Standard Solution Methods Research Articles

SOLUTION OF OPTIMAL CONTROL PROBLEMS ON A PARALLEL MACHINE USING THE EPSILON METHOD

SUMMARYUsing Balakrishnan's epsilon formulation and the Rayleigh‐Ritz method with a Walsh function basis, optimal control problems are transformed from the standard two‐point boundary value problem form to an unconstrained non‐linear programming problem. The resulting matrix‐vector equations describing the optimal solution can be very easily implemented on say an IBM PC running Matlab or MatrixX on the one hand and have standard parallel solution methods for implementation on parallel computers on the other. Some computational results performed on an Intel hypercube are presented to illustrate that considerable computational savings can be realized by using the proposed algorithm.

Tabu search for nonlinear and parametric optimization (with links to genetic algorithms)

In spite of the widespread importance of nonlinear and parametric optimization, many standard solution methods allow a large gap between local optimality and global optimality, inviting consideration of metaheuristics capable of reducing such gaps. We identify ways to apply the tabu search metaheuristic to nonlinear optimization problems from both continuous and discrete settings. The step beyond strictly combinatorial settings enlarges the domain of problems to which tabu search is typically applied. We show how tabu search can be coupled with directional search and scatter search approaches to solve such problems. In addition, we generalize standard weighted combinations (as employed in scatter search) to include structured weighted combinations capable of satisfying specified feasibility conditions (e.g., mapping weighted combinations of scheduling, partitioning and covering solutions into new solutions of the same type). The outcome suggests ways to exploit potential links between scatter search and genetic algorithms, and also provides a basis for integrating genetic algorithms with tabu search.

BUU dynamics of unbound spherical nuclei

Abstract Within the nuclear Boltzmann-Uhling-Uhlenbeck model, we investigate the dynamical evolution of spherical calcium and gold nuclei that have been agitated into unbound configurations by either compression or heating. Using a modified pseudo-particle method that preserves the spherical symmetry, we find that the conversion of the compressional energy into radial motion is only weakly dissipative and, remarkably, for a range of initial compressions between density doubling and tripling the nucleus expands to a quasi-stationary unstable bubble-like configuration. The same processes are also studied with the standard method of solution in which perfect symmetry is absent and it is shown that while the bubbles then clusterize into bound fragments, the qualitative character of the outcome is different and sensitive to the employed number of pseudo-particles per nucleon, a purely numerical parameter. Our studies suggest that for suitable initial compressions there exists a specific nuclear multifragmentation process in which the decompression leads to an unstable hollow configuration that subsequently clusterizes into massive fragments.

Epsilon-ritz method for solving optimal control problems: Useful parallel solution method

Using Balakrishnan's epsilon problem formulation (Ref. 1) and the Rayleigh-Ritz method with an orthogonal polynomial function basis, optimal control problems are transformed from the standard two-point boundary-value problem to a nonlinear programming problem. The resulting matrix-vector equations describing the optimal solution have standard parallel solution methods for implementation on parallel processor arrays. The method is modified to handle inequality constraints, and some results are presented under which specialized nonlinear functions, such as sines and cosines, can be handled directly. Some computational results performed on an Intel Sugarcube are presented to illustrate that considerable computational savings can be realized by using the proposed solution method.

An Efficient Solution Technique for the Radiative Transfer Equation

The discretization of the multidimensional radiative transfer equation results in a very large linear system of equations. The standard solution method used by astrophysicists is a simplified fixed-point iteration (called approximate Λ-iteration),which becomes slow for the problems most interesting in astrophysics. As an alternative, a class of (block) conjugate gradient-like methods is proposed, together with special preconditioners for coping with the nonsymmetry and the bad condition number. The efficiency of these new solvers in combination with a finite-element upwind scheme on problem-adapted nonuniform grids is demonstrated by several test calculations for physically relevant data.

Lumped-parameter representation of fluid-system transients

Flow and pressure transients in closed pipes are described by non-linear hyperbolic partial differential equations. In many control applications a discrete-space lumped-parameter ordinary differential equation model is required. With such representations, care is needed to ensure that they retain essential physical characteristics of the phenomena described. Intentionally or otherwise, most of these fluid-system models are related to the numerical discretisation technique known as the Method of Lines. Aspects of implementing this are investigated, including the form of the basic equations and variables, representation of pipe friction, choice of interpolating polynomials in space and method of integration in time. These are compared with the standard Method of Characteristics solution for the original partial differential equations to eliminate any potential numerical instabilities, parasitic transient solutions or inconsistencies between steady and transient states which might influence adversely system control or stability studies. The Method of Lines is most appropriate for parabolic problems, and many existing applications to fluid transients are found to be satisfactory only where changes are gradual and continuous. The preferred formulation is free from this restriction, and applied to a stability investigation (prediction of compressor surging) is shown to perform reliably.

