(Slide 24)

The chain models formalism can be viewed as a generalization of the theory of electrical circuits. Like circuit theory, the theory of chain models is based on concepts from algebraic topology, which provides a rigorous mathematical foundation. Chain models are a generalization in two ways. First, they are not restricted to electronics -- we can build chain models of products that include physical phenomena such as heat transfer, rigid body dynamics, fluid flow, elasticity, and so forth. Like an electrical circuit, with resistors, capacitors, transistors, and so forth, a chain model is constructed of primitive components, but in this case, the correspond to physical phenomena in the different physical domains. For instance, a fluid element is a component that represents fluid flowing in a particular region of space, just as a resistor models the electrons flowing through a particular electronic component. The second way in which chain models generalize electrical circuits is that they can represent geometric information. This allows them to represent behavior that depends more critically on spatial characteristics -- behavior that is modeled by distributed parameter systems. These spatial systems are among the most demanding domains in engineering, and include phenomena such as fluid flow, heat transfer, elasticity and plasticity. In order to model these more difficult domains, chain models incorporate higher dimensional analogs of the 0 and 1 dimensional components of electrical circuits. By rules similar to those used in constructing a circuit out of electrical components, a chain model may be constructed out of components such as fluid volumes, resistors, elastic solids, and so forth. As is the case with electrical circuits, building a chain model results in a mathematical model as well -- a set of equations that can be used to analyze the behavior of the system.