Showing papers on "Boundary value problem published in 2003"

The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors.

Abstract: The equation for the magnetic lead field for a given magnetoencephalography (MEG) channel is well known for arbitrary frequencies omega but is not directly applicable to MEG in the quasi-static approximation. In this paper we derive an equation for omega = 0 starting from the very definition of the lead field instead of using Helmholtz's reciprocity theorems. The results are (a) the transpose of the conductivity times the lead field is divergence-free, and (b) the lead field differs from the one in any other volume conductor by a gradient of a scalar function. Consequently, for a piecewise homogeneous and isotropic volume conductor, the lead field is always tangential at the outermost surface. Based on this theoretical result, we formulated a simple and fast method for the MEG forward calculation for one shell of arbitrary shape: we correct the corresponding lead field for a spherical volume conductor by a superposition of basis functions, gradients of harmonic functions constructed here from spherical harmonics, with coefficients fitted to the boundary conditions. The algorithm was tested for a prolate spheroid of realistic shape for which the analytical solution is known. For high order in the expansion, we found the solutions to be essentially exact and for reasonable accuracies much fewer multiplications are needed than in typical implementations of the boundary element methods. The generalization to more shells is straightforward.

R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type

Abstract: Introduction Notations and conventions $\mathcal R$-Boundedness and Sectorial Operators: Sectorial operators The classes ${\mathcal{BIP}}(X)$ and $\mathcal H^\infty(X)$ $\mathcal R$-bounded families of operators $\mathcal R$-sectorial operators and maximal $L_p$-regularity Elliptic and Parabolic Boundary Value Problems: Elliptic differential operators in $L_p(\mathbb{R}^n E)$ Elliptic problems in a half space: General Banach spaces Elliptic problems in a half space: Banach spaces of class $\mathcal{HT}$ Elliptic and parabolic problems in domains Notes References.

Effective slip in pressure-driven Stokes flow

Abstract: Nano-bubbles have recently been observed experimentally on smooth hydrophobic surfaces; cracks on a surface can likewise be the site of bubbles when partially wetting fluids are used. Because these bubbles may provide a zero shear stress boundary condition and modify considerably the friction generated by the solid boundary, it is of interest to quantify their influence on pressure-driven flow, with particular attention given to small geometries. We investigate two simple configurations of steady pressure-driven Stokes flow in a circular pipe whose surface contains periodically distributed regions of zero surface shear stress. In the spirit of experimental studies probing slip at solid surfaces, the effective slip length of the resulting flow is evaluated as a function of the degrees of freedom describing the surface heterogeneities, namely the relative width of the no-slip and no-shear stress regions and their distribution along the pipe. Comparison of the model with experimental studies of pressure-driven flow in capillaries and microchannels reporting slip is made and a possible interpretation of the experimental results is offered which is consistent with a large number of distributed slip domains such as nano-size and micron-size nearly flat bubbles coating the solid surface. Further, the possibility is suggested of a shear-dependent effective slip length, and an explanation is proposed for the seemingly paradoxical behaviour of the measured slip length increasing with system size, which is consistent with experimental results to date.

Solving ODEs with MATLAB

Abstract: From the Publisher: This book is for people who need to solve ordinary differential equations (ODEs), both initial value problems (IVPs) and boundary value problems (BVPs) as well as delay differential equations (DDEs). These topics are usually taught in separate courses of length one semester each, but solving ODEs with Matlab provides a sound treatment of all three in about 250 pages. The chapters on each of these topics begin with a discussion of "the facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed - but only the methods most widely used. Although the treatment of each method is brief and technical issues are minimized, the issues important in practice and for understanding the codes are discussed. Often solving a real problem is much more than just learning how to call a code. The last part of each chapter is a tutorial that shows how to solve problems by means of small but realistic examples.

