Time-domain modeling of elastic and acoustic wave ... - TSpace

Time-domain modeling of elastic and acoustic wave propagation in unbounded media, with application to metamaterials

by

Hisham Assi

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering collaborative program with the Institute of Biomaterial and Biomedical Engineering University of Toronto

c Copyright 2016 by Hisham Assi

Abstract Time-domain modeling of elastic and acoustic wave propagation in unbounded media, with application to metamaterials Hisham Assi Doctor of Philosophy Graduate Department of Electrical and Computer Engineering collaborative program with the Institute of Biomaterial and Biomedical Engineering University of Toronto 2016 Perfectly matched layers (PML) are a well-developed method for simulating wave propagation in unbounded media enabling the use of a reduced computational domain without having to worry about spurious boundary reflections. Many PML studies have been reported for both acoustic waves in fluids and elastic waves in solids. Nevertheless, further studies are needed for improvements in the fields of formulation, stability, and inhomogeneity of PMLs. This thesis introduces new second-order time-domain PML formulations for modeling mechanical wave propagation in unbounded solid, fluid, and coupled fluid-solid media. It also addresses certain stability issues, and demonstrates application of these formulations. Using a complex coordinate stretching approach a PML for the time-domain anisotropic elastic wave equation in two dimensional space is compactly formulated with two secondorder equations along with only four auxiliary equations. This makes it smaller than existing formulations, thereby simplifying the problem and reducing the computational burden. A simple method is proposed for improving the stability of the discrete PML problem for a wide range of otherwise unstable anisotropic elastic media. Specifically, the value of the scaling parameter was increased thereby moving unstable modes out of the discretely resolved range of spatial frequencies. ii

A new second-order time-domain PML formulation for fluid-solid heterogeneous media is presented. This formulation satisfies the interface coupling boundary conditions which were chosen such that they can be readily integrated into a weak formulation of the complete fluid-solid problem and which can be used in a finite element method (FEM) analysis. Numerical FEM results are given to establish the accuracy of the formulations and to provide examples of their application. In particular, numerical examples are shown to validate the elastic wave PML formulation and to illustrate the improved stability that can be achieved with certain anisotropic media that have known issues. In addition, the effectiveness of the fluid-solid PML is numerically demonstrated for absorbing all kinds of bulk waves, as well as surface and evanescent waves. Finally, the new formulations were used to predict the transient response of a solid phononic structure consisting of a superfocusing acoustic lens.

iii

To the man whose heart was a jasmine flower: to my father.

iv

Acknowledgements First and foremost, I would like to thank my supervisor, Prof. Richard Cobbold for his great support throughout my doctorate study. I cannot thank him enough for his confidence in me to investigate and develop my own research directions under his continuous encouragement and guidance. I will always remember his beautiful enthusiasm for new and good scientific ideas. I am very grateful that Prof. Cobbold gave me the opportunity to be one of his students. I am also grateful to Prof. Adrian Nachman and Prof. George Eleftheriades, for advising me on my thesis, and for sparing their precious time to be members in my supervisory committee. Prof. Eleftheriades and Prof. Nachman were also my teachers in graduate courses; I am lucky to have such great humans and scientists in my academic life. In addition, I would like to thank Prof. Mary Pugh from the Department of Mathematics for her helpful advice on the idea of discrete stability. Furthermore, I would like to thank my colleagues for the supportive friendly environment: Masoud Hashemi, Amir Manbachi, Luis Aguilar, and all the members of the Ultrasound Group. Finally, I will be always grateful to my mother, my wife, Falastin, and my sister, Rinad, for their love, support, and high expectations. For Anat, my little daughter, thank you for being such a beautiful and lovely girl while I am writing my thesis.

v

Contents

Abstract

ii

Dedication

iv

Acknowledgements

v

List of Figures

xvi

List of Tables

xvi

Nomenclature

xvii

1 Introduction

1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Development of perfectly matched layers . . . . . . . . . . . . . . . . . .

4

1.3

Thesis objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 Theoretical background

9

2.1

Acoustic waves in fluids . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Elastic waves in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.1

Elastic wave equations . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2.2

Material properties . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Coupled acoustic-elastic modeling . . . . . . . . . . . . . . . . . . . . . .

14

2.3

vi

2.4

Plane waves and dispersion relations . . . . . . . . . . . . . . . . . . . .

15

2.5

Complex coordinates stretching . . . . . . . . . . . . . . . . . . . . . . .

19

3 PML formulation for elastic waves propagation

24

3.1

The PML formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.2

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3

Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.4

Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4 Stability of the PML formulation 4.1

4.2

4.3

37

Plane-wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.1.1

Stability conditions for the continuous problem . . . . . . . . . .

38

4.1.2

Discrete stability and scaling parameter

. . . . . . . . . . . . . .

40

4.1.3

Stability analysis examples . . . . . . . . . . . . . . . . . . . . . .

43

Numerical FEM results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.2.1

Quadratic Lagrange finite element . . . . . . . . . . . . . . . . . .

45

4.2.2

Other discretization methods . . . . . . . . . . . . . . . . . . . .

49

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

5 PML formulation for fluid-solid media 5.1

5.2

5.3

51

PML formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5.1.1

PML formulation for fluid media . . . . . . . . . . . . . . . . . .

52

5.1.2

The coupling boundary conditions in the PML . . . . . . . . . . .

54

5.1.3

Complete formulation of fluid-solid PML . . . . . . . . . . . . . .

57

Numerical methods and results . . . . . . . . . . . . . . . . . . . . . . .

58

5.2.1

Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.2.2

Numerical examples

. . . . . . . . . . . . . . . . . . . . . . . . .

63

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

vii

6 PMLs and modeling metamaterials

69

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

6.2

Lens design and properties . . . . . . . . . . . . . . . . . . . . . . . . . .

70

6.3

Time-domain simulations and results . . . . . . . . . . . . . . . . . . . .

72

6.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

7 Summary and conclusions

80

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

7.2

Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

7.3

Limitations and future work . . . . . . . . . . . . . . . . . . . . . . . . .

83

Appendix A Classical PML stability conditions

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96

List of Figures 1.1

(A) An example of a slab of a solid metamaterials superlens whose unit cell is composed of four brass cylinders embedded in an Al-SiC foam matrix with a vacuum cylindrical cavity in the center. The lens, which was designed to operate in water, has a frequency domain simulation response as shown in (B). The lens design and the figures are adapted from Zhou et al. [1], and reproduced with permission. . . . . . . . . . . . . . . . . . .

1.2

3

Illustrating the use of a perfectly matched layer (PML) for achieving nearperfect modeling of the solution to the unbounded wave radiation problem (after Johnson [2] ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

5

Slowness curves for all the materials whose properties are given in Table 2.1. The slowness and group velocity vectors are indicated for selected points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

18

Illustrating the effect of complex coordinate stretching for a 1D harmonic wave, shown in (A), propagating into a PML. The point x = x0 marks the beginning of the PML; the shaded region in (B), (C), and (D). (B) Shows the case where the damping coefficient β > 0, and the scaling coefficient α = 1. In (C) the same value of β as in (B) was used while α > 1. (D) Illustrates the case of an evanescent wave with α > 1 and β = 0. . . . . .

ix

21

3.1

Snapshot images for the excitation waveform shown in panel (A). Panels (B), (C) and (D) are snapshot images showing the amplitude of the displacement for a transient longitudinal wave propagating in the isotropic solid medium listed in Table 2.1. The radiation originates from a surface of a 1 mm diameter cylinder that radial with the displacement profile shown in (A). Marked on the time axis of (A) are the times at which the snapshots in (B), (C) and (D) are taken. Note that (B) and (C) have linear scales, while (D) is in dB’s. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

31

Validation results: the three points À, Á, and Â marked on Figure 3.1 (B) are the locations in the physical domain where the displacements were both simulated and analytically calculated. These three points are at (1.6 mm, 1.2 mm), (4.8 mm, 0 mm), and (4.8 mm, 4.8 mm) respectively. The solid line is the theoretical and the dashed line is from the FEM simulation. (A) and (B) show the two components of the displacement field at point À. (C) Displacement field at point Á. (D) Showing both components of the displacement field at point Â. . . . . . . . . . . . . . . . . . . . . . .

3.3

33

Showing the evolution of energy in the physical domain, as represented p by k u21 + u22 k∞ for the isotropic medium (Material I). In (A) the effects of changing the mesh size in (3.11) are investigated. Panel (B) shows the effects of changing the scaling parameter. . . . . . . . . . . . . . . . . . .

x

34

4.1

Illustrating the effect of incorporating the scaling coefficient α1 on the roots of (4.3) for the continuous, constant coefficient problem. The color maps show the maximum imaginary part of the roots, max(= {ω}), for Material III. For the classical case of α1 = 1 in the left panel, for a case in which α1 = 1/7 in the central panel, and for the case of α1 = 7 in the right panel. The arrow points to the value of π/h0 , where h0 is the mesh size used in numerical simulations as described in (3.11). . . . . . . . . .

4.2

44

Snapshot images showing the waveforms, originating from same cylinder as shown in Figure 3.1, but propagating in three different anisotropic solid media, namely III, IV, and V as specified in Table 2.1. In the middle column the snapshot times were chosen to approximately correspond to the quasi-shear wave being absorbed by the PML. The color maps on the third column are in dB’s. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

46

Showing the evolution of energy in the physical domain as represented p by k u21 + u22 k∞ for scaling parameters of 1, < 1, and > 1. In (A), for Material III using default mesh size corresponding to N = 5. In (B), for Material V using mesh sizes corresponding to N = 5 (solid lines) and N = 10 (dotted lines) and α02 = 1 for all five curves. . . . . . . . . . . .

4.4

47

Propagation snapshots as in Figure 4.2 for Materials III and V, after introducing scaling coefficients of α01 = 10, and α02 = 30 for Material III, and α01 = 10, and α02 = 1. These images should be compared to the

4.5

corresponding panels in Figure 4.2. . . . . . . . . . . . . . . . . . . . . . p Evolution of energy in the physical domain as represented by k u21 + u22 k∞

48

for Material V. The results were obtained by using cubic and quartic Lagrange and Hermite finite elements. For (A), α01 = α02 = 1, while in (B), α01 = 20 and α02 = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

50

5.1

Illustrating the problem concerned with the PML close to the fluid-solid interface. The fluid domain, ΩF , has the boundary ∂ΩF = ΓF ∪ Γ and the solid domain has the boundary ∂ΩS = ΓS ∪ Γ, where Γ = ∂ΩF ∩ ∂ΩS (the red line) is the fluid-solid interface, and ΓF and ΓS are the outer boundaries of the fluid and solid domains respectively. The inner dotted square, centered at the origin with dimensions of 2x0 , is the physical domain surrounded by a PML of thickness d. . . . . . . . . . . . . . . . .

5.2

56

Snapshot at t = 4.5µs showing the field distribution in both the solid and fluid regions caused by a line source at the point S in the fluid close to the fluid-solid interface, with a transient excitation source given by (5.11). The color scale shows the pressure in the fluid and σ22 in the solid, assuming a normalized source. The boundary of the PML is shown as a dotted line, while the solid line is the fluid-solid interface. To illustrate the agreement between the analytical and FEM solutions in the fluid, the top half has been split to show the FEM and analytical results. Points A at (9 mm, 1 mm) and B at (3 mm, 3 mm) are the locations where the waveforms are shown in the next figure. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

61

Waveforms at the points A and B in Figure 5.2 showing the agreement between the FEM and analytically calculated pressure waveforms. The normalized mean absolute errors are: 6.7 × 10−4 for (A) and 1.0 × 10−3 for (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

62

5.4

Snapshots illustrating the effectiveness of the PML in absorbing different kinds of bulk and surface waves for an irregular fluid-solid interface. The gray solid line is the interface between the fluid (upper half) and the solid (lower half). At 2.5µs the P-wave is being absorbed by the lower boundary. At 3.5µs the S-wave is being absorbed by the same boundary while the leaky Rayleigh waves have already been absorbed by the side boundaries. Scholte waves and the incident pressure waves can be seen as being absorbed at 4.5µs. The 10µs panel shows that the energy remaining in the computational domain is very small. . . . . . . . . . . . . . . . . .

5.5

64

Snapshots illustrating the effectiveness of the PML in absorbing strong evanescent waves. This was achieved by setting the fluid-solid interface very close to the PML in the solid. In the left column no coordinate scaling was used, while for the snapshots corresponding to the same times, as shown in the right column, the scaling parameter of α02 = 99 was used in the solid medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

66

(A) The normalized total energy in the physical domain as given by (5.15) and (5.16) for the simulation in Figure 5.4 is represented by the red thin line. The thick cyan line represents the same results, but using scaling parameters α0j = 99, j = 1, 2, in both the fluid and the solid domains. (B) The normalized total energy in the physical domain of the solid for both simulations shown in the left and right columns of Figure 5.5, but with an additional case that corresponds to α02 = 9. . . . . . . . . . . . .

xiii

67

6.1

Properties of the phononic crystal lens shown in Figure 1.1. (A) The band structure of an infinite crystal with a unit cell shown in Figure 1.1 (A). The sixth branch (the blue line) with negative slopes in both the ΓX and the ΓM directions, is used for the lens operation. This branch has the same phase velocity as water (red dotted line) at the point marked by p, corresponding to frequency of 37.65 kHz. (B) Equifrequency contours plot showing that the negative slope of the sixth branch is present in all directions, and is isotropic around the 37.65 kHz frequency. (Reproduced, with permission, from Zhou et al. [1]) . . . . . . . . . . . . . . . . . . . .

6.2

71

Snapshots of simulations of a radiating line source in water close to an infinite slab of solid Al-SiC foam are shown in the left column. In the right column a periodic structure of 8×50 unit cells, similar to the one shown in Figure 1.1 (A), were inserted in the foam to generate a metamaterials lens. The simulation snapshots in the right column illustrate that the group velocity in the metamaterial is much less than in the surrounding water. The final panel at the 2 ms snapshot shows focusing of the source: details of the doted red square region is given in the next figure. . . . . .

6.3

74

(A) Enlarged view of the last snapshot in the right column of Figure 6.2 to better illustrate the field distribution in the periodic structure. (B) Enlarged view 5 µs after (A). This corresponds to just less than 1/5 of wave period at 37.65 kHz. It can be seen that there is a forward phase shift in the water, while it is translated equally backwards in the periodic structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

75

The normalized total energy, as given by (5.15) in fluids and (5.16) in solids, in the whole physical domain for the simulation shown in Figure 6.2. 76

xiv

6.5

When the metamaterials is extended into the PML region, the solution becomes unstable early in the simulation. Note that the color map scale in (B) is six orders of magnitude greater than in other snapshots. This marks a limitation of the PML. . . . . . . . . . . . . . . . . . . . . . . .

