A modified diffusion equation for room-acoustic predication

This letter presents a modified diffusion model using an Eyring absorption coefficient to predict the reverberation time and sound pressure distributions in enclosures. While the original diffusion model [Ollendorff, Acustica 21, 236-245 (1969); J. Picaut et al., Acustica 83, 614-621 (1997); Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] usually has good performance for low absorption, the modified diffusion model yields more satisfactory results for both low and high absorption. Comparisons among the modified model, the original model, a geometrical-acoustics model, and several well-established theories in terms of reverberation times and sound pressure level distributions, indicate significantly improved prediction accuracy by the modification.     

On open boundary conditions for long wave equation in the hodograph plane

We discuss implications of the seaward boundary conditions used in initial-boundary value problem formulation of nonlinear shallow-water wave propagation over a linear slope. We first demonstrate the reflection of wave velocity in the case of Dirichlet condition and that of water elevation in the case of Neumann condition. We then show that linear superposition of the two boundary conditions results in much less reflection at the artificial boundary. We also propose a new boundary condition of mixed type and compare its results with that of the aforementioned conditions.     

Rate processes on fractals: Theory,simulations, and experiments

Heterogeneous kinetics are shown to differ drastically from homogeneous kinetics. For the elementary reaction A + A products we show that the diffusion-limited reaction rate is proportional tot ^– h[A]² or to [A]^x, whereh=1- d _s/2, X=1+2/d _s =(h-2)(h-1), andd _s is the effective spectral dimension. We note that ford = d _s =1, h =1/2 andX = 3, for percolating clustersd _s = 4/3,h = 1/3 andX = 5/2, while for dust ds <1, 1 >h > 1/2 and >X > 3. Scaling arguments, supercomputer simulations and experiments give a consistent picture. The interplay of energetic and geometric heterogeneity results in fractal-like kinetics and is relevant to excitation fusion experiments in porous membranes, films, and polymeric glasses. However, in isotopic mixed crystals, the geometric fractal nature (percolation clusters) dominates.     

On conservation laws and boundary conditions for “short waves” equation

In this paper we study local conservation laws for the equation of short waves in the form of a variational problem. We analyze an infinite symmetry group of the equation and generate a finite number of conservation laws corresponding to given infinite symmetries through appropriate boundary conditions.     

Calculating the pressure in simulations using periodic boundary conditions

Because it is not immediately clear how to write down a proper Hamiltonian for a system in periodic boundary conditions, particularly with Coulombic interactions, we consider a large, finite array of copies of a basic simulation cell containingN particles with some interaction between them. We also putN independent copy particles in each of the copy cells of the array and write down a constrained Lagrangian for the whole system. Constraints on the velocities of the particles of the whole array together with an appropriate initial condition implement the periodic structure in the cells of the array of copies. We derive a Hamiltonian for the whole system with constraints and then derive the equations of motion and a virial expression for the pressure tensor in terms of the forces on the system. In the limit as the array of cell copies becomes large, the equations of motion become the standard ones used in periodic-boundaryconditions simulations. The method also provides an unequivocal algorithm for the pressure in this limit in terms of a virial expression. Particular attention is paid to the case of Coulombic interactions.     

Entropic, LES and boundary conditions in lattice Boltzmann simulations of turbulence

Large scale (1600³-grid) entropic lattice Boltzmann (ELB) simulations are performed on the 27-bit model at sufficiently high Reynolds numbers to find intermittency corrections to the Kolmogorov k ^-5/3 inertial spectrum. Even though the transport coefficients in ELB and in the Large Eddy Simulation (LES) lattice Boltzmann schemes have very different origins, there are strong similarities in their turbulence statistics from 512³-grid simulations. A new LB moment-space boundary condition algorithm is tested on the 2D backstep problem, with excellent agreement with experimental data even up to a Reynolds number of 800.     

Non-periodic boundary conditions for molecular simulations of condensed matter

A formally exact effective boundary potential is defined that can be combined with hard-wall boundary conditions to allow computer simulations to be performed that mimic an infinite system. The effective potential accounts for the interactions that would be present across the boundary if the system were truly infinite. The exact many-body potential is approximated by one- and two-body potentials. The algorithm is implemented and tested for a Lennard–Jones fluid. The advantages of the present formulation over the widely used periodic boundary conditions are discussed.     