Non-saturating magnetoresistance in heavily disordered semiconductors

Abstract: The resistance of a homogeneous semiconductor increases quadratically with magnetic field at low fields and, except in very special cases, saturates at fields much larger than the inverse of the carrier mobility, a number typically of the order of 1 T (refs 1, 2). A surprising exception to this behaviour has recently been observed in doped silver chalcogenides3,4,5, which exhibit an anomalously large, quasi-linear magnetoresistive response that extends down to low fields and survives, even at extreme fields of 55 T and beyond. Here we present a simple model of a macroscopically disordered and strongly inhomogeneous semiconductor that exhibits a similar non-saturating magnetoresistance. In addition to providing a possible explanation for the behaviour of doped silver chalcogenides, our model suggests potential routes for the construction of magnetic field sensors with a large, controllable and linear response.

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

Abstract: The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.

Fractal mobile/immobile solute transport

Abstract: [1] A fractal mobile/immobile model for solute transport assumes power law waiting times in the immobile zone, leading to a fractional time derivative in the model equations. The equations are equivalent to previous models of mobile/immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times. The solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase. In this regard, the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase. The solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers, including power law breakthrough curves at late time, and power law decline in the measured mobile mass. The MADE site mobile tritium mass decline is consistent with a fractional time derivative of order γ = 0.33, while Haggerty et al.'s [2002] stream tracer test is well modeled by a fractional time derivative of order γ = 0.3.

Multireflection boundary conditions for lattice Boltzmann models

Abstract: We present a general framework for several previously introduced boundary conditions for lattice Boltzmann models, such as the bounce-back rule and the linear and quadratic interpolations. The objectives are twofold: first to give theoretical tools to study the existing link-type boundary conditions and their corresponding accuracy; second to design boundary conditions for general flows which are third-order kinetic accurate. Using these new boundary conditions, Couette and Poiseuille flows are exact solutions of the lattice Boltzmann models for a Reynolds number Re=0 (Stokes limit) for arbitrary inclination with the lattice directions. Numerical comparisons are given for Stokes flows in periodic arrays of spheres and cylinders, linear periodic array of cylinders between moving plates, and for Navier-Stokes flows in periodic arrays of cylinders for Re<200. These results show a significant improvement of the overall accuracy when using the linear interpolations instead of the bounce-back reflection (up to an order of magnitude on the hydrodynamics fields). Further improvement is achieved with the new multireflection boundary conditions, reaching a level of accuracy close to the quasianalytical reference solutions, even for rather modest grid resolutions and few points in the narrowest channels. More important, the pressure and velocity fields in the vicinity of the obstacles are much smoother with multireflection than with the other boundary conditions. Finally the good stability of these schemes is highlighted by some simulations of moving obstacles: a cylinder between flat walls and a sphere in a cylinder.

Theory of low-frequency magnetoelectric coupling in magnetostrictive-piezoelectric bilayers