6.6

77

The same simulation as in the right column of Figure 6.2 was repeated but with a smaller physical domain that places the edge of the periodic structure at the inner edge of the PML, as shown by the snapshot insert. The normalized total energy in the metamaterials lens for this case, E2 , is compared to energy in the lens for the simulations in Figure 6.2, E1 . The lower plot shows the deviation between E1 and E2 . . . . . . . . . . . . .

xv

78

List of Tables 2.1

Elasticity coefficients, along with the calculated minimum and the maximum phase velocities for the materials examined. . . . . . . . . . . . . .

xvi

14

Nomenclature List of symbols αj (xj )

The scaling parameter in the jth direction, page 20

βj (xj )

The damping parameter in the jth direction, page 20

δij

The Kronecker delta function, page 11

ηij

An arbitrary non-zero symmetric tensor, page 13

ΓX, ΓM

The principal symmetry directions of the first Brillouin zone, page 71

Γik (k)

Components of the Christoffel’s tensor, page 16

={•}

The imaginary part of a complex value, page 38

λ

Wavelength, page 2

λ(x)

The first Lamé coefficients, page 11

C

The set of complex numbers, page 15

R

The set of real numbers, page 15

µ(x)

The second Lamé coefficients, page 11

ω

The angular temporal frequency, page 15

xvii

φ, φi , ψi , and ψij Test functions, page 58 0 such that 2 X i,j,k,l=1

ηij Cijkl ηkl ≥

2 X

ηij ηij

(2.10)

i,j=1

for any non-zero symmetric tensor ηij .

2.2.2

Material properties

All solid media considered in this thesis are orthotropic, which is a special case of anisotropic media whose axes of symmetry coincide with x1 and x2 . It should be noted that isotropic materials are special case of the orthotropic materials. For simplicity and consistency with the notation commonly used [46], the indices replacement: 11 → 1, 22 → 2, 12 → 3, and 21 → 3 is used when presenting the elasticity properties. Hence, Hooks law for orthotropic media reduces to      1 0 C12   ∂u ∂x1  σ1  C11 0      σ3   0 C33 C33 0   ∂u1       ∂x2   =  . σ   0 C  ∂u  0   3    ∂x12  33 C33      ∂u2 σ2 C21 0 0 C22 ∂x2

(2.11)

Since C12 = C21 , only four independent elasticity coefficients are needed to describe the elasticity of orthotropic solids. The values of these coefficients for the five materials studied in chapter 3, chapter 4, as well as in section 2.4, are displayed in Table 2.1, where, for simplicity, the values of the elasticity coefficients are displayed assuming ρS = 1. Material I is isotropic (C11 = C22 = C12 + 2C33 ) while the others are anisotropic. In particular, Materials II, III, IV are identical to media II, III, IV as specified by Bécache et al. [35], and media V, which was also studied in [16, 29], corresponds to a zinc crystal.

14

Chapter 2. Theoretical background

Table 2.1: Elasticity coefficients, along with the calculated minimum and the maximum phase velocities for the materials examined.

Material I II III IV V

2.3

C11 7.8 20 4 10 16.5

C22 7.8 20 20 20 6.2

C33 2 2 2 6 3.96

C12 3.8 3.8 7.5 2.5 5

cmin 1.4 1.4 0.78 2.4 1.8

cmax 2.8 4.5 4.5 4.5 4.1

Coupled acoustic-elastic modeling

In order to model the wave propagation in a domain that contains both fluid and solid media, one intuitive solution would be to use the wave equation, (2.9), throughout the entire domain and to set the shear modulus (µ) to zero for the fluid media, which makes λ the fluid bulk modulus. The question as to why this is not appropriate can be readily answered by noting the ellipticity constraint, (2.10), which implies for the isotropic case that the second Lamé coefficient, µ, needs to be strictly positive [47, 48]. Moreover, by using the displacement equation in both the solid and fluid domains, the continuity of the particle velocity is enforced throughout, which is not the case at the fluid-solid interface where the fluid can perfectly slip and only the normal component of the particle velocity is continuous [41]. Therefore, to model heterogeneous fluid-solid media, (2.2) should be used in the fluid domain and (2.9) in the solid domain, along with the appropriate coupling boundary conditions at the interface between the two media. Perfect slip can be modeled by imposing the following kinematic boundary condition (2.2) at the fluid-solid interface: 2 X j=1

nFj

2 X 1 ∂p ∂ 2 uj =− nFj 2 , ρF ∂xj ∂t j=1

(2.12)

where nF is an outward directed unit vector normal to the fluid boundaries. In addition,


15

the traction needs to be continuous at the interface. Specifically, the normal stress needs to be equal and opposite to the pressure, i.e. 2 X i,j,k,l=1

nSi nSj σij = −p,

(2.13)

while the tangential stress needs to vanish at the interface. This can be achieved by imposing the following boundary condition on (2.9) at the solid-fluid interface: 2 X

nSj Cijkl

j,k,l=1

∂uk = −nSi p, ∂xl

(2.14)

where nS is an outward directed unit vector normal to the solid boundaries. Note that nS = −nF at the fluid-solid interface. It should be noted that having (2.12) and (2.14) in this form, corresponds to Neumann boundary conditions for the two second-order wave equations, i.e. (2.2) and (2.9) respectively. This simplifies modeling the problem, especially in finite elements schemes, as used in this thesis.

2.4

Plane waves and dispersion relations

In this section, plane wave analysis will be used to analyze the wave propagation characteristics in anisotropic solids, and more importantly, to establish the background needed to study the PML stability in chapter 4. Specifically, harmonic plane wave solutions of the form: u = A exp [i(k · x − ωt)],

(2.15)

will be considered for the elastic wave equation as given by (2.9). In the above equation, A ∈ C2 is the constant amplitude polarization vector, k ∈ R2 is the wavevector, ω ∈ C is the angular frequency, and i2 = −1. Assuming constant material properties, substituting (2.15) into (2.9) yields an alge-

16


braic equation, known as the Christoffel’s equation, that is given by 2 X k=1

ω 2 δik − Γik Ak = 0,

(2.16)

where δik is the Kronecker delta function and

Γik (k1 , k2 ) =

2 X

Cijkl kj kl ,

(2.17)

j,l=1

is the symmetric Christoffel’s tensor or operator. Without loss of generality, it was assumed that ρS = 1. Equation (2.16) is an eigenvalue problem, where ω 2 and A are, respectively, the eigenvalue and eigenvector of the Christoffel’s tensor. Dividing (2.16) by the wavenumber squared, |k|2 = k12 + k22 , yields the same eigenvalue problem but with k1 k2 ω , instead of ω, and the propagation direction, K = |k| , |k| = the phase velocity, c = |k| cos θ, sin θ , instead of k. Hence, solving (2.16) provides the possible phase velocities or characteristic speeds and the corresponding wave polarizations for each propagation direction for a given material. Equation (2.16) has a nontrivial solution only when

F0 (ω, k) = det ω 2 δik − Γik (k) = 0.

(2.18)

The above equation is referred to as the dispersion relation between k and ω or the characteristic polynomial of the Christoffel’s operator. For an orthotropic medium this equation can be written as F0 (ω, k1 , k2 ) = ω 4 − ω 2 (C11 + C33 ) k12 + (C33 + C22 ) k22 + C11 C33 k14 + C22 C33 k24 + C11 C22 − c212 − 2C12 C33 k12 k22 = 0 .

(2.19)

If the elasticity tensor satisfies the ellipticity condition as given by (2.10), then for

17


any orthotropic medium, this translates to

2 . C11 > 0, C22 > 0, C33 > 0, and C11 C22 > C12

(2.20)

Hence, Γik (k) is symmetric positive definite for any k 6= 0 , and therefore its two eigenvalues, 2 ω1,2 (k)

= Γ11 + Γ22 ±

q

(Γ11 − Γ22 )2 + 4Γ212 ,

(2.21)

are positive and the two corresponding eigenvectors are orthogonal. Consequently, the four roots of (2.19), namely,

ωP (k) = ±

q ω12 (k),

ωS (k) = ±

q ω22 (k),

(2.22)

and their corresponding phase velocities, ±cP (θ) and ±cS (θ), are all real. In the above equation, the subscripts, P and S refer to the primary (fast) wave, and secondary (slow) wave respectively. Although the polarization vectors that corresponds to these two modes are mutually orthogonal (as mentioned above), they are not necessarily polarized along or perpendicular to the wavevector, k, as in the special case of isotropic media. Hence, for the general anisotropic case, the two modes are referred to as quasi-longitudinal and quasi-shear. The four roots in (2.22) will be referred to as the physical modes. If, in addition the following condition

C11 6= C33 and C22 6= C33 ,

(2.23)

is satisfied, then the four roots of (2.19) will be distinct enabling the group velocity, which specifies the direction of energy transport, to be defined by

V g = ∇k ω = −

∇k F0 (ω, k) . ∂F0 (ω, k) /∂ω

(2.24)

18


II

III

s2

I

−0.7

0

0.7

−0.7

0

0.7

−1.1

V

IV

0 s1

1.1

Slow wave

s2

Fast wave Group velocity Slowness vector

−0.4

0 s1

0.4

0.5

0 s1

0.5

Figure 2.1: Slowness curves for all the materials whose properties are given in Table 2.1. The slowness and group velocity vectors are indicated for selected points.

Finally, the slowness vector defined by S =

k1 k2 , ω ω

provides a convenient means for

understanding and visualizing dispersion relations. Graphs of

F0 (1, S1 S2 ) = 0,

(2.25)

which follows from the homogeneity of (2.19) in k and ω, are called the slowness curves. Such curves for the materials in Table 2.1 are shown in Figure 2.1. The inner curves correspond to the fast wave (the longitudinal or quasi-longitudinal) and the outer curves to slow waves (shear or quasi-shear). From (2.24), it should be noted that the direction


19

of the group velocity is normal to these curves. In addition, the phase velocity in each propagation direction can be obtained from the slowness curves, c(S) = ±1/ |S|. The maximum and minimum phase velocity for a given material are denoted by cmax , and cmin and these values are also listed in Table 2.1. It should also be mentioned that Bécache et al. [35] found that the stability of the split-field classical PML depends on the shape of slowness curves as will be discussed in chapter 4.

2.5

Complex coordinates stretching

Complex coordinate stretching for constructing a PML for a given wave equation was introduced by Chew et al. in 1994 [21]. Subsequently, this process was recognized as analytic continuation of each spatial coordinate to the complex plane [49–51]. If the medium sufficiently far from the origin, where the PML is added (see Figure 1.2), is linear and homogeneous, then the radiation solution in this region can be written as a superposition of harmonic plane waves [2]. Therefore, any radiation that is incident on the PML in the jth direction is analytic in xj and are subject to analytic continuation in the complex plane of xj , giving rise to the reflectionless and damping properties of the PML. To obtain a PML formulation for a given wave equation, the complex coordinate ˜ : R2 → C2 . Since x ˜ = x in stretching can expressed as a coordinate transform: x 7→ x ˜ appears the physical domain and the PML region is assumed to be homogeneous, then x only in the form of spatial partial derivatives in the PDEs. Given a field variable u, then using the chain rule: 2

X ∂u ∂ x˜k ∂u = , ∂xj ∂ x ˜ k ∂xj k=1

(2.26)

20


which reduces to ∂u ∂u ∂ x˜j = , ∂xj ∂ x˜j ∂xj

(2.27)

since x˜j depends only on xj . As a result, defining the complex stretch function by

sj (xj ) =

∂ x˜j ∂xj

(2.28)

suffices to perform transformation: ∂ 1 ∂ = . ∂ x˜j sj (xj ) ∂xj

(2.29)

A two-parameter complex stretch function introduced by Fang and Wu [22] in their generalized PML (GPML) is used in this work. This function is given by

βj (xj ) sj (xj ) = αj (xj ) 1 + i , ω

(2.30)

where the βj ≥ 0 is the damping parameter responsible for damping the propagating wave inside the PML. In this equation, the scaling parameter, αj > 0, is responsible for either stretching (αj > 1) or compressing (0 < αj < 1) the coordinate. It should be noted that in the physical domain, where x˜j (xj ) = xj , βj = 0 and αj = 1. Other complex stretch functions have been widely used. One of which is the one-parameter classical stretch function which is the same as (2.30) but in the absence of αj . Another form is that introduced by Kuzuoglu and Mittra [23] in which the frequency in (2.30) is shifted in the imaginary direction by the value of a third parameter (i.e. ω is replaced by ω + i γj ), leading to PML formulations that are usually called complex frequency shift PML (CFS-PML) . To illustrate the effect of the complex coordinate stretching, consider the simple case of the 1D oscillatory solution with wavenumber, k, and a constant wave speed c = ω/k.

21


1

1

Amplitude

A

B

0

0

−1

−1 x=x0

1

1

Amplitude

C

D

0

−1

0 Spatial variable, x

Spatial variable, x

Figure 2.2: Illustrating the effect of complex coordinate stretching for a 1D harmonic wave, shown in (A), propagating into a PML. The point x = x0 marks the beginning of the PML; the shaded region in (B), (C), and (D). (B) Shows the case where the damping coefficient β > 0, and the scaling coefficient α = 1. In (C) the same value of β as in (B) was used while α > 1. (D) Illustrates the case of an evanescent wave with α > 1 and β = 0.

Dropping the time dependence, the propagating solution can be expressed as exp(ikx) and this is shown in Figure 2.2 (A). In the complex stretched coordinates, x˜, this solution becomes Zx h i h Zx i exp(−ik˜ x) = exp −ik α (´ x) d´ x × exp −c α (´ x) β (´ x) d´ x . 0

0

(2.31)

22


In the absence of the scaling parameter, i.e., for α = 1, the first term is the original solution, while because β > 0, the second term corresponds to damping as shown in Figure 2.2 (B). Note that this damping term is wavenumber independent thanks to the presence of ω in the stretch function in (2.30). By introducing a scaling parameter such that α > 1, both the oscillation and the damping of propagating waves will be increased as shown in Figure 2.2 (C). In this case, the real grid is stretched resulting in an apparent increase in the number of cycles, which is equivalent to increasing the spatial frequency, k, in the original coordinate. As subsequently shown in chapter 4, this concept can be used to improve the discrete stability of PMLs. For evanescent waves, the complex-stretched solution is

h

exp(−k˜ x) = exp −k

Zx

i

h

α (´ x) d´ x × exp −ic

0

Zx

i α (´ x) β (´ x) d´ x .

(2.32)

0

In this case, the first term in is associated with the amplitude, while the second term affects the oscillation. As a result, when α > 1, the damping of the evanescent waves will increase [22, 28] as shown in Figure 2.2 (D). This effect will be investigated in chapter 5 using full 2D time-domain simulations. Appropriate choices are now needed for the stretch function parameters αj (xj ) and βj (xj ). Despite the absence of a rigorous methodology for their choice [14, 17], polynomial functions are often used as indicated below:

αj (xj ) =

   1 m

if |xj | < x0

|xj | − x0   1 + α0j − 1 if x0 ≤ |xj | ≤ x0 + d, d    0 if |xj | < x0 βj (xj ) = n |xj | − x0   β0j if x0 ≤ |xj | ≤ x0 + d, d

(2.33a)

(2.33b)


23

where d is the thickness of the PML, 2x0 is the dimension of the square physical domain centered at the origin (see Figure 1.2), m and n are the polynomial orders, and α0j and β0j are constants that represent the maximum values of αj and βj respectively. The values of these two parameters need to be specified. It is helpful to express the value of β0j in terms of a desired amplitude reflection coefficient (Rj ) caused by the reflection from the outer boundary of the PML. It can be shown that for normal incidence and assuming αj = 1, β0j =

cmax (n + 1) ln (1/Rj ) . 2d

(2.34)

On the other hand, the choice of α0j in (2.33a) depends on the desired scaling (stretching or compression) of the original coordinate. The scaling of the original coordinate is simply the derivative of the real part of x˜j with respect to xj , which is equal to αj (xj ). Hence, the value of α0j is simply the maximum scaling of the original coordinate in the jth direction. The orders of the polynomial functions, m and n, in (2.33a) and (2.33b) can be any integer. Quadratic polynomials will be used in this work unless mentioned otherwise. Note that both parameters in Equation 2.33 are continuous when m, n ≥ 1 and differentiable for m, n ≥ 2. As a final comment on complex coordinate stretching, we recommend that only the damping parameter, βj , be used, unless there is a need for the scaling parameter, αj , as in the case of unstable media or strong evanescent waves. The reason for that is the fact that increasing the scaling parameter (αj > 1) causes more reflection to the physical domain as shown in chapter 3. We are not sure whether this is merely numerical reflection due to steeper change in the PDE coefficients, or it is an intrinsic issue regrading the validity of scaling the real coordinates, which might be related to the break at the interface in Figure 2.2 (D).

