A recently introduced nonlocal peridynamic theory removes the obstacles present in classical continuum mechanics that limit the prediction of crack initiation and growth in materials. It is also applicable at different length scales. This study presents an alternative approach for the derivation of peridynamic equations of motion based on the principle of virtual work. It also presents solutions for the longitudinal vibration of a bar subjected to an initial stretch, propagation of a pre-existing crack in a plate subjected to velocity boundary conditions, and crack initiation and growth in a plate with a circular cutout. Furthermore, damage growth in composites involves complex and progressive failure modes. Current computational tools are incapable of predicting failure in composite materials mainly due to their mathematical structure. However, the peridynamic theory removes these obstacles by taking into account non-local interactions between material points. Hence, an application of the peridynamic theory to predict how damage propagates in fiber reinforced composite materials subjected to mechanical and thermal loading conditions is presented. Finally, an analysis approach based on a merger of the finite element method and the peridynamic theory is proposed. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This proposed approach demonstrates the capability of the PD approach to assess the durability of complex composite structures.

Thesis advisor: Krzysztof Kempa / Nonlocal effects, the wavenumber dependence in a medium's response to external disturbance, is treated in this thesis. Numerical computation methods to include nonlocal effects in plasmonic nanostructures’ electromagnetic response are discussed, and applications of plasmonics to a few other fields are elaborated. First, a computation scheme is proposed to extend conventional finite-difference time-domain (FDTD) methods to nonlocal domain. An effective film whose response is derived from Feibelman's d-function formalism is to replace the highly non-uniform metal surfaces in simulations. It successfully produces numerical results of plasmonic resonance shift and field enhancement which agrees with the experimental data to first order. This scheme is still classical, thus very fast compared to the other first principle quantum methods such as density functional theory. Then electron's scattering rate in an effective medium with plasmonic nanostructures embedded-in, in random phase approximation, is developed, with the wavenumber dependence in the medium’s response accounted. Utilizing this calculation scheme of electron’s scattering rate, further specific applications are following. We show by simulation of the plasmonic nanostructures and calculation of the electron scattering rates that hot-electron plasmon-protection (HELPP) effects can protect the extra energy of hot electrons from being dissipated as heat. This can be a prototype of the 3rd generation solar cells. In another application, we investigate the electron polar-optical-phonon (POP) scattering in heavily-doped semiconductors when plasmonic nanostructures are embedded-in. We show that electron-POP scattering can be significantly suppressed compared to that of bulk semiconductors. In the third application, we propose the plasmonic multiple exciton generation (PMEG) scheme, with simulations and calculations, showing that the efficiency of multiple exciton generation in conventional semiconductors could be enhanced significantly with proper designed plasmonic nanostructures embedded-in or attached-adjacent. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Thesis advisor: Krzysztof Kempa / In this work, we first examine nonlocal behavior in plasmonic systems and develop or expand upon models that enable calculation of higher-order, nonlocal responses for systems with novel geometries. The effects of nonlocality, i.e., spatial dispersion, are prominent in nanostuctures with small feature sizes, and accurate calculations of the nonlocal response of nanostructures are increasingly important for the study of novel physics at the nanoscale. Next, we consider a specific biological system, double-stranded DNA, and investigate the nonlocal and nonlinear model that describes its dynamics. We consider the regime of strong driving with THz radiation and study the parameter-space where molecular damage occurs, motivated by the prospect of using selective damage for potential novel therapies. In a related study, we also consider the possibility of generating THz radiation through the nonlinear, difference-frequency response of a plasmonic system. Plasmonic difference-frequency generation could enable deep penetration of THz signals into the body and, therefore, these projects are intimately connected. Ultimately, these two regimes of behavior, nonlocality and nonlinearity, represent rich areas for applicable research. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

This study presents peridynamic field equations for mechanical deformation, thermal diffusion, moisture diffusion, electric potential distribution, porous flow and atomic diffusion in either an uncoupled or a coupled manner. It is a nonlocal theory with an internal length parameter. Therefore, it can capture physical phenomenon for the problems which include non-local effects and are not suitable for classical theories. Moreover, governing equations of peridynamics are based on integro-differential equations which permits the determination of the field variable in spite of discontinuities. Inherent with the nonlocal formulations, the imposition of the boundary conditions requires volume constraints. This study also describes the implementation of the essential and natural boundary conditions, and demonstrates the accuracy of their implementation. Solutions coupled field problems concerning plastic deformations, thermomechanics, hygrothermomechanics, hydraulic fracturing, thermal cracking of fuel pellet and electromigration are constructed. Their comparisons with the finite element predictions establish the validity of the PD field equations for coupled field analysis.

