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Energy-Based Modeling and Structure-Preserving Discretization of Physical Systems

M. H. M Rashid

TL;DR

The paper develops a unified energy-based framework that generalizes port-Hamiltonian systems to handle constraints and high-index differential-algebraic equations, with a focus on preserving energy balance and dissipation through both modeling and discretization. It introduces structure-preserving regularization to reduce index, proves exponential stability under coercivity, and provides nonlinear structure-preserving discretizations (midpoint and discrete gradient) that maintain the energy-dissipation structure at the discrete level. The theory is demonstrated across diverse applications, including poroelasticity, nonlinear circuits, constrained mechanics, phase-field models, and quantum-thermodynamic memory systems, illustrating robust numerical behavior and physically meaningful long-time dynamics. The work lays a solid foundation for computational physics and engineering, with future directions toward adaptive time stepping, stochastic port-Hamiltonian systems, and model reduction to tackle large-scale multiphysics problems.

Abstract

This paper develops a comprehensive mathematical framework for energy-based modeling of physical systems, with particular emphasis on preserving fundamental structural properties throughout the modeling and discretization process. The approach provides systematic methods for handling challenging system classes including high-index differential-algebraic equations and nonlinear multiphysics problems. Theoretical foundations are established for regularizing constrained systems while maintaining physical consistency, analyzing stability properties, and constructing numerical discretizations that inherit the energy dissipation structure of the continuous models. The versatility and practical utility of the framework are demonstrated through applications across multiple domains including poroelastic media, nonlinear circuits, constrained mechanics, and phase-field models. The results ensure that essential physical properties such as energy balance and dissipation are maintained from the continuous formulation through to numerical implementation, providing robust foundations for computational physics and engineering applications.

Energy-Based Modeling and Structure-Preserving Discretization of Physical Systems

TL;DR

The paper develops a unified energy-based framework that generalizes port-Hamiltonian systems to handle constraints and high-index differential-algebraic equations, with a focus on preserving energy balance and dissipation through both modeling and discretization. It introduces structure-preserving regularization to reduce index, proves exponential stability under coercivity, and provides nonlinear structure-preserving discretizations (midpoint and discrete gradient) that maintain the energy-dissipation structure at the discrete level. The theory is demonstrated across diverse applications, including poroelasticity, nonlinear circuits, constrained mechanics, phase-field models, and quantum-thermodynamic memory systems, illustrating robust numerical behavior and physically meaningful long-time dynamics. The work lays a solid foundation for computational physics and engineering, with future directions toward adaptive time stepping, stochastic port-Hamiltonian systems, and model reduction to tackle large-scale multiphysics problems.

Abstract

This paper develops a comprehensive mathematical framework for energy-based modeling of physical systems, with particular emphasis on preserving fundamental structural properties throughout the modeling and discretization process. The approach provides systematic methods for handling challenging system classes including high-index differential-algebraic equations and nonlinear multiphysics problems. Theoretical foundations are established for regularizing constrained systems while maintaining physical consistency, analyzing stability properties, and constructing numerical discretizations that inherit the energy dissipation structure of the continuous models. The versatility and practical utility of the framework are demonstrated through applications across multiple domains including poroelastic media, nonlinear circuits, constrained mechanics, and phase-field models. The results ensure that essential physical properties such as energy balance and dissipation are maintained from the continuous formulation through to numerical implementation, providing robust foundations for computational physics and engineering applications.

Paper Structure

This paper contains 11 sections, 12 theorems, 113 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Energy-Based Modeling Framework
  4. Energy Dissipation Property
  5. Structure-Preserving Discretization
  6. Midpoint Rule Discretization
  7. Discrete Gradient Method
  8. Structure-Preserving Interconnections
  9. Energy-Based Modeling Framework: Theoretical Foundations
  10. Examples Illustrating the Main Results
  11. Conclusion and Future Work

Key Result

Lemma 2.2

The energy function $H(z_{1}, z_{2})$ satisfies the dissipation inequality: In particular, for vanishing inputs ($u = 0$), the system is dissipative: $\frac{d}{dt}H \leq 0$.

Theorems & Definitions (27)

  • Remark 2.1
  • Lemma 2.2: Energy Dissipation Altmann2025
  • proof
  • Lemma 2.3: Discrete Energy Dissipation, Midpoint Rule Altmann2025Gonzalez1996
  • Lemma 2.4: Discrete Energy Dissipation, Discrete Gradient Method Altmann2025McLachlan1999
  • Lemma 2.5: Structure-Preserving Interconnection Altmann2025
  • Theorem 3.1: Structure-Preserving Regularization
  • proof
  • Corollary 3.2: Exponential Stability of Regularized System
  • proof
  • ...and 17 more