Chapter 3 PML formulation for elastic waves propagation1 Based on the background provided in the last chapter, a new second-order time-domain perfectly matched layer formulation is obtained for modeling elastic wave propagation in unbounded two dimensional space. The formulation uses a complex coordinate stretching approach with a two-parameter stretch function. The final system, consisting of just two second-order displacement equations along with four auxiliary equations, is smaller than existing formulations, thereby simplifying the problem and reducing the computational cost. Following a brief description of the numerical methods, evidence for the validity of this formulation is provided by comparison of the numerical results obtained using Material I in Table 2.1 with analytical solutions. 1

The material in this chapter is part of a paper published in Mathematics and Mechanics of Solids (DOI: 10.1177/1081286515569266)

24

Chapter 3. PML formulation for elastic waves propagation

3.1

25

The PML formulation

With the help the background in chapter 2, our time-domain PML formulation for modeling wave propagation in unbounded solids can be introduced using the complex coordinates stretching. All parameters, namely, ρS , Cijkl , αj , βj , and, hence, sj , were assumed to be space dependent throughout the derivation leading to a variable-coefficient PML formulation. Since the stretch function also depends on the frequency, the first part of the derivation is in the frequency domain. First, we take Fourier transforms of Newton’s second law (2.3) and Hooks law (2.6), ˜ : R2 → C2 , and then transform the spatial coordinates using complex stretching, x 7→ x as introduced in section 2.5. These steps lead to:

2

(−iω) uî ρS =

2 X ∂σ îj j=1

∂ x˜j (3.1)

σ îj =

2 X

Cijkl

k,l=1

∂ uˆk , ∂ x˜l

ˆ denotes the Fourier transform in time. We proceed by splitting each of the where stress field components in (3.1) into two non-physical components, σij = σij1 + σij2 , while keeping the displacement field components unsplit, yielding: 2 X ∂ 2 (−iω) uî ρS = ∂ x˜j j=1

σ îjl =

2 X k=1

Cijkl

2 X l=1

! σ îjl (3.2)

∂ uˆk . ∂ x˜l

The need to solve this differential equation along contours in the complex plane can be avoided by inverse transforming the complex-stretched coordinates back to the original spatial coordinates using (2.29). This is followed by multiplying the first equation by

26


s1 s2 and the second by sl , leading to: (−iω)2 uî ρS s1 s2 =

2 X j=1

sl σ îjl =

2 X

Cijkl

k=1

Note that while

s1 (x1 ) s2 (x2 ) sj (xj )

s1 s2 , sj

2 X s1 s2

∂ ∂xj

l=1

sj

! σ îjl (3.3)

∂ uˆk . ∂xl

= s2 (x2 ) when j = 1 and it equals to s1 (x1 ) when j = 2. Therefore,

and similarly

α1 α2 αj

and

β1 β2 , βj

are variable in space, they do not depend on

xj and can thus be placed either inside or outside the xj derivative which is the case in the first equation above. To facilitate the derivation of our formulation, it is helpful to introduce the variable ςijl (x, t)

Z =

t

σijl (x, τ ) dτ.

(3.4)

0

Taking the Fourier transform for both sides yields

ςîjl (x, ω) =

σ îjl (x, ω) +πσ îjl (x, 0)δ(ω). −iω

(3.5)

However, only the first term is relevant to any PML formulation constructed using the stretch function (2.30), which is not defined for the static case of ω = 0 [17]. Consequently, substituting σ îjl = −iω ςîjl into (3.3) and expanding s1 and s2 using (2.30), results in α1 α2 ρ S

" 2 # 2 X X β β α α ∂ 1 2 1 2 (−iω)2 + (−iω) (β2 + β1 ) + β1 β2 uî = −iω + ςîjl α ∂x β j j j j=1 l=1

αl (−iω +

βl ) ςîjl

=

2 X k=1

Cijkl

∂ uˆk . ∂xl (3.6)

Note that replacing the non-physical variable σijl does not affect the formulation since any variables other than the displacement are auxiliary variables that do not necessarily

27


have a physical meaning. The time-domain form of this equation can now be obtained by taking its inverse Fourier transform (−iω ⇒ ∂/∂t), leading to ρS

" 2 X 2 ∂ 2 ui ∂ui 1 ∂ X + β1 β2 ui = + (β2 + β1 ) ∂t2 ∂t α ∂xj l=1 j=1 j

∂ςijl β1 β2 l + ς ∂t βj ij

!#

(3.7) ∂ςijl ∂t

+ βl ςijl =

2 X Cijkl ∂uk k=1

αl ∂xl

.

By substituting ∂ςijl /∂t from the second equation into the first and simplifying, yields ρS

∂ 2 ui ∂t2

+ (β1 + β2 )

∂ui + β1 β2 ui ∂t

  2 2 2 X X X C β1 β2 1 ∂  ijkl ∂uk = + − βl ςijl  αj ∂xj αl ∂xl βj j=1

l=1

k,l=1

2 X ∂ςijl Cijkl ∂uk + βl ςijl = . ∂t αl ∂xl k=1

(3.8)

Noting that if j 6= l, then

β1 β2 −βl βj

= βl −βl = 0, so that only four of the eight components

of ςijl , namely, ςijj remain in the first equation. These four variable are needed to solve for the displacement field and will be considered as auxiliary variables denoted by wij ≡ ςijj . Thus, our time-domain PML formulation consists of two second-order displacement field equations and four auxiliary equations that can be expressed as ρ˜S

∂ 2 ui ∂ui +b + c ui 2 ∂t ∂t

2 X ∂ = ∂xj j=1

2 X

∂uk C˜ijkl + aj wij ∂xl k,l=1

!

(3.9) ∂wij + βj wij = ∂t where ρ˜S (x) = α1 α2 ρS , C˜ijkl (x) =

2 X k=1

Cijkj ∂uk , αj ∂xj

α1 α2 Cijkl , αj αl

aj (x) =

α1 α2 αj

β1 β2 βj

− βj , b(x) = β1 + β2 ,

and c(x) = β1 β2 . It should be noted that the number of equations in (3.9) is less than that present in the classical form and the convolutional form (typically 10 and 13 equations respectively [26]) and other time-domain PML formulations [15, 16, 29]. Note that, in the


28

physical domain, the two equations in (3.9) are decoupled, and the first one is identical to the original equation as given by (2.9). Since the new PML formulation in (3.9) is introduced in a compact form, it is appropriate to show a detailed expansion of this formulation. For the case of orthotropic media discussed in subsection 2.2.2, (3.9) consists of the following two second-order equations, α1 α2 ρS

∂u1 ∂ 2 u1 + (β + β ) + β β u = 1 2 1 2 1 ∂t2 ∂t

∂ α2 C11 ∂u1 ∂u2 ∂ ∂u2 α1 C33 ∂u1 + C12 + α2 (β2 − β1 ) w11 + C33 + + α1 (β1 − β2 ) w12 ∂x1 α1 ∂x1 ∂x2 ∂x2 ∂x1 α2 ∂x2 α1 α2 ρS

∂u2 ∂ 2 u2 + (β1 + β2 ) + β1 β2 u2 = ∂t2 ∂t

α2 C33 ∂u2 α1 C33 ∂u2 ∂u1 ∂ ∂u1 ∂ + C33 + α2 (β2 − β1 ) w21 + C12 + + α1 (β1 − β2 ) w22 ∂x1 α1 ∂x1 ∂x2 ∂x2 ∂x1 α2 ∂x2

along with the following four auxiliary equations: C11 ∂u1 ∂w11 + β1 w11 = ∂t α1 ∂x1 C33 ∂u1 ∂w12 + β2 w12 = ∂t α2 ∂x2 ∂w21 C33 ∂u2 + β1 w21 = ∂t α1 ∂x1 ∂w22 C22 ∂u2 + β2 w22 = ∂t α2 ∂x2

3.2

Numerical Methods

To model an infinite 2D medium, in this and the next chapter, we assumed a physical domain of 1.0 cm2 centered at the origin and surrounded by a 1.0 mm thick PML. The source of excitation is a 1 mm diameter infinite cylinder embedded in medium and centered at the origin. The boundary of the cylinder is assumed to vibrate either normally or tangentially with a displacement whose normalized time-dependence is given by the


29

first derivative of a Gaussian, i.e., √ 2 2 u0 (t) = − 2e πf0 (t − t0 ) e−π f0 (t−t0 ) ,

(3.10)

where f0 is the dominant frequency and t0 is a source delay time. For all numerical simulations f0 = 1500 Hz and t0 = 1 ms. With these values, 90% of the energy of the signal is contained below the frequency fc = 1900 Hz. A finite element method was used to numerically solve for the field distribution in solid media. Specifically, the weak form of (3.9) was used to model the problem by using the “Mathematics Model” in the FEM software COMSOL Multiphysics. This form of the formulation is discussed in chapter 5. Quadratic Lagrange finite elements, which is the default in COMSOL, were used. A square mesh was used for the PML region, but we retained a triangular shape in the physical domain. The choice of an appropriate mesh size is governed by the shortest wavelength of significance for the propagating pulse, i.e., cmin . fc

In particular, the mesh size was taken to be:

h0 =

1 cmin . N fc

(3.11)

Since quadratic Lagrange finite elements were used in our simulations, which are secondorder accurate, this mesh size corresponds to 2N degrees of freedom. Note that not only the slowest wave, cmin , but also the high frequency part, fc , of this wave were used in (3.11). Hence, the dominant-frequency waves will be represented by more than 2N degrees freedom per wavelength. The second-order generalized alpha method (using ρ∞ = 0.75, see [52]) was used for time discretization. Being an implicit method it is not as sensitive to the choice of time

30


step as are explicit methods. To make the best use of the mesh, a time step size of

ts =

0.9 cmin N cmax fc

(3.12)

was used. This is just less than the time needed for the fastest wave to travel along the smallest mesh dimensions, i.e,

h0 . cmax

The computational domain is bounded by the outer edge of the PML and the cylinder surface. Dirichlet boundary conditions were used throughout: specifically, u = 0 on the outer boundary of the computational domain, and u = −u0 (t) n on the surface of the cylinder, where n is a unit vector outward normal to cylinder surface. However, because the unstable modes are usually the quasi-shear modes [35], the media was excited by tangential vibrations of the cylinder surface in chapter 4 where the stability of the PML is studied. In this case, the Dirichlet boundary conditions on the cylinder surface are: u1 = u0 (t) n2 , and u2 = −u0 (t) n1 . For the initial conditions, all the field variables and their first time derivatives were set to zero at t = 0. For all FEM simulations, the reflection coefficients in (2.34) were chosen to be Rj = 1 × 10−6 , and m = n = 2 were used in (2.33a) and (2.33b), unless stated otherwise.

3.3

Validation

To test both the validity of our PML formulation and the accuracy of our simulations, we made use of the exact solution for an infinitely long cylinder of radius a and centered at the origin that radially vibrates a normalized disturbance into an unbounded isotropic solid consisting of Material I in Table 2.1. The choice of such a material is governed by the fact that it is easier to get analytical solutions for isotropic media, and that isotropic media has no stability issues in PMLs. FEM Simulations of wave propagation in this isotropic media using a PML are presented in Figure 3.1, where snapshots of the

31


propagating pulse described by (5.11) are shown for three instants of time. The mesh size was chosen according to (3.11), with N = 9.

A

5 ↓B

↓C

D↓

0

①

②

0.5

−5

−1 0

2 4 Time (ms)

20

−5

x1 (mm)

0

5

1

0

D

C

x2 (mm)

5

1

③

B

x 2 (mm)

Normlized velocity

1

0.5

−5 −5

x1 (mm)

5

0

−50

−5

x1 (mm)

5

−100 dB

Figure 3.1: Snapshot images for the excitation waveform shown in panel (A). Panels (B), (C) and (D) are snapshot images showing the amplitude of the displacement for a transient longitudinal wave propagating in the isotropic solid medium listed in Table 2.1. The radiation originates from a surface of a 1 mm diameter cylinder that radial with the displacement profile shown in (A). Marked on the time axis of (A) are the times at which the snapshots in (B), (C) and (D) are taken. Note that (B) and (C) have linear scales, while (D) is in dB’s.

It is known that for the isotropic case, compressional sources generate only primary waves with wave speed cp in homogeneous solids [46]. Hence, the field everywhere has


polarization in the radial direction and depends only on r =

32

p x21 + x22 . By making use

of the results given in section 2.7 of [46], the frequency domain expression for the field can be shown to be given by:

u(r, ω) =

(1)

ω cP

r

(1)

ω cP

a

H1 H1

(1)

where H1

ˆ r

(3.13)

is Hankel function of the first kind, and ˆ r = (cos θ, sin θ) is the radial unit

vector. By multiplying (3.13) with the Fourier transform of the source, (5.11), then taking the inverse Fourier transform of the resultant, the time-domain analytical solution was obtained to be compared with the FEM results. As shown in Figure 3.2 the agreement is excellent, thereby providing good evidence for the effectiveness of our PML formulation in simulating unbounded media and the correctness of the FEM model. The late-time stability of the model was also verified, by running the simulation for 50000 time steps. Another important measure of the effectiveness of the PML can be obtained by looking at the manner in which the energy in the physical domain evolves in time so as to ensure that the amount of energy reflected back into the physical domain is small. Figure 3.3 shows how the energy, as represented by the maximum magnitude of the displacement,

p

u21 + u22 , evolves in time. Besides providing additional validation, this figure also ∞

shows the effects of changing the mesh size and the scaling parameter. In panel (A) four curves exist: one is theoretical, while the other three show the FEM results using our PML for three different mesh size corresponding to N =9, 5, 3 in (3.11). The deviation between the theoretical and PML curves represents the error in the PML simulations which comes from two sources. One is due to discretization of the continuous problem as exists in any numerical calculations, while the other is caused by the PML and the choice of its parameters. The results in (A) provide further evidence that our PML finite element model has negligible error. These results also shows that the value of N =5,

33

①

A

0.2 0 −0.2

Theory

u2 Normlized

u1 Normlized


0 Theory FEM

0

2

4

FEM

0

②

C

−0.2

Theory

4

u1 ,u2 Normlized

u1 Normlized

0.2

2

0

−0.2

FEM

0

①

B

0.2

2

4

③

D

0.1 0 −0.1

Theory FEM

0

Time (ms)

2

4

Time (ms)

Figure 3.2: Validation results: the three points À, Á, and Â marked on Figure 3.1 (B) are the locations in the physical domain where the displacements were both simulated and analytically calculated. These three points are at (1.6 mm, 1.2 mm), (4.8 mm, 0 mm), and (4.8 mm, 4.8 mm) respectively. The solid line is the theoretical and the dashed line is from the FEM simulation. (A) and (B) show the two components of the displacement field at point À. (C) Displacement field at point Á. (D) Showing both components of the displacement field at point Â.

corresponding to 10 degrees of freedom for the shortest wavelength, is an appropriate choice. In fact, for this choice the maximum error expressed in terms of the normalized maximum deviation from the analytical solution was found to be 0.082. Of course, the average error can be expected to be less. Note that this represents the total error including the PML errors. The value of N = 5 will be used for all of our simulations in this chapter and the next chapter, unless stated otherwise. Increasing the scaling parameter leads to an increase in rates of change of the PML

34


0

0

10

2

10

↵ 01 = 1 ↵ 01 = 5 ↵01 = 10

2

↵01 = 20

Energy

10

10 Theory N=9 N=5 N=3

10

4

10

4

A 10

B

6

0

5

10

15

20

10

time (ms)

6

0

5

10

15

20

time (ms)

Figure 3.3: Showing the evolution of energy in the physical domain, as represented by p k u21 + u22 k∞ for the isotropic medium (Material I). In (A) the effects of changing the mesh size in (3.11) are investigated. Panel (B) shows the effects of changing the scaling parameter.

equation’s coefficients, which upon discretization, can be expected to increase the numerical reflection from the PML. The second panel (B) examines the extent of this increase by showing how the energy evolution depends on α1 using values ranging from 1 to 20. It is evident that the error is increased with α1 . Nevertheless, introducing a scaling parameter can be useful for some applications, like improving the PML stability as shown in the next chapter. Hence, there is a tradeoff between the desired improvements caused by increasing the scaling parameter and the numerical error. As subsequently shown, the penalty can be quite acceptable for potentially unstable materials.

