Chapter V. Basic Research
Army Science and Technology Master Plan (ASTMP 1997)

1. Mathematical Sciences

a. Strategy

Mathematics plays an essential role in modeling, analysis, and control of complex phenomena and systems of critical interest to the Army. To achieve Army goals, research in several areas is important:

(1) Applied analysis

(2) Computational mathematics

(3) Probability and statistics

(4) Systems and control

(5) Discrete mathematics.

An investment strategy meeting with participants from ARO, ARL, RDECs, COE-WES, CAA, DUSA-OR, and academia identified several exciting research areas that will have significant impact on future Army technologies. Based on these recommendations, research priorities inside these areas are listed below.

b. Major Research Areas

Applied Analysis

Physical modeling and mathematical analysis for nonlinear ordinary and partial differential, difference, and integral equations for:

Computational Mathematics

Probability and Statistics

Systems and Control

Discrete Mathematics

c. Other Research Areas

As noted above, mathematical modeling is increasingly being identified as critical for progress in many areas of Army interest. The mathematical and scientific tasks in these areas of interest are frequently of significant complexity. As a result, researchers from two or more areas of mathematics must often collaborate among themselves and with experts from other areas of science and engineering to achieve Army goals. Some examples of cross-cutting areas of research include the break-up of liquid droplets in high-speed air flow (for determination of the dispersion of chemical or biological agents spilled from intercepted theater-range missiles), computational methods for penetration mechanics, and automatic target recognition. For example, promising approaches to computer vision for automatic target recognition require research in a wide range of areas including constructive geometry, numerical methods, stochastic analysis, Bayesian statistics, probabilistic algorithms, and distributed parallel computing.

d. Benefits of Research

With the change from a predictable large threat to numerous and often unpredictable regional threats, the need for more flexibility in Army systems and more rapid development of these systems increases. As the cost of physical experimentation increases, the role of mathematical modeling becomes more important. Mathematical modeling is a major factor in assuring that a system is well designed and that it will work once built. In all of the following areas, mathematics is a fundamental tool required by the Army of the present and the future:

Two of these areas bear further comment. Advances in analytical and computational fluid dynamics are required to understand detonation (see Figure V-5).

Figure V-5. Density gradient soon after detonation of an explosive in a gas. The quasi-spherical density contours produced by many previous computational techniques were non-physical. More accurate highly convoluted contours are produced by new computational techniques developed under ARO sponsorship.

Advances in modeling and computational capabilities are needed to support stochastic modeling and simulation of combat to assess changes in doctrine and tactics and to determine the cost effectiveness of new systems on the battlefield (see Figure V-6).

Figure V-6. Focusing Various Mathematical Inputs Toward Potential Military Applications Through Distributed Simulation

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