Numerical solution of minimax problems of optimal control,part 2

In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved numerically.It is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10^–3 if the single-subarc approach is employed. These relative differences are of order 10^–4 or better if the multiple-subarc approach is employed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

Classical trajectory analysis in atom—triatom collisions: Continuous quantization and scaling behaviour

A method of analysing classical trajectory data, based on recently derived scaling principles, is applied to a model atom-triatom collinear collision system. Apart from the utility of the scaling idea in extending trajectory computations, the analysis of the scaling coefficients in terms of transition probabilities increases the scope of the classical scaling theory as a means of obtaining (at the very least) qualitative quantum-mechanical information from classical trajectories. As an useful adjunct, the method of continuous quantization is applied to generate approximate transition probabilities. These results are semiquantitative; thus a combination of classical scaling and continuous quantization affords a powerful means of modeling complex collision cases with a minimum of computational effort.     

Numerical analysis of thermosolutal flows in a cavity with gravity modulation effects

 In this work, a two-dimensional thermosolutal convection flow under a sinusoidal gravity modulation (g-jitter) field is studied to understand the effects of the periodic source on flow field, as well as heat and mass transfer mechanisms. A semi-implicit projection finite element method is adopted to solve the transient Navier–Stokes, energy and species concentration equations. The fingering regime and the diffusive regime are explored for a series of gravity modulation frequencies. Two types of flow evolution, synchronous and subharmonic responses, are obtained for different frequencies. Distribution of unstable responses for the singly unstable condition is in agreement with the literature predict. The results show that heat and mass transfer rates are affected by the response type. For a subharmonic variation flow field, the overall Nusellt number and Sherwood number exhibit larger values. In addition, the augmenting condition in combining thermally driving force and solutally driving force is analyzed for a practical crystal material and displays a different response distribution from those in the fingering regime and the diffusive regime. Received on 19 February 2001 / Published online: 29 November 2001     

Simple method for the chromatographic separation and determination of mono-, di- and tricyclic aromatic hydrocarbons in heavier petroleum fractions

Summary The method described is a simple procedure for separating gas oil boiling range petroleum fraction into its aromatic hydrocarbons of the mono-, di- and trinucleartype. This is accomplished by gradient elution through an alumina adsorption column under established/standardised conditions. Characterisation is performed by UV-absorption. The method can be used also for investigating aromatic hydrocarbon structures from other petroleum fractions.

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Einfaches Verfahren zur chromatographischen Trennung und Bestimmung von mono-, di- und tricyclischen aromatischen Kohlenwasserstoffen in schwereren Erdölfraktionen

Zusammenfassung Zur Trennung der Erdölfraktion vom Siedebereich des Gasöls in die ein-, zwei- und dreikernigen aromatischen Bestandteile wird eine Gradientenelution an einer Aluminiumoxidsäule unter standardisierten Bedingungen angewendet. Zur Charakterisierung dienen UV-spektrometrische Messungen. Das Verfahren kann auch zur Untersuchung von Aromaten aus anderen Erdölfraktionen eingesetzt werden.

     

Laminar flow and heat transfer over a two-dimensional triangular step

The velocity correction algorithm is used in the finite element method to solve forced convection problems between parallel plates with a triangular step, for Reynolds numbers up to 1000. Equal-order interpolation functions for velocity, pressure and temperature are used. The solutions show a smooth variation of pressure. The streamfunction, isotherms, isobars and velocity profiles are presented for a typical Reynolds number of 500. The skin friction and heat transfer results are presented for Reynolds numbers up to 1000.