Master of Science / Department of Mathematics / Bacim Alali / Nonlocal vector calculus, introduced in generalizes differential operators' calculus to nonlocal calculus of integral operators. Nonlocal vector calculus has been applied to many fields including peridynamics, nonlocal diffusion, and image analysis. In this report, we present a vector calculus for nonlocal operators such as a nonlocal divergence, a nonlocal gradient, and a nonlocal Laplacian. In Chapter 1, we review the local (differential) divergence, gradient, and Laplacian operators. In addition, we discuss their adjoints, the divergence theorem, Green's identities, and integration by parts. In Chapter 2, we define nonlocal analogues of the divergence and gradient operators, and derive the corresponding adjoint operators. In Chapter 3, we present a nonlocal divergence theorem, nonlocal Green's identities, and integration by parts for nonlocal operators. In Chapter 4, we establish a connection between the local and nonlocal operators. In particular, we show that, for specific integral kernels, the nonlocal operators converge to their local counterparts in the limit of vanishing nonlocality.

e van der Waals (vdW) interactions, or dispersion forces, are crucial in many chem- ical, physical and biological processes and received much attention from developers of density functional theory (DFT) methods. e most popular non-empirical DFT method for treating vdW interactions is the vdW density functional by Dion et al. (vdW-DF). Despite its success, vdW-DF is not accurate enough for many chemical applications. Here, we investigate two possible ways how to improve its accuracy. First, we reoptimize the only weakly speci ed parameter of vdW-DF for several semi-local functionals. On the S benchmark database set, we nd that revPBE is the best performer, decreasing the error from . % to . %. Second, a system-speci c but very accurate (∼ . kcal/mol) DFT correction scheme is proposed for precise calcula- tions of adsorbent−adsorbate interactions by combining vdW-DF and the empirical DFT/CC correction scheme. e new approach is applied to small molecules (CH , CO , H , H O, N ) interacting with a quartz surface and a lamella of UTL zeolite. e very high accuracy of the new scheme and its relatively easy use and numerical stability compared to the earlier DFT/CC scheme o er a straightforward solution for obtaining reliable predictions of adsorption energies.

This dissertation is about the partial differential equation (PDE) based image denoising models. In particular, we are interested about speckle noise images. We provide the mathematical analysis of existing speckle denoising models and propose three new models based on total variation minimization methods. The first model is developed using a new speckle noise model and the solution of associated numerical scheme is proven to be stable. The second one is a speckle version of Chambolle algorithm and the convergence of the numerical solution was proved under certain assumptions. The final model is a nonlocal PDE based speckle denoising model derived by combining the excellent noise removal properties of the nonlocal means algorithm with the PDE models. We enhanced the computational efficiency of this model by adopting the Split Bregman method. Numerical results of all three models show that they compare favorably to the conventional models.

In this paper, the monotone method is extended to the initial-boundary value problems of nonlocal PDE system of first order, both quasi-monotone and non-monotone. A comparison principle is established, and a monotone scheme is given.

