Showing papers on "Boundary value problem published in 2008"

Multiple integrals in the calculus of variations

Abstract: Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value problems.- A variational method in the theory of harmonic integrals.- The -Neumann problem on strongly pseudo-convex manifolds.- to parametric Integrals two dimensional problems.- The higher dimensional plateau problems.

Nonlinear fluid dynamics from gravity

Abstract: Black branes in AdS5 appear in a four parameter family labeled by their velocity and temperature. Promoting these parameters to Goldstone modes or collective coordinate fields—arbitrary functions of the coordinates on the boundary of AdS5—we use Einstein's equations together with regularity requirements and boundary conditions to determine their dynamics. The resultant equations turn out to be those of boundary fluid dynamics, with specific values for fluid parameters. Our analysis is perturbative in the boundary derivative expansion but is valid for arbitrary amplitudes. Our work may be regarded as a derivation of the nonlinear equations of boundary fluid dynamics from gravity. As a concrete application we find an explicit expression for the expansion of this fluid stress tensor including terms up to second order in the derivative expansion.

Formulation of the k-w Turbulence Model Revisited

Abstract: This paper presents a reformulated version of the author'sk-ω model of turbulence. Revisions include the addition of just one new closure coefficient and an adjustment to the dependence of eddy viscosity on turbulence properties. The result is a significantly improved model that applies to both boundary layers and free shear flows and that has very little sensitivity to finite freestream boundary conditions on turbulence properties. The improvements to the k-ω model facilitate a significant expansion of its range of applicability. The new model, like preceding versions, provides accurate solutions for mildly separated flows and simple geometries such as that of a backward-facing step. The model's improvement over earlier versions lies in its accuracy for even more complicated separated flows. This paper demonstrates the enhanced capability for supersonic flow into compression corners and a hypersonic shock-wave/ boundary-layer interaction. The excellent agreement is achieved without introducing any compressibility modifications to the turbulence model.

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

Abstract: The equations of motion of the Euler–Bernoulli and Timoshenko beam theories are reformulated using the nonlocal differential constitutive relations of Eringen [International Journal of Engineering Science 10, 1–16 (1972)]. The equations of motion are then used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Numerical results are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies of carbon nanotubes.

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Abstract: We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

Hybrid functionals applied to extended systems

Abstract: We present an overview of the description of structural, thermochemical, and electronic properties of extended systems using several well known hybrid Hartree–Fock/density-functional-theory functionals (PBE0, HSE03, and B3LYP). In addition we address a few aspects of the evaluation of the Hartree–Fock exchange interactions in reciprocal space, relevant to all methods that employ a plane wave basis set and periodic boundary conditions.

The gravity dual of a p-wave superconductor

Abstract: We construct black hole solutions to the Yang-Mills equations in an AdS4-Schwarzschild background which exhibit superconductivity. What makes these backgrounds p-wave superconductors is that the order parameter is a vector, and the conductivities are strongly anisotropic in a manner that is suggestive of a gap with nodes. The low-lying excitations of the normal state have a relaxation time which grows rapidly as the temperature decreases, consistent with the absence of impurity scattering. A numerical exploration of quasinormal modes close to the transition temperature suggests that p-wave backgrounds are stable against perturbations analogous to turning on a p+ip gap, whereas p+ip-wave configurations are unstable against turning into pure p-wave backgrounds.

Surface Effect on the Elastic Behavior of Static Bending Nanowires

Abstract: The surface effect from surface stress and surface elasticity on the elastic behavior of nanowires in static bending is incorporated into Euler-Bernoulli beam theory via the Young-Laplace equation. Explicit solutions are presented to study the dependence of the surface effect on the overall Young's modulus of nanowires for three different boundary conditions: cantilever, simply supported, and fixed-fixed. The solutions indicate that the cantilever nanowires behave as softer materials when deflected while the other structures behave like stiffer materials as the nanowire cross-sectional size decreases for positive surface stresses. These solutions agree with size dependent nanowire overall Young's moduli observed from static bending tests by other researchers. This study also discusses possible reasons for variations of nanowire overall Young's moduli observed.

