Approximate Analytical Approach Research Articles

Disintegration of Large Meteoroids in Earth's Atmosphere: Theoretical Models

Disintegration of large meteoroids, 1 m to 1 km in size, when affected by aerodynamic forces in flight is considered in this paper. Arguments are adduced that ablation is of secondary importance in comparison with mechanical processes of deformation and fragmentation. 2D hydrodynamic simulations using the free-Lagrangian method and the Eulerian method with a volume-of-fluid front tracking procedure have been carried out. The cosmic body was treated as a fluid, with the equation of state of water, moving through gas of appropriate density. We find that disintegration is more complex than simple models based on the estimate of lateral expansion due to differential ram pressure across a meteoroid make it. Rayleigh-Taylor instabilities strongly deform the body and it breaks up in the center. The outer radius of an originally spherical or cylindrical body agrees with the analytic models of spreading. However, a body of accidentally aerodynamic shape does not have its cross section significantly enlarged. A sandbag model has been developed in which a heavily dispersed meteoroid is represented as a conglomeration of noncolliding particles moving through the atmosphere. The particles transfer energy and impulse to the atmosphere and are enclosed by a single bow shock. Calculations show that a spherical swarm of particles takes a conical form but lateral expansion agrees with the above-mentioned simple theoretical models. The approximate analytical approach of a spreading fragmented impactor has got additional support: integration of the drag, ablation, and radiation equations produces results which are in a good agreement with light flashes registered by DoD satellites.

A simple approach to obtain tight upper bounds for the asymptotic queueing behavior of statistical multiplexers with heterogeneous traffic

A statistical multiplexer with infinite buffer capacity and a finite number of independent (not necessarily identical) 2-state MMBP (Markov Modulated Bernoulli Process) traffic sources is considered in this paper, which is modeled as a discrete-time single-server queueing system with correlated arrivals. A simple analytic approach is presented to obtain an upper bound for the tail distribution of the buffer contents. This upper bound is good and even tight in many cases, as shown by the numerical results. Compared to a previously reported exact and general solution technique, our approximate analytic approach is very easy to use and not limited by the system size, and costs nearly no CPU-time. So it is useful in practice. This analytic method can also be easily extended to general m-state Markov modulated arrival processes (e.g., m-state MMBP's).

Theoretical investigation of the dynamic process of the illumination of GaAs.

The dynamic process of light illumination of GaAs is studied numerically in this paper to understand the photoquenching characteristics of the material. This peculiar behavior of GaAs is usually ascribed to the existence of EL2 states and their photodriven metastable states. To understand the conductivity quenching, we have introduced nonlinear terms describing the recombination of the nonequilibrium free electrons and holes into the calculation. Though some photoquenching such as photocapacitance, infrared absorption, and electron-paramagnetic-resonance quenching can be explained qualitatively by only considering the internal transfer between the EL2 state and its metastability, it is essential to take the recombination into consideration for a clear understanding of the photoquenching process. The numerical results and approximate analytical approach are presented in this paper for the first time to our knowledge. The calculation gives quite a reasonable explanation for n-type semiconducting GaAs to have infrared absorption quenching while lacking photoconductance quenching. Also, the calculation results have allowed us to interpret the enhanced photoconductance phenomenon following the conductance quenching in typical semi-insulating GaAs and have shown the expected thermal recovery temperature of about 120 K. The numerical results are in agreement with the reported experiments and have diminished some ambiguities in previous works.

Topological defects, correlation functions, and power-law tails in phase-ordering kinetics.

The correlation functions ${\mathit{C}}_{\mathrm{\ensuremath{\varphi}}}$ and ${\mathit{C}}_{\mathrm{\ensuremath{\varphi}}}^{2}$, associated with the order-parameter field \ensuremath{\varphi}\ensuremath{\rightarrow}(r,t) and its square, respectively, are discussed using heuristic arguments and an approximate analytical approach. Topological defects (walls, strings, monopoles) in the field, seeded by a quench from the high- to the low-temperature phase, lead to singular short-distance behavior in the scaling functions, and power-law tails in the corresponding structure factors. For superfluid helium, the structure factor ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}^{2}$(k,t) is measurable in principle using small-angle scattering (whereas ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$ is inaccessible). It is predicted to exhibit a power-law tail, \ensuremath{\sim}[${\mathit{a}}^{4}$/L(t${)}^{2}$](lnka${)}^{2}$/k, where L(t) is the characteristic scale at time t after the quench and a is the core size of a vortex line. Correlation functions for the defect density are also discussed.

