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A novel energy-based modeling framework

R. Altmann, P. Schulze

TL;DR

This work introduces an energy-based modeling framework for constrained dynamical systems, providing an alternative to port-Hamiltonian formulations. It centers on an energy function $H(z_1,z_2)$ with a state partition $z=[z_1; z_2; z_3]$ and a structure matrix pair $(\mathbf{J},\mathbf{R})$ that ensure dissipation and enable power-preserving interconnections. The authors prove automatic energy dissipation, discuss structure-preserving interconnections, and develop time-stepping schemes (midpoint and discrete gradient) that preserve dissipativity. Ten diverse examples, from poroelasticity and DAEs to circuits and PDEs like Cahn–Hilliard, illustrate the framework's broad applicability and practical relevance for energy-based modeling and numerical simulation.

Abstract

We introduce an energy-based model, which seems especially suited for constrained systems. The proposed model provides an alternative to the popular port-Hamiltonian framework and exhibits similar properties such as energy dissipation as well as structure-preserving interconnection and Petrov-Galerkin projection. In terms of time discretization, the midpoint rule and discrete gradient methods are dissipation-preserving. Besides the verification of these properties, we present ten examples from different fields of application.

A novel energy-based modeling framework

TL;DR

This work introduces an energy-based modeling framework for constrained dynamical systems, providing an alternative to port-Hamiltonian formulations. It centers on an energy function with a state partition and a structure matrix pair that ensure dissipation and enable power-preserving interconnections. The authors prove automatic energy dissipation, discuss structure-preserving interconnections, and develop time-stepping schemes (midpoint and discrete gradient) that preserve dissipativity. Ten diverse examples, from poroelasticity and DAEs to circuits and PDEs like Cahn–Hilliard, illustrate the framework's broad applicability and practical relevance for energy-based modeling and numerical simulation.

Abstract

We introduce an energy-based model, which seems especially suited for constrained systems. The proposed model provides an alternative to the popular port-Hamiltonian framework and exhibits similar properties such as energy dissipation as well as structure-preserving interconnection and Petrov-Galerkin projection. In terms of time discretization, the midpoint rule and discrete gradient methods are dissipation-preserving. Besides the verification of these properties, we present ten examples from different fields of application.
Paper Structure (17 sections, 5 theorems, 65 equations)

This paper contains 17 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Energy-Based Modeling Framework
  3. Energy dissipation and interconnections
  4. Petrov--Galerkin projection
  5. Discretization in time
  6. Collection of Examples
  7. Example I: pH systems and gradient systems
  8. Example II: poroelasticity
  9. Example III: a class of index-$1$ DAEs
  10. Example IV: semi-explicit index-$2$ DAEs
  11. Example V: DC power network
  12. Example VI: nonlinear circuit model
  13. Example VII: constrained mechanical systems
  14. Example VIII: vibrating string
  15. Example IX: viscoelastic Stokes problem
  16. ...and 2 more sections

Key Result

Lemma 2.3

The energy satisfies $\frac{\mathrm{d}}{\mathrm{d}t} H \le \langle y, u\rangle$. In particular, system eq:model:inputoutput is energy dissipative for $u = 0$.

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: energy dissipation
  • proof
  • Lemma 2.4: structure-preserving interconnection
  • proof
  • Lemma 2.5: structure-preserving Petrov--Galerkin projection
  • proof
  • Remark 2.6
  • Lemma 2.7: discrete energy dissipation, midpoint rule
  • ...and 5 more