Quantum thermal transport in nanostructures

Abstract: In this colloquia review we discuss methods for thermal transport calculations for nanojunctions connected to two semi-infinite leads served as heat-baths. Our emphases are on fundamental quantum theory and atomistic models. We begin with an introduction of the Landauer formula for ballistic thermal transport and give its derivation from scattering wave point of view. Several methods (scattering boundary condition, mode-matching, Piccard and Caroli formulas) of calculating the phonon transmission coefficients are given. The nonequilibrium Green's function (NEGF) method is reviewed and the Caroli formula is derived. We also give iterative methods and an algorithm based on a generalized eigenvalue problem for the calculation of surface Green's functions, which are starting point for an NEGF calculation. A systematic exposition for the NEGF method is presented, starting from the fundamental definitions of the Green's functions, and ending with equations of motion for the contour ordered Green's functions and Feynman diagrammatic expansion. In the later part, we discuss the treatments of nonlinear effects in heat conduction, including a phenomenological expression for the transmission, NEGF for phonon-phonon interactions, molecular dynamics (generalized Langevin) with quantum heat-baths, and electron-phonon interactions. Some new results are also shown. We briefly review the experimental status of the thermal transport measurements in nanostructures.

Vibration of Plates

Abstract: Plates are integral parts of most engineering structures and their vibration analysis is required for safe design. Vibration of Plates provides a comprehensive, self-contained introduction to vibration theory and analysis of two-dimensional plates. Reflecting the author's more than 15 years of original research on plate vibration, this book presents new methodologies and demonstrates their effectiveness by providing comprehensive results. The text also offers background information on vibration problems along with a discussion of various plate geometries and boundary conditions, including the new concepts of Boundary Characteristic Orthogonal Polynomials (BCOPs).

Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory

Abstract: We study boundary conditions in N=4 super Yang-Mills theory that preserve one-half the supersymmetry. The obvious Dirichlet boundary conditions can be modified to allow some of the scalar fields to have a ``pole'' at the boundary. The obvious Neumann boundary conditions can be modified by coupling to additional fields supported at the boundary. The obvious boundary conditions associated with orientifolds can also be generalized. In preparation for a separate study of how electric-magnetic duality acts on these boundary conditions, we explore moduli spaces of solutions of Nahm's equations that appear in the presence of a boundary. Though our main interest is in boundary conditions that are Lorentz-invariant (to the extent possible in the presence of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations of Neumann boundary conditions. We make preliminary comments on the action of electric-magnetic duality, deferring a more serious study to a later paper.

Boundary conditions for Dirac fermions on a terminated honeycomb lattice

Abstract: We derive the boundary condition for the Dirac equation corresponding to a tight-binding model on a two-dimensional honeycomb lattice terminated along an arbitrary direction Zigzag boundary conditions result generically once the boundary is not parallel to the bonds Since a honeycomb strip with zigzag edges is gapless, this implies that confinement by lattice termination does not, in general, produce an insulating nanoribbon We consider the opening of a gap in a graphene nanoribbon by a staggered potential at the edge and derive the corresponding boundary condition for the Dirac equation We analyze the edge states in a nanoribbon for arbitrary boundary conditions and identify a class of propagating edge states that complement the known localized edge states at a zigzag boundary

A unified approach to boundary value problems

Abstract: This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle. The author s thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated radon transform and the Dirichlet-to-Neumann map for a moving boundary; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: Preface; Introduction; Chapter 1: Evolution Equations on the Half-Line; Chapter 2: Evolution Equations on the Finite Interval; Chapter 3: Asymptotics and a Novel Numerical Technique; Chapter 4: From PDEs to Classical Transforms; Chapter 5: Riemann Hilbert and d-Bar Problems; Chapter 6: The Fourier Transform and Its Variations; Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging; Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary; Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs; Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval; Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain; Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains; Chapter 13: Formulation of Riemann Hilbert Problems; Chapter 14: A Collocation Method in the Fourier Plane; Chapter 15: From Linear to Integrable Nonlinear PDEs; Chapter 16: Nonlinear Integrable PDEs on the Half-Line; Chapter 17: Linearizable Boundary Conditions; Chapter 18: The Generalized Dirichlet to Neumann Map; Chapter 19: Asymptotics of Oscillatory Riemann Hilbert Problems; Epilogue; Bibliography; Index.

