Wave Propagation in Poroelastic Materials

Transcription

1 Wave Propagation in Poroelastic Materials M. Yvonne Ou Department of Mathematical Sciences University of Delaware Newark, DE ructure Interaction Probs, May Acknowledgement: Research funded by NSF-DMS Math. Biology Grant and NIH CBER PILOT Grant , and collaborators E. Cherkaev (UUtah), R. Lemoine & R. LeVeque (UW), S. Nintcheu(ORNL), M. Mak & M. Xu (Chinese U. of HK)

2 Outline Motivation Biot s equation for poroelasticity Balance form and finite volume method Numerical results of a plane strain problem using CLAWPACK Application to cancellous bones in literature Numerical simulation for effective elastic moduli by boundary element method (BEM) Conclusion and Future Work

3 Images of cancellous bone and cortical bone Figure:

4 Main interest: Two classes of inverse problems for composite materials To infer statistical information from measurement of effective properties: Dehomogenization To infer effective properties from waves requires building an effective wave solver for the governing equation and a better inversion scheme [Buchanan-Gilbert-MYO-211].

5 Effective permittivity with isotropic constituents [Golden-Papanicolaou-83] The complex permittivity of the medium is modeled by a (spatially) stationary random field (x ; ), x 2 R d and 2, where is the set of all realizations of the random medium, (x ; ) = 1 1 (x ; ) (x ; ): (1) E (x ; ): electric field, D(x ; ): displacement field e k : unit vector in direction k, < >: ensemble average over (PDE) r D = ; r E = ; < E >= e k : (2) Effective complex permittivity < E > def = < D >

6 Integral representation formula (IRF) Theorem (Golden-Papanicolaou-83) Let be one of the diagonal component of the effective permittivity tensor and s def = 1=(1 h); h def = 1 = 2. Then F (s) def = 1 (h)= 2 is analytic off [; 1] in the s plane with the integral representation F (s) = 1 Z 1 = 2 dz s z ; (3) where the positive measure on [; 1] is the spectral measure of the self-adjoint operator 1, where = r( ) 1 r (Helmholtz projector). Separation of contrast (h) and information of microstructure (). This provides a characterization of microstructure (Cherkaev and Zhang). If the is known, then computation of the effective property amounts to only plug in the contrast s into IRF.

7 Numerical results for measure Uniqueness of from data on an arc on the!-plane [E. Cherkaev 21] p=.11 p=.25 p= Why Moments?.25 p=.43 p=.25 p= Figure: Reconstructions of the spectral function of Maxwell-Garnett composite using quadratic regularization (left figure) and nonnegativity constraint (right figure). [Cherkaev-MYO-8]

8 First few moments Table: n calculated for Maxwell-Garnett composite solving regularized problem for the spectral function. Volume fractions: 11%; 25%; 43%. [Cherkaev-MYO-8] Exact N.neg Tik Exact N.neg Tik Exact N.neg Tik Note: Tik: Tikhonov; n-neg: Non-negativity

9 Relations between the moments of and the microstructure [Golden-Papanicolaou-83] m(h) := = 2 ; h := 1 = 2, analytic outside ( 1; ]. Expanding m at h = 1 and F (s) at s = 1 Z ( 1) n 1 1 m (n) (1) = z n 1 (dz ) =: (n 1) : n! Differentiating m at h = 1 m (1) = volume fraction of region occupied by 1 : (n 1) is related to the integral of n-point correlation functions, n = 1; 2; 3; : : :. If the permittivity 1 or 2 depends on parameters such as frequency or temperature, we may reconstruct the moments from information of 1, 2 and.

