Exact Numerical Integration Research Articles

Two-dimensional excitons in magnetic fields.

We have obtained the energies and the wave functions for two-dimensional (2D) excitons in magnetic fields for the ground and several excited states, using an exact numerical integration of the Schr\"odinger equation for 2D excitons in a magnetic field. The results of the exact calculation for the ground-state energy and wave functions are in excellent agreement with those of variational and perturbation calculations except at intermediate fields where \ensuremath{\gamma}=(\ensuremath{\Elzxh}${\ensuremath{\omega}}_{c}$/2R) is of order 10. Here, R is the exciton Rydberg unit and ${\ensuremath{\omega}}_{c}$ is the cyclotron frequency of the exciton. Results are presented for the position of the exciton peak and the magnetoabsorption as a function of the field for 2D excitons.

Stretching a surface in a rotating fluid

A surface is stretched in a rotating fluid. The solution to the governing set of nonlinear differential equations depends on a parameter λ which is the ratio of the rotation rate to the stretching rate. Perturbation solutions for small and large λ compare well with exact numerical integration.

The boundary layer due to a moving heated line on a horizontal surface

A line heat source lies on an adiabatic horizontal surface. The governing equations under Boussinesq approximation show nonexistence of horizontal boundary layers if the source is still. Boundary layer solutions exist only when the source is moving laterally on the bottom surface with a certain minimum speed. Perturbation solutions for weak heat input agree well with exact numerical integration. The velocity and temperature profiles show similarity. Nonexistence and nonuniqueness are found.

Overpressure during emergency relief venting in bubbly and churn‐turbulent flow

AbstractVessel overpressure behavior during emergency relief venting in both bubbly and churn‐turbulent flow regimes was obtained analytically. The transient predictions are shown to be in good agreement with the more exact numerical integration scheme. The applications of current results in relief vent sizing for both reactor vessels and storage tanks are demonstrated via numerical examples.

Crushing of an Elastic-Perfectly Plastic Ring or Tube Between Two Planes

A thin ring is crushed between two rigid planes. Due to plastic deformation the ring does not recover its original shape when the compression is removed. For an elastic-perfectly plastic flexural material, the ring undergoes two to five different stages. The mathematical problem is formulated and solved by exact numerical integration and accurate analytical approximations.

Instabilities of the Ginzburg-Landau equation: periodic solutions

The evolution of spatially periodic unstable solutions to the Ginzburg-Landau equation is considered. These solutions are shown to remain pointwise bounded (Lagrange stable). The first step in the route to chaos is limit cycle behavior. This is treated by perturbation theory and shown to result in a factorable form. Agreement between the perturbation result and an exact numerical integration is shown to be excellent.

Reassessment of the theory of stimulated Raman scattering.

We analyze a modification of the standard theory of stimulated Raman scattering (SRS) first proposed by Sparks, and show that it incorporates a possibly important physical effect, but that its original formulation is incorrect. The analysis is based on an exact numerical integration of the coupled equations of the modified theory, the results of which we compare with both the conventional theory of SRS and with one set of experimental data. We suggest a reformulation of the modified theory that leads to a gain which is in somewhat better agreement with the data than is the conventional theory.

Semiclassical solutions of the Zaharov-Shabat scattering problem for phase modulated potentials

The Zakharov-Shabat scattering problem can be solved in the semiclassical limit using WKB-type expansions. The method has been used to determine analytically the effect of phase modulation on soliton creation for pulses evolving under the nonlinear Schrödinger equation. Closed form expressions are obtained for square potentials modulated by simple polynomial phase functions. Comparison with results of exact numerical integration shows good agreement.

A Simulation Study of Recursive Parameter Estimation Techniques for a Stochastic-dynamic Flood Routing Model

Recursive parameter estimation of a real-time, stochastic, flood-routing model is extensively studied. Following common hydrologic practice, flood routing is based on a series of nonlinear reservoirs. In order to utilize linear Kalman filtering and optimal estimation techniques] for parameter estimation, statistical linearization of the nonlinear functions of the parameters is performed. The methodology is suitable for nonlinear multidimensional functions of random nonstationary processes. The linearization coefficients depend not only on the parameter means but on the whole probability density of the parameters. Implementation of the parameter estimation procedure requires the determination of the expected values of nonlinear multi-dimensional functions. A simple analytical method, called the Taylor-Gauss approximation, developed for the determination of the expected values, is compared to exact numerical integration procedures. The recursive estimation methodology based on statistical linearization is compared to an extended Kalman filter procedure based on ordinary linearization. Comparison of the different recursive parameter estimation techniques is done through a numerical simulation experiment. Sensitivity analysis with respect to meas-urement errors and sampling frequency is performed. Evaluation of the recursive parameter estimation techniques is based both on measures of accuracy and on measures of the CPU cost of each methodology.

The liquid–vapor interface of a binary liquid mixture near the consolute point

The liquid–vapor interface above mixtures of isopropanol (i-C3H7OH) and perfluoromethylcyclohexane (C7F14) has been studied in the vicinity of the consolute point (Tc=363 K). As three-phase coexistence is approached, the excess fluorocarbon adsorbed at this interface increases; the adsorption is expected to diverge at Tc for a mixture of the critical composition. A simple model of the interface which incorporates the adsorption anomaly is compared with our ellipticity measurements. Both the model and our data yield ellipticities which have a finite maximum at 0.1 K above Tc. (In general, the ellipticity is not a monotonic function of the adsorption.) The calculation of the ellipticity uses an exact numerical integration of Maxwell’s equations for a model dielectric constant vs height profile. The model dielectric constant profile for the critical composition is consistent with a short-ranged density vs height profile between the vapor and the liquid mixture as well as a much longer ranged composition vs height profile expected at a boundary of a mixture near its consolute point. The liquid–vapor correlation length (0.6 nm) was estimated from the ellipticity data far from Tc. The correlation length for the composition profile was found by combining our interfacial tension data taken below Tc with a compilation based on two-scale-factor universality. Ellipticity data for noncritical compositions are presented; however, their analysis will be presented elsewhere.