Thermodynamic signature of growing amorphous order in glass-forming liquids

Abstract: That the dynamical properties of a glass-forming liquid at high temperature are different from behaviour in the supercooled state has already been established. Numerical simulations now suggest that the static length scale over which spatial correlations exist also changes on approaching the glass transition. Supercooled liquids exhibit a pronounced slowdown of their dynamics on cooling1 without showing any obvious structural or thermodynamic changes2. Several theories relate this slowdown to increasing spatial correlations3,4,5,6. However, no sign of this is seen in standard static correlation functions, despite indirect evidence from considering specific heat7 and linear dielectric susceptibility8. Whereas the dynamic correlation function progressively becomes more non-exponential as the temperature is reduced, so far no similar signature has been found in static correlations that can distinguish qualitatively between a high-temperature and a deeply supercooled glass-forming liquid in equilibrium. Here, we show evidence of a qualitative thermodynamic signature that differentiates between the two. We show by numerical simulations with fixed boundary conditions that the influence of the boundary propagates into the bulk over increasing length scales on cooling. With the increase of this static correlation length, the influence of the boundary decays non-exponentially. Such long-range susceptibility to boundary conditions is expected within the random first-order theory4,9,10 (RFOT) of the glass transition. However, a quantitative account of our numerical results requires a generalization of RFOT, taking into account surface tension fluctuations between states.

Straight velocity boundaries in the lattice Boltzmann method

Abstract: Various ways of implementing boundary conditions for the numerical solution of the Navier-Stokes equations by a lattice Boltzmann method are discussed. Five commonly adopted approaches are reviewed, analyzed, and compared, including local and nonlocal methods. The discussion is restricted to velocity Dirichlet boundary conditions, and to straight on-lattice boundaries which are aligned with the horizontal and vertical lattice directions. The boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes. This procedure makes it possible to compare boundary conditions on an equal footing, although they were originally derived from very different principles. It is concluded that all five boundary conditions exhibit second-order accuracy, consistent with the accuracy of the lattice Boltzmann method. The five methods are then compared numerically for accuracy and stability through benchmarks of two-dimensional and three-dimensional flows. None of the methods is found to be throughout superior to the others. Instead, the choice of a best boundary condition depends on the flow geometry, and on the desired trade-off between accuracy and stability. From the findings of the benchmarks, the boundary conditions can be classified into two major groups. The first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations. Boundary conditions in this group are found to be exceptionally accurate at low Reynolds number. Boundary conditions of the second group replace all variables on boundary nodes by new values. They exhibit generally much better numerical stability and are therefore dedicated for use in high Reynolds number flows.

Two-relaxation-time Lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions

Abstract: We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach [16, 18] is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions. We propose second and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the effective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived permeability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations [5, 46] are improved with respect to this property. PACS: 47.10.ad, 47.56+r, 02.60-x

