the influence of the number N of eigenvalue data, here we will focus on the most two challenging cases. The first one (s

L¼ 0:35; c

L¼ 0:1; t ¼ 0:1), seeFig. 2, corresponds to small mass increase located in a small interval, which allows to

assess the sensitivity of the method to the identification of the mass perturbation just as it occurs in the nanorod. The second case (s

L¼ 0:35;cL¼ 0:1; t ¼ 1, seeFig. 3) deals with mass variation having the same support as before, but with an abrupt

increase of mass with large L1amplitude.

The analysis ofFig. 2shows that the identified coefficient agrees well with the exact one, and accuracy of reconstruction rapidly improves as N increases. Similar properties can be deduced from the analysis ofFig. 3, apart from the oscillatory char-acter of the reconstructed coefficient, which is now more evident for N¼ 6; 9, whereas it becomes almost negligible when N¼ 15.

For the sake of completeness, it should be noted that part of our results involve not necessarily small mass variations, see, for example,Fig. 4. This would suggest that the proposed reconstruction method has some unexpected potential, in spite of the fact that the proof of convergence presented in Section4requires to work in a sufficiently small neighborhood of the referential nanorod.

Finally, some synthetic numerical information concerning the sequence of iterations is reported inTable 2. Few iterations are sufficient to satisfy the convergence criterion, e.g., less than five in these cases. The quantity e defined in(63), which can be considered at each iteration step as the average discrepancy between identified and target eigenvalues, is reduced by at least 3 4 orders of magnitude with respect to the initial value. The errors on the mass coefficient, either evaluated on L2

or L1norm, are reduced significantly through the iterations, albeit to a lesser extent than e. In particular, the relative error in L1 norm is even more than one order of magnitude less than the initial value, so confirming the accuracy of the reconstruction of regular mass variations. It should also be noted that the matrix Að Þj remains well conditioned during the iterations, with

Fig. 1. Mass density per unit lengthq¼qð Þ to be identified in 0;x L 2


. (a) Smooth mass changes as in(109); (b) discontinuous mass changes as in(110); (c) overlapping mass changes as in(111).

Table 1

First 25 eigenvalues of the clamped–clamped uniform unperturbed nanorod in (1)–(3), with physical parameters as in Section5.1. Comparison between analytical (kn, column 2) and numerical (kFEMn ) values obtained with Ne¼ 100; 200; 400 equally spaced finite elements. Percentage errors: en¼ 100  kFEMn  kn

 

=knfor Ne¼ 100 (column 3), Ne¼ 200 (column 4), Ne¼ 400 (column 5).

n _knð_rad=sÞ2 _Ne¼ 100 Ne¼ 200 Ne¼ 400 1 _{3:556  10}þ12 _{1:4  10}6 _{1:4  10}6 _{1:4  10}6 2 _{1:426  10}þ13 _{1:4  10}6 _{1:4  10}6 _{1:4  10}6 3 _{3:220  10}þ13 _{1:5  10}6 _{1:5  10}6 _{1:5  10}6 4 _{5:755  10}þ13 _{5:0  10}7 _{5:0  10}7 _{5:0  10}7 5 9:055  10þ13 _{1:6  10}6 _{5:2  10}7 _{5:2  10}7 6 1:315  10þ14 _{8:1  10}6 _{5:2  10}7 _{5:2  10}7 7 _{1:807  10}þ14 _{1:3  10}5 _{2:4  10}6 _{2:4  10}6 8 _{2:387  10}þ14 _{2:6  10}5 _{1:2  10}6 _{1:2  10}6 9 _{3:059  10}þ14 _{5:3  10}5 _{3:9  10}6 _{6:6  10}7 10 _{3:829  10}þ14 _{9:9  10}5 _{4:9  10}6 _{2:8  10}7 11 _{4:703  10}þ14 _{1:7  10}4 _{1:1  10}5 _{2:6  10}7 12 _{5:688  10}þ14 _{2:9  10}4 _{1:7  10}5 _{1:3  10}6 13 _{6:792  10}þ14 _{4:6  10}4 _{2:7  10}5 _{2:5  10}6 14 8:022  10þ14 _{7:0  10}4 _{4:4  10}5 _{2:4  10}6 15 9:389  10þ14 _{1:0  10}3 _{6:4  10}5 _{3:9  10}6 16 1:090  10þ15 _{1:5  10}3 _{9:1  10}5 _{8:2  10}6 17 _{1:257  10}þ15 _{2:1  10}3 _{1:3  10}4 _{9:6  10}6 18 _{1:440  10}þ15 _{2:9  10}3 _{1:8  10}4 _{1:4  10}5 19 _{1:642  10}þ15 _{3:9  10}3 _{2:4  10}4 _{1:5  10}5 20 _{1:862  10}þ15 _{5:2  10}3 _{3:2  10}4 _{2:0  10}5 21 _{2:103  10}þ15 _{6:8  10}3 _{4:2  10}4 _{2:4  10}5 22 _{2:365  10}þ15 _{8:8  10}3 _{5:4  10}4 _{3:2  10}5 23 _{2:651  10}þ15 _{1:1  10}2 _{6:9  10}4 _{4:2  10}5 24 2:961  10þ15 _{1:4  10}2 _{8:7  10}4 _{5:6  10}5 25 3:297  10þ15 _{1:7  10}2 _{1:1  10}3 _{6:7  10}5

