1. Mathematical Sciences
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
Physical modeling and mathematical analysis for nonlinear ordinary and partial differential, difference, and integral equations for:
- Advanced materials, including smart materials and structure and advanced composites.
- Fluid flow, including flow around rotors, missiles, and parachutes, combustion, detonation and explosion, two-phase flow, and granular flow.
- Nonlinear dynamics for optics, dielectrics, electromechanics and other nonlinear systems; physics-based mathematical models of human dynamics.
- Rigorous numerical methods for fluid dynamics, solid mechanics, material behavior, and simulation of large mechanical systems.
- Optimization: large-scale integer programming, mixed-integer programming, and nonlinear optimization.
Probability and Statistics
- Stochastic analysis and applied probability: stochastic differential equations and proc-esses, interacting particle systems, probabilistic algorithms, stochastic control, large deviations, simulation methodology, image analysis.
- Statistics: analysis for very large data sets or very small amounts of data from nonstandard distributions, point processes, Bayesian methods, integration of statistical procedures with scientific and engineering information, Markov random models, cluster analysis.
Systems and Control
- Mathematical system theory and control theory: control in the presence of uncertainties, robust and adaptive control for multi-variable and nonlinear systems, system identification and its relation to adaptive control, hybrid control, H-infinity control, nonholonomic control.
- Foundations of intelligent control systems: discrete event dynamical systems, hybrid systems, learning and adaptation, distributed communication and control, and intelligent control systems.
- Computational geometry, logic, network flows, graph theory, combinatorics.
- Symbolic methods: computational algebraic geometry for polynomial systems, discrete methods for combinatorial optimization, symbolic methods for differential equations, mixed symbolic-numerical methods, parallel symbolic sparse matrix methods, algorithmic methods in symbolic 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:
- Design of advanced materials and novel manufacturing processes
- Behavior of materials under high loads, failure mechanics
- Structures, including flexible and adaptable structures
- Fluid flow, including reactive flow
- Power and directed energy
- Microelectronics and photonics
- Automatic target recognition
- Soldier and aggregates of soldiers as systems: behavioral modeling, performance, mobility, heat-stress reduction, camouflage (visible, IR), chemical and ballistic protection.
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
Click here to view enlarged version of image.