Formulation Of Boundary Conditions Research Articles

Radiation boundary conditions for time-dependent waves based on complete plane wave expansions

We develop complete plane wave expansions for time-dependent waves in a half-space and use them to construct arbitrary order local radiation boundary conditions for the scalar wave equation and equivalent first order systems. We demonstrate that, unlike other local methods, boundary conditions based on complete plane wave expansions provide nearly uniform accuracy over long time intervals. This is due to their explicit treatment of evanescent modes. Exploiting the close connection between the boundary condition formulations and discretized absorbing layers, corner compatibility conditions are constructed which allow the use of polygonal artificial boundaries. Theoretical arguments and simple numerical experiments are given to establish the accuracy and efficiency of the proposed methods.

Design of a two‐step pulsed pressure‐swing adsorption‐based oxygen concentrator

AbstractA two‐step pulsed pressure‐swing adsorption (PPSA) process has been modeled to assess the extent to which an oxygen concentrator might be miniaturized for medical applications. The process consists of a single bed of packed adsorbent particles that is alternately pressurized and depressurized at the feed end. An enriched oxygen product is withdrawn at ambient pressure from the product end when the bed is pressurized at the feed end. The product end remains closed during depressurization. The model development addresses the manner in which axial dispersion enters into the describing equations and the formulation of proper boundary conditions, both of which have not been handled rigorously in some prior modeling studies. The describing equations are solved using COMSOL® Multiphysics software. The effect on the performance of the adsorption time, desorption time, bed length, particle diameter, and imposed pressure drop across the bed have been investigated. An interesting novel result is that for a chosen particle size, bed length, and applied pressure drop, there is an optimum combination of adsorption and desorption times that maximizes the product purity. The results suggest that there are operating windows for both 5A and partially Ag‐exchanged Li‐substituted 13X zeolite adsorbents wherein the product oxygen purity is greater than 90%. At a given product flow rate within this operating window, the extent of miniaturization is limited by the (maximum) cycling frequency that is practically achievable. Sizing of an oxygen concentrator for personal medical applications is also discussed. A principal conclusion is that a compact oxygen concentrator capable of producing a highly oxygen‐enriched product is possible using commercially available adsorbents and implementable operating conditions. © 2009 American Institute of Chemical Engineers AIChE J, 2010

Bi-polyanalytic functions on the upper half plane

The equation for , is investigated in the upper half plane. A Cauchy integral representation formula is obtained explicitly. Different forms of boundary conditions originating from the well-known Schwarz, Dirichlet and Neumann problems from complex analysis are studied. These boundary value problems are solved in the upper half plane.

Mixed boundary conditions for piezoelectric plates

For plate bending and stretching problems in piezoelectric materials, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate mixed boundary conditions accurate to all order. A decay analysis technique is used to establish necessary conditions that the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate. For the case of axisymmetric bending and stretching of a circular plate, these decaying state conditions are obtained explicitly for the first time when the mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution.

On the stability and dissipation of wall boundary conditions for compressible flows

AbstractCharacteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.

Geometrization of metric boundary data for Einstein’s equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.

Consistent Boundary Conditions for 2D and 3D Lattice Boltzmann Simulations

Consistent formulations of 2D and 3D pressure and velocity boundary conditions along both the stationary and non-stationary plane wall and corner for lattice Boltzmann simulations are proposed. The unknown distribution functions are made function of local known distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definitions of den- sity and momentum. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. Discrete macroscopic equa- tion is also derived for steady fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate cen- tral difference equation predicts continuous distribution across the boundary pro- vided the boundary unknown distribution functions satisfy the macroscopic quan- tity. Three different local known distribution functions are experimented to assess both this observation and the applicability of the present formulation, and are scru- tinized by calculating two-dimensional Couette-Poiseuille flow, Couette flow with wall injection and suction, lid-driven square cavity flow, and three-dimensional square duct flow. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known distri- bution functions is as expected negligible, which are consistent with the results from the derived discrete macroscopic equation.

Nonnormal Multidecadal Response of the Thermohaline Circulation Induced by Optimal Surface Salinity Perturbations

Abstract Optimal perturbations of sea surface salinity are obtained for an idealized North Atlantic basin using a 3D planetary geostrophic model—optimality is defined with respect to the intensity of the meridional overturning circulation. Both optimal initial and stochastic perturbations are computed in two experiments corresponding to two different formulations of the surface boundary conditions: the first experiment uses mixed boundary conditions (i.e., restoring surface temperature and prescribed freshwater flux), whereas the second experiment uses flux boundary conditions for both temperature and salinity. The latter reveals greater responses to both initial and stochastic perturbations that are related to the existence of a weakly damped oscillatory eigenmode of the Jacobian matrix, the optimal perturbations being closely related to its biorthogonal. The optimal initial perturbation induces a transient modification of the circulation after 24 yr. The spectral response to the optimal stochastic perturbation reveals a strong peak at 35 yr, corresponding to the period of this oscillatory eigenmode. This study provides an upper bound of the meridional overturning response at multidecadal time scales to freshwater flux perturbation: for typical amplitudes of Great Salinity Anomalies, initial perturbations can alter the circulation by +2.25 Sv (1 Sv ≡ 106 m3 s−1; i.e., 12.5% of the mean circulation) at most; stochastic perturbations with amplitudes typical of the interannual variability of the freshwater flux in midlatitudes induce a circulation variability with a standard deviation of 1 Sv (i.e., 5.5% of the mean circulation) at most.

