Approximate Boundary Conditions Research Articles

Modeling of the nonlinear vibrations of a stiffened moderately thick plate

We consider a multi-structure consisting of a plate reinforced by a thin stiffener on a portion of its boundary. The model we consider for this structure (viewed as a heterogeneous plate) is nonlinear and takes into account the transverse shear effects. Our aim is to model this junction and reproduce the effect of the thin stiffener by means of approximate boundary conditions on the junction region.

Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations

We consider one-dimensional fractional sub-diffusion equations on an unbounded domain. For a problem of this type for which an exact or approximate artificial boundary condition is available we reduce it to an initial-boundary value problem on a bounded domain. We then analyze the numerical solution of the problem by polynomial and nonpolynomial spline methods. The consistency and the Von Neumann stability analysis of these methods are also discussed. Numerical experiments clarify the effectiveness and order of accuracy of the proposed methods.

On the Accuracy of Asymptotic Solutions for TM Waves Diffracting on an Impedance Wedge

The contribution focuses on the accuracy of two asymptotic solutions aimed at representing the electromagnetic field scattered by penetrable wedges. One is a heuristic manipulation of the solution for the perfect electrical conductor, and the other one is a more rigorous coefficient based on approximate boundary conditions. The results presented here extend those proposed by other authors by illustrating the accuracy of such solutions at the edge of validity of the uniform theory of diffraction. In particular, they show that the heuristic formulation can be freely applied in similar conditions, while the other might not always lead to accurate predictions.

Analysis of an arbitrary-profile, cylindrical, impedance reflector surface illuminated by an E-polarized complex line source beam

Electromagnetic scattering from a cylindrical reflector surface having an arbitrary conic section profile is studied. We assumed an electrically thin layer antenna illuminated by a complex line source in E-polarization mode. Our boundary value formulation, without loss of generality, involves an integral equation approach having impedance-type thin-layer boundary conditions. For simplicity, we also considered both faces of the reflector of the same uniform impedance value. Our computation employs the Method of Analytical Regularization (MAR) technique: the integral equations are converted into the discrete Fourier transform domain yielding two coupled dual series equations, which are then solved by the Fourier inversion and Riemann Hilbert Problem techniques. We demonstrate the accuracy and the convergence behaviors of our numerically solved MAR results that can serve as an accurate benchmark for comparison with widely used results obtained by approximate boundary conditions.

Structure of the disturbed region of the atmosphere after the nuclear explosion in Hiroshima

An attempt is undertaken to describe the development of the disturbed region of the atmosphere caused by the nuclear explosion over Hiroshima on August 6, 1945. Numerical simulation of the phenomenon is performed using the dynamic equations for a nonconducting inviscid gas taking into account the combustion of urban buildings, phase changes of water, electrification of ice particles, and removal of soot particles. The results of the numerical calculation of the development of the disturbed region indicate heavy rainfall, the formation of a storm cloud with lightning discharges, removal of soot particles, and the formation of vertical vortices. The temporal sequence of these meteorological phenomena is consistent with the data of observations. Because of the assumptions and approximations used in solving the problem, the results are of qualitative nature. Refinement of the results can be obtained by a more detailed study of the approximate initial and boundary conditions of the problem.

Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons

Li's Fourier factorization rules [J. Opt. Soc. Am. A13, 1870 (1996)] should be applied to achieve a fast convergence rate in the analysis of diffraction gratings with the Fourier modal method. I show, however, that Li's inverse rule cannot be applied for periodic patterns of graphene when the conventional boundary condition is used. I derive an approximate boundary condition in which a nonzero but sufficiently small height is assumed for the boundary. The proposed boundary condition enables us to apply the inverse rule, leading to a significantly improved convergence rate. A periodic array of graphene ribbons is in fact a special type of finite-conductivity strip grating, and thus the proposed approach is also applicable to these kinds of structures.

Effects on criticality of selected scattering phase functions in neutron transport equation using the Chebyshev approximation

Abstract Criticality and diffusion problems have been widely studied in neutron transport theory using polynomial expansion techniques in different geometries. In this study, instead of conventional scattering functions, two different phase function will be used in slab geometry transport equation by using the TN approximation and Marshak boundary conditions. Critical half thicknesses of the slab will be calculated. The numerical results obtained will be presented in tables.