Fig. 2. Reconstruction of smooth mass changes as in(109), withs L¼ 0:35;

c

Fig. 3. Reconstruction of smooth mass changes as in(109), withs L¼ 0:35;

c

L¼ 0:10, t ¼ 1:00, using the first N ¼ 6; 9; 12; 15 eigenfrequencies.

Fig. 4. Reconstruction of smooth mass changes as in(109), withs

condition number

_j

Að Þj¼ kAð Þj_{k k A} ð Þj1_{k ranging between 30 and 120 in all the cases studied. Here,}

kAð Þj_{k ¼ max}

jyj¼1jAð Þjyj, where jyj ¼ ffiffiffiffiffiffiffiffiffipy yis the Euclidean norm of the vector y2 RN.

5.3.2. Identification of discontinuous mass densities

The mass density to be determined is assumed equal to

Table 2

Some results of the reconstruction of smooth mass changes as in(109)versus iteration number j (up to convergence), with (a):s

L¼ 0:35;cL¼ 0:10; t ¼ 0:10 (Fig. 2); (b): s L¼ 0:35; c L¼ 0:10; t ¼ 1:00 (Fig. 3); (c): s L¼ 0:35; c

L¼ 0:30; t ¼ 1:00 (Fig. 4), using the first N¼ 6 (columns 2  5), N ¼ 15 (columns 6  9) eigenfrequencies. The quantity e is defined in(63); eL2¼

kqidentqexactk_L2 kqexactk_L2 , eL1¼kq

ident_qexact_k L1 kqexactkL1 , whereq

ident_¼qident_{ð Þ,x q}exact_¼qexact_{ð Þ are the identified and the exactx} mass density per unit length, respectively.jAð Þj

is the condition number of the matrix Að Þj

. The unperturbed nanorod corresponds to j¼ 0.