3.4

Summary and discussion

Using a PML approach with complex coordinate stretching we have formulated a model that describes 2D time-domain wave propagation in an unbounded linear anisotropic solid. This formulation preserves the second-order form of the original wave equation,


35

with variable coefficients, while using a small number of equations as compared to other formulations. Specifically, classical first-order PML formulations for time-domain elastic wave propagation in 2D are usually described using ten split-field auxiliary equations with the need to compute the five stress-velocity total variables as the sum of two splitfield auxiliary variables. Our formulation contains two second-order equations along with four auxiliary equations which, to the best of knowledge, is the smallest number so far reported to describe wave propagation in solids using a time-domain PML formulation. Furthermore, our approach simplifies the description of the problem and, as discussed below, can be expected to reduce the computational resources needed for discrete implementations, and can be expected to have improved stability. While the computational burden varies from one numerical scheme to another, for a given numerical scheme, a small formulation reduces this burden [26]. In second-order formulations for the same problem the computational burden varies with the number of equations needed. This could vary from fourteen [29] to the ten-equation formulation introduced by Li et al. [53], who argued that their formulation needs less numerical resources than classical first-order PML formulations. This provides evidence that our six-equation formulation should be advantageous in this regard. There is good evidence that first-order PML formulations are less robust than secondorder formulations. Specifically, Kreiss and Duru [37] have carefully compared the properties of first and second-order formulations and have concluded that the second-order PML has much better discrete stability than the first-order one. They found that secondorder PML exhibits growth only if unstable modes are well resolved by the discrete grid, while the first-order counterpart is unstable for most resolutions [37]. These results are in agreement with the findings that first-order PML formulations are only weakly well-posed [15, 35], while some second-order PML formulations are strongly well-posed [29]. There are other advantages for choosing second-order formulations. The second-order


36

displacement elastic wave equation emerges directly from Newton’s second law [29], unlike the fist-order stress-velocity elastic wave equation which introduces a new non-physical wave mode with zero velocity [15, 29]. Moreover, the second-order PML formulations are more readily implemented in common numerical schemes that are based on second-order displacement wave equations [26, 53].

Chapter 4 Stability of the proposed PML formulation1 The stability of the PML formulation, as presented in the last chapter, will be examined with the help of a plane-wave analysis. Such an analysis simplifies the stability study by turning the PDEs into algebraic equations that leads to a dispersion relation as discussed in section 2.4. Moreover, thanks to the superposition principle, any given solution can be represented as a superposition of harmonic plane waves that satisfies the dispersion relation. For the continuous (non-discrete) case, it is shown that by increasing the scaling parameter in the stretch function any existing instability is moved to higher spatial frequencies. Since discrete models cannot resolve frequencies beyond a certain limit, this can lead to significant computational stability improvements. Numerical FEM results are shown to illustrate the improved discrete stability that can be achieved with certain anisotropic media that have known stability issues. 1

The material in this chapter is part of a paper published in Mathematics and Mechanics of Solids (DOI: 10.1177/1081286515569266)

37

Chapter 4. Stability of the PML formulation

4.1

38

Plane-wave analysis

The PML formulation, as given by (3.9), has variable coefficients, however, to study the stability it is helpful to assume constant coefficients, which allows use of the wellestablished methods based on plane wave analysis [15, 16, 29, 35]. In this section, the stability conditions for the continuous problem will be established first. Subsequently, for materials that violate these conditions, we propose the use of the scaling parameter to improve the stability of the discrete problem. Finally, use of the plane wave analysis is illustrated by considering the unstable materials in Table 2.1.

4.1.1

Stability conditions for the continuous problem

Although PMLs were introduced for the purpose of simulating discrete problems, it is helpful to begin by studying the stability of the continuous problem. As stated in section 2.4, for the general case, the roots of a given dispersion relation can be complex, i.e. ω(k) = < {ω(k)} + i = {ω(k)}. For this general case, the plane wave solution, as given by (2.15), is

u = A exp [= {ω} t] exp [i (k · x − < {ω} t)] .

(4.1)

Thus, the sign of the imaginary part of ω determines the stability of the given problem. Specifically, if = {ω} > 0, the solution grows exponentially with time. Alternatively if = {ω (k)} 6 0, then the solution is stable.

∀ k ∈ R2 ,

(4.2)

Previous studies [15, 29, 35] have shown that instability

starts in one or both directions of the PML, but not in the corner region where the full PML equation is involved. For our purpose, just one direction will be considered, namely,


39

the x1 direction where β2 = 0 and α2 = 1. For this case, the dispersion relation for our PML formulation, (3.9), can be shown to be F 1 (ω, k1 , k2 , β1 , α1 ) ≡ F0

k1 ω, k2 (ω + iβ1 ) = 0, (ω + iβ1 ) ω, α1

(4.3)

where F0 is given by (2.19). While a closed form solution for (4.3) does not in general exist, information about the roots can be obtained. Firstly, equation (4.3) is an 8th order polynomial in iω with real coefficients and, according to the complex conjugate root theorem, its roots come in complex conjugate pairs. Therefore, (4.3) has four pairs of roots, ω (k1 , k2 , β1 , α1 ), each of which has the same imaginary part while the real parts differ only in sign. Moreover, since the roots of a polynomial are continuous function of its coefficients [54], the roots will be continuous functions of k1 , k2 , β1 , and α1 . Using the implicit function theorem, it can be shown that they are also differentiable in the vicinity of the simple roots. In addition, if ω(k1 , k2 , β1 , α1 ) is a root of (4.3), then ω(−k1 , k2 , β1 , α1 ) , ω(−k1 , −k2 , β1 , α1 ) and ω(k1 , −k2 , β1 , α1 ), are also roots, i.e., the roots are symmetric about the k1 and k2 axes. This allows us to consider just the first quarter of the k−space. To find the conditions for which the roots of (4.3) satisfy (4.2), we first consider the case of the classical stretch function for which α1 = 1. For this case (4.3) is identical to the equation for F˜pml as given by Bécache et al. [35] as part of the dispersion relation of the classical split-field PML (see their equation (64)). Using perturbation techniques, they studied the stability of F1 (ω, k1 , k2 , β1 , 1) and found all the necessary and sufficient stability conditions in terms of the elasticity coefficients as summarized in appendix A. Most importantly, Bécache et al. found that the necessary stability condition violating which causes the most severe instabilities, namely condition (A.3), depends on the shape of slowness curves as stated in Theorem 2 of their work [35], i.e., Definition 4.1 (Geometric stability condition). For the PML to be stable in the xj


40

direction it is necessary that all points on the slowness curves satisfy

Sj × Vgj > 0. This means that the jth component of the group velocity needs to be in the same direction as the jth component of the slowness vector, as can be readily identified on the slowness curves of Figure 2.1. This geometric stability condition was also found to be necessary for other PML formulations [15, 29]. To find the stability condition for our PML formulation, we need to consider (4.3) for the general case of α1 6= 1. By inspection, it is evident that the roots of F1 (ω, α1 k1 , k2 , β1 , α1 ) = 0 are the same as the roots of F1 (ω, k1 , k2 , β1 , 1)=0. This implies that: Corollary 4.1. If there exist an unstable pair of roots of (4.3) in the vicinity of the point (k1 , k2 ) = (k10 , k20 ) for a fixed value of β1 and α1 = 1, changing the scaling parameter, α1 , from unity will merely cause this unstable pair of roots to move to the vicinity of the point (α1 k10 , k20 ) in k−space, leading to: Corollary 4.2. For the continuous (non-discrete) case, the necessary and sufficient condition for the stability of our constant coefficient problem, as defined by (3.9), are exactly the same as that reported by Bécache et al. [35] for their split-field system (see Appendix A), including the geometric stability condition.

4.1.2

Discrete stability and scaling parameter

It has been shown that the discrete stability of numerical solutions can be improved if the unstable continuous modes are not well resolved by the discrete mesh [15, 29, 37], especially for second-order formulations [37]. Hence, an intuitive solution to improve the discrete stability would be to increase the mesh size to a limit where it can’t resolve


41

waves corresponding to unstable modes [15]. Nevertheless, there is limit on increasing the mesh size for a given problem. Therefore, to achieve discrete stability to a wider range of materials, there is a need for a method that improves the stability without changing the mesh size. As will be seen, we propose a simple effective method to achieve that after discussing what has already been done in the field. Bécache et al. [35] gave examples of unstable materials that violate one or more of the conditions given in appendix A, such as Materials IV and III in Table 2.1. They also showed, using numerical simulation, that the corresponding discrete problems are also unstable. Subsequently, Appelö et al. [15] showed that by adding a frequency shift parameter to the stretch function the unstable modes can be stabilized, but not if the material violates the geometric stability condition. This is the case even for the continuous problem before discretization. They also showed that for the discrete case, when instability occurs beyond the spatial frequency (wavevector) range resolved by the mesh, then the numerical solutions will be stable. In particular, using a big enough mesh size h, they found Material IV to be numerically stable provided that any unstable roots that are present satisfy k1 , k2 > π/h, even without using the frequency shift parameter. Material III could not be stabilized because of the limit on the mesh size. Duru et al. [29] introduced a formulation using both a frequency shift and a scaling parameter (in addition to the damping parameter) and showed that by reducing the scaling parameter from unity (i.e., α1 < 1) and using a positive frequency shift parameter, the stability of discrete solutions could be improved for media that violate the geometric stability condition. We will show that the stability of discrete solutions can be improved by increasing the scaling parameter from unity without the need of a frequency shift parameter. If the unstable roots of our dispersion relation (4.3) for a given material are in a higher wavevector range such that the waves corresponding to these roots cannot be resolved by the mesh, then discrete models can be expected to be stable. On the other hand, if


42

the waves corresponding to the unstable continuous roots are resolvable, by increasing α1 the roots can be shifted to a higher wavevector range according to Corollary 4.1, which might improve the discrete stability. Such an improvement will occur if the roots have a negative or even a very small positive imaginary part for the lower wavevectors. To investigate this, we return to the dispersion relation given by (4.3) and let ξ = k1 / α1 (remember α1 > 0). For a fixed positive value of β1 , the roots of the dispersion relation are continuous functions in terms of ξ and k2 , as discussed earlier. Noting that decreasing the value of ξ is equivalent to increasing α1 or decreasing k1 , so that as ξ → 0 the dispersion relations becomes: (ω + iβ1 )4 ω 2 − C22 k22

ω 2 − C33 k22 = 0,

(4.4)

which admits no solution, ω(k2 ), with a positive imaginary part. In fact, two of the four pairs of roots of (4.4) have the imaginary parts equal to −β1 , while the imaginary parts of the other two pairs are equal to zero. Since the roots of the dispersion relation are continuous functions in term of ξ = k1 /α1 , by considering the first quarter of the k−space and for a fixed value of β1 > 0, it follows that: Lemma 4.1. For any given values k0 and ω0 > 0, there exist a value, α0 , such that all roots of (4.3) for k1 < k0 have imaginary parts = {ω} < ω0 for α1 > α0 . Because the growth rate is governed by the value of = {ω}, this lemma expresses the fact that any desired growth rate greater than zero can be achieved for all solutions that correspond to wavevectors (k1 < k0 , k2 ) by a sufficient increase in the scaling parameter. It should be noted that decay rates (ω0 < 0) can be achieved if the unstable pair of roots happens to be one of the pairs whose imaginary parts approach −β1 in (4.4). Hence, the lemma assumes the worst case scenario . In fact, the results to be presented in section 4.2 provide evidence that increasing the scaling parameter improves the stability even for materials that violate the geometric stability condition.


4.1.3

43

Stability analysis examples

To illustrate the results above, the imaginary parts of the roots, of (4.3) were numerically obtained, using MATLAB, over an appropriate range of wavevectors for these materials in Table 2.1 that violates one or more of the conditions in appendix A. The maximum value of the imaginary parts of the roots, max(= {ω (k)}) for each k were plotted over the first quarter of the k−space. As discussed earlier, the stretch function parameters were assumed to be constant. For all cases, the value β1 was taken to be that corresponding to a reflection coefficient R1 = 1 × 10−6 (see (2.34)). Considered first is Material III which severely violates the geometric stability condition in both directions (Definition 4.1 and Figure 2.1). In Figure 4.1, which contains three panels that display max(= {ω (k)}): the first panel uses the classical stretch function, α1 = 1, the center panel corresponds to α1 = 1/7 , while the right panel uses α1 = 7. Note that the wavevector k, can be represented in a polar form, i.e., wavenumber, |k|, and angle θ, where k1 = cos(θ) and k2 = sin(θ). It can be inferred from Corollary 4.1 and Lemma 4.1 that increasing the scaling parameter, α1 , moves any existing instability to a higher wavenumber and smaller θ, while decreasing α1 moves the instability to a smaller wavenumber and greater θ. The center and right panels show exactly these expectations. Nonetheless, the continuous problem is still unstable because the positive imaginary part was merely shifted. However, our interest is in discrete solutions. For a simple 1D discrete problem with a uniform mesh size h0 , the highest spatial frequency that can be well resolved is π/h0 . This value can be thought of as the value k0 in Lemma 4.1. Using a value of h0 that follows (3.11), the arrows in the graphs of Figure 4.1 point to this threshold for guidance. It is clear from the left panel of Figure 4.1 that unstable roots with positive imaginary parts are present in the wavevectors range that can be resolved by discrete models. Hence, it can be expected that the FEM simulations will be unstable for this case. For the center

44


↵1 = 1

↵1 = 7

↵1 = 1/7

k2 (1/mm)

60

40

20

0

0

20

40

60

0

20

k1 (1/mm)

40

60

0

20

k1 (1/mm)

40

60

k1 (1/mm)

max(={!}) (1/ms) 0

1

2

3

4

5

6

7

Figure 4.1: Illustrating the effect of incorporating the scaling coefficient α1 on the roots of (4.3) for the continuous, constant coefficient problem. The color maps show the maximum imaginary part of the roots, max(= {ω}), for Material III. For the classical case of α1 = 1 in the left panel, for a case in which α1 = 1/7 in the central panel, and for the case of α1 = 7 in the right panel. The arrow points to the value of π/h0 , where h0 is the mesh size used in numerical simulations as described in (3.11).

panel it can be seen that the roots are moved to lower spatial frequencies but are still in the resolvable range. On the other hand, in the right panel, the unstable roots are mostly shifted beyond the wavevector threshold of π/h0 . An improvement in the stability can be expected because the maximum value of ={ω} has been substantially decreased from 6.0 (1/ms), in the left panel, to 0.24 (1/ms) for wavevectors below k1 = π/h0 by setting α1 = 7 as expected in Lemma 4.1. Similar results were also obtained for the x2 direction but, because the violation in the x2 direction for this material is very severe, a higher value for α2 was needed to ensure stability over the same range of wavevectors. Similar plane-wave analyses were performed for Materials IV and V. For Material IV, even with α1 = 1, the unstable pair of roots were found to occur at higher wavevectors than those that can be numerically resolved and hence, it should be stable. For Material


45

V, the unstable pair were in the range that can be numerically resolved for α1 = 1, suggesting the possibility of numerical instability. These predictions will be tested in discrete time-domain schemes in the next section.

