A spectral finite element

March 8, 2016 13:19 Naili˙CMBBE˙revised

Computer Methods in Biomechanics and Biomedical Engineering

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Computer Methods in Biomechanics and Biomedical Engineering Vol. 00, No. 00, xxx 0000, 1–18

RESEARCH ARTICLE Ultrasonic wave propagation in viscoelastic cortical bone plate coupled with fluids: A spectral finite element study Vu-Hieu Nguyen and Salah Naili Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France (submitted) This work deals with the ultrasonic wave propagation in the cortical layer of long bones which is known as being a functionally-graded anisotropic material coupled with fluids. The viscous effects are taken into account. The geometrical configuration mimics the one of axial transmission technique used for evaluating the bone quality. We present a numerical procedure adapted for this purpose which is based on the spectral finite element method. By using a combined Laplace-Fourier transform, the vibroacoustic problem may be transformed into the frequency-wavenumber domain in which, as radiation conditions may be exactly introduced in the infinite fluid halfspaces, only the heterogeneous solid layer needs to be analyzed using finite element method. Several numerical tests are presented showing very good performance of the proposed approach. We present some results to study the influence of the frequency on the FAS velocity in (visco)elastic bone plate.

Keywords: ultrasound; cortical bone; axial transmission; spectral finite element method; vibroacoustic

1.

Introduction

In recent years, quantitative ultrasound (QUS) has demonstrated its promising potential in assessment of in vivo bone characteristics. An advantage of QUS over X-ray techniques is its ability to give some information about the elastic properties and defects of bones. Moreover, ultrasound is non-ionizing and the ultrasonic apparatus is relatively inexpensive in comparison with X-ray techniques and can be made portable. For measuring in vivo properties of the cortical layer of long bones, a so-called “axial transmission” (AT) technique has been developed (Lowet and Van der Perre 1996). The axial transmission technique uses a set of ultrasonic transducers (transmitters and receivers) placed in the same side on a line in contact with the skin along the bone axial axis. The transmitter emits an ultrasound pulse wave (around 250 KHz-2 MHz) that propagates along the cortical layer of bones. The analysis of the signals received at the receivers can allow the quantification of the geometrical information as well as mechanical characteristics of the cortical bone at the measured skeletal site (Nicholson et al. 2002; Moilanen 2008). At the macroscopic scale, porosity in the radial direction (which is defined in the bone’s cross-section plan) is heterogeneous. The observations at all ages and for ∗ Corresponding

author. Email: [email protected]

ISSN: 1741-5977 print/ISSN 1741-5985 online c 0000 Taylor & Francis

DOI: 10.1080/1741597YYxxxxxxxx http://www.informaworld.com

March 8, 2016 13:19 Naili˙CMBBE˙revised 2


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V.-H. Nguyen and S. Naili

both genders show that the mean porosity in the endosteal region (inner part of the bone) is significantly higher than the one in the periosteal region (outer part of the bone) (Bousson et al. 2001; Thomas et al. 2005; Sansalone et al. 2010). This statement may be explained by the fact that cortical bone is affected by age-related bone resorption and osteoposis, causing reduction of bone shell’s thickness as well as increase of porosity, namely in the endosteal region. Moreover, the macroscopic mechanical properties of bones have been shown to strongly depend on its porosity (Dong and Guo 2004; Baron et al. 2007). As a consequence, cortical bone would naturally be considered as a functionally graded material. The studies of ultrasonic wave propagation in elastic plates with gradient of physical properties have been dedicated in several works. For the application to cortical bone, Baron and Naili (2010) have studied the reflection coefficients of a fluid-loaded functionally graded anisotropic bone plate using an asymptotic approach in the frequency domain. The dynamic responses of cortical bone in time domain have also been studied using isotropic bone model (e.g. Bossy et al. (2004)) or using an anisotropic and heterogeneous bone model (e.g. Ha¨ıat et al. (2009)), showing that the heterogeneous of material bone properties change considerably the velocity of first arriving signal (FAS) which is defined from the earliest signals recorded at the receivers. Its velocity measured in the time domain was shown to be able to discriminate healthy subjects from osteoporotic patients in several studies. Therefore, the FAS velocity is now considered as a relevant index of bone properties (Barkmann et al. 2000; Hans et al. 1999; Stegman et al. 1995). Furthermore, cortical bone may be considered as elastic medium with random properties. Some probabilistic models have been proposed in order to consider this aspect (Macocco et al. 2005, 2006; Desceliers et al. 2009). Due to its structural and physical characteristics, cortical bone is a strong attenuation medium causing the losses of ultrasonic wave propagation (Lakes et al. 1986; Langton et al. 1990). The attenuation of ultrasonic waves propagating through bone medium may be caused by the viscoelastic of the bone solid or by scattering effects due to the pores (Sasso et al. 2007, 2008). It has been shown that the broadband ultrasonic attenuation, which is defined as the slope of the curve of frequency dependent attenuation coefficient, is significantly dependent on the bone microstructure, as well as to the bone physical properties such as mass and bone mineral densities (Sasso et al. 2008). The influence of viscous effect on ultrasonic response obtained with an AT technique has recently been studied (Naili et al. 2010) showing that the velocity of FAS is significantly modified when the viscous effects are taken into account. However, the viscous effects due to ultrasonic broadband frequencies have not been studied because considering high frequency signal increases the computational cost considerably. Mechanical modeling of this experiment deals with considering a model describing vibro-acoustic interactions of a solid waveguide (which represents the cortical bone) coupled with two fluid media (which represents soft tissues of each side of solid layer such as skin or marrow). The cortical bone may be described as platelike or cylindrical-like structures. This technique of nondestructive testing used to evaluate the material properties requires careful analysis of the reflections, conversion modes and interferences of longitudinal and shear waves within the bone structure. Many studies have focused on the modelling of guided waves in long bones by using fluid-loaded (multilayer) plate models. The understanding of wave phenomena involved in the multilayer structures has been considered in the frequencydomain by Baron and Naili (2008, 2010); Nayfeh (1995); Shuvalov et al. (2006) or in the time-domain by Bossy et al. (2002); Grimal and Naili (2006); Desceliers et al.