j e _j_Að Þj e_L2 eL1 e _j_Að Þj e_L2 eL1 (a) 0 2:40  103 _{1:30  10}þ1 _{1:92  10}2 _{9:09  10}2 _{1:44  10}3 _{3:10  10}þ1 _{1:92  10}2 _{9:09  10}2 1 3:11  105 _{1:31  10}þ1 _{1:04  10}2 _{3:95  10}2 _{3:84  10}5 _{3:12  10}þ1 _{1:58  10}3 _{5:34  10}3 2 5:12  108 _{1:41  10}þ1 _{1:04  10}2 _{3:88  10}2 _{3:35  10}7 _{3:64  10}þ1 _{1:31  10}3 _{4:43  10}3 (b) 0 _{2:33  10}2 _{1:30  10}þ1 _{1:81  10}1 _{5:00  10}1 _{1:29  10}2 _{3:10  10}þ1 _{1:81  10}1 _{5:00  10}1 1 _{2:52  10}3 _{1:31  10}þ1 _{1:00  10}1 _{2:43  10}1 _{2:23  10}3 _{3:12  10}þ1 _{5:70  10}2 _{1:47  10}1 2 9:00  105 _{2:36  10}þ1 _{9:68  10}2 _{2:02  10}1 _{2:63  10}4 _{8:16  10}þ1 _{1:14  10}2 _{2:55  10}2 3 3:37  107 _{2:60  10}þ1 _{9:68  10}2 _{2:02  10}1 _{7:77  10}6 _{1:14  10}þ2 _{7:17  10}3 _{1:50  10}2 (c) 0 _{6:42  10}2 _{1:30  10}þ1 _{2:82  10}1 _{5:00  10}1 _{3:64  10}2 _{3:10  10}þ1 _{2:82  10}1 _{5:00  10}1 1 _{1:24  10}2 _{1:31  10}þ1 _{9:46  10}2 _{1:74  10}1 _{7:03  10}3 _{3:12  10}þ1 _{9:57  10}2 _{1:84  10}1 2 _{9:08  10}4 _{2:44  10}þ1 _{1:53  10}2 _{3:54  10}2 _{6:14  10}4 _{5:89  10}þ1 _{1:44  10}2 _{3:08  10}2 3 2:20  105 _{3:25  10}þ1 _{9:97  10}3 _{2:79  10}2 _{1:54  10}5 _{7:92  10}þ1 _{4:64  10}3 _{1:78  10}2 4 4:89  108 _{3:30  10}þ1 _{9:96  10}3 _{2:80  10}2 _{9:11  10}8 _{8:11  10}þ1 _{4:62  10}3 _{1:75  10}2

Fig. 5. Reconstruction of discontinuous mass changes as in(110), withs L¼ 0:15;

c

Fig. 6. Reconstruction of discontinuous mass changes as in(110), withs L¼ 0:15;

c

L¼ 0:10, t ¼ 1:00, using the first N ¼ 6; 9; 12; 15 eigenfrequencies.

Fig. 7. Reconstruction of discontinuous mass changes as in(110), withs L¼ 0:35;

c

q

ð Þ ¼x

q

0þ

q

0t

v

sc 2;sþ

c 2

½ ; ð110Þ

where s; c; t have the same meaning as in the previous section, seeFig. 1(b). Since the mass density

_q

has jump discontinuities at x¼ s c

2, whereas the approximating functionsUnð Þ are smoothx

functions of the axial coordinate x, it is expected that the reconstruction may fail near these points. Numerical simulations confirm this undesired behavior. Some representative cases are collected inFigs. 5 and 6.

These cases correspond to perturbations both located near the left end of the nanorod (s

L¼ 0:15) and with small support

(c

L¼ 0:1), but having either small (t ¼ 0:1, case i)) or large (t ¼ 1:0, case ii)) intensity, respectively. In case i) (seeFig. 5), the

results are accurate enough for N¼ 15, whereas the oscillations of the identified mass coefficient have appreciable ampli-tude in case ii) (seeFig. 6), and propagate in the remaining part of the interval 0;L

2



. The support of the perturbation is slightly overestimated and, as it was expected, L1estimates fail near the jumps.

Fig. 7collects the results for large mass variations, both in L2_{and in L}1

norm, fors

L¼ 0:35;cL¼ 0:3; t ¼ 1:0. The support of

the mass variation still is well estimated, even if oscillations with significant amplitude occur both within the support and in

Fig. 8. Reconstruction of discontinuous mass changes as in(110), withs L¼ 0:35;

c

L¼ 0:30, t ¼ 1:00, using the first N ¼ 20; 25 eigenfrequencies.

Fig. 9. Reconstruction of overlapping mass changes as in(111), withs

L¼ 0:25; t ¼ 0:50; s1

the remaining part of the interval 0_;L 2



. The number of iterations needed to satisfy the stopping criterion(63)is slightly big-ger than the smooth case, being however always less than 10.

Finally, it can be shown that results generally improve by considering larger N, say N¼ 20; 25. In order to reduce the numerical approximation error on higher order eigenvalues, all these cases have been developed on a numerical model of the nanorod having Ne¼ 400 equally spaced finite elements. As it can be seen by comparingFigs. 7 and 8, the increase of

Neshows beneficial effects, such as more accurate estimate of the actual support of the mass variation and reduction of

the amplitude of the spurious oscillations of the identified coefficient around the exact value. However, the identification of large mass variations with left (respectively, right) jump position close to the left end (respectively, to the mid point) of the nanorod still remains problematic to some extent.

