Zeroth-order Solution Research Articles

Modeling permeability in imperfectly layered porous media. II. A two-dimensional application of block-scale permeability

In the previous paper (Zijl and Stam, 1992), a theory has been developed to calculate the nine components of the three-dimensional intrinsic permeability tensor on the scale of a grid-block from a local-scale, predominantly layered subsurface. The resulting block-scale expressions can be written as a perturbation series of which the first term, or zeroth-order solution, coincides with the conventionally applied arithmetic and harmonic averages over the layers of the subsurface. The derived expressions permit the calculation of the diagonal and off-diagonal terms of the permeability tensor. In the present paper, these expressions will be applied in some numerical examples. Two basic two-dimensional hypothetical permeability distributions are adopted, and the various terms of the theoretical expressions are calculated. The results will be used to derive guidelines to discern the situations where higher order solutions can be neglected, and where conventional harmonic and arithmetic averages give a good estimate of the permeability on grid-block scale.

Free convection through a vertical channel in a rotating porous medium

The paper investigates the combined effect of viscous and Darcy dissipations in the case of a fully developed natural convection flow of an incompressible viscous fluid between two heated vertical walls in a rotating porous system. A perturbation method has been employed to obtain the solution of governing non-linear coupled equations. The zeroth order solution corresponds to the flow in the absence of viscous and Darcy dissipations. The effect of Darcy and Ekman numbers on the field variables and the mass flow rate has been discussed.

Transient radiation of a single-cycle sinusoidal pulse from a thin dipole

The behavior of the radiated electric field from a half-wave dipole in free space excited by a single-cycle sinusoidal voltage is studied. The zero-order approximate solution for the current along the antenna in the frequency domain is used, so that the time-dependent current as well as the radiated electric field can be calculated and interpreted easily. Under a suitable matched condition, the time dependent current along each half-section of the dipole is made up of a current pulse traveling from the feed-point and a reflected one from the end-point. However, the radiated electric field consists of five overlapping signals exhibiting that radiation takes place only from the discontinuities of the dipole antenna. Both current and the radiated electric field are extended in time due to the reflection of current from the end-points of the dipole antenna. It appears that the spectral bandwidth of the radiated electric field is narrower than that of the exciting voltage pulse. This phenomenon may be due to the filtering effect of the half-wave dipole.

Sequential decomposition and policy iteration schemes for M-player games with partial weak coupling

We formulate two general classes of M-player deterministic and stochastic nonzero-sum games where the players can be placed into two groups such that there are strong interactions within each group and a weak interaction between the two groups. This weak interaction is characterized in terms of a small parameter ε which, when set equal to zero, leads to two independent nonzero-sum games. Under the Nash equilibrium solution concept both within and in between the groups, we study the merits of an iterative method for the construction of the equilibrium by solving simpler problems at each stage of the iteration. In this iterative scheme, the zeroth order solution is the Nash equilibrium of the two independent games obtained by setting ε = 0, whereas the higher-order solutions are Nash equilibria of quadratic games, even though the original problem may have nonquadratic cost functions.

Analytical expansions of torque-free motions for short and long axis modes

Torque-free motion of a rigid body is integrable and its solution is expressed in terms of elliptic functions and elliptic integrals. The conventional analytical expression of the solution, however, is complicated and not suitable for hand-calculation. Recently the rotational motions of small celestial bodies in the solar system are frequently investigated by numerically integrating the equations of motion instead of using the analytical solution, since the numerical evaluation of the analytical and exact solution is a little bit difficult. As the observational accuracy of the rotational motions of the small bodies in the solar system is quite low, what we need for the reduction of these observations are rough estimates of the period of Eulerian motion ( or the free precession period) and the amplitudes of the main periodic terms. Here we give simple analytical expansions of torque-free motions for short- and long-axis modes, which are correct up to the second-order of a small parameter. These expressions include only trigonometric functions and are easily evaluated by hand calculation for estimates of the essential quantities from which we can determine a global rotational motion of the torque-free motion. They can also be used as the zero-th order solution in a perturbation method, when the motion is perturbed by external torques.