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(2008); Ha¨ıat et al. (2009). To study multilayer structures in the time-domain, there are mainly two approaches. The first one involves using (semi-)analytical methods such as the generalized ray/Cagniard-de-Hoop technique (Grimal and Naili 2006) or the direct stiffness matrix method (Kausel 2006). The second one involves using numerical methods such as the finite difference method (Virieux 1986; Bossy et al. 2002; Kampanis et al. 2008) or the finite element method (FEM) (Protopappas et al. 2007; Ha¨ıat et al. 2009). Although the analytical methods are attractive to obtain reliable transient responses of the structure, numerical methods are often more efficient to treat problems with inhomogeneous materials or complex geometries. However, most numerical methods require important computational costs, especially for problems in the high-frequency domain. Moreover, absorbing boundary conditions are required when considering unbounded domains (Thompson and Huan 2000; Givoli 2004; Kampanis et al. 2008). When considering waveguides with geometrical and mechanical properties which are constant only along one or two directions, the Hybrid Numerical Method (see e.g. Han et al. (2001a,b); Liu and Xi (2002)), alternatively called Spectral Finite Element Method (SFEM, see e.g. Gopalakrishnan et al. (2008); Marzani (2008)), have been employed. The key point of this method consists in using a hybrid algorithm which begins by employing the Fourier transform (with respect to time and to the longitudinal direction of the waveguide) to transform problem into the frequencywavenumber domain. Then, the wave equations in the spectral domain governed in a cross-section (or even a 1D domain in the case of infinite plates or axisymmetric wave-guides), which may actually have inhomogeneous material properties, can be easily handled using the finite element method (Han et al. 2001a,b; Marzani 2008; Desceliers et al. 2008). In this work, we propose to use the spectral finite element method to study timeresponse of ultrasonic wave in AT test by also considering viscous effects in both fluid and solid layers. In order to take into account the full coupling effects between the bone plate and the unbounded fluid and to reduce the computational cost, the the wave equations in the fluid will not be solved using FEM (Desceliers et al. 2008) but will be analytically handled. Moreover, the Laplace transform is proposed to be employed to the time variable instead of the Fourier transform, allowing to have good numerical stability even for simulations of long duration tests. The present paper is organized as follows. After this introduction as background, Section 2 gives the governing equations of the vibro-acoustic problem for modelling the bone-fluid interaction. Next, the sections 3 and 4 develop the equations in the Laplace-Fourier domain and provide the spectral finite element formulation in this domain. The inverse Laplace transform technique to come back into time domain is also discussed. Then, some numerical tests are presented in Section 5 for the validation of the proposed procedure as well as for considering the viscosity effects to the FAS velocity at different signal frequencies. Last, some conclusions and discussions will be presented in Section 6. 2.

Statement of the problem

In this section we describe a two-dimensional (2D) idealized model of the AT test performed on cortical bone. The geometrical configuration consists of a plate of infinite extent (representing the cortical bone layer) and two semi-infinite fluid media (representing soft tissues) (see Fig. 1). A source of pressure and a line of the receivers are placed in the upper fluid medium. The characteristics of waves scattered from the bone depend on the material and geometrical properties of the bone.



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2.1

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Description of the problem

Let R(O; e1 , e2 ) be the reference Cartesian frame where O is the origin and (e1 , e2 ) is an orthonormal basis for this space. The coordinates of a point x in R are specified by (x1 , x2 ) and the time is denoted by t. We consider an infinite solid layer occupying the domain Ωb that represents the cortical bone plate with a constant thickness h (Ωb = {x(x1 , x2 ); 0 ≥ x2 ≥ −h}). This bone plate is loaded on its upper and lower surfaces by two fluid halfspaces. The upper fluid domain is denoted by Ωf1 (Ωf1 = {x(x1 , x2 ); x2 ≥ 0}) and the lower one is denoted by Ωf2 (Ωf2 = {x(x1 , x2 ); x2 ≤ −h}). The plane interfaces between bf the bone (Ωb ) and the fluids (Ωf1 and Ωf2 ) are denoted by Γbf 1 and Γ2 , respectively (see Fig. 1). Γf∞ 1 Fluid 1 (Ωf1 ) e2 e1

source (xs1 , xs2 )

receivers Γbf 1

nbΓ1

O Solid (Ωb ) Fluid 2 (Ωf2 )

Γbf 2

nbΓ2

Γf∞ 2 Figure 1. The trilayer model for ultrasound axial transmission test

Both fluid domains Ωf1 and Ωf2 are considered as homogeneous fluid media and the solid plate is assumed to be an anisotropic and heterogeneous viscoelastic solid. More precisely, according to the cortical bone properties, we assume that mechanical properties of the solid may be varied in the depth defined by e2 -axis but they are homogeneous along the longitudinal defined by e1 -axis. The system is excited by an acoustic source at a point xs = (xs1 , xs2 ) located in the upper fluid domain Ωf1 . 2.2

Governing equations

The governing equations for the wave propagation in a coupled fluid-solid system can be found in many references (see e.g. Carcione (2001)). Here we only outline the main equations serving to solve the considered trilayer problem. In what follows, we denote respectively by ∇ and ∇2 the operators of gradient and Laplace with respect to x and by the superposed dot the differentiation with respect to time t. 2.2.1

Wave equations in the fluid domains Ωf1 and Ωf2

Let us first consider the fluid domain Ωf1 . In this domain, a dissipation model for the fluid acoustic is introduced. This dissipation mechanism is assumed to be small and due only to viscosity. The theory derived is considered as a theory linear of viscosity without memory. In this case, the linearized equation of the wave propagation reads: 1 ˙ p¨1 − ∇2 (p1 + γ1 p˙ 1 ) = Q, c21

∀ x ∈ Ωf1 ,

(1)



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where p1 (x, t) is the acoustic pressure in the domain Ωf1 ; Q(x, t) is the acoustic f source density; the constant c1 is the wave velocity p in Ω1 at rest and depends on its bulk modulus K1 and mass density ρ1 : c1 = K1 /ρ1 ; γ1 denotes the viscosity coefficient. The impulsive point source acting at the point xs = (0, xs2 ) that can be described as: Q˙ = ρ1 F (t) δ(x1 , x2 − xs2 ),

(2)

where F (t) is a given real scalar function depending only on the time t and δ(x1 , x2 ) is the Dirac’s delta function. Similarly to the description in the domain Ωf1 , the pressure field p2 in the fluid domain Ωf2 reads: 1 p¨2 − ∇2 (p2 + γ2 p˙ 2 ) = 0, c22

∀ x ∈ Ωf2 ,

(3)

p where c2 = K2 /ρ2 is the wave velocity in Ωf2 in which K2 and ρ2 are its bulk modulus and mass density at rest, respectively; γ2 denotes the viscosity coefficient. 2.2.2

Dynamic equations in the solid layer Ωb

We denote by u(x, t) = {u1 , u2 }T the time-dependent vector of displacement at a point x located in the solid domain Ωb . The dynamic equations describing the motion in the solid domain Ωb are given by: ¨ − LT s = 0, ρu

