Exact Numerical Method Research Articles

The influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity

Based on Sherief ‘s generalized thermodiffusion theory with one relaxation time, the dynamic response of an infinite body with a cylindrical cavity whose surface suffers thermal shock is studied by using finite element method. In order to avoid losing precision in the application of integrated transformation method, finite element equations are solved directly in time domain. The temperature, displacement, stress, diffusion concentration as well as chemical potential are obtained. The results show that the present method is an effective and exact numerical analysis method for the generalized thermoelasticity problem and the generalized thermodiffusion problem. The effects of the diffusional item on temperature, displacement and stress in elastic body are studied simultaneously.

Ground-state properties of interacting two-component Bose gases in a one-dimensional harmonic trap

We study ground-state properties of interacting two-component boson gases in a one-dimensional harmonic trap by using the exact numerical diagonalization method. Based on numerical solutions of many-body Hamiltonians, we calculate the ground-state density distributions in the whole interaction regime for different atomic number ratio, intra- and inter-atomic interactions. For the case with equal intra- and inter-atomic interactions, our results clearly display the evolution of density distributions from a Bose condensate distribution to a Fermi-like distribution with the increase of the repulsive interaction. Particularly, we compare our result in the strong interaction regime to the exact result in the infinitely repulsive limit which can be obtained by a generalized Bose-Fermi mapping. We also discuss the general case with different intra- and inter-atomic interactions and show the rich configurations of the density profiles.

Biased random walks on a lattice: Exact numerical method to study the effect of alternating fields in disordered and asymmetric systems of obstacles

The migration of a particle in a system of obstacles under the action of an external field is often modeled using lattice Monte Carlo algorithms. For example, such simulation methods have been used to study the electrophoresis of charged molecules in sieving gels and the separation of particles using ratchet systems. In the case of constant fields or low-frequency alternating fields, the Monte Carlo simulation method can be mapped onto a numerical or algebraic matrix problem that can be solved exactly. In this Rapid Communication, we generalize this matrix approach to treat periodic time-dependent fields. The evolution of the spatial distribution function during a period is computed using a sequence of transfer matrices, and a steady-state closure relation allows us to calculate the exact mean velocity of the particle during a complete cycle. As an example, we examine the properties of a simple spatially asymmetric ratchet system in the presence of periodic alternating fields (symmetric and asymmetric) as well as random telegraph signals.

Finite element modeling of acoustic scattering from rough interfaces.

The finite element method is applied to the problem of acoustic scattering from rough surfaces. This method has the advantage that the surface can have almost any form whereas analytic approximation methods generally require constrained surface roughness parameters. Scattering from a rough pressure release boundary and from a rough interface between two fluid media is considered in two dimensions. The problem domain is truncated with perfectly matched layers. These greatly decrease the number of degrees of freedom in the finite element problem while minimizing unwanted reflections of outbound energy back into the physical domain. The Helmholtz–Kirchhoff integral is used to extend the solution to points outside the modeled domain. Using finite elements as well as an exact integral equation method, solutions are obtained for multiple realizations of the rough scattering surface. The scattering strengths of energy reflected back into the water and of energy penetrating the sea floor in the case of bottom scattering are calculated over a range of scattering angles. The two exact numerical methods show good agreement. Results are compared with those of analytic approximation methods. [Work sponsored by ONR, Ocean Acoustics.]

Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA

In this paper, a fast exact numerical method, based on the method of moments, is developed to calculate the scattering by an object above a rough surface. N. Déchamps et al. have recently developed the PILE (Propagation-Inside-Layer Expansion) method for a stack of two one-dimensional rough interfaces separating homogeneous media. This method allows us to calculate separately and exactly the multiple scattering contributions inside the layer. This is done with a decomposition by block of the impedance matrix (the inverse of the impedance matrix of each interface and two coupling matrices are involved). The purpose of this paper is to extend the PILE method to the more general case of two illuminated surfaces and to apply it to an object located above a rough surface. In addition, to invert a matrix of large size, the Forward–Backward Spectral Acceleration (FB-SA) approach of complexity 𝒪(N) proposed by Chou and Johnson is applied. The new method, extended PILE combined with FB-SA, is tested on Perfectly Conducting (PC) circular and elliptic cylinders located above a rough surface (dielectric or PC) obeying a Gaussian process with Gaussian and exponential height autocorrelation functions.

Repulsive and attractive Casimir forces in a glide-symmetric geometry

We describe a three-dimensional geometry in which both attractive and repulsive Casimir forces arise using ordinary metallic materials, as computed via an exact numerical method (no uncontrolled approximations). The geometry consists of a zipperlike, glide-symmetric structure formed of interleaved metal brackets attached to parallel plates---because of the interleaving pattern, a net repulsive force can arise from a combination of attractive interactions. Depending on the separation, the perpendicular force between the plates and brackets varies from attractive (large separations) to repulsive (intermediate distances) and back to attractive (close separations), with one point of stable equilibrium in the perpendicular direction. This geometry was motivated by a simple intuition of attractive interactions between surfaces, and so we also consider how a rough proximity-force approximation of pairwise attractions compares to the exact calculations.

Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method

This paper presents three-dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the B-spline Ritz method based on the theory of elasticity. The proposed method is formulated by the Ritz procedure with a triplicate series of B-spline functions as amplitude displacement components. The geometric boundary conditions are numerically satisfied by the method of artificial spring. To demonstrate the convergence and accuracy of the present method, several examples with various boundary conditions are solved, and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories. Rapid, stable convergences as well as high accuracy are obtained by the present method. The effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated. The results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods.

Sequence effects on the forced translocation of heteropolymers through a small channel

By using a recently developed Monte Carlo algorithm and an exact numerical method, we calculate the translocation probability and the average translocation time for charged heterogeneous polymers driven through a nanopore by an external electric field. The heteropolymer chains are composed of two types of monomers (A and B) which differ only in terms of their electric charge. We present an exhaustive study of chains composed of eight monomers by calculating the average translocation time associated with the 256 possible arrangements for various ratios of the monomer charges (lambda(A)lambda(B)) and electric field intensities E. We find that each sequence leads to a unique value of the translocation probability and time. We also show that the distribution of translocation times is strongly dependent on the two forces felt by the monomers ( approximately lambda(A)E and approximately lambda(B)E). Finally, we present results that highlight the effect of having repetitive patterns by studying the translocation times of various block copolymer structures for a very long chain composed of N=2(18) monomers (all with the same number of A and B monomers).

How to constrain inflationary parameter space with minimal priors

We update constraints on the Hubble functionH(ϕ) during inflation, using the most recent cosmic microwave background (CMB) and largescale structure (LSS) data. Our main focus is on a comparison between various commonlyused methods of calculating the primordial power spectrum via analytical approximationsand the results obtained by integrating the exact equations numerically. In each case, weimpose naive, minimally restrictive priors on the duration of inflation. We find that thechoice of priors has an impact on the results: the bounds on inflationary parameterscan vary by up to a factor of two. Nevertheless, it should be noted that, withinthe region allowed by the minimal prior of the exact method, the accuracy ofthe approximations is sufficient for current data. We caution, however, that acareless minimal implementation of the approximative methods allows models forwhich the assumptions behind the analytical approximations fail, and recommendusing the exact numerical method for a self-consistent analysis of cosmologicaldata.

Long-time electron spin storage via dynamical suppression of hyperfine-induced decoherence in a quantum dot

The coherence time of an electron spin decohered by the nuclear spin environment in a quantum dot can be substantially increased by subjecting the electron to suitable dynamical decoupling sequences. We analyze the performance of high-level decoupling protocols by using a combination of analytical and exact numerical methods, and by paying special attention to the regimes of large interpulse delays and long-time dynamics, which are outside the reach of standard average Hamiltonian theory descriptions. We demonstrate that dynamical decoupling can remain efficient far beyond its formal domain of applicability, and find that a protocol exploiting concatenated design provides best performance for this system in the relevant parameter range. In situations where the initial electron state is known, protocols able to completely freeze decoherence at long times are constructed and characterized. The impact of system and control nonidealities is also assessed, including the effect of intrabath dipolar interaction, magnetic field bias and bath polarization, as well as systematic pulse imperfections. While small bias field and small bath polarization degrade the decoupling fidelity, enhanced performance and temporal modulation result from strong applied fields and high polarizations. Overall, we find that if the relative errors of the control pulse flip angles do not exceed 3%, decoupling protocols can still prolong the coherence time by up to 2 orders of magnitude.

Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA

A fast, exact numerical method based on the method of moments (MM) is developed to calculate the scattering from an object below a randomly rough surface. Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)] have recently developed the PILE (propagation-inside-layer expansion) method for a stack of two one-dimensional rough interfaces separating homogeneous media. From the inversion of the impedance matrix by block (in which two impedance matrices of each interface and two coupling matrices are involved), this method allows one to calculate separately and exactly the multiple-scattering contributions inside the layer in which the inverses of the impedance matrices of each interface are involved. Our purpose here is to apply this method for an object below a rough surface. In addition, to invert a matrix of large size, the forward-backward spectral acceleration (FB-SA) approach of complexity O(N) (N is the number of unknowns on the interface) proposed by Chou and Johnson [Radio Sci.33, 1277 (1998)] is applied. The new method, PILE combined with FB-SA, is tested on perfectly conducting circular and elliptic cylinders located below a dielectric rough interface obeying a Gaussian process with Gaussian and exponential height autocorrelation functions.

Exact finite-difference schemes for first order differential equations having three distinct fixed-points

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixed-points.