4.2

Numerical FEM results

In this section, finite element analyses are used to demonstrate the improvement in discrete stability, as expected from Lemma 4.1 for the materials studied using plane wave analysis in subsection 4.1.3. It should be noted that while constant coefficients were assumed for the plane wave analysis, for the discrete FEM simulation, αj and βj are functions of xj as shown in (2.33).

4.2.1

Quadratic Lagrange finite element

In this subsection, second-order Lagrange finite elements are used for space discretization. Figure 4.2 shows the FEM result for the three unstable materials using the classical stretch function, i.e., without introducing any scaling coefficients. This was achieved by setting α0j = 1 in (2.33a). Each row in this figure shows three snapshots for the wave propagating in Materials III, IV, and V, respectively. In the final column, to more clearly demonstrate the ability of the PML to successfully remove virtually all vestiges of the waveform in the physical domain, we have chosen to use a dB scale and a sufficiently long time interval relative to the excitation duration. For all three cases shown in Figure 4.2, the second column shows that much of the energy has left the physical domain in less than 5 ms. On the other hand, it is evident from the third column that most of the energy for Material IV has been dissipated after 30 ms (594 time steps for Material IV) and there is no evidence of instability, but for Materials III and IV it can be seen that serious instability is present. Material III shows serious instabilities that appear to start

46


0.5

A

0

0

x2 (mm)

dB

3.5 (ms) 0.5

D

900

30.0 (ms)

1

5

0

E

F

0

−5

0

0

2.0 (ms)

dB

0.5

G

−100

30.0 (ms)

3.9 (ms) 1

5 x2 (mm)

C

0

2.0 (ms)

Material IV

1100

B

−5

Material V

30.0 (ms)

4.3 (ms) 1

5 x2 (mm)

Material III

time = 2.0 (ms)

0

H

I

0

−5 −5

0 x1 (mm)

5

0

−5

0 x1 (mm)

5

0

−5

0 x1 (mm)

5

dB

−100

Figure 4.2: Snapshot images showing the waveforms, originating from same cylinder as shown in Figure 3.1, but propagating in three different anisotropic solid media, namely III, IV, and V as specified in Table 2.1. In the middle column the snapshot times were chosen to approximately correspond to the quasi-shear wave being absorbed by the PML. The color maps on the third column are in dB’s.

after arrival of the slow wave in the PML (∼ 4ms). The questions as to whether and how the scaling parameter can be used to improve the discrete stability are examined in Figure 4.3. Panel (A) compares the energy for Material III with three cases: α0j = 1, α0j < 1, and α0j > 1 for Material III. For this medium, higher order polynomial values were used (m = n = 8 in (2.33)) to achieve a

47


smoother transition in the PDE coefficients at the interface between the physical domain and the PML. It is clear that for a material with well established instability problems, stabilization can be achieved by increasing the scaling parameter. Similar results are shown in panel (B) for Material V which has less severe instabilities than Material III. While it is clear that using the scaling parameter improves the stability even for α0j < 1, by using α0j > 1 further improvement in stability is achieved with less sensitivity to the choice of the mesh size. Note that when a finer mesh is used, instabilities can be expected to reappear or to increase [15, 29]. 3

1

10

10

↵01 = 1, ↵02 = 1

A

↵01 = 0.1, ↵02 = 1/30

2

10

B

0

10

↵01 = 10, ↵02 = 30 1

10

1

10

2

10

3

10

4

10

5

10

6

Energy

10

0

10

10

↵01 = 1, N = 5

1

10

2

10

3

0

5

10

15

time (ms)

20

25

0

↵01 = 0.1, N = 5 ↵01 = 10, N = 5 ↵01 = 0.1, N = 10 ↵01 = 10, N = 10 5

10

15

20

25

time (ms)

Figure 4.3: Showing the evolution of energy in the physical domain as represented by p k u21 + u22 k∞ for scaling parameters of 1, < 1, and > 1. In (A), for Material III using default mesh size corresponding to N = 5. In (B), for Material V using mesh sizes corresponding to N = 5 (solid lines) and N = 10 (dotted lines) and α02 = 1 for all five curves.

Figure 4.4 shows propagation snapshots for Materials III and V at the same times as in Figure 4.2, but with the values α01 = 10, α02 = 1 for V, and α01 = 10, α02 = 30 for III. Comparison of these two figures shows the effect of increasing the scaling parameter on the stability. While the instabilities disappeared for all directions in Material V and in the x1 direction for Material III, some instability remained in the x2 direction causing some energy to be reflected back to the physical domain. This is likely due to the severity of

48


4.3 (ms) 0.5

A

C

0

0

0

2.0 (ms)

dB

0.5

D

−100

30.0 (ms)

3.9 (ms) 1

5 x2 (mm)

0

B

−5

Material V

30.0 (ms)

1

5 x2 (mm)

Material III

time=2.0 (ms)

0

E

F

0

−5 −5

0 x1 (mm)

5

0

−5

0 x1 (mm)

5

0

−5

0 x1 (mm)

5

dB

−100

Figure 4.4: Propagation snapshots as in Figure 4.2 for Materials III and V, after introducing scaling coefficients of α01 = 10, and α02 = 30 for Material III, and α01 = 10, and α02 = 1. These images should be compared to the corresponding panels in Figure 4.2.

the violation of the geometric stability in the x2 direction for this material. Nevertheless, comparing Figure 4.2 (C) and Figure 4.4 (C), and noting the use of dB scales, the use of a higher value for the scaling coefficient results in a major improvement in stability for Material III. We could find no specific guidelines for the best choice of the scaling parameter besides solving for the roots of the continuous problem for different values of alpha, as we have done in Figure 7. Nonetheless, we performed simulations for several materials (more than reported herein), and noticed that unless the material has severe instability (like Material III in x2 -direction) values for α0j greater than 10 were not needed. In fact, values as small as α0j = 2 might be sufficient for some cases.


4.2.2

49

Other discretization methods

Lemma 4.1 provides evidence that the lower wavevectors parts of the solution are either totally stable or have very low growth rates. This behavior can be extended to higher wavevectors parts of the solution by increasing the scaling parameter. A sufficient increase can cause any existing instability to move to even higher wavevectors parts of the solution that are not well resolved by discrete models. This can lead to the stability of the discrete solutions. The results presented in subsection 4.2.1 provide evidence that this is the case, at least for second-order Lagrange finite element discretization. The question arises as to whether this same improvement can be achieved with other discrete schemes is examined below. Other authors have shown that the discrete stability improves when the unstable modes are not well resolved using finite difference schemes [15, 29, 37]. But, to provide further evidence that under resolution can lead to improved discrete stability we have chosen to repeat the simulations for Material V (zinc crystal) that were presented in the previous subsection, but which are now calculated with both higher order and different finite element types (shape functions). The results are presented in Figure 4.5 where the energy evolution is shown for both 3rd and 4th order Lagrange and Hermite finite elements: (A) in the absence of any scaling (α01 = α02 = 1), and (B) with the scaling parameters α01 = 20 and α02 = 5. For the same mesh size, the higher order shape functions that interpolate the solution with more degrees of freedom, better resolve the parts of the solution with higher spatial frequencies. As expected, this leads to a more prominent appearance of high frequency instabilities as shown in Figure 4.5 (A), where the energy in the physical domain after 30 ms for the 4th order shape functions is around seven orders higher than that for the 2nd order Lagrange case shown in Figure 4.3 (B). Although there exist some differences between the Lagrange and Hermite finite elements results, Figure 4.5 (B) clearly demon-

50


strates that by appropriately increasing the scaling parameters stability can be achieved for both finite elements types and orders.

A

, ,

0

5

10

15

20

( )

25

, ,

30

103 102 101 100 10 1 10 2 10 3 10 4 10 5

B

, ,

0

5

10

15

20

( )

Figure 4.5: Evolution of energy in the physical domain as represented by k

25

, ,

30

p u21 + u22 k∞ for

Material V. The results were obtained by using cubic and quartic Lagrange and Hermite finite elements. For (A), α01 = α02 = 1, while in (B), α01 = 20 and α02 = 5.

4.3

Summary

A simple method has been proposed to improve the stability of the discrete PML problem for otherwise unstable anisotropic elastic media. This was achieved by increasing the value of the scaling parameter sufficiently to move the unstable modes out of the discretely resolved range of spatial frequencies. Also, we have demonstrated that the applicability of our method for different finite element methods. While all discrete models are unable to well resolve the components of the solution that are higher than certain frequencies, the behavior of discrete solutions, including their stability, may vary between numerical methods; a topic that requires detailed study, but which is beyond the scope of this thesis.

Chapter 5 PML formulation for wave propagation in unbounded fluid-solid media

1

As discussed in previous chapters, wave propagation in an infinite medium can be numerically simulated by surrounding a finite region by a perfectly matched layer. When the medium is heterogeneous consisting of both solids and liquids, difficulty arises in specifying the properties of the PML especially because parts of it lie at the solid-fluid interface. Such a situation could arise in many important fields including marine seismology where water is in contact with earth, biomedical ultrasound where soft tissue is in contact with bone, and acoustic metamaterials. In this chapter, we present a second-order time-domain PML formulation for fluid-solid heterogeneous media in two dimensions that satisfies the interface coupling boundary condition throughout the computational domain. Prior publications that studied PML’s for heterogeneous media assumed that the 1

The material in this chapter was accepted as a paper in The Journal of the Acoustical Society of America.

51

Chapter 5. PML formulation for fluid-solid media

52

differing media consisted of two or more media of the same type (i.e., solid-solid or fluidfluid) but having differing properties [17, 38, 39]. Here, a PML formulation is introduced for the problem of an unbounded heterogeneous medium consisting of both solids and fluids, as illustrated in Figure 5.1 . Results are given to establish the accuracy of the formulation and to provide examples of its application. In particular, the effectiveness of the PML in absorbing all kinds of bulk waves, as well as surface and evanescent waves is studied.

5.1

PML formulations

For the solid region of a heterogeneous medium the PML formulation derived in chapter 3 can be used. But as noted in section 2.3, this same equation can not be used in the fluid region by putting the shear modulus to zero. To overcome this problem, a new secondorder time-domain PML formulation for acoustic wave propagation is formulated for use in the fluid region. In addition, formulations for the fluid-solid coupling boundary conditions, namely (2.12) and (2.14), in the PML are presented. This enables the full PML formulation for the heterogeneous fluid-solid problem to be presented. The weak form of this full formulation, as used in FEM, is also presented.

5.1.1

PML formulation for fluid media

A time-domain PML formulation that preserves the single second-order from of the acoustic wave equation is introduced for modeling wave propagation in unbounded fluids. The derivation starts by applying a time-domain Fourier transform to (2.1) together with the complex coordinate stretching transform, as described in (2.29), leading to:

53


−iω pˆ = −K

2 X vj 1 ∂ˆ s ∂xj j=1 j

(5.1)

∂ pˆ −iω vˆj ρF = − , sj ∂xj where the hat •ˆ represents the Fourier transform. Multiplying the first equation by −iω s1 s2 , the second by sj , and expanding sj according to (2.30), yields 2 X ∂ α1 α2 β1 β2 α1 α2 (−iω) + (−iω) (β1 + β2 ) + β1 β2 pˆ = −K −iω + vˆj ∂x α β j j j j=1

2

(−iω + βj ) vˆj = −

1 ∂ pˆ . ρF αj ∂xj (5.2)

The time-domain equations can be obtained by taking the inverse Fourier transform of (5.2) (−iω → ∂/∂t). Moreover, substituting ∂vj /∂t from the second equation into the first leads to the second-order time-domain PML formulation for the pressure field given by 1 ˜ K

X 2 ∂ 2p ∂p ∂ 1 ∂p +b + cp = − aj v j ∂t2 ∂t ∂xj ρFj ∂xj j=1

(5.3)

−1 ∂p ∂vj , + βj vj = ∂t ρF αj ∂xj ˜ where K(x) =

K , α1 α2

ρFj (x) =

α2j ρF α1 α2

, aj (x) =

α1 α2 αj

β1 β2 βj

− βj , b(x) = β1 + β2 , and c(x) =

β1 β2 . Unlike the original pressure field equation as given by (2.2), the particle velocity field does not cancel and will be considered as auxiliary variables for the second-order pressure equation. Hence, this PML time-domain formulation consists of one secondorder equation in pressure and two auxiliary equations. It should be noted that in the physical domain where αj = 1 and βj = 0, the two equations in (5.3) are decoupled, and the first one is identical to the original equation as given by (2.2) as would be expected. If a source term F (x, t), is added to RHS of the original problem, this will simply lead to adding the same term to the RHS of the first


54

equation in the PML formulation as given by Eq.(5.3), because F (x, t) has its support only in the physical domain where the stretch function equals to unity. Similarly, a load vector, Fj (x, t) can be added to the right-hand side of the first equation in (3.9) to model excitation source in solids.