Asymptotic solutions to weakly coupled stochastic teams with nonclassical information

The authors develop a new iterative approach toward the solution of a class of two-agent dynamic stochastic teams with nonclassical information when the coupling between the agents is weak, either through the state dynamics or through the information channel. In each case, the weak coupling is characterized in terms of a small (perturbation) parameter. When this parameter value (say, in ) is set equal to zero, the original fairly complex dynamic team, with a nonclassical information pattern, is decomposed into or converted to relatively simple stochastic control or team problems, the solution of which makes up the zeroth-order approximation (in a function space) to the team-optical solution of the original problem. The fact that the zeroth-order solution approximates the optimal cost up to at least O( in ) is shown. It is also shown that approximations of all orders can be obtained by solving a sequence of stochastic control and/or simpler team problems. >

Long-wavelength instabilities in three-layer flow down an incline

This paper reports a long-wavelength instability which has not previously been identified for three-layer free-surface flow down an inclined plane. The instability is identified in the zeroth-order asymptotic solution in wave number, indicating that neither inertial nor finite wavelength effects are necessary to induce instabilities in three-layer systems. Various neutral stability boundaries are presented which demonstrate the effect of viscosity stratification, density stratification, and layer thicknesses. It is found that destabilization occurs in cases where the middle-layer viscosity (for equal densities in each layer) or density (for equal viscosities in each layer) is smaller than those of the adjacent layers. The regions of instability afford a smooth transition between neutrally stable regions of the parameter space where the in-phase and out-of-phase characteristics of the interfaces differ.

Spin—orbit effects in the Br atom in the framework of the no-pair theory

New programs have been developed which enable us to calculate the two-electron terms of the no-pair Hamiltonian. We present all-electron calculations for the Br atom with non-relativistic and relativistic Hamiltonians. The MRD-CI method is used to determine excitation energies from the 2P u ground state to the first excited states 4P g and 2P g. The zero-field splitting of the ground state is investigated by means of first-order perturbation theory. Non-relativistic and relativistic spin—orbit Hamiltonians are applied in the perturbation calculations, whereby the spin-free CI wavefunctions have been used as zero-order solutions of the unperturbed system. By comparing the different results for the spin—orbit matrix elements, we are able to separate effects arising from the relativistic wavefunction and effects caused by the relativistic spin—orbit Hamiltonian.

Interrelated Rotordynamic Effects of Cylindrical and Conical Whirl of Annular Seal Rotors

A comprehensive approach for computing the dynamic coefficients of an annular seal is presented. The coefficients are partly those associated with a uniform lateral eccentricity mode of the rotor (known as the cylindrical whirl mode) and with an angular eccentricity (which gives rise to a conical whirl type). The rotor excitation effects in both cases are treated as interrelated by recognizing the fluid-exerted moments resulting from the lateral eccentricity and the net fluid force resulting from the angular eccentricity. In all cases, the rotor is assumed to undergo a whirling motion around the housing centerline. The computational procedure is a finite-element perturbation model in which the zeroth-order undisplaced-rotor flow solution in the clearance gap is obtained through a Petrov-Galerkin approach. Next, the rotor translational and angular eccentricities, considered to be infinitesimally small, are perceived to cause virtual distortions of varied magnitudes in the finite element assembly which occupies the clearance gap. Perturbations in the flow variables including, in particular, the rotor surface pressure, are then obtained by expanding the finite-element equations in terms of the rotor eccentricity components. The fluid-exerted forces and moments are in this case computed by integration over the rotor surface, and the full matrix of rotordynamic coefficients, in the end, obtained. The computational model is verified against a bulk-flow model for a sample case involving a straight annular seal. Choice of this sample model for validation was made on the basis that no other existing model has yet been expanded to account for the mutual interaction between the cylindrical and conical rotor whirl, which is under focus in this study.

The exact bound-state ansaetze as zero-order approximations in perturbation theory II. An illustration:V(r)=r 2+fr 2/(1+gr 2)

Challenging Gallas double-well interactions are shown tractable by the quasi Rayleigh Schroedinger perturbation theory of paper I, and a good precision is obtained in the first-order numerical test. The related technicalities are clarified: The existence of the suitable zero-order solutions withf ≠ 0 andg ≠ 0 is proved and a few closed formulas for corrections are given and discussed.