5.3.3. Identification of overlapping added masses

The identification method has also been tested on mass distributions having less schematic profile than those considered in the previous sections. As an example, here we report some representative results related to two overlapping added mass profiles, one being regular (and belonging to the class of Section5.3.1) and the other one having triangular shape, with a jump type discontinuity. More precisely, the mass density to be determined has the expression

q

ð Þ ¼x

q

0þ

q

0max t cos 2

p

ðx sÞ c  

v

sc 2;sþ2c ½ ; _ct1₁ðx sð1 c1ÞÞ

v

½s1c1;s1   ; ð111Þ

Fig. 10. Reconstruction of overlapping mass changes as in(111), withs

L¼ 0:15; t ¼ 0:50; s1

where s½1 c1; s1  0;2L



and t1

q

0; t1> 0, is the support and the maximum value of the triangular added mass profile,

respectively; see Fig. 1(c). Some of the results are summarized in the sequel. They have been obtained using

Ne¼ 200; 400 equally spaced finite elements, for N ¼ 6; 9; 12; 15 and N ¼ 20; 25, respectively. In order to simplify the

presen-tation of the results, the parameters c and c1have been assumed equal to 0:2L. It should be noted that, depending on the

values of s1and t1, the coefficient

q

ð Þ inx (111)can be either continuous or discontinuous. The latter case occurs, for example,

when s12 s c2; s þc2

 

and t1is small enough with respect to t.

The determination of continuous mass coefficient turns out to be very accurate even when only the first 9 12 eigenfre-quencies are used in identification, as it was found for the class of smooth variations considered in Section5.3.1. We refer to

Fig. 9for a typical result.

In case of discontinuous coefficient, as it was already noticed in Section5.3.2, spurious oscillations occur near the jump, with amplitude which turns out to be proportional to the intensity of the jump. As a consequence, identification of the smooth portion of the mass coefficient may become inaccurate for small values of t. At least 15 20 first eigenfrequencies seem to be needed to obtain acceptable accuracy in these cases, see, for example,Fig. 10. The method also shows good ability in identifying mass variations with disjoint supports, particularly when the values of t1and t are close, see, for example, Fig. 11.

Fig. 11. Reconstruction of overlapping mass changes as in(111), withs

L¼ 0:15; t ¼ 0:50; s1

5.3.4. Physical post-filtering

In this section we show that, when the method is combined with additional information about the unknown mass coef-ficient, such as monotonicity or a priori information on the support of the mass variation, the reconstruction may further improve, leading to good uniform approximation of the solution. More precisely, we have considered in the sequel the fol-lowing a priori information:

F1) The mass variation is positive, e.g., the condition(14)holds.

As remarked in Section2, this information is simply available from the physics of the problem. Actually, our identification algorithm produces a function

q

ð Þ which may oscillate near the unperturbed linear mass densityx

q

0. Basing on

assump-tion F1, we filter the results of identification by setting the mass density to be

q

0whenever the reconstructed value of

q

ð Þx

is smaller than

q

0.

F2) There are situations in which it is a priori known that a single localized mass variation occurs in the nanosensor, that

is, the support of the mass variation rð Þ is a closed (and possible small) interval compactly contained in 0;x L 2



. In this case, under the assumption that the reconstructed coefficient

q

ð Þ is a good uniform approximation of the actualx mass, one can determine the interval of maximum mass increase and neglect all the other possible regions on which the mass density increases.

Fig. 12. Filtering effects on identification. Reconstruction of discontinuous mass changes as in(110), withs L¼ 0:15;

c

L¼ 0:10, t ¼ 1:00, using the first N ¼ 15 eigenfrequencies.

Fig. 13. Noise effects on identification of smooth mass changes. Upper row: mass changes as in(109), withs

L¼ 0:35;cL¼ 0:10, t ¼ 0:10. Lower row: mass changes as in(109), withs

L¼ 0:35;cL¼ 0:30; t ¼ 1:00.

Fig. 14. Noise effects on identification of discontinuous mass changes. Upper row: mass changes as in(110), withs L¼ 0:15;

c

L¼ 0:10, t ¼ 0:10. Lower row: mass changes as in(110), withs

L¼ 0:35; c