Abstract: A theoretical model is presented for low-frequency magnetoelectric (ME) effects in bilayers of magnetostrictive and piezoelectric phases. A novel approach, the introduction of an interface coupling parameter k, is proposed for the consideration of actual boundary conditions at the interface. An averaging method is used to estimate effective material parameters. Expressions for ME voltage coefficients ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}=\ensuremath{\delta}E/\ensuremath{\delta}H,$ where $\ensuremath{\delta}E$ is the induced electric field for an applied ac magnetic field $\ensuremath{\delta}H,$ are obtained by solving elastostatic and electrostatic equations. We consider both unclamped and rigidly clamped bilayers and three different field orientations of importance: (i) longitudinal fields $({\ensuremath{\alpha}}_{E,L}^{\ensuremath{'}})$ in which the poling field E, bias field H, and ac fields $\ensuremath{\delta}E$ and $\ensuremath{\delta}H$ are all parallel to each other and perpendicular to the sample plane, (ii) transverse fields $({\ensuremath{\alpha}}_{E,T}^{\ensuremath{'}})$ for in-plane H and $\ensuremath{\delta}H$ parallel to each other and perpendicular to out-of-plane E and $\ensuremath{\delta}E,$ and (iii) in-plane longitudinal fields $({\ensuremath{\alpha}}_{E,IL}^{\ensuremath{'}})$ for all the fields parallel to each other and to the sample plane. The theory predicts a giant ME coupling for bilayers with cobalt ferrite (CFO), nickel ferrite (NFO), or lanthanum strontium manganite (LSMO) for the magnetostrictive phase and barium titanate (BTO) or lead zirconate titanate (PZT) for the piezoelectric phase. Estimates of ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}$ are carried out as a function of the interface coupling k and volume fraction \ensuremath{ u} for the piezoelectric phase. In unclamped samples, ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}$ increases with increasing k. The strongest coupling occurs for equal volume of the two phases for transverse and longitudinal cases, but a maximum occurs at $\ensuremath{ u}=0.1$ for the in-plane longitudinal case. Upon clamping the bilayer, the ME effect is strengthened for the longitudinal case and is weakened for the transverse case. Other important results of the theory are as follows. (i) The strongest ME coupling is expected for the in-plane longitudinal fields and the weakest coupling for the (out-of-plane) longitudinal case. (ii) In ferrite-based composites, ${\ensuremath{\alpha}}_{E,T}^{\ensuremath{'}}$ and ${\ensuremath{\alpha}}_{E,IL}^{\ensuremath{'}}$ are a factor of 2--10 higher than ${\ensuremath{\alpha}}_{E,L}.$ (iii) The highest ME voltage coefficients are expected for CFO-PZT and the lowest values are for LSMO-PZT. Results of the present model are compared with available data on the volume and static magnetic field dependence of ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}.$ We infer, from the comparison, ideal interface conditions in NFO-PZT and poor interface coupling for CFO-PZT and LSMO-PZT.

Multipole method for microstructured optical fibers. I. Formulation

Abstract: We describe a multipole method for calculating the modes of microstructured optical fibers. The method uses a multipole expansion centered on each hole to enforce boundary conditions accurately and matches expansions with different origins by use of addition theorems. We also validate the method and give representative results.

The non-linear dependence of flux on black hole mass and accretion rate in core-dominated jets

Abstract: We derive the non-linear relation between the core flux F v of accretion-powered jets at a given frequency and the mass M of the central compact object. For scale-invariant jet models, the mathematical structure of the equations describing the synchrotron emission from jets enables us to cancel out the model-dependent complications of jet dynamics, retaining only a simple, model-independent algebraic relation between F v and M. This approach allows us to derive the F v -M relation for any accretion disc scenario that provides a set of input boundary conditions for the magnetic field and the relativistic particle pressure in the jet, such as standard and advection-dominated accretion flow (ADAF) disc solutions. Surprisingly, the mass dependence of F v is very similar in different accretion scenarios For typical flat-spectrum core-dominated radio jets and standard accretion scenarios, we find F v ∼ M 1 7 / 1 2 . The 7-9 orders of magnitude difference in black hole mass between microquasars and active galactic nuclei (AGN) jets imply that AGN jets must be about 3-4 orders of magnitude more radio-loud than microquasars, i.e. the ratio of radio to bolometric luminosity is much smaller in microquasars than in AGN jets. Because of the generality of these results, measurements of this F v -M dependence are a powerful probe of jet and accretion physics. We show how sour analysis can be extended to derive a similar scaling relation between the accretion rate m and F v for different accretion disc models. For radiatively inefficient accretion modes, we find that the flat-spectrum emission follows F v α (Mm) 1 7 / 1 2 .

A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation

Abstract: SUMMARY The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second-order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite-element method, the spectral-element method and some finite-difference methods. We illustrate the efficiency of this second-order perfectly matched layer based upon 2-D benchmarks with body and surface waves.

Molecular scale contact line hydrodynamics of immiscible flows.

Abstract: From extensive molecular dynamics simulations on immiscible two-phase flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line. In particular, the behavior at high capillary numbers, leading to the breakup of the fluid-fluid interface, is accurately predicted.