Simplified analysis of buried reinforced concrete arches under simulated nuclear loads

An approximate analytical approach is modified and applied for the study of shallow-buried reinforced concrete arches under severe dynamic loads. The approach contains a closed form solution for wave propagation in soil, transfer functions for simulation of soil-structure interaction, and a layered beam finite element for representing the structure. Several modifications were incorporated in the original approach, for improving the applied loads and for insuring cross-sectional equilibrium in the analysis. The approach has been used for the investigation of four structures that were tested previously by other researchers, and the numerical results were compared with experimental data and corresponding solutions from advanced numerical codes. The results represented reasonably well the general behavior of the structures under consideration, but several serious limitations of the proposed approach were noted. The comparison between experimental and numerical results provided useful observations on the effectiveness of this approach, and recommendations were presented for future applications.

Thermal/structural dynamic analysis via approximate analytical approach

The present paper uses a classical Galerkin weighted residual formulation to obtain the approximate analytical solution of a thermally loaded beam executing free flexural vibrations. The approach used is one where the time variable is considered in the same manner as the spatial variable, and is included in the basis functions. The basis functions used in the approach are polynomials obtained from the terms of a power series, with the condition of nullity on the boundary. This choice simplifies the algebraic manipulations considerably and yields close form expressions for components of the system matrix. The latter also simplifies the numerical computation of coefficients of the approximating polynomial. The approach provides benefits in terms of increased accuracy and lower computational costs.

Venting of runaway reactions with gas generation

AbstractPressure relief venting of a runway reaction with gas generation is examined in terms of the overall system response and the associated venting requirement. This article presents an exact formulation as well as an approximate analytical approach. The latter is shown to be particularly useful in vent‐sizing applications. Using an aqueous hydrogen peroxide decomposition example, the analytical results are demonstrated to agree well with the exact numerical results over a wide range of overpressure. In addition, the analytical result is shown to reduce to the correct limit for a pure vapor system and offers a useful vent‐sizing equation for a pure‐gassy system.

Theory of the threshold field for the depinning transition of a charge density wave

The Hamiltonian of an elastic string pinned by random potentials is often used to describe the depinning transition of a charge density wave in the presence of impurity pinning. The properties of the pinned states show close analogies to those of glassy systems, while the depinning transition resembles a dynamical critical phenomenon. Here we develop an approximate analytical approach based on the quasi-harmonic treatment of the nonlinear pinning potentials together with an expression of the correlation functions to lowest order. This method allows a nonperturbative treatment of a problem with many nonlinear degrees of freedom and quenched disorder. In this way it is possible to characterize the pinned configurations and to compute the threshold field as a function of the coupling constant of the model. The results are in very good agreement with the computer simulations for strong and intermediate pinning and provide an approximate but quantitative method for the determination of the pinned-unpinned phase diagram.

An Analytical Approach to the Problem of Mass Loss from Wolf-Rayet Stars

We investigate the dependence of the wind parameters on the properties of the parent star by means of an approximate analytical approach, largely based upon dimensional arguments (Bandiera and Turolla, 1990). The main assumption we use are.

Interpretation of Wellblock Pressures in Numerical Reservoir Simulations Part 3—Off-Center and Multiple Wells Within a Wellblock

SummaryIn reservoir simulation by finite differences, well models are used to relate the wellblock pressure, the actual wellbore pressure, and the flow rate of the well. These well models are based on an "equivalent wellblock radius, " r0. Two methods for obtaining r0 exist: (1) by numerical experiments, and (2) by an approximate analytical approach that assumes that the pressures of the blocks surrounding the wellblock satisfy the exact radial solution. By definition, the numerical approach gives the correct r0. The more commonly used analytical approach frequently gives a good approximation but, in some cases, it does not. Hence, an analytically derived r0 should always be checked by a numerical calculation. In this paper, two important well geometries are investigated numerically: a single well arbitrarily located within an isolated wellblock and two or more wells within a single wellblock. The numerical r0 for a single well is found to be independent of location, a result that contradicts the prediction of the analytical approach.

Calculation Of Flowing Well Pressures In Reservoir Simulation Using Nine-Point Differencing

