Thesis (Ph. D.)--California Institute of Technology, 1977. UM #77-24,050. / Advisor names found in the Acknowledgments pages of the thesis. Title from home page (viewed 03/09/2010). Includes bibliographical references.

The time-dependent elastic wave equation is the foundation of seismology. It is very useful in studying the wave propagation in elastic solids. Simulation of Rayleigh waves, which is governed by the equation, requires high accuracy solutions. Finite difference method have been widely used for the simulation of Rayleigh waves. However, it is not obvious how to effectively impose the free surface boundary condition on a curved surface. On the other hand, discontinuous Galerkin methods are more flexible in handling complex geometries. / In this thesis, we develop a class of discontinuous Galerkin methods for time-dependentelastic wave equation in isotropic homogeneous medium. This method is explicit, locally and globally energy conserving. Also, the L² convergence of this method is optimal and the convergence in energy norm is independent of Poisson's ratio. / Besides, we apply our method to simulate Rayleigh waves on curved free surfaces. We also impose a perfectly matched layer to absorb the outward waves. Numerical examples show that our method can accurately capture features of Rayleigh waves even in a domain with high Poisson's ratio. / 時間依賴型彈性波動方程」是地震學的基礎。這組方程對於波在彈性固體中傳播的研究非常有用。雷利波是由這個方程所控制。模擬雷利波須要有非常準確的解。有限差分法廣泛地應用在雷利波的模擬上，可是如何有效地施加自由表面邊界條件於曲面上的方法並不明顯。另一方面，間斷伽遼金方法能更靈活地處理複雜的幾何形狀。 / 在這篇論文中，我們發展了一類間斷伽遼金方法去求「在均勻各向同性的介質上的時間依賴型彈性波動方程」的解。我們將表明，這種方法是顯式的，局部及全域能量守恆的，而它的收斂是最優的和獨立於泊松比的。 / 除此之外，我們運用這個方法來模擬雷利波在具有起伏的自由表面的半空間模型的傳播。我們會使用完美匹配層去吸收朝外的波動。數值算例反映，即使在高柏松比的介質中，這個方法也可以準確地捕捉雷利波的特徵。 / Lam, Chi Yeung. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 44-47). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Bloch analysis was originally developed by Felix Bloch to solve Schrödinger's equation for the electron wave function in a periodic potential field, such as that found in a pristine crystalline solid. His method has since been adapted to study elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In this thesis, we detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. We show conclusively that translational invariance is not a proper justification for invoking the existence of a "propagation constant," and that in nonlinear media this results in a flawed analysis. Next, we propose a general framework for applying Bloch analysis in damped systems and investigate the effect of damping on dispersion curves. In the context of Schrödinger's equation, damping is absent and energy is conserved. In the damped setting, application of Bloch analysis is not straight-forward and requires additional considerations in order to obtain valid results. Results are presented in which the approach is applied to example structures. These results reveal that damping may introduce wavenumber band gaps and bending of dispersion curves such that two or more temporal frequencies exist for each dispersion curve and wavenumber. We close the thesis by deriving conditions which predict the number of wavevectors at each frequency in a dispersion relation. This has important implications for the number of nearest neighbor interactions that must be included in a model in order to obtain dispersion predictions which match experiment.

The analysis of wave propagation in solids with complex microstructures, and local heterogeneities finds extensive applications in areas such as material characterization, structural health monitoring (SHM), and metamaterial design. Within continuum mechanics, sources of heterogeneities are typically associated to localized defects in structural components, or to periodic microstructures in phononic crystals and acoustic metamaterials. Numerical analysis often requires computational meshes which are refined enough to resolve the wavelengths of deformation and to properly capture the fine geometrical features of the heterogeneities. It is common for the size of the microstructure to be small compared to the dimensions of the structural component under investigation, which suggests multiscale analysis as an effective approach to minimize computational costs while retaining predictive accuracy. This research proposes a multiscale framework for the efficient analysis of the dynamic behavior of heterogeneous solids. The developed methodology, called Geometric Multiscale Finite Element Method (GMsFEM), is based on the formulation of multi-node elements with numerically computed shape functions. Such shape functions are capable to explicitly model the geometry of heterogeneities at sub-elemental length scales, and are computed to automatically satisfy compatibility of the solution across the boundaries of adjacent elements. Numerical examples illustrate the approach and validate it through comparison with available analytical and numerical solutions. The developed methodology is then applied to the analysis of periodic media, structural lattices, and phononic crystal structures. Finally, GMsFEM is exploited to study the interaction of guided elastic waves and defects in plate structures.