A high-resolution code for turbulent boundary layers

A new high-resolution code for the direct simulation of incompressible boundary layers over a flat plate is described. It can accommodate a wide range of pressure gradients, and general time-dependent boundary conditions such as incoming wakes or wall forcing. The consistency orders of the advective and pressure-correction steps are different, but it is shown that the overall resolution is controlled by the higher-order advection step. The formulation of boundary conditions to ensure global mass conservation in the presence of arbitrary forcing is carefully analyzed. Two validation boundary layers with and without a strong adverse pressure gradient are presented, with maximum Reynolds numbers Re θ ≈ 2000 . They agree well with the available experiments. Turbulent inflow conditions for the zero-pressure case are implemented by a recycling method, and it is shown that at least the initial 300 momentum thicknesses have to be discarded before the effect of the artificial inflow is forgotten. It is argued that this is not a defect of the method used to generate the inflow, but a property of the boundary layer.

Duct Orifices, Part II: Circular Ducts

Duct Orifices, Part II: Circular Ducts Part I of the present paper has demonstrated the application of mode analysis to the evaluation of sound fields at flat duct orifices of some shapes by subdivision of the sound field area in zones and formulation of the boundary conditions for sound pressure and normal particle velocity at different zone limits, respectively. The present Part II of the paper will apply that method to orifices of circular ducts.

Modelling of graded index waveguide fabricated by ion exchange on Er3+ doped glass

A general numerical tool, based on thermal diffusion equation and full-vectorial eigen-mode equation, has been presented for the systematic analysis of graded index channel waveguide fabricated by ion exchange on Er3+ doped glass. Finite difference method with full-vectorial formulation (FV-FDM) is applied to solving the full-vectorial modes of graded index channel waveguide for the first time. The coupled difference equations based on magnetic fields in FV-FDM are derived from the Taylor series expansion and accurate formulation of boundary conditions. Hybrid nature of vectorial guided modes for both pump (980 nm) and signal light (1550 nm) are demonstrated by the simulation. Results show that the fabrication parameters of ion exchange, such as channel opening width and time ratio of second step to first step in ion exchange, have large influence on the properties of waveguide. By optimizing the fabrication parameters, maintenance of monomode for signal light and improvement of the gain dynamics can be achieved in Er3+ doped waveguide amplifier (EDWA) fabricated by ion exchange technique. This theoretical model is significant for the design and fabrication of EDWA with ion exchange technique. Furthermore, a single polarization EDWA, which operates at wavelength from 1528 nm to 1541 nm for HE polarization, is numerically designed.

On the convergence of compact difference schemes

Difference schemes that are compact in space, i.e., schemes constructed on a two- or three-point stencil in each spatial direction, are more efficient and convenient for boundary condition formulation than other high-order accurate schemes. Originally, these schemes were developed primarily to obtain smooth solutions. In the last two decades, compact schemes have been actively used to compute gas dynamic flows with shock waves. However, when a numerical solution with guaranteed accuracy is desired, the actual properties of difference schemes have to be known in the calculation of solutions with discontinuities. For some widely used compact schemes, this issue has not yet been well studied. The properties of compact schemes constructed by the method of lines are examined in this paper. An initial-boundary value problem for the linear heat equation with discontinuous initial data is used as a test problem. In the method of lines, the spatial derivative in the heat equation is approximated on a two-point stencil according to a fourth-order accurate compact differentiation formula. The resulting evolution system of ordinary differential equations is solved using various implicit one-step two- and three-stage schemes of the second and third order of accuracy. The relation between the properties of the stability function of a scheme and the spatial monotonicity of the numerical solution is analyzed. In computations over long time intervals, the compact schemes are shown to be superior to traditional schemes based on the second-order accurate three-point approximation of the spatial derivative.

Comparative study of thermal post-buckling analysis of uniform slender & shear flexible columns using rigorous finite element and intuitive formulations

Thermal post-buckling analysis of uniform, isotropic, slender and shear flexible columns is presented using a rigorous finite element formulation and a much simpler intuitive formulation. The ends of the columns are axially restrained to move and consequently any temperature rise above the stress free condition of the column produces an equivalent constant compressive mechanical load that causes the column to buckle at a critical temperature. Further increase in temperature beyond critical temperature results in the thermal post-buckling phenomenon. As a result of constraints imposed on the axial displacement at the ends of the column, the post-buckling phenomenon is governed by the von-Karman strain displacement relation applicable to one dimensional problems. Empirical formula for ratio of nonlinear axial load to critical load (equivalent constant mechanical load for a given temperature rise) as a function of the central deflection are obtained using both the rigorous finite element and intuitive formulations for various boundary conditions. The boundary conditions considered are the classical such as hinged–hinged, clamped–clamped and clamped–hinged conditions and nonclassical boundary conditions like the hinged–guided or the clamped–guided conditions. Post-buckling analysis results pertaining to nonclassical boundary conditions are meagre in the literature. It is observed that results obtained from both the formulations are in excellent agreement for all boundary conditions considered. Also the accuracy and simplicity of the intuitive formulation is aptly demonstrated to slender and shear flexible columns.