10 Reconstruction of Moments [Cherkaev-MYO-8, MYO-1] F (s) = Z 1 dz s z = s + 1 s s 3 + :::; n = Z 1 z n dz Reconstruction of coefficients of analytic functions in Hardy space H 2 in the unit circle jj < 1, = 1=s. 1 Polynomial interpolation of the kernel function s k z using n nodes z i, i ; k = 1; : : : ; n on [; 1] this results in an inversion scheme for the weights of a finite sum of Dirac measures supported at z i whose moments are close to those of. [L=M ] Padé approximation of the effective function F (s) this results in an inversion scheme for the coefficients of Padé approximants whose first L + M + 1 Taylor coefficients agree with those of F (s) at s = 1. Analytic continuation of F (s) into jsj < 1 with simple poles on [; 1].

11 Biot s equations for waves in poroelastic materials M. A. Biot, J. Acous. Socie. Amer. (1956a, 1956b). Three pieces of physics 1 Strain-stress relation 2 Kinetic energy 3 Energy dissipation u: displacement vector for solid U : displacement vector for fluid * Stress acting on the solid part of each face: ij, i ; j = 1; 2; 3 * Stress tensor acting on the fluid part of each face: s ij, s = p, : porosity. * Strain tensor in the solid: e ij = 1 2 j * Strain tensor in the fluid: =r U i ), i ; j = 1; 2; 3 Coupled stress-strain relation ij = 2N e ij + ij (Ae + Q), s = Q + Re. Here e def = r u.

12 Kinetic energy T = j _uj ( _u _U ) j U _ j 2 Dissipation function D = bj _u _U j 2 = b[( u_ 1 U_ 1 ) 2 + ( u_ 2 U_ 2 ) 2 + ( u_ 3 U_ 3 ) 2 ] *^b is frequency-dependent for high frequency! >! c Balance P of force in x 1 -direction Solid j = j U _ U _ _ u 1 Dynamic equations N r 2 u + r[(a + N )e + 2 ( 11 u + 12 U ) (u U ) r[qe + 2 ( 12 u + 22 U (u U ) Two longitudinal waves by taking r on both sides to get wave equations r 2 (Pe + 2 ( 11 e + 12 ) (e ) r 2 (Qe + 2 ( 12 e (e ) *P=A+2N

13 Biot s 1962 formulation in JAP Stress-strain relations = C u ɛ = [ 11 ; 22 ; 33 ; 23 ; 13 ; 12 ; p] t ɛ = [ 11 ; 22 ; 33 ; 2 23 ; 2 13 ; 2 12 ; ] t = div [(U u)] : variation of Fluid content Equations of j ( ij ) t v i + t q i + f i ; i = 1; 2; 3 v solid velocity, q = (@ t U v): fluid velocity rel. to solid, = (1 ) s + f Dynamic Darcy s i p = t v i + i t q i (t) i (t): viscodynamic operator

14 J. Carcione, JASA 1996! c := min i m, m i i := f T i i For! <! c : i (t) = m i (t) + H (t) i p = t v i + m t q i + i q i For!! c, approximate by the generalized Zener kernel 2 XL i 3 i (t) = 1 e t il 5 H i p = t v i + i L i l=1 il il il ; il : relaxation times; il il 1 i q i i L i XL i 1 il A qi + il t e 1 il = e 1 il + il L i il 1 XL i i e il l=1 il q i il

15 A transversely isotropic case: plane strain problem Stress-strain t xx = c u x v x + c u z v z + 1 M (@ x q x z q z ) t s 11 t zz = c u x v x + c u z v z + 3 M (@ x q x z q z ) t s 33 t xz = c u 55 (@ z v x x v z ) t s 55 t p = 1 x v x 3 z v z M (@ x q x z q z ) t s f (7) Equations of motion Darcy s law for low frequency t v x + t q x x xx z xz t v z + t q z x xz z zz x p = t v x + m t q x z p = t v z + m t q z + q x (1) 1 q z (11) 3

16 Balance Law where A t g + A@ x g + B@ z g = Dg t s (12) t g = xx zz xz v x v z p q x q z t s t s t s t s t s f c11 u 1 M c13 u 3 M c55 u m 1 41 f 41 m M M f f 43 1 C A (13)