Thesis (M. S.)--Mechanical Engineering, Georgia Institute of Technology, 2010. / Committee Chair: Dr. Michael Leamy; Committee Member: Dr. Karim Sabra; Committee Member: Dr. Aldo Ferri. Part of the SMARTech Electronic Thesis and Dissertation Collection.

D. Tech. Mathematical Technology. / Investigates hyperbolic and pseudohyperbolic equations and the results are applied to higher-order rod approximations for the propagation of the longitudinal stress waves in elastic rods. The main objectives of this thesis are: 1. Provide a unified approach to the derivation of the families of one-dimensional hyperbolic differential equations simultaneously with the associated natural and essential boundary conditions describing longitudinal vibration of finite length rods. 2. Establish a new theoremto shorten the derivation of equations of motion and the corresponding boundary conditions, modelling longitudinal wave propagation in the rod. 3. Prove that, when deriving the higher-order rod equations, the lower-order are still included, thus increasing the number of deformations in the rod or the accuracy of the model. 4. Provide mathematical tools for the classification of the obtained equations. 5. Compare the accuracy of the above-mentioned vibration theories in elastic rods based on the investigation of their frequency spectrums which are not available in the literature. 6. Show how two of the above vibration theories, the Rayleigh-Bishop and Mindlin-Herrmann theories, can be applied to predict wave propagation in a piezoelectric circular cylinder and isotropic conical rod. In both cases a numerical example is given as a simulation of the solution.7. Find general methods for solving problems of longitudinal vibration of finite length rods for all of the above-mentioned theories.

This study extends a recently-developed [1] cellular automata (CA) elastodynamic modeling approach to arbitrary two-dimensional geometries through development of a rule set appropriate for triangular cells. The approach is fully object-oriented (OO) and exploits OO conventions to produce compact, general, and easily-extended CA classes. Meshes composed of triangular cells allow the elastodynamic response of arbitrary two-dimensional geometries to be computed accurately and efficiently. As in the previous rectangular CA method, each cell represents a state machine which updates in a stepped-manner using a local "bottom-up" rule set and state input from neighboring cells. The approach avoids the need to develop partial differential equations and the complexity therein. Several advantages result from the method's discrete, local and object-oriented nature, including the ability to compute on a massively-parallel basis and to easily add or subtract cells in a multi-resolution manner. The extended approach is used to generate the elastodynamic responses of a variety of general geometries and loading cases (Dirichlet and Nuemann), which are compared to previous results and/or comparison results generated using the commercial finite element code, COMSOL. These include harmonic interior domain loading, uniform boundary traction, and ramped boundary displacement. Favorable results are reported in all cases, with the CA approach requiring fewer degrees of freedom to achieve similar or better accuracy, and considerably less code development.

