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 propa- gation 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 second- order 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 me- dia 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 envi- ronment: 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
Acknowledgementsv
List of Figures xvi
List of Tables xvi
Nomenclature xvii
1 Introduction1 1.1 Motivation...... 2 1.2 Development of perfectly matched layers...... 4 1.3 Thesis objectives and outline...... 7
2 Theoretical background9 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 2.3 Coupled acoustic-elastic modeling...... 14
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 37 4.1 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 4.2 Numerical FEM results...... 45 4.2.1 Quadratic Lagrange finite element...... 45 4.2.2 Other discretization methods...... 49 4.3 Summary...... 50
5 PML formulation for fluid-solid media 51 5.1 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 5.2 Numerical methods and results...... 58 5.2.1 Validation...... 59 5.2.2 Numerical examples...... 63 5.3 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 96
viii 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 ma- trix 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...... 3 1.2 Illustrating the use of a perfectly matched layer (PML) for achieving near- perfect modeling of the solution to the unbounded wave radiation problem (after Johnson [2] )...... 5
2.1 Slowness curves for all the materials whose properties are given in Ta- ble 2.1. The slowness and group velocity vectors are indicated for selected points...... 18 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...... 21
ix 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 dis- placement 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 snap- shots in (B), (C) and (D) are taken. Note that (B) and (C) have linear scales, while (D) is in dB’s...... 31 3.2 Validation results: the three points , , and 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 . (C) Displacement field at point . (D) Showing both components of the displacement field at point ...... 33 3.3 Showing the evolution of energy in the physical domain, as represented
p 2 2 by u + u ∞ for the isotropic medium (Material I). In (A) the effects k 1 2k of changing the mesh size in (3.11) are investigated. Panel (B) shows the effects of changing the scaling parameter...... 34
x 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)...... 44 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...... 46 4.3 Showing the evolution of energy in the physical domain as represented
p 2 2 by u + u ∞ for scaling parameters of 1, < 1, and > 1. In (A), for k 1 2k 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...... 47 4.4 Propagation snapshots as in Figure 4.2 for Materials III and V, after in-
troducing 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...... 48
p 2 2 4.5 Evolution of energy in the physical domain as represented by u + u ∞ k 1 2k for Material V. The results were obtained by using cubic and quartic La-
grange and Hermite finite elements. For (A), α01 = α02 = 1, while in (B),
α01 = 20 and α02 = 5...... 50
xi 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...... 56 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...... 61 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)...... 62
xii 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...... 64 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...... 66 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...... 67
xiii 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])...... 71 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...... 74 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...... 75 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. 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...... 77 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...... 78
xv List of Tables
2.1 Elasticity coefficients, along with the calculated minimum and the maxi- mum phase velocities for the materials examined...... 14
xvi Nomenclature
List of symbols
th αj(xj) The scaling parameter in the j direction, page 20
th βj(xj) The damping parameter in the j 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 Lam´ecoefficients, page 11
C The set of complex numbers, page 15