CONDUCTIVE MEDIUM MODELING WITH AN AUGMENTED GIBC FORMULATION

This paper describes an augmented generalized impedance boundary condition (AGIBC) formulation for accurate and e-cient modeling of conductive media. It is a surface integral equation method, so that it uses a smaller number of unknowns. The underlying GIBC provides a rigorous way to account for the skin efiect. Combining with the novel augmentation technique, the AGIBC formulation works stably in the low-frequency regime. No loop- tree search is required. The formulation also allows for its easy incorporation of fast algorithms to enable the solving of large problems with many unknowns. Numerical examples are presented to validate the formulation.

Faster 3D finite element time domain – Floquet absorbing boundary condition modelling using recursive convolution and vector fitting

A recursive convolution (RC) based on the vector fitting (VF) method and triangular temporal basis functions is employed to compute the Floquet absorbing boundary condition (FABC) formulation in the vector 3D finite element time-domain (FETD) modelling of doubly periodic structures. This novel implementation (VF-RC-FETD-FABC) results in significantly lower computation time requirements than the standard convolution (SC) implementation. The numerical examples presented show a reduction in computation time requirements by a factor of at least 3.5. In addition, a time window in excess of 50 000 time steps is recorded over which practically stable results are obtained. This temporal window is sufficiently large to allow the modelling of many practical problems. The results from four such problems are presented, confirming the accuracy and speed of the VF-RC-FETD-FABC software.

Simulation and Modeling the Mechanical Behavior of Textile Reinforced Composites by Combining the Binary Model and X‐FEM

AbstractThis paper presents a new efficient modeling strategy based on the combination of Binary Model and X–FEM. It is applied to represent the inner architecture of textile reinforced composites where the fabric is characterized by a complex geometry. Homogenization methods are used to compute the effective elastic material properties. Thereby, the discrete formulation of periodic boundary conditions is adapted. Finally, the results in terms of effective material properties reveal a good agreement with parameters obtained by experimental tests. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling and numerical simulation of secondary settlers: A Method of Lines strategy

In this paper, attention is focused on a parabolic partial differential equation (PDE) modeling sedimentation in a secondary settler and the proper formulation of the problem boundary conditions (i.e., the conditions prevailing at the feed, clear water and sludge outlets). The presence of a diffusion term in the equation not only allows the reproduction of experimental observations, as reported in a number of works, but also makes the numerical solution of the initial-boundary value problem significantly easier than the original conservation law (which is a nonlinear hyperbolic PDE problem requiring advanced numerical techniques). A Method of Lines (MOL) solution strategy is then proposed, based on the use of finite differences or spectral methods, and on readily available time integrators. The efficiency and flexibility of the general procedure are demonstrated with various numerical simulation results.

Boundary conditions for the bending of a piezoelectric beam

For beam bending in transversely isotropic piezoelectric media, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the beam of general edge geometry and loadings. By generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed edge-data of the beam must satisfy these conditions in order that they could generate a decaying state within the beam. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions for the case of bending deformation of piezoelectric beam are derived explicitly. They are then used for the correct formulation of boundary conditions for the beam theory solution (or the interior solution). Besides, an analytical solution of elastic beam is formulated to verify validity of our boundary conditions. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two sets of mixed edge-data are obtained for the first time.

Measurement of the thermal contact parameters at a workpiece – tool interface in a HSM process

In high speed machining process, the workpiece-tool thermal contact is not well known. We propose an experimental approach founded on an original measurement principle that allows accurate estimation of the thermal contact parameters. These latter are three: the thermal contact resistance, the generated heat flux density and the partition coefficient of this latter. In this paper we present the principle of measurement of these parameters considering the global theoretical model and its difficult and propose a simpler alternative model. This latter consider separately both sub domains to estimate the thermal superficial conditions. That allows determining the three parameters of contact allowing a precise formulation of the boundary condition at the workpiece-tool interface. The purpose of this simplified model is to uncorrelate the estimates of generated heat flux and the partition coefficient.

Quadrature-free spline method for two-dimensional Navier-Stokes equation

In this paper, a quadrature-free scheme of spline method for two-dimensional Navier-Stokes equation is derived, which can dramatically improve the efficiency of spline method for fluid problems proposed by Lai and Wenston (2004). Additionally, the explicit formulation for boundary condition with up to second order derivatives is presented. The numerical simulations on several benchmark problems show that the scheme is very efficient.