Exergetic Port-Hamiltonian Systems Modeling Language

TL;DR

EPHS address the challenge of modeling complex multiphysical systems by introducing a compositional, thermodynamically consistent language that unifies energy-based storage, reversible Dirac-structure interchanges, and irreversible Onsager exchanges within a graphical framework. The approach leverages interconnection patterns built from an operad of undirected wiring diagrams to ensure power preservation and thermodynamic compliance, offering a flexible path from simple primitives to complex composites. By situating EPHS among bond graphs, port-Hamiltonian systems, and metriplectic/GENERIC formalisms, the paper highlights its modularity, hierarchical composition, and explicit exergy semantics, demonstrated through a shunt motor example. The work lays the groundwork for category-theoretic formalization, software implementation, and potential integration with discretization and machine learning, enabling scalable, thermodynamically sound modeling of engineered and physical systems.

Abstract

Mathematical modeling of real-world physical systems requires the consistent combination of a multitude of physical laws and phenomenological models. This challenging task can be greatly simplified by hierarchically decomposing systems into ultimately simple components. Moreover, the use of diagrams for expressing the decomposition helps make the process more intuitive and facilitates communication, even with non-experts. As an important requirement, models have to respect fundamental physical laws such as the first and the second law of thermodynamics. While some existing modeling frameworks make such guarantees based on structural properties of their models, they lack a formal graphical syntax. We present a compositional and thermodynamically consistent modeling language with a graphical syntax. In terms of its semantics, we essentially endow port-Hamiltonian systems with additional structural properties and a fixed physical interpretation, ensuring thermodynamic consistency in a manner closely related to the metriplectic or GENERIC formalism. While port-Hamiltonian systems are inspired by graphical modeling with bond graphs, neither the link between the two, nor bond graphs themselves, can be easily formalized. In contrast, our syntax is based on a refinement of the well-studied operad of undirected wiring diagrams. By combining a compositional, graphical syntax with an energy-based, thermodynamic approach, the presented modeling language simplifies the understanding, reuse, and modification of complex physical models.

Figures (12)

• Figure 1: Graphical display of an interconnection pattern with different syntactic elements shown in different colors. The outer box shown in purple represents the interface of the resulting composite system. The three protruding lines represent its ports called $\mathtt{f.p}$, $\mathtt{f.m}$ and $\mathtt{b_k}$. The two inner boxes shown in red represent (the interface of) the subsystems called $\mathtt{ke}$ (storage of kinetic energy) and $\mathtt{sa}$ (self-advection of kinetic energy). Both have ports called $\mathtt{p}$ (momentum) and $\mathtt{m}$ (mass). Box $\mathtt{sa}$ has another port called $\mathtt{b_k}$ (advection of kinetic energy across the boundary of the spatial domain). The common prefix $\mathtt{f}$ (fluid) of the outer ports $\mathtt{f.p}$ (fluid momentum) and $\mathtt{f.m}$ (fluid mass) combines them into a multiport. The black dots represent junctions where power is exchanged among the connected ports. The dashed line indicates that only information about the state (fluid mass) is exchanged via the outer port $\mathtt{f.m}$, with no energy transfer occurring.
• Figure 2: Hierarchical specification of an electro-magneto hydrodynamics (EMHD) model 2024LohmayerKrausLeyendecker: The fluid model f shown in (b) includes the kinetic energy system kin depicted in (a), as well as four additional models not detailed here: an internal energy system int, a thermal conduction model th, and volume and shear viscosity models vol and shr. The electro-magneto hydrodynamics model emhd shown in (c) includes the fluid model f depicted in (b), as well as three additional models not detailed here: an electromagnetic system em, an electric conduction model el, and a model for the electro-kinetic coupling ekc.
• Figure 3: Informal Venn diagram illustrating that EPHS integrates ideas from graphical and energy-based modeling of physical systems with bond graphs (see, e.g., 1961Paynter2010Borutzky), the metriplectic/GENERIC framework for nonequilibrium thermodynamics (see, e.g., 1984Morrison2005Oettinger), port-Hamiltonian theory for open, dissipative systems (PHS) (see, e.g., 2014SchaftJeltsema), and applied category theory (ACT) research on the formalization of graphical languages and compositional dynamical systems (see, e.g., 2019FongSpivak2013Spivak2015BaezErberle2020Libkind2023Myers).
• Figure 4: Interconnection pattern of an isothermal oscillator model. Box $\mathtt{pkc}$ represents the reversible coupling of the potential energy domain represented by the junction on its left and the kinetic energy domain represented by the junction on its right. Box $\mathtt{env}$ represents the isothermal environment, which absorbs the heat that is generated by the damping. The outer box represents the interface of the composite system. Its port $\mathtt{p}$ exposes the kinetic energy domain, allowing for an external forcing of the oscillator model.
• Figure 5: Interconnection pattern of an isothermal oscillator model. Whenever all ports connected to a certain junction have the same name, we write the name only once at the junction.

Theorems & Definitions (2)

• proof
• proof