Discontinuities In Boundary Conditions Research Articles

Planar Gas Expansion under Intensive Nanosecond Laser Evaporation into Vacuum as Applied to Time-of-Flight Analysis.

A computational investigation of the dynamics of gas expansion due to intense nanosecond laser evaporation into vacuum has been carried out. The problem is solved in a one-dimensional approximation, which simplifies calculations and at the same time allows one to analyze the main features of the expansion dynamics. For analysis we use three different approaches. Two of them are based on kinetic analysis via the direct simulation Monte Carlo (DSMC) method and numerical solution of the model Bhatnagar-Gross-Krook (BGK) equation. The third one focuses on derivation of an analytical continuum solution. Emphasis is placed on the analysis of the velocity distribution function and the average energy of particles passing through the time-of-flight detector on the normal to the evaporation surface, which is important for interpreting experimental measurements. The formulated problem is quite difficult as the considered flow is time-dependent, contains discontinuities in boundary conditions and involves large variations of local Knudsen numbers as well as steep gradients of the velocity distribution function. Data were obtained on the particle energy in the time-of-flight distribution for the range of regimes from the free molecular flow to continuum one. The maximum attainable average energy of particles in the time-of-flight distribution is determined. The non-monotonicity of the energy increase was found, which is explained based on analysis of the velocity distribution of particles.

An accuracy-preserving numerical scheme for parabolic partial differential equations subject to discontinuities in boundary conditions

In this paper, we develop a method to alleviate the loss of accuracy that occurs when parabolic partial differential equations (PDEs), subject to discontinuous boundary conditions, are solved numerically. Employing the Keller box finite-difference method, we consider a benchmark case involving the linear one-dimensional transient heat equation, subject to discontinuous heat flux at one of the boundaries. The method we develop constitutes an improvement on earlier work which involved dropping a spatial grid point at each time step; moreover, we demonstrate that our new approach is able to maintain second-order accuracy for the solution and the numerical scheme, whereas it is in general not even possible to calculate the accuracy for either using the earlier approach. Furthermore, our method can also be used in principle for discontinuities in non-linear steady-state boundary-layer problems, such as those that occur in fluid mechanics and electrochemistry; some examples of these are given.

Discontinuous isogeometric boundary element (IGABEM) formulations in 3D automotive acoustics

The isogeometric boundary element method (IGABEM) is a technique that employs non-uniform rational B-splines (NURBS) as basis functions to discretise the solution variables as well as the problem geometry in a boundary element formulation. IGABEM has shown improved convergence properties over the conventional boundary element method (BEM) algorithms. However, in acoustics, IGABEM has only been applied to problems with simple smooth boundary conditions. In most real-world engineering design and analysis acoustic problems, geometric corners and discontinuities in boundary conditions can give rise to more complexity in the solution field that may be more efficiently modelled using a discontinuous approach.In the current work, we develop a discontinuous IGABEM formulation based on discontinuous elements and a suitable collocation scheme. Continuous and discontinuous formulations are compared. In this paper, a three dimensional model with different sets of boundary conditions is presented to explore the conditions under which a discontinuous formulation outperforms the continuous IGABEM. A simple car passenger compartment model characterised by panels with piecewise continuous impedance boundaries is presented to illustrate the potential of the proposed method for integrated engineering design and analysis.

Investigation of wave processes in a rectangular region with discontinuous boundary conditions

In the work in linear formulation the problem of propagation of nonstationary stress waves in an elastic single-supported construction, which is a rectangular strip, is solved. Formulated in terms of stresses and velocities the mixed problem is modeled numerically by means of an explicit difference scheme through counting, based on the method of spatial characteristics. The wave process is caused by applying of an external dynamic load on the front boundary of the rectangular region, and the lateral boundaries of the region are stress-free. At the lower boundary of the rectangular region, inhomogeneous boundary conditions are given, where the middle part of the boundary is rigidly fixed, and its lateral sections are free of stresses. Namely at the points of their conjugation, where the boundary conditions change jump-like, the method is proposed for obtaining resolving equations for finding required functions. The concentration of dynamic stresses in the neighborhood of discontinuity of boundary conditions is investigated. The results of the study are brought to numerical solution.DOI: http://dx.doi.org/10.5755/j01.mech.24.5.20102

Some models of viscous gas dynamics

In this paper, the problems containing two or more small parameters are investigated. In the analysis of such problems, a diagrammatic method is used to visualize the structure of perturbed flow regions, and to estimate the similarity parameters and mathematical models depending on the limiting transitions. The interaction between the laminar flows and transonic flows is considered in the context of the problem containing a small parameter inverse to the Reynolds number, the Mach number other than 1 and the pressure perturbation amplitude of the initiating viscous-inviscid interaction processes. The results of the flow analysis in a laminar boundary layer under the assumption of discontinuity of boundary conditions, for example, discontinuity of the surface velocity, are presented. For such problems a diagram of possible embedded perturbed flow regions is plotted. In addition, the results of studying the nonstationary processes of interaction between the flow in the boundary layer and the external supersonic flow are discussed.

Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions

A method is proposed to improve the accuracy of the numerical solution of elliptic problems with discontinuous boundary conditions using both global and local meshless collocation methods with multiquadrics as basis functions. It is based on the use of special functions which capture the singular behavior near discontinuities in boundary conditions. In the case of global collocation, the method consists in enlarging the functional space spanned by the RBF basis functions, while in the case of local collocation, the method consists in modifying appropriately the problem in order to eliminate the singularities from the formulation. Numerical results for benchmark problems such as a stationary heat equation in a box (harmonic) and Stokes flow in a lid-driven square cavity, show significant improvements in accuracy and in compliance with the continuity equation.

A refined stress-strain analysis in the load transfer zone of flat specimens of high-strength unidirectional composites in uniaxial tension 1. Theoretical analysis

With the use of an analytical approach developed, the influence of distribution of a tangential load on the stress concentration in uniaxially tensioned flat specimens of high-strength unidirectional composites near the grips of a testing machine is evaluated. In view of singularity of the analytical solution derived at the points of discontinuity of boundary conditions, for estimating the stress concentration, it is suggested to employ the averaged value of longitudinal stresses, which is calculated by means of an improper integral across the thickness of a near-surface layer.

A singularity correction procedure for digital simulation of potential-step chronoamperometric transients in one-dimensional homogeneous reaction–diffusion systems

Transient solutions for potential-step chronoamperometry, obtained by conventional finite-difference simulations, are usually grossly inaccurate in the neighbourhood of temporal discontinuities in boundary conditions. In the case of one-dimensional homogeneous reaction–diffusion systems this problem can be overcome by applying a singularity correction procedure proposed in this work. The procedure is a modification of the approach recently suggested by Flyer and Fornberg [J. Comput. Phys. 184 (2003) 526] for the solution of partial differential equations with incompatible initial/boundary conditions. It consists of decomposing any concentration profile into a sum of two components: an analytically obtainable one that satisfies discontinuous boundary conditions but ignores homogeneous reaction(s), and a numerically obtainable part that satisfies continuous boundary conditions and takes the homogeneous reactions into account. A similar decomposition can also be applied to the electric current. Computational tests performed for four simple examples of electrochemical systems indicate that the most accurate results (close to the discontinuity) are obtained by combining the decomposition of the concentrations, applied at all discrete time levels, with the decomposition of the current.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

Abstract Local, difficult-to-solve spatio-temporal structures such as extremely thin reaction layers at the electrodes, thin moving reaction fronts, fast transient variations, isolated temporal discontinuities in boundary conditions, edge effects, etc. are characteristic of the initial boundary value and boundary value problems occurring in electrochemical kinetic modelling. Further progress in the simulation methodology is hindered by the lack of automatic solution techniques for such problems. Based on a critical consideration of various adaptive grid strategies for partial differential equations reported in the scientific literature, a new finite-difference adaptive grid strategy has been formulated, that is especially designed for electrochemical kinetic simulations, and is currently limited to models in one-dimensional space geometry. The intention is to enable an automatic solution of the governing partial/ordinary differential equations to a prescribed accuracy, without any a priori knowledge about the spatio-temporal location of the emerging solution structures. In view of the importance of the solution (concentration) gradients for the electrochemical theory, simultaneous control of the spatial errors of the solutions and their spatial gradients is included in the strategy. Spatial grid adaptation is based on a local uniform grid refinement, using overlapping grid patches. Temporal step selection uses a recent control theoretic algorithm, combined with a simple method of detecting temporal discontinuities of the boundary conditions. A third-order accurate, implicit and L-stable Rosenbrock time-stepping scheme ROWDA3 is used to enable efficient and non-oscillatory temporal integration in the time intervals where a regular solution occurs. A second-order accurate Lawson–Morris–Gourlay extrapolation scheme is used to enable a reliable time-stepping at temporal discontinuities.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to six representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve local spatial and temporal solution structures at the electrodes. The models describe: the square-wave controlled potential transient for an uncomplicated heterogeneous charge transfer reaction, the potential step method and linear potential sweep voltammetry for the standard catalytic reaction mechanism with a pseudo-first-order homogeneous reaction, linear potential sweep voltammetry for an EC reaction mechanism involving a follow-up homogeneous dimerisation, steady-state voltammetry at a rotating disc electrode, and steady-state voltammetry at a spherical microelectrode. These examples are characterised by extremely thin reaction layers and other instances of local spatial regions of considerable variations of the solution, including transient effects caused by temporal discontinuities of boundary conditions. The strategy creates dynamically spatio-temporal grids well adapted to such solution structures, without any a priori knowledge about their location. The accuracy of the numerical solutions proves well controlled by means of error tolerance parameters. Numerical difficulties observed with the previously described adaptive moving grid technique do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of thin reaction layers associated with fast second-order homogeneous reactions, and boundary layers associated with hydrodynamic electroanalytical methods, for which the computational effort appears too large.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 3. An adaptive moving grid—adaptive time step strategy for problems with discontinuous boundary conditions at the electrodes