17 B = c13 u 1 M c33 u 3 M c55 u m 1 41 m 3 43 f 43 3 M M f 41 f f 4 D = 11 f C A 1 C A (14) (15)

18 Finite Volume Method Normal and transverse Riemann problem solver Godunov operator splitting with 4t = O(1 5 ) MC limiter Preliminary results 1. Materials with different porosities. Left: :2, Right: :5 2. AMRCLAW zz and p excited at point source by Ricker pulse with peak frequency of 3135Hz, sandstone and shale

19 e228 A. Hosokawa / Ultrasonics 44 (26) e227 e231 Outline Dehomogenization Ultrasound Image processing Computation Parameters of bovine cancellous bone models in the viscoelastic FDTD Ultrasound in cancellous bone and method relevant effective 2.1. Viscoelastic FDTD method Trabecular thickness a parameters s.15 mm Pore space between trabeculae af 1.35 mm e s - r - - t e F ðjþ ¼ 1 jt ðjþ ; 2. FDTD methods The wave propagation in a 2D viscoelastic medium is Trabecular length ls 4.5 mm governed by the following equations: sonics Hosakawa 44 (26) e227 e231 26, Ultrasonics e229 q o_u i ot ¼ os ii oi þ os These values were uniformly distributed in the range of ±25%. xy oj ; ð1aþ Table 3 os ii Þ Biot s parameters of bovine cancellous ot þ c iis bone ii ¼ðkþ2lÞ o_u i models oi þ k o_u j oj with ; parallel ð1bþ and perpendicular trabecular orientations os xy Þ ot þ c Parallel s xy xy ¼ l o_u x oy þ o_u y ; ð1cþ ox Perpendicular Young s modulus of solid bone E s where i, 22. j = x, y GPa (i 5 j), u i are the 22. particle GPa displacement Þ Poisson s ratio of solid bone m s components.32 of the medium in the.23 i-direction (an overdot Bulk modulus of bone marrow K f denotes 2. a timegpa derivative), s 2. GPa Density of solid bone q s 196 kg/m 3 ii are the normal stress components in the i-direction, s 196 kg/m 3 Þ Density of bone marrow q f 93 kg/m 3 xy is the shear stress component on the x y plane, c 93 kg/m 3 ii and c xy are, respectively, the resistance Poisson s ratio of trabecular frame m b coefficients for the normal and shear stress components related to the attenuation coefficients of the longitudinal and Variable r Þ Viscosity of bone marrow g 1.5 P s 1.5 P s shear waves, q is the medium density and k, l are Lamé Parameter n coefficients. For Eqs. (1a) (1c), central differencing with Pore size a 1.35 mm 1.35 mm staggered grids is Permeability k used to constitute a leap-frog system [8] for _u Porosity b i and s.83 ii, s xy..83 Resistance coefficients c ii (i = x,y) A schematic 1. drawing 1 6 of the cancellous 1. bone 1 model 6 in the Resistance coefficient c viscoelastic xy 1. FDTD 1 method 7 [9] is shown 1. in Fig The skeletal frame of cancellous bone was made of numerous trabecular rods. The porosity b was adjusted by changing the trabecular thickness a s and pore space between trabeculae a f, and the trabecular orientation was adjusted by changing M. Y. Ou (UDel) the trabecular length Poroelasticity l s and angle h. In order to form an Table 1 Fig. 2. Geometric drawings of bovine cancellous bone models in the viscoelastic FDTD method, with (a) parallel and (b) perpendicular trabecular orientations. Table 2