∀ x ∈ Ωb ,

(4)

where ρ denotes the mass density of the solid; the vector s contains the components of the stress tensor σ by using Voigt’s notation: s = {σ11 , σ22 , σ12 }T and the operator L is defined as follows:

L = L1 ∂1 + L2 ∂2 ,



1  where L1 = 0 0

 0 0, 1



0  L2 = 0 1

 0 1, 0

(5)

where ∂1 and ∂2 designate the partial derivative operators with respect to x1 and x2 , respectively. The viscosity model for the solid layer with frequency-independent coefficients is based on the linear theory of viscoelasticity without memory. This model represents a generalization to three dimensions and to arbitrary material anisotropy of the one-dimensional, isotropic Kelvin-Voigt continuum model. The model of KelvinVoigt may be regarded as consisting of a linearly elastic element (spring) with elasticity c in parallel with a linearly viscous element (dashpot) with viscosity E. In this case, the constitutive equation for anisotropic viscoelastic solid reads:

˙ s = C e + E e,



 c11 c12 c16 C =  c12 c22 c26  , c16 c26 c66



 η11 η12 η16 E =  η12 η22 η26  , η16 η26 η66

(6)

where e = {ǫ11 , ǫ22 , 2ǫ12 }T = L u is the strain vector using Voigt’s notation; C and E are the elasticity and viscosity tensors, respectively. Note that all material properties of the solid depend only on x2 , i.e. ρ = ρ(x2 ), C = C(x2 ) and E = E(x2 ).



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2.2.3

Interface and boundary conditions

bf The interface conditions at the fluid-solid interfaces Γbf 1 and Γ2 may be described by the same way. The condition required for the continuity of normal velocities at the interfaces reads:

¨ · nbf ∇(pα + γα p) ˙ · nbf α = −ρα u α ,

∀ x ∈ Γbf α

(α = 1, 2)

(7)

whereas the condition of the continuity of normal stresses at the interfaces requires: bf σnbf α = −(pα + γα p˙ α )nα ,

∀ x ∈ Γbf α

(α = 1, 2)

(8)

bf where nbf α (α = 1, 2) is the unit normal vector of the interface Γα pointing out of Ωb (see Fig. 1). Finally, the boundary condition of two fluid halfspace at infinity reads:

pα → 0, 2.2.4

∀ x → Γfα∞ (α = 1, 2).

(9)

Boundary value problem in term of (p1 , u, p2 ).

This paragraph rewrites all previous equations in a more compact form. The equations in terms of (p1 , u, p2 ) of the coupled fluid-solid problem may be represented as follows: – Dynamic equilibrium equations: 1 p¨1 − ∇2 (p1 + γ1 p˙ 1 ) = ρ1 F (t) δ(x1 , x2 − xs2 ), c21

¨ − LT s = 0, ρu

1 p¨2 − ∇2 (p2 + γ2 p˙ 2 ) = 0, c22

∀ x ∈ Ωf1 ,

(10)

∀ x ∈ Ωb ,

(11)

∀ x ∈ Ωf2 ,

(12)

˙ where s = C L u + E L u. – Interface and boundary conditions:  1  ∂2 (pα + γα p˙ α ) = −¨ u2 , ρα t = { 0, −(pα + γα p˙ α )}T  pα = 0,

∀ x ∈ Γbf α

(13)

∀ x ∈ Γfα∞ .

(14)

where t := {σ12 , σ22 }T = LT2 s. Note that for the present configuration (see Fig. 1), the unit normal vectors are bf T defined by the relations nbf 1 = −n2 = {0, 1} . 3.

Equations in the Laplace-Fourier (s − k1 ) domain

The boundary value problem given by Eqs. (10)-(14) deals with solving a system of linear partial differential equations for which the coefficients are constants in the longitudinal direction defined by e1 -axis. Here we propose to solve the system as follows: (i) the system of equations is firstly transformed into wavenumberfrequency domain by using a Fourier transform with respect to x1 combining to a



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Laplace transform with respect to t; (ii) in the wavenumber-frequency domain, the equations in both fluid domains are analytically solved giving impedance boundary conditions for the solid layer that may be solved by the finite element method; (iii) the space-time solution is finally obtained by performing two inverse transforms. The general form of a Laplace-Fourier transform (LF) applied to a real-valued function y(x1 , x2 , t) denoted by y˜(k1 , x2 , s) is defined as:

y˜(k1 , x2 , s) :=

Z

0

∞ Z +∞

−ik1 x1

y(x1 , x2 , t)e

−∞

dx1

e−st dt,

(15)

√ where i is the imaginary unit defined by i = −1; s ∈ C is the complex Laplace variable; k1 ∈ R is the real Fourier variable representing the wavenumber in e1 -axis; R and C denote the sets of all the real and complex numbers, respectively. In (s − k1 ) domain, the time derivative and the spatial differential operation with ∂(⋆) respect to x1 can be replaced by ∂(⋆) ∂t → s(⋆) and ∂x1 → ik1 (⋆), respectively.

3.1

Transformed problem in (s − k1 ) domain for the fluids Ωf1 and Ωf2

By applying the Laplace-Fourier transform (15) to Eqs. (10), (13) and (14), we obtain a boundary-valued differential equation of p˜1 with respect to only x2 :

s2 + k12 c¯21

p˜1 − ∂2 p˜1 = ρ¯1 F˜0 (s)δ(x2 − xs2 ),

for x2 > 0,

(16)

∂2 p˜1 = −¯ ρ1 s 2 u ˜2 ,

at x2 = 0

(17)

p˜2 → 0,

when x2 → +∞,

(18)

where c¯21 = c21 (1+ sγ1 ) and ρ¯1 = ρ1 /(1+ sγ1 ). Solution to the system of equations (16)-(18) leads to express p˜1 in semi-explicit form as:

p˜1 = −

ρ¯ 1 ρ¯1 ˜ −α1 (xs2 −x2 ) s 1 2˜ + e−α1 (x2 +x2 ) + F0 e s U21 e−α1 x2 , for 0 ≤ x2 ≤ xs2 , 2 α1 α1

p˜1 = −

1 ρ¯1 ˜ ρ¯1 2 ˜ −α1 x2 s s F0 eα1 (x2 −x2 ) + e−α1 (x2 +x2 ) + , s U21 e 2 α1 α1

(19)

where α1 :=

q

s2 c¯21

for x2 ≥ xs2 , (20)