Abstract A quantity of considerable importance in flow rate predictions in reservoir simulation is the equivalent well block radius (r o) because it directly relates the numerically calculated grid block pressure to the actual well flowing pressure. Simulation of steam and miscible floods generally requires nine-point differencing. Previous work on approximating ro for nine-point differencing has either severe limitations or can give erroneous results and, hence, incorrect rate predictions. This paper presents, for the first time, a rigorous, closed form expression for ro for an isolated well, valid for any nine-point method. This expression is based on mathematical analysis and confirmed by careful numerical testing. It identically reduces to the expression for five-point differencing (obtained earlier by Peaceman) as a special case. Extension is made to the anisotropic case and confirmed numerically. Results for some additional well-geometries are also presented for the case of nine-point differencing. These include: arbitrary location of a well in the well block, multiple wells within a single block, a well in an edge or comer block. Finally, effects of heterogeneity on predicting flow rates are considered. Introduction In the numerical simulation of reservoir flow, it is usually necessary to use grid blocks whose areal dimensions are much larger than the diameter of a well. Consequently, the numerically calculated well block pressure, po is substantially different from the actual flowing bottomhole pressure of the well, pwf) as shown in Figure 1. Well inflow models are used to relate the well block pressure to the actual wellbore pressure and flow rate through the concept of an equivalent well block radius(1) (ro)), defined by Equation (1) Available In Full Paper. Equation (1) can be expressed as a rate prediction equation: Equation (2) Available In Full Paper. Equation (1) indicates that ro, may be interpreted as the distance from the well at which the steady-state flowing pressure equals the numerically calculated well block pressure, as shown in Figure 1. Peaceman(1) showed that for an isolated vertical well centred in its grid block in a homogeneous and isotropic medium, Equation (3) Available In Full Paper. when conventional five-point differencing is employed over a square grid. Peaceman obtained Equation (3) in two different ways. The first was by numerical experiments; the second by an approximate analytical approach that assumes that the pressures in the grid blocks immediately adjacent to the well block all satisfy the exact radial flow solution. For the square grid case, the first method yielded: Equation (4) Available In Full Paper. while the second method, which is henceforth referred to as the radial flow method, yielded Equation (5) Available In Full Paper. In a second paper, Peaceman(2) pointed out that the radial flow assumption is not generally valid and can result in significant error. For non-square (i.e. rectangular) grids, the radial flow method was found to predict: Equation (6) Available In Full Paper. whereas numerical experimentation showed that the correct expression is Equation (7) Available In Full Paper.

New theoretical model of stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite

A new analytical method has been developed that can predict the stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite associated with either a single matrix crack or a fibre break. Account is taken of thermal residual stresses arising from a mismatch in thermal expansion coefficients between the fibre and matrix. In addition Poisson ratio mismatches are also taken into account. The theoretical approach retains all relevant stress and displacement components, and satisfies exactly the equilibrium equations, the interface conditions and other boundary conditions involving stresses. Two of the four stress-strain-temperature relations are satisfied exactly, whereas the remaining two are satisfied in an average sense. The required non-interface displacement boundary conditions are also satisfied in an average sense. The general representation is used to solve three types of stress transfer problem. A matrix crack and a broken fibre are analysed for the case when there is perfect bonding between fibre and matrix. The third type of problem takes account of frictional slip at the interface governed by the Coulomb friction law. The approximate analytic approach described in this paper, and the preliminary numerical predictions presented, indicate that the stress transfer between fibres and matrix in a unidirectional fibre-reinforced composite, loaded in tension, can now be investigated theoretically in more detail than before. The paper includes some discussion of singularities in the stress fields, which are smoothed by the averaging techniques employed in the analysis. The analytical approach has enabled the development of a micro-mechanical model of frictional slip at the fibre-matrix interface based on the Coulomb friction law, which is more realistic than assuming that the interfacial shear stress is a constant. Predictions are presented of the stress distributions along the fibre-matrix interface and, in particular, it is shown how the length of the frictional slip zone is related to applied strain, friction coefficient, fibre volume fraction and the difference between the test and ‘manufacturing’ temperatures. An indication is given of many other areas of composite modelling where the new theory will be applied.

Strong adiabatic shock waves in arbitrarily inhomogeneous media. Analytic approach

The approximate analytic approach of Laumbach and Probstein to the calculation of the motion of shock waves in arbitrarily inhomogeneous media is widely used in astrophysical applications. Its accuracy and region of applicability are discussed in the present paper. It is noted that accelerated shock waves are not described satisfactorily in this approach, and some modifications are proposed, specifically, the calculation of the shock wave trajectory is improved, the quadratic approximation is calculated, and its accuracy is estimated. An approximate method permitting the explicit finding of the flow function is proposed. A point explosion in the Gaussian disk of a galaxy is calculated and the degree of collimation of the resulting flow is estimated.

A Low-Current Arc Touching an Insulation Barrier in the Contact Gap

The case of a low-current arc interacting with an insulation surface is of special interest for switching technology because of low arc mobility in a self-created magnetic field. The results of an analytical and experimental investigation of the insulation barrier's damage in the contact zone due to a low-current arc on the surface are reported. Both the arc and the arc-track shapes are estimated as well as the propagating depth of the transformed and ablated insulation layers. In addition, calorimetric measurements of the energy exchange between the arc and the barrier are presented. The propagation power flow is estimated as a value of 5-6 MW/m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . An approximate analytical approach is discussed using some of the experimentally estimated data. The main result is the recommended time limit for the interaction between a low-current arc and an insulation barrier.

