Previous Approximate Solutions Research Articles

The equivalent method for arbitrary pulses loaded on the elastic-brittle panel

The risk of urban blasts is rising nowadays. As the most fragile component in modern buildings, glass may cause casualties when it fails in a blast. Thus, being able to accurately predict the average splash speed of fragments is of great significance. In almost all the published studies on glass blast-resistance, the blast pulses are treated as monotonously decaying pulses, which is somewhat oversimplified for the pulses with complex shapes in urban blasts.In this study, for the pulses applied to the elastic panels, the applicability of the Youngdahl Equivalent Method (Y-EM), which is usually adopted in plastic analysis, is checked by employing our previous finite deformation approximate solutions. It is confirmed that Y-EM is applicable for the pulses loaded on an elastic-brittle panel when the finish time in Y-EM is set as the time at which the deflection reaches its maximum, rather than the duration of the external pulse. Furthermore, based on the combination of Y-EM and saturation analysis, an equivalent method for arbitrary pulses loaded on the elastic-brittle panel is proposed. Adopting this method, the average splash speed of glass fragments is predicted, which is proved to be accurate by comparison with the published experimental data. As the theoretical basis of the prediction, the pulse equivalent method proposed here is also proved to be effective.

Approximate solutions for the Bagley-Torvik fractional equation with boundary conditions using the Polynomial Least Squares Method

In this paper we apply the recently introduced Polynomial Least Squares Method (PLSM) to compute approximate analyticalpolynomial solutions for the Bagley-Torvik fractional equation with boundary conditions. The Bagley—Torvik equation may be used to model the motion of real physical systems such as the motion of a thin rigid plate immersed in a Newtonian fluid. In order to emphasize the accuracy of PLSM, we included a comparison with previous approximate solutions obtained for the Bagley-Torvik fractional equation by means of other approximation method.

The effect of a weak nonlinearity on the lowest cut-off frequencies of a cylindrical shell

The plane strain problem for a thin circular cylindrical shell is considered within the framework of the Sanders–Koiter theory. The relative shell thickness and displacement amplitude are chosen to be of the same asymptotic order. The leading nonlinear correction to the lowest cut-off frequencies is derived using the method of multiple scales. In contrast to the traditional two-mode Galerkin expansions assuming inextensibility of the shell transverse cross section, the developed fourth-order asymptotic scheme operates with five angular modes. The obtained results reveal asymptotic inconsistency of previous approximate solutions to the problem.

A new iterative method for solving non-square systems of linear equations

This technical note presents a new iterative procedure for solving systems of m linear equations in n variables under a sufficient condition that is practical. We show how this procedure may utilize elementary row operations to meet its sufficient condition. In this iterative procedure, the approximate solution obtained in each iteration is a convex combination of some l∞-norm projections of the previous approximate solution. Under a regularity condition, this procedure converges quadratically. Application examples are given that show how this procedure can generate desired non-basic solutions and how it can aid Fourier–Motzkin elimination method in solving linear programming problems.

Theoretical study on the stability of practical polarization transformers

Abstracts Stability of practical polarization transformer (PPT) proposed by Huang is studied for the application of all-fiber optical current transformers (AFOCTs). The previous zero-order approximate solution ignores inter-mode coupling, which is one of the factors causing AFOCT instability. This paper develops a concise approximate solution to take into account the coupling between two local eigen-modes in the PPT. It is shown that power exchange of the two modes is determined by the internal coupling coefficient. Guidelines for designing spin rate profiles are given and verified with numerical results.