Les phénomènes de diffraction et de diffusion des ondes jouent un rôle important dans l'interprétation de la coda des sismogrammes. Par conséquent, une compréhension approfondie des mécanismes de diffraction et de leurs influences sur la propagation des ondes est une étape fondamentale vers l'identification des propriétés statistiques d'un milieu aléatoire. Cette thèse porte sur la diffraction des ondes élastiques dans des milieux aléatoirement hétérogènes avec un comportement local isotrope. On s'intéresse au régime où: La longueur d'onde est du même ordre de grandeur que la longueur de corrélation, la longueur d'onde est petite comparé à la distance de propagation (haute-fréquence) et l'amplitude des fluctuations est petite. Une approche cinétique basée sur les équations de transfert radiatif des ondes élastiques est adoptée. La première partie de cette thèse décrit une analyse détaillée de l'influence de la structure de corrélation sur les paramètres de diffraction et sur l'établissement d'un régime de diffusion. La seconde partie présente les simulations éléments spectraux à grande échelle des ondes élastiques afin d'observer numériquement l'apparition d'un régime d'équipartition. Des analyses théoriques ainsi que des simulations montrent également une nouvelle approche pour l'identification des propriétés statistiques du milieu. / Scattering and diffusion phenomena play a crucial role in the interpretation of the coda part ofseismograms. Consequently, a profound understanding of scattering mechanisms and their effectson wave propagation is a fundamental step towards the identification of the statistical propertiesof random media. The focus of this work is on the scattering of elastic waves in a randomly heterogeneousmedia with locally isotropic material behavior. The weakly heterogeneous regime isconsidered, in which the wave length is similar to the correlation length, the wave length is smallcompared to the propagation length (high frequency) and the amplitude of the heterogeneities issmall. A kinetic framework based on the transport equations of elastic waves is adopted. Thefirst part of the thesis describes a detailed analysis of the influence of the correlation structure onthe scattering parameters and on the arising of the diffusion regime. The second part presentslarge scale spectral element simulations of elastic waves to observe numerically the onset of theequipartitioning regime. The theoretical analyses and simulations also reveal a novel approach toidentify local properties of the heterogeneous medium.

The present research is concerned with developing analysis methods for analyzing and exploring finite-amplitude elastic wave propagation through periodic media. Periodic arrangements of materials with high acoustic impedance contrasts can be employed to control wave propagation. These systems are often termed phononic crystals or metamaterials, depending on the specific design and purpose. Design of these systems usually relies on computation and analysis of dispersion band structures which contain information about wave propagation speed and direction. The location and influence of complete (and partial) band gaps is a particularly interesting characteristic. Wave propagation is prohibited for frequencies that correspond to band gaps; thus, periodic systems behave as filters, wave guides, and lenses at certain frequencies. Controlling these behaviors has typically been limited to the manufacturing stage or the application of external stimuli to distort material configurations. The inclusion of nonlinear elements in periodic unit cells offers an option for passive tuning of the dispersion band structure through amplitude-dependence. Hence, dispersion analysis methods which may be utilized in the design of nonlinear phononic crystals and metamaterials are required. The approach taken herein utilizes Bloch wave-based perturbation analysis methods for obtaining closed-form expressions for dispersion amplitude-dependence. The influence of material and geometric nonlinearities on the dispersion relationship is investigated. It is shown that dispersion shifts result from both self-action (monochromatic excitation) and wave-interaction (multi-frequency excitation), the latter enabling dynamic anisotropy in periodic media. A particularly novel aspect of this work is the ease with which band structures of discretized systems may be analyzed. This connection enables topology optimization of unit cells with nonlinear elements. Several important periodic systems are considered including monoatomic lattices, multilayer materials, and plane stress matrix-inclusion configurations. The analysis methods are further developed into a procedure which can be implemented numerically with existing finite-element analysis software for analyzing geometrically-complex materials.

This work is a numerical and analytical study of wave motion through dynamic materials (DM). This work focuses on showing several results that greatly extend the applicability of the checkerboard focusing effect. First, it is shown that it is possible to simultaneously focus dilatation and shear waves propagating through a linear elastic checkerboard structure. Next, it is shown that the focusing effect found for the original €œperfect€� checkerboard extends to the case of the checkerboard with smooth transitions between materials, this is termed a functionally graded (FG) checkerboard. With the additional assumption of a linear transition region, it is shown that there is a region of existence for limit cycles that takes the shape of a parallelogram in (m,n)-space. Similar to the perfect case, this is termed a €œplateau€� region. This shows that the robustness of the characteristic focusing effect is preserved even when the interfaces between materials are relaxed. Lastly, by using finite volume methods with limiting and adaptive mesh refinement, it is shown that energy accumulation is present for the functionally graded checkerboard as well as for the checkerboard with non-matching wave impedances. The main contribution of this work was to show that the characteristic focusing effect is highly robust and exists even under much more general assumptions than originally made. Furthermore, it provides a tool to assist future material engineers in constructing such structures. To this effect, exact bounds are given regarding how much the original perfect checkerboard structure can be spoiled before losing the expected characteristic focusing behavior.