5.1.2

The coupling boundary conditions in the PML

It should be recalled that the fluid-solid coupling boundary conditions,(2.12) and (2.14), are stated in a form that corresponds to Neumann boundary conditions for the two second-order wave equations,(2.2) and (2.9) respectively. This formulation helps simplifying the modeling problem. One of the main advantages of our PML formulations, as given by (5.3) and (3.9), is that they preserve the standard second-order form of the wave equation. Hence, we need to formulate the boundary condition so as to preserve this simplicity. For example, rather than merely applying the complex coordinate transform to the original boundary condition, (2.12), we need to design the boundary condition for P2 1 ∂p − a v in terms of the state variables in n (5.3) to state the value of j j j=1 Fj ρF ∂xj j

the solid at the fluid-solid interface. It can be deduced from (5.2), and (5.3) that 2 X j=1

nFj

1 ∂ pˆ − aj vˆj ρFj ∂xj

=

2 X j=1

nFj

α1 α2 αj

β1 β2 −iω + vˆj . βj

(5.4)

In the physical domain, where αj = 1 and βj = 0, the RHS of the above equation P2 becomes vj , due to the continuity of the normal velocity at the fluidj=1 nFj (−iω)ˆ P2 2 solid interface this equals to ˆj . But in the PML region, where βj > 0 j=1 nFj (−iω) u and αj can differ for unity, it is a non-trivial problem to find the value of the RHS of (5.4). Nevertheless, to maintain the validity of complex coordinate stretching approach, the fluid-solid interface, denoted by Γ in Figure 5.1, needs to intersect the PML in a

55


direction that lies perpendicular to the layer and parallel to the coordinate axis. This simplifies the problem because in the PML region parallel to the xl axis, where nFj = 1 if j = l and is equal to zero if j 6= l, the continuity of the normal velocity at the interface reduces to vˆl = −iω uˆl . Hence, the RHS of (5.4) reduces to α1 α2 αl

β1 β2 −iω + βl

α1 α2 vˆl = αl

β1 β2 2 (−iω) − iω uˆl . βl

(5.5)

This result, along with the case in the physical domain as discussed above, enables (5.4) to be written as 2 X

1 ∂ pˆ − aj vˆj ρFj ∂xj

nFj

j=1

=

2 X

nFj

j=1

α1 α2 αj

β1 β2 2 (−iω) − iω uˆj . βj

(5.6)

By taking the inverse Fourier transform of (5.6), the time-domain of the boundary condition governing (5.3) throughout the fluid-solid interface can be obtained as 2 X

nFj

j=1

1 ∂p − aj v j ρFj ∂xj

2 X

α1 α2 =− nFj αj j=1

∂ 2 uj β1 β2 ∂uj + ∂t2 βj ∂t

.

(5.7)

Using similar approach, it can be shown that the boundary conditions on (3.9) throughout the fluid-solid interface can be expressed as: 2 X j=1

where P (x, t) =

Rt 0

nSj

2 X

∂uk C˜ijkl + aj wij ∂xl k,l=1

! = −nSi α1 α2 (p + b P ) ,

(5.8)

p(x, τ ) dτ needs to be evaluated only at the fluid-solid interface in

the PML region where b = β1 + β2 doesn’t vanish. Note that in the physical domain where αj = 1 and βj = 0, (5.7) reduces to the prefect slip boundary condition of (2.12), while (5.8) reduces to the continuity of traction boundary condition given by (2.14), as expected. It should also be noted that the two boundary conditions above are presented

56


in a form that enables these conditions to be easily integrated in the weak form of final PML formulation as presented in the next subsection.

F

PML

nF

⌦F (Fluid)

PML

nS

⌦S (Solid)

S

Figure 5.1: Illustrating the problem concerned with the PML close to the fluid-solid interface. The fluid domain, ΩF , has the boundary ∂ΩF = ΓF ∪Γ and the solid domain has the boundary ∂ΩS = ΓS ∪ Γ, where Γ = ∂ΩF ∩ ∂ΩS (the red line) is the fluid-solid interface, and ΓF and ΓS are the outer boundaries of the fluid and solid domains respectively. The inner dotted square, centered at the origin with dimensions of 2x0 , is the physical domain surrounded by a PML of thickness d.

57


5.1.3

Complete formulation of fluid-solid PML

Using the results in section 3.1 along with those derived in the previous subsections, the complete formulation of the problem as described in Figure 5.1 can be obtained. Assuming that the source term is in the fluid side, and that the field variables and their first time derivatives vanish at t = 0, then the problem to be solved for t > 0 is:  X 2  1 ∂ 2p 1 ∂p ∂p ∂    + cp = − aj vj + F +b   ˜ ∂t2 ∂t ∂x ρ ∂x K  j F j j  j=1     ∂vj −1 ∂p   + βj vj =    ∂t ρF αj ∂xj        !  2 X  2 2  X  ∂ui ∂ ui ∂ ∂uk   ρ˜S +b + c ui = + aj wij C˜ijkl  2  ∂t ∂t ∂xj k,l=1 ∂xl   j=1    2  X  ∂wij Cijkj ∂uk   + β w = j ij  ∂t αj ∂xj k=1     2  2 2  X X  1 ∂p ∂ u β β ∂u α α j 1 2 j 1 2   nFj − aj vj = − + nFj  2  ρ ∂x α ∂t βj ∂t  F j j j  j=1 j=1   !  2 2  X X  ∂uk   + aj wij = −nSi α1 α2 (p + b P ) nSj C˜ijkl    ∂x l  j=1 k,l=1           p (x, t) = 0        ui (x, t) = 0

x ∈ ΩF x ∈ ΩF

x ∈ ΩS x ∈ ΩS

(5.9)

x∈Γ x∈Γ x ∈ ΓF x ∈ ΓS .

In addition to this second-order differential formulation, a weak formulation of the problem can be written as # Z "X 2 1 ∂p ∂φ 1 ∂ 2p ∂p − aj vj + +b + c p φ − F φ dΩF ˜ ∂t2 ρFj ∂xj ∂xj K ∂t j=1

ΩF

+

Z X 2 Γ

α1 α2 nFj αj j=1

(5.10a)

∂ 2 uj β1 β2 ∂uj + ∂t2 βj ∂t

φ dΓ = 0,

58


Z "X 2 ΩS

j=1

!

2 X

2

#

∂φi ∂ ui ∂ui ∂uk + aj wij + ρ˜S + c ui φi dΩS +b C˜ijkl 2 ∂xl ∂xj ∂t ∂t k,l=1 Z + nSi α1 α2 (p + b P ) φi dΓ = 0,

(5.10b)

Γ

along with the auxiliary equations, Z

∂vj 1 ∂p + βj vj + ∂t ρF αj ∂xj

ψj dΩF = 0,

(5.10c)

ΩF

Z ΩS

2

X Cijkj ∂uk ∂wij + βj wij − ∂t αj ∂xj k=1

! ψij dΩS = 0.

(5.10d)

In the above set of equations, φ(x), φi (x), ψi (x), and ψij (x) are test functions. These functions and their first derivatives are assumed to be square integrable functions that also satisfy the Dirichlet boundary condition of the corresponding field. In mathematical notation, the previous statement for the test function φ, for example, can be written as: 1 (ΩF ). φ ∈ H0,Γ F

5.2

Numerical methods and results

A finite element method was used to numerically solve the problem illustrated in Figure 5.1 and described in (5.9). Specifically, the weak formulation as described in (5.10) was used to model the problem by using the “Mathematics Model” in the FEM software COMSOL Multiphysics. Our excitation source consisted of a monopole source in the fluid with a time dependence corresponding to the first derivative of a Gaussian with a peak amplitude of unity and a space dependence of f (x). Specifically, the forcing function is given by √ 2 2 F (x, t) = − 2e πf0 (t − t0 ) e−π f0 (t−t0 ) f (x),

(5.11)

59


where f0 is the dominant frequency and t0 is a source delay time. For all numerical simulations, f0 = 1.5 MHz and t0 = 1 µs. Moreover, f (x) is a line source for the validation subsection and is a Gaussian function for the examples subsection. For the purpose of choosing appropriate spatial and time discretization, it is helpful to define the minimum and maximum characteristic wave speeds associated with the heterogeneous medium by cmin and cmax . For the space discretization we used second-order Lagrange finite elements. A square mesh was used for the PML region, but the default triangular shape was retained in the physical domain. The choice of an appropriate mesh size is governed by the shortest wavelength of significance for the propagating pulse, i.e., cmin /f0 . In particular, the mesh size was taken to be

h0 =

cmin , N f0

(5.12)

which, for the second-order shape functions used, corresponds to 2N degrees of freedom per wavelength. The value N = 10 was used for all simulations given in this chapter. For time discretization, a second-order generalized alpha method was used with ρ∞ = 0.75, as defined by Chung et al.[52]. When compared to explicit methods, this implicit method is less sensitive to the time step chosen. As a result, a time step size of 0.9h0 /cmax was used, which is just less than the time needed for the fastest wave to travel through the smallest mesh dimension. For all FEM simulations the value of β0j was chosen to correspond to a reflection coefficient: Rj = 1 × 10−6 in (2.34). In addition, α0j = 1 and m = n = 2 were used in (2.33), unless stated otherwise.

5.2.1

Validation

There are two aspects of the new formulation that need to be verified. First, it is necessary to ensure that the PML formulation near the fluid-solid interface, as given in


60

(5.9) and (5.10), describes the original unbounded problem. Second, validation is needed for the finite element model used in the simulation. Both aspects can be tested by comparing our PML results with those obtained from an established analytical solution of the corresponding unbounded problem. Such a solution for layered media can be obtained using the Cagniard–de Hoop method [55, 56], which has been used to obtain a closed-form Green’s functions based on integral transform techniques [57–64]. For the purpose of validation, an infinite domain was considered wherein the upper half space is fluid and lower is solid, and the excitation consists of an infinite monopole line source in the fluid and parallel to the fluid-sold interface. The pressure field spacetime Green’s function of this configuration as derived by de Hoop et al. [58] was convolved with the excitation source, as given in (5.11), and compared to our FEM solution. Fu et al. [65] showed good agreement between the experimental measurements obtained for cylindrical wave propagation close to a liquid-solid interface with those numerically calculated from the time-space Green’s function given by de Hoop et al. [58]. The same media and values as in their work were chosen, namely, water (ρF = 1000 kg/m3 , cF = 1480 m/s) and K9 glass (an optical borosilicate crown glass, ρS = 2530 kg/m3 , cp = 5690 m/s, cs = 3460 m/s). The physical domain was chosen to be a square centered at the origin with a side length of 2 cm, surrounded by a 2 mm wide PML, and excited by a line source that follows (5.11), with f (x) = δ(x − x0 ), where x0 =(0 mm, 1 mm) as shown in Figure 5.2. The mesh size and the time step were calculated as given earlier with cmin = cF , and cmax = cp . The scaling parameter, αj was set to unity throughout, i.e., no scaling to the real coordinate was introduced. The FEM solutions were obtained for p(x, t) in the fluid medium and u(x, t) in the solid medium. Nevertheless, the pressure field, p, in the fluid medium and the stress component,

− σ22 = −

2 X k,l=1

C22kl

∂uk , ∂ x˜l

(5.13)


61

in the solid medium will be shown in all snapshots illustrated in this chapter. It should be noted that −σ22 was chosen because it has the same value as the pressure at an interface that is parallel to x1 .

Analytical

FLUID

FEM

ide

Inc

ec

fl Re

ad

2

He

SOLID

S-

wa

wa

ted

Scholte

L. Rayleigh

P-

nt

FEM

ve

ve

1

Figure 5.2: Snapshot at t = 4.5µs showing the field distribution in both the solid and fluid regions caused by a line source at the point S in the fluid close to the fluid-solid interface, with a transient excitation source given by (5.11). The color scale shows the pressure in the fluid and σ22 in the solid, assuming a normalized source. The boundary of the PML is shown as a dotted line, while the solid line is the fluid-solid interface. To illustrate the agreement between the analytical and FEM solutions in the fluid, the top half has been split to show the FEM and analytical results. Points A at (9 mm, 1 mm) and B at (3 mm, 3 mm) are the locations where the waveforms are shown in the next figure.

62


A snapshot of the solution at t = 4.5µs is shown in Figure 5.2. To visually show the agreement with the analytical solution, we took advantage of the symmetry in the problem and replaced FEM solution in right side of the physical domain of the fluid medium (x1 , x2 > 0) by the analytical solutions. Good agreement between two solutions is evident. Additional evidence for the agreement can be deduced from the fact that our simulations exhibited the various bulk and surface waves that are expected to be present. Specifically, Figure 5.2 shows the presence of the incident, the reflected, and the head waves in the fluid medium. In addition, it shows both of the primary (P), and the secondary (S) waves in the solid medium, as well as Scholte and leaky Rayleigh waves at the interface.

( )

60 40

FEM-PML Analytical

A

FEM-PML Analytical

B

20 0 20 40 0

2

4

6

8

10 12 14

0

2

4

6

8

10 12 14

Figure 5.3: Waveforms at the points A and B in Figure 5.2 showing the agreement between the FEM and analytically calculated pressure waveforms. The normalized mean absolute errors are: 6.7 × 10−4 for (A) and 1.0 × 10−3 for (B).

For a quantitative validation of the formulation and the simulations, the analytical and simulated pressure field wavefronts at two points in the physical domain were compared as marked in Figure 5.2: point (A), which is close to both the PML and the fluidsolid interface, and point (B). Figure 5.3 provides a comparison of the computed time

63


dependence with that obtained from simulations. These results together with snapshot image provide good evidence for the validity of our model.

5.2.2

Numerical examples

Two numerical examples are presented here for an irregular fluid-solid interface and details of the field when the interface is very close to the boundary layer. The first example shows the effectiveness of fluid-solid PML in absorbing the longitudinal waves in the fluid regions, both the P- and S-waves in solids, together with absorption of the surface waves in their propagation direction along the interface. While in the second example, the effectiveness of the PML in absorbing the surface wave in the evanescence direction, perpendicular to the fluid-solid interface, is investigated. All parameters are the same as those used in the validation subsection, except for the computational domain dimensions, the scaling parameter, and f (x). A more realistic spatial distribution than the Dirac delta function was used for the source, namely, (x1 − x01 )2 + (x2 − x02 )2 1 , f (x) = 2 exp −π r0 r02

(5.14)

where r0 was chosen to be 1/40 of the physical domain width and x0 is the center of the source. The source, f (x), was set to zero for kx − x0 k2 > r0 . All the example simulations were run for a total of 30 µs, and the snapshot results, as discussed below, are shown in Figure 5.4 and Figure 5.5. To provide a quantitative measure of the effectiveness of the PML the total instantaneous energy in the physical domain was calculated. In the fluid physical domain this energy is given by 1 EF (t) = 2

Z " ΩF

# 2 p ρF vj2 + dΩF , K j=1 2 X

(5.15)

64


while that in the solid physical domain it is given by Z "

1 ES (t) = 2

ρS

2 2 X ∂uj ∂t

j=1

ΩF

# ∂ui ∂uk + dΩF . Cijkl ∂xj ∂xl i,j,k,l=1 2 X

(5.16)

In both equations, the first term represents the kinetic energy, while the second term is the potential energy. The results for both examples are shown in Figure 5.6.

= .

5

= .

(

)

40

0 5 = .

(

)

5

0

=

0

40

5 5

(

0

)

5

5

(

0

)

5

Figure 5.4: Snapshots illustrating the effectiveness of the PML in absorbing different kinds of bulk and surface waves for an irregular fluid-solid interface. The gray solid line is the interface between the fluid (upper half) and the solid (lower half). At 2.5µs the P-wave is being absorbed by the lower boundary. At 3.5µs the S-wave is being absorbed by the same boundary while the leaky Rayleigh waves have already been absorbed by the side boundaries. Scholte waves and the incident pressure waves can be seen as being absorbed at 4.5µs. The 10µs panel shows that the energy remaining in the computational domain is very small.


65

Example 1 For this example the physical domain is a 1 cm square centered at the origin and surrounded by a 1 mm PML, with a source centered at x0 = (0 mm, 0.5 mm), and irregular fluid-solid interface. The first three snapshot simulations shown in Figure 5.4 demonstrate the absorption of all kinds of bulk and surface waves by the PML, while the last snapshot shows that only a very small amount of energy is reflected to the physical domain after 10 µs. In Figure 5.6 (A), the thin (red) line shows how the total energy in the physical domain evolves over time and provides quantitative evidence on the effectiveness of the PML.