Approximate Analytic Expression of Surface Charge Density/Surface Potential Relationship for a Spherical Colloidal Particle Immersed in a General Electrolyte Solution

We implement a variational iteration method for solving a Poisson–Boltzmann equation (PBE) describing a spherical colloid immersed in a general electrolyte solution. In this method, a general Lagrange multiplier is introduced to construct correction functional for the problem, and the multiplier is identified optimally via a variational method; a linearization solution of the original PBE is chosen as an initial approximation for the iteration. To proceed with the iteration analytically, we approximate the highly nonlinear term of the PBE by a polynomial of proper order. Due to presence of an exponent integral function in the first-order iteration solution, higher-order iteration solutions are analytically unavailable currently. Based on the first-order iteration solution, analytic expression for surface charge density/surface potential relationship is acquired. The present analytic solution contrasts sharply with previous ones by two striking features: (1) the present one surpasses well beyond previous approximate solutions in that general electrolyte type can be dealt with in a unified way, and (2) valid application scope of the present expression is in small κa domain and thus is complementary with those of previous ones.

Analysis of models for induced gas flow in the unsaturated zone

Accurate description of induced gas flow in an unsaturated zone is indispensable for characterizing the unsaturated zone. Hantush's approximation and a finite difference approximation of the leakage term are frequently employed in modeling the induced gas flow in an unsaturated zone underlying a leaky confining layer. However, no studies have been conducted on the errors that result from these approximations. This study presents a physically more rigorous model based on mass conservation in both the upper leaky confining layer and the lower permeable layer (the unsaturated zone). Flow in the upper layer is assumed to be vertical, and in the lower layer, it is two‐dimensionally axisymmetric, with gas pressure and flux continuity at the interface of the two layers. A new steady state solution for a well with a finite radius and two new transient solutions for a well with an infinitely small radius and with a finite radius are developed for this rigorous model. In addition, another new transient solution is derived using the finite difference approximation of the leakage term for the same problem. The developed solutions are compared with previous approximate solutions. Results indicate that Hantush's approximation overestimates the gas pressure at shallower depths and underestimates it at deeper depths under the condition of gas injection. This error changes with the hydrogeological parameters and well configurations. The finite difference approximation works reasonably well under the usual field conditions. Neglecting the well radius for a small radius well does not induce much error.

Application of He’s variational iteration method to the estimation of diaphragm deflection in MEMS capacitive microphone

In this paper, He’s variational iteration method (VIM) is used to analyze the deflection of poly silicon diaphragm of Micro Electro Mechanical Systems (MEMS) capacitive microphone. The residual stresses in the material used to make the diaphragm change the vibrational characteristics of the microphone diaphragm and consequently influence the microphone’s first resonant frequency, cutoff frequency and sensitivity. The most successful devices use poly silicon as a diaphragm material, because of its residual stress is controllable by high-temperature annealing after ion implantation by boron or phosphorous. External acoustic force causes to deflect the diaphragm of the structure and VIM is a powerful analytical method to predict the structural behavior and the microphone performance. Comparison of this new method with the previous approximate solution [1], is applied to assure us about the accuracy of solution.

Adhesion-Induced Instability in Asperities

Adhesive forces between two approaching asperities will deform the asperities, and under certain conditions this will result in a sudden runaway deformations leading to a jump-to-contact instability. We present finite element-based numerical studies on adhesion-induced deformation and instability in asperities. We consider the adhesive force acting on an asperity, when it is brought near a rigid half-space, due to van der Waals interaction between the asperity and the half-space. The adhesive force is considered to be distributed over the volume of the asperity (body force), thus resulting in more realistic simulations for the length scales considered. Iteration scheme based on a “residual stress update” algorithm is used to capture the effect of deformation on the adhesion force, and thereby the equilibrium configuration and the corresponding force. The numerical results are compared with the previous approximate analytical solutions for adhesion force, deformation of the asperity and adhesion-induced mechanical instability (jump-to-contact). It is observed that the instability can occur at separations much higher, and could possibly explain the higher value of instability separation observed in experiments. The stresses in asperities, particularly in case of small ones, are found to be high enough to cause yielding before jump-to-contact. The effect of roughness is considered by modeling a spherical protrusion on the hemispherical asperity. This small-scale roughness at the tip of the asperities is found to control the deformation behavior at small separations, and hence are important in determining the friction and wear due to the jump-to-contact instability.

