The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations. / Ph. D.

<p> This thesis proposes to demonstrate, by means of numerieal examples, the applicability of the approximate solution for shallow, spherical, calotte shells enclosing polygonal bases for the purposes of practical design.</p> <p> The theoretical solution is based on a collocation procedure by means of which prescribed boundary conditions are satisfied at discrete boundary points and is derived from the general theory of MUSHTARI and VLASOV in which the transverse shear deformation of the shell is neglected in comparison with its transverse bending and extensional surface deformation.</p> / Thesis / Master of Engineering (MEngr)

The focus of this dissertation is analyzing and manipulating acoustic wave propagation in metamaterials, which can be used to assist the design of acoustic devices. Metamaterials are artificial materials, which are arranged in certain patterns at a scale smaller than the wavelength and can exhibit properties beyond those naturally occurring materials. With metamaterials, novel phenomena, such as focusing, super absorption, cloaking and localization of ultrasound, are theoretically proposed and experimentally verified. In recent years, a planar version of metamaterials, often called meta-surfaces, has attracted a great deal of attention. Meta-surfaces can control and manipulate the amplitude, phase, and directions of waves. In this dissertation, we conducted a systematic study by deriving the effective medium theories (EMTs), and developing the theoretical and numerical models for our proposed designed metamaterial. Very recently, acoustic meta-surfaces have been used in the design of acoustic lenses, which can achieve various functionalities such as focusing and collimation. In the designs of acoustic lenses, impedance is an important issue because it is usually difficult to make the impedance of the lens equal to that of the environment, and mismatched impedance is detrimental to the performance of the acoustic lens. We developed an EMT based on a coupled-mode theory and transfer matrix method to characterize the propagation behavior and, based on these models, we report two designs of acoustic lenses in water and air, respectively. They are rigid thin plates decorated with periodically distributed sub-wavelength slits. The building block of the acoustic lens in water is constructed from coiling-up spaces, and that of the acoustic lens in air is made of layered structures. We demonstrate that the impedances of the lenses are indeed matched to those of the background media. With these impedance-matched acoustic lenses, we demonstrate acoustic focusing and collimation, and redirection of transmitted acoustic energy by finite-element simulations. In the framework of the hidden source of the volume principle, an EMT for a coupled resonator structure is derived, which shows that coupled resonators are characterized by a negative value of the effective bulk modulus near the resonance frequency and induce flat bands that give rise to the confinement of the incoming wave inside the resonators. The leakage of sound waves in a resonance-based rainbow trapping device prevents the sound wave from being trapped at a specific location. Based on our EMT, we report a sound trapping device design based on coupled Helmholtz resonators, loaded to an air waveguide, to effectively tackle the wave leakage issue. We show that a coupled resonators structure can generate dips in the transmission spectrum by an analytical model derived from Newton’s second law and a numerical analysis based on the finite-element method. We compute the transmission spectra and band diagram from the effective medium theory, which are consistent with the simulation results. Trapping and the high absorption of sound wave energy are demonstrated with our designed device.

The research detailed in this study focuses on the determination of an appropriate objective function to aid parameter estimation when simulating areas influenced by tidally varying flows. Three objective functions that are measures of how well the model results match field data at several locations and times were tested. A set of test cases is developed to represent tidally influenced systems and allow for the testing of the objective functions. These objective functions were tested by computing their values and comparing them for the various estimated parameters. Based on results of the first method of testing a further analysis was performed using PEST, an automatic parameter estimation tool. A weighted least squares of the velocity and water surface values with a weight function on the velocity term based on the shallow water equations is found to be a reasonable objective function at this point in the research.

No description available.

No description available.

No description available.

No description available.

Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.

A novel acoustic sensor has been developed, capable of remotely monitoring the free surface ‘fingerprint’ of shallow flows. Temporal and spatial properties of this pattern are shown to contain information regarding the nature of the flow itself. The remote measurement can thereby be used to infer the bulk flow properties such as depth, velocity, and the hydraulic roughness of the pipe. The instrument is non-invasive and is also low cost, low maintenance, and low power. Such a device will allow for widespread monitoring of flow conditions in drainage networks, enabling pro-active maintenance and reliable real-time control.