Second-mode attenuation and cancellation by porous coatings in a high-speed boundary layer

AbstractNumerical simulations of the linear and nonlinear two-dimensional Navier–Stokes equations, and linear stability theory are used to parametrically investigate hypersonic boundary layers over ultrasonic absorptive coatings. The porous coatings consist of a uniform array of rectangular pores (slots) with a range of porosities and pore aspect ratios. For the numerical simulations, temporally (rather than spatially) evolving boundary layers are considered and we provide evidence that this approximation is appropriate for slowly growing second-mode instabilities. We consider coatings operating in the typical regime where the pores are relatively deep and acoustic waves and second-mode instabilities are attenuated by viscous effects inside the pores, as well as regimes with phase cancellation or reinforcement associated with reflection of acoustic waves from the bottom of the pores. These conditions are defined as attenuative and cancellation/reinforcement regimes, respectively. The focus of the present study is on the cases which have not been systematically studied in the past, namely the reinforcement regime (which represents a worst-case scenario, i.e. minimal second-mode damping) and the cancellation regime (which corresponds to the configuration with the most potential improvement). For all but one of the cases considered, the linear simulations show good agreement with the results of linear instability theory that employs an approximate porous-wall boundary condition, and confirm that the porous coating stabilizing performance is directly related to their acoustic scattering performance. A particular case with relatively shallow pores and very high porosity showed the existence of a shorter-wavelength instability that was not initially predicted by theory. Our analysis shows that this new mode is associated with acoustic resonances in the pores and can be more unstable than the second mode. Modifications to the theoretical model are suggested to account for the new mode and to provide estimates of the porous coating parameters that avoid this detrimental instability. Finally, nonlinear simulations confirm the conclusions of the linear analysis; in particular, we did not observe any tripping of the boundary layer by small-scale disturbances associated with individual pores.

Line-source excited impulsive EM field response of thin plasmonic metal films

In this paper, reflection against and transmission through thin plasmonic metal films, basic building blocks of many plasmonic devices, are analytically investigated directly in the time domain for an impulsive electric and magnetic line-source excitation. The electromagnetic properties of thin metallic films are modeled via the Drude model. The problem is formulated with the help of approximate thin-sheet boundary conditions and the analysis is carried out using the Cagniard–DeHoop technique. Closed-form space-time expressions are found and discussed. The obtained time-domain analytical expressions reveal the existence of the phenomenon of transient oscillatory surface effects along a plasmonic metal thin sheet. Illustrative numerical examples of transmitted/reflected pulsed fields are provided.

Approximation error method for full-wave tomography

In ultrasound tomography (UT), the speed of sound (SOS) is reconstructed based on ultrasound measurements made on the surface of the object. As a part of the reconstruction process, which includes the solution of the inverse problem, propagation of acoustic signals in the medium is simulated using a forward model. Consequently, modeling errors can generate artifacts into reconstructed SOS. Accurate full-wave models can be computationally heavy and thus impractical in many real applications. On the other hand, approximate models typically lead to less accurate reconstructions. In this study, measurement noise and modeling errors of UT are modeled in Bayesian framework, and a numerical method that takes both errors into account is developed. The performance of the method is investigated by numerical simulations in which artifacts generated by a fast but less accurate forward model and approximate boundary conditions are compensated.

Including fluid shear viscosity in a structural acoustic finite element model using a scalar fluid representation

An approximate boundary condition is developed in this paper to model fluid shear viscosity at boundaries of coupled fluid–structure system. The effect of shear viscosity is approximated by a correction term to the inviscid boundary condition, written in terms of second order in-plane derivatives of pressure. Both thin and thick viscous boundary layer approximations are formulated; the latter subsumes the former. These approximations are used to develop a variational formation, upon which a viscous finite element method (FEM) model is based, requiring only minor modifications to the boundary integral contributions of an existing inviscid FEM model. Since this FEM formulation has only one degree of freedom for pressure, it holds a great computational advantage over the conventional viscous FEM formulation which requires discretization of the full set of linearized Navier–Stokes equations. The results from thick viscous boundary layer approximation are found to be in good agreement with the prediction from a Navier–Stokes model. When applicable, thin viscous boundary layer approximation also gives accurate results with computational simplicity compared to the thick boundary layer formulation. Direct comparison of simulation results using the boundary layer approximations and a full, linearized Navier–Stokes model are made and used to evaluate the accuracy of the approximate technique. Guidelines are given for the parameter ranges over which the accurate application of the thick and thin boundary approximations can be used for a fluid–structure interaction problem.