A new solution of transient confined–unconfined flow driven by a pumping well

A new solution of transient confined–unconfined flow driven by a pumping well is developed and compared to previous approximate solutions of Moench and Prickett [Moench AF, Prickett TA. Radial flow in an infinite aquifer undergoing conversion from artesian to water table conditions. Water Resour Res 1972;8:494–9] and Hu and Chen [Hu L, Chen C. Analytical methods for transient flow to a well in a confined–unconfined aquifer. Ground Water 2008;46(4):642–6]. The problem is rewritten in dimensionless form with the Boltzmann transform. The nonlinear equation for flow in the unconfined zone is solved with the Runge–Kutta method. Position of the conversion interface is determined with an iteration scheme. This study shows that the confined–unconfined flow depends on three dimensionless parameters that represent the confined–unconfined storativity ratio ( a D ), the ratio of the initial hydraulic head over the aquifer thickness ( f i ), and the dimensionless pumping rate ( q D ). The rate of expansion of the unconfined zone increases with q D , but decreases with a D and f i . Differences between the two previous approximate solutions and the new solution of this study are observable in the estimated position of the conversion interface and the drawdown–time curves. The new solution can be applied to estimate the time for confined–unconfined conversion to occur (critical conversion time), and the time when the pumping well becomes dry (critical drying time). The critical conversion time is found to be very sensitive to the initial hydraulic head. The critical drying time is often much larger than the critical conversion time and may never be observed during a finite pumping period.

The Steady‐State Analytical Solution of the Global Atmospheric Electricity Model on the Charging of Thunderstorms in Lower Atmosphere

AbstractGlobal atmospheric electricity model is the circuit between atmosphere and surface. In the lower atmosphere, the main sources of electric current within the model are thunderstorms, which are considered as vertical electric dipoles. Based on the current continuity equation, using spherical harmonic function expansion, with the lower boundary as the surface atmospheric electric potential and the upper boundary condition as the electric potential of the bottom of the ionosphere, and non‐homogeneous current source function of the vertical dipole assumptions for the charging of thunderstorms, we derive the steady‐state analytical solution of the global atmospheric electricity model in the lower atmosphere. It is proved that the solution can be written as the sum of limited polynomial series. The conclusions are consistent with the previous theoretical analysis and experimental observation. It is an improvement to the previous approximate solution, and the formula in this article can be used in numerical calculation and simulation of atmospheric electric parameters in the future.

Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities

We reformulate the strong-interaction limit of electronic density functional theory in terms of a classical problem with a degenerate minimum. This allows us to clarify many aspects of this limit, and to write a general solution, which is explicitly calculated for spherical densities. We then compare our results with previous approximate solutions and discuss the implications for density functional theory.

An Analytic Model to Account for Quantum–Mechanical Effects of MOSFETs Using a Parabolic Potential Well Approximation

An analytic model to account for the quantum-mechanical effects (QMEs) of the MOSFETs using a parabolic potential well approximation is presented in this paper. Based on the solution of the coupled Schroumldinger and Poisson equations following the Wentzel-Kramer-Brillouin method, a transcendental equation of the subband energy level has been rigorously derived to obtain an approximate analytic solution for the subband energy levels and the inversion charge centroid. Calculated results from the obtained analytical solution are compared with the previous approximate solutions reported in the literature and the numerically simulated data. A good agreement between the analytical and numerical is obtained, proving the validity of the analytic modeling of QMEs

Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers

An improved harmonic balance (HB) method is been presented to construct more accurate approximations to frequencies of oscillators with non-linear fractional powers. Unlike the classical HB method, linearization is carried out prior to harmonic balancing thus resulting in simple linear algebraic equations instead of complicated nonlinear algebraic equations. Hence, we are able to establish the approximate frequencies for the oscillators more directly. These approximate results are valid for various fractional powers including the limiting case of vanishing power. Comparing with previous approximate solutions, the approximate solutions derived here are more accurate with respect to established exact solutions.

