Dirichlet Boundaries Research Articles

A criterion for physically acceptable extra dimensions with boundaries

We present a criterion for deciding which compact extra-dimensional spaces yield physically reliable Newton’s law corrections. We study compact manifolds with boundary and without boundary. The boundary conditions which we use on the boundaries are Dirichlet or Neumann. We find that compact connected Riemannian manifolds with Dirichlet boundaries are completely excluded as extra-dimensional spaces.

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.

BOUNDARY EFFECTS IN STRING THEORY AND THE EFFECTIVE STRING COUPLING

The effect of the ideal boundaries of infinite-genus surfaces and Dirichlet boundaries on the value of the string coupling is evaluated. It is demonstrated that the value coincides with the unified gauge coupling and it is not altered by the inclusion of curved four-dimensional spacetimes in the path integral for quantum gravity or boundaries with fractional dimension and nonzero harmonic measure.

Small time approximations of Green's functions for one‐dimensional heat conduction problems with convective boundaries

AbstractSmall time approximations are obtained for Green's functions in one‐dimensional heat conduction problems with convective boundaries. The method of images similar to those for cases with Dirichlet or Neumann boundaries is used. Multiple reflections in this method generate infinite series representations for the Green's functions whose general terms involve convolution integrals with integrable singularities. We truncate the series after single and double reflections (n = 1, 2) and transform the convolution integrals to known closed‐form expressions. Numerical results are presented for n = 1, 2 and compared with exact results obtained using the classical separation of variables. It is seen that results with n = 1, 2 provide accurate approximations for dimensionless time α t / L2 up to 0.3 and complementary to those obtained by separation of variables.

Green's Function for the Damped Wave Equation on a Finite Interval Subject to Two Robin Boundary Conditions

ABSTRACTThe Green's function for the damped wave equation on a finite interval subject to two Robin boundary conditions is studied. The problem for a semi-infinite interval with one Robin boundary is considered first by the Laplace transform method to establish the reflection principle at a Robin boundary. It is seen that the reflected wave generated by an exiting wave at a Robin boundary is a convolution involving the exiting wave and some kernel function. This generalizes the well known classical results for reflections at Dirichlet and Neumann boundaries. The reflection principle at a Robin boundary is then used for the finite interval case by considering multiple reflections and this leads to an infinite sequence of waves. In order to justify the results obtained here we also study the finite interval case by the Laplace transform method. The Laplace transform of the Green's function is expanded, for large values of the transform parameter, into an infinite series of negative exponentials and then inverted term by term. Agreement is reached between these two approaches.

Stability of a Chebychev pseudospectral solution of the wave equation with absorbing boundaries

Stability of the pseudospectral Chebychev collocation solution of the two-dimensional acoustic wave problem with absorbing boundary conditions is investigated. The continuous one-dimensional problem with one absorbing boundary and one Dirichlet boundary has previously been shown to be far from normal. Consequently, the spectrum of that problem says little about the stability behavior of the solution. Our analysis proves that the discrete formulation with Dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries, either for one-dimensional or two-dimensional wave propagation, is not far from normal. The near-normality follows from the near-normality of the second-order derivative pseudospectral differential operator. Further, the nearness to normality is independent of the boundary discretization. Stability limits on the timestep are, however, dependent on the boundary operator, with an explicit Euler method having the most restrictive condition. The Crank-Nicolson implementation has a stability limit the same as the Dirichlet formulation. Furthermore, in this case the restriction scales by 1 2 in moving from one dimension to two dimensions, exactly as in the central finite difference approximation. Numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis, for both Chebychev- and modified-Chebychev-implementations. We conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution.

Boundaries Immersed in a Scalar Quantum Field

We study the interaction between a scalar quantum field , and many different boundary configurations constructed from (parallel and orthogonal) thin planar surfaces on which is constrained to vanish, or to satisfy Neumann conditions. For most of these boundaries the Casimir problem has not previously been investigated. We calculate the canonical and improved vacuum stress tensors and of ; for each example. From these we obtain the local Casimir forces on all boundary planes. For massless fields, both vacuum stress tensors yield identical attractive local Casimir forces in all Dirichlet examples considered. This desirable outcome is not a priori obvious, given the quite different features of and . For Neumann conditions. and lead to attractive Casimir stresses which are not always the same. We also consider Dirichlet and Neumann boundaries immersed in a common scalar quantum field, and find that these repel. The extensive catalogue of worked examples presented here belongs to a large class of completely solvable Casimir problems. Casimir forces previously unknown are predicted, among them ones which might be measurable.

Infinite-genus surfaces and the universal Grassmannian

Correlation functions can be calculated on Riemann surfaces using the operator formalism. The state in the Hilbert space of the free field theory on the punctured disc, corresponding to the Riemann surface, is constructed at infinite genus, verifying the inclusion of these surfaces in the Grassmannian. In particular, a subset of the class of O HD surfaces can be identified with a subset of the Grassmannian. The concept of flux through the ideal boundary is used to study the connection between infinite-genus surfaces and the domain of string perturbation theory. The different roles of effectively closed surfaces and surfaces with Dirichlet boundaries in a more complete formulation of string theory are identified.

Casimir effect for soft boundaries.