˜21 is the LF-transform of solid layer’s vertical displacement + k12 ; U

˜ at the upper fluid-solid interface Γbf ˜2 (k1 , 0, s). 1 : U21 := u Similarly, the solution of p˜2 in (s − k1 ) domain may be expressed as follows: p˜2 = − where α2 =

q

s2 c¯22

ρ¯2 2 ˜ α2 (x2 +h) s U22 e , α2

˜22 := u + k12 and U ˜2 (k1 , −h, s).

for x2 ≤ −h,

(21)



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Transformed problem in (s − k1 ) domain for the solid Ωb

Applying the LF-transform (15) to the system of equations associated with the solid (11) leads to: ˜ = 0, ˜ − ik1 LT1 + LT2 ∂2 s ρs2 u

(22)

˜ − ik1 AT3 ∂2 u ˜ − ∂2˜t = 0, s2 A1 + k12 A2 u

(23)

¯ (ik1 L1 + L2 ∂2 ) u ¯ = C + s E. For each couple of values (s, k1 ) ∈ ˜=C ˜ and C where s C × R, the equation (22) is a system of partial differential equations of u˜ with respect only to x2 :

where

˜t = (ik1 A3 + A4 ∂2 )˜ u

(24)

and the 2-by-2 matrices A1 , A2 , A3 and A4 are defined by:

ρ A1 = 0

0 , ρ

(25)

¯ L 1 = L T C L 1 + s L T E L 1 = Ae + s Av , A2 = LT1 C 2 2 1 1 T ¯ T e v T A3 = L 2 C L 1 = L 2 C L 1 + s L 2 E L 1 = A3 + s A3 , ¯ L2 = LT2 C L2 + s LT2 E L2 = Ae4 + s Av4 . A4 = LT2 C

(26) (27) (28)

in which the 2-by-2 matrices A1 , Ae2 , Ae3 , Ae4 , Av2 , Av3 and Av4 depend only on the physical parameters. The boundary conditions (13) in (s − k1 ) domain read: ˜t(0) =

0 , −(1 + sγ1 )˜ p1 (0)

˜t(−h) =

0 , −(1 + sγ2 )˜ p2 (−h)

(29)

in which the solution of p˜1 (0) on Γbf ˜2 (−h) on Γbf 1 and the solution of p 2 can be determined by using Eqs. (20) and (21), respectively: ρ¯1 ˜ −α1 xs2 ˜21 , − s2 U F0 e α1 ρ¯2 ˜ p˜2 (−h) = − s2 U 22 . α2

(30)

p˜1 (0) = −

(31)

Thus, the boundary conditions (29) can be rewritten as: ˜t(0) = F0 − P1 u ˜ (0) ,

˜t(−h) = P2 u ˜ (−h)

(32)

where F0 =

0 ρ1 ˜ −α1 xs2 F0 e α1

!

,

"

0 P1 = 0

0

#

ρ1 2 , s α1

"

0 P2 = 0

0

#

ρ2 2 . s α2

(33)



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4.

Spectral finite element formulation and time-space solution

4.1

Weak formulation of the solid layer

The weak formulation of the boundary value problem (23) and (32) may be now introduced using a classic procedure (see e.g. Bathe (1996)). Let C ad be the the admissible function space constituted by all sufficient smooth ˜ (x2 ) : ] − h, 0[→ C2 . The conjugate transpose complex-valued functions x2 → δu ˜ is denoted by δu ˜ ∗. of δu ˜ ∗ ∈ C ad and inteUpon integrating the equation (23) against a test function δu grating by parts, we obtain: Z

0

˜ δu

∗

2

s A1 +

k12 A2

−h

−

ik1 AT3 ∂2

˜ dx2 + u

Z

0 −h

∗ 0 ˜ ∗˜t dx2 − δu ˜ ˜t −h = 0. ∂2 δu

(34)

The last terms associated with boundary conditions given in the equation (34) may be calculated by using the relations (32). The weak formulation of the 1Dboundary problem (23) reads: for all k1 fixed in R and for all s fixed in C, find ˜ (k1 , x2 , s) ∈ C ad such that: u Z

0

˜ δu −h

∗

2

s A1 +

k12 A2

−

ik1 AT3 ∂2

˜ dx2 + u

Z

0

˜ ∗ (ik1 A3 + A4 ∂2 ) u ˜ dx2 ∂2 δu −h

˜ ∗ (0) P1 u ˜ (0) + δu ˜ ∗ (−h) P2 u ˜ (−h) = δu ˜ ∗ (0) F0 , + δu

4.2

˜ ∈ C ad . (35) ∀ δu

Finite element formulation

We proceed by introducing a finite Selement mesh of the domain [−h, 0] which contains nel elements Ωe : [−h, 0] = e Ωe (e = 1, ..., nel ). By the Galerkin finite ˜ and δu ˜ in each element e are approximated element method, both functions u using the same shape function: ˜ (x2 ) = Ne Ue , u

˜ (x2 ) = Ne δUe , δu

∀ x2 ∈ Ω e ,

(36)

where Ne is the shape function, Ue and δUe are the vectors of nodal solutions ˜ and δu ˜ within the element Ωe , respectively. Replacing Eq. (36) into (35) of u and assembling the elementary matrices, we obtain a static-like linear system of equations: K + KΓ U = F,

(37)

where U is the global nodal displacement vector; K is the global “stiffness matrix” of the solid; KΓ represents the coupled operator between fluid and solid; the vector F is the external force vector. For all couple (s, k1 ) fixed in C × R, these quantities



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may be expressed by: K = s2 K1 + k12 Ke2 + ik1 Ke3 + Ke4 + sk12 Kv2 + isk1 Kv3 + sKv4 , ρ2 2 ρ1 2 Γ s , 0, ..., 0, s , K = Diag 0, α1 α2 T ρ1 ˜ −α1 xs2 , , 0, ..., 0 F = 0, F0 e α1

(38) (39) (40)

e,v e,v where the matrices K1 , Ke,v 2 , K3 and K4 are independently defined to (s, k1 ):

K1 =

[Z

Ωe

e

Ke,v 3

=

[Z e

NTe A1 Ne dx2 ,

Ωe

Ke,v 2 =

[Z e

o n T N dx2 , N′ e Ae,v e 3 a

Ωe

Ke,v 4

NTe Ae,v 2 Ne dx2 ,

=

[Z e

Ωe

(41)

T

′ N′ e Ae,v 4 N e dx2 , (42)

in which the notation {⋆}a is devoted for the anti-symmetric part of the {⋆} and {⋆}′ means the differentiation with respect to x2 . 4.3