Incorporation of Exact Boundary Conditions into a Discontinuous Galerkin Finite Element Method for Accurately Solving 2D Time-Dependent Maxwell Equations

A scheme that discretizes exact absorbing boundary conditions (EACs) to incorporate them into a time-domain discontinuous Galerkin finite element method (TD-DG-FEM) is described. The proposed TD-DG-FEM with EACs is used for accurately characterizing transient electromagnetic wave interactions on two-dimensional waveguides. Numerical results demonstrate the proposed method's superiority over the TD-DG-FEM that employs approximate boundary conditions and perfectly matched layers. Additionally, it is shown that the proposed method can produce the solution with ten-eleven digit accuracy when high-order spatial basis functions are used to discretize the Maxwell equations as well as the EACs.

Modeling Active Electrolocation in Weakly Electric Fish

In this paper, we provide a mathematical model for the electrolocation in weakly electric fishes. We first investigate the forward complex conductivity problem and derive the approximate boundary conditions on the skin of the fish. Then we provide a dipole approximation for small targets away from the fish. Based on this approximation, we obtain a noniterative location search algorithm using multifrequency measurements. We present numerical experiments to illustrate the performance and the stability of the proposed multifrequency location search algorithm. Finally, in the case of disk- and ellipse-shaped targets, we provide a method for reconstructing separately the conductivity, the permittivity, and the size of the targets from multifrequency measurements.

Theoretical investigation of 2D periodic nanoplasmon structures

The problem of diffraction of electromagnetic waves by 2D periodic metal gratings is solved with allowance for the finite permittivity of a metal in the optical band. The developed mathematical model is based on the solution of the vector integro-differential equation of diffraction by 3D dielectric bodies by means of the Galerkin method. It is noted that the dependence of the scattered field amplitude on the wavelength has a resonance character and that the resonance wavelengths can substantially exceed the dimensions of a grating cell. The application of the method of approximate boundary conditions for the calculation of gratings containing nanodimensional metal layers is justified. It is demonstrated that a grating with small reflection and transmission factors under the plasmon-resonance conditions can be created.

Accurate and Efficient Simulation of Multiphase Flow in a Heterogeneous Reservoir With Error Estimate and Control in the Multiscale Finite-Volume Framework

Summary The multiscale finite-volume (MSFV) method is designed to reduce the computational cost of elliptic and parabolic problems with highly heterogeneous anisotropic coefficients. The reduction is achieved by splitting the original global problem into a set of local problems (with approximate local boundary conditions) coupled by a coarse global problem. It has been shown recently that the numerical errors in MSFV results can be reduced systematically with an iterative procedure that provides a conservative velocity field after any iteration step. The iterative MSFV (i-MSFV) method can be derived with an improved (smoothed) multiscale solution to enhance the localization conditions, with a Krylov subspace method [e.g., the generalized-minimal-residual (GMRES) algorithm] preconditioned by the MSFV system, or with a combination of both. In a multiphase-flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure), which is necessary to achieve the desired accuracy in the saturation solution. In this work, we extend the i-MSFV method to sequential implicit simulation of time-dependent problems. To control the error of the coupled saturation/pressure system, we analyze the transport error caused by an approximate velocity field. We then propose an error-control strategy on the basis of the residual of the pressure equation. At the beginning of simulation, the pressure solution is iterated until a specified accuracy is achieved. To minimize the number of iterations in a multiphase-flow problem, the solution at the previous timestep is used to improve the localization assumption at the current timestep. Additional iterations are used only when the residual becomes larger than a specified threshold value. Numerical results show that only a few iterations on average are necessary to improve the MSFV results significantly, even for very challenging problems. Therefore, the proposed adaptive strategy yields efficient and accurate simulation of multiphase flow in heterogeneous porous media.