R The set of real numbers, page 15
µ(x) The second Lam´ecoefficients, page 11
ω The angular temporal frequency, page 15
xvii φ, φi, ψi, and ψij Test functions, page 58
The real part of a complex value, page 38 <{•}
ρF (x) The fluid mass density, page 10
ρS(x) The fluid mass density, page 11
σij(x, t) Components of the symmetric stress tensor, page 11 x˜ Complex stretched spacial coordinates, page 19
εj βj relative to the wavenumber, page 98
εkl(x, t) Components of the symmetric strain tensor, page 11
R t ςij(x, t) 0 σij(x, τ) dτ, page 26 a Lattice constant, page 70 a The radius of a vibrating cylinder, page 32
2 Aj Components of the polarization vector: A C , page 15 ∈
α1α2 β1β2 aj(x) βj , page 27 αj βj − b(x) β1(x1) + β2(x2), page 27 c Phase velocity, or wave’s characteristic speed, page 16
c(x) β1(x1) β2(x2), page 27
cP , cS Phase velocities for primary and secondary waves in solids, page 17
cmax The maximum phase velocity in all directions, page 19
cmin The minimum phase velocity in all directions, page 19
xviii Cijkl(x) Components of the fourth order elasticity tensor, page 11 d The thickness of the PML, page 23
EF (t) Total instantaneous energy in fluids, page 64
ES(t) Total instantaneous energy in solids, page 64
F (x, t) Force term; RHS of acoustic wave equation, page 10
f0 The dominant frequency of the source, page 29
Fi(x, t) Force vector; RHS of elastic wave equation, page 12
h0 The maximum mesh size, page 29 i The imaginary unit: i2 = 1, page 15 − i, j, k, l Indices for the two dimensional space, page 9
K(x) The adiabatic bulk modulus, page 10
2 kj Components of wavevector: k R , page 16 ∈
kj Kj = |k| Components of the propagation direction:(cos θ, sin θ), page 16 m, n The orders of the polynomial functions, page 23
nFj Outward normal unit vector to fluids boundaries, page 15
nSj Outward normal unit vector to solids boundaries, page 15
R t P (x, t) 0 p(x, τ) dτ, page 55 p(x, t) The deviation from the ambient pressure, page 10
th Rj The amplitude reflection coefficient in j direction, page 23
xix kj Sj = ω Components of the slowness vector, page 18
∂x˜j sj (xj) The complex stretch function: , page 20 ∂xj
t Time variable, page 10
t0 The delay time of the source, page 29
ts Time step, page 30
ui(x, t) Components of particle displacement vector, page 11
u0 (t) The normalized time-dependent displacement source, page 29
Vg Components of group velocity: Vg = kω, page 18 j ∇
vi(x, t) Components of the particle velocity vector, page 10
wij The auxiliary variables in the elastic PML equation, page 27
2 xj Components of space variable vector: x R , page 10 ∈ List of abbreviations
2D Two-dimensional space, page 6
CFS-PML Complex Frequency Shift PML, page 20
FEM Finite Element Method, page iii
GPML Generalized PML, page 20
PDE Partial Differential Equations, page 1
PML Perfectly matched layer, page ii
RHS Right-hand side, page 10
xx Chapter 1
Introduction
Numerical simulations of wave propagation are important for the study of waves and their applications in science, engineering, and medicine. Analytical methods have their limitations, particularly when the propagation medium is inhomogeneous or consists of mixed media (e.g. fluids and solids). Because numerical solutions of partial differential equations (PDE) require that the region of interest be divided into a finite number of nodes, it is generally difficult to simulate the wave behavior in a finite region without em- ploying boundaries that prevent spurious wave reflections back into the computational region. Special methods have been developed to characterize the properties of these boundaries which, ideally, should act as perfect absorbers for waves of any frequency and angles of incidence. This can be achieved by terminating the computational domain with a perfectly matched layer, wherein the waves rapidly decay with negligible reflec- tions, thereby making the PML an effective technique for modeling electromagnetic and mechanical wave phenomena. Propagation in two different types of media, specifically for acoustic waves in fluids and elastic waves in solids, needs careful attention. For both types of waves many PML related studies have been reported in the literature. Neverthe- less, further studies are needed to simplify the method and to better address issues like fluid-solid coupling, surface wave propagation, anisotropy, stability, periodic structures,
