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Time-Domain Modeling of Elastic and Acoustic Wave Propagation in Unbounded Media, with Application to Metamaterials

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Time-Domain Modeling of Elastic and Acoustic Wave Propagation in Unbounded Media, with Application to Metamaterials 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 , Á, 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 Á. (D) Showing both components of the displacement field at point Â....................... 33 3.3 Showing the evolution of energy in the physical domain, as represented p 2 2 by u
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