Showing papers on "Boundary value problem published in 2000"

Spectral Methods in MATLAB

Abstract: Preface 1 Differentiation matrices 2 Unbounded grids: the semidiscrete Fourier transform 3 Periodic grids: the DFT and FFT 4 Smoothness and spectral accuracy 5 Polynomial interpolation and clustered grids 6 Chebyshev differentiation matrices 7 Boundary value problems 8 Chebyshev series and the FFT 9 Eigenvalues and pseudospectra 10 Time-stepping and stability regions 11 Polar coordinates 12 Integrals and quadrature formulas 13 More about boundary conditions 14 Fourth-order problems Afterword Bibliography Index

Congested traffic states in empirical observations and microscopic simulations

Abstract: We present data from several German freeways showing different kinds of congested traffic forming near road inhomogeneities, specifically lane closings, intersections, or uphill gradients. The states are localized or extended, homogeneous or oscillating. Combined states are observed as well, like the coexistence of moving localized clusters and clusters pinned at road inhomogeneities, or regions of oscillating congested traffic upstream of nearly homogeneous congested traffic. The experimental findings are consistent with a recently proposed theoretical phase diagram for traffic near on-ramps [D. Helbing, A. Hennecke, and M. Treiber, Phys. Rev. Lett. 82, 4360 (1999)]. We simulate these situations with a continuous microscopic single-lane model, the ``intelligent driver model,'' using empirical boundary conditions. All observations, including the coexistence of states, are qualitatively reproduced by describing inhomogeneities with local variations of one model parameter. We show that the results of the microscopic model can be understood by formulating the theoretical phase diagram for bottlenecks in a more general way. In particular, a local drop of the road capacity induced by parameter variations has essentially the same effect as an on-ramp.

A Posteriori Error Estimation in Finite Element Analysis

Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A MATLAB differentiation matrix suite

Abstract: A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

Approximation of boundary element matrices

Abstract: This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance.

Robust Computational Techniques for Boundary Layers

Abstract: Introduction to Numerical Methods for Problems with Boundary Layers Numerical Methods on Uniform Meshes Layer Resolving Methods for Convection-Diffusion Problems in One Dimension The Limitations of Non-Monotone Numerical Methods Convection-Diffusion Problems in a Moving Medium Convection-Diffusion Problems with Frictionless Walls Convection-Diffusion Problems with No Slip Boundary Conditions Experimental Estimation of Errors Non-Monotone Methods in Two Dimensions Linear and Nonlinear Reaction-Diffusion Problems Prandtl Flow past a Flat Plate-Blasius' Method Prandtl Flow past a Flat Plate-Direct Method References.

Contact-line dynamics of a diffuse fluid interface

Abstract: An investigation is made into the moving contact line dynamics of a Cahn–Hilliard–van der Waals (CHW) diffuse mean-field interface. The interface separates two incompressible viscous fluids and can evolve either through convection or through diffusion driven by chemical potential gradients. The purpose of this paper is to show how the CHW moving contact line compares to the classical sharp interface contact line. It therefore discusses the asymptotics of the CHW contact line velocity and chemical potential fields as the interface thickness e and the mobility κ both go to zero. The CHW and classical velocity fields have the same outer behaviour but can have very different inner behaviours and physics. In the CHW model, wall–liquid bonds are broken by chemical potential gradients instead of by shear and change of material at the wall is accomplished by diffusion rather than convection. The result is, mathematically at least, that the CHW moving contact line can exist even with no-slip conditions for the velocity. The relevance and realism or lack thereof of this is considered through the course of the paper.The two contacting fluids are assumed to be Newtonian and, to a first approximation, to obey the no-slip condition. The analysis is linear. For simplicity most of the analysis and results are for a 90° contact angle and for the fluids having equal dynamic viscosity μ and mobility κ. There are two regions of flow. To leading order the outer-region velocity field is the same as for sharp interfaces (flow field independent of r) while the chemical potential behaves like r−ξ, ξ = π/2/max{θeq, π − θeq}, θeq being the equilibrium contact angle. An exception to this occurs for θeq = 90°, when the chemical potential behaves like ln r/r. The diffusive and viscous contact line singularities implied by these outer solutions are resolved in the inner region through chemical diffusion. The length scale of the inner region is about 10√μκ – typically about 0.5–5 nm. Diffusive fluxes in this region are O(1). These counterbalance the effects of the velocity, which, because of the assumed no-slip boundary condition, fluxes material through the interface in a narrow boundary layer next to the wall.The asymptotic analysis is supplemented by both linearized and nonlinear finite difference calculations. These are made at two scales, experimental and nanoscale. The first set is done to show CHW interface behaviour and to test the qualitative applicability of the CHW model and its asymptotic theory to practical computations of experimental scale, nonlinear, low capillary number flows. The nanoscale calculations are carried out with realistic interface thicknesses and diffusivities and with various assumed levels of shear-induced slip. These are discussed in an attempt to evaluate the physical relevance of the CHW diffusive model. The various asymptotic and numerical results together indicate a potential usefullness for the CHW model for calculating and modelling wetting and dewetting flows.

