Christopher Bailey-Kellogg, Feng Zhao
Many important physical phenomena, such as temperature distribution, air flow, and acoustic waves, are described as continuous, distributed parameter fields. Analyzing and controlling these physical processes and systems are common tasks in many scientific and engineering domains. However, the challenges are multifold: distributed fields are conceptually harder to reason about than lumped parameter models; computational methods are prohibitively expensive for complex spatial domains; the underlying physics imposes severe constraints on observability and controllability. This paper develops an ontological abstraction and a structure-based design mechanism, in a framework collectively known as spatial aggregation (SA), for reasoning about and synthesizing distributed control schemes for physical fields. The ontological abstraction models a physical field as a hierarchy of networks of spatial objects. SA applies a small number of generic operators to a field to compute concise structural descriptions such as iso-contours, gradient trajectories, and influence graphs. The design mechanism uses these representations to find feasible control configurations. We illustrate the mechanism using a thermal control problem from industrial heat treatment and demonstrate that the active exploitation of structural knowledge in physical fields yields a significant computational advantage.