20 to the trabecular alignment, and Fig. 4(b) shows the waveforms in the perpendicular direction. The experimental waveforms in the parallel and perpendicular directions Hosakawa s result Outline Fig. Dehomogenization 4 shows the viscoelastic FDTD (upper) Ultrasound and Biot s it is shown in Fig. Image 4(b) that, processing using Biot s FDTD method, Computation FDTD (middle) results of the pulsed waveforms propagating the simulated waveform in the perpendicular direction is through bovine cancellous bone, together with the not in good agreement with the experimental waveform. experimentally observed results (lower). Fig. 4(a) shows The slower speed of the fast wave and the higher attenuation the waveforms for the propagation in the direction parallel of the slow wave, which are the propagation characteristics in the perpendicular direction, could be analyzed by increasing value of the parameter n and by decreasing value of the permeability k because n and k are inversely propor- Amplitude (arb. units) Amplitude (arb. units) Time (2 μs/div.) (a) Time (2 μs/div.) (b) Fig. 4. Viscoelastic FDTD (upper), Biot s FDTD (middle) and experimental (lower) results for observations of pulsed waveforms propagating through bovine cancellous bone in the directions (a) parallel and (b) perpendicular to the trabecular alignment.

21 The fluid water characteristics are: bulk modulus K f 2.28 GPa, density f 1 kg m 3, viscosity 1 3 kgms 1. The incident experimental signal generated by the transducer is shown in Fig. 3 and its spectrum in Fig. 4. Fellah et. al. 24, 26,JASA Outline Dehomogenization Ultrasound Image processing Computation Ultrasound in cancellous bone -continue- TABLE I. Biot s model parameters of cancellous bone. Biot s model parameters of cancellous bone M1 M2 M3 Thickness L cm Density s (kg/m 3 ) Porosity Tortuosity Viscous characteristic length m Bulk modulus of the elastic solid Ks (GPa) Bulk modulus of bone skeletal frame Kb (GPa) Shear modulus of the frame N (GPa) FIG. 17. Comparison between experimental transmitted signal solid lin and simulated FIG. 15. transmitted Experimental signal setup dashed for ultrasonic line for bone measurements. sample M2. 7 J. Acoust. Soc. Am., Vol. 116, No. 1, July 24 fluid phases of Fellah theet porous al.: Ultrasonic material propagation and thus in cancellous on the oth bo parameters whichncts were kept constant Wkspduring on this Fluid-St study. In Fellah 26,, 1,, K b and N are sought after by solving the inverse problem.

22 VI. CONCLUSION Outline a fast wave, and adehomogenization part is transmitted as a slow wave. When Ultrasound Image processing Computation any of these components, traveling at different speeds, hit the In this paper, analytical calculus of the reflection and second surface, a similar effect takes place: part is transmitted into the fluid, part is reflected as a fast or slow wave. elastic frame is established. This calculus is based on Biot s transmission coefficient of a slab of cancellous bone with an Fellah et al s results Sample characteristics are measured using standard theory modified by the Johnson et al. model to describe the methods 11,16,17,19,25 27 Appendix B and given in Table I. viscous interaction between fluid and structure. Numerical Solid The line: fluid water Experimental characteristics transmitted are: bulk signal; modulus Dashed K line: Simulated transmitted signal f simulations of transmitted waves in the time domain were 2.28 GPa, density f 1 kg m 3, viscosity 1 3 kgms 1. The incident experimental signal generated by the transducer is shown in Fig. 3 and its spectrum in Fig. 4. TABLE I. Biot s model parameters of cancellous bone. Biot s model parameters of cancellous bone M1 M2 M3 Thickness L cm Density s (kg/m 3 ) Porosity Tortuosity Viscous characteristic length m nts. Bulk modulus of the elastic solid Ks (GPa) Bulk modulus of bone skeletal frame Kb the other(gpa) tudy. Shear modulus FIG. 16. of the Comparison frame N (GPa) between experimental 2.6 transmitted 1.7 signal.35 solid line FIG. 17. Comparison between experimental transmitted For longitudinal signal solid waves, line the relation A5 and simulated transmitted signal dashed line for bone sample M1. and simulated transmitted signal dashed line for bone sample M2. s x, P 2N 2 s f 7 J. Acoust. Figures Soc. 16, Am., 17, Vol. and 116, 18 No. show 1, Julythe 24comparison between Fellah et al.: Ultrasonic propagation in cancellous bone x 2 x 2 cal simuperiments signals dashed line for bone samples M1, M2, and M3, experimental transmitted signals solid line and simulated Using relations 12 14, metrics A respectively. The experimental transmitted wave forms are s x, Z c uency of traveling through the cancellous bone in the same direction a 558PR as the trabecular alignment x direction. For some situations, Z re amplihigh fre- Z fast and slow waves are superimposed, depending on the coupling between the two porous medium phases, so that this le in this is due to the modified Biot s parameters for cancellous bone, Z removed which as mentioned in Sec. IV, play an important role in the setup is arrival time of the two waves. were ma- M. Y. The Ou experimental (UDel) data and FIG. theoretical 18. Comparison prediction between Poroelasticity experimental are transmitted signal solid line where the coefficients Z 1 ( ) and Z 2 (