Isotropic angle-domain elastic reverse-time migration

Abstract: Multicomponent data usually are not processed with specifically designed procedures but with procedures analogous to those used for single-component data. In isotropic media, the vertical and horizontal components of the data commonly are taken as proxies for the P- and S-wave modes, which are imaged independently with the acoustic wave equations.This procedure works only if the vertical and horizontal components accurately represent P- and S-wave modes, which generally is not true. Therefore, multicomponent images constructed with this procedure exhibit artifacts caused by incorrect wave-mode separation at the surface.An alternative procedure for elastic imaging uses the full vector fields for wavefield reconstruction and imaging. Thewavefieldsarereconstructedusingthemulticomponentdata as a boundary condition for a numerical solution to the elastic wave equation. The key component for wavefield migration is theimagingcondition,whichevaluatesthematchbetweenwavefields reconstructed from sources and receivers. For vector wave fields, a simple component-by-component crosscorrelation between two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes. We can separate elastic wavefields after reconstruction in the subsurface and implement theimagingconditionascrosscorrelationofpurewavemodesinstead of the Cartesian components of the displacement wavefield.Thisapproachleadstoimagesthatareeasiertointerpretbecause they describe reflectivity of specified wave modes at interfaces of physical properties.As for imaging with acoustic wavefields, the elastic imaging condition can be formulated conventionally crosscorrelation with zero lag in space and time and extendedtononzerospaceandtimelags.Theelasticimagesproduced by an extended imaging condition can be used for angle decomposition of primary PP or SS and converted PS or SP reflectivity. Angle gathers constructed with this procedure have applicationsformigrationvelocityanalysisandamplitude-variation-with-angleanalysis.

Efficient Generation of Inflow Conditions for Large Eddy Simulation of Street-Scale Flows

Abstract: Using a numerical weather forecasting code to provide the dynamic large-scale inlet boundary conditions for the computation of small-scale urban canopy flows requires a continuous specification of appropriate inlet turbulence. For such computations to be practical, a very efficient method of generating such turbulence is needed. Correlation functions of typical turbulent shear flows have forms not too dissimilar to decaying exponentials. A digital-filter-based generation of turbulent inflow conditions exploiting this fact is presented as a suitable technique for large eddy simulations computation of spatially developing flows. The artificially generated turbulent inflows satisfy the prescribed integral length scales and Reynolds-stress-tensor. The method is much more efficient than, for example, Klein’s (J Comp Phys 186:652–665, 2003) or Kempf et al.’s (Flow Turbulence Combust, 74:67–84, 2005) methods because at every time step only one set of two-dimensional (rather than three-dimensional) random data is filtered to generate a set of two-dimensional data with the appropriate spatial correlations. These data are correlated with the data from the previous time step by using an exponential function based on two weight factors. The method is validated by simulating plane channel flows with smooth walls and flows over arrays of staggered cubes (a generic urban-type flow). Mean velocities, the Reynolds-stress-tensor and spectra are all shown to be comparable with those obtained using classical inlet-outlet periodic boundary conditions. Confidence has been gained in using this method to couple weather scale flows and street scale computations.

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

Abstract: In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded domains. We present in de- tail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimen- sions and the cubic nonlinear case.

Setting the boundary free in AdS/CFT

Abstract: We describe a new class of boundary conditions for AdSd+1 under which the boundary metric becomes a dynamical field. The key technical point is to show that contributions from boundary counter-terms in the bulk gravitational action render such fluctuations normalizable. In the context of AdS/CFT, the simplest version of Neumann boundary conditions for AdS promotes the CFT metric to a dynamical field but adds no explicit gravitational dynamics; the gravitational dynamics is just that induced by the conformal fields. Other AdS boundary conditions couple the CFT to a gravity theory of choice. We use this correspondence to briefly explore the coupled CFT + gravity theories and, in particular, for d = 3 we show that coupling topologically massive gravity to a large N CFT preserves the perturbative stability of the theory with negative (three-dimensional) Newton's constant.

Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems

Abstract: Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.

Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements

Abstract: Boundary Value Problems.- Function Spaces.- Variational Methods.- Variational Formulations of Boundary Value Problems.- Fundamental Solutions.- Boundary Integral Operators.- Boundary Integral Equations.- Approximation Methods.- Finite Elements.- Boundary Elements.- Finite Element Methods.- Boundary Element Methods.- Iterative Solution Methods.- Fast Boundary Element Methods.- Domain Decomposition Methods.