Diffraction By an Anisotropic Impedance Half Plane

For a plane wave incident at an oblique (skew) angle on an anisotropic impedance half plane, the second order difference equation for the angular spectra is converted to a pair of inhomogeneous non-singular integral equations and then solved. The exact expression for the scattered field is determined and compared with a previous approximate solution.

New formula to calculate the skin effect in isolated tubular conductors at low frequencies

The exact solution of the AC/DC resistance ratio of an isolated tubular conductor involves the calculation of values from 12 Kelvin functions. Previous approximate solutions are complicated, and graphical solutions require interpolation from sets of curves. A simplified solution, based on a new formula, is presented. The values of the resistance ratio derived from this formula are within ±1% of those derived from the exact solution in the practical ranges 0.05 ≤ (thickness/diameter) ≤ 0.40 and 0 < √(frequency/DC resistance) ≤ 3000, except for t/D2 = 0.40 and √(f/Rdc) > 2000. For practical tubular conductors, the formula can be used for frequencies up to about 10 kHz, depending on the thickness t and the outer diameter D2. The application of the skin effect formula to a steel-cored conductor can result in significant error.

Two-Dimensional Analysis Model for Tapered Pulse Tubes

A two-dimensional model for pulse tubes with tapered cross-section was proposed. Net enthalpy flow and steady mass streaming were investigated by two-dimensional analysis of mass, momentum and energy equations of the gas as well as energy conservation of the tube wall. Steady mass flux profiles show good agreement with the previous approximate solution. It was shown that steady mass streaming can be reduced by tapering a pulse tube and by increasing the length of a pulse tube. Effects of the velocity phase angle and frequency on steady mass streaming were shown

Analysis of corrugated surface wave antenna using hybrid MOM/UTD technique

A new efficient technique for the analysis of corrugated surface wave antennas with infinite ground plane fed by a parallel-plate waveguide is presented, termed the hybrid method of moments/uniform geometrical theory of diffraction (MOM/UTD) technique. The relative power pattern obtained using the method is compared with the patterns obtained from the previous approximate solution and experiment. The power pattern of the presented method agrees more closely with that of the experiment, compared with that of the previous approximate solution.

Unified QCD description of hadron and jet multiplicities

The evolution equation for parton multiplicities in quark and gluon jets which takes into account the soft gluon interference is solved numerically using the initial conditions at threshold. If the k_T-cutoff Q_c is lowered towards the hadronic scale Q_0 of a few hundred MeV, the jets are fully resolved into hadrons. Both hadron and jet multiplicities in e+e- annihilation are well described with a common normalization. Evidence is presented within this perturbative approach that the coupling \alpha_s(k_T) rises by an order of magnitude when approaching the low energy region. The ratio of hadron multiplicities in gluon and quark jets is found smaller than in previous approximate solutions of the evolution equation.

New analytical expressions for dark current calculations of highly doped regions in semiconductor devices

We studied highly doped quasi-neutral regions of semiconductor devices with position dependent doping concentration in the absence of illumination. An important parameter of a highly doped region is its dark current. To clarify how the doping profile influences the dark current, simple analytical expressions are useful. To this end, we first transformed the transport equations to a simple dimensionless form. This enables us to write already existing analytical expressions in an elegant way. It is demonstrated how, from any analytical dark current expression, a direct counterpart can be derived. Next, we derived a dimensionless form for a nonlinear first-order differential equation for the effective recombination velocity. Starting from the analytical solution of this differential equation for uniformly doped regions and using linearization techniques, we obtained two new simple and accurate expressions for the dark current. The expressions are valid for general doping profiles with different minority carrier transparencies. The exact solution is included between both new approximate solutions. The new expressions are compared with previous approximate solutions.