Example 2 Although PMLs are designed to terminate propagating waves, it might be desired to terminate strong evanescent waves. In addition, it has been shown that evanescent waves can cause spurious reflections from the PML interface when discrete schemes are used [66]. Using the frequency shift parameter in the CFS-PML formulations can lead to an improved behavior in the presence of strong evanescent waves [13, 25]. For formulations that use only two parameters stretch function (GPML), as for our case, it has previously been shown that for the continuous (non-discrete) problem, increasing the scaling parameter improves the absorption of the evanescent waves in elastic media [28]. In the example shown here, we show that this is also the case numerically, i.e., for the discrete case. The physical domain in this example is a rectangle centered at the origin with a width of 20 mm and a height of 4 mm surrounded and by a 1 mm PML, while the center of the source is at x0 = (−9 mm, 0 mm). As shown by the line at x2 = −1 mm in Figure 5.5, the fluid-solid interface is set very close to the PML. To ensure that αj will only increase the decay of evanescent waves and rule out its effect on damping propagating waves,

66


βj can be replaced by βj /αj . Note that such a replacement makes the second term of Rx x) d´ x , hence increasing αj will not affect the damping of the (2.31) equal to exp −c β (´ 0

propagating waves. Three snapshots of the field distribution are shown in Figure 5.5 for the case of no scaling in the left column, while the right column corresponds to the case of α02 = 99 in the solid domain. By inspecting the lower boundary where the surface waves are being absorbed in their evanescence directions, it is evident that increasing the scaling parameter improves the effectiveness of the PML.

(

)

(

)

(

)

=

10

(

0

)

10

=

2 0 2

30

2 0 2 2 0 2

0

10

(

0

)

10

30

Figure 5.5: Snapshots illustrating the effectiveness of the PML in absorbing strong evanescent waves. This was achieved by setting the fluid-solid interface very close to the PML in the solid. In the left column no coordinate scaling was used, while for the snapshots corresponding to the same times, as shown in the right column, the scaling parameter of α02 = 99 was used in the solid medium.

In order to provide a more quantitative measure of this improvement, the manner in which total energy in the solid’s physical domain evolves in time for α02 = 1, 9, and 99 is shown. Figure 5.6 (B) provides evidence for the improvement in the PML absorption of evanescent waves caused by increasing the scaling parameter. To make sure that this improvement was due to the better absorption of the evanescent waves in particular, the

67


simulation in Example 1, where there is no prominent evanescent waves close to PML, was repeated using α01 = α02 = 99. The cyan (thick) line in Figure 5.6 (A), which represents the energy evolution for this case, shows that increasing the scaling parameter can degrade the PML absorption. This further supports the fact that the improvement seen in Figure 5.6 (B) is due to improved absorption of the evanescent waves.

()

100 10

1

10

2

10

3

10

4

10

5

10

6

= =

= = =

A 0

5

B 10

15

20

( )

25

30

0

5

10

15

20

( )

25

30

Figure 5.6: (A) The normalized total energy in the physical domain as given by (5.15) and (5.16) for the simulation in Figure 5.4 is represented by the red thin line. The thick cyan line represents the same results, but using scaling parameters α0j = 99, j = 1, 2, in both the fluid and the solid domains. (B) The normalized total energy in the physical domain of the solid for both simulations shown in the left and right columns of Figure 5.5, but with an additional case that corresponds to α02 = 9.

5.3

Summary

A time-domain PML formulation has been introduced for terminating a region consisting of fluid and solid domains close to their interface. For the fluid domain, a PML formulation consisting of a single second-order pressure equation along with two auxiliary equations, was introduced. The second-order PML formulation, as previously presented


68

in chapter 3, was used for the solid domain. Coupling boundary conditions for the fluidsolid interface were carefully derived for both the physical domain and the PML region, without affecting the perfect matching characteristics. These boundary conditions were chosen such that they can be directly integrated into a weak formulation of the complete fluid-solid problem and which can be readily used in FEM analyses. Numerical simulations showed the validity of the formulation and the effectiveness of the PML in absorbing bulk waves, surface waves, and strong evanescent waves with the help of changes to the scaling parameter.

Chapter 6 Application of PML to modeling the transient response of a metamaterial acoustic lens 6.1

Introduction

It is the purpose of this chapter to demonstrate that the approach developed in the earlier chapters can be successfully applied to determine the transient response of a metamaterial acoustic lens. As noted in chapter 1, the development and application of acoustic metamaterials is a rapidly developing research field that has many applications. Especially of interest to our group is the development of a super-lens with sub-wavelength imaging properties that could offer potential improvements in medical diagnostic and therapeutic ultrasound. Acoustic metamaterials are, commonly, phononic crystals whose periodicity is smaller than the operating wavelength. In fact, the first simulation and experimental demonstration of such a lens appears to be that described in 2008 by Sukhovich et al. [67] and which was improved and detailed in subsequent papers [68, 69]. They used a

69

Chapter 6. PMLs and modeling metamaterials

70

flat lens consisting of a periodic array of 1mm diameter stainless-steel rods immersed in methanol. Subsequently, other groups have made use of more complex periodic 2D and 3D structures [70] whose structural periodicity is in the order of magnitude of the incoming acoustic radiation. It should be noted that the operation environment for these lenses is usually water, as opposed to the free space for the electromagnetic case. An acoustic phononic crystal contains periodically structured solid or fluid inclusions within either a solid [1, 71] or a fluid [67, 68] host matrix. When the matrix is solid the resulting structure is generally referred to as an elastic phononic crystal. This is more suitable for ultrasonic imaging applications and it supports more complex modes that arise from the coupling of longitudinal and transverse waves [1, 71]. One such lens that operates in water was introduced by Zhou et al. [1], details of which are shown in Figure 1.1, and this was chosen as an example in order to illustrate the time-domain simulation method developed in earlier chapters.

6.2

Lens design and properties

The solid lens design of Figure 1.1, was inspired by a four-particle unit cell model previously proposed by Lai et al. [72] to create a locally resonant effect for longitudinal wave modes [1]. As illustrated in Figure 1.1 (A), the lens is a periodic structure whose unit cell is composed of four brass cylinders embedded in an Al-SiC foam background matrix with a cylindrical cavity arranged in the center. The lattice constant for the square unit cell is a = 5 mm, the diameter of the cavity is 0.8 mm, the brass cylinders are 1 mm in diameter and are located 1 mm from the center. Both the foam and brass are assumed to be isotropic solids. The foam has a mass density, ρS = 72 kg/m3 , a primary phase velocity, cp = 1144 m/s, and a secondary phase velocity, cs = 611 m/s. While for the brass: ρS = 8500 kg/m3 , cp = 4652 m/s, and cs = 2113 m/s. Water was taken to have a mass density ρF = 1000 kg/m3 , and phase velocity, cF = 1490 m/s.

k2 a/⇡

cell. The as phenomenon very similar modulusmode, jeff which are found to be negative, shown inisFigs. 2(d) to that observe 16 tions corresponds to the longitudinal is oftointeraccording surface integration method. The16predicted hybrid elastic solids, which have been recognized and 2(e). Though not based on isaplotted reliable assumption, est in this work. The displacement amplitude distribution ofdirection branch in the CX by the cross Fig. 2(a) the materials with doublyinnegative mass density and bul and(marked shows the agreement with the the bandformation structure results. the eigenstate at the C point in this branch with effective medium analysis maylus. explain of neg-in the foam corres In our model, the wavelength Both indicate of the retrieved mass density qeff and bulk “A”) is plotted in Fig. 2(b),ative wherecurvature the arrows the effective wave of the longitudinal branch and to the C point frequency is times modulus j are found to be negative, as shown in Figs. 2(d)6more eff Chapter PMLs modeling metamaterials 71 bigger than th direction of6.motion. Theand eigenstate is clearly a monopolar and 2(e). Though not based on a reliable the constant; we areassumption, not convinced that this value fall response as evidenced by the collective motion of fouranalysis may explain the formation of negeffective medium the effective medium description of metam ative curvature of the longitudinal wave branch and more However, if the effective medium model is forced for the structure, effective parameters can be ac Μ integration method.16 The p according to surface A branch in the CX direction is plotted by the cross in B and shows the agreement with the band structure p Both of the retrieved effective mass density qeff a modulus jeff are found to be negative, as shown in F andΓ 2(e). ThoughΧnot based on a reliable assumpt effective medium analysis may explain the formation ative curvature of the longitudinal wave branch an FIG. 2. (a) The band structure of a square lattice of the infinite crystal (solid line), the dispersion curve of the water (dashed line), and the dispersion curve given by effective medium prediction (cross); (b) and (c) the displacement distributions of the unit cell corresponding to the eigenstates marked in square lattice of and the“B,” infinite crystal (solid effective mass density qeff (a) with “A” respectively; the retrieved (d) and effective bulkand and shear jeff and leff (e). water (dashed line), themoduli dispersion

-

k a/⇡

1 re of a FIG. 3. The eigenfrequency contour plot of the solid phononic crystal. of the Figure 6.1: Properties of the phononic crystal lens shown in Figure 1.1. (A) The band dium prediction (cross); (b) and (c) the in displaceThis article is copyrighted as indicated the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39 On: Thu, 12 Feb 2015 20:56:45 t cell corresponding to the eigenstates marked in crystal structure of an infinite with a unit cell shown in Figure 1.1 (A). The sixth branch ctively; the retrieved effective mass density qeff ear moduli jeff and leff(the (e). blue line) with negative FIG. 3. The eigenfrequency contour the solid phononic slopes in both the ΓX and the plot ΓMofdirections, is usedcrystal. for the

lens operation. Thisisbranch the sameat:phase velocity as water (red dotted line) at the to IP: cated in the article. Reuse of AIP content subject has to the terms http://scitation.aip.org/termsconditions. Downloaded 142.150.190.39 On: Thu, 12 Feb 2015 20:56:45of 37.65 kHz. (B) Equifrequency contours plot point marked by p, corresponding to frequency FIG. 2. (a)that The band of aslope square of lattice the infinite crystal showing the structure negative theofsixth branch is(solid present in all directions, and is isotropic line), the dispersion curve of the water (dashed line), and the dispersion curve given effective medium prediction (Reproduced, (cross); (b) and (c) with the displacearound theby37.65 kHz frequency. permission, from Zhou et al. [1]) ment distributions of the unit cell corresponding to the eigenstates marked in (a) with “A” and “B,” respectively; the retrieved effective mass density qeff (d) and effective bulk and shear moduli jeff and leff (e).

FIG. 3. The eigenfrequency contour plot of the solid phononic

Based on finite element simulations, Zhou et al. calculated the band structure (the

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. D 142.150.190.39 On: Feb 2015 20:56:45 dispersion relation) of the infinitely periodic crystal inThu, the12principal symmetry directions

(ΓX and ΓM) of the first Brillouin zone in the reciprocal (wavevector) space [1]. This band structure is shown in Figure 6.1 (A), where the first two branches starting from the zero wavevector point, Γ, correspond to quasi-shear and quasi-longitudinal modes. They noted that the sixth branch with a negative slope in both principal directions has a polarization in the longitudinal direction: this is the one of primary interest for the lens operation. The dispersion relation for water, indicated by the red dotted lines, is superimposed on the dispersion relation for the crystal. A surface plot of the frequency in the vicinity of the sixth branch is shown in Figure 6.1 (B). It covers the first Brillouin zone of the wavevector space in all directions, i.e., − πa ≤ k1 , k2 ≤ πa .


72

Optimal focusing by a flat metamaterials lens is achieved when the phase velocities, ω , |k|

in the water and metamaterials have the same magnitude [5]. This corresponds to

the point p in Figure 6.1 (A), where the targeted sixth branch of the band structure meets the water line at the frequency f0 = 37.65 kHz. Moreover, the equi-frequency contour at 37.65 kHz was found to be nearly circular as shown in Figure 6.1 (B), i.e., the lens is isotropic around f0 = 37.65 kHz, which is another requirement for optimal focusing. More importantly, Figure 6.1 shows that, for the sixth branch, the wavevector, ∂ω ∂ω k and the group velocity, Vg = ( ∂k , ), have opposite directions for all k, which is 1 ∂k2

a fundamental requirement for the presence of negative refraction and superfocusing in metamaterials. While the group velocity in water has the same magnitude as the phase velocity, the group velocity in the lens is significantly slower than the phase velocity, which can be readily implied from the slope of the sixth branch. This means that the energy propagation in the lens can be expected to be very slow in comparison to that in water. This, among other phenomenon, cannot be readily observed in the commonly used frequency domain simulations such as the one in Figure 1.1 (B), which demonstrates the focusing at the ideal frequency f0 = 37.65 kHz. In practice, excitation signals are transients so that time-domain simulations should provide a much more complete picture of the focussing properties.

6.3

Time-domain simulations and results

All the models presented in this thesis are 2D formulations, and the original design of the lens described by Zhou et al. is a 2D phononic crystal [1]. Therefore, materials are assumed invariant and infinite in the x3 direction and all the elements from which the phononic crystal is formed have a cylindrical shape and all lie along the x3 axis. In all of the simulations in this section, we model a solid slab that is infinite in the x1 direction and has a thickness of 4 cm in x2 direction with the lower edge on the x1 axis, and is


73

surrounded by water. For the computational domain, both the x1 and x2 directions were terminated by PML walls of thickness d. In the first simulation to be described, the slab consists of just the homogeneous Al-SiC foam, while in subsequent simulations the brass cylinders and cavities were inserted in the foam as shown inFigure 1.1 and these enable the lens to be represented as proposed Zhou et al. [1]. The finite element method wave used to solve this fluid-solid PML problem based on (5.10). The maximum mesh size, the time step, the stretch function parameters, and all other numerical settings are the same as those used in section 5.2, unless mentioned otherwise. The minimum mesh size was chosen to be small enough so as to capture the boundaries of the cylindrical inserts in the lens. A line source was assumed to be situated in water 2 mm above the slab at x0 = (0 cm, 4.2 cm), and has a Gaussian modulated cosine time dependence given by: " 2 # πf0 B √ (t − t0 ) , p0 (t) = cos [2πf0 (t − t0 )] exp − 2 ln 2

(6.1)

where the central frequency,f0 , was chosen to be 37.65 kHz, corresponding to the point p in Figure 6.1 (A). The full width at half maximum (FWHM = f0 B) was chosen to be 1.88 kHz by setting B = 0.05. The time delay was chosen to be t0 = 0.7 ms and the simulations were run for a total of 10 ms.

Lens situated in the physical domain The left column of Figure 6.2 shows snapshots of the simulations for the case for which the slab of solid consists only of the foam matrix, while the right column shows the corresponding snapshots when the metamaterial periodic structure (8 × 50 cells) is included, but is well within the physical domain. The very slow group velocity and the focusing can be seen in the simulations. While the energy emitted from the line source diverged

74


quickly out of the physical domain in the left column, it moved slowly in the lens and converged to a focus on the other side of the lens.

time = 0.3 ms

x2 (cm)

18

4 0

5 -15

time = 1 ms

x2 (cm)

18

4 0

0

-15

18

time = 2 ms

x2 (cm)

-5

4 0

-15 -18

0

x1 (cm)

15

-18

0

15

x1 (cm)

Figure 6.2: Snapshots of simulations of a radiating line source in water close to an infinite slab of solid Al-SiC foam are shown in the left column. In the right column a periodic structure of 8×50 unit cells, similar to the one shown in Figure 1.1 (A), were inserted in the foam to generate a metamaterials lens. The simulation snapshots in the right column illustrate that the group velocity in the metamaterial is much less than in the surrounding water. The final panel at the 2 ms snapshot shows focusing of the source: details of the doted red square region is given in the next figure.