Combustion conditions of volatile materials decomposing in condensed and gas phases

A classical model of combustion conditions is presented in which conversion occurs only in the gas phase. This paper presents an approximate analytical approach to such a model of combustion, allowing investigations to be undertaken over a wide range of parameter variation and the combustion characteristics to be estimated sufficiently and simply, which is important for a multiparameter process. The conceptual aspect of the method of determing the combustion characteristics is described; the theoretical pressure dependences of the combustion characteristics are shown.

Accuracy of dipole localization with a spherical homogeneous model.

Dipole localization methods (DLM's) with a spherical, homogeneous, isotropic model were applied to the problem of locating and characterizing simulated dipole sources of the brainstem acoustic evoked response (BAER) in cats. Dipole source parameters considered were chosen to be consistent with measurements of gross potential within the brainstem during the BAER. The steepest ascent method was used to solve the least-squares minimization problem on a set of noiseperturbed surface voltages to obtain parameters of a single assumed dipole source. The magnitudes of errors in dipole postion and in dipole moment vectors were calculated for two surface voltage location sets, two assumed dipole source locations, and a range of surface signal-to-noise ratios. An approximate analytic approach to the simulation results attributed DLM errors to an apparent "noise dipole" calculated as the dipole term in the multipole expansion of the added surface noise. The standard deviation of the "noise dipole" magnitude was directly proportional to the standard deviation of surface noise voltage and inversely proportional to the root of the number of surface voltages. This analytic result was in general agreement with the mean of the dipole moment parameter errors in the simulation study. It was found that recalculation of the surface voltage set from the solution dipole of the simulation problem or from the "noise dipole" of the analytic treatment resulted in an improvement of signal-to-noise ratio at the surface.

Satellite Librational Damping Using a Quasi-Feedback Solar Controller

The feasibility of developing a simple quasi-feedback solar attitude controller is explored. The case of arbitrarily-shaped satellites carrying two pairs of solar surfaces in circular orbits is considered. An approximate analytical approach is used to synthesize the suitable open-loop control relations enabling decay of the satellite librations. The quasi-feedback control scheme suggested here is found to be quite effective in damping the pitch motion. The proposed control approach appears to be particularly attractive as it requires knowledge of the librational angle and velocity only at a fixed satellite position in orbit.

Quark fragmentation and trigger side momentum distributions in high-p T processes

It has been widely argued that the experimental evidence concerning the momentum accompanying highpT triggers is a grave problem for models which take the trigger hadron to be a quark fragment. It is claimed that the trigger hadron takes much too large a fraction (z c ) of the jet momentum for the trigger side jet to be a quark. The jet momentum is not directly measured, but deduced from the derivative of the momentum (p x ) accompanying the trigger with respect to the trigger transverse momentum-p T t . This argument is shown to be unsafe. Using both an approximate analytic approach to illustrate the physics and subsequently a full numerical computation it is proved that the deduction of the fractional momentum accompanying the trigger, 1/z c −1, fromdp x /dp T t is not correct. Further we show that models—specifically that of Feynman and Field—which do take the trigger to be a quark fragment are essentially in agreement with the data on trigger side momentum distributions. A surprising prediction of our analysis is thatp x should be approximately constant forp T t ≧6GeV/c.

New Solar Attitude Control Approach for Satellites in Elliptic Orbits

The investigation explores the feasibility of developing a simple solar attitude controller. The case of gravityoriented satellites carrying two sets of solar surfaces in elliptic orbits has been considered. An approximate analytical approach has been used to evolve the suitable control criteria for regulating the movement of solar surfaces so as to counter the adverse effect of eccentricity normally responsible for a considerable deterioration in the attitude control characteristics of the conventional passive or semipassive methods. The solar controller thus developed is found to be quite effective in limiting the librational amplitude of motion. It is felt that the simplicity of the proposed controller configuration and ease of its operation make it particularly attractive for several satellite applications requiring modest attitude accuracies.

Reflection, Refraction, and Absorption of Elastic Waves at a Frictional Interface: SH Motion

The reflection, refraction, and absorption of obliquely incident plane harmonic antiplane strain (SH) waves at a frictional interface between dissimilar semi-infinite elastic solids is investigated by an approximate analytical approach. The frictional stress at the interface is assumed to depend on the normal stress and the relative slip across the interface, but remains otherwise arbitrary throughout the analysis. General expressions are developed for the transmission and reflection coefficients, and the partitition of incident wave energy into reflection, transmission, and absorption. The special case of bonding by Coulomb friction is examined in detail as an example of application of the general procedure. An exact solution is also presented for the case of bonding by Coulomb friction, and a comparison between approximate and exact solutions provides an indication of the accuracy of the approximate method of analysis.