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. 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.

Surrogate Modeling for Physical Systems with Preserved Properties and Adjustable Tradeoffs

“…In a recent article [34], common reference semantics for lumped parameter system modeling were presented, based on algebraic topological foundations of network theory [27,7], which can serve as a unifying abstraction of system modeling languages such as Modelica [11], Simulink [10], linear graphs [28], and bond graphs [23], etc. A key advantage of using this abstraction is the ability to automatically map a given topological structure for the LPM (e.g., a circuit graph or mass/spring/damper network) to a set of governing ODEs.…”

“…They also include constitutive laws associated with lumped components such as springs and dampers in mechanical systems, resistors and capacitors in analog circuits, conductors in heat transfer, and so on. The recipe for generating the ODEs from system topology in [34] is given by Tonti diagrams [33] of network theory (Fig. 4 (c)).…”

“…4 (a), we first convert the LPM from the domain-specific format (e.g., Modelica) to the domain-agnostic canonical form (i.e., oriented cell complex) as shown in Fig. 4 (b), using the semantics provided in [34]. Each 1−cell is associated with a symbolic constitutive relation and given an initial value to the constitutive parameter.…”

“…Each 1−cell is associated with a symbolic constitutive relation and given an initial value to the constitutive parameter. After selecting a state variable of interest, the state equations (i.e., system of second-order ODEs/DAEs) are generated by tracing groups of paths along the Tonti diagram of network theory with the appropriate physical types [34]. There are a total of 8 different options for state variables, each of which corresponds to a different groups of paths [34].…”

“…After selecting a state variable of interest, the state equations (i.e., system of second-order ODEs/DAEs) are generated by tracing groups of paths along the Tonti diagram of network theory with the appropriate physical types [34]. There are a total of 8 different options for state variables, each of which corresponds to a different groups of paths [34]. Once the system of ODEs/DAEs are assembled, we use ordinary least squares regression-Figure 4: The workflow for the data-driven approach to LPM construction.…”