Fast Hybrid Solution of Algebraic Systems

The error and timing of solvers consisting of both analog and digital circuitry for sparse linear systems of equations are proposed and analyzed. High speed but low precision is obtained from the analog circuits. This is combined with low speed but high precision from the digital circuits. The hybrid circuit should be faster than digital circuits alone. As a preconditioner to standard iterative solution methods, the hybrid circuit makes the cost of the preconditioning step negligible. The hybrid circuit is also applied to a standard multilevel algorithm.

RCS of waveguide cavities: a hybrid boundary-integral/modal approach

A hybrid technique combining the flexibility of the boundary-integral method and the efficiency of the modal approach is presented. The technique is used for analyzing the electromagnetic scattering from open waveguide cavities. The generalized scattering matrices of the arbitrary waveguide discontinuity regions are determined using a boundary-integral equation formulation. The scattering matrix of the composite structure is found by cascading the scattering matrices of individual sections. The radar cross section of the open cavity is then calculated by a reciprocity formulation in conjunction with the Kirchoff's approximation. Excellent agreements with the standard method of moments solution and with experimental data for offset bend structures are found. Examples of cavities with arbitrarily shaped and absorber-coated terminations are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved method for calculating booster to launch pad interface transient forces

This paper presents a methodology for the calculation of the transient interface forces between a launch vehicle and its launch pad. The method presented is closed form and, thus, is mathematically exact. The method uses variable constraint equations to couple the launch vehicle to the pad. The resulting interface forces are monitored each time step, and as the launch vehicle separates from the pad, the constraint equations are modified accordingly. The only assumption in the method is that the forces acting on the booster and pad, including the subject interface forces, vary linearly between integration time steps. This is the same assumption used in standard linear transient loads analyses. In addition, the nature of the constraint equations allows a great reduction in the number of coupled equations to be solved. Solving the resulting small system of equations yields the interface forces, which are then used to solve for the booster and pad response by the standard linear transient solution method. Use of constraint equations also frees the analyst from calculating coupled boosterpad systems modes, which are a necessity in some methodologies currently in use.

Stereological analysis of creep cavities on polished sections

An analysis is presented of the stereological problem of estimating the distribution of sizes and shapes of creep cavities from cavity measurements taken on polished plane sections, subject to the assumptions that the cavities may be represented as collections of oblate spheroids and that the distribution of orientations of the cavities is known a priori. The problem is cast in the form of a double integral equation whose solution yields the desired distribution. An iterative numerical solution technique for this equation is proposed which seems to yield marked improvements over standard solution methods.

Use of preconditioned bi-conjugate gradient method in modeling two-phase fluid flow in porous media

We have found the preconditioned bi-conjugate gradient method superior to the standard conjugate gradient method for iterative solution of linear systems occurring in solving the finite difference form of partial differential equations describing multi-dimensional two-phase flow in porous media. The matrices involved in these systems are typically non-symmetric and not positive definite. Fully vectorizable preconditioning was effective in our solutions using the bi-conjugate gradient method, whereas more extensive, non-vectorizable preconditioning is often necessary when using the conjugate gradient method.

Numerical solution of symmetric integral equations with kernels non-smooth on the diagonal

A standard method for solution of a syrmnetric integral equation in an interval with a kernel K(x,t) ∊ C(IxI) is to solve the eigenproblem with wjn>0 and xjn∊I, j=l,...,n. For kernels with discontinuous derivatives on the diagonal x=t we use a variant of Gregory's quadrature fonnula to improve the error estimate of (1) from 0 (n-2) at most to 0(n-4).

Matrix-Free Methods for Stiff Systems of ODE’s

We study here a matrix-free method for solving stiff systems of ordinary differential equations (ODE’s). In the numerical time integration of stiff ODE initial value problems by BDF methods, the resulting nonlinear algebraic system is usually solved by a modified Newton method and an appropriate linear system algorithm. In place of that, we substitute Newton’s method (unmodified) coupled with an iterative linear system method. The latter is a projection method called the Incomplete Orthogonalization Method (IOM), developed mainly by Y. Saad. A form of IOM, with scaling included to enhance robustness, is studied in the setting of Inexact Newton Methods. The implementation requires no Jacobian matrix storage whatever. Tests on several stiff problems, of sizes up to 16,000, show the method to be quite effective and much more economical, in both computational cost and storage, than standard solution methods, at least when the problem has a certain amount of clustering in its spectrum.

PREDICTION OF HEAT AND FLUID FLOW IN COMPLEX GEOMETRIES USING GENERAL ORTHOGONAL COORDINATES

A method is presented for solving the equations of motion in orthogonal curvilinear coordinates using a computational grid that conforms to the boundaries and is orthogonal in the interior. Details of the derivation are provided. The particular formulation proposed maintains simplicity, clarity, and flexibility by special treatment of the stresses and diffusive fluxes. Because the equations are simple generalizations of those that arise when specific analytic coordinates are used, standard solution methods are applicable. The method is demonstrated for two problems, one of which involves both heat and fluid flow.

A New Standard Substance for Calibration of Circular Dichroism: Ammonium d-10-Camphorsulfonate

Non-hygroscopic water-soluble crystals of ammonium d-10-camphorsulfonate(ACS) were prepared and proposed as a practical secondary standard substance for the calibration of circular dichroism(CD) to replace deliquescent d-10-camphorsulfonic acid(CSA). ACS gave the same CD, ORD and molar rotation, [M]D, as CSA within an error of 1%. The molar ellipticity, [θ]290.5+7910 (c=0.06, water) was obtained by the CSA standard solution method, by which CD and [M]D were critically measured with a CD apparatus and a high precision polarimeter, respectively. The [M]25D+52.06 or [α]25D+20.88 (c=6, R.S.D. 0.1%) was used to check the optical purity.

Determination of iodine value fatty acids by a flow-injection method

An automated method for determining the iodine value for various industrial fatty acids is described. The method is based on flow-injection techniques and the chemistry is a modified version of the standard Wij's solution method. The system is fully computerized with an AIM-65 microcomputer. Two detection system, u.v.-visible absorbance and potentiometric sensing, are compared. Results comparable to the conventional method are obtained. Continuous-run sample throughput rates of 12 h −1 are eralistic.

Evaluation of a Test Strip Used to Monitor Food Processing Sanitation

A test strip, which contained a small absorbent pad on the end of a plastic strip for detection of microbial contamination of liquids and surfaces, was compared to standard rinse solution and surface contact methods. Bottles and food contact surfaces were unclean, cleaned or cleaned and sanitized before being evaluated with test methods. Results from test strips correlated well with those of the standard rinse solution method for bottles that were clean and/or sterilized, but not for those that were heavily contaminated. When test strips were used on cleaned contact surfaces, counts were one log cycle greater than those of contact plate or swab methods; however, the three surface methods correlated well for surfaces that were cleaned and sanitized. To insure that the contact methods were recovering microorganisms, surfaces were spread with known levels of Escherichia coli and Staphylococcus aureus. Results correlated well for low levels of contamination, but not for levels greater than 1 × 104 organisms/25 cm2 Overall, test strips could be used for quick indication of sanitation of cleaned and sanitized food contact surfaces and containers if special precautions and limitations were understood.

An Extended Programming Experience

The paper summarizes a twenty-year experience with the computational aspects of the transportation problem of linear programming. The problem is completely described and standard solution methods are summarized. The order bounds for these computations are examined. Attention then turns to some sample research problems, including shortening the search for a move, solving a condensed problem, choosing a starting basis, and several recently developed dual methods. Brief reference is made to the difficulties which arise in publishing unorthodox approaches such as those discussed in the paper.

Argon/propane ionization-chamber dosimetry for mixed x-ray/neutron fields.

The photoneutrons produced by high-energy x-ray machines can diffuse through the mazes usually employed at the treatment-room entrance and readily penetrate the lead-lined doors used for x-ray shielding. The measurement of these neutrons in the presence of x-rays and the determination of dose equivalent poses a problem for which there is currently no standard method of solution. In order to separate x-ray dose from neutron dose, the author employed an ionization chamber alternately filled with argon or propane. The response characteristics of this chamber to x-ray and neutrons are described. Quality factors were determined from a calculated neutron spectrum. As a result of these measurements, a 10-in. polyethylene door was added to the entranceway of a 25-MV linear accelerator.

Effects of phosphorus oxychloride and phenylphosphonic dichloride on the radical polymerization of methyl methacrylate and styrene

AbstractEffects of pentavalent phosphorus compounds on the radical polymerization of methyl methacrylate (MMA) and styrene (St) were studied. Phosphorus oxychloride (Cl3PO) and phenyl‐phosphonic dichloride (C6H5Cl2PO) were used. Polymerization was carried out in benzene at 50°C by the standard solution method, α,α′‐azobisisobutyronitrile (AIBN) being used as the initiator. In the polymerization of MMA, both phosphorus compounds increased the rate of polymerization. NMR spectral data suggested that this increasing effect was due to the complex formation between each phosphorus compound and MMA monomer. In the case of polymerization of St, NMR data also indicated the formation of a complex between the phosphorus compound and St monomer. Both phosphorus compounds showed an increasing effect for the rate of polymerization. Though these increasing effects could be explained by the complex formation, the polymerization of St in the presence of Cl3PO was especially found to be due to the cationic polymerization initiated simultaneously by Cl3PO in addition to the radical polymerization. These phosphorus compounds acted as chain‐transfer agents in both polymerization systems. The parameters (Qtr,etr) which indicate the reactivity of a chain‐transfer agent were calculated from the observed values of chain‐transfer constant for both polymerization systems.