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.

The elastic wave equation describes the propagation of elastic disturbances produced by seismic events in the Earth or vibrations in plates and beams. The acoustic wave equation governs the propagation of sound. The description of the wave fields resulting from an initial configuration or time dependent forces is a valuable tool when gaining insight into the effects of the layering of the Earth, the propagation of earthquakes or the behavior of underwater sound. In the most general case exact solutions to both the elastic wave equation and the acoustic wave equation are impossible to construct. Numerical methods that produce approximative solutions to the underlaying equations now become valuable tools. In this thesis we construct numerical solvers for the elastic and acoustic wave equations with focus on stability, high order of accuracy, boundary conditions and geometric flexibility. The numerical solvers are used to study wave boundary interactions and effects of curved geometries. We also compare the methods that we have constructed to other methods for the simulation of elastic and acoustic wave motion.

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.

<p> In this thesis, both direct and inverse elastic scattering problems are considered. For a given incident wave, the direct problem is to determine the displacement of wave field from the known structure, which could be an obstacle or a surface in this thesis; The inverse problem is to determine the structure from the measurement of displacement on an artificial boundary.</p><p>In the second chapter, we consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.<br></p><p>In chapter 3, we extend the argument developed in chapter 2 to elastic surface grating problem, where the surface is assumed to be periodic and elastic rigid; Then, we treat the obstacle scattering in three dimensional space; The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Finally, in chapter 4, a rigorous mathematical model and an efficient computational method are proposed to solve the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangle slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near field data conversion and utilized the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem; Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction; Numerical examples are presented to demonstrate the effectiveness of the proposed methods for solving the questions mentioned above.<br></p>

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.

This thesis analyses the governing equation of elastic wave propagation by the finite difference method , and considered absorbing boundary condition and the material damping to simulate behavior of wave propagation. Otherwise, we combined with the mechanics of the soil pore water pressure raised by shear stress effected repeatedly and the soil property is changed by water pressure effected to simulate physical phenomenon in half-space, and probe into the soil liquefaction process during different force types. Using the developed numerical wave propagation model probe into reducing vibration by dug trench and filler trench, and analyzed data by 1/3 octave band method. This thesis discuss with reducing vibration effect by different trench disposed¡Bdifferent filler material property, complex filler, and extending the force source pile length.

We are concerned with the high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveforms of the medium's response to drive the inverse problem. Specifically, we use a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to the proposed inversion schemes. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). The PML is a buffer zone that surrounds the domain of interest, and enforces the decay of outgoing waves. In order to resolve the forward problem, we present a hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. We discuss several time-integration schemes, including an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We report numerical results demonstrating stability and efficacy of the forward wave solver, and also provide examples attesting to the successful reconstruction of the two Lame parameters for both smooth and sharp profiles, using synthetic records. We also report the details of two field experiments, whose records we subsequently used to drive the developed inversion algorithms in order to characterize the sites where the field experiments took place. We contrast the full-waveform-based inverted site profile against a profile obtained using the Spectral-Analysis-of-Surface-Waves (SASW) method, in an attempt to compare our methodology against a widely used concurrent inversion approach. We also compare the inverted profiles, at select locations, with the results of independently performed, invasive, Cone Penetrometer Tests (CPTs). Overall, whether exercised by synthetic or by physical data, the full-waveform inversion method we discuss herein appears quite promising for the robust subsurface imaging of near-surface deposits in support of geotechnical site characterization investigations.

Tampriosios bangos (arba akustinės ar bet kokios kitos bangos) sklidimo tyrimai yra svarbūs tokiose srityse kaip seismologija arba neardantis medžiagos testavimas. Tamprioje srityje šis reiškinys aprašomas tampriosios bangos dinamine diferencialine lygtimi. Tačiau šios lygties sprendimas naudojant tokius skaitinius metodus kaip baigtiniai elementai reikalauja sritį padalinti į milijonus elementų. Naujų skaičiavimo technologijų kaip bendros paskirties grafiniai procesoriai (GPU) atsiradimas skaičiavimų laiką leidžia ženkliai sumažinti, tačiau algoritmai turi būti specialiai pritaikomi. Todėl šiame darbe koncentruojamasi į trumpos tampriosios bangos baigtinių elementų modelio sukūrimą ir algoritmų tobulinimą naudojant GPU bei pagrindinį procesorių (CPU). Lygties integravimui buvo pasirinktas centrinių skirtumų metodo (CSM) schema. Ši integravimo schema buvo modifikuota taip, kad būtų galima išskirti tris integravimo algoritmo etapus: išorinės jėgos įvertinimas, elementų deformacijos sąlygotų jėgų įvertinimas bei magų poslinkių, greičių ir jėgų perskaičiavimas. Remiantis strategija pasiūlyta [1] šaltinyje, buvo sukurti lygiagretūs algoritmai 2 ir 3 etapo skaičiavimams atlikti. Toliau antrojo etapo algoritmas buvo optimizuotas 2 kartus. Pirmiausia buvo atsisakyta elementų mazgų indeksų masyvo: tai skaičiavimo laiką sumažino 20%. Po to algoritmas buvo modifikuotas taip, kad elementus būtų galima apdoroti blokais kaip siūloma [12] ir [22] šaltiniuose. Skaičiavimo laiką tai leido... [toliau žr. visą tekstą] / Understanding elastic wave (or acoustic or any other type of wave for that matter) phenomenon is of great importance in areas such as seismology or non destructive testing (NDT). This phenomenon in case of elastic environment is described by dynamic elastic differential equations. However, computational models like finite element method consumes huge amounts of computational power as even for relatively small problems require dividing area of interest into millions of elements. In the advent of general purpose GPU computing new opportunities for speeding up computations as well as challenges for developing high performance algorithms suited for new kinds of processors arise. Therefore this work concentrates on developing a finite element based short elastic wave propagation model on GPU as well as CPU. Central difference explicit wave equation integration scheme has been chosen. It then was slightly modified in order to separate integration algorithm into three phases: external force evaluation, evaluation of forces that occur due to stresses of elements and recalculation of node shifts, speeds and forces. A parallel algorithm has been developed for executing third and seconds phases, based on strategy suggested in [1]. Then the algorithm of the second phase has been optimized 2 times: at first the array of element node indices was eliminated yielding 20% performance boost, then modifications have been made to process elements in blocks by using strategy described at [22]... [to full text]