1 Chapter 1. Introduction 2 and backward waves in PMLs.
1.1 Motivation
Some of the above mentioned issues, especially periodic structures and backward waves, are of special importance for studying wave propagation in metamaterials. These are artificial materials that are not usually found in nature and which are engineered to provide unique properties over a limited range of frequencies [3]. Metamaterials can be designed to manipulate electromagnetic, acoustic, or elastic waves in ways not normally encountered, enabling new functionalities such as invisibility cloaking based on coordi- nate transformations and equivalent materials [4], as well as for focusing through the use of a metamaterial slab designed to achieve subwavelength imaging [5]. Achieving subwavelength resolution (super-resultion) in ultrasound imaging systems using acoustic metamaterials could potentially be valuable for ultrasound medical diagnostics as well as for improving the resolution of acoustic microscopy. A metamaterials lens can be realized through the use of periodically structured inclusions of one or more different materials in- serted in a host, or background material. An example is that shown in Figure 1.1 (A) [1] where the lens consists of a periodic array of unit cells. Inclusion and host can be either fluids or solids, sometimes, the resulting material is described as acoustic or elastic due to the fact that its host is fluid or solid respectively. In order to test the focusing properties of a proposed metamaterials lens, prior to fabrication, numerical simulations are needed. In the case of the lens proposed by Zhou et al. [1], their frequency domain simulation is shown in Figure 1.1 (B). As can be seen, for a point source placed very close to the surface of the lens, the resulting image indicates that the lateral spatial resolution is 0.41λ, where λ is the wavelength corresponding to the operating frequency in water. Our interest in such lenses persuaded us to develop the numerical tools for enabling simulation of the transient performance of lenses which Chapter 1. Introduction 3 could contain both fluid and solid media.
Lens
A 1 mm B
5 mm
Figure 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.
Many of the mathematical and numerical models used for modeling the propagation of mechanical waves have significant underlying assumptions and simplifications. These include the assumptions of linearity enabling frequency domain simulations to be per- formed, the use of the pressure acoustic wave equations for both fluids and solids [6], and the use of a large computational domain to avoid reflections from the boundaries. Following our initial motivation to have accurate yet effective simulation tools to test metamaterials lenses, time-domain models are considered. These have the advantage of being suitable for broadband and nonlinear simulations. In addition, some wave phenom- ena that are important in studying metamaterials like group velocity and phase direction can be readily observed in time-domain simulations. Time-domain modeling is generally more complex than that in the frequency-domain. Reducing this complexity by producing compact and effective time-domain models is desirable. Moreover, to accurately model a wide variety of proposed metamaterials designs, both the acoustic and the elastic wave Chapter 1. Introduction 4 equations need to be considered together with the appropriate coupling conditions. Fi- nally, the computational domains need to be terminated by boundaries that are effective over a wide range of frequencies and all possible incident angles. Hence, the proposed models are time-domain PML formulations for modeling mechanical wave propagation in unbounded media. In the next section, a brief review concerning the perfectly matched layers is presented, paying close attention to issues related to this thesis.
1.2 Development of perfectly matched layers
Since their introduction by B´erenger in 1994 [7], perfectly matched layers (PML) have been widely used for simulating wave propagation in unbounded media. Such layers are illustrated in Figure 1.2 for the case of 2D source distribution and inhomogeneities sur- rounded by a portion of the entire homogeneous propagation medium. Spurious wave reflections from the computational domain boundaries are avoided by adding a fictitious layer in which the waves rapidly decay causing negligible reflections regardless of the frequency and incident angle. This makes PML an effective mean for modeling a variety of wave phenomena. After their application to electromagnetic waves, many PML for- mulations have been introduced for acoustic wave propagation in fluids [8–10] and elastic wave propagation in solids [11–17]. B´erengershowed that by adding specific conductivity parameters to Maxwell’s equa- tions, perfect matching and decaying of the propagating waves in the PML could be achieved [7]. An alternative method is to assume that the material contained within the PML is a uniaxial anisotropic media [18–20], generally referred to as the uniaxial PML approach. A third method, with greater generality and flexibility, is the complex coordi- nate stretching approach [21]. In fact, the conductivity parameter introduced by B´erenger [7] can be thought of as a damping parameter in a stretch function that transforms the spatial coordinate in the layer to the complex plane. Subsequently, more parameters Chapter 1. Introduction 5 were added to the complex stretch function, namely, a scaling parameter[22, 23] and a frequency-shift parameter[23], mainly with the aim of making the PML method causal [23]. Although the original PML was subsequently found to be causal [16, 24], other benefits accrued from this new multi-parameter stretch function. These parameters can be used to improve the absorption of the evanescent waves and the absorption at grazing incident angles [25–28], and to improve the stability in the PML for anisotropic elastic media [16, 29, 30].