Perturbation solution to the nonlinear problem of oblique water exit of an axisymmetric body with a large exit-angle

In this paper, a nonlinear, unsteady 3-D free surface problem of the oblique water exit of an axisymmetric body with a large water exit-angle was investigated by means of the perturbation method in which the complementary angle α of the water exit angle was chosen as a small parameter. The original 3-D problem was solved by expanding it into a power series of α and reduced to a number of 2-D problems. The integral expressions for the first three order solutions were given in terms of the complete elliptic functions of the first and second kinds. The zeroth-order solution didn't turn out to be a linear problem as usual but a nonlinear one corresponding to the vertical water exit for the same body. Computational results were presented for the free surface shapes and the forces exerted up to the second order during the oblique water exit of a series of ellipsoids with various ratios of length to diameter at different Froude numbers.

Localized flute vortices in plasmas

The properties of localized vortex solutions of the nonlinear flute mode equations are investigated. It is shown that such solutions must satisfy the condition that the integral of the density perturbation over the plane transverse to the magnetic field must vanish. Monopole vortex solutions with discontinuous vorticity are obtained by perturbation analysis, treating the plasma inhomogeneities as small parameters of the first order, and using a circularly symmetric solution with arbitrary radial profile as the zeroth order solution. The results are illustrated by the so-called "rider" solution, and the stability properties of the solution are discussed.

Error analysis of limiting-case solutions to the steady-state-biofilm model

Simple analytical solutions for the limiting cases of deep, fully penetrated, first-order and zero-order biofilms are evaluated for accuracy by comparison with the full solution of the steady-statebiofilm model. The fundamental dimensionless parameter S min ∗ is critical to the evaluation. Deep solutions are accurate when S S ∗/S min ∗ in large enough, but fully penetrated solutions rarely are accurate. First-order and zero-order solutions are accurate when S S ∗ is less than about 3 × 10 −3 and greater than about 10 2, respectively, Most of the expected range of S min ∗ values is accurately represented only by the full solution, unless S S ∗/S min ∗ is significantly greater than 1.0. This analysis emphasizes that the limiting-case solutions have restrictive ranges of applicability for steady-state biofilms.

Approximate optimal atmospheric guidance law for aeroassisted plane-change maneuvers

In this paper, we develop an approximate optimal guidance law for the aeroassisted plane-change problem through the use of an expansion of the Hamilton-Jacobi-Bellman equation. This guidance law maximizes the final velocity of the re-entry vehicle while meeting terminal constraints on altitude, flight-path angle, and heading angle. The expansion is made with respect to a small parameter that arises naturally as the ratio of the atmospheric scale height to the radius of the Earth. The integrable zeroth-order solution obtained when the small parameter is set to zero corresponds to a solution of the problem where the aerodynamic forces dominate the orbital forces (all nonaerodynamic forces). Higher-order terms in the expansion are determined from the solution of linear, partial differential equations requiring only quadratures. Results from this guidance law, using only a first-order correction term, are quite impressive. Comparisons are made to a numerical optimal solution and another approximate analytical solution for this problem.

On the nth-order structure of solitonic wormholes

The structure of the equations describing a solitonic wormhole in the Einstein–Yang–Mills–Higgs system for an arbitrary compact and connected gauge group 𝒢 and representation of an arbitrary Higgs field Q is studied. The general structure of these equations and its use in deriving the first-order equations, which result by applying the operator ON=limN→0 ∂/∂N to the original equations, where N is the lapse function, is discussed. It is also possible to write down the nth-order equations, which are obtained by applying n times the operator ON, for a certain class of solutions. These nth-order equations are then specialized to the model with 𝒢=SU(2) and Q in the adjoint representation. These nth-order equations on the background of a non-Abelian zero-order solution, which can be gauge transformed to satisfy the ’t Hooft ansatz at the internal infinity of the hole, are solved. The technique used to solve the system is that of harmonic expansion, yielding an algebraic system of equations which can be interpreted as an eigenvalue problem. The eigenvalues are functions of the parameters of the model and zero-order quantities. Thus, depending on the values of these parameters, nontrivial nth-order solutions may exist or not. Those cases in which there exist nontrivial solutions are stated. Only for those cases can it be expected that global non-Abelian solitonic wormholes are found.