Computation of time-space solution

For fixed values of (s, k1 ) in the Laplace-Fourier transformed domain, the solution ˜ may be computed by solving the system of linear equations in the complex of u domain (37). The solutions for p˜1 and p˜2 in two fluid domains may be then determined by using equations (19) and (20), respectively. In order to obtain the spatio-temporal solution, we need to perform a numerical inverse Laplace-Fourier transform. Note that the complex inverse Laplace transform will be used, hence we don’t have the well-known ill-posed problem due to the real inverse Laplace transform (see e.g Davies and Martin (1979)). In this paper, the inverse Fourier transform is computed by using the usual Fast Fourier Transform (FFT) technique. The inverse Laplace transform is carried out using the Convolution Quadrature Method (CQM) which has proved to be a very efficient technique for computing the time response solution in many dynamic problems (Schanz and Antes 1997). Let N1 and ∆k1 respectively the natural number and step of the wavenumber k1 using for the FFT procedure. The space-solution is reconstructed upon on a broad spectral band [−k1max , k1max ] where k1max = N1 ∆k1 /2. The space interval is then ] , xmax calculated by ∆x1 = 2π/(N1 ∆k1 ) corresponding to a space span [−xmax 1 1 max with x1 = π/∆k1 . Using the CQM, the solution in the time domain, e.g. of p1 (x, t), corresponding to a time-dependent excitation function f0 (t) is evaluated by a convolution integral: p1 (x, t) = =

Z Z

t

π1 (x, τ )f0 (t − τ )dτ

0 t 0

L−1 [˜ π1 (x, s)]f0 (t − τ )dτ,

(43)

where π1 is the fundamental solution that corresponds to a Dirac distribution excitation (i.e. the Green function) and π ˜1 is its Laplace transform. The CQM supplies a stable quadrature formula to approximate this convolution integral in the



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time-domain and the required computational time with this formula is attractive: p1 (x, n∆t) =

n X

ωn−m (x, ∆t) f0 (m∆t) ,

m=0

(n = 0, 1, ..., Nt ) ,

(44)

where ∆t is the time step defined by ∆t = T /Nt in which T designates the duration of the phenomenon studied and Nt is a natural number. The weights ωn (x, ∆t) are determined in terms of π ˜1 (x, s): ωn (x, ∆t) =

Nt −1 R−n X 1 iℓ N2π −inℓ N2π t , γ Re t π ˜1 x, e Nt ∆t

(45)

ℓ=0

where γ(z) = 3/2 − 2z + z 2 /2 and√R is the radius of a circle in the domain of analyticity of π ˜1 . Choosing RNt = ǫ, formula (45) yields an error in ωn of order O(ǫ), where the “big-oh” notation was used. Furthermore, due to the exponential term in formula (45), the approximate convolution integral in equation (44) may be efficiently computed using the FFT technique. Remark 1 : To reduce the computational cost, the matrices K1 , K2 , K3 and K4 may be computed once before the loop on (s, k1 ). For each value of s and k1 , the global matrix K is obtained by performing the summation given in (38). Moreover, one may note that: K(−k1 , s) = KT (k1 , s) and U(s∗ ) = U∗ (s). 5.

Numerical tests

5.1

5.1.1

Validation

Numerical and physical parameters

This section presents some numerical tests describing an in vivo ultrasound test on human cortical long bones. The acoustic source (Eq. (2)) located at the position (xs1 , xs2 ) = (0, 2) (in mm) in the upper fluid domain Ωf1 has the time-history function given by: 2

F (t) = F0 e−4(fc t−1) sin(2πfc t),

(46)

where F0 = 1 m.s−2 and the fc is the central frequency. For all tests presented below, the total duration of the simulation will be taken by T = 2 × 10−5 s. For the validation test, a source with the central frequency fc = 1 MHz is applied. Figure 2 depicts the time function of this signal F (t) and its spectrum which contains a frequency bandwidth about 0 − 2.5 MHz (obtained by Fourier transform). Both fluid domains Ωf1 and Ωf2 are assumed to be identical and are considered as an idealized fluid, i.e. γ1 = γ2 = 0. Here, the mechanical properties of the fluids are given by ρ1 = ρ2 = 1 000 kg.m−3 et c1 = c2 = 1 500 m.s−1 . The cortical bone has a constant thickness h = 4 mm. The bone is assumed to be a transversely isotropic viscoelastic medium for which the elastic and viscoelastic tensors are respectively defined by 5 independent constants (c16 = c26 = 0 and η16 = η26 = 0). For the numerical tests, we used the data given in Ha¨ıat et al. (2009) in which the elastic and viscosity tensors were derived from Dong and Guo (2004); Baron et al. (2007) (for elastic tensor) and from Sasso et al. (2007) (for viscosity tensor). According



12

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V.-H. Nguyen and S. Naili 2

50

1.5 FFT[F(t)]

F(t)

10

100

0

−50

−100 0

x 10

1

0.5

1

2

3 Time (s)

4

0 0

5

1

−6

x 10

2 3 Frequency (Hz)

4

5 6

x 10

Figure 2. Time-history function of the source (left) and its spectrum (right)

to the variations within the physiological range given in these references, Tab. 1 presents the maximal, minimal and reference (mean) values of elastic and viscosity parameters. Table 1.

Mechanical properties of cortical bone obtained from Ha¨ıat et al. (2009); Naili et al. (2010)

Values Ref. Min. Max.

ρs (kg.m−3 )

c11 (GPa)

c22 (GPa)

c12 (GPa)

c66 (GPa)

η11 (Pa.s)

η22 (Pa.s)

η12 (Pa.s)

η66 (Pa.s)