Figure 1.2: Illustrating the use of a perfectly matched layer (PML) for achieving near-perfect modeling of the solution to the unbounded wave radiation problem (after Johnson [2] ).
Many PML formulations have been introduced for elastic wave propagation [11–17] as well as for general hyperbolic equations [31]. Amongst these, the split-field formu- Chapter 1. Introduction 6 lations usually involves a single parameter stretch function and are typically described by systems of first order equations with double the number of physical equations such as those used by B´erenger[7]. Unsplit field formulations use the physical field variables along with extra auxiliary variables that are typically needed to obtain the time-domain equations from the frequency-domain equations. Mathematically, it has been proven that certain unsplit field PMLs are strongly well-posed, while the split-field type is only weakly well-posed [29, 32, 33]. The use of multi-parameter stretch function usually requires a convolution to obtain a time-domain formulation, leading to the name convolutional PML [34], as used in many unsplit field models. The majority of these formulations use a large number of equations (10 or more for the 2D case) to describe elastic wave propagation in the PML which adversely affects the computational time and resources. Stability is another known issue in PMLs [15, 16, 29, 35–37], especially for some anisotropic solids. Several methods for addressing this problem have been proposed [15, 16, 29]. In par- ticular, by controlling the stretch function parameters and the mesh size in numerical calculations, the discrete stability has been shown to be improved for certain cases where the corresponding continuous problem was unstable [15, 29, 37]. The PML technique is based on an underling assumption that the medium close to the PML is homogeneous. This enables analytic continuation of the radiation solution over the new complex spatial coordinate without any reflection. Nevertheless, since the new space variable in each direction is only a function of the original space variable in the same direction, PMLs can terminate unbounded heterogeneous media if the interface between them is perpendicular to the PML. In fact, PML studies for two heterogeneous fluids [38] and two or more solid media [17, 39] have been reported. But, to the best of our knowledge, no PML formulation has been reported that appropriately terminates the fluid-solid domains near their interface. The two media need to be modeled by different equations with appropriate interface coupling boundary conditions. It should be noted Chapter 1. Introduction 7 that accurate modeling of the fluid-solid coupling requires careful consideration even in the absence of a PML [38, 40–44].
1.3 Thesis objectives and outline
The body of this thesis begins in chapter 2 with a review of the theoretical background needed for understanding the methods presented in the subsequent three chapters. The objectives and outline of these three chapters together with the simulation results of chapter 6, are as follows:
1. To develop an effective second-order time-domain PML formulation for elastic wave propagation in anisotropic solids with a small number of equations. Second-order equations emerge directly from Newton’s second law and are more robust as com- pared to the first order velocity-stress system of equations [37]. Moreover, the second-order equations are more readily implemented in common numerical schemes [26]. Such a formulation is introduced in chapter 3. It consists of just two second- order displacement equations along with four auxiliary equations, which is smaller than any existing formulation, thereby simplifying the problem and reducing the computational burden. Numerical simulations are used to validate the PML for- mulation.
2. To improve the discrete stability of the PML for some anisotropic materials that have stability issues. Although second-order formulations have better stability than classical split-field formulations, PML instabilities are still evident for a class of solid media. To improve the discrete stability for those media, a simple approach is proposed in chapter 4, in which the scaling parameter of the complex stretch function is increased. The applicability of the proposed method is demonstrated by numerical simulations. Chapter 1. Introduction 8
3. To present a rigorous second-order PML formulation for terminating heterogeneous fluid-solid media close to their interface. A second-order time-domain PML for- mulation is proposed in chapter 5 with fluid-solid coupling boundary conditions that can be readily integrated into the weak formulation of the partial differen- tial equations. For numerical schemes like FEM, such a formulation is particularly appropriate. Excluding the auxiliary equations, the number of equations and the second-order form of the original problem is preserved in this formulation. Numer- ical simulations are used to illustrate the effectiveness of the new PML formulation to absorb bulk, evanescent and surface waves.
