An Application for a Heterogeneous System

We will conclude by asking if there are problems that can use such systems. We will answer the question by giving an example typical of many real-world problems.

The scientific application is the aeroelastic analysis of an airframe. The computational problem is complicated because there are two distinct but coupled subproblems to be solved. The real aerodynamic design problem is not just to compute airflow over a plane in the context of a static frame but to design planes the way they are flown, which is dynamically. As soon as you do that, you get into the coupled problem of the fluid flow and the airframe structure. There is a structural engineering problem, which is represented by the finite-element analysis of the fuselage and the wing. There is a fluid-dynamics problem, which is the flow of air over the wing surface. Finally, there is a real interaction between these two problems. Obviously, the lift depends on the wing surface, but correspondingly, the fluid flow can cause vibrations in the wing.

One can represent the computation schematically as a two-dimensional (surface) structural problem with O (N² ) degrees of freedom or as a three-dimensional fluid problem with, typically, O (N³ ) degrees of freedom. Here N measures the spatial resolution of each problem.

Typically, the fluid computation can be solved by explicit methods, but the finite-element problem requires implicit solution techniques. The fluid models require solution time proportional to the number of degrees of freedom. The finite-element problems are typically sparse matrix problems, and it is hard to do better than O (N³ ) on solution times. Thus, we have a computation that is O (N³ ) for both the fluid and structure components.

Thus, the two computational phases are much the same in overall computational complexity. However, the communication of data, the interaction between the two phases, is related to the surface, only, and is therefore an O (N² ) data-transfer problem. Therefore, provided one is solving a large enough problem so that the O (N² ) communication cost is negligible, one can imagine solving the fluid part on a computer that is most effective for fluid computation and the structural part on a machine that is most effective for structural problems.

We have been studying this problem quite actively at the University of Colorado (Farhat et al. 1991, Saati et al. 1990, and Farhat et al. 1990) and have found that the fluid problem is best done on a machine like the

Connection Machine, on which one can take advantage of the SIMD architecture and work with grids that are logically rectangular. The structural problem is best done on machines that can handle irregular grids effectively, for example, a CRAYY-MP. Thus, ideally we would like to solve the whole problem on a heterogeneous system that includes both Cray and CM-2 machines. One should also remember that both phases of this computation have heavy visualization requirements. Thus, both systems need to be tightly coupled to high-end graphics systems. Furthermore, if a record is to be saved of a simulation, then high-speed access to a disk farm is mandatory because of the huge volumes of data generated per run. A fully configured heterogeneous environment is therefore essential.