Nonequilibrium quantum dissipation in spin-fermion systems

Dissipative processes in non-equilibrium many-body systems are fundamentally different than their equilibrium counterparts. Such processes are of great importance for the understanding of relaxation in single molecule devices. As a detailed case study, we investigate here a generic spin-fermion model, where a two-level system couples to two metallic leads with different chemical potentials. We present results for the spin relaxation rate in the nonadiabatic limit for an arbitrary coupling to the leads, using both analytical and exact numerical methods. The non-equilibrium dynamics is reflected by an exponential relaxation at long times and via complex phase shifts, leading in some cases to an "anti-orthogonality" effect. In the limit of strong system-lead coupling at zero temperature we demonstrate the onset of a Marcus-like Gaussian decay with {\it voltage difference} activation. This is analogous to the equilibrium spin-boson model, where at strong coupling and high temperatures the spin excitation rate manifests temperature activated Gaussian behavior. We find that there is no simple linear relationship between the role of the temperature in the bosonic system and a voltage drop in a non-equilibrium electronic case. The two models also differ by the orthogonality-catastrophe factor existing in a fermionic system, which modifies the resulting lineshapes. Implications for current characteristics are discussed. We demonstrate the violation of pair-wise Coulomb gas behavior for strong coupling to the leads. The results presented in this paper form the basis of an exact, non-perturbative description of steady-state quantum dissipative systems.

Application of artificial neural networks to approximation and identification of sea-keeping performance of a bulk carrier in ballast loading condition

Application of artificial neural networks to approximation and identification of sea-keeping performance of a bulk carrier in ballast loading condition This paper presents an application of artificial neural networks to approximation and identification of additional wave-generated resistance, slamming and internal forces depending on ship motion and wave parameters. The analysis was performed for a typical bulk carrier in ballast loading conditions. The investigations were carried out on the basis of ship response data calculated by means of exact numerical methods. Analytical functions presented in the form of artificial neural networks were analyzed with a view of their accuracy against standard values. Possible ways of application of the artificial neural networks were examined from the point of view of accuracy of approximation and identification of the assumed ship response parameters.

Superposition rule for light scattering by a composite particle

A superposition rule for light scattering by composite particles is presented that expresses the scattering amplitude of a composite particle as a superposition of that of the host particle and those of the shadowed inclusions. The superposition rule is derived for a soft composite particle but also provides insight into light scattering by a general composite scatterer. Favorable comparison with an exact numerical method demonstrates the usefulness of the rule in analyzing light scattering by composite particles such as biological cells.

Schottky gate effects on transport properties of Al0.35Ga0.65N/GaN heterostructures

AbstractThe effects of the Schottky gate on transport properties of Al0.35Ga0.65N/GaN heterostructure‐based high electron mobility transistor parameters such as the two‐dimensional electron gas density, two‐dimensional electron mobility and current–voltage characteristics were determined by using exact numerical methods including transferred electron effects from 2‐dimensional to 3‐dimensional states. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computer simulation of muon spin evolution

This paper describes an exact numerical method for simulating the response of the muon polarisation in a variety of experimental situations. It models the muon together with nearby spins such as electrons and nuclei, interacting via dipolar, hyperfine and quadrupolar interactions. It also takes account of Zeeman interaction with both static and RF magnetic fields, and spin flips and site changes. For the simple case of a static environment a Hamiltonian is calculated and diagonalised. External RF fields are modelled by calculating an effective Hamiltonian for the evolution over one RF cycle. For spin flips and site changes the evolution of the density matrix is used, and the solution contains exponential terms representing the relaxation rate. The program uses Monte-Carlo averaging to model powder samples. It can plot out results as a function of field, frequency or hop rate, or vary any parameter to fit experimental data. The output can be as a time domain signal, a frequency spectrum or a calculated quantity such as integral asymmetry, relaxation rate or linewidth.

Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].

ＢＦ‐ｓｐｌｉｎｅ Ｒｉｔｚ法を用いた長方形Ｍｉｎｄｌｉｎ板の振動解析

This paper presents a numerical method, the BF-spline Ritz method, for free vibration analysis of rectangular Mindlin plates with arbitrary boundary conditions. The proposed method utilizes admissible functions comprising the B-spline functions multiplied by a boundary function to define the Ritz trial function for the transverse displacement and rotations of the Mindlin plates. The geometric boundary conditions of the plate at edges are automatically satisfied using boundary functions. To demonstrate the validity and accuracy of the BF-spline Ritz method, several examples are solved, and results are compared with those obtained by exact and other numerical methods. Excellent convergence and high accuracy are obtained by the present method regardless of arbitrary boundary conditions. The effects of thickness-width ratio, aspect ratio and boundary conditions on frequency parameters of rectangular Mindlin plates are shown in tabular forms to serve as benchmark data for future development of new numerical methods.

Exact numerical simulation of power-law noises

Many simulations of stochastic processes require colored noises: I describe here an exact numerical method to simulate power-law noises: the method can be extended to more general colored noises, and is exact for all time steps, even when they are unevenly spaced [as may often happen for astronomical data; see, e.g., N. R. Lomb, Astrophys. Space Sci. 39, 447 (1976)]. The algorithm has a well-behaved computational complexity, it produces a nearly perfect Gaussian noise, and its computational efficiency depends on the required degree of noise Gaussianity.