Topological Semantics for Lumped Parameter Systems Modeling

Abstract

Behaviors of many engineering systems are described by lumped parameter models that encapsulate the spatially distributed nature of the system into networks of lumped elements; the dynamics of such a network is governed by a system of ordinary differential and algebraic equations. Languages and simulation tools for modeling such systems differ in syntax, informal semantics, and in the methods by which such systems of equations are generated and simulated, leading to numerous interoperability challenges. Logical extensions of SysML aim specially at unifying a subset of the underlying concepts in such languages. We propose to unify semantics of all such systems using standard notions from algebraic topology. In particular, Tonti diagrams classify all physical theories in terms of physical laws (topological and constitutive) defined over a pair of dual cochain complexes and may be used to describe different types of lumped parameter systems. We show that all possible methods for generating the corresponding state equations within each physical domain correspond to paths over Tonti diagrams. We further propose a generalization of Tonti diagram that captures the behavior and supports canonical generation of state equations for multi-domain lumped parameter systems. The unified semantics provides a basis for greater interoperability in systems modeling, supporting automated translation, integration, reuse, and numerical simulation of models created in different authoring systems and applications. Notably, the proposed algebraic topological semantics is also compatible with spatially and temporally distributed models that are at the core of modern CAD and CAE systems.