4. To demonstrate the use of our tools in modeling metamaterials. This is the topic covered in chapter 6, where the transient response of a metamaterial acoustic lens is presented and discussed.
The final chapter summarizes the contributions of this thesis, discusses current limi- tations of PMLs, and proposes future research directions. Chapter 2
Theoretical background
The purpose of this chapter is to provide the background needed for subsequent chapters. Based on the physics, the first two sections briefly describe the mathematical models for wave propagation in linear inviscid fluids and in linear elastic solids, respectively. Accurate modeling of the fluid-solid coupling requires that careful account be taken of the coupling boundary conditions at the fluid-solid interface and these are considered in section 2.3. Many of wave propagation properties for a given problem can be gained by considering plane wave solutions to the corresponding wave equation. These include, among others, the dispersion relation, the slowness curves, the wave polarizations, and the stability of the solutions. The latter is of special importance for this thesis and will be used in chapter 4 to study the stability of PMLs for anisotropic solids. Section 2.4 briefly covers the plane wave analysis, specially for the case of anisotropic solids. Finally, the complex coordinate stretching, which is the most general method to formulate PMLs for given wave equations, is presented in section 2.5. Since just two dimensional space (2D) is considered in this work, all the indices i, j, k, l = 1, 2.
9 Chapter 2. Theoretical background 10
2.1 Acoustic waves in fluids
Assuming that the fluid is inviscid, the state variables are of first order of smallness, and the state changes are adiabatic [45], conservation of mass and conservation of momentum lead, respectively, to: 2 ∂p X ∂vj = K ∂t ∂x − j=1 j (2.1) ∂v 1 ∂p j = , ∂t −ρF ∂xj
2 where t is time, x R is the space variable, vi(x, t) are the components of the fluid’s ∈ particle velocity, p(x, t) is the deviation from the ambient pressure, K(x) is the bulk modulus, and ρF (x) is the fluid mass density. The above system of equations is the first-order form of the acoustic wave equation which consists of three equations for the 2D case. In addition, a second-order scalar wave equation for the pressure field can be obtained from (2.1), namely 2 1 ∂2p X ∂ 1 ∂p = 0. (2.2) K ∂t2 ∂x ρ ∂x − j=1 j F j
This equation requires that both ρF , and K be strictly positive function of x. Moreover, a monopole source can be modeled by adding a load function, F (x, t) to the right-hand side (RHS) of (2.2).
2.2 Elastic waves in solids
Wave propagation in linear elastic solids can be described using Newton’s second law, and Hook’s law along with the linear approximation of the strain. These lead to the following three equations: Chapter 2. Theoretical background 11
2 2 ∂ ui X ∂σij ρ = , (2.3) S ∂t2 ∂x j=1 j
2 X σij = Cijkl εkl, (2.4) k,l=1 1 ∂uk ∂ul εkl = + , (2.5) 2 ∂xl ∂xk
where, ui(x, t) are the components of particle displacement vector, σij(x, t) , and εkl(x, t) are the components of the symmetric stress and strain tensors respectively. Moreover,
ρS(x) is the solid mass density and Cijkl(x) are the components of the fourth order elasticity tensor. The symmetries of the strain and stress tensors enable this elasticity
tensor to have the properties expressed by: Cijkl = Cijlk = Cjikl, and Cijkl = Cklij. Using the linear approximation of the strain, equation (2.5), together with the symmetry properties of the elasticity tensor, Hook’s law in (2.4) can be expressed as
2 X ∂uk σij = Cijkl . (2.6) ∂xl k,l=1