The finite-difference adaptive moving grid strategy suggested in Parts 1 and 2 for the solution of electrochemical kinetic one-dimensional partial differential equations has been further extended to allow temporal grid adaptation as well as time-stepping across discontinuities in boundary conditions. This extension aims at the development of a general simulation algorithm, capable of a largely automatic solution of a variety of kinetic problems. The validity of the strategy developed has been tested on three examples of the modelling of electrochemical transients: square-wave-controlled potential transient in pure diffusion conditions, potential-step transient for the catalytic electrode reaction mechanism with a fast homogeneous reaction and linear potential scan voltammetric transient for an electrode reaction accompanied by a fast follow-up homogeneous dimerization reaction.

The elastic-plastic wrinkling of rectangular plates subjected to locally uniform tension

After overcoming the difficulty with discontinuity in boundary conditions, the pre-buckling elastic stress field for a rectangular plate subjected to locally uniform tension is given. An iteration procedure for solving an elastic-plastic plane stress problem is developed, so as to obtain the pre-buckling elastic-plastic stress field for the same plates. The effects of the aspect ratio of plates, tensile non-uniformity and material properties upon the stress field are also discussed. By assuming the shape of wrinkling region and the wrinkling mode in it, the dependences of critical wrinkling load and the size of wrinkling region upon the material and geometrical parameters are revealed. It is found that wrinkling is dominant by a combined parameter λ E = 1 12(1 − v 2) E σ s t a in elastic case or λ p = 1 9 E σ s t a in plastic case.

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.

A Method for Analyzing the Heat Insulating Properties of the Laminar Compressible Boundary Layer

In some cooling problems associated with high energy flows it may be convenient to localize strongly the cooling, as for example by injecting a coolant through an upstream porous strip, and to depend on the insulating properties of the boundary layer to reduce, or to eliminate completely the need for further cooling on the surface downstream of the highly cooled section. This upstream cooling technique may be of interest in connection with optical windows in hypersonic wind tunnels, and on radomes, wings, and bodies of high-speed aircraft and missiles. In this paper a method for investigating the insulating properties of a laminar compressible boundary layer on a two-dimensional surface with zero heat transfer is presented. The physical situation considered thus corresponds to the case in which the heat transfer downstream of the strongly cooled section is completely eliminated. Of practical concern is how the temperature of the uncooled surface varies in the downstream direction from its low initial value and thus how the low energy layer established by the upstream cooling insulates the downstream surface. The Karman integral method extended to both the momentum and energy partial differential equations of the boundary layer has been used. The station, at which cooling and/or injection ceases, corresponds to a discontinuity in boundary conditions and thus in solutions. At this point the flux of mass, momentum, and energy within the boundary layer has been made continuous by the introduction of three additional parameters in the velocity and stagnation enthalpy profiles. Thus the velocity and stagnation enthalpy profiles have both been taken as sixth degree polynomials. The resulting two integral-differential equations are then solved for two unknown functions of the distance along the wall. These two functions are related to the boundary-layer thickness and to the wall temperature. Initial conditions corresponding to a given initial wall temperature and an initial boundary-layer thickness are prescribed. Exact closed-form solutions for the case of zero axial pressure gradient are obtained. For flows with significant pressure gradients, numerical solutions are required in general. Several numerical examples of practical interest are presented. H M P R S T T t u U V W

X-Ray Absorption Corrections for Single Crystals

It is shown in this paper that exact expressions may be derived for the absorption factor for the intensity of Bragg reflection by a small euhedral crystal completely bathed in a parallel x-ray beam. The expression is a sum of terms evaluated for subdivisions of the crystal determined according to discontinuities in boundary conditions imposed by the distribution of edges between the plane faces of the crystal. For a prismatic crystal of constant cross section, general expressions are derived for five general types of subdivisions in terms of the linear absorption coefficient and certain linear dimensions. Other expressions are also given for other types of subdivisions which arise in certain cases. The application of the method, including corrections for secondary extinction, is illustrated by calculations on tetragonal barium titanate.