The results of such simulations depend heavily on how the elements are assembled into the final shape, or mesh. Coming up with a better mesh may result in a better answer, and adaptive methods automate the process of constructing a more accurate mesh. Demkowicz, a professor in the Department of Aerospace Engineering and Engineering Mechanics and the Texas Institute for Computational and Applied Mathematics (TICAM) has built his career on studying adaptive methods.
Adaptive methods can be applied to many types of problems, and Demkowicz's efforts have ranged from the pattern that electromagnetic waves make when hitting the human head to the acoustics of submarines. There are two adaptations these methods can implement--changing the mesh size and changing the order of approximation. The mesh size is usually referred to by the variable h, and the order of approximation with the variable p, so these methods are usually called h-adaptive and p-adaptive, respectively.
A typical adaptive method simulation begins with an initial mesh on which the computer solves the problem. Next, an error estimate is calculated--finding the right error calculation is a subject of research in its own right--and the error distribution is estimated for the entire mesh. Finally, the mesh is modified by either changing the size of the mesh or by changing the order of approximation, and the process is repeated.
However, certain common types of engineering problems--specifically problems with singular solutions--require methods that automatically vary both the size and order of approximation--called hp-adaptive methods. Under the leadership of Tinsley Oden, TICAM's director, Demkowicz and colleagues wrote the first codes to implement hp-adaptive methods. The codes eventually resulted in PHLEX, the first hp-adaptive finite element commercial software, developed at the Computational Mechanics Company. Applications included a variety of complex problems in solid and fluid mechanics, focusing especially around supersonic compressible flows.
In the early 1990s, Oden and Demkowicz began working on wave propagation problems. The U.S. Navy, for example, funded acoustics studies of submarines using hp-adaptive techniques. By 1996, the acoustics work had progressed far enough for the team to begin looking at electromagnetics--a harder problem because solving Maxwell's equations is computationally more complex. "With a little luck, we proposed a method that has so far been very successful," Demkowicz said.
With the electromagnetics codes available, DARPA funded studies of integrated circuits and how electromagnetic waves interfered with the circuitry of combined digital and analog chips. In another project, the Air Force supports studies of antenna problems and large 3-D scattering (radar) simulations.
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ounds and electronic signals travel in the form of waves, traveling around and through objects in their path. The path that sound waves follow around the human head and into the ears helps a person locate the source of a sound. Electromagnetic waves, which have much greater energy than acoustic waves, often travel through the head, bouncing off the skull and brain. To study how the waves of sound and electronic signals interact with the irregular shape of the human head, Leszek Demkowicz and colleagues at the University of Texas are creating programs to track the path of these waves through a model of the human head and other objects.
Figure 1: An Adaptive Mesh 