周期最適化による最小燃料巡航

Application of periodic optimization to minimum fuel aircraft trajectories is discussed. The most part of a long range trajectory is a cruise segment, which is assumed to be a steady flight in many papers. The steady condition is one of the constraints which lower optimality. In this paper, the optimum cruise segment is considered as an unsteady flight. The cruise segment is devided to subarcs in which same control sequences are used. The solution of minimum fuel problem for the subarc optimizes a total trajectory. The periodic optimization has some advantages in an operation. Zero-order approximate solution is presented as a maximum thrust ascent and a minimum thrust descent. This solution indicates the limitation of the exact one when the final range is specified infinite. Numerical results show the periodic cruise can save fuel 3-4% decrease.

Acoustic modes in optical fiberlike waveguides

A perturbation analysis of guided and leaky modes in fiber acoustic waveguides with core and cladding parameters that are slightly different is presented. The perturbing parameter is the shear-velocity difference between the core and cladding material epsilon(s). Acoustic fields and eigenvalues are expanded in power series of epsilon (s)(1/2) for radial and flexural modes and in powers of epsilon(3) for torsional modes. Expansion of leaky longitudinal modes is also in terms of epsilon(s), but the nature of perturbation analysis for these modes is somewhat different from that of guided modes. Zero-order solutions for all types of modes are obtained, and some important higher-order effects are discussed. Common features of optical and acoustic modes in weakly guiding fibers are addressed. It is shown that with respect to zero-order solutions of guided modes, optical and acoustic fibers have identical propagation characteristics. Exact and zero-order propagation characteristics for several lower-order shear-type modes are calculated and compared.

The role of hybridization in perturbative bond theories: The existence of exact strictly localized orbitals in small molecules

The significance of hybrid atomic orbitals is discussed in the context of perturbation theories of the chemical bond. It is emphasized that the correct choice of hybrids is essential in obtaining sufficiently accurate zeroth order (strictly localized) solution, representing a “Pauling-point” in chemical-bond theories. It is proved that for some small molecules, e.g. for first-row hydrides in small basis sets, there may exist a set of non-orthogonal exactly localized (two-centre) occupied molecular orbitals which span the same subspace and have the same energy as do the canonical MOs, but have no tails on other atoms. Although this statement cannot be generalized to larger molecules or larger basis sets, it still suggests that the apparent interbond delocalization (tails) predicted by various perturbative schemes does not necessarily correspond to physical interbond interaction, but, to a large extent, it may be a consequence of using poor atomic hybrids. As examples, non-orthogonal strictly localized MOs are reported for CH 4 and NH 3 in the STO-3G basis set.

Electric field generation in hollow dielectric waveguide lasers I: General theory of unperturbed symmetric waveguide configuration

Starting with material equations, the stationary nonlinear field equation of TE-polarized electromagnetic fields is established within the framework of a semiclassical theory. In contrast with linear-mode concepts, this equation is employed to construct approximate solutions after assuming both a symmetric slab configuration and relatively weak pumping parameters. Constants of both integration and propagation are fixed within the final seminumerical analysis through crossing conditions. To investigate the influence of material and pumping parameters on integrated power flow and shape-of-wave solutions, the theory is illustrated by applying it to the zero-order (basic) solution.

Electric field generation in hollow dielectric waveguide lasers II: First-order perturbation theory in the presence of waveguide imperfections due to rough boundaries

After developing formulas for the scattered fields in the first Born approximation, we treat the interaction of primary and scattered waves within a twofold perturbation theory exceeding the first Born approximation. In this model scattering causes an increase of the decomposition of population inversion in the interior (|x| d/2). The relationship between the corresponding total scattering intensities and the most important roughness properties is exhibited by numerical estimates of simplified formulas. Finally, the scattering-induced effect on the primary power flow, integrated with respect to x, is investigated seminumerically for the zero-order primary-field solution.