For the 2D problem, the elasticity tensor, Cijkl, has 16 components. Fortunately, due to the symmetries mentioned above, these reduce to six independent parameters, at most. For the special case of isotropic solids, the elasticity tensor can be described by two independent parameters like the Lam´ecoefficients, λ(x) and µ(x). In term of these two coefficients, the elasticity tensor can be written as:
Cijkl = λ δijδkl + µ(δikδjl + δilδjk), (2.7)
where δij is the Kronecker delta function. Chapter 2. Theoretical background 12
2.2.1 Elastic wave equations
Based on the above physics laws, different wave equation formulations for linear elas- tic solid can be derived. By introducing the particle velocity field variable vi(x, t) =
∂ui(x, t)/∂t, a velocity-stress first-order formulation for the wave equation can be ob- tained from (2.3) and (2.6) as
2 ∂vi 1 X ∂σij = ∂t ρ ∂x F j=1 j (2.8) 2 ∂σij X ∂vk = Cijkl . ∂t ∂xl k,l=1
In this formulation, five first-order equations are needed to describe the problem. Namely, two velocity vector components, vi, and four stress tensor components, σij. Due to the symmetry in the stress tensor (σij = σji) these reduce to three components. It should be noted that first-order systems are traditionally used in PML formulations which is not the case in this work. A second-order formulation of the elastic wave equation can be readily obtained by substituting the stress definition from (2.6) into (2.3) yielding
2 2 2 ! ∂ ui X ∂ X ∂uk ρS 2 Cijkl = 0. (2.9) ∂t − ∂xj ∂xl j=1 k,l=1
In this formulation, only two second-order equations are needed to describe the elastody- namic problem in term of the displacement field. The source of energy that excites the elastic medium can be added as a load vector, Fi(x, t) , to the right-hand side (RHS) of
(2.9). Note that, in (2.9), ρS needs to be strictly positive function of x, and Cijkl needs Chapter 2. Theoretical background 13 to be elliptic, i.e., there exist > 0 such that
2 2 X X ηij Cijkl ηkl ηijηij (2.10) ≥ i,j,k,l=1 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
∂u1 σ1 C11 0 0 C12 ∂x1 ∂u1 σ3 0 C33 C33 0 ∂x2 = . (2.11) ∂u2 σ3 0 C33 C33 0 ∂x1 ∂u2 σ2 C21 0 0 C22 ∂x2
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 B´ecache et al. [35], and media V, which was also studied in [16, 29], corresponds to a zinc crystal. Chapter 2. Theoretical background 14
Table 2.1: Elasticity coefficients, along with the calculated minimum and the maximum phase velocities for the materials examined.
Material C11 C22 C33 C12 cmin cmax I 7.8 7.8 2 3.8 1.4 2.8 II 20 20 2 3.8 1.4 4.5 III 4 20 2 7.5 0.78 4.5 IV 10 20 6 2.5 2.4 4.5 V 16.5 6.2 3.96 5 1.8 4.1
2.3 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 Lam´ecoefficient, 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 2 2 X 1 ∂p X ∂ uj n = n , (2.12) Fj ρ ∂x Fj ∂t2 j=1 F j − j=1
where nF is an outward directed unit vector normal to the fluid boundaries. In addition, Chapter 2. Theoretical background 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 nS nS σij = p, (2.13) i j − i,j,k,l=1
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 ∂uk nSj Cijkl = nSi p, (2.14) ∂xl − j,k,l=1
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 charac- teristics 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- Chapter 2. Theoretical background 16 braic equation, known as the Christoffel’s equation, that is given by
2 X 2 ω δik Γik Ak = 0, (2.16) − k=1 where δik is the Kronecker delta function and
2 X Γik(k1, k2) = Cijklkjkl, (2.17) j,l=1 is the symmetric Christoffel’s tensor or operator. Without loss of generality, it was
2 assumed that ρS = 1. Equation (2.16) is an eigenvalue problem, where ω and A are, respectively, the eigenvalue and eigenvector of the Christoffel’s tensor. Dividing (2.16) by the wavenumber squared, k 2 = k2 +k2, yields the same eigenvalue problem but with | | 1 2 ω k1 k2 the phase velocity, c = |k| , instead of ω, and the propagation direction, K = |k| , |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