Minimal kinematic boundary conditions for simulations of disordered microstructures

In simulations of representative volume elements (RVEs) of materials with disordered microstructures, commonly used rigid and periodic boundary conditions (BCs) introduce additional constraints, causing: (i) boundary effects, (ii) unrealistic stiff response, (iii) artificial wavelengths in the solution fields, and (iv) suppression of solutions with localized deformation that otherwise may occur in the simulation. In this paper we define the minimal kinematic boundary conditions such that only the desired overall strain is imposed on the RVE, with no other undesirable constraints. We prove that such BCs result in a unique solution for the linear elastic case, and that the uniqueness for nonlinear problems is dependent on the pointwise positive definiteness of the incremental stiffness tensor. Upon incorporating the minimal BCs into the finite element framework, we consider, as an example, two-dimensional, linear elastic, disordered polycrystals and perform a systematic study of the effects of boundary conditions while varying the RVE size and controlling the sampling error. The results demonstrate that the minimal BCs, applicable to a RVE of any shape, are superior to other BCs, in that they give more realistic overall behaviour, reduce the required size of the RVE, and eliminate the superficial wavelengths in the solution field, ubiquitous in simulations with other boundary conditions.     

Lateral heat diffusion in layered structures: Theory and photothermal experiments

It is very important to understand how the heat diffuses in the case of nano objects and of layered structures. We will demonstrate that the lateral diffusion could be very different in the case of a good conductor layer deposited on a bad one and for the reverse (bad one on good substrate). Moreover, we will show how a thermoreflectance experiment can reveal both the thermal conductivity and the thermal diffusivity of thin layers deposited on substrates.     

Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.     

On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces

This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded bodyD with general boundary conditions. First, mildL ¹-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with anH-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weakL ¹-solutions are proved by compactness, using theH-theorem from the cutoff case. Finally, anH-theorem is given also for the infinite-range case.     

Determination of the defining boundary in nuclear magnetic resonance diffusion experiments

While nuclear magnetic resonance diffusion experiments are widely used to resolve structures confining the diffusion process, it has been elusive whether they can exactly reveal these structures. This question is closely related to x-ray scattering and to Kac's "hear the drum" problem. Although the shape of the drum is not "hearable," we show that the confining boundary of closed pores can indeed be detected using modified Stejskal-Tanner magnetic field gradients that preserve the phase information and enable imaging of the average pore in a porous medium with a largely increased signal-to-noise ratio.     

Fractional diffusion equation in half-space with Robin boundary condition

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.     

Characteristic boundary conditions for direct simulations of turbulent counterflow flames

Improved Navier–Stokes characteristic boundary conditions (NSCBC) are formulated for the direct numerical simulations (DNS) of laminar and turbulent counterflow flame configurations with a compressible flow formulation. The new boundary scheme properly accounts for multi-dimensional flow effects and provides nonreflecting inflow and outflow conditions that maintain the mean imposed velocity and pressure, while substantially eliminating spurious acoustic wave reflections. Applications to various counterflow configurations demonstrate that the proposed boundary conditions yield accurate and robust solutions over a wide range of flow and scalar variables, allowing high fidelity in detailed numerical studies of turbulent counterflow flames.     

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

Because of the long range nature of hydrodynamic interactions, the problem of boundary conditions on a finite simulation cell of a hydrodynamically dense suspension of particles in Brownian motion is quite as complicated as the analogous problem in simulation of the statistical mechanics of charged and dipolar systems. One resolution of this problem is to use periodic boundary conditions and to view them as a way of describing a physical system composed of a large spherical array of periodic replicas of the simulation cell. The hydrodynamic interactions are calculated using the quasi-static linearized Navier-Stokes equation. This requires that the suspending fluid velocity remains small throughout the array. That the sum of the particle velocities in the simulation cell be zero is insufficient to force boundedness of the fluid velocity as the array becomes large. Boundedness in the array of the suspending fluid velocity is achieved if a rigid wall boundary condition is applied at the outer edge of the array as the array becomes large. With this condition the net particle velocity equals zero condition is not needed. The condition allows lattice sum representations for the suspending fluid velocity to be derived. These lattice sums are absolutely and rapidly convergent and periodic. Representations of the velocity in the array with boundary condition allow calculation of mobility tensors which are also periodic and can be evaluated numerically in tolerable amounts of computer time. A major effect of these calculations is to identify the physical model system corresponding to a truly periodic fluid velocity and mobility tensor as a large array with rigid wall boundary condition.     

Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations.

Percolation objects were fabricated based on computer-generated, two- or three-dimensional templates. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by NMR imaging and, in the presence of a pressure gradient, NMR velocity mapping. The fractal dimension, the correlation length, and the percolation probability were evaluated both from the computer-generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e., between a dynamic and a structural parameter. We examined interdiffusion between two compartments initially filled with H2O and D2O, respectively, by proton imaging. The results confirm the theoretical expectation. As a third transport mechanism, thermal convection in percolation clusters of different porosities was studied with the aid of NMR velocity mapping. The velocity distribution is related to the convection roll size distribution. Corresponding histograms consist of a power law part representing localized rolls, and a high-velocity cut-off for cluster-spanning rolls. The maximum velocity as a function of the porosity clearly visualizes the percolation transition.     

Exact solution for the fractional cable equation with nonlocal boundary conditions

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.