The kinetic effects in an inductively coupled plasma (ICP) due to thermal motion of particles modified by self-consistent magnetic fields are studied by using a particle-in-cell (PIC) simulation. In the low pressure, low frequency regime, electron mean free paths are large relative to device size and the trajectories are strongly curved by the induced radio frequency (RF) magnetic field. This causes problems for linear theories, which ignore the influence of the magnetic field on the particles, and are therefore unable to recover effects accumulated along each nonlinear path.<p>The tools to perform high-performance parallel PIC simulations of inductively coupled plasmas were developed to allow rapid scanning of a broad range of the input parameters, such as wave amplitude, frequency, and plasma temperature. Different behavioural regimes are identified by observing the resultant variations in the skin depth, surface impedance, and ponderomotive force (PMF). At low electron-neutral collision rates, these are shown to include the local collisionless regime, the anomalous skin effect regime, and the nonlinear regime.<p>The local collisionless regime occurs at high driving frequencies and is characterized by plasma behaviour independent of both the driving frequency and amplitude: a short skin depth, low energy absorption, and strong PMF. The anomalous skin effect regime occurs at low frequencies and low amplitudes: the plasma varies with driving frequency, but not driving amplitude, the skin depth increases with frequency, the plasma is much more absorptive in the anomalous regime than in the local regime, and the PMF increases with frequency. The nonlinear regime occurs at low frequencies and high amplitudes: the plasma varies with driving amplitude, but not frequency, the skin depth decreases with amplitude, there is low energy absorption, and the PMF increases with wave amplitude.<p>The simulation runs in four modes: linear collisionless, linear collisional, nonlinear collisionless, and nonlinear collisional. The linear modes, in which the particles ignore the magnetic field, are used to validate the results against theory, while the nonlinear modes are used to test actual plasma behaviour. In linear collisionless mode, the plasma was found to exhibit only the local collisionless and anomalous skin effect regimes, as expected by theories. In nonlinear collisionless mode, the plasma exhibits the nonlinear regime in addition to the regimes found in linear mode. Finally, the nonlinear regime disappears in nonlinear collisionless mode because the curved paths caused by the magnetic field are disrupted by collisions.<p>Finally, the regime boundaries are investigated as a function of temperature. Since the plasma properties vary continuously, a boundary exists where two regimes share the same characteristics. From linear theories, it is known that the division between the local collisionless and anomalous skin effect regimes moves to higher frequencies as the plasma temperature is increased. When nonlinear fields are present, this still occurs, but in conjunction with the boundary between the local collisionless and nonlinear regimes moving to higher wave amplitudes. Temperature also effects the boundary between the anomalous skin effect and nonlinear regimes, causing the minimum frequency of the anomalous skin effect regime to be reduced at low wave amplitudes.

The kinetic effects in an inductively coupled plasma (ICP) due to thermal motion of particles modified by self-consistent magnetic fields are studied by using a particle-in-cell (PIC) simulation. In the low pressure, low frequency regime, electron mean free paths are large relative to device size and the trajectories are strongly curved by the induced radio frequency (RF) magnetic field. This causes problems for linear theories, which ignore the influence of the magnetic field on the particles, and are therefore unable to recover effects accumulated along each nonlinear path.<p>The tools to perform high-performance parallel PIC simulations of inductively coupled plasmas were developed to allow rapid scanning of a broad range of the input parameters, such as wave amplitude, frequency, and plasma temperature. Different behavioural regimes are identified by observing the resultant variations in the skin depth, surface impedance, and ponderomotive force (PMF). At low electron-neutral collision rates, these are shown to include the local collisionless regime, the anomalous skin effect regime, and the nonlinear regime.<p>The local collisionless regime occurs at high driving frequencies and is characterized by plasma behaviour independent of both the driving frequency and amplitude: a short skin depth, low energy absorption, and strong PMF. The anomalous skin effect regime occurs at low frequencies and low amplitudes: the plasma varies with driving frequency, but not driving amplitude, the skin depth increases with frequency, the plasma is much more absorptive in the anomalous regime than in the local regime, and the PMF increases with frequency. The nonlinear regime occurs at low frequencies and high amplitudes: the plasma varies with driving amplitude, but not frequency, the skin depth decreases with amplitude, there is low energy absorption, and the PMF increases with wave amplitude.<p>The simulation runs in four modes: linear collisionless, linear collisional, nonlinear collisionless, and nonlinear collisional. The linear modes, in which the particles ignore the magnetic field, are used to validate the results against theory, while the nonlinear modes are used to test actual plasma behaviour. In linear collisionless mode, the plasma was found to exhibit only the local collisionless and anomalous skin effect regimes, as expected by theories. In nonlinear collisionless mode, the plasma exhibits the nonlinear regime in addition to the regimes found in linear mode. Finally, the nonlinear regime disappears in nonlinear collisionless mode because the curved paths caused by the magnetic field are disrupted by collisions.<p>Finally, the regime boundaries are investigated as a function of temperature. Since the plasma properties vary continuously, a boundary exists where two regimes share the same characteristics. From linear theories, it is known that the division between the local collisionless and anomalous skin effect regimes moves to higher frequencies as the plasma temperature is increased. When nonlinear fields are present, this still occurs, but in conjunction with the boundary between the local collisionless and nonlinear regimes moving to higher wave amplitudes. Temperature also effects the boundary between the anomalous skin effect and nonlinear regimes, causing the minimum frequency of the anomalous skin effect regime to be reduced at low wave amplitudes.