75


To better illustrate the field distribution and wave propagation in the periodic structure, a zoomed view of the last snapshot in the right column of Figure 6.2 is presented in Figure 6.3 (A). As can be seen from Figure 6.3 (B), 5 µs later, the phase moved in the opposite direction of the energy in the lens, while it moved the same distance along the direction of the energy in the water. This matches the theoretical expectation as noted in the previous section. Specifically, that since the lens and the water have the same phase velocity in the vicinity of f0 , while the wavevector and the group velocity are in opposite directions in the lens, backward waves are produced accompanied by negative refraction and focusing. time = 2 ms

x2 (cm)

5

5 μs later

5

4

4

0

0

5

0

A

B -5

-5

0

x1 (cm)

5

-5

0

5

x1 (cm)

Figure 6.3: (A) Enlarged view of the last snapshot in the right column of Figure 6.2 to better illustrate the field distribution in the periodic structure. (B) Enlarged view 5 µs after (A). This corresponds to just less than 1/5 of wave period at 37.65 kHz. It can be seen that there is a forward phase shift in the water, while it is translated equally backwards in the periodic structure.

To provide a quantitative measure of the effectiveness of the PML, the total instantaneous energy in the physical domain for the two simulations in Figure 4.2 is presented in Figure 6.4 as a function of time. While it is evident that the energy leaves the phys-

76


ical domain faster in homogeneous foam case than for the lens case, the energy profiles decrease continuously following the excitation, indicating no significant reflection from the PML.

()

100

10

1

10

2

10

3

Foam only Periodic structure

0

2

4

( )

6

8

10

Figure 6.4: The normalized total energy, as given by (5.15) in fluids and (5.16) in solids, in the whole physical domain for the simulation shown in Figure 6.2.

Lens extended into the PML The PML technique is based on an underling assumption that the medium in the PML is homogeneous along the direction of the stretched coordinate, as discussed earlier in this thesis. Theoretically, this implies that the PML would not act as a perfect absorber if the periodic structure is extended into PML region. Nevertheless, encouraged by the electromagnetic results reported by Oskooi et al. [73] which showed that extending a photonic crystal into PML is practically possible, we decided to repeat our simulations as described in the previous section but with the lens extended into the PML. The results are presented in Figure 6.5 clearly shows that the solution ”blows up” marking a

77


limitation of the PMLs in this case. time = 0.1 ms

time = 0.2 ms

15.5

x2 (cm)

1 ⇥ 106

1

4 0

0

0

A

B -1 0

12.5

x1 (cm)

0

12.5

1 ⇥ 106

x1 (cm)

Figure 6.5: When the metamaterials is extended into the PML region, the solution becomes unstable early in the simulation. Note that the color map scale in (B) is six orders of magnitude greater than in other snapshots. This marks a limitation of the PML.

Lens terminated at the edge of the PML Finally, it should be noted that since we could not extend the lens inside the PML, it would be convenient to test if the lens could be placed at the inner edge of the PML without causing instability or significant reflection into the physical domain. To test this possibility, the physical domain of the simulations in the right column in Figure 6.2 was shrunk such that the edge of the (8× 50 cells) lens is at the edge of the PML as shown in the snapshot inside Figure 6.6. The way the total energy evolves in time within the lens, for this case, E2 and the original case, E1 , are both presented in this figure. The deviation between the two energies is three orders of magnitude less that maximum energy. Hence, it is appropriate for the periodic metamaterials to be placed at the edge of the PML: this should save computational resources by shrinking the computational domain and thereby avoiding possible distortion from the edges of the lens.


78

Figure 6.6: The same simulation as in the right column of Figure 6.2 was repeated but with a smaller physical domain that places the edge of the periodic structure at the inner edge of the PML, as shown by the snapshot insert. The normalized total energy in the metamaterials lens for this case, E2 , is compared to energy in the lens for the simulations in Figure 6.2, E1 . The lower plot shows the deviation between E1 and E2 .

6.4

Summary

The time-domain PML models introduced in this thesis were used to model the wave propagation in an infinitely long solid slab surrounded by water. By inserting a previously designed periodic structure into part of the solid slab, a phononic crystal metamaterials lens was produced. The transient response of this lens due to a line source in


79

the water was simulated to show focusing of the source on the other side of the lens and other theoretically predicted phenomena of acoustic metamaterials. When assuming that the phononic crystal slab is infinite by extending the periodic structure into the PML, severe instability occurs in the simulations marking a limitation of PMLs. A discussion concerning this and other limitations of the PML together with possible future work for improvement is presented in the final chapter.

Chapter 7 Summary and conclusions Described in this thesis are new and effective second-order time-domain PML formulations for modeling wave propagation in an unbounded solid, fluid, and coupled fluid-solid media. These formulations are used to model the transient response of acoustic metamaterials, isotropic and anisotropic solids and unbounded fluid-solid structures. Also addressed in this thesis are issues concerned with the computational stability of PMLs.

7.1

Summary

Motivated by the need to simulate and test the response of proposed designs for metamaterial acoustic lenses, the first chapter reviews the current state of the PMLs methods for modeling wave propagation. It was pointed out that there is a need for improvements in the fields of formulation, stability, and inhomogeneity of PMLs. In chapter 2, following a brief theoretical review of mechanical waves propagation in fluids and solids, the method of complex coordinate stretching for creating PML’s with the desired properties is described. The essence of the contributions of this thesis is presented in the subsequent chapters. Chapter 3 addresses the problem of 2D wave propagation in an unbounded linear

80

Chapter 7. Summary and conclusions

81

anisotropic solid using a PML approach. A time-domain second-order PDE has been derived using complex coordinate stretching. An advantage of this PML formulation is the small number of equations. Specifically, two second-order equations along with four auxiliary equations which is the smallest number so far reported to describe wave propagation in solids using a time-domain PML formulation. It is argued that this simplifies the problem and reduces the computational resources needed. In chapter 4, a stabilization method is proposed to improve the stability of the discrete PML problem for a wide range of otherwise unstable anisotropic elastic media. This was achieved by increasing the value of the scaling parameter sufficiently to move the unstable modes out of the discretely resolved range of spatial frequencies. The applicability of this proposed method was demonstrated for different finite element methods. Chapter 5 introduces a new time-domain PML formulation for terminating a region consisting of fluid and solid domains close to their interface. For the fluid domain, a PML formulation consisting of a single second-order pressure equation along with two auxiliary equations was proposed. The second-order PML formulation, as previously presented in chapter 3, was used for the solid domain. Coupling boundary conditions for the fluid-solid interface were carefully derived for both the physical domain and the PML region, without affecting the perfect matching characteristics. These boundary conditions were chosen such that they can be directly integrated into a weak formulation of the complete fluid-solid problem and which can be readily used in a FEM analysis. Numerical simulations showed the validity of the formulation and the effectiveness of the PML in absorbing bulk waves, surface waves, and strong evanescent waves with the help of the scaling parameter. The complexity of phononic crystals metamaterials, their remarkable properties, and the need to accurately simulate their behavior, makes them good candidates for testing the time-domain PML models introduced in this thesis. In chapter 6, the transient


82

response of a phononic crystal metamaterial lens due to a line source in the water was simulated showing the focusing of the energy at the other side of the lens as well as other theoretically predicted phenomena of acoustic metamaterials. Assuming the lens to be infinite, it was shown that by extending the periodic structure into the PML, severe instability occurs thereby identifying a limitation of PMLs. This behavior was expected and will be further discussed in section 7.3 following a brief summary of the contributions of this thesis.

7.2

Thesis contributions

• Introduction of an effective second-order time-domain PML formulation for elastic wave propagation in 2D anisotropic solids. To the best of our knowledge it is the formulation with the least number of equations. The formulation was verified and was shown to be more robust than classical first-order formulations. • Proving that increasing the scaling parameter of the complex stretch function shifts all modes, including the unstable ones, to a higher wavevectors domain. It was also shown that the modes corresponding to the low wavevector domain have negative or very low positive imaginary parts making them more stable. • Proposing that increasing the scaling parameter can be used to improve the discrete stability of the PML. This was shown to be true for different finite elements types and orders. • Introduction of a second-order time-domain PML formulation for 2D acoustic wave propagation in fluids with a small number of equations. • First PML formulation for the fluid-solid coupled problem in which the interface in both the physical domain and in the PML are properly specified. Such a formulation


83

is needed in many fields including marine seismology and biomedical ultrasound. • Showing that by only increasing the scaling parameter, a clear improvement is achieved in evanescent wave absorption in a 2D discrete setting. The improvement was previously predicted but only for analytical 1D cases. • Demonstration of the application of the new formulation to predict the transient response of a solid phononic structure consisting of a superfocusing acoustic lens.

7.3

Limitations and future work

PML formulations While they compactly preserve the second-order forms of the original problems, all the PML formulations introduced in this thesis are for wave propagation in 2D space. The fact that the space is only stretched in 2D makes it easier to obtain PDEs that are second-order in time. Hence, it might be difficult to extend the same methodology to obtain 3D formulations (as the reviewer of one of our papers stated). However, our initial assessment indicates that the same methodology can be used to develop 3D formulations. Specifically, we have already performed an extension to 3D for the acoustic equation. The second-order PML formulation has, as expected, three auxiliary equations with an extra cost: the need for the field variables histories (time integrals) in the corner regions of the PMLs. On the other hand, all the PML formulations introduced in this thesis are in Cartesian coordinates. Hence, one of the future directions could be to use the same methodology to obtain time-domain PML formulations in polar, cylindrical, and spherical coordinates.


84

Validity and stability of PML PMLs functionality assumes material homogeneity along the stretched coordinate, which enables analytic continuation of the radiation solution along the new complex coordinate without any reflection. In addition, by requiring the damping parameter to be positive, there is an underlying assumption that all the modes of the radiation solution along the stretching direction correspond to forward waves, thus the geometric stability condition in Definition 4.1. These two assumptions explain many of the validity and stability aspects of the PMLs. The periodic structures of acoustic metamaterials violate both of the above mentioned assumptions: they are inhomogeneous in the stretching direction and they have backward waves over a range of frequencies, hence, the severe instability that was seen in Figure 6.5. For the electromagnetic case and in the first band of the band structure where all the modes correspond to forward waves with a fixed phase velocity, Oskooi et al. [73] showed that extending a photonic crystal into the PML is practically possible in discrete schemes by introducing a smooth high order profile for the damping parameter. They clarified that the absorbing layers are theoretically no longer PMLs. Contrary to their case, we have an elastic wave equation and backward waves are present corresponding to the sixth branch of the band structure. We repeated the simulation in Figure 6.5 with high order polynomials describing the damping parameter as well as with a negative value damping parameter. However, there was no improvement in the results. It would be interesting to investigate whether the encouraging results seen by Oskooi et al. could be achieved for phononic crystals in fluids or solids even for the low frequencies in the first band. For homogeneous materials with backward waves, there would be no problem if all the modes correspond to backward waves. Only the damping parameter would need to be negative and there should be no stability issues. But in reality, only the modes over a range of wavevectors, as seen in chapter 4, or over a range of frequencies, as in the


85

case of homogenized metamaterials, correspond to backward waves. We did introduce a practical solution for the stability of the first type where backward waves occur over a range of wavevectors for a class of anisotropic solid materials and demonstrated the discrete stability improvement using different FEM simulations. While all discrete models are unable to properly resolve components of the solution that are beyond specific spatial frequencies, the behavior of discrete solutions, including their stability, may vary between numerical methods. Hence, the improvement in discrete stability by increasing the scaling parameter needs to be further investigated for different numerical methods and other PML formulations. On the other hand, future investigations could tackle the PML problem for homogenized metamaterials where backward waves correspond to a range of frequencies. Since, the complex coordinate stretching is performed in the frequency domain, and the complex stretch function is a function of frequency, the damping parameter could be made frequency dependent with negative values for the backward waves frequency range and positive for the other ranges. This approach worked for a homogeneous electromagnetic metamaterial that is described by the Drude model, as recently reported by Bécache et al. [74], following earlier works on PMLs for metamaterials [75–77]. Once a homogenization model is chosen for the acoustic metamaterials, it would be interesting to test if the same approach would work to formulate a PML for homogenized acoustic metamaterials.

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Appendix A Classical PML stability conditions In their detailed and rigorous work [35], Bécache et al. established the stability conditions for the classical split-field PML formulation for orthotropic solids in term of elasticity coefficients, and also introduced the geometrical stability criterion. This appendix summarizes the results of their work that are relevant to this thesis. Namely, the necessary and sufficient conditions for the stability of the polynomial F1 (ω, k1 , k2 , β1 , 1), which turns out to be the same conditions for our PML formulations as stated in Corollary 4.2. Using the following normalized variables: k = (cos θ, sin θ) , |k|

εj =

βj , |k|

and c (θ, εj ) =

ω (k,βj ) , |k|

(A.1)

the dispersion relation F1 (ω, k1 , k2 , β1 , 1) = 0 can be written in terms of the phase velocity and propagation direction, as

F0 [(c + iε1 ) c, c cos θ, (c + iε1 ) sin θ] = 0.

(A.2)

Bécache et al. [35] argued that for high frequencies (|k| β1 , i.e. ε1 → 0), the roots of (A.2) are a perturbed version of its known roots for the case of ε1 = 0, which are the

96

Appendix A. Classical PML stability conditions

97

four real-valued physical modes that correspond to (2.22) and four zero-valued modes. Hence, they used the perturbation techniques to obtain the following results: 1. The necessary conditions for stability of (A.2) at high frequencies are (a) In the vicinity of the four physical modes, the necessary stability condition is (Theorem 3 in [35]) h

(C12 + C33 )2 − C11 (C22 − C33 )

ih i (C12 + C33 )2 + C33 (C22 − C33 ) 6 0.

(A.3)

This is an equivalent form of the geometric stability condition as stated in Definition 4.1, but in term of the elasticity coefficients. Violating this condition usually leads to more severe instability than violating other condition [15, 29, 35, 53]. For example, material III in Table 2.1 violates this condition in both directions. (b) In the vicinity of the four non-physical modes, it is necessary that the two following inequalities are satisfied (Lemma 4 in [35])

(C12 + 2C33 )2 6 C11 C22 2 (C12 + C33 )2 6 C11 C22 + C33

(A.4a) (A.4b)

for the PML in the x1 direction to be stable. For example, material IV in Table 2.1 violates this condition. Conditions in (a) and (b) above are considered necessary but not sufficient stability conditions since they were obtained for high frequency (ε1 1) case only. 2. If the roots of (A.2) are stable for high frequencies (ε1 → 0), then the sufficient conditions for the stability of the PML would be the ones that guarantee that these roots, c (θ, ε1 ), stay in the “good” side of the complex plane and never meet the real axis as ε1 increases. This is satisfied if and only if both inequality (A.4a) and one of

Appendix A. Classical PML stability conditions

98

the following condition are satisfied (Lemma 5 in [35]):

(C12 + C33 )2 6 (C11 − C33 ) (C22 − C33 ) 2 (C11 + C33 ) (C12 + C33 )2 6 (C11 − C33 ) C11 C22 − C33 .

(A.5a) (A.5b)

The stability conditions for the x2 direction can be obtained simply by permuting C11 and C22 in the inequalities (A.3), (A.4), and (A.5) above.

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