Bonita V. Saunders

The effect of gravity modulation on thermosolutal convection in an infinite layer of fluid

In a gravitational field, the opposing effects of components of different diffusivities, for example, temperature and solute, in the density profile in a fluid may produce convective instabilities that exhibit a broad range of dynamical behavior. The effect of time periodic vertical gravity modulation on the onset of these instabilities in an infinite horizontal layer with stress free boundaries is examined. This work is viewed as a first step in expanding previous results in solidification to the full problem of characterizing the effects of gravity modulation in thermosolutal convection during the directional solidification of binary alloys. Calculations carried out both with and without steady background acceleration are presented, the latter results being relevant to microgravity conditions.

Design of a digital mathematical library for science, technology and education

The concept of a digital library is of proven worth because of its ability to provide dramatic capabilities that are impossible with traditional print media. We are interested in providing such capabilities for scientific, technical and educational users of mathematical reference data. Our attention is focused on the highly specialized field of mathematics that is concerned with the properties, application and computation of the elementary and higher mathematical functions. Calling upon domain experts worldwide for assistance, the National Institute of Standards and Technology is conducting an ambitious project to construct, ab initio, a comprehensive and authoritative Web resource on this subject. The need to make effective use of the latest developments in digital library research is a major focus, as is the development of content. We discuss our approach to such difficulties as the representation, display and manipulation of symbolic expressions, numerical data and graphical visualizations, and we describe a prototype Web site that has been constructed to test, evaluate and advance the NIST Digital Library of Mathematical Functions project.

Gravitational modulation of thermosolutal convection during directional solidification

During directional solidification of a binary alloy at constant velocity, thermosolutal convection may occur due to the temperature and solute gradients associated with the solidification process. For vertical growth in an ideal furnace (lacking horizontal gradients) a quiescent state is possible. The effect of a time-periodic vertical gravitational acceleration (or equivalently vibration) on the onset of thermosolutal convection is calculated based on linear stability using Floquet theory. Numerical calculations for the onset of instability have been carried out for a semiconductor alloy with Schmidt number of 10 and Prandtl number of 0.1 with primary emphasis on large modulation frequencies in a microgravity environment for which the background gravitational acceleration is negligible. The numerical results demonstrate that there is a significant difference in stability depending on whether a heavier or lighter solute is rejected. For large modulation frequencies, the stability behavior can be described by either the method of averaging or an asymptotic resonant mode analysis.

Digital Repository of Mathematical Formulae

The purpose of the NIST Digital Repository of Mathematical Formulae (DRMF) is to create a digital compendium of mathematical formulae for orthogonal polynomials and special functions (OPSF) and of associated mathematical data. The DRMF addresses needs of working mathematicians, physicists and engineers: providing a platform for publication and interaction with OPSF formulae on the web. Using MediaWiki extensions and other existing technology (such as software and macro collections developed for the NIST Digital Library of Mathematical Functions), the DRMF acts as an interactive web domain for OPSF formulae. Whereas Wikipedia and other web authoring tools manifest notions or descriptions as first class objects, the DRMF does that with mathematical formulae. See http://gw32.iu.xsede.org/index.php/Main_Page .

A boundary conforming grid generation system for interface tracking

Abstract The development of an algebraic grid generation system to track a solid-liquid interface during directional solidification of a binary alloy is discussed. A single mapping, constructed with tensor product B-splines, is proposed for calculations of both shallow and deep solidification cells. The initial spline coefficients for the coordinate mapping are modified to minimize a discrete functional that regulates the smoothness and orthogonality of the mesh. The use of transfinite blending function interpolation to obtain an initial grid is examined.