In quantum field theory with confining ``hard'' (e.g., Dirichlet) boundaries, the latter are represented in the Schr\"odinger equation defining spatial quantum modes by infinite step-function potentials. One can instead introduce confining ``soft'' boundaries, represented in the mode equation by some smoothly increasing potential function. Here the global Casimir energy is calculated for a scalar field confined by harmonic-oscillator (HO) potentials in 1, 2, and 3 dimensions. Combinations of HO and Dirichlet boundaries are also considered. Some results differ in sign from comparable hard-wall ones.

Strong magnetic fields, Dirichlet boundaries, and spectral gaps

We consider magnetic Schrodinger operators $$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$ inL 2(R n ), where $$\vec a \in C^1 (R^n ;R^n )$$ and λeR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with $$\vec a$$ , and $$M_{\vec a} = \{ x;\vec a(x) = 0\}$$ , we prove that $$H(\lambda \vec a)$$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set $$M\backslash M_{\vec a}$$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $$M\backslash M_{\vec a}$$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of $$H(\lambda \vec a)$$ . We finally address the question of absolute continuity of $$\vec a$$ for periodic $$H(\vec a)$$ .

Dirichlet strings

Strings propagating along surfaces with Dirichlet boundaries are studied in this paper. Such strings were originally proposed as a possible candidate for the QCD string. Our approach is different from previous ones and is simple and general enough, with which basic problems can be easily addressed. The Green function on a surface with Dirichlet boundaries is obtained through the Neumann Green function of the same surface, by employing a simple approach to Dirichlet conditions. An easy consequence of the simple calculation of the Green function is that in the simplest model, namely the bosonic Dirichlet string, the critical dimension is still 26, and the tachyon is still present in the spectrum, while the scattering amplitudes differ dramatically from those in the usual string theory. We discuss the high-energy, fixed-angle behavior of the four-point scattering amplitudes on the disk and the annulus. We argue for general power-like behavior of arbitrary-high-energy, fixed-angle scattering amplitudes. We also discuss the high-temperature property of the finite-temperature partition function on an arbitrary surface, and give an explicit formula of the one on the annulus.

Variational formulation of the boundary element method in transient heat conduction

Transient heat conduction, as a prototype of diffusion problems, is considered over homogeneous domains with mixed boundary conditions (on temperature and flux). Distributed time dependent discontinuities of flux and of temperature are taken as unknown sources over the Dirichlet (given temperature) and Neumann (given flux) boundaries, respectively. The boundary integral operator thus arising, is shown to be symmetric in space and time with respect to a time-convolutive bilinear form. A variational principle is then derived which characterizes the boundary solution in time. Discretizations in space and time lead to linear equations with a symmetric coefficient matrix; solution procedures are briefly discussed.

DAMPING COEFFICIENT OF STRUCTURAL FOUNDATION ANALYZED BY 3-D FEM WITH NON-REFLECTING BOUNDARY

The Dynamic FEM is frequently used for analyzing wave propagation problems. In case of infinite or semi-infinite media, the artificial boundaries reflect the waves. The present authors could solve the problem by applying Smith-Cundall's method. It superimposes two types of waves reflected from Dirichlet's and Neumann's boundaries. In this study, the radiation damping of structure models has been analyzed using this Finite Element Method.

3-D DYNAMIC ANALYSIS OF GROUND MOTION BY FEM WITH NON-REFLECTING BOUNDARY

Dynamic Finite Element Method is frequently used in analyzing wave propagation problems. In the case of infinite or infinite half media, the presence of artificial boundaries introduces wave reflections from boundaries. Authors tried to solve the problem applying Smith-Cundall's method and extended the method to three dimensional problem. Smith-Cundall's method is to solve the problem by superposing two types of reflected wave from Dirichlet's and Neumann's boundaries. This method is theoretically complete. Authors made clear the weak point of the method, however, the influence of the weak point on the computed results is small, therefore, the results by this method is reliable. This method treats problems in time domain, so nonlinear problem such as liquefaction can be solved by this method in near future.

A Method of Correcting the Superfish-Calculated Value of ZT2/Q for Cavities with Truncated Beam Tubes

The performance of an accelerating cavity can be characterized by the values of ZT/sup 2//Q for each of the cavity's resonant modes. For cylindrically symmetric structures, the program SUPERFISH can be used to find the field patterns of the azimuthally - invariant (''accelerating'') modes. However, SUPERFISH is limited to lossless regions of space entirely enclosed by Dirichlet or Neumann boundaries i.e., standing-wave solutions, and so cannot correctly treat the physical case of a cavity with open beam tubes through which energy can propagate. Instead, it is necessary to truncate the beam tubes with perfect reflectors, which introduce large amplitude standing-waves in the beam tubes, and consequently large errors in the calculated accelerating voltage and ZT/sup 2//Q. In the past, these errors have been avoided by employing long tapered beam tubes. This technique requires a considerable amount of additional computer memory, and cannot be used when memory is at a premium. In this paper, the authors describe a new technique which evaluates the errors in the accelerating voltage due to the truncated beam tubes. The method requires no special geometries or time-consuming calculations, and the results are in excellent agreement with other cavity calculations.