1722 1660 1753

23.05 17.6 29.6

15.1 11.8 25.9

8.7 5.1 11.1

4.7 3.3 5.5

157 39 521

109 44 334

121 39 131

18 0 36.2

Let us consider a homogeneous bone layer which has the “mean values” of elastic and viscosity tensors given in Tab. 1. According to these parameters presented, the phase velocities of compressional waves in Ωb propagating along e1 - and e2 -axes are cp1 = 3 698 m.s−1 and cp2 = 2 984 m.s−1 , respectively. As we have mentioned before, wave attenuation in the bone tissue may be caused by two different physical effects: (i) transmission of mechanical energy to heat due to the viscoelasticity of the medium and (ii) scattering of the energy at the local heterogeneities (so called Bloch dispersion, see for instance Andrianov et al. (2008)). The latter leads to the appearance of pass and stop frequency bands. It is theoretically known (Andrianov et al. 2008) that the first stop band threshold appears at λ = 2ℓ, where λ is the wavelength and ℓ is the typical size of the microstructure. For the case in consideration, the smallest wavelength in cortical bone corresponds to the minimum shear wavelength which is about 1.4 mm (cs ∼ 1 400 m.s−1 and f = 1 MHz). This wavelength is much higher than the size of the pore in cortical bone (about 100 µm if we consider that the pore’s size is equivalent to a typical diameter of Haversian canals). As a consequence, at the given frequency, the Bloch dispersion would acceptably be neglected. The numerical parameters used for simulation have been chosen in a similar way to the one required for a common procedure of dynamic finite element analysis (see e.g. Bathe (1996)). The space step ∆x is about the 1/10 shortest wavelength and the time step is chosen by checking the Courant-Friedrichs-Lewy condition. In order to obtain a bounded-value solution in space on the e1 -axis, the domain size should be chosen sufficiently large to avoid the fact that fastest waves pass over the boundaries after the duration T . To carry out the finite element analysis on the e2 -axis, the bone plate’s thickness is discretized into 8 three-noded quadratic isoparametric Lagrangian elements. The numerical values of parameters required for the finite element analysis as well as for the inverse Laplace-Fourier transform



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are given in Tab. 2.

Table 2.

Parameters used for numerical simulation

nel

order

∆x1 (µm)

N1

T (µs)

Nt

ǫ (Eq. (45))

8

3

75

2 048

20

2 048

10−12

5.1.2

Comparison between SFEM and dynamic FEM solutions

x1 = 0.02 m Elastic Viscoelastic

p1(x1,t) (kPa)

0.02

0

−0.02

−0.04

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2 −5

x 10

x1 = 0.02 m

−3

2

1 time (s)

x 10

Elast−SFEM Elast−FEM Visco−SFEM Visco−FEM

p1(x1,t) (kPa)

1 0 −1 −2 −3 −4

0.85

0.9

0.95

1 time (s)

1.05

1.1

1.15

1.2 −5

x 10

Figure 3. Upper graph: Signal p1 (t) measured at the point (20,2)(mm); lower graph : zoom of the same signal and comparison with FE results

Figure 3 (upper graph) presents the numerical solution of p1 measured at the position (x1 , x2 ) = (20, 2) (mm) by using the proposed spectral finite element procedure. Both elastic and viscoelastic cases have been considered. The first peak corresponds to the first wave propagating directly in the solid plate and called First Arriving Signal (FAS). The second one corresponds to the first wave reflected from the fluid-solid upper interface. The third one, that has biggest amplitude, corresponds to the direct wave propagating in the fluid. These results show that taking into account the viscosity in the bone plate highly weakens the p1 signal. For validation purposes, we also provide the solutions of both cases obtained from conventional finite element method (Naili et al. 2010). We may find that the SFEM and FEM results match each to other very well in both elastic and viscoelastic cases. Note that while the dynamic finite element method needs about 2 hours for simulation on a common desktop PC (using the software COMSOL Multiphysics (2008)), the proposed spectral finite element procedure needs only about 200 seconds on the same PC.



14

5.1.3

Nguyen-


Convergence and stability analyses

In order to perform a convergence analysis of the proposed method, we introduce 1 (x) that is defined by: a function Lp2,T v u Nt u X p1 (p1 (x, n∆t))2 , L2,T (x) = t∆t

(47)

n=0

then the relative error at a point x is estimated by: p L 1 (x) − Lp1 ,ref (x) 2,T 2,T relative error = , p1 ,ref L (x)

(48)

2,T

p1 ,ref (x) is the reference value obtained from the FE analysis. Figure 4 where L2,T presents convergence studies with respect to different parameters by calculating relative errors at two points R1 and R2 . In Fig. 4a, the parameters N1 = 2 048 and nel = 8 which determine the discretization in space are fixed and the effect of the total number of time step Nt (that is related to time step size ∆t) on the convergence is shown. Next, Fig. 4b shows the effect of sampling number N1 , which is associated with the wavenumber k1 , by fixing Nt = 2 048 and nel = 8. Last, Fig. 4c presents the relative errors versus the number of finite elements nel when using N1 = 2 048 and Nt = 2 048. These studies have shown that the proposed procedure using spectral finite elements has very good convergence. (a) N1 = 2048; nel = 8

(b) Nt = 2048; nel = 8

0.3 x1 = 0.002m relative error

0.2 0.15 0.1

0.15 0.1 0.05

0

0

11 log2(Nt)

12

13

x1 = 0.002m

0.25

x1 = 0.02m

0.2

0.05 10

x1 = 0.002m

0.25

x1 = 0.02m

9

0.3

relative error

0.25 relative error

(c) N1 = 2048; Nt = 2048

0.3

x1 = 0.02m

0.2 0.15 0.1 0.05

9

10

11 log2(N1)

12

13

0

2

4 6 8 number of elements

Figure 4. Convergence studies: numerical errors versus (a) time sample number (Nt ), (b) space sample number (N1 ), (c) finite element discretization (nel )

5.2

Effects of the frequency and viscosity on the FAS velocity

The FAS (First Arriving Signal) velocity has been shown to be an important parameter to quantify the bone characteristics when the axial transmission technique is used. In this section, we present some numerical results to study the influence of the emitted signal’s frequency on the FAS velocity. The effect of the bone’s viscosity is taken into account. In order to determine the FAS velocity, the p1 -signal is captured at an array of 14 sensors, each 0.8 mm apart. The distance from the emitted source to the first sensor is 11 mm. At each sensor, the first zero-crossing location of measured p1 -signal may be determined and the FAS velocity is evaluated by the slope of the the linear regression of these location over 14 sensors. The result obtained on the 14 sensors is shown in Fig. 5. Figure 6 compares the FAS velocities, denoted VF , according to different central frequencies (ranging from 500 kHz to 5 MHz). Both elastic and viscoelastic

10



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sensor number #

15

10

5

0

0

0.5

1 time (s)

1.5

2 −5

x 10

Figure 5. Seismograph of p1 (t) measured at 14 sensors: the red line is the linear regression of zero-crossing positions.

