Showing papers on "Boundary value problem published in 1999"
TL;DR: In this paper, a general boundary condition that accounts for the reduced momentum and heat exchange with wall surfaces is proposed and its validity is investigated and it is shown that it is applicable in the entire Knudsen range and is second-order accurate in Kn in the slip flow regime.
Abstract: Rarefied gas flows in channels, pipes, and ducts with smooth surfaces are studied in a wide range of Knudsen number (Kn) at low Mach number (M) with the objective of developing simple, physics-based models. Such flows are encountered in microelectromechanical systems (MEMS), in nanotechnology applications, and in low-pressure environments. A new general boundary condition that accounts for the reduced momentum and heat exchange with wall surfaces is proposed and its validity is investigated. It is shown that it is applicable in the entire Knudsen range and is second-order accurate in Kn in the slip flow regime. Based on this boundary condition, a universal scaling for the velocity profile is obtained, which is used to develop a unified model predicting mass flow rate and pressure distribution with reasonable accuracy for channel, pipe, and duct flows in the regime (0 Kn). A rarefaction coefficient is introduced into this two-parameter model to account for the increasingly reduced intermolecular collisions...
1,106 citations
01 Jan 1999
TL;DR: In this paper, the authors derived two fundamental theorems of finite-difference methods: finite difference methods beyond Scalar Wave Equations (SFE) and finite volume methods.
Abstract: Introduction Basic Finite-Difference Methods Beyond Scalar Wave Equations Series-Expansion Methods Finite Volume Methods 6 Semi-Lagrangian Methods Physically Insignificant Fast Waves Non-reflecting Boundary conditions Appendix: Derivations of two fundamental theorems.
877 citations
TL;DR: In this article, the full development and analysis of four models for the transversely vibrating uniform beam are presented, including the Euler-Bernoulli, Rayleigh, shear and Timoshenko models.
833 citations
TL;DR: In this paper, the authors consider the case of a drop of liquid in equilibrium with its vapor on the solid substrate, and show that when the contact angle is large enough, the boundary condition can drastically differ from a no-slip condition.
Abstract: It is well known that, at a macroscopic level, the boundary condition for a viscous fluid at a solid wall is one of ``no slip.'' The liquid velocity field vanishes at a fixed solid boundary. We consider the special case of a liquid that partially wets the solid (i.e., a drop of liquid, in equilibrium with its vapor on the solid substrate, has a finite contact angle). Using extensive molecular dynamics simulations, we show that when the contact angle is large enough, the boundary condition can drastically differ (at a microscopic level) from a no-slip condition. Slipping lengths exceeding 30 molecular diameters are obtained for a contact angle of 140\ifmmode^\circ\else\textdegree\fi{}, characteristic of mercury on glass. This finding may have important implications for the transport properties in nanoporous media under such ``nonwetting'' conditions.
701 citations
TL;DR: In this paper, one-particle quantum scattering theory on an arbitrary finite graph with n open ends is formulated and discussed, where the Hamiltonian is defined to be (minus) the Laplace operator with general boundary conditions at the vertices.
Abstract: We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.
654 citations
TL;DR: In this paper, the authors investigated the details of the bulk-boundary correspondence in Lorentzian signature anti-de Sitter space and provided an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates.
Abstract: We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti--de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates. For ${\mathrm{AdS}}_{3},$ the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\mathrm{SL}(2,R{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SL}(2,R{)}_{R}$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincar\'e coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.
628 citations
TL;DR: In this article, the similarity solutions describing the steady plane (flow and thermal) boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution are examined both analytically and numerically.
Abstract: The similarity solutions describing the steady plane (flow and thermal) boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution are examined both analytically and numerically. The mass- and heat-transfer characteristics of these boundary layers are described and compared with the results of earlier authors, obtained under the more familiar power-law boundary conditions.
617 citations
TL;DR: The basic issues for an accurate representation of the relevant electrostatic interactions are introduced and the Ewald summation methods, the fast particle mesh methods, and the fast multipole methods are discussed.
Abstract: Current computer simulations of biomolecules typically make use of classical molecular dynamics methods, as a very large number (tens to hundreds of thousands) of atoms are involved over timescales of many nanoseconds. The methodology for treating short-range bonded and van der Waals interactions has matured. However, long-range electrostatic interactions still represent a bottleneck in simulations. In this article, we introduce the basic issues for an accurate representation of the relevant electrostatic interactions. In spite of the huge computational time demanded by most biomolecular systems, it is no longer necessary to resort to uncontrolled approximations such as the use of cutoffs. In particular, we discuss the Ewald summation methods, the fast particle mesh methods, and the fast multipole methods. We also review recent efforts to understand the role of boundary conditions in systems with long-range interactions, and conclude with a short perspective on future trends.
592 citations
TL;DR: This paper provides a detailed convergence analysis of the multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale.
Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.
573 citations
01 Jul 1999
TL;DR: An algorithm for fast, physically accurate simulation of deformable objects suitable for real time animation and virtual environment interaction and how to exploit the coherence of typical interactions to achieve low latency is presented.
Abstract: We present an algorithm for fast, physically accurate simulation of deformable objects suitable for real time animation and virtual environment interaction. We describe the boundary integral equation formulation of static linear elasticity as well as the related Boundary Element Method (BEM) discretization technique. In addition, we show how to exploit the coherence of typical interactions to achieve low latency; the boundary formulation lends itself well to a fast update method when a few boundary conditions change. The algorithms are described in detail with examples from ArtDefo, our implementation.
561 citations
TL;DR: In this article, the authors describe probes of anti-endash de Sitter spacetimes in terms of conformal field theories on the AdS boundary, showing a scale-radius duality.
Abstract: We describe probes of anti{endash}de Sitter spacetimes in terms of conformal field theories on the AdS boundary. Our basic tool is a formula that relates bulk and boundary states{emdash}classical bulk field configurations are dual to expectation values of operators on the boundary. At the quantum level we relate the operator expansions of bulk and boundary fields. Using our methods, we discuss the CFT description of local bulk probes including normalizable wave packets, fundamental and D-strings, and D-instantons. Radial motions of probes in the bulk spacetime are related to motions in scale on the boundary, demonstrating a scale-radius duality. We discuss the implications of these results for the holographic description of black hole horizons in the boundary field theory. {copyright} {ital 1999} {ital The American Physical Society}
TL;DR: In this paper, the authors derived stable and accurate interface conditions based on the SAT penalty method for the linear advection?diffusion equation, which are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator.
Book•
29 Jan 1999
TL;DR: The setting of problems for the one-dimensional Viscoelastic system is described in this article. But the setting of the three-dimensional system is not discussed in this paper.
Abstract: Preliminaries Some Definitions C0-Semigroup Generated by Dissipative Operator Exponential Stability and Analyticity The Sobolev Spaces and Elliptic Boundary Value Problems Linear Thermoelastic Systems The Setting of Problems for the One-Dimensional Thermoelastic System The Exponential Stability for the Dirichlet Boundary Conditions at Both Ends The Exponential Stability for the Stress-Free Boundary Conditions at Both Ends The Exponential Stability for the Stress-Free Boundary Conditions at One End The Thermoelastic Kirchhoff Plate Equations Linear Viscoelastic System Linear Viscoelastic System Wave Equation with Locally Distributed Damping Linear Viscoelastic System with Memory The Linear Viscoelastic Kirchoff Plate with Memory Linear Thermoviscoelastic Systems Linear One-Dimensional Thermoviscoelastic System Linear Three-Dimensional Thermoviscoelastic System with Memory Elastic Systems with Shear Damping Shear Diffusion Equations Laminated Beam with Shear Damping Linear Elastic Systems with Boundary Damping Second-Order Hyperbolic Equation Euler-Bernoulli Beam Equation Uniformly Stable Approximations Main Theorem Approximations of the Thermoelastic System Approximation of the Viscoelastic System Bibliography
TL;DR: In this article, the authors investigate the thermal conductivity of silicon nanowires based on molecular dynamics simulations and find that the simulated thermal conductivities of nanowire with square cross sections are about two orders of magnitude smaller than those of bulk Si crystals in a wide range of temperatures (200 −500 K) for both rigid and free boundary conditions.
Abstract: We investigate the thermal conductivity of silicon nanowires based on molecular dynamics (MD) simulations. The simulated thermal conductivities of nanowires with square cross sections are found to be about two orders of magnitude smaller than those of bulk Si crystals in a wide range of temperatures (200–500 K) for both rigid and free boundary conditions. A solution of the Boltzmann transport equation is used to explore the possibility of explaining the MD results based on boundary scattering.
TL;DR: It is shown that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices, and the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions.
Abstract: Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.
TL;DR: In this article, a three-dimensional model of droplet impact onto asymmetric surface geometries is developed, based on RIPPLE, and combines a fixed-grid control volume discretization of the flow equations with a volume tracking algorithm to track the droplet free surface.
Abstract: A three-dimensional model has been developed of droplet impact onto asymmetric surface geometries. The model is based on RIPPLE, and combines a fixed-grid control volume discretization of the flow equations with a volume tracking algorithm to track the droplet free surface. Surface tension is modeled as a volume force acting on fluid near the free surface. Contact angles are applied as a boundary condition at the contact line. The results of two scenarios are presented, of the oblique impact of a 2 mm water droplet at 1 m/sec onto a 45° incline, and of a similar impact of a droplet onto a sharp edge. Photographs are presented of such impacts, against which the numerical results are compared. The contact angle boundary condition is applied in one of two ways. For the impact onto an incline, the temporal variation of contact angles at the leading and trailing edges of the droplet was measured from photographs. This data is applied as a boundary condition to the simulation, and an interpolation scheme propos...
TL;DR: The theory of exact boundary conditions for constant coefficient time-dependent problems is developed in detail, with many examples from physical applications as discussed by the authors, and an illustrative numerical example is given.
Abstract: We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient time-dependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to compare different accurate treatments, and an illustrative numerical example is given. We close with a discussion of some important problems that remain open.
TL;DR: In this paper, effective boundary conditions for the linear electrodynamics of single and multishell carbon nanotubes (CN's) are derived using the dynamic conductivity of CN's, which is obtained for different CN's (zigzag, armchair, and chiral) in the frame of semiclassical as well as quantum-mechanical treatments.
Abstract: Effective boundary conditions, in the form of two-sided impedance boundary conditions, are formulated for the linear electrodynamics of single- and multishell carbon nanotubes (CN's). The impedance is derived using the dynamic conductivity of CN's, which is obtained for different CN's (zigzag, armchair, and chiral) in the frame of the semiclassical as well as quantum-mechanical treatments. Propagation of surface waves in CN's is considered. The phase velocities and the slow-wave coefficients of surface waves are explored for a wide frequency range, from the microwave to the ultraviolet regimes. Relaxation is shown to qualitatively change the dispersion characteristics in the low-frequency limit, thereby rendering the existence of weakly retarded plasmons impossible. A dispersionless propagation regime is shown possible for the surface waves in the infrared regime. Attenuation and retardation in metallic and semiconductor CN's are compared.
TL;DR: In this paper, a Corrective Smoothed Particle Method (CSPM) is proposed to solve the problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothing Particle Hydrodynamics (SSPH).
Abstract: Combining the kernel estimate with the Taylor series expansion is proposed to develop a Corrective Smoothed Particle Method (CSPM). This algorithm resolves the general problem of particle deficiency at boundaries, which is a shortcoming in Standard Smoothed Particle Hydrodynamics (SSPH). In addition, the method’s ability to model derivatives of any order could make it applicable for any time-dependent boundary value problems. An example of the applications studied in this paper is unsteady heat conduction, which is governed by second-order derivatives. Numerical results demonstrate that besides the capability of directly imposing boundary conditions, the present method enhances the solution accuracy not only near or on the boundary but also inside the domain. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.
TL;DR: In this paper, the authors developed effective field theories (EFTs) for the scattering of two particles at wavelengths which are large compared to the range of their interaction, and showed that the renormalized EFT is equivalent to the effective range expansion, to a Schrodinger equation with a pseudo-potential, and to an energy expansion of a generic boundary condition at the origin.
TL;DR: In this paper, a finite difference formulation is applied to track solid?liquid boundaries on a fixed underlying grid, where the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked.
TL;DR: In this article, the authors present a generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years, which is based on volumetric averaging of various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure.
Abstract: This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.
TL;DR: It is shown that counterions form a strongly correlated liquid at the surface of the macroion that leads to additional attraction of counterions to the surface, which is absent in conventional solutions of the Poisson-Boltzmann equation.
Abstract: Screening of a strongly charged macroion by multivalent counterions is considered. It is shown that counterions form a strongly correlated liquid at the surface of the macroion. Cohesive energy of this liquid leads to additional attraction of counterions to the surface, which is absent in conventional solutions of the Poisson-Boltzmann equation. Away from the surface this attraction can be taken into account by a new boundary condition for the concentration of counterions near the surface. The Poisson-Boltzmann equation is solved with this boundary condition for a charged flat surface, a cylinder, and a sphere. In all three cases, screening is much stronger than in the conventional approach. At some critical exponentially small concentration of multivalent counterions in the solution, they totally neutralize the surface charge at small distances from the surface. At larger concentrations they invert the sign of the net macroion charge. The absolute value of the inverted charge density can be as large as 20% of the bare one. In particular, for a cylindrical macroion it is shown that for screening by multivalent counterions, predictions of the Onsager-Manning theory are quantitatively incorrect. The net charge density of the cylinder is smaller than their theory predicts and inverts the sign with a growing concentration of counterions. Moreover, the condensation loses its universality and the net charge linear density depends on the bare one.
TL;DR: In this paper, a numerical model for simulating wave interaction with porous structures is presented, which calculates the mean flow outside of porous structures based on the Reynolds averaged Navier-S...
Abstract: This paper presents a numerical model for simulating wave interaction with porous structures. The model calculates the mean flow outside of porous structures based on the Reynolds averaged Navier-S...
TL;DR: In this paper, boundary conformal field theory is used to study the changes of boundary conditions generated by marginal boundary fields, where the deformation parameters may be regarded as continuous moduli of D-branes.
TL;DR: In this paper, a class of L 1 vector fields, called divergence-measure vector fields (DMEFs), are analyzed and the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of vector fields are established.
Abstract: We analyze a class of L 1 vector elds, called divergence-measure elds. We establish the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of L 1 elds. Then we apply this theory to analyzeL 1 entropy solutions of initial-boundary-value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class ofm m systems with afne characteristic hypersurfaces. The analysis inL 1 also extends toL p .
TL;DR: In this article, the area compressibility modulus (CA) of a hydrated lipid bilayer was calculated from the simulations of eight molecular dynamics simulations, differing only in the applied surface tension, γ, defining the boundary conditions of the periodic cell.
Abstract: Eight molecular dynamics simulations of a hydrated lipid bilayer have been carried out differing only in the applied surface tension, γ, defining the boundary conditions of the periodic cell. The calculated surface area per molecule and deuterium order parameter profile are found to depend strongly on γ. We present several methods to calculate the area compressibility modulus, KA, from the simulations. Equivalence between the constant area and constant surface tension ensembles is investigated by comparing the present simulations with earlier work from our laboratories and we find simulation results to depend much more strongly on the specified surface area or surface tension than on the ensemble employed.
TL;DR: In this paper, the authors investigated time harmonic Maxwell equations in heterogeneous media, where the permeability μ and the permittivity e are piecewise constant and the associated boundary value problem can be interpreted as a transmission problem.
Abstract: We investigate time harmonic Maxwell equations in heterogeneous media, where the permeability μ and the permittivity e are piecewise constant. The associated boundary value problem can be interpreted as a transmission problem. In a very natural way the interfaces can have edges and corners. We give a detailed description of the edge and corner singularities of the electromagnetic fields.