On Symmetric Buckling of a Finite Flat-Lying Heavy Sheet

An elastic sheet with non-negligible density and finite length lies horizontally on the ground. The ends are clamped and subjected to compressive forces. Depending on the force, the sheet may be regarded as “long” or “short” with different characteristics. The critical buckling load, redefined as the force below which the sheet will always return to the horizontal state under any finite disturbance, is higher than the Euler buckling load of a weightless sheet. When deflections are small and finite the sheet is stable for given end displacement, but is unstable for given force. Approximate analytic solutions compare well with the results of exact numerical integration.

An exact numerical time integration of scalar equations for undamped structural systems

AbstractAn exact numerical time integration of scalar equations for undamped structural systems is presented. Typical numerical examples are included to illustrate the use of the proposed procedure.

Buckling and postbuckling of the lying sheet

The buckling and postbuckling of a finite, heavy elastic sheet is studied. The sheet originally lies on a horizontal plane, one end clamped and one end subjected to a compressive load. Depending on the magnitude of the normalized force F, the lying sheet may be regarded “long” or “short”. Stability analyses in both cases show the buckling behavior is quite different from the Euler column theory. A modified critical buckling load is defined. The analytic results compare well with exact numerical integration.

Physical adsorption of gases on heterogeneous surfaces. Series expansion of isotherms using central moments of the adsorption energy distribution

An alternative approach to the use of the usual integral equation for heterogeneous adsorption is developed. It provides a means of obtaining an explicit result for arbitrary local isotherms and distributions of energy of adsorption. The approximation is compared with exact numerical integrations, using the Langmuir local isotherm and a normal energy distribution. The series fails to converge for energy distributions that are very wide. Truncation of the expansion after the first-order term yields an equation which is capable of fitting experimental isotherm data.

An improved treatment of transverse shear in the mindlin-type four-node quadrilateral element

An improved displacement methodology based on Mindlin theory is developed and applied to a four-node, twelve degrees-of-freedom quadrilateral element. Special attention is directed toward enhancement of the transverse shear energy representation. Particularly, a non-uniform order kinematic interpolation scheme is accompanied by what is termed element appropriate shear correction factors. The element matrices are consistently derived from Hamilton's variational principle using exact numerical integration throughout. The element stiffness matrix possesses correct rank and is well conditioned over the entire Mindlin theory range of thickness-to-length ratios. Exceptionally good element performance is demonstrated by solutions to a set of static and vibrational problems.

Ultrafast non-radiative decay via conical intersections of molecular potential-energy surfaces: C 2H 4+

Conical intersections of molecular potential-energy surfaces are shown to lead to ultrashort (decay time ⪡ 1 ps) non-radiative decay of electronically excited molecules. In this work we calculate the ultrashort decay time for the Ā-X̄ conical intersection in the ethylene radical cation with the aid of ab initio derived parameters and obtain a value of 0.7 × 10 −14 s. We perform the calculations by exact (within the model hamiltonian adopted) numerical integration of the time-dependent Schrödinger equation. The results are compared to those obtained within a Green's function approach. They show a pronounced mutual influence of the vibrational levels of the upper potential-energy surface on the non-radiative decay time. By analyzing the time evolution of the molecular wavefunction on a microsecond time scale we give arguments for a strong quenching of the radiative rate. This rationalizes the absence of detectable emission in C 2H 4 − and other radical cations. The importance of conical intersections and their impact on molecular dynamics in open-shell systems is pointed out.

Lifting of Heavy Elastic Sheet

A sheet of elastic material, lying on the ground is lifted from one edge by a vertical force F. The nonlinear heavy elastica equations for large deflections are solved by perturbation method for small F and exact numerical integration for general F. An asymptotic solution for large F is also found. The results can be applied to the design and lifting of heavy elastic sheets, such as carpeting and sheet metal, as well as to the design and lifting of heavy beams, rods, and cables.

Lifting a heavy elastic sheet or rod from an incline

The lifting of a heavy elastica from an inclined plane is studied through perturbation and exact numerical integration. The maximum moment and maximum tensile force sustained are found as a function of lifting force and inclination angle. Buckling or bifurcation phenomena occurs when the inclination is -90°.

Directivity index of arrays of random line arrays

The directivity index of a random array of random line arrays is analyzed statistically. Expressions are obtained for the ensemble average and variance of the directivity index for the general case of an array of random line arrays. These expressions are applied to the specific case of line arrays distributed with uniform pdf in a circular region, and both the average and the variance of the directivity index are approximated in closed form. The closed‐form expressions are evaluated for a variety of parameter combinations to show the effects of number of elements per line, array dimensions, and array steering. The closed‐form expression for directivity index is validated by comparison of its calculated results with those of a more exact numerical integration technique.

ON THE BIFURCATION SOLUTIONS OF AN AXIALLY ROTATING ROD

A straight flexible rod is rotated axially from one end. The problem depends on a parameter J which represents the relative importance of centrifugal effects to flexural rigidity. There exist numerous bifurcation branches with the first branch most important. The problem is solved by matched asymptotic expansions for large J. The results compare well with exact numerical integration. We find the straight rod is stable for J<1.87510.