(A) Homogeneous bone plate (ref. values)

(B) Homogeneous bone plate (min. values)

4100 Elastic Viscoelastic

3350

3900

FAS (m.s−1)

FAS (m.s−1)

4000

3400

3800 3700

3300 3250 3200

3600 3500 0

1

2 3 frequency (MHz)

4

3150 0

5

(C) Homogeneous bone plate (max. values) Elastic Viscoelastic

4600 FAS (m.s−1)

FAS (m.s−1)

2 3 frequency (MHz)

4

5

4800

4600 4400 4200

Elastic Viscoelastic

4400 4200 4000

4000 3800 0

1

(D) Linearly graded bone plate

5000 4800

Elastic Viscoelastic

1

2 3 frequency (MHz)

4

5

3800 0

1

2 3 frequency (MHz)

4

5

Figure 6. Dependency of FAS velocities on elastic and viscoelastic properties of bone as well as on the frequency of emitted signal

cases have been studied. For each case, we consider 4 set of bone properties: (A) homogeneous with reference values, (B) homogeneous with minimum values, (C) homogeneous with maximum values and (D) linearly graded with maximum values at upper interface and minimum values at lower interface. Note that elastic constants of both elastic and viscoelastic cases are the same but the viscosity constants are set to be zero for the elastic case. When considering the elastic cases, one may check that the FAS velocity shown in Fig. 6 has good agreement with statement made in other works in the literature (Ha¨ıat et al. 2009; Naili et al. 2010). The quantity VF depends strongly on the elasticity coefficients of cortical bone. The comparison between two curves of the



16

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elastic case in Figs. 6(C) and 6(D) shows that the effect of functionally graded bone material is significant in low frequency range [500 kHz − 2 MHz]. However, this effect is no longer important in the higher frequency range as the FAS velocity depends only on the axial wave celerity at the upper surface of the bone plate. Note that similar remarks have been stated in Pham et al. (2009). The FAS velocities obtained by using the viscoelastic model are more important the ones obtained by using the elastic model. Moreover, these differences considerably increase with higher emitted frequencies. However, when the bone’s viscosity is important (cases (C) and (D)), we cannot determine the FAS velocity for high frequencies (fc = 4 MHz and fc = 5 MHz respectively) because of the very strong attenuations in the bone plate. 6.

Conclusion

Mechanical modeling of QUS using axial transmission technique deals with a timedomain vibro-acoustic problem. This configuration may be modeled by a twodimensional trilayer medium which consists of a solid layer (representing cortical bone layer) sandwiched between two fluid media (representing soft tissues such as skin or marrow). The bone material is assumed to be an anisotropic viscoelastic solid and the fluids are idealized as two fluid halfspaces. The numerical scheme derived from the proposed spectral finite element formulation has been shown to be not only very stable for the simulation of the propagation of ultrasonic waves through a functionally graded anisotropic elastic bone plate immersed in a fluid, but also very competitive in computational time. Some advantages of this method may be outlined. Firstly, in the Laplace-Fourier domain, the presence of two halfspace domains may be represented by exact radiation conditions derived from the analytical solutions of the waves in the fluids. Consequently, the finite element modelling is carried out uniquely for the solid domain. Secondly, thanks to the fact that only one-dimension problem is constructed in (s − k1 ) domain, the memory required for computation is very small in comparison with the one required for two-dimensional finite element analysis in the time-domain. It has been shown that the Convolution Quadrature Method is an efficient and stable technique in order to evaluate the time-domain solutions for the transient high-frequency problem presented here. The proposed method allows us to consider the anisotropy as well as the heterogeneity of the solid material. Using this method, numerical studies of ultrasonic wave propagation in cortical bone may be easily carried out, eventually in the very high frequency domain. The FAS velocity is strongly influenced by viscous effects of the solid layer. Moreover, high frequency ultrasonic waves may be totally attenuated by viscosities in the bone. The numerical results have shown that the spatial gradient of material properties with depth has a significant influence on the behavior of ultrasonic waves propagating in cortical bones. In the scope of this paper, we only presented some preliminary results for illustration purposes. It worths to mention that the linear theory of viscoelasticity without memory (i.e. viscosity is assumed to be frequency-independent) was used for the study. In certain situations, the viscosity can depend on the frequency (see Sasso et al. (2007)). In such a case, the magnitude of tensor E increases with the increase of the frequency. To take into account the frequency-dependent viscosity of cortical bone as well of the marrow (see e.g. (Bryant et al. 1989; Naili et al. 2010)), one can propose a physically reasonable simplification which accept that the viscosity depend only on the central frequency of the emitted signal. Hence, the solution obtained in the present paper is a model one and in reality we must