Homotopy of exact coherent structures in plane shear flows

Abstract: Three-dimensional steady states and traveling wave solutions of the Navier–Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton’s method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation (homotopy) from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Reτ=44.2 for Lx+=273.7 and Lz+=105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of ...

Surface roughness and hydrodynamic boundary slip of a newtonian fluid in a completely wetting system.

Abstract: The influence of surface roughness on the boundary condition for the flow of a Newtonian fluid near a hard wall has been investigated by measurement of the hydrodynamic drainage force. The degree of slip is found to increase with surface roughness. This leads to the conclusion that in most practical situations boundary slip takes place, leading to a reduction of the drainage force.

A Dilatonic deformation of AdS(5) and its field theory dual

Abstract: We find a nonsupersymmetric dilatonic deformation of AdS5 geometry as an exact nonsingular solution of the type-IIB supergravity. The dual gauge theory has a different Yang-Mills coupling in each of the two halves of the boundary spacetime divided by a codimension one defect. We discuss the geometry of our solution in detail, emphasizing the structure of the boundary, and also study the string configurations corresponding to Wilson loops. We also show that the background is stable under small scalar perturbations.

On the calculation of diffusion coefficients in confined fluids and interfaces with an application to the liquid-vapor interface of water

Abstract: We propose a general methodology for calculating the self-diffusion tensor from molecular dynamics for a liquid with a liquid-gas or liquid-solid interface. The standard method used in bulk fluids, based on computing the mean square displacement as a function of time and extracting the asymptotic linear time dependence from this, is not valid for systems with interfaces or for confined fluids. The method proposed here is based on imposing virtual boundary conditions on the molecular system and computing survival probabilities and specified time correlation functions in different layers of the fluid up to and including the interfacial layer. By running dual simulations, one based on MD and the other based on Langevin dynamics, using the same boundary conditions, one can fit the Langevin survival probability at long times to the MD computed survival probability, thereby determining the diffusion coefficient as a function of distance of the layers from the interface. We compute the elements of the diffusion tensor of water as a function of distance from the liquid vapor interface of water. Far from the interface the diffusion tensor is found to be isotropic, as expected, and the diffusion coefficient has the value $D\approx$ .22\AA$^2$/psec in agreement with what is found in the bulk liquid. In the interfacial region the diffusion tensor is axially anisotropic, with values of $D_{\parallel}\approx$. 8\AA$^2$/psec and $D_{\perp}\approx$. 5\AA$^2$/psec for the components parallel and normal the interface surface respectively. We also show that diffusion in confined geometries can be calculated by imposing appropriate boundary conditions on the molecular system and computing time correlation functions of the eigenfunctions of the diffusion operator corresponding to the same boundary conditions.

Finite-difference frequency-domain algorithm for modeling guided-wave properties of substrate integrated waveguide

Abstract: In multilayer microwave integrated circuits such as low-temperature co-fired ceramics or multilayered printed circuit boards, waveguide-like structures can be fabricated by using periodic metallic via-holes referred to as substrate integrated waveguide (SIW). Such SIW structures can largely preserve the advantages of conventional rectangular waveguides such as high-Q factor and high power capacity. However, they are subject to leakage due to periodic gaps, which potentially results in wave attenuation. Therefore, such a guided-wave modeling problem becomes a very complicated complex eigenvalue problem. Since the SIW are bilaterally unbounded, absorbing boundary conditions should be deployed in numerical algorithms. This often leads to a difficult complex root-extracting problem of a transcend equation. In this paper, we present a novel finite-difference frequency-domain algorithm with a perfectly matched layer and Floquet's theorem for the analysis of SIW guided-wave problems. In this scheme, the problem is converted into a generalized matrix eigenvalue problem and finally transformed to a standard matrix eigenvalue problem that can be solved with efficient subroutines available. This approach has been validated by experiment.