A Boundary Condition Capturing Method for Multiphase Incompressible Flow

Abstract: In l6r, the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid compressible Euler equations. In l11r, related techniques were used to develop a boundary condition capturing approach for the variable coefficient Poisson equation on domains with an embedded interface. In this paper, these new numerical techniques are extended to treat multiphase incompressible flow including the effects of viscosity, surface tension and gravity. While the most notable finite difference techniques for multiphase incompressible flow involve numerical smearing of the equations near the interface, see, e.g., l19, 17, 1r, this new approach treats the interface in a sharp fashion.

D-Branes And Mirror Symmetry

Abstract: We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in nonlinear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around � �

Boundary Liouville field theory. 1. Boundary state and boundary two point function

Abstract: Liouville conformal field theory is considered with conformal boundary. There is a family of conformal boundary conditions parameterized by the boundary cosmological constant, so that observables depend on the dimensional ratios of boundary and bulk cosmological constants. The disk geometry is considered. We present an explicit expression for the expectation value of a bulk operator inside the disk and for the two-point function of boundary operators. We comment also on the properties of the degenrate boundary operators. Possible applications and further developments are discussed. In particular, we present exact expectation values of the boundary operators in the boundary sin-Gordon model.

On the plasticity of single crystals: free energy, microforces, plastic-strain gradients

Abstract: This study develops a general theory of crystalline plasticity based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. A viscoplastic regularization of the basic equations that obviates the need to determine the active slip systems is developed. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, the special case of single slip is discussed. Specific solutions are presented: one a single shear band connecting constant slip-states; one a periodic array of shear bands.

The Properties of Ferroelectric Films at Small Dimensions

Abstract: ▪ Abstract This paper reviews the literature on size effects in ferroelectric materials, with an emphasis on thin film perovskite ferroelectrics. The roles of boundary conditions, defect chemistry, electrode interfaces, surface layers, and microstructure in controlling the measured properties of ferroelectric films, as well as the observed deviation from bulk properties are discussed. Examples of the manifestation of size effects in terms of the low and high field dielectric properties, the piezoelectric effect, and the leakage behavior of films are given.

Sensitivity and power deposition in a high-field imaging experiment.

Abstract: Image signal-to-noise ratio and power dissipation are investigated theoretically up to 400 MHz. While the text is mathematical, the figures give insights into predictions. Hertz potential is introduced for probe modeling where charge separation cannot be ignored. Using a spherical geometry, the potential from current loops that would produce a homogeneous static B1 field is calculated; at high frequency it is shown to create an unnecessarily inhomogeneous field. However, a totally homogeneous field is shown to be unattainable. Boundary conditions are solved for circularly polarized fields, and strategies for limited shimming of the sample B1 field are then presented. A distinction is drawn between dielectric resonance and spatial field focusing. At high frequency, the region of maximum specific absorption is shown to move inside the sample and decrease. From the fields in both rotating frames, the signal-to-noise ratio is derived and compared with the traditional, low-frequency formulation. On average, it is mostly found to be slightly larger at high frequency. Nevertheless, the free induction decay is sometimes found to be annulled.

Simulating surface tension with smoothed particle hydrodynamics

Abstract: A method for simulating two-phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high-order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid-based volume of fluid method for two-dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two-phase system and readily extend to three-dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free-surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.