Computer Methods in Biomechanics and Biomedical Engineering REFERENCES

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use different values of Eta for different frequencies. Note that the proposed formulation, which is established in the frequency domain before inverting back to the time domain, has the potential to take into account the effect of nonlinear frequency-dependent viscoelasticity. The detail investigation of behaviour of heterogeneous viscoelastic cortical bone under broadband frequency ultrasounds will be presented in a forthcoming paper. References Andrianov I, Bolshakov V, Danishevs’kyy V, Weichert D. 2008. Higher-order asymptotic homogenization and wave propagation in periodic composite materials.. Proc. R. Soc. A 464:1181–1201. Barkmann R, Kantorovich E, Singal C, Hans D, Genant HK, Heller M, Gluer CC. 2000. A new method for quantitative ultrasound measurements at multiple skeletal sites. J. Clin. Densitometry 3:1–7. Baron C, Naili S. 2008. Elastic wave propagation in a fluid-loaded anisotropic waveguide with laterally varying properties. C.R. M´ ecanique 336:772–730. ———. 2010. Propagation of elastic wavesin a fluid-loaded anisotropic functionally graded waveguide: Application to ultrasound characterization. J. Acoust. Soc. Am 127(3):1307–1317. Baron C, Talmant M, Laugier P. 2007. Effect of porosity on effective diagonal stiffness coefficients (cii ) and anisotropy of cortical bone at 1 MHz: A finite difference time domain study. J. Acoust. Soc. Am. 122:1810–1817. Bathe KJ. 1996. Finite Element Procedure. Prentice Hall. Bossy E, Talmant M, Laugier P. 2002. Effect of bone cortical thickness on velocity measurements using ultrasonic axial transmission: A 2D simulation study. J. Acoust. Soc. Am. 112:297–307. ———. 2004. Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models. J. Acoust. Soc. Am. 115:2314–2324. Bousson V, Meunier A, Bergot C, Vicaut E, Rocha M, Morais M, Laval-Jeantet A, Laredo J. 2001. Distribution of intracortical porosity in human midfemoral cortex by age and gender. J. Bone Miner Res. 16:1308–11317. Bryant J, David T, Gaskell P, King S, G. L. 1989. Rheology of bovine marrow. Proc. Inst. Mech. Eng. 203:771–775. Carcione J. 2001. Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous Media. Elsevier. COMSOL Multiphysics. User’s Guide. 2008. Davies B, Martin B. 1979. Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of Computational Physics 33(1):1–32. Desceliers C, Soize C, Grimal Q, Haiat G, Naili S. 2008. A time-domain method to solve transient elastic wave propagation in a multilayer medium with a hybrid spectral-finite element space approximation. Wave Motion 45(4):383 – 399. Desceliers C, Soize C, Grimal Q, Talmant M, Naili S. 2009. Identification of an anisotropic elasticity tensor for an elastic layer using transient wave propagation in a fluid-solid multilayer – Model with uncertainties and experiments.. J. Acoust. Soc. Am. 125(4):2027–2034. Dong NX, Guo EX. 2004. The dependence of transverse isotropic elasticity of human femoral cortical boneon porosity. J. Biomechanics 37(8):1281–1287. Givoli D. 2004. High-order local non-reflecting boundary conditions: a review. Wave Motion 39(4):319 – 326. Gopalakrishnan S, Chakraborty A, Roy Mahapatra D. 2008. Spectral Finite Element Method. Spinger. Grimal Q, Naili S. 2006. A theoretical analysis in the time-domain of wave reflection on a bone plate. J. Sound Vib. 298:12–29. Ha¨ıat G, Naili S, Grimal Q, Talmant M, Desceliers C, Soize C. 2009. Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: Application to axial transmission. J. Acoust. Soc. Am. 125(6):4043–4052. Han X, Liu GR, Lam KY. 2001a. Transient waves in plates of functionally graded materials. International Journal for Numerical Methods in Engineering 52(8):851–865. Han X, Liu GR, Xi ZC, Lam KY. 2001b. Transient waves in a functionally graded cylinder. International Journal of Solids and Structures 38(17):3021 – 3037. Hans D, Srivastav SK, Singal C, Barkmann R, Njeh CF, Kantorovich E, Gluer CC, Genant HK. 1999. Does combining the results from multiple bone sites measured by a new quantitative ultrasound device improve discrimination of hip fracture?. J. Bone Miner. Res. 14(4):644–651. Kampanis NA, Dougalis VA, Ekaterinaris JA, editors. 2008. Effective Computational Methods for Wave Propagation. Chapman & Hall/CRC. Kausel E. 2006. Fundamental Solution in Elastodynamics. Cambridge University Press. Lakes R, Yoon HS, Katz JL. 1986. Ultrasonic wave propagation and attenuation in wet bones. J. Biomed. Eng. 8:143–148. Langton CM, Ali A, Riggs CM, Evans JA, Bonfield W. 1990. A contact method for the assessment of ultrasonic velocity and broadband attenuation in cortical and cancellous bone. Clin. Phys. Physiol. Meas. 11:243–249. Liu G, Xi ZC. 2002. Elastic Waves in Anisotropic Laminates. CRC Press. Lowet G, Van derPerre G. 1996. Ultrasound velocity measurements in long bones: measurement method and simulation of ultrasound wave propagation. Journal of Biomechanics 29:1255–1262. Macocco K, Grimal Q, Naili S, Soize C. 2005. Probabilistic modelling of an ultrasonic setup: Calculation of the dispersion on wave measurement. C. R. Mec. 333:565–573.

March 8, 2016 13:19 Naili˙CMBBE˙revised 18


Nguyen-

REFERENCES

———. 2006. Elastoacoustic model with uncertain mechanical properties for ultrasonic wave velocity prediction: application to cortical bone evaluation. J. Acoust. Soc. Am. 119:729–740. Marzani A. 2008. Time-transient response for ultrasonic guided waves propagating in damped cylinders. International Journal of Solids and Structures 45(25-26):6347 – 6368. Moilanen P. 2008. Ultrasonic guided waves in bone. IEEE Transactions of Ultrasonics, Ferroelectrics, and Frequency Control 55(6):1277–1286. Naili S, Vu MB, Grimal Q, Talmant M, Desceliers C, Soize C, Ha¨ıat G. 2010. Influence of viscoelastic and viscous absorption on ultrasonic wave propagation in cortical bone: Application to axial transmission. J. Acoust. Soc. Am. 127(4):2622–2634. Nayfeh HA. 1995. Wave Propagation in Layered Anisotropic Media. Elsevier. Nicholson P, Moilanen P, Karkkainen T, Timonen J, Cheng S. 2002. Guided ultrasonic waves in long bones: modelling, experiment and in vivo application. Physiological Measurement 23(4):755–768. Pham TL, Moinonzio JG, Talmant M, Laugier P. 2009. Impact of a multi-frequency sequence of measurement on first arriving signal velocity on a bone plate modelIn: IEEE International Ultrasonics Symposium Proceedings. Protopappas VC, Kourtis IC, Kourtis LC, Malizos KN, Massalas CV, Fotiadisa DI. 2007. Threedimensional finite element modeling of guided ultrasound wave propagation in intact and healing long bones. J. Acoust. Soc. Am. 121(6):3907–3921. Sansalone V, Naili S, Bousson V, Bergot C, Peyrin F, Zarka J, Laredo JD, Ha¨ıat G. 2010. Determination of the heterogeneous anisotropic elastic properties of human femoral bone: from nanoscopic to organ scale. J. Biomech. 43:1857–1863. Sasso M, Ha¨ıat G, Yamato Y, Naili S, Matsukawa M. 2007. Frequency Dependence of Ultrasonic Attenuation in Bovine Cortical Bone: An In Vitro Study. Ultrasound in Medicine & Biology 33(12):1933 – 1942. ———. 2008. Dependence of ultrasonic attenuation on bone mass and microstructure in bovine cortical bone. J. Biomech. 41(2):347 – 355. Schanz M, Antes H. 1997. Application of ’Operational Quadrature Methods’ in Time Domain Boundary Element Methods. Meccanica 32:179–186. Shuvalov A, Poncelet O, Deschamps M. 2006. Analysis of the dispersion spectrum of fluid-loaded anisotropic plates: leaky-wave branches. Journal of Sound and Vibration 296(3):494 – 517. Stegman MR, Heaney RP, Travers-Gustafson D, Leist J. 1995. Cortical ultrasound velocity as an indicator of bone status. Osteoporos. Int. 5(5):349–533. Thomas C, Feik SA, Clement J. 2005. Regional variation of intracortical porosity in the midshaft of the human femur: age and sex differences. J. Anat. 206:115–125. Thompson LL, Huan R. 2000. Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation. Computer Methods in Applied Mechanics and Engineering 187(1-2):137 – 159. Virieux J. 1986. P-SV-wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51(4):889–901.

A spectral finite element

A spectral finite element

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