23 Translation between microstructure and effective parameters Numerical simulation from images of cancellous bone to some effective properties. Ongoing numerical projects focusing on cancellous bones. Step 1 Image segmentation and 3D reconstruction from -CT scans (a) raw -CT scan (b) digitalized data (c) 3-D structure Figure: -CT scans are provided by Prof. L. Cardoso from CUNY Med. Center

24 Numerical projects for effective parameters of cancellous bone, SCBLAP with Dr. S. Nincheu at ORNL Step 2 Meshing and solving (BVP) by BEM Advantage Meshing only the interface and outer surface, not the entire domain. (a) meshed trabeculae (b) electric potential Figure: Our preliminary results

25 Effective dielectric tensor ɛ (BVP) ( Div [(ɛ1 1 (x ) + ɛ 2 2 (x ))r] = ; = P d j =1 E j x j (16) with the constant vector E = e j, the j -th unit vector in Cartesian coordinates ((e j ) i = ij ) followed by evaluating the integral Z ɛ def 1 ik = [ 1 1 (x ) (x )]@ i dx (17) jj 1 Z = jj [ 1 1 n 1 ds n 2 ds ] (18) 2

26 Illustration of ideas To solve Laplace equation: 4u = in a simply connected bounded domain with boundary and constant boundary elements. Key: Green s identity ( R R u(x); x 2 G(x; y)t(y)d y H(x; y) n(y)u(y)d y = ; x 2 R 3 n R G(x; y) = 1 x y 4kx yk, H(x; y) = 4kx yk 3, 1 Triangulate into N non-overlapping elements j, j = 1; ; N. 2 Define g j (x) := G(x; y)d y and h j (x) := H(x; y) n(y)d y. j j 3 Discretized system is Gftg = Hfug, with u = (u 1 ; ; u N ) T and u = (t 1 ; ; t N ) T G ij = lim x!x j g j (x ), H ij = lim x!x j h j (x ), x j 2 j, j = 1; ; N. 4 Substitute in boundary data and rearrange to obtain R Afz g = fbg with fz g containing those u i and t j which are unknown, and fbg with the boundary data. 5 Solve the linear system to obtain u j and t j, j = 1; ; N. 6 For any x 2, we have u(x) = j=1 j=1 NX t j g j (x) NX u j h j (x)

27 Outline Dehomogenization 3. Real Trabecula Marrow Ultrasound Image processing Computation and A small piece of trabecula is used in this test case. The following are the results Figure (viii) Trabecula and Marrow Number of Iteration jj 11 jj 12 jj 13 Energy Norm Figure (ix) Cross-sectional View at z = 15 = Trabecula, Red = Marrow) (Blue

28 Future Work To generalize dehomogenization schemes for viscoelastic composites by using the 2-parameter IRF [MYO-211] To develop a better image processing scheme in order to handle bone sample of realistic size. To derive integral representation formula for permeability tensor. To compare the simulated results with lab measurement. To build highly parallel numerical codes for effective parameters computation for bones Numerical analysis of the FVM and BEM codes.

29 Thank you.