Boundary value problems for fractional diffusion equations

Abstract: The fractional diffusion equation is solved for different boundary value problems, these being absorbing and reflecting boundaries in half-space and in a box. Thereby, the method of images and the Fourier–Laplace transformation technique are employed. The separation of variables is studied for a fractional diffusion equation with a potential term, describing a generalisation of an escape problem through a fluctuating bottleneck. The results lead to a further understanding of the fractional framework in the description of complex systems which exhibit anomalous diffusion.

Impedance-based health monitoring of civil structural components

Abstract: This paper presents experimental evidence on the use of the impedance-based health-monitoring technique on components typical of civil structures. The basic principle behind this technique is to utilize high-frequency structural excitations (typically >30 kHz) through a surface-bonded piezoelectric sensor/actuator to detect changes in structural point impedance due to the presence of damage. Real-time damage detection on composite-reinforced concrete walls was investigated and the capability of this technique to detect imminent damage, well in advance of actual failure, was confirmed. Concepts that directly applied to this technique itself, such as effects of boundary condition changes and the effects of temperature changes, were also investigated. Experimental investigations were carried out on a 1/4-scale bridge element and a pipe joint commonly found in civil structures, to verify robustness of the technique to changes in environmental conditions. Data collected from the tests demonstrate the capabilit...

Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations

Abstract: Introduction Singularities of solutions to equations of mathematical physics: Prerequisites on operator pencils Angle and conic singularities of harmonic functions The Dirichlet problem for the Lame system Other boundary value problems for the Lame system The Dirichlet problem for the Stokes system Other boundary value problems for the Stokes system in a cone The Dirichlet problem for the biharmonic and polyharmonic equations Singularities of solutions to general elliptic equations and systems: The Dirichlet problem for elliptic equations and systems in an angle Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle The Dirichlet problem for strongly elliptic systems in particular cones The Dirichlet problem in a cone The Neumann problem in a cone Bibliography Index List of symbols.

The Topological Asymptotic for PDE Systems: The Elasticity Case

Abstract: The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole. In this paper, such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique. The method is general and can be easily adapted to other linear PDEs and other types of boundary conditions.

On the interface boundary condition of Beavers, Joseph, and Saffman

Abstract: We consider the laminar viscous channel flow over a porous surface. It is supposed, as in the experiment by Beavers and Joseph, that a uniform pressure gradient is maintained in the longitudinal direction in both the channel and the porous medium. After studying the corresponding boundary layers, we obtain rigorously Saffman's modification of the interface condition observed by Beavers and Joseph. It is valid when the pore size of the porous medium tends to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem.

Neural-network methods for boundary value problems with irregular boundaries

Abstract: Partial differential equations (PDEs) with boundary conditions (Dirichlet or Neumann) defined on boundaries with simple geometry have been successfully treated using sigmoidal multilayer perceptrons in previous works. The article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the exact satisfaction of the boundary conditions. The method has been successfully tested on two-dimensional and three-dimensional PDEs and has yielded accurate results.

Optimal Control of Distributed Systems: Theory and Applications

Abstract: The existence of solutions to optimal control problems Optimality system for optimal control problems The solvability of boundary value problems for a dense set of data The problem of work minimization in accelerating still fluid to a prescribed velocity Optimal boundary control for nonstationary problems of fluid flow and nonhomogeneous boundary value problems for the Navier-Stokes equations The Cauchy problem for elliptic equations in a conditionally well-posed formulation The local exact controllability of the flow of incompressible viscous fluid Bibliography Index.

Mechanical Response of Polymers: An Introduction

Abstract: Preface 1. Discussion of response of a viscoelastic material 2. Constitutive equations for one-dimensional response of viscoelastic materials: mechanical analogs 3. Constitutive equations for one-dimensional linear response of a viscoelastic material 4. Some features of the linear response of viscoelastic materials 5. Histories with constant strain or stress rates 6. Sinusoidal oscillations 7. Constitutive equation for three dimensional linear isotropic viscoelastic materials 8. Axial load, bending and torsion 9. Dynamics of bodies with viscoelastic support 10. Boundary value problems for linear isotropic viscoelastic materials 11. Influence of temperature Appendices References Index.

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

Abstract: The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth

Abstract: Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth