Van Snyder's Papers

Mathematical Software

Algorithm 531: Contour Plotting, ACM Transactions on Mathematical Software 4, 3 (September 1978) pp 290-294. Also https://dl.acm.org/doi/10.1145/355791.355800

Given a two-dimensional array of samples of a surface, contour values, and a subroutine to draw lines, the subroutine GCONTR determines sequences of points in the plane which may be used to draw contours through equal values of the surface.

A contour plotting algorithm may be constructed by following contours from some starting point until they either close or intersect a boundary, or by examining each cell of the grid in turn and drawing all contours found within the cell. The advantages of contour following are that contour labeling is relatively easy and that the pen of an incremental plotter does not move about as much without writing anything. the advantages of methods which draw all contours found inside a cell are that less auxiliary storage is needed and that each cell can be completely processed before going on to the next, thereby allowing generation of contours over a much larger array than can be accomodated in memory at one time.

GCONTR is of the type which follows contours. On a representative problem, a progam using GCONTR generated about 11,000 plotter commands with about 57,000 commands generated by a program using a cellular method. For this problem, grid lines, user identification, and contour values required about 36,000 additional commands.

Complicated and comprehensive software that is meant to execute in a non-interactive or semi-interactive mode needs to be configured to carry out the desired tasks, needs to carry out those tasks efficiently, needs to be extensible to take on additional ambitions, and needs to be maintainable. All of these goals can be advanced by posing the software as a language interpreter. Herein, we describe the application of that principle to data analysis software for the Microwave Limb Sounder instrument on the Earth Observing System Aura satellite.

A method to compute explicit solutions of homogeneous triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients is described. It is suitable for the limited case of well separated eigenvalues, or for multiple zero eigenvalues provided the entire column corresponding to a zero eigenvalue is zero. The solution for the case of constant inhomogeneity is described. The method requires only to compute a constant matrix using a simple recurrence. Computing the solutions of the system from that matrix, for values of the independent variable, requires to exponentiate only the diagonal of a matrix. It is not necessary to compute the exponential of a general triangular matrix. Although this work was motivated by a study of nuclear decay without fission or neutron absorption, which is used throughout as an example, it has wider applicability.

Algorithm 982: Explicit solutions of triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients provides an explicit solution for an homogeneous system, and a brief description of how to compute a solution for the inhomogeneous case. The method described is not directly useful if the coefficient matrix is singular. This remark explains more completely how to compute the solution for the inhomogeneous case, and for the singular coefficient matrix case.

An implementation of approximations for Fresnel integrals and associated functions is described. The approximations were originally developed by W. J. Cody, but a Fortran implementation using them has not previously been published.

The subprogram contains the lines

              WANTG = NEEDG(MODE+4) .AND. .NOT. HAVEG
              IF (WANTG) THEN
                 IF (X .LE. 1.9D0) THEN
  

The third line should be testing on the variable AX.

Without this change, symmetry is accurate only to nine digits.

Algorithm 723: Fresnel Integrals has been improved to provide more precise results for x >> 0.

There are mistakes and typographical errors in Remark on Algorithm 723: Fresnel Integrals, which appeared in ACM Transactions on Mathematical Software 22, 4 (December 1996). This remark corrects those errors. The software provided to Collected Algorithms of the ACM was correct.

This note gives the gravitational potential of the disk $\{(x,\; y,\; z):\; x2+y2\le \; \rho 2,\; z=0$ and the gravitational field at the point $(x,\; y,\; z)$. Formulas for a ring can be obtained as the difference of our results for two different values of ρ. Results are obtained in terms of elliptic integrals and we indicate how these functions can be computed efficiently. Formulas necessary for the computation of partial derivatives are also given.

A discussion and analysis of a well known but frequently-discounted hashing algorithm is presented. We believe its use is preferred in most hashing applications.

Microwave Limb Sounder

MLS is a passive instrument that measures thermal microwave emission from the limb of the atmosphere at 128 antenna pointings, ranging from the surface of the Earth to 100 kilometers altitude at the limb, in about 1,000 channels, every 26 seconds.

The Level 2 step in the data analysis sequence uses those microwave power spectra as input and inverts the radiative transfer equation to calculate temperature, water vapor, and the abundance of about fifteen minor atmospheric constituents, at 72 pressure levels between altitudes of eight and 80 kilometers, on 3,500 vertical profiles, every day.

I'm a co-author of many of the papers listed on the MLS web page

Nuclear Power

We propose a new form for metallic nuclear reactor fuel, consisting of finely-divided particles mixed with sodium for thermal bond. This results in greater fuel density than with solid slugs fabricated at 75% initial smear density. Larger surface-to-volume ratio allows more fission product gases and metallic fission products to diffuse out of fuel particles, resulting in less swelling, greater burnup before processing, a simple preliminary spent-fuel processing step, and the possibility to postpone pyroelectric processing until several cycles of burnup and preliminary processing. Less frequent pyroelectric processing, simple preliminary processing, and larger surface-to-volume ratio reduce total processing cost. Preliminary processing produces separate fission products, in metallic rather than salt or mineral form, in particular producing caesium and strontium separately, thereby simplifying and reducing storage cost. Intrinsically structurally weak fuel would not rupture fuel pin cladding by swelling. Expense and complexity of the process would be offset by reduced total system cost.

Finely-Divided Metal as Nuclear Reactor Fuel recommended that experiments ought to be conducted to determine the relationship between the volumetric density of fuel particles in sodium, and thermal conductivity. That work had already been done in the more general context of two-phase liquid-solid systems.

In EBR-II, fuel was contained within fuel pins, and surrounded by coolant. We propose to invert that relationship in a future LMFBR: Motile fuel, composed of fine metallic particles mixed with sodium for thermal bond, is contained within a hexagonal unit cell penetrated by coolant tubes. A plenum above the fuel is connected to an exhaust port, to remove fission-product gases, especially the powerful neutron poison xenon, and maintain pressure within the unit cell in the range of the pressures in the coolant tubes or surrounding environment. It is possible to increase fuel volume fraction, improve neutron economy, increase breeding ratio, and reduce or eliminate fuel/container mechanical interaction. Passive safety implications of this proposal have not been assessed.

In April 1961, Atomic Power Development Associates produced Summary of the APDA Fuel Development Programs. Chapter XVII described a Paste Fuel Concept. The report noted that there are "advantages inherent in a mobile-fueled reactor." Patent number 3,169,117, entitled Nuclear Reactor Paste Fuel Composition, was issued on May 9, 1961. In May 1964, Argonne National Laboratory produced Catalog of Nuclear Reactor Concepts. The chapter concerning paste fuels concluded "The few paste-fuel concepts developed to date and the present early stages of such developments show that considerably more work probably will be required before the paste-fuel concept can be considered for commercial development." This monograph enlarges upon and quantifies the APDA concept, which appears not to have been pursued. Additional passive safety concepts, that might also eliminate the need for control assemblies, are described. Several important consequences of continuously-processed fuel that are not discussed in the APDA report are described, in particular, that the "iodine pit" startup control instability can be eliminated.

Using new ideas and novel combinations of old ideas, it is possible to construct an inherently and passively safe compact boiling sodium reactor. Fuel is metal nitride ceramic particles. Liquid sodium coolant enters at the bottom, flows around fuel particles, and sodium vapor exits at the top, circulated by its boiling, not by pumps. The device consists of a reaction region, connected to a storage region in which criticality is impossible. It has no moving parts other than paste fuel, coolant, and pumps and valves to control gas pressures above the regions. There are no control rods: Power output depends upon the amount of fuel in the reaction region, which is controlled by gas pressure or electromagnetic pumps. If the gas pressure or electromagnetic pump fails, fuel flows passively under the influence of gravity from the reaction region to the storage region and the reaction stops. The reaction region is small. The volume of circulating coolant is small. Power output can be changed rapidly. If electricity is produced by magnetohydrodynamic generators, the entire plant has essentially no moving parts. External electrical power is not necessary for safety. Small volumes of fuel slurry are removed continuously for processing, which eliminates the ``iodine pit'' startup control instability. There is no need to shut down the reactor for refueling. The capacity factor should well exceed 95%.

Present methods to separate caesium and strontium from other fission products use expensive and hazardous solvents such as calixarene ethers, bis-1,2-dicarbollyl cobalt titanium in nitrobenzene, and an aqueous phase containing polyethelene glycol and 1,2-cyclohexyldiaminetetraacetic acid. A simple aqueous process can separate caesium and strontium from other fission products that remain in electrolyte after pyroelectric refining. All steps operate at atmospheric pressure. Chemical steps operate at room temperature. The only consumed inputs are heat and ammonium sulfate. Inexpensive initial inputs are required, but they are conserved. There are no hazardous or expensive solvents, and no wastes other than fission products. The final waste forms have significantly higher fission-product density than in the zeolite waste form produced at Experimental Breeder Reactor II. Essentially all electrolyte is recovered. The objective is not to obtain pure caesium and strontium, but rather to produce other waste forms that do not contain them. Whether this process is useful depends upon the relationship of its expense to the cost savings that result from storing caesium and strontium separately from other fission products.