Viscous boundary layer correction on a pressure-field acoustic model

Fluid viscosity plays an important role in many acoustics and structural acoustics problems. For example, using an inviscid approximation to the flow of fluid-loaded micro-electro-mechanical systems and micro-scale biological structures results in large errors in the predicted response. Using a linearized Navier--Stokes solution, however, increases the number of unknowns by at least a factor of three compared to an inviscid approximation where pressure is the only degree of freedom. In this work, an approximate boundary condition is developed to include fluid viscosity for coupled fluid-structure systems. The viscous effect is included as a correction term to the inviscid boundary condition, written in terms of second order in-plane derivatives of pressure. This is the key step enabling the development of a variational formulation that is directly amenable for approximation in a finite element method (FEM) code as only a minor modification to existing structural acoustic code. Hence, this approach retains the great computational advantage over the conventional viscous FEM formulation. We show results demonstrating the accuracy of the approximate boundary condition as compared to the full three dimensional Navier-Stokes solution.

Numerical study of acoustic multiperforated plates

It is rather classical to model multiperforated plates by approximate impedance boundary conditions. In this article we would like to compare an instance of such boundary conditions obtained through a matched asymptotic expansions technique to direct numerical computations based on a boundary element formulation in the case of linear acoustic.

Quantum chemical modeling of the structure formation and proton transfer in 2-hydroxybenzenesulfonic, 4-hydroxy-1,3-benzenedisulfonic, and 1,3-benzenedisulfonic acids

Hydrate clusters of 2-hydroxybenzenesulfonic and 1,3-benzenedisulfonic acids were calculated in terms of the density functional theory (DFT) by the B3LYP/6-31G** method. The process of water adsorption on the crystal surface of 4-hydroxy-1,3-benzenedisulfonic acid dihydrate was simulated using the generalized gradient approximation (DFT/PBE) and periodic boundary conditions. For the model system (OHC6H4SO3 −)·H5O2 +, the activation barriers for the proton transfer were calculated depending on the distance between the O atoms and the deviation of the proton from the O...O bond line. The presence of one H2O molecule per SO3H group is energetically most favorable for the formation of clusters of 1,3-benzenedisulfonic acid containing a stoichiometric amount of water. The simulation of the hydration of 4-hydroxy-1,3-benzenedisulfonic acid dihydrate (OHC6H3(SO3H)2·2 H2O + n H2O, n = 1–3) showed that the superstoichiometric H2O molecule is adsorbed on the crystal surface of this dihydrate with energy release of 0.75–0.95 eV. The position of this water molecule is less favorable in the bulk than on the surface.

NUMERICAL STUDY OF THE INTERACTION BETWEEN POSITIVE AND NEGATIVE MASS OBJECTS IN GENERAL RELATIVITY

Based on Baumgarte–Shapiro–Shibata–Nakamura formalism and moving puncture method, we demonstrate the first numerical evolutions of the interaction between positive and negative mass objects. Using the causal property of general relativity, we set our computational domain around the positive mass black hole while excluding the region around the naked singularity introduced by the negative mass object. Besides the usual Sommerfeld numerical boundary condition, an approximate boundary condition is proposed for this nonasymptotically-flat computational domain. Careful checks show that either boundary condition introduces smaller error than the numerical truncation errors. This is consistent with the causal property of general relativity. Except for the numerical truncation error and round-off error, our method gives an exact solution to the full Einstein's equation for a portion of spacetime with two objects whose masses have opposite signs. So our method opens the door for numerical explorations with negative mass objects. Based on this method, we investigate the Newtonian limit of spacetime with two objects whose masses have opposite sign. Our result implies that this spacetime does have a Newtonian limit which corresponds to a negative mass particle chasing a positive mass particle. This result sheds some light on an interesting debate about the Newtonian limit of a spacetime with positive and negative point masses.

Establishing size effects in discontinuous fibre composites using 2D finite element analysis

Size effects are observed within discontinuous fibre composites, such that the material properties change with the specimen volume. Representative volume elements (RVEs) are commonly used to simulate random fibre architectures for finite element analysis of discontinuous fibre composites. A series of simplified 2D RVE models have been created and studied in this paper, in order to determine the relationship between the critical RVE size and fibre length and volume fraction. All models are subjected to periodic boundary conditions, but average properties are extracted from an inner region offset from the model boundary by a distance equivalent to two fibre lengths. According to Saint–Venant’s principle, this offset removes the uncertainty associated with the approximate boundary conditions. A statistical stopping criterion has been adopted to determine the number of realisations required to achieve a representative set of elastic properties for each fibre architecture. The critical RVE side length is shown to be approximately four times the fibre length when considering convergence of the tensile and shear stiffnesses for the range of fibre lengths and volume fractions studied.