Web-based 3D visualization in a digital library of mathematical functions

The National Institute of Standards and Technology (NIST) is developing a digital library of mathematical functions to replace the widely used National Bureau of Standards Handbook of Mathematical Functions [Abramowitz and Stegun 1964]. The NIST Digital Library of Mathematical Functions (DLMF) will provide a wide range of information about high level functions for scientific, technical and educational users in the mathematical and physical sciences. Clear, concise 3D visualizations that allow users to examine poles, zeros, branch cuts and other key features of complicated functions will be an integral part of the DLMF. Specially designed controls will enable users to move a cutting plane through the function surface, select the surface color mapping, choose the axis style, or transform the surface plot into a density plot. To date, Virtual Reality Modeling Language and Extensible 3D (VRML/X3D) standards have been used to implement these capabilities in more than one hundred 3D visualizations for the DLMF. We discuss the development of these visualizations, focusing on the design and implementation of the VRML code, and show several examples.

Interactive 3D visualization of mathematical functions using VRML

Although there have been no major revisions of the National Bureau of Standards Handbook of Mathematical Functions [1] since its first publication in 1964, it continues to be widely used by the mathematical and scientific community. As a result, the National Institute of Standards and Technology (NIST), the successor organization to NBS, is engaged in a large scale project to update and expand the handbook and disseminate it on the World Wide Web as the NIST Digital Library of Mathematical Functions (DLMF). A key feature of the DLMF will be dynamic 3D visualizations that illustrate the unique features of special functions and provide the user with interactive capabilities. This paper discusses results to date in using the Virtual Reality Modeling Language (VRML) to create such visualizations. We show screens from a mockup version of the DLMF to demonstrate the use of VRML to create clear and informative displays of complicated mathematical functions that permit user interaction. We also briefly examine some preprocessing issues such as obtaining accurate data and the use of “contour” meshes to clip surface data before conversion to VRML format.

Semantic Preserving Bijective Mappings of Mathematical Formulae Between Document Preparation Systems and Computer Algebra Systems

Document preparation systems like Open image in new window offer the ability to render mathematical expressions as one would write these on paper. Using Open image in new window , Open image in new window , and tools generated for use in the National Institute of Standards (NIST) Digital Library of Mathematical Functions, semantically enhanced mathematical Open image in new window markup (semantic Open image in new window ) is achieved by using a semantic macro set. Computer algebra systems (CAS) such as Maple and Mathematica use alternative markup to represent mathematical expressions. By taking advantage of Youssef’s Part-of-Math tagger and CAS internal representations, we develop algorithms to translate mathematical expressions represented in semantic Open image in new window to corresponding CAS representations and vice versa. We have also developed tools for translating the entire Wolfram Encoding Continued Fraction Knowledge and University of Antwerp Continued Fractions for Special Functions datasets, for use in the NIST Digital Repository of Mathematical Formulae. The overall goal of these efforts is to provide semantically enriched standard conforming MathML representations to the public for formulae in digital mathematics libraries. These representations include presentation MathML, content MathML, generic Open image in new window , semantic Open image in new window , and now CAS representations as well.

Portable vectorized software for Bessel function evaluation

A suite of computer programs for the evaluation of Bessel functions and modified Bessel functions of orders zero and one for a vector of real arguments is described. Distinguishing characteristics of these programs are that (a) they are portable across a wide range of machines, and (b) they are vectorized in the case when multiple function evaluations are to be performed. The performance of the new programs are compared with software from the FNLIB collection of Fullerton on which the new software is based.

Dynamic 3D visualizations of complex function surfaces using X3DOM and WebGL

In 1997 the National Institute of Standards and Technology (NIST) embarked on a huge project to replace one of the most cited resources for mathematical, physical and engineering scientists, the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Abramowitz and Stegun 1964], originally released by the National Bureau of Standards (NBS) in 1964. The 1997 project, designed to update and modernize the handbook, culminated in May 2010 with the launch of a freely available website, the NIST Digital Library of Mathematical Functions [DLMF] (http://dlmf.nist.gov/), and its print companion, the NIST Handbook of Mathematical Functions [Olver et al. 2010]. While the presence of graphics was sparse in the original handbook, the new resource contains more than 600 illustrations of high level mathematical functions, including close to 200 interactive 3D visualizations on the website. We provide the motivation for the visualization work through the